## Abstract

For white-light spectral interferometry, the phase information is usually retrieved via the Fourier transform method and the temporal phase-shifting method. In comparison, the synchronous phase-shifting method can be used to synchronously acquire interferometric signals with good accuracy and reduced noise. Therefore, it has potential for online measurement and is suitable for application in precision industries and for ultrahigh-speed measurement. In this work, a white-light spectral interferometer for synchronous phase shifting based on polarization interference was built, and the two-step phase-shifting algorithm was used to retrieve phase information. A variety of spectral interferometric signals were simulated based on the mathematical model of the two-step phase-shifting algorithm to illustrate the effects of differences in intensity and envelope shape, random noise, and phase-shift error on measurement of the absolute distance. Measurements of the absolute distance were conducted, and they indicated that the system had high accuracy.

Published by The Optical Society 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.

## 1. INTRODUCTION

Optical interferometry is widely used for surface topography, distance, and film thickness measurements because of its high accuracy and resolution, non-destructiveness, and rapid measurement [1–9]. Depending on the light source used in the system, interferometric techniques can mainly be divided into monochromatic interferometry and white-light interferometry. Compared with monochromatic interferometry, white-light interferometry [10,11] extends the measurement range by solving the phase ambiguity, which is caused by the arctangent function used in the phase-retrieval algorithm [12]; however, this method requires time-consuming mechanical scanning that makes the measurements more sensitive to environmental noise. If the interferometric signal is received by the spectrometer, then a white-light spectral interferometry [13] is realized. Compared with other interferometric techniques, this method solves the issue of phase ambiguity and avoids mechanical scanning in the vertical direction because of the spectral characteristics, ultimately resulting in improved measurement efficiency.

For white-light spectral interferometry, the phase information is usually
retrieved using the Fourier transform method and the phase-shifting method
[14,15]. Because only a single-shot spectral interferometric signal is
collected, the Fourier transform method can realize rapid measurement.
However, because of spectral leakage, this method is not suitable for
high-precision measurement. Furthermore, the results obtained using the
Fourier transform method are related to the wavelength bandwidth and
window function used [14], and the
distance sign (positive or negative) cannot be determined. The
phase-shifting method, which can be divided into the temporal
phase-shifting method and the synchronous phase-shifting method, retrieves
phase information by introducing the controlled phase shift into the
interferometric signals and has high precision [16]. For the temporal phase-shifting method, the
controlled phase shift is introduced by the phase shifter, resulting in
the interferometric signals being acquired successively and not
simultaneously. Therefore, this method could introduce errors during
successive measurements and is not suitable for dynamically changing
samples. In contrast, the synchronous phase-shifting method acquires
interferometric signals synchronously, which has promise for realizing
online measurement and is suitable for application in precision industries
and for ultrahigh-speed measurement. Smythe and Moore [17] developed an instantaneous
phase-measuring interferometer that used a four-camera system and a
digital video interface for display. Sivakumar *et
al.* [18] also used four
cameras and attached three nonpolarization beam splitters (BSs) together
to simultaneously acquire phase-shifting interferometric images. Their
research illustrated that when the vibration frequency is lower than
1000 Hz, the effects of vibration are insignificant for this method.
However, the use of four cameras is expensive, and it is difficult to
maintain consistency. Although some methods can acquire the phase-shifting
interferometric images using only one camera, they usually require special
optical components [19–21].
Shaked *et al.* [22] suggested dual interference channel quantitative
phase microscopy using a Wollaston prism; the phase information was
retrieved via the Hilbert transform. Guo *et
al.* [23] implemented a
simultaneous phase-shifting interferometer and used the two-step
phase-shifting algorithm to retrieve phase information; however, for this
method, the reference intensity needed to be measured and stored in
advance. Li *et al.* [24] used a spatial light modulator (SLM) to achieve the
background deduction of interferogram and then retrieve the phase
information. To date, there has been minimal research on the synchronous
phase-shifting method for spectral interferometry.

