Measurement of highly reflective surface shape using wavelength tuning Fizeau interferometer and polynomial window function

Precision Engineering 45 (2016) 187–194

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Precision Engineering journal homepage: www.elsevier.com/locate/precision

Measurement of highly reflective surface shape using wavelength tuning Fizeau interferometer and polynomial window function Yangjin Kim ∗ , Naohiko Sugita, Mamoru Mitsuishi Department of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

a r t i c l e

i n f o

Article history: Received 27 October 2015 Received in revised form 7 February 2016 Accepted 16 February 2016 Available online 3 March 2016 Keywords: Interferometry Surface shape Phase shifting algorithm Phase error

a b s t r a c t In this study, a 5N − 4 phase shifting algorithm comprising a polynomial window function and a discrete Fourier transform is developed to measure interferometrically the surface shape of a silicon wafer, with suppression of the coupling errors between the higher harmonics and the phase shift error. A new polynomial window function is derived on the basis of the characteristic polynomial theory by locating five multiple roots on the characteristic diagram. The characteristics of the 5N − 4 algorithm are estimated with respect to the Fourier representation in the frequency domain. The phase error of the measurements performed using the 5N − 4 algorithm is discussed and compared with those of measurements obtained using other conventional phase shifting algorithms. Finally, the surface shape of a 4-in. silicon wafer is measured using the 5N − 4 algorithm and a wavelength tuning Fizeau interferometer. The accuracy of the measurement is discussed by comparing the amplitudes of the crosstalk noise calculated by other algorithms. The uncertainty of the entire measurement was 34 nm, better than that of any other conventional phase shifting algorithms. © 2016 Elsevier Inc. All rights reserved.

1. Introduction Silicon wafers have been widely used in the semiconductor industry because of their excellent performance and the ease of fabricating integrated circuits on their surface and controlling the value of resistance [1]. The surface shape of a silicon wafer must be measured precisely when estimating the performance of semiconductor devices [2]. The semiconductor industry uses atomic force microscopy (AFM) to measure the surface shape of a silicon wafer; however, using AFM to measure the entire surface shape is time consuming. Wavelength tuning Fizeau interferometry is another method for measuring the surface shape distribution of a silicon wafer. In wavelength tuning interferometry, phase shifting is used to vary the phase difference between a sample beam and a reference beam, and the signal irradiance is acquired at equal phase difference intervals [3]. The phase distribution of a fringe pattern can be calculated with a phase shifting algorithm. When using wavelength tuning interferometry to measure the surface shape of a silicon wafer, not only the phase shift errors that occur during wavelength tuning and the harmonics resulting from the high reflectivity of the surface, but also the coupling errors between the phase shift errors and the higher harmonics

∗ Corresponding author. Tel.: +81 358416357. E-mail address: [email protected] (Y. Kim). http://dx.doi.org/10.1016/j.precisioneng.2016.02.011 0141-6359/© 2016 Elsevier Inc. All rights reserved.

must be considered because the surface reflectivity is high (30%) [4]. These systematic errors influence the calculated phase and appear as crosstalk noise obtained by subtracting successive results. Many studies [5–23] have reported on error-compensating algorithms that can eliminate the effect of systematic errors. Systematic approaches to the design of such phase shifting algorithms include averaging over successive samples [5,9,11], using a Fourier representation [7] or analytical expansion [10,16,17,19,20], using data-sampling windows [12,18], and characteristic polynomial theory [13,21–23]. The prominent Schwider–Hariharan 5 sample algorithm [5,6] can compensate for phase shift miscalibration but not for coupling errors. The 2N − 1 algorithm, developed by Surrel [13], uses characteristic polynomial theory to compensate for phase shift miscalibration and the coupling error between the higher harmonics and the phase shift miscalibration; however, this algorithm cannot compensate for the nonlinearity in the phase shift error. Hibino et al. [16], who derived two kinds of 19 sample algorithms [19,20] by considering the refractive index dispersion in the transparent plate, proposed a phase shifting algorithm that can compensate for the coupling error. However, Hibino algorithms do not satisfy the condition for maximum fringe contrast [22]. Phase shifting algorithms should satisfy the maximum fringe contrast condition when a highly reflective surface such as that of a silicon wafer is measured [22]. We developed the 4N − 3 algorithm that can compensate for up to second-order nonlinearity and coupling errors [23]. The surface shape and variation in the optical

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3 = 1 r1 r2 .

