1. Introduction

Interferometry is used to measure small deformations and displacements in surfaces under test. Many interferometric methods exist, but one of the most precise and in common use nowadays is phase shifting interferometry. Phase-Shifting Interferometry is not a specific optical hardware configuration, but rather a data collection and analysis series of methods that can be applied to a great variety of testing situations. In phase shifting interferometry [1] a series of interferograms is recorded while the reference phase of the interferometer changes to produce an optical path difference.

Fringe pattern results are analyzed by several methods. Some techniques are limited by the number of images that are taken, 3, 5, etc. There is a 2+1 steps algorithm [2, 3, 4], 5-steps Schwider-Hariharan [1, 5], 3-steps algorithm, [6, 7]. Morimoto [8] has proposed a Phase-Shifting Interferometry method which detected the maximum brightness changes in a period and get the phase distribution.

In the general case N interferograms are used and each pixel of the N images is processed by a least-squares algorithm. The general least-squares solution for a series of N interferograms recorded at phase shifts was first discussed by Brunning [9] and more rigorously by Greivenkamp [10]. Recently, Wang [11] proposed an advanced random phase-shifted algorithm to extract phase distributions. The algorithm is based on a least-squares iterative procedure. Surrel [12] discusses how a specific number N of interferograms can influence the onset of undesirable harmonics in the solution for the phase.

A common problem associated with any least-square method is the sensitivity to outlier data fluctuations. Discordant points change dramatically the fitting process because the weight increase as the distance square from the fitting. Due to this inconvenience we decided to study another method to recover the phase which uses a different approach to measure the similarity of two functions: the correlation.

The use of correlations is not new into interferometry. For example, Hedser [13] has used it to find the size of the phase shifting step. Yasuno [14] introduced a novel signal processing method to analyze interferograms that is based on phase-resolved correlation in which one dimensional interferogram is resolved into a three dimensional intensity distribution in position-frequency phase space. Chen [15] developed a computationally efficient algorithm to recover the phase using phase-shift interferometry with imprecise phase shifts. This algorithm calculates the correlation between the intensity of two pixels. However, to build such correlation typically it is required the sampling of hundreds of different interferograms and such sampling is not done.

To our knowledge, up until now there has not been a direct use of correlation to recover the phase. We therefore, explore a correlation algorithm to recover the phase from fringe patterns generated by phase-shifting inteferometry. In our case the correlation is to be calculated for each pixel of the images, between the theoretical function describing the intensity of the phase shifting interferogram and the observed intensity. We study how the algorithm performs through numerical simulations where we take into account the presence of random noise in the reference phase. We apply the algorithm to real interferograms and show that the results are in agreement with numerical simulations predictions. With this method we recover the phase with an accuracy of less than 2%.

In Section 2 we present the description of the correlation algorithm. Section 3 shows the numerical simulation results and the results obtained with real interferograms with a comparison among them. Finally, in Section 4 we discuss the results and we give the conclusions in Section 5.

2. Correlation algorithm

The basic equation of phase shifting interferometry is,

where a(x,y) is the fringe contrast, b(x,y) is the average intensity and α_l is the phase-shifting given as ${\alpha }_l=\frac{2\pi }{N}l$. At each shift α_l we get an interferogram I_l(x,y), and we can get up to N such interferograms. Hence, for each pixel (x,y) we have N intensity values that should describe a cosine function. We can use a correlation function to measure how close this cosine law is followed by the data and obtain the phase from this measurement.

In the following we will be exploring the correlation for each pixel (x,y) of the image and we will omit the notation (x,y) for brevity. The correlations will be analyzed in the z direction that corresponds to the interferogram number or phase step.

The correlation function compares two functions f(z) and g(z) as one of them is shifted by an amount ξ with respect to the other in the z direction:

We will then let the function f(z) be determined by the observed data points while g(z) is a simple cosine function, that is:

where I(z) represent one of the N interferograms, $\overline{I\left(z\right)}$ is the average intensity of the N images, z = −α_l and $\xi =\frac{2\pi }{m}k$, with k between 1 and m, m is the number of points where the two functions are to be compared.

