## Abstract

From a sequence of simultaneous multi-wavelength phase-shifting interferograms (SMWPSIs), a novel single-wavelength phase retrieval method based on the least-squares iterative algorithm is proposed and utilized in dual-wavelength interferometry. Firstly, only one time phase-shifting procedure implements the phase shifts of all illumination wavelengths simultaneously, and then the accurate wrapped phases of each single-wavelength can be respectively retrieved from SMWPSIs by the least-squares iterative operation, so the phase of synthetic wavelength can be obtained by the subtraction easily. Using the proposed method, both the simulation and the experimental results demonstrate that the optical setup is simpler; the requirements for the displacement of the phase-shifting device and the number of the captured interferograms are smaller compared to the traditional phase-shifting multi-wavelength interferometry or off-axis multi-wavelength interferometry. Even in the case that the phase-shifts are unknown, the wrapped phases and the phase-shifts of each single-wavelength can be obtained by the proposed method.

© 2014 Optical Society of America

## 1. Introduction

Optical phase measuremet is very important in a number of engineering applications and science research. Interferometry or holography is a well known technique in optical phase measuremet with the advantages of noncontact, nondestruction, high accuracy and resolution [1–3]. Though single-wavelength interferometry(SWI) method is a better solution for the high accuracy surface measuement, it can be used only in the smooth surface with wavelength scale variation due to its measuring range between two adjacent pixals is less than half of a wavelength. Otherwise, the complicated phase unwrapping operation is still required in SWI. To adress this, multi-wavelength interferometry(MWI) is proposed and a lot of studies have been done [4–8]. Taking the dual-wavelength interferometry (DWI) as an example, the longer synthetic wavelength ${\Lambda}_{12}$ is constructed by two single-wavelength ${\lambda}_{1}$ and ${\lambda}_{2}$(${\Lambda}_{12}={\lambda}_{1}{\lambda}_{2}\text{\hspace{0.05em}}/\text{\hspace{0.05em}}\text{\hspace{0.05em}}\left|{\lambda}_{1}-{\lambda}_{2}\right|$) [9], so MWI or DWI should be a better candidate in the development and application of interferometry. Though multi-wavelength phase-shifting interferometry is a well-known method to overcome 2π ambiguity of single wavelength interferometry, the noise of phase retrieval is multiplied by synthetic wavelength. So synthetic wavelength measurement is suitbale for sorting the phase values of single-wavelength into the correct 2π interval. The height measurement is still based on the phase value related a single-wavelength [10].

In recent years, many MWI methods are developed [10–19]. Since the phase of synthetic wavelength can be obtained by a simple subtraction between the wrapped phases of single-wavelength easily, so the wrapped phase retrieval of single-wavelength is an essential content in MWI. Multi-wavelength in-line phase-shifting interferometry [10–12] is a powerful method with the advantages of the higher measuring accuracy [11], a better result in the case that the interference fringe contrast is poor and a higher space-bandwidth product of CCD sensor due to the in-line interferometry system is employed [13]. However, this method needs multi-wavelength illumination lasers and implements the phase-shifts of each wavelength in turn, so not only the phase-shifting procedure becomes very time-consuming and complicated, but its measuring accuracy will be more sensitive to the performance of the displacement actuator and the variation of the experimental environment. Single-shot multi-wavelength off-axis digital holography [14–16] can retrieve the wrapped phase of each wavelength from only one-frame off-axis interferogram by Fourier transform, so it is suitable for real-time and dynamic phase measurement. But it needs the complicated experimental setup and has the lower system space-bandwidth product, and its measuring accuracy is easily influenced by the inherent errors, such as the energy leakage of data discontinuity, which usually appears in the FFT-based spatial filtering technique. A approach named as the simultaneous two-wavelength Doppler phase-shifting digital holography is proposed by D. Barada [17], in which the phase shifts is produced by a reference mirror moving with a uniform velocity, and a sequence of simultaneous phase-shifting dual-wavelength interferograms (SPSDWIs) are captured by a monochrome CCD. Then the wrapped phases of each single-wavelength are retrieved from the complex amplitude located in the spectral peak by performing the temporal Fourier transform for each pixel of interferograms. Using this approach, the optical setup is greatly simplified and the noise induced by the environmental disturbance can be restrained effectively. However, to ensure the spectral peak separation, it is required that the moving distance of the reference mirror and the number of the captured interferograms should be large enough. In [18], a color CCD is introduced for recording simultaneous phase-shifting multi-wavelength interferograms, in which 3-group single-wavelength phase-shifting interferograms can be obtained by direct color separation. However, if the wavelength difference between illumination lasers is too small, this approach will not work well. Recently, a two-step demodulation algorithm based DWI is reported [19–21], in which only 5-frames SPSDWIs captured by a monochrome CCD are enough for the phase retrieval of single-wavelength, but this approach has an major drawback about the requirement of the special phase shifts.

