Topic 2: Scalar Diffraction Aim: Covers Scalar Diffraction theory to - - PDF document

Modern Optics

T H E U N I V E R S I T Y O F E D I N B U R G H

Topic 2: Scalar Diffraction

Aim: Covers Scalar Diffraction theory to derive Rayleigh-Sommerfled

• diffraction. Take approximations to get Kirchhoff and Fresnel approx-

imations. Contents:

• 1. Preliminary Theory.
• 2. General propagation between two planes.
• 3. Kirchhoff Diffraction
• 4. Fresnel Diffraction
• 5. Summary

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T H E U N I V E R S I T Y O F E D I N B U R G H

Scalar Wave Theory

Light is really a vector electomagnetic wave with E and B field linked by Maxwell’s Equations. Full solution only possible in limited cases, so we have to make as- sumptions and approximations. Assume: Light field can be approximated by a complex scalar potential. (am- plitude) Valid for: Apertures and objects

NOT Valid for: Very small apertures, Fibre Optics, Planar Wave Guides, Ignores Po- larisation. Also Assume: Scalar potential is a linear super-position of monochromatic compo-

• nents. So theory only valid for Linear Systems, (refractive index does

not depend on wavelength).

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T H E U N I V E R S I T Y O F E D I N B U R G H

Scalar Potentials

For light in free space the E and B field are linked by

∇

∇

∇

ε0µ0 ∂E ∂t ∇

∂t

which, for free space, results in the “Wave Equation” given by

∇2E

c2 ∂2E ∂t2

Assume light field represented by scalar potential Φ (r ;t

also obey the “Wave Equation”, so:

∇2Φ

c2 ∂2Φ ∂t2

Write the Component of scalar potential with angular frequency ω as

Φ (r ;t

then substituting for Φ we get that,

where κ

So that u

Wave Theory).

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T H E U N I V E R S I T Y O F E D I N B U R G H

Diffraction Between Planes

u(x,y;0) u(x,y;z)

1

P z y z y x P x

In P0 the 2-D scalar potential is:

u

where in plane P1 the scalar potential is:

u

where the planes are separated by distance z. Problem: Given u0

is any plane P1 separated from P0 by z. Looking for a 2-D Scalar Solution to the full wave equation.

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T H E U N I V E R S I T Y O F E D I N B U R G H

Fourier Transform Approach

Take the 2-D Fourier Transform in the plane, wrt x ;y. so in plane P1 we get

U

u

since we have that

u

U

then if we know U

u

The amplitude u

U

These terms are Orthogonal (from Fourier Theory), So each of these terms must individually obey the Helmholtz Equa- tion. Note: the term:

U

in a Plane Wave with Amplitude U

where κ

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T H E U N I V E R S I T Y O F E D I N B U R G H

Propagation of a Plane Wave

From Helmholtz Equation we have that

Noting that the exp

∂2U

∂z2

λ2

• U

so letting

γ2

λ2

then we get the relation that

∂2U ∂z2

With the condition that

U

we get the solution that

U

This tells us how each component of the Fourier Transform propa- gates between plane P0 and plane P1, so:

u

U0

which is a general solution to the propagation problems valid for all

z.

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T H E U N I V E R S I T Y O F E D I N B U R G H

Free Space Propagation Function

Each Fourier Component (or Spatial Frequency) propagates as:

U

Define: Free Space Propagation Function

H

so we can write:

U

Look at the form of H

u2

λ2

Case 1: If u2

H

Phase Shift so all spatial frequency components passed with Phase Shift. Case 2: If u2

H

So Fourier Components of U0

Negative Exponential. (evanescent wave)

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T H E U N I V E R S I T Y O F E D I N B U R G H

Frequency Limit for Propagation

In plane P1 where z

remove high frequency components.

U

u2

so Fourier Transform of u

Maximum spatial frequency when u

ing with period λ.

d θ

d sinθ

for d

f d

No diffraction when d

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T H E U N I V E R S I T Y O F E D I N B U R G H

Convolution Relation

We have that

U

so by the Convolution Theorem, we have that

u

With u

We then have that

h

exp

u2

where

γ

1 λ2

“It-Can-Be-Shown” (with considerable difficulty), that

h

λ2 ∂ ∂z

κr

• where be have that

r2

and

κ

λ

we therefore get that

h

λ

κr

• z

r

r

which is known as “The Impulse Response Function for Free Space Propagation”

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T H E U N I V E R S I T Y O F E D I N B U R G H

Point Object

u(x,y;z) y x z a x,y δ( ) y x r P P

1

In P0 we have a Delta Function, so:

u0

So in plane P1 we have

u

ah

a λ

κr

• z

r

r

where r is the distance from:

So the intensity in plane P1 is given by:

i

r2

κ2r2

• z

r

where the λ2 has been incorperated into the constant b.

