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Multi-mode interferometric measurement system based on wavelength modulation and active vibration resistance

Tong Guo, Xinyuan Guo, and Yangyang Wei

Open Access Open Access

Abstract

Optical interferometers are widely used in the measurement of micro- and nanoscale surface topography. However, their accuracy and resolution can be seriously affected by environmental noise. We present a multi-mode interferometric measurement system based on wavelength modulation and active vibration resistance. This supports two measurement modes: wavelength-scanning interferometry, which is suitable for structured surfaces, and wavelength-tuning interferometry, which is suitable for smooth continuous surfaces. In addition, the system can measure the optical path difference of the current position in real time, which is convenient for making system adjustments and improving the measurement accuracy. The proposed system was used to measure 1.806 µm and 43.2 nm step height standards. Under different degrees of vibration, the measured heights in the two modes agreed well with the calibrated values.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

With the rapid development of ultra-precision processing, micro- and nanoscale metrology is becoming increasingly important [1,2]. Optical interferometry has been widely explored for application to surface measurements because it has the advantages of being a non-contact and high-accuracy technique [3]. Various interferometry methods for surface inspection have been developed for different applications, including phase-shift interferometry (PSI), white-light vertical scanning interferometry (VSI), and wavelength-modulation interferometry. Among these, PSI is very suitable for precise measurements owing to its high resolution and accuracy. However, it suffers from the problem of phase ambiguity, which limits its maximum measurable range to quarter of the light wavelength [4]. Although VSI solves the phase-ambiguity problem and extends the measurement range, this method requires time-consuming mechanical scanning processes that make the measurements more sensitive to environmental disturbances [5]. In addition, PSI and VSI both use ceramic piezoelectric transducers (PZT) to achieve phase shift; however, using a mechanical phase shift introduces additional errors, and the measurement speed is limited by the PZTs, which are easily affected by external environmental noise [6]. Wavelength-modulation interferometry [7–10] achieves a phase shift by changing the wavelength of the light source. This leads to the advantages of a fast measurement speed and no mechanical scanning, and it has thus the potential for online and high-precision measurements. Kuwamura and Yamaguchi [11] first proposed the use of diode lasers for wavelength scanning, but this technique has the problems of a narrow wavelength-modulation range and nonlinear wavenumber scanning. Ruiz et al. [12] reported the use of a tunable external cavity diode laser to achieve wavelength-scanning interferometry (WSI), and this allows a wider modulation range without a mode jump. Jiang et al. [13] used a white-light source and acousto-optic tunable filtering (AOTF) to realize wavelength-scanning measurements. Their technique has a large wavelength-modulation range, a fast speed, and a linear wavenumber, making it very suitable for wavelength modulation.

Interferometry is susceptible to environmental noise, such as vibration, air turbulence, and thermal effects. Most interferometric systems can therefore only be used for measurements in a laboratory environment [14]. Over the last few decades, different vibration-compensation techniques have been presented, and these can be divided into active and passive approaches. In terms of interferometric methods with mechanical scanning, passive approaches are often used for vibration compensation. For example, Tereschenko et al. [15,16] reconstructed correct signals using a point-wise distance-measuring interferometer and then used linear interpolation or trigonometric approximation to obtain the actual surface. Liu et al. [17] proposed an iterative method to solve the phase, and this can compensate for the influence of vibration on the phase-shift interferometry. However, active approaches have higher accuracy. Wavelength-modulation interferometry is very suitable for active vibration resistance as it does not involve mechanical scanning. Tereschenko et al. [18] reported an anti-vibration servo-control system that can compensate for the vibration in real time by changing the sweep speed of the AOTF. Muhamedsalih et al. [19] adopted a common optical path design, and used PZTs and hardware proportional–integral controllers to compensate for the influence of vibration in WSI.

