### 1 Introduction

*N*+1)-fringe algorithm [9] using a Fourier representation [20]. However, this algorithm does not suppress the intensity modulation. Surrel developed a complex polynomial theory to derive the windowed phase-modulated algorithm [12] that can suppress the linear miscalibration of phase modulation and compounding error. However, Surrel’s algorithm cannot suppress the nonlinear intensity modulation.

### 2 19-fringe Phase-modulated Algorithm

### 2.1 Complex Polynomial of Phase-modulated Algorithm

*A*

*,*

_{m}*j*

*, and*

_{m}*g*

*are the amplitude, phase, and fringe contrast of the*

_{m}*m*th harmonics and

*a*

*is a phase-modulated parameter. We can calculate the target phase*

_{r}*j*

*using the phase-modulated algorithm.*

_{m}*d*= 2p/

*N*rad to separate the reference phases, where

*N*is an integer, an

*M*-fringe phase-modulated algorithm is defined by [3]

*I*(

*a*

*) is the*

_{r}*r*th irradiance signal defined by Eq. (1) and

*a*

*and*

_{r}*b*

*are the*

_{r}*r*th sampling components. The design of a phase-modulated algorithm refers to the determination of sampling components

*a*

*,*

_{r}*b*

*, and sampling number*

_{r}*M*.

*a*

*is a polynomial function of the ideal phase-modulated value*

_{r}*a*

_{0}

*, including the phase-modulation error*

_{r}*e*:

*a*

_{0}

*= 2p[*

_{r}*r*− (

*M*+ 1)/2]/

*N*,

*e*

_{0}is linear miscalibration of the phase modulation,

*e*

*(1 ≤*

_{q}*q*≤

*p*) is the

*q*th nonlinearity of phase modulation, and

*p*is the maximum order of the nonlinearity.

*P*(

*x*) of a phase-modulated algorithm [12], defined in the following Eq. (4), to determine the phase-modulated algorithm.

*x*= exp(

*imd*) and

*i*is the imaginary unit.

*m*th 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

*e*

_{0}[9,12].

### 2.2 Derivation of 19-fringe Phase-modulated Algorithm

*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].

i. The

*m*th harmonics of the signal irradiance.ii. The linear miscalibration and nonlinearity of the phase modulation.

iii. The compounding errors between the higher harmonics and linear miscalibration of phase modulation.

iv. The first order nonlinearity of the intensity modulation while modulating the wavelength.

*N*(

*N*= 8) [16].

*m*= 1 (

*x*= exp(

*id*)), to compensate for the

*m*th 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.

*z*= exp(

*i*p/4). The sampling components

*a*

*and*

_{r}*b*

*of the 19-fringe algorithm can be calculated by expanding Eq. (6) and separating the real and imaginary numbers and given to*

_{r}### 2.3 Fourier Representations of Phase-modulated Algorithms

*n*is the frequency variable.

*F*

_{1}and

*F*

_{2}are purely imaginary and real functions based on the symmetric properties of the sampling components

*a*

*and*

_{r}*b*

*[10,15]. Fig. 2 represents the*

_{r}*iF*

_{1}and

*F*

_{2}of the 19-fringe phase-modulated algorithms.

*iF*

_{1}and

*F*

_{2}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*/

*n*

_{1}= 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*/

*n*

_{1}= 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*/

*n*

_{1}= 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].

*n*/

*n*

_{1}= 0 to compensate for the linear modulation of intensity [14]. The sampling functions shown in Fig. 2 have zero gradients at

*n*/

*n*

_{1}= 0. In addition, the second order derivatives of the sampling functions shown in Fig. 2 are zero at the

*n*/

*n*

_{1}= 0, which indicates that 19-fringe algorithm possesses a compensation ability for the first order nonlinearity of intensity modulation.

### 3 Error Analysis

*A*

*of the*

_{m}*m*th harmonics can be expressed as

*d*

_{0}is the coefficient of intensity modulation and

*d*

*(*

_{p}*p*1) is the

*p*th nonlinearity coefficient of intensity modulation. Fig. 3 shows the calculated phase error for nonlinearity of intensity defined by Eq. (11) based on the phase-modulated algorithms listed in Table 1. The phase errors shown in Fig. 3 indicate that the difference between the calculated phase without intensity modulation and the calculated phase containing nonlinear intensity using Eqs. (2) and (11).

*d*

_{1}to a value less than 0.1, the 19-fringe algorithm achieves precise measurement with sub-nanometer accuracy. The phase errors by the Schwider-Hariharan 5-fringe [7,8] and Larkin-Oreb

*N*+1 [9] algorithms are the largest since these algorithms do not possess the compensation ability for intensity modulation. Hibino’s 11-fringe, Surrel’s 2

*N*–1, and Onodera’s 6-fringe algorithms can compensate for the linear modulation of intensity, but cannot compensate for the nonlinearity of intensity modulation. Therefore, these three algorithms show larger errors than that of 19-fringe algorithm.

### 4 Experiment

### 4.1 Wavelength-modulating Fizeau Interferometer

*L*= 2 mm. Fig. 4 depicts the experimental setup for surface assessment of the fused silica plate using the wavelength-modulating Fizeau interferometer.

^{−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.

### 4.2 Results and Discussion

*l*as follows:

*l*/20 and 10

^{−7}, respectively, resulting in an overall measurement accuracy of 34 nm.

*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.

### 5 Conclusion

*l*/20~30 nm.