Abstract

In this study, a 6N – 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 6N – 5 algorithm were estimated by connection with the Fourier representation in the frequency domain. The phase error of the measurements performed using the 6N – 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 6N – 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 4N – 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 6N – 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 6N – 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 6N – 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 6N – 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.

 figure: Fig. 1

Fig. 1 Three-surface laser interferometry.

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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]

I(αr)=A0+m=1Amcos(νmtφm)=A0+A1cos(αrφ1)+A2cos(2αrφ2)+A3cos(3αrφ3)+,
where αr is a phase shift parameter, Am, νm and φm are the amplitude, frequency and phase of the mth harmonic component, respectively, and A0 is the DC component. When the wavelength is scanned from λ1 to λ2 (λ1 < λ2), the mth harmonic component frequency νm is proportional to the optical path difference between the interfering beams [11]. When the distance of the air-gap is set as 3nT, the frequency ν1 corresponding to the optical thickness and the frequency ν3 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

φm=arctanr=1MbrI(αr)r=1MarI(αr),
where ar, br are the sampling amplitudes and I(αr) is the rth sampled signal irradiance given by Eq. (1). When the phase shift is nonlinear, each phase shift value αr is a function of the phase shift parameter. The phase shift value for the rth sample can be expressed as a polynomial function of the unperturbed phase shift value α0r as [12]
αr=α0r[1+ε0+ε1α0rπ+ε2(α0rπ)2++εp(α0rπ)p],
where p is the maximum order of the nonlinearity, ε0 is the error coefficient of the phase shift miscalibration, εq (1 ≤ qp) is the error coefficient of the qth nonlinearity of the phase shift, and α0r = 2π[r – (M + 1)/2]/N is the unperturbed phase shift [13].

The calculated phase error Δφ is a function of the amplitude ratios Am/A1 and the error coefficient εq of phase shift and can be Taylor-expanded as [12,13]

Δφ=ο(Ak)+ο(εq)+ο(Akεq),
for k = 2, 3, …, m and q = 0, 1, …, p. In Eq. (4), ο(Ak), ο(εq) and ο(Akεq) 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.

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 ε0 of Eq. (4), Surrel proposed a 2N – 1 algorithm by locating double roots on the characteristic diagram [17]. Using Surrel’s characteristic polynomial theory, a 4N – 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 6N – 5 phase shifting algorithm, which can compensate for the error of the phase shift ο(ε4) up to the 4th order nonlinearity and the coupling errors up to the ο(Amε4) of Eq. (4). The characteristic polynomial of 6N – 5 algorithm can be given by

P(x)=[Psync(x)]6=[1+x+x2++xN1]6=r=16N5wrxr1,
where Psync(x) is the characteristic polynomial of the synchronous detection [17,18]. P(x) of Eq. (5) has 6N – 5 coefficients and the general term of these coefficients is in the form of the polynomial of r.

 figure: Fig. 2

Fig. 2 (a) 6N – 5 algorithm (N = 12) obtained by locating six multiple roots on the characteristic diagram, and (b) rotation of 6N – 5 algorithm to extract the mth arbitrary harmonic signal phase. Six circled dot indicates a six multiple root.

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

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

ar=2Nwrcos2mπN[r(3N2)],
br=2Nwrsin2mπN[r(3N2)],
where wr is the polynomial window function defined by the following Eqs. (8)–(13).

  • i. 1 ≤ rN
    wr=1120r(r+1)(r+2)(r+3)(r+4),
  • ii. N + 1 ≤ r ≤ 2N
    wr=124r5+(N4512)r4+(N22+2N3524)r3+(N323N2+214N2512)r214(N2)2(N24N+1)r+120N(N1)(N2)(N3)(N4),
  • iii. 2N + 1 ≤ r ≤ 3N – 2
    wr=112r5+(N+56)r4+(92N28N+3512)r3+(192N3+27N221N+256)r2+(394N438N3+1894N220N+2)r120N(N1)(N2)(79N2153N+48),
  • iv. 3N – 1 ≤ r ≤ 4N – 5
    wr=112r5+(32N56)r4+(212N2+12N3512)r3+(712N363N2+632N256)r2+(2314N4+142N34414N2+30N2)r+120N(N1)(731N31579N2+906N144),
  • v. 4N – 4 ≤ r ≤ 5N – 5
    wr=124r5+(N+512)r4+(192N28N+3524)r3+(892N3+57N221N+2512)r2+(4094N4178N3+3994N220N+1)r120N(N1)(1829N32261N2+854N96),
  • vi. 5N – 4 ≤ r ≤ 6N – 5
    wr=1120(6Nr)(6Nr1)(6Nr2)(6Nr3)(6Nr4).

