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Abstract
In classical interferometric null test measurements, the measurement and reference beam path should be the same. A difference in the beam paths results in the so called retrace error. One very common approach to avoid retrace errors is to adapt the measurement wavefront to the reference wavefront with a computer generated hologram (CGH), which is costly and time consuming. A much more flexible approach is to do non nulltest measurement in combination with mathematical treatment of retrace errors. Most of such methods are based on iterative optimization or calibration of the nominal optical design of the interferometer. While this may be a convenient solution in the context of research, the more common use may be limited due to the need of the optical design of all interferometer components. In many cases, the optical designs of standard off the shelf optical assemblies are not available or disclosed by the manufacturer. This is especially true for transmission spheres of interferometers. We introduce the so called Black Box Model (BBM), used in the well known Tilted Wave Interferometry (TWI), as a mathematical model to account for retrace errors in interferometry. The Black Box Model is based on point characteristic functions which are adapted to the result and behavior of a real interferometer by calibration. With an extended calibration method, the need of a specific optical design of the interferometer is no longer necessary. Thus the method is attractive for a wide field of use in interferometry with standard off the shelf components.
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1. Introduction
Retrace errors in interferometric measurements are systematic errors that can be observed if the path of the object wave through the system differs from the path of the reference wavefront [1]. Essentially two error modes occur in case of non equal paths [2–4]: A shearing (or mapping) effect, mainly caused by the imaging part of the interferometer, and a phase error. The latter is caused because two different paths are affected by different parts of the interferometer which may be influenced by distinct local errors.
The consequence in the existence of retrace errors is that they either have to be avoided, or corrected. Avoiding is straightforward if the SUT is spherical or flat (because typical reference surfaces of interferometers are either spherical or flat). In case of nonspherical SUT (aspheres and freeform-surfaces), a well-established method in avoiding retrace errors is to adapt the measured wavefront to the reference wavefront with a computer generated hologram (CGH) [5]. But this method is time consuming, expensive and not flexible regarding different types of aspherical forms.
In 2004, Gappinger and Greivenkamp presented a numerical correction method for retrace errors based on a lens design model [6]. They used an iterative reverse optimization procedure to adapt the initial lens design model to measured calibration data. With their calibrated model, they were able to calibrate a nonnull interferometric measurement with a departure of more than 200 $\lambda$ in wavefront to an accuracy of 0.02 $\lambda$ in rms and 0.16 $\lambda$ in PV. Weibo also used ray tracing as a correction method for retrace errors [7]. He did a fit of the intermediate wavefront results to Zernike polynomials and correction of the mapping errors with a limited number of coefficients. A combination of reverse optimization procedure with ray tracing and correction by Zernike coefficients was developed by Liu [8,9] in 2014. In simulations it was shown that the method can reconstruct an error in a SUT of about 15 $\lambda$ in PV and 2 $\lambda$ in RMS to an reconstruction error far below $\lambda$/10.
Retrace errors are also intrinsic in stitching interferometry. Shahinian presented a method to compensate for them by determining the artifacts from measured data at local slopes [10]. This kind of stitching interferometry is an additional way in accurate asphere and freeform measurement. Also an auxiliary interferometric profiler can be used to get additional information related to field- and object orientation-dependent systematic errors [11].
Another possibility in treatment of retrace error is the use of point characteristic functions (PCF) according to Hamiltonian optics [12]. Characteristic functions are functions which describe a real system exactly in result and behavior. This concept allows to account for retrace path errors and retrace element errors in the interferometer independent of the measured surface. It was developed at our institute since 2006 [13] and first publications where published in conjunction with the tilted wave interferometer (TWI) [14–16]. TWI is a sophisticated method to measure aspheres and freeform surfaces and is strongly affected by retrace errors. As an example of use for point characteristic functions, we will present details of the TWI throughout the next sections. The application of PCF to classic Fizeau interferometers was also done by Yiwei [17] in 2015. Like in state of the art of TWI, he used ray tracing to generate a start system, which is transformed into a point characteristic function.
