## Abstract

The interferometric surface measurement of single or stacked parallel plates presents considerable technical difficulties due to multiple-beam interference. To apply phase-shifting methods, it is necessary to use a pathlength-dependent technique such as wavelength scanning, which separates interference signals from various surfaces in frequency space. The detection window for frequency analysis has to be optimized for maximum tolerance against frequency detuning due to material dispersion and scanning nonlinearities, as well as for suppression of noise from other frequencies. We introduce a new class of phase-shifting algorithms that fulfill these requirements and allow continuous tuning of phase detection to any frequency of interest. We show results for a four-surface stack of near-parallel plates, measured in a Fizeau interferometer.

©2004 Optical Society of America

## 1. Introduction

Phase-shifting interferometry of two surfaces for optical testing has been the subject of intensive research in the last decade, is well understood today, and can deal with a wide range of error sources [1–4]. Recently there has been some interest in measuring objects with more than one optical surface [5]. This leads to a multitude of interference patterns from the various cavities that are formed in the beam path, and the complexity of the problem increases quickly with the number of surfaces. If *n* surfaces are involved, there will be *n*(*n*-1)/2 interference patterns of first order — i.e. with each beam being reflected just once — and many more second- and third-order interference signals. In these cases, phase-shifting by varying the optical path length of the reference beam is no longer suitable, because the interference signals from all the surfaces involved will change at the same temporal rate as the phase is shifted, and thus cannot be separated from each other. Hence, the multiple-beam interference noise will severely degrade the quality of the measurement [6].

An alternative method is white light interferometry [7,8] which separates the desired interference signal from noise components by spatially localizing each interference fringe system within the small coherence length of the source. The typical measurement accuracy is of the order of one part in a thousand of the coherence length, which usually translates to 20–30 nanometers.

Another approach is wavelength-scanning interferometry [9–18] in which the interference signal is isolated in frequency space because the modulation frequency of the interference fringes depends on the optical path difference between the test surface and the reference surface. Provided the relative lengths of the cavities are set up properly, the necessary separation of signals in frequency space can be achieved [10,13,14,16–18].

Whereas conventional phase-shifting interferometry requires stable phase detection only at a single signal frequency, the new challenge for wavelength-scanning interferometry is simultaneous phase detection at several signal frequencies, with good suppression of noise from neighboring frequency signals. The goal for the design of an appropriate phase-shifting formula is therefore to compensate for any phase-shift miscalibration (also known as detuning of signal frequency), and to suppress, ideally to zero, the contributions from all other signal frequencies. In the design of the measurement setup, the preferred signal spectrum would consist of a combination of “harmonics” whose frequency ratios are very nearly integers. This can be achieved by appropriately adjusting a geometric parameter, such as the air-gap distance between the reference surface of the interferometer and one of the test surfaces. In practice, however, the signal frequency cannot be precisely determined *a priori*, partly because the optical thickness of the test object is often unknown and partly because the single-mode wavelength tuning range of the light source is usually limited by nonlinearities and mode-hops. Thus, the ideal case of sharp peaks at exact harmonic ratios in the spectrum of the signal frequencies is unlikely to occur. Rather, the new phase-shifting formula should efficiently suppress all unwanted frequencies, including nearly integer-multiple “harmonics”, multiples that are detuned due to dispersion, and other fractional-frequency contributions.

Apodization of the data-sampling time window [19,20] for discrete Fourier analysis [21] is a systematic and inherent tool for the suppression of harmonic signals in wavelength scanning interferometry [16]. However, most of the known windows do not allow for compensation of phase-shift miscalibration. Table 1 shows common windows and their sensitivity to phase-shift miscalibration and suppression characteristics at harmonic frequencies.

In two-surface interferometry, earlier research [22,23] has led to phase-shift algorithms with five to eight samples and an approximation to the von Hann window. These happen to be insensitive to phase-shift miscalibration when the phase-shift interval between successive samples is π/2. For multiple-surface interferometry however, it is required that the number of samples be arbitrary and generally much larger than ten, depending on the frequency content of the interferogram. Also, to detect the phases of individual frequency components, the nominal phase shift per sample must be highly flexible. To date, no phase-shift formula has been reported in the literature that combines: arbitrary number of samples; insensitivity to phase-shift miscalibration; tunability to a frequency of choice; and good suppression of signals at higher (or lower) harmonic frequencies. Phase-shifting algorithms with triangular windows are insensitive to phase-shift miscalibration [2], however, their spectral transfer functions have substantial amplitudes in the sidelobes around the detection frequency, and are thus sensitive to higher or lower signal frequencies when these are not exactly at integer ratios.