(5)

The fringe contrast  m for each successive harmonic of order m √ decreases in strength by the multiplicative factor − r1 r2 . The phase distribution can be determined using a phase shifting algorithm. Consider an M-sample phase shifting algorithm, where the reference phases are separated by M − 1 equal intervals of ı = 2␲/N rad and N is an integer. A general expression for the calculated phase in this algorithm is given by [24]

M ∗

ϕ = arctan Fig. 1. Laser Fizeau interferometer.

thickness of a lithium niobate (LNB) crystal wafer were measured simultaneously using wavelength tuning and the 4N − 3 algorithm. However, the ripples that result from the residual phase shift error and coupling errors were clearly observed in the measured surface shape and the variation in the optical thickness [23]. The ripples that resulted from the imperfect suppression of the coupling errors were observed when one subtracted the successively measured surface shapes or the variations in optical thickness. We developed and present here a new 5N − 4 phase shifting algorithm that comprises a polynomial window function and a discrete Fourier transform (DFT) term to measure the surface shape of a silicon wafer with suppression of the coupling errors. The characteristics of the 5N − 4 algorithm are discussed with respect to the Fourier representation of the phase shifting algorithm in the frequency domain. We show that our 5N − 4 algorithm yields the smallest phase error compared with those of five conventional phase shifting algorithms. Finally, the surface shape of a 4-in. silicon wafer was measured using a wavelength tuning Fizeau interferometer and the 5N − 4 algorithm. The accuracy of the measurement of the surface shape was 2.2 nm. The accuracy of the conventional phase shifting algorithms also is discussed with respect to the crosstalk noise. 2. 5N − 4 phase shifting algorithm 2.1. Laser Fizeau interferometer A laser Fizeau interferometer (Fig. 1) allows the interference of multiple reflections between a sample surface and a reference surface by virtue of the high degree of coherence of the light. Let the reference and sample surface reflectivities be r1 and r2 , respectively. The observed signal irradiance I(˛r ) in the interference fringe pattern that occurs during phase shifting is given by [4,17] I (˛r ) =

∞ 



= I0 1 +

∞ 

 m cos (ϕm − m˛r )

(1)

m=1

= I0 + I0 1 cos (ϕ1 − ˛r ) + I0 2 cos (ϕ2 − 2˛r ) + · · ·, where ˛r is the phase shift parameter and Am and ϕm are the amplitude and phase of the mth harmonic component, respectively. The DC component I0 of the signal irradiance and the fringe contrast  m of the mth harmonic component are given by [4] I0 =

r1 + r2 − 2r1 r2 , 1 − r1 r2

2 (1 − r1 ) (1 − r2 ) √ r1 r2 , r1 + r2 − 2r1 r2 √ 2 = −1 r1 r2 , 1 =

(2) (3) (4)

br I (˛r )

a I r=1 r (˛r )

,

(6)

where ar and br are the rth sampling amplitudes and I(˛r ) is given by Eq. (1). When the phase shift is nonlinear, each ˛r value is a function of the phase shift parameter and can be expressed as a polynomial function of the unperturbed phase shift value ˛0r [16]:



˛r = ˛0r [1 + ε (˛0r )] = ˛0r 1 + ε0 + ε1 +ε2

 ˛ 2 0r



˛0r 

+ · · · + εp

 ˛ p 0r



,

(7)

where p is the maximum order of the nonlinearity, ε0 is the error coefficient of the phase shift miscalibration, εq (1 ≤ q ≤ p) is the error coefficient of the qth nonlinearity of the phase shift, and ˛0r = 2␲[r − (M + 1)/2]/N. The phase error ϕ in the calculated phase is a function of the amplitude ratio Am /A1 and the error coefficient εq and can undergo a Taylor expansion as follows:







