Analytical Jacobian and its application to tilted-wave interferometry

Ines Fortmeier; Manuel Stavridis; Axel Wiegmann; Michael Schulz; Wolfgang Osten; Clemens Elster

doi:10.1364/OE.22.021313

1. Introduction

Aspherical or freeform surfaces enable the correction of several wavefront aberrations using a single surface and are a powerful tool in optical design [1, 2]. However, to achieve good performance, highly accurate fabrication is needed, which is limited by the capabilities of metrology systems intended for the measurement of such surfaces. The most common measurement technique to test such surfaces is to use interferometric measurement systems and correct the usually spherical test wavefront to the null test condition by a compensator element, usually a computer generated hologram (CGH) [2–4]. However, the highly accurate testing of aspheres or freeform surfaces by such compensator elements is very costly and inflexible because every surface requires its particularly designed and fabricated CGH. Beyond this, CGH production errors have to be taken into account for a realistic uncertainty estimation. Furthermore, locally (almost) null test configurations such as stitching [5] or scanning along the optical axis [6] exist. For these systems the null condition is valid for sub apertures of the specimen. To cover the whole specimen, measurement system and/or specimen have to be moved relative to each other. In recent years, several measurement systems working in the non-null test configuration were proposed [7–14]. These measurement techniques do not use a compensator element. Thus, the measurement configuration and the analysis method have to handle high deviations from the null test condition, often leading to vignetting effects, subsampling effects or retrace errors.

The TWI [7] also works in the non-null test configuration. This technique combines a special interferometric measurement setup with numerical simulations and mathematical evaluation procedures to overcome the mentioned challenges introduced by the violation of the null-test condition. The basic idea is that small changes of the specimen induce characteristic changes in the optical path lengths (OPLs) between a point source and a detector pixel. This relationship is approximated by a linear model which is utilized to reconstruct the specimen from several measured OPLs. The linear model is established by simulating the OPLs for variations of an approximation of the specimen under test. The approximation of the specimen under test may not be highly accurate which affects the linear model established on its basis. Therefore, the method is applied iteratively. First sensitivity analyses have proven the TWI to be a very promising technique for the flexible and accurate measurement of aspherical surfaces [15, 16]. Furthermore, effects of alignment on the measurement accuracy of the TWI have been investigated and were verified by experimental results [17, 18].

The linear model used by the TWI is represented by the Jacobian matrix consisting of the partial derivatives of the OPLs with respect to the parameters of the specimen. Typically, the specimen is parametrized in terms of Zernike polynomials whose coefficients then serve as the parameters of the specimen. Often, the entries of the Jacobian matrices are approximated by carrying out a numerical differentiation [19]. Numerical differentiation is frequently used for calculating the derivative of OPLs with respect to system design parameters of complex optical systems, cf., e.g., [20]. However, the accuracy achieved may depend critically on the chosen step length [15, 20]. Furthermore, in order to calculate the Jacobian with respect to p parameters, p additional evaluations of the underlying function are required at least. For the TWI each of these function evaluations amounts to one simulation of the whole experiment, determining (numerically) the OPLs for all source-pixel pairs. A typical calculation time for this is in the order of some seconds (13,350 rays). Since the number of the Zernike coefficients of the specimen is quite high (more than 100 usually), a single evaluation of the Jacobian matrix by numerical differentiation would take up to several minutes and dominates by far the calculation time for the TWI. This is not only a severe drawback for practical applications, it also limits systematic studies required to achieve an optimal design of the TWI.

The analytical determination of Jacobian matrices for general optical systems has long been researched [21–25]. Usually ray-tracing is used to follow the path of a ray from a point source through an optical system. If a parameter of the optical system is perturbed the ray’s endpoint after passing the system will also be perturbed. In these cases, the Jacobian matrix calculation is very complex.

In interferometric applications, the OPLs are measured between fixed start- and endpoints (source-pixel pairs). The OPLs of these source-pixel pairs represent a point characteristic [26] of the optical system. In order to determine the OPLs of these rays by simulation, a ray-aiming process is necessary, to find the ray path between each source-pixel pair. A ray-aiming process consists of several ray-tracing steps: The start direction of the ray at the point source is optimized until the ray arrives at the specified pixel. For such applications, the influence of perturbations of the optical system on the OPL of the source-pixel pairs is of interest. The analytical Jacobian matrix calculation for these point characteristic functions can be simplified dramatically by applying common perturbation theory [26–28].

