Improvement of the fringe analysis algorithm for wavelength scanning interferometry based on filter parameter optimization

Tao Zhang; Feng Gao; Hussam Muhamedsalih; Shan Lou; Haydn Martin; Xiangqian Jiang

doi:10.1364/AO.57.002227

1. INTRODUCTION

With the rapid development of the manufacturing process, a variety of surface textures and structures are emerging, bringing in huge inspection requirements, among which optical interferometry plays an important role, especially for micro-/nanoscale applications which require high precision without destruction. There are many types of interferometers such as phase shifting interferometer (PSI), vertical scanning interferometer (VSI), and wavelength scanning interferometer (WSI), with the merits of high-precision and nondestructive measurement and are widely used in various industries such as semiconductor, micro fluidics, and photovoltaic films [1–4]. However, monochromatic interferometry like PSI can only be used to measure smooth surfaces because of the well-defined $2 π$ phase ambiguity problem. VSI adopts mechanical scanning to make precise measurement with the expense of time [5,6]. WSI does not suffer from these two shortcomings, i.e., the $2 π$ phase ambiguity problem and mechanical scanning, and thus has the potential of online and large vertical range measurement, attracting much attention from both academia and industry [7–10]. The fringe analysis algorithm is a crucial part in WSI which has a tremendous impact on the performance of the instrument. The phase slope method and height estimation through phase algorithm are widely used in interferometry. Takeda et al. proposed the phase slope method based on Fourier transform [11,12]. Height estimation through the phase algorithm was originally used in white light interferometry [13,14]. Moschetti et al. introduced it into wavelength scanning interferometry and achieved large improvement [15]. The algorithms were explained thoroughly in their papers except the details of the filter design. Among the measurement practice, it is found that WSI is vulnerable to environmental noise. It makes accurate measurements in a well-controlled measurement laboratory environment. However, when the environmental disturbance increases, the measurement accuracy decreases rapidly. When the signal quality lowers to some extent, large deviation up to 10s of μm appears. This paper proposes an approach to improve the performance by optimizing the filter employed in the fringe analysis algorithms based on simulation. The interference signal is analyzed, and a mathematic model is built. The simulation shows large improvement and can be achieved by optimizing the filter design. Experimental validation has been conducted to verify the simulation results as well. Finally, an adaptive fringe analysis algorithm which selects the most suitable algorithm and the optimized parameters according to the estimated SNR of the captured signal to guarantee the optimal result is proposed.

2. PRINCIPLE

Traditionally, the fringe analysis process is as illustrated in Fig. 1. The captured intensity signal is Fourier transformed and filtered, and then inverse Fourier transform is conducted before the phase information is extracted and analyzed to acquire the height information. The paper focuses on the filter design of the fringe analysis algorithm for wavelength scanning interferometry highlighted in Fig. 1 with the green block and is tested on the setup shown in Fig. 2. The measurement system is comprised of three parts: the light source, the console, and the interferometer. The white light from a halogen lamp is filtered by the acousto-optical tunable filter (AOTF) into monochromatic light, which is then guided through the fiber to illuminate the interferometer. By changing the frequency of the driving RF signal continuously, wavelength scanning is achieved. The wavelength scanning and frame capturing process is controlled and synchronized by the PC. For each measurement, 256 interferograms are captured.

Fig. 1. Traditional fringe analysis process.

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Fig. 2. Diagram of the system setup.

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For each pixel across all the captured frames, an intensity signal similar to a periodic sinusoidal distribution is attained for the corresponding point on the surface being inspected. Figure 3(a) shows an example. Equation (1) gives the mathematical representation for the intensity signal,

I_{x y} (λ) = a_{x y} (λ) + b_{x y} (λ) \cos (\frac{4 π h_{x y}}{λ} + ϕ_{0 x y}),

where

a_{x y} (λ)

and

b_{x y} (λ)

refer to the background intensity and fringe visibility, respectively. Normally the variation rates of

a_{x y} (λ)

,

b_{x y} (λ)

, and

ϕ_{0 x y}

are much lower compared to the variation

\frac{4 π h_{x y}}{λ}

, where

λ

refers to the scanned wavelength,

x

and

y

are the row and column indices for the pixel of the areal CMOS sensor of the camera, respectively. The central wavelength of the illumination light diffracted through the AOTF can be determined by the equation:

