## Abstract

Standard phase-shifting interferometry (PSI) generally requires collecting at least three phase-shifted interferograms to extract the physical quantity being measured. Here, we propose the application of a simple two-frame PSI for the testing of a range of optical surfaces, including flats, spheres, and aspheres. The two-frame PSI extracts modulated phase from two randomly phase-shifted interferograms using a Gram-Schmidt algorithm, and can work in either null testing or non-null testing modes. Since only two interferograms are used for phase demodulation and the phase shift amount can be random, requirements on environmental conditions and phase shifter calibration are greatly relaxed. Experimental results of three different mirrors suggest that the two-frame PSI can achieve comparable measurement precision with conventional multi-frame PSI, but has faster data acquisition speed and less stringent hardware requirements. The proposed two-frame PSI expands the flexibility of PSI and holds great potential in many applications.

© 2016 Optical Society of America

## 1. Introduction

High-precision testing of optical surfaces like flats, spheres, and aspheres is of critical importance for achieving optimum performance of various kinds of optical systems [1–6], such as microlithographic projection lens systems [3], inertial confinement fusion (ICF) systems [4], and large astronomical telescopes [5]. Many different measurement techniques, including profilometry [2], interferometry [7–11], and wavefront sensing [12], have been developed over the past several decades. Among these, profilometry and interferometry are the two most commonly used techniques. Profilometry usually performs the measurement by scanning a contact stylus across the surface under test to give either cross-section or whole-aperture surface variations [2]. The technique has the disadvantages of low speed, low accuracy, and possibly causing damage to the surface, and, therefore, is typically used for unpolished optical surface testing. Interferometry uses light to contactlessly sense surface figure deformation and compares it with light wavelength [7, 9–11]. It can accurately reproduce whole surface figure map and is currently the standard technique for the testing of polished optical surfaces and whole optical systems.

Phase-shifting interferometry (PSI), a major breakthrough in the field of interferometry, provides a unique tool to measure optical surfaces with unprecedented accuracy [7]. To enable accurate digital phase demodulation, PSI usually employs a piezoelectric transducer (PZT) to push the reference mirror to produce a series of phase-shifted interferograms with known phase shift amounts. Well-developed demodulation algorithm [8], such as three-, four-, and five-bucket algorithms, are then used to extract the modulated phase. However, these standard procedures require at least three interferograms as input and the phase shifts between them should be known beforehand [8]. These requirements impose critical demands on the accuracy of the phase shifter and the stability of the environment. Using fewer numbers of interferograms with unspecified or random phase shifts may greatly reduce hardware and environmental requirements, and, therefore, is highly desirable.

In essence, a single interferogram already contains all information being measured. Thus single-frame interferometry has attracted much interest in the past several years [13–19]. But the problem of single-frame interferometry is that it is impossible to resolve the sign of the modulated phase from only one interferogram due to the even property of the cosine function [20], and, therefore, is not suitable for optical surface testing. Increasing another frame of interferogram helps resolve the sign ambiguity [21]. From this point, two-frame PSI might be a good compromise between single-frame interferometry and multi-frame phase-shifting interferometry in terms of being able to resolve phase sign ambiguity using the minimum number of interferograms and being able to reduce hardware and environmental requirements.

In this study, we propose the application of a two-frame PSI for testing optical surfaces using only two-frame randomly phase-shifted interferograms. The two-frame PSI employs a recently developed Gram-Schmidt (GS) demodulation algorithm by Vargas et al. [21] for phase extraction, and can work in either null testing or non-null testing mode. The presented two-frame PSI is able to achieve comparable measurement precision and repeatability as standard PSI techniques, but has faster data acquisition speed, lower hardware requirements, and greater measurement flexibility. The principle, examples, discussion, and conclusion are presented below.

