Optics
Optics
Optics
PY3101
M.P. Vaughan
University College Cork (2014)
Contents
I
Introduction to Optics
1 Overview
1.1
General remarks . . . .
1.2
Learning objectives . .
1.3
A short history of optics
1.4
Applications of optics .
1.5
Course overview . . . .
1.6
References . . . . . . .
7
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9
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27
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CONTENTS
II
Wave Optics
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67
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5 Diffraction
5.1
General remarks . . . . . . . . . . . . .
5.2
Learning objectives . . . . . . . . . . .
5.3
Light passing through a narrow aperture
5.4
Single slit diffraction . . . . . . . . . . .
5.5
Diffraction limited imaging . . . . . . . .
5.6
Multiple slit diffraction . . . . . . . . . .
5.7
Diffraction gratings . . . . . . . . . . . .
5.8
Diffraction around objects . . . . . . . .
5.9
Summary . . . . . . . . . . . . . . . . .
5.10
References . . . . . . . . . . . . . . . .
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III
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Electromagnetic Waves
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CONTENTS
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
General remarks . . . . .
Learning objectives . . .
Linear polarisation . . . .
Jones matrices . . . . . .
Elliptically polarised light
Wave plates . . . . . . .
Analysis of polarised light
Summary . . . . . . . . .
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183
Geometrical Optics
187
9 Fermats Principle
9.1
General remarks . . . . . .
9.2
Learning objectives . . . .
9.3
Geometric wavefront . . . .
9.4
Fermats Principle . . . . .
9.5
Perfect mirrors . . . . . . .
9.6
Perfect lenses . . . . . . .
9.7
Curvature . . . . . . . . . .
9.8
Parameterisation of a curve
9.9
Summary . . . . . . . . . .
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CONTENTS
10.9
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
Crystal Optics
11 Crystal Symmetry
11.1
General remarks . . . . . . . . . . . .
11.2
Learning objectives . . . . . . . . . .
11.3
Group theory . . . . . . . . . . . . . .
11.4
Symmetry of a square . . . . . . . . .
11.5
Point groups in 2D . . . . . . . . . . .
11.6
Point groups in 3D . . . . . . . . . . .
11.7
Symmetry of the electric susceptibility
11.8
Principal crystal axes . . . . . . . . .
11.9
Symmetry operations . . . . . . . . .
11.10 Summary . . . . . . . . . . . . . . . .
235
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280
CONTENTS
Part I
Introduction to Optics
1. Overview
1.1
General remarks
1.2
Learning objectives
1.3
Traditionally, the subject of optics has dealt with the nature of visible light.
Applications of optical principles may be traced back into antiquity, with references to the use of lenses (or water filled vessels) for magnification and
burning glasses and the polishing of metal to obtain a mirrored surface.
From a modern perspective, we would recognise these as applications of
geometrical optics, although it was not until the considerations of the Ancient Greeks that any kind of theoretical understanding of these phenomena was attempted.
9
10
CHAPTER 1. OVERVIEW
(a)
(b)
(c)
Figure 1.1: (a) Detail from Raphaels The School of Athens depicting Euclid
(with dividers bending over) [image in public domain]. (b) Illustration of Hero
of Alexandria from a 1688 German translation of Heros Pneumatics [image
in public domain]. (c) Image of Alhazen as shown on the obverse of the
1982 Iraqi 10 dinar note.
1.3.1
One of the earliest surviving theoretical text is Euclids Optics (circa 300
BC). It is perhaps not too surprising for the Father of Geometry, that Euclid
gave a geometrical account of light, in which he viewed light as a cone of
rays with the eye at the vertex. His account then explains the phenomenon
of perspective and unseen objects. However, the idea that light travels in
straight lines remained an unexplained assumption.
Later, around 40 AD in the work Catoptrica, Hero (or Heron) of Alexandria was able to show geometrically that the path of a ray reflected from a
plane to an observation point is the shortest possible path the light could
have taken, subject to the constraint that the ray must touch the plane. In
retrospect, we might call this the principle of least distance and bears close
resemblance to a modern explanation. Hero, however, did not appreciate
that light travels at different speeds in different media and it turns out that
the correct principle would be one of least time. Nowadays, this would often
be described in terms of the optical path length, which has dimensions of
distance but remains directly proportional to the time taken.
1.3.2
With the demise of the Greek civilisation, the torch of philosophical enquiry
was taken up in the Islamic world. The mathematical advances of in number bases, algebra and the development of algorithmic methods are well
known, if not always attributed to Islamic thinkers.
One important early contributor to the field of optics was Ibn Sahl (c. 940
11
- 1000) who, in his treatise On Burning Mirrors and Lenses (984) explains
the focussing of light by curved mirrors and lenses. Significantly, he also
gives the first detailed formulation of what is now known as Snells Law of
refraction
Another major protagonist was the polymath Alhazen (c. 965 - c. 1040).
Between 1011 to 1021 Alhazen wrote a seven volume treatise on optics
Kitab al-Manazir (Book of Optics). In the course of this work, Alhazen
conducted many experiments on the rectilinear (straight line) propagation of
light, reflection and refraction. This early adoption of the Scientific Method
makes Alhazens work significant in the more general history of Science.
However, in terms of optics, his main contribution is often held to be his
detailed description of the human eye.
1.3.3
The Enlightenment
(a)
(b)
(c)
(d)
12
CHAPTER 1. OVERVIEW
the path taken between two points by a ray of light is the path that can
be traversed in the least time
This is closely related to Heros Principle of least distance and, like that
earlier concept, is a variational principle that finds the optimum path as
an extremum of some observable. A modern version of Fermats Principle
multiplies the time of travel by the wave speed at a given time to obtain the
optical path length in dimensions of length.
(a)
(b)
13
(c)
1.3.4
Wave optics
Such was Newtons intellectual influence at the height of his powers that the
corpuscular theory of light tended to hold sway simply because Newton said
it was so. Indeed, it is with some irony that when Augustin-Jean Fresnel
(1788 - 1827) submitted a paper to the French Academy of Sciences in
1818 expounding the wave view of light, Poisson, a member of the judging
committee and supporter of the corpuscular theory, attempted to disprove
Fresnels work by arguing that this would imply the appearance of a bright
spot in the shadow of an illuminated sphere. In fact, such a bright spot does
indeed exist, now known as the Fresnel bright spot.
Although a major figure in the development of wave optics, Fresnels
work was slightly pre-dated by that of Thomas Young (1773 - 1829). In fact
it was Young who provided the first conclusive evidence for the wave nature
of light in his now famous double slit experiment in 1803 (see Chapter 5).
The double slit experiment showed that light clear exhibited the wave phenomenon of diffraction, an effect totally inexplicable in terms of corpuscles
(particles).
Fresnel independently discovered Youngs work, adding substantially to
the theory of diffraction (Chapter 5) as well being one of the first people
to appreciate the importance of the polarisation of light. He also derived
the equations that now bear his name for the reflectance and transmission
ratios of light crossing between different media (Chapter 8).
14
CHAPTER 1. OVERVIEW
(a)
(b)
(c)
Figure 1.4: (a) Portrait of Micheal Faraday by Thomas Phillips, 1842 [image
in public domain]. (b) Engraving of James Clerk Maxwell by G. J. Stodart
from a photograph by Fergus of Greenack [image in public domain]. (c)
Photograph of Albert Einstein during a lecture in Vienna in 1921 [image in
public domain].
1.3.5
Electromagnetism
15
demonstrated the generation of radio frequency EM waves via the changing electric field of oscillating charge in a dipole antenna [2]. The SI unit of
frequency is now named the hertz (Hz) in commemoration of this achievement.
At the time, it was generally believed that all wave propagation must
take place in some kind of physical medium. The proposed medium for
light was called the luminous ether. However, since the speed of light c
was found to be a constant, it was assumed that this must be relative to the
ether. Moreover, since the Earth is moving in an orbit around the Sun, it
would seem that the Earth itself must sometimes have a relative speed to
the reference frame of the ether.
As a test of these ideas, in 1887 Michaelson and Morley attempted to
measure the relative motion of the Earth against the luminous ether by
measuring the speed of light at different points in its orbit using an interferometer . To their surprise, Michaelson and Morley discovered that the
speed of light was always the same, which seemed a very counter-intuitive
result. Hendrik Antoon Lorentz (1853 - 1928) modelled this using by developing his Lorentz transformations and attempted to explain the phenomenon in terms of contraction of the ether. However, his transformations also implied a bizarre dilation of the temporal dimension that he could
not explain.
It was left to Albert Einstein (1879 - 1955) to complete the electromagnetic explanation of light. In 1905, he published his paper on Special Relativity based on the mathematics of the Lorentz transformations [3]. He dispensed with the idea of absolute space and time, arguing that only relative
motion was physically meaningful. In this way, he showed that the concept
of the luminous ether was superfluous to requirements. Later, Einsteins
former Mathematics teacher Hermann Minkowski (1864 - 1909) formalised
Einstein theory, showing that his concepts of relativity could interpreted in
terms of a four-dimensional continuum, now known simply as spacetime.
1.3.6
Quantum mechanics
At the turn of the century, it seemed that classical physics was about to mop
up all the remaining problems of physics. Only a few loose threads hung
from the tapestry, which were about to be pulled on.
One of these outstanding problems was the explanation of blackbody
radiation. A blackbody is a body that both absorbs all the radiation incident
on it and emits radiation with a spectral density characteristic of its temperature. This was the problem that Max Planck (1858 - 1947) was concerning
himself with in 1894. At the time, the spectral density could be explained in
both the high and low frequency limits but no single law for the spectrum of
radiation over all ranges was known (Chapter 2).
16
CHAPTER 1. OVERVIEW
(a)
(b)
(c)
Figure 1.5: (a) Photograph of Max Planck in 1933 [image in public domain].
(b) The official 1921 Nobel Prize in Physics photograph of Albert Einstein,
which he won for his explanation of the photoelectric effect [image in public
domain]. (c) Photograph of Niels Bohr around 1922 from the Nobel Prize
Biography.
Planck eventually found that the only consistent explanation he could
come up with required that the light must be emitted in discrete quanta.
These quanta would then have an energy = hf , where h is Plancks constant and f is the frequency of the light. This apparently went against all
the conventional wisdom that light was in fact a form of electromagnetic radiation and seemed to be resurrecting the corpuscular theory of light again.
Needless to say, Plancks paper [4] was not well-received.
One person who did take Plancks ideas seriously was Albert Einstein,
who in 1905 realised that he could explain the photoelectric effect by extending Plancks ideas to the notion that the electromagnetic field itself was
quantised [5]. In the photoelectric effect, electrons are liberated from an
illuminated sample of metal. However, in an apparent contradiction with the
classical theory of light, the energy of these electrons is not proportional to
the intensity of the radiation. Rather, greater intensity just leads to a greater
number of electrons. On the other hand, the energy of the electrons is dependent on the frequency of light. Einstein therefore postulated that these
liberated electrons were ionised by the absorption of a quantum of light with
an energy = hf . The intensity of the radiation field is then just proportional to the number of light quanta in it. These quanta were later dubbed
photons.
This dual explanation of light has given rise to the use of the expression wave-particle duality. However, this is possibly a misleading way of
putting things. It suggests that light is sometimes a wave and sometimes
a particle. In fact, all the particle-like effects associated with photons can
be reproduced by the construction of a wave packet, which can be made
17
18
1.4
CHAPTER 1. OVERVIEW
Applications of optics
The applications of optical theory are many and varied. Here we list just a
taster of possible applications.
1.4.1
Ophthalmetry
Preceding the invention of telescopes and microscopes was the use of
lenses for correcting deficiencies in vision. This area is known as ophthalmetry and has traditionally made great use of geometrical optics. More
recently, there has been growing use of lasers in corrective surgery.
Photography
Another area where optimal imaging is a necessary requirement is in photography. Compound photographic lens of all types are designed for different purposes. Additionally, high quality lenses often incorporate antireflective (AR) coatings (Chapter ??) and/or polarising filters (Chapter 7).
1.4.2
Spectroscopy
Spectroscopy concerns the spectral analysis of light. Many different techniques exist for achieving this, including diffraction gratings, prisms and
resonant cavities.
19
Chemical analysis
Different chemical substances may often be identified by discrete peaks in
their emission or absorption spectra. Typically, a substance may be heated
to a gaseous phase and white light shone through it. Analysing the spectrum of the light in the direction of the light may reveal dark lines or absorption resonances, where the energy, and hence frequency, of the photons
matched a difference in the electron energy levels of the chemical. In other
directions, bright lines may be seen where these excited states drop down
to lower energy levels, emitting a photon in the process. This absorption
and emission lines are typically unique for a given chemical substance, so
that they provide a chemical signature.
Astronomy
Although modern astronomy exploits all wavelengths of the electromagnetic spectrum, optical telescopes were of crucial importance in its development. Moreover, spectroscopic analysis of the light can provide information
about the temperature of stars (see Chapter 2) and the chemical make-up
of extra-terrestrial matter.
1.4.3
Photonics
20
CHAPTER 1. OVERVIEW
Depending on the design of the device, this can either make the semiconductor conductive, allowing a photocurrent to flow or create a voltage
that can be used to provide electrical power. The former case is called
photoconductivity. The latter is the case for photovoltaic cells (i.e. solar
cells).
Lasers
Lasers in general are discussed briefly in Chapter 2. Semiconductor lasers
have found many applications, such as in fibre optic communications, reading CD and DVD drives and scanning technologies. A major advantage of
such devices is their small size, cheapness and versatility. For instance,
a block of semiconducting material naturally acts as a resonant cavity and
may be readily adapted with Bragg reflectors (Chapter ??) to provide the
correct wavelength selectivity.
1.4.4
Fibre optics
21
Waveguiding
A fundamental optical application in fibre optic systems is that of waveguiding, the process by which the optical signal is contained within the fibre (or
other optical waveguide). The basic physical principle involved here is that
of total internal reflection (Chapters 4 and 8), although a rigorous treatment
of the problem requires a solution of Maxwells equations (Chapter 6).
Optical modulation
Another important aspect of fibre optic communications is the question of
how to modulate the light. In practice, one of the best ways of achieving
this is to modulate the light after it has been generated by exploiting the
electrooptic effect. This is an effect in which the electrical susceptibility
of a material is changed by the application of an electric field (taken to be
static on the time scale of the optical oscillations). Both linear and nonlinear
effects exist and are used.
Optical amplifiers
When transmitting light over many miles of fibre optic cable, optical attenuation becomes a problem. In these circumstances, some form of amplification must be used to boost the signal. Such amplifiers are essentially
lasers operated under the conditions that they only emit when a signal is
passed through them. One possible technology is the semiconductor optical amplifier (SOA). However, the preferred technology is at present the
erbium doped fibre amplifier (EDFA). This is a special fibre doped with erbium providing energy levels for laser operation.
Wavelength converters
One use of SOAs is in there use as wavelength converters for wavelength
division multiplexing (WDM) systems. In such devices, a signal of one
wavelength may create a population inversion for the amplification of a signal of another.
Nonlinear effects
Due to the high intensity of laser light confined to the small cross-sectional
area of the fibre optic core, nonlinear effects in fibre optics often become
significant. One of the most important of these is the third order effect
known as four wave mixing, in which three frequency components of the
electrical polarisation combine to produce a fourth. Although we mention
22
CHAPTER 1. OVERVIEW
nonlinear optics at several points in the text, a proper study of these effects
is beyond the scope of this course.
1.4.5
Computer graphics
In some senses, the technology of computer graphics or computer generated imagery (CGI) may be thought of as bringing the entirety of optical
understanding together, since the purpose is often to simulate reality as
accurately as possible. Since the subject also brings in a hefty dose of
computer science and computational physics, it is a huge topic in its own
right. Here we mention a couple of aspects of the technology.
Ray tracing
In optical texts, ray tracing usually refers to a technique of approximating the
wave nature of light by following an optical ray in the propagation direction
of the wavefront. It is often used to analyse the focussing of light through
lenses or reflected from mirrors in geometrical optics.
In computer graphics, the term has a closely related meaning but usually refers to the rendering of a 2D screen by following rays into a 3D virtual
space. This involves various aspects including
the perspective projection of the virtual 3D image on to the 2D screen
the reflection or refraction of a ray
the diffuse scattering of a ray
All of these areas require a firm understanding of optics.
Scattering theory (rendering algorithms)
Scattering theory is an area that becomes increasingly complicated as one
delves into it. Fundamentally, the way in which light is scattered from an
object is straight forward. The incoming light induces electric dipole oscillations in the material medium which are then re-radiated. The details of
these oscillations for a given material, however, may become very complicated. Nevertheless, modern CGI attracts a high level of financial investment and, as a result, techniques have become very sophisticated.
1.5
Course overview
23
I Introduction to Optics
A general introduction to optics, including a brief history of its development and discussion of the complementary understandings provided
by electromagnetism on the one one hand and quantum mechanics
on the other. Since the course material focusses on the electromagnetic picture, a brief introduction to the physics of waves is also given,
with the emphasis and the universality of the mathematical description of waves. Thus, the same form of wave equation that is used for
electromagnetism can also be used to model the elastic vibrations of
a continuous solid. A first categorisation of media is also given:
Linear / nonlinear
Isotropic / anisotropic
Homogeneous / inhomogeneous
In this section, we describe linearity mathematically in terms of linear and nonlinear differential equations (wave equations in our case).
Since we also introduce the general concepts of polarisation and refractive index, we also explain isotropy as the independence of the
refractive index on polarisation direction. In crystal optics, this is often not the case and we have an anisotropic medium.
II Wave Optics
In the section of wave optics we introduce the powerful concept of the
Huygens-Fresnel Principle, based on Huygens Principle with the additional concepts of interference worked out in such detail by Fresnel.
Using this principle, we obtain the Laws of Optical Propagation
The Law of Rectilinear Propagation (light travels in straight lines)
The Law of Reflection
The Law of Refraction (Snells Law)
This is applied to the important topic of diffraction, a characteristically
wave-like phenomenon and providing (via Youngs double slit experiment) the most convincing evidence for the wave description of light.
Of practical importance for spectroscopic analysis is the diffraction
grating, which is also introduced.
III Electromagnetic Waves
The central exposition of optics given in this text revolves around
Maxwells electromagnetic equations. We begin this topic with a discussion of EM wave propagation in a both a vacuum and a material
medium. Here we meet the crucially important topic of the electric
24
CHAPTER 1. OVERVIEW
susceptibility tensor, which gives the frequency response of a medium
to an applied electric field. This response manifests in the form of the
electrical polarisation of the medium. We spend quite some time discussing the susceptibility tensor and its frequency dependence (which
gives rise to a frequency dependent refractive index). Much of this
discussion is to provide a firm basis for our later description of crystal
optics.
We then move on to discuss the optical polarisation (not to be confused with the electrical polarisation of a medium). Here, we introduce
the formal apparatus of the Jones vector to describe the polarisation
and the Jones matrix to describe the effect on the polarisation of an
optical element. Initially, these concepts are dealt with formally in the
main, awaiting the section on crystal optics for a full explanation of
the physics.
Light propagation between different media is also covered at some
length. This is begun with coverage of Fresnels equations. Using the
understanding we glean from this, we then look at the technologically
important area of thin-film interference.
IV Crystal Optics
In this section, we consider wave propagation in anisotropic materials.
We begin with a short account of group theory and crystal symmetry
to provide both a means of categorisation and a powerful analytic tool
for determining the optical properties of particular crystal systems.
This is achieved by applying symmetry arguments to the form of the
susceptibility tensor.
Finding the solutions for wave propagation in an anisotropic medium,
we encounter the highly useful concept of the index ellipsoid. Using
this, we may analyse simple cases of wave propagation. When we do,
we encounter the phenomenon of birefringence, in which orthogonal
components of the optical field can move out of phase with each other.
Birefringence can be exploited to change the polarisation state of light
from, say, linearly polarised to being elliptically polarised. Technologically, this allows the construction of optical elements known as
wave plates (considered formally earlier in the chapter on polarisation). Wave plates are frequently used in photonic applications, such
as optical modulators and in 3D glasses, to name just two possibilities.
Closely related to birefringence is the concept of pleochroism (or
more often the limiting case of dichroism. This is the selective absorption of light depending on polarisation direction. One of the main
uses of dichroism is in linear polarisers. In more aesthetic applica-
1.6. REFERENCES
25
tions, dichroic glass is often used in jewellery due to the deep and
varied colours it produces.
A further class of crystal optics concerns optically active materials
in which a linear state of polarisation undergoes a rotation about its
propagation axis. A well-known case of this is Faraday rotation, in
which a magnetic field induces the rotation. This effect has technological applications in the fabrication of optical isolators for fibre optic
systems as well. It is also occurs naturally in, for instance, light from
astronomical objects passing through strong magnetic fields.
V Geometrical Optics
In the final section, we turn to geometrical optics. In some ways,
this may be considered as far more elementary than the other topics
covered in this course but it is as a result of that understanding that we
can see how the assumptions of geometrical optics may be justified
(and its limitations).
Here, our theoretical underpinning is that of Fermats Principle. This
is a remarkably powerful tool and has a firm theoretical justification.
The main limitation of geometrical optics is that it employs ray-tracing
throughout, which, although a very useful concept, remains an idealism due to the effects of diffraction. Nevertheless, we are again able
to derive the Laws of Optical Propagation on the basis of Fermats
Principle, which is expressed in terms of the optical path length.
With the theoretical foundations established, we then move on to
analyse both perfect imaging (in which light is perfectly imaged onto
a point or plane and imaging via spherical mirrors and lenses. The
paraxial approximation for small angles is introduced, which makes
the analysis of thin lenses mathematically tractable. We also see how
lens may be combined and how images are constructed.
In addition, we also consider the types of optical aberrations encountered and strategies for alleviating them.
1.6
References
26
REFERENCES
Additional reading
[7] Grant R. Fowles, Introduction to Modern Optics, Dover Publications
(1989)
[8] Eugene Hecht, Optics, Addison-Wesley (2001)
[9] Max Born and Emil Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, Cambridge
University Press (1999)
2.1
General remarks
2.2
Learning objectives
28
2.3
2.3.1
Although Thomas Youngs double slit experiment of 1803 offered what appeared to be incontrovertible evidence for the wave nature of light, an explanation of the underlying physics was not forthcoming until Maxwells
proof [1] that the modified equations of electromagnetism yielded a wave
equation for the propagation of electric and magnetic fields. Moreover,
Maxwell showed that these waves would travel with the speed of light, which
in turn could be calculated from fundamental constants. This prediction was
decisively confirmed by Hertz [2], who generated radio waves from oscillating electric charge, having all the required characteristics of Maxwells
electromagnetic waves (EM waves).
In Fig. 2.1, we illustrate the spectrum of electromagnetic radiation (sometimes referred to as Maxwells rainbow), from long wavelength radio waves
29
Figure 2.1: The electromagnetic spectrum in terms of wavelength and frequency, ranging from radio frequency (RF) to gamma rays (-rays).
2.3.2
RF radiation
At the low frequency end of radio frequency (RF) radiation, we have the
ranges of extremely low frequency (ELF), from about 3 Hz to 3 kHz, and
very low frequency (VLF) between about 3 kHz to 30 kHz. Such radiation is often bounded within the Earths atmosphere as a standing wave
with the surface of the Earth and the ionosphere forming a resonant cavity
(the ionosphere is the region of the atmosphere between 85 and 600 km
above sea level in which the atmospheric molecules are largely ionized by
bombardment by solar radiation). Natural sources of this frequency of radiation include the movement of charge associated with lightening strikes.