ϕ = ϕ∗ − ϕ1 = o (Ak ) + o εq + o Ak εq ,

(8)

for k = 2, 3, . . ., m and q = 0, 1, . . ., p. In Eq. (8), о(Ak ), о(εq ), and о(Ak εq ) are the error in the harmonics, the phase shift error, and the coupling error between the higher harmonics and the phase shift error, respectively. For example, о(ε0 ) is the phase shift miscalibration and о(A2 ε1 ) is the coupling error between the second harmonic and first-order nonlinearity of the phase shift error. When measuring the surface shape of a highly reflective sample such as a silicon wafer, the coupling error is large because the higher harmonics components considerably influence the calculated phase distribution, even though the phase shift miscalibration is extremely small [4]. 2.2. Derivation of 5N − 4 phase shifting algorithm The characteristic polynomial P(x) proposed by Surrel [13] is defined as

Am cos (ϕm − m˛r )

m=1

r=1 M

P (x) =

M 

(ar + ibr ) xr−1 ,

(9)

r=1

where i is the imaginary number, x = exp(imı), and ı = 2/N. Surrel noted that the locations and multiplicities of the roots of the polynomial in the characteristic diagram determine the sensitivity of the algorithm to higher harmonics and phase shift miscalibration [13]. First, to suppress the mth harmonic component, the characteristic polynomial of the phase shifting algorithm should have single roots in the characteristic diagram, as shown in Fig. 2(a) [13]. This is the synchronous detection algorithm proposed by Bruning [3] and it does not compensate for о(εq ) and о(Ak εq ) specified in Eq. (8). To suppress o(ε0 ) and o(Am ε0 ), the double roots should be located on the characteristic diagram as shown in Fig. 2(b), the 2N − 1 algorithm proposed by Surrel [13], which uses the triangular window

Y. Kim et al. / Precision Engineering 45 (2016) 187–194

189

Fig. 3. Shape of polynomial window function defined by Eqs. (11a)–(11e).

(iii) 2N + 1 ≤ r ≤ 3N − 4 wr =



1 4 5 3 r + − N+ 4 2 2

 25

Fig. 2. Root multiplicity on the characteristic diagram (N = 6). (a) Bruning’s synchronous detection [3], (b) Surrel’s 2N − 1 algorithm [13], (c) Hanayama’s 2N − 1 algorithm [21], and (d) 5N − 4 algorithm.

+ − +

function [26] and the DFT term. Hanayama [21] developed a modified window function to suppress the sidelobe level of Surrel’s 2N − 1 algorithm in the frequency domain by adding the compensation term to the triangular window function, as seen in Fig. 2(c). This 2N − 1 algorithm of Hanayama imperfectly compensates for the coupling error because the single roots are at m = 0 and m = 2 and it cannot compensate for the bias modulation of intensity [15]. By locating the five multiple roots of the characteristic diagram shown in Fig. 2(d), the phase shifting algorithm can compensate for up to third-order nonlinearity of phase shift о(ε3 ) and coupling errors о(Am ε3 ). In this case, the characteristic polynomial shown in Fig. 2(d) is given by



P (x) = Psync (x)



= 1 + x + x2 + · · · + xN−1



5 (10)

5N−4

=

wr xr ,

r=1

(i) 1 ≤ r ≤ N 1 r (r + 1) (r + 2) (r + 3) , 24

(11a)

(ii) N + 1 ≤ r ≤ 2 N 1 wr = − r 4 + 6 +

5

−

 65

5 2

N3 −

 655 24



 55

N − 1 r3 + −

4

r2

r

(11c)



N2 +



45 11 N− 4 6

165 2 55 N + N−1 r 4 4

N4 +





r2 (11d)



195 3 605 2 15 N − N + N , 4 24 4

(v) 4N − 3 ≤ r ≤ 5N − 4 1 (5N − r) (5N − r − 1) (5N − r − 2) (5N − r − 3) . 24

(11e)