The least-squares based advance iterative algorithm(AIA) is firstly proposed by Wang [22, 23]. In the case that the phase shifts are unknown, both the accurate phase and phase shifts can be retrieved through using only 3-frame phase-shifting interferograms, AIA has been successfully utilized in SWI.

In this study, by capturing a sequence of SMWPSIs, we propose a novel single-wavelength phase retrieval method based on the least-squares iterative algorithm. Using this method, only one time phase-shifting procedure implements the phase shifts of all illumination wavelengths simultaneously, and then the accurate wrapped phases of each single-wavelength can be respectively retrieved from SMWPSIs by the iterative operation, so the phase of synthetic wavelength can be obtained by the subtraction easily. As an example, this proposed method is utilized in DWI to extract the wrapped phases and the phase-shifts of each single-wavelength simultaneously. Following, we first introduce the principle of the proposed method, then present the simulation and the experimental results.

## 2. Principle

In multi-wavelength phase-shifting interferometry, a single-wavelength interferogram can be described by the intensity of each pixel as

where*l*,

*m*and

*n*respectively denote the wavelength, the pixel position and the sequence number of phase-shifting interferogram;${a}_{lm}$and${b}_{lm}$represent the background and the modulation, respectively;${\phi}_{lm}$denote the measured phase and ${\delta}_{ln}$ is the phase-shifts of the

*n*th phase-shifting interferogram .

Assuming multi-wavelength illumination lasers simultaneously go through the same in-line phase-shifting interferometry system, a sequence of simultaneous multi-wavelength phase-shifting interferograms (SPSMWIs) are captured by a monochrome CCD camera. Thus, the total intensity of the *n*th interferogram can be expressed as the incoherent superposition of all single-wavelength interferograms

_{,}$c{\text{'}}_{1ln}^{}={b}_{lm}\mathrm{cos}{\delta}_{ln}$and$c{\text{'}}_{2ln}^{}=-{b}_{lm}\mathrm{sin}{\delta}_{ln}$. Note that Eq. (3) is employed for the phase retrieval by the intensity change of different interferogram in the same pixel, and Eq. (4) is used for the phase shifts extraction by the intensity change of all pixels in the same interferogram.

Assuming ${i}_{mn}^{}$ represents the practical intensity of interferogram; the least-squares error ${E}_{n}$ accumulated from the same pixel of all interferograms in Eq. (3) can be expressed as

*n*th interferogram in Eq. (4) can be expressed as

*L*, both$A$ and $A\text{'}$ are the real symmetric matrix of$(2L+1)\times (2L+1)$;$C$, $C\text{'}$, $I$ and $I\text{'}$ are the real matrix of $(2L+1)\times 1$. In Eq. (9),$A$ can be expressed as

*T*denotes the corresponding transposed matrix. Thus,

*C*can be determined from Eq. (9) byAnd then the wrapped phase of single-wavelength can be obtained bySimilarly,$C\text{'}$can be determined from Eq. (10)And then the phase-shifts of single-wavelength can be obtained by

From the above derivation, it is presented that both the wrapped phases and the phase-shifts of each single-wavelength can be simultaneously determined by the least-square iterative algorithm. If the phase-shifts of each single-wavelength are known, the wrapped phases of each single-wavelength can be directly calculated by Eqs. (17) and (18). Otherwise, based on the iterative operation in Eqs. (17)-(20) until the phase-shifts reach the convergence limit, the accurate wrapped phases and phase-shifts of each wavelength also can be obtained. Of course, there are several aspects should be illuminated: (1) To guarantee that both$A$ and$A\text{'}$ are nonsingular matrixes, it is required that the number of the captured interferograms must be more than 2*L* + 1; (2) In Eqs. (17) and (18), it is required that the background intensity and the modulation in all illumination lasers are unchanged during the phase–shifting procedure; (3) In Eqs. (19) and (20), it is required that the modulation of each pixel in all illumination lasers are unchanged during the phase–shifting procedure. For convenience, we give a detail procedure of the phase retrieval by the least-square iterative algorithm as following:

