JDWUAH Optoelectronics Ch1 PDF
JDWUAH Optoelectronics Ch1 PDF
JDWUAH Optoelectronics Ch1 PDF
1
JDW, ECE Fall 2009
Course Textbook and Topics Covered
Prentice-Hall Inc.
© 2001 S.O. Kasap
ISBN: 0-201-61087-6
http://photonics.usask.ca/
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JDW, ECE Fall 2009
Additional Supportive Material
• The breadth of this course is larger than a single textbook
• Certain sections will have added material presented in class from the following
textbooks
3
JDW, ECE Fall 2009
Course Programmatics
• Course will be taught on slides posted on Angel after each class
– Students are expected to take their own notes based on class presentation
– Figures and key points will be provided after on slides
• Additional material from each subject topic will be posted on Angel as optional
reading for anyone interested
• Course will focus on key concepts and equations that describe them
– First principle derivations will NOT be required unless they are critical for student
development
– Most homework and test assignments will can be answered by understanding the question
and applying a formula
• Class project (25% of the total grade)
– Students will be asked to work in teams to research a particular topic in
optoelectronics
– Teams will turn in a term paper no less than 10 pages (1½ space)
– Teams will present the topic in 8 min presentations
• Homework will be due at the beginning of class every Tuesday (15%)
• Two in class exams and 1 comprehensive final (60%)
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JDW, ECE Fall 2009
Introduction to Optoelectronics
• Definition of Optoelectronics
– Sub field of photonics in which
voltage driven devices are used
to create, detect, or modulate
optical signals using quantum
mechanical effects of light on
www.udt.com
semiconductors materials http://www.led.scale-
– Examples of optoelectronic
train.com/blue0603led.php
Ex Eo cos(t kz o )
E and B have constant phase
in this xy plane; a wavefront
z E
E
k
Propagation
B
Ex
Ex = Eo sin(t–kz)
A plane EM wave travelling along z, has the same E x (or By) at any point in a
given xy plane. All electric field vectors in a given xy plane are therefore in phase.
The xy planes are of infinite extent in the x and y directions.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
9
JDW, ECE Fall 2009
Optical Field
• Use of E fields to describe light
– We know from Electrodynamics that a time varying H field
results in time varying E fields and vise versa
– Thus all oscillating E fields have a mutually oscillating H field
perpendicular to both the E field and the direction of
propagation
– However, one uses the E field rather than the H field to describe
the system
• It is the E field that displaces electrons in molecules and ions in the
crystals at optical frequencies and thereby gives rise to the
polarization of matter
• Note that the fields are indeed symmetrically linked, but it is the E
field that is most often used to characterize the system
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JDW, ECE Fall 2009
Optional Plane Wave Representations
• 1-D solution • General solution
E x Eo cos(t kz o ) E E (r , t ) Eo cos(t kz o )
jo j (t k r )
E x E ( z , t ) Eo cos(t kz o ) E (r , t ) Re[ Eo e e ]
j (t k r )
E (r , t ) Re[ Ec e ]
Since cos( ) Re[ei ]
Where k r k x x k y y k z z
E ( z , t ) Re[ Eo e jo e j (t kz ) ]
E ( z , t ) Re[ Ec e j (t kz ) ] k = wave vector whose magnitude is 2/λ
y Direction of propagation
k
r E(r,t)
r
z
O
P E
k r
P
O
z
A perfect plane wave A perfect spherical wave A divergent beam
(a) (b) (c)
In radians!!!!!!
(a) r
2w
(a) Wavefronts of a Gaussian light beam. (b) Light intensity across beam cross
section. (c) Light irradiance (intensity) vs. radial distance r from beam axis (z).
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall) 14
JDW, ECE Fall 2009
Example 1
• Consider a HeNe laser beam at 633 nm with a spot size of 10
mm. Assuming a Gaussian beam, what is the divergence of
the beam?
Beam divergence
4 4( 633 10 9
m)
2 8.06 10 5
rad 0.0046 o
(2wo ) (10 103 m)
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JDW, ECE Fall 2009
Frequency Dependent Permittivity
• Materials do not often demonstrate a single degree of polarization along any one direction
across the entire frequency range.
• In fact the frequency dependence of permittivity is what gives rise to properties such as
absorption within a solid and allows one to see objects in “color”.
