In phase shifting interferometry, the fringe contrast is preferred to be at a maximum when there is no phase shift error. In the measurement of highly-reflective surfaces, the signal contrast is relatively low and the measurement would be aborted when the contrast falls below a threshold value. The fringe contrast depends on the design of the phase shifting algorithm. The condition for achieving the fringe contrast maximum is derived as a set of linear equations of the sampling amplitudes. The minimum number of samples necessary for constructing an error-compensating algorithm that is insensitive to the jth harmonic component and to the phase shift error is discussed. As examples, two new algorithms (15-sample and (3N – 2)-sample) were derived that are useful for the measurement for highly-reflective surfaces.
© 2014 Optical Society of America
Phase shifting interferometry has been used for testing the surface shape and the optical thickness variation of optical components. In phase shifting interferometry , the phase difference between a sample wave front and a reference wave front is changed linearly or stepwise, and the resulting signal irradiance distribution is stored at each step, or bucket , in the frame memory of a computer. The phase can be calculated from the arctangent of the ratio between the two combinations of the observed signal irradiances, according to the phase shifting algorithm used. The fringe contrast that determines the maximum measurable range is commonly used for noise reduction and phase unwrapping. When the contrast at a measuring position is smaller than a threshold value, commercial interferometers would omit this position as an “outside or undetectable position” of the observing aperture. Figures 1(a) and 1(b) show the interferograms in the tests of a highly-reflective mirror (R = 60% at wavelength 632.8 nm) and a fused silica plate (R = 4% at wavelength 632.8 nm), respectively. In case of highly reflective surfaces, such as the surface of semiconductors, the contrast of the fundamental signal is relatively low and, thus, should be kept at a maximum. The fringe contrast is generally changed by the magnitude of the phase shift error. The contrast seems to increase or decrease depending on the phase shift error and it is not necessarily at a maximum when there is no phase shift error.
Phase shift errors are the most common sources of systematic errors in phase shifting measurements . Phase shifts also become spatially non-uniform in testing a high numerical-aperture spherical concave, which results in an inevitable error. Many phase shifting algorithms have been developed that eliminate the phase shift errors [3–19]. However, the condition of maximum fringe contrast has so far not been taken into account when designing these algorithms. It is interesting to note that synchronous detection [1, 20] gains or loses the contrast, depending on the sign of the phase shift miscalibration. In several algorithms, such as 4-bucket , Larkin and Oreb (N + 1) , Schmit and Creath 5 , and Hibino 11 , the contrast is not optimized, while other algorithms such as Scwider-Hariharan 5 [5, 7], Hibino 7 , de Groot 7 and 13 [12, 16], and Surrel (2N – 1)  the contrast is maximized.
In this paper, we derive the condition that the fringe contrast is at a maximum when there is no phase shift error. We discuss the minimum number of samples necessary for constructing the new algorithm that satisfies the contrast maximum condition and has immunity to the jth harmonic components and phase shift miscalibration. We will also derive the maximum contrast condition in the Fourier description  and evaluate the conventional algorithms visually. As examples, two new algorithms, 15-sample and (3N – 2)-sample algorithms were derived that are insensitive to the 6th and (N – 2)th order harmonic signals, respectively.
2. Fringe contrast maximum condition
2.1 Fringe contrast of phase shifting algorithm
Consider the signal irradiance I(x, y, αr) at a designated point (x, y) with a sinusoidal periodic waveform as a function of a phase shift parameter αr:
The rigorous expressions for fringe contrast of 3-sample , 4-sample , and 5-sample algorithms [23, 24] have already been reported. Spatio-temporal fringe contrast has also been reported . However, these studies did not discuss the condition of maximum fringe contrast.
Consider an M-sample phase shifting algorithm, where the reference phases are separated by M – 1 equal intervals of 2π/N rad, where N is an integer. Generally, best phase shift step of N is N = j + 2, when j is the harmonic order [13, 14]. A general expression for the calculated phase φ* is
If we note that the numerator and denominator of Eq. (2) are proportional to the sine and cosine of phaseφ*, respectively, the fringe contrast γ is given by
When there is no phase shift error, the coefficient A reduces to the irradiance I0. The coefficient A changes very slowly and slightly depending on the phase shift error.
