Resolving capacity of the circular Zernike polynomials

M. V. Svechnikov; N. I. Chkhalo; M. N. Toropov; N. N. Salashchenko

doi:10.1364/OE.23.014677

1. Introduction

Zernike polynomials are widely used to describe the optical surfaces and wavefronts of optical systems. This popularity is due to the orthogonality and symmetry properties of these functions, as well as direct correspondence of individual Zernike polynomials to balanced classical aberrations [1, 2]. Many optical systems have a rotation axis and the working area of mirrors and lenses is of a circular or annular shape. The possibility of selecting, from the total deformation, the wavefront aberrations of certain types and aberrations with a rotation axis of a certain order and symmetric and antisymmetric distortions greatly simplifies the analysis of optical systems and makes it easy to separate the impacts of various factors on the quality of the wavefronts.

The properties and use of Zernike polynomials to describe real systems are considered in a large number of works. Consistent derivation of Zernike polynomials as a system that is orthogonal on the unit circle and has certain symmetry of properties is described in [1]. With regard to detailed consideration of the properties of Zernike polynomials, links to key works on this topic are adduced in [3], and there are also expressions for the annular orthogonal polynomials. Orthogonal systems based on circular Zernike polynomials have been built for more complex areas, such as rectangles, ellipses, and regular hexagons [4,5]. A lot of research is devoted to the computation of Zernike polynomials and the numerical expansion of a function defined in a discrete set of points [3,6,7]. The connection between Fourier coefficients and the coefficients of expansion in Zernike polynomials is shown in [8,9].

In all the above studies, despite their diversity, the ability of Zernike polynomials to display a relief of a certain lateral scale is not considered. At the same time, Zernike polynomials are used not only for the observation of certain kinds of aberrations but also for complete representation of the wavefront, which may have a very complex shape after passing through the optical system. A low quality of approximation of complex relief by Zernike polynomials compared with interpolation of splines was noted in [10]. Rimmer and Wyant [11] give the following considerations regarding the order of polynomials which should be an approximation of the wavefront: “The order to which the data must be fit depends on the nature of the data. The goal is to get a good representation of the data without fitting noise as well. This can usually be done by fitting to successively higher orders and stopping when there is no further significant reduction in the rms residual error”. It is difficult to quarrel with them, since this statement expresses the purpose of fitting the data from a common-sense perspective. In paper [12], the authors also emphasize the dependence of the relief representation accuracy on a set of Zernike polynomials. They provide the residual root mean square (RMS) using 9, 16, 25, 36, 45, 100, and 230 of the first terms of the wavefront expansion and the maps of mistakes that contain the high-frequency part of the relief. The authors of [13] also criticize the widespread approach that consists of using 36-polynomial fitting of the wavefront. In [14], there are shown the spatial spectra of a surface approximated by Zernike polynomials of orders 10, 20, 30, and 40, but there are no any quantitative conclusions about the dependence of the displayed spectrum on the order of polynomials.

Thus, it can be said that the question of the required number of approximating polynomials for fitting wavefronts has not yet been systematically analysed. In this paper, we attempt to fill this gap by answering the following questions:

•What is the minimum number of polynomials that is necessary to adequately display a local deformation of a certain size?
•What is the resolvable scale of relief approached by a certain number of Zernike polynomials?
•How does the resolvable scale depend on the type of relief?
•What is the relationship between the number of approximating polynomials and the displayed frequency spectrum of an arbitrary topography?
•What is the connection between the mean square error of approximation and the number of approximating polynomials?

This kind of research is meaningful for any system of approximating functions. However, since the most commonly used ones in optical applications are the circular Zernike polynomials, and, based on them, annular orthogonal polynomials, this work is devoted to research and demonstration of the frequency properties of circular Zernike polynomials.

2. Definitions and notation

In this paper we use the following notation:

Z_{n}^{m} (r, θ) = \sqrt{\frac{2 (n + 1)}{π (1 + δ_{m}^{0})}} R_{n}^{m} (r) Θ^{m} (θ) - circle Zernike polynomials,

δ_{m}^{0}

– Kronecker’s delta,

Θ^{m} (θ) = {\begin{matrix} \cos | m | θ & (m > 0) \\ 1 & (m = 0) \\ \sin | m | θ & (m < 0) \end{matrix} - angular functions,

R_{n}^{m} (r) = \sum_{s = 0}^{(n - | m |) / 2} \frac{{(- 1)}^{s} (n - s)!}{s! (\frac{n + m}{2} - s)! (\frac{n - m}{2} - s)!} r^{n - 2 s} - radial polynomials,

W (\vec{r}) = \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} Z_{n}^{m} (r, θ) - expansion of function

c_{n m} = \iint_{r \leq 1} W (\vec{r}) Z_{n}^{m} (r, θ) d^{2} \vec{r} - expansion coefficients of W (\vec{r}),

We will call the set of polynomials of order n₀

{Z_{n}^{m} (r, θ), \forall n \leq n_{0}}

and denote it by Zn₀. For example, the commonly used 36-term Zernike polynomials set, presented, in particular, in [3], will be denoted by Z7. The number of members in a set of order n₀ is calculated as follows:

N_{n_{0}} = \frac{(n_{0} + 1) (n_{0} + 2)}{2} .