Extraterrestrial sources of ELF also include astronomical objects such as
neutron stars with very high magnetic fields.
Moving towards higher frequencies, we have long wave and medium
wave radio. These frequencies my be amplitude modulated for carry acoustic signals (i.e. AM radio). Above this in the very high and ultra high frequencies (VHF and UHF), signals may be frequency modulated (i.e. FM)
with the higher frequencies traditionally used from television broadcasting.
Radio waves are typically generated via oscillating charge in an antenna
of some kind (the most well known type being the dipole antenna. These
oscillations are driven by electronic circuitry, typically an electronic oscillator
coupled to a power amplifier.
30
2.3.3
Microwave radiation
Above radio waves, the next major band of the electromagnetic spectrum
is dubbed microwaves, lying between about 0.3 GHz and 300 GHz. This
range of radiation is often generated by very fast charge oscillations in devices such as the magnetron (often used to power microwave ovens) or, in
solid state devices via the Gunn diode. Microwaves are also widely used in
free-space telecommunications.
2.3.4
31
forts to understand blackbody radiation [3]. Armed with this additional understanding, we shall be in a better position to offer insight into the nature
and generation of the thermal infra-red, optical and higher frequency radiation.
2.4
Blackbody radiation
2.4.1
Stars as blackbodies
2.4.2
Plancks Law
The problem that Planck set out to resolve was that the spectral profile of
blackbody radiation could not be modelled by existing laws. According to
32
Class
T (K)
Colour
O
B
A
F
G
K
M
33000
10000 33000
7500 10000
6000 7500
5200 6000
3700 5200
2400 3700
blue
blue-white
white
yellow-white
yellow
orange
red
(2.1)
(2.2)
This meant that the spectral density of a blackbody could now be given
in terms of Plancks Law. As a function of frequency and temperature, the
intensity I is given by
33
I (f, T ) =
1
2hf 3
,
2
hf
/k
T 1
B
c e
(2.3)
hf
kB T
(2.4)
and finding
dI (x)
= 0.
dx
(2.5)
x (ex 1) xex = 0,
(2.6)
This yields
which must be solved numerically (e.g. using the bisection method). Note
the solution at x = 0 is not the maximum and should be avoided. To 6 DP,
we then find a value of xmax = 2.821439. From Eq. (2.4) we then find
fmax
= 58.8 GHz K1 .
T
In terms of wavelength, we have f = c, so
max =
c
fmax
5.1 103 m K
.
T
(2.7)
(2.8)
2.898 103 m K
,
T
(2.9)
(2.10)
34
Figure 2.2: Generalised schematic of a fixed-point blackbody. The blackbody radiation is emitted from the central cavity, held at a constant temperature by the freezing of the very pure metal surrounding it.
Given the precise description of the spectral radiation from a blackbody
in terms of fundamental constants (h and c), blackbodies are a prime candidate for the first link in the calibration chain for optical sources and detectors. In fact, at the National Physical Laboratory in the UK, where the SI
unit of luminous intensity the candela is maintained, artificial blackbodies
are used that can be held very precisely at a known temperature. A generalised sketch of a fixed-point blackbody (i.e. operating at a fixed reference
temperature) is shown in Fig. 2.2.
As an example, the copper point blackbody (CuBB) operates by melting very pure copper surrounding the cavity itself and then cooling it below
its freezing temperature. Due to the purity of the copper, it will under-cool
without freezing as there are few seed sites to initialise the crystallisation.
As it does freeze, the temperature rises back up to the freezing point temperature (1.358 103 K) where it remains stable for some time. By Wiens
Law, the maximum emission at this temperature is at 2.134 m.
2.4.3
Plancks idea was later extended by Einstein in his explanation of the photoelectric effect [3] in 1905.
35
As discussed in Chapter 1, in the photoelectric effect, electrons are liberated from an illuminated sample of metal. However, whereas the classical
theory would predict that the energy of these electrons should proportional
to the intensity of the radiation, instead, the energy is related to the frequency of the light. Einstein explained this by assuming that the radiation
field itself was quantized. He then formulated his explanation in terms of
the maximum energy of the liberated electrons
max = hf ,
(2.11)
where is known as the work function of the metal. That is, the energy
that the electron has to expend to be released from the metal. These light
quanta later became dubbed photons.
2.5
36
2.6
A great deal is made in the literature of the expression wave-particle duality. This suggests a a paradoxical picture of things in which quantum entities
are sometimes like waves and sometimes like particle. This popularist interpretation is open to severe criticism, although there remains a sense in
which this description contains meaningful physics.
Despite the two apparently contradictory explanations of light emerging
around the turn of the century, there was no formal link between them. This
comes from de Broglies hypothesis in 1924 that matter also has a wavelike
nature. Although principally applied to so-called matter waves the theory
is rooted in the physics of waves and Special Relativity.
According the Special Relativity, the energy and momentum of a particle
travelling with a speed v are
= m0 c2
(2.12)
p = m0 v,
(2.13)
and
v2
1 2
c
1/2
(2.14)
(2.15)
(2.16)
hvp
.
m0 c2
(2.17)
f =
and
=
c2
= ,
p
v
(2.18)
h
.
p
37
(2.19)
(2.20)
2.7
(2.21)
2 1 = h
.
(2.22)
where
2.7.1
(2.23)
The ways in which light may be emitted or absorbed via the electron-photon
interaction can be described by one of three categories:
1. absorption by which the energy of an incoming photon is absorbed by
an electron, which is then excited to a higher energy state. The photon may be thought of as being destroyed by this process. Note that
since electrons are fermions and, as such, subject to the PEP, there
must be an empty higher state of the right energy for this processes
to occur.
2. spontaneous emission by which an excited electron spontaneously
decays to a lower energy state with the emission of a photon with
the energy difference. This process may be viewed as one of photon
creation. Again, the conditions of the PEP apply.
38
2.7.2
(2.24)
(2.25)
2.8
39
(2.26)
In order that this yields a probability, the integral of this expression over all
space must be finite. That is, normalising the integral to unity, we must have
Z
| (r)|2 d3 r = 1.
(2.27)
Suppose, now, that (r) has some exact wavelength k. In this case,
the squared modulus of (r) will have the same value everywhere and the
integral over all space will be infinite. Thus, such a wavefunction cannot
represent a particle.
The resolution of this problem is the construct a localised wave packet
from k-states. The more localised we make the packet, the greater the
number of k-states we require. Thus, improving our knowledge of the position, means losing information about the exact momentum. Formally, the
relation between the uncertainty in position x and that in momentum px
is given by
xpx
h
.
2
(2.28)
Note that the concept of the wave packet retains our particle-like description
within the scope of wave mechanics.
Another important example is the energy-time uncertainty principle
40
t
h
.
2
(2.29)
This may be interpreted as saying that the energy of system may vary by
an arbitrary amount over a small enough time period. It is this uncertainty
principle that lies at the heart of quantum tunnelling and makes possible
the fusion reactions fuelling the Sun.
2.8.1
Vacuum fluctuations
Another fascinating consequence of energy-time uncertainty is that of vacuum fluctuations. If the vacuum had an exact energy (for instance 0), this
would be a violation of the HUP, so the actual energy is therefore not defined. This means that pairs of particles and anti-particles may pop out
of the vacuum with any energy provided they appear and disappear again
within a short enough time. Now the photon is its own antiparticle, so in
terms of radiation, vacuum fluctuations are the appearance and disappearance of photon pairs. Since such particles are transient, we often refer to
them as virtual particles.
Vacuum or quantum fluctuations are not theoretically unproblematic.
The possibility of the spontaneous appearance of particles of any energy
leads to the ungainly notion that the energy of the vacuum is infinite. This
means that ad hoc methods have to be employed to renormalise the field
to make it finite. These methods essentially involve adding an infinite constant to the energy - a mathematically invalid procedure!
However, strong evidence exists for the the existence of vacuum fluctuations. For our current purposes, the fluctuations suggest an interpretation of spontaneous emission. Earlier, we stated that there were three
distinct processes for the electron-photon interaction: absorption, spontaneous emission and stimulated emission. In fact, we may think of this as
reducing to just two with spontaneous emission being emission stimulated
by the vacuum fluctuations. According to this picture of things, each possible photon state of the vacuum may be called a mode and the vacuum
fluctuations as a single photon per mode.
Spontaneous emission is then the coupling of the process of emission
with the rise from the vacuum of the single virtual photon state. Now, the
process of emission can occur only in one way - from an excited state
to a lower state having a difference in energy, momentum and spin equal
to that of the virtual particle. On the other hand, the virtual particle may
emerge as the end result of a very large number (approaching infinity) of
different processes. Thinking of this in terms of probability, the entropy
change involved with such emission will be extremely large. In other words,
this is an irreversible process.
2.9
41
Thermal infra-red
The discussion of blackbody radiation did not broach the particular mechanisms by which the electromagnetic waves are generated. The far, mid
wavelength and short wavelength infra-red (FIR, MWIR and SWIR) regions
of the EM spectrum between about 300 GHz to 100 THz are often termed
thermal radiation. This range of radiation is generally associated with the
kinetic energies of material bodies, where the emitted EM radiation is due
to energy loss in inelastic collisions. This may involve changes in the linear
kinetic energy or in the vibrational energy of atomic oscillations.
The average energy exchanged in such interactions is the thermal energy 21 kB T , giving a definition of the temperature. This then corresponds
to the peak of the spectral energy distribution. However, there is also a
large spread of kinetic and vibrational energies, giving a wide distribution of
radiation.
2.10
Optical sources
2.10.1
Incandescent sources
2.10.2
Another decoherent source of light is the light emitting diodes (LEDs). This
is a semiconductor device in which the emission of photons is via electrons
dropping down into lower energy states. However, unlike the incandescent
source, this is not a thermal source of radiation, with the frequency of emitted light lying in a relatively narrow band. The LED is therefore far more
efficient at converting the driving input energy (an electric current) into the
required frequency range. Hence, visible light LEDs lack the hot infrared
emission associated with incandescent bulbs and feel cool.
LEDs are have similarities to semiconductor lasers. The major difference between these two types of device is that for LEDs, the emission is
spontaneous, whereas in a laser, we have stimulated emission (see Chapter 2).
42
The common feature for semiconductor devices is that the excited energy levels may be filled electronically (note in Chapter 2 we briefly discuss
the optical pumping of the higher energy levels in a laser).
2.10.3
Lasers
Figure 2.3: Energy level diagram for a ruby laser (a three-level system).
The chromium ions are pumped optically from the ground state G to excited
and 4000 A via discharge from a
states 2 and 3 at wavelengths 5500 A
xenon flash lamp. The excited states 2 and 3 are very short-lived, with
lifetimes of 108 s 109 s. These then decay very quickly to the
metastable state 1 ( 3 ms). The lasing transition then takes place from
43
2.11
Optical absorption
p = dx.
The probability p that the photon is not absorbed at x + dz is then
p (x + dx) = p (x) (1 dx) .
(2.31)
(2.32)
(2.33)
So the probability that the photon is absorbed at the end of this flight is
given by
P (x) dx = ex dx.
(2.34)
xP (x) dx =
xex dx,
0
0
Z
x
1
= xe
+
ex dx = .
0
hxi =
(2.36)
44
Name
Symbol
Value
Boltzmann
Planck
Dirac
Stefan-Boltzmann
kB
h
h
2.12
Here we briefly mention the very short wavelength end of the EM spectrum.
Whilst ultraviolet (UV) is often taken to be part of the optical spectrum, X
and -rays may be distinguished by the method of generation. Whilst radiation in the optical spectrum originates from charge transfer between electronic energy levels in materials, high energy rays typically originate from
energy transitions within the nucleus of an atom. Some frequencies of Xrays may also be generated in this way, although usually we define X-rays
to produced in other ways, such as Bremsstrahlung (braking radiation)
produced when electrons are rapidly decelerated by positively charged nuclei in a material.
Both X-rays and -rays have a plethora of applications, particularly in
medical physics in radiography and radiotherapy.
2.13
Summary
(2.37)
2.13. SUMMARY
45
Plancks Law
1
2hf 3
.
2
hf
/k
T 1
B
c e
(2.38)
2.898 103 m K
.
T
(2.39)
I (f, T ) =
Stefan-Boltzmann Law
L = AT 4 .
(2.40)
(2.41)
p=h
k.
(2.42)
46
REFERENCES
Heisenberg Uncertainty Principle
xpx
t
h
.
2
h
.
2
(2.43)
(2.44)
vacuum fluctuations
Optical sources of radiation
incandescent sources
light emitting diodes
lasers
Optical absorption
Absorption coefficient . 1/ is the average path length of a photon
through a material.
2.14
References
3.1
General remarks
3.2
Learning objectives
48
3.3
3.3.1
A ubiquitous concept within physics is that of the simple harmonic oscillator. This type of oscillator occurs, for instance, whenever some physical
displacement in a field or material medium is resisted by a restoring force
that is proportional to the displacement. Another way of putting this, which
is of general significance, is that this restoring force is linear in the displacement.
Although modern physics informs us that light is not a displacement in
a material medium (it was originally supposed that this was the case, the
medium being called the ether), we may still view light as being a displacement of a field. For the time being, and through much of this book, we shall
refer to this as the optical field. We may then apply much of the same
intuition, and indeed mathematics, that we glean the elastic displacement
of a physical material in mechanics.
49
A good deal of insight may be initially gained from a mechanical example. An elementary but universal concept is that of the ideal spring. By
definition, an ideal spring is a mathematical model of a real spring to which
Hookes law applies. This just says that the restoring force is proportional
to the extension of the spring. The constant of proportionality is then known
as the spring constant, which we shall denote by K. The force exerted by
the spring is then proportional to the displacement u
F = Ku.
(3.1)
d2 u
= Ku.
dt2
(3.2)
(3.3)
where
=
K
m
1/2
(3.4)
3.3.2
2
.
T0
(3.5)
So how does the concept of the simple harmonic oscillator relate to optics?
As we shall see in Chapter 6, light may be understood as an electromagnetic oscillation propagating through space. A possible source of such a
disturbance is an oscillating electric dipole, in which we have two charges
of opposite sign separated by some distance z. Although this is not the
place for a detailed treatment of electric dipoles, we wish to argue for an
approximate model of a dipole that may be treated as a harmonic oscillator.
Classically, the dipole is visualized as two point charges displaced from
one another. In reality, however, in the absence of any other restraints,
two such classical charges, q and q, would be drawn together with an
increasing force, given by Coulombs Law
50
F =
q2
,
40 z 2
(3.6)
where the constant 0 is known as the permittivity of free space and z is,
as defined above, the distance between the charges (we assume, for simplicity, that the material medium has a relative permittivity of unity). This is
symptomatic of a general problem of classical physics in that it fails to explain the stability of matter composed, as it is, of equal amounts of negative
and positive charge.
This problem is solved very elegantly in quantum theory, where the concept of the point-like particle evaporates and is replaced with a wave description of particles. For our present purposes, this means that we may
imagine a negatively charged electron, not as a point particle, but rather
a cloud of charge, with the highest charge densities spread out over surfaces in space. For instance, an s-type electronic state can be imagined
approximately as a sphere of charge.
The positively charged protons will also be smeared out in space, although, due to there greater mass, the uncertainty principle will allow them
to be far more localized. We will therefore continue to imagine a proton as
a point-like particle.
Figure 3.1: An electric dipole modelled as a sphere of radius R and constant negative charge density displaced from a positive point charge by u.
.
51
4u3
.
3
(3.7)
The force between the positive charge and electron cloud is therefore
F =
q
qQ
u.
=
40 u2
30
(3.8)
Hence, this model of an electric dipole yields the same force law as the
ideal spring with K = q/ (30 ).
3.3.3
In the case of the mass on an ideal spring, the mass will have its maximum
kinetic energy as it passes through its minimum of potential energy, at which
point there is no force acting on it. Thereafter, the restoring force acts to
reduce the kinetic energy, converting it to the potential energy stored in the
spring. For a displacement u, the potential energy U is given by
Z
U =
0
F du0 = K
Z
0
u0 du0 = 21 Ku2 + U0 ,
(3.9)
(3.10)
where A is the amplitude of the oscillation. Thus, the energy of the oscillator is proportional to square of the amplitude. An analogous result holds
quite generally for linear oscillations, i.e. oscillations in which force is proportional to a displacement. Whilst the particular physics of the optical field
are quite different, we shall find that the energy of waves in this field is also
proportional to the square of the amplitude.
52
3.4
3.4.1
z dz
2 u (z)
Tz
Tz
=
[u (z + dz) u (z)]
[u (z) u (z dz)] ,
2
t
dz
dz
(3.11)
so
2 u (z + dz)
=
t2
Tz
z
(3.12)
53
2u
=
t2
Tz
z
2u
.
z 2
(3.13)
3.4.2
(3.15)
(3.16)
=
= v
t
t
(3.17)
and
2
=
2
t
t
= v2
2
.
2
(3.18)
Similarly, we have
=
=
z
z
(3.19)
2
2
= 2,
2
z
(3.20)
2
2
=
= 0.
x2
y 2
(3.21)
so
whilst
(3.22)
54
which gives
1/2
T
.
v=
(3.23)
We may therefore write the wave equation in the more general form
2 u =
1 2u
.
v 2 t2
(3.24)
This is the general form of the wave equation for a linear medium. Note that,
although was derived for the oscillations of an elastic medium, the particular physics of the system have been washed out, leaving just the abstract
mathematical form. This equation may then be used to to find the wave
solutions for diverse physical systems, so long as the harmonic oscillator
remains a valid model. In Chapter 6, we shall find that the same equation
governs the electromagnetic oscillations of light in a linear medium.
3.4.3
The polarisation
The vectorial nature of the displacement implies that that u may be in the
direction of the propagation (longitudinal) or transverse to it. This constitutes the polarisation of the medium. In general, there are various cases to
consider, although we shall find later that not all applied to the optic field.
Unpolarised
means that the displacement is randomly orientated in space.
Longitudinal polarisation
In this case, the displacement is in the propagation direction . As an
example, sound waves in fluids and solids are longitudinal. However,
electromagnetic waves generally are not.
Transverse polarisation
In this case, the displacement is perpendicular to the propagation
direction. For example, surface waves on a liquid are transverse.
Electromagnetic waves (constituting light) are generally transverse.
Moreover, the transverse polarisation may be
linearly polarised
in which the displacement always lies in the same plane, or
elliptically polarised
in which the polarisation rotates around the direction of propagation.
55
cos
sin
,
(3.25)
where |u0 | cos and |u0 | sin are the x and y components of E0 respectively, and is the angle the polarisation vector makes with the x-axis.
3.4.4
1 2f
=0
v 2 t2
(3.26)
(3.27)
(3.28)
56
L (f + g) = L (g) + L (f ) .
As an example, let
L (u) =
du
.
dx
(3.30)
Then
L (f + g) =
df
dg
d
(f + g) =
+
.
dx
dx dx
(3.31)
(3.32)
Now
L (f + g) = (f + g)2 = f 2 + 2f g + g 2 6= L (f ) + L (g) .
(3.33)
du
.
dx
(3.34)
df
dg
+
dx dx
6= f
df
dg
+g ,
dx
dx
(3.35)
57
3.5
3.5.1
Properties of a medium
In the derivation of the wave equation Eq. (3.24), we assumed that the
forces on an element of the displaced field or medium depended only on
the linear displacement. This yielded a constant phase velocity v. Although
the details of different systems may be very different, this result will remain
generally true where-ever this modelling assumption is valid.
In reality, of course, additional physical effects may emerge that will
change this fundamental result. Invariably, this will lead to a phase velocity
that depends in some way on these extra factors. Since the details are likely
to be particular to the physical system, we make no attempt to quantify
them here. Later, in Chapter 6, we shall see how the simple model of wave
propagation becomes modified in optics.
For now, let us denote the constant phase velocity found in Eq. (3.24)
by c. In optics, this will be correspond to the speed of light in a vacuum. As
we shall see, this is a universal constant, emerging from the two other constants of nature the permittivity and permeability of free space. In analogy
with the spring constant, we might think of these constants as describing
the fundamental stiffness of space.
In a material medium, other factors come into play. Rather than plunge
into the details of these forces, for the time being we merely subsume their
effect into a material parameter known as the refractive index n. The modified wave speed v is then
v=
c
.
n
(3.36)
n2 2 u
.
c2 t2
(3.37)
58
3.6
Plane waves
3.6.1
The wavefront
Locally, each part of the propagating wave solution to Eq. (3.24) will behave like a simple harmonic oscillator, each vibrating with the frequency of
the source. Points in space having the same phase within their cycle of
oscillation constitute a wavefront.
As the phase of each point in space changes, the wavefront must then
propagate outwards from the source. The speed of the wave front is then
the phase velocity v featured in the wave equation. The distance between
successive wavefronts in the direction of propagation is, of course, the
wavelength , related to the phase velocity by
v=
= f =
.
T
2
(3.38)
A slightly more useful description is given via the definition of the wavevector k. The magnitude of the wavevector is related to the wavelength via
|k| = k =
2
,
(3.39)
.
k
(3.40)
3.6.2
59
Plane wavefronts
Figure 3.3: Illustration of a section of plane wave (shown in two dimensions). The arrow shows the direction of propagation. Note that the example shown here is for a transversely polarised wave - i.e. the displacement
of the wave is at right angles (or transverse) to the direction of propagation.
The wavefronts are defined by the regions of constant phase, which here
are the planes perpendicular to the direction of motion (only one dimension
of these planes is illustrated here but we see that the wavefronts are along
straight lines perpendicular to the propagation direction).
A very useful mathematical description of a wave is given by the plane wave
(see Fig. 3.3). Such planar wavefronts parallel to the direction of motion.
Hence, if, for example, the wave is traveling in the z direction, all points in
the x y plane at a given value of z have the same phase (an therefore
define a wavefront). Using a general complex form, we may express a
plane wave as
E (r, t) = E0 ei(krt) ,
(3.41)
(3.42)
2 = k 2 ,
(3.43)
= i,
t
(3.44)
60
and
2
= 2 .
(3.45)
t2
Note, however that these substitutions are only valid for single plane wave
solutions. Assuming this is the case, the wave equation may now be written
k2 u = 2
n2
u,
c2
(3.46)
= .
k
n
3.6.3
(3.47)
Fouriers Theorem
The plane wave is a somewhat fictionalized entity since it takes as its source
an infinite two-dimensional plane. In practice, such waves may be taken
to approximate spherical waves radiating from a point source as the distance to the source approaches infinity. However, a particular utility of plane
waves lies in their mathematical versatility by virtue of Fouriers Theorem.
Fouriers Theorem tells us that any periodic function (subject to certain mathematical constraints) may be represented by a sum of sinusoidal
terms with the same periodicity. Moreover, as we take the limit of the period
T , we find that we may represent a given function by an integral of
sinusoids. However, this is precisely what the plane waves are. This means
that, in principle, we can represent any solution of the wave function by an
integral over plane waves.
