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Retrace error calibration for interferometric measurements using an unknown optical system

R. Beisswanger, C. Pruss, and S. Reichelt

Open Access Open Access

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.

Published by Optica Publishing Group under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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Data availability

Data that support the plots within this paper are available from the corresponding author upon reasonable request

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Figures (15)
Fig. 1.
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

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Fig. 2.
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.

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Fig. 3.
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]).

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Fig. 4.
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.

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Fig. 5.
Fig. 5. Flow chart to calibrate a BBM based interferometer model with unknown design of the TS.

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Fig. 6.
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.

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Fig. 7.
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).

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Fig. 8.
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.

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Fig. 9.
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).

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Fig. 10.
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’

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Fig. 11.
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.

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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.

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Fig. 13.
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).

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Fig. 14.
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.

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Fig. 15.
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.

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Tables (2)

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Table 1. Convergence results of simulated calibration for Twyman-Green type TWI and quadratic fit: Simulation is limited by numerical noise.

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Table 2. Convergence results from calibration: Simulation is limited by numerical noise. Real calibration has typical values of calibrations with known design.

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

W ^ Q ( M , N , X , Y ) = ∑ i j Q i , j Z i ( M , N ) Z j ( X , Y )
W ^ P ( x , y , m , n ) = ∑ k l P k , l Z k ( x , y ) Z l ( m , n )
W ^ R ( M , N , m , n ) = ∑ h R h Z h ( M , N , m , n )
b = b Q ( W ^ Q ( M , N , X , Y ) ) + b P ( W ^ P ( x , y , m , n ) ) + b SUT ( p , S ) − b R ( M , N , m , n )
b = A ⋅ x
x = A − 1 ⋅ b r e a l
β ′ = Δ y ′ Δ y = f T S f C o l ∝ S
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