Modulation Transfer Function (MTF) is an important method of describing the performance of an optical system. A consequence of applying Fourier theory to image forming optical systems, MTF describes the contrast in the image of a spatial frequency presented in the scene being viewed.

Spatial frequency is measured in cycles per mm in focal image space, and in cycles per milliradian or other angular unit in afocal spaces. "Cycles" is the correct term for sinusoidal MTF response. The term "line pairs" is often used, but strictly speaking line pairs only applies to bar, as opposed to sinusoidal targets. ZEMAX can compute MTF for both bar and sinusoidal targets.



The bitmap image analysis window above shows 'good' image quality¹ in the section showing the three boys: but how good is good? The contrast ratio in the various line images provides quantitative data on the quality of the image formed. MTF describes the transfer of modulation from the object to the image as a function of spatial frequency and is commonly used to specify lens performance, and as optimization and tolerancing targets during lens design.



The MTF plot for this lens, above, shows the contrast ratio for any spatial frequency up to the maximum that the lens can support. In this case, we choose to limit the spatial frequencies shown to a maximum of 100 cycles/mm.  Also shown for reference is the diffraction limited performance of an aberration-free lens of the same f/#. Once you are familiar with MTF plots, you can assess image quality at a glance.

[1] Note that the apparent image quality is reduced by screen resolution and the compression of the image. The original image as produced by ZEMAX is of photographic quality. See How to Produce Photo-Realistic Output Images for full details.

As RMS wavefront error goes to zero, the MTF approaches its diffraction limited performance. The default RMS wavefront merit function is therefore an excellent place to start when optimizing a system to have a desired spatial frequency response. You can then switch to explicit MTF optimization operands as needed for a final adjustment of system performance.

MTF calculations take longer than wavefront calculations, and so initial wavefront optimization is strongly preferred. In addition, when system performance is not already good, direct MTF optimization can stagnate. For example, if the lens used to form the images on the previous page is simply defocused, the MTF can drop to zero and then rise again, as shown below.



This is referred to as spurious resolution, and can cause the local optimizer to stagnate, as the MTF must get worse before it can get better. Although Hammer and Global Search can handle this, it is almost always best to optimize on wavefront error until all spatial frequencies of interest are within the first minimum of the MTF plot. Only then should you optimize for specific MTF targets.

The MTF algorithm used by the Analysis feature, which produces a graph of MTF for all spatial frequencies that the lens can support, is based upon Fraunhofer diffraction theory. The method involves a fast Fourier transform of a grid of rays traced to the pupil (hence the name FFT MTF). The resulting MTF is the modulation as a function of spatial frequency for a sine wave object, although optionally the real, imaginary, phase, and square wave response is available.

When it comes to optimization, usually only specific spatial frequencies are required, and it is unnecessary to compute the MTF at all supported frequencies. Therefore, The MTF* operands which compute MTF at a specific spatial frequency (see the User's Guide for full documentation) support a GRID parameter that can switch between the grid method used by the graphics window and a fast, sparse-sampling method that is the default and strongly recommended for most (almost all) optimization cases.

In a manner similar to Gaussian Quadrature, the sparse sampling computation converges very, very quickly, and computes the MTF to arbitrary precision with vastly fewer rays than the grid method.  And, most importantly, it is fully accurate in all cases where Fraunhofer theory is applicable.

This table demonstrates the convergence of the two methods, and the time taken to converge, as the sampling is increased for a calculation of the polychromatic MTF at 50 lp/mm, on-axis in the double Gauss sample file provided with ZEMAX:



Here is the same data, this time for the edge of the field of view:



Note that generally speaking, 1% convergence is adequate for the purposes of optimization and tolerancing. Experimental methods of measuring MTF are generally not repeatable below 0.1% in any case. Extreme precision is not required for good optimization results; three significant figures is usually adequate. However, both algorithms will converge to arbitrary precision with adequate sampling, and the fast algorithm will do so many orders of magnitude faster when extreme precision is needed.

The grid algorithm will converge faster in only one known case: where aberrations are very large and the resulting MTF very low: less than about 5%. Note that MTF is not normally used to specify performance, much less be used as an optimization or tolerancing target, in this regime. ZEMAX automatically traps this condition and switches to the grid method in this case. Note also that the geometric MTF calculation is a better choice in this regime.

