## Abstract

In this study, a 6*N* – 5 phase shifting algorithm comprising a polynomial window function and discrete Fourier transform is developed for the simultaneous measurement of the surface shape and optical thickness of a transparent plate with suppression of the coupling errors between the higher harmonics and phase shift error. The characteristics of the 6*N* – 5 algorithm were estimated by connection with the Fourier representation in the frequency domain. The phase error of the measurements performed using the 6*N* – 5 algorithm is discussed and compared with those of measurements obtained using other algorithms. Finally, the surface shape and optical thickness of a transparent plate were measured simultaneously using the 6*N* – 5 algorithm and a wavelength tuning interferometer.

© 2015 Optical Society of America

## 1. Introduction

The surface shape and optical thickness variation of transparent plates are fundamental characteristics of the optical devices used in the semiconductor and display industries. To reduce the cost of the measurement and estimation processes, it is desirable to measure the surface shape and optical thickness variation of a large diameter transparent plate simultaneously. Many approaches have been developed for this simultaneous measurement.

One such approach reported by several authors involves the use of white light interferometry [1,2]. In this technique, the diameter of an observing aperture is restricted to < 1 cm because the accurate translation of a large reference mirror along the optical axis is difficult. When the thickness of a measurement sample increases to more than a few millimeters, the rapid increase in the coherence length degrades the measurement resolution. Wavelength tuning interferometry has also been used for the measurement of optical thickness of the transparent plate [3–6]. A lateral shearing interferometer with a wavelength tuning laser diode is applied in the simultaneous measurement of the thickness and refractive index of the transparent plate [3]. The geometric thickness and refractive index are separated after the optical thickness is measured independently by wavelength tuning and sample location. However, these techniques are not suitable for measuring the thickness distribution because these methods assume that the sample has a spatially uniform thickness.

The surface shape and optical thickness variation of mask blank glass were measured simultaneously using the wavelength tuning interferometry and the phase shifting technique [6]. In this case, the measurement accuracy of wavelength tuning interferometry strongly depends on the phase shifting algorithm used. However, the phase shifting algorithm employed in [6] can only compensate for the linear miscalibration of phase shift [7]. To compensate for the residual phase shift error, a 4*N* – 3 algorithm was developed to measure the surface shape of a highly reflective sample [8]. However, the ripples resulting from the phase shift error and coupling errors were clearly observed on the measured surface shape and optical thickness. These coupling errors occur during the measurement of the surface shape of a multiple-surface interferometer because of higher harmonics resulting from inner reflections. However, there is no report which has achieved the simultaneous measurement of surface shape and optical thickness with strong suppression of the ripples.

In this study, a new 6*N* – 5 phase shifting algorithm, comprising of the polynomial window function and discrete Fourier transform (DFT) term, is developed to measure the surface shape and optical thickness variation of a transparent plate with suppression of the coupling errors. The characteristics of the 6*N* – 5 algorithm are discussed in connection with the Fourier representation of the phase shifting algorithm in the frequency domain. It is shown that developed polynomial window function yields the smallest phase error among the conventional window functions. Finally, the surface shape and optical thickness variation of a transparent plate were measured simultaneously using the wavelength tuning Fizeau interferometer and 6*N* – 5 algorithm. The experimental results indicate that the optical thickness variation and surface shape measurement accuracy for the transparent plate is 1.8 nm and 3.2 nm, respectively.

## 2. Derivation of the 6*N* – 5 phase shifting algorithm

#### 2.1 Laser Fizeau interferometer

Consider a three-surface interference system consisting of a reference surface and transparent plate, as shown in Fig. 1. In Fig. 1, *L* is the air-gap distance and *nT* is the optical thickness of the measurement transparent sample.

The signal irradiance is formed by the multiple-beam interference between the reflection beams from the reference surface and the sample surfaces. The irradiance signal during wavelength tuning is given as [9,10]

*α*is a phase shift parameter,

_{r}*A*,

_{m}*ν*and

_{m}*φ*are the amplitude, frequency and phase of the

_{m}*m*

^{th}harmonic component, respectively, and

*A*

_{0}is the DC component. When the wavelength is scanned from

*λ*

_{1}to

*λ*

_{2}(

*λ*

_{1}<

*λ*

_{2}), the

*m*

^{th}harmonic component frequency

*ν*is proportional to the optical path difference between the interfering beams [11]. When the distance of the air-gap is set as 3

_{m}*nT*, the frequency

*ν*

_{1}corresponding to the optical thickness and the frequency

*ν*

_{3}corresponding to the surface shape can be separated in the frequency domain using a phase shifting algorithm [11].

