Diffraction Integral

In the last section about Fresnel zones and the zone plate we have considered how different path contribute to the intensity at a point on the optical axis. We would like to generalize this idea to an integral formulation allowing us to calculate any kind of diffraction pattern.

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Fig.: Diffraction integral.

Assume we have a light source S as in the image above, which sends out a spherical wave (does not need to be a spherical wave). The spatial amplitude of this wave at the point P(x,y) at a tiny aperture element dσ

Us(x,y)=U0(x,y)eiϕ(x,y)

where

U0=AR=Ag2+x2+y2

and

ϕ(x,y)=(kR)

This is the amplitude of the Huygens wave, which eminates from the point P(x,y) to propagate towards the screen at P(x,y). This Huygens wave contributes a fraction of an amplitude dUp to the total amplitude in point P(x,y), which is given by

dUp=CUsdσreikr

with C=icos(θ)/λ as found in a more detailed calculation and called the obliquity factor.

The total amplitude at the point P(x,y) is then given by the integral over all contributions.

Up=CUseikrrdxdy

with dxdy=dσ. The integral runs over all positions in the aperature plane (x,y) where we have an opening. This integral is called the Fresnel-Kirchhoff diffraction integral and allows us to calculate complicate scalar diffraction patterns.

Fresnel Approximation

The diffracion integral does not always need to be calculated in completely, but we may use approxaimations to obtain diffraction patterns in different regimes. The first approximation, we would like to have a short look at is the Fresnel approximation, which yields the diffraction pattern in the near field.

The distance r from the point P(x,y) to the point P(x,y) can be written as

r=z02+(xx)2+(yy)2z0(1+(xx)22z02+(yy)22z02+)

The second line assumes that cos(θ)=z0/r1 and C=i/λ. We thus consider small diffraction angles. Using this approximation we find for the amplitude of the wave at a point P(x,y)

U(x,y,z0)=ieikz0λz0Us(x,y)exp[ik2z0((xx)2+(yy)2)]dxdy

As the integration is over x and y, we may draw out all screen coordinate elements such that

U(x,y,z0)=ieikz0λz0eik2z0(x2+y2)Us(x,y)eik2z0(x2+y2)eik2z0(xx+yy)dxdy

This is the Fresnel approximation.

Fraunhofer Approximation

If we further assume that the aperture is small as compared to the distance at which we observe the diffraction pattern, we can further simplify the Fresnel approximation to yield the Fraunhofer approximation giving the diffraction patter in the far field. The condition is

z01λ(x2+y2)

In this case we can neglect the term

eik2z0(x2+y2)1

which results in

U(x,y,z0)=ieikz0λz0eik2z0(x2+y2)Us(x,y)eik2z0(xx+yy)dxdy

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Fig.: Diffraction pattern of a slit in the near field (Fresnel diffraction, left) and the far field (Fraunhofer diffraction, right).

While these formulas provide the mathematical tools, we may obtain a more intuitive idea about the different approximation in the following way. Consider the image below, where we would like to know about the diffraction intensity of a slit of width b at the optical axis at a distance D.

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Fig.: Illustration of the importance of additional geometrical path length difference for the discrimination of Fresnel (near-field) and Fraunhofer (far-field) diffraction.

The waves from the center of the slit and the edge have to travel towards that point a different pathlength, whcih we may calculate to

Δs=b24+D2D=Db24D2+1D

We may develop the square root into a Taylor series and obtain

Δs=b28Db4128D3+O(4)b28D

The second order correction term b28D decreases quadratic with the distance D of the point, which means that at large distances, we can safely assume Δs=0 on the axis, i.e. all waves arriving at that point have to travel the same distance. This corresponds to the far-field approximation. To be more specific we require

b28D2<λ8

or

b2λD<1

to be fullfilled to be in the far field.

F=b2λD{1,Fraunhofer1,Fresnel1,Full vector

This number F is called the Frensel number and gives us an idea by how far the dimensions of the opening contribute to the diffraction pattern rather than the direction of the wave propagation only.

Babinet’s Principle

The above considerations of diffraction have some intruiging consequence. Consider the two apertures in the image below.

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Fig.: Two complementary apertures, which have the same diffraction pattern in the far field.

The left aperture will create in the far field an amplitude distribution Uh, while the inverse aperture on the right will cause an amplitude Ud. If we combine both amplitudes in the far field, we obtain a total amplitude distribution

U=Uh+Ud

In the case when we have two complementary apertures, that total amplitude has to be zeor, when hole and dot are placed at the same position. We therefore obtain

Uh=Ud

and therefore

Ih=Id

This is the Principle of Babinet which states:

Babinet’s Principle

The far field diffraction intensity distribution of complementary apertures is the same.

The images below show an experimental demonstration of Babinet’s principle on a slit and a wire.

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Fig.: Babinet’s principle demonstrated experimentally on a slit (left) and a wire (right).