λ = α Δ n \frac{ν_{a}}{f_{a}},

where

α

is a constant that depends on the incident angle of the entrance light beam,

Δ n

is the birefringence crystal refractive index,

ν_{a}

and

f_{a}

are the propagation velocity, and the frequency of the applied acoustic wave, respectively. The wavenumber

k

is the reciprocal of the wavelength. Because the wavelength scanning range is limited, the parameters

α

,

Δ n

, and

ν_{a}

are seen as constants, which means, the wavenumber

k

changes linearly during the wavelength scanning process because

f_{a}

changes linearly. The intensity signal can be represented as

I_{x y} (k) = a_{x y} (k) + b_{x y} (k) \cos (4 π k h_{x y} + ϕ_{0 x y}) = a_{x y} (k) + \frac{b_{x y} (k)}{2} e^{j ϕ_{0 x y}} e^{j 4 π h_{x y} k} + \frac{b_{x y} (k)}{2} e^{- j ϕ_{0 x y}} e^{- j 4 π h_{x y} k},

I_{x y} (k) = a_{x y} (k) + c_{x y} (k) e^{j 4 π h_{x y} k} + c_{x y}^{*} (k) e^{- j 4 π h_{x y} k},

c_{x y} (k) = \frac{b_{x y} (k)}{2} e^{j ϕ_{0 x y}},

FT (I_{x y} (k)) = A (f) + C (f - 2 h_{x y}) + C^{*} (f + 2 h_{x y}) = A (f) + C (f - OPD) + C^{*} (f + OPD),

where FT means Fourier transform, and OPD refers to the optical path difference between the reference arm and the measurement arm. It is seen from Eq. (6) that there are only several pulses in the frequency domain of the intensity signal. In this context, as a function of the wavenumber,

I_{x y} (k)

is in the spatial frequency domain. Fourier transform of the function leads to a function in the spatial domain. Therefore, the coordinate

f

in Eq. (6) is a spatial coordinate. This also explains why the shift between the frequency components is a spatial length (OPD). When the OPD is large enough, the effective signal

C (f - OPD)

or

C^{*} (f + OPD)

component can be extracted from the low-frequency component

A (f)

with a band-pass filter. This also explains one reason why the OPD should be large enough when performing the measurement with WSI, because otherwise the three components would overlap, making it impossible to separate. After

C (f - OPD)

or

C^{*} (f + OPD)

is extracted, the inverse Fourier transform (IFT) is made. Take

C (f - OPD)

as an example (

C^{*} (f + OPD)

is similar):

IFT (C (f - OPD)) = c_{x y} (k) e^{j 4 π h_{x y} k} = \frac{1}{2} b_{x y} (k) e^{j (4 π h_{x y} k + ϕ_{0 x y})} .

Fig. 3. Signal in (a) time (wavenumber) domain and (b) frequency domain (before removal of the DC component).

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Then the natural logarithm is applied, and the phase information $ϕ (k)$ is acquired by taking the imaginary part:

\ln (c_{x y} (k) e^{j 4 π h_{x y} k}) = \ln (\frac{1}{2} b_{x y} (k)) + j (4 π h_{x y} k + ϕ_{0 x y}),

ϕ (k) = imag (\ln (\frac{1}{2} b_{x y} (k)) + j (4 π h_{x y} k + ϕ_{0 x y})) = 4 π h_{x y} k + ϕ_{0 x y} .

After that, the phase is unwrapped and the ratio between the phase change ( $Δ ϕ$ ) and the wavenumber change ( $Δ k$ ) contains the height information because the variation rate of $ϕ_{0 x y}$ is much lower:

h_{x y} = \frac{1}{4 π} \frac{Δ ϕ}{Δ k} .

Above is how the phase slope method analyzes the fringes [11,12]. The height estimation through the phase algorithm is similar, but it analyzes the phase more thoroughly, i.e., it considers the dispersion and reflection effect as well [15]. The phase is decomposed as below:

ϕ (k) = 4 π k h_{x y} + τ (k - k_{0}) + γ_{0},

where

τ

represents the dispersion difference between the measurement arm and the test arm,

γ_{0}

refers to the phase bias on reflection. Thus, the fringe pattern phase at the initial wavenumber is known with

2 π

phase ambiguity as

ϕ (k_{0}) = 4 π k_{0} h_{amb} .