It should be noted that different authors use different definitions of circular polynomials [1,3,8,9]. Therefore, when using multiple sources at the same time it is necessary to watch closely for the factor in front of the radial and angular functions and before the expansion coefficients.

3. Expansion of test functions

In the beginning, we will represent a specified relief in the form of a polynomial expansion of various orders, up to 200, and watch the dependence of the RMS residual error on the order of expansion and the type of relief. As an expanded function (relief) we will take a kinds of “primitives”, namely single Gaussian peaks of various sizes located in different parts of the unit circle, solitary strips of different widths and locations, and rings of different widths and diameters.

3.1 Gaussian peak at the centre

The first structure whose expansion we will consider is a single peak described by a Gaussian function (Fig. 1). Initially, we set the 2σ-width of the peak equal to 0.64 (about a third of the diameter of the circle) and arrange our peak in the centre of the unit circle.

Fig. 1 The function to be expanded. σ = 0.32; the peak is located in the centre of the unit circle.

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The expansion converges to the true relief with an increase in the number of Zernike polynomials involved in the representation of the original function. Figure 2 shows an example of this process for three different expansion orders: 3, 7, and 12. It can be seen from the series of images Fig. 2(b) and 2(c) that the difference between a function and its expansion becomes more high frequency and decreases in magnitude with increasing expansion order. The colour map of heights represents the relative scale, where the red and blue colours correspond to the highest and lowest points, respectively.

Fig. 2 (a) Presentation of the original function by polynomial sets Z3, Z7, and Z12; (b) the difference between the original function and its corresponding expansion; (c) the central cross-section of the function and its expansions. Residual RMS, specified in column (c), is the RMS of difference.

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Hereinafter 100% RMS means the RMS of the initial relief. The rate of errors decreases with increases in the order of expansion, as illustrated in Fig. 3(a).

Fig. 3 Dependence of the residual deformation on the order of expansion (a); the expansion coefficients on the respective set of Zernike polynomials (b).

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Figure 3(b) shows which terms of the expansion give the most significant contribution to the approximating function.

As can be seen in Fig. 2(c), use of the popular 36-term set (Z7) for representation of the peak with a width of one third of the diameter of the frame leads to an RMS error of more than 18%.

Now we will reduce the peak width and monitor the growth of the order of polynomials needed to represent functions with the same error.

Figure 4 shows the dependence of the residual RMS on the order of polynomials for different widths of Gaussian peaks. For these dependencies, the following can be concluded: to display a narrow peak with the same relative error and width, the order of the polynomials used should be increased by the same factor as the factor by which the width of the peak is reduced. While this is an “empirical” observation, a theoretical explanation of this fact and the specific form of the dependence of the residual RMS on the order of polynomials will be given below.

Fig. 4 Dependence of the expansion error of five Gaussian peaks, of corresponding σ, on the order of polynomials.

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Figure 5 shows the observed relief when peaks of different widths are fitted by polynomials of different orders. Horizontally arranged images are of the same peak; in the vertical direction, the peak width decreases.

Fig. 5 Representation of tapering peaks by polynomials of increasing orders. Above each expansion the polynomial set used is indicated; under each expansion, its RMS error is indicated (columns 5(a), 5(b), and 5(c)). Under the original maps (column 5(d)) the σ-widths of the Gaussian peaks are indicated (the circle is unitary).

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3.2 Gaussian peak at the edge

Now we arrange our peak at a distance of 0.75 from the centre of the unit circle. In this case the expansion will contain functions depending on the angle. The widths of the peaks we take are the same as in the previous simulation: σ = 0.32, 0.16, 0.08, 0.04, and 0.02.

Figure 6, like Fig. 5, shows the effects of insufficient approximation. Table 1 compares the approximation error of identical peaks located in the centre and at a distance of 0.75 from the centre of the unit circle. This comparison shows that when using the same order of approximating polynomials, relief elements which are closer to the edge are better resolved than those elements that are closer to the centre. The ratio of the Gaussian peaks’ RMS error lies between 1.1 and 2.

Fig. 6 Representation of tapering peaks by polynomials of increasing orders. Above each expansion the polynomial set used is indicated, and under each expansion, its RMS error is indicated (columns 6(a), 6(b), and 6(c)). Under the original maps (column 6(d)), the σ-widths of the Gaussian peaks are indicated.

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Table 1. Comparison of approximation errors for the centred and shifted peaks. Residual RMS is given as a percentage of the total RMS of the relief.

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3.3 Rings and stripes with a Gaussian profile

A set of polynomials that fits a single peak of width S sufficiently well will also be good enough for the approximation of any relief whose smallest size is ≥ S. This is easy to understand if we express the relief in the form of a sum of peaks, each of which will be adequately displayed in the approximation.

Let us check this assumption on the following structures: rings with various widths and radii and single stripes of different widths and at different distances from the centre of the circle. All of these elements are shown with Gaussian profiles in Fig. 7.

Fig. 7 Test functions. Rings and stripes have a Gaussian profile with a height of 1 and the corresponding σ. Such sigmas (as well as σ = 0.32 and 0.16) describe the single peaks discussed earlier. The parameter R denotes the smallest distance from the maximum of the structure to the centre of the unit circle.

