2.1

Merit function

It is often desirable to state the performance of an optical system by a single number. This immediately allows ranking of different optical systems, optimization of an optical system during its design, or finding the optimum state of an active or adaptive optics system. Examples of performance metrics that deliver a single numerical value are the Strehl ratio (S), the RMS wavefront error (σ), and image-sharpness metrics. Examples of performance metrics that deliver more than a single number, and thus contain more information, are the point spread function (PSF), the modulation transfer function (MTF), wavefront maps, and spot diagrams.

Since all single-number performance metrics lack detailed information about the performance of an optical system, the question arises, which single-number metric is most suitable for a certain imaging scene (e.g., for Earth observation typical imaging applications are urban areas, forests, and maritime surveillance) and for a certain task (e.g. tracking fast moving objects), and what its limitations are. An image-based metric can be applied in different image regions and thus achieve optimal performance for different field angles. For example, when trying to resolve a double star, the region of interest will be a small region of the image. Thus the active optics should correct a narrow field of view. On the other hand, when observing star clusters and nebulas, a larger image region should be corrected, and the correction of the active optics should be balanced over a wide field of view.

We use the image-sharpness metric S₁ introduced by Muller and Buffington¹. This is a single-number metric: S₁: = ∫ I(x, y)² dx dy, where I(x, y) is the irradiance at the point (x, y) of the image plane. Our merit function (MF) is a discretized adaptation of S₁:

In the above formula x and y are the axes of the image plane, N_x and N_y are the numbers of pixels in each axis, and I is the pixel value. The minus sign converts the sharpness maximization to a minimization problem. MF balances the influences from the contrast of all spatial frequencies.

2.2

Control domain

In optical systems design, the optimization variables are parameters of the optical elements, e.g., material, diameter, thickness, surfaces’ curvatures, and position. Examining the performance of sets of values for all the parameters, the designer tries to optimize a merit function. In an adaptive optics system, the adaptive element offers additional degrees of freedom to compensate for aberrations. The optimization variables are the inputs of the adaptive element, i.e., the voltages of the electrodes for a deformable mirror or a spatial light modulator. But the relation of the voltages of a deformable mirror with the optical performance is complex. For this reason, we wish to transform the space of the voltages of the deformable mirror to wavefront shapes, which can be expressed, e.g., as Zernike modes. Assuming a linear deformable mirror^†, this is done with the influence functions. Using the Zernike modes as variables, we gain insight into the optimization procedure and can possibly reduce the number of the variables in order to speed up the optimization. The Zernike modes used in this paper are listed in the Appendix. Having selected the merit function and the control domain, an algorithm to search the space of Zernike modes should be designed.

3.1

Simulation method

We simulate the pupil wavefront of an imaging system with a uniformly circular aperture in MATLAB over a 300 pixels × 300 pixels grid. We obtain the PSF by using the 2-D fast Fourier transform of the wavefront. We choose the width of the diffraction-limit PSF to be 40 pixels, significantly larger than the required width of 5 pixels, according to the Nyquist sampling theorem. To reduce the computational cost, we limit the total grid of the PSF to 1201 pixels × 1201 pixels, 30 times larger than the diffraction-limited PSF. The error caused by this truncation is negligible. Finally, we generate the MTF of the system by the 2-D fast Fourier transform of the PSF.

We assume that the corrective active element is placed in a plane conjugate to the pupil and that it is controlled with Zernike modes. We normalize the aberrated PSFs to the maximum of the diffraction-limited PSF, to allow comparison among PSFs with different aberrations. Throughout the paper we use the Zernike notation of Wyant and Creath³. We call Z_i the i-th Zernike mode and Z_i its coefficient.