Ray tracing requires an exact optical design model of the instrument and all of its optical components, which is not always available for off the shelf components. Especially the designs of assemblies like transmission spheres (TS) are usually proprietary to the manufacturer. This limits most of the methods for retrace error correction to the case where the optical design of the interferometer is completely available. Besides the lack of design data, ray tracing needs special software and hardware including support and licences. The operation of the software requires trained staff.
We present a method which enables the use of point characteristic functions without the need of ray tracing software and the knowledge of complete optical design of the interferometer. For a better understanding, we present this in two steps: First, the calibration of PCF is explained with ray tracing, but with unknown design of the transmission sphere. The next step abandons ray tracing as an initial step for calibrating a PCF. We present a calibration routine which calibrate the system automatically after mounting a TS with unknown design to the interferometer, even if the focal length is unknown.
Results and methods are presented for the case of the TWI-System. Because of decentered point sources in TWI, retrace errors are intrinsic in TWI. This implies that correction (this means calibration) is mandatory. But that also means, that steep aspheres can be measured, which are not measurable by conventional interferometers based on a single central source. A comparison to the uncorrected case is not possible based on TWI. The example is used to show the potential of the PCF. Comparison with and without retrace error correction can be done in less sophisticated single source systems, and the adaption of the presented method is straightforward.
2. Tilted wave interferometry
TWI uses decentered point sources, arranged in an array (PSA: Point Source Array) to generate tilted waves in measurement space and hence provide a large angular spectrum to enable full field measurement [18,13]. A sketch of the TWI in Fizeau configuration is shown in Fig. 1. The point sources are collimated with a collimator lens Col and transformed by a Transmission Sphere TS to wavefronts compatible to measure aspheres. The measurement wavefront is reflected by the SUT and imaged with an Imaging Optic (IO) to the camera (Cam). An aperture in the Fourier plane of the wavefront limits the angular spectrum and hence the interferometric fringe density to below the Nyquist limit.
Since TWI is an intrinsically non null measurement method, calibration of the interferometer is required to separate interferometer errors from deviations of the SUT to its nominal design [19,15]. After calibration, all possible rays through the system can be calculated by the point characteristic functions. Hence, the calculation of a nominal SUT design in the calibrated system include the same retrace error like the measured SUT, and they vanish in the difference between nominal calculation and real measurement. The remaining wavefront describes the deviation of the real SUT to its nominal design.
Fig. 1. Sketch of a TWI in Fizeau configuration. Each point source generates none (if angular range is incompatible), one or more interferograms on an aspherical SUT. PSA: Point-Source-Array, BS: Beam-Splitter (50:50), Col: Collimating Optic, TS: Transmission Sphere, AP: Aperture in Fourier domain, IO: Imaging Optic, Cam: Camera
3. Black box model to calibrate point characteristic functions
The point characteristic functions are expressed as a Black Box Model (BBM) which is an abstraction of the interferometer. In TWI, they describe the interferometer as a set of surfaces $\hat {W}$, which can be expressed by mathematical functions in the context of Hamiltonian’s optics and perturbation methods in optics [16]. Ray tracing is used to trace rays between some of the surfaces to a theoretical design description of an ideal SUT. This data of an ideal SUT in a given BBM is compared to the real measured SUT. Provided that, due to the calibration, the BBM represents the real system to below the maximum accepted uncertainty, the deviation of the real SUT to its design is available in the accepted uncertainty. Figure 2 shows a sketch to visualize the BBM. The so called Q black box describes ray data from a single point source A(M,N) (with M and N being the two dimensional coordinates of the corresponding point source) to the virtual surface B(X,Y) (X and Y are the coordinates for a single ray) in measurement space. Because a best fit sphere is subtracted, B(X,Y) is a plane and serves as starting point for the ray-tracing to the SUT. Similarly, the P black box describes the surface from measurement space to the camera of the interferometer. Each ray arriving at the P surface C(x,y) is mapped to a camera pixel D(m,n) (m and n being the coordinates of the camera pixel). The black boxes are described in a double Zernike expansion [20]. With Z being the Zernike polynomials, Zi (Eq. (1)) and Zk (Eq. (2)) are describing the black boxes over the field of all point sources, whereas Zj and Zl are describing the black boxes of the point sources itself. Because of the double expansion, the corresponding coefficients of the polynomials are arranged in matrices (Qij and Pkl). An example of such matrices are shown in Fig. 3. The mathematical description of the charakteristic functions for Q and P ($\hat {W}_Q$, $\hat {W}_P$) is
In interferometry, a beam is measured against a reference beam. Accordingly, there is a R black box, which describes a beam from a point source A(M,N) via the reference surface to a camera pixel D(m,n). The mathematical description of the characteristic function for the reference $\hat {W}_R$ is a single expansion Zernike polynomial series because it describes only one source with rotational symmetry.
