Contrast Optimization: A faster and better technique for optimizing on MTF ABSTRACT Keywords: INTRODUCTION THEORY

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1 Contrast Optimization: A faster and better technique for optimizing on MTF Ken Moore, Erin Elliott, Mark Nicholson, Chris Normanshire, Shawn Gay, Jade Aiona Zemax, LLC ABSTRACT Our new Contrast Optimization technique allows for robust and efficient optimization on the system MTF at a given spatial frequency. The method minimizes the wavefront differences between pairs of rays separated by a pupil shift corresponding to the targeted spatial frequency, which maximizes the MTF. Further computational efficiency is achieved by using Gaussian Quadrature to determine the pattern of rays sampled. Examples are given to demonstrate the advantages of the technique. Keywords: MTF, Optimization, Imaging systems 1 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. Our new 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. Others have proposed similar, efficient optimization MTF methods based on wavefront variance.[1][2] (Those efficient methods were found just before computers became fast and inexpensive, and were apparently neglected as modern optical design software progressed.) 2 THEORY A given point on the MTF can be calculated as the autocorrelation of the complex pupil function (the complex wavefront in the exit pupil).[3] Most optical software uses this calculation method because it is more computationally efficient than calculating the full MTF as the FFT of the PSF intensity. The autocorrelation calculation is essentially a convolution. This involves shifting the complex pupil, multiplying the shifted and unshifted pupils, summing the terms, and taking the modulus to get the MTF. The amount of pupil shift corresponds to the spatial frequency ( ) of the final MTF value. For a circular pupil, a shift of = D/2 causes the shifted and unshifted apertures to just touch and the MTF value to go to zero. That shift corresponds to the cutoff frequency. Smaller shifts are linearly scaled from this value using Equation 2. (1) (2)

2 Considering a generic 3x3 pupil function, we can write down the shifted and unshifted pupil functions, p(x,y) and p(x-,y). Carrying out the convolution calculation and taking the modulus results in a series of cosines. (3) The arguments of the cosines is the phase difference in the pupils across a distance. To maximize the MTF values, then, the argument of the cosines should be minimized. In other words, the phase differences across any distance in the pupil should be minimized to maximize the MTF. (4) This equation can be used to build an optimization merit function. Multiple sets of two rays, separated by in the pupil, are traced to find the phase difference between the two rays. The optimization will attempt to drive all phase differences to zero. This is similar to reducing wavefront slope, but it is carried out at the particular spatial frequency corresponding to. 3 IMPLEMENTATION Equation 5 was used to create a default Merit Function in OpticStudio. The user selects Contrast Optimization 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, as shown in Figure 1, which are designed to be used in pairs. MECS traces three rays: an unshifted ray, a ray shifted by in the S direction in the pupil, and a ray shifted by 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 difference 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.[4] 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. In the 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. (5)

3 Figure 1: Implementation of Contrast optimization in OpticStudio. 4 CONTRAST LOSS PLOTS The optimization method is easy to understand in a visual way. Consider a wavefront that contains coma in a vertical orientation, as shown in Figure 2. If we represent the phase in the pupil by the orientation of the vector, the MTF calculation multiplies the two vectors. Minimizing the angle between the vectors, then, maximizes the MTF. (a) Figure 2: (a) A wavefront containing coma. (b) The wavefront s representation as vectors whose orientation represents the phase of the wavefront. The unshifted pupil overlaid with a pupil shifted in the vertical direction. The MTF is maximized when the vectors have the same orientation (or phase) so that the product of the vectors is maximized. To take advantage of the visual representation, we have added a Contrast Loss plot to OpticStudio. An early prototype is shown below in Figure 3. The plot allows the user to see exactly where in the pupil is causing the most loss of MTF. The magnitude of each vector is ½ (1 cos( 1-2 )). When the phase difference is zero, the magnitude of the vector is zero, so longer vectors indicate more loss of contrast. The orientation of each vector is the average phase: ( )/2, so that the underlying wavefront shape is still visible. (b)

4 Figure 3: An early prototype of the Contrast Loss plot. The magnitude of each vector is ½ (1 cos( 1-2)). The orientation of each vector is the average phase: ( 1 + 2)/2. 5 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, as shown in Figure 4. The design was done in visible wavelengths. The entrance pupil diameter is 33.3 mm. Figure 4: A starting point with plane parallel plates for a standard Double Gauss design. 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 Table 1. Contrast Optimization achieved comparable times to RMS Wavefront optimization. Sampling: 3 rings x 6 arms Sampling: 6 rings x 6 arms Sampling: 9 rings x 6 arms Contrast Optimization at 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 at 20 cycles/mm (failed) (failed) sec* Table 1: Optimization times in seconds for various optimization techniques. *Done after additional optimization on spot size. 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.

5 The Contrast Optimization and RMS Wavefront optimizations achieved similar results. One of the optimized systems is shown in Figure 5. The corresponding MTF curves are shown in Figure 6. Figure 5: The Double Gauss example system after optimization with Contrast Optimization. Figure 6: MTF of a system optimized using Digital Contrast Optimization at a spatial frequency of 20 cycles/mm, at a wavelength of nm. 6 BEST FOCUS EXAMPLE A simple example of finding best focus shows that Contrast Optimization will correctly trade higher spatial frequencies for performance at the specified frequency. We created a simple case to find the best focus position for a simple plano-convex singlet. Using three different Merit Functions, each optimization will choose a slightly different position within the spherical caustic as shown below in Figure 7. Figure 7: Finding best focus of a convex-plano singlet (F/5, = 500 nm, stop diameter of 11 mm) with three optimization methods: Contrast optimization, RMS Spot Size optimization, and RMS Wavefront optimization. The resulting focus distances are 54.64, 54.53, and mm..