Here, we report a white-light spectral interferometer based on polarization interference to record synchronous phase-shifting signals and use the two-step phase-shifting algorithm to retrieve phase information. The system setup, the extraction process of phase information, and the realization of the phase shift are analyzed in detail. Furthermore, a variety of spectral interferometric signals were simulated based on the mathematical model of the two-step phase-shifting algorithm to illustrate the effects of differences in intensity and envelope shape, random noise, and phase-shift error on the measurement of the absolute distance. Measurements of the absolute distance were used to verify the accuracy of the developed system. Furthermore, the five-step phase-shifting algorithm and the Fourier transform method were also used to measure the absolute distance for comparison.

## 2. SYSTEM SETUP

Figure 1 shows a schematic diagram of the Linnik microscopic white-light spectral interferometer used in this work, including the light source unit, the Linnik interference unit, the detection unit, the synchronous phase-shifting unit, and the processing and control unit. The light source was a halogen lamp with a central wavelength of 680 nm (FWHM, 190 nm) and an operating wavelength range of 400–1300 nm (OSL2, Thorlabs) that illuminated the system after collimation; it was then passed through a linear polarizer to match the intensity of the measuring and reference beams. The polarizing beam splitter (PBS) cube divided the incident light into two beams, which then enter the measurement and reference optical paths through two objective lenses. A 10X double-objective lens with a numerical aperture (N.A.) of 0.28 was used in this work. The quarter-wave plates (QWP1, QWP2) between the PBS and the objective lenses could change the polarization direction of the measuring and reference beams, as indicated by the red arrows in Fig. 1, allowing for the beams to affect the other units of the system. Subsequently, the measuring and reference beams are transmitted and reflected by the PBS again, and then incident on the quarter-wave plate (QWP3), form two circularly polarized lights with opposite rotation directions. Then, a portion of the measuring and reference beams interfere at the linear polarizer (Analyzer) and is imaged using a charge-coupled device (CCD) camera (A102k, Basler) for observation. The other portion of the beams interfere at the linear polarizers (Analyzer1 and Analyzer2) and is collected by two spectrometers (HR2000+, Ocean Optics) via optical fibers after passing through the BS cube. The lead zirconate titanate piezoelectric ceramics (PZT) scanner (P-622.ZCD, PI) allowed for verification of the accuracy of the system by conducting the absolute distance experiments. The function of the stepping motor is to perform a mechanical scanning process to obtain the surface profile or topography.

## 3. MEASUREMENT PRINCIPLE

#### A. Phase Extraction

The white-light spectral interferometric signals can be expressed as

where ${I_r}$ and ${I_m}$ are the reference and measuring beams, respectively, and $\varphi $ is the phase difference, which can be described using the following equation where $d$ is the optical path difference (OPD) between the reference beam and the measuring beam, also known as the absolute distance, $k = 1/ \lambda $ is the wavenumber, $\phi $ is the unwrapped phase, and $m$ is the interference order. Then, the OPD $d$ can be calculated usingThe phase difference $\phi $ can be retrieved using the phase-shifting algorithm, which introduces the controlled phase shift into the interferometric signals,

where $n$ is the number of phase shifts and $\Delta \delta $ is the phase shift. In this work, a two-step phase-shifting algorithm with a $\pi /{2}$ phase shift was applied.#### B. Phase Shift Realization

In this work, the synchronous phase-shifting method was mainly realized via the PBS cube, the QWP, and the linear polarizer. The Jones matrix can be used to express the polarization state of the beam after passing through some optical elements [25,26],

where ${G_i}$ is the Jones matrix of the optical element.The p-polarized beam ${E_{10}}$ and s-polarized beam ${E_{20}}$ transmitted and reflected by the PBS cube (PBS in Fig. 1) can be expressed as

Then, the beam passes through the QWP3 (Fig. 1), which is aligned with the fast axis at 45° with respect to the plane of polarization emerging from the PBS, and can be expressed as

The phase change of the interferometric signal is twice the change of the angle $\alpha $. Therefore, the angle between the two linear polarizers in our system (Analyzer1 and Analyzer2 in Fig. 1) should be $\pi /{4}$ to obtain a $\pi /{2}$ phase shift in the two-step phase-shifting algorithm.