Hence, labeling the plane waves by wavevector, the general solution of
the wave equation may be written
Z
E (r, t) = Ek ei(krk t) d3 k.
(3.48)
3.7
Group velocity
Very often, the refractive index will depend on the frequency of the oscillation. This means that, in turn, the phase velocity will also depend on
frequency. Hence, the components of the general solution to the wave
equation, Eq. (3.48), will tend to drift apart - a phenomenon known as dispersion.
With no unique phase velocity, the question of the how fast a wave
signal actually propagates naturally arises. Consider the general solution
wave expression given by Eq. (3.48). We may think of this as describing a
61
wave packet, which must travel with some average velocity. We call this the
group velocity vg (k0 ) associated with some average wavevector k0 .
3.7.1
If the domain of k does not exceed by too much some average wavevector k0 , then to a good approximation we can write k as a Taylor series
expanded to first order
k k0 + k k0 (k k0 ) ,
(3.49)
k = x
+y
+z
kx
ky
kz
(3.50)
(3.51)
where
(t) = (k k0 k0 k0 ) t
(3.52)
z(r, t) = r k k0 t.
(3.53)
and
Hence E (r, t) is modulated by a time-dependent phase ei(t) whilst being translated with a velocity k k0 . This is the group velocity . Dropping
the 0 subscript on k, this is
vg (k) = k k .
3.7.2
(3.54)
What, then, is the physical significance of the group velocity? The question
is of particular importance in optics when, under certain conditions, the
phase velocity may exceed the speed of light c. However, Einsteins Special
Theory of Relativity tells us that nothing can travel faster than the speed of
light. Does this not imply a contradiction?
62
3.8
Summary
d2 u
= Ku.
dt2
(3.55)
(3.56)
where
=
K
m
1/2
(3.57)
(3.58)
3.8. SUMMARY
63
2 u =
n2 2 u
,
c2 t2
(3.59)
64
2
.
(3.60)
.
k
(3.61)
Plane waves
A plane wave solution of the wave equation may be written as
E (r, t) = E0 ei(krt) ,
(3.62)
Ek ei(krk t) d3 k.
(3.63)
Group velocity
The group velocity is given by
vg (k) = k k .
(3.64)
Part II
Wave Optics
65
4.1
General remarks
68
4.2
Learning objectives
4.3
69
Spherical waves
(4.1)
C
.
4r2
(4.2)
(4.3)
E0 i(tkr)
e
.
(4.4)
r
We require one further condition on the form of a spherical wave, namely
that it must remain finite at r = 0. This condition is met by setting the cos
component of the complex exponential to zero and imposing the form
E (r, t) =
E0
sin (t kr) .
(4.5)
r
A plot of this function at a given time t is shown in Fig. 4.1.
Alternatively, we could allow the temporal factor to remain complex and
put
E (r, t) =
E0 it
e sin (kr) .
(4.6)
r
The particular choice of the form may be made based on mathematical
convenience.
E (r, t) =
70
Figure 4.1: Illustration of a spherical wave (shown propagating in twodimensions). This wave propagates outwards in all directions from the central peak. The form shown here is given by Eq. (4.5).
4.4
(4.7)
71
Figure 4.2: Illustration of Huygens Principle in the general case. The initial
wavefront at t = t0 acts as a source of secondary, spherical wavelets. After
an infinitesimal time t, the wavelets have acquired radii r = vt, which,
in the isotropic and homogeneous case, are all equal. The new wavefront
at t = t0 + t is then tangential to each wavelet (note that only two such
wavelets are shown for the sake of clarity).
Figure 4.2 illustrates Huygens Principle in the general case. The initial wavefront at t = t0 acts as a source of secondary, spherical wavelets
(of the form given by Eq. (4.5)). After an infinitesimal time t, the wavelets
have acquired radii r = vt. Since we are restricting our consideration to
isotropic, homogeneous media, these radii are all equal. The new wavefront at t = t0 + t is then tangential to each wavelet (note that Fig. 4.2 only
shows two such wavelets for the sake of clarity).
We may note that the two wavefronts have a shortest distance between
them r. It follows then that the vector from a point on the initial wavefront
along this shortest distance is at right angles to the wavefront. Picturing this
wavefront as a surface of constant phase embedded in three-dimensional
space, it should be appreciated that this vector is then parallel with the
gradient of the phase - i.e. points in the direction of the fastest change in
phase.
A line crossing a wavefront perpendicularly to it is known as a ray. In
the isotropic case, a ray also indicates the direction of wave propagation.
The relationship between wavefronts and rays is further elucidated in the
simplified cases of spherical and planar waves. These are discussed in the
next sub-sections.
72
4.4.1
Spherical wavefronts
(a)
(b)
4.4.2
Planar wavefronts
Figure 4.4 illustrates Huygens Principle for plane waves. In Fig. 4.4 (a),
we see the initial wavefront acting as a source of secondary, spherical
wavelets. Since the wavelets all have the same radii and the new wavefront must be tangential to these, we see that the wavefronts must all be
parallel to each other. Moreover, these wavefronts move in the direction of
(a)
73
(b)
4.5
4.5.1
Here, we reiterate the conclusion of the last section for an infinite planewave propagating in a isotropic and homogeneous medium. The direction
of propagation of the wavefront will be orthogonal to the plane. Since ev-
74
4.5.2
Reflection
(a)
(b)
Figure 4.5: (a) A plane wave approaching the boundary between two homogeneous media with refractive indices n1 and n2 at time t = t1 , just as
the edge of the wavefront touches the boundary at A. (b) At a later time
t = t2 , the wavefront emanating from point A has a radius equal to the
length of BD. The reflected wavefront now lies along the line CD.
Figure 4.5 (a) shows the wavefront of a plane-wave approaching the boundary between two homogeneous media with refractive indices n1 and n2 .
The direction of propagation is at an angle of i to the normal of the boundary. This is the angle of incidence. At the time t = t1 shown, the wavefront
has just reached the boundary at point A.
At a later time t = t2 , the point on the wavefront at B reaches the
boundary at D, shown in Fig. 4.5 (b). The length of BD is then
BD =
cT
,
n1
(4.8)
AC = BD.
75
(4.9)
The reflected wavefront must therefore pass through D and be tangential to the spherical wavefront centred on A. The direction of propagation
then makes an angle r to the normal. This is the angle of reflection.
From elementary trigonometry, in Fig. 4.5 (a) we have
BD = AD sin i
(4.10)
AC = AD sin r .
(4.11)
Equation. (4.9) tells us that we may equate equations (4.10) and (4.11),
giving
sin i = sin r
(4.12)
i = r .
(4.13)
or, simply,
In other words:
The angle of incidence is equal to the angle or reflection.
4.5.3
Refraction
Figure 4.6 (b) shows the transmitted wave propagating in the medium with
refractive index n2 . During the time T , the spherical wavefront at A has
propagated a radial distance of
AE =
cT
= AD sin t .
n2
(4.14)
n1
= AD sin t .
n2
(4.15)
(4.16)
76
(a)
(b)
Figure 4.6: (a) A plane wave approaching the boundary between two homogeneous media with refractive indices n1 and n2 at time t = t1 . (b) At a
later time t = t2 , the wavefront emanating from point A within the second
medium n2 has propagated a radial distance of AE. The new plane-wave
wavefront then passes through D and E.
4.6
4.6.1
(4.17)
77
transmission is not met (since such cases would require sin 1 > 1).
The minimum angle c at which light approaching from the medium with
the higher refractive index (n2 ) propagates exactly along the interface without being transmitted into the medium of refractive index n1 is known as the
critical angle . For this, we require 1 = /2, so sin 1 = 1 and
1 n1
.
(4.18)
c = sin
n2
If the angle of incidence (from the n2 medium) is greater than c , there
appears to be no transmission and the ray is entirely reflected at the interface. This is known as total internal reflection. In fact, we shall see in
Chapter 8 that there is, in fact, a decaying wave transmitted into the n1
known as the evanescent wave.
4.6.2
NA = n0 sin 0 ,
where 0 is the maximum angle of incidence for light entering the system
from a medium of refractive index n0 that the guide will accept (light approaching at angles greater than this will be lost in the cladding layers).
Using Snells law, we see from Fig. 4.8 that the numerical aperture is given
by
n0 sin 0 = n2 sin = n2 sin (/2 c ) .
(4.20)
1/2
(4.21)
n1
n2
2 !1/2
.
(4.22)
78
Hence, taking n2 inside the outer brackets, the required result for the numerical aperture is
n0 sin 0 = n22 n21
1/2
(4.23)
Figure 4.8: Construction for the numerical aperture (NA) of a slab waveguide consisting of cladding layers with refractive index n1 and a guide layer
of refractive index n2 . Light enters the guide from a medium of refractive
index n0 .
4.7
4.7.1
The Principle of Linear Superposition asserts that, for a linear medium, two
separate waveforms may be added together linearly to produce a single
displacement of the field. In terms of electromagnetic waves, this means
that the electric and magnetic fields associated with the propagating optical
field add linearly via normal vector addition. (It should be pointed out that
this no longer applies in the field of non-linear optics, where the induced
electrical polarisation (or magnetisation) of a material medium is no longer
a linear function of the applied fields).
This linear superposition can lead to enhancement or cancellation of
the field - known respectively as constructive and destructive interference.
More generally, the phenomena constitutes what we call interference effects.
For interference effects to appear over observable spatial or temporal
intervals, the underlying waveforms must be coherent. That is, there must
be some fixed relative phase between them. In practice, of course, such
coherence does not persist indefinitely and over all space. We therefore
speak of the degree of coherence, which may be quantified in terms of interference effects. Moreover, if we are considering the coherence of waves
displaced in space, we talk of spatial coherence (i.e. the waveforms at different points in space stay in phase). At a particular point in space, we are
79
4.8
Huygens-Fresnel Principle
Figure 4.9: Geometric construction for the superposition of waves over the
x y plane at point z, via the Huygens-Fresnel Principle.
With these points in place, we are now in a position to state the HuygensFresnel Principle. This is a subtle modification of Huygens Principle to
explicitly include the effects of interference via the principle of superposition
and may be stated as follows.
The Huygens-Fresnel Principle
For light of a given frequency, every point on a wavefront acts as a
secondary source of spherical wavelets with the same frequency and
the same initial phase. The wavefront at a later time and position is
then the linear superposition of all of these wavelets.
4.8.1
We saw previously that plane-wave propagation, a hence the Law of Rectilinear Propagation, could be analysed in terms of Huygens Principle. However, that principle does not forbid propagation of an arbitrary wavefront in
both the forwards and backwards direction. To see how this problematic,
80
E0 sin (krz )
dA,
rz
(4.24)
where E0 is the optical field per unit area, k is the wave vector and
rz = x2 + y 2 + z 2
1/2
(4.25)
1/2
(4.26)
and
dEP =
E0 sin (krz )
rdrd.
rz
(4.27)
sin (krz )
rdr.
rz
(4.28)
rdr
.
rz
(4.29)
2E0
[ cos (krz )]
r=0 .
k
(4.30)
Note that
drz = r r2 + z 2
1/2
dr =
Hence,
2E0
EP =
(4.31)
4.9. SUMMARY
4.8.2
81
Let us now reconsider the problem of plane wave propagation from a central wave front. Allowing for interference, wavefronts that travelled some
arbitrary distance z in either direction before returning, would yield a phase
EP (z) = E0 cos (2kz) .
Integrating this over all possible values of z gives
Z
Z
cos (2kz) dz.
EP (z) dz = E0
(4.32)
(4.33)
r=0
r=0
(4.34)
In other words, the spurious back propagation of the wave front is cancelled
out via interference.
4.9
Summary
Spherical waves
Spherical waves have the form
E (r, t) =
E0
sin (t kr)
r
(4.35)
E (r, t) =
E0 it
e sin (kr) ,
r
(4.36)
or
82
REFERENCES
The law of rectilinear propagation
In a homogeneous medium, light travels in straight lines.
The law of reflection
In a homogeneous incident medium, the angle of incidence equals
the angle of reflection.
The law of refraction
When light passes from a homogeneous medium with refractive
index ni into another homogeneous medium with refractive index
nt , the angles of incidence and refraction, i and t , are given
by Snells Law
ni sin i = nt sin t .
Coherence
Two waveforms are said to be coherent if there is a fixed relative
phase between them.
The Huygens-Fresnel Principle
The Huygens-Fresnel Principle may be stated as
For light of a given frequency, every point on a wavefront acts
as a secondary source of spherical wavelets with the same frequency and the same initial phase. The wavefront at a later
time and position is then the linear superposition of all of these
wavelets.
The Huygens-Fresnel Principle introduces interference that cancels
out the spurious back propagation of a wave front allowed by Huygens Principle alone.
4.10
References
5. Diffraction
5.1
General remarks
In Chapter 4 we considered the phenomenon of the interference of coherent waves. A term that may, in principle, be used synonymously with
interference is diffraction. However, in practice, diffraction is more often
used to describe interference effects that exhibit clearly delineated regions
of constructive or destructive interference. In this chapter, we shall focus
on the diffraction of light as it passes through an array of narrow apertures
in a screen. The same analysis may also be used to describe diffraction
from scattering features on a surface. In both cases, we often refer to the
screen (or surface) as a diffraction grating.
The diffraction grating was discovered by the Scottish mathematician
James Gregory around 1670, a year or so after Newtons prism experiments, by passing sunlight through a birds feather. In a letter to John
Collins [1], Gregory recommends his experiment for Newtons consideration:
If ye think fit, ye may signify to Mr. Newton a small experiment, which
(if he know it not already) may be worthy of his consideration. Let in
the suns light by a small hole to a darkened house, and at the hole
place a feather, (the more delicate and white the better for this purpose,) and it shall direct to a white wall opposite to it a number of
small circles and ovals, (if I mistake them not,) whereof one is somewhat white, (to wit, the middle, which is opposite to the sun,) and all
the rest severally coloured.
Another example of diffraction in nature (best explained together with
the closely related phenomenon of thin-film interference) is the iridescence
of a butterflies wing, the colours of which are seen to change as the wing is
moved.
Many other examples and applications can be mentioned, some of which
will be covered in this chapter.
5.2
Learning objectives
84
CHAPTER 5. DIFFRACTION
Analysis of Fraunhofer diffraction
Single-slit
Double-slit
Fraunhofer diffraction from a circular aperture
The Airy disc
Rayleigh criterion
Analysis of multiple slit diffraction in the far field
The grating equation
Use of diffraction gratings in monochromators.
5.3
85
wavelet at the slit combines to produce the resultant optical field EP . Now
at the aperture, all points will be in phase. In a medium of constant refractive index, the phase at P from each wavelet will be a function of the
geometric path alone. The resultant interference at P will then be a linear
superposition of the phases of each contribution.
(a)
(b)
86
CHAPTER 5. DIFFRACTION
region, where the diffraction pattern varies considerably with increasing distance from the aperture. This region is characterised by Fresnel diffraction.
Further from the screen as the radial distance from the aperture R becomes
very large compared to D, we have the far field region, in which the diffraction pattern settles down to a constant profile. This is Fraunhofer diffraction,
which we shall be analysing in the next sections. A more precise criterion
for the onset of Fraunhofer diffraction is given in Sec. 5.4.
5.4
Figure 5.3: Geometry of single slit diffraction in the far field for light normally
incident on the screen.
In this section, we shall analyse far-field (Fraunhofer) diffraction from a single slit in a screen. Figure 5.3 shows the geometry for a slit of width D,
through which monochromatic light of wavelength passes. In this case,
the light approaching the screen on the left is normally incident to it. That
is, the wavefronts of the incident light are parallel to the screen. Note that
since the electric field EP is for a point lying on a semicircle a distance R
from the centre of the slit, the angle must take a value on the interval
[/2, /2].
We can characterise the electric field at the slit by defining the field
strength per unit length EL , so that the actual field at x is dE(x) = EL dx.
The contribution to the electric field EP is then just the spherical wave emanating from the infinitesimal region dx
dEP =
87
EL
sin [t kr (x)] dx,
r(x)
(5.2)
where k = 2/. The total field EP is then the integral of Eq. (5.2) over the
slit width D
Z
D/2
EP =
D/2
EL
sin [t kr(x)] dx.
r(x)
(5.3)
From Fig. 5.3 we see that r(x) is given by the cosine rule
r2 (x) = R2 + x2 2Rx cos
,
(5.4)
so
2x
x2
sin
r(x) = R 1 + 2
R
R
1/2
.
(5.5)
+ ...,
2
8
(5.6)
so
(
)
2
1 x2
2x
1 x2
2x
r(x) = R 1 +
sin
sin + . . . ,
2 R2
R
8 R2
R
x2
x
x2
2
= R 1+
sin
sin + . . . ,
2R2 R
2R2
x
x2
2
= R 1 sin +
cos + . . . .
(5.7)
R
2R2
5.4.1
We may make the condition for Fraunhofer diffraction a little more precise
by inspecting the phase term kr(x) in more detail. From Eq. (5.7), we have
kx2
cos2 + . . . .
(5.8)
2R
Now the third term in Eq. (5.8) takes its maximum value when x = D/2
and = 0. That is
kr(x) = kR kx sin +
kD2
D2
=
.
(5.9)
8R
4R
The condition that this term makes a negligible contribution to the phase is
88
CHAPTER 5. DIFFRACTION
D2
.
4R
(5.10)
.
R
D
(5.11)
5.4.2
(5.12)
In Eq. (5.3), it is only the phase of the sin function that is particularly
sensitive to variations in r(x) and for this we use the approximation above.
For the 1/r(x) term, we may simply substitute 1/R. With these approximations, Eq. (5.3) becomes
Z
D/2
EP =
D/2
EL
sin [t kR + kx sin ] dx.
R
(5.13)
(5.14)
so, performing the integration over the x-dependent part of the complex
exponent, we have
Z
D/2
ikx sin
e
D/2
eikx sin
dx =
ik sin
D/2
=D
D/2
sin
,
(5.15)
where
=
kD
sin .
2
(5.16)
EL D sin
sin (t kR) .
R
(5.17)
EL D 2 sin 2 2
sin (t kR) .
I (, t) |EP | =
R
2
89
(5.18)
(5.19)
Figure 5.4 shows plots of Eq. (5.19) for /D ratios of 1/2 and 1/4. Both
curves exhibit a large central peak of intensity surrounded by smaller peaks
going out to larger angles. The zeros between the peaks occur at values of
kD
sin = m,
(5.20)
2
where m is an integer. Hence, the zeros around the central peak are given
by
=
.
(5.21)
D
Note that this result is only valid for D. For ratios of /D > 1, never
reaches due to the limits on . Hence Eq. (5.19) has no zeros. In this
case, we have for /D 1
sin =
sin 2
sin 2
= lim
= 1.
lim
0
/D
(5.22)
Taken with the result of Eq. (5.22), Eq. (5.21) therefore gives a measure
of the angular spread of the central diffraction peak. For /D 1, the
diffraction pattern approaches that for a point source, i.e. a spherical wave,
and the intensity varies little across all values of . As becomes less than
D, the central peak becomes narrower, approaching a delta function for
/D 1.
5.5
5.5.1
The analysis for a circular aperture proceeds along the same lines as that
for the single slit, except that we now need to integrate over an area to get
the total contribution to the field. Taking the origin to be at the centre of the
aperture and the z-axis to be perpendicular to plane of the aperture, the
problem may be rendered in spherical polar coordinates, with being the
polar angle.
The result for the intensity is then found to be
90
CHAPTER 5. DIFFRACTION
Figure 5.4: Far field diffraction patterns for /D = 1/2 and /D = 1/4.
2
,
(5.23)
Figure 5.5 (a) shows an example of the diffraction pattern for a circular
aperture. The central bright peak is known as the Airy disc, whilst the rings
around it are referred to as Airy rings.
where D is now the diameter of the aperture and J1 is the first order Bessel
function.
5.5.2
.
D
(5.24)
This has the same form as Eq. (5.21) for the single slit and the same observations about the spread of the central peak apply here.
Note that a distant point source will have the same diffraction pattern
as that for a circular aperture. Thus, Eq. (5.24) may be used to establish
limitations for the ability to resolve two such point sources, based on the
overlap of the central peaks. Using the small angle approximation, E. (5.24)
becomes
(a)
91
(b)
(c)
Figure 5.5: (a) Airy rings for a diffraction from a circular aperture. (b) The intensity profiles for two resolvable distant point sources. (c) Merged intensity
profiles for unresolvable distant point sources.
min 1.22
.
D
(5.25)
The Rayleigh criterion for the resolution of two points is then that the angular separation between them must be greater than min . This is illustrated
in Figs. 5.5 (b) and (c).
5.6
5.6.1
Figure 5.6 shows the geometry for diffraction through multiple slits. The
analysis of this situation in the far field proceeds similarly to that of single
slit diffraction. Taking N = 2 then produces Youngs well-known case for
the double slit.
For N slits, the total contribution to the field EP is
EP =
N
1 Z na+D/2
X
n=0
dEP .
(5.26)
naD/2
(5.27)
92
CHAPTER 5. DIFFRACTION
Figure 5.6: Geometry for multiple slit diffraction in the far field.
N
1 ikx sin na+D/2
X
e
n=0
ik sin
naD/2
N
1
X
n=0
N
1
X
eikna sin D
ein2 D
n=0
sin
,
(5.28)
where we have pulled out the factor for the single slit and defined
=
ka
sin
2
(5.29)
kD
sin .
2
(5.30)
and
=
The remaining factor in Eq. (5.28) is a geometric progression with common factor ei2
SN =
N
1
X
n=0
Multiplying this by
ei2
gives
ei2n .
(5.31)
93
Figure 5.7: Far field diffraction patterns for a single slit with /D = 1/2
(dashed line) and a multiple slit with a/D = 5/2 and N = 10.
SN ei2 =
N
X
ei2n .
(5.32)
n=1
(5.33)
SN
=
=
1 ei2N
1 ei2
eiN eiN eiN
sin N
= ei(N 1)
.
ei (ei ei )
sin
(5.34)
The phase factor ei(N 1) may be dropped out of Eq. (5.34) once the
squared modulus is taken. Note that since
sin N
= N,
(5.35)
0 sin
it is useful to explicitly incorporate a normalising factor 1/N into this ratio
(rather than to implicitly include it in I (0)). Hence, the intensity takes the
form
lim
94
CHAPTER 5. DIFFRACTION
I () = I (0)
sin N
N sin
2
sin
2
.
(5.36)
An example of this diffraction pattern is shown in Fig. 5.7 against the pattern
for a single slit with /D = 1/2 (dashed line) for values of a/D = 5/2 and
N = 10.
As /D becomes very large, the enveloping single slit pattern broadens
out and the interference pattern becomes a series of sharp peaks. Now it
can be shown that the maxima of
sin N
N sin
2
(5.37)
a
ka
sin m =
sin m = m,
2
(5.38)
(5.39)
This is the grating equation for a transmission grating with light of normal
incidence (see Fig. 5.8).
5.6.2
95
Off-axis incidence
In the foregoing analysis, the incident light was normal to the grating (which
we may refer to as on-axis). In the more general case, the incident light
may be at an angle to the grating normal (i.e. off-axis). This is illustrated
in Fig. 5.9. Note that the angles i0 and i00 are equal, giving the magnitude
of the angle between the incident light and the grating normal. Measuring
the angle anti-clockwise from the z-axis, we have
i = i0 .
