Interferometric profile measurement of optical-thickness by wavelength tuning with suppression of spatially uniform error

Yangjin Kim; Kenichi Hibino; Mamoru Mitsuishi

doi:10.1364/OE.26.010870

1. Introduction

Optical thickness variation is the fundamental factor to be considered when designing and estimating transparent optical devices of semiconductor and display equipment [1, 2].

Phase-shifting interferometry [3–5] has been widely used for measuring the surface profile and thickness variation of transparent plates. Phase-shift miscalibration, nonlinearity, and environmental vibration are the most important sources of systematic errors of the calculated phase [6–9]. For precision thickness measurement, these errors should be compensated using an error-compensating phase-shifting algorithm. There are several methods to design the error-compensating algorithm based on linear equations [9, 10], the averaging method [11], the characteristic polynomial theory [12], and Fourier analysis [13, 14]. Hibino et. al developed several error-compensating algorithms based on linear equations by using the error expansion method [9]. Surrel derived the 2N – 1 triangular windowed algorithm [12], which can compensate for the phase-shift miscalibration and coupling errors between the higher harmonic components and phase-shift miscalibration, by using the characteristic polynomial theory. de Groot developed a 7-sample algorithm that can compensate for phase-shift nonlinearity using a data sampling window [15].

However, these algorithms focused on eliminating the spatially nonuniform error, even though the nonlinearity of phase shift can also cause the spatially uniform DC error. Generally, the nonlinearity of phase shift is caused by the strain sensor used to calibrate the piezoelectric modulator. Furthermore, calculations using zeroth-order Zernike polynomials have shown that a uniform DC drift is involved in the piston [16]. This uniform drift has received little attention because it does not result in the deformation of the entire configuration. However, when profiling the absolute optical thickness of transparent parallel plates using wavelength tuning [17], the uniform drift can cause a serious error because of final thickness variation, which is defined by the product of the calculated phase and synthetic wavelength, which is much greater than the used wavelength.

In this study, a new 9-sample phase-shifting algorithm is developed, which can suppress not only the spatially nonuniform error but also the spatially uniform DC drift. First, the equation for eliminating the spatially uniform error (Zernike piston) is derived using the Taylor error series as well as the symmetric and asymmetric properties of sampling amplitudes of the algorithm. Next, the 9-sample algorithm is derived using linear equations, and the characteristics of the algorithm were visualized and discussed through Fourier representation. Finally, the optical-thickness variation of a transparent parallel fused silica optical flat was measured by using the 9-sample algorithm and a Fizeau interferometer, and the error was discussed by comparing the 9-sample algorithm with the conventional phase-shifting algorithms.

2. 9-sample phase-shifting algorithm

2.1 Wavelength-tuning phase-shifting interferometry

Consider a simple two-surface interferometer for profiling the optical-thickness variation of a transparent parallel plate as shown in Fig. 1. When profiling the optical-thickness variation, the reference surface is not necessary.

Fig. 1 Two-surface interferometer.

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When wavelength-tuning is used as the method of phase shift, the intensity of the fringe pattern during wavelength-tuning can be expressed as

I (α_{r}) = I_{0} [1 + γ \cos (α_{r} - φ)],

where I₀ is the DC component of intensity, γ is the fringe contrast, α_r is the phase-shift variable of r, and φ is the target phase corresponding to the optical-thickness variation. Further, n and T are the refractive index and geometric thickness of the plate, respectively. The target phase φ can be calculated by the M-sample phase-shifting algorithm as follows:

φ = Arc \tan \frac{\sum_{r = 1}^{M} b_{r} I (α_{r})}{\sum_{r = 1}^{M} a_{r} I (α_{r})} .

In Eq. (2), a_r and b_r are the sampling amplitudes that determine the characteristics of phase-shifting algorithm. When the phase-shift variable α_r is nonlinear during wavelength-tuning, each phase shift can be expressed as

α_{r} = α_{0 r} [1 + ε_{0} + ε_{1} (\frac{α_{0 r}}{π}) + ε_{2} {(\frac{α_{0 r}}{π})}^{2} + \dots],

where ε₀ is the coefficient of linear phase-shift miscalibration, ε_q (q ≥ 1) is the coefficient of the qth nonlinearity of phase-shift error, α₀_r = 2π[r – (M + 1)/2]/N is the ideal phase-shift variable without error (N is called the phase step number and is an integer). In Eq. (3), ε_q is the source of the spatially uniform error caused by the piston.