The 5N − 4 algorithm comprises a polynomial window function and DFT term. Fig. 3 shows the shape of the window function defined by Eqs. (11a)–(11e). 3. Characteristics of the 5N − 4 phase shifting algorithm

where Psync (x) is the characteristic polynomial of the synchronous detection and wr is the polynomial window function defined as

wr =



45 11 N+ 4 4

75 3 385 2 15 N + N − N , 4 24 4

1 wr = − r 4 + 6

+ −

4

N2 −

N4 −

24

2

 35

105 2 55 3 N − N+ 4 4 2

 155

+

r3 +

N3 +

(iv) 3N − 3 ≤ r ≤ 4N − 4

wr =

5

2



5



 5

15 11 N − 1 r3 + − N2 + N− 6 4 4 6



15 2 55 N − N + N−1 r 6 4 12 3

5 N (N − 1) (N − 2) (N − 3) , 24



3.1. Fourier representation of the 5N − 4 algorithm The phase shifting algorithm can be visualized and better understood by using a Fourier representation of the sampling amplitudes of the algorithm [7,25]. The sampling functions in the numerator and denominator of the M-sample algorithm given by Eq. (6) are defined in the frequency domain as, respectively, f1 (˛) =

M 

br ı (˛ − ˛r ) ,

(12)

ar ı (˛ − ˛r ) ,

(13)

r=1

r2 (11b)

f2 (˛) =

M  r=1

where ı(˛) is the Dirac delta function. Using the sampling functions and Parseval’s equation, the phase shifting algorithm given by Eq.

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Fig. 4. Sampling functions for (a) synchronous detection (N = 24) [3], (b) Larkin–Oreb N + 1 algorithm (N = 24) [7], (c) Surrel’s 2N − 1 algorithm (N = 12) [13], (d) Hibino 19 algorithm [19], (e) Hanayama’s 2N − 1 algorithm (N = 12) [21], and (f) 5N − 4 algorithm.

(6) can be rewritten as

purely imaginary and purely real functions, respectively [10,16], and are expressed as

∞

M ∗

ϕ = arctan

b I r=1 r (˛r ) M a I r=1 r (˛r )



∞ = arctan

−∞ ∞

= arctan

F1 () J () d

F −∞ 2

() J () d

,



f −∞ 1 ∞ f −∞ 2

(˛) I (˛) d˛

F1 () =

(˛) I (˛) d˛

M 

br exp (−˛r ) = −i

r=1

(14)

where F1 (), F2 (), and J() are the Fourier transforms of f1 (˛), f2 (˛), and the signal irradiance I(˛), respectively. Because of the symmetrical and asymmetrical properties of the sampling amplitudes ar and br and the phase shift parameter ˛r , F1 and F2 must be

F2 () =

M  r=1

M 

br sin (˛r ) ,

(15)

r=1

ar exp (−˛r ) =

M 

ar cos (˛r ) .

(16)

r=1

Table 1 presents the conventional phase shifting algorithms that can be used to measure the surface shape and its characteristics in the frequency domain and their ability to compensate for phase errors. Fig. 4 shows the sampling functions iF1 and F2 of the six conventional phase shifting algorithms listed in Table 1. The

Y. Kim et al. / Precision Engineering 45 (2016) 187–194

191

Table 1 Characteristics of phase shifting algorithms.

Synchronous detection Larkin–Oreb N + 1 Surrel 2N − 1 Hibino 19 Hanayama 2N − 1 New 5N − 4

Reference

Image number

Sidelobe level [%]

Compensation ability Phase shift error

Coupling error

[3] [7] [13] [19] [21] –

24 25 23 19 23 26

32.693 43.758 6.036 5.403 1.436 0.060

None о(ε0 ) о(ε0 ) о(ε0 ) о(ε0 ) Up to о(ε3 )

