The surface shape and optical thickness variation of a lithium niobate (LNB) wafer were measured simultaneously using a wavelength-tuning interferometer with a new phase-shifting algorithm. It is necessary to suppress the harmonic signals for testing a highly reflective sample such as a crystal wafer. The LNB wafer subjected to polishing, which is in optical contact with a fused-silica (FS) supporting plate, generates six different overlapping interference fringes. The reflectivity of the wafer is typically 15%, yielding significant harmonic signals. The new algorithm can flexibly select the phase-shift interval and effectively suppress the harmonic signals and crosstalk. Experimental results indicated that the optical thickness variation of the LNB wafer was measured with an accuracy of 2 nm.
© 2014 Optical Society of America
Lithium niobate wafers (no = 2.2865 at a wavelength of 632.8 nm) are transparent uniaxial crystals that are widely used in optical modulators such as waveguides and second-harmonic generators because their refractive index can be altered considerably by superimposed voltage and temperature changes. A crystal parallel-polished to less than 1 mm in thickness together with coated dielectric films and a transparent electrode on both sides is widely used in solar spectroscopy as a Fabry-Perot interferometer for the near-infrared to visible wavelength range. To realize a high finesse number, the surface shape and optical thickness deviation of the wafer should be strictly monitored with nanometer-scale uncertainty.
In the conventional process, the wafer is in optical contact with the supporting plate during polishing and then removed from the plate for the measurement. The repeated removal from the plate can increase the risk of damage to the wafer. It is therefore desirable for the wafer to measure the surface shape and optical thickness deviation simultaneously with the supporting plate together.
One such approach reported by several authors involves white light interferometry [1–3], wherein the surface shape and optical thickness variation are measured simultaneously by alternatively separating the interference fringe caused by multiple surfaces on an optical axis, using the characteristics that interference fringe is localized only within short coherence length. This method allows the coherence length to be as small as approximately 2 μm, which is the limit of the visible region when using a white-light LED as a light source. Moreover, attempts to increase the measurement sensitivity by more than 1 nm have been made by applying a Fourier analysis to an interference fringe waveform and detecting the phase of some wavelength components . However, when a sample’s thickness is greater than a millimeter, the measurement accuracy decreases because the zero position of the optical path difference of transmitted light differs with each wavelength owing to the sample dispersion. For example, when illuminating a 1-mm-thick fused silica plate with red light having a spectral bandwidth of 100 nm, the equiphase surfaces for each wavelength shift by a maximum of 3.3 μm. Furthermore, the size of the observing aperture is restricted to 1 cm in diameter owing to the difficulty of obtaining an accurate translation of a large reference mirror along the optical axis.
Using a wavelength-tuning interferometer [4–9], the restriction of the aperture size can be resolved, and the same measurement accuracy is obtained, independent of the sample thickness. However, the observed image comprises combinations of different interference fringes because of the use of monochromatic light. In order to measure the surface shape and optical thickness variation simultaneously, it is necessary to separate the phases of these interference fringes. If the source wavelength is scanned linearly in time, the phase of each interference fringe changes with a different frequency, proportional to the optical path difference of the two interfering beams. In the case of a scanning width less than 1 nm, the surface shape and optical thickness variation were determined with nanometer-scale accuracy [7–9].
In order to correctly extract the phase of a specific interference signal from a multiple-surface interferogram, it is necessary to use a special phase-shifting algorithm that can be tuned flexibly to each fringe in frequency domain and reduce the effect of the nonlinearities of the detector and phase modulation [10, 11]. Unlike two-beam interference, strong crosstalk occurs in multiple-beam interference in the presence of the nonlinearity. In conventional techniques, window functions such as Hann or Hamming have been used in the discrete Fourier analysis  to suppress the effect of harmonic signals. However, in order to avoid a phase extraction error for these window functions, the signal frequency should be known precisely. Although it is not difficult to measure the period of an interference fringe in two-beam interferometry, it is rather difficult to detect the single period of the fringes in multiple-beam interferometry. Otherwise, we should scan the source wavelength sufficiently that all the fringes are well separated in frequency domain. Usually, both of these are difficult, and we must assume that the sampling and the signal frequencies are slightly detuned.
In this paper, we start with a triangle window function that can reduce a phase-shift miscalibration but has little effect on the harmonic suppression. We modified the triangle window to reduce the nonlinearity in the phase shift and enhance the harmonic suppression effect. A general expression of a (4N – 3)-sample algorithm was derived using a characteristic polynomial theory . As an example, a new 61-sample (N = 16) algorithm was used for the measurement of a lithium niobate (LNB) wafer 5 mm thick and 74 mm in diameter, in contact with a fused silica (FS) supporting plate. The surface shape and optical thickness variation of the wafer and those of the FS plate were measured simultaneously.