In this paper, we describe a multi-mode interferometric measurement system based on wavelength modulation and active vibration resistance, which can simultaneously achieve two measurement modes—wavelength-scanning interferometry (WSI) and wavelength-tuning interferometry (WTI)—in the presence of external vibration. The wavelength-scanning process is achieved by using AOTF. A digital signal processor (DSP) is used to stabilize the interferometer during the measurement process.

2. Measuring system and principle

2.1 Measuring system

As shown in Fig. 1, the multi-mode interferometric measurement system is composed of two light sources, a Linnik interferometer, and a vibration-resistance system. The two light sources are a laser-driven light source (LDLS) and a super-luminescent diode (SLD) with a center wavelength of 830 nm. Wavelength scanning is achieved by changing the driving frequency of the AOTF. The model of AOTF is TF560-280-1-5-NT2 (Gooch & Housego, UK). Its scanning range is 420–700 nm, and the wavelength resolution is smaller than 0.3 nm. The SLD is the light source of the reference interferometer, which is used for active vibration resistance. The light beams from the LDLS and the SLD enter the Linnik interferometer simultaneously, and they are each split into two beams using a hot mirror (FT2). The light from the main interferometer is incident on the digital camera for imaging. The output of the reference interferometer is acquired by the photodetector (PD). The analog signal from the PD is digitized by an analog-to-digital converter and input to an embedded controller (DSP) for processing. The model of the main microcontroller is TMS320F28335 (Texas Instruments, USA), and its output is used to control a piezoelectric ceramic actuator PZT1 (Pst150VS12, CoreMorrow, China), which can compensate for vibration in real time with a response time of 200 µs. PZT2 is a piezoelectric ceramic actuator with capacitive sensor model P-622.ZCD (Physik Instrumente, Germany), which is used for large range scanning. The open loop stroke of PZT2 is 250 µm.

 figure: Fig. 1.

Fig. 1. Schematic diagram of interferometric measurement system based on wavelength modulation. LDLS: laser-driven light source. SLD: super-luminescent diode. BS: beam splitter. COL: collimator. OL: 5 ${\times} $ objective lens. OF: multimode optical fiber. FT1/FT2: hot mirrors. FL: focusing lens. PZT1: piezoelectric ceramic actuator for vibration resistance. PZT2: piezoelectric ceramic actuator for scanning. PD: photodetector.

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2.2 Measuring principle

In this section, the principles of WSI and WTI, the rough acquisition of the measurement position, and active vibration resistance are introduced.

2.2.1 Wavelength-scanning interferometry

In WSI, the digital camera captures multiple images with different wavelengths, and each pixel of the images captured by WSI is independent of every other pixel. By analyzing all of the captured pixels, the surface topography of the sample can be obtained. Figure 2(a) shows an example. The intensity signal at any pixel is expressed as:

$$\begin{aligned} {I_{xy}}({{k_i}} )&= {a_{xy}}({{k_i}} )+ {b_{xy}}({{k_i}} )\cos ({4\pi {k_i}{h_{xy}}} )\\&= {a_{xy}}({{k_i}} )+ \frac{{{b_{xy}}({{k_i}} )}}{2}({{e^{j{\varphi_{xy}}(i )}} + {e^{ - j{\varphi_{xy}}(i )}}} ) \end{aligned},$$
where ${I_{xy}}$ is the intensity value captured by the camera; i is the frame number of the image; x and y are the pixel coordinates in the horizontal and vertical directions of the digital camera, respectively; ${k_i}$ is the wavenumber of the i-th image; ${a_{xy}}({{k_i}} )$ is the background intensity; ${b_{xy}}({{k_i}} )$ is the fringe visibility; ${h_{xy}}$ is the height at a pixel; and ${\varphi _{xy}}(i )= 4\pi {k_i}{h_{xy}}$.

 figure: Fig. 2.

Fig. 2. WSI signal in (a) the time domain, and (b) the frequency domain.

