The Modulation Transfer Function (MTF) provides a measure of the contrast in the output image as a function of spatial frequency for a sine wave input (object). This article explains why FFT MTF and Huygens MTF yield different results in system with tilted image surfaces.

**Authored By Mark Nicholson**

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## Introduction

The Modulation Transfer Function (MTF) provides a measure of the contrast in the output image as a function of spatial frequency for a sine wave input. The MTF is calculated by taking the Fourier transform of the Point Spread Function (PSF) using either the Fast Fourier Transform (FFT) PSF or the Huygens PSF.

The difference in computation planes between the FFT and Huygens PSF creates a significant difference in results for optical systems with tilted image surfaces. The Huygens calculation automatically accounts for the image surface tilt while the FFT calculation does not. OpticStudio can only correct the FFT MTF if there is a local shift of the image plane in one direction - otherwise, Huygens should be used. This article will present examples of both situations.

## Coordinate systems and single axis tilts

The Huygens MTF calculation is derived from the Huygens PSF. This calculation is performed on a plane that is normal to the image surface. Because the computation plane is in image space coordinates, it automatically compensates for local coordinates shifts such as tilted image surfaces. For this reason, there is no requirement that the chief ray must be normal to the image surface and this computation can be used for systems with tilted image surfaces.

The FFT MTF calculation is derived from the FFT PSF. This calculation is done in pupil space coordinates which results in a computation plane that is normal to the chief ray. For this reason, rotating the image surface will have no effect on the orientation of the computed MTF, but tilting the image surface will.

For an image surface tilted in either the X or Y direction, OpticStudio is able to correct the FFT calculation by applying a vector shift in the direction of the tilt. The images below represent a comparison of results of the FFT and the Huygens computation for a paraxial system with the image surface set at focus (the system is available for download as an article attachment). In this case, the image surface is tilted 60 degrees about X. Because of the correction factor applied by OpticStudio for the FFT calculation, both the Huygens and FFT calculations surface plot and cross sections are correct.

As shown below, the resulting FFT Surface MTF and the Huygens Surface MTF, as well as their complementary cross sections have comparable results.

## Complex tilted image surface

For systems with more complex tilts, such as an image surface tilted in two or more dimensions, OpticStudio is still able to correctly compensate the FFT MTF cross section plots using the single component vector shift, but not the FFT MTF surface plots. For complex image surface rotations, the vector shift needed to correct the FFT Surface is no longer uniform over an entire surface but becomes a complex function of position. Below are examples of FFT and Huygens computations for a paraxial system with the image surface positioned at paraxial focus. The image surface is tilted 60 degrees about X and then tilted 45 degrees about Y.

It is clear that the MTF cross sections still agree, but the FFT Surface MTF is no longer appropriately scaled.

KA-01496

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