In this paper, we start with the triangular window and modify it to improve the suppression of harmonic signals while preserving the insensitivity to phase-shift miscalibration. The resultant new window shows much improved suppression of harmonic signals, similar to the von Hann window. The new “2*N*-1” sample algorithm is applied to interferometric testing of a stacked-plate array consisting of five surfaces. The shape of the test surface closest to the reference surface and variations in the optical thickness of each layer of the test object are measured. This enables the shape of all surfaces of the test object to be calculated if its refractive index is known.

## 2. Multiple-surface interferometry

We first specify the experimental configuration, in order to discuss the fringe modulation frequencies in the Fizeau interferometer [24,25]. Fig. 1 shows the measurement setup for testing multiple surfaces of a transparent object.

The reference surface and a stack of nearly-parallel polished BK7 glass surfaces *S*
_{1} through *S*
_{4} are illuminated by a collimated monochromatic beam from a tunable laser diode. The test object is aligned so that its surfaces are also roughly parallel with the reference surface *Ref*. To record phase-shifted interferograms, the wavelength is scanned linearly over a width Δ*λ* from a central value *λ*
_{0}. The reflections between all surfaces of the test stack and from the reference surface generate interference fringes which are imaged on and captured by the CCD detector.

In wavelength scanning, the modulation frequency of each interference fringe system is approximately proportional to the optical path difference between the constituent pair of interfering beams [13]. However, the frequency is not exact because the dispersion of the test object shifts this modulation frequency [17]. We denote the central wavelength of our phase-shifting sequence by *λ*
_{0}, and the refractive index of the test plates at that wavelength by *n*
_{0}. To arrive at a frequency table for the various fringe modulation frequencies, we label the external air gap distance (distance between *Ref* and *S*
_{1}) by *a*, the optical thickness of the top glass plate (distance between *S*
_{1} and *S*
_{2}) by *p*
_{1}, the distance between *S*
_{2} and *S*
_{3} (internal air gap) by *d*, and the optical thickness of the bottom glass plate (distance between *S*
_{3} and *S*
_{4}) by *p*
_{2}. Assuming that the test object dimensions are fixed, the only quantity we can influence is the air gap *a*. Since we want to detect the phases of all dominant frequency components simultaneously, it is necessary to choose an appropriate value of the air gap so that the resultant frequency ratios are as close as possible to integers.

The modulation frequency *ν*
_{1}, which represents the interference signal between the reflections from *Ref* and *S*
_{1}, is given by

where *dλ/dt* is the wavelength-scanning rate. With *a*=10.1 mm, and taking the refractive index *n*
_{0}=1.51 and dispersion coefficient (*λ*
_{0}/*n*
_{0})(*dn/dλ*)_{0}=-0.01 of the BK7 glass at wavelength *λ*
_{0}=690 nm into consideration, the dominant frequencies due to first-order reflections are listed in Table 2. For our test object, *d*=20 mm and *p*
_{1}=*p*
_{2}=30.6 mm.

If the available wavelength tuning range were large enough, we could always make the frequency ratios into integers (e.g. 1000: 4026: 6006:… in Table 2). However, with single-mode scanning, the available tuning range is limited, as is the number of intensity samples for practical reasons. Therefore, in most cases we will have to allow fractional values in the ratios of signal frequencies.

From Table 2, we can graph the expected frequency spectrum as shown in Fig. 2; the relative amplitudes depend on the number of reflecting surfaces in the signal path, and therefore the signal-to-noise ratio depends on which frequency component is being measured. There are two doublet peaks at *ν*/*ν*
_{1} ≈ 3 and *ν*/*ν*
_{1} ≈ 5; these come from the two glass plates being of equal thickness, and inevitably, the symmetrical arrangement of these with respect to *d*. Those signal pairs cannot be used to detect the phases, since wavelength-scanning interferometry cannot separate signals with the same frequency. Note that, although we cannot detect the phases of these two signals involving the optical thickness variations of the two glass plates directly, we can measure the thickness variations indirectly by detecting the phases of other frequency components, as will be shown below.

It is obvious from this spectrum that the frequency-filtering properties of the phase-shifting algorithm need to be very selective, as there are many strong frequencies throughout the spectrum and most of these are unwanted signals which must be regarded as noise. Moreover, the algorithm is required to accommodate frequency detuning of the signal, or in other words, phase-shift miscalibration, since all signals are not always distributed at exact integer multiples of *ν*
_{1}, and we want to be able to measure all surfaces using a single wavelength-shifting sequence. We will now consider how such a selective phase-shifting formula can be designed.