In this paper we derive an analytic expression for Jacobian matrices of interferometer systems that allows their numerical evaluation using a single calculation of the OPLs of the interferometer system. We do this by applying optical perturbation methods and by extending previous work on the derivation of analytical derivatives of ray path data in optical systems [26–28]. Stone [27] and Rimmer [28] derived such wavefront aberrations due to small parameter perturbations of a single surface of an optical system. Their methods can be applied to determine partial derivatives of OPLs with respect to arbitrary system design parameters such as the position and tilt of a surface, a parameter of a surface parametrization such as a Zernike coefficient, a lens radius, or an aspherical coefficient. In extension to this, we determine the partial derivatives also for perturbations of complete lens systems, consisting of several surfaces, e.g. the tilt of an objective lens. All calculations can be carried out using the knowledge of the ray path through the system only. Thus, after a single simulation of the ray paths all partial derivatives can be calculated immediately. In this way, the Jacobian matrix is calculated exactly and the overall calculation time of the interferometer can be reduced significantly. Another advantage of the analytical Jacobian is its application in the calibration step of the TWI method.

Perturbation theory was already mentioned by Garbusi [8] in order to justify the calibration and reconstruction principle of the TWI. Nevertheless these methods have not been applied for the calculation of the Jacobian matrices. Numerical differentiation was used instead. We show that computation times can be short when applying the methods proposed by Stone and Rimmer to the tilted-wave interferometer. This is an important achievement for the TWI development, since the time between the execution of a measurement and the display of the result is reduced significantly. Therefore, the proposed method leads to an important benefit for practical applications. Furthermore, the fast calculation of the Jacobian matrices is essential for systematic studies of the optical design and measurement setup. We demonstrate that the Jacobian matrix of the surface reconstruction can be used as a tool to determine an optimal specimen position for its reconstruction.

The paper is organized as follows. The setup and the idea of the tilted-wave interferometer are presented in section 2. For better comprehension of the analytical Jacobian calculation method, common formulas to calculate the Jacobian matrix entries from optical ray path data are briefly introduced in section 3. In section 4, we apply the analytical Jacobian to the application of the TWI and present some results. Finally we draw some conclusions.

2. The tilted-wavefront interferometer (TWI)

The setup of the TWI is based on a Twyman-Green interferometer and is shown in Fig. 1(a). A two-dimensional point source array is used for the illumination of the specimen under test [7]. The point-source array consists of 17 times 17 point sources that are arranged in the focal plane of the collimator. In the measurement setup these point sources are realized by a special component consisting of a combination of a micro lens array and a pinhole array. The wavefronts originating from these point sources lead to spherical waves after the objective respectively plane waves after the collimator, which are tilted differently to each other. Therefore, the specimen is illuminated simultaneously by several test wavefronts, each tilted in a different way (see Fig. 1(b)). For each point on the surface of the specimen at least one of these wavefronts will compensate its local slope in such a way, that the reflected wavefront is almost parallel to the reference wavefront. The application of multiple tilted wavefronts enables a measurement of the entire specimen without vignetting effects [7]. An additional beam stop in the Fourier plane of the imaging optics limits the fringe density at the detector and thus avoids subsampling effects. The setup is used to measure the OPLs of a chosen sample of source-pixel combinations for the surface under test. From the differences of these OPLs to OPLs simulated for a start solution of the topography under test, the deviation of the surface from this start solution is then reconstructed.

Fig. 1 a) Optical design of the TWI. b) Two bundles of rays, each starting at a different point source, are traced through the optical system of the TWI. Due to the different tilts of the wavefronts leaving the objective lens, different parts of the specimen under test can be measured. The rays of the reference arm are not shown here.

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2.1. Surface reconstruction

In order to reconstruct the low spatial frequency errors of the surface under test, the deviation of the surface from an approximation T₀ of the specimen is parametrized. To this end, a Zernike polynomial function is employed,

z (x, y) = T_{0} (x, y) + \sum_{j = 1}^{m} c_{j} Z_{j} (x, y) .

The approximation T₀ of the specimen may equal the design function of the surface under test, or an intermediate solution in the iterative application of the TWI method. The c_j represent the coefficients of the Zernike polynomial function that have to be determined in the reconstruction process, and Z_j are corresponding Cartesian Zernike polynomials. The OPLs L = (L₁, L₂,...)^T are measured for the surface under test and compared to those OPLs L₀ that are calculated numerically assuming the surface under test to equal T₀. From the differences ΔL = L − L₀, the Zernike coefficients c = (c₁, c₂,...)^T from (1) are determined by solving the inverse problem

Δ L = Jc,

where J denotes the Jacobian matrix

J_{i j} = \frac{\partial L_{i}}{\partial c_{j}} .

The entries of the Jacobian matrix J consist of the partial derivatives of the complete OPLs for each source-pixel combination i with respect to the parameters c that describe the deviation of the specimen from T₀.

The method is usually applied iteratively [7, 8], where for the first iteration the design topography is used for T₀. In subsequent iterations the current solution is taken for T₀.