2. Multiple-surface interferometer
We specify the experimental configuration in order to discuss the fringe modulation frequencies in the Fizeau interferometer. Figure 1 shows the measurement setup for testing multiple surfaces of the measurement sample comprising an LNB wafer and an FS supporting plate. The source is a tunable diode laser with a Littman external cavity (New Focus TLB-6900) comprising a grating and a cavity mirror . The source wavelength is scanned linearly from 632.8 nm to approximately 0.23 nm using a piezo-electric transducer (PZT). The beam is transmitted using an isolator and divided into two by a beam splitter: one beam goes to a wavelength meter (Anritsu MF9630A), and the other is incident on a single-mode fiber. The focused output beam is reflected by a polarization beam splitter. The linearly polarized beam is then transmitted to a quarter-wave plate to become a circularly polarized beam. This beam is collimated to illuminate the reference surface and the measurement sample. The reflections from the multiple surfaces of the measurement sample and reference surface travelback along the same path, and then they are transmitted through the quarter-wave plate again to attain an orthogonal linear polarization. The resulting beams pass through the polarization beam splitter and combine to generate an interferogram on the screen with a resolution of 640 × 480 pixels. The sample is placed horizontally on a mechanical stage, with an air-gap distance of L.
The irradiance signal observed by the CCD detector is formed by the multiple-beam interference between the reflection beams from the reference surface and the sample surfaces. The irradiance signal of a function of time t is given as
Figure 2 shows the geometric layout of the reference surface and the LNB wafer that adheres to the FS supporting plate. The LNB wafer of thickness T1, diameter 74 mm, and refractive index n1 = 2.27, which adheres to the FS plate of thickness T2, diameter 80 mm, and refractive index n2 = 1.45, is separated from the reference surface by the air gap distance L. The average distance between the LNB wafer and the FS supporting plate in the optical contact was measured as 2~3 nm. For details, the p and s polarization beams of a linearly polarized HeNe laser were incident 45 degrees to the contacted sample, and the intensities of both reflections from the LNB top and the LNB-FS boundary surfaces were separated and measured, respectively. As the distance increases, the reflection from the boundary increases, which can be calculated to estimate the average distance. Therefore, the air layer of the boundary part of the LNB wafer and FS supporting plate may be ignored.
According to Eq. (2), the modulation frequency changes with respect to three parameters: the optical thickness of LNB wafer, optical thickness of FS supporting plate, and air gap distance. The modulation frequency of each interferogram observed on the screen is given as
Although the air gap distance L can be set arbitrarily, it is generally impossible to make the modulation frequencies of all of the components correspond to a simple integer ratio because the thickness of the LNB wafer and FS supporting plate cannot be chosen freely. However, it is possible to set the air gap distance such that the specific signal frequency does not overlap with the frequency of the other main signals. If the air gap distance is set as 61.3 mm, the ratio of the optical thickness of the air gap distance to that of LNB wafer and FS supporting plate is set to approximately 5:1:2. In this case, six types of interference fringes are formed by the direct reflection beam from the three surfaces of the measurement sample and the reference surface. Table 1 shows the relative frequencies of the interference fringes. All six signals shown have dominant amplitudes ranging from 0.3 to 1, with arbitrary units.
3. Flexible phase-shifting algorithm
Systematic approaches for deriving a phase-shifting algorithm have been proposed by several authors based on characteristic polynomials , the averaging method , the addition of auxiliary functions [15, 16], simultaneous linear equations [17, 18], the extended averaging method , the and recursive method . Hibino et al. derived two 19-sample algorithms [8, 9] that have a very small sensitivity to noise frequencies in the range of 2ν1–10ν1 and tolerate some detuning of their signal frequencies. However, we must be more flexible with the basic phase-shift interval, that we can sample appropriately all frequencies of interest. A flexible algorithm that compensates for a certain order of the phase-shift error and the harmonic signals up to the order j, offer many insights into the performance of the algorithms, such as the systematic and random errors by the algorithms as a function of the number of samples .
3.1 Characteristic polynomial and sampling function of phase-shifting algorithm
A general M-sample phase-shifting algorithm is given byEq. (1).