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

 figure: Fig. 3

Fig. 3 Shape of the polynomial window function (N = 12) defined by Eqs. (8)–(13).

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3. Characteristics of the 6N – 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

F1(ν)=r=1Mbrexp(iαrν),
F2(ν)=r=1Marexp(iαrν).

For the symmetrical property of the sampling amplitudes, F1 and F2 are purely imaginary and purely real functions, respectively [12]. Figure 4(a) shows the sampling functions iF1 and F2 of the 6N – 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.

 figure: Fig. 4

Fig. 4 Sampling functions of the 6N – 5 algorithm: (a) m = 1 and (b) m = 3 (N = 12).

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In the frequency domain, the sidelobe amplitude of the 6N – 5 algorithm is suppressed by approximately 0.012%, which is better than the sidelobe suppressions for the 2N – 1 algorithm proposed by Surrel (6.036%) [17] and 4N – 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 6N – 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 6N – 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(Amε0), the sampling functions should have the 0 gradients at m = 2, 3, …, N – 2 [12,13]. The sampling functions of 6N – 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(Amε4) 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.

Tables Icon

Table 1. Representative Phase-Shifting Algorithm

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 σmis resulting from the phase shift miscalibration is given by

σmis=122|iF1(ν)F2(ν)1|,
where iF1 and F2 are the sampling functions defined by Eqs. (14) and (15). The RMS phase error σcou resulting from the coupling error between the mth harmonic and phase shift miscalibration is given by
σcou=12m=2γmγ1[iF1(mν)iF1(ν)]2+[F2(mν)F2(ν)]2,
where γm is the fringe contrast of the mth harmonic, which is determined by the reflectivities of the reference and sample surfaces [10,26].

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

σ=σmis2+σcou2,
where σmis and σcou are obtained from Eqs. (16) and (17).

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.

 figure: Fig. 5

Fig. 5 RMS phase errors of phase shifting algorithms comprising the windows listed in Table 1 and DFT terms as functions of phase shift miscalibration ε0 when the reflectivities of the reference and sample surfaces are 4%.

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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(A2). When using the wavelength tuning diode laser, the phase shift miscalibration ε0 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.

 figure: Fig. 6

Fig. 6 Wavelength tuning Fizeau interferometer used to measure optical thickness variation of a BK7 transparent plate. PBS denotes polarization beam splitter; QWP is quarter-wave plate; HWP is half-wave plate.

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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 δλ/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 6N – 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.

 figure: Fig. 7

Fig. 7 (a) Laboratory photo of BK7 transparent parallel plate in wavelength tuning Fizeau interferometer and (b) raw interferogram at a wavelength of 634.6 nm.

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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 ν1, 3ν1, and 4ν1 corresponding to the optical thickness of sample, surface shape, and rear surface shape, respectively, can be separated using the 6N – 5 algorithm in the frequency domain. The necessary range for wavelength tuning δλ was calculated as

δλ=λ24πnTδφ0.0997nm.

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 φ1 and φ3 corresponding to the optical thickness variation and surface shape of a BK7 plate, respectively, were calculated using the 6N – 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.

 figure: Fig. 8

Fig. 8 Measured (a) optical thickness variation and (b) surface shape of a BK7 plate at a wavelength of 634.6 nm.

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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 6N – 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 6N – 5 phase shifting algorithm, comprising a new polynomial window function and DFT term, was developed based on the characteristic polynomial theory. The 6N – 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 6N – 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 6N – 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.

References and links

1. T. Fukano and I. Yamaguchi, “Simultaneous measurement of thicknesses and refractive indices of multiple layers by a low-coherence confocal interference microscope,” Opt. Lett. 21(23), 1942–1944 (1996). [CrossRef]   [PubMed]  

2. M. Haruna, M. Ohmi, T. Mitsuyama, H. Tajiri, H. Maruyama, and M. Hashimoto, “Simultaneous measurement of the phase and group indices and the thickness of transparent plates by low-coherence interferometry,” Opt. Lett. 23(12), 966–968 (1998). [CrossRef]   [PubMed]  

3. G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42(19), 3882–3887 (2003). [CrossRef]   [PubMed]  

4. K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. 43(6), 1241–1249 (2004). [CrossRef]   [PubMed]  

5. S. Kim, J. Na, M. J. Kim, and B. H. Lee, “Simultaneous measurement of refractive index and thickness by combining low-coherence interferometry and confocal optics,” Opt. Express 16(8), 5516–5526 (2008). [CrossRef]   [PubMed]  