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Table 2 shows the approximation error of all the structures considered. These numbers show that, indeed, from the perspective of fitting, a single peak is the worst among the structures that have the same minimum size. Also, all the modelled structures are better resolved when located close to the edge of the area.

Table 2. Comparison of the fitting errors of peaks, rings, and stripes with a Gaussian profile. Residual RMS values are given as a percentage of the total RMS of the relief.

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All the results presented in this section are obtained by numerical expansion of test functions with Gaussian profiles. Further, before obtaining analytical expressions for the dependences of the error of approximation of local deformations on the polynomial order, we consider an arbitrary topography and introduce an appropriate way to describe the surface.

4. Description of arbitrary surface

Requirements for the wavefront deformations are usually formulated in a form that is independent of the particular implementation of these deformations. For example, the Marechal criterion ([1], p. 528) states that in order to achieve diffraction-limited resolution of an optical system, the standard deviation of the wavefront from a sphere whose centre is in the diffraction focus should not exceed λ/14, where λ is the working wavelength. The diffraction quality means Strehl ratio greater than 0.8. Similar requirements can be formulated for individual aberrations ([1], p. 532), and the requirements are much milder for some of these.

Relief of optical (and any other) surfaces can be very complicated. Although the shapes of mirrors and substrates can be reproduced from sample to sample with sufficient accuracy during forming and polishing, the relief on a smaller scale, for example, with a lateral size of less than a millimetre, turns out to be random. However, the characteristics of such a random topography can be determined and specified by surface-processing technology.

Instead of working with the function describing the relief, we can work with its Fourier transform; these representations are equivalent. However, if the topography requirements are formulated in terms of the RMS deviation, the complete information about the relief is redundant. In this case, a simplified description of the surface is possible with the use of the so-called PSD (Power Spectral Density) function, which is the Fourier transform of the autocorrelation function of the relief. PSD describes the statistical properties of the relief and makes it possible to find the RMS deviation in any range of spatial frequencies. PSD is typically used to describe the roughness of the surface on millimetre to nanometre scale. Nevertheless, the distinction between the shape and roughness is rather relative and the approach to the surface description can be shared. For example, in [15], the shape of a telescope mirror is presented as a PSD function, along with the roughness. Some information about using PSD can be derived from [16].

In the Appendix, we derive the relationship between the PSD function and the Fourier transform of relief and express the PSD of relief in terms of its Zernike expansion coefficients. In this section, we use the ready-made formulas.

Let us define $P S D_{2 D} (ν)$ as the power spectral density, averaged over all directions, which depends only on the modulus of the spatial frequency. Effective relief deformation in the spectral range ${ν_{\min}, ν_{\max}}$ can be found by the formula

σ_{e f f}^{2} = 2 π \int_{ν_{\min}}^{ν_{\max}} P S D_{2 D} (ν) ν d ν .

The spectral function itself can be found on a relief $W (\vec{r})$ as follows:

P S D_{2 D} (ν) = \frac{1}{S} \frac{1}{2 π} \int_{0}^{2 π} {| F (\vec{ν}) |}^{2} d φ,

where

F (\vec{ν}) = \iint W (\vec{r}) e^{2 π i \vec{ν} \vec{r}} d^{2} \vec{r}

is the Fourier transform of the surface.

If the surface is given in the form of an expansion in Zernike polynomials,

W (\vec{r}) = \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} Z_{n}^{m} (r, θ),

where

c_{n m}

are the expansion coefficients, then

P S D_{2 D} (ν) = \frac{2 π}{S} \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} n' = n \\ n' + = 2 \end{array}}^{\infty} (2 - δ_{n}^{n'}) B_{n}^{n'} U_{n} (ν) {(U_{n'} (ν))}^{*},

where

U_{n} (ν) = {(- 1)}^{n / 2} \sqrt{2 (n + 1)} \frac{J_{n + 1} (2 π ν)}{2 π ν},

B_{n}^{n'} = \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} c_{n' m},

S

is the area of the unit circle,

δ_{n}^{n'}

is the Kronecker delta, and

J_{n}

is the Bessel function of the first kind of order n.

On the basis of the form of functions $U_{n} (ν)$ , notably the presence of Bessel functions, it can be said that the n-th radial Zernike polynomial contains almost no frequencies $\leq n / 2 π$ as a Bessel function of n-th order is negligibly small if the argument is substantially less than n. This allows us to formulate a criterion for a sufficient number of polynomials in the expansion of an arbitrary function. The condition is as follows: for reliable representation of relief with spatial frequency $\leq ν$ , it is enough to have a set of polynomials of order $\geq 2 π ν$ . In this criterion, the spatial frequency is indicated in the inverse radius of the circle (we have a unit circle). We will illustrate the criterion by the fitting of a randomly generated map by polynomials of different orders.

Figure 8 illustrates the representation of complex relief by polynomials of different orders, and RMS errors are also shown. Figure 9 shows the spectra of these representations and threshold frequencies of $n / 2 π$ , where n is the order of the expansion. It can be seen that in the frequency range below the threshold, the spectrum coincides almost perfectly with the spectrum of the original relief, and after the threshold the frequency begins to subside quickly. In addition, the relation $ν_{\max} ~ n$ confirms the observation made for local deformations: The number of times we reduce the element of relief is the same as the number of times we should increase the order of polynomials to display this element with the same error.