(5.40)
(5.41)
Figure 5.9: The condition for constructive interference for off-axis incident
light passing through a transmission grating.
5.7
Diffraction gratings
For a reflection grating, the incident light arrives on the same side as the
diffracted light. Here, the sources of interfering spherical waves will be
perturbations on the grating that give rise to scattering.
Equation (5.41) still holds in this case. However, since i is now negative, the sign is usually brought out of the sinusoidal term to give
96
CHAPTER 5. DIFFRACTION
m = a (sin m sin i ) .
(5.42)
97
5.7.1
Resolving power
R=
(5.43)
(5.44)
98
CHAPTER 5. DIFFRACTION
5.7.2
Monochromators
Czerny-Turner monochromator
Figure 5.12 shows a schematic of a Czerny-Turner monochromator. Polychromatic light (light containing many frequencies) enters the monochromator via a slit at A. This is then focused into plane waves by a concave mirror
known as the collimator at B and directed towards the diffraction grating at
C. Ideally, the collimator should have a parabolic surface to perfectly image the input light without aberration. At the grating, only light meeting the
diffraction condition for the input angle i and output angle m are diffracted
in a collimated beam towards the centre of a second mirror at D known as
the camera. The camera provides the inverse operation of the collimator,
focusing the light to a point at the exit slit E. Wavelengths that did not satisfy the diffraction condition for these angles are reflected away from the
output path. The output light will therefore only contain components within
a narrow wavelength range.
The slits and mirrors are kept fixed in this design, whilst the grating
may be rotated to meet the diffraction condition for different wavelengths.
Thus the wavelength (or frequency) decomposition of the input light may be
99
ascertained from the output of a detector placed at the exit slit. Calibration
of the monochromator is achieved using monochromatic light of a known
wavelengths and adjusting the rotation of the grating until the maximum
intensity output is obtained.
Optical spectrum analyser
5.8
We have seen how a wave may spread out as it is diffracted through a narrow aperture. An analogous situation pertains when light encounters an
opaque object: the wavefronts may now diffract around the object. Some
100
CHAPTER 5. DIFFRACTION
(a)
(b)
Figure 5.14: (a) Diffraction around the side of an object according to Huygens Principle. (b) Including interference via the Huygens-Fresnel Principle.
insight into this phenomenon is furnished by Huygens Principle, as illustrated in Fig. 5.14 (a). Here we see how light at the edge of the object,
acting as a point source of a wavelet, propagates around the object. This
however, is not the full story. According to such a picture , all waveforms
ought to be diffracted around the object by /2. The fact that this does not
happen is explained via interference.
In Fig. 5.14 (b) we see a similar picture to the earlier diagrams of light
passing through a slit, in which the condition for constructive interference
at P is met by the wavelength of light being greater than the maximum path
difference
~
> AB
(5.45)
.
Now as is reduced, so must the maximum path difference to maintain
constructive interference. Hence, the angle will also tend to be reduced,
limited the angular spread of the wavefront.
For an object in a train of wavefronts, this means that if the size of the
object is comparable to the wavelength, the waves will tend to diffract right
around the object, as illustrated in Fig. 5.15.
5.9
Summary
5.9. SUMMARY
101
(a)
(b)
Figure 5.15: (a) Waves encountering an object with dimensions that are
greater than the wavelength. (b) Diffraction around an object of dimensions
comparable with the wavelength.
Far field or Fraunhofer diffraction
The diffraction pattern settles down to a constant profile.
(5.46)
where
=
kD
sin ,
2
(5.47)
.
R
D
(5.48)
102
CHAPTER 5. DIFFRACTION
The Airy disc
The intensity for far-field diffraction from a circular aperture of
diameter D is given by
2
,
(5.49)
.
D
(5.50)
sin N
N sin
2
sin
2
.
(5.51)
ka
sin ,
2
(5.52)
kD
sin ,
2
(5.53)
(5.54)
5.10. REFERENCES
103
R=
(5.55)
Monochromators
Diffraction gratings may be used in monochromators, such as the
Czerny-Turner design, to measure the frequency composition of polychromatic light.
Optical spectrum analyser
The monochromator may be integrated into an automated unit known
as an optical spectrum analyser.
5.10
References
104
REFERENCES
Part III
Electromagnetic Waves
105
6.1
General remarks
Alongside the quantum mechanical explanation of light, Maxwells prediction of the generation of electromagnetic (EM) waves is key to our understanding of optical phenomena. Furnishing Thomas Youngs conclusive observations of the wave nature of light with a physical explanation, Maxwell
predicted [1] in 1865 that electric and magnetic fields would mutually induce each other, propagating with a constant speed in a vacuum given by
c = (0 0 )1/2 , reproducing the measured speed of light. In turn, 0 and
0 are both fundamental physical constants, determining the response of
the vacuum to electric and magnetic fields respectively. Maxwells prediction was then later confirmed by Hertz in a paper of 1892 [2] reporting the
generation of radio waves in the laboratory setting.
Although it was initially believed that EM waves would require some
kind of physical medium for their propagation (the luminous ether), this
was later shown by Einstein to be superfluous to requirements in his 1905
paper on Special Relativity [3]. Einstein was also a leading figure in the
early development of quantum theory. Despite the apparently contradictory
nature of the so-called wave-particle duality of light, the two pictures are,
in fact, complimentary to one another. In this Chapter, we shall reiterate the
relation between the classical theory of optical absorption and the quantum
mechanical explanation.
A crucial aspect of this Chapter is the introduction to the electric susceptibility tensor. Although we shall currently limit our consideration to linear,
isotropic and homogeneous media, the explanation of optical phenomena
in anisotropic media (to be covered in the Chapters on Crystal Optics) will
depend heavily on our understanding of the susceptibility tensor.
Lastly in this Chapter, we shall consider the flux of electromagnetic energy. We specify this in terms of the Poynting vector. Again, we shall
anticipate subtleties in our understanding of the energy flux to arise in the
context of anisotropic media.
107
108
6.2
Learning objectives
6.3
Maxwells equations
D = f ,
(6.1)
B = 0,
(6.2)
B
E =
,
t
D
H = jf +
,
t
(6.3)
(6.4)
The first of these, Eq. (6.1), is Gauss Law, stating that the divergence
of the electric displacement D is equal to the free charge density f . In
other words, the free charges are the sources or sinks D. The electric
displacement is related to the electric field E and the electrical polarisation
P of a material medium due to E via
D = 0 E + P,
(6.5)
109
(magnetic monopoles). Pictorially, this means that every field line of the
magnetic field eventually joins up with itself in a loop.
Equation (6.3) is Faradays Law of electromagnetic induction, which
says that a changing magnetic field induces a non-conservative electric
field. A conservative field F is a vector field related to a scalar potential
via F = . However, the curl of this would be , which is
identically zero. Meanwhile, E is also identically zero, meaning that
the induced electric field has no source or sinks and that the field lines all
join up in loops.
The fourth equation, Eq. (6.4) is Amperes Law modified by the addition of the displacement current density , which Maxwell realised must be
present to meet the requirement of charge conservation. Amperes law relates the curl of the field vector H to the free current density jf . H is related
to B and the magnetisation M of a material medium by
H=
1
B M,
0
(6.6)
6.4
6.4.1
(6.7)
and
H=
1
B.
0
(6.8)
In addition, the free charge density f and free current density jf will both
be zero. Maxwells equations therefore reduce to
E = 0,
(6.9)
B = 0,
(6.10)
B
,
E =
t
E
B = 0 0
,
t
Taking the curl of Eq. (6.11), we have
(6.11)
(6.12)
110
E=
( B)
.
t
(6.13)
2E
.
t2
(6.14)
(6.15)
2E
.
t2
(6.16)
A similar argument leads to the wave equation for the magnetic field
2 B = 0 0
2B
.
t2
(6.17)
Comparison with the general wave equation Eq. (3.24) of Chapter 3, reveals
that the speed of light in a vacuum is
c = (0 0 )1/2 ,
(6.18)
6.4.2
We shall assume plane wave solutions of Eq. (6.16) or Eq. (6.17) of the
form
E (r, t) = E0 ei(krt)
(6.19)
(6.20)
i,
t
(6.21)
111
2
2
t2
(6.22)
ik .
(6.23)
and
(6.24)
which gives
c=
,
|k|
(6.25)
k
,
(6.27)
6.5
6.5.1
(6.28)
where E is the electric susceptibility tensor. Equation (6.28) may be referred to as a constitutive equation for the polarisation in terms of E . However, the form of this equation does not make obvious that the susceptibility
tensor characterises the frequency response of the material to an applied
field. The temporal response of the medium may be encapsulated in a response function R (t) and the electrical polarisation then rendered in the
form
112
R (t ) E ( ) d.
P (t) = 0
(6.29)
(6.30)
(6.32)
(6.33)
to obtain
Z
P (t) = 0
E E eit d.
(6.34)
P eit d,
(6.35)
P (t) =
P = 0 E E .
Hence, we regain Eq. (6.28), which we now see is actually in the frequency
domain. However, to avoid excessive use of subscripts, we shall usually
take this to be tacit.
6.5.2
Linearity
In general, we may express E in tensorial form via
P i = 0
X
ij
(1)
ij Ej + 0
X
ijk
(2)
ijk Ej Ek + 0
X
ijkl
(3)
ijkl Ej Ek El + . . .
(6.37)
113
(1)
Here, ij depends only linearly on the applied field whilst the components
with superscripts (n) for n > 1 are non-linear in E. We shall assume
throughout that these non-linear terms are small and neglect them. Thus,
we assume that the material is linear and put
(1)
(6.38)
(E )ij = ij .
In the case of very high intensity light, the electric field is large and these
non-linear terms may need to be incorporated. This is the subject matter of
non-linear optics.
There is also a constitutive equation analogous to Eq. (6.28) for the
magnetisation of the material
M=
B
B.
0
(6.39)
E 0
0
(6.40)
E = 0 E 0 .
0
0 E
Note that, if I is the identity matrix, this relation is just
E = E I.
(6.41)
Wave solutions
In a dielectric, i.e. an electrically insulating medium, we may take f and jf
to be zero. Equations (6.1) and (6.4) now become
D=0
(6.42)
and
H=
In the meantime, Eq. (6.5) becomes
D
.
t
(6.43)
114
D = 0 (I + E ) E = 0 E,
(6.44)
1
1
(I B ) B =
B.
0
0
(6.45)
(6.46)
and
B = 0 0
E
.
t
(6.47)
v = (0 0 )1/2 =
c
.
n
(6.49)
where
n = ()1/2
(6.50)
6.6
(6.51)
pi =
(6.52)
qi ui ,
where qi is the charge displaced from equilibrium by ui for the ith dipole.
The equation of motion of the system may now be written as
X 2
X d2 pi
1 dpi
qi E0 it
2
+ i pi +
=
e ,
(6.53)
dt2
dt
mi
i
where i , qi and mi are the resonant frequency, charge and mass of the
ith oscillator respectively and is the driving frequency. The time constant
i characterises the dampening force acting on the ith oscillator and, as we
shall see, is associated with the absorption of a quantum of radiation.
Applying Eq. (6.28) to Eq. (6.53) gives
X 2
1
qi E0 it
2
0 E E + E + 0 E =
e .
(6.54)
mi
i
mi
i
X
i
qi2 /mi
.
i2 2 + i/i
(6.56)
116
6.6.1
Relative permittivity
Figure 6.1: Curves showing the real and imaginary parts of the relative
permittivity near an absorption resonance (at i ).
The relative permittivity is given in terms of E by
= 1 + E .
(6.57)
Expressing this in terms of real and imaginary parts, this may be written
= 1 + Re (E ) + iIm (E ) ,
= 1 i2 ,
(6.58)
1
0
X
i
Ci i2 2
2
i2 2 + 2 /i2
(6.59)
and
2 = 1
0
X
i
Ci /i
.
2
i2 2 + 2 /i2
(6.60)
117
That is, the material medium shows a strong propensity to absorb some of
the energy of the optical field at this frequency.
The frequency dependence of leads to a frequency dependence of
the refractive index and, in turn, the wave speed. This leads to the phenomenon of dispersion, in which the different frequency components of the
electromagnetic field in a medium spread out from one another due to their
different wave speeds.
6.6.2
Sellmeiers equation
Ci
.
i2 2
(6.61)
Ci 2i 2
.
(2c)2 2 2i
(6.62)
X
i
X
i
X Ai 2
.
2 2i
i
(6.63)
6.7
6.7.1
Optical loss
The absorption coefficient
118
Figure 6.2: Graph of the refractive index of borosilicate crown glass using
Sellmeiers equation with empirically fitted parameters.
charge. This may be formulated via consideration of the complex part of
the relative permittivity as given in Eq. (6.58). If the relative permittivity is
complex, then the refractive index will also become complex. We may write
this as
n = n0 i.
(6.65)
(6.66)
c
c
=
.
n
n0 i
(6.67)
(6.68)
119
2z
I (z) |E (z, t)| = |E0 | exp
.
c
2
(6.69)
(6.70)
where
=
2
c
(6.71)
is the absorption coefficient. Note that this is exactly the same result obtained in Chapter ?? for the photon description of light.
6.7.2
Photon absorption
The process of optical absorption just described has an immediate quantum mechanical interpretation. Recall that the energy of a photon is given
by = h. In terms of resonant frequency, we may consistently interpret
the energy difference between electronic states as
=h
i .
(6.72)
6.8
Time symmetry
120
(6.74)
i = 1 + 2 .
(6.75)
where
(This arises out of the conservation of energy, where the energy of a photon
is
h). For the linear susceptibility, we would have, by analogy
(1) (i ; j ) ,
(6.76)
i = j .
(6.77)
where
Since we shall normally limit our consideration to the linear case, this
notation is rather too cumbersome for general use. However, we will find it
useful for making clear the implications of time reversibility. As a first step,
however, let us write the angular frequency in a more general form as a
complex entity
j = j,R + ij,I
(6.78)
and substitute this into Eq. (6.33) to obtain, for the linear susceptibility
Z
(1)
(i ; j ) =
R ( ) ei(j,R +ij,I ) d
(6.79)
(i ; j ) =
R (t) ei(j,R ij,I ) d.
(6.80)
121
(6.81)
(6.82)
If we now re-enforce the constraint that should only have real values, we
then have
(1) (i ; j ) = (1) (j ; i ) .
(6.83)
(1) (i ; j ) Ej eit ,
(6.84)
(1) (i ; j ) Ej eit .
(6.85)
(6.86)
However, since
Pi eit
= 0
(1) (i ; j ) Ej eit ,
(6.87)
this implies
(1) (i ; j ) = (1) (i ; j ) .
(6.88)
(6.89)
In other words
In a lossless material, the linear electric susceptibility tensor is a symmetric matrix.
This result has a number of useful consequences. We will encounter one of
this in Section 6.10 on the Poynting vector. Further consequences of more
general symmetries will be explored in Chapter 11.
122
6.9
Dispersion
6.9.1
The rainbow
Figure 6.3: Light from the Sun transmitting into a spherical droplet of water,
reflecting once and transmitting at an angle 0 to the horizontal.
Perhaps the most beautiful common example of dispersion is that of the
rainbow, often seen on rainy days when the Sun also appears. Figure 6.3
shows the case for a single frequency component of the light from the Sun
transmitting into a spherical droplet of water, reflecting once and transmitting out again at an angle 0 to the horizontal.
6.9. DISPERSION
123
(6.90)
Note that 0 does not vary monotonically with the input angle 1 but has
a maximum value. To determine this, we may differentiate Eq. (6.90) with
respect to 1 and set the result equal to zero. That is,
d2
d0
=4
2 = 0.
d1
d1
(6.91)
(6.92)
So, since
1
2 = sin
n1
sin 1 ,
n2
(6.93)
we have
d2
d1
i1/2 ,
(6.94)
1/2
(6.95)
= 2,
which yields
1 = sin1
"
4
3
1
3
n2
n1
2 #1/2
(6.96)
2 = sin1
Thus,
n1
sin 1
n2
"
#
n 4 1 n 2 1/2
1
2
.
= sin1
n2 3 3 n1
(6.97)
124
0,max = 4 sin
n1
2 sin1 () .
n2
(6.98)
where
"
4 1
=
3 3
n2
n1
2 #1/2
.
(6.99)
(6.100)
Figure 6.4: Since blue light has a greater refractive index than red in water,
it is refracted to a greater degree and emerges at a shallower angle to the
horizontal. Light from many droplets then builds up a pattern seen as a bow
from the ground, with red light at the outer rim.
Dispersion of water
In the optical range, water exhibits normal dispersion, which means that violet light ( 400 nm) sees a higher refractive index than red light ( 700 nm).
As a consequence, violet (and then blue) light is refracted to a greater degree than red by a spherical water droplet, as shown in Fig. 6.4. As a
result, the deviation of red light has the greatest angle to the horizontal.
125
From a particular viewing point, an observer will see light refracted from
many droplets with the net appearance of the rainbow with red light lying
around the outer rim.
In fact, if the viewer was at a high enough altitude (in practice, in an
aircraft), the rainbow would appear as a complete circle. On ground level,
however, the lower portion of the circle is cut off and we just see a bow.
6.10
u=
1
2
(E D + B H) ,
(6.101)
(6.102)
and S is the Poynting vector giving the flow of energy crossing unit area.
Hence, u/t is the rate at which the energy density decreases.
Taking the time derivative of Eq. (6.102), S is the divergence of the
energy flow (the volume integral of this being the energy flux through the
surface of the volume) and jf E is the work done on any free charges by
the electric field.
The time derivative of Eq. (6.102) is
u
1 E
D B
H
=
D+E
+
H+B
.
(6.103)
t
2 t
t
t
t
We shall be assuming a linear and homogeneous medium but allowing it
to be anisotropic. We shall therefore need to make use of the constitutive
equations, Eqs. (6.44) and (6.45) for the electric displacement D and field
vector H respectively.
Since the susceptibility tensors have no time dependence we have, from
Eq. (6.44),
D
E
= 0 (I + E )
.
(6.104)
t
t
Now, the ith component of the matrix product of E and the time derivative
of E is
X
Ej
E
E
=
ij
.
(6.105)
t i
t
j
Hence
126
X
Ej
E X
E
E E
=
Ei E
=
Ei ij
.
t
t i
t
i
(6.106)
ij
E
E X Ej
=
ji Ei =
E E.
t
t
t
(6.107)
ij
D
t
= 0 E
(6.108)
H
B
=
H.
(6.109)
t
t
Using Eqs. (6.108) and (6.109), Eq. (6.103) for the time derivative of the
energy density becomes
B
D B
u
=E
+
H.
(6.110)
t
t
t
We now apply Faradays law, Eq. (6.3), and Maxwells modified form of
Amperes law, Eq. (6.4), to give
u
t
= ( E) H E ( H jf ) ,
= ( E) H E ( H) + jf E.
(6.111)
(6.112)
(6.113)
(6.114)
This is the instantaneous Poynting vector, being the energy flowing across
unit area per unit time.
6.11. SUMMARY
127
Isotropic medium
In an isotropic medium, S is in the direction of the wave-vector k (although
this is not true in the anisotropic case). It may then be shown that
2
S = kE
0
0
1/2
(6.115)
hSi =
1 2
2 kE0
0
0
1/2
,
(6.116)
6.11
Summary
(6.117)
(6.118)
(I B ) = 1 .
(6.119)
and
128
c
,
n
(6.120)
where
n = ()1/2
(6.121)
(6.122)
is satisfied.
Optical loss in media
The imaginary part of the refractive index implies a loss of energy
from the optical field. The intensity of the field then decays exponentially as
I (z) = I (0) ez ,
(6.123)
where
=
2
c
(6.124)
(6.125)
6.12. REFERENCES
6.12
129
References
130
REFERENCES
7. Polarisation
7.1
General remarks
7.2
Learning objectives
132
CHAPTER 7. POLARISATION
Circular polarisation
= /2 right circularly polarised
= /2 left circularly polarised
Elliptical polarisation
Jones matrix
Linear polariser
Rotation of a state of polarisation by an angle
Wave plates
Birefringence
Half-wave plate
Quarter-wave plate
General retardation plate - phase shift =
Analysis of polarised light
Malus Law
7.3
7.3.1
Linear polarisation
The Jones vector
(7.1)
where E0 is a Jones vector containing the details of the polarisation. A principle advantage of using the Jones vector notation is that the exponential
factor multiplying it (sometimes referred to as the propagator ) often cancels
from expressions, leaving the description of the light entirely in terms of the
polarisation.
For the time being, we shall assume that we have linearly polarised
light, although we shall see that other polarisation states are possible. Taking the wave direction to be along the z-axis for definiteness, we have, in
the present case
133
E0 = |E0 |
cos
sin
(7.2)
where is the angle of E0 relative to the x-axis. Note that we did not need
to specify the direction of E in the z-direction since this is zero by definition.
It should also be noted that the column vector (which we may refer to in
) is normalised. That is, the dot product of p
with itself is
short-hand as p
unity. In terms of matrix multiplication, this can be represented by multplying
=
p
p
px
py
px
py
= p2x + p2y = 1.
(7.3)
1
(i) = 0, so E0 = |E0 |
. This is linearly x-polarised light.
0
0
. This is linearly y-polarised light.
(ii) = /2, so E0 = |E0 |
1
7.3.2
Linear polarisers
134
CHAPTER 7. POLARISATION
Figure 7.1: A linear polariser based on dichroism. The EM radiation approaches a grid of horizontal conductors which absorb the field in that direction. Linearly polarised light is then transmitted perpendicularly to this
(along the transmission axis (TA) of the polariser).
Close to total attenuation of the EM field might be achieved via some
cross-hatched network of wires, provided the spacings between them were
on a commensurate scale to the wavelength of light. Such meshing may be
seen threading the windows of microwave ovens, for instance.
Materials for optical polarisers
Such an effect may also be achieved for optical wavelengths. One possible
candidate is tourmaline, a crystal boron silicate mineral containing various
impurities. This mineral belongs to the trigonal crystal system (see Chapter 11). A major limitation of tourmaline as an effective polariser is that it is
highly colour dependent.
A better choice is herapathite or iodoquinine sulfate. The story goes
that this material was discovered by accident when a student added iodine
to the urine of a dog that had been fed quinine (!) This lead to the formation
of green crystals that turned out to polarise light. However, it turns out that
it is very difficult to grow large crystals of herapathite.
Polaroid sheets
Progress was initially made by embedding small crystals of herapathite
into a polymer. This was the first Polaroid sheet known as J-sheet. A
135
cos
sin
,
(7.4)
Linearly x-polarised
E0 = E0
1
0
,
(7.5)
136
CHAPTER 7. POLARISATION
Linearly y-polarised
E0 = E0
0
1
.
(7.6)
Figure 7.4: Sketch of a linear polariser with its transmission axis (TA) at an
angle to the Ex -axis. Initially unpolarised light approaches the polariser,
which then only passes light polarised along the direction of its transmission
axis.
7.4
Jones matrices
The Jones calculus may be used to formally model the optical elements
that prepare or change a state of polarisation. Since these operators must
act on a column vector, such optical elements are represented by matrices.
7.4.1
Linear polariser
Firstly, let us consider an element that passes only light polarised in the
Ex -direction. Formally, we are looking for an operator that satisfies the
mapping
E0x
E0y
E0x
0
.