Surrel proposed the characteristic polynomial which determines the characteristics of the phase-shifting algorithm. All of the properties of any phase-shifting algorithm can be deduced by examination of the roots of the characteristic polynomial. For example, the synchronous detection algorithm proposed by Bruning  has the single roots on the characteristic diagram. The characteristic polynomial of the synchronous detection algorithm is called the “DFT polynomial.” In addition, Surrel proposed a 2N – 1 algorithm  that can compensate for phase-shift miscalibration by locating the double root on the characteristic diagram when N is the integer. The sampling amplitudes of the 2N – 1 algorithm comprise the triangle window function and discrete Fourier transform term defined by
The phase-shifting algorithm can be visualized and well understood if we take a Fourier representation of the sampling functions of the algorithm [15, 23]. The sampling functions at the frequency domain of the numerator F1 and denominator F2 of an algorithm given by Eq. (4) are defined as17, 18, 24, 25].
Figure 3 shows the sampling functions iF1 and F2 of Surrel’s 2N – 1 algorithm (N = 16). The sidelobe amplitude of this algorithm is approximately 5.253%, which is not suitable for the measurement of a highly reflective sample such as an LNB wafer surface because to suppress harmonics effectively is the important factor when measuring a highly reflective sample.
3.2 Polynomial window function
In order to measure the highly reflective sample surface precisely, it is important to suppress the harmonic signals more effectively than Surrel’s 2N – 1 algorithm and compensate for notonly the phase-shift miscalibration but also the nonlinearity of the phase-shift error. Generally, the phase-shift error can be defined as
By locating the quadruple roots on the characteristic diagram as shown in Fig. 4(a), we can generate the 4N – 3 phase-shifting algorithm, which has the compensation ability for the 2nd-order ε2 nonlinearity of the phase-shift error. By rotating the characteristic diagram according to –mδ (δ = 2π/N) as shown in Fig. 4(b), we obtain an algorithm that can extract the mth arbitrary harmonic signal phase.
The sampling amplitudes of the 4N – 3 algorithm are the product of a new window function and the sinusoidal terms of the discrete Fourier transform, which are given as
Figure 5(a) shows the sampling functions of the 4N – 3 algorithm that detect the fundamental frequency ν = 1 corresponding to the optical thickness variation of the LNB wafer. Figure 5(b) shows the sampling functions at the frequency ν = 5, which corresponds to the surface shape of the LNB wafer. In the frequency domain, the sidelobe amplitude of the
4N – 3 algorithm is suppressed by approximately 0.239%, which is better than the sidelobe suppressions for the 2N – 1 algorithms proposed by Surrel (5.253%)  and Hanayama (1.380%) [26, 27] when N = 16. The highest sidelobe amplitude of 4N – 3 algorithm is approximately 0.747% when N = 5 and decreases with the increase in N.
Triangle window function was generated by self-convolution of rectangle window function . New 4N – 3 algorithm can be also derived from N-step square-window algorithm by 4 successive convolutions with itself using frequency transfer function [29–31].
3.3 Error analysis
To illustrate the theoretical merits of the new window function, Table 2 shows the amplitudes of the highest sidelobes for the sampling functions compared with several types of conventional windows. Except for the Blackman window function, the new window function exhibits the smallest sidelobe amplitude at the harmonic frequencies.
In addition, phase measurement errors were calculated as a function of parameters ε0, ε1, and ε2 for several window envelopes (see Eq. (7)). Figure 6 shows the calculated PV phase errors in the fundamental frequency (m = 1) for N = 16 with the new window (61-sample), the Hann window (61-sample), the Hamming window (64-sample) and the Blackman window (61-sample). Figures 6(a)-6(c) show the errors caused by errors ε0, ε1, and ε2, respectively. We observe that the new window function has better suppression ability than all of the conventional windows. New window function has higher sidelobes than Blackman window function, however, the new window function offers better immunity up to the 2nd-order nonlinearity ε2 in the phase shift. We can therefore conclude that the new window functionexhibits a greatly improved compensation ability for the phase-shift nonlinearity, and it has a comparable suppression ability to the harmonic signals obtained using the conventional Blackman window function.
The surface shape and optical thickness variation of an LNB wafer 5 mm thick and 74 mm in diameter were measured. The wafer was adhered to an FS supporting plate 16.5 mm thick and 80 mm in diameter. The optical setup is shown schematically in Fig. 1. The measurement sample had four reflecting surfaces, including the reference surface of the interferometer. The air-gap distance L was set to 61.3 mm, approximately five times the optical thickness of the LNB wafer. With these parameters, the ratios of the modulation frequencies in the harmonic spectrum become almost integer values, as shown in Table 1. The source wavelength was measured using a wavelength meter, which was calibrated using a stabilized HeNe laser with an accuracy of δλ/λ~10−7 at a wavelength of 633 nm.