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The Fourier transform amplitude of the intensity signal ${I_{xy}}({{k_i}} )$ is shown in Fig. 2(b), and it can be expressed as:

$$FFT({{I_{xy}}({{k_i}} )} )= A(f )+ B({f - OPD} )+ B({f + OPD} ),$$
where FFT indicates the fast Fourier transform; $A(f )$ is the offset of the signal; the two peaks B correspond to the amplitude of the cosine function; and $\textrm{OPD}$ is the optical path difference of the tested sample surface. This explains why there are three peaks in the FFT. To prevent the three peaks from overlapping, the measurement position of the WSI must deviate from zero optical path difference. In section 2.2.3, we proposed a method to roughly obtain the current measurement position.

We perform the inverse Fourier transform (IFT) of $B({f - \textrm{OPD}} )$:

$$IFT({B({f - OPD} )} )= {b_{xy}}({{k_i}} ){e^{j{\varphi _{xy}}(i )}}/2.$$

Then, we take the logarithm of Eq. (3) and take its imaginary part to obtain the wrapped phase:

$${\varphi _{xy}}(i )= imag\left( {ln \left( {\frac{{{b_{xy}}({{k_i}} )}}{2}{e^{j{\varphi_{xy}}(i )}}} \right)} \right) = 4\pi {k_i}{h_{xy}}.$$

Finally, the phase is unwrapped, and the slope method can be used to obtain the height information:

$${h_{xy}} = \frac{{\Delta \varphi }}{{4\pi \Delta k}}.$$

To improve the measurement accuracy and reduce the influence of noise, the absolute distance can be calculated from the ratio of the absolute phase to the corresponding wavenumber. We finally get $h_{xy}^{\prime}$ using:

$${h_{xy}}^{\prime} = \frac{{{\phi _{xy}}}}{{{k_i}}} = \frac{{{\varphi _{xy}} + 2m\pi }}{{{k_i}}} = \frac{{{\varphi _{xy}} - 2\pi int\left( {\frac{{{\phi_{xy}} - 2{h_{xy}}{k_i}}}{{2\pi }}} \right)}}{{{k_i}}},$$
where ${\phi _{xy}}$ and ${\varphi _{xy}}$ are the absolute phase and wrapped phase of wavelength $\lambda $, respectively; ${h_{xy}}$ is the height value calculated by the slope method; and m is the interference order.

2.2.2 Wavelength-tuning interferometry

WTI is a technique that achieves a phase shift by changing the wavelength. In WTI, the phase-shifted interference signal of the t-th step can be expressed as:

$${I_t}({x,y} )\; = {a_\textrm{t}}({x,y} )+ {b_\textrm{t}}({x,y} )\cos \left( {\frac{{4\pi h({x,y} )}}{{{\lambda_t}}}} \right),$$
where ${I_t}({x,y} )$ is the intensity value captured by digital camera; x and y are the pixel coordinates in the horizontal and vertical directions of the camera, respectively; ${\lambda _t}$ is the wavelength of the t-th phase shift; ${a_t}({x,y} )$ is the background intensity; ${b_t}({x,y} )$ is the fringe visibility; and ${h_{xy}}$ is the height.

Assuming that the initial output wavelength of the AOTF is ${\lambda _0}$ and the wavelength step is $\mathrm{\Delta }\lambda $, the wavelength value after the phase shift at the t-th step can be expressed as:

$$\phi _t\left( {x,y} \right) = \displaystyle{{4\pi h\left( {x,y} \right)} \over {\lambda _t}} = \displaystyle{{4\pi h\left( {x,y} \right)} \over {\lambda _0 + t\Delta \lambda }}.$$

