## Abstract

This paper presents the derivation of a *17*-sample phase-shifting algorithm that can compensate the miscalibration and first-order nonlinearity of phase shift error, coupling error, and bias modulation of the intensity and satisfy the fringe contrast maximum condition. The phase error of measurements performed using the *17*-sample algorithm is discussed and compared with those of measurements obtained using other algorithms. Finally, the optical thickness variation of a BK7 optically transparent plate obtained using a wavelength tuning Fizeau interferometer and the *17*-sample algorithm are presented. The experimental results indicate that the optical thickness variation measurement accuracy for the BK7 plate was 3 nm.

© 2015 Optical Society of America

## 1. Introduction

Phase-shifting interferometry has been used to estimate the surface shapes and optical thickness variations of optical devices for the last few decades [1]. In phase-shifting interferometry [2], the phase difference between a sample beam and a reference beam is varied by phase shifting, and the irradiance signal distribution is acquired at each step or bucket [3] in the frame memory of a computer. The phase distributions of interference fringe patterns can be calculated by using phase-shifting algorithms.

Phase-shifter miscalibration and nonlinearity are the most significant sources of error in high-precision optical interferometers [3]. When measuring highly reflective samples using phase-shifting Fizeau interferometry, the effects of the higher harmonics increase [4], and the coupling error between the higher harmonics and phase-shift error can be the source of systematic error in the evaluated phases. Another possible source of systematic error is the bias modulation of intensity. This problem can occur when laser diodes whose powers may be modulated due to current variations are involved. In addition, the fringe constant maxima resulting from phase-shifting algorithms should occur when there is no phase-shift miscalibration, which is called the fringe contrast maximum condition [5].

Many studies have been reported on error-compensation algorithms [6–22]. The Schwider-Hariharan *5*-sample algorithm [6, 7] can compensate for phase-shift miscalibration but not for coupling error. Larkin and Oreb derived a symmetric (*N* + 1)-sample algorithm [8] based on a Fourier representation [23], and Schmit and Creath developed *5*-sample and *6*-sample algorithms [11] based on the extended averaging method. However, these algorithms do not satisfy the fringe contrast maximum condition [5]. Surrel developed a windowed discrete Fourier transform (DFT) phase-shifting algorithm [13] that can compensate for phase-shift miscalibration and coupling error using a characteristic polynomial. However, this algorithm cannot compensate for nonlinearity in the phase-shift error [12, 17]. Onodera derived a *6*-sample algorithm insensitive to bias modulation of the intensity [14], and Surrel described this insensitivity using characteristic polynomial theory [16]. A *12*-sample algorithm proposed by Surrel [15] can compensate for linear miscalibration, first-order phase-shift nonlinearity, and coupling error; however, this algorithm dose not satisfy the fringe contrast maximum condition.

We already developed a 4*N* – 3 algorithm [22] that can compensate for up to second-order phase-shift nonlinearity and for coupling between the harmonics and phase-shift error. However, the 4*N* – 3 algorithm is difficult to utilize for actual measurements in the manufacturing industry. In [22], 61 images were acquired for measurement, though it is difficult to acquire so many images in the glass manufacturing industry. In actual interferometric measurements, it is preferable to use as few images as possible to reduce cost, time, and technical problems.

This paper presents the derivation of a *17*-sample phase-shifting algorithm that can compensate for miscalibration and first-order nonlinearity in the phase shift, coupling between the harmonics and phase-shift miscalibration, and bias modulation of the intensity and can satisfy the fringe contrast maximum condition. It is shown that the *17*-sample algorithm yields the smallest phase error among the conventional phase-shifting algorithms, except for the 4*N* – 3 algorithm. Finally, the optical thickness variation of a BK7 optical parallel plate is presented, as measured using a wavelength tuning Fizeau interferometer and the *17*-sample algorithm.