A general *M*-sample phase shifting algorithm is given by

*a*,

_{r}*b*are the sampling amplitudes and

_{r}*I*(

*α*) is the

_{r}*r*

^{th}sampled signal irradiance given by Eq. (1). When the phase shift is nonlinear, each phase shift value

*α*is a function of the phase shift parameter. The phase shift value for the

_{r}*r*

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

*α*

_{0}

*as [12]*

_{r}*p*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 [13].

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

*A*

_{1}and the error coefficient

*ε*of phase shift and can be Taylor-expanded as [12,13]

_{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 error between the harmonics and phase shift error. When measuring the surface shape and optical thickness of a transparent plate using wavelength tuning interferometry, the coupling errors become the critical factor because of the surface reflectivity and the inner reflections.

_{k}ε_{q}#### 2.2 The 6N – 5 phase shifting algorithm

Systematic approaches for deriving error-compensating algorithms have been proposed by several authors based on an analytical expansion of the phase error [12], an averaging method of successive samples [14,15], a Fourier representation [16], and a characteristic polynomial theory [17].

Surrel proposed the characteristic polynomial theory to design and estimate the phase shifting algorithms. All properties of any phase shifting algorithm can be deduced from the locations and multiplicities of the roots of the characteristic polynomial. For example, the synchronous detection proposed by Burning [18] has single roots on the characteristic diagram [17], and does not have the ability to compensate for the phase shift error and coupling error. To compensate for the phase shift miscalibration *ε*_{0} of Eq. (4), Surrel proposed a 2*N* – 1 algorithm by locating double roots on the characteristic diagram [17]. Using Surrel’s characteristic polynomial theory, a 4*N* – 3 algorithm that can compensate for up to the 2nd order nonlinearity and coupling error was developed by locating quadruple roots on the diagram [8].

By locating six multiple roots on the characteristic diagram as shown in Fig. 2(a), we can generate the 6*N* – 5 phase shifting algorithm, which can compensate for the error of the phase shift *ο*(*ε*_{4}) up to the 4th order nonlinearity and the coupling errors up to the *ο*(*A _{m}ε*

_{4}) of Eq. (4). The characteristic polynomial of 6

*N*– 5 algorithm can be given by

*P*(

_{sync}*x*) is the characteristic polynomial of the synchronous detection [17,18].

*P*(

*x*) of Eq. (5) has 6

*N*– 5 coefficients and the general term of these coefficients is in the form of the polynomial of

*r*.

By rotating the characteristic diagram according to –*mδ* (*δ* = 2π/*N*) as shown in Fig. 2(b), we can obtain the general 6*N* – 5 phase shifting algorithm that can extract the *m*^{th} arbitrary harmonics signal phase [8].

The sampling amplitudes of the 6*N* – 5 algorithm are the product of a polynomial window function and a DFT term, and are given as

*w*is the polynomial window function defined by the following Eqs. (8)–(13).

_{r}- ii.
*N*+ 1 ≤*r*≤ 2*N*$$\begin{array}{l}{w}_{r}=-\frac{1}{24}{r}^{5}+\left(\frac{N}{4}-\frac{5}{12}\right){r}^{4}+\left(-\frac{{N}^{2}}{2}+2N-\frac{35}{24}\right){r}^{3}+\left(\frac{{N}^{3}}{2}-3{N}^{2}+\frac{21}{4}N-\frac{25}{12}\right){r}^{2}\\ -\frac{1}{4}{\left(N-2\right)}^{2}\left({N}^{2}-4N+1\right)r+\frac{1}{20}N\left(N-1\right)\left(N-2\right)\left(N-3\right)\left(N-4\right),\end{array}$$ - iii. 2
*N*+ 1 ≤*r*≤ 3*N*– 2$$\begin{array}{l}{w}_{r}=\frac{1}{12}{r}^{5}+\left(-N+\frac{5}{6}\right){r}^{4}+\left(\frac{9}{2}{N}^{2}-8N+\frac{35}{12}\right){r}^{3}\\ +\left(-\frac{19}{2}{N}^{3}+27{N}^{2}-21N+\frac{25}{6}\right){r}^{2}+\left(\frac{39}{4}{N}^{4}-38{N}^{3}+\frac{189}{4}{N}^{2}-20N+2\right)r\\ -\frac{1}{20}N\left(N-1\right)\left(N-2\right)\left(79{N}^{2}-153N+48\right),\end{array}$$ - iv. 3
*N*– 1 ≤*r*≤ 4*N*– 5$$\begin{array}{l}{w}_{r}=-\frac{1}{12}{r}^{5}+\left(\frac{3}{2}N-\frac{5}{6}\right){r}^{4}+\left(-\frac{21}{2}{N}^{2}+12N-\frac{35}{12}\right){r}^{3}\\ +\left(\frac{71}{2}{N}^{3}-63{N}^{2}+\frac{63}{2}N-\frac{25}{6}\right){r}^{2}+\left(-\frac{231}{4}{N}^{4}+142{N}^{3}-\frac{441}{4}{N}^{2}+30N-2\right)r\\ +\frac{1}{20}N\left(N-1\right)\left(731{N}^{3}-1579{N}^{2}+906N-144\right),\end{array}$$ - v. 4
*N*– 4 ≤*r*≤ 5*N*– 5$$\begin{array}{l}{w}_{r}=\frac{1}{24}{r}^{5}+\left(-N+\frac{5}{12}\right){r}^{4}+\left(\frac{19}{2}{N}^{2}-8N+\frac{35}{24}\right){r}^{3}\\ +\left(-\frac{89}{2}{N}^{3}+57{N}^{2}-21N+\frac{25}{12}\right){r}^{2}+\left(\frac{409}{4}{N}^{4}-178{N}^{3}+\frac{399}{4}{N}^{2}-20N+1\right)r\\ -\frac{1}{20}N\left(N-1\right)\left(1829{N}^{3}-2261{N}^{2}+854N-96\right),\end{array}$$