3.2

RMS wavefront error and Strehl ratio

The RMS wavefront error (σ) is a common merit function in optical systems design. For small aberrations, the RMS wavefront error is directly related to the Strehl ratio and to the modulation transfer function (MTF). Maréchal formulated the following relation with the Strehl ratio: S ≈ [1 − 2π²σ²/λ²]², where λ is the wavelength. Shannon formulated an empirical formula with the MTF⁴: MTF(ν) = DTF(ν){1 − (σ/0.18)²[1 − 4(ν − 0.5)²]}, where ν is the normalized spatial frequency and DTF(ν) the diffraction-limited MTF. The Zernike modes are balanced with respect to the RMS wavefront error for every aberration. Therefore, for small aberrations, the Zernike modes are also balanced with respect to the Strehl ratio and to the MTF. This means that, as long as the total aberration is sufficiently small, adding any aberration to the wavefront leads to deterioration of all image quality metrics: increase of the RMS wavefront error, decrease of the Strehl ratio, and decrease of the MTF for all spatial frequencies.

For large aberrations, the Strehl ratio and the MTF can be multiple-valued for the same RMS wavefront error^5,6. Higher RMS wavefront error may thus lead to lower or higher Strehl ratio depending on the aberration modes contributing to the aberration. The same is also true for the MTF⁴. The term “large aberrations” commonly refers to σ > λ/4 or S < 0.4. To illustrate the complex relation between the RMS wavefront error and the Strehl ratio for large aberrations, we calculate them when defocusing (Zernike mode Z₃) in the presence of different values of astigmatism 0° (Zernike mode Z₄) and show the results in Fig. 2. The RMS wavefront error (Fig. 2a) increases monotonically when the aberration increases, because the Zernike modes are balanced with respect to the RMS wavefront error. On the other hand, the Strehl ratio (Fig. 2b) decreases monotonically for increasing |z₃| only as long as z₄ ≤ 0.4λ (dark blue, orange and yellow curves). For z₄ ≥ 0.6λ (violet, green and light blue curves), the Strehl ratio has two maxima away from z₃ = 0. The multiple-valued Strehl ratio with respect to the RMS wavefront error is shown in Fig. 2c for σ > 0.2λ.

Figure 2.

a) The total RMS wavefront error (σ) when varying defocus (Z₃) for different values of astigmatism 0° (Z₄) is monotonically increasing. b) The Strehl ratio (S) may decrease or increase adding defocus (Z₃) in the presence of astigmatism 0° (Z₄). c) Combined representation of plots a) and b). The Strehl ratio becomes multiple-valued with respect to the total RMS wavefront error for σ > 0.2λ. Each curve has a different constant value z₄, and |z₃| increases for increasing total RMS wavefront error.

3.3

Merit function for combinations of Zernike modes

We recently showed⁷ that for large aberrations the Zernike modes are not orthogonal to each other with respect to the merit function defined by (1). Here, we further explore this dependence by running simulations for combinations of two Zernike mode aberrations. Incoherent images of extended objects can be generated by the 2-D convolution of an extended object and the PSF. Using an extended object would restrict the validity of the results in the spatial frequencies that are present in the object. However, the PSF contains all spatial frequencies and can be used to draw conclusions for every spatial frequency that may be present in the object. Therefore, we calculate the merit function for the 1201 pixels × 1201 pixels image of the PSF. The combinations of Zernike modes shown in this paper are characteristic examples to investigate the physical causes for the non-orthogonality of the Zernike modes with respect to the merit function. To this end, we use wavefront maps, the PSF, and the MTF.

In section 3.3.1 we discuss the combination of defocus (Z₃) and astigmatism 0° (Z₄). In section 3.3.2 we discuss the combination of astigmatism 0° (Z₄) and secondary astigmatism 0° (Z₁₁), as an example for the combination of Zernike modes with the same azimuthal order. The next two sections discuss combinations of trefoil 0° (Z₉): the section 3.3.3 with coma x (Z₆), and the section 3.3.4 with astigmatism 0° (Z₄). The global minimum (optimum) of the merit function is for zero aberration, as expected, in all cases. We show that for large aberrations, the merit function can be improved by adding a Zernike mode, despite the fact that this increases the RMS wavefront error.