Step1. Set up the initial phase-shifts of each single-wavelength and the convergence threshold value. To reduce the number of the iterative operation and ensure the high calculation accuracy, it is required that the initial phase-shifts should be set as far as possible the same with the actual phase-shifts. In general, the accurate wrapped phases of each single-wavelength can be obtained when the phase-shifts of each wavelength reach the convergence threshold value. The convergence condition of the phase-shifts can be expressed as

*ε*denotes the preset convergence threshold value;

*k*is the number of the iterative operation

_{.}If the convergence conditions of the phase-shifts at each wavelength are satisfied, the iterative operation will be broken.

Step2. Calculate the wrapped phase of each single-wavelength by Eqs. (17) and (18).

Step3. Calculate the phase-shifts of each single-wavelength by Eqs. (19) and (20).

Step4. If the condition of Eq. (21) is satisfied, the wrapped phase of single-wavelength can be determined in Step 1. Otherwise, repeat step2 and step3.

## 3. Numerical simulation

Numerical simulation is carried out to verify the performance of the proposed method, in which two illumination lasers with wavelength of *λ*_{1} = 532nm and *λ*_{2} = 632.8nm are employed to record a sequence of SPSDWIs. To simulate the practical experimental environment, the measured phase, background intensity and modulation of the simulated SPSDWIs are from the experimental interferograms. In addition, the preset phase-shifts are used as the theoretical value and 5-frame SPSDWIs are chosen in this numerical simulation. The convergence threshold value is set as$\epsilon \le 0.01$.

Figure 1(a) shows one of experimental single-wavelength phase-shifting interferograms with size of 256 × 256 pixels at 532nm, its wrapped phase retrieved by 5-step phase-shifting algorithm; background and modulation are respectively given in Figs. 1(b)-1(d). Similarly, we also give the corresponding results at 632.8nm, as shown in Figs. 1(e)-1(f).

Figure 2 presents a sequence of 5-frame simulated SPSMWIs, in which the measured phase, background and modulation of each wavelength are obtained from the experimental interferograms, and the phase-shifts are man-made. We can see that these simulated SPSMWIs are nearly the same with the above experimental interferograms. In Figs. 2(a)-2(e), the phase-shifts are set as 0, 1.571, 3.142, 4.712 6.283rad at 632.8nm and 0, 1.868, 3.737, 5.605, 7.474rad at 532nm, respectively.

Following, using the above 5-frame SPSDWIs and the proposed algorithm, Figs. 3(a) and 3(b) respectively give the retrieved wrapped phases at 532nm and 632.8nm, and the corresponding differences between the retrieved wrapped phases and the reference phases are respectively shown in Figs. 3(c) and 3(d). In addition, the initial phase-shifts are set as 0.300, 2.168, 4.037, 5.905, 7.774rad at 532nm and 0.300, 1.870, 3.442, 5.012, 6.5832rad at 632.8nm, and the convergence threshold value is $\epsilon $ = 0.01rad, and the number of the iterative operation is equal to 7 when the convergence threshold value is satisfied. It is found that the root-mean-square error (RMS) of the difference between the reference phases and the retrieved wrapped phases are less than 0.03rad at 532nm or 632.8nm, indicating that the accurate wrapped phases of single-wavelength have been obtained by the proposed algorithm.

Table 1 gives the extracted phase-shifts and the difference between the extracted phase-shifts and the preset value. It is found that the RMS of the difference between the extracted phase-shifts and the preset value are less than 0.02 rad at 532nm or 632.8nm, indicating that the accurate phase-shifts of each single-wavelength is also obtained by the proposed algorithm.

Figure 4 shows the phase map of synthetic wavelength, in which Gauss smoothing operation is used. Subsequently, by a simple subtraction between Figs. 3(a) and 3(b), Fig. 4(a) gives the phase of synthetic wavelength by proposed algorithm, and the reference phase of synthetic wavelength is obtained by a subtraction between Figs. 1(b) and 1(f), as shown in Fig. 4(b). Moreover, Fig. 4(c) presents the phase difference between Figs. 4(a) and 4(b), and it is found that the RMS of the difference between the reconstructed phase and the reference phases is only 0.03rad.