None None о(Am ε0 ) None о(Am ε0 ) (imperfect) Up to о(Am ε3 )

characteristics of a phase shifting algorithm can be deduced from the sampling functions in the frequency domain [10,15,16,22,27]. The synchronous detection algorithm, the Larkin–Oreb N + 1 algorithm, and the Hibino 19 algorithm do not satisfy the fringe contrast maximum condition because there are gradients in the frequency domain [22]. In contrast, the 5N − 4 algorithm satisfies the fringe contrast maximum condition because the sampling functions of this algorithm have a zero gradient at the fundamental frequencies. The Hanayama 2N − 1 algorithm cannot compensate for the bias modulation because its sampling functions have nonzero gradients at m = 0, as shown in Fig. 4(e). Fig. 4(f) shows that the sidelobe level of sampling functions of the 5N − 4 algorithm is barely visible (∼0.06%) in the frequency domain. In addition, the 5N − 4 algorithm can compensate for the bias modulation because the sampling functions have zero gradients at zero frequency [15]. To compensate for the coupling errors о(Am ε0 ), the sampling functions should have zero gradients at m = 2, 3, . . ., m − 2 [16,27], as does the 5N − 4 algorithm. Furthermore, the sampling functions of the 5N − 4 algorithm, up to derivatives of the third order, have zero gradients, which means that the algorithm can compensate for coupling errors up to о(Am ε3 ) of Eq. (8).

because the reference surface of a spherical sample is curved and the phase shift that occurs along the axial translation is not uniform over the field of view [28,29] and is estimated to be approximately 30% [30]. Fig. 5 also shows that the 5N − 4 algorithm can suppress the rms phase error better than the other algorithms, even for a phase shift miscalibration of 30%. 4. Interferometric measurement of the surface shape of a silicon wafer 4.1. Wavelength tuning Fizeau interferometer Fig. 6 shows the optical setup used to measure the surface shape of a highly reflective silicon wafer in a wavelength tuning Fizeau interferometer. The temperature inside the laboratory for

3.2. Ability to suppress the coupling error de Groot and Hibino [4,17] studied the coupling errors between the higher harmonics and the phase shift miscalibration thoroughly. de Groot [4] analyzed the calculated phase error between the higher harmonics and the phase shift miscalibration. The root mean square (rms) phase error  mis that results from the phase shift miscalibration is given by mis =





1  iF1 ()  − 1 , √  2 2 F2 ()

(17)

where iF1 () and F2 () were defined by Eqs. (15) and (16). The rms phase error  cou that results from the coupling error between the mth harmonic and the phase shift miscalibration is given by [4] 1  m 2 1 ∞

cou =

  iF

m=2

(m) iF1 () 1

2

+

F

(m) F2 () 2

2

,

(18)

where  m is the fringe contrast of the mth harmonic as defined by Eqs. (2)–(5). Therefore, the net rms error is given by [4]



=

2 + 2 . mis cou

(19)

Fig. 5(a) shows the solutions of Eq. (19) based on the phase shifting algorithms listed in Table 1, for reference and sample reflectivities of 4%. The corresponding results for a reference surface reflectivity 4% and a sample reflectivity 30% is shown in Fig. 5(b). Comprehensive numerical simulations for all algorithms confirm these results up to the fourth Fizeau harmonic. Fig. 5 shows that even when ε0 is 0, the rms phase error of the Hibino 19 algorithm is not zero because this algorithm cannot compensate for the phase shift errors. When a Fizeau interferometer is used to measure a spherical surface, ε0 is particularly important

Fig. 5. Root mean square phase errors of the phase shifting algorithms listed in Table 1 as functions of phase shift miscalibration when the reference surface reflectivity is 4% and the silicon wafer surface reflectivity is (a) 4% and (b) 30%. “New” means the 5N − 4 algorithm.

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Fig. 7. Raw interferogram at a wavelength of 633 nm.