## 3. Flexible phase extraction method

There are several techniques to derive or refine a phase-shifting algorithm. These include characteristic polynomials [2], recursive methods [3], simultaneous equations [4], averaging [6], least-squares techniques [26], extended averaging [27], and addition of auxiliary functions [28,29]. We combine the approaches in Refs. 2 and 28 to derive the new formulae, and evaluate their sampling functions in the frequency domain [28–30]. We have previously derived two 19-sample phase-shifting algorithms [18] which have very small sensitivity to the noise frequencies in the range from 2*ν*
_{1} to 10*ν*
_{1} and tolerate some detuning of their signal frequencies. However, we now need to be more flexible with the basic phase-shift interval, so that we can sample appropriately all frequencies of interest; that is, the phase shift *δ*_{k}
between successive intensity samples must be less than 180° for all *ν*_{k}
. For example, if the frequency of the signal whose phase we want to measure is 9*ν*
_{1}, the phase increment in that frequency must be *δ*
_{9} < 180°, from which it follows that *δ*
_{1}<20°. We start with a generic 2*N*-1 sample formula and a phase shift of *δ*=2π/*N*, so that the total phase shift will be π(4-4/*N*), i.e. a little less than two fringe periods.

#### 3.1 Simple triangular window function

A phase shift formula for detecting the phase of the *k*
^{th} harmonic component can be written as

where *k* is the nearest integer to *ν*/*ν*
_{1}, *α*_{r}
=(*r* - *N*)*δ* is the *r*
^{th} phase-shift in the external air gap, *I*(*x, y*, *α*_{r}
) is the *r*
^{th} irradiance image, and *a*_{r}
and *b*_{r}
are the sampling amplitudes tuned to frequency *k*, confined by a triangular window function *w*(*r*) that is symmetrical about *N* [2]:

$$=2-\frac{r}{N}\phantom{\rule{.5em}{0ex}}\mathrm{for}\phantom{\rule{.5em}{0ex}}N+1\le r\le 2N-1.$$

With Eq. (3), we can express the sampling coefficients as

$${b}_{r}=-w\left(r\right)\mathrm{sin}\phantom{\rule{.2em}{0ex}}k{\alpha}_{r}.$$

Using these weights, we can tune the response of the phase detection to the *k*
^{th} harmonic. The frequency response of the algorithm can be analyzed through its filter functions, defined by [30]

$${F}_{b}\left(\nu \right)=\sum _{r=1}^{2N-1}{b}_{r}\phantom{\rule{.2em}{0ex}}\mathrm{exp}\left(i{\alpha}_{r}\frac{\nu}{{\nu}_{1}}\right),$$

where the modulation frequency *ν*
_{1} in the external air gap is defined by Eq. (1). For a triangular window in a symmetrical representation [31–33] according to Eqs. (2), (3), and (4), *F*_{a}
(*ν*) and *F*_{b}
(*ν*) are purely real and purely imaginary functions, respectively. However, in order to assess the noise in the sidelobes, Fig. 3 shows |*F*_{a}
(*ν*)| and |*F*_{b}
(*ν*)| on a logarithmic scale, with sixty-three samples (*N*=32), and the phase detection tuned to the fourth harmonic signal (*k*=4); note however that *N*=32 allows *k* to take any value between 1 and 15.

At the detection frequency *kν*
_{1}, *F*_{a}
and *iF*_{b}
have matched gradients, which ensures the insensitivity of the algorithm to frequency detuning at this signal frequency. Conventional data windows such as von Hann or Hamming do not have matched gradients of *F*_{a}
and *iF*_{b}
at the detection frequency (with incidental exceptions for *k*=*N*/4, which translates to a phase shift of π/2 per sample — see the discussion of Table 1 above). At all other frequencies *m*
*ν*
_{1}, *m*∊IN_{0}, *m*≠*k*, up to the Nyqvist frequency of *N*
*ν*
_{1}/2, |*F*_{a}
(*ν*)| and |*F*_{b}
(*ν*)| have zero magnitude and zero slope. This combination ensures the cancellation of harmonics, even in the presence of slight detuning. The signal levels between the zeroes are considerably lower than for a “single-cycle” triangular window.

The maximum level of the sidelobes at (*k*±1)*ν*
_{1}, neighboring the detection frequency *kν*
_{1} is -27 dB or 4.5 %, which is still significantly larger than -32 dB and -43 dB for the von Hann and Hamming windows, respectively. Since this noise level in the worst case causes a phase measurement error larger than that from double-reflection noise within the glass cavity, it cannot be neglected. Therefore we will modify the window in the next section, so that it gives better suppression of the sidelobes in the vicinity of the detection frequency.