## 4. Experimental research

A Mach-Zehnder interferometer based dual-wavelength in-line phase-shifting interference system, as shown in Fig. 5, is built up to verify the proposed method. A diode-pumped solid-state laser with wavelength λ1 = 532nm (green light) and a He-Ne laser with wavelength λ2 = 632.8nm (red light) are utilized as illumination sources, and a PZT is employed as the phase-shifting inducer. And the interferograms are captured by monochrome CCD with size of 768 × 576 pixels (7.68mm × 5.76mm).

Firstly, to present the actual accuracy of the proposed method, two object beams go through an optical flat and a microscope, and interference with the reference beam to form the interference ring, a sequence of single-wavelength phase-shifting interferograms, as well as the corresponding SPSDWIs are respectively captured. The results obtained from SPSDWIs are compared with the corresponding results obtained by the traditional phase-shifting DWI. Figures 6(a) and 6(b) respectively give one of single-wavelength phase-shifting interferograms at 532nm or 632.8nm. And the wrapped phases of single-wavelength retrieved by the AIA algorithm [22,23] are used as the reference. In addition, Fig. 6(c) also shows one of SPSDWIs.

Following, Figs. 7(a) and 7(b) respectively show the wrapped phase retrieved from 16-frame single-wavelength phase-shifting interferograms at 532nm and 632.8nm, and Fig. 7(c) gives the phase map of synthetic wavelength obtained from Figs. 7(a) and 7(b). In contrast, the wrapped phase at 532nm and 632.8nm retrieved from 16-frame SPSDWIs are respectively shown in Figs. 7(d) and 7(e), and Fig. 7(f) gives the phase map of synthetic wavelength obtained from Fig. 7(d) and 7(e). It is found that the RMS of the difference between Figs. 7(a) and 7(d), 7(b) and 7(e), 7(c) and 7(f) are respectively 0.036rad, 0.051rad, 0.056rad, indicating that the results by the proposed method are nearly the same with the traditional phase-shifting DWI method. Finally, we also compare the uncertainty of the results by the proposed method and the traditional phase-shifting DWI method. As shown in Fig. 8, the blue, red, green curves respectively represent the height distribution obtained by the proposed method, the traditional DWI method at 532nm and 632.8nm, in which Fig. 8(a) gives the height distribution of the 200th line in Fig. 7(a)-7(c) and Fig. 8(c) shows the height distribution of the 200th line in Fig. 7(d)-7(f). In addition, Figs. 8(b) and 8(d) respectively show the magnification of the black dotted line area Figs. 8(a) and 8(c). Since there is no the requirement for the phase unwrapping, it is observed that the DWI based on the subtraction of phase values related to different wavelengths extend the unambiguity range while the uncertainty of the synthetic wavelength height measurement is increased. From the above results, we can see that the uncertainty of the experimental results is much higher compared to the simulation results. Although we have tried our best to make the simulation interference pattern close to the experimental interferograms, many uncertainty factors in the experiment cannot be simulated, such as the air turbulence, mechanical vibration, interferometry random noise. All these factors will lead to that the phase shifts are slightly changed with the position of interferogram. Meanwhile, the background variation of interferogram usually exists in the experiment. Therefore, the noise in the captured experimental interferogram will be higher than that in the simulated interference fringe, and this will lead to the uncertainty of the experimental result is much higher than the simulation result.

Next, a vortex phase plate (RPC photonics corp., VPP-1c) with 1.53um height is chosen as the measured object. Firstly, two lasers simultaneously go through the above in-line interferometry system to form dual-wavelength interferograms. Secondly, by introducing phase-shifting technique, a sequence of SPSDWIs is captured by a monochrome CCD with size of 768 × 576 pixels (7.68mm × 5.76mm). In this study, to eliminate the additional phase introduced by the microscope, a sequence of 5-frame SPSDWIs with and without the measured object are respectively captured, as shown in Figs. 9 and 10, in which the phase-shifts are uniformly distributed in 2$\pi $rad at 632.8nm. Thus, using the double-exposure method, we first retrieve the additional phase and the total wrapped phase of each wavelength from Figs. 9 and 10, respectively. And then the actual wrapped phases of each wavelength can be obtained after removing the additional phase.