## 4. SIMULATION AND ANALYSIS

#### A. Mathematical Model of Two-Step Phase-Shifting Algorithm

The spectral interferometric signals of the two-step phase-shifting algorithm with a $\pi /{2}$ phase shift can be expressed as

The spectral interferometric signals of the two-step phase-shifting algorithm simulated in Fig. 2 are in good agreement, but this would be almost impossible in practical experiments. Practical measurement of the absolute distance would be influenced by differences in intensity and envelope shape, random noise, and phase-shift error of the spectral signals because of errors in the beam-splitting ratio by the BS, differences between the spectrometers, and the wavelength-dependent transmission and reflection of the optical elements. Figure 3 shows the recorded spectral interferometric signals, and it can be seen that there are significant differences between the spectral signals. Therefore, we simulated a variety of spectral interferometric signals based on the mathematical model of the two-step phase-shifting algorithm in the following sections to illustrate the influences.

#### B. Influence of Differences in Intensity and Envelope Shape

### 1. Spectral Intensity Difference

Spectral interferometric signals of the two-step phase-shifting algorithm with a $\pi /{2}$ phase shift at the absolute distance position of 10 µm were simulated to demonstrate the influence of spectral intensity difference; for this, different intensities were used, while the envelope shapes were uniform. The processed spectral signals after subtracting the mean envelope and the unwrapped phase are shown in Fig. 4. Significant nonlinearities can be observed in the plots of the unwrapped phase and the wavenumber, which may affect measurement of the absolute distance.

Additionally, the absolute distances at 20 different positions in 1 µm increments were simulated. The simulation results for the absolute distance in Fig. 5 show that the intensity difference may lead to inaccurate measurement of the absolute distance. (The errors indicate the difference between the calculated value and the ideal simulation value.) It can also be seen from Fig. 5 that it has a larger influence on measurement results close to the absolute distance of zero. The reason is that the nonlinearity of the unwrapped phase has a certain periodicity (see Fig. 4), so the errors also have a certain periodicity (see Fig. 5). The nonlinearity of the unwrapped phase close to the absolute distance of zero contains fewer periods than the nonlinearity far away from the absolute distance of zero. That is to say, the results close to the absolute distance of zero are more susceptible to the nonlinearity of the unwrapped phase. To sum up, the spectral intensity difference needs to be minimized to improve the measurement accuracy of the absolute distance.

### 2. Spectral Envelope Shape Difference

Spectral interferometric signals of the two-step phase-shifting algorithm with a $\pi /{2}$ phase shift at the absolute distance position of 10 µm were simulated for two cases to demonstrate the effects of difference in the envelope shape. For the first case, the envelope shapes of the spectral signals were uniform and Gaussian with a central wavelength of 640 nm. In the second case, the envelope shapes were both Gaussian but with different central wavelengths (640 and 680 nm). Figures 6(a) and 6(c) show the processed spectral signals after subtracting the mean envelope, and Figs. 6(b) and 6(d) indicate that there were nonlinearities in the plots of the unwrapped phase and the wavenumber, which may affect measurement of the absolute distance. The envelope shape difference is also an intensity difference, and it can be seen from Figs. 6(b) and 6(d) that the nonlinearities are more obvious for the part with the larger intensity difference.

Additionally, the absolute distances at 20 different positions in 1 µm increments were simulated. The simulation results for the absolute distance in Fig. 7 show that the envelope shape difference may also lead to inaccurate measurement of the absolute distance and has a larger influence on measurement results close to the absolute distance of zero. Therefore, the spectral envelope shape difference also needs to be minimized to improve the measurement accuracy of the absolute distance.