#### 3.2 Characteristic polynomial of a phase-detection formula

Here we briefly describe the use of the characteristic polynomial theory [2] to define the notation of an algorithm, clarify its relationship to the Fourier interpretation [30], and develop the polynomial representation of sampling functions in a triangular window.

The characteristic polynomial *P*(*z*) of a phase-shifting formula is defined by

where *z*=exp(*iδν/ν*
_{1}), *ν* can take any value between zero and the Nyqvist frequency and the complex constant *c* is chosen so that the sampling amplitudes *a*_{r}
and *b*_{r}
are in symmetrical/antisymmetrical representation — in other words, *F*_{a}
(*ν*) and *F*_{b}
(*ν*) are purely real and purely imaginary, respectively. With our definitions, *c*=exp(*i*(*N*-1)*δ*). The operation in Eq. (6) is known as a *z* transform; it maps the frequency axis to the upper-half circumference of a unit circle in the complex plane, and represents the filter functions as complex quantities. Note that this expression evaluates all coefficient pairs at all signal frequencies. Since it is a complex polynomial of degree (2*N*-2) in *z*, it has (2*N*-2) zeroes, or roots, where multiple roots lead to special properties (see below). These correspond to the zeroes of the spectral transfer functions *F*_{a}
and *F*_{b}
by virtue of the relationship

i.e., for all *ν* where *P*(exp(*iδν/ν*
_{1}))=0, the phase-extraction formula will produce an output of zero. Between the zeroes, *F*_{a}
(*ν*) and *F*_{b}
(*ν*) are also given by the value of the polynomial for the pertinent *ν*. From Eqs. (3), (4) and (6), it can be shown that the polynomial for a triangular window is given by

where we have abbreviated *ζ*=exp(*iδ*), and the normalization factor 2*ζ*
^{2k}/*N*
^{2} is chosen to satisfy the condition

$$=2,$$

i.e., at the detection frequency, the filter functions have unity magnitude, are in exact quadrature [30] and cause no offset in the detected signal phase. Fig. 4 shows the positions of polynomial roots on the unit circle.

Since the filter functions associated with a triangular window have zeroes with zero slope at integral harmonic frequencies *m*
*ν*
_{1}, *m≠k*, the characteristic polynomial has double roots at *z*=exp(*imδ*) for *m*∊{1, 2, …, *k*-1, *k*+1, *k*+2, …, *N*/2}. The polynomial also has a double root at zero frequency (*m*=0, *z*=1); this cancels the bias intensity and slight variations of it, which happens even with modern constant-current tunable laser diodes. The positions of the other double roots are symmetrically distributed on the unit circumference with respect to the real axis, so that every double root has a complex conjugate. This ensures that harmonics above the Nyqvist frequency are suppressed as well. The exception to the symmetry is at the complex conjugate of the detection frequency, *z*=exp(-*ikδ*); of course the polynomial must not have a root at *z*=exp(*ikδ*), because that is the frequency to be detected.

This arrangement of roots of the polynomial ensures that the filter functions remain very small between their zeroes (except of course at *z*=exp(*ikδ*)), which helps to suppress noise from higher (or lower) harmonic signals.

Moreover, for the compensation of phase-shift miscalibration, the filter functions need to have matched gradients at the detection frequency [4,28–31]. Surrel [2] showed that this is equivalent to a double root of the characteristic polynomial at *z*=exp(-*ikδ*), so that we can express the condition as

$$=0,$$

where we have used the symmetries *F*_{a}
(-*ν*)=*F*_{a}
(*ν*) and *F*_{b}
(-*ν*)=-*F*_{b}
(*ν*). This condition is fulfilled for the triangular window, as Fig. 4 shows. However, in the polynomials of other windows that are better suited to suppress sidelobes of the signal (such as the von Hann and Hamming windows), only a single root appears at *z*=exp(-*ikδ*) and any phase-shift miscalibration cannot be compensated.

#### 3.3 Modified triangular window

Although a clever choice of the air gap can make the frequencies of signal components into integers (see Table 2), we have no guarantee that this will always be possible with all noise contributions. If a noise peak happens to be at, say, 2.5 or 3.5 times the detection frequency, the noise would cause substantial cross-talk when the triangular window is used. Our aim is now to reduce the highest sidelobes around the detection frequency while preserving the stability against signal detuning.