The steps for the reconstruction of a specimen under test using the TWI can be summarized as follows:

Simulate OPLs L₀ between chosen source-pixel pairs for the approximated topography of the specimen under test T₀.
Measure the OPLs L between the chosen source-pixel pairs for the real specimen under test.
Determine the Jacobian matrix J consisting of the partial derivatives of the OPLs with regard to small perturbations of the surface under test.
Solve the inverse problem Δc = J⁻¹(L − L₀).
To account for non-linearities of the system, use the solution as a new design specimen $T_{0_{k + 1}} (x, y) = T_{0_{k}} (x, y) + \sum_{j = 1}^{m} c_{j_{k}} Z_{j} (x, y)$ , sub index k indicating the iteration step, and repeat steps 1 to 4 until the residual is small.

2.2. Jacobian matrix from numerical differentiation

In order to determine the entries of the Jacobian matrix from numerical differentiation, forward numerical differential quotients are used according to

J_{i j} = \frac{L_{i} (c_{j} + Δ c_{j}) - L_{i} (c_{j})}{Δ c_{j}} .

In this way, the calculation of the Jacobian matrix requires m simulations of the complete experiment (including all source-pixel combinations), where m is the number of Zernike coefficients c_j. For highly-accurate reconstructions a large number of Zernike coefficients is needed (e.g. typically more than 100), which implies large calculation times and makes updating the Jacobian matrix in each iteration inconvenient. Furthermore, systematic optimization of the geometry and parameters of the setup cannot be carried out within reasonable times. Another disadvantage of the numerical approximation Eqs. (4) to the Jacobian is that the step sizes Δc_j, j = 1, 2,... have to be chosen carefully in order to avoid errors, cf. [15].

3. Analytical Jacobian of OPLs

In this manuscript we apply optical perturbation theory to the TWI and give some examples of the resulting benefits. We first repeat the main ideas of the theory and then determine, in extension to this, the partial derivatives also for perturbations of complete lens systems, consisting of several surfaces, e.g. the tilt of an objective lens. Although some of those results are not applied in this paper, they are given for completeness of the theory and will be helpful in other cases, e.g. to perform a sensitivity analysis of the complete system.

Therefore, in this section the relation between partial derivatives of OPLs with respect to different parameters of the optical system and nominal ray aiming data are presented. An OPL always refers to the optical length of a path through the system, connecting two points of the optical system. Therefore, such a quantity is also called a point characteristic function [26]. The optical path through an optical system is determined by Fermat’s principle, which states that the OPL connecting two points of an arbitrary optical system is stationary [29]. This is used for determining the derivatives of the OPLs with respect to any perturbation of the system. To calculate the derivatives the optical path data (intersection points, direction vectors) are needed which can be determined by ray-tracing and ray-aiming through the optical system. Once this knowledge is available the sought derivatives of the OPLs can be calculated analytically in terms of the formulas presented in the subsequent subsections.

3.1. Translation of a single surface

The OPL change ΔL between two points of an optical system with regard to a small shift Δs^T = (Δx, Δy, Δz) of the position of a single surface of the optical system can be calculated from the ray-aiming data of the nominal system, by applying basic perturbation theory [26], as shown e.g. by Stone [27]:

Δ L = n_{r} (e_{x} Δ x + e_{y} Δ y + e_{z} Δ z) - n_{r}^{'} (e_{x}^{'} Δ x + e_{y}^{'} Δ y + e_{z}^{'} Δ z),

where e^T = (e_x, e_y, e_z) and e′^T = (e′_x, e′_y, e′_z) are the corresponding direction vectors of the ray arriving at the surface and accordingly leaving the surface and n_r and n′_r represent the refraction indices of the media corresponding to the rays (see Fig. 2(a)). Therefore, the partial derivatives of the OPL with respect to a shift of a single surface of the optical system can be expressed by

(\begin{matrix} \frac{\partial L}{\partial x} \\ \frac{\partial L}{\partial y} \\ \frac{\partial L}{\partial z} \end{matrix}) = (\begin{matrix} n_{r} e_{x} - n_{r}^{'} e_{x}^{'} \\ n_{r} e_{y} - n_{r}^{'} e_{y}^{'} \\ n_{r} e_{z} - n_{r}^{'} e_{z}^{'} \end{matrix}) .

Fig. 2 (a) Schematic sketch of a small shift of a surface: Small perturbations Δs induce a path length change of ΔL = n_r(e_xΔx + e_yΔy + e_zΔz) − n′_r(e′_xΔx + e′_yΔy + e′_zΔz), see also [26], [27]. (b) Tilt of a single surface: Definition of the rotation axes. (c) Schematic sketch of a ray path through an optical system consisting of several surfaces. Since a translation of the element group leads to the same perturbation of all surfaces of the element, the resulting OPL perturbation of the paths inside the element group cancel each other out.

