Nature and characteristics of light

 Light is a transverse electromagnetic radiation

- Light travels in vacuum at a constant speed carrying EM
energy and this speed c is a universal constant c ≈ 3× 108 m/s =
300,000 km/s
- The sun is 149, 000, 000 km away from the earth
- If you could drive to the sun, it would take you 177 years to get
there!
- But for light, it takes just 8 minutes arrive at earth!
• The speed of light changes when it travels in non-vacuum
media such as air (0.003% slower) or glass (30% slower)
• Refractive index of a medium n is defined as the speed of
light in vacuum c divided by its speed v in the medium.
Dual nature of light
Photoelectric effect
Two concerns observed in Photoelectric effect which
contradict wave theory:
1.More intense radiation (larger-amplitude waves) did not
cause emitted electrons to have more energy.
2.The energy of the emitted electron was dependent on the
wavelength of the light, not the amplitude of the wave.
• Light knocked the electrons out of the metal surface as if the
light were made of particles - photons.
• There is a minimum energy threshold for an electron to
escape from the metal- characteristic escape energy (CEE).
• Photons with frequencies below a given threshold eject no
electrons, no matter how intense the light.
• Photons with frequencies above the threshold do eject
electrons, no matter how low the intensity.
• The energy of the released electrons can be calculated from:
Example: Calculate the threshold wavelength of light needed
to just release electrons from gold (Given pgold = 7.68 × 10–19
J = 4.79 eV )
Solution:

• Any light having longer wavelength than 260 nm will not

have sufficient energy to be able to eject an electron out from
the gold metal.
Wave Model
• The wave model describes light as an oscillating transverse
Electric Field and Magnetic field.
• Wavelength(λ) , Period(τ), Frequency(f )
• See the following one-dimensional representation of the
electromagnetic wave
Properties of EM waves
 Basic Optics
Basics
Optics

Geometrical Wave
optics optics

Helps to understand Helps to understand

• The basics of light reflection and • The phenomena of Interference,
refractions Diffraction, Polarization
• The use of simple optical elements • The use of thin film coatings on
(mirrors, prisms, lenses and fibers) mirrors to enhance or suppress
reflection
• The operation of grating, wave
plates, polarizers, interferometers,
etc
I. Geometrical Optics
Upon finishing this title or part, you will be able to
• Distinguish between light rays and light wavefronts
• State the law of reflection and show how it applies to light
rays at mirrors
• Define index of refraction and give typical values for glass,
water, and air
• State Snell’s law of refraction and show how it applies to
light rays at lenses
• Describe total internal reflection and show how it is used in
fibers
• Use graphical method to characterize the images formed
mirrors and lenses
• Use Gaussian Optics and equations to determine location,
size and orientations of images formed by mirrors and lenses
• Use the Lensmaker’s equation to determine the focal length
of a lens
• Use the ray-transfer matrix to analyze propagation of light in
an optical system
• Describe the f-number and numerical aperture of a lens and
explain how they control image brightness in Cameras and
Microscopes
• Describe image aberration and explain the different types of
aberrations
Light rays and wavefronts
Consider a water wave created on a quiet pond by a bobbing
cork.