Fig. 6. Wavelength tuning Fizeau interferometer for measuring the surface shape of a 4-in. silicon wafer. PBS, polarizing beam splitter; QWP, quarter-wave plate; HWP, half-wave plate.

our experiment was 20.5 ◦ C (±0.1 ◦ C). Fizeau interferometer has the smallest influence of air turbulence among the conventional interferometer and the intensity modulation by the residual air turbulence can be compensated by 5N − 4 algorithm. The light source is a tunable diode laser with a Littman external cavity (New Focus TLB–6300–LN) comprising a grating and a cavity mirror. The source wavelength is scanned linearly over time from 632.8 to 638.4 nm by translating the cavity mirror using a piezoelectric (PZT) transducer and a picomotor at constant speed [31]. An isolator transmits the beam that is then split by a beam splitter: one beam goes to a wavelength meter (Anritsu MF9630A) that is calibrated by a stabilized HeNe laser with an accuracy of ı / ∼ 10−7 at a wavelength of 632.8 nm; the other is incident on the interferometer. A polarizing beam splitter (PBS) reflects the focused output beam. The linearly polarized beam is then transmitted to a quarter-wave plate (QWP) to form a circularly polarized beam that is collimated by a collimator lens to illuminate the reference surface and the sample. The reflected light from the multiple surfaces of the sample and the reference travels backward along the same path and is then transmitted back through the QWP to be orthogonally linearly polarized. The resulting beams pass through the PBS and combine to generate a fringe pattern on the screen with a resolution of 640 × 480 pixel. The sample is horizontal on a mechanical stage with an air gap L between it and the reference surface.

Fig. 8. Intensity change at one point on the silicon wafer during wavelength tuning.

scanning process, the signal interference fringes shifted by five periods to yield a total shift of 10 rad. The phase shift for each step was equal to /3 for the fundamental component of the surface of the silicon wafer. Because the PZT response (fine scanning mode) of the cavity of the source laser was approximately 3% nonlinear, a quadratic voltage was applied incrementally to the PZT and the nonlinearity decreased to 1% of the total phase shift. Fig. 8 shows the intensity change during wavelength tuning at one point on the silicon wafer. There was the small intensity modulation during wavelength tuning, but the phase error resulted from intensity

4.2. Results and discussion We measured the surface shape of a 4-in. silicon wafer (surface reflectivity 30%) using a wavelength tuning interferometer with an air gap L ∼ 3 mm between the sample and the reference and the 5N − 4 algorithm. Fig. 7 shows the raw interferogram of a silicon wafer at a wavelength of 633.0 nm. The wavelength scanning range ı needed to measure the surface shape is calculated as ı = −

2 ıϕ ≈ 0.336 nm. 4L

(20)

The wavelength was finely scanned from 633.0812 to 633.4198 nm and 26 interference images were recorded at equal intervals. For each experiment, it took 30 s to acquire 26 images. The slow image acquisition rate resulted mainly from the slow data transfer rate from CCD camera to the computer. During the

Fig. 9. Surface shape of a 4-in. silicon wafer at a wavelength of 633 nm.

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193

Fig. 10. Crosstalk noise obtained by subtracting successive results that are calculated using (a) synchronous detection (N = 24), (b) Larkin–Oreb N + 1 algorithm (N = 24), (c) Surrel 2N − 1 algorithm (N = 12), (d) Hibino 19 algorithm, (e) Hanayama 2N − 1 algorithm (N = 12), and (f) 5N − 4 algorithm (N = 6).

modulation can be ignored because the 5N − 4 algorithm has the compensation ability for the intensity modulation. The phase distribution p was calculated using the 5N − 4 algorithm: p=

1 arctan 2

26







w I sin r=1 r r 26 w I cos r=1 r r



r/3

,

(21)

r/3

where Ir is the intensity of the rth recorded image as defined by Eq. (1) and the window function wr is defined by Eqs. (11a)–(11e). The phase distribution p calculated using Eqs. (11a)–(11e) and (21) is the phase value of image I13 (=I14 ). Fig. 9 shows the measured surface shape of a 4-in. silicon wafer. Fig. 9 shows that the surface of the silicon wafer is convex with an amplitude of 1.13 ␮m. The size of the error in the repeatability between two measurements taken 3 days apart was approximately 2.198 nm. The uncertainty in the measurement of the reference surface was approximately /20 = 32 nm, so the uncertainty in the measurement of the surface shape was approximately 34 nm. We also calculated the surface shape of a silicon wafer using the phase shifting algorithms listed in Table 1. Fig. 10 shows the crosstalk noise obtained by subtracting successive results that were calculated using those phase shifting algorithms. In general, the configuration of the crosstalk noise is similar to that of the raw interferogram shown in Fig. 7. Table 2 gives the amplitudes of