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T H E U N I V E R S I T Y O F E D I N B U R G H

Shape of Function

The shape of i

0.2 0.4 0.6 0.8 1 1.2

• 4
• 3
• 2
• 1

1 2 3 4 i(x,1) i(x,2) i(x,3) i(x,4)

for z

Compare with a isolated 3-D point source, spherical expanding waves, so intensity in plane at distance z of:

I

r2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 4
• 3
• 2
• 1

1 2 3 4 I(x,1) I(x,2) I(x,3) I(x,4)

Note We have a 2-D Delta Function, (whole in a screen), and NOT an 3-D point source.

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T H E U N I V E R S I T Y O F E D I N B U R G H

Full Expression

The full convolution expression is

u

P

u0

where s;t are variables in plane P0 The h

so l is Distance from

u(s,t;0) z u(x,y;z) l s t P

1

P z x y Full Expression is

u

λ

P

u0

κl

l

l

• dsdt

Rayleigh-Sommerfeld Diffraction Equation

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T H E U N I V E R S I T Y O F E D I N B U R G H

Kirchoff Diffraction

Look at Impulse Response Function:

h

λ

κr

• z

r

r

Most Practical cases, P0 and P1 separated by MANY wavelength,

Approximate the term:

κr

• ı

so that

h

ıλ

r

r

Look at Terms

exp

r

Spherically expanding wave from point

z r

“Obliquity Factor” which forces propagation in z direction. (In Good- man’s notation the obliquity is written as a cos() term). The other terms are just constants.

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T H E U N I V E R S I T Y O F E D I N B U R G H

Note: For 3-D point source, get expanding spherical wave obliquity factor results from 2-D source in a plane. Impulse Response Function: Spherical Wave with directional weight- ing term. Model: Each point in P0 acts a source of impulse response functions, that sum is P1.

h

r

This is Hygen’s Secondary Wavelet (weighted by obliquity factor).

Hygen’s Secondary Wavelets

Model: Each point on the wavefront gives rise to Spherical Waves,

h

r

Add postulate that “wave propagate in positive z direction”. Kirchhoff

r

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T H E U N I V E R S I T Y O F E D I N B U R G H

Kirchhoff Diffraction Integral

In Rayleigh-Sommerfeld integral,

z

l

make the approximation that

κl

• ı

so we get that

u

ıλ

P

u0

l

l

• dsdt

which is valid (1%), z

Typical starting point for optical calculations. Look at z=l factor:

P l z

1

P x s z θ

so that

z l

Same expression as in books [eg. Goodman page 52, equation (3- 51)]

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T H E U N I V E R S I T Y O F E D I N B U R G H

Fresnel Approximation

Assume extend of P0 and P1

x z x z P P r

1 So we have that

We have Kirchhoff impulse response

h

ıλ

r

r

Which we can write as:

h

where the amplitude

A

ıλ z r2

Since x&y

r

• 1

z2

2

2z

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T H E U N I V E R S I T Y O F E D I N B U R G H

Amplitude Term: First Order approx:

r

A

ıλz

Phase Term: Second Order approx:

r

• z

2z exp (ıκr

• exp

• ı κ

2z

• So the Fresnel Approximation is that

h

ıλz exp

• ı κ

2z

• Fresnel Approximations

1) Replace Spherical waves by Parabolic Waves 2) Ignore Obliquity factor. NOTE: only valid

This is also known as “Paraxial Approximation”. Useful in many prac- tical systems. Use in the rest of the course

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T H E U N I V E R S I T Y O F E D I N B U R G H

Fresnel Diffraction

We have that

u

exp

ıλz

u0

• ı κ

2z

• (x

• dsdt

This we can expand to get the

1

1 ıλz exp (ıκz)

2

exp

• ı κ

2z

• Z

4

u0

• ı κ

2z

• exp
• ıκ

z

• dsdt

3

Look the FOUR terms

• 1. Absolute amplitude and phase, depends only on z (constant

which is not normally important).

• 2. Parabolic phase term, no effect on intensity.
• 3. Fourier Transform scaled by κ=z
• 4. Scalar distribution in P0 weighted by parabolic phase term.

Fresnel Diffraction

term [+ Extra phase and constants]

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T H E U N I V E R S I T Y O F E D I N B U R G H

In Fourier Space

In Fourier Space we have

U

so we have that

H

ıλz exp

exp

• ı κ

2z

• exp

This can be separated into 2 integrals in x and y, and with identity

π b exp

• a2

4b

• It-Can-be-Shown that

H

• ıπλz(u2

• which is again a parabolic term.

Fresnel diffraction

term.

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T H E U N I V E R S I T Y O F E D I N B U R G H

Summary

Scalar Wave solution for prapagation between two plane: Full Solution

If distance between planes

Scaler Kirchoff Diffraction This is Hygen’s Secondary Spherical wavelets plus obliquity factor due to 2-D whole in plane. If distance between planes Large and planes are Small make small angle approximation to get Fresnel Diffraction which replaces Spherical waves with Parabolic and ignores obliquity factor. Fresnel Diffraction will be used for the rest of this course.

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