2.2 Derivation of 9-sample algorithm

A new phase-shifting algorithm was derived by focusing on the elimination of the first-order nonlinearity of phase-shift error, corresponding to ε₁ in Eq. (3). By substituting Eqs. (1) and (3) into Eq. (2) and expanding the numerator and denominator into a Taylor series, the phase error can be expresses as

Δ φ = Arc \tan \frac{\sum_{r = 1}^{M} b_{r} I (α_{r})}{\sum_{r = 1}^{M} a_{r} I (α_{r})} - φ = π (X ε_{0} \sin 2 φ + Y ε_{1} + Z ε_{1} \cos 2 φ + \dots),

where the coefficients X, Y, and Z are defined as

X = \frac{1}{2} \sum_{r = 1}^{M} (\frac{α_{0 r}}{π}) (a_{r} \sin α_{0 r} + b_{r} \cos α_{0 r}) = 0,

Y = - \frac{1}{2} \sum_{r = 1}^{M} {(\frac{α_{0 r}}{π})}^{2} (a_{r} \cos α_{0 r} + b_{r} \sin α_{0 r}) = 0,

Z = \frac{1}{2} \sum_{r = 1}^{M} {(\frac{α_{0 r}}{π})}^{2} (a_{r} \cos α_{0 r} - b_{r} \sin α_{0 r}) = 0,

where the symmetric sampling amplitude a_r = a_M _{+ 1 –} _r and asymmetric sampling amplitude b_r = -b_M _{+ 1 –} _r are used [10]. The two equations below are also used for suppressing the mth harmonic components [10, 12] (m is the harmonic order):

\sum_{r = 1}^{M} a_{r} \sin (m α_{0 r}) = \sum_{r = 1}^{M} b_{r} \cos (m α_{0 r}) = 0,

\sum_{r = 1}^{M} a_{r} \cos α_{0 r} = \sum_{r = 1}^{M} b_{r} \sin α_{0 r} = δ (m, 1) .

The error coefficient X of Eq. (5) can be removed by aligning the phase-shift variable to set α₀₍_M _{+ 1)/2} = 0. The error term Zε₁cos2φ is a spatially uniform error because it varies with the calculated phase φ. The 7-sample algorithm derived by de Groot can eliminate the term Z. The spatially nonuniform error is proportional to the error coefficient Y. Therefore, to eliminate the spatially uniform error, Eq. (6) should be considered when designing the error-compensating algorithm.

In the conventional phase-shifting algorithms that have the classical discrete Fourier sampling windows a_r = w_rcosα₀_r and b_r = w_rsina₀_r [18], the error coefficient Y is rewritten as

Y = - \frac{1}{2} \sum_{r = 1}^{M} {(\frac{α_{0 r}}{π})}^{2} w_{r} .

The coefficient Y cannot be zero, because the general sampling windows w_r have positive values. Therefore, it is necessary to design a new sampling window that extends to the negative region.

In this study, a new 9-sample phase-shifting algorithm is derived. To suppress the second order harmonic component, the phase-step number should be 4 or greater [10, 12]. Further, it is preferable to use the lowest possible number of images; therefore, the phase step number was set as 4. In this case, the ideal phase-shift variable is defined as

α_{0 r} = \frac{π}{2} (r - \frac{M + 1}{2}) .

Considering the symmetric and asymmetric properties of sampling amplitudes, the sampling window of the 9-sample algorithm is given by

w_{r} = [w_{1}, w_{2}, w_{3}, w_{4}, w_{5}, w_{4}, w_{3}, w_{2}, w_{1}] .

To eliminate the uniform DC drift and suppress the second order harmonics, the following equations should be satisfied.

\sum_{r = 1}^{9} α_{0 r} w_{r} \sin (2 α_{0 r}) = 0,

\sum_{r = 1}^{9} α_{0 r}^{2} w_{r} = 0,

\sum_{r = 1}^{9} α_{0 r}^{2} w_{r} \cos (2 α_{0 r}) = 0,

\sum_{r = 1}^{9} w_{r} = 2,

\sum_{r = 1}^{9} w_{r} \cos (m α_{0 r}) = 0 (m = 1, 2, 3),

Equations (13)–(15) are obtained from Eqs. (5)–(7), respectively, and Eqs. (16) and (17) are obtained from Eqs. (8) and (9) by using the formula of angle sum and difference identities of the trigonometric functions. As mentioned above, by aligning the origin of the phase-shift variable as the center of the fringe pattern, Eq. (13) becomes the identical equation. Moreover, Eq. (17) (m = 1 and 3) are identical. There are five unknown variables in Eq. (12) and five linear equations (Eqs. (14) – (17)). Therefore, the sampling window value w_r can be determined uniquely. By solving the linear equations, a new sampling window can be obtained as follows:

w_{r} = [- \frac{1}{16}, - \frac{1}{16}, \frac{1}{4}, \frac{9}{16}, \frac{5}{8}, \frac{9}{16}, \frac{1}{4}, - \frac{1}{16}, - \frac{1}{16}] .