Similar to the approach that was begun in Ref. [28] and extended in Ref. [29], we design auxiliary sampling functions that we can add as a linear combination to the original ones, to tweak the behavior of the filter functions. We want to leave the response peak as it is, but its side lobes should decrease; therefore the auxiliary function needs no sensitivity at *kν*
_{1}, and its own sidelobes should counterbalance those of the original filter functions. The set of correction coefficients, ${a}_{r}^{c}$
and ${b}_{r}^{c}$
, should then have polynomials with a double root at *kν*
_{1}. This ensures that the original sensitivity peak remains unchanged, and the change to its sides will be symmetrical; this is not possible with a single root at the detection frequency, since that would correspond to a zero transition, and hence introduce asymmetry. Further, we require single roots at (*k*±1)*ν*
_{1} (so the filter functions ${F}_{a}^{c}$
and ${F}_{b}^{c}$
will provide a negative response or “undershoot” after the zero transition). The pertinent characteristic diagram is shown in Fig. 5.

Thus, the required auxiliary polynomial will have the same roots as the original one of Fig. 4, except at the detection frequency *z*=exp(*ikδ*) and at the two neighboring harmonic frequencies *z*=exp(*i*(*k*±1)*δ*). The specific form of the correction polynomial is then given by

where the factor -2*ς*
^{2k} sin*δ*/*Nπ* was chosen so that the derivatives of the sampling functions ${\mathit{\text{dF}}}_{a}^{c}$
/*dν* and ${\mathit{\text{idF}}}_{b}^{c}$
/*dν* are real and equal to unity at frequency *ν*=(*k*-1) *ν*
_{1}.

Fig. 6 (a) shows *F*_{a}
(*ν*) and *F*_{b}
(*ν*) of the original triangular envelope and Fig. 6 (b) shows the corrections ${F}_{a}^{c}$
(*ν*) and ${F}_{b}^{c}$
(*ν*).

The sampling functions of the correction window have the inverse sign of the original functions at all frequencies (except in the immediate vicinity of the detection frequency, where (*k*-1) *ν*
_{1}<*ν*<(*k*+1) *ν*
_{1}), so that *all* sidelobes will decrease in amplitude. We can now combine *P*(*z*) with *P*_{c}
(*z*) according to

(with *ε* real) and tweak ε until the sidelobes are minimized. Since a linear combination of Eqs. (8) and (11), generated from the added amplitudes *a*_{r}
+${\epsilon a}_{r}^{c}$
, *b*_{r}
+${\epsilon b}_{r}^{c}$
, will always have double roots at the conjugate signal frequency *z*=exp(-*ikδ*), the new compound formula will compensate for the phase-shift miscalibration, too. The best value for *ε* depends on the phase divisor *N* and the detection order *k*; however, for *N*>20, the optimum is steady between 0.22 and 0.23. Therefore we choose this parameter as *ε*=*π*/14=0.224. When *ε* is inserted in Eqs. (11) and (12), the optimized data window for the new algorithm becomes

$$=2-\frac{r}{N}+\frac{1}{14}\mathrm{sin}\left(\frac{2\pi \phantom{\rule{.2em}{0ex}}r}{N}\right)\phantom{\rule{.9em}{0ex}}\mathrm{for}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}N+1\le r\le 2N-1\phantom{\rule{.2em}{0ex}},$$

Fig. 7 shows the filter functions for the new sampling window of Eq. (13). Comparing Fig. 7 to Fig. 3, the highest peaks around the detection frequency are decreased significantly, from -27 dB to -38 dB, or from 4.5% to 1.2% of the main peak. The extra zero transitions between (*k*±1)*ν*
_{1} and (*k*±2)*ν*
_{1} reveal that each of the original double roots at (*k*±1)*ν*
_{1} has now been split into two single roots. All the other sidelobes are also attenuated by several dB since ${\mathrm{F}}_{\mathrm{a}}^{\mathrm{c}}$ and ${F}_{b}^{c}$
have the inverse sign of the original *F*_{a}
and *F*_{b}
at all noise frequencies.

#### 3.4 Error Analysis

To weigh the theoretical merits of the new formula, Table 3 shows the error-compensating capability of phase-shifting algorithms with several types of window functions.

All windows, except for the rectangular one, have very small amplitudes at the harmonic frequencies. This comes at the sacrifice of increasing the width of the detection peak, and affects the sensitivity of the filter to random noise. The sensitivity to random noise can be estimated by the sum of the squared sampling amplitudes [34] (called equivalent noise bandwidth, ENBW, in electronics [19]), which is defined by

For a large number of samples, the sensitivity decreases as 1/√*N* because the response of the formula is re-normalized (cf. Eqs. (8) and (9)), but the differences between the envelopes stay the same. Table 3 lists the relative sensitivities to random noise for large *N*. The noise sensitivity of the new algorithm increases by 7% compared to the triangular window.