## 2. Derivation of *17*-sample phase shifting algorithm

#### 2.1 Phase-shifting algorithm and characteristic polynomial

In a laser Fizeau interferometer, the signal irradiance *I*(*α _{r}*) in the interference fringe pattern during phase shifting is given by [24]

*α*is a phase-shift parameter, and

_{r}*A*,

_{m}*φ*, and

_{m}*γ*are the amplitude, phase, and fringe contrast of the

_{m}*m*

^{th}harmonic component. Considering an

*M*-sample phase-shifting algorithm, where the reference phases are separated by

*M*– 1 equal intervals of

*δ*= 2π/

*N*rad and

*N*is an integer, a general expression for the calculated phase in this algorithm is given by

*a*and

_{r}*b*are the

_{r}*r*

^{th}sampling amplitudes, and

*I*(

*α*) is given by Eq. (1).

_{r}When the phase shift is nonlinear, each phase-shift value *α _{r}* is a function of the phase-shift parameter. The phase shift value for the r

^{th}sample can be expressed as a polynomial function of the unperturbed phase-shift value

*α*

_{0}

*as [17]*

_{r}*p*(

*p*≤

*m*– 1) is the maximum order of the nonlinearity,

*ε*

_{0}is the error coefficient of the phase-shift miscalibration,

*ε*(1 ≤

_{q}*q*≤

*p*) is the error coefficient of the

*q*

^{th}nonlinearity of the phase shift, and

*α*

_{0}

*= 2π[*

_{r}*r*– (

*M*+ 1)/2]/

*N*is the unperturbed phase shift.

The error Δ*φ* in the calculated phase is a function of the amplitude ratios *A _{m}*/

*A*

_{1}and the error coefficients

*ε*and can be Taylor-expanded as

_{q}*k*= 2, 3, …,

*m*and

*q*= 0, 1, …,

*p*. In Eq. (4),

*ο*(

*A*),

_{k}*ο*(

*ε*), and

_{q}*ο*(

*A*) denote, respectively, the error in the harmonics, the phase-shift error, and the coupling between the harmonics and the phase-shift error.

_{k}ε_{q}Systematic approaches for deriving error-compensation algorithms have been proposed by several authors based on the averaging method [6, 11], linear equations [10, 17], and characteristic polynomial theory [13]. Surrel proposed using characteristic polynomial theory to design and estimate the phase-shifting algorithms [13, 15, 16]. The characteristic polynomial *P*(*x*) of a phase-shifting algorithm is defined by

*i*is the imaginary unit, and

*x*is defined as

*x*= exp(

*imδ*).

Surrel reported that the polynomial root locations and multiplicities in the characteristic diagram indicate the insensitivity of the higher harmonics and phase-shift errors. To suppress the *m*^{th} harmonic component, the characteristic polynomial of the phase-shifting algorithm should have single roots in the characteristic diagram [2, 13]. Similarly, to compensate for phase-shift miscalibration, the characteristic polynomial should have a double root at *m* = −1 in the characteristic diagram [8, 9, 13]. For example, the 2*N* – 1 windowed algorithm proposed by Surrel [13] can compensate for harmonics up to the (*N* – 2)^{th}, as well as for phase-shift miscalibration.

#### 2.2 17-sample phase-shifting algorithm

To utilize interferometric measurements in the actual manufacturing industry, the phase-shifting algorithm should have at least the characteristics listed below.

- A. Insensitivity to
i. the

*m*^{th}harmonic components*ο*(*A*)_{m}ii. the phase-shift miscalibration

*ο*(*ε*_{0}) and first-order nonlinearity*ο*(*ε*_{1})iii. the coupling error between the harmonics and phase-shift miscalibration

*ο*(*A*_{m}ε_{0})iv. the bias modulation of intensity

- B. Satisfaction of the fringe contrast maximum condition

To ensure fulfillment of the above requirements, the characteristic diagram proposed by Surrel [13] was used, and to measure the optical thickness deviation of the transparent plate, the phase division number *N* was set to 8 [18].