The new sampling window has negative values at both ends and is symmetric about the center value as in the conventional window. From the obtained sampling window and the relations a_r = w_rcosα₀_r and b_r = w_rsina₀_r, the new 9-sample algorithm can be derived as follows:

φ = Arc \tan \frac{I_{2} + 9 I_{4} - 9 I_{6} - I_{8}}{I_{1} + 4 I_{3} - 10 I_{5} + 4 I_{7} + I_{9}} .

2.3 Characteristics of 9-sample algorithm

A phase-shifting algorithm can be visualized and well understood by using the Fourier representation of sampling amplitudes [13, 19]. The sampling functions F₁(ν) and F₂(ν), which can be derived from the Fourier transform of sampling amplitudes a_r and b_r, respectively, are given by

F_{1} (ν) = \sum_{r = 1}^{M} b_{r} \exp (- i α_{0 r} ν) = - i \sum_{r = 1}^{M} b_{r} \sin (α_{0 r} ν),

F_{2} (ν) = \sum_{r = 1}^{M} a_{r} \exp (- i α_{0 r} ν) = \sum_{r = 1}^{M} a_{r} \cos (α_{0 r} ν) .

Figure 2 shows the sampling functions iF₁ and F₂ of the 9-sample algorithm and other conventional phase-shifting algorithms.

Fig. 2 Sampling functions of (a) 9-sample algorithm, (b) de Groot 7-sample algorithm [14] (c) Hibino 7-sample algorithm [9], and (d) Surrel 2N – 1 algorithm (N = 6) [11].

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In Fig. 2, the peak at the fundamental frequency of 9-sample algorithm is flatter than those of other algorithms. The peak configuration at the fundamental frequency is affected by the negative regions of the sampling window defined by Eq. (18).

Next, the compensation ability for the phase-shift nonlinearity ε₁ was investigated. The phase-shift nonlinearity during wavelength-tuning can be estimated as ~3% (ε₁ = ~0.03), and phase-shift miscalibration ε₀ is set to moderate value of 1 and 0.05 [10]. Table 1 lists the phase error calculated using above four algorithms (Fig. 2) when they are applied to measure the absolute optical thickness [17] (synthetic wavelength λ_s ~43 mm).

Table 1. RMS error at λ_s = 43 mm.

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Table 1 indicates that the new 9-sample algorithm shows a better performance compared to the other conventional algorithms. The algorithms developed by Hibino and Surrel do not have compensation ability for the phase-shift nonlinearity e1 of Eq. (3). Therefore, even though there is no phase-shift miscalibration, the RMS error calculated by the Hibino 7-sample and Surrel 2N – 1 algorithms are larger than those calculated by the 9-sample and de Groot 7-sample algorithms.

The 7-sample algorithm derived by de Groot shows almost same error as the 9-sample algorithm. Both these algorithms eliminate the error coefficient Z in Eqs. (4) and (7). However, the de Groot 7-sample algorithm does not eliminate the error coefficient Y of Eq. (6), but the uniform DC drift corresponding to the coefficient Y does not appear in the RMS phase error. In contrast, the new 9-sample algorithm eliminates not only the coefficient Z but also the coefficient Y. Therefore, for measuring the absolute optical thickness, the 9-sample phase-shifting algorithm is considered to show the best performance.

3. Verification experiment of 9-sample algorithm

3.1 Wavelength-tuning Fizeau interferometer

The optical-thickness variation of a fused silica optical plate with 100-mm diameter and 8-mm thickness was measured using the 9-sample algorithm and wavelength-tuning Fizeau interferometer shown in Fig. 3. The temperature inside the laboratory was 20.5°C ( ± 0.1°C). The measurement sample was ground to a thickness less than 1.5 μm.

Fig. 3 Wavelength-tuning Fizeau interferometer. PBS denotes polarization beam splitter; QWP is quarter-wave plate; HWP is half-wave plate.