Figure 3 shows the shape of the polynomial window function defined by Eqs. (8)–(13).

## 3. Characteristics of the 6*N* – 5 phase shifting algorithm

#### 3.1 Fourier representation of the 6N – 5 algorithm

The phase shifting algorithm can be visualized and well understood if we take a Fourier representation of the sampling amplitudes of the algorithm [19]. The sampling functions at the frequency domain of the numerator and denominator of an algorithm are defined as

For the symmetrical property of the sampling amplitudes, *F*_{1} and *F*_{2} are purely imaginary and purely real functions, respectively [12]. Figure 4(a) shows the sampling functions *iF*_{1} and *F*_{2} of the 6*N* – 5 algorithm (*N* = 12) that detect the fundamental frequency *ν* = 1 corresponding to the optical thickness of the transparent sample in Fig. 1. Figure 4(b) shows the sampling functions at the frequency *ν* = 3, which corresponds to the surface shape in Fig. 1.

In the frequency domain, the sidelobe amplitude of the 6*N* – 5 algorithm is suppressed by approximately 0.012%, which is better than the sidelobe suppressions for the 2*N* – 1 algorithm proposed by Surrel (6.036%) [17] and 4*N* – 3 algorithm (0.256%) [8] when *N* = 12.

The characteristics of a phase shifting algorithm can be deduced from the sampling functions on the frequency domain [12,13,20–22]. From Fig. 4, the 6*N* – 5 algorithm satisfies the fringe contrast maximum condition because the sampling functions of this algorithm have the 0 gradients at the target frequencies [21]. The 6*N* – 5 algorithm also has the compensation ability for the bias modulation because the sampling functions have the 0 gradients at the frequency value of 0 [22]. To compensate for the coupling errors *ο*(*A _{m}ε*

_{0}), the sampling functions should have the 0 gradients at

*m*= 2, 3, …,

*N*– 2 [12,13]. The sampling functions of 6

*N*– 5 algorithm have the 0 gradients at

*m*= 2, 3, …,

*N*– 2, and furthermore, the sampling functions up to derivatives of the 5th order have the 0 gradients, which means the compensation ability for the coupling errors up to

*ο*(

*A*

_{m}ε_{4}) of Eq. (4).

#### 3.2 RMS phase error analysis

To illustrate the theoretical merits of the new polynomial window function, Table 1 shows the amplitudes of the highest sidelobes for the sampling functions compared with several types of conventional windows [25]. The amplitude of the highest sidelobe of the new polynomial window function shows the smallest value among the conventional window functions.

When measuring the surface shape and optical thickness of a transparent plate, not only should the phase shift error but also the coupling errors between the higher harmonics and phase shift error be considered. The coupling errors have been studied thoroughly by Hibino and de Groot [9,26]. In [26], de Groot analyzed the calculated phase error due to phase shift miscalibration and also the error caused by coupling errors between the higher harmonics and phase shift miscalibration. The RMS phase error *σ _{mis}* resulting from the phase shift miscalibration is given by

*iF*

_{1}and

*F*

_{2}are the sampling functions defined by Eqs. (14) and (15). The RMS phase error

*σ*resulting from the coupling error between the

_{cou}*m*

^{th}harmonic and phase shift miscalibration is given by

*γ*is the fringe contrast of the

_{m}*m*

^{th}harmonic, which is determined by the reflectivities of the reference and sample surfaces [10,26].