(7.7)
137
This mapping will then involve multiplying the left-hand term by a matrix
representing the optical element. For simple elements, it is often straightforward to find the elements of the matrix by inspection. Thus, we see that
Eq. (7.7) is satisfied by
E0x
1 0
E0x
=
.
(7.8)
0
0 0
E0y
Denoting this matrix by Px , the operation of an x-linear polariser is then
encapsulated by
Px =
1 0
0 0
(7.9)
(7.10)
0 0
0 1
(7.11)
In the more general case, we consider a linear polariser with its transmission axis at an angle to Ex -axis. We may specify this via a unit vector
parallel to the transmission axis,
cos
=
.
(7.12)
p
sin
The amplitude of the transmitted light is then
= E0x cos + E0y sin
|E1 | = E0 p
(7.13)
) p
.
E1 = (E0 p
(7.14)
and
.
E0y
E0x cos sin + E0y sin2
(7.15)
cos2
cos sin
cos sin
sin2
.
(7.16)
138
7.4.2
CHAPTER 7. POLARISATION
Rotator
E0x
0
E0x cos
E0x sin
(7.17)
cos 0
sin 0
R,x =
(7.18)
However, in the general case, we would also require that y-linearly polarised be mapped according to
0
E0y
E0y sin
E0y cos
(7.19)
0 sin
0 cos
(7.20)
We may obtain both operations simultaneously by adding these two matrices, giving
(7.21)
R = R,x + R,y
or
R =
cos sin
sin cos
.
(7.22)
Hence, this is the required rotation matrix (c.f. Appendix A.2). Note that the
inverse operation would be a rotation by .
7.4.3
Combining matrices
Note that although R rotates the electric field vector E by within a given
coordinate system, it may also be interpreted as rotating the coordinate
system by (without physically changing E) and vice versa for R . Let
us take the latter assumption and apply it to the case of a linear polariser
with its transmission axis at an angle to the Ex axis.
We first transform to the coordinate system Ex0 -Ey0 which is rotated from
Ex -Ey by . This is accomplished via the rotator R .
139
E00 = R E0 .
(7.23)
(7.24)
(7.25)
M = R Px R .
Substituting from Eqs. (7.103) and (7.9), we have
cos sin
sin cos
1 0
0 0
cos sin
sin cos
cos sin
cos sin
,
=
0
0
sin cos
cos2
cos sin
= P ,
=
cos sin
sin2
M =
,
(7.27)
as found earlier.
We see here an example of the way in which operations performed one
after another may be encapsulated by multiplying the matrices for the individual processes together. However, it is important that this multiplication
is carried out in order (later operations acting on the left), since matrix multiplication is not commutative.
7.5
7.5.1
140
CHAPTER 7. POLARISATION
|E0x |
eix
0
+ |E0y |
0
eiy
ei(tkz) ,
(7.28)
(7.29)
Ee
= |E0 |
cos
i
e sin
ei(tkz) ,
(7.30)
where, again, is the angle between the electric field and the x-axis. Absorbing the common phase shift into the electric field vector, the Jones
vector for elliptical polarisation is
E0 = |E0 |
cos
i
e sin
.
(7.31)
We may pause to identify some special cases of Eq. 7.31 that reduce back
down to linear polarisation.
141
cos
sin
E0 = |E0 |
(7.32)
(7.33)
so
E0 = |E0 |
cos
sin
(7.34)
7.5.2
Circular polarisation
Let us now consider the cases where = /2 and impose the condition
= /4, so that cos = sin = 21/2 .
(iii) = /2. This gives
ei/2 = i,
(7.35)
so
|E0 |
E0 =
2
1
i
(7.36)
1
i
ei(tkz) .
(7.37)
cos (t kz)
sin (t kz)
(7.38)
142
CHAPTER 7. POLARISATION
Thus, in space the electric field vector, E, rotates around the propagation axis (z-axis) in the same direction as a right-handed screw.
Hence, this is known as right circular polarisation. Note that looking
along the approaching wave towards the origin, an observer would
see E at a fixed point in space rotating clockwise in time.
(7.39)
so
|E0 |
E0 =
2
1
i
.
(7.40)
This reverses the sign of y-component from the previous case. Hence,
the electric field rotates around the propagation axis in the opposite
direction to right circular polarisation. Moreover, at a given point in
space, E rotates anti-clockwise in time. This is therefore known as
left circular polarisation.
Summarising the results for circularly polarised light,
right circular polarised light
E0
E+ =
2
1
i
(7.41)
7.5.3
1
i
.
(7.42)
The results of the previous section held for the case where imposed the
condition that amplitudes of the x and y components of the electric field
were equal. If we now relax this condition, the resultant polarisation is no
longer circular. Instead, for phase shifts of = /2, we will have
cos
E = |E0 |
ei .
(7.43)
i sin
where = t kz. Considering just the real part, we then have
Re [E] = |E0 |
143
cos cos
i sin sin
,
(7.44)
which just traces out an ellipse in either a clockwise ( = /2) or anticlockwise ( = /2) direction. Putting
Ey
E0y
2
+
Ex
E0x
2
= 1.
(7.45)
This is an ellipse in standard form, so its principle axes are aligned to the
x and y coordinate axes. Hence the maximum and minimum values of Ex
are E0x and similarly for Ey .
7.5.4
In the most general case, the phase shift is arbitrary and there is no
restriction on the relative amplitudes of E0x and E0y . The electric field
vector may now be written as
E0x
E=
ei ,
(7.46)
ei E0y
where = t kz as before. The real part of E is
E0x cos
E=
.
E0y cos ( + )
(7.47)
Firstly, let us consider the rotation of the field vector around the propagation direction. From Eq. (7.47), we see that the real part of E makes an
angle
1 E0y cos ( + )
= tan
(7.48)
E0x cos
to the x-axis. Taking the derivative with respect to time, we have
144
CHAPTER 7. POLARISATION
= 2
2 cos2 ( + ) .
2
t
E0x cos + E0y
(7.49)
(7.50)
Ey = E0y cos ( + ) .
(7.51)
and
This gives
Ex
= cos and
E0x
145
Ey
= cos cos sin sin ,
E0y
(7.52)
so
Ey
Ex
(7.53)
Ey
E0y
2
2
Ex
cos +
E0x
Ex 2
+
cos2 = sin2 sin2
E0x
= 1 cos2 sin2 .
Ey
E0y
(7.54)
Ey
E0y
2
+
Ex
E0x
2
2
Ey
E0y
Ex
E0x
cos = sin2 .
(7.55)
(7.57)
(7.58)
and
Ex Ey = Ex02 Ey02 cos sin + Ex0 Ey0 cos2 sin2 .
(7.59)
146
CHAPTER 7. POLARISATION
Since the ellipse is in standard form in the Ex0 -Ey0 coordinate system, the
cross-terms Ex0 Ey0 must disappear. From Eq. (7.55), we therefore require
!
1
cos 2
1
2
2
cos = 0,
(7.60)
sin 2
2
E0x E0y
E0y
E0x
where we have used the identities
2 cos sin = sin 2
(7.61)
(7.62)
and
Hence, we have
tan 2 =
2E0x E0y
2 E 2 cos .
E0x
0y
(7.63)
2 (E0y /E0x )
1 (E0y /E0x )2
cos .
(7.64)
Defining tan = E0y /E0x and using the double angle identity for tan, this
may be expressed as
tan 2 =
2 tan
cos = tan 2 cos .
1 tan2
(7.65)
7.5.5
Limiting cases
We may now apply the special conditions considered in the previous subsections to our general formulation for the polarisation.
(i) = 0. In this case cos = 1, sin = 0 and Eq. (7.55) reduces to
Ey
Ex
E0y
E0x
2
= 0,
(7.66)
yielding
Ey =
E0y
Ex .
E0x
(7.67)
147
This is the equation of a straight line with gradient equal to the ratio of
the amplitudes of the y and x components of the field. We therefore
have linear polarisation as expected. Note that this implies that it is
the phase shift that gives rise to the elliptical shape that E traces out.
(ii) = . In this case cos = 1, sin = 0. Thus Eq. (7.55) gives us
Ey =
E0y
Ex .
E0x
(7.68)
Again, we have linear polarisation with the negative of the gradient for
the = 0 case.
(iii) = /2. Now cos = 0 and sin = 1. Equation. (7.55) therefore
reduces to Eq. (7.45), the equation of an ellipse in standard form.
Ey
E0y
2
+
Ex
E0x
2
= 1.
(7.69)
(7.70)
7.6
Wave plates
7.6.1
Birefringence
A retardation or wave plate is an optical element that produces some retardation between the orthogonal components of the wave. The physical origin
of this retardation is due to the phenomenon of birefringence, in which the
components of the wave see a different refractive index (and hence have
a different phase velocity) depending on the orientation of the polarisation
within some anisotropic material. The subject of anisotropy is dealt with in
detail in Part V on Crystal Optics. Meanwhile, we shall describe a simplified
picture for the sake of insight.
Just as a linear polariser has a particular transmission axis, so we can
define two transmission axes perpendicular to the propagation direction for
a wave plate. These are known as the fast and slow axes, corresponding
to the wave speeds of the transmitted orthogonal components. Thus, the
phase of the wave along the fast axis is ahead of that along the slow axis.
The speed of the waves is, of course, determined by the refractive index that it sees. This is determined by a construction known as the index
148
CHAPTER 7. POLARISATION
ellipsoid. This concept will be fully developed in Chapter 12. For now, we
shall just consider an ellipsoid defined by the equation
x2
y2
z2
+
+
= 1.
n2o
n2o n2e
(7.71)
where no and ne are called the ordinary and extraordinary refractive indices
respectively. The fact that there are only two special refractive indexes
reflects the fact that the crystal type Eq. (7.71) describes has only one
optical axis. Hence, such a material is described as being uniaxial.
Consider that case of two orthogonally polarised plane waves, Ee and
Eo propagating with the same wavevector k in this crystal, as shown in
Fig. 7.7. The plane perpendicular to k intersects the index ellipsoid in an
ellipse. One of the axes of this ellipse is always equal to the ordinary refractive index no , the other is dependent on the extraordinary refractive index
ne and the angle of k to the extraordinary axis. Thus, one of the plane
waves will travel with a wave speed vo = c/no , the other with wave speed
vo () = c/n (). These are known as the ordinary and extraordinary waves
respectively.
Figure 7.7: The index ellipsoid for a uniaxial crystal projected in 2D showing two orthogonally polarised plane waves, Ee and Eo propagating with
the same wavevector k. The plane perpendicular to k intersects the index
ellipsoid in an ellipse. One of the axes of this ellipse is always equal to the
ordinary refractive index no , the other is dependent on the extraordinary
refractive index ne and the angle of k to the extraordinary axis.
149
(7.72)
For a wave with extraordinary and ordinary components, Ee and Eo respectively, propagating in a direction r, we may write
n()r
Ee = E0 exp i t
(7.73)
c
and
h
no r i
Eo = E0 exp i t
(7.74)
c
where we have used k = /v = n/c. The second of these equations may
be re-written
n()r
Eo = E0 exp i t
c
exp i [n() no ] r .
c
(7.75)
7.6.2
n()
r.
c
(7.76)
Half-wave plate
150
CHAPTER 7. POLARISATION
1 0
0 1
(7.79)
Thus, this has the effect of reversing the direction of y-component sin
(where is the initial angle of the polarisation to the x-axis). In other words,
the half-wave plate has the effect of rotating a state of linear polarisation
by 2 through the x-axis. Note that this is equivalent to a rotation of 20
through the y-axis, where 0 is the angle of the electric field vector to the yaxis. Thus, whilst the fast axis may be aligned in either the x or y directions,
we may say unambiguously that
a half-wave plate has the effect of rotating a linear state of polarisation
by an angle 2 through the fast (or slow) axis, where is the initial
angle of the electric field vector to the fast (or slow) axis.
7.6.3
Quarter-wave plate
1 0
0 i
.
(7.80)
(7.81)
Comparing this result to Eq. (7.43), we see that this yields elliptically polarised light. In this case, the principle axes of the ellipse are aligned with
the Ex and Ey axes. In the specific case where
cos = sin ,
we have = /4 and Eq. (7.81) reduces to
|E0 |
1
E1 =
.
2 i
(7.82)
(7.83)
This is the Jones vector for circularly polarised light. Specifically, as given
in the last section, for = /2 we have right circular polarisation and for
= /2 we have left circular polarisation. Thus
151
7.6.4
7.6.5
1 0
0 ei
.
(7.84)
3D glasses
152
CHAPTER 7. POLARISATION
then pass one of these projections and block the other. This is achieved
by first passing the light through a quarter wave plate and then a linear
polariser, the linear polarisers being rotated at 90 to each other in each
lens.
The two projections are right and left circularly polarised. Passing each
through a quarter wave plate gives
E0
M/2 E+ =
2
E0
M/2 E =
2
1 0
0 i
1 0
0 i
1
i
E0
=
2
1
1
(7.85)
and
1
i
E0
=
2
1
1
(7.86)
Note that Eq. (7.85) yields linearly polarised light aligned along the y = x
axis, whilst Eq. (7.86) gives linearly polarised light along the y = x axis.
Hence, if this is followed by a linear polariser with its transmission axis
aligned along y = x, the combination will pass left circularly polarised light
and block right circularly polarised light.
For the other lens, the opposite effect may be achieved either with a
M/2 quarter wave plate and the same orientation linear polariser or the
same wave plate and an orthogonally orientated linear polariser. Note, that
due to the rotational symmetry of the circularly polarised light, tilting the
glasses does not effect the analysis of the light (an advantage over just
using linear polarisers).
Figure 7.9: Analysis of circularly polarised light into linearly polarised light.
7.7
7.7.1
153
cos 1 cos t
sin 1 cos (t + )
.
(7.87)
cos 2
sin 2
.
(7.88)
The analyser will only pass the component of light polarised in this direction.
Thus, the amplitude of the transmitted light Ep is found by taking the dot
product
(7.89)
Putting
(7.91)
2
2
E0x
cos2 t cos2 2 + E0y
cos2 (t + ) sin2 2 +
154
CHAPTER 7. POLARISATION
|Ep |
=
=
Z 2/
|Ep |2 dt
2 0
1 2
2
E0x cos2 2 + E0y
sin2 2 + 2E0x E0y cos 2 sin 2 cos .
2
(7.92)
(7.93)
This is known as Malus Law for the transmission of linearly polarised light.
For circularly polarised light, r = 1 and = /2. In this case, we have
I () = I0 cos2 + sin2 = I0 .
(7.94)
7.8
Summary
Linear polarisation
Light polarised in a fixed direction all along the propagation direction
is said to be linearly polarised. For light travelling in the z-direction,
the general expression for linearly polarised light is
E0 = E0
cos
sin
(7.95)
Linearly x-polarised
E0 = E0
1
0
0
1
(7.96)
(7.97)
Linearly y-polarised
E0 = E0
7.8. SUMMARY
155
Retardation
In anisotropic media, orthogonal components of the light may see
different refractive indices. This introduces a phase shift between the
components known as the retardation .
Circular polarisation
If the y component of a linearly polarised plane wave is multiplied by
a factor ei/2 , then it will acquire a phase shift of = /2. When
E0x = E0y , this leads to circularly polarised light. There are two cases
to consider:
= /2 right circularly polarised
In this case, the electric field vector at a particular point along the
z-axis rotates in a clockwise direction in the Ex -Ey plane. This is
known as right circularly polarised light.
E0
E+ =
2
1
i
(7.98)
1
i
.
(7.99)
Elliptical polarisation
In the general case, light is elliptically polarised, with the components
of the electric field Ex and Ey satisfying
Ey
E0y
2
+
Ex
E0x
2
2
Ey
E0y
Ex
E0x
cos = sin2 .
(7.100)
The rotation of the electric field vector about the propagation direction
depends on the retardation .
Case: 0 < < . Rotation is clockwise.
Case: < < 0. Rotation is anti-clockwise.
156
CHAPTER 7. POLARISATION
The Jones vector for the general case is
E0 = |E0 |
cos
i
e sin
(7.101)
Jones matrix
A Jones matrix represents the operation of an optical element on a
state of polarisation (represented by a Jones vector). Examples covered are
Linear polariser (with the transmission axis at an angle to the
Ex axis)
P =
cos2
cos sin
cos sin
sin2
.
(7.102)
cos sin
sin cos
.
(7.103)
Half-wave plate
M =
1 0
0 1
(7.104)
Quarter-wave plate
M/2 =
1 0
0 i
.
(7.105)
1 0
0 ei
.
(7.106)
7.8. SUMMARY
157
(7.107)
(7.108)
158
CHAPTER 7. POLARISATION
8.1
General remarks
8.2
Learning objectives
Boundary conditions
The boundary conditions of the electromagnetic fields at interfaces
between media of different refractive indices.
159
160
8.3
Boundary conditions
We shall consider the boundary conditions that must prevail when an electromagnetic wave crosses the boundary between media of different refractive index. To facilitate this, we shall make use of Stokes theorem
Z
I
( F) dA =
F dl,
(8.1)
where the line integral on the right-hand-side is around the curve C enclosing the area A.
161
Figure 8.1: Line integral taken around a section of the interface between
two media of different refractive index, showing the components of the electric field parallel to the boundary.
8.3.1
(8.2)
(8.3)
Z
E dl =
B
dA.
t
(8.4)
Now let us perform the line integral on the left-hand-side of Eq. (8.4)
around a rectangular loop set into the interface between two media, as
illustrated in Fig. 8.1. Since the integration is taken in the anti-clockwise
direction, as the height h of the rectangle tends to zero, we have
I
lim
E dl = E2k E1k l.
(8.5)
h0 C
Here E1k and E2k are the components of the electric field either side of the
interface parallel to it and l is the length of the rectangle. Now as h tends
to zero, so does the area of the rectangle enclosed by the line integral. So,
162
B
dA = 0
t
(8.6)
E2k E1k l = 0.
(8.7)
E2k = E1k
(8.8)
lim
h0 A
8.3.2
(8.10)
8.3.3
(8.11)
Let us perform the surface integral over the surface of cylindrical volume
sunk into the boundary between two media of different refractive index, as
illustrated in Fig. 8.2. As the height h of the cylinder is taken to zero, we
have
163
Figure 8.2: A cylindrical volume sunk into the boundary between two media
of different refractive index, showing the field lines of D passing through the
circular ends of the volume.
Z
lim
h0 A
Z
D dA =
Z
D dA1
A1
D dA2 .
(8.13)
A2
This integrations over the opposite ends of the cylinder have opposite signs
since the normal to the surface always points outwards whereas D points
into the volume on one side of the boundary and outwards on the other.
Now, in the absence of charge, Eq. (8.11) tells us that
Z
Z
D dA1
D dA2 = 0.
(8.14)
A1
A2
Therefore, keeping the circular ends of the cylinder parallel to the boundary,
we have
D1 A1 = D2 A2 ,
(8.15)
(8.16)
164
Again, this result requires modification if the free charge density is not
zero and we have
D1 D2 = f ,
(8.17)
8.3.4
Since magnetic field lines are always closed, the Maxwell equation
B=0
(8.18)
is always valid. Applying the same argument as used for the electric displacement, we then have
B is continuous across a boundary.
Note that this is always true.
8.4
Figure 8.3: Illustration of reflection and refraction in terms of the wavevectors of the incident, reflected and transmitted waves, ki , kr and kt respectively.
165
Armed with the conceptual tools of the previous subsections, we may now
return again to the topic of light propagation between different media. Figure 8.3 illustrates the familiar concept of reflection and refraction at the
interface between different media pictured, in this case, in terms of the
wave-vectors k (so we may think of this as a visualization of k-space).
The subscripts i, r and t label the wave-vectors of the incident, reflected
and transmitted waves respectively. The plane of incidence is defined by
the incident wave-vector ki and the normal to the boundary n. Since this
defined as being the kx ky plane, ki has no kz -component.
We now apply the boundary condition that the component of the electric
field parallel to the interface Ek is continuous. Therefore
Ek = Eik ei(i tki r) + Erk ei(r tkr r) = Etk ei(t tkt r) .
(8.19)
If this is to hold true at all times, the angular frequencies must all be equal,
i.e.
i = r = t .
(8.20)
(8.21)
Thus
We have already chosen our coordinate system so that ki has no kz component. If we also define the interface between the media, in real
space, to be at y = 0, then the parallel component of any wave-vector
will have no y-component (since this would be transverse to the boundary).
Applying these conditions to Eq. (8.21) and re-arranging, we have
Eik eikix x = Etk eiktx x eiktz z Erk eikrx x eikrz z
(8.22)
Now, for the boundary conditions to hold, any phase due to the kz component on any wave must be common to all terms. For this to be true, these
components must all be zero. Hence
Eik eikix x = Etk eiktx x Erk eikrx x .
(8.23)
(8.24)
(8.25)
166
where n1 is the refractive index of the incident medium and k0 is the free
space wave-vector. Since the reflected wave travels in the same medium,
we also have
|kr | = n1 |k0 | = |ki | .
(8.26)
(8.27)
where n2 is the refractive index of the second medium. Inspecting Fig. 8.3
and using the result of Eq. (8.24) for the equality of the kx components, we
see that
kix = |ki | sin i = kir = |kr | sin r .
(8.28)
(8.29)
i = r ,
(8.30)
which implies
(8.31)
(8.32)
8.5
(8.33)
Fresnel equations
167
8.5.2
s-polarised light
Figure 8.5 illustrates s-type polarisation (with the electric field E normal to
the page) showing the magnetic field B, with the incident light taken to be
from the medium with refractive index n1 . From the continuity of the Hk at
the boundary, we have
Hsi cos i Hsr cos r = Hst cos t .
Similarly, the continuity of B gives
(8.34)
168
Figure 8.5: s-type polarisation (with the electric field E normal to the page)
showing the magnetic field B.
(8.35)
(8.36)
(8.37)
and
Now, we may use Snells Law to substitute for sin 2 so that Eq. (8.37)
becomes
n1
.
n2
Eliminating Bst from Eqs. (8.36) and (8.38) gives
(8.38)
2
cos 1
n2
(Bsi Bsr )
= (Bsi + Bsr ) .
1
cos 2
n1
(8.39)
or
Bsi
2 cos 1 n2
1 cos 2 n1
= Bsr
2 cos 1 n2
+
1 cos 2 n1
.
(8.40)
(8.41)
169
where n0i = ni /i . Now, from Eqs. (6.26) and (6.51) of Chapter 6, we have
|B| =
nk0
|E| .
(8.42)
(8.43)
rs =
(8.44)
cos 2
1
Bst
2
cos 1
Bsi Bst
(8.45)
n1
.
n2
(8.46)
1 cos 2
n1
= Bst
Bsi ,
2 cos 1
n2
(8.47)
which gives
2Bsi = Bst
n1 1 cos 2
+
n2 2 cos 1
.
(8.48)
and therefore
2(n2 /1 ) cos 1
Bst
= 0
.
Bsi
n1 cos 1 + n02 cos 2
(8.49)
(8.50)
Thus,
ts =
Est
2n01 cos 1
= 0
.