The new window offers immunity to phase-shift miscalibration, and shows a suppression of noise from higher harmonics that is comparable to the von Hann window. To assess the immunity to phase-shift miscalibration (i.e. signal detuning), let us assume that the frequency of detection signal has a slight detuning, expressed by *ν*_{k}
=*ν*
_{1}(*k*+*ε*
_{1}), where integer *k* is the detection order and *ε*
_{1} is an error coefficient less than unity. For simplicity, we also assume that there are no noise signals at the other frequencies. Phase measurement errors were calculated as a function of parameter *ε*
_{1} for several window envelopes. Fig. 8 shows the calculated peak-to-valley (PV) errors in the fundamental frequency detection (*k*=1) for *N*=50 with the new window (99-sample), the Hann window (99-sample) and the Hamming window (100-sample), where we define the envelopes by

and

In Fig. 8, the PV error for the new window has a roughly quadratic dependence on the error parameter, while the error dependence for the other windows is approximately linear. These behaviors were expected from the discussion in Section 3.3. For higher detection orders (*k*≥2), the PV errors generally become smaller than those for the fundamental-order detection.

Additional phase errors can be caused by nonlinearity in the wavelength scanning. When the scanning has a quadratic nonlinearity, the signal frequency *ν*_{k}
is no longer constant and is given by

where *ε*
_{2} is the nonlinear coefficient. If we have a noise component at frequency *ν*_{m}
=*mν*
_{1} with the same amplitude, the error in the detected phase can be calculated to give

$$=\frac{k{\epsilon}_{2}}{2\pi}\sum _{m}\left\{\begin{array}{c}\left({{\mathit{iF}}_{b}}^{\left(2\right)}\left(m{\nu}_{1}\right)+{{F}_{a}}^{\left(2\right)}\left(m{\nu}_{1}\right)\right)\mathrm{cos}\left({\phi}_{k}-{\phi}_{m}\right)\\ +\left({{\mathit{iF}}_{b}}^{\left(2\right)}\left(m{\nu}_{1}\right)-{{F}_{a}}^{\left(2\right)}\left(m{\nu}_{1}\right)\right)\mathrm{cos}\left({\phi}_{k}+{\phi}_{m}\right)\end{array}\right\}$$

$$=\frac{\pi \phantom{\rule{.2em}{0ex}}k{\epsilon}_{2}}{2}\sum _{m}\left\{f(k,m)\mathrm{cos}\left({\phi}_{k}-{\phi}_{m}\right)+g(k,m)\mathrm{cos}\left({\phi}_{k}+{\phi}_{m}\right)\right\},$$

where we abbreviate ${F}_{a\mathit{,}b}^{\left(2\right)}$
=${\mathit{\text{dF}}}_{a\mathit{,}b}^{2}$
/*dν*
^{2} and the coupling constants *f* and *g* are defined by

$$g(k,m)=-\frac{1}{{N}^{2}}\sum _{r=1}^{2N-1}{\left(r-N\right)}^{2}w\left(r\right)\mathrm{cos}\frac{2\pi \phantom{\rule{.2em}{0ex}}r}{N}\left(k+m\right),$$

and the summations of *m* are over all signal components. From Eq. (18), a quadratic nonlinearity couples frequency components — in other words, it creates crosstalk. From Eq. (19), we see that the coupling constants *f*(*k, m*), *g*(*k, m*) depend only on the frequency difference |*k-m*| or sum *k*+*m*. With a detailed calculation for several window functions *w*(*r*), it can be shown that *f* (1, 2) is the dominant coefficient, the coefficients are steady for *N*≥8, and the error caused by a quadratic nonlinearity comes mainly from coupling between neighboring frequency components. Table 4 shows the values of the largest coupling constants for several windows.

Note also that *k* does not necessarily have to be an integer. If the signal frequencies are non-integers, the detection can be tuned to those as well by inserting the appropriate *k* into Eq. (4). This moves all the zeroes of *F*_{a}
and *F*_{b}
away from the exact harmonics, so that (most importantly) the insensitivity to bias intensity is lost. However, since the amplitudes of *F*_{a}
and *F*_{b}
remain fairly small throughout the spectrum, most of the noise still stays between -40 dB and -60 dB, so that, even though exact cancellation of harmonics is not occurring anymore, the “continuous” tuning should be usable under practical conditions.

