During the wavelength modulating, the signal irradiance of the fringe pattern in a laser Fizeau interferometer is defined by [22,23]

where A_m, j_m, and g_m are the amplitude, phase, and fringe contrast of the mth harmonics and a_r is a phase-modulated parameter. We can calculate the target phase j_m using the phase-modulated algorithm.

When using equal intervals of d = 2p/N rad to separate the reference phases, where N is an integer, an M-fringe phase-modulated algorithm is defined by [3]

where I(a_r) is the rth irradiance signal defined by Eq. (1) and a_r and b_r are the rth sampling components. The design of a phase-modulated algorithm refers to the determination of sampling components a_r, b_r, and sampling number M.

When nonlinearity occurs during wavelength modulating, the real phase-modulated value a_r is a polynomial function of the ideal phase-modulated value a_0r, including the phase-modulation error e:

where a_0r = 2p[r − (M + 1)/2]/N, e₀ is linear miscalibration of the phase modulation, e_q (1 ≤ q ≤ p) is the qth nonlinearity of phase modulation, and p is the maximum order of the nonlinearity.

For designing an error-compensating phase-modulated algorithm, many authors have proposed several systematic approaches based on the averaging method [7,11], a linear equation [10,15], and complex polynomial theory [12,14, 18,19]. Surrel developed the complex polynomial P(x) of a phase-modulated algorithm [12], defined in the following Eq. (4), to determine the phase-modulated algorithm.

where x = exp(imd) and i is the imaginary unit.

In Surrel’s theory [12], the insensitivity of the phase-modulated algorithm to the higher harmonics and phase-modulation errors is determined by the properties of the polynomial roots in the complex diagram. For instance, the phase-modulated algorithm that has single roots on the complex diagram can suppress the mth harmonics [3,12]. In addition, the phase-modulated algorithm that has a double root at m = −1 is insensitive to linear miscalibration of the phase modulation e₀ [9,12].

To compensate for linear modulation of intensity, Onodera developed a 6-fringe algorithm [13] based on the least-square fitting method [21]. Surrel described this insensitivity to the intensity modulation using complex polynomial theory [14], where

From Eq. (5), insensitivity to the linear intensity modulation during wavelength modulating can be satisfied when the complex polynomial of a phase-modulated algorithm has a double root at m = 0. To compensate for the first order nonlinearity of intensity modulation, a triple root should be located at m = 0 on the complex diagram of the phase-modulated algorithm. Surrel also derived a 5-fringe algorithm that has similar characteristics to those of Onodera’s 6-fringe algorithm [14]. However, complex polynomials of these two algorithms do not have a root at m = 2, which means that these two algorithms are vulnerable to the second harmonics [14]. For this reason, the highly reflective surface of the silicon wafers or transparent parallel plates cannot be assessed by Onodera’s 6-fringe and Surrel’s 5-fringe algorithms [22,24].

In this study, Surrel’s complex polynomial theory is used to derive new phase-modulated algorithms. The roots of the complex polynomial are determined considering the following requirements:

Compensation for:

Satisfaction of the fringe contrast maximum condition.

To assess both surface and optical thickness of a transparent sample, we assign eight to the phase division number N (N = 8) [16].

Fig. 1 represents the root location and multiplicity of the complex polynomial of the 19-fringe algorithm. First, the complex diagrams of the algorithm should contain the single roots, except at m = 1 (x = exp(id)), to compensate for the mth harmonics [3,12]. In addition, the complex diagrams of the algorithm should contain the double roots, except at m = 1, to compensate for the linear miscalibration of phase modulation and compounding errors between the higher harmonics and linear miscalibration [12,25]. The complex diagram of the algorithm should have a triple root at m = −1 to compensate for the first order nonlinear phase modulation [25]. Furthermore, the complex diagram of the algorithm should contain a triple root located at m = 0 to compensate for the first order nonlinearity of the intensity modulation [25]. Finally, by locating the triple root at m = 2 and 3 on the complex diagram, the fringe contrast maximum condition can be satisfied [18]. To derive the 19-fringe phase-modulated algorithm, we expand the complex polynomial shown in Fig. 1.

Based on Fig. 1, the complex polynomial P(x) of 19-fringe algorithm is given by

where z = exp(ip/4). The sampling components a_r and b_r of the 19-fringe algorithm can be calculated by expanding Eq. (6) and separating the real and imaginary numbers and given to

We can visualize the phase-modulated algorithms by using Fourier representation [9,20]. To visualize the algorithms in the frequency domain, we define the sampling functions for the sampling components of the phase-modulated algorithm expressed in Eqs. (9) and (10).

where n is the frequency variable. F₁ and F₂ are purely imaginary and real functions based on the symmetric properties of the sampling components a_r and b_r [10,15]. Fig. 2 represents the iF₁ and F₂ of the 19-fringe phase-modulated algorithms.