6. K. Hibino, Y. Kim, S. Lee, Y. Kondo, N. Sugita, and M. Mitsuishi, “Simultaneous measurement of surface shape and absolute optical thickness of a glass plate by wavelength tuning phase-shifting interferometry,” Opt. Rev. 19(4), 247–253 (2012). [CrossRef]  

7. R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004). [CrossRef]  

8. Y. Kim, K. Hibino, R. Hanayama, N. Sugita, and M. Mitsuishi, “Multiple-surface interferometry of highly reflective wafer by wavelength tuning,” Opt. Express 22(18), 21145–21156 (2014). [CrossRef]   [PubMed]  

9. K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999). [CrossRef]  

10. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry,” Appl. Opt. 54(13), 4207–4213 (2015). [CrossRef]  

11. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000). [CrossRef]   [PubMed]  

12. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997). [CrossRef]  

13. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Measurement of optical thickness variation of BK7 plate by wavelength tuning interferometry,” Opt. Express 23(17), 22928–22938 (2015). [CrossRef]   [PubMed]  

14. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993). [CrossRef]  

15. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995). [CrossRef]   [PubMed]  

16. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992). [CrossRef]  

17. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef]   [PubMed]  

18. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974). [CrossRef]   [PubMed]  

19. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990). [CrossRef]  

20. Y. Surrel, “Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts: comment,” J. Opt. Soc. Am. A 15(5), 1227–1233 (1998). [CrossRef]  

21. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Design of phase shifting algorithms: fringe contrast maximum,” Opt. Express 22(15), 18203–18213 (2014). [CrossRef]   [PubMed]  

22. R. Onodera and Y. Ishii, “Phase-extraction analysis of laser-diode phase-shifting interferometry that is insensitive to changes in laser power,” J. Opt. Soc. Am. A 13(1), 139–146 (1996). [CrossRef]  

23. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995). [CrossRef]   [PubMed]  

24. L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003). [CrossRef]   [PubMed]  

25. F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978). [CrossRef]  

26. P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014). [CrossRef]   [PubMed]  

27. K. Liu and M. G. Littman, “Novel geometry for single-mode scanning of tunable lasers,” Opt. Lett. 6(3), 117–118 (1981). [CrossRef]   [PubMed]  

28. P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014). [CrossRef]   [PubMed]  

References

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  1. T. Fukano and I. Yamaguchi, “Simultaneous measurement of thicknesses and refractive indices of multiple layers by a low-coherence confocal interference microscope,” Opt. Lett. 21(23), 1942–1944 (1996).
    [Crossref] [PubMed]
  2. M. Haruna, M. Ohmi, T. Mitsuyama, H. Tajiri, H. Maruyama, and M. Hashimoto, “Simultaneous measurement of the phase and group indices and the thickness of transparent plates by low-coherence interferometry,” Opt. Lett. 23(12), 966–968 (1998).
    [Crossref] [PubMed]
  3. G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42(19), 3882–3887 (2003).
    [Crossref] [PubMed]
  4. K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. 43(6), 1241–1249 (2004).
    [Crossref] [PubMed]
  5. S. Kim, J. Na, M. J. Kim, and B. H. Lee, “Simultaneous measurement of refractive index and thickness by combining low-coherence interferometry and confocal optics,” Opt. Express 16(8), 5516–5526 (2008).
    [Crossref] [PubMed]
  6. K. Hibino, Y. Kim, S. Lee, Y. Kondo, N. Sugita, and M. Mitsuishi, “Simultaneous measurement of surface shape and absolute optical thickness of a glass plate by wavelength tuning phase-shifting interferometry,” Opt. Rev. 19(4), 247–253 (2012).
    [Crossref]
  7. R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
    [Crossref]
  8. Y. Kim, K. Hibino, R. Hanayama, N. Sugita, and M. Mitsuishi, “Multiple-surface interferometry of highly reflective wafer by wavelength tuning,” Opt. Express 22(18), 21145–21156 (2014).
    [Crossref] [PubMed]
  9. K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
    [Crossref]
  10. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry,” Appl. Opt. 54(13), 4207–4213 (2015).
    [Crossref]
  11. P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
    [Crossref] [PubMed]
  12. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
    [Crossref]
  13. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Measurement of optical thickness variation of BK7 plate by wavelength tuning interferometry,” Opt. Express 23(17), 22928–22938 (2015).
    [Crossref] [PubMed]
  14. J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
    [Crossref]
  15. J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995).
    [Crossref] [PubMed]
  16. K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992).
    [Crossref]
  17. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
    [Crossref] [PubMed]
  18. J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
    [Crossref] [PubMed]
  19. K. Freischlad and C. L. Koliopoulos, “Fourier description of digital phase-measuring interferometry,” J. Opt. Soc. Am. A 7(4), 542–551 (1990).
    [Crossref]
  20. Y. Surrel, “Phase shifting algorithms for nonlinear and spatially nonuniform phase shifts: comment,” J. Opt. Soc. Am. A 15(5), 1227–1233 (1998).
    [Crossref]
  21. Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Design of phase shifting algorithms: fringe contrast maximum,” Opt. Express 22(15), 18203–18213 (2014).
    [Crossref] [PubMed]
  22. R. Onodera and Y. Ishii, “Phase-extraction analysis of laser-diode phase-shifting interferometry that is insensitive to changes in laser power,” J. Opt. Soc. Am. A 13(1), 139–146 (1996).
    [Crossref]
  23. P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995).
    [Crossref] [PubMed]
  24. L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003).
    [Crossref] [PubMed]
  25. F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
    [Crossref]
  26. P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014).
    [Crossref] [PubMed]
  27. K. Liu and M. G. Littman, “Novel geometry for single-mode scanning of tunable lasers,” Opt. Lett. 6(3), 117–118 (1981).
    [Crossref] [PubMed]
  28. P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014).
    [Crossref] [PubMed]

