The apodization factor refers to the rate of decrease of the beam amplitude as a function of radial pupil coordinate. The beam amplitude is normalized to unity at the center of the pupil. The amplitude at other points in the entrance pupil is given by apodization factor. This article will show the process and results for changing the apodization type of a system. the example file could be download here.
Apodization in OpticStudio
Consider this optical system (which can be downloaded from the link at our server), in which a paraxial lens surface is used to represent some highly corrected optical system, and is uniformly illuminated over its entire aperture:
The FFT Point Spread Function is the classic Airy profile, which follows from the fact that the Fourier transform of a circular top-hat function is a Bessel function.
Note that this plot has been logarithmically scaled to enhance the low-intensity “feet” of the diffraction pattern. Now we change to using a Gaussian apodization function so that the beam intensity varies over the aperture:
In this case, the apodization factor of 3 means that the entrance pupil diameter represents sqrt(3) 1/e2 beam widths, and the layout plot looks like so:
Now relatively little energy is truncated at the aperture stop, and since the Fourier transform of a Gaussian is also a Gaussian, the FFT PSF looks like so:
and we have ‘apodized’ or cut the feet off of, the diffraction pattern. Note that the lens has not changed: just the illumination as a function of pupil. In this case, there is little energy at the edge of the lens, and so little energy is diffracted away from the main lobe of the beam.
The term apodization is also sometimes applied to spatial filtering, in which a pinhole aperture allows only the central lobe of the Airy profile to pass, and ‘cuts the feet’ from the diffraction pattern.
However, in optical design jargon, the term “Apodization’ has grown to mean any function that describes the illumination of the pupil, whether it results in a single-lobed diffraction pattern or not. It is important because aberrations vary over the pupil, and so the variation of ‘brightness’ over the pupil represents the importance of a particular part of the pupil.
Apodization Functions
OpticStudio supports the following apodization functions:
Uniform, which means rays are distributed uniformly over the entrance pupil, simulating uniform illumination. This is usually the case for distant objects.
Gaussian, which imparts an amplitude variation over the pupil that is Gaussian in form. The apodization factor refers to the rate of decrease of the beam amplitude as a function of radial pupil coordinate and can be used to study the effects of truncated Gaussian amplitude variations. If the apodization factor is zero, then the pupil illumination is uniform. If the apodization factor is 1.0, then the beam amplitude has fallen to the 1 over e point at the edge of the entrance pupil (which means the intensity has fallen to the 1 over e squared point, about 13% of the peak). The apodization factor can be any number greater than or equal to 0.0. Values larger than about 4.0 are not recommended. This is because most computations will sample too few rays to produce meaningful results if the beam amplitude falls too quickly at the edges of the pupil.
Cosine cubed, which simulates the intensity fall-off characteristic of a point source illuminating a flat plane. Cosine cubed apodization is only useful and should only be used for point sources or field points close to the axis when compared to the entrance pupil diameter. The name ‘cosine cubed’ is used because, for a point source, the intensity of a ray illuminating a differential area on a plane is given by
where θ is the angle between the z axis and the ray intersecting the entrance pupil, and the relative intensity at the center of the pupil is 1.0.
A surface transmittance can be used to define arbitrary surface apodizations. The transmittance function can be any formula based upon the ray coordinates, direction cosines, surface parameters, or other data; or may be derived from a look-up table, or any other method that can be implemented within the DLL.