The configuration of the sampling functions iF₁ and F₂ represent the error-compensating ability of the phase-modulated algorithm [9,10,14,15,18,25]. Since the sampling functions of the developed algorithm shown in Fig. 2 have zero gradients at the n/n₁ = 1, this algorithm can suppress the linear miscalibration of phase modulation and satisfy the fringe contrast maximum condition [9,18]. In addition, Fig. 2 shows that the second order derivatives of the sampling functions are zero at the n/n₁ = 1, which indicates that 19-fringe algorithm is insensitive to the first order nonlinear phase modulation. The sampling functions of the algorithm shown above have zero gradients n/n₁ = 2, 3, …, N–2, which indicates a compensation ability for the compounding errors between the higher harmonics and linear miscalibration of phase modulation [15,25].

Finally, the sampling functions should have zero gradients at n/n₁ = 0 to compensate for the linear modulation of intensity [14]. The sampling functions shown in Fig. 2 have zero gradients at n/n₁ = 0. In addition, the second order derivatives of the sampling functions shown in Fig. 2 are zero at the n/n₁ = 0, which indicates that 19-fringe algorithm possesses a compensation ability for the first order nonlinearity of intensity modulation.

By combining a wavelength-modulating Fizeau interferometer with the new19-fringe phase-modulated algorithm, we assess the surface of an 80-mm-diameter, 8-mm-thick fused silica plate. The fused silica plate is placed horizontally on a mechanical stage with an air-gap distance of L = 2 mm. Fig. 4 depicts the experimental setup for surface assessment of the fused silica plate using the wavelength-modulating Fizeau interferometer.

We maintain the laboratory temperature at 20.5°C, and use a tunable diode laser source with a Littman external cavity (New Focus TLB–6300–LN). During the surface assessment, we linearly modulate the wavelength of the diode laser source from 632.8 nm to 638.4 nm by using a picomotor and PZT [26]. The beam from the diode laser source is split by a beam splitter. Two divided beams are directed to a wavelength meter (Anritsu MF9630A) and the Fizeau interferometer. The accuracy of the wavelength meter is calibrated with 10⁻⁷ at a wavelength of 632.8 nm. The beam incident to the interferometer illuminate the surfaces of the reference and parallel plate through the collimator lens and the reflected beams from these surfaces generate the fringe pattern. We acquire the interferogram using the CCD camera with a resolution of 640 × 480 pixels.

Fig. 5 depicts a photo of the transparent fused silica plate in the wavelength-modulating Fizeau interferometer and a raw interferogram of the sample at a wavelength of 632.8 nm. We calculated the necessary range of wavelength modulating dl as follows:

We finely modulate the wavelength from 632.8198 nm to 633.1189 nm, and acquire 19 interferograms at regular wavelength intervals in 10 s. Due to the nonlinearity of the diffraction grating in the laser source, the nonlinear response of ~3% occur in the PZT. For linearly modulating the wavelength of the laser source, we incrementally apply a quadratic voltage to the PZT so that the nonlinearity decrease to 1% of the total phase modulation.

We assess the surface of the fused silica plate using the acquired interferograms and proposed 19-fringe phase-modulated algorithm, as depicted in Fig. 6. The assessed surface has a concave configuration with an amplitude of ~2.2 mm.

The repeatability error of the surface assessment is 4.042 nm. The uncertainty of the reference surface and the wavelength meter are l/20 and 10⁻⁷, respectively, resulting in an overall measurement accuracy of 34 nm.

Next, the wavelength is modulated with intensity modulation, as shown in Fig. 7(a). We assess the surface of the transparent plate using the phase-modulated algorithms listed in Table 1. The surfaces of transparent plate assessed by theses algorithms other than 19-fringe algorithm contain significant ripples due to the sensitivity to the nonlinearity of intensity modulation. As a representative of these results, Fig. 7(b) shows the surface of the transparent plate assessed by the Larkin-Oreb N+1 algorithm [9]. In Fig. 7(b), considerable ripples can be observed because this algorithm does not possess the compensation ability for the intensity modulation. For intensity modulation, as shown in Fig. 7(a), the surface determined by the 19-fringe algorithm shows the same configuration with no ripples observed. Due to the substantial ripples, the repeatability error of the surface assessment by the Larkin–Oreb algorithm is ~885 nm.