Fig. 8 Approximations of random relief by Zernike polynomials of different orders.

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Fig. 9 The spectra of the initial relief and its expansions, shown in Fig. 8. The red dashed lines indicate the $n / 2 π$ frequencies for the corresponding order n. A unit of frequency is the inverse radius of the circle; a unit of $P S D_{2 D} (ν)$ is the relief height unit in the fourth degree.

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The problem of a narrow spectral range of expansion was tackled by the authors of [17]. With the aim of measuring a relief in the scale range of 1 mm to 10 μm and using the approximation by polynomials of the order 68, they were faced with rapid falling of the PSD to higher frequencies. To overcome this limitation, they had to rebuild the optics to work with a smaller frame. The criterion proposed above allows us to say a priori which spectral range will be available after the approximation and thus to choose the best approach to solving the problem.

5. The dependence of the approximation errors on the order of approximation

The connection between the maximal spatial frequency and the order of polynomials allows the RMS error of approximation to be estimated relatively easily using the PSD function.

Since R M S_{a p p r o x}^{2} \approx 2 π \int_{0}^{ν_{\max} = n / 2 π} P S D_{2 D} (ν) ν d ν,

then R M S_{e r r o r}^{2} \approx 2 π \int_{ν_{\max} = n / 2 π}^{\infty} P S D_{2 D} (ν) ν d ν .

Then the relationship in which we are interested will be written as

\frac{R M S_{e r r o r}}{R M S_{r e l i e f}} (n) \approx \sqrt{\frac{2 π \int_{n / 2 π}^{\infty} P S D_{2 D} (ν) ν d ν}{\frac{1}{S} \iint W {(\vec{r})}^{2} d^{2} \vec{r}}} = \sqrt{\frac{\int_{0}^{2 π} \int_{n / 2 π}^{\infty} {| F (ν, φ) |}^{2} ν d ν d φ}{\iint W {(\vec{r})}^{2} d^{2} \vec{r}}} .

We write the sign of approximate instead of exact equality, because in reality the radial polynomials of order n $R_{n}^{m} (r)$ contain frequencies greater than $n / 2 π$ . For this reason, the PSD function does not fall sharply to zero but subsides gradually as illustrated in Fig. 9. The dropping “tail” contains part of the roughness, but in most cases it is much smaller than the total relief deformation. This is not always the case; for example, if the relief is fully represented by polynomials of a very low order, such as 0 or 1, then almost all of the deformation will be contained in a decreasing “tail”: $R M S_{t a i l} \approx R M S_{r e l i e f} .$

If $R M S_{t a i l} < < R M S_{e r r o r},$ then the formula (13) describes this approximation error. Nevertheless, it is possible that $R M S_{t a i l} > R M S_{e r r o r},$ for example, if the surface is almost entirely represented by a set of Zernike polynomials of order n (n is not small). In this case, the error of approximation is extremely small and the formula (13) is an upper bound.

Thus, in the case of “insufficient approximation” (when $R M S_{t a i l} < < R M S_{e r r o r}$ ), (13) works well. Let us obtain the dependence of the approximation error on the order of polynomials for single peaks located in the centre of the unit circle.

5.1 Gaussian peak

W (\vec{r}) = e^{\frac{- r^{2}}{σ^{2}}}

We use [18] (Formula 11.4.29):

F (ν, φ) = \int_{0}^{2 π} \int_{0}^{1} W (r) e^{2 π i \vec{ν} \vec{r}} r d r d φ \approx 2 π \int_{0}^{\infty} e^{\frac{- r^{2}}{σ^{2}}} J_{0} (2 π ν r) r d r = π σ^{2} e^{\frac{- {(2 π ν)}^{2} ν^{2}}{4}},

\frac{R M S_{r e s i d u a l}}{R M S_{p e a k}} \approx \sqrt{\frac{2 π \int_{n / 2 π}^{\infty} π σ 4 e^{\frac{- {(2 π ν)}^{2} ν^{2}}{2}} ν d ν}{\frac{σ^{2}}{2}}} = e^{\frac{- n^{2} σ^{2}}{4}} .

In deriving (15), we assumed that σ < 1 and it is possible to replace the upper limit of radius integration at ∞.

5.2 The trigonometric peak

W (\vec{r}) = {\begin{matrix} \frac{1}{2} (\cos (\sqrt{π} \frac{r}{σ}) + 1) & r \leq \sqrt{π} σ \\ 0 & r > \sqrt{π} σ \end{matrix}

F (ν, φ) = π \int_{0}^{\sqrt{π} σ} (\cos (\sqrt{π} \frac{r}{σ}) + 1) J_{0} (2 π ν r) r d r .

The integral (18) and $\frac{R M S_{r e s i d u a l}}{R M S_{p e a k}}$ were found numerically.

5.3 Rectangular peak

W (\vec{r}) = {\begin{matrix} 1 & r \leq σ \\ 0 & r > σ \end{matrix}

F (ν, φ) = 2 π \int_{0}^{σ} J_{0} (2 π ν r) r d r = \frac{σ}{ν} J_{1} (2 π ν σ)

(see [18], formula 11.3.20),

\frac{R M S_{r e s i d u a l}}{R M S_{p e a k}} \approx \sqrt{\frac{2 π \int_{n / 2 π}^{\infty} \frac{σ^{2}}{π ν} J_{1}^{2} (2 π ν σ) d ν}{\frac{σ^{2}}{2}}} = \sqrt{J_{0}^{2} (σ n) + J_{1}^{2} (σ n)} .