• Wave front is defined as a locus of points that connect

identical wave displacements-that is identical positions above
or below the normal surface of the quiet pond.
• A ray is a line perpendicular to a series of successive wave
fronts specifying the direction of energy flow in the wave.
• This shows plane wave fronts of light bent by a lens into
circular (spherical in 3D) wave fronts that then converge onto
a focal point F.
• The same diagram shows the light rays corresponding to
these wave fronts, bent by the lens to pass through the same
focal point F.
• This figure shows clearly the connection between actual
waves and the rays used to represent them.
Light rays
• In the study of geometrical optics, we find it acceptable to
represent the interaction of light waves with plane and
spherical surfaces - with mirrors and lenses - in terms of light
rays.
• When light waves propagate through and around objects
whose dimensions are much greater than the wavelength, the
wave nature of light is not readily discerned, so that its
behavior can be adequately described by rays obeying a set of
geometrical rules. This model of light is called ray
optics/geometrical optics.
• Strictly speaking, ray optics is the limit of wave optics when
the wavelength is infinitesimally small.
• With the useful geometric construct of a light ray, we can
illustrate propagation, reflection, and refraction of light in
clear, uncomplicated drawings.
Light behavior at the interface between two media
When light is incident on an interface between two optical
media, four things can happen to the incident light:-
• It can be partly or totally reflected at the interface
• It can be partly transmitted via refraction at the interface and
enter the second medium
• It can be scattered in random directions at the interface
• It can be partly absorbed in either medium
1. Reflection
Light reflection from optical surfaces
In this basic geometrical optics module, we shall
• Consider only smooth surfaces that give rise to specular
(regular, geometric) reflections (Figure - a).
• Ignore ragged, uneven surfaces that give rise to diffuse
(irregular) reflections (Figure - b).
The Law of reflection
1. The angle θi the incident ray makes with the e normal (line
perpendicular to the surface) is equal to the angle θr that the
reflected ray makes with the normal
2. The incident ray, reflected ray, and normal always lie in the
same plane.
2. Refraction
Light refraction through optical surfaces
• Is the bending of light rays at an interface between two
optical media
• Optical medium is characterized a constant called index of
refraction (n)
• The index of refraction for any transparent optical medium is
defined as the ratio of the speed of light in a vacuum to the
speed of light in the medium

where c- speed of light in vacuum, v - speed of light in the

medium, n-index of refraction of the medium
• The greater the index of refraction of a medium, the lower the
speed of light in that medium and the more light is bent in
going from air into the medium.
Snell’s law: the law of refraction
• When light travels from a medium having lower n to another
medium with higher n, light bends towards the normal, and
vice versa
Snell’s law

 If n 1 < n2, θ2 < θ1 - Refracted ray bends towards the

normal
 If n1 > n2, θ2 > θ1 - Refracted ray bends away from the
normal
Example: In a handheld optical instrument used under water,
light is incident from water (n = 1.33) onto the plane surface of
flint glass (n = 1.66) at an angle of incidence of 45°.
(a) What is the angle of reflection of light off the flint glass?
(b) Does the refracted ray bend toward or away from the
normal?
(c) What is the angle of refraction in the flint glass?
Critical angle & total internal reflection
• Consider light ray traveling from a medium of higher index
(ex. glass) to one of lower index (ex. air)– refracted ray bends
away from the normal.
• The incident angle at which the refracted ray makes an angle
of 90° is called critical angle θc.
• For any incident angle θ1 > θc, total internal reflection occurs
and the light stays in the same medium
• Compared with ordinary reflection from mirrors, the
sharpness and brightness of totally internally reflected light
beams is enhanced considerably.
Example: Optical Fibers
- Make use of the concept of total internal reflection
- Are widely used in communication, sensing and imaging
Exercise: A step-index fiber 0.0025 inch in diameter has a core
index of 1.53 and a cladding index of 1.39. See the following
drawing. What is the maximum acceptance angle θm for a
cone of light rays incident on the fiber face such that the
refracted ray in the core of the fiber is incident on the cladding
at the critical angle?
Solution:

