For imaging systems, the performance specification is often MTF at a given spatial frequency. This is especially important for systems with digital detectors, where spatial frequencies beyond a certain value are not needed and mid-range frequency performance is desired. However, traditional methods of optimizing directly on MTF is challenging and computationally expensive. OpticStudio's Contrast Optimization capability largely solves these problems by providing an alternative method for optimizing MTF.

**Authored By Erin Elliott, Jade Aiona**

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## Introduction

For imaging systems, the performance specification is often MTF at a given spatial frequency. This is especially important for systems with digital detectors, where spatial frequencies beyond a certain value are not needed and mid-range frequency performance is desired.

Optimizing directly on MTF is difficult, though. The MTF calculation is computationally expensive. The MTF tends to be badly behaved in the early stages of a design, so that other optimization methods are needed until the system is very close to its final form.

Contrast Optimization largely solves these problems. Instead of calculating the full MTF, we measure the phase differences in the exit pupil across a distance corresponding to the desired spatial frequency in the MTF. This quantity can be used to construct a merit function during optimization that has a minimum in the same location as the MTF. The method is much faster and more well-behaved than a direct optimization on the MTF value.

## Contrast Optimization Default Merit Function

Professional and Premium users of OpticStudio can access a default merit function that uses the above method to optimize for MTF. The user selects the Image Qulaity criterion “Contrast” in the Merit Function Wizard, specifies the spatial frequency of interest, and sets the weight of the sagittal MTF value versus the tangential. The wizard creates a Merit Function using the MECS and MECT operands which are designed to be used in pairs. MECS traces three rays: an unshifted ray, a ray shifted by d in the S direction in the pupil, and a ray shifted by d in the T direction in the pupil. MECS then reports the optical path difference (OPD) in waves for the sagittal pair of rays. MECT reports the OPD for the tangential pair of rays.

The rays can be distributed using a simple grid or using Gaussian Quadrature, which allows effective sampling of the pupil with a minimum number of rays.

The optimization attempts to drive all the wavefront differences in the MECS and MECT operands to zero. This maximizes the MTF at the spatial frequency corresponding to the pupil shift, `d`.

This Merit Function contains much more information than a traditional MTF optimization. A traditional MTF optimization would contain just two operands: the MTF value in the sagittal direction, and the MTF value in the tangential direction. As the optimization changes the values of the system variables, the MTF value can either go up or down, but the optimizer has no insight into why those values go up or down, or which parts of the pupil are problematic. With Contrast Optimization, much more information is available to the optimizer. It can “see” which parts of the pupil are causing the most problems for the MTF.

## Contrast Loss Map

The optimization method is easy to understand in a visual way by using the Contrast Loss Map, which allows the user to see exactly which areas in the pupil are causing the most loss of MTF.

Larger circles indicate a region in the pupil that is contributing to a lower MTF. The magnitude of each circle is *½ {1 – cos[**q**(x) – **q**(x-**d**) ]}*, where *[**q**(x) – **q**(x-**d**) ]* is the phase difference between the shifted and unshifted rays. When the phase difference is zero, the size of the circle is zero. When the phase difference is 90°, the size of the circle is ½, and when the phase difference is 180°, the size of the circle is 1. The clock hand within each circle just shows the average phase: *[**q**(x) – **q**(x-**d**) ]*/2, so that the underlying wavefront shape is visible in the plot.

## Double Gauss example

Optimization of a Double Gauss design shows that Contrast Optimization can be comparable in speed to wavefront optimization, and that Contrast Optimization is more robust than directly optimizing on the MTF. We began with a standard Double Gauss design. The starting point was a system of plane parallel plates, with a radius of curvature solve on the last surface to force an F/3 system. The design was done in visible wavelengths. The entrance pupil diameter is 33.3 mm.

The remaining radii of curvature and spacings were set to variables for a Damped Least Squares optimization. The optimization was carried out in three ways: using RMS wavefront error as the criterion, using Contrast Optimization, and by directly optimizing on the MTF. The results are given in the table below. Contrast Optimization achieved comparable times to RMS Wavefront optimization.

Optimization method |
Sampling:3 rings x 6 arms |
Sampling:6 rings x 6 arms |
Sampling:9 rings x 6 arms |

Contrast Optimization (20 cycles/mm) |
4.6 sec | 5.8 sec | 9.8 sec |

RMS wavefront optimization |
5.5 sec | 5.1 sec | 9.4 sec |

Direct MTF optimization (20 cycles/mm) |
FAILED | FAILED | 144.6 sec |

Because of the poor performance in the starting system, direct optimization on the MTF is not possible. Optimization on RMS spot size was carried about before attempting to optimize directly on the MTF. The Contrast Optimization and RMS Wavefront optimizations achieved similar results. One of the optimized systems and the corresponding MTF curves are shown below.

## Shape factor example

Optimizing a singlet on its shape factor confirms that the Contrast Optimization merit function has the same minimum as an MTF optimization. The data also shows that Contrast optimization has a smoother parameter space and will travel more directly to the best solution compared to other optimization methods.

We used an F/10 singlet with a 10 mm beam diameter at a wavelength of 500 nm, and considered the on-axis field point only. The variables in the system were the shape factor of the lens and the distance to the image plane. We expect to see a shape factor of near 0.7 after the optimization.

Because of the spherical aberration in the system, the MTF is poorly behaved. Above a spatial frequency of about 15 cycles/mm, MTF is in the domain of spurious resolution where optimization to increase the MTF actually produces lower performance.

The graph above shows the value of the Contrast Optimization merit function along the right axis, and the distance of the MTF from the diffraction-limited value along the left axis, as a function of shape factor of the singlet. At both 10 and 25 cycles/mm, the merit function value has a minimum in the same location as the MTF value. The merit function value has a quadratic shape, giving a smoother parameter space for the optimization and a more well-defined minimum. The MTF value, in contrast, is fairly flat within the spherical caustic. RMS spot size and RMS wavefront values are similarly flat within the caustic. The Contrast Optimization merit function value is closely related to the derivative of the wavefront, so the behavior is not unexpected.

We can also see that at 25 cycles/mm, the MTF value is not strictly decreasing, which will cause problems for a damped least squares optimization. Both of the above observations suggest that Contrast optimization will move more quickly to the ideal value during an optimization.

Also note in the graph above that the value of the Contrast Optimization merit function does not give the value of the MTF. After carrying out a Contrast Optimization, it will likely be necessary to retrieve the final MTF value from a direct MTF calculation. This can be easily done with a macro so that it doesn’t slow down the optimization itself.

## References

- K. E. Moore, E. Elliott, et. al. "Digital Contrast Optimization - A faster and better method for optimizing system MTF," in
*Optical Design and Fabrication 2017 (Freeform, IODC, OFT)*, OSA Technical Digest (online) (Optical Society of America, 2017), paper IW1A.3.

KA-01644

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