2.2 Condition for fringe contrast maximum
When the phase shift is nonlinear, each phase shift value αr is a function of the phase shift parameter. The phase shift value for the rth sample can be denoted by a polynomial of the unperturbed phase shift value α0r as
The fringe contrast γ should have a maximum when there is no phase shift error. Here we consider the case in which there is only a miscalibration error ε1.Eq. (6) to first order. If we note that the contrast becomes an extremum at ε1 = 0, the derivative of the contrast with respect to error ε1 can be written as8, 10, 14, 20, 26] ofEqs. (10) and (11) for m = 1 also define the normalization of the sampling amplitudes ar and br.
Note that Eq. (8) needs to be zero for an arbitrary value of phase φ. This condition is identical to the following equations.
2.3 Number of samples necessary for constructing the algorithm
Here we discuss the minimum number of samples necessary for constructing the algorithm that satisfies the contrast maximum condition and has immunity to the jth harmonic component, phase shift miscalibration, and the coupling error between the harmonic signal and phase shift error. When a phase shifting algorithm is insensitive to harmonic components of the signal up to the jth order, the sampling amplitudes ar and br satisfy Eqs. (9)–(12), a total of 4j + 2 equations. Equations (9)–(12) are the common conditions for synchronous detection . The number of samples necessary to eliminate the harmonic signals up to the jth order has already been investigated by several authors [10, 13, 14, 20]. The minimum number of samples is j + 2, when the interval equals 2π/(j + 2) rad.
The necessary condition for constructing an algorithm that is insensitive to the pth nonlinearity of the phase shift error and the coupling of phase shift was already investigated by Hibino et al . When the error coefficient ε1 is spatially uniform and the system is not sensitive to a fixed dc component in the calculated phase, the conditions for the sampling amplitudes ar and br to eliminate the phase shift miscalibration are the following 3 equations [14, 26] as
If we assume the following symmetric and asymmetric properties of sampling amplitudes and phase shift parameter αr asEquations (15) and (16) are reduced to trivial equations. Also, Eq. (17) will be the trivial equation from Eqs. (13) and (14). The number of independent equations among Eqs. (18)–(21) is equal to j – 1, when the phase shift interval is 2π/(j + 2) rad .
Finally, the total number of samples necessary for the immunity to the jth harmonic component, phase shift miscalibration and the coupling error, and satisfying the contrast maximum condition, is then calculated to give
Surrel reported that (2N – 2) samples are necessary for constructing an algorithm that is insensitive to the jth harmonic component, phase shift miscalibration, and their coupling error . For computational efficiency, Surrel added one more sample to his 2N – 2 algorithm and derived another 2N – 1 algorithm. It will be shown in Sect. 4, this 2N – 1 algorithm is one of the solutions that satisfy the contrast maximum condition.
3. Fourier representation of contrast maximum
The contrast maximum condition can be visualized if we take a Fourier representation of the sampling functions of the algorithm [8, 11–14, 21]. The sampling functions of the numerator and the denominator of a phase shifting algorithm given by Eq. (2) are defined byEqs. (26) and (27).Eq. (22).
Freischlad and Koliopoulos showed that for any phase shifting algorithm the Fourier transforms of two sampling functions have matched values at the fundamental frequency . Larkin and Hibino have shown that when a phase shifting algorithm is insensitive to a miscalibration error of phase shift, the first derivatives of the Fourier transforms of the sampling functions also have matched values at the fundamental frequency [8, 10, 14]. In Sect. 2.3, we have derived new conditions, Eqs. (13) and (14), for a phase shifting algorithm to show the maximum value of fringe contrast when there is no phase shift miscalibration. Here, we derive a Fourier representation of these new conditions.
Comparing Eqs. (28) and (29) with Eqs. (13) and (14), we find that the condition is identical to the requirement that the first derivative of the sampling functions iF1 and F2 are zero at the fundamental frequency.