Esi
n1 cos 1 + n02 cos 2
(8.51)
170
(8.52)
(8.53)
(8.54)
(8.55)
and
(8.56)
n1
n2 1 1
= Ept
.
n2
n 1 2
(8.57)
171
Hence
(Epi + Epr ) = Ept
n 2 1
n0
= Ept 20 ,
n 1 2
n1
(8.58)
n0
cos 1
= (Epi + Epr ) 10 ,
cos 2
n2
(8.59)
Epr
n0 cos 1 n01 cos 2
= 20
.
Epi
n2 cos 1 + n01 cos 2
(8.60)
(8.61)
or
tp =
Ept
2n01 cos 1
= 0
.
Epi
n2 cos 1 + n01 cos 2
(8.62)
Figure 8.7: Graph of the s and p-polarised reflection and transmission coefficients for light incident from the lower n1 medium. Note that rp goes
through zero at the Brewster angle B .
172
8.5.3
Alternative forms
Eliminating the refractive indices using Snells Law the Fresnel equations
for the reflection and transmission coefficients for s and p-polarised light
may be given in the following alternative forms.
s-polarised
Reflection
rs =
n1 cos i n2 cos t
sin (i t )
Esr
=
=
,
Esi
n1 cos i + n2 cos t
sin (i + t )
(8.63)
Transmission
ts =
Est
2n1 cos i
2 cos i sin t
=
=
,
Esi
n1 cos i + n2 cos t
sin (i + t )
(8.64)
Epr
n2 cos i n1 cos t
tan (i t )
,
=
=
Epi
n2 cos i + n1 cos t
tan (i + t )
(8.65)
p-polarised
Reflection
rp =
Transmission
tp =
Ept
2n1 cos i
2 cos i sin t
=
=
.
Epi
n2 cos i + n1 cos t
sin (i + t ) cos (i t )
(8.66)
8.5.4
173
Brewster angle
tan (i t )
sin (i t ) cos (i + t )
=
.
tan (i + t )
sin (i + t ) cos (i t )
(8.67)
t = sin
n1
sin B
n2
=
B .
2
(8.68)
sin B = sin
B = sin cos B + sin B cos = cos B .
n2
2
2
2
(8.69)
Rearranging and taking the arctangent of both sides then gives us our required result
1
B = tan
n2
n1
.
(8.70)
Road glare
The phenomenon of polarisation by reflection at the Brewster angle gives
us a strategy for reducing glare. For example, Fig. 8.8 illustrates the case on
a sunny day when light reflected from the road or bonnet of a car can have
an adverse effect on visibility for the motorist. Most of the reflected light will
be s-polarised (aligned parallel to the ground). Polarising sunglasses are
usually designed with the transmission axis in the vertical direction. Hence,
such glasses will cut out much of the glare.
Occasionally, car wind screens are also treated with a polarising film.
So long as the transmission axes of the screen and glasses are aligned,
there will be little problem. However, if the driver were to rotate his or her
lenses by 90 then the polarisers would become crossed and almost all the
light would be blocked!
174
Figure 8.8: On a sunny day, there may be significant glare from the road or
the bonnet of a car. Since this is reflected light, it will tend to be s-polarised.
This glare can then be greatly reduced by wearing polarising sunglasses to
block the s-polarised component.
(a)
(b)
Figure 8.9: An example of the low reflection coefficient near the Brewster
angle for p-polarised light. In (a), the window is practically opaque due to
the bright reflection from the Sun. In (b), however, a polarising filter is used
on the camera to block s-polarised light. Since the amplitude of p-polarised
light is already low, most of the light seen from the window is transmitted
through it from the room inside.
8.6
175
Time reversibility
In Eqs. (8.63) to (8.66), the light begins in the medium labelled with the i
subscript and is transmitted into the medium labelled by t. Let us consider
the case where time is reversed, so that the light begins in the medium
labelled t and is transmitted to the i medium. To evaluate the reflection
and transmission coefficients, we simply need to swap the positions of the
angles i and t . We shall denote the time-reversed coefficients using prime
(0 ) notation.
For reflection, on time reversal we have
sin (t i )
= rs
sin (t + i )
(8.71)
tan (t i )
= rp .
tan (t + i )
(8.72)
rs0 =
and
rp0 =
We see that, in general
r0 = r.
(8.73)
sin2 (i t )
sin2 (i + t ) sin2 (i t )
=
,
2
sin (i + t )
sin2 (i + t )
giving
1 rs2 =
Similarly, we have
tan2 (i t )
tan2 (i + t ) tan2 (i t )
=
,
tan2 (i + t )
tan2 (i + t )
sec2 (i + t ) sec2 (i t )
,
tan2 (i + t )
cos2 (i t ) cos2 (i + t )
,
cos2 (i t ) sin2 (i + t )
1 rp2 = 1
=
=
(8.74)
176
giving
1 rp2 =
(8.75)
(8.76)
2 cos i sin t
.
sin (i + t ) cos (i t )
(8.77)
2 cos t sin i
sin (t + i )
(8.78)
ts =
and
tp =
So
t0s =
and
2 cos t sin i
.
sin (t + i ) cos (t i )
Hence, for s-polarised waves
t0p =
t0 t = 1 r2 .
8.7
(8.79)
(8.80)
(8.81)
(8.82)
Stokes treatment
The results of the previous section may also be derived following a somewhat more intuitive approach via the Stokes treatment. In Fig 8.10 (a), we
see the reflection and transmission of an initial ray of amplitude E0 . The
time-symmetric counterpart of this situation is shown in Fig 8.10 (b). In
the latter case, there are two initial waves: one with amplitude rE0 incident
from the n1 side of the boundary and another with amplitude tE0 incident
from the n2 side.
Figure 8.11 shows the combined effect of these two situations. We note
the following:
(a)
177
(b)
Figure 8.10: (a) The reflection and transmission of an initial ray of amplitude
E0 . (b) The time-symmetric counterpart to (a).
178
rt + r0 t E0 = 0,
(8.83)
r0 = r.
(8.84)
r2 + t0 t E0 = E0 ,
(8.85)
t0 t = 1 r2 .
(8.86)
which implies
which implies
8.8
Irradiance
1 2
2 kE0
0
0
1/2
.
(8.87)
This may be written in terms of the speed of light c = (0 0 )1/2 and the
refractive index n = ()1/2 to obtain
hSi =
nE02
k.
20 c
(8.88)
Note that this gives the intensity of the light or, equivalently, the irradiance.
We can therefore write Eq. (8.88) as
hSi = I k.
8.8.1
(8.89)
Figure 8.12 shows a beam of light incident on the boundary between media
of refractive indices n1 and n2 . Also shown is the reflected beam with r =
i as required by the Law of Reflection. The intensity over the surface at A
with normal vector n is given by
i n = Ii cos i .
IA = Ii k
This also equals
(8.90)
8.8. IRRADIANCE
179
r n = Ir cos r .
IA = Ir k
(8.91)
We may then define the reflectance R as the ratio of the reflected and
incident intensities.
Ir cos r
Ir
R=
=
=
Ii cos i
Ii
E0r
E0i
2
(8.92)
Recalling the results of the previous section, we may express this in terms
of the reflection coefficient r
R = r2 .
(8.93)
E0t
E0i
2
.
(8.94)
180
Figure 8.13: Chart of the intensity reflectance for s and p-polarised light.
Here, the refractive index of the incident medium n1 = 1 and the transmitted
medium has n2 = 1.33 (note that this is the refractive index of water).
Again, we may express this in terms of the transmission coefficient t
T =
n2 cos t 2
t .
n1 cos i
(8.95)
(8.96)
Ir cos r
It cos t
+
.
Ii cos i
Ii cos i
(8.97)
Thus,
R + T = 1.
(8.98)
181
Figure 8.14: View of a water surface from a low angle. Most of the incident
light at such an angle is reflected.
8.9
1/2
(8.99)
kty = kt cos t = kt
n1
1
n2
!1/2
2
sin t
(8.100)
n2
= sin c ,
n1
(8.101)
182
kty = ikt
n1
n2
2
!1/2
2
sin t 1
ity .
(8.102)
(8.103)
(8.104)
(8.105)
which becomes
Since we have set the problem up so that the transmitted wave travels in
the ve y-direction, this represents an exponentially decaying wave. This
is known as the evanescent wave.
8.9.1
(a)
(b)
8.10. SUMMARY
183
Figure 8.16: Fibre optic coupler. Note that (as is common) the second input
port has been removed.
Another use of couplers is in interferometry, when optical couplers are
used as the basis of the fibre optic Mach-Zehnder interferometer . In this
case, the power from an input fibre is divided equally between to output
fibres. A phase shift is then applied to one arm of the Mach-Zehnder interferometer before the power is recombined (with interference effects) at a
second coupler.
8.10
Summary
Boundary conditions
The boundary conditions of the electromagnetic fields at interfaces
between media of different refractive indices.
Ek is continuous across a boundary
Hk is continuous across a boundary.
When jf 6= 0, we have the modified result that
H2k H1k = jf ,
D is continuous across a boundary.
(8.106)
184
(8.107)
n1 cos i n2 cos t
sin (i t )
=
.
n1 cos i + n2 cos t
sin (i + t )
(8.108)
Transmission
ts =
2n1 cos i
2 cos i sin t
=
.
n1 cos i + n2 cos t
sin (i + t )
(8.109)
n2 cos i n1 cos t
tan (i t )
.
=
n2 cos i + n1 cos t
tan (i + t )
(8.110)
p-polarised
Reflection
rp =
8.10. SUMMARY
185
Transmission
tp =
2n1 cos i
2 cos i sin t
=
.
n2 cos i + n1 cos t
sin (i + t ) cos (i t )
(8.111)
Brewster angle
When the angle of incidence equals the Brewster angle, the reflected light is entirely s-polarised.
B = tan1
n2
n1
.
(8.112)
(8.113)
t0 t = 1 r 2 .
(8.114)
Stokes treatment
Alternative derivation of the results of time reversibility via the Stokes
treatment.
Irradiance
Analysis of power reflection and transmission between different media.
Reflectance
R = r2 .
(8.115)
Transmittance
T =
n2 cos t 2
t .
n1 cos i
(8.116)
186
(8.117)
Part IV
Geometrical Optics
187
9. Fermats Principle
9.1
General remarks
9.2
Learning objectives
190
9.3
Geometric wavefront
Figure 9.1: The locus of points with equal optical path-length at time t for
rays emanating from a single point at time t0 constitutes a geometric wavefront.
Figure 9.1 shows the locus of points on rays emanating from a single
point at time t0 at a later time t. These points are all in phase with one
another and constitute a geometric wavefront. Putting T = t t0 , we may
then multiply T by c to express the propagation time in dimensions of space
(r) = c (t t0 ) .
(9.1)
The quantity (r) is known as the optical path length and is a function of
distance. Now, if dS/dt = v = c/n, then
Z r
(r) =
n(x, y, z) dS.
(9.2)
0
191
9.4
Fermats Principle
The path taken between two points by a ray of light is the path that
can traversed in the least time.
So, if T () is the time taken to traverse a path that depends in some way
on , then we require
dT
= 0.
(9.4)
d
More generally, however, T will depend on the function that defines the
path-length. Suppose we denote such a function by f . We should then
express T as
(9.5)
T = T [f ] ,
9.4.1
Rectilinear propagation
which leads to
y0 = n
"
1+
dy
dx
2 #1/2
dx.
(9.7)
192
T
= 0,
f (x)
(9.8)
9.4.2
Calculus of variations
EulerLagrange equation
Given a functional of the form
Z
T [f ] =
x2
L x, f (x), f 0 (x) dx,
(9.9)
x1
.
f (x)
f
dx f 0
(9.10)
Setting the right hand side of this to zero yields the Euler-Lagrange equation.
The shortest optical path length between two points
For an optical path between two points (x1 , y1 ) and (x2 , y2 ) in the x y
plane, the total length is given by
Z x2
1/2
[f ] = n
1 + f 02
dx.
(9.11)
x1
Hence
1/2
L x, f (x), f 0 (x) = 1 + f 02
,
(9.12)
L
=0
f
(9.13)
d
f0
d L
=
.
dx f 0
dx [1 + f 02 ]1/2
(9.14)
and
Setting this last result to zero and integrating this with respect to x gives,
after re-arranging,
A
f (x) =
1A
0
1/2
= m,
(9.15)
193
(9.16)
which is the equation of a straight line. The value of m and c may then be
found by applying the boundary conditions that the line must pass through
(x1 , y1 ) and (x2 , y2 ).
Thus we have shown that for an isotropic and homogeneous medium,
the shortest optical path length between two points is a straight line.
9.4.3
9.4.4
Reflection
194
a point in the boundary plane between media with refractive indices n1 and
n2 , as shown in Fig. 9.2. The geometric path-lengths between the points A
and B and the point at (x, 0) in the plane are given by
q
2
SA = x2 + yA
(9.17)
and
q
2.
(xB x)2 + yB
(9.18)
x = SA sin i
(9.19)
xB x = SB sin r .
(9.20)
SB =
Note that
and
(9.21)
Differentiating this with respect to x and using Eqs. (9.19) and (9.20),
xB x
x
d
= n1
= n1 (sin i sin r ) .
(9.22)
dx
SA
SB
Hence, when d/dx = 0
sin i = sin r
(9.23)
i = r ,
(9.24)
or
9.4.5
Refraction
In the case of transmitted light (see Fig. 9.3), Eqs. (9.17) to (9.20) still hold
(after substituting t for r . The optical path-length is now
= n1 SA + n2 SB
(9.25)
x
xB x
d
= n1
n2
= n1 sin i n2 sin t .
dx
SA
SB
(9.26)
and
195
n1 sin i = n2 sin t ,
(9.27)
9.5
9.5.1
Perfect mirrors
Imaging rays from a point onto a plane
Figure 9.4 shows a sketch of a mirror that perfectly images all rays from the
origin onto a plane wave, for instance, the planar wavefront at y = A shown.
In other words, all such rays must have the same optical path-length . The
general form for in term of y = y(x) (the equation of the surface of the
mirror) will then be
= n (S + A y) = B,
(9.28)
where B is a constant. Since all rays travel in the same refractive index n,
we may put B 0 = B/n and expand S to give
x2 + y 2 = B 0 A + y = y + C,
(9.29)
(9.30)
196
Figure 9.4: Sketch of a perfect mirror that images all rays from a point (at
the origin) onto a plane wave.
y=
C
x2
.
2C
2
(9.31)
x2
+ y0
4y0
(9.32)
9.5.2
Figure 9.5: Partial sketch of a mirror that perfectly images light from the
point A to the point B (and vice versa).
197
Consider a mirror that perfectly images light from one point onto another
point, as implied by Fig. 9.5. We require the equation for the surface of a
mirror that perfectly images light from one point onto another point. Hence
all optical path-lengths from A to B via y(x), as shown in Fig. ??, must be
equal. Label the point where the curve meets the positive x axis C and
define this to be at x = R1 . Referring to Fig. ??, a general optical path
length is then given by
1/2
1/2
= nS1 + nS2 = n [d + x]2 + y 2
+ n [x d]2 + y 2
.
(9.33)
(9.34)
Since the refractive index is the same for all optical path-lengths, this
cancels out. Hence, equating Eqs. (9.33) and (9.34) and rearranging, we
have
[d + x]2 + y 2
1/2
1/2
= 2R1 [x d]2 + y 2
.
(9.35)
[x d]2 + y 2
1/2
= R1
xd
.
R1
(9.37)
x2 d2
.
R12
(9.38)
Adding 2xd to both sides and moving all constant terms to the right-handside gives
d2
d2
2
2
2
x 1 2 + y = R1 1 2 .
(9.39)
R1
R1
Finally, defining R22 = R12 1 d2 /R12 and dividing through by this gives
x2
y2
+
= 1.
R12 R22
(9.40)
198
9.6
Perfect lenses
Figure 9.7 shows a sketch of a lens that perfectly images rays from an
external point at A onto a plane-wave within the lens. We therefore seek
the equation of the surface of the lens that gives the same optical pathlength for all such rays.
The optical path-length for the ray propagating from A along the x-axis
to a wave-front at x = C is
A0C = n1 d1 + n2 C.
(9.41)
(9.42)
199
Figure 9.7: Sketch of a perfect lens that images rays from an external point
at A onto a plane-wave within the lens.
1/2
S1 = [x + d1 ]2 + y 2
.
(9.43)
ABC = A0C
(9.44)
We therefore require
Substituting for both sides of this equation from Eqs. (9.41) and (9.42), we
have
1/2
n1 [x + d1 ]2 + y 2
+ n2 (C x) = n1 d1 + n2 C.
(9.45)
(9.46)
(9.47)
(9.48)
200
x2 2xx + x2
(9.49)
+ 2n1 d1 (n2 n1 ) x
2n1 d1 (n2 n1 ) x n21 y 2 = 0.
(9.50)
(9.51)
which gives
x =
n1 d1
.
n2 + n1
(9.52)
Substituting this back into Eq. (9.50) and moving the constant terms to the
right-hand-side gives
n2 n1
n22 n21 x2 n21 y 2 = n21 d21
.
n2 + n1
(9.53)
(9.54)
This is then the equation of the surface of the lens in standard form.
9.7
Curvature
An infinitesimal curve segment dS may be related to the angle the gradient of the curve at that point makes with x-axis (i.e. tan = dy/dx). This is
illustrated in Fig. 9.8. The curvature of a surface at a given point may then
be defined as
d
1
,
dS
R
(9.55)
9.7. CURVATURE
201
9.7.1
Curvature of a circle
Figure 9.9: Sketch illustrating the curvature of a circle in terms of an infinitesimal arc-length and the angle that it subtends.
In the case of a circle of radius R (see Fig. 9.9), d is just the angle
202
= lim
(9.56)
9.8
Parameterisation of a curve
In Fig. 9.10, we see that the curvature of a general curve y(x) may be
matched at a given point (x, y) to that of a circle with radius R. Hence, the
curvature of y(x) at (x, y) is 1/R.
In practice, calculation of the curvature may be achieved by parameterising a curve. For a curve lying in the x y plane, x and y may be given in
terms of a parameter t
x = x(t),
y = y(t).
(9.57)
dx
= y0
dt
(9.58)
and similarly for higher derivatives. Using the chain rule, the curvature is
then given by
=
d
d dt
=
.
dS
dt dS
(9.59)
dy
dy dt
y0
=
= 0,
dx
dt dx
x
(9.60)
y0
x0
(9.61)
so
1
= tan
.
Recalling that
1
d tan1
= 1 + 2
d
and using the chain rule, we have
(9.62)
203
Figure 9.10: Sketch of a general curve showing how the radius of curvature
at a given point is related to the radius of a circle sharing a common tangent.
d
dt
0 2 !1
d (y 0 /x0 )
y
=
1+ 0
dt
x
00
0 2 !1
y
x00 y 0
y
=
02
1+ 0
0
x
x
x
x0 y 00 x00 y 0
.
x02 + y 02
(9.63)
Meanwhile, we have
dS = dx2 + dy 2
1/2
(9.64)
so
dS
=
dt
dx
dt
2
dy
+
dt
2 !1/2
= x02 + y 02
1/2
(9.65)
Hence, combining Eqs. (9.63) and (9.65) according to Eq. (9.59), we arrive
at the general expression for the curvature
=
x0 y 00 x00 y 0
(x02 + y 02 )3/2
(9.66)
204
9.8.1
Parameterisation of a circle
(9.67)
cos2 t + sin2 t = 1,
(9.68)
Since
x(t) = R cos t + xC ,
(9.69)
y(t) = R sin t + yC
giving
x0 = R sin t, y 0 = R cos t,
x00 = R cos t, y 00 = R sin t.
(9.70)
=
=
(9.71)
as found earlier.
9.8.2
Parameterisation of a parabola
(9.72)
(9.73)
y 00
(x02
y 02 )3/2
=
2a
2
1 + [2at + b]
3/2 .
(9.74)
205
1
,
R
(9.75)
giving
R=
1
2a
(9.76)
for the radius of curvature. For the equation of the parabolic mirror we
derived earlier, we found
a=
1
4y0
(9.77)
(9.78)
206
9.8.3
Parameterisation of a hyperbola
Since the standard form for a hyperbola is given by Eq. (9.49), we may use
the following parameterisation:
x = a cosh t, y = b sinh t,
x0 = a sinh t, y 0 = b cosh t,
x00 = a cosh t, y 00 = b sinh t.
(9.79)
ab
3/2 .
a2 sinh t + b2 cosh2 t
2
(9.80)
When t = 0, we have
x = a,
y = 0,
(9.81)
a
n1
=
.
b2
(n2 n1 ) d1
(9.82)
(n2 n1 ) d1
.
n1
(9.83)
9.8.4
Parameterisation of an elllipse
(9.84)
y(t) = R2 sin t
(9.85)
and
y 0 = R2 cos t
(9.86)
y 00 = R2 sin t
(9.87)
and
x00 = R1 cos t,
207
(9.88)
R1 R2
R1
3/2 = 2
2
R
2
R2
(9.89)
R1 R2
R2
= 2,
3/2
R1
R12
(9.91)
R12
.
R2
(9.92)
208
9.9
Summary
Fermats Principle
Light traverses the route between two points for which the optical path
length is a minimum.
The laws of geometric optics
Fermats Principles may be applied to derive:
The law of rectilinear propagation
In a homogeneous medium, light travels in straight lines.
The law of reflection
In a homogeneous incident medium, the angle of incidence equals
the angle of reflection.
The law of refraction
When light passes from a homogeneous medium with refractive
index ni into another homogeneous medium with refractive index
nt , the angles of incidence and refraction, i and t , are given
by Snells Law
ni sin i = nt sin t .
Perfect imaging
Imaging from a point or plane to a point or plane such that all rays
have the same optical path-length.
Perfect mirrors
Parabolic
A parabolic mirror perfectly images rays from a point (at the focus
of the parabola) onto a plane (and vice versa).
Elliptical
An elliptical mirror perfectly images rays from a point (at one
focus of the ellipse) onto another point (at the other focus).
Perfect lenses
Hyperbolic
A hyperbolic lens perfectly images rays from a particular point
outside the lens onto a plane inside the lens.
Curvature
A curve may be characterised by its curvature at a given point
9.9. SUMMARY
209
d
1
,
dS
R
(9.93)
210
10.1
General remarks
Having looked at the special cases of perfect imaging, we now turn our
attention to the case of spherical lenses and mirrors. We saw, in the previous chapter, that mirrors and lenses with conic section profiles may be
approximated by a circle or sphere over a small angle, when we match the
curvatures. This approximation means that in the paraxial approximation
of small angles we may get very nearly perfect imaging. As the paraxial
approximation begins to fail, we start to encounter spherical aberrations,
due to the fact that a sphere cannot image perfectly.
First, we shall take a look at spherical lenses generally, before applying
the paraxial and thin lens approximations to obtain the thin lens equation.
This introduces the concept of the focal length. We follow this with an analysis of thin lenses in combinations.
We then consider spherical mirrors, where a similar treatment to that
used for lenses is employed for finding the analogous equations and focal
lengths. Then, using the methods so far developed, the rules of image
construction for convex and concave lenses are described.