### 3. Normalization of Spectral Signal

The differences in the intensity and envelope shape of the spectral interferometric signals may affect measurement of the absolute distance; hence, they need to be minimized to improve the measurement accuracy. The optical elements in the system need to be adjusted to ensure the spectral signals are consistent, but it is difficult to obtain consistent spectral signals only via hardware adjustment. Therefore, a normalization process for the spectral signals is needed to minimize the influence of the differences in intensity and envelope shape. Some of the simulated spectral signals with differences in the intensity and envelope shape from Sections 4.B.1 and 4.B.2 have been normalized. The processed spectral signals and unwrapped phase with and without the normalization process are shown in Fig. 8. It can be seen that the normalization process can reduce the nonlinearities between the unwrapped phase and the wavenumber, i.e., improve the measurement accuracy.

Additionally, the absolute distances at 20 different positions in 1 µm increments were simulated. The simulation results for the absolute distance with and without the normalization process in Fig. 9 demonstrate that normalization can effectively minimize the influence of differences in the intensity and envelope shape of the spectral signals and improve the measurement accuracy of the absolute distance, especially for measurement results close to the absolute distance of zero.

#### C. Influence of Random Noise

The spectral interferometric signals may contain random noise because of vibration of the experimental setup and an unsteady ambient temperature. The spectral interferometric signals of the two-step phase-shifting algorithm with a $\pi /{2}$ phase shift at the absolute distance position of 10 µm were simulated; their envelope shapes were both Gaussian but had different central wavelengths (640 and 680 nm). Gaussian noise with a signal-to-noise ratio (SNR) of 50, 40, and 30 dB was added to the spectral signals to illustrate the influence of random noise. Figure 10 shows that there were serious errors in the wrapped phase of the spectral signals on adding Gaussian noise with an SNR of 30 dB. The reason for this is that the spectral signals have spikes caused by the random noise, which may affect the extraction of the relative maximum and minimum points for obtaining the envelopes.

The absolute distances at 20 different positions in 1 µm increments were simulated, and the locally weighted regression smoother filtering method [27] was used to perform a filtering process. The simulation results for the absolute distance with and without the filtering process in Fig. 11 shows that the filtering process may improve the measurement accuracy of the absolute distance. Furthermore, it can be seen that the different absolute distance positions have different sensitivities to the random noise, and measurement results close to the absolute distance of zero are more sensitive to random noise. Therefore, the spectral signals need to be smoothed before extracting the envelopes to reduce the influence of the spikes and improve the robustness of the algorithm.

#### D. Influence of Phase-Shift Error

The phase shift of $\pi /{2}$ in our system was obtained with an angle of $\pi /{4}$ between the two linear polarizers (Analyzer1 and Analyzer2 in Fig. 1); they usually have slight deviations because of installation errors of the optical elements and other factors. We simulated the spectral interferometric signals of the two-step phase-shifting algorithm with a phase shift that had a certain deviation around $\pi /{2}$ to illustrate the influence of phase-shift error. Figure 12 shows that the plots of the unwrapped phase and the wavenumber had slight nonlinearities when there was a certain deviation in the phase shift.

The absolute distances at 20 different positions in 1 µm increments were simulated; the results are shown in Fig. 13. It can be seen that the measurement of the absolute distance is not sensitive to the phase-shift error within a certain range, and the influence was in the nanometer or even subnanometer scale. This can be attributed to two reasons. One would be that the spectral signals were simultaneously acquired, and most of the common phase noise is eliminated [22]. The second is that the wave plates in the system are achromatic, i.e., the influence of the phase-shift error caused by factors such as installation errors at each wavelength is constant within the wavelength bandwidth used. Therefore, it has minimal effect on the slope of the linear function of the unwrapped phase $\phi $ and the wavenumber $k$. However, for the temporal phase-shifting method, the phase-shift error is mainly influenced by the nonlinearity and hysteresis of the phase shifter, which has different effects on each spectral signal and each wavelength. While some errors can be reduced by increasing the number of phase shifts or improving the algorithm, these methods require a large amount of data acquisition time or have substantial computational loads [28,29].