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3.2. Partial derivatives with respect to arbitrary system design parameters

In those cases where the optical surfaces can be described by spatial functions of the form z = f (x, y, p_k), e.g. for spheres, aspheres, planes or polynomial functions, the partial derivatives of the complete OPL with respect to any system parameter p_k, can be calculated by applying the chain rule

\frac{\partial L}{\partial p_{k}} = \frac{\partial L}{\partial z} \frac{\partial z}{\partial p_{k}},

where the partial derivative

\frac{\partial L}{\partial z}

is considered at the intersection point of the ray with the surface whose parameter p_k is perturbed. The chain rule can be applied provided that the description of the surface is of the form z = f (x, y, p_k) and that the function is continuously differentiable. Therefore, the analytical calculation of the partial derivative of the OPL with respect to any parameter of a surface such as the radius, an aspherical parameter, or a Zernike coefficient, is straightforward. We note that partial derivatives of the OPL with respect to arbitrary surface parameters have already been derived by Stone using a first order Taylor series approximation [27].

3.3. Tilt of a single surface

The partial derivative of the complete OPL with respect to a tilt of a surface requires special treatment. A tilt induces perturbations that depend on the distance of the ray from the center of rotation (see Fig. 2(b)). Therefore, the perturbation has to be projected onto the normal of the surface at the intersection point of the ray [28]. The aberration of the complete OPL induced by the tilt of a single surface can be expressed by [28]

Δ L = Δ s^{T} n (n_{r} e^{T} n - n_{r}^{'} {e^{'}}^{T} n),

where n is the normal to the surface at the intersection point of the ray, s is the perturbation of the surface resulting from an infinitesimal tilt along one of the coordinate axes, e and e′ are the corresponding direction vectors of the ray arriving and accordingly leaving the surface, and n_r and n′_r represent the refraction indices of the media. The partial derivative of the OPL with respect to a tilt of a surface along the x-axis is then given by

\frac{\partial L}{\partial α} = ({(\frac{\partial R_{α}^{- 1}}{\partial α} \cdot p)}^{T} n) (n_{r} e^{T} n - n_{r}^{'} {e^{'}}^{T} n),

where p is the point of intersection of the nominal ray with the surface that is tilted, R_α is the rotation matrix given for a rotation about the x-axis, n is the normal to the surface at the intersection point of the ray, e and e′ are the corresponding direction vectors of the ray arriving and accordingly leaving the surface, and n_r and n′_r represent the refraction indices of the media. Rotations with respect to a tilt angle β or γ around the y-axis or the z-axis, respectively, are treated analogously. We note that partial derivatives of the OPL with respect to a tilt of a single surface have already been derived by Rimmer [28].

3.4. Perturbation of an optical element group

We apply the above stated basic equations of perturbation theory to derive partial derivations of the OPL with respect to a translation or tilt of an optical element group consisting of several surfaces. First, we consider the translation of an optical element group. All surfaces of the shifted element group are translated by the same shift Δs. Hence, the perturbations of the paths inside the optical element cancel each other out so that only the optical path perturbations of the arriving and leaving part of the ray path have to be taken into account, see Fig. 2(c). This simplification can be deduced from Buchdahl’s basic perturbation theory [26]. Therefore, the partial derivative of the complete OPL with respect to the translation of the element can be calculated using Eqs. (6), where n_r is the refraction index of the medium corresponding to the ray arriving at the element group, n′_r is the refraction index of the medium corresponding to the ray leaving the element group, and e and e′ are the direction vectors of the rays arriving and leaving the element group.

In contrast, the different perturbations at the surfaces inside the element do not cancel each other out if the element group is tilted. When the tilt of an element group consisting of N elements is considered, the perturbations induced by each surface l of the element group have to be summed up, leading to

\frac{\partial L}{\partial α} = \sum_{l}^{N} {(\frac{\partial L}{\partial α})}_{l},

where the subindex l indicates the surface l of the element group.

3.5. Perturbations of elements with multiple ray passage

In interferometric measurement setups, the complete optical path often passes through several optical elements twice, once from the source to the surface under test and once reflected from the surface under test to the detector. In order to calculate the partial derivatives of the complete OPLs with respect to perturbations in the position or tilt of an element or element group through which rays pass more than once, the complete optical path can be divided into sub-paths. Since the complete optical path between two points is composed of all sub-paths, the partial derivative of the complete path is also composed of the sum of the partial derivatives of all sub-paths. In dividing the complete optical path into N sub-paths of length L_i and considering the partial derivative with respect to the parameter p_k of the corresponding element, the partial derivative of the complete path length L can be expressed by

\frac{\partial L}{\partial p_{k}} = \sum_{i = 1}^{N} \frac{\partial L_{i}}{\partial p_{k}} .

4. Application to the TWI

In this section, the calculation of partial derivations of the OPLs with respect to any system parameter is applied by using the knowledge of the nominal optical ray paths.

4.1. Jacobian matrix from nominal ray path data

Using the optical ray path data of the system all the Jacobian matrix entries (Eqs. (4)) can be calculated by applying Eqs. (6) and Eqs. (7) leading to

\frac{\partial L_{i}}{\partial c_{j}} = \frac{\partial L_{i}}{\partial z} \frac{\partial z}{\partial c_{j}} .