The observed function f(z) should in turn be a function of the unknown phase of the object, according to Eq. (1) f (z)=a Cos(ϕ+z), and thus the correlation of f(z) and g(z) may be written,

where $\Delta =\frac{2\pi }{N}$.

The correlation attains its maximum when the argument of both cosine functions are equal, that is when k=k_max such that,

so that the phase is given by,

Therefore, by finding the point k_max where the maximum correlation is attained, we can measure the phase ϕ.

3. Numerical simulations

The correlation algorithm that has been proposed, finds the phase displacement at which the observed intensity can be better described by a cosine function. This measure is carried out independently for each pixel, so the performance of the method can be evaluated for a single pixel. We will then carry out numerical simulations on “images” of one single pixel. The number of such images N will be modified to evaluate the precision of the method in the presence of varying amounts of noise. Even in the case of zero noise the uncertainty in the phase value δϕ is given (see Eq.7) by,

for $\delta k=\frac{1}{2}$.

In order to have an intrinsic precision of the method of $\frac{\lambda }{100}$ (δϕ=0.06 rad), we will then require m=50 for the numerical simulations.

A common source of error in phase shifting interferometry is the phase-shifting device, which can introduce either systematic or random variations to a desired position. An error in the position of the reference will produce variations in the intensity that are considered as noise in the interferograms. We carried out numerical simulations introducing a position error given by,

where η is a random number, taken from a Gaussian distribution with zero mean and unity standard deviation, ε is the uncertainty in position in nm, error is added to the phase, so Eq. (1) is re-written as,

For the experiments we arbitrary took a(x,y)=100, Ī=1000, λ=0.5 µm an optical path difference of OPD=0.38 µm corresponding to ϕ=4.77 rad. We considered several values for the uncertainty in position: ε=5, 10, 20, 50, y 100 nm. For each case of ε we simulated N interferograms, with N=4, 6, 10, 15, 20, 25, 30, 40, 50, 60, 70, 80, 90 and 100, according to Eq. (10). In each case we found the maximum k_max from the correlation of this data with the expected cosine given by Eq. (4). We further repeated each experiment 100 times corresponding to different realizations of the random number η. Therefore we obtained 100 different phase values ϕ, which allowed us to calculate the mean value, and more important, the RMS dispersion σ.

As an example of the results that were obtained, Fig. 1 presents the correlation algorithm results for the case of N=15 interferograms. The graphs on (a) show the intensity values (vertical axis) as a function of the phase-shifting amount indicated in the horizontal axis. Asterisks indicate the simulated data points, and the continuum line is the cosine function of Eq. (4) corresponding to the value of k_max found with the algorithm; (b) shows the results of the correlation (vertical axis) as a function of the displacement position k (horizontal axis). We can localize the position k_max where the correlation between the functions is maximum. We can notice that although the intensity dispersion increases greatly when the position uncertainty ε increases, the correlation algorithm works well giving correct results for the phase.

After repeating this process 100 times for each set of values, we obtain the standard deviations σ in the determination of the phase value presented in Table 1. For each value N and position uncertainty ε the table indicates σ measured in percentage. This measurement will be referred to as phase error, and is a measurement of the precision of the algorithm.

 

Fig. 1. Results for N=15 interferograms. (a) Asterisk indicates the Intensity as a function of phase shifting. (b) Shows the correlation results as a function of k and indicates the value k_max where the maximum is located. The recovered phase is 4.77 rad and the cosine function corresponding to it is shown with a solid line in (a)

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The results of the table may be seen graphically. Figure 2 shows the phase error (σ[%]) that can be obtained for a given number N of interferograms. Different sets of points indicate different values of the position uncertainty ε. Only the cases that have phase error less than 5 % are shown. This occurs for any N>4 value when ε=5, 10, 20, 50 nm, and for N>60 when ε=100 nm.

From this graph, we note that the phase error σ decreases when the number N increases for any value of ε in a predictable way. This will allow us to choose the number N of interferograms needed to recover the phase of a test surface as a function of the required precision and of the accuracy of the mechanism to shift the phase in the particular interferometer that is going to be used.

Table 1. Phase error.

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Fig. 2. Phase error as a function of the number of interferograms. The position uncertainty ε=5, 10, 20, 50 and 100 nm is indicated on each curve.