Table 2 Characteristics of phase shifting algorithms included in this study. Phase shifting algorithm

Image number

Amplitude of crosstalk noise [nm]

Synchronous detection Larkin Oreb N + 1 Surrel 2N − 1 Hibino 19 Hanayama 2N − 1 New 5N − 4

24 25 23 19 23 26

30.978 22.693 8.345 24.697 10.689 2.198

the calculated crosstalk noise of the phase shifting algorithms in Table 1. Fig. 10 and Table 2 show that the amplitude of the crosstalk noise calculated using the 5N − 4 algorithm is the smallest of the six phase shifting algorithms. The synchronous detection algorithm had the largest amplitude because it cannot compensate for о(εq ) and о(Am εq ) in Eq. (8). The Larkin–Oreb N + 1 and Hibino 19 algorithms also have large crosstalk noise amplitudes because they cannot compensate for the coupling errors between the higher harmonics and phase shift error. Small crosstalk noise also was observed in the noise distribution calculated by the 5N − 4 algorithm. These small errors were the result of intensity variation during wavelength tuning and from residual phase shift error and coupling errors. 5. Conclusion We developed the 5N − 4 phase shifting algorithm to measure the surface shape of a highly reflective silicon wafer while suppressing the coupling errors between the higher harmonics and the phase shift errors. The 5N − 4 algorithm comprises a new polynomial window function and a DFT term. The new polynomial window function was calculated by locating the five multiple roots on the characteristic diagram. The characteristics of the 5N − 4 algorithm were discussed with respect to the Fourier representation of the algorithm in the frequency domain compared with other conventional phase shifting algorithms. The phase error calculated using the 5N − 4 algorithm is the smallest of the six phase shifting algorithms listed in Table 1. Finally, we measured the surface shape of a 4-in. silicon wafer using the 5N − 4 algorithm and a wavelength tuning Fizeau interferometer. In addition, the amplitude of the crosstalk noise of the 5N − 4 algorithm was smaller than those of the other five algorithms. The total uncertainty in the measurement was approximately 34 nm because the uncertainty in the measurement of the reference surface was 32 nm.

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References [1] Venkatesh VC, Inasaki I, Toenshof HK, Nakagawa T, Marinescu ID. Observations on polishing and ultraprecision machining of semiconductor substrate materials. CIRP Ann—Manuf Technol 1995;44:611–8. [2] Gui C, Elwenspoek M, Tas N, Gardeniers JGE. The effect of surface roughness on direct wafer bonding. J Appl Phys 1999;85:7448–54. [3] Bruning JH, Herriott DR, Gallagher JE, Rosenfeld DP, White AD, Brangaccio DJ. Digital wavefront measuring interferometer for testing optical surfaces and lenses. Appl Opt 1974;13:2693–703. [4] de Groot PJ. Correlated errors in phase-shifting laser Fizeau interferometry. Appl Opt 2014;53:4334–42. [5] Schwider J, Burow R, Elssner KE, Grzanna J, Spolaczyk R, Merkel K. Digital wavefront measuring interferometry: some systematic error sources. Appl Opt 1983;22:3421–32. [6] Hariharan P. Digital phase-stepping interferometry: effects of multiply reflected beams. Appl Opt 1987;26:2506–7. [7] Larkin KG, Oreb BF. Design and assessment of symmetrical phase-shifting algorithms. J Opt Soc Am A: Opt Image Sci Vision 1992;9:1740–8. [8] Surrel Y. Phase stepping: a new self-calibrating algorithm. Appl Opt 1993;32:3598–600. [9] Schwider J, Falkenstörfer O, Schreiber H, Zöller A, Streibl N. New compensating four-phase algorithm for phase-shift interferometry. Opt Eng 1993;32(8):1883–5. [10] Hibino K, Oreb BF, Farrant DI, Larkin KG. Phase-shifting for nonsinusoidal waveforms with phase shift errors. J Opt Soc Am A: Opt Image Sci Vision 1995;12:761–8. [11] Schmit J, Creath K. Extended averaging technique for derivation of error-compensating algorithms in phase shifting interferometry. Appl Opt 1995;34:3610–9. [12] de Groot P. Derivation of algorithms for phase-shifting interferometry using the concept of a data-sampling window. Appl Opt 1995;34:4723–30. [13] Surrel Y. Design of algorithms for phase measurements by the use of phase stepping. Appl Opt 1996;35:51–60. [14] Onodera R, Ishii Y. Phase-extraction analysis of laser-diode phase-shifting interferometry that is insensitive to changes in laser power. J Opt Soc Am A: Opt Image Sci Vision 1996;13:139–46. [15] Surrel Y. Design of phase-detection algorithms insensitive to bias modulation. Appl Opt 1997;36:805–7.