The first term is the partial derivation of the OPL with respect to a shift of the surface along the optical axis z, see section 3. The second term of Eqs. (12) represents the derivative of the surface with respect to the coefficient of the parameterization given by the corresponding Zernike polynomial

\frac{\partial z}{\partial c_{j}} = Z_{j} (x, y),

cf. Eqs. (1). Therefore, only one simulation of the experiment is necessary to calculate the complete Jacobian matrix.

4.2. Simulation environment and ray-tracing

The simulation of the TWI has been achieved by a simulation environment that was developed at PTB and implemented in MATLAB. The basic simulation environment has already been used earlier to successfully perform virtual experiments for different optical measurement systems [15, 16, 30]. Recently, ray-tracing and ray-aiming procedures were added so that optical systems can be modeled without using any commercial optical design software. The ray-tracing algorithm is realized as a sequential ray-tracing, tracing a ray through the optical system by maintaining the optical laws of reflection and refraction. Thus, all intermediate results, and in particular those of the ray-aiming process, are available and enable the application of the above-proposed formulas for the calculation of the required Jacobian matrix.

4.3. Results

In order to illustrate the use of the analytic Jacobian, we present results obtained for simulated TWI data. Figure 3 shows the specimen under test which was obtained by the topography given in Tab. 1 (design 1). An additional perturbation using 153 Zernike polynomials (see Fig. 4, left) was added to the design topography to simulate measurement data. The OPLs were calculated using 13,350 source-pixel combinations. From the ray-aiming data (direction vectors, intersection points at the surface under test) the entries of the Jacobian matrix were calculated applying Eqs. (12). Thus, the corresponding Jacobian matrix has a size of 13,350 rows and 153 columns (see Fig. 5, middle). For comparison, the Jacobian matrix entries were also determined using numerical differentiation (see Fig. 5, left), where the step size Δc_j was set to 10 nm. The OPL data of the simulated measurement were then perturbed by white Gaussian noise with an absolute standard deviation of 5 nm and used to reconstruct the specimen under test. The TWI reconstruction was carried out iteratively using 2 iterations. The Jacobian was calculated using the ray-aiming data and this was repeated in each iteration. Figure 4 (right) shows the remaining errors of the obtained reconstruction which are in the order of a few nanometers only. Note that we neglected specimen alignment errors and assumed a perfect calibrated optical TWI system (see section 4.4). Furthermore, we assumed that the OPLs can be determined in an absolute manner. In further improved scenarios, more iteration steps are expected.

Fig. 3 Profile of specimen under test (design surface 1).

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Table 1. Design parameters of the aspherical surfaces under test used for the simulations.

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Fig. 4 Deviation of the specimen from its design and residual of reconstruction result.

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Fig. 5 Comparison between the Jacobian matrices established by numerical differential quotients and by calculation from nominal ray path data.

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Next we show in Fig. 5 (right) a comparison of the Jacobian matrix entries (at the first iteration of the TWI reconstruction procedure) with that obtained by numerical differentiation. The entries have an rms deviation of 1.3 · 10⁻⁵, which confirms that the step size chosen in the numerical differentiation has been chosen appropriately. Note that this accuracy cannot be immediately confirmed without the analytical Jacobian matrix.

Finally, we illustrate the difference in computing time when calculating the Jacobian according to the analytical calculation scheme with respect to the numerical differentiation. Table 2 shows a comparison of the computing time for a simple reconstruction example. For this simulated case the gain is 2 orders of magnitude.

Table 2. Computing time for a simple example performed on a computational server for a typical TWI reconstruction problem (13,350 rays, 153 Zernike coefficients, design surface with parameters described in Tab. 1, no parallel jobs).

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4.4. TWI system calibration

Measured OPLs are also disturbed by alignment and production errors of the optical surfaces of the TWI setup. For these reasons also the model of the TWI in the simulations is calibrated to better reflect real measurements. To this end, measurements are performed with well-known specimens (e.g. calibration spheres) in different specimen positions. These measurements are then used to establish a correction function that shall account for the difference of the computer model and the real optical system. Again, the perturbation theory is used and this correction model is constructed with the help of Jacobian matrices [8]. During this calibration procedure many more source-pixel pairs are considered, so that each simulation of the experiment takes even more time than each simulation of the experiment in the reconstruction step. Furthermore, a very high number of parameters is used to correct the model (usually more than 1,000). Table 3 shows a comparison of the computing time for a simple TWI calibration example. In this case the difference in calculation time is 4 orders of magnitude.

Table 3. Computing time for a simple example performed on a computational server for a typical TWI calibration problem (63,152 rays, 1,368 parameters, surface under test: sphere with radius R = 10 mm, 259 sphere positions, no parallel jobs).