In case of TWI in Fizeau configuration, four separate R black boxes are needed [21–23]. To simplify the discussion, we limit the discussion to one R black box without loss of generality. A beam b through the BBM system can be expressed as
Fig. 2. Scheme of the Black Box Model of the Fizeau type TWI. Ray-tracing is applied between B(X,Y), SUT and C(x,y). The illustration is an "unfolded" abstraction of a beam path which is reflected at the SUT. Beam propagation is calculated with the polynomial point-to-point description from the reference surfaces A(M,N) to B(X,Y) and from C(x,y) to D(m,n). Propagation from B(X,Y) via the SUT to C(x,y) is done with raytracing. The R-Blackbox represents the reference beam-path of the interferometer whose results are added to the Q and P results.
Fig. 3. Polynomial coefficients of the two dimensional Zernike description of the BBM. The polynomial description of R is one dimensional. The format of the matrices are the same for Twyman-Green and Fizeau configuration. In case of Fizeau configuration four R Matrices are used (for details see [21]).
3.1 Calibration of a black box model
Calibration of the BBM based TWI implies adaption of the design based (ideal) Surfaces A,B,C and D to the real condition, including the real interferometer errors. This is done by calculating interferograms from the ideal interferometer and an ideal SUT and comparing them to measured interferograms from the real interferometer. For the calibration of the interferometer calibration spheres are used. Their radius of curvature is chosen depending on the interferometer objective, in our case, e.g., R = 40 mm. Details can be found in [24]. The Zernike parameters in the Q, P an R matrices of the BBM Model are then adapted iteratively according to the difference between ‘calculated from design’ and measurement. To solve linear dependencies because of ambiguities, a special calibration routine is needed [19,15].
This adaption of the interferometer to real conditions needs a start system. In the state of the art, design software like Zemax [25] or Quadoa [26] is used to trace rays through the representative parts of the TWI (assemblies for Q, P and R configuration, lower part of Fig. 2). This requires the complete optical design model. Subsequently the rays (optical path length) are fitted and transformed to the double Zernike expansion Q, P and R. If x is a vector of the parameters which describe the BBM (coefficients Qij, Pkl Rg and further parameters), rays can be calculated with the forward calculation
The equation is a linear transformation with A as Jacobian matrix. With breal being real measured rays of the interferometer, Eq. (5) can be inverted to the backward condition
Using an iterative least squares solving approach, the parameters x can be determined out of the real beams breal.
4. Generation of a start system by scaling the optical design
Like in other interferometric measurement techniques (e.g., Computer Generated Hologram (CGH) interferometry [27]), the f-number of the TS has to be compatible to the the best fit sphere (BFS) of the SUT. Thus, a change of the TS and a recalibration of the TWI might be necessary if the best fit sphere is not compatible to the f-number. A recalibration process uses the previous mathematical interferometer description (which is the BBM) for the TWI with the corresponding TS. If a new TS is used with the TWI for the first time, no previous BBM exists, and with an initial calibration, a new BBM has to be created. In the state of the art, this is only possible if the complete optical design of the TWI, including the TS, is available.
In this section, the calibration of a TWI with an unknown optical design of the TS, using an arbitrary design, is explained. Based on this, the calibration of a TWI without design is explained in section 5.