Figure 10 shows all three of these peaks.

Fig. 10 Gaussian, trigonometric, and rectangular peaks.

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Approximation errors of all three peaks are plotted in Fig. 11 on logarithmic and linear scales. The graphs show the real error, obtained by numerical expansion, and the theoretical calculations worked out above. The theoretical and practical results are in very good agreement for all n from 0 to 200.

Fig. 11 Dependence of the approximation error of three types of peaks on the order of Zernike polynomials. The peaks are located in the centre of the unit circle. Markers denote the values obtained by numerical expansion; solid lines represent the theoretical estimate obtained above. Figure 11(a) and 11(b) show the same content on logarithmic and linear scales. σ = 0.08

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Figure 11(a) clearly shows that despite the fact that the trigonometric and Gaussian peaks are very similar (see. Figure 10), the rate of reduction of the error depending on the order of polynomials differs significantly starting from some n. The rectangular peak, from the position of representation by Zernike polynomials, is significantly worse than the Gaussian and trigonometric ones. Thus, the smoother the peak is and the fewer steep fronts it contains, the smaller the number of polynomials required for its approximation.

In the earlier section on the examples of stripes, rings, and peaks with Gaussian profiles, we showed that the local deformation is better represented by a finite set of polynomials which are located closer to the edge of the circle. Thus, the theoretical estimates of $\frac{R M S_{r e s i d u a l}}{R M S_{p e a k}}$ made for the centred local deformation can serve as upper bounds for the same deformation shifted to the edge.

6. Conclusion

The spatial scale that is displayed in the approximation of relief by a finite set of Zernike polynomials essentially depends on the order of the approximating set. Depending on the particular technical task, requirements can be formulated on this scale, and in this case it is necessary to carry out an approximation to meet these requirements. To satisfy the spatial frequency demands $ν_{\max}$ (in “real” units: cm⁻¹, mm⁻¹, and so on), it is sufficient to use a set of polynomials of the order $\geq 2 π R ν_{\max}$ (here R is the radius of the circle on which the approximation is carried out).

If it is necessary to be able to represent single peaks adequately, as a priori criteria, we offer the formula (13) for estimation of the RMS error of approximation. Such a need, in particular, can occur during the preparation of optical substrates for precise ion-beam etching. As long as the intensity distribution of the ion beam is close to Gaussian, formula (16) $\frac{R M S_{r e s i d u a l}}{R M S {}_{p e a k}} = e^{\frac{- σ^{2} n^{2}}{4}}$ can be used to determine the required order of approximation for a specified error. Here σ is the standard deviation of the Gaussian function (the width of the ion beam in radii of the circle), and n is the order of the approximating set. While the shape of the mirror can be arbitrarily complex, the formula (16) makes it possible to use all the possibilities of the ion beam without creating an excessively detailed map of the surface.

When the requirements for lateral resolution are high, it is necessary to use a very large number of Zernike polynomials for approximation, as this number increases as the square of the resolution ability. The higher the polynomial order is, the more sensitive are the coefficients of expansion to errors in the original data. The time taken to calculate the coefficients also becomes considerable. Thus, to work with a wide range of lateral scales, we recommend using other methods of approximating the surface that are better suited to this task.

Appendix

In this appendix, we express the PSD function in terms of the Zernike expansion coefficients of the relief. Let us consider the surface to be restricted by domain S (its area is also denoted by S), out of this domain the relief $W (\vec{r}) \equiv 0$ . By definition,

P S D_{2 D} (\vec{ν}) = \iint C (\vec{ρ}) e^{2 π i \vec{ν} \vec{ρ}} d^{2} \vec{ρ},

where C (\vec{ρ}) = \frac{1}{S} \iint W (\vec{r} + \vec{ρ}) W (\vec{r}) d^{2} \vec{r}

is the autocorrelation function of the surface and the integration is performed over the whole space. With this definition, it is clear that

C (0) = σ^{2}

, where σ is the RMS deviation. Since the correlation function is expressed in terms of PSD by the inverse Fourier transform

C (0) = σ^{2} = \iint P S D_{2 D} (\vec{ν}) e^{- 2 π i \vec{ν} * 0} d^{2} \vec{ν} = \iint P S D_{2 D} (\vec{ν}) d^{2} \vec{ν},

the integral relationship between RMS and PSD is determined. Now we express the PSD as the Fourier transform of the relief and for this we substitute the autocorrelation function in the integral:

P S D_{2 D} (\vec{ν}) = \iint C (\vec{ρ}) e^{2 π i \vec{ν} \vec{ρ}} d^{2} \vec{ρ} = \iint {\frac{1}{S} \iint W (\vec{r} + \vec{ρ}) W (\vec{r}) d^{2} \vec{r}} e^{2 π i \vec{ν} \vec{ρ}} d^{2} \vec{ρ} .