• Thus, the maximum acceptance angle is 39.7° and the acceptance cone is twice
that, or 2 θm = 79.4°.
• The acceptance cone indicates that any light ray incident on the fiber face
within the acceptance angle will undergo total internal reflection at the core-
cladding face and remain trapped in the fiber as it propagates along the fiber.
Light refraction in Prism
• Glass prisms are often used to bend light in a given direction
Light refraction in Prism: Dispersion
• The refractive index n slightly varies with wavelength –
dispersion.
• The shorter the wavelength the higher the refractive index
- For example, the index of refraction for flint glass is about
1% higher for blue than for red light.
- Hence blue bends more than the red light – giving rise to
color separation.
• Widely used in photonics systems (spectrometers,
monochromators, etc)
Light refraction: Rainbow
• Light of different colors have slightly different refractive
indices in water and therefore show up at different positions
in the rainbow.
Image formation
I. with Mirrors
II. with Lenses
I. Image formation with Mirrors
• Mirrors are everywhere - in homes, auto headlamps,
astronomical telescopes, and laser cavities, etc.
• Plane and spherical mirrors are used to form 3D images of
3D objects.
• If the size, orientation, and location of an object relative to a
mirror are known, its image obtained
- Graphically (using the law of reflection and ray tracing)
- Analytically (using formulas)
A. Images formed with plane mirror
1. Graphical ray-trace method (plane mirror)
• The eye sees a point image at S′ in exactly the same way it would
see a real point object placed there.
• Since the actual rays do not exist below the mirror surface, the
image is said to be a virtual image.
• The image S′ cannot be projected on a screen as in the case of a
real image.
• An extended object, such as the arrow in Figure (b) is imaged
point by point by a plane mirror surface in similar fashion.
• Each object point has its image point along its normal to the
mirror surface.
• In Figure 2-14c, where the mirror does not lie directly below the
object, the mirror plane may be extended to determine the
position of the image as seen by an eye positioned to receive
reflected rays originating at the object.
• Figure 2-14d illustrates multiple images of a point object O
formed by two perpendicular mirrors.
B. Images formed with spherical mirrors
1. Graphical ray-trace method (Spherical mirrors)
To employ the method of ray tracing, we agree on the
following:
- The axis of symmetry normal to the mirror surface is its
optical axis.
- The point where the optical axis meets the mirror surface is
the vertex V.
- Light will be incident on a mirror surface initially from the
left.
- The point on the optical axis located half way from the vertex
is called the focal point F.
To locate an image graphically, we use two points common to
each mirror surface:
1. Center of curvature C and
2. Focal point F
• Consider parallel rays impinging on the spherical mirrors.
• Applying the law of reflection, we get that :
- The reflected ray from a concave mirror passes through a focal point F.
- The reflected ray from a convex mirror appears to come from a focal point F
behind the mirror.
• Notice that a line drawn from the center of curvature C to any point on the mirror
is a normal line and thus bisects the angle between the incident and reflected rays.
Key rays used in ray-tracing
• The figure below shows the three key rays labeled 1, 2 and 3 for each mirror
• They are used to locate an image point corresponding to a given object point.

Real image can be formed on the screen located there whereas virtual image
cannot.
2. Analytical Method
Derivation of mirror formulas for image location
• Consider a concave mirror with that images a point on the optical axis as shown
below.
• s and s’ are measured relative to the mirror vertex V.
• We seek a relationship between s and s’ that depends on only the radius of
curvature r of the mirror.
Similarly, for convex mirror,
Sign convention for Mirrors
• By adopting the following sign convention, the same formula can be used for both
mirrors.

• Object and image distances s and s’ are both positive when located to the left of
the vertex V and both negative when located to the right.
• The radius of curvature r is positive when the center of curvature C is to the left of
the vertex V (concave mirror) and negative when C is to the right (convex mirror).
• Vertical dimensions are positive above the optical axis and negative below.
Magnification of Mirror image
• Magnification m is defined as the ratio of image height hi to object height ho:

• Taking sign convention into account, magnification m can be expressed in terms

of the object distance s and image distance s’ as:

 +ve m value implies the image is erect.

 -ve m value implies the image is inverted.
Example:

Hint: Use the general formula with appropriate mirror sign convention
Solution:
II. Image formation with Lenses
• A lens is made up of a transparent refracting medium, generally of some type of
glass, with spherically shaped surfaces on the front and back.