2015 (2)

2014 (4)

2012 (1)

K. Hibino, Y. Kim, S. Lee, Y. Kondo, N. Sugita, and M. Mitsuishi, “Simultaneous measurement of surface shape and absolute optical thickness of a glass plate by wavelength tuning phase-shifting interferometry,” Opt. Rev. 19(4), 247–253 (2012).
[Crossref]

2008 (1)

2004 (2)

2003 (2)

2000 (1)

1999 (1)

K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
[Crossref]

1998 (2)

1997 (1)

1996 (3)

1995 (2)

1993 (1)

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[Crossref]

1992 (1)

1990 (1)

1981 (1)

1978 (1)

F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

1974 (1)

Brangaccio, D. J.

Bruning, J. H.

Burke, J.

Coppola, G.

Creath, K.

de Groot, P.

de Groot, P. J.

De Nicola, S.

Deck, L. L.

Fairman, P. S.

Falkenstörfer, O.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[Crossref]

Farrant, D. I.

Ferraro, P.

Freischlad, K.

Fukano, T.

Gallagher, J. E.

Hanayama, R.

Y. Kim, K. Hibino, R. Hanayama, N. Sugita, and M. Mitsuishi, “Multiple-surface interferometry of highly reflective wafer by wavelength tuning,” Opt. Express 22(18), 21145–21156 (2014).
[Crossref] [PubMed]

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

Haruna, M.

Hashimoto, M.

Herriott, D. R.

Hibino, K.

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry,” Appl. Opt. 54(13), 4207–4213 (2015).
[Crossref]

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Measurement of optical thickness variation of BK7 plate by wavelength tuning interferometry,” Opt. Express 23(17), 22928–22938 (2015).
[Crossref] [PubMed]

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Design of phase shifting algorithms: fringe contrast maximum,” Opt. Express 22(15), 18203–18213 (2014).
[Crossref] [PubMed]

Y. Kim, K. Hibino, R. Hanayama, N. Sugita, and M. Mitsuishi, “Multiple-surface interferometry of highly reflective wafer by wavelength tuning,” Opt. Express 22(18), 21145–21156 (2014).
[Crossref] [PubMed]

K. Hibino, Y. Kim, S. Lee, Y. Kondo, N. Sugita, and M. Mitsuishi, “Simultaneous measurement of surface shape and absolute optical thickness of a glass plate by wavelength tuning phase-shifting interferometry,” Opt. Rev. 19(4), 247–253 (2012).
[Crossref]

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. 43(6), 1241–1249 (2004).
[Crossref] [PubMed]

K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
[Crossref]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts,” J. Opt. Soc. Am. A 14(4), 918–930 (1997).
[Crossref]

Iodice, M.

Ishii, Y.

Kim, M. J.

Kim, S.

Kim, Y.

Koliopoulos, C. L.

Kondo, Y.