We change the order of integration and use the fact that the integrals are infinite:

\begin{array}{l} P S D_{2 D} (\vec{ν}) = \frac{1}{S} \iint {\iint W (\vec{r} + \vec{ρ}) e^{2 π i \vec{ν} (\vec{r} + \vec{ρ})} d^{2} \vec{ρ}} W (\vec{r}) e^{- 2 π i \vec{ν} \vec{r}} d^{2} \vec{r} = \\ = \frac{1}{S} \iint {\iint W (\vec{r} + \vec{ρ}) e^{2 π i \vec{ν} (\vec{r} + \vec{ρ})} d^{2} (\vec{ρ} + \vec{r})} W (\vec{r}) e^{- 2 π i \vec{ν} \vec{r}} d^{2} \vec{r} = \\ = \frac{1}{S} F (\vec{ν}) \iint W (\vec{r}) e^{- 2 π i \vec{ν} \vec{r}} d^{2} \vec{r} = \frac{1}{S} F (\vec{ν}) F^{*} (\vec{ν}) = \frac{1}{S} {| F (\vec{ν}) |}^{2}, \end{array}

where F (\vec{ν}) = \iint W (\vec{r}) e^{2 π i \vec{ν} \vec{r}} d^{2} \vec{r}

is the Fourier transform of the relief and the asterisk means complex conjugation.

Suppose now that the area S is a circle of unitary radius and the relief is represented as a set of Zernike polynomials:

W (\vec{r}) = \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} Z_{n}^{m} (r, θ) .

Then

F (\vec{ν}) = \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} \iint_{r \leq 1} Z_{n}^{m} (r, θ) e^{2 π i ν r} d^{2} r .

Take the integral

\iint_{r \leq 1} Z_{n}^{m} (r, θ) e^{2 π i \vec{ν} \vec{r}} d^{2} \vec{r}

in polar coordinates.

\iint_{r \leq 1} Z_{n}^{m} (r, θ) e^{2 π i \vec{ν} \vec{r}} d^{2} \vec{r} = \sqrt{\frac{2 (n + 1)}{π (1 + δ_{m}^{0})}} \iint_{r \leq 1} R_{n}^{m} (r) Θ^{m} (θ) e^{2 π i ν r \cos (θ - φ)} d^{2} \vec{r},

where

φ

is the angular coordinate of

\vec{ν}

and

ν = | \vec{ν} |

. Using the relation [18] (formulas 9.1.44 and 9.1.45),

\cos (z \cos θ) = J_{0} (z) + 2 \sum_{k = 1}^{\infty} {(- 1)}^{k} J_{2 k} (z) \cos 2 k θ,

\sin (z \cos θ) = 2 \sum_{k = 0}^{\infty} {(- 1)}^{k} J_{2 k + 1} (z) \cos (2 k + 1) θ,

where

J_{n}

is the Bessel function of the first kind of order n, we obtain

\int_{0}^{2 π} Θ^{m} (θ) e^{2 π i ν r \cos (θ - φ)} d θ = 2 π {(- 1)}^{| m | / 2} J_{| m |} (2 π ν r) Θ^{m} (φ) .

Now, using the relation \int_{0}^{1} R_{n}^{m} (r) J_{| m |} (k r) r d r = {(- 1)}^{\frac{n - | m |}{2}} \frac{J_{n + 1} (k)}{k}

(see [1], p. 525), we obtain

\iint_{r \leq 1} Z_{n}^{m} (r, θ) e^{2 π i \vec{ν} \vec{r}} d^{2} \vec{r} = 2 π \sqrt{\frac{2 (n + 1)}{π (1 + δ_{m}^{0})}} {(- 1)}^{n / 2} \frac{J_{n + 1} (2 π ν)}{2 π ν} Θ^{m} (φ)

\begin{array}{l} F (\vec{ν}) = 2 π \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} \sqrt{\frac{2 (n + 1)}{π (1 + δ_{m}^{0})}} (^{- 1) n / 2} \frac{J_{n + 1} (2 π ν)}{2 π ν} Θ^{m} (φ) = \\ = 2 π \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} U_{n} (ν) \frac{Θ^{m} (φ)}{\sqrt{π (1 + δ_{m}^{0})}} . \end{array}

Here we have introduced the notation

U_{n} (ν) = {(- 1)}^{n / 2} \sqrt{2 (n + 1)} \frac{J_{n + 1} (2 π ν)}{2 π ν} .

So, PSD is expressed in terms of Zernike expansion coefficients as follows:

P S D_{2 D} (ν, φ) = \frac{4 π^{2}}{S} {| \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} U_{n} (ν) \frac{Θ^{m} (φ)}{\sqrt{π (1 + δ_{m}^{0})}} |}^{2} .