3.3.1

Defocus and astigmatism

Figure 3 shows our merit function calculated for the PSF under the same conditions as in Fig. 2, that is when defocusing in the presence of different values of astigmatism 0°. The merit function has a single minimum at z₃ = 0 as long as z₄ ≤ 0.2λ, but has two minima for opposite values of z₃ when z₄ ≥ 0.4λ. Although the progression of the curve resembles that of the Strehl ratio (Fig. 2b), it is different, because the Strehl ratio is just the maximum of the PSF, i.e., the value at a single point, whereas the merit function takes into account the whole PSF.

Figure 3.

The merit function for the same conditions as in Fig. 1, that is when varying defocus (Z₃) for different values of astigmatism 0° (Z₄).

We examine two aberrations, marked as “Aberration 1” and “Aberration 2” in Fig. 3. They both have z₄ = 0.6λ. Aberration 2 has additional defocus z₃ = 0.3λ and lower (better) merit function than aberration 1. Figure 4 shows the wavefront maps, the PSF and the MTF for these aberrations.

Figure 4.

Plots for the aberrations 1 and 2 marked in Fig. 3. The aberrations have the same value of astigmatism 0° (Z₄), but different values of defocus (Z₃). From upper left to bottom right the plots show: the wavefront, the PSF, the PSF profiles in x and y axes, and the MTF in x and y axes.

The Zernike modes of defocus (Z₃) and astigmatism 0° (Z₄) contribute to the first-order field-independent aberration of focus³: . If only Z₃ and Z₄ exist in the system, the first-order field-independent focus becomes zero when |z₃| = z₄/2, the ratio of the coefficients for the aberration 2. Then the system suffers only from first-order field-independent astigmatism (W₂₂).

Adding defocus of |z₃| = z₄/2 in the presence of astigmatism 0° slightly increases the width of the PSF in one axis of the image plane (the axis x for the aberration 2). This leads to deterioration of the contrast and of the resolution for spatial frequencies oriented in the direction of this axis. At the same time, this significantly shrinks the PSF in the other axis of the image plane, leading to diffraction-limited contrast and resolution for spatial frequencies oriented in the direction of that axis (the axis y for the aberration 2). This principle is applied to astigmatic systems which are focused differently depending on the object.

3.3.2

Zernike modes with the same azimuthal order

In our recent publication⁷, we showed that for large aberrations the Zernike modes for defocus (Z₃) and spherical aberration (Z₈) are not orthogonal to each other with respect to the merit function. Here, we show another example for combination of Zernike modes with the same azimuthal order, Z₄ and Z₁₁, i.e., the Zernike modes for astigmatism 0° and secondary astigmatism 0°, both of azimuthal order of +2. Figure 5 shows the merit function calculated for the PSF, when varying z₄ in the presence of different values of z₁₁.

Figure 5.

The merit function when varying astigmatism 0° (Z₄) for different values of secondary astigmatism 0° (Z₁₁).

For z₁₁ = 0 there is a single minimum for the merit function at z₄ = 0. For z₁₁ > 0 the global minimum shifts towards positive values of z₄. Due to our step size for the Zernike coefficients, we first resolve this shift of the global minimum when z₁₁ = 0.4λ (yellow curve). In Fig. 6 we examine the aberrations marked as “Aberration 1” and “Aberration 2” in Fig. 5. Both have z₁₁ = 0.6λ (violet curve), but aberration 2 has additional astigmatism 0° z₄ = 0.6λ and lower (better) merit function than aberration 1.

Figure 6.

Plots for the aberrations 1 and 2 marked in Fig. 5. The aberrations have the same value of secondary astigmatism 0° (Z₁₁), but different values of astigmatism 0° (Z₄). From upper left to bottom right the plots show: the wavefront, the PSF, the PSF profile, and the MTF. The table below the plots records several metrics to compare the optical performance for the two aberrations. Improved (deteriorated) performance for each metric is marked with green (red).