4.1 Scaling of the transmission sphere
The f-number is the focal length divided by aperture diameter. Hence, compatibility of the f-number to the best fit sphere implies compatibility to these two dimensions. In an optical design, the f-number can be easily adapted by scaling the TS with a scale factor S. Though this scaling is not a perfect adaption to the situation, it will be sufficient for the start system for the calibration procedure. Furthermore, because typical TS for interferometry do have very similar design targets, it is possible to use a design of a typical TS and scale it to achieve the desired f-number. The resulting scaled optical design can be used to create a start system to calibrate a TWI with a TS of different optical design. The additional ‘error load’ introduced by the difference in optical design is usually low frequent, rotational invariant and very well reproducible by Zernike coefficients and therefore calibratable with the presented BBM method.
A scheme of an essential part of a TWI is given in Fig. 4. Simplified it can be considered as a imaging system consisting of the two lenses Col and TS. Decentering the point source in field space by $\Delta$y results in a displacement of the focal point by $\Delta$y′ in measurement space. The so called magnification $\beta$′ is determined by Eq. (7) and is proportional to the scaling S of the TS.
Changing the TS has an effect on $\beta$′, which is an important parameter essentially for two reasons: First, in a simplified view, the calibration routine needs for every point source a confocal interferogram. This is acquired by placing a sphere in each confocal position of the corresponding source, whereas the position of the sphere is dependent on the systems magnification. Second, the mathematical scaling of the BBM reference surfaces are dependent on $\beta$′. The calibration algorithm is much more robust if the nominal scaling is consistent with the real scaling.
Fig. 4. Sketch of the TWI to illustrate the effect of focal length of the Transmission Sphere. BS = Beam Splitter, Col = Collimating lens, IO = Imaging Optic for the camera, Cam = Camera, Ap = Aperture to limit spatial frequency.
To achieve the desired magnification in the optical design, the scaling factor S is of interest. If the focal length of the TS and the collimator Col is known, S can be calculated with Eq. (7). The proportional factor is available if the focal length is plotted against S and a linear fit.
If the focal length is not known, S can be determined by a set of measurements. This is what is described in the following section.
4.2 Determining the scaling factor S
The focal length of an optical system, like the TS of the TWI, is defined as the distance from the optical principal plane to the focal point [28]. The principal plane may be located somewhere inside the TS or even outside and is available by the optical design, which we assume to be unknown. Since it is not straightforward to measure precisely the location of the principal plane of an unknown TS design, the focal length may be hard to determine by measuring the distance to the focal point. Moreover, misalignments in the interferometer may lead to a curved wavefront emerging from the collimator, which may change the effective focal length of the TWI. In the following we show a way to determine S by measurement in the interferometer if the focal length is not available to the desired accuracy. Although the process may look complicated, the benefit becomes obvious below.
The left side of the flow chart (Fig. 5) describes preparatory work including design software, but arbitrary (typical) design of the TS. This preparatory work is entirely nominal, and the interferometer part without TS is also a nominal optical design. The presented method is used in combination with real measured vaules. Taking multiple measurements and linear fitting increase accuracy. Linearity between decentration of the sphere and the tilt coefficient of the Zernike description is only given in a certain range of decentration, depending on the radius of the sphere and the TS. But this is no limitation, because the linear range (determined empirically) is sufficient for the process and symmetry around zero displacement can be used. If nonlinearity becomes obvious, the maximum displacement must be reduced. A remaining mismatch of the scalefactor is tolerated by the calibration algorithm of the BBM. A general statement of the tolerance of the calibration algorithm and the achievable accuracy would go beyond the scope and is subject for future investigations.
With the work in advance (had to be done once), it is possible to automate the calibration of the interferometer with a unknown TS with a software routine based on the flow chart on the right side. Although the algorithm is quite complex, it is assumed to be very comfortable for the end user if it is automated by a script. Intermediate results are shown in the Figs. 6 and 7.
Fig. 6. Determination of the slope of the Zernike tilt vs. the decentration of a sphere with different scaling factors S. In a next step, S is plotted over the slopes.
Fig. 7. Determining the scaling factor S from a decentered sphere. S is plotted over the slopes and fitted with a polynomial of 2nd order (left). This relation between scalefactor and slope allows direct determination of the appropriate S with slopes from measurement of a real system (right).