• Lenses are at the heart of many optical devices (cameras, microscopes, binoculars
and telescopes).
• The law of reflection determines the imaging properties of mirrors.
• Snell’s law of refraction determines the imaging properties of lenses.
• Lenses are used primarily for image formation with visible light, but also for
ultraviolet and infrared light.
• A ray incident on the lens refracts at the front surface (according to Snell’s law)
propagates through the lens, and refracts again at the rear surface (as shown).
Types of lenses
Lenses are broadly classified into two major groups:
1. Converging/positive lenses
2. Diverging/negative lenses
Lenses can also be classified as:
1. Thick lens
2. Thin lens
Thick lens
• The thickness of a lens is not negligible compared with the radius of curvature of
its faces.
• Ray-tracing techniques and lens-imaging formulas are more complicated for thick
lenses.
• Computer programs are often developed to trace the rays through the lens.
Thin lens – focus of this course
• The thickness of a lens is small compared with the radii of curvature of its
surfaces.
• Ray-tracing techniques and lens formulas are relatively simple for thin lenses.
• Gaussian/paraxial approximations can be used (light cone within 20 deg is
considered).
• Most optical systems are analyzed using thin lens assumptions – we will study
only thin lenses.
Focal points of thin lenses
• As in mirrors, the focal points of lenses are defined in terms of their effect on
parallel light rays and plane wave fronts.
- For the positive lens, refraction of the light causes it to converge to focal point F
(real image) to the right of the lens.
- For the negative lens, refraction of the light causes it to diverge as if it is coming
from focal point F (virtual image) located to the left of the lens.

How converging/diverging of plane wave takes place?

• Light travels more slowly in the lens medium than in the surrounding air, so the
thicker parts of the lens retard the light more than do the thinner parts.
• Recall that in mirrors light remains only in one side – have single focal point.
• In lenses, light can approach from either side of the lens.
• Hence unlike mirrors, thin lenses have two focal points symmetrical located on
each side of the lens.
Image formation with Lenses
1. Graphical method using ray-tracing
• We make use of three key points for the lens and associate each of them with a defining
ray.
–Three points: front focal point F, back focal point F’ and Vertex V
–Three defining rays: shown in the figure below labeled as 1, 2 & and 3
–In fact only two rays are enough to locate the image
Train of lenses
• For accuracy in drawing, a common practice used is to show the positive lens as a
vertical line with normal arrowheads and the negative lens as a vertical line with
inverted arrowheads, and to show all ray bending at these lines.
• The figure shows a ray trace through an “optical system” made up of a positive
and a negative lens.
2. Analytical Method -Thin Lens
• As in mirrors, formulae can be used to locate images formed by lenses
• The lens formula is derived by combining Snell’s Law with Gaussian Optics
Deriving the lens formula using the ray-transfer matrix method
• Consider a light ray entering an optical system from the left at P1(y1,α1) and leaving the
optical system at P2(y2,α2) as shown below.
• The optical system can be represented by a 2х2 ray-transfer matrix M.
1. Translational Matrix- it is about propagation in a homogeneous medium
2. Refraction Matrix- at Plane surface
3. Refraction Matrix- at Convex Spherical Surface (R is +ve)
4. Refraction Matrix- at Concave Spherical Surface (R is -ve)
For thick lens (t ≠ 0)
For thin lens (t = 0)(t is small compared to R1 and R2)
• For thin lens (t = 0)
Condition of image formation
• For an image to be formed, all the rays from P should meet at P’.
• In that case y’ should be independent of α. Hence B = 0
Image magnification m produced by thin lens

Where h’- is the transverse size of the image, h- is the transverse size of the object & s and
s’ are object and image distances respectively.
Sign convention