Equation (8) can be approximated as:

$${\phi _t}({x,y} )\approx \frac{{4\pi h({x,y} )}}{{{\lambda _0}}} - \frac{{4\pi h({x,y} )}}{{{\lambda _0}^2}}t\Delta \lambda = {\phi _0}({x,y} )+ {\delta _t},$$
where ${\phi _0}({x,y} )$ is the initial phase and ${\delta _t}$ is the value of the phase shift. Taking Eq. (9) into Eq. (7), we obtain:
$${\textrm{I}_{\textrm{xy}}}({{\textrm{k}_\textrm{t}}} )= {\textrm{a}_{\textrm{xy}}}({{\textrm{k}_\textrm{t}}} )+ {\textrm{b}_{\textrm{xy}}}({{\textrm{k}_\textrm{t}}} )\cos ({{\varphi_0}({\textrm{x,y}} )}+ {\mathrm{\delta}_\textrm{t}} ),$$
where ${k_t}$ is the wavenumber of the t-th phase shift. From Eq. (9), it can be easily obtained that the t-th phase shift is determined by the height information $h({x,\; y} )$ and the wavelength step. However, in the actual measurement process, $h({x,y} )$ is unknown and different heights $h({x,y} )$ correspond to different phase-shift values ${\delta _t}$. Therefore, the phase-shift error is obvious. To reduce the influence of the phase-shift error, a fixed four-step iterative algorithm based on least squares is used for phase extraction. The precise phase is not required in this method. The true phase shift and the measured phase can be found through multiple iterations.

Expanding the cosine function in Eq. (10), we can get:

$${I_{xy}}({{k_t}} )= {a_{xy}}({{k_t}} )+ {b_{xy}}({{k_t}} )cos{\phi _0}({x,y} )cos{\delta _t} - {b_{xy}}({{k_t}} )sin{\phi _0}({x,y} )sin{\delta _t}.$$

We take the minimum sum of squared errors between the theoretical value of interference light intensity ${I_{xy}}^{\prime}({{k_t}} )$ and the measured value ${I_{xy}}({{k_t}} )$ as the iteration criterion:

$$E = \mathop \sum \limits_t^M \mathop \sum \limits_{x = 1}^m \mathop \sum \limits_{y = 1}^n {[{{I_{xy}}^{\prime}({{k_t}} )- {I_{xy}}({{k_t}} )} ]^2} = erro{r_{min}},$$
where M is the number of iterations; m and n are the total numbers of pixels in the x and y directions, respectively; and $erro{r_{min}}$ is the minimum error.

If the phase shift ${\delta _t}$ is known at the beginning of the iteration, Eq. (10) can be simplified as an equation related only to ${\delta _t}$:

$${I_{xy}}({{k_t}} )\; = {a_{xy}}({{k_t}} )+ {B_{xy}}({{k_t}} )cos {\delta _t} + {C_{xy}}({{k_t}} )sin\; {\delta _t},$$
where ${B_{xy}}({{k_t}} )= {b_{xy}}({{k_t}} )\textrm{cos}{\phi _0}({x,y} )$ and ${C_{xy}}({{k_t}} )={-} {b_{xy}}({{k_t}} )\textrm{sin}{\phi _0}({x,y} )$ can be calculated by the least-squares algorithm. The initial phase can be expressed as:
$${\phi _0}({x,y} )= arctan \frac{{ - {C_{xy}}({{k_t}} )}}{{{B_{xy}}({{k_t}} )}}.$$

Taking the calculated ${\phi _0}({x,y} )$ as a known value, the phase shift can be obtained by the same method. We convert Eq. (10) into a form related only to ${\phi _0}({x,y} )$:

$${I_{xy}}({{k_t}} )\; = {a_{xy}}({{k_t}} )+ {B_{xy}}^{\prime}({{k_t}} )cos {\phi _0}({x,y} )+ {C_{xy}}^{\prime}({{k_t}} )sin{\phi _0}({x,y} ),$$
where ${B_{xy}}^{\prime}({{k_t}} )= {b_{xy}}({{k_t}} )\textrm{cos}{\delta _t}$ and ${C_{xy}}^{\prime}({{k_t}} )={-} {b_{xy}}({{k_t}} )\textrm{sin}{\delta _t}$ can be calculated by the least-squares algorithm. The phase shift can be expressed as:
$${\delta _t} = arctan \frac{{ - {C_{xy}}^{\prime}({{k_t}} )}}{{{B_{xy}}^{\prime}({{k_t}} )}}.$$

Finally, the initial phase ${\phi _0}\; ({x,\; y} )$ and the phase shift ${\delta _t}$ can be solved through iteration.