The unambiguous height is:

h = \frac{1}{4 π} \frac{ϕ (k_{0})}{k_{0}} + \frac{m}{2 k_{0}},

where

m = round [\frac{1}{4 π} (\frac{Δ ϕ}{Δ k} - τ - \frac{ϕ (k_{0}) - γ_{0}}{k_{0}}) 2 k_{0}] .

Height estimation through the phase algorithm is not independent because the phase slope is also needed for determining the fringe order as shown in Eq. (14). Height estimation through the phase algorithm is reported to be able to improve the accuracy by nearly a factor of 10 [15].

It is seen from Eq. (3) that the signal $I_{x y} (k)$ is a narrow-band, close to single-frequency sinusoidal signal. This means, in the frequency domain, the effective signal is close to only one pulse. However, because the actual signal sequence is limited, there is spectral leakage, but the band is still narrow. The period of the $I \sim k$ curve is:

K = \frac{k_{e} - k_{s}}{N_{c}} = \frac{1}{f},

where

K

means the period in the curve, while

k_{e}

and

k_{s}

represent the maximum and minimum wavenumber, respectively, in the scanning process.

N_{c}

refers to the number of cycles, while

f

represents the frequency in the curve. The sample frequency is:

f_{s} = \frac{N_{frm}}{k_{e} - k_{s}},

where

N_{frm}

represents the captured frame number in the process, while

k_{e}

and

k_{s}

refer to the maximum and minimum wavenumber, respectively, in the scanning process. Therefore,

\frac{f}{f_{s}} = \frac{N_{c}}{N_{frm}} .

Thus, the center of the effective signal in the frequency domain can be determined, i.e., the $N_{c}$ frequency bin in the FFT result (counting from 0, $N_{c} + 1$ if counting from 1). For example, Fig. 3(a) is the signal of a pixel captured in the experiment. There are approximately six cycles in the scanning process, so the signal lies in the 6th component in the frequency domain. The actual result completely agrees, as shown in Fig. 3(b). In this application, an ideal bandpass filter is adopted because for other types of filters, the edge of the pass-band is not steep enough to eliminate the noise. Because the signal has a narrow band, the pass-band of the filter should not be too wide to eliminate the noise as much as possible. However, when the pass-band is narrow, the filter ringing becomes large. The optimal results are achieved by compromising between the noise and the ringing. Therefore, the width of the filter window should be relatively narrow when the signal has low quality to eliminate the noise. The filter should have a relatively wide window when the signal quality is high to reduce the filter ringing. This paper mainly focuses on finding a balance between the noise and the ringing of the filter to optimize the results.

3. SIMULATION AND EXPERIMENTAL RESULTS

The fringe analysis algorithms are simulated by Matlab which provides signal generators to accurately output signal with specified SNRs. Here the SNR acts as an input parameter to control the quality of the generated signal, namely the noise level. Additive white Gaussian noise is added to the signal. The simulation is carried out to determine the optimized parameters of the filter for the signal with different levels of SNR exhaustively. A lookup table (LUT) is created to record the algorithm with better performance and the optimized parameters corresponding to a signal with different SNRs exhaustively. The assessment is based on the overall error and the robustness, which is assessed by the difference between the peak value and the mean value of the error. After that, the algorithm and the parameters are selected according to the LUT based on the estimated SNR of the captured signal adaptively when performing fringe analysis. In the time domain, only a rectangle window is used in this paper, although other types of windows such as Hamming, Hann, and Gaussian windows are helpful to reduce the ripple caused by spectral leakage because the simulation shows that other types of windows affect the robustness of the algorithm [16]. Because the filter is an ideal bandpass filter, it is denoted by the pass-band window in the Fourier domain. The representation window (a:b) means the pass-band is from frequency bin a to frequency bin b. The DC component of the fringe pattern is removed by mean subtraction. The amplitude of the error ringing decreases with the increase of the optical path difference (OPD). When the OPD is small, even if the signal quality is very high, the deviation is large because of the filter ringing. For this reason, when doing the measurement, the OPD should be relatively large within the depth of focus (DOF) in case the contrast of the fringe gets too low. So the small OPD should be avoided in the measurement practice and is omitted in the simulation. Only the height (equal to half of the OPD) above 5 μm is considered in this paper.