In the following, a simulation of a calibration with unknown Zemax design is presented. Data shown in the Figs. 6 and 7 are an example for a simulation of a TWI in Twyman-Green configuration. In this simulation, we scaled a Zemax based design of a TWI with a TS of focal length 122 mm to get a start system with focal length of 170 mm and 254 mm respectively, whereas the scaling factors for the target start systems where determined as described. The target systems (170 mm and 254 mm, considered as "real" systems in this simulations) have different designs of TS than the scaled design (122 mm). Results of the simulations are shown below.
5. Calibration without optical design
As described above, a BBM derived from a typical nominal design TS is compatible to calibrate a TWI with a different design of the TS as long as the focal length and the accompanied $\beta$′ are close together. So far, the BBM is derived from an known optical design model, which has to be manually scaled. Moreover corresponding software (Zemax, CodeV, Quadoa,…) has to be used, which may not be available to everyone. A much more elegant way is to derive a BBM out of existing BBM’s. To do this, several BBM’s are derived from nominal designs and certain scaling factors. By subtracting the first BBM without scaling from the other scaled BBM’s, the mathematical development of the BBM with the scaling factors ($\delta _{\mathrm {BBM}}/\delta _{\mathrm {scaling}}$) is available. Figure 8 shows a sketch of the process.
The BBM consists of the two dimensional Zernike descriptions Q, P and R, which in turn consist of scalar coefficients. Some coefficients exhibit a strong response on scaling the TS, whereas some coefficients respond very weak. Because optic components for typical TS are designed and manufactured to have very smooth surfaces, scalar coefficients for higher order polynomials (i.e., high frequency errors) tend to respond weak to scaling. Typically, alignment and design dependent errors are of low spatial frequency or radial symmetric. Hence, their polynomial coefficients dominate in the response. In the first step, a cutoff regarding the response is done (here: cutoff < $10^{-8}$ mm). The dependence of coefficients with low response on S tend to be noisy and are set to zero. This implies that these coefficients will be zero in the start system for calibration. The value for the cutoff $10^{-8}$ mm is somehow arbitrary. Investigations regarding our systems showed that this value can easily be raised to above $10^{-6}$ mm without any effect on the calibrated systems. However, this depends heavily on the situation, i.e., the difference between the modeled and the real system, and how far the system is subsequently extrapolated. It is assumed, that lowering the threshold takes more coefficients into account and is only detrimental regarding computing time (which is in our case only a few seconds). Exploring the limits, respectively the minimum coefficients needed for the BBM, is a topic for future research work.
Fig. 8. Illustration of the process to determine the BBM with the scaling factor of the Transmission Sphere: A BBM consists of the Q, P and R descriptions (Fig. 3) and some additional numeric values. Several BBM’s are derived from optical designs of a TWI with scaled TS. Out of the difference of the BBM’s to the unscaled BBM, the evolvement of the BBM with the scale factor is available.
In the second step, a linear or quadratic fit of the remaining coefficients is done with the boundary condition, that the coefficient of determination $R^2$ of the fit must be above a certain threshold. In this discussion the threshold for $R^2$ is 0.9, and all coefficient developments below the threshold are excluded from the following steps (i.e., set to zero). Again, the threshold for $R^2$ is somehow arbitrary but reasonable and strongly dependent if the development is with linear or quadratic accuracy.
Quadratic fit provide a very good accuracy to describe the development of the coefficients with scaling. Based on the assumption that the slope of the coefficient developments have constant algebraic signs, it is not expected that higher fit orders will result in relevant improvements in the results.
Depending on the situation, linear fit might also be possible to develop the systems coefficients for a suitable start system. Because of the higher residual fitting errors, the quadratic fit should be preferred. In this work we demonstrate that the calibration of a BBM is possible with a start system out of a linear fit with the accompanied higher error load. In chapter 6.2 results of a real calibration and measurement, based on linear coefficient development, is shown.
