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July 2003 Chuck DiMarzio, Northeastern University 11270-07-1
ECEG105Optics for Engineers
Course NotesPart 7: Diffraction
Prof. Charles A. DiMarzio
Northeastern University
Fall 2007
August 2007
July 2003 Chuck DiMarzio, Northeastern University 11270-07-2
Diffraction Overview• General Equations
• Fraunhofer
– Fourier Optics
– Special Cases
– Image Resolution
– Diffraction Gratings
– Acousto-Optical Modulators
• Fresnel
– Cornu Spiral
– Circular Apertures
• Summary
It's All About /D
August 2007
?/D
D
July 2003 Chuck DiMarzio, Northeastern University 11270-07-3
Difraction: Quantum Approach
• Uncertainty
• Photon Momentum
• Uncertainty in p
ΔxΔp≥h
Δk x≥2πΔx
p=h
2πk
• Angle of Flight
• For a Better Result
– Use Exact PDF
– Gaussian is best
• Satisfies the equality
• Minimum-uncertainty wavepacket
sin θ =k x
k
Δsin θ =2πΔxk
= 2πλΔx2π
= λΔx
July 2003 Chuck DiMarzio, Northeastern University 11270-07-4
Quantum Diffraction Examples
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6
-4
-2
0
2
4
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6
-4
-2
0
2
4
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6
-4
-2
0
2
4
6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-6
-4
-2
0
2
4
6
200 Random Paths
Aperture1
Aperture2
Aperture5
Aperture10
July 2003 Chuck DiMarzio, Northeastern University 11270-07-5
Maxwell’s Eqs & Diffraction∇×E=−∂B
∂t
z
x
y
z-component of curl is zero
y-component of curl is zerox-component is notE in y direction, B in -x directionPropagation in z direction
zx
y
z-component of curl is not zeroif E changes in x direction
Now, B has a z component, soPropagation is along both z and x
July 2003 Chuck DiMarzio, Northeastern University 11270-07-6
Summary of Diffraction MathMaxwell’sEquations
HelmholtzEquation
Green’sTheorem
KirchoffIntegralTheorem
Fresnel-KirchoffIntegralFormula
FresnelDiffraction
Fourier Transforms
HankelTransforms
MieScattering
YeeNumericalMethods
All ScalarWaveProblems
Spheres
Scalar Fields
GeneralProblems
FieldsFar FromAperture
r>>λ
Obliquity=2,ParaxialApproximation
ShadowsandZonePlates
x,ySeparableProblems
CircularApertures
FraunhoferConditions
PolarSymmetry
“SimpleSystems”
July 2003 Chuck DiMarzio, Northeastern University 11270-07-7
Kirchoff Integral Theorem (1)
• General Wave Probs.
– Solve Maxwell's Eqs.
– Use Boundary Conditions
– Hard or Impossible
• Kirchoff Integral Approach
– Algorithmic
– Correct (Almost)
• Based on Maxwell's Equations
• Scalar Fields
– Complete• Amplitude and Phase
– Amenable to Approximation
– Comp. Efficient?
– Intuitive• Similar to Huygens
July 2003 Chuck DiMarzio, Northeastern University 11270-07-8
Kirchoff Integral Theorem (2)• The Idea
– Consider Point of Interest
– Correlate Wavefronts• “Best Wavefront”
– Converging Uniform Spherical Wave
• Actual Wavefront
• The Mathematics
– Start with Converging Spherical Wave
– Green's Theorem
– Helmholtz Equation• Ties to Maxwell's
Equations (Scalar Field)
– Various Approximations
– Numerical Techniques
• Results
– Fresnel Diffraction
– Fraunhofer Diffraction
July 2003 Chuck DiMarzio, Northeastern University 11270-07-9
Kirchoff Integral Setup
P
Surface A0
Surface A
The Goal: A Green’s FunctionApproach.
U x,y,z =∫ G x,y,z,x 1 ,y1 ,z 1 U x 1 ,y1 ,z 1 dV 1
July 2003 Chuck DiMarzio, Northeastern University 11270-07-11
Helmholtz-Kirchoff Integral
P
Surface A0
Surface A
P
Surface A
r’ r
n
A0
U=U0eikr'
r
July 2003 Chuck DiMarzio, Northeastern University 11270-07-15
Integral Expressions
(Hankel Transform)
July 2003 Chuck DiMarzio, Northeastern University 11270-07-16
Fraunhofer and Fresnel
z
z
• Fraunhofer works
– in far field or
– at focus.
• Fresnel works
– everywhere else.
– For example, it predicts effects at edges of shadows.