## 2. Two-frame PSI for optical surface testing

The proposed two-frame PSI is a reduced form of standard PSI. Therefore, all conventional configurations are principally applicable to the two-frame PSI. Figure 1 shows its Twyman-Green layout [22, 23]. The emitted beam from a He-Ne laser at λ = 632.8 nm is first collimated by a beam expander, and then divided by a beam splitter into two parts. The reflected part (red in color) propagates to a reference mirror and is then reflected back to serve as the reference beam. The transmitted part (blue in color), after passing through a compensation lens, is incident onto the optical surface under test, phase modulated by surface figure deformation, and then reflected back into the interferometer. The reference beam and the measurement beam meet at the beam splitter, interfere with each other, and produce fringe patterns with periodic intensity modulation, which are imaged onto the CCD (charge-coupled device) detector for data acquisition. By pushing the reference mirror using a PZT phase shifter, two-frame interferograms are acquired. The modulated phase is then extracted using the GS technique, unwrapped, fitted, and calibrated to give the figure map of the surface under test. Here we will put emphases on the issues of phase demodulation from two-frame interferograms and system error calibration.

#### 2.1 Phase demodulation of two-frame randomly phase-shifted interferograms

Many different algorithms [20, 21, 24–30] for two-frame interferograms demodulation have been developed. Pioneering work includes the asynchronous self-tuning (AST) algorithm [25], the optical flow (OF) algorithm [26, 27], the Gram-Schmidt (GS) orthogonalization algorithm [21], the continuous wavelet transform (CWT) algorithm [28], the Hilbert-Huang transform (HHT) based algorithm [29, 30], etc. Due to its simplicity and robustness, the GS algorithm is adopted here to demonstrate the validity of the two-frame PSI. The core idea of the GS algorithm is based on the assumption that interferograms can be essentially treated as vectors. In this way, two-frame interferograms can be orthogonalized using the classic GS process [31, 32] to produce two signals in quadrature. Detailed description of the algorithm can be found in [21]. Here we briefly outline its principle.

Two-frame randomly phase-shifted fringe patterns can be formulated as

where*x*and

*y*are the spatial coordinates;

*a*and

*b*are the background and the modulation terms, respectively;

*ϕ*and

*δ*are the phase map and the phase shift to be recovered. Usually,

*a*is a smoothly-varying signal and can be removed by high-pass filtering [33]. The filtered version of the interferograms become as By treating the two phase-shifted interferograms as two linearly independent vectors, they can be orthogonalized using the GS orthogonalization process [21, 31]. Denote the first vector

**u**

_{1}asIts orthognal vector

**u**

_{2}can be obtained by projecting the vector

**I**′ 2 onto the space spanned by

**u**

_{1}, that is,

**I**′

_{2},

**u**

_{1}〉 denotes the inner product of

**I**′

_{2}and

**u**

_{1}[31]. Using the approximation ∑{[cos

*ϕ*(

*x*,

*y*)]

^{2}cos

*δ*(

*x*,

*y*)} ≫ ∑[cos

*ϕ*(

*x*,

*y*)sin

*ϕ*(

*x*,

*y*)sin

*δ*(

*x*,

*y*)] (usually true for fringe patterns with more than one fringes [21]), Eq. (7) can be simplified asUsing this derivation, the orthogonal vector

**u**

_{2}[Eq. (6)] becomesThe normalized vectors of

**u**

_{1}and

**u**

_{2}can be written as

**u**

_{1}|| ≈||

**u**

_{2}||. In this way, the phase modulo 2π can be directly computed from the two orthonormal bases asThe GS demodulation algorithm is an effective technique for phase demodulation from two randomly phase-shifted fringe patterns. It works well for any phase shifts except π rad, in which case the two fringe patterns are linearly dependent and orthogonalization becomes impossible. After demodulation, the phase goes through regular phase unwrapping [8] and wavefront fitting [34] procedures, and is finally calibrated for surface figure reconstruction, as discussed below.

#### 2.2 Surface figure reconstruction

The two-frame PSI for optical surface testing can work in two different modes, i.e., the null testing mode and the non-null testing mode, depending on whether the compensation optics can fully compensate for the surface being tested (see Fig. 2). Surface figure map reconstruction of the two cases slightly differs due to the presence of retrace error in non-null testing [9, 22, 35].