Let us simplify this expression. To begin we make an assumption about surface isotropy. This allows us to use PSD averaged over all directions:

$P S D_{2 D} (ν, φ) \to P S D_{2 D} (ν) = \frac{1}{2 π} \int_{0}^{2 π} P S D_{2 D} (ν, φ) d φ$ . As a rule, surface roughness is characterized by this function that depends only on the modulus of the spatial frequency. The method of practical computation of $P S D_{2 D} (ν)$ in the case where the relief is set in a square grid is given in [19].

\begin{array}{l} P S D_{2 D} (ν) = \frac{4 π^{2}}{S} \sum_{n = 0}^{\infty} \sum_{n' = 0}^{\infty} \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} \sum_{\begin{array}{l} m' = - n' \\ m' + = 2 \end{array}}^{n'} c_{n m} c_{n' m'} U_{n} (ν) {(U_{n'} (ν))}^{*} \times \\ \times \frac{1}{π \sqrt{(1 + δ_{m'}^{0}) (1 + δ_{m}^{0})}} \frac{1}{2 π} \int_{0}^{2 π} Θ^{m} (φ) Θ^{m'} (φ) d φ . \end{array}

The orthogonality of trigonometric functions

Θ^{m} (φ)

implies that

\frac{1}{π \sqrt{(1 + δ_{m'}^{0}) (1 + δ_{m}^{0})}} \int_{0}^{2 π} Θ^{m} (φ) Θ^{m'} (φ) d φ = δ_{m}^{m'}

. Now m is contained only in the coefficients of expansion

c_{n m}

and

c_{n' m}

. We separate out the summation over m:

B_{n}^{n'} = \sum_{\begin{array}{l} m = - \min (n, n') \\ m + = 2 \end{array}}^{\min (n, n')} c_{n m} c_{n' m}

. Since the expansion coefficients are real,

B_{n}^{n'}

is real too.

Up to this moment, the formula for the spectral power density is as follows:

P S D_{2 D} (ν) = \frac{2 π}{S} \sum_{n = 0}^{\infty} \sum_{n' = 0}^{\infty} B_{n}^{n'} U_{n} (ν) {(U_{n'} (ν))}^{*} .

Let us continue the simplification.

\begin{matrix} \sum_{n = 0}^{\infty} \sum_{n' = 0}^{\infty} B_{n}^{n'} U_{n} (ν) {(U_{n'} (ν))}^{*} = \sum_{n = 0}^{\infty} B_{n}^{n} U_{n} (ν) {(U_{n} (ν))}^{*} + \sum_{n = 0}^{\infty} \sum_{n' > n}^{\infty} B_{n}^{n'} U_{n} (ν) {(U_{n'} (ν))}^{*} + \\ + \sum_{n' = 0}^{\infty} \sum_{n > n'}^{\infty} B_{n}^{n'} U_{n} (ν) {(U_{n'} (ν))}^{*} = \sum_{n = 0}^{\infty} B_{n}^{n} U_{n} (ν) {(U_{n} (ν))}^{*} + \sum_{n = 0}^{\infty} \sum_{n' > n}^{\infty} B_{n}^{n'} U_{n} (ν) {(U_{n'} (ν))}^{*} + \\ + \sum_{n = 0}^{\infty} \sum_{n' > n}^{\infty} B_{n'}^{n} {(U_{n} (ν))}^{*} U_{n'} (ν) = \sum_{n = 0}^{\infty} B_{n}^{n} U_{n} (ν) {(U_{n} (ν))}^{*} + 2 Re \sum_{n = 0}^{\infty} \sum_{n' > n}^{\infty} B_{n}^{n'} U_{n} (ν) {(U_{n'} (ν))}^{*} . \end{matrix}

as

B_{n}^{n'} = B_{n'}^{n}

. Functions

U_{n} (ν) {(U_{n'} (ν))}^{*}

are purely real if n and n’ are of the same parity and purely imaginary in the opposite case. Imaginary terms can be discarded from the sum (since their real part = 0) and we get the final expression:

P S D_{2 D} (ν) = \frac{2 π}{S} \sum_{n = 0}^{\infty} \sum_{\begin{array}{l} n' = n \\ n' + = 2 \end{array}}^{\infty} (2 - δ_{n}^{n'}) B_{n}^{n'} U_{n} (ν) {(U_{n'} (ν))}^{*},

Where

B_{n}^{n'} = \sum_{\begin{array}{l} m = - n \\ m + = 2 \end{array}}^{n} c_{n m} c_{n' m}

,

U_{n} (ν) = {(- 1)}^{n / 2} \sqrt{2 (n + 1)} \frac{J_{n + 1} (2 π ν)}{2 π ν}

,

c_{n m}

are the Zernike expansion coefficients,

S

is the area of the unit circle, and

J_{n}

is the Bessel function of order n.

Acknowledgments

This work was supported by Russian Foundation for Basic Research, grants 13-02-00377, 14-02-00549 and 15-42-02139.

References and links

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7. B. Honarvar Shakibaei and R. Paramesran, “Recursive formula to compute Zernike radial polynomials,” Opt. Lett. 38(14), 2487–2489 (2013). [CrossRef] [PubMed]

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9. G.-M. Dai, “Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,” Opt. Lett. 31(4), 501–503 (2006). [CrossRef] [PubMed]

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11. M. P. Rimmer and J. C. Wyant, “Evaluation of large aberrations using a lateral-shear interferometer having variable shear,” Appl. Opt. 14(1), 142–150 (1975). [CrossRef] [PubMed]

12. X. Hou, F. Wu, L. Yang, and Q. Chen, “Experimental study on measurement of aspheric surface shape with complementary annular subaperture interferometric method,” Opt. Express 15(20), 12890–12899 (2007). [CrossRef] [PubMed]

13. C. J. Progler and A. K. K. Wong, “Zernike coefficients: are they really enough?” Proc. SPIE 4000, 40–52 (2000). [CrossRef]

