Fresnel Diffraction

This program numerically evaluates the Fresnel diffraction integral to compute the amplitude/phase distribution of the EM field in a plane that is a fixed distance behind an aperture, whose transmission function is given.

The program makes use of the convolution theorem and fast Fourier transform to compute more efficiently.

Image Format

The program operates with images in the Portable PixMap (PPM, netpbm) format. An input image describes the transmission function of the aperture: brightness determines the amplitude of the transmitted light, and hue determines the phase shift.

The output image similarly uses brightness and hue to encode the distribution of the diffracted wave. Phase shift is calculated relative to an undiffracted wave that had travelled the same distance.

Compiling

To compile you would need a C compiler and libfftw3 (for fast Fourier transforms). Then the program can be built with make.

Running

The program expects a newline-separated sequence of inputs on stdin. An input looks like:

<input.ppm> <output.ppm> <lambda> <distance> <flags>

Here <input.ppm> is the input image (see Image Format) describing the transfer function of the aperture, <output.ppm> is the output image where the distribution of the diffracted light would be written, <lambda> is light wavelength in pixels, <distance> is the distance between the aperture and the screen in pixels.

<flags> is zero or more letters:

• i (intensity): instead of an amplitude-phase distribution, the resulting image will contain a black and white distribution of intensity (squared amplitude).
• s (split view): render intensity in the top half of the image and amplitude-phase in the bottom half.
• 'n' (normalize): normalize the output (make the maxmimum magnitude 1) before rendering.
• x: assume the input image to be periodic in the X direction. This causes light to "wrap around" the left/right sides of the screen. Otherwise the image is assumed to be of finite size, surrounded by an infinite plane that blocks all light.
• y: assume the input image to be periodic in the Y direction.
• m (measure): spend some time figuring out what's the best way to do the Fourier transform, so that multiple inputs of the same input size can be processed faster.
• p (patient): spend even more time figuring out how to do the Fourier transform. This is rarely useful.

Integration

We calculate the Fresnel diffraction integral with dx = dy = 1 pixel. Such integration breaks down if distance to a point on the aperture varies too much between adjacent screen pixels, in particular if sqrt(x^2+d^2)/(lambda x) < 1 pixel, where x is the characteristic size of the screen.