### A. Null testing mode

Null testing is usually used for the measurement of planar, spherical, and aspherical surfaces by employing well-calibrated null optics, such as reference flats [7], reference spheres [7], or computer-generated holograms (CGHs) [36–38], as the compensation elements. In null testing, the reference null optics fully compensates for the surface under test, and incident rays and return rays travel through the same path (i.e., the null testing condition [35,39]). In this way, intrinsic aberrations of the interferometer itself are minimized and high-precision measurements can be achieved. Figure 2(a) shows a specific null testing example, in which a transmission sphere (an aplanatic lens) is employed as the null optics and its focal point coincides with the curvature center of the spherical mirror. The resultant interferograms on the image plane principally have zero fringes, but, due to misalignments, practical interferograms typically have several straight fringes, as shown in Fig. 2(b).

In null testing, the figure map *W* of the surface being measured can be reconstructed by simply removing the misalignment error from the demodulated phase *ϕ*, that is,

*W*

_{CCD}=

*ϕ*is the experimental wavefront on the CCD;

*a*,

*b*,

*c*, and

*d*are misalignment coefficients, representing piston,

*x*-tilt,

*y*-tilt, and defocus of the wavefront

*W*

_{CCD}, and can be computed through wavefront fitting [34].

### B. Non-null testing mode

Due to its simplicity and high precision, null testing is currently the standard mode used in optical surface testing. However, the null testing condition is required to be fully satisfied in this case or significant measurement errors will be involved. This puts strict requirements on the design, fabrication, and calibration of the compensation null optics. While a compensation flat or a compensation sphere can be used as the null optics for a range of planar or spherical surfaces (as long as the numerical apertures match), a compensation null corrector for aspherical surface is one-to-one, and its design, fabrication, and calibration can be time-consuming and costly. Testing an aspherical surface in a non-null manner helps ease the problem, and expands the measurement capability and flexibility of null interferometers [22, 35, 39]. Non-null interferometric testing (especially for aspheric optics) has thus drawn much attention [9, 22, 23, 35, 39]. Different from null testing, due to the violation of the null-testing condition, the incident rays and return rays no longer follow the same paths in non-null testing and retrace error exists. The resultant interferograms on the image plane principally have many circular closed fringes, as shown in Fig. 2(a).

To correctly reconstruct the figure map of the surface being measured in non-null testing, both retrace error and misalignment error need to be calibrated. Typical retrace error calibration techniques involve system modelling and reverse optimization [22, 39]. In our previous work, we developed a simple theoretical reference wavefront (TRW) technique for retrace error removal based on interferometer simulation [22, 23]. The TRW method suggests that, by accurately modelling the interferometer in lens design softwares using surface data of each elements (such as curvatures, thicknesses, refractive indexes, and conic constants), retrace error *W*_{retrace} can be numerically computed by tracing a grid of rays through the reference arm and the test arm of the interferometer and calculating the optical path difference [22, 23]. In this way, the figure map *W* of the surface under test can be simply obtained as

*W*

_{CCD}without invoving any complex optimization process. When the retrace error is zero, the above equation reduces to Eq. (13), which indicates that null testing can be regarded as a special case of non-null testing or non-null testing can be considered as a generation of null testing.

Figure 3 shows the general workflow of the proposed two-frame PSI working in the null testing and the non-null testing modes.

## 3. Simulation and experiments

#### 3.1 Error analysis of the Gram-Schmidt algorithm

The GS demodulation algorithm mainly has three error sources, i.e., low-pass filtering [Eqs. (3) and (4)] and two approximations [Eqs. (8) and (12)]. Low-pass filtering typically uses a low-pass Gaussian filter to remove the smoothly-varying background term *a*(*x*, *y*) of the interferograms. This operation usually will not induce significant errors and is well accepted [17, 21, 26, 33]. The two approximations are considered to be the dominant error sources of the algorithm. Whether they hold or how true they are depends on the number of fringes in the interferograms. The more the fringes are, the smaller the demodulation error. The more-than-one fringes criterion is a basic rule of thumb from a statistic sense and it ensures the validity of the two approximations to a great extent.

To investigate the dependence between phase demodulation error and the fringe number *n*, we conducted a group of numerical experiments. In the experiments, the interferograms were simulated according to Eqs. (1) and (2), where the background term, the modulation term, and the phase shift were set to *a*(*x*, *y*) = 0, *b*(*x*, *y*) = 1, *δ* = 0.9π, and the phase was modelled as

*n*is a value controlling the fringe number of the interferogram (e.g.