Figure 1 shows the design of a phase-shifting algorithm that satisfies the above requirements. First, to suppress the *m*^{th} harmonic components *ο*(*A _{m}*) (

*m*≤ 6) of Eq. (4), the single roots should be located on the characteristic diagram, except at

*m*= 1 (

*x*= exp(

*iδ*)), as shown in Fig. 1(a). The phase-shifting algorithm of Fig. 1(a) is the synchronous detection method derived by Bruning [2]. To suppress the linear miscalibration

*ο*(

*ε*

_{0}) and first-order nonlinearity of the phase-shift

*ο*(

*ε*

_{1}) in Eq. (4), a triple root should be located at

*m*= −1 (

*x*= exp(-

*iδ*)) on the characteristic diagram, as shown in Fig. 1(b). The (

*N*+ 1)-sample algorithm derived by Surrel [9], which is insensitive to linear miscalibration of the phase shift, has a double root at

*m*= −1 on the characteristic diagram. Next, to suppress the couplings

*ο*(

*A*

_{m}ε_{0}) between the

*m*

^{th}harmonics and the phase-shift miscalibration

*ε*

_{0}, double roots should be located at

*m*= ± 2, ± 3, and 4 (

*x*= exp( ± 2

*iδ*),

*x*= exp( ± 3

*iδ*), and

*x*= exp(4

*iδ*), respectively) on the characteristic diagram. The couplings

*ο*(

*A*

_{5}

*ε*

_{0}) and

*ο*(

*A*

_{6}

*ε*

_{0}) can be compensated for by designing double roots at

*m*= −2 and

*m*= −3 respectively, as in Fig. 1(c) [13]. The intensity modulation bias during phase shifting can be suppressed by including a double root at

*m*= 0 (

*x*= 1) on the characteristic diagram [16], as in Fig. 1(d). Finally, to satisfy the fringe contrast maximum condition, a triple root should be located at

*m*= 3 (

*x*= exp(3

*iδ*)) on the characteristic diagram, as in Fig. 1(e). The fringe contrast maximum can be obtained by locating the roots on the characteristic diagram so that the symmetrical condition about the line connecting the point of

*m*= 1 and

*m*= 5 of Fig. 1(e) with locating the double root or more on the position of

*m*= −1.

Based on Fig. 1(e), the characteristic polynomial *P*(*x*) of the new *17*-sample phase shifting algorithm is given by

*ζ*= exp(

*i*π/4). Expanding the characteristic polynomial defined by Eq. (6) and arranging the coefficients, the sampling amplitudes

*a*and

_{r}*b*of the

_{r}*17*-sample phase shifting algorithm can be obtained from

#### 2.3 Fourier representation of phase-shifting algorithm

The phase-shifting algorithm can be visualized and well understood by obtaining Fourier representations of the sampling functions of the algorithm [23]. The sampling functions in the frequency domain for the numerator and denominator of the *M*-sample algorithm given by Eq. (2) are respectively defined as

*ν*is the frequency. To ensure symmetric sampling amplitudes,

*F*

_{1}and

*F*

_{2}must be purely imaginary and purely real functions, respectively [10, 17]. The design requirements mentioned above can also be represented using the sampling functions in the frequency domain, as shown below [5, 8, 10, 13, 16, 17].

- A. Insensitivity to
i. the

*m*^{th}harmonic components [10, 13]:$$i{F}_{1}\left(\nu \right)={F}_{2}\left(\nu \right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\nu =0,2,\cdots ,6\right).$$ii. the miscalibration and first-order nonlinearity of the phase-shift [8, 10, 13, 17]:

$${\frac{\text{d}i{F}_{1}\left(\nu \right)}{\text{d}\nu}|}_{\nu =1}={\frac{\text{d}{F}_{2}\left(\nu \right)}{\text{d}\nu}|}_{\nu =1},$$$${\frac{{\text{d}}^{2}i{F}_{1}\left(\nu \right)}{\text{d}{\nu}^{2}}|}_{\nu =1}={\frac{{\text{d}}^{2}{F}_{2}\left(\nu \right)}{\text{d}{\nu}^{2}}|}_{\nu =1}.$$iii. the coupling between the harmonics and phase-shift miscalibration [17]:

$$\frac{{\text{d}}^{2}i{F}_{1}\left(\nu \right)}{\text{d}{\nu}^{2}}=\frac{{\text{d}}^{2}{F}_{2}\left(\nu \right)}{\text{d}{\nu}^{2}}=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\nu =2,3\cdots ,6\right).$$iv. the bias modulation of the intensity [14, 16]:

- B. Satisfaction of the fringe contrast maximum condition [5]:

*iF*

_{1}and

*F*

_{2}of the derived

*17*-sample algorithm.