Therefore, the net RMS error is given by [26]

where*σ*and

_{mis}*σ*are obtained from Eqs. (16) and (17).

_{cou}Figure 5 shows the solution of Eq. (18) based on the window functions listed in Table 1 for the reflectivities of the reference sample surfaces at 4%, similar to Fig. 1.

From Fig. 5, the new polynomial window function shows the smallest RMS phase error among the conventional window functions listed in Table 1. The Blackman window shows a nonzero RMS phase error when there is no phase shift miscalibration because the algorithm comprising Blackman window and DFT does not have the ability to compensate for the error of the 2nd harmonic component *ο*(*A*_{2}). When using the wavelength tuning diode laser, the phase shift miscalibration *ε*_{0} should be considered as 10%. The developed polynomial window shows the smallest error under 0.01 nm among the window functions listed in Table 1 even though there is the phase shift miscalibration of ± 10%.

## 4. Simultaneous measurement of surface shape and optical thickness variation

#### 4.1 Wavelength tuning Fizeau interferometer

Figure 6 shows the optical setup for measuring the surface shape and 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 by 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 beams 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

The surface shape and optical thickness variation of a transparent BK7 plate 8 mm thick and 100 mm in diameter was measured using the wavelength tuning Fizeau interferometer, as shown in Fig. 6, and the 6*N* – 5 algorithm defined by Eqs. (6)–(13). Figure 7(a) shows a laboratory photo of the BK7 transparent parallel plate in the wavelength tuning Fizeau interferometer, and Fig. 7(b) shows an observed raw interferogram of the measurement sample at a wavelength of 634.6 nm.

By setting the air gap distance *L* = 36.36 mm to be three times the optical thickness of the BK7 transparent plate *nT*, three main signal frequencies *ν*_{1}, 3*ν*_{1,} and 4*ν*_{1} corresponding to the optical thickness of sample, surface shape, and rear surface shape, respectively, can be separated using the 6*N* – 5 algorithm in the frequency domain. The necessary range for wavelength tuning δ*λ* was calculated as

The wavelength was finely scanned from 634.6023 nm to 635.7017 nm and 67 interferograms were recorded at equal intervals. During the wavelength tuning process, the signal interference fringes corresponding to the optical thickness changed four periods of 12π radians while the signal corresponding to the surface shape changed twelve periods of 36π. The phase shift for each step was π/6 for the optical thickness fringes of the BK7 plate. Because there was a nonlinearity of approximately 3% in the PZT response of the source laser cavity, a quadratic voltage increment was applied to the PZT to make the resultant wavelength tuning 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 *φ*_{1} and *φ*_{3} corresponding to the optical thickness variation and surface shape of a BK7 plate, respectively, were calculated using the 6*N* – 5 algorithm defined by Eqs. (6)–(13). Figure 8(a) and (b) show the measured optical thickness variation and surface shape of a BK7 plate, respectively.

In Fig. 8, the optical thickness of the BK7 plate appears to have a concave configuration of 1.9 μmPV, while the surface shape has a convex configuration of 1.0 μmPV. The ripples were hardly observed in the measured results, which means that the 6*N* – 5 algorithm has an excellent compensation ability for the phase shift errors and the couplings errors between the higher harmonics and the phase shift errors.

The repeatability errors of the measurement of the optical thickness variation and surface shape, i.e., the RMS of the difference between a pair of measurements taken with the interval of 3 days, were approximately 1.784 nm and 3.109 nm, respectively. The measurement uncertainty of the reference surface is approximately *λ*/20 = 32 nm. Therefore, the measurement uncertainties of the optical thickness and surface shape were approximately 1.8 nm and 35 nm, respectively.

## 5. Conclusion

In this study, a 6*N* – 5 phase shifting algorithm, comprising a new polynomial window function and DFT term, was developed based on the characteristic polynomial theory. The 6*N* – 5 algorithm has the compensation ability for up to the 5th order nonlinearity of the phase shift error, and for the coupling errors between the higher harmonics and up to the 5th order nonlinearity of the phase shift error. The characteristics of the 6*N* – 5 algorithm were estimated by connecting with the Fourier representation in the frequency domain. The developed new polynomial window shows the smallest RMS phase error among the conventional window functions listed in Table 1. Finally, the surface shape and optical thickness variation of a transparent BK7 plate were measured using the 6*N* – 5 algorithm and wavelength tuning Fizeau interferometer. The measurement uncertainty of the optical thickness and surface shape were 1.8 nm and 35 nm, respectively.

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