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4.5. Optimal positioning of the surface under test

The clearly reduced computing time for the analytical determination of the Jacobian matrices allows for systematic studies of the TWI method. For example, by applying methods of optimal design, the Jacobian matrix of the TWI reconstruction step can be used to find an optimal position of the surface under test in the test area. In this selected example the calculation of the Jacobian matrix for the TWI reconstruction, and therefore Eqs. (6) and Eqs. (7) of section 3, are applied. The Zernike coefficients c, that describe the deviation of the specimen under test from its design, are estimated from Eqs. (2) in a least-squares sense. Accounting for the measurement uncertainties for the measured OPL change ΔL, the expression

χ^{2} = {(J \hat{c} - Δ L)}^{T} V_{Δ L}^{- 1} (J \hat{c} - Δ L)

has to be minimized to solve the inverse problem, where ΔL is a set of measurement data. For the Zernike coefficients c, we then obtain the estimate ĉ. Assuming the measurements for ΔL to be uncorrelated with variances

σ_{Δ L_{i}}^{2}

, the covariance matrix V_ĉ of the obtained parameter estimate ĉ is given by

V_{\hat{c}} = {(J^{T} V_{Δ L}^{- 1} J)}^{- 1},

when

V_{Δ L} = diag (σ_{Δ L_{1}}^{2}, \dots, σ_{Δ L_{n}}^{2})

. In the homoscedastic case, i.e. for

σ_{Δ L_{1}}^{2} = \dots = σ_{Δ L_{n}}^{2} = σ

, we obtain

V_{\hat{c}} = σ^{2} {(J^{T} J)}^{- 1} .

The diagonal entries of V_ĉ are the squared uncertainties u²(ĉ_j) of the estimated Zernike coefficients. Minimizing the sum of them will lead to optimal specimen positions for the surface reconstruction according to the ’A-optimality’ criterion in optimal design of experiments [31]. In order to demonstrate the method, two different specimen designs with aspherical parameters given in Tab. 1 and a measured aperture of 44 mm (design surface 1) and 28.6 mm (design surface 2) were used. For this study, we again assumed a perfectly calibrated optical TWI system (see section 4.4) and supposed that the OPLs can be determined in an absolute manner.

The specimen position was changed along the optical axis z between ±3 mm in 61 steps, starting at a distance of 5 mm (design surface 1) and 20 mm (design surface 2) from the last surface of the objective lens (see Fig. 1). These start positions were approximated from the difference between the back focal length of the objective lens f and the best fit radius of the aspherical surface r_fit: z_start = f − r_fit. The resulting uncertainty matrix V_ĉ was determined for each position, assuming a standard deviation of the Gaussian noise in the measurement data of σ = 5 nm. The diagonal entries of the uncertainty matrix then correspond to the squared uncertainties of the Zernike coefficients c. The resulting uncertainties for each Zernike coefficient c_j are plotted in Fig. 6(a) for design surface 1 and in Fig. 6(b) for design surface 2. Each row of the plots corresponds to one measurement position. The positions are numbered from 1 to 61, corresponding to a deviation of −3 mm (position 1) and 3 mm (position 61) from the initial position of the surface under test. Each column corresponds to one Zernike coefficient. Therefore, the color of each entry displays the uncertainty of an estimated Zernike coefficient for a defined specimen position. Figures 6(a) and (b) show, that there exist positions where the uncertainties are smaller than for other specimen positions. There are certain Zernike coefficients that tend to have a higher uncertainty in some measurement positions. These Zernike coefficient seem to be very sensitive to the distribution of the measurement points at the specimen. Nevertheless, the uncertainty of the single coefficients amounts to less than a nanometer, which shows that the reconstruction works fairly well for the simulated cases. The structure of the estimated parameter uncertainties strongly depends on the design of the surface under test. Nevertheless, one can conclude that specimen positions at distances from the last surface of the objective lens smaller than z_start (rows 1–30 in Fig. 6) are beneficial for surface reconstruction.

Fig. 6 Optimal specimen positioning: The uncertainty values for each estimated parameter (column) of the specimen reconstruction are shown for different surface positions (row) for design surface 1 (a) and design surface 2 (b). The color of each entry displays the uncertainty of an estimated Zernike coefficient for a defined specimen position.

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5. Conclusions

The tilted-wave interferometer (TWI) is a novel measurement principle for the measurement of aspherical surfaces. The technique is promising but computationally very demanding. At present, the calculation of the required Jacobian matrices is a central task. The analytic calculation of the Jacobian matrix described here guarantees short computation times, and in this way fosters the practical application of this novel measurement principle. Furthermore, the short computation time allows for systematic studies of the TWI method, e.g. studies concerning the optimal experimental design, as shown in this paper. The described calculation of Jacobian matrices can also be applied for the sensitivity analysis of the OPLs with respect to arbitrary system parameters, such as position, alignment, tilt or radius of a single surface or lens system, and a polynomial or aspherical coefficient of a surface.

Acknowledgments

This work is part of the EMRP project IND 10 “Form metrology - Optical and tactile metrology for absolute form characterization”. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. Furthermore the authors would like to thank Goran Baer for fruitful discussions concerning tilted-wave interferometry.