As shown in Fig. 2, the gradients of the sampling functions *iF*_{1} and *F*_{2} at the fundamental frequency *ν* = 1 are zero, confirming the fulfillment of requirements A-ii and B. The values and gradients of the sampling functions at *ν* = 2, 3, …6 are zero, indicating the satisfaction of requirements A-i and A-iii. Finally, the gradients of the sampling functions at *ν* = 0 are zero, indicating the fulfillment of requirement A-iv.

## 3. Estimation of *17*-sample phase-shifting algorithm

When estimating the phase error by using the phase-shifting algorithm, not only the phase-shift error defined by Eq. (3), but also the coupling error, should be considered [24, 25]. de Groot analyzed the calculated phase error due to phase-shift miscalibration and also the error introduced by coupling between the higher harmonics and phase-shift miscalibration [24]. The RMS phase error *σ _{mis}* resulting from the phase-shift miscalibration is given by

The RMS phase error *σ _{cou}* resulting from the coupling between the

*m*

^{th}harmonics and phase-shift miscalibration is given by

*γ*is the fringe contrast of the

_{m}*m*

^{th}harmonics. Therefore, the net RMS error is given by [24]where

*σ*and

_{mis}*σ*are obtained from Eqs. (17) and (18).

_{cou}Figures 3(a) and 3(b) show the solutions of Eq. (19) based on the phase-shifting algorithms that are listed in Table 1, for a reference surface reflectivity of 4% and sample reflectivities of 4% (BK7) and 30% (silicon wafer). And Fig. 4 shows the characteristic diagram of the phase-shifting algorithms listed in Table 1.

When measuring a spherical surface using a Fizeau interferometer, phase-shift miscalibration is particularly important because of the curvature of the reference surface for a spherical sample and because the phase shift caused by axial translation is not uniform throughout the field of view [26]. In this case, the phase-shift miscalibration *ε*_{0} was previously estimated to be approximately ± 30% [27]. Figure 3 shows that the *17*-sample algorithm can suppress the RMS phase error better than any other algorithm can, even for a phase-shift miscalibration of ± 30%.

When a phase-shift miscalibration is 30%, the synchronous detection algorithm proposed by Bruning [2] shows the largest RMS errors because it cannot compensate for phase-shift and coupling errors. Larkin-Oreb *N* + 1 algorithm has the compensation ability for the phase shift miscalibration *ε*_{0} of Eq. (3), however this algorithm does not have the compensation ability for the coupling errors shown in Fig. 4(c). Surrel *12*-sample algorithm has the similar characteristics with the developed *17*-sample algorithm, having the compensation ability for the phase shift error up to the 1st order nonlinearity *ε*_{1} of phase shift and coupling errors shown in Fig. 4(a) and (d). 2*N* – 1 algorithm developed Hanayama has the imperfect compensation ability for the coupling error because this algorithm has the single root of the position *m* = 2 on the characteristic diagram shown in Fig. 4(e).

## 4. Experimental measurement of optical thickness variation

#### 4.1 Wavelength tuning Fizeau interferometer

The optical thickness variation of a BK7 plate 12 mm thick and 80 mm in diameter was measured using a wavelength tuning Fizeau interferometer and the *17*-sample algorithm. Fizeau interferometer has the smallest influence of air turbulence among the conventional interferometer. Figure 5 shows the optical setup for measuring the optical thickness variation of the BK7 transparent plate using the Fizeau interferometer. The temperature inside the laboratory was 20.5 °C, and the source was a tunable diode laser with a Littman external cavity (New Focus TLB–6300–LN) consisting of a grating and a cavity mirror. The source wavelength was scanned linearly in time from 632.8 nm to 638.4 nm, translating the cavity mirror at a constant speed using a piezoelectric (PZT) transducer and picomotor [28].