A set of initial phase shifts is required for the iterative algorithm. An accurate initial value for the phase shift can reduce the time required for iteration and improve accuracy. However, the amount of phase shift is related to the current measurement position of the sample. In actual tests, it is difficult to obtain the current position. Therefore, a real-time method to roughly obtain the current measurement position is proposed here.

2.2.3 Rough acquisition of measurement position

The current measurement position can be calculated from the interference signal of the SLD from the reference interferometer. If the spectrum of the SLD light source is Gaussian and the effect of the numerical aperture of the objective lens is ignored, the interference signal can be expressed as:

$$I(z )= exp\left[ { - 2{\pi^2}{{\left( {\frac{z}{{{L_c}}}} \right)}^2}} \right]cos\left( {\frac{{4\pi z}}{{{\lambda_0}}} + \psi } \right),$$
where z is the current measurement position; ${L_c}$ is the coherence length of the light source; $\psi $ is the initial phase; and ${\lambda _0}$ is the center wavelength of the light source. The upper envelope of the interference signal can be approximately extracted from the maximum point in the signal. The upper envelope signal can be obtained by fitting the normalized data by the Gaussian function ${I_{\textrm{env}}}(z )$:
$${I_{env}}(z )= exp\left[ { - 2{\pi^2}{{\left( {\frac{z}{{{L_c}}}} \right)}^2}} \right].$$

The current position of the sample can then be estimated based on the maximum point output by the photodetector.

The absolute intensity ${I_m}$ is affected by the power of the SLD light source and the change of tilt angle. As shown in Fig. 3(a), envelopes 1, 2, and 4 have different light-source power levels, while envelopes 2 and 3 have different tilt angles and the same light-source power level. The power of the light source and the tilt angle will only change the amplitude of the output from the detector. The shape of the upper envelope is only related to the scanning position and the shape of the light source. The envelope is essentially unaffected by the light-source power and tilt angle after normalization, as shown in Fig. 3(b).

 figure: Fig. 3.

Fig. 3. Extracted envelopes with different light-source powers and sample tilt angles: (b) before normalization, and (b) after normalization.

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2.2.4 Principle of vibration compensation

In the system presented in this paper, vibration resistance is achieved by using a reference interferometer. The main interferometer shares its optical path with the reference interferometer, so both are affected by the same environmental vibrations. The specific control principle is shown in Fig. 4. The output of the reference interferometer is collected by the photodetector and sent to the DSP controller. To meet the nanometer-level precision control requirements, the DSP controller includes a 16-bit analog-to-digital converter (AD7606, Analog Devices, USA), a TMS320F28335 microcontroller, and a 16-bit digital-to-analog converter (DAC8552, Texas Instruments, USA).

 figure: Fig. 4.

Fig. 4. Schematic diagram of closed-loop control system.

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To realize real-time compensation for the influence of external vibration on the interferometer, a proportional–integral algorithm is applied in the DSP controller to process the signal data, and the processing result is sent to the PZT1 actuator.

3. Experimental analysis

3.1 Test of rough acquisition of measurement position

Firstly, the SLD envelope is calibrated. This process only needs to be done once, and in this process, PZT2 is used for large-range (about 40 µm) scanning. After normalizing the intensity of the output signal of the photodetector, the interference signal and its upper envelope are as shown in Fig. 5(a), and then a Gaussian function is used to fit the upper envelope of the signal. The fitting result is shown in Fig. 5(b). The correlation coefficient of this fit is 0.9994.

 figure: Fig. 5.

Fig. 5. Process of obtaining the rough height value: (a) SLD interference signal and its upper envelope, and (b) comparison before and after upper envelope fitting.