## 5. EXPERIMENTAL RESULTS

#### A. Absolute Distance Measurement

Measurements of the absolute distance were performed to verify the accuracy of the system, and a plano mirror of the same material as the reference mirror was used as a sample. The spectral interferometric signals of the two-step phase-shifting algorithm were collected from the spectrometers via optical fibers, as shown in Fig. 14(a). Figure 14(b) shows the processed spectral signals after the filtering and normalization processes. The wrapped and unwrapped phases were retrieved using the phase extraction algorithm, as shown in Figs. 14(c) and 14(d), respectively. Simultaneously, a PZT scanner (P-622.ZCD, PI) was used to drive the reference mirror and move it to 20 different positions along the optical axis in 1 µm increments to measure the absolute distances. The measurement results are shown in Fig. 15, and they confirmed the high accuracy of the measurement system for measuring the absolute distance. Figure 15(b) shows the differences in the absolute distance between adjacent positions and was calculated for comparison with 1 µm increments. It is worth noting that movement errors by the PZT scanner may also affect the measurements, but they are difficult to avoid or eliminate.

#### B. Comparison with Other Measurement Methods

The five-step phase-shifting algorithm and the Fourier transform method were also used to measure the absolute distance for comparison under the same experimental conditions as in Section 5.A, and the plots of the differences in the absolute distance between adjacent positions are shown in Fig. 16. Because the acquisition of five spectral signals effectively reduces the influence of factors such as intensity fluctuations and the phase-shift error, the five-step phase-shifting algorithm has a higher accuracy. The Fourier transform method has a relatively low accuracy because of spectral leakage, and the calculated results are related to the wavelength bandwidth and the window function used (the hamming window and a wavelength bandwidth of 535–760 nm were used in this work). Furthermore, the distance sign (positive or negative) cannot be determined. The two-step phase-shifting algorithm also has a relatively high accuracy. The spectral interferometric signals of the two-step phase-shifting algorithm in this work were acquired synchronously, so this method is suitable for application in precision industries and for ultrahigh-speed measurement.

## 6. CONCLUSION

In summary, a white-light spectral interferometer was presented in this work based on polarization interference to record synchronous phase-shifting signals. The system setup, the extraction process of phase information, and the realization of the phase shift were analyzed in detail. A variety of spectral interferometric signals were simulated based on the mathematical model of the two-step phase-shifting algorithm, and the following conclusions were drawn:

- (1) The differences in spectral intensity and envelope shape may lead to inaccurate measurement of the absolute distance, especially for measurements close to the absolute distance of zero; however, these errors can be minimized by the normalization process.
- (2) Random noise can cause serious errors in the measurement of absolute distance. To improve the measurement accuracy and the robustness of the algorithm, the spectral signals need to be smoothed to reduce the influence of the spikes caused by noise.
- (3) The measurement of the absolute distance is not sensitive to the phase-shift error within a certain range and is in the nanometer or even subnanometer scale.

In addition, measurements of the absolute distance were carried out to verify the accuracy of the measurement system using a PZT scanner. The results of the absolute distance indicated that the proposed system has high measurement accuracy. Furthermore, the five-step phase-shifting algorithm and the Fourier transform method were also used to measure the absolute distance for comparison. The results demonstrated that the five-step phase-shifting algorithm and the two-step phase-shifting algorithm have relatively high accuracy. Furthermore, the spectral interferometric signals for the two-step phase-shifting algorithm in this work were acquired synchronously; therefore, this method can be used in precision industries and for ultrahigh-speed measurement. Future work will involve the measurements of film thickness and of dynamically changing samples to verify and further explore the application of this study.

## Funding

National Key Research and Development Program of China (2018YFB1107600); 111 Project Fund (B07014); Engineering and Physical Sciences Research Council (EP/P006930/1).

## Disclosures

The authors declare no conflicts of interest.

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