Figure 2 shows the sampling functions of conventional algorithms: (a) synchronous detection (4-sample) , (b) Schwider-Hariharan 5-sample [5, 7], (c) Larkin-Oreb N + 1 algorithm (N = 6) , (d) Surrel 2N – 1 algorithm (N = 6) , (e) Hibino 11-sample algorithm , and (f) de Groot 13-sample algorithm . The sampling functions have zero slopes at the fundamental frequency (ν = 1), in Figs. 2(b), 2(d) and 2(f). These three algorithms have maximum contrast when there is no phase shift miscalibration.
Figure 3 shows the fringe contrast for the Schwider-Hariharan 5-sample algorithm [5, 7] as a function of phase shift miscalibration ε1. The contrast decreases by 21% for ε1 = ± 0.3. A miscalibration of ε1 = –0.3 is common in the spherical concave test, in which the phase shift of the oblique ray component becomes smaller than that of the axial ray. We can see that the fringe contrast maximum condition is satisfied in this algorithm.
4. Characteristic polynomial and fringe contrast maximum condition
4.1 An example of 15-sample algorithm
In this section, we discuss the fringe contrast maximum condition in the characteristic polynomial representation . First we derive an algorithm, as an example, that satisfies the fringe contrast maximum condition for the case of j = 6 and p = 1. From Eq. (23), this algorithm consists of the 15-sample algorithm.Eq. (30) into Eqs. (9)–(14) and Eqs. (18)–(21), we obtain 16 equations for sampling amplitudes. The solutions are unique and given by
The resultant algorithm is14].
Surrel showed that when a phase shifting algorithm has matched slopes at the fundamental frequency (ν = 1), the characteristic polynomials have double roots at ζ−1 = exp(−2πi/N) on the unit circle . Algorithms that satisfy the fringe contrast maximum condition have matched zero slopes and, thus, have double roots at ζ−1. In contrast, algorithms which have double roots at ζ−1 do not necessarily satisfy the contrast maximum condition.
However, it is worth to note that when the algorithm has double roots or triple roots on the all positions on the unit circle ζ-m (m = …, −2, −1, 0, 2…) in the characteristic diagram, the sampling functions in Fourier space become symmetric around the fundamental frequency (ν = 1). Then the sampling functions have zero slopes at the fundamental frequency and automatically satisfy the contrast maximum condition. This conclusion is confirmed by the above 15-sample algorithm.
In the next subsection, we derive another example of the 3N – 2 algorithm that has triple roots and confirm this conclusion.
4.2 Characteristic polynomials and 3N – 2 algorithm
The second example is a (3N – 2)-sample algorithm (j = N – 2, p = 2) which has triple roots at all the positions on the unit circle in the characteristic diagram . By expanding the characteristic polynomials, a new windowed DFT phase shifting algorithm is obtained by the Eq. (33).
Figure 4 shows the Fourier transforms iF1 and F2 for (a) 15-sample and (b) (3N – 2)-sample algorithms (N = 15). We can observe that both algorithms have zero slopes at the fundamental frequency. Note that we did not require the contrast maximum condition in deriving the 3N – 2 algorithm. However, from Fig. 3, we can observe that the algorithm has matched zero slopes at the fundamental frequency. Therefore, we can conclude that when the algorithm has double or multiple roots on the all positions of the unit circle in the characteristic diagram, the algorithm satisfies the contrast maximum condition.
The condition for a phase shifting algorithm to satisfy the fringe contrast maximum requirement was derived as a set of linear equations of the sampling amplitudes. We discussed the minimum number of samples necessary for constructing the error-compensating algorithm that satisfies the contrast maximum condition and has immunity to the jth harmonic components and phase shift miscalibration. The maximum contrast condition was seen to require zero-derivatives for the sampling functions in the Fourier description. The relation between the contrast maximum condition and characteristic polynomial representation was also discussed. As examples, two algorithms, 15-sample and (3N – 2)-sample, were derived that are useful for the measurement of the highly-reflective surfaces.
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