6 The MTF curves for each case are shown in Figure 8. The Contrast optimization, carried out at 70 cycles/mm, has selected a solution near that of wavefront optimization. It is reducing the wavefront as measured across an interval of (D/2). The Contrast optimization solution improves the performance slightly at the desired frequency at the expense of higher frequencies. If the Contrast optimization is carried out at very low spatial frequencies, the result agrees with the RMS Spot Size optimization, since in that case, the Contrast optimization is sampling nearly instantaneous wavefront slope just as the rays in a spot diagram do. Figure 8: MTF curves for the best focus positions found using three different Merit Functions. 7 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 suggests that Contrast optimization has a smoother parameter space and will travel more directly to the best solution compared to other optimization types. 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 (see Figure 9). 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.

7 Figure 9: The MTF of a singlet with a shape factor of -3, and with a shape factor of Figure 10 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 welldefined minimum. The MTF value, in contrast, is fairly flat within the spherical caustic. (RMS spot size and RMS wavefront values, not shown, 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 Figure 10 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. Figure 10: The value of the Contrast Optimization merit function (right axis), and distance of the MTF from the diffraction-limited value (left axis), as a function of shape factor for an F/10 singlet.

8 8 TWO-MIRROR TELESCOPE EXAMPLE Optimizing a two-mirror telescope shows that Contrast Optimization can move to a solution more efficiently than other optimization techniques. We started with an F/20 two-mirror telescope with a 4 primary mirror at a wavelength of 500 nm (Figure 11). The system was optimized on-axis. After adding field angles out to 0.25, the system was dominated by coma. The system was then optimized across all field angles with variables on the radius of curvature of the primary mirror, the conic constants of both mirrors, and the image plane location. Optimizations were carried out using merit functions based on RMS wavefront, RMS spot size, and Contrast Optimization. The best solution for the two-mirror telescope is the Ritchey-Chretien form, which corrects the coma and spherical aberration to produce a system limited by astigmatism.[5] Astigmatism is less damaging to the system than coma because it is a function of the field angle squared instead of field angle cubed. The Contrast Optimization was carried out at 10 cycles/mm (for an aperture shift of = 0.1 (D/2)). Because Contrast Optimization reduces the wavefront slope across a shift of, it correctly finds the lowest-slope solution and eliminates the coma term, as shown in Table 2. RMS Wavefront and Spot Size merit functions, in contrast, contain no direct information about wavefront slope. Those optimizations leave slightly more residual coma than the Contrast Optimization. Figure 11: An F/20 two-mirror telescope and spot diagrams, optimized for on-axis performance, with a full field of view of 0.5. Z6 X-astigmatism (Z6 6 1/2 2 cos (2 )) Z7 Y coma (Z7 8 1/2 (3 3 2 ) sin ) before optimization after Contrast Optimization after RMS Spot Size optimization after RMS Wavefront Error optimization Table 2: Coma and astigmatism before and after each type of optimization, for the two-mirror telescope. In our trials, Contrast Optimization went directly to the correct solution in just a single cycle. The variable values in RMS Wavefront and Spot Size optimization wandered around just a little, as shown for the radius of curvature of the primary mirror in Table 3. The other variables in the system showed the same behavior. This again suggests that the Contrast Optimization merit function creates a simplified parameter space compared to other optimization types. The RMS Wavefront and Spot Size optimizations have multiple possible solutions, but there is only one solution that has the lowest slope at the spatial frequency corresponding to. Note that the two-mirror telescope example does not make full use of the capabilities of Contrast Optimization. Because the resulting system is diffraction-limited, Contrast Optimization in this case does not make any trades between MTF performance at different spatial frequencies. In a diffraction-limited system, the relationship between spatial frequencies

9 in the MTF is known, and Contrast Optimization would find the same result regardless of which spatial frequency is optimized. In other words, in this example, Contrast Optimization is just used as a way to reduce wavefront slope. primary mirror radius of curvature (mm) optimization loop # RMS Wavefront optimization RMS Spot Size optimization Contrast optimization mm Table 3: The primary mirror radius of curvature over 10 optimization cycles, for various optimization techniques. Contrast optimization finds a solution in a single cycle and remains there. 9 SUMMARY The Contrast Optimization technique allows for fast and robust optimization at specific spatial frequencies in the MTF. Merit function values are minimum in the same location as the MTF maxima. The merit function values tend to have a smooth shape with well-defined minima, which helps an optimization move quickly to the best solution. The technique can also be used as a general way to select solutions with low wavefront slopes. [1] W. B. King, Correlation between the relative modulation function and the magnitude of the variance of the waveaberration difference function, J. Opt. Soc. Am., 59, 692 (1969). [2] L. G. Seppala, [The Formulation of a Merit Function to Improve the Modulation Transfer Function of Lenses], University of Rochester, Rochester, NY (1974). [3] J.D. Gaskill, [Linear Systems, Fourier Transforms, and Optics], John Wiley & Sons Inc., New York(1978). [4] G.W. Forbes, Optical system assessment for design: numerical ray tracing in the Gaussian pupil, JOSA A, Vol. 5, No. 11, pages (1988). [5] D.J. Schroeder, [Astronomical Optics], Academic Press, Inc, San Diego (1987).