The addition of astigmatism 0° increases the wavefront deviation at the edges of the aperture, but smoothens the wavefront in the central part. This becomes obvious in Fig. 7 that shows the wavefront profiles for the aberrations 1 and 2 along the red dotted lines in Fig. 6. The wavefront with only secondary astigmatism 0° (aberration 1) has smaller variance, but the addition of astigmatism 0° (aberration 2) leads to a flatter wavefront in the central region of the aperture. We can calculate the Zernike modes over a smaller radius. Using the formulas for scaling the Zernike modes from the aperture where they are defined (radius r) to a smaller radius⁸, we find that for the aberration 2 there exists a radius r′ = 0.86r on which . On this radius the scaled Zernike modes comprise only . For comparison, for the aberration 1 the scaled Zernike modes on the radius r′ comprise and . The calculations are shown in the Appendix.

Figure 7.

The wavefront profiles for the aberrations 1 and 2 along the red dotted lines of Fig. 6. The addition of astigmatism 0° (blue curve – aberration 2), in the presence of secondary astigmatism 0°, leads to a flatter wavefront in the central region of the aperture (marked as 2r′). For r′ = 0.86r the wavefront comprises a smaller value of secondary astigmatism 0°, and the astigmatism 0° is zero.

In the image plane, the relative heights of the side lobes of the PSF decrease from 41% to 13%. Consequently, energy is squeezed into the central lobe, slightly increasing its width but also increasing its peak (the Strehl ratio). The narrower central lobe of the PSF for the aberration 1 can be interpreted as a higher resolution limit. This is valid, provided that the detection routine doesn’t misinterpret the side lobes (with 41% relative height) as distinct objects. Finally, the MTF for the aberration 1 falls fast for low spatial frequencies until 30% of the diffraction-limited cutoff frequency (ν_cut) and rises again with a peak near the diffraction-limited contrast at about 50% of the ν_cut. The MTF for the aberration 2 is in general smoother and achieves better contrast at low and mid spatial frequencies.

3.3.3

Trefoil and coma

The Zernike mode of trefoil 0° is Z₉ = ρ³ cos(3ϑ) = 4ρ³ cos³ ϑ − 3ρ³ cos ϑ. The first term (4ρ³ cos³ ϑ) is the fifth-order aberration of trefoil. The second term (3ρ³ cos ϑ) is the third-order coma (neglecting the field dependence) and is added to make the Zernike mode orthogonal to the lower order modes on the unit circle and to minimize the RMS wavefront error. The fifth-order aberration ρ³ cos³ ϑ (neglecting the field dependence) is called “trefoil” when studying the wavefront. It is also called “elliptic coma”, based on the image plane intensity: the circles that appear in the image spot for third-order coma turn into ellipses when fifth-order aberration of trefoil is added^9,10. This relation between trefoil and coma is revealed when we vary the Zernike coma x (Z₆) in the presence of different values of Zernike trefoil 0° (Z₉). Figure 8 shows the merit function calculated for the PSF.

Figure 8.

The merit function when varying coma x (Z₆) for different values of trefoil 0° (Z₉).

For z₉ ≤ 0.2λ there is a single minimum for the merit function at z₆ = 0. For z₉ > 0.4λ the global minimum shifts towards positive values of z₆. In Fig. 9 we examine the aberrations marked as “Aberration 1” and “Aberration 2” in Fig. 8. They both have z₉ = 0.8λ, but aberration 2 has additional coma x z₆ = 0.7λ and lower (better) merit function than aberration 1.

Figure 9.

Plots for the aberrations 1 and 2 marked in Fig. 8. The aberrations have the same value of trefoil 0° (Z₉), but different values of coma x (Z₆). From upper left to bottom right the plots show: the wavefront, the PSF, the PSF profiles in x and y axis, and the MTF in x and y axis. The table below the plots records several metrics to compare the optical performance for the two aberrations. Improved (deteriorated) performance for each metric is marked with green (red).