A constant window (2:25) was adopted in Ref. [12], therefore it is used for comparison in this paper. Figures 4 and 5 are the simulation results processed with the window (2:25) and the optimized windows by phase slope method and height estimation through the phase algorithm, respectively. The vertical axes (error) represent the difference between the determined height and the true height. Figures 4(a) and 5(a) are the simulation results of a high quality signal with 50 dB SNR. The results show the window (2:25) has nearly optimal performance because the error mainly comes from the ringing of the filter and with the increase of the window width, the ringing declines. Therefore, when the signal has high quality, a relatively wide window should be used to reduce the ringing. Also, the performance of the height estimation through the phase algorithm is much higher than the phase slope method. In Fig. 4(a), errors up to 10 s of nanometers are seen, whereas in Fig. 5(a) the errors are below three nanometers. Figures 4(b) and 5(b) are the simulation results of relatively low-quality signal with 15 dB SNR, the results show the noise has a large impact on the performance so the optimal window width is narrower so that the noise is eliminated. Especially for the phase slope method, when the window is wide, the elimination of the noise is too weak, causing a large error of up to several μm. In contrast, when the window is narrow, the noise is effectively filtered, and the error is still not too large (below 200 nm). The performance of height estimation through a phase algorithm is still much higher than the phase slope method. The improvement brought by the filter design is also noticeable. Both in Figs. 4(b) and 5(b), the optimized window reduces more noise and results in better accuracy. Figures 4(c) and 5(c) are the simulation results of a very low-quality signal with only 5 dB SNR. The results show the noise is much more serious than the filter ringing. When the window is wide, the noise is too severe, causing deviation up to 10 s of μm for both of the algorithms. Meanwhile, when the optimized window is adopted, the noise is effectively eliminated, and the error is much smaller. The measurement system can still achieve submicrometer accuracy for both algorithms. In both Figs. 4(c) and 5(c), the optimized window helps to reduce the errors from up to 10 s of μm to hundreds of nms. The error of the height estimation through a phase algorithm is of the order of half a wavelength mainly due to erroneous determination of the fringe order. The performance of the phase slope method is close to, and sometimes even higher than the height estimation through a phase algorithm. Because the algorithm in Ref. [12] employs a constant relatively wide window, the measurement system works very well when the SNR is high ( $> 20 dB$ ), when the SNR decreases to 20 dB, it works very well for most of the time but has very low possibility to get μm level deviation for some special cases. When the SNR decreases to 10 dB, the algorithm almost does not work because the deviation is too large. In contrast, the simulation shows the proposed approach can still reach submicrometer accuracy even when the SNR drops to 5 dB, which means the proposed method noticeably improves the immunity to the noise and the stability.

Fig. 4. Simulation results of the signal with different SNR processed by phase slope method with different windows, with the window (2:25) on the left and optimized window on the right. (a) 50 dB SNR. (b) 15 dB SNR. (c) 5 dB SNR.

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Fig. 5. Simulation results of the signal with different SNR processed by height estimation through phase algorithm with different windows, with the window (2:25) on the left and optimized window on the right. (a) 50 dB SNR. (b) 15 dB SNR. (c) 5 dB SNR.