An example of linear extrapolation of a single coefficient is shown in Fig. 9. The coefficient is the 2nd in field (column) and 32th in source (row) out of the double Zernike Q coefficients. The linear fit is based on five scaling steps of the TS and extrapolation is to a scaling of 1.1. A scaled Zemax design model is used to estimate the residual fitting errors of the linear extrapolated BBM. In case of the coefficient (2 32), the error is 11.3%. An example of the effect of residual fitting errors in the BBM is shown in Fig. 10 for the case of a sphere in confocal position in a Fizeau type TWI. The errors in terms of Zernike coefficients are shown in Fig. 14. As shown in section 6, this development error converge to zero during calibration with the BBM (to the degree of numerical accuracy) and real calibration with the combined error load converge to the same magnitude of remaining calibration error as usual real non extrapolated calibrations.
Fig. 9. Linear extrapolation of coefficients vs. scale factor. Right: example of a linear fit of the coefficient 32 (source), 2 (field) and the error due to linear extrapolation (compared to scaled optical design).
Fig. 10. Nominal phase images of a sphere close to the confocal position of the TWI (Fizeau type TWI). (a) Nominal image calculated from a BBM derived from the target design. (b) Image from an linear extrapolated BBM. (c) Difference between ‘from optical design’ and ‘from extrapolation’
6. Simulation and measurement results
In this section, we show a simulated calibration of a Twyman-Green TWI with interpolated BBM based on quadratic fit. Subsequently we present a simulated and a real calibration of a Fizeau type TWI based on linear extrapolation of the BBM. The linear extrapolation demonstrates that residual fitting errors do not have an effect to the calibration results in this example. It is advised to use quadratic fit in practice, since the computing time is negligible with nowadays computers.
6.1 Simulation of Twyman-Green type TWI and quadratic fit
In simulation, we use the extra- or interpolated BBM system as start system for a calibration (as would be in real calibration). The target system is a scaled Zemax design with different optical design of the TS (Design B) than the design used to generate the basis for extra or interpolated BBM (Design A). Figure 11 shows the simulation steps: BBM models are generated from a scaled Zemax system to generate the basis for extra- or interpolation. The Zemax design target system (Design B), considered as "real" system is used to generate "measured" interferograms, which are used as input data for calibration of the extrapolated system A′. After calibration, the resulting BBM B′ equals (to numerical precicion) B in the result and behavior.
Fig. 11. Simulation process of a calibration with extra- or intrapolated BBM. The optical Design A is used to generate an extra- or intrapolated BBM A′ with suitable focal length f′. After calibration, the BBM B′ is equal (to numerical uncertainties) to the BBM B from the target Design B.
Simulation for Twyman-Green TWI was done with a Zemax design model of focal length 122 mm. The target systems are scaled design models (f = 170 mm, S170 = 1.28 and f = 254 mm, S254 = 1.76). Scaling factors are determined as it would be done in a real system by measurement (Fig. 6 and 7). An example of quadratic fitted coefficients (P black box) is shown in Fig. 12.
Fig. 12. Example of development of BBM coefficients with scaled TS for Twyman-Green type TWI. Left: Coefficients which develop > $10^{-8}$ per scaling step (in absolute values, one step is 10% of dimensional scaling). Middle: Location of coefficients which develop between $10^{-4}$ and $10^{-3}$ (in absolute values). Right: Quadratic fit of coefficients. The scaling factor matching the target systems are 1.28 for TS 170 and 1.76 for TS 254.
With quadratic fit, most of the coefficients above the cutoff threshold can be fitted with $R^2$ close to one. Figure 13 shows the difference of the confocal positions between optical design and interpolated system. As can be seen in Fig. 14, the difference is dominated by spherical aberrations of higher order (indicated by the red arrows), which is due to a slight mismatch in focus of the TS. The convergence of the simulated calibration is very high and limited by numerical noise and the remaining calibration error is below 1 nm in RMS (Table 1). Since the algorithmic part of simulated calibration (and measurement) is identical to the real calibration, the input data format is the same, namely unsigned 8-Bit. Hence, the numerical noise is $\varepsilon = \lambda$/256. In the present case of $\lambda$ = 633 nm (HeNe laser) the numerical noise is $\varepsilon \approx \pm$ 2.5 nm ($\pm$ due to unsigned). Because real noise in the present measurement systems is much higher, we limit simulation to this accuracy. The influence of noise and other sources of uncertainty in real situations is studied in detail in [23,24,30,31]
Table 1. Convergence results of simulated calibration for Twyman-Green type TWI and quadratic fit: Simulation is limited by numerical noise.