Imagine using a 35-mm SLR camera lens at full aperture, say f/1.8. The lens aberrations are most severe at full aperture. As the lens is stopped down, aberrations are reduced and the image quality is improved. The improvement in image quality does not last forever: at some setting of the aperture, diffraction from the aperture prevents you from gaining any further improvement in image quality.

For cases where there are many waves of aberration, the Geometric MTF calculation and GMT* optimization operands are recommended. It is based on a Fourier transform of the geometric spot data, and uses a Gaussian quadrature sampling scheme.

The primary advantage to using the geometric MTF is very high speed, compared to the diffraction calculation. The geometric MTF is very accurate for systems with large aberrations, where the diffraction MTF calculation would need enormous sampling to converge. The geometric calculation is typically factors of 100 or more faster in this regime.

The Geometric MTF calculation can even account for scattering from optical surfaces, which reduces MTF by increasing the background illumination.

The GMTF is very useful in Global Search, where one wants to efficiently search all parameter space to identify regions where good designs can be found.

The diffraction calculation should always be used when diffraction effects are important.

FFT-based methods of computing the Point Spread Function and MTF are well known, and are based on Fraunhofer diffraction theory. The primary assumptions made are:

• The f/# is large enough so that scalar diffraction theory applies
• The region over which the diffraction PSF has significant energy is small compared to the distance from the exit pupil of the optical system to the image surface
• The exit pupil is not significantly distorted with respect to the entrance pupil. This means a uniform distribution of rays on the entrance pupil remains reasonably uniform on the exit pupil
• The sampling is set high enough to accurately model the PSF

Many, but not all, optical imaging systems meet the simplifying assumptions necessary for the Fraunhofer diffraction theory used by the FFT MTF algorithm.

The Huygens MTF is not based upon the FFT. The only assumptions are that the the f/# is large enough so that scalar diffraction theory applies and that the sampling is set high enough to accurately model the PSF 

You can read more about the Huygens calculation in the article What is a Point Spread Function.

Virtually all imaging systems meet the simplyifying assumptions necessary for computing the Huygens PSF. The Huygens MTF is generally slower than the FFT (Fraunhofer) MTF, but more accurate for those cases where the FFT MTF assumptions do not apply.

There is another case where only the Huygen's calculation can be used: if the chief ray cannot be traced through the system, then a chief-ray centered reference sphere cannot be created. The reference sphere is essential for many wavefront calculations. In this case, the Huygens PSF and MTF calculations can be used. One such case is this multiple mirror telescope, in which the chief ray has no path the the image surface.



Because the chief ray cannot trace to the image surface, OPD cannot be calculated, and any parameter derived from OPD cannot be calculated:



The Huygens' PSF and MTF calculations work perfectly, as they are not dependent on any one ray used as a reference:





This file is part of the standard ZEMAX distribution, and can be found at {ZEMAXroot}\Samples\Non-sequential\Miscellaneous\Multiple mirror telescope.ZMX.

A final benefit of the Huygen's method is that the MTFs over multiple configurations can be summed. This is also of use in multiple mirror telescopes, but particularly in the case of very long baseline instruments, where the primary mirrors are separated by large distances. In this case it is not desired to use a common entrance pupil, because too few rays hit the primary mirrors to be efficient. In this specific case the Huygens' MTF calculation is uniquely capable.

To optimize or tolerance using Huygens' MTF, use the MTH* operands.

The Modulation Transfer Function is an important parameter in optical system design. ZEMAX offers several algorithms for computing it.

• Fraunhofer-theory based FFT methods are the most common approach. ZEMAX computes the MTF for all spatial frequencies supported by a lens (though it will optionally only display those frequencies of interest) in a graph of MTF versus frequency. When optimizing or tolerancing on MTF, only the spatial frequency of interest is computed, which is much faster and requires far fewer rays for a given level of precision. The grid method of computing MTF is available as an option, if required.
   
• For conventional optical systems with large aberrations the Geometric MTF computes an approximate MTF with great speed. It is a good choce for "roughing in" a design for best MTF as fast as one can optimize RMS spot radius.
   
• For systems which do not meet the assumptions of Fraunhofer theory, or in which the chief ray cannot be traced, the Huygens' calculation is available, and this provides a robust calculation of MTF. The only disadvantage of the Huygens method is speed.