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4. Experimental results

We tested the correlation algorithm with real interferograms of a flat surface under test. The interferograms were taken with a Linnik interferometer [16] with the goal to measure the micro-roughness that results from an experimental polishing process. The phase-shifting was produced with a micro-steper motor of 25 nm/step resolution. Nine interferograms were acquired spaced every 2 steps between them. Figure 3 show some of them.

Figure 4 shows the intensity variations for one arbitrary pixel of the acquired interferograms. To apply the correlation algorithm it is necessary to identify the number of interferograms required to complete a period, which call N_c here. The solid line in Fig. 4 shows a cosine function fitting from which we identify Nc=4.2, and notice that N_c can be an fractional number and that N can be larger than N_c resulting in a better precision in the phase value. We applied the correlation algorithm with ${\alpha }_l=\frac{2\pi }{N_c}l$ for Eq. (4). The actual number of images N taken for the process was varying between 4 and 9. The resulting phase image for the case N=9 (in 2π module) is shown in Fig. 5. The image is later unwrapped using an algorithm given by Salas [17]:

where []2π indicates the wrapping operation, Φ_o is the object’s phase, Φ_r is the reference phase. After applying the algorithm we obtained the unwrapped image of the object phase shown in Fig. 6.

 

Fig. 3. Interferograms of sample surface acquired with a Linnik interferometer. (a) 0 steps, (b) 2 steps, (c) 4 steps.

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Fig. 4. Intensity variations for one arbitrary pixel from the nine interferograms. Continuum curve represent the fitting to obtained Intensity values.

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Fig. 5. Resultant phase image that results from the correlation algorithm for N=9. Phase wrapped image.

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Fig. 6. (a) Unwrapped phase image. The box indicates a region to be analyzed to measure the (b) noise and the (c) micro-roughness.

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4.1. Image analysis

The unwrapped image is used to analyze and identify the noise and the micro-roughness (see Fig. 6). If we focus on regions with a size of 1.1×1.1 µm (5×5 pixels) we notice sharp structures with high frequency pixel to pixel variations. Furthermore, these structures change when the number of images N into the correlation algorithm is changed from 4 to 9, indicating that they are not real structures but phase fluctuations that measure the precision of the correlation algorithm with the present experimental setup. We then measure the variance of such structures σ to determine the phase error. Table 2 present these results and makes a comparison with the numerical simulation results for ε=5 nm and ε=10 nm values. We notice that the behavior of real interferograms is similar to numerical results. The noise that is produced by the uncertainty of the shift position of the step-motor of between 5 and 10 nm, gives a similar phase error as the real interferograms. This suggests that the uncertainty of the steper motors used in the real experiments is on the order of one half step, which is expected. We also conclude from this table that the phase can be measured with precision better than 1% in al cases of N=4, ..9.

Figure 7 presents the curves whose values are in Table 2, points indicate the real interferograms results and continuum lines indicate the numerical results.

Smooth variations due to the micro-roughness left by the polishing process are noticed in larger image sections; at scales of 4.4×4.4 µm. In Fig. 6 we mark a small region with a box, indicating a typical 4.4 µm region (20×20 pixels) to be analyzed for micro-roughness. We took 15 such similar regions, and we found a micro-roughness value of 11.2±2.8 nm. The result is in good agreement with the one obtained with a mechanical profile, which indicates an average value of 15 nm roughness given by reference [18].

Table 2. Comparison of phase error from real interferograms results with numerical experiment results.

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Fig. 7. Phase error as a function of the number of interferograms for numerical simulations (connect dots) and experimental results(stars).

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5. Conclusions

The correlation algorithm works for real interferograms, the results are better than 2% in the phase value accuracy. Results show how the precision improves as the number of interferograms increases, even when this number is greater than the necessary to have a full period of fringes. From the results of Table 2, we can know how many interferograms are necessary according the required precision and the uncertainty of the positioning device used for the phase shifting. We were able to measure micro-roughness in a test sample, in good agreement with other methods. We also measured the obtained phase precision and showed that it corresponds to the positioning uncertainties of the micro-steper motor that was used for phase shifting.