Subsequently, the height of vortex phase plate can be determined, in which the convergence threshold value is set as 0.01rad. Using an approximate evaluation, the initial phase-shifts of the iterative operation are set as 0, 1.87, 3.74, 5.61, 7.47rad at 532nm and 0, 1.57, 3.14, 4.71, 6.28rad at 632.8nm. Next, by performing the iterative operation in Fig. 9, the additional phases at 532nm and 632.8nm are respectively presented in Fig. 11(a)-11(b), and the extracted phase-shifts are equal to 0, 1.97, 4.01, 5.95, 7.89rad at 532nm and 0, 1.62, 3.33, 4.92, 6.49rad at 632.8nm, respectively. We can see that the average ratio of the phase-shifts at 532nm and 632.8nm is equal to 1.211, and the difference between the average ratio of the phase-shifts and the ratio of wavelengths (632.8nm/532nm = 1.189) is only 0.02. Moreover, from Fig. 10, we can obtain the total wrapped phases of vortex phase plate including the additional phase at 532nm and 632.8nm, as shown Figs. 11(c)-11(d), the extracted phase-shifts are 0, 2.11, 4.05, 6.02, 7.96rad at 532nm and 0, 1.76, 3.42, 5.06, 6.70rad at 632.8nm. The average ratio of the phase-shifts at 532nm and 632.8nm is 1.190, and the difference between the average ratio of the phase-shifts and the ratio of wavelengths (632.8nm/532nm = 1.189) is only 0.001. Results above exhibit that the accurate phase-shifts of each single-wavelength have been obtained by the proposed method.

Following, Figs. 12(a) and 12(b) respectively give the actual wrapped phases of vortex phase plate at 532nm and 632.8nm after removing the additional phase. Then, as shown in Fig. 12(c), the height map of vortex phase plate can be obtained by the following expression. Figure 12 (d) shows the height distribution of the 200th line in Fig. 12(c).

Finally, we increase the number of the captured SPSDWIs to 20 frames, in which the phase-shifts are also uniformly distributed in 10$\pi $at 632.8nm. Figures 13(a) and 13(b) respectively give the actual wrapped phases of vortex phase plate at 532nm and 632.8nm after removing the additional phase. Finally, as shown in Fig. 13(c), the height map of vortex phase plate can be obtained by the Eq. (22). Figure 13(d) shows the height distribution of the 200th line in Fig. 13(c). As shown in Fig. 13, we can see that the random noise appeared in Fig. 12 has been restrained effectively, indicating that the ability of noise reduction with the proposed method is greatly improved as the number of the captured interferograms and the phase-shifting period increase.

Clearly, by combining a sequence of 5-frame experimental SPSDWIs with the least-squares iterative algorithm, only one time phase-shifting procedure implements the phase shifts of all illumination wavelengths simultaneously, and then the accurate wrapped phases of each single-wavelength can be respectively retrieved from SMWPSIs by the iterative operation, so the phase of synthetic wavelength can be obtained by a simple subtraction between the wrapped phases of single-wavelength. Specially, it is found that along with the number of the captured interferograms and the phase-shifting period are increased, using the proposed method, the random noise induced by the external disturbance is greatly reduced, so the accuracy of phase retrieval is greatly improved.

## 5. Conclusion

In this study, by capturing a sequence of simultaneous multi-wavelength phase-shifting interferograms (SMWPSIs), a novel single-wavelength phase retrieval method based on the least-squares iterative algorithm is proposed and utilized in DWI. Firstly, by one time phase-shifting procedure, the phase shifts of all illumination wavelengths can be implemented simultaneously and a sequence of SPSDWIs are captured by a monochrome CCD. And then the accurate wrapped phase of each single-wavelength can be respectively retrieved from SMWPSIs, so the phase of synthetic wavelength can be obtained by a subtraction between the wrapped phases of single-wavelength easily. Compared with the Fourier transform based off-axis interferometry [16], in the proposed approach, it is exhibited that not only the frequency spectrum of the measured object is expanded, but the noise induced by the external disturbance is decreased. Moreover, since only one time phase-shifting procedure implements the phase shifts of all illumination wavelengths simultaneously, so the measuring optical setup is simpler and the condition requirements for the displacement of the phase-shifting device and the number of the captured interferograms are smaller compared to the traditional phase-shifting DWI [11] and Doppler phase-shifting DWI [17]. Specially, along with the number of the captured interferograms and the phase-shifting period are increased, it is found that the accuracy of phase retrieval is greatly improved. In a word, due to the advantages of convenient performance, rapid speed and high accuracy, this proposed approach will supply a useful tool for supply a useful tool for the single-wavelength phase retrieval of DWI or MWI.

## Acknowledgments

This work is supported by National Nature Science Foundation of China grants (61177005, 61275015 and 61475048).

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