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Some experiments have been carried out to verify the simulation results. A step-height standard manufactured by VLSI Standards, Inc. was measured to assess the measurement noise of the system processed by different algorithms. The top surface area without step-height features is used to act as a standard flat. The specimen used an ultra-smooth quartz photomask blank to assure a very flat and smooth working surface and parallelism of the top and bottom surfaces within a few seconds of arc [17]. The measurement noise (denoted as $N_{m}$ ) is estimated with the root mean squared value of the subtraction of two repeated measurements divided by the square root of two. The results are shown in Fig. 6. The results show the optimized filter design can only improve the performance slightly when the signal has high quality. Also, the height estimation through a phase algorithm has much lower noise than the phase slope method. The results agree with the simulation results exactly. Another step height specimen with an array of squares manufactured by focused ion beam (FIB) in the lab adopted for calibration in Ref. [4] has been measured by WSI processed with different algorithms. As a comparison, it is also measured with a Taylor Hobson CCI 3000. The results are shown in Fig. 7. As an example, the reference CCI result shows that the depth of the top-left square is 248 nm (conforming to ISO 5436-1). The result acquired by the phase slope method using the window (2:25) shows the depth of the same square is 6.554 μm, which is incorrect but reproduces the large errors in the simulation for the low-quality signal accurately [Fig. 4(c)]. The result obtained by the phase slope method with the optimized parameters is 255 nm, which agrees with the CCI result very well, and the deviation is within the vertical resolution limit of the WSI. Similar improvement of the results of other squares is obtained as well. The fringe phase-derived results are omitted here because the results are sunk into the ghost steps due to the low-quality signal, which also agrees with the simulation accurately [18].

Fig. 6. Measurement noise (a) of a standard flat, (b) processed by phase slope method with the window (2:25) and optimized window, respectively; (c) and (d) processed by height estimation through phase algorithm with the window (2:25) and optimized window, respectively.

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Fig. 7. Measurement result of a specimen with different methods. (a) The result measured with Taylor Hobson CCI 3000. (b) The result measured with WSI and processed by phase slope method with the window (2:25). (c) The result measured with WSI and processed with the optimized parameters.

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In the practical application, the position of the effective signal is usually the maximum item in the frequency domain excluding the DC component. Therefore, an adaptive algorithm which is able to improve the robustness of the WSI can be achieved based on the preacquired LUT mentioned above once the SNR of the captured signal can be estimated accurately to make sure the optimal result is always achieved. As illustrated in Fig. 8, the signal quality (SNR) is firstly estimated, and then the phase analysis algorithm and filter parameters are both chosen according to the estimated signal quality based on the lookup table acquired by the simulation to obtain the optimal results, such as the one shown in Table 1.

Fig. 8. Adaptive phase analysis algorithm for wavelength scanning interferometry.

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Table 1. Example Lookup Table Obtained by the Simulation

View Table

4. CONCLUSION

This paper focuses on the fringe analysis algorithm for wavelength scanning interferometry (WSI), optimizing the parameters for the filter to acquire the optimal result. The characteristics of the signal are analyzed in both time (wavenumber) and frequency domain. The signal is found to be narrow-band (near single frequency) and the central frequency is calculated theoretically. Therefore, the position of the pass-band of the filter is determined. The width of the filter is optimized with the simulation by compromising between eliminating the noise and the filter ringing. The experimental results agree with the simulation accurately, which means the proposed method effectively enables the WSI to work in a much noisier environment.

To sum up, when the signal quality is high, both the phase slope method and height estimation through the phase algorithm can be adopted to analyze the fringes and the proposed method only improves slightly. Height estimation through the phase algorithm has much higher performance than the phase slope method. However, when the signal has low quality, height estimation through the phase algorithm is seriously troubled by ghost steps. But the phase slope method with the optimized parameters can still work and produce decent results with huge improvement. The proposed method shows the potential of improving the performance by integrating the two algorithms based on the signal quality to acquire the optimal results once the signal SNR can be estimated accurately and is listed as part of the future work. The method can be extended to spectrally resolve white light interferometry (SRWLI) as well because the fringe analysis algorithm for the SRWLI is similar to the WSI [19].

Funding

Engineering and Physical Sciences Research Council (EPSRC) (EP/I033424/1, EP/P006930/1).

REFERENCES

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SNR of the Captured Signal	Preferred Algorithm^a	Half Window Width
$SNR \geq 50$	2	13
$50 > SNR \geq 35$	2	10
$35 > SNR \geq 25$	2	6
$25 > SNR \geq 14$	2	3
$14 > SNR \geq 3$	1	2
$SNR < 3$	1	1

Improvement of the fringe analysis algorithm for wavelength scanning interferometry based on filter parameter optimization

Abstract

1. INTRODUCTION

2. PRINCIPLE

3. SIMULATION AND EXPERIMENTAL RESULTS

4. CONCLUSION

Funding

REFERENCES

Cited By

Figures (8)

Tables (1)

Equations (17)

Applied Optics