August 2007
July 2003 Chuck DiMarzio, Northeastern University 11270-07-17
Fraunhofer Diffraction• Equations
• A Hint of Fourier Optics
• Numerical Computations
• Special Cases (Gaussian, Uniform)
• Imaging
• Brief Comment on SM and MM Fibers
• Gratings
• Brief Comment on Acousto-Optics
August 2007
July 2003 Chuck DiMarzio, Northeastern University 11270-07-18
Fraunhofer Diffraction (1)
Very ImportantParameter
July 2003 Chuck DiMarzio, Northeastern University 11270-07-24
Numerical Computation (2)
• Quadratic Phase of Integrand
– Near Focus (z=f): Not a problem
– Otherwise
• Many cycles in integrating over aperture
• Contributions tend to cancel, so
• roundoff error becomes significant
• but geometric optics is pretty good here,
– except at edges.
– We will approach this problem later.
July 2003 Chuck DiMarzio, Northeastern University 11270-07-29
Imaging: Rayleigh Criterion
R/d0 is f#
August 2007
July 2003 Chuck DiMarzio, Northeastern University 11270-07-30
Single-Mode Optical Fiber
Beam too Large(lost power at edges)
Beam too Small(lost power through cladding)
July 2003 Chuck DiMarzio, Northeastern University 11270-07-31
Diffraction Grating
i
d
ReflectionExample
d
July 2003 Chuck DiMarzio, Northeastern University 11270-07-32
Grating Equation
-100 0 100 200-1
-0.5
0
0.5
1sin(d)
sin(i)
degrees
-sin(i) n=0
-1
-2
12
3
4
5
-3ReflectedOrders
TransmittedOrders
July 2003 Chuck DiMarzio, Northeastern University 11270-07-33
Grating Fourier AnalysisGrating Diffraction Pattern
Slit
Convolve
Sinc
Multiply
Repetition Pattern
Multiply Convolve
Apodization
Result
Result
July 2003 Chuck DiMarzio, Northeastern University 11270-07-34
Grating for Laser Tuning
f
Gain
f
Cavity Modes
i
August 2007
July 2003 Chuck DiMarzio, Northeastern University 11270-07-35
Monochrometer
i
sin
n=1 n=2 n=3
Aliasing
August 2007
July 2003 Chuck DiMarzio, Northeastern University 11270-07-36
Acousto-Optical Modulator
Absorber
Sound Source
• Acoustic Wave:
– Sinusoidal Grating
• Wavefronts as Moving Mirrors
– Signal Enhancement
– Doppler Shift
• Acoustic Frequency Multiplied by Order
August 2007
More Rigorous Analysis is Possible but Somewhat Complicated
July 2003 Chuck DiMarzio, Northeastern University 11270-07-37
Fresnel Diffraction
• Fraunhofer Diffraction Assumed:– Obliquity = 2– Paraxial Approximation– At focus or at far field
• Relax the Last Assumption– More Complicated Integrals– Describe Fringes at edges of shadows
July 2003 Chuck DiMarzio, Northeastern University 11270-07-39
Cornu Spiral
C(u), Fresnel Cosine Integral
S(u)
, Fre
snel
Sin
e In
tegr
al
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-5<u<5
u=0
u=1u=2
July 2003 Chuck DiMarzio, Northeastern University 11270-07-40
Using the Cornu Spiral
C(u), Fresnel Cosine Integral
S(u)
, Fre
snel
Sin
e In
tegr
al
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
a=1
July 2003 Chuck DiMarzio, Northeastern University 11270-07-41
Small Aperture
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-6 -4 -2 0 2 4 6 8-3
0
0.2
0.4
0.6
0.8
1
1.2
1.4
=500 nm, 2a=100m, z=5m.Fraunhofer Diffraction would have worked here.
position, mm
July 2003 Chuck DiMarzio, Northeastern University 11270-07-42
Large Aperture
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
-0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.0250
0.5
1
1.5
2
2.5
3
=500 nm, 2a=1mm, z=5m.
position, m
July 2003 Chuck DiMarzio, Northeastern University 11270-07-43
Circular Aperture
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
FresnelCosineIntegrand
Outputof FresnelZonePlate
kr/2z
kr/2z
July 2003 Chuck DiMarzio, Northeastern University 11270-07-44
Phase in Pupil (1)
Linear PhaseShift is tilt
D/2
Quadratic PhaseShift is focus
July 2003 Chuck DiMarzio, Northeastern University 11270-07-45
Phase in Pupil (2)
Quartic Phaseis SphericalAberration
Fresnel Lens has wrapped quadraticphase
Atmoshperic Turbulencecan be modeled as randomphase in the pupil plane
July 2003 Chuck DiMarzio, Northeastern University 11270-07-46
Summary of Diffraction MathMaxwell’sEquations
HelmholtzEquation
Green’sTheorem
KirchoffIntegralTheorem
Fresnel-KirchoffIntegralFormula
FresnelDiffraction
Fourier Transforms
HankelTransforms
MieScattering
YeeNumericalMethods
All ScalarWaveProblems
Spheres
Scalar Fields
GeneralProblems
FieldsFar FromAperture
r>>λ
Obliquity=2,ParaxialApproximation
ShadowsandZonePlates
SeparableProblems
CircularApertures
FraunhoferConditions
PolarSymmetry
“SimpleSystems”