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Secondly, the maximum value of the current tilt angle and the SLD light source power need to be obtained. Again, PZT2 is applied for large-range scanning, which is completed in about 3 s. The DSP controller collects and processes data and sends the maximum value to the PC. This process only needs to be carried out once when changing the power of the light source or changing the tilt angle of the sample.

Finally, PZT1 is applied for small-range scanning to acquire the maximum value of the interference signal at the current position. The scanning range is approximately equal to half of the center wavelength of the SLD. The maximum value is sent to the PC, which calculates and displays the optical path difference of the current position in real time. The coherence length of the SLD light source used is about 40 µm. Therefore, the measuring range is about ±20 µm.

To verify its accuracy, the position calculated by this method was compared with the position calculated by WSI at the same position, and the results are shown in Table 1. The test deviation of the two methods was less than 1 µm, which meets the test requirements.

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Table 1. Comparison of rough acquisition method and WSI calculation position.

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To verify the repeatability and applicability of the method, interference envelopes collected at different times, different tilt angles, different light-source powers, and different starting scanning positions were compared. Figure 6 shows 12 sets of envelopes. Within the range of ±20 µm in the measurement position, the difference in the envelope position is less than 1 µm.

 figure: Fig. 6.

Fig. 6. Interference envelopes of SLD extracted under different conditions.

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3.2 Single-frequency vibration-resistance accuracy test

To verify the performance of the vibration-resistance system, a signal generator was used to output a sinusoidal signal to the PZT stage to generate single-frequency sinusoidal vibration. The PZT stage is an S-303 compact piezoelectric phase shifter (Physik Instrumente, Germany). Its resonance frequency is 25 kHz, its driving voltage range is 0–10 V, and its open-loop displacement stroke is about 3 µm.

By recording the change in the output voltage of the photodetector before and after the vibration resistance is switched on, the control accuracy can be calculated, as shown in Eq. (19):

$$\frac{{{h_{error}}}}{{\Delta V}} = \frac{{{h_{pv}}}}{{{V_{pv}}}} = \frac{{\lambda \pi }}{{4\pi {V_{pv}}}} = \frac{\lambda }{{4{V_{pv}}}},$$
where ${h_{\textrm{error}}}$ is the error between the actual height and the target height after stabilization; $\mathrm{\Delta }V$ is the output voltage change of the stable photodetector; ${h_{pv}}$ and ${V_{pv}}$ respectively represent the height and voltage change when the phase change is π; and $\lambda $ is the center wavelength of the SLD light source.

A 16-bit digitizer (USB-6346, National Instruments, USA) was applied for data acquisition. The sampling frequency was 2 kHz, and the sampling duration was 3 s. Figure 7 shows a comparison of the output from the photodetector before and after vibration resistance was turned on in the presence of vibration of amplitude 200 nm and frequency 5 Hz. At 1.795 s, the vibration-resistance system was turned on. After about 3 ms, the system becomes stable, and the voltage changes from 870.1 mV to 22.9 mV. The peak–valley (PV) value calculated using Eq. (19) changes by about 5.3 nm after stabilization.

 figure: Fig. 7.

Fig. 7. Output of the optical detector before and after the system is stabilized when a single-frequency sinusoidal vibration is applied: (a) overall output, and (b) zoomed-in section.

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The control accuracy was tested with a vibration frequency of 5 Hz at different amplitudes (20 nm–800 nm) and when the amplitude was 100 nm at different vibration frequencies (1 Hz–40 Hz). As shown in Fig. 8, the residual error value after stabilization increases with the increase of vibration frequency or amplitude.

 figure: Fig. 8.

Fig. 8. Residual change after the engagement of the vibration-resistance system when single-frequency sinusoidal vibration is applied: (a) constant frequency and varying amplitude, and (b) constant amplitude and varying frequency.

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When the vibration resistance is turned on, the system can be stabilized after about 3 ms. As the vibration amplitude and frequency are increased, the residual error increases. With a vibration frequency