• Most materials of optical interest have absorption bands in which the permittivity, and thus
the refractive index, changes drastically. Shifting of these constants by doping the material,
(or adding large magnetic fields) has allowed for the development of bandgap
semiconductors with specific optical properties for optical generation and detection.
• Consider the simplest expression used to calculate the permittivity
r 1 N / o
Where N is the number of polarizable molecules per unit volume, and is the
polarizability per molecule.
If I can inject or remove the relative N value in a solid, then one can change the permittivity
of that solid and therefore its electronic and optical properties.
If the solid is a stack of semiconductor materials with different N values that respond
optically when biased, then one can create an optoelectronic device!!!
If the polarizability, , is frequency dependent (and it is), then our optoelectronic device will
work over a particular frequency range which can be engineered for the spectral band of
interest!!!! 18
JDW, ECE Fall 2009
Group Velocity
• First and foremost: THERE ARE NO PERFECT
MONOCHROMATIC WAVES in practice +
• There are always bundles of waves with slightly
different frequencies and wave vectors –
• Assume the waves travel with slightly different
frequencies, ω+ω and ω - ω Emax Emax k
• The wave vectors are therefore represented by
K + k and k - k
• The combined transform generates a wave Wave packet
packet oscillating at a mean beat frequency ω
that is amplitude modulated by a slowly time Two slightly different wavelength waves travelling in the same
varying field at ω direction result in a wave packet that has an amplitude variation
which travels at the group velocity.
• The maximum amplitude moves with a wave
vector k © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
• The velocity of the packet is called the group For video of wave packets
velocity and is defined as
with and without v = vg:
d
vg http://newton.ex.ac.uk/tea
d ching/resources/au/phy11
• The group velocity defines the speed at which 06/animationpages/
the energy is propagated since it defines the
speed of the envelope of the amplitude
variation 19
JDW, ECE Fall 2009
Example: Group Velocity
+
–
Emax Emax k
Wave packet
n 1.48
• By definition, the group velocity is then
Ng
d c c 1.47
vg
dk dn Ng 1.46
n n
d 1.45
• We define Ng as the group index of the
medium. 1.44
500 700 900 1100 1300 1500 1700 1900
• We now have a way to determine the effect
Wavelength (nm)
of the medium on the group velocity at
different wavelengths (frequency Refractive index n and the group index Ng of pure
dependence!!!) SiO 2 (silica) glass as a function of wavelength.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
• The refractive index, n, and group index, Ng,
depend on the permittivity of the material, r • Ng and n are frequency (wavelength) dependent
• We define a dispersive medium is a medium • Notice the minima for Ng at 1300 nm.
in which both the group and phase velocities • Ng is wavelength independent near 1300
depend on the wavelength. nm
• All materials are said to be dispersive over • Light at 1300 nm travels through SiO2 at
particular frequency ranges the same group velocity without
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dispersion
JDW, ECE Fall 2009
Example: Effects of a Dispersive Medium
• Consider 1 um wavelength light propagating through SiO2
• At this wavelength, Ng and n are both frequency dependent with no
local minima
• Thus the medium is dispersive
• Now we must ask the question, are the group and phase velocities of
the propagating wave packet the same?
• Phase Velocity
1.49
dz c m m
v 3x108 / 1.450 2.069 x108 1.48
dt n s s Ng
1.47
d c m m 1.45
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JDW, ECE Fall 2009
Energy Flow in EM Waves
• Let us recall that there is indeed a B field in the EM wave.
• Recall from electrostatics that
c
E x vBy B y
n
• where v r o o
1
Speed of light in the medium
n r
• As the EM wave propagates along the direction k, there is an energy flow in that direction
– Electrostatic energy density 1 E 2
r o x
2 Where both these values are equal
– Magnetostatic energy density 1 B y2 o H y2
2o 2
• The Energy flow per unit time per unit area, S, is defined as the Poynting Vector
Avt r o E x2
S Eo H o
v r o E x2 v 2 r o E x By
At
2 2 c2 n2
S v r o E B v r o o E H 2 2 E H E H
n c
Irradiance
• Magnitude of the Pointing Vector is called the irradiance
• Note that because we are discussing sinusoidal waveforms, that the instantaneous irradiance
of light propagating in phase is taken from the instantaneous amplitude of E and B
respectively
S v r o E x By
2
• Instantaneous irradiance can only be measured if the power meter responds more quickly
than the electric field oscillations.
• As one might imagine, at optical frequencies, all practice measurements are made using the
average irradiance.