References and links

1. G. Schulz, Aspherical Surfaces, (North-Holland Physics Publishing, Amsterdam, 1988), Chap. IV, pp. 349–415.

2. B. Braunecker, R. Hentschel, and H. J. Tiziani, eds., Advanced Optics Using Aspherical Elements, (SPIE Press, 2008), pp. 292–307.

3. B. Dörband and H. J. Tiziani, “Testing aspheric surfaces with computer-generated holograms: analysis of adjustment and shape errors,” Appl. Opt. 24, 2604–2611 (1985). [CrossRef] [PubMed]

4. C. Pruss, S. Reichelt, H. J. Tiziani, and W. Osten, “Computer-generated holograms in interferometric testing,” Optical Engineering , 43, 2534–2540 (2004). [CrossRef]

5. J. Fleig, P. Dumas, P. E. Murphy, and G. W. Forbes, “An automated subaperture stitching interferometer workstation for spherical and aspherical surfaces, ” Proc. SPIE 5188, 296–307 (2003). [CrossRef]

6. M. F. Kuechel, “Absolute measurement of rotationally symmetrical aspheric surfaces,” OSA Optical Fabrication and Testing, Rochester, OFTuB5 (2006). [CrossRef]

7. E. Garbusi, C. Pruss, and W. Osten, “Interferometer for precise and flexible asphere testing,” Opt. Lett. 33, 242973– 2975 (2008). [CrossRef]

8. E. Garbusi and W. Osten, “Perturbation methods in optics: application to the interferometric measurement of surfaces,” J. Opt. Soc. Am. A 26, 122538–2549 (2009). [CrossRef]

9. Y. Liu, G. Lawrence, and C. Koliopoulos, “Subaperture testing of aspheres with annular zones,” Appl. Opt. 27, 4504–4513 (1988). [CrossRef] [PubMed]

10. R. O. Grappinger and J. E. Greivenkamp, “Iterative reverse optimization procedure for calibration of aspheric wavefront measurements on a nonnull interferometer,” Appl. Opt. 43, 5152–5161 (2004). [CrossRef]

11. J. E. Greivenkamp and R. O. Grappinger, “Design of a nonnull interferometer for aspheric wave fronts,” Appl. Opt. 43, 5143–5151 (2004). [CrossRef] [PubMed]

12. C. Tian, Y. Yang, T. Wei, and Y. Zhuo, “Nonnull interferometer simulation for aspheric testing based on ray tracing,” Appl. Opt. 50, 3559–3569 (2011). [CrossRef] [PubMed]

13. D. Liu, Y. Yang, C. Tian, Y. Luo, and L. Wang, “Practical methods for retrace error correction in nonnull aspheric testing,” Opt. Express 17, 7025–7035 (2009). [CrossRef] [PubMed]

14. J. Pfund, N. Lindlein, and J. Schwider, “Nonnull testing of rotationally symmetric aspheres: a systematic error assessment,” Appl. Opt. 40, 439–446 (2001). [CrossRef]

15. I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Sensitivity analysis of tilted-wave interferometer asphere measurements using virtual experiments,” in Modeling Aspects in Optical Metrology IV, B. Bodermann, K. Frenner, and R. M. Silver, eds., Proc. SPIE8789, 878907 (2013). [CrossRef]

16. I. Fortmeier, M. Stavridis, A. Wiegmann, M. Schulz, G. Baer, C. Pruss, W. Osten, and C. Elster, “Results of a Sensitivity Analysis for the Tilted-Wave Interferometer,” in Fringe 2013, 7th International Workshop on Advanced Optical Imaging and Metrology, W. Osten, ed., (Springer, 2014), pp. 701–706.

17. G. Baer, J. Schindler, C. Pruss, and W. Osten, “Correction of misalignment introduced aberration in non-null test measurements of free-form surfaces,” J. Europ. Opt. Soc. Rap. Public. 8, 13074 (2013). [CrossRef]

18. G. Baer, J. Schindler, C. Pruss, and W. Osten, “Comparison of alignment errors in asphere metrology between an interferometric null-test measurement and a non-null measurement with the tilted-wave-interferometer,” Proc. SPIE 8884, Optifab 2013, 88840T (2013). [CrossRef]

19. R. L. Burden and J. D. Faires, Numerical Analysis, 9th edition, (Brooks/Cole, 2011), Chap. 4.1.

20. H. Gross, Handbook of Optical Systems, Volume 3: Aberration Theory and Correction of Optical Systems, (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2007), Chap. 32, pp. 307–309.