## 4. Experiments

The test setup is shown schematically in Fig. 1. The test object consists of a stack of three near-parallel BK7 plates (*n*
_{0}=1.51) of thickness *T*=20 mm and diameter 150 mm. The middle plate has a 100 mm diameter hole in the center, so it acts only as a spacer for the other two plates. Therefore, we have four reflecting surfaces in the object, plus the reference surface of the interferometer. The air-gap distance *a* was set to 10.1 mm. With these parameters, the ratios of the modulation frequencies in the harmonic spectrum become almost integer values, as listed in Table 2. We also placed a wedged glass flat of smaller diameter beside the object, with its top surface in the same horizontal plane as object surface *S*
_{1}, so that we could calibrate the phase-shift increment in the air gap by means of two-beam interference. Fig. 9 shows a laboratory photo of the test object in the wavelength scanning Fizeau interferometer [24] and a sample interferogram.

The source is a tunable diode laser with a Littman-Metcalf cavity [35] whose central wavelength is *λ*
_{0}=690.0 nm. The wavelength was tuned by rotating the external cavity grating by piezo-actuator. The total tuning range of the laser was 0.248 nm, of which we utilized some 0.095 nm. The laser was operated around the center of its tuning band to minimize hysteresis effects in the piezo-translator that drives the laser tuning; but even so, distinct nonlinearities were found. Strictly speaking, we want the wavelength shift to be nonlinear, as equal phase shifts are generated by equal increments of wavenumber, not wavelength; but since the tuning range is only 0.014% of the wavelength, the error due to this approximation becomes negligible. We have compensated the quadratic part of the nonlinearity by driving the piezo with voltages following the inverse quadratic function, but there still remained a small nonlinearity of around 1% in the scanning.

#### 4.1 Results

The minimum necessary tuning range, Δ*λ*, is defined by

This tuning range was used and led to consistent measurement results. Our particular test setup however, offers the possibility to separate the signal frequencies further by doubling the phase shift in the air gap, so that the wavelength is scanned by 2Δ*λ* and the interference fringes formed in the air gap *a* modulate through almost four periods during the scanning. The ratios of the dominant signal frequencies *ν*
_{1}: *ν*
_{2}: *ν*
_{3}: *ν*
_{4}:….: *ν*
_{10} then become approximately integer values of 2: 8: 12: 18:…: 6 (see Table 2). This cannot be done with *N*=32, because *ν*
_{4} would be above the Nyqvist frequency. Therefore, we used *N*=50, which requires the recording of 99 interferograms. These modifications demonstrate how the formula can be adapted to a wide range of experimental conditions. The phase detection formula was given by Eq. (2), with the new window of Eq. (13) replacing the coefficients of Eq. (4). Frequency components *ν*_{5}
and *ν*
_{10}, and *ν*
_{6} and *ν*
_{9}, respectively, are doublets, and no meaningful signal can be detected at these frequencies. The phases *φi*(*x,y*) associated with the remaining *ν*
_{1} through *ν*
_{4} and *ν*
_{7} and *ν*_{8}
frequency components were all measured by the new algorithm, as defined by Eqs. (2), (4), and (13), and (see Table 2) set to (*N*=50, *k*=2), (*N*=50, *k*=8), (*N*=50, *k*=12), (*N*=50, *k*=18), (*N*=50, *k*=16), and (*N*=50, *k*=4), respectively.

Fig. 10 shows examples of directly measured as well as deduced figure maps of *p*
_{1}, *d*, *p*
_{2}, and of the entire glass stack. All phase maps *φ*
_{1} through *φ*
_{4}, and *φ*_{7}
and *φ*_{8}
(where we drop spatial dependencies for convenience) were used in the evaluation.

The internal air gap *d* can be measured in two possible ways: either directly through *φ*_{8}
, or as a difference between measurements of *S*
_{3} and *S*
_{2} (corresponding to *φ*_{3}
and *φ*_{2}
, respectively). We have already discussed that the thickness variations of the top plate (associated with frequency *ν*_{5}
) cannot be detected directly, because the thickness signal of the second plate (frequency *ν*_{10}
) overlaps with it. However, the variation in optical thickness *p*
_{1} of the top plate can be calculated from the difference *φ*
_{2}-*φ*
_{1}. Similarly, the variation in optical thickness of the second plate *p*
_{2} cannot be measured directly from the phase of the *ν*
_{10} component, but can be calculated from *φ*
_{4}-*φ*
_{3}, or even from *φ*_{7}
-*φ*_{8}
-(*φ*_{2}
-*φ*_{1}
) — see Table 2. Finally, the entire glass stack can be measured in several ways; the last row of Fig. 10 shows some examples and compares the resulting maps obtained.