10.2
Learning objectives
212
Image construction
Magnification
Monochromatic aberration
Third order aberration
Spherical aberration
Coma
Astigmatism
Field curvature
Distortion
10.3
Spherical lenses
10.3.1
213
(10.1)
(10.2)
and
(10.3)
Although we do not expect a spherical lens to image perfectly, reasonable imaging may be possible to a good approximation. The requirement
214
for perfect imaging is that all the rays have equal optical path-length. That
is, we require to be a constant
d
= 0.
d
(10.4)
(10.5)
where
dl0
d
(so + R) R sin
(so + R) R sin
l0
1/2
(10.6)
and, similarly,
dli
(si R) R sin
=
.
d
li
(10.7)
(si R)
(so + R)
= n2
,
l0
li
.
l0
li
li
l0
10.3.2
(10.8)
(10.9)
It is important to note that lo and li appearing in Eq. (10.9) are still functions
of and the associated distances s0 or si . It is therefore not possible to
determine either s0 or si analytically. However, we may make progress by
employing the paraxial approximation. The is a small angle approximation
in which the sinusoidal functions are taken to be approximately equal to the
first terms of their Taylor series expansions. Hence, for small , we may put
sin
(10.10)
cos 1.
(10.11)
and
215
(10.12)
(10.13)
and
n1 n2
+
s0
si
=
1
(n2 n1 ) .
R
(10.14)
This is the general equation for a spherical lens surface (note that we could
have obtained the same result by applying Snells law directly to this problem).
Let us now consider the case when so . Equation (10.14) then
becomes
n2
1
= (n2 n1 ) ,
si
R
(10.15)
n2
R fi ,
n2 n1
(10.16)
where we have defined the focal length inside the lens fi . Similarly, when
si , we get
so =
n1
R fo ,
n2 n1
(10.17)
10.3.3
Special cases
(10.18)
216
(a)
(b)
(c)
Figure 10.3: Special cases of Eq. (10.14): (a) si < 0. In this case n2 /n1
is not large enough to refract the transmitted ray below the horizontal and
si marks a virtual image to the left of the lens. (b) When R , the
curvature of the lens becomes zero (the lens surface becomes a flat plane).
(c) so = si . This can only be the case when the source of the ray is inside
the lens and |so | = |R|.
giving
si =
n2
so ,
n1
(10.19)
= (n2 n1 ) ,
so
s0
R
(10.20)
R = so .
(10.21)
so
217
10.4
Thin lenses
Figure 10.4: A thick lens of thickness d and spherical surfaces with radii of
curvature R1 and R2 .
Figure 10.4 shows a thick lens of thickness d and spherical surfaces with
radii of curvature R1 and R2 . We require an approximate model for which
we can let d 0, in other words, a thin lens. To achieve this, we begin by
applying Eq. (10.14) to each surface in turn. For the surface with radius R1 ,
we have
n1
n2
1
+
(n2 n1 ) ,
(10.22)
=
s01 si1
R1
where s01 and si1 are shown in Fig. 10.5. Note that si1 is the distance to a
virtual image and is negative.
We obtain a similar result for R2 , except that si2 is now in the medium
with refractive index n1 whilst so2 is in the medium with refractive index n2 .
Thus, we have
n2
1
n1
+
=
(n2 n1 ) .
(10.23)
si2 so2
R2
Combining Eqs. (10.22) and (10.23), we have
n1
1
1
+
si2 so1
=
1
1
R1 R2
(n2 n1 ) n2
1
1
+
si1 so2
.
(10.24)
218
Figure 10.5: The lens of Fig 10.4 showing the imaging distances associated
with each surface.
n1
1
1
+
si2 so1
=
1
1
R1 R2
(n2 n1 )
n2 d
.
si1 (d si1 )
(10.25)
Finally, taking the limit d 0 and putting si2 = si and so1 = so , we obtain
1
1
n2 n1
+ =
so si
n1
1
1
R1 R2
.
(10.26)
1
1
R1 R2
.
(10.27)
This is known as the lens makers formula. From this it follows that
1
1
1
+ = ,
so si
f
(10.28)
10.5
Consider two thin lenses separated by a distance d. Our objective is to obtain find expressions for the effective focal lengths of this combination. We
shall refer to these as the front and back focal lengths, ff and fb respectively.
219
Using the Gaussian lens formula for the first lens, we have
1
1
1
+
= ,
so1 si1
f1
(10.29)
so1 f1
.
so1 f1
(10.30)
(10.31)
(d si1 ) f2
.
d si1 f2
(10.32)
si2 =
Now so2 = d si1 , so
si2 =
f2 (so1 [d f1 ] f1 d)
.
so1 (d f1 f2 ) f1 (d f2 )
(10.33)
If we now take the limit of so1 , si2 becomes the back focal length
fb . We therefore have
fb =
f2 (d f1 )
.
(d f1 f2 )
(10.34)
Similarly, so1 becomes ff as si2 . This will be the case when the
denominator on the right-hand-side of Eq. (10.33) tends to zero. Hence,
we have
ff =
10.5.1
f1 (d f2 )
.
d f1 f2
(10.35)
f2 f1
= ff ,
f1 + f2
(10.36)
(10.37)
220
1 X 1
.
=
f
fi
(10.38)
10.6
Spherical mirrors
Figure 10.6: A ray of light reflecting off the surface of a spherical mirror.
10.6.1
With reference to Fig. 10.6, the conventions for mirror calculations are taken
to be
221
Figure 10.6 shows a ray of light reflecting off the surface of a spherical
mirror. Note that, since si is the left of the centre of curvature of the mirror,
by convention it is taken to be negative. is the angle of reflection, so must
be equal to the angle of incidence. Hence, from Fig. 10.6,
= + .
(10.39)
The reflected ray makes an angle with the horizontal. The incident ray
makes an angle with the horizontal and meets the reflected ray at an
angle of 2. Thus, from the figure, we must have
2 = ( + ) ,
(10.40)
2 = + .
(10.41)
which gives
(10.42)
= 2.
(10.43)
Hence
From Fig. 10.6, we see that h = R sin . For small we have sin ,
and hence
h
.
R
(10.44)
222
At the same time, the length of the chord subtended by approaches h and
becomes normal to the horizontal. Hence
h
so
and, noting that si is negative by convention,
h
.
si
Hence, substituting these expressions into Eq. (10.43) gives
1
1
2
+ = .
so si
R
(10.45)
(10.46)
(10.47)
(10.49)
This is the same expression as the Gaussian lens formula for a spherical
lens.
10.7
Image construction
For a spherical lens, the following general guide for image construction
should be applied (all rays propagate from left to right)
223
The tip of the image will then be the intersection of the rays described
above.
10.7.1
Convex lens
yi
.
yo
(10.50)
Note, since yi is negative, so is MT (the image is inverted). On the righthand-side of the lens, we find similar triangles giving the relation
yo
yi
= .
f
xi
(10.51)
xi
.
f
(10.52)
(10.53)
f
.
xo
(10.54)
which gives
MT =
224
f 2 = xo xi .
10.7.2
(10.55)
Concave lens
225
The tip of the image occurs where the rays from the tip of the object
intersect on the left-hand-side, shown in the figure at i. In this case, xo is
the distance between the object and f on the right of the lens whilst xi is
the distance between the image and f on the left of the lens. The distances
xo , yo and so for the object and xi , yi and si for the image are shown on the
graph.
The magnification is given by
M=
yi
yo
(10.56)
(10.57)
M=
(10.58)
yi
yo
= ,
f
xi
(10.59)
giving
M=
yi
xi
= .
yo
f
(10.60)
f
xi
= ,
xo
f
(10.61)
226
f 2 = xo xi .
(10.62)
This is the same result as for the convex lens (f > 0).
10.8
Monochromatic aberration
10.8.1
Third-order aberration
In the earlier analysis of spherical lenses, we applied the paraxial approximation to obtain Eq. (10.14) from the exact expression Eq. (10.9). Specifically, this meant assuming is close to zero and making the approximations
sin
(10.63)
cos 1.
(10.64)
and
3
+ ...
3!
(10.65)
cos = 1
2
+ ....
2!
(10.66)
and
These higher-order terms add in correcting terms as is allowed to increase. Due to the power of 3 in the sin expansion, the second terms in
each expansion is referred to as a third-order correction. The effect of these
terms is to introduce third-order aberrations from the paraxial treatment of
spherical lenses.
Employing the third-order terms, a corrected version of Eq. (10.14) may
then be obtained. First, we remind ourselves of the correct expressions for
lo and li appearing in Eq. (10.9). We have
lo2 = (so + R)2 + R2 2 (so + R) R cos
(10.67)
(10.68)
and
227
lo2 s2o
(10.69)
li2 s2i .
(10.70)
and
We now use
cos 1
2
,
2
(10.71)
leading to
lo so
(so + R) 2
1+
R
s2o
1/2
(10.72)
and
li si
(R si ) 2
1+
R
s2i
1/2
.
(10.73)
n1 n2
+
l0
li
R
n2 si n1 so
,
li
l0
(R si ) 2 1/2
n2 1 +
R
s2i
(so + R) 2 1/2
n1 1 +
R
,
s2o
R 2 2 n1 1
1
n2 1
1
+
(n2 n1 ) +
+
,
2
so R so
si R si
=
where n2 is the refractive index of the lens and n1 is the refractive index of
the surrounding medium. Thus, our new expression is
n1 n2
+
s0
si
=
1
R2 n1 1
1
n2 1
1
(n2 n1 ) +
+
+
.
R
2
so R so
si R si
(10.74)
Note that the R2 term gives a measure of the displacement of the intersection of the ray with the lens from the optical axis. Thus, in the third-order
treatment, the new term increases in proportion with the square of the angular displacement.
228
10.8.2
Spherical Aberration
1
1
+
R so
.
(10.75)
n1
.
(n2 n1 )
(10.76)
1 (n2 n1 ) R2
=
+
R
n1
2so
1
1
+
R so
1
.
(10.77)
(10.78)
229
10.8.3
Coma
Coma (see Fig. 10.12) is a type of aberration in which off-axis points develop a comet-like tail, due to the variable magnification through the optical system. It affects both lenses and mirrors. In particular, Fig. 10.12
illustrates the problem for a parabolic mirror. For on-axis planar wavefronts,
230
10.8.4
Astigmatism
231
10.8.5
Field Curvature
Field curvature is an aberration that occurs when light that is focused onto
a curved surface is projected onto a planar screen. This is illustrated in
Fig. 10.14.
10.8.6
Distortion
232
Figure 10.14: Illustration of field curvature, showing the curved image (dotted line). This cannot be projected onto the plane screen without warping
the image.
10.9. SUMMARY
233
(a)
(b)
(c)
10.9
Summary
n1 n2
+
s0
si
=
1
(n2 n1 ) .
R
(10.79)
1
1
R1 R2
.
(10.80)
1
1
R1 R2
.
(10.81)
(10.82)
234
Monochromatic aberration
Third order aberration
Spherical aberration
Coma
Astigmatism
Field curvature
Distortion
Part V
Crystal Optics
235
11.1
General remarks
11.2
Learning objectives
Group theory
Symmetry of a square
Point groups in 2D
Point groups in 3D
Symmetry of the electric susceptibility
237
238
11.3
Group theory
11.3.1
Definition of a group
Closure
For any two elements of G, xi and xj , if
xi xj = xk ,
then xk is also in G.
Associativity
For all elements of G,
(xi xj ) xk = xi (xj xk ) .
239
11.4
Symmetry of a square
240
Figure 11.1: Lines of reflection symmetry for a square. Note that reflections
are often indicated by the symbol by convention.
11.4.1
Reflection lines
Figure 11.1 shows the reflection lines of symmetry for a square. In any
particular case, the symmetry operation i is then just a reflection in the
corresponding axis. As can be seen, there are four possible reflection operations that leave the square unchanged. These are detailed in Fig. 11.2,
where the coloured dots have been added for reference only.
11.4.2
Rotations
11.4.3
Combining operations
Symmetry operations may be combined. Consider the operation of a rotation by /2 followed by a reflection in the horizontal line. Letting R be the
combined operation, we would write this as
R = h C/2 .
(11.1)
241
Figure 11.2: Reflection symmetry operations for a square. Note that the
dots are added for reference only.
Figure 11.3: Rotation symmetry operations for a square. Note that the dots
are added for reference only.
242
(11.2)
h C/2 = d2 .
I
I
C3/2
C
C/2
h
v
d1
d2
C3/2
C
C/2
h
v
d1
d2
11.4.4
C/2
C/2
I
C3/2
C
d2
d1
h
v
C
C
C/2
I
C3/2
v
h
d2
d1
C3/2
C3/2
C
C/2
I
d1
d2
v
h
h
h
d2
v
d1
I
C
C/2
C3/2
v
v
d1
h
d2
C
I
C3/2
C/2
d1
d1
h
d2
v
C3/2
C/2
I
C
d2
d2
v
d1
h
C/2
C3/2
C
I
Matrix representations
(11.5)
we see that
h =
1 0
0 1
.
(11.6)
243
(a)
(b)
(c)
(d)
Figure 11.4: .
From Fig. 11.4 (b) for v we have x x and y remains unchanged.
Hence, we deduce
v =
1 0
0 1
(11.7)
0 1
1 0
(11.8)
and
d2 =
0 1
1 0
.
(11.9)
For an arbitrary rotation (see Appendix A.2), we may define the rotation
matrix
244
C =
cos sin
sin cos
(11.10)
For the particular symmetry rotations of the square, we then just substitute for
0 1
1 0
1 0
0 1
C/2 =
C =
(11.11)
(11.12)
and
C3/2 =
0 1
1 0
(11.13)
1 0
0 1
.
(11.14)
h C/2 = d2 .
(11.15)
I=
Combining operators
We saw previously that, for instance,
We can perform this using the matrix representation via matrix multiplication
h C/2 =
1 0
0 1
0 1
1 0
=
0 1
1 0
= d2 .
(11.16)
245
(11.17)
11.5
Point groups in 2D
Table 11.2: Multiplication table for C4 .
I
C3/2
C
C/2
I
I
C3/2
C
C/2
C/2
C/2
I
C3/2
C
C
C
C/2
I
C3/2
C3/2
C3/2
C
C/2
I
11.5.1
Families of groups
In 2D, the types of point groups occur in two families of groups. Using
Schonflies
notation
Cyclic groups
Cn - groups of n-fold rotation
Dihedral groups
Dn - groups of n-fold rotation and reflection
The point group of the square is therefore known as D4 . It is said to have
order 8 corresponding to the 8 symmetry operations. The 4-fold rotation
group C4 is then said to be a subgroup of D4 (note from the multiplication
table that C4 does indeed form a group).
246
11.5.2
Group generators
Generators of C4
Consider the repeated operation of C/2
C/2 I = C/2 ,
2
C/2 C/2 = C/2
= C ,
3
C/2 C = C/2
= C3/2 ,
4
= I.
C/2 C3/2 = C/2
11.6
Point groups in 3D
(11.18)
247
(11.19)
Cn is the same as the family of rotation groups in 2D, where the nfold rotation is now around a rotation axis. Of the other groups, we shall
only be concerned with Oh , having the full octahedral symmetry . This is
the symmetry of the cube and has order 48 (i.e. there are 48 symmetry
operations).
As a few examples, the simple cubic lattice has the same symmetry.
The number of symmetries may be reduced by the configuration of the
basis atoms. Zinc blende has the point group Td . This group has order 24.
Diamond structure has the point group Oh - the symmetry of the cube.
11.7
From thermodynamic considerations (time symmetry), it was shown in Chapter (6) that the electric susceptibility tensor must be symmetric, i.e. that the
elements satisfy
ij = ji .
(11.20)
Another way describing this symmetry is to say that E is equal to its transpose (see Appendix A.2). Thus, we may re-write Eq. (11.20) as
E = TE .
(11.21)
(11.22)
248
ij =
1, if i = j,
0, if i 6= j.
(2)
(3)
x1
x1
x1
(2)
(3)
X = x(1)
x2
x2
2
(1)
(2)
(3)
x3
x2
x2
(11.23)
(11.24)
and
(1)
0
0
= 0 (2)
0 ,
0
0 (3)
(11.25)
(11.26)
(11.27)
11.8
(11.28)
(11.29)
(11.30)
249
(11.31)
(11.32)
0E
(11.33)
Essentially what we have done is to transform from an original coordinate system O to a new system O0 in which the elements of the electric
susceptibility tensor are diagonal. The Cartesian axes of this new coordinate system then correspond to the principal axes of the crystal.
11.8.1
The transformation to the new coordinate system was obtained via the matrix operation U1 . According to Eulers rotation theorem we may change
from one Cartesian system to any other by some sequence of (up to three)
rotations. Therefore, we may interpret U1 as the product of these rotations. There are different possible choices for the rotations we might apply
but we shall use the following:
(1) A rotation around the z-axis by an angle
As a matrix operation, this rotation is given by
R(z)
cos sin 0
= sin cos 0 .
0
0
1
(11.34)
(x)
1
0
0
= 0 cos sin .
0 sin cos
(11.35)
250
R(z)
cos sin 0
= sin cos 0 .
0
0
1
(11.36)
(z)
U1 = R(z)
R R .
11.9
Symmetry operations
11.9.1
(11.37)
So far, we have just considered symmetry operations in the abstract. However, consideration of these operations yields physical information about
the crystal structure. In particular
A physical property of a system must reflect the symmetry of the system,
The physical properties we shall be concerned with are those of the
electric susceptibility tensor.
11.9.2
Let us now consider further rotations applied to the diagonalised susceptibility tensor corresponding to symmetry operations of the particular crystal
structure. It turns out that there are seven crystal systems (see Table 11.3)
associated with particular sets of symmetry operations.
Let us assume that we have already diagonalised E and consider
some transformation represented by the matrix T. Applying this to Eq. (11.28)
gives
TD = 0 T (I + E ) T1 TE.
(11.38)
(11.39)
In other words, dropping the prime notation for the diagonalised matrix, E
has the form
x 0 0
E = 0 y 0 .
0
0 z
251
(11.40)
11.9.3
Isotropic systems
Cubic symmetry
Figure 11.5: Illustration of the C4 symmetry for a cubic system. Note that
the cube has three axes with this symmetry.
As a particular example of how we may make use of point group symmetries, we shall investigate the optical properties of materials with cubic symmetry. Let us consider the C4 group associated with some axis of rotational
symmetry (a subgroup of Oh .
For definiteness, let us take the z-axis to be the axis of symmetry for definiteness. Earlier we saw that a C/2 is a generator of C4 . From Eq. (11.34)
we then see that this rotation is given by
0 1 0
C/2 = 1 0 0 ,
(11.41)
0 0 1
252
C/2
1
= C3/2
0 1 0
= 1 0 0 .
0 0 1
(11.42)
0 1 0
0 1 0
x 0 0
= 1 0 0 0 y 0 1 0 0
0 0 1
0 0 1
0
0 z
0 1 0
0 x 0
0
0
= 1 0 0 y
0 0 1
0
0
z
y 0 0
= 0 x 0 .
0
0 z
0E
(11.43)
Comparing this with the untransformed matrix, we see that x and y have
changed places. For a crystal with this symmetry, the physical properties
of the system must not change under this operation. Therefore, it must at
least be the case that
x = y .
(11.44)
o 0 0
E = 0 o 0 .
(11.45)
0 0 o
Here we have used the o subscript to stand for ordinary, in line with convention.
11.9.4
Uniaxial systems
Tetragonal symmetry
The tetragonal crystal system unit cell has just one square face and only
one axis to which the C4 symmetry symmetry applies (see Fig. 11.6 (a)).
(a)
253
(b)
Figure 11.6: Illustration of the (a) single C4 symmetry for a tetragonal system. Note that only the square face has this line of symmetry perpendicular
to it. (b) The C6 symmetry for the hexagonal system.
We can therefore only apply the transformation of Eq. (11.43) once and
thus only enforce the condition that two of the diagonal elements are equal.
If we take these to be the x and y elements, the susceptibility tensor takes
the form
o 0 0
E = 0 o 0 ,
0 0 e
(11.46)
where the e subscript stands for extraordinary. Crystals of this type are
given the optical classification uniaxial, as the direction associated with e
is taken to be the optical axis of the system.
Hexagonal symmetry
The hexagonal symmetry has an axis with C6 symmetry, as shown in Fig. 11.6
(b) (i.e. a 6-fold rotation symmetry). In this case, the generator of the group
is the rotation matrix
254
C/3
1
1
=
3
2
0
3 0
1 0 ,
0 1
(11.47)
0E =
1
3 0
1
3 1 0
4
0
0 1
3 0
1
1
3 1 0
4
0
0 1
+ 3y
1 x
3 (y x )
4
0
x 0 0
1 3 0
0 y 0 3
1
0
0
0 z
0
0
1
3x 0
x
3y
y
0
0
0
z
3 (y x ) 0
(11.48)
3x + y
0 .
0
z
255
Crystal system
Optical classification
Triclinic
Monoclinic
Biaxial
Orthorhombic
Trigonal
Tetragonal
Uniaxial
Hexagonal
Cubic
Isotropic
and types of rotation symmetries are the same as for the tetragonal system.
Thus, the susceptibility tensor takes the form
o 0 0
E = 0 o 0 ,
0 0 e
11.9.5
(11.49)
Biaxial systems
x 0 0
E = 0 y 0 ,
0
0 z
(11.50)
256
11.10
Summary
Group theory
Definition of a group
A group is a set of elements which can be combined by some
operation.
There are then four conditions that must be met to define a
group.
Identity
Invertibility
Closure
Associativity
Symmetry of a square
The symmetry operations may be represented by matrices.
The set of symmetry operations forms a group.
There are 8 symmetry operations for the square
Reflection
There are four axes of reflection: two diagonal d1 and d2 , horizontal h and vertical v .
d1 =
h =
(11.51)
.
0 1
1 0
d2 =
v =
0 1
1 0
1 0
0 1
1 0
0 1
(11.52)
(11.53)
(11.54)
Rotations
There are four rotations around the centre point of the square of
n/2, n {1, 2, 3, 4}. Note that the rotation of 2 does nothing.
C/2 =
0 1
1 0
.
(11.55)
11.10. SUMMARY
257
C =
1 0
0 1
C3/2 =
C2 =
0 1
1 0
1 0
0 1
(11.56)
(11.57)
.
(11.58)
Point groups in 2D
A point group is a group of symmetries that keep at least one point
fixed.
Families of groups
In 2D, the types of point groups occur in two families of groups.
Cyclic groups
Cn - groups of n-fold rotation
Dihedral groups
Dn - groups of n-fold rotation and reflection
Group generators
Generators of C4
C/2 I = C/2 ,
2
C/2 C/2 = C/2
= C ,
3
C/2 C = C/2
= C3/2 ,
4
C/2 C3/2 = C/2
= I.
Generators of D4
C/2 h = d1 ,
2
C/2 d1 = C/2
h = v ,
3
C/2 v = C/2
h = d2 ,
4
C/2 d2 = C/2
h = h .
258
Point groups in 3D
In 3D, there are 7 axial groups
Cn , S2n , Cnh , Cnv , Dn , Dnd , Dnv
(11.59)
(11.60)
Cn is the same as the family of rotation groups in 2D, where the n-fold
rotation is now around a rotation axis.
Oh , has the full octahedral symmetry. This is the symmetry of the
cube and has order 48 symmetry operations.
Zinc blende has the point group Td . This group has 24 symmetry
operations.