21. D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. 47, 913–925 (1957). [CrossRef]

22. D. P. Feder, “Differentiation of ray-tracing equations with respect to construction parameters of rotationally symmetric optics,” J. Opt. Soc. Am. 58, 1494–1505 (1968). [CrossRef]

23. T. B. Andersen, “Optical aberration functions: derivatives with respect to axial distances for symmetrical systems,” Appl. Opt. 21, 1817–1823 (1982). [CrossRef] [PubMed]

24. P. D. Lin, “Design of optical systems using derivatives of a rays: derivatives of variable vector of spherical boundary surfaces with respect to system variable vector,” Appl. Opt. 52, 7271–7287 (2013). [CrossRef] [PubMed]

25. P. D. Lin, “Analysis and design of prisms using the derivatives of a ray. Part II: the derivatives of boundary variable vector with respect to system variable vector,” Appl. Opt. 52, 4151–4162 (2013). [CrossRef] [PubMed]

26. H. A. Buchdahl, An Introduction to Hamiltonian Optics, (Dover Publications, Inc., 1993), pp. 8–11.

27. B. D. Stone, “Perturbations of optical systems,” J. Opt. Soc. Am. A 14, 2837–2849 (1997). [CrossRef]

28. M. Rimmer, “Analysis of Perturbed Lens Systems,” Appl. Opt. 9, 533–537 (1970). [CrossRef] [PubMed]

29. M. J. Kidger, Fundamental Optical Design, (SPIE Press, 2002), pp. 4–5.

30. A. Wiegmann, M. Stavridis, M. Walzel, F. Sievert, T. Zeschke, M. Schulz, and C. Elster, “Accuracy evaluation for sub-aperture interferometry measurements of a synchrotron mirror using virtual experiments,” Precision Engineering 35, 183–190 (2011). [CrossRef]

31. F. Pukelsheim, Optimal Design of Experiments (Classics in Applied Mathematics, Vol. 50), (Society for Industrial and Applied Mathematics, 2006).

parameter	design 1	design 2

aperture	60 mm	45 mm
measured aperture	46 mm	28.6 mm
k	−0.09	−1
R	34 mm	24.86 mm
A₄	5 · 10⁻⁶ mm⁻³	3 · 10⁻⁶ mm⁻³
A₆	7 · 10⁻¹⁰ mm⁻⁵	−1.8 · 10⁻¹⁰ mm⁻⁵
A₈	−5 · 10⁻¹² mm⁻⁷	−8 · 10⁻¹³ mm⁻⁷
A₁₀	3 · 10⁻¹⁵ mm⁻⁹	1.35 · 10⁻¹⁶ mm⁻⁹
A₁₂		−9.37 · 10⁻¹⁹ mm⁻¹¹
A₁₄		1.68 · 10⁻²¹ mm⁻¹³
A₁₆		−8.72 · 10⁻²⁵ mm⁻¹⁵

job description	analytical Jacobian method	numerical Jacobian method	time factor

simulate single experiment	2.2 seconds	2.2 seconds	1
calculate Jacobian	0.5 seconds	330 seconds (≈ 2.2 · 153 seconds)	660
solve inverse problem	0.14 seconds	0.14 seconds	1

job description	analytical Jacobian method	numerical Jacobian method	time factor

simulate single experiment	65 seconds	65 seconds	1
calculate Jacobian	2.0 seconds	24 hours (≈ 65 · 1, 368 seconds)	44,460

parameter	design 1	design 2

aperture	60 mm	45 mm
measured aperture	46 mm	28.6 mm
k	−0.09	−1
R	34 mm	24.86 mm
A₄	5 · 10⁻⁶ mm⁻³	3 · 10⁻⁶ mm⁻³
A₆	7 · 10⁻¹⁰ mm⁻⁵	−1.8 · 10⁻¹⁰ mm⁻⁵
A₈	−5 · 10⁻¹² mm⁻⁷	−8 · 10⁻¹³ mm⁻⁷
A₁₀	3 · 10⁻¹⁵ mm⁻⁹	1.35 · 10⁻¹⁶ mm⁻⁹
A₁₂		−9.37 · 10⁻¹⁹ mm⁻¹¹
A₁₄		1.68 · 10⁻²¹ mm⁻¹³
A₁₆		−8.72 · 10⁻²⁵ mm⁻¹⁵

job description	analytical Jacobian method	numerical Jacobian method	time factor

simulate single experiment	2.2 seconds	2.2 seconds	1
calculate Jacobian	0.5 seconds	330 seconds (≈ 2.2 · 153 seconds)	660
solve inverse problem	0.14 seconds	0.14 seconds	1

Analytical Jacobian and its application to tilted-wave interferometry

Abstract

1. Introduction

2. The tilted-wavefront interferometer (TWI)

2.1. Surface reconstruction

2.2. Jacobian matrix from numerical differentiation

3. Analytical Jacobian of OPLs

3.1. Translation of a single surface

3.2. Partial derivatives with respect to arbitrary system design parameters

3.3. Tilt of a single surface

3.4. Perturbation of an optical element group

3.5. Perturbations of elements with multiple ray passage

4. Application to the TWI

4.1. Jacobian matrix from nominal ray path data

4.2. Simulation environment and ray-tracing

4.3. Results

4.4. TWI system calibration

4.5. Optimal positioning of the surface under test

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Tables (3)

Equations (16)

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