# Calculating Diffraction Patterns

This is a great book on optics:

> Goodman, Joseph W.  *Introduction to Fourier Optics*.
> Roberts & Company Publishers, 2005.

Page numbers below refer to pages in this book.

## Rayleigh-Sommerfeld Integral

We want to calculate the scalar field created by light coming through
an aperture.  The scalar field is a complex-valued scalar function
over space that captures the necessary information about propagating
light, given certain assumptions (p. 35).

The Huygens-Fresnel principle says that the field at a point can be
calculated by summing contributions from spherical waves emanating
from each point in the aperture.  Mathematically (p. 52):

     %% U(P_0) = \frac{1}{j\lambda} \iint_\Sigma U(P_1)
     %% \frac{\exp(jkr_{01})}{r_{01}}\cos \theta\,ds

where:

* `%P_1%` is a point in the aperture `%\Sigma%`
* `%P_0%` is the point where we wish to calculate the field `%U%`
* `%\lambda%` is the wavelength of the incoming (monochromatic) light
* `%k = 2\pi/\lambda%`
* `%r_{01}%` is the distance between `%P_1%` and `%P_0%`
* `%\theta%` is the angle made with the aperture normal by the ray from `%P_1%` to `%P_0%`
* `%ds%` is an infinitesimal area element of the aperture
* `%j=\sqrt{-1}%`

## Propagation of the Angular Spectrum

A faster way to do calculations on propagating light is to decompose
the scalar field into plane-waves propagating in different directions.
When an x-y slice of such a plane-wave is viewed, it will have a
frequency between `%0%` and `%1/\lambda%`.

Writing the field in terms of its x-y Fourier transform (p. 60):

     %% U(x,y,z) = \iint_{-\infty}^{\infty} A(f_X, f_Y ; z)
     %% exp\left[j 2\pi (f_X x + f_Y y)\right]\,df_X\,df_Y

The transfer function for propagating waves across a distance `%z%` in
Fourier space is then (p. 61):

     %% H(f_X,f_Y) = \begin{cases}\exp\left[j 2\pi
     %% \frac{z}{\lambda}\sqrt{1-(\lambda f_X)^2-(\lambda f_Y)^2}\right]
     %% & \text{if $\sqrt{f_X^2 + f_Y^2} < \frac{1}{\lambda}$} \\ 0
     %% & \text{otherwise} \end{cases}

Note that the angular spectrum approach and the Rayleigh-Sommerfeld
integral yield identical predictions of the diffracted field (p. 61).

## A Spherical Thin Lens

A lens simply introduces a phase factor proportional to its thickness
at a given `%(x,y)%` position on its plane.  In the paraxial
approximation (rays close to parallel with the optical axis) it can be
written (p. 101):

     %% \exp\left[-j\frac{k}{2f}(x^2+y^2)\right]

where `%f%` is the focal distance.  Note that the refractive index of
the lens is not needed, as it is included in `%f%`.  Goodman
introduces `%f%` after making the paraxial approximation, but by
undoing the small-angle approximation we get the presumably more
accurate:

     %% \exp\left[-jk\left(f-\sqrt{f^2-x^2-y^2}\right)\right]

## Implementation

To calculate the diffraction pattern created on the film by a plane
wave incident on an aperture containing a lens:

* Choose a set of x-y values (or just x-values, neglecting y to get a
  1-D pattern) spaced no closer than half a wavelength; spacing them
  farther apart seems to work, to a point; the values need not extend
  beyond the aperture in magnitude

* Calculate the scalar field in the plane of the aperture by
  evaluating the plane wave, which represents an infinitely distant
  point source; points not in the aperture are set to 0

* Apply the phase shift from the lens

* Perform an FFT to work in angular spectrum space; make sure you know
  what signed frequency each element of the result represents

* To propagate by some z-distance, apply the propagation phase `%(3)%`
  in Fourier space

* To apply an occluding mask, perform an IFFT followed by a binary
  mask followed by an FFT

* You may then propagate again, mask again, etc.

* When you get to the film plane, perform an IFFT and take the squared
  magnitude to get the intensity pattern on the film

* This intensity pattern is the point response to the distant source;
  assuming your distant sources are incoherent, the intensity patterns
  add linearly, and the pattern on the film is linear in the
  intensities of the sources (p. 135)

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