Why Optics?

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ECEG105Optics for EngineersCourse NotesPart 7
Diffraction
Prof. Charles A. DiMarzio Northeastern
University Fall 2007
August 2007
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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
??/D
D
August 2007
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Difraction Quantum Approach

• Uncertainty
• Photon Momentum
• Uncertainty in p

• Angle of Flight
• For a Better Result
• Use Exact PDF
• Gaussian is best
• Satisfies the equality
• Minimum-uncertainty wavepacket

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Quantum Diffraction Examples
200 Random Paths
Aperture 1 ?
Aperture 5 ?
Aperture 2 ?
Aperture 10 ?
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Summary of Diffraction Math
Scalar Fields
Maxwells Equations
Greens Theorem
Fresnel Diffraction
Obliquity2, Paraxial Approximation
Helmholtz Equation
Simple Systems
Shadows and Zone Plates
Yee Numerical Methods
rgtgt?
Fraunhofer Conditions
Fresnel- Kirchoff Integral Formula
Kirchoff Integral Theorem
x,y Separable Problems
Spheres
Fourier Transforms
Mie Scattering
Polar Symmetry
All Scalar Wave Problems
Fields Far From Aperture
Hankel Transforms
General Problems
Circular Apertures
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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

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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

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Kirchoff Integral Setup
Surface A
Surface A0
P
The Goal A Greens Function Approach.
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Kirchoff Integral Thm. Solution
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Helmholtz-Kirchoff Integral
Surface A
Surface A0
P
Surface A
A0
r
r
P
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H-K Integral Approximations
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Some Approximations
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Integral Expressions
(Hankel Transform)
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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

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Fraunhofer Diffraction (1)
Very Important Parameter
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Fraunhofer Diffraction (2)
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Fraunhofer Lens (1)
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Numerical Computation (1)
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Circular Aperture, Uniform Field
D
h
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Diffraction Grating
Reflection Example
d
?i
?d
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Grating Equation
sin(?d)
5
1
4
3
0.5
sin(?i)
2
1
0
n0
-sin(?i)
-0.5
-1
-2
-1
-100
0
100
200
-3
Reflected Orders
Transmitted Orders
degrees
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Grating for Laser Tuning
Gain
f
Cavity Modes
f
?i
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Monochrometer
?i
Aliasing
n1
n2
n3
sin?
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Acousto-Optical Modulator

• Acoustic Wave
• Sinusoidal Grating
• Wavefronts as Moving Mirrors
• Signal Enhancement
• Doppler Shift
• Acoustic Frequency Multiplied by Order

Sound Source
Absorber
More Rigorous Analysis is Possible but Somewhat
Complicated
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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

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Rectangular Aperture
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Small Aperture

• ?500 nm, 2a100?m, z5m.
• Fraunhofer Diffraction would have worked here.

0.8
1.4
0.6
1.2
0.4
1
0.2
0.8
0
0.6
-0.2
0.4
-0.4
0.2
-0.6
-0.8
0
-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
position, mm
-3
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Large Aperture
?500 nm, 2a1mm, z5m.
0.8
3
0.6
2.5
0.4
2
0.2
0
1.5
-0.2
1
-0.4
0.5
-0.6
-0.8
0
-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
0.025
position, m
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Summary of Diffraction Math
Scalar Fields
Maxwells Equations
Greens Theorem
Fresnel Diffraction
Obliquity2, Paraxial Approximation
Helmholtz Equation
Simple Systems
Shadows and Zone Plates
Yee Numerical Methods
rgtgt?
Fraunhofer Conditions
Fresnel- Kirchoff Integral Formula
Kirchoff Integral Theorem
Spheres
Fourier Transforms
Separable Problems
Mie Scattering
Polar Symmetry
All Scalar Wave Problems
Fields Far From Aperture
Hankel Transforms
General Problems
Circular Apertures