*n*= 1 for one fringe).

Figure 4 shows the peak-to-valley (PV) and root-mean-square (RMS) values of phase demodulation errors using the GS algorithm with respect to the fringe number *n*. Apparently, the errors decrease significantly with the increase of the fringe number *n*. For *n* = 1, the PV and RMS errors are 0.186 rad and 0.065 rad, respectively, which are already negligibly small for most applications. Therefore, the more-than-one fringe condition could be used as a rule of thumb to judge whether two-frame phase-shifted interferograms could be processed by the GS algorithm. Figure 5 specifically gives an example demonstrating the demodulation result and error when only one fringe (*n* = 1) is present in the interferograms.

#### 3.2 Null testing of a spherical mirror

To demonstrate the feasibility of the proposed two-frame PSI, we performed a series of null and non-null testing experiments on three different optical surfaces, one concave spherical mirror and two concave paraboloidal mirrors. Their specifications are summarized in Table 1.

The concave spherical mirror with a radius of 500.0 mm and a clear aperture of 80.0 mm (i.e., R-number 6.25) was first measured in the null testing mode using a commercial laser interferometer (Zygo, Middlefield, CT, USA). The compensation null optics employed in the experiment was a standard Zygo transmission sphere with an F-number 5.0 and is capable of covering the whole aperture of the spherical mirror. After alignment, the PZT was allowed to scan 12 steps to generate 13-frame fringe patterns with an equal phase shift 2π/13. Two images from the 13-frame fringe patterns were randomly selected and used as input for the GS demodulation. The retrieved phase *ϕ* is shown in Fig. 6(c). A standard phase unwrapping [8] and wavefront fitting [34] procedure was then applied to estimate the misalignment coefficients *a*, *b*, *c* and *d* from the demodulate phase *ϕ*. The finally reconstructed surface figure map *W* with misalignment error removed is presented in Fig. 6(d). Its PV and RMS values are quantified to be 0.098λ and 0.015λ, respectively.

To examine the validity and accuracy of the result, the 13-frame measurement result from the Zygo interferometer with piston, tilt, and defocus removed was exported for comparison and is shown in Fig. 6(e). Its PV and RMS values are 0.088λ and 0.014λ, respectively. As we can see, although the two-frame result looks a bit rougher and has slightly larger PV and RMS values, it is highly consistent with the Zygo 13-frame result.

#### 3.3 Non-null testing of two paraboloidal mirrors

The two-frame PSI was further demonstrated by measuring two concave paraboloidal mirrors using a custom-made Twyman-Green interferometric system (see Fig. 1). The two mirrors have different aperture sizes and radii of curvature, and the maximum departures from their best-fit spheres (BFS) are 2.2 μm and 14.5 μm (see Table 1). A simple biconvex singlet was used as the compensation lens for both of them. The radii, thickness, and refractive index of the singlet were calibrated before the experiment, and can be found in Table 2.

The paraboloidal mirror with a 2.2 μm asphericity was first measured. Since the singlet was not specifically designed to compensate for the mirror, the null-testing condition could not be met and retrace error existed. As expected, the interferograms on the image plane had many closed fringes [see Figs. 7(a) and 7(b)]. To calibrate the retrace error for correct surface figure reconstruction, the interferometer, including both the reference arm and the test arm, was modeled in the lens design software Zemax (Zemax, LLC, Kirkland, WA, USA) based on measured surface data of each element. After numerically tracing a grid of rays through the two arms, the optical path difference between them was computed and treated as the retrace error [22,23], which was wrapped and is shown in Fig. 7(c). The surface data of the test arm used in the modelling were summarized in Table 2.