The beam was transmitted using an isolator and was divided into two by a beam splitter: one beam was sent to a wavelength meter (Anritsu MF9630A), which was calibrated using a stabilized HeNe laser with an accuracy of δ*λ*/*λ* ~10^{−7} at a wavelength of 632.8 nm, and the other was incident on the interferometer. The focused output beam was reflected by a polarization beam splitter. The linearly polarized beam was then transmitted to a quarter-wave plate to form a circularly polarized beam. This beam was collimated to illuminate the reference surface and measurement sample. The reflections from the measurement sample and reference surfaces were sent back along the same path and were then transmitted through the quarter-wave plate again to achieve orthogonal linear polarization. The resulting beams passed through the polarization beam splitter and combined to generate a fringe pattern on the screen, with a resolution of 640 × 480 pixels.

#### 4.2 Results and error analysis

Figure 6(a) shows a laboratory photo of the BK7 transparent plate in the wavelength tuning Fizeau interferometer, and Fig. 6(b) shows an observed raw interferogram of the sample at a wavelength of 632.8 nm.

The necessary range for wavelength scanning δ*λ* was calculated as

The wavelength was finely scanned from 632.9123 nm to 632.9619 nm, and 17 interference images were recorded at equal intervals. Each experiment took 15 seconds to acquire 17 images. The slow image acquisition rate mainly resulted from the slow data transfer rate from the charge-coupled-device camera to the computer. During the scanning process, the signal interference fringes shifted by three periods, for a total shift of 6π radians. The phase shift for each step was π/4 due to the optical thickness of the BK7 plate. Because there was an approximately 3% nonlinearity in the PZT response, a quadratic voltage was incrementally applied to the PZT so that the resulting wavelength scanning would be linear. The nonlinearity of PZT resulted from the nonlinear behavior of the diffraction grating in the tunable laser. Consequently, the nonlinearity decreased to 1% of the total phase shift. The phase distribution was calculated using the *17*-sample phase-shifting algorithm. Figure 7 shows the measured optical thickness variation of the BK7 transparent plate.

In Fig. 7, it is evident that the optical thickness variation on the right side of the BK7 plate is larger than that on the left side. The repeatability error of the optical thickness variation was 2.889 nm. This error resulted from residual nonlinearity beyond the *ε*_{2} and *ε*_{3} phase-shift-error terms in Eq. (3) and from coupling error beyond the *ο*(*A _{m}ε*

_{2}) and

*ο*(

*A*

_{m}ε_{3}) terms in Eq. (4). To estimate the performance of the

*17*-sample algorithm, the measurement repeatability errors of the other algorithms listed in Table 1, as well as that of the 4

*N*– 3 algorithm [22], were investigated, as shown in Table 2.

The measurement repeatability error of the synchronous detection algorithm is 52.198 nm, which is the largest value among the errors shown in Table 2, because synchronous detection cannot compensate for phase-shift or coupling errors. The measurement repeatability error of the *17*-sample algorithm is the smallest, except for the 2.019 nm error of the 4*N* – 3 algorithm (image number *M* = 61). However, as mentioned above, the 4*N* – 3 algorithm is difficult to utilize for actual measurements in the glass industry because of its long intensity-change period.

## 5. Conclusion

In this paper, the derivation of a *17*-sample phase shifting algorithm was presented. The new algorithm can compensate for phase-shift miscalibration and first-order nonlinearity, coupling between the harmonics and phase-shift miscalibration, and bias modulation of the intensity and can satisfy the fringe contrast maximum condition. It was shown that the *17*-sample algorithm yielded the smallest phase error among the conventional phase-shifting algorithms. Finally, the optical thickness variation of a BK7 optical parallel plate was presented, as measured using a wavelength tuning Fizeau interferometer and the *17*-sample algorithm. The measurement repeatability error of the *17*-sample algorithm was lower than those achieved by any of the other algorithms listed in Table 1.

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