# What is the difference between the FFT and Huygens PSF?

This article describes how the Huygens PSF and FFT PSF are calculated and gives applications of each algorithm. Also included is a procedure to test the validity of the FFT algorithm and a list of system requirements for using the FFT.

Authored By Betsy Goodwin

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## Introduction

The Point Spread Function (PSF) of an optical system is the image that results at the image surface from a single point source radiated from object space. Aberrations and diffraction cause spreading, resulting in an image that is not a perfect point, despite arising from a point source. OpticStudio has two diffraction-based PSF calculations: the Fast Fourier Transform (computed in pupil space) and the Huygens (computed in image space).

This article discusses and compares the Huygens and the Fast Fourier Transform (FFT) PSF calculations.

## Huygens PSF

The Huygens PSF calculation is performed on a plane that is normal to the image surface at the point where the chief ray intersects the image surface.  It is valid for nearly all optical systems, including systems designed in Non-Sequential mode, and is more robust in the presence of local coordinate shifts. The only assumption made for this calculation is that there is adequate pupil sampling.

The Huygens PSF of an optical system is computed by tracing a grid of rays from the source point to the image surface. For each ray, the amplitude, coordinates, direction cosines, and optical path difference (OPD) are used to compute the complex amplitude of the plane wave incident at every point on the image space grid. A coherent sum for all rays is performed at every point in the image space grid, and the intensity of each point is the square of the resulting complex amplitude sum.

This technique considers each point on a wavefront to be a perfect point source with an amplitude and phase. Each point source radiates a spherical "wavelet". As the point source radiation propagates through space, the diffraction of the wavefront is given by the complex sum of all the spherical wavelets radiated. In OpticStudio, this is performed by converting each ray into a planar wavefront and interfering all of them at the image plane; the result is the Huygens PSF.

Below are example Huygens PSF calculations from the Cooke 40-degree field sample file. This file is attached to this article. Below is the Huygens Surface PSF and underneath it is the Huygens False Color PSF. To compute a Huygens PSF for a Sequential system in OpticStudio, choose Analyze...Image Quality...PSF...Huygens PSF.  ## FFT PSF

Although the FFT PSF calculation is computationally faster than the Huygens PSF calculation, some assumptions must be made in order to perform the calculation, which limit its domain of validity.

The FFT method uses the Fourier transform of the complex amplitude of the wavefront, measured in the exit pupil on a plane perpendicular to the chief ray to calculate the diffraction PSF of the optical system. The amplitude and phase in the exit pupil are computed for a grid of rays, a FFT is performed in the far field, and the diffraction image intensity is computed. The FFT PSF of an optical system is computed by tracing a grid of rays from the source point to the image plane, and then back-propagating this grid to the exit pupil. For each ray, the amplitude and optical path difference are used to compute the complex amplitude of a point on the wavefront. The FFT of this grid is squared to yield the real valued PSF.

In order for the FFT PSF to yield valid results, some additional assumptions must hold true (relative to the Huygens calculation). To perform the FFT PSF a system must be F/1.5 or slower (i.e. the F/# must be greater than 1.5), the image surface must lie in the far field of the system, and the chief ray must be nearly normal to the image surface. Further, to be accurate, the FFT calculation assumes that pupil aberrations are minimal, such that a uniform distribution of rays in cosine space is formed on the exit pupil.

To compute an FFT PSF for a Sequential system, choose Analyze...Image Quality...PSF...FFT PSF. Note that the FFT PSF cannot be computed for NSC systems. Below are examples of the FFT PSF and the Huygens PSF calculations from the second configuration in the archive file found at the beginning of this article. This optical system has a tilted image surface which violates the assumption that the chief ray is normal to the image surface. While the FFT PSF results are inaccurate for this system, the Huygens PSF is correct.  ## FFT validity test

To verify that rays have a reasonably linear relationship between the entrance pupil and exit pupil coordinates, in both position and angle space, a quick test can be performed to confirm that an optical system falls into the validity of the FFT PSF.

To verify the uniformity of rays in cosine space falling on the image plane, click on Analyze...Image Quality...Rays and Spots...Standard Spot Diagram. Choose “Settings” at the top of the active window and the check the “Direction Cosines” box. If the resulting graph is relatively uniform, the FFT PSF will yield a valid result. If the rays are highly aberrated, the Huygens analysis must be used.

Below is a picture of a Spot Diagram analysis shown in cosine space that resulted in a uniform grid of rays for the Cooke triplet example file found at the beginning of this article. This is an example of a uniform grid of rays on the image surface in cosine space. This verifies that the FFT PSF will yield valid results for this optical system. If the image surface of this system were tilted 45 degrees about Y and 45 degrees about Z (as in configuration 2 in the attached file), and the cosine space spot diagram were recomputed, the results would yield the “non-uniform” spot diagram shown below. In this case, the Huygens PSF must be used. KA-01494