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As the Fizeau interferometer, FUJINON G102 was used without a He-Ne laser. The source laser is a tunable diode laser of a Littman external cavity (New Port TLB–6300–LN) [20] that can be scanned linearly in time from 632.8 nm to 638.4 nm. The source laser beam is divided into two directions by a beam splitter. One beam is transmitted to a wavelength meter (Anritsu MF9630A), which was calibrated using a stabilized He-Ne laser (632.8 nm) with an uncertainty of 10⁻⁷. The other beam is transmitted to the Fizeau interferometer. Inside the interferometer, the beam is collimated to illuminate the fused silica plate, and the reflection from the front and back surfaces of the plate return along the same path. The resulting beams pass through a polarized beam splitter and combine to generate an interferogram on a screen of a resolution of 640 × 480 pixels.

Figures 4(a) and 4(b) show an experimental photo and raw fringe pattern at 632.8 nm, respectively.

Fig. 4 (a) Experimental photo and (b) raw fringe pattern of the fused silica plate at 632.8 nm.

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3.2 Results and discussion

The necessary wavelength scanning range δλ was calculated as

δ λ = - \frac{λ^{2}}{4 π n T} δ φ \approx - 0.01716 nm .

The wavelength was scanned finely from 632.8029 nm to 632.8201 nm by using the piezoelectric transducer of the tunable diode laser. The exact wavelength was measured using the wavelength meter with an uncertainty of 10⁻⁷. Every time the phase changed by π/4, nine interferograms were obtained for calculating the phase corresponding to the optical-thickness variation. The experiment took 5 s to acquire the nine interferograms. The phase distribution was calculated by the 9-sample algorithm defined by Eq. (19). Figure 5 shows the calculated optical-thickness variation.

Fig. 5 Calculated optical-thickness variation of a fused silica transparent plate at 632.8 nm.

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The experiments were performed 15 times for three days. The maximum value of 1.886 μm is the average over 15 experiments. The standard deviation for 15 experiments was 5.198 nm. This standard deviation resulted from the residual phase-shift error and thickness fluctuation due to temperature change ( ± 0.1°C). The edge part of the measurement sample (~3 mm to the radius direction) was excluded from the calculation analysis because there is a scattering effect of light at the edge part. The lower part of the sample was thicker than the other part. From Fig. 5, we can see that the transparent plate was ground around the center.

The optical-thickness deviation of the same fused silica plate was measured by the de Groot 7-sample algorithm and by wavelength-tuning with the same process of the 9-sample algorithm. The average maximum value and standard deviation were 1.888 μm and 7.342 nm, respectively. We can infer that the uniform DC drift error of the 7-sample algorithm was approximately 2 nm.

4. Conclusion

The conventional phase-shifting algorithms do not consider the uniform DC drift error in the calculated phase distribution, because it does not change the entire configuration. In this study, a new 9-sample phase-shifting algorithm was developed, which can eliminate the effect of the uniform DC drift. The error term corresponding to the uniform DC drift error was derived using the Taylor expansion as well as the symmetric and asymmetric properties of sampling amplitudes. The 9-sample algorithm can eliminate this error term and suppress the second-order harmonic component. The characteristics of the 9-sample algorithm was investigated using the Fourier representation, and the calculated phase error was compared with those from other conventional algorithms. As a verification experiment, the optical-thickness variation of a transparent parallel plate was measured using the 9-sample algorithm and wavelength-tuning Fizeau interferometer. The uniform DC drift error was estimated as 2 nm by comparing the result with that obtained using the 7-sample algorithm

Funding

National Research Foundation of Korea (2017R1C1B2010496).

References and links

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Algorithm	RMS error [nm]
Algorithm	ε₀ = 0 and ε₁ = 0.03	ε₀ = 0.05 and ε₁ = 0.03
New 9-sample	12.710	14.103
de Groot 7-sample	14.158	17.772
Hibino 7-sample	48.433	68.985
Surrel 2N – 1 (N = 6)	41.330	53.808

Interferometric profile measurement of optical-thickness by wavelength tuning with suppression of spatially uniform error

Abstract

1. Introduction

2. 9-sample phase-shifting algorithm

2.1 Wavelength-tuning phase-shifting interferometry

2.2 Derivation of 9-sample algorithm

2.3 Characteristics of 9-sample algorithm

3. Verification experiment of 9-sample algorithm

3.1 Wavelength-tuning Fizeau interferometer

3.2 Results and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (5)

Tables (1)

Equations (22)

Optics Express