14. C. Bond, P. Fulda, L. Carbone, K. Kokeyama, and A. Freise, “Higher order Laguerre-Gauss mode degeneracy in realistic, high finesse cavities,” Phys. Rev. D Part. Fields Gravit. Cosmol. 84(10), 102002 (2011). [CrossRef]

15. D. Martínez-Galarce, R. Soufli, D. L. Windt, M. Bruner, E. Gullikson, S. Khatri, E. Spiller, J. C. Robinson, S. Baker, and E. Prast, “Multisegmented, multilayer-coated mirrors for the Solar Ultraviolet Imager,” Opt. Eng. 52(9), 095102 (2013). [CrossRef]

16. E. Sidick, “Power spectral density specification and analysis of large optical surfaces,” Proc. SPIE 7390, 73900L (2009). [CrossRef]

17. M. V. Svechnikov, N. I. Chkhalo, M. N. Toropov, N. N. Salashchenko, and M. V. Zorina, “Application of point diffraction interferometry for middle spatial frequency roughness detection,” Opt. Lett. 40(2), 159–162 (2015). [CrossRef] [PubMed]

18. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Formulas, Graphs, and Mathematical Tables (Dover, 1972).

19. B. N. J. Persson, O. Albohr, U. Tartaglino, A. I. Volokitin, and E. Tosatti, “On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion,” J. Phys. Condens. Matter 17(1), R1–R62 (2005). [CrossRef] [PubMed]

	σ = 0.32		σ = 0.16		σ = 0.08		σ = 0.04		σ = 0.02
Polynomial order	Z3	Z12	Z7	Z25	Z12	Z50	Z25	Z100	Z50	Z200
Centred peak	63.8%	0.9%	65.8%	1.4%	72.9%	1.3%	76.3%	1.6%	76.3%	1.7%
Peak is shifted to 0.75 from centre	49.2%	0.8%	51.3%	0.7%	65.4%	0.7%	66.4%	0.8%	67.8%	0.9%
$\frac{R M S_{r e s i d u a l} (c e n t e r e d)}{R M S_{r e s i d u a l} (s h i f t e d)}$	1.3	1.125	1.28	2	1.11	1.85	1.15	2	1.125	1.89

	σ = 0.08			σ = 0.04			σ = 0.02
Polynomial order	Z7	Z20	Z40	Z14	Z40	Z80	Z28	Z80	Z160
Centred peak	90.2%	45.9%	5.9%	90.2%	49.3%	6.8%	91.4%	51%	7.2%
Shifted peak	84%	36%	3.6%	86.35%	38.3%	3.9%	87.7%	39.5%	4.1%
Centred stripe	72.2%	27.9%	2.8%	72.2%	30.4%	3.2%	74%	31.8%	3.5%
Shifted stripe	53.6%	10.4%	0.17%	58.3%	11%	0.13%	59.9%	11.4%	0.13%
Small ring	65.7%	27.1%	2%	70.4%	26.9%	2.3%	71.5%	28.9%	3.1%
Large ring	58.6%	10.6%	0.17%	56.1%	10.2%	0.13%	61.3%	11.8%	0.1%

	σ = 0.32		σ = 0.16		σ = 0.08		σ = 0.04		σ = 0.02
Polynomial order	Z3	Z12	Z7	Z25	Z12	Z50	Z25	Z100	Z50	Z200
Centred peak	63.8%	0.9%	65.8%	1.4%	72.9%	1.3%	76.3%	1.6%	76.3%	1.7%
Peak is shifted to 0.75 from centre	49.2%	0.8%	51.3%	0.7%	65.4%	0.7%	66.4%	0.8%	67.8%	0.9%
$\frac{R M S_{r e s i d u a l} (c e n t e r e d)}{R M S_{r e s i d u a l} (s h i f t e d)}$	1.3	1.125	1.28	2	1.11	1.85	1.15	2	1.125	1.89

	σ = 0.08			σ = 0.04			σ = 0.02
Polynomial order	Z7	Z20	Z40	Z14	Z40	Z80	Z28	Z80	Z160
Centred peak	90.2%	45.9%	5.9%	90.2%	49.3%	6.8%	91.4%	51%	7.2%
Shifted peak	84%	36%	3.6%	86.35%	38.3%	3.9%	87.7%	39.5%	4.1%
Centred stripe	72.2%	27.9%	2.8%	72.2%	30.4%	3.2%	74%	31.8%	3.5%
Shifted stripe	53.6%	10.4%	0.17%	58.3%	11%	0.13%	59.9%	11.4%	0.13%
Small ring	65.7%	27.1%	2%	70.4%	26.9%	2.3%	71.5%	28.9%	3.1%
Large ring	58.6%	10.6%	0.17%	56.1%	10.2%	0.13%	61.3%	11.8%	0.1%

Resolving capacity of the circular Zernike polynomials

Abstract

1. Introduction

2. Definitions and notation

3. Expansion of test functions

3.1 Gaussian peak at the centre

3.2 Gaussian peak at the edge

3.3 Rings and stripes with a Gaussian profile

4. Description of arbitrary surface

5. The dependence of the approximation errors on the order of approximation

5.1 Gaussian peak

5.2 The trigonometric peak

5.3 Rectangular peak

6. Conclusion

Appendix

Acknowledgments

References and links

Cited By

Figures (11)

Tables (2)

Equations (42)

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