Fig. 13. Difference of the confocal position between optical design (left) and interpolated system (right) in case of a Twyman-Green type TWI and quadratic interpolation. Trefoil aberration in the interferogram (a) and (b) is a result of the extended beamsplitter in the interferometer (thick wedged glass plate).
Fig. 14. Zernike coefficients of the differences of the confocal positions shown in Fig. 10(c) and 13(c). The RMS deviation is 11.3 nm for linear Fizeau TWI extrapolation and 7.7 nm for quadratic Twyman-Green interpolation. Index of coefficients is according to Noll [29]. Red arrows indicate rotational symmetric spherical aberrations of higher order.
6.2 Calibration of a Fizeau type TWI and linear extrapolation
Simulation of calibration of a Fizeau type TWI was done in similar conditions than in the real calibration. The design of the scaled TS differs from the design of the target TS and the scaling factor is 1.28 (the same scaling factor is also used in Twyman-Green simulation). Coefficients are calculated by linear extrapolation. Figure 10 shows the deviation between extrapolated and target system for a position close to confocal. The remaining error of the simulated calibration (Table 2) is as low as in simulation of the quadratic interpolated Twyman-Green TWI. Based on the simulated calibration, a simulation of measurement was done for an asphere of the same type than in the real measurement (coefficients shown in the left of Fig. 15). With a reconstruction error of about 1 nm in RMS (which is limited by numerical noise), no effect of the high linear coefficient fit error before the calibration can be observed in the measurement reconstruction. Real calibration was done with a TS without design data. The TS is a four inch F/# = 2.4 TS from Mahr with a residual wavefront error of $\lambda$/20 and a Back Focal Length (BFL) of 247 mm. The developments of the BBM coefficients are determined from values of an arbitrary design of a TS with focal length of 160 mm (it is unknown how close the designs of the real TS and the used design TS are). To match the focal length of the real TS, the BBM is scaled by a factor of 1.28. Calibration of the real system was done with two different wavelengths ($\lambda _{1}$ = 632.8 nm , $\lambda _{2}$ = 637.9 nm). The remaining calibration error of < 12 nm in RMS and about 100 nm in PV is in the range of conventional state of the art calibrations with known TS. High values for PV is due to the high sensitivity of PV to real noise, where a single pixel (missed by several noise filters) has high influence. Figure 15 shows a measurement of an asphere in the real calibrated TWI with a linear extrapolated start system. Coefficients of the design data, according to [32], are shown on the left.
Table 2. Convergence results from calibration: Simulation is limited by numerical noise. Real calibration has typical values of calibrations with known design.
Fig. 15. Measurement of an asphere with a Fizeau type TWI calibrated with a linear extrapolated start system. Left: Theoretical design data. Right: Deviation of the measured real SUT from the theoretical design.
7. Summary
We have introduced the black box model (BBM) as a possibility to correct for retrace errors in non nulltest interferometry. An extended calibration method allows to calibrate a BBM without the need of a specific optical design of the interferometer. This was demonstrated by calibrating a BBM with a transmission sphere (TS), with unknown optical design, in simulation and in a real system.
A development base for extra- or interpolation of the BBM can be created with a typical but arbitrary optical design of a TS. With work in advance, direct scaling of the BBM is possible. This allows calibration of the interferometer without the need of knowledge of the optical design and the accompanied software.
An important parameter for the presented calibration approach is the focal length of the interferometer. A method is presented to determine the focal length from a few measurements. The related scaling factor of the TS, needed to find a good starting point for the calibration, can be determined in a similar way. The presented sequences can be automated so that after placing a sphere in confocal position the system is ready for calibration.
The possibilities of the BBM where demonstrated by simulation and measurements in case of the TWI. Calibration errors in simulation are limited by numerical noise. Residual fitting errors of BBM coefficients, due to linear extrapolation, do not have any effects in simulation and real calibration and measurement.
Disclosures
The authors declare that there are no conflicts of interest related to this paper.
Data availability
Data that support the plots within this paper are available from the corresponding author upon reasonable request
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