In the experiment, we acquired a total of 13-frame phase-shifted interferograms. The phase shift amounts between two consecutive images were unknown. Two, three, and thirteen images were randomly selected out of the 13-frame interferograms and demodulated using the 2-frame GS algorithm, 3-frame advanced iteration algorithm (AIA) [40], and 13-frame AIA [40], respectively (three is the minium number of images that the AIA can process). The retrieved phases *ϕ* of the three groups are shown in the first row of Fig. 8. Same post data processing procedure [Eq. (14)] was then applied to the retrieved phases by removing the computed retrace error [Fig. 7(c)] and the fitted misalignement error, and the correspondingly reconstructed surface figure maps *W* of the three groups are presented in the second row of Fig. 8. As indicated by the results, the retrieved phase and recontructed surface figure by the 2-frame method have no obvious differences from those by the 3-frame and 13-frame methods, and are essentially very close. The quantified PV values of the surface figure maps using 2-frame, 3-frame, and 13-frame interferograms are 0.749λ, 0.707λ, and 0.692λ, respectively, and the corresponding RMS values are 0.122λ, 0.119λ, and 0.113λ. To further compare the reconstruction results, the 2-frame and the 3-frame results were subtracted from the 13-frame result (reference), and the difference maps are shown in Figs. 6(g) and 6(h). As we can see, the 2-frame and the 3-frame results agree with the 13-frame result very well except for some high-frequency differences, which is considered to be reasonable due to weaker averaging effect when using fewer number of interferograms.

The paraboloidal mirror with a 14.5 μm asphericity was also measured by the two-frame PSI employing the same singlet as the compensation lens. Similarly, the singlet also cannot fully compensate for the mirror and a large retrace error existed. Two randomly phase-shifted interferograms with very dense fringes, as shown in Figs. 9(a) and 9(b), were acquired and demodulated using the GS algorithm. The retrieved phase and computed retrace error from system modelling are shown in Figs. 9(c) and 9(d), respectively. Figure 9(e) illustrates the reconstructed surface figure *W* after removing the retrace error and the misalignment error. The PV and RMS values are quantified to be 0.248λ and 0.027λ, respectively.

Two comparison experiments were performed to validate the results. The first experiment was conducted on the same setup. Keeping all other conditions the same, the mirror under test was rotated 180 degrees and two-frame fringe patterns were collected for data processing. The second experiment was carried out on a custom-made auto-collimation setup (see Chapter 12.4.3 in [1]) using a Zygo interferometer. In this case, 13-frame interferograms were acquired and used for data processing. The measurement results of the two comparison experiments are shown in Figs. 9(f) and 9(g), respectively. By comparing the results, we can see that the 0 degree result is essentially in very good agreement with the 180 degree and the auto-collimation results, and the numbered features in the 0 degree result well reproduced in both two comparison results. The quantified PV values for Figs. 9(f) and 9(g) are 0.275λ and 0.275λ, respectively, and RMS values are 0.030λ and 0.033λ. The maximum PV and RMS deviations of the three experiments are estimated to be smaller than λ/30 and λ/150, respectively. The proposed two-frame PSI exhibits excellent repeatability and accuracy.

Table 3 summarizes the PV and RMS values of all experiments presented above.

## 4. Conclusion

In summary, we proposed the application of a two-frame phase-shifting interferometry (PSI) for the measurement of optical surfaces. The two-frame system employs a Gram-Schmidt demodulation algorithm to retrieve modulated phase from as few as two randomly phase-shifted fringe patterns, and can work in either null testing or non-null testing mode. While, in null testing, only misalignment error needs to be removed for surface figure reconstruction [Eq. (13)]; in non-null testing, retrace error also needs to be calibrated [Eq. (14)], which can be realized by numerically tracing a grid of rays through the interferometer and subtracting theoretical optical path difference from experimental wavefront. Null testing is regarded as a special case of non-null testing when the retrace error is zero. Experimental results of three mirrors suggest that by only employing two randomly phase-shifted fringe patterns, the proposed two-frame PSI could achieve comparable measurement precision as conventional multi-frame PSI, but has faster data acquisition speed and less stringent hardware requirements.

The proposed two-frame PSI expands the measurement flexibility of conventional PSI, and is suitable for low-cost applications. This is because the phase shift between two interferograms can be random and strict calibration of the phase shifter is not required. Moreover, since only two interferograms are used, the demands on data transmission, storage, and computation are greatly reduced. Therefore, a simpler data acquisition and processing system instead of a regular desktop can be used, which is beneficial to cost reduction and instrument miniaturization.

## Funding

This work was supported by National Natural Science Foundation of China (NSFC) under Grant Nos. 60877043 and 61575061.

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