Adding positive coma x in the presence of trefoil 0° (aberration 2) makes the wavefront resemble to a single ripple in the u axis. The wavefront is practically uniform in one axis, the v axis of the coordinate system of the aperture. The wavefront actually becomes completely independent of v when z₆ = z₉, in which case the wavefront is W = z₉(4u³ − 2u) and two of the three ripples of the wavefront vanish.

In the image plane, adding coma x increases the Strehl ratio and shrinks the PSF, for both x and y axes. The PSF width in the y axis decreases until the diffraction-limited width. Finally, at the cost of the contrast reduction for low spatial frequencies oriented in the x axis, the MTF increases at mid and high frequencies for spatial frequencies oriented in both x and y axes. It even reaches the diffraction limit MTF for spatial frequencies oriented in the y axis in the case of positive coma x of the same magnitude as the trefoil 0° (z₆ = z₉).

3.3.4

Trefoil and astigmatism

Apart from “trefoil” and “elliptic coma”, the fifth-order aberration ρ³ cos³ ϑ (neglecting the field dependence) is also called “triangular astigmatism” in part of the literature¹⁰. This is connected to the image plane intensity: in the presence of third-order astigmatism adding fifth-order aberration of trefoil turns the image spot into a triangle. This motivated us to research the combination of the Zernike modes of astigmatism and trefoil. We varied the trefoil 0° (Z₉) in the presence of different values of astigmatism 0° (Z₄) and show the merit function calculated for the PSF in Fig. 10.

Figure 10.

The merit function when varying trefoil 0° (Z₉) for different values of astigmatism 0° (Z₄).

For z₄ ≤ 0.2λ there is a single minimum for the merit function at z₉ = 0. For z₄ ≥ 0.6λ two equal minima appear, for opposite values of z₉. In Fig. 11 we examine the aberrations marked as “Aberration 1” and “Aberration 2” in Fig. 10. They both have z₄ = 0.6λ, but aberration 2 has additional trefoil 0° z₉ = 0.4λ and lower (better) merit function than aberration 1.

Figure 11.

Plots for the aberrations 1 and 2 marked in Fig. 9. The aberrations have the same value of astigmatism 0° (Z₄), but different values of trefoil 0° (Z₉). From upper left to bottom right the plots show: the wavefront, the PSF, the PSF profiles in x and y axis, and the MTF in x and y axis. The table below the plots records several metrics to compare the optical performance for the two aberrations. Improved (deteriorated) performance for each metric is marked with green (red).

The addition of trefoil 0° partially compensates the ripple caused by the astigmatism 0° at one half of the aperture. For the aberration 2 with positive trefoil 0°, it’s the left half of the aperture (π/2 ≤ θ ≤ 3π/2, negative u). The wavefront aberration is W = z₄Z₄ + z₉Z₉ = (z₄ + z₉u)(u² − υ²) − 2z₉uυ² and its v-derivative is 𝜕W/𝜕υ = − 2υ(z₄ + 3z₉u). We notice that for u = ±0.5, the v-dependence of the wavefront vanishes when z₄/z₉ = ∓3/2. This is equal to the ratio of the Zernike coefficients of Z₄ and Z₉ for the aberration 2 (0.6λ/0.4λ).

In the image plane, the PSF shrinks and its peak intensity increases. Ripples with relatively small peaks appear, the highest being 17% of the peak intensity. Finally, the MTF with additional trefoil 0° is slightly lower for spatial frequencies up to about 25% of the ν_cut, but is significantly higher for mid spatial frequencies between 25% and 70% of the ν_cut. This leads to higher resolution. Setting the limiting resolution at about 10% contrast, the cutoff frequency is about 0.3ν_cut for z₄ = 0.6λ (aberration 1) and increases to 0.5ν_cut by adding trefoil 0° .