[16] Hibino K, Oreb BF, Farrant DI, Larkin KG. Phase-shifting algorithms for nonlinear and spatially nonuniform phase shifts. J Opt Soc Am A: Opt Image Sci Vision 1997;14:918–30. [17] Hibino K. Error-compensating phase measuring algorithms in a Fizeau interferometer. Opt Rev 1999;6:529–38. [18] de Groot P. Measurement of transparent plates with wavelength-tuned phaseshifting interferometry. Appl Opt 2000;39:2658–63. [19] Hibino K, Oreb BF, Fairman PS. Wavelength-scanning interferometry of a transparent parallel plate with refractive-index dispersion. Appl Opt 2003;42:3888–95. [20] Hibino K, Oreb BF, Fairman PS, Burke J. Simultaneous measurement of surface shape and variation in optical thickness of a transparent parallel plate in wavelength-scanning Fizeau interferometer. Appl Opt 2004;43:1241–9. [21] Hanayama R, Hibino K, Warisawa S, Mitsuishi M. Phase measurement algorithm in wavelength scanned Fizeau interferometer. Opt Rev 2004;11: 337–43. [22] Kim Y, Hibino K, Sugita N, Mitsuishi M. Design of phase shifting algorithms: fringe contrast maximum. Opt Express 2014;22:18203–13. [23] Kim Y, Hibino K, Hanayama R, Sugita N, Mitsuishi M. Multiple-surface interferometry of highly reflective wafer by wavelength tuning. Opt Express 2014;22:21145–56. [24] Creath K. Phase measurement interferometry techniques. In: Wolf E, editor. Progress in optics. Amsterdam: North Holland Elsevier; 1988. p. 351–93. [25] Freischlad K, Koliopoulos CL. Fourier description of digital phase-measuring interferometry. J Opt Soc Am A: Opt Image Sci Vision 1990;7:542–51. [26] Harris FJ. On the use of windows for harmonic analysis with the discrete Fourier transform. Proc IEEE 1978;66:51–83. [27] Kim Y, Hibino K, Sugita N, Mitsuishi M. Measurement of optical thickness variation of BK7 plate by wavelength tuning interferometry. Opt Express 2015;23:22928–38. [28] Moore RC, Slaymaker FH. Direct measurement of phase in a spherical-wave Fizeau interferometer. Appl Opt 1980;19:2196–200. [29] de Groot P. Phase-shift calibration errors in interferometers with spherical Fizeau cavities. Appl Opt 1995;34:2856–63. [30] Creath K, Hariharan P. Phase-shifting errors in interferometric tests with highnumerical-aperture reference surfaces. Appl Opt 1994;33:24–5. [31] Liu K, Littman MG. Novel geometry for single-mode scanning of tunable lasers. Opt Lett 1981;6:117–8.