Figure 7(a) shows a laboratory photo of the measurement sample in the wavelength-tuning Fizeau interferometer, and Fig. 7(b) shows an observed raw interferogram of the sample at the wavelength of 632.8 nm.
The necessary range for the wavelength scanning δλ is calculated as
The wavelength was finely scanned from 632.8427 nm to 632.9076 nm, and 61 interference images were recorded at equal wavelength intervals. One experiment takes 1 minute to acquire 61 interference images. The slow rate is mainly due to a slow data transfer rate from the camera to PC. During the scanning, the signal interference fringes corresponding to the optical thickness of the LNB wafer changed by four periods of 8π radians. In contrast, the fringes corresponding to the air gap changed by twenty periods of 40π radians. The amount of the phase shift for each step was equal to 4π/16 for the optical thickness fringes of the LNB wafer. Because there was an approximately 3% nonlinearity in the PZT response, a quadratic voltage was incrementally applied to the PZT so that the resultant wavelength scanning would be linear. As a result, the nonlinearity decreased to 1% of the total phase shift. The phase was calculated using the algorithm of Eqs. (8)–(9).
Figures 8(a) and 8(b) show the phase distributions of φ1 and φ2, which correspond to the optical thickness of the LNB wafer and FS supporting plate, respectively. Figures 8(c) and 8(d) show the phase distributions of φ5 and φ6, which correspond to the surface shape of the LNB wafer and FS supporting plate, respectively. In Fig. 8(a), the optical thickness of the LNB wafer gradually decreases to the bottom right, whereas the optical thickness of the FS supporting plate decreases to the bottom left. The top surface of the sample seems concave of 474 nmPV, which is observed in Fig. 8(c). In these figures, we can observe that the entire sample is distorted, becoming concave, by the bonding stress between the LNB and FS plates while each plate has a slightly wedged shape.
The residual ripples observed in Fig. 8(b)–8(d) are caused mainly by the interference noise of multiple-reflection beams which has the same frequency and partly by a crosstalk noise from other frequencies. The amplitude of these ripples could not be reduced and seems almost steady even if other window functions are used. It is difficult to separate the target signal and these multiple-reflection signals in frequency space theoretically. If a number is put to each reflecting surface as 1: reference surface, 2: top of LNB wafer, 3: boundary of LNBwafer and FS supporting plate, and 4: bottom of the FS supporting plate like Fig. 2, the interference noise of multiple-reflection can be found as follows.
- (1) Fig. 8(b): optical thickness variation of FS supporting plate (m = 2)
The interference between the double reflection inside the wafer (3 → 2 → 3) and the dominant reflection from the wafer (surface 2) produces a harmonic component of frequency m = 2, which is overlapped with the optical thickness signal of FS supporting plate (m = 2).
- (2) Fig. 8(c): surface shape of LNB wafer (m = 5)
The double reflection (4 → 2 → 4) and dominant reflection from the border (surface 3) produces a harmonic component of m = 5 which is overlapped with the surface shape of the wafer (m = 5).
- (3) Fig. 8(d): surface shape of FS supporting plate (m = 6)
The double reflection (4 → 2 → 4) and the dominant reflection from the wafer top (surface 2) produces a harmonic component of m = 6 which is overlapped with the surface shape of the FS supporting plate (m = 6).
Figure 9(a) and 9(b) shows the phase distributions before unwrapping, which correspond to the surface shape of LNB wafer obtained by 4N – 3 algorithm with polynomial window function and Larkin-Oreb N + 1 algorithm (N = 16) , respectively. The residual error amplitude of the phase obtained by 4N – 3 algorithm with new polynomial window function is much smaller than that of the phase obtained by Larkin-Oreb N + 1 algorithm. The residual errors observed in the phase obtained by Larkin-Oreb algorithm mainly come from the cross-talk from the neighboring frequency components.
The repeatability of the phase measurement of the optical thickness variation, i.e., the root-mean-square (rms) of the difference between a pair of measurements taken successively within a few seconds, was approximately 2 nm.
We derived a new 4N – 3 phase shifting algorithm that can compensate up to the 2nd-order nonlinearity in the phase shift, efficiently suppress the effect of higher and lower harmonic noise components, and extract the phase of an arbitrary order in the multiple-interference system. The new window function exhibits superior harmonic noise suppression compared with most conventional windows. Moreover, the new window exhibits better compensation for the phase-shift error than any conventional windows. This new algorithm was applicable to a sample with a high reflective index owing to the suppression ability of harmonic noise components. Using this new algorithm, the optical thickness variation of an LNB wafer adhered to an FS plate was measured with an accuracy of 2 nm.
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