### 1 Introduction

*N*– 6 [23] and 8

*N*– 7 [24] algorithms, for interferometric surface measurements. These algorithms show superior error-suppression ability but require a large number of interferograms for surface profiling (more than 20-frame interferograms). In the recorded interferograms, the environmental effects, such as temperature variance [25] and floor vibration [26], can be accumulated, and cause systematic errors in the measurement results [7]. In addition, phase-tuning algorithms applied to industrial applications have used four [27] and five [28,29] interferograms because using more than 10-frame interferograms for surface assessment is time-consuming.

### 2 Irradiance Signal in Wavelength-Modulated Interferometry

*S*

_{xy}_{0}is the DC component, and

*S*

*,*

_{xyz}*a*

*, and*

_{xz}*f*

*are the amplitude, phase-tuned parameter, and phase of harmonic component, respectively. In coefficients of the irradiance signal, the subscript*

_{yz}*x*indicates the

*x*-th phase-tuned interferogram (

*x*= 1, 2, …,

*M*),

*y*is a pixel location on the interferogram (

*y*= 1, 2, …,

*L*), and

*z*is the order of the harmonic component (

*z*= 1, 2, …,

*P*).

*f*

_{y}_{1}.

### 3 Harmonic Phase-Iterative Method

### 3.1 Determination of Phase and Phase-Tuned Parameter

*ML*available equations and

*ML*+

*MLP*+

*MP*+

*LP*unknowns, corresponding to

*S*

_{xy}_{0},

*S*

*,*

_{xyz}*a*

*, and*

_{xz}*f*

*, respectively. For instance, since the parameter*

_{yz}*S*

_{xy}_{0}is related to the phase-tuned interferograms (

*x*= 1, 2, …,

*M*) and pixel location on the interferograms (

*y*= 1, 2, …,

*L*), the maximum number of cases of

*S*

_{xy}_{0}is

*ML*. Since the number of unknowns is larger than that of usable equations, partial least-squares fitting is used in phase extraction [18].

*S*

_{xy}_{0}and

*S*

*, defined in Eq. (1), contain only pixel-to-pixel variation. Based on this assumption, the irradiance signal can be given to*

_{xyz}*A*

*,*

_{y}*B*

*, and*

_{yz}*C*

*correspond to*

_{yz}*S*

_{y}_{0},

*S*

*cos*

_{yz}*f*

*, and*

_{yz}*S*

*sin*

_{yz}*f*

*, respectively. When the phase-tuned parameter*

_{yz}*a*

*is the given value, Eq. (2) consists of*

_{xz}*L*+ 2

*LP*unknowns. The unknowns can be calculated if

*M*≥ 1+ 2

*P*because Eq. (2) is composed of

*ML*available equations. Based on this condition, the number of frames is greater than or equal to five for phase extraction with suppression of the second harmonics (

*M*= 5 and

*P*= 2). To determine the coefficients

*A*

*,*

_{y}*B*

*, and*

_{yz}*C*

*, the least-squares error associated with the frame number*

_{yz}*E*

*is given to*

_{y}*I*

_{xy}*is the experimental irradiance signal. Through least-squares fitting,*

^{e}*A*

*,*

_{y}*B*

*, and*

_{yz}*C*

*were determined by minimizing the*

_{yz}*E*

*:*

_{y}##### (6)

*B*

*and*

_{yz}*C*

*are determined in terms of the pixel position and harmonic component. The phase of the*

_{yz}*z*-th harmonics is calculated as:

*S*

_{xy}_{0}and

*S*

*, defined in Eq. (1), contain only frame-to-frame variation. From this assumption, Eq. (1) can be rewritten as*

_{xyz}##### (10)

*A′*

*,*

_{x}*B′*

*, and*

_{xz}*C′*

*correspond to*

_{xz}*S*

_{x}_{0},

*S*

*cos*

_{xz}*a*

*, and -*

_{xz}*S*

*sin*

_{xz}*a*

*, respectively. When phase*

_{xz}*f*

*is the known value, Eq. (10) consists of*

_{yz}*M*+ 2

*MP*unknowns. The unknowns can be determined if

*L*

^{3}1 + 2

*P*because Eq. (10) is comprised of

*ML*usable equations. For determining coefficients

*A′*

*,*

_{x}*B′*

*, and*

_{xz}*C′*

*, the least-squares error associated with the pixel*

_{xz}*E′*

*is expressed as*

_{x}*A′*

*,*

_{x}*B′*

*, and*

_{xz}*C′*

*are determined by minimizing the*

_{xz}*E′*

*with the least-squares fitting:*

_{x}##### (14)

*B′*

*and*

_{xz}*C′*

*are obtained in terms of each phase-tuned interferogram and harmonic component, respectively. The phase-tuned parameter can then be calculated as follows:*

_{xz}### 3.2 Convergence Formula of Phase-Iterative Method

*a*

*=*

_{xz}*za*

_{x}_{1}and

*f*

*=*

_{yz}*zf*

_{y}_{1}. Owing to these relationships, a harmonic-related error can occur in the least-squares fitting. This error can cause the divergence of the least-square fitting in Eqs. (6) and (14). To suppress the harmonic-related error, the harmonic convergence formula, expressed following Eq. (18), is proposed.

