eceg105 optics for engineers course notes part 7: diffraction

$: ECEG105 Optics for Engineers Course Notes Part 7: Diffraction$
July 2003 Chuck DiMarzio, Northeastern University 11270-07-1

ECEG105Optics for Engineers

Course NotesPart 7: Diffraction

Prof. Charles A. DiMarzioNortheastern University

Fall 2007

August 2007


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 λ/D

August 2007

?λ/D

D


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


Quantum Diffraction Examples

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 16

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 16

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 16

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 16

4

2

0

2

4

6

200 Random Paths

Aperture1 λ

Aperture2 λ

Aperture5 λ

Aperture10 λ


Maxwell’s Eqs & Diffraction

∇×E=−∂B∂t

z

x

y

z-component of curl is zero

y-component of curl is zerox-component is not

E 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


Summary of Diffraction Math

Maxwell’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”


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


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


Kirchoff Integral Setup

P

Surface A0

Surface A

The Goal: A Green’s FunctionApproach.

U x,y,z =∫G x,y,z,x1 ,y1 ,z1 U x1 ,y1 ,z1 dV1


Kirchoff Integral Thm. Solution


Helmholtz-Kirchoff Integral

P

Surface A0

Surface A

P

Surface A

r’ r

n

A0

U=U0eikr'

r


H-K Integral Approximations


Some Approximations


Paraxial Approximationx1 x

z


Integral Expressions

(Hankel Transform)


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


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


Fraunhofer Diffraction (1)

Very ImportantParameter


Fraunhofer Diffraction (2)


Fraunhofer Lens (1)


Fraunhofer Lens (2)

z

z


Fraunhofer Diffraction Summary

z

z


Numerical Computation (1)


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.


Circular Aperture, Uniform Field

D

h


Square Aperture, Uniform Field

z

D


No Aperture, Gaussian Field

D


Fraunhoffer Examples


Imaging: Rayleigh Criterion

R/d0 is f#

August 2007


Single-Mode Optical Fiber

Beam too Large(lost power at edges)

Beam too Small(lost power through cladding)


Diffraction Grating

θi

θd

ReflectionExample

d


Grating Equation

100 0 100 2001

0.5

0

0.5

1sin(θd)

sin(θi)

degrees

-sin(θi) n=0

-1

-2

12

3

4

5

-3ReflectedOrders

TransmittedOrders


Grating Fourier AnalysisGrating Diffraction Pattern

Slit

Convolve

Sinc

Multiply

Repetition Pattern

Multiply Convolve

Apodization

Result

Result


Grating for Laser Tuning

f

Gain

f

Cavity Modes

θi

August 2007


Monochrometer

θi

sinθ

n=1 n=2 n=3

Aliasing

August 2007


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


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


Rectangular Aperture


Cornu Spiral

C(u), Fresnel Cosine Integral

S(u

), F

resn

el

Sin

e I

nte

gra

l

0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.80.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

-5<u<5

u=0

u=1u=2


Using the Cornu Spiral

C(u), Fresnel Cosine Integral

S(u

), F

resn

el

Sin

e I

nte

gra

l

0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.80.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

a=1


Small Aperture

0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.80.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

6 4 2 0 2 4 6 83

0

0.2

0.4

0.6

0.8

1

1.2

1.4

λ=500 nm, 2a=100µm, z=5m.Fraunhofer Diffraction would have worked here.

position, mm


Large Aperture

0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.80.8

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

λ=500 nm, 2a=1mm, z=5m.

position, m


Circular Aperture

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

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 51

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


Phase in Pupil (1)

Linear PhaseShift is tilt

φ

D/2

Quadratic PhaseShift is focus


Phase in Pupil (2)

Quartic Phaseis SphericalAberration

Fresnel Lens has wrapped quadraticphase

Atmoshperic Turbulencecan be modeled as randomphase in the pupil plane


Summary of Diffraction Math

Maxwell’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”

eceg105 optics for engineers course notes part 7: diffraction

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