K. Hibino, Y. Kim, S. Lee, Y. Kondo, N. Sugita, and M. Mitsuishi, “Simultaneous measurement of surface shape and absolute optical thickness of a glass plate by wavelength tuning phase-shifting interferometry,” Opt. Rev. 19(4), 247–253 (2012).
[Crossref]

Larkin, K. G.

Lee, B. H.

Lee, S.

K. Hibino, Y. Kim, S. Lee, Y. Kondo, N. Sugita, and M. Mitsuishi, “Simultaneous measurement of surface shape and absolute optical thickness of a glass plate by wavelength tuning phase-shifting interferometry,” Opt. Rev. 19(4), 247–253 (2012).
[Crossref]

Littman, M. G.

Liu, K.

Maruyama, H.

Mitsuishi, M.

Mitsuyama, T.

Na, J.

Ohmi, M.

Onodera, R.

Oreb, B. F.

Rosenfeld, D. P.

Schmit, J.

Schreiber, H.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[Crossref]

Schwider, J.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[Crossref]

Streibl, N.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[Crossref]

Sugita, N.

Surrel, Y.

Tajiri, H.

Warisawa, S.

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

White, A. D.

Yamaguchi, I.

Zöller, A.

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[Crossref]

Appl. Opt. (11)

G. Coppola, P. Ferraro, M. Iodice, and S. De Nicola, “Method for measuring the refractive index and the thickness of transparent plates with a lateral-shear, wavelength-scanning interferometer,” Appl. Opt. 42(19), 3882–3887 (2003).
[Crossref] [PubMed]

K. Hibino, B. F. Oreb, P. S. Fairman, and J. Burke, “Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer,” Appl. Opt. 43(6), 1241–1249 (2004).
[Crossref] [PubMed]

Y. Kim, K. Hibino, N. Sugita, and M. Mitsuishi, “Surface profile measurement of a highly reflective silicon wafer by phase-shifting interferometry,” Appl. Opt. 54(13), 4207–4213 (2015).
[Crossref]

P. de Groot, “Measurement of transparent plates with wavelength-tuned phase-shifting interferometry,” Appl. Opt. 39(16), 2658–2663 (2000).
[Crossref] [PubMed]

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[Crossref] [PubMed]

J. H. Bruning, D. R. Herriott, J. E. Gallagher, D. P. Rosenfeld, A. D. White, and D. J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13(11), 2693–2703 (1974).
[Crossref] [PubMed]

J. Schmit and K. Creath, “Extended averaging technique for derivation of error-compensating algorithms in phase-shifting interferometry,” Appl. Opt. 34(19), 3610–3619 (1995).
[Crossref] [PubMed]

P. de Groot, “Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window,” Appl. Opt. 34(22), 4723–4730 (1995).
[Crossref] [PubMed]

L. L. Deck, “Fourier-transform phase-shifting interferometry,” Appl. Opt. 42(13), 2354–2365 (2003).
[Crossref] [PubMed]

P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014).
[Crossref] [PubMed]

P. J. de Groot, “Correlated errors in phase-shifting laser Fizeau interferometry,” Appl. Opt. 53(19), 4334–4342 (2014).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

J. Schwider, O. Falkenstörfer, H. Schreiber, A. Zöller, and N. Streibl, “New compensating four-phase algorithm for phase-shift interferometry,” Opt. Eng. 32(8), 1883–1885 (1993).
[Crossref]

Opt. Express (4)

Opt. Lett. (3)

Opt. Rev. (3)

K. Hibino, “Error-compensating phase measuring algorithms in a Fizeau interferometer,” Opt. Rev. 6(6), 529–538 (1999).
[Crossref]

K. Hibino, Y. Kim, S. Lee, Y. Kondo, N. Sugita, and M. Mitsuishi, “Simultaneous measurement of surface shape and absolute optical thickness of a glass plate by wavelength tuning phase-shifting interferometry,” Opt. Rev. 19(4), 247–253 (2012).
[Crossref]

R. Hanayama, K. Hibino, S. Warisawa, and M. Mitsuishi, “Phase measurement algorithm in wavelength scanned Fizeau interferometer,” Opt. Rev. 11(5), 337–343 (2004).
[Crossref]

Proc. IEEE (1)