Diamond structure has the point group Oh .
Symmetry of the electric susceptibility
From time symmetry, the electric susceptibility tensor must be symmetric, i.e. that the elements satisfy
ij = ji .
(11.61)
(11.62)
(11.63)
(11.64)
11.10. SUMMARY
259
o 0 0
E = 0 o 0 .
0 0 o
(11.65)
o 0 0
E = 0 o 0 .
0 0 e
(11.66)
x 0 0
E = 0 y 0 .
0
0 z
This is a biaxial system.
(11.67)
260
12.1
General remarks
So far, we have concentrated on isotropic media. Some mention of uniaxial anisotropic media previously proved necessary in the context of wave
plates and the phenomenon of birefringence. Here, however, we shall approach the problem with a more rigorous and comprehensive treatment.
The starting point for our analysis is with the electric susceptibility tensor. Firstly, we shall consider EM wave propagation in an anisotropic media,
how the optical vibrations are resolved into modes. We shall see that there
are up to two possible optic axes, encompassing the three cases of
isotropic media
uniaxial anisotropic media (one optic axis)
biaxial anisotropic media (two optic axes)
During the course of this analysis, we shall meet the index ellipsoid,
which enables us to picture the modes of vibration. We shall also provide a
second, and somewhat easier, derivation of the index ellipsoid in terms of
the electric displacement vector.
A somewhat non-intuitive consequence of anisotropy is that of Poynting
walk-off. This is the phenomenon in which the energy of the EM radiation,
given in terms of the Poynting vector S, travels in a different direction to
the wavevector k. An analysis of this effect followed, giving the angular
displacement between S and k.
12.2
Learning objectives
261
262
12.3
12.3.1
= (I + E ) ,
2D
,
t2
(12.2)
where we have made the common assumption that the magnetic permeability may be modelled as a scalar. Since does not depend on t, we may
employ the usual vector identity to write Eq. (12.2) as
( E) 2 E = 0 0
2E
.
t2
(12.3)
Assuming that we have aligned our Cartesian coordinates with the principal
axes of the crystal (see Chapter 11), we may write this out in component
form as
2 Ei
( E) 2 Ei = 0 0 i 2 .
xi
t
(12.4)
2
Ei .
c2
(12.5)
(12.6)
263
(12.8)
where ai is the ith Cartesian component of the unit vector in the k direction.
Moreover, since all powers of n are even, we obtain a quadratic in n2
an4 + bn2 + c = 0,
(12.9)
a = n2x a2x + n2y a2y + n2z a2z ,
(12.10)
b = n2x n2y 1 a2z + n2x n2z 1 a2y + n2y n2z 1 a2x
(12.11)
and
c = n2x n2y n2z .
12.3.2
(12.12)
Optic axes
The solutions for n2 correspond to two modes of vibration, whereby different components of the polarisation see different refractive indices. These
modes of vibration may be visualised by means of the index ellipsoid, illustrated in Fig. 12.1. Each principal axis of the crystal is associated with a
refractive index ni . We may then construct an ellipsoid in index space with
semi-axes nx , ny and nz .
Let us now consider some arbitrary wavevector k. Taking the intersection of the plane perpendicular to k with the index ellipsoid defines an
ellipse, as shown in Fig. 12.1. The semi-axes of this ellipse give refractive
indices n0 and n00 , which correspond to the two modes of vibration D0 and
D00 .
In general, an ellipsoid has two circular cross-sections (see Fig. 12.2).
In the case of just two distinct semi-axes, we have a spheroid and there
is just one circular cross-section. The normals to these cross-sections are
known as the optic axes of the crystal. This then explains the nomenclature
of the optical classes
For biaxial crystals, there are two optic axes (these are shown as N1
and N2 in Fig. 12.2).
For uniaxial crystals have only one optic axis (taken, by convention,
to be along the z-axis).
264
Figure 12.1: The index ellipsoid for an anisotropic material, aligned to the
principal axes of the crystal. The plane perpendicular to the k direction
intersects the index ellipsoid in an ellipse. The half-length of the principal
axes of this ellipse correspond to the refractive indices, n00 and n0 , of the
two modes of vibration for the optical field.
Figure 12.2: The normal to the plane intersecting the index ellipsoid in a
circular cross-section is an optic axis of the crystal. In such a direction,
the optical field sees only one refractive index and hence there is only one
mode of vibration. Illustrated is the general case with two optic axes N1
and N2 .
265
Uniaxial crystals
For certain special k directions, Eq. (12.9) will have repeated roots. In these
cases, the optical field will only see one refractive index and, hence, there
will only be one mode of vibration. These directions are known as the optic
axes of the crystal and are determined by the condition
b2 4ac = 0.
(12.13)
(12.14)
b = n2o n2o 1 a2z + n2e 1 + a2z ,
(12.15)
c = n4o n2e
(12.16)
2
n2o n2e
2
(12.17)
For n2o 6= n2e , the discriminant is zero when az = 1, i.e. when k is parallel
with the z-axis. Thus, this is the optic axis of the crystal (see Fig. 12.3).
The solutions for n2 are then
h
i 1
n2 = n2e 1 + (ne /no )2 1 a2z
(12.18)
n2 = n2o .
(12.19)
and
Hence, we see that one refractive index depends on az whilst the other is
constant. If k makes an angle with the z-axis, then we have a2z = cos2
and Eq. (12.18) may be re-written making the -dependence explicit as
h
i
1
.
n2 = n2e 1 + (ne /no )2 1 cos2
(12.20)
266
Figure 12.3: The index ellipsoid of a uniaxial crystal. Note that the circular
cross-section lies in the z = 0 plane. Hence, the optic axis is along the
z-axis. For a general k direction, we find that one of the vibrational modes
has a constant refractive index whilst the other has a -dependent refractive
index.
12.4
Poynting walk-off
(12.21)
(12.22)
i,
t
(12.23)
ik .
(12.24)
and
267
D = ik D = 0.
(12.25)
(12.26)
(12.27)
ik E = i0 H.
(12.28)
(12.29)
(12.31)
12.5
1
2
(Dx Ex + Dy Ey + Dz Ez ) .
(12.32)
268
uE =
1
2
Dy2
Dx2
D2
+
+ z
0 x 0 y
0 z
!
.
(12.33)
(12.34)
in terms of the refractive indices associated with each axis. This is known
as the index ellipsoid and can be used to determine the refractive index
that a component of a wave actually sees.
we take the intersection of the plane
Given the propagation direction k,
and the index ellipsoid. This gives us an ellipse that
perpendicular to k
determines the possible refractive indices that the components of the wave
see.
12.5.1
Uniaxial crystals
In the case that two of the axes of the index ellipsoid are the same, we have
a uniaxial crystal. By convention, we usually take the equivalent axes to be
in the x and y directions. So, putting nx = ny = no and nz = ne , we have
x2
y2
z2
+
+
= 1.
n2o
n2o n2e
(12.35)
269
Figure 12.5: The index ellipsoid for a uniaxial crystal projected in 2D.
From this, we may find the -dependent refractive index from elementary
trigonometry and the equation of the index ellipsoid.
intersects the index ellipsoid in an ellipse.
The plane perpendicular to k
The half length of the axis of this ellipse lying in the x-y plane is no , regard The half length of the other axis is ne (), which
less of the direction of k.
(see Fig. 12.3).
does depend on k
Now for any given propagation direction, we may have a component of
D that lies in the x-y plane and thus sees a refractive index of no . We call
any wave polarised in this direction the ordinary wave. Waves polarised
will see a
in the orthogonal direction within the plane perpendicular to k
refractive index ne () that depends on the angle between the z-axis and
This component is called the extraordinary wave.
k.
Since the index ellipsoid has rotational symmetry about the z-axis, we
can take x = 0 with no loss of generality. We can then find value of ne ()
from the y and z coordinates of the intersection of the ellipse in the z-y
plane (see Fig. 12.5).
x = 0,
y = n() cos ,
z = n() sin .
Substituting these into Eq. (12.35), we have
(12.36)
270
n2 ()
n2 ()
2
cos
+
sin2 = 1,
n2o
n2e
(12.37)
giving
n2 () =
cos2 sin2
+
n2o
n2e
1
.
(12.38)
(12.39)
as found earlier.
12.6
Birefringence
12.6.1
Uniaxial crystal
(12.40)
12.6. BIREFRINGENCE
271
and
h
no r i
Eo = E0 exp i t
(12.42)
c
where we have used k = /v = n/c. The second of these equations may
be re-written
n()r
Eo = E0 exp i t
exp i [n() no ] r .
c
c
(12.43)
12.6.2
Double refraction
272
12.7
Summary
Modes of vibration
In an anisotropic media there are, in general, two modes of vibration.
These are orthogonal modes that see two different refractive indices.
For wave-vector
k = k (ax ex + ay ey + az ez ) ,
(12.45)
the refractive indices of these modes are found from the solutions of
the quadratic for n2
an4 + bn2 + c = 0,
(12.46)
a = n2x a2x + n2y a2y + n2z a2z ,
(12.47)
where
b = n2x n2y 1 a2z + n2x n2z 1 a2y + n2y n2z 1 a2x
(12.48)
and
c = n2x n2y n2z .
(12.49)
Optic axes
In any crystal, there is at least one k direction for which the optical
field sees only one refractive index (and there is therefore only one
mode). Such a direction is called an optic axis of the crystal.
Biaxial crystals
In biaxial crystals, there are two optic axes.
Uniaxial crystals
In uniaxial crystals, there is only one optic axis.
Modes of vibration in a uniaxial crystal
In a uniaxial crystal, we have nx = ny = no and nz = ne (note that,
by convention, the optic axis is taken to be along the z-axis). The
refractive indices of the modes may be found from
h
i 1
n2 = n2e 1 + (ne /no )2 1 a2z
(12.50)
12.7. SUMMARY
273
and
n2 = n2o .
(12.51)
Note that
(1) One refractive index is constant, whilst the other depends only
on az (i.e. the angle between k and the z-axis)
(2) When az = 1, we have n2 = n2o for both modes (z is the optic
axis)
(3) These expressions may be found far more easily via consideration of the index ellipsoid.
The index ellipsoid
From the energy density for the electric field
uE = 12 D E =
1
2
(Dx Ex + Dy Ey + Dz Ez )
(12.52)
(12.53)
(12.54)
Using the index ellipsoid, we may more easily find the refractive indices of the extraordinary and ordinary waves given above.
274
Birefringence
The birefringence is defined in terms of the retardation or phase shift
acquired between two orthogonal components of light seeing different
refractive indices in an anisotropic crystal. Hence, if the retardation is
(r) =
n
r,
c
(12.55)
A.1
Geometric progression
n1
X
ari .
(A.1)
i=0
Note that each term in the series is increased by a factor r. Such a series
is known as a geometric progression.
Expanding the summation, we have
Sn = a + ar + ar2 + . . . + arn1 .
(A.2)
(A.3)
or
Sn =
a (1 rn )
.
1r
(A.4)
A.2
A.2.1
a
.
1r
(A.5)
Matrices
Transpose of a matrix
275
(A.6)
276
A.2.2
ij
= (A)ji .
(A.7)
Complex conjugation
A11 A21 A31
A11 A12 A13
A21 A22 A23 = A12 A22 A32 .
A13 A23 A33
A31 A32 A33
(A.8)
A.2.3
ij
= (A)ji .
(A.9)
Symmetric matrix
A.2.4
ij
(A.10)
Hermitian matrix
Note that if all the elements of this matrix are real, then the matrix is symmetric.
A.2.5
Rotation matrices
A.2. MATRICES
(a)
277
(b)
(c)
Figure A.1: General rotation matrices in 3D for (a) a rotation around the z
axis; (b) a rotation around the x axis and (c) a rotation around the y axis.
R = R3 R2 R1 ,
(A.12)
where
R1 is the rotation by 1 around the z-axis shown in Fig. A.1 (a)
R2 is the rotation by 2 around the x-axis shown in Fig. A.1 (b)
R3 is the rotation by 3 around the y-axis shown in Fig. A.1 (c)
Figure A.2: A general rotation around the z axis of the point P from orginal
coordinates (x, y) to new coordinates (x, y)
Considering R1 , it is clear that the z-coordinate is left unchanged. This
situation is depicted in Fig. A.2, viewed from along the z-axis. Here we see
278
the rotation of the point P from orginal coordinates (x, y) to new coordinates
(x, y). Note that if this rotation is performed continually as goes from 0 to
2, we would see P rotated around the origin in a complete circle.
From Fig. A.2, we see that the new coordinates are given by
x0 = x cos y sin ,
y 0 = x sin + y cos ,
z 0 = z.
Thus, in matrix form we have
0
x
cos sin 0
x
y 0 = sin cos 0 y .
z0
0
0
1
z
Reintroducing the 1 subscript, we have defined the rotation matrix
cos 1 sin 1 0
R1 = sin 1 cos 1 0 .
0
0
1
(A.13)
(A.14)
Similarly, we have
1
0
0
R2 = 0 cos 2 sin 2
0 sin 2 cos 2
(A.15)
and
cos 3 0 sin 3
1
0
R3 = 0
sin 3 0 cos 3
A.3
A.3.1
(A.16)
Vector calculus
Double curl identity
The expression
E
(A.17)
( E)k
( E)j .
xj
xk
(A.18)
279
( E)i =
xj
Ej
Ei
xi
xj
xk
Ei Ek
xk
xi
.
(A.19)
( E)i =
=
=
2 Ej
2 Ek
2 Ei 2 Ei 2 Ei
2 Ei
+
+
,
xi xi xi xj
xi xk
x2i
x2j
x2k
Ei Ej
Ek
2 Ei 2 Ei 2 Ei
+
+
,
xi xi
xj
xk
x2i
x2j
x2k
( E) 2 Ei .
xi
(A.20)
E = ( E) 2 E.
(A.21)
Hence
Index
p-polarised, 167, 172
reflection coefficient, 172
transmission coefficient, 172
s-polarised, 167, 172
reflection coefficient, 172
transmission coefficient, 172
3D glasses, 151152
absorption coefficient, 43, 117119
Airy disc, 90
Airy rings, 90
Alhazen, 11
AM, see amplitude modulated
Amperes Law, 109
amplitude, 51
amplitude modulated, 29
Ancient Greeks, 10
angle
of incidence, 75
of reflection, 75
angular frequency, 49, 62
anisotropic, 133, 147
anisotropic medium, 58, 63
anti-reflective coatings, 18
associativity, 238
astigmatism, 230
Augustin-Jean Fresnel, 13
axial groups, 246
biaxial crystal, 263
Biaxial systems, 255
birefringence, 147149
double refraction, 271
uniaxial crystal, 270271
blackbody radiation, 3134
Bohr, Niels, 17
Boltzmanns constant, 32
boundary conditions, 160164
Bremsstrahlung, 44
Brewster angle, 173174
calculus of variations, 192193
Catoptrica, 10
circle of least confusion, 229
closure, 238
coherence, 7881
coma, 229230
complex conjugate, 133
computer generated imagery, 22
computer graphics, 22
conservation of energy, 37
constitutive equation, 111
copper point blackbody, 34
corpuscular theory of light, 12
Coulombs Law, 49
critical angle, 7677, 181
crossed polarisers, 173
crystal structure, 250
crystal system
trigonal, 134
curvature, 200, 202
of a circle, 201
of a hyperbola, 206
of a parabola, 204
cyclic groups, 245
de Broglie wavelength, 17, 3637
11
Descartes, Rene,
Diamond structure, 247
dichroic sheet, 133
dichroism, 133136
diffraction, 8486
280
INDEX
around objects, 99100
circular aperture, 8991
far field, 86
far field approximation, 88
Fraunhofer, 86
Fresnel, 86
limited imaging, 8991
multiple slit, 91
near field, 85
single slit, 8689
diffraction grating
order, 94
resolving power, 97
diffraction gratings, 9599
dihedral groups, 245
dispersion, 60, 117, 122125
anomalous, 117
normal, 117
displacement current density, 109
distortion, 231
barrel, 231
moustache, 231
pincushion, 231
divergence theorem, 162
double slit experiment, 13
281
energy density, 125
Enlightenment, The, 1112
erbium doped fibre amplifier, 21
Euclid, 10
Optics, 10
Eulers rotation theorem, 249
Euler-Lagrange equation, 192
evanescent wave, 77, 182
extraordinary refractive index, 265
extremely low frequency, 29
282
INDEX
generator, 246
non-abelian, 239
group theory, 238239
group velocity, 6062
definition, 61
rotator, 138
Jones vector, 110, 132133
circular polarisation, 141142
elliptical polarisation, 139140
Kitab al-Manazir, 11
Heisenbergs Uncertainty Principle, 17,Kronecker delta, 247
30, 39
laws of wave propagation, 7375
herapathite, 134
rectilinear propagation, 7374, 80
Hero of Alexandria, 10
reflection, 7475, 166
Hertz, Heinrich, 14
refraction, 75, 166
homogeneity, 67
lens
homogeneous medium, 58, 63
concave, 213, 224
Hookes law, 49
convex, 213, 223
Huygens Principle, 12, 7071
hyperbolic, 200
Huygens, Christiaan, 12, 67
sign conventions, 212
Huygens-Fresnel Principle, 7881
spherical, 212
thin, 217
Ibn Sahl, 10, 12
lens makers formula, 218
ideal spring, 4849
lenses and mirrors, 18
identity element, 238
light emitting diodes, 20, 41
image construction, 222
linear analyser, 153154
index ellipsoid, 263, 267268
linear chain of harmonic oscillators,
uniaxial crystal, 268270
5253
infra-red
linear differential equation, 55
near, 29
linear differential equations, 55
inhomogeneous medium, 58, 63
Linear medium, 57, 63
intensity, 69, 178
linear operator, 56
interference, 7881
linear operators, 5556
interferometer, 15
linear polarisers, 133136
inverse element, 238
linearity, 5556, 67
invertibility, 238
Lippershey, Hans, 11
iodoquinine sulfate, 134
long wave, 29
irradiance, 178180
Lorentz factor, 36
isotropic, 58, 252
Lorentz transformations, 15
isotropic medium, 58, 63
Lorentz, Hendrik Antoon, 15
isotropic systems, 251252
luminosity, 33
isotropy, 67
INDEX
Malus Law, 154
matrices, 275278
complex conjugation, 276
Hermitian, 276
rotation, 276278
symmetric, 276
transpose, 275276
matrix mechanics, 39
Maxwells equations, 108109
plane wave solutions, 110111
Maxwells rainbow, 28
Maxwell, James Clerk, 14
medium wave, 29
Michaelson and Morley, 15
microscope, 18
Minkowski, Hermann, 15
mirror
parabolic, 196
sign conventions, 220
mirrors
spherical, 220
modes of vibration, 263
monochromators, 9899
negative uniaxial, 149
Newton, Isaac, 11
non-linear optics, 113
nonlinear, 56
nonlinear optics, 56
normal dispersion, 124
numerical aperture, 77
optic axis
anisotropic, 263265
optical axis, 253
optical classes, 251
optical coupling, 182183
optical loss, 117119
optical spectrum, 29
optics
applications, 1822
history, 917
ordinary refractive index, 265
parameterisation, 202
283
circle, 204
ellipse, 206
hyperbola, 206
parabola, 204
paraxial approximation, 214, 226
Pauli Exclusion Principle, 35
perfect lenses, 198
perfect mirrors, 195
permeability, 57
of free space, 109
relative, 114
permittivity, 57
of free space, 108
relative, 114
phase velocity, 36, 5354, 58, 70
anisotropic, 262
photoconductivity, 20
photocurrent, 20
photoelectric effect, 16, 34
photonics, 1920
photons, 16
photovoltaic cells, 20
Plancks constant, 16, 32
Plancks Law, 32
Planck, Max, 15
plane wave, 59
plane waves, 5860
plane-wave, 191
pleochroism, 133
point group, 245
in 2D, 245246
in 3D, 246247
polarisation, 13, 5455
circular, 141142
elliptical, 139147
left circular, 142
linear, 132136
right circular, 142
polarising filters, 18
Polaroid, 134
H-sheet, 135
J-sheet, 134
polyhedral groups, 247
positive uniaxial, 149
284
Poynting vector, 125127, 178, 267
Poynting walk-off, 266267
Poyntings theorem, 125
principal axes, 248250, 262
Principle of Linear Superposition, 78
79
principle of linear superposition, 55
propagator, 132
quantum mechanics, 1517
radio waves, 28
radius of curvature, 200
rainbow, 122125
ray tracing, 22
Rayleigh criterion, 9091
RC time constant, 30
rectilinear propagation, 191
reflectance, 178180
reflecting telescope, 12
reflection, 193, 240
reflection coefficients, 167174
reflection grating equation, 96
refraction, 194
refractive index, 5762, 114
relative permittivity, 116
resonant angular frequency, 49
restoring force, 48
retardation, 139140, 149, 271
RF, see radio waves
rotation, 240
rotation matrix, 138
ruby laser, 42
sagittal plane, 230
Schonflies
notation, 245
Sellmeiers equation, 117
semiconductor lasers, 20, 41
semiconductor optical amplifier, 21
simple harmonic oscillator, 4851
energy, 51
slow axis, 147
Snells Law, 75, 166, 168, 170, 172
Snellius, Willebrord, 12
soft focus imaging, 229
INDEX
solar cells, 20
spacetime, 15
Special Relativity, 15, 107
spectroscopy, 1819
speed of light
in a vacuum, 110
spherical aberration
longitudinal, 228
transverse, 229
spherical waves, 69
spheroid, 263
spin, 38
spontaneous emission, 20, 41
spring constant, 49
Stefan-Boltzmann Law, 33
stereoscopic, 151
stimulated emission, 20, 41
Stokes theorem, 160
Stokes treatment, 176178
subgroup, 245
surface charge density, 164, 184
surface current, 162
symmetry
combining operations, 240242
cubic, 251252
hexagonal, 253254
matrix representations, 242245
monoclinic, 255
of a square, 239245
orthorhombic, 255
tetragonal, 252253
triclinic, 255
trigonal, 254255
symmetry operations, 250255
tangential plane, 230
telescope, 18
temporal response, 111
the speed of light in a vacuum, 57
thin lens equation, 218
time symmetry, 119121
total internal reflection, 21, 7678,
181182
tourmaline, 134
INDEX
transmission axis, 133, 134
transmission coefficients, 167174
transmittance, 178180
transverse electric, 167
transverse magnetic, 167
UHF, see ultra high frequency
ultra high frequency, 29
ultraviolet catastrophe, 32
Uncertainty Principle
energy-time, 119
uniaxial crystal, 263, 265
uniaxial systems, 252255
vacuum fluctuations, 40
vector calculus, 278279
double curl, 278279
very high frequency, 29
very low frequency, 29
VHF, see very high frequency
virtual particles, 40
VLF, see very low frequency
wave equation, 5256, 109111
anisotropic, 262263
wave optics, 13
wave packet, 16, 61
wave plate, 147152
general retardation, 151
half-wave, 141, 149150
quarter-wave, 150151
wavefront, 58
planar, 7273
spherical, 72
waveguide
slab, 7778
waveguiding, 21
wavelength, 58
wavelength division multiplexing, 21
wavevector, 58
Wiens Displacement Law, 33
work function, 35
Young, Thomas, 13
Zinc blende, 247
285