1
• The average irradiance is I S avg S v r o Eo2
2
1c 1 m
r o Eo2 cn o Eo2 1.33 108 nEo2
2n 2 s
1 2 1 2
v r o Eo Bo v r o o E H
2 2
1 c
2
1 c 2 n2 1
r o o E H 2 2 E H E H
2 n2 2n c 2
Example: Electric and Magnetic Fields in Light
• The intensity (irradiance) of the red laser from a He-Ne laser at a certain location was
measured to about 1mW/cm2.
• What are the magnitudes of the electric and magnetic fields?
sin t v2 n1 kr
Ai Br
A light wave travelling in a medium with a greater refractive index ( n1 > n2) s uffers
reflection and refraction at the boundary.
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JDW, ECE Fall 2009 © 1999 S.O. Kas ap,Optoelectronics (Prentice Hall)
Total Internal Reflection
• If n1 > n2 then transmitted angle > incidence angle.
• When t =90o, then the incidence angle is called the critical angle
sin c 2
n
n1
• When i > c there is
– no transmitted wave in medium
– Total internal reflection occurs
– an evanescent wave propagates along the boundary (i.e. high loss electric field
propagating along the surface)
Transmitted
(refract ed) light
kt
t n2 Evanescent w ave
n 1 > n2
ki i i kr c c i >c
TIR
Incident Reflected
light light
(a) (b) (c)
Light wave travelling in a more dense medium strikes a less dens e medium. Depending on
the incidence angle with respect to c, which is determined by the ratio of the refractive
indices , the wave may be transmitted (refracted) or reflected. (a) i < c (b) i = c (c) i
> c and total internal reflection (TIR).
r nt 1 r 1 t
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JDW, ECE Fall 2009
Internal Reflection
• Light traveling from a more dense medium Magnitude of reflection coefficients Phase changes in degrees
c
into a less dense one ( n2 < n1 ) 1 180
TIR
is the angle at which r becomes zero and • Reflected waves at angles greater than p
TE/TM polarization begins to occur are linearly polarized because they
tan p
n2 1 contain field oscillations that are
n1 1.44 contained within a well defined plane
p 35o perpendicular to the plane of incidence
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JDW, ECE Fall 2009 AND the plane of propagation
Phase Change in TIR
• For p < i < c, Fresnel’s eqn. gives r < 0.
Predicts a phase shift of 180o
• For i ≥ c, Fresnel’s eqn. gives r and r = 1 TIR =
such that the reflected wave has the same Total Internal Reflection
amplitude as the incident wave and TIR
occurs
Magnitude of reflection coefficients Phase changes in degrees
• For i > c we have r = 1, but the phase c
change, and are derived from 1
(a)
180
TIR
0.9 (b)
0.8 120
1 sin 2 i n 2 0.7
60
tan 0.6 p
2 cos i 0.5 0
0.4 c
| r | 60
1 sin i n 0.3 //
2 2
tan 0.2 p
| r // | 20
2 2 n 2 cos i 0.1
0 80
0 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 90
Et ( y, z , t ) e 2 y e j (t kiz z )
kiz ki sin i evanescent wave vector
2
2n2 n1
2 sin 2 i 1 attenuation coefficient
n2
• The penetration depth of the electric field into medium 2 is
1 Et =e-1
2
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JDW, ECE Fall 2009
External Reflection
• Light traveling from a less dense medium
into a more dense one ( n2 > n1 )
1
• At normal incidence, both Fresnel External reflection
0.8
coefficients for r and r are negative
0.6
• External reflection for TM and TE at 0.4
p r//
normal incidence generates a 180 degree 0.2
phase shift. This phase shift is observed 0
at all angles for TE waves and up to p for -0.2
TM waves -0.4
• Also, r goes through zero at the -0.6
r
Brewster angle, p -0.8
-1
• At p the reflected wave is polarized in 0 10 20 30 40 50 60 70 80 90
the E component only. Thus Light
Incidence angle, i
incident at p or higher in angle does not
generate a phase shift in reflection for The reflection coefficients r// and r vs. angle
TM waves. of incidence i for n1 = 1.00 and n2 = 1.44.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
• Transmitted light in both internal and
external reflection does NOT experience
a phase shift 33
JDW, ECE Fall 2009
Example: Evanescent Wave
• TIR from a boundary n1 > n2 generates an evanescent wave in medium 2 near the boundary
• Describe the evanescent wave characteristics and its penetration into medium 2
j (t kt r )
Et t Eio e
2
2n2 n1
kt A2 2 sin 2 i 1
n2
kt r yk t cos t zkt sin t
Penetration depth
sin t n1 sin i 1 1
n2
2
Apply Snell’s
Law at c > i
cos t 1 sin 2 t jA2 Additionally, TIR allows us to calculate t
Eto 2 cos i
j (t zkt sin t jykt A2 ) t
Et t Eio e Eio 2
n
cos i 2 sin 2 i
Et t Eio e ykt A2 e j (t zkt sint ) n1
t to e j Complex transmission value with
For TIR imaginary phase constant
kt sin t ki sin i k z
Et t Eio e ykt A2 e j (t k z z ) 34
JDW, ECE Fall 2009
Example: Internal Reflection
• Reflection of light from a less dense medium
• Wave is traveling in a glass of index n1 = 1.450
• Wave becomes incident on a less dense medium of index n2 = 1.430
sin c
n2 1.430
• What is the minimum incidence angle for TIR? c 80.47o
n1 1.450
• What is the phase change in the reflected wave at an incidence angle of 85 degrees?