*z*is the preset threshold of the iterative calculation and

*k*is the number of iterative cycles.

*a*

*. Phase-tuned parameter and phase can be determined via phase-iterative method without harmonic-related error when all calculated*

_{xz}*a*

*(*

_{xz}*x*= 1, 2, …,

*M*and

*z*= 1, 2, …,

*P*) satisfy above formula. For instance, when the irradiance signal includes up to the second harmonic (

*P*= 2), the phase-iterative method requires at least five interferograms to identify the unknowns (

*z*= 1, 2 and

*x*= 1, 2, …, 5). To determine the phase-tuned parameter and phase,

*a*

_{x}_{1}and

*a*

_{x}_{2}should satisfy Eq. (18).

*P*. The properties of the proposed technique were determined before conducting the iterative cycle. The proposed convergence condition and technique enable the proposed iterative method to be resistant to harmonic-related errors, thereby enhancing the ability of the iterative calculation.

### 3.3 Phase-Extraction Process

**Step 1**. The pixel arrangements of the pixel-selection technique are preset in accordance with the number of unknowns.**Step 2**. Using the pixel-selection technique and pixel-to-pixel variation, the phase*f*_{yz}on the selected pixels was calculated with Eqs. (5) and (9) using the phase-tuned parameter^{k}*a*_{xy}^{k}^{– 1}calculated. For the first iterative cycle, the phase-tuned parameter of the first and*z*-th harmonics are predetermined as follows: 0 <*a*_{x}_{1}^{0}< 2p and*a*_{x}_{z}^{0}=*za*_{x}_{1}^{0}.**Step 3**. Using the pixel-selection technique and frame-to-frame variation, the phase-tuned parameter*a*_{xy}on the selected pixels was calculated with Eqs. (13) and (17) using the phase calculated in previous step.^{k}**Step 4**. Results of the iterative calculation are estimated by the harmonic convergence formula. If the calculated phase-tuned parameter satisfies the predetermined threshold (e.g.,*z*= 10^{−5}rad),*a*_{xz}is the final parameter. Otherwise, the phase-tuned parameter must be recalculated by repeating steps 2 and 3 until recalculated value satisfies the convergence formula.^{k}**Step 5**. Using all pixels of the original interferograms and pixel-to-pixel variation, the target phase*f*_{y}_{1}was calculated with Eqs. (5) and (9) using the final phase-tuned parameter*a*_{xz}. Then, the surface of the glass plate was determined using an unwrapping technique.^{k}

### 4 Numerical Simulation

*N*

_{x}_{0}and

*N*

*, which are arbitrary values between 0.9 and 1, and the pixel-to-pixel variations are express by*

_{x}*g*and

*h*(−1 ≤

*g*≤ 1, −1 ≤

*h*≤ 1). In addition, the phase-tuned parameter contains phase-tuned error and is defined as

*a*

_{0}

_{g}_{1}is the ideal phase-tuned parameter, and

*y*

_{0}and

*y*

_{1}are linear and nonlinear phase-tuned error coefficients where

*y*

_{0}= 0.05 and

*y*

_{1}= 0.03. Then,

*a*

_{0}

_{g}_{1}is adjusted as follows:

*a*

_{0}

_{x}_{1}= [0 1 2 3 4] for phase-iterative methods and

*a*

_{0}

_{x}_{1}= p(

*x*– 1)/2 for phase-tuning method (

*x*= 1, 2, …,

*M*). Moreover, each interferogram contained white Gaussian noise with a mean of zero to consider the environmental effects. The reference phase and interferogram of the simulation is depicted in Fig. 4.

*z*was 10

^{−5}rad. In addition, HPIA used 40 × 40 pixels of the pixel-selection technique.

### 5 Experiment

### 5.1 Wavelength-Modulated Fizeau Interferometer

^{−7}accuracy. The beam sent to the interferometer illuminates the reference and glass plate surfaces by the collimator lens, and the interference of two reflected beams from the reference and glass plate surfaces generates the interferogram. The accuracy of the reference surface was approximately

*λ*/20 (~32 nm). While changing wavelengths using the wavelength-modulated technique, a charge-coupled device (CCD) camera saves the interferograms ith 640 × 480 pixel resolution and 25 mm pixel size. Figs. 7 and 8 show the glass plate on the mechanical stage and the raw interferogram of the glass plate observed by the CCD camera.

### 5.2 Surface Assessment of Transparent Glass Plate

*z*≥ 3), residual nonlinearity of the wavelength-modulated technique, and the characteristics of the CCD camera, such as pixel resolution and pixel size. We calculated the total accuracy of experiments, 32.1 nm, by the root square sum of the accuracy of the reference surface of 32 nm and the standard deviation of surface assessment of 1.885 nm.