The repeatability of the phase measurement — the root-mean-square (rms) of the difference of a pair of measurements taken successively within a few seconds — was about 2 nm; however, from Fig. 10 (a) and (b), and particularly from Fig. 10 (e)–(g) it is evident that there are mismatches between different measurements. These are due to systematic errors caused by multiple reflection, residual spectral cross-talk from other signal frequencies, and nonlinearity in the wavelength scanning. To obtain an estimate of these errors, we evaluated the differences between nominally equal measurements, some of which are shown in Fig. 10 above. We found a root-mean-square (rms) error of about 12 nm and a peak-to-valley (PV) error of about 65 nm, so that we can state an accuracy of *λ*/10. The errors are not negligible and we need to analyze their sources in more detail.

#### 4.2 Discussion of errors

So far, we have neglected the phase errors caused by multiple-reflection beams. There are approximately a hundred combinations of interference fringes of order *R* caused by three-time reflections between the five surfaces, where *R*=0.04 is the reflectance of the test surfaces. The frequencies of these noise signals range from *ν*
_{1} to 18*ν*
_{1}. Since the new window efficiently suppresses all higher harmonic components up to the Nyqvist frequency of 12.5*ν*
_{1} (bear in mind that we have doubled the basic phase increment), the dominant contribution to the error comes from noise signals that coincide with the detection frequency. For example, in the detection of the *ν*
_{8} (≈2*ν*
_{1}) component, there are six different components of order *R* which have the same frequency as *ν*
_{8}. Since each noise component can give a PV phase error of ≈*R*
*λ*
_{0}/2*π*, the multiple-reflections can in the worst case cause a PV error of 27 nm in the phase detection of *φ*
_{8}.

The detuning of the signal frequency from the integral harmonic frequencies is designed to be at most 1%, as shown in Table 2. However, there always exists a miscalibration in the wavelength scanning rate. Even a miscalibration of a few percent will cause a significant detuning in the high-order harmonic detection, such as the detection of the 16^{th}-order *ν*
_{7} component. We calibrated the tuning rate using the reference fringe pattern (see Fig. 9), so that the detuning of all detection frequencies was less than 10%. From Fig. 8, the new algorithm is able to compensate for first-order (linear) detuning of that extent. The phase error caused by detuning is a second-order effect of *λ*
_{0}/4000, which gives an error of ≈0.15 nm. In addition to these conventional phase-shift errors, the nonlinearity also broadens all frequency peaks across the spectrum and hence causes crosstalk between the signal and the noise components. We discussed this error above, and in an earlier work about three-surface interferometry [18].

Similarly, the PV phase error caused by a small quadratic nonlinearity can be estimated by Eq. (18) and Table 4. The PV error caused by crosstalk from the neighboring frequency components at *ν*
_{k-1} and *ν*
_{k+1} is estimated to be ≈0.5 *f* (1,3) *kε*
_{2}
*λ*
_{0},. For a one-percent nonlinearity (*ε*
_{2}=0.01), the PV error is smaller than 0.31 nm.

All these systematic errors caused by multiple-reflection beams were found to be a significant contribution to the phase measurements, and lead to PV errors of some tens of nanometers. We expect that indirect measurements increase the PV error by a factor that is approximately equal to the number of measurements used; however the phases are not statistically independent, being functions of the four parameters *a*, *p*
_{1}, *d*, and *p*
_{2}. In agreement with this, additional tests confirmed that the error figure depends strongly on the number of surfaces involved.

## 5. Conclusion

We have introduced a new class of highly versatile phase-shifting formulae which compensates for the detuning of the signal frequency and efficiently suppresses the effect of higher (or lower) harmonic noise components. A standard triangular window envelope for intensity sampling was modified by adding a complementary function generated from characteristic polynomials, to offer better suppression of higher harmonics while keeping the immunity to signal frequency detuning. The resultant algorithm allows the detection and discrimination of a multitude of interference components and can be implemented with an arbitrary number of phase steps, so that there is no theoretical limit to the highest detectable signal frequency. While the designed error suppression is exact at integer harmonics only, the phase detection is continuously tunable between zero and the Nyqvist frequency, with little deterioration of the noise-rejection properties.

A transparent cavity with four nearly parallel optical surfaces, plus the interferometer’s reference surface, was measured by the new algorithm. The resulting interference pattern consists of ten first-order components that have different temporal frequencies of fringe modulation under wavelength scanning. The experimental results show that the new phaseshifting method allows some signal frequency detuning and can efficiently separate the phases of distinct frequency components. The top surface shape and variations in the optical thickness of the three layers of a test object were measured with an error of some tens of nanometers, which is mainly caused by the interference noise from multiple reflections between the object surfaces.

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