Just as for mirrors, we must agree on a sign convention to be used in the application of
the above thin lens formulae.
• Light travels initially from left to right toward the lens.
• Object distance s is positive for real objects located to the left of the lens and negative
for virtual objects located to the right of the lens.
• Image distance s’ is positive for real images formed to the right of the lens and negative
for virtual images formed to the left of the lens.
• The focal length f is positive for a converging lens and negative for a diverging lens.
• The radius of curvature r is positive for a convex surface and negative for a concave
surface.
• Transverse distances (h’ and h’) are positive above the optical axis and negative below.
Example 1
A double-convex thin lens used as a simple “magnifier” has a front surface with a radius of
curvature of 20 cm and a rear surface with a radius of curvature of 15 cm. The lens material
has a refractive index of 1.52. Answer the following questions to learn more about this
simple magnifying lens.
(a) What is its focal length in air?
(b) What is its focal length in water (n = 1.33)?
(c) Does it matter which lens face is turned toward the light?
(d) How far would you hold an index card from this lens to form a sharp image of the sun
on the card?
Solution
c) For thin lens, it does not matter (check by calculation) but for thick lens it does matter.
d) The light from the sun (parallel when it arrives the earth) will focus at a distance equal to
the focal length from the lens.
Example 2

Locate the final image, determine its size and state whether it is real or virtual, erect or
inverted.
Solution
Object – image relationships
Image formation in a microscope
Location of real and virtual images in a light microscope marked s through w
• Note that the specimen at s lies just in front of objective, resulting in a real, magnified
image at t in the eyepiece.
• The primary image at t lies just inside the focus of the eyepiece, resulting in diverging
rays at u .
• The cornea and lens of the eye form a real image of the object on the retina at v which is
because of the diverging angle at u perceives the object as a magnified virtual image at w
.
Imaging through multiple lenses
i) Graphical method
• Parallel-ray method, find intermediate image, use it as object for next lens
ii) Mathematical method
• Find intermediate image, use it as object for the next lens
iii) Combinations of thin lenses
• In contact
• Separated
Example: Two separated +ve Lenses
i) Graphical Method (Parallel - ray method)
Needed information
• Focal lengths of lenses
• Location of lenses
• Location of object
Step 1: Ignore the second lens and trace at least two of the key rays from tip of object
Step 2: Image (real) from step 1 becomes Object (virtual) for step 2. Now ignore Lens1 and
repeat step 1
ii) Mathematical method

• Given f1, f2, d, and s1, find s2’

• Apply imaging formula to first lens to find the intermediate image in that lens Find
object distance for second lens (negative means virtual object)
• Find object distance for second lens (negative means virtual object)

• Use imaging formula again to find final image

• Magnification with multiple lenses

- The magnification is by definition the image size divided by the object size.
Combination of multiple lenses
a) Thin lenses in contact
• Can be applied to multiple thin lenses in contact as well
• As always, be careful with sign convention
b) Thin lenses in separation
F - number and Numerical aperture of Lenses
• The size of a lens determines its light gathering power and, consequently, the brightness
of the image it forms.
• Two commonly used indicators of this special characteristic of a lens are called the f-
number (f/#) and the numerical aperture (NA).

Camera lens Objective lens

F-number of lenses
• F-number (f/#) is defined as follow:
• Since the total exposure (joules/cm2) on the film/camera is the product of the Intensity
(watt/cm2) and the exposure time or shutter speed (seconds), a desirable film exposure
can be obtained in a variety of ways.
Numerical Aperture (NA) of Lenses
• The numerical aperture is another important lens design parameter, related directly to
how much light the lens gathers
• The concept of NA finds immediate application in the design of the lens for a microscope
and optical fiber
• Because light-gathering capability is crucial for such optical devices/instruments.
• Let θ be the maximum half-angle of the cone of light collected by the lens, D the physical
diameter of the lens and n be the refractive index of the medium b/n the lens and the
object (specimen).
• The Numerical aperture (NA) is defined as:

• The higher the NA, the more light can be collected by the lens.
NA of Microscope Objective Lens
The NA of a microscope objective lens can be increased by increasing the
1. Refractive index n of the medium between the lens and the specimen and
2. Diameter D of the lens
NA and oil immersion