F. J. Harris, “On the use of windows for harmonic analysis with discrete Fourier transform,” Proc. IEEE 66(1), 51–83 (1978).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Three-surface laser interferometry.
Fig. 2
Fig. 2 (a) 6N – 5 algorithm (N = 12) obtained by locating six multiple roots on the characteristic diagram, and (b) rotation of 6N – 5 algorithm to extract the mth arbitrary harmonic signal phase. Six circled dot indicates a six multiple root.
Fig. 3
Fig. 3 Shape of the polynomial window function (N = 12) defined by Eqs. (8)–(13).
Fig. 4
Fig. 4 Sampling functions of the 6N – 5 algorithm: (a) m = 1 and (b) m = 3 (N = 12).
Fig. 5
Fig. 5 RMS phase errors of phase shifting algorithms comprising the windows listed in Table 1 and DFT terms as functions of phase shift miscalibration ε0 when the reflectivities of the reference and sample surfaces are 4%.
Fig. 6
Fig. 6 Wavelength tuning Fizeau interferometer used to measure optical thickness variation of a BK7 transparent plate. PBS denotes polarization beam splitter; QWP is quarter-wave plate; HWP is half-wave plate.
Fig. 7
Fig. 7 (a) Laboratory photo of BK7 transparent parallel plate in wavelength tuning Fizeau interferometer and (b) raw interferogram at a wavelength of 634.6 nm.
Fig. 8
Fig. 8 Measured (a) optical thickness variation and (b) surface shape of a BK7 plate at a wavelength of 634.6 nm.

Tables (1)

Tables Icon

Table 1 Representative Phase-Shifting Algorithm

Equations (19)

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I( α r )= A 0 + m=1 A m cos( ν m t φ m ) = A 0 + A 1 cos( α r φ 1 )+ A 2 cos( 2 α r φ 2 )+ A 3 cos( 3 α r φ 3 )+,
φ m =arctan r=1 M b r I( α r ) r=1 M a r I( α r ) ,
α r = α 0r [ 1+ ε 0 + ε 1 α 0r π + ε 2 ( α 0r π ) 2 ++ ε p ( α 0r π ) p ],
Δφ=ο( A k )+ο( ε q )+ο( A k ε q ),
P( x )= [ P sync ( x ) ] 6 = [ 1+x+ x 2 ++ x N1 ] 6 = r=1 6N5 w r x r1 ,
a r = 2 N w r cos 2mπ N [ r( 3N2 ) ],
b r = 2 N w r sin 2mπ N [ r( 3N2 ) ],
w r = 1 120 r( r+1 )( r+2 )( r+3 )( r+4 ),
w r = 1 24 r 5 +( N 4 5 12 ) r 4 +( N 2 2 +2N 35 24 ) r 3 +( N 3 2 3 N 2 + 21 4 N 25 12 ) r 2 1 4 ( N2 ) 2 ( N 2 4N+1 )r+ 1 20 N( N1 )( N2 )( N3 )( N4 ),
w r = 1 12 r 5 +( N+ 5 6 ) r 4 +( 9 2 N 2 8N+ 35 12 ) r 3 +( 19 2 N 3 +27 N 2 21N+ 25 6 ) r 2 +( 39 4 N 4 38 N 3 + 189 4 N 2 20N+2 )r 1 20 N( N1 )( N2 )( 79 N 2 153N+48 ),
w r = 1 12 r 5 +( 3 2 N 5 6 ) r 4 +( 21 2 N 2 +12N 35 12 ) r 3 +( 71 2 N 3 63 N 2 + 63 2 N 25 6 ) r 2 +( 231 4 N 4 +142 N 3 441 4 N 2 +30N2 )r + 1 20 N( N1 )( 731 N 3 1579 N 2 +906N144 ),
w r = 1 24 r 5 +( N+ 5 12 ) r 4 +( 19 2 N 2 8N+ 35 24 ) r 3 +( 89 2 N 3 +57 N 2 21N+ 25 12 ) r 2 +( 409 4 N 4 178 N 3 + 399 4 N 2 20N+1 )r 1 20 N( N1 )( 1829 N 3 2261 N 2 +854N96 ),
w r = 1 120 ( 6Nr )( 6Nr1 )( 6Nr2 )( 6Nr3 )( 6Nr4 ).
F 1 ( ν )= r=1 M b r exp( i α r ν ) ,
F 2 ( ν )= r=1 M a r exp( i α r ν ) .
σ mis = 1 2 2 | i F 1 ( ν ) F 2 ( ν ) 1 |,
σ cou = 1 2 m=2 γ m γ 1 [ i F 1 ( mν ) i F 1 ( ν ) ] 2 + [ F 2 ( mν ) F 2 ( ν ) ] 2 ,
σ= σ mis 2 + σ cou 2 ,
δλ= λ 2 4πnT δφ0.0997nm.

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