2
sin i n
2
sin 85
1.43
o
2 2
1.45 116.45o
tan 1.61447
2 cos i cos 85o
|| sin 2 i n 2 1 || 62.1o
tan 2 tan
2 n cos i
2
n 2
• What is the penetration depth of the evanescent wave into medium 2 when the incidence
angle is 85o? 2
2n2 n1
2 sin 2 i 1 1.28 106 / m
n2
1 7.8 10 7 m 35
2
JDW, ECE Fall 2009
Intensity, Reflectance, and Transmittance
• Relative (%) intensity of the reflected light traveling through the media – Reflectance
2
2
Ero Ero|| 2
R r
2
R|| 2
r||
2
Eio Eio||
2
n n
R R R|| 1 2
n1 n2
• Relative (%) intensity of the transmitted light traveling through the media – Transmittance
2 2
n2 Eto n 2 n2 Eto|| n
T 2 t
2
2 T|| 2 t||
n1
2
n1 Eio n1 Eio|| n1
4n1n2
T T T||
n1 n2 2
• Sum of the transmittance and reflectance in any conserved system must equal 1
R T 1
36
JDW, ECE Fall 2009
Example: Internal and External Reflection
• Light propagates at normal incidence from air, n = 1, to glass with a refractive index of 1.5.
What is the reflection coefficient and the reflectance w.r.t to the incident beam?
n1 n2 1 1.5
r r 0.2 2
R|| r|| 0.04 or 4%
n1 n2 1 1.5
• Light propagates at normal incidence from glass, n = 1.5, to air with a refractive index of 1.0.
What is the reflection coefficient and the reflectance w.r.t to the incident beam?
n1 n2 1.5 1
r r 0.2 2
R|| r|| 0.04 or 4%
n1 n2 1.5 1
• What is the polarization angle of the in the external reflection for the air to glass interface
described by the first question above? How would one make a polaroid device (device that
polarizes light based on the polarization angle)?
tan p
n2 At an incidence angle of 56.3o the reflected light will be polarized with
1.5
n1 an E to the plane of incidence. Transmitted light will be partially
polarized. By using a stack of N glass plates, one can increase the
p 56.3o
polarization of the transmitted light by a factor of N
37
JDW, ECE Fall 2009
Reflectance at Different Angles of Incidence
• Light propagates at 30o incidence from air, n = 1, to glass with a refractive index of 1.5. What
is the reflection coefficient and the reflectance w.r.t to the incident beam?
n1
sin i sin t Ero cos i n sin i
2 2
Snell’s Law r
n2 Eio cos i n 2 sin 2 i
t = 19.5o
replace n1 n1 cos i
n2 n2 cos t
cos i 0.866
cos t 0.943
n1 n2 0.866 1.414 2
r r 0.24 R|| r|| 0.058 or 5.8%
n1 n2 0.866 1.414
38
JDW, ECE Fall 2009
Example: Antireflection Coatings on Solar Cells
A
B
2n2 d m
2d m 4n
2
– Thus the thickness of the coating must be multiples of the quarter wavelength of light
propagating through it.
– Also, to obtain a good degree of destructive interference, the amplitudes of the A and B
waves must be comparable. Thus we need n2 n1n3
– This yields a reflection coefficient between the air and coating that is equal to that
between the coating and the semiconductor. In our case n2 should equal 1.87 which is
close to that of Si3N4 at 1.9. 40
JDW, ECE Fall 2009
Example: Antireflection Coatings on Solar Cells
A
B
C
1 2 1 2 (nm)
n1 n2 n1 n2 330 550 770
o
Schematic illus tration of the principle of the dielectric mirror with many low and high
refractive index layers and its reflectance. 42
JDW, ECE Fall 2009 © 1999 S.O. Kasap, Optoelectronics (P rent ice Hall)
Multiple Interference and Optical Resonators
Relative intensity
M1 M2 m=1
A 1 f R ~ 0. 8
m=2 R ~ 0. 4
m
B
L m=8
m - 1 m m + 1
(a) (b) (c)
Schematic illus tration of the Fabry-Perot optical cavity and its properties. (a) Reflected
waves interfere. (b) Only standing EM waves, modes, of certain wavelengths are allowed
in the cavity. (c) Intens ity vs. frequency for various modes. R is mirror reflectance and
lower R means higher loss from the cavity.
© 1999 S.O . Kas ap,Optoelectronics (Prentice Hall)
43
JDW, ECE Fall 2009
Multiple Interference and Optical Resonators
• Since the electric field at the mirrors must be zero, we can only fit integer multiples of
half wavelengths into the cavity of length L
m L m 1,2,3...
2
• Each cavity mode is defined by the m value, or the number of the half wavelengths that
constructively interfering within the cavity.
• Resonant frequencies are the beat oscillation frequencies resonant in the cavity
Relative intensity
M1 M2 m=1
c
vm m mv f A 1 f R ~ 0. 8
2L m=2 R ~ 0. 4
m
c
vf B
2L L m=8
m - 1 m m + 1
• Free spectral range (a) (b) (c)
Schematic illus tration of the Fabry-Perot optical cavity and its properties. (a) Reflected
vm vm1 vm v f waves interfere. (b) Only standing EM waves, modes, of certain wavelengths are allowed
in the cavity. (c) Intens ity vs. frequency for various modes. R is mirror reflectance and
lower R means higher loss from the cavity.
44
© 1999 S.O . Kas ap,Optoelectronics (Prentice Hall)
JDW, ECE Fall 2009
Fabry-Perot Optical Resonator
• Simple optical cavity that stores radiation energy only at certain frequencies
• Assume a wave A travels within the cavity and is reflected back and forth as wave B.
• The field and intensity of the cavity are:
A
Ecavity A Ar 2e 2 jkL Ar 4 j 4kL Ar 6e6kL ... Ecavity
1 r 2e jkL
2 Io Io
I Ecavity I max
( I R) 4 R sin 2 (kL)
2
( I R) 2
• Spectral width of the cavity: vf R
vm F
F 1 R
Where F is called the Finesse of the cavity which is the ratio of mode separation to spectral
width. Thus as losses decrease, finesse increases. Also larger finesses lead to sharper
mode peaks
Partially reflecting plates
Transmitted light
(1 R) 2
Input light Output light I trans I incident
( I R) 2 4 R sin 2 (kL)
L
m - 1 m
Fabry-Perot etalon
• Consider a Fabry-Perot optical cavity of air length = 100 microns with mirrors that have a 0.9
reflectance.
• Calculate the cavity mode nearest to 900 nm.
m
2L
2 100 10 6 m 222.22
900 10 9 m
m
2 L 2 100 10 6 m
900.90nm
m 222
• Calculate the separation of the modes and the spectral width of each mode.
c 3 108 m / s
vm v f 1.5 1012 Hz
6
2 L 2 100 10 m
R 0.9
F 29.8
1 R 1 0.9
v
vm f 5.03 1010 Hz
F
c
m 2
vm m vm 0.136nm ½ bandwidth of resonator output
vm c
46
JDW, ECE Fall 2009
Optical Tuning
Reflected
B = Low refractive index
Reflected
transparent film ( n 2)
B Virtual reflecting plane
n2 light
y
n1
Penetration depth, n2 n1 FT IR Incident
T IR n1
z i > c light
n 1 > n2 i > c
i r T ransmitted
A Incident
z A light (b)
(a) C A
Incident Reflected Glass prism
light light
The reflected light beam in total internal reflection appears to have been laterally s hifted by (a) A light incident at the long face of a glas s pris m suffers TIR; the prism deflects the
an amount z at the interface. light.
(b) Two pris ms separated by a thin low refractive index film forming a beam-splitter cube.
© 1999 S.O. Kasap, Optoelectronics (P rentice Hall) The incident beam is s plit into two beams by FTIR.
© 1999 S.O. Kasap, Optoelectronics (P rentice Hall)
• Coherence time: l (a) A sine wave is perfectly coherent and contains a well-defined frequency o. (b) A finite
t wave train lasts for a duration t and has a length l. Its frequency s pectrum extends over
c = 1/t. It has a coherence time t and a coherence length l. (c) White light exhibits
practically no coherence.
• Spatial coherence measures the extent at © 1999 S.O. Kasap, Optoelectronics (P rent ice Hall)
l 0.6mm
incoherent beam.
Diffract ed beam
Circular aperture
A light beam incident on a s mall circular aperture becomes diffracted and its light
intensity pattern after passing through the aperture is a diffraction pattern with circular
bright rings (called Airy rings ). If the screen is far away from the aperture, this would be a
Fraunhofer diffraction pattern.
© 1999 S.O. Kas ap,Optoelectronics (Prentice Hall)
49
JDW, ECE Fall 2009
Introduction to Diffraction
Incident plane wave
A secon dary
wave so urce
An oth er ne w
wa vefront (diff ract ed)
Ne w
wa vefront z
(a) (b)
(a) Huygens-Fresnel
_______________principles states that each point in the aperture becomes a source of
secondary waves (spherical waves ). The s pherical wavefronts are separated by . The new
wavefront is the envelope of the all these s pherical wavefronts . (b) Another possible
wavefront occurs at an angle to the z-direction which is a diffracted wave.
50
JDW, ECE Fall 2009
Introduction to Diffraction
Incident y
Incident
• Light emitted from a point source light wave
Y light wave Screen
y R = Large
E y e jk sin
y
y a
E ( ) C
y 0
y e jk sin
A
a b
c
y
1
Ce jk sin a sin ka sin y z
E ( ) 2 z ysin
1
ka sin Light intensity
2 (a) (b)
• The single slit diffraction equation yields an (a) The aperture is divided into N number of point sources each occupyingy with
amplitude y. (b) The intensity distribution in the received light at the screen far away
intensity
2 from the aperture: the diffraction pattern
1
Ca sin ka sin © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
I ( ) 2 I (0) sin c 1 ka sin
1 2
ka sin
2
• With zero intensity points at
m
sin
a
m 1,2,...
sin 1.22 where D is the diameter of the aperture
D
51
JDW, ECE Fall 2009
Image Resolution
y
• Consider 2 nearby coherent sources are
S1 A2
S1 imaged through an aperture of diameter D
s • The two sources have an angular
S2 S2 separation of .
A1 L
I
• As the points get closer together
Screen
– angular separation becomes narrower
Resolution of imaging systems is limited by diffraction effects. As points S1 and S2
get closer, eventually the Airy disks overlap so much that the resolution is lost. – diffraction patterns overlap more
© 1999 S.O. Kasap ,Optoelectronics (Prentice Hall) • According to the Rayleigh criterion, the two
spots are just observable when the
principle maximum of one diffraction
pattern coincides with the minimum of
b
another
a • This minimum is obtained by the angular
radius of the Airy disk
The rectangular aperture of dimensions a b on the left sin 1.22
D
gives the diffraction pattern on the right.
where D is the diameter of the aperture
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)
52
JDW, ECE Fall 2009
Diffraction Gratings
First-order
• Bragg Diffraction Condition Zero-order
Incident m = 1 First-order
light wave
d sin m
m = 0 Zero-order
Incident
First-order
light wave
m 0,1,2,3,...
m = -1 First-order
• For light incident at an angle (a) Transmission grating (b) Reflect ion grating
d (sin m sin i ) m (a) Ruled periodic parallel scratches on a glass serve as a transmission grating. (b) A
reflection grating. An incident light beam results in various "diffracted" beams. The
zero-order diffracted beam is the normal reflected beam with an angle of reflection equal
m 0,1,2,3,... to the angle of incidence.
(a) (b )
(a) A diffraction grating with N slits in an opaque scree. (b) The diffracted light
pattern. There are distinct beams in certain directions (schematic) Blazed (echelette) grating.
© 1999 S.O. Kasap, Optoelectronics (Prentice Hall) 53
JDW, ECE Fall 2009 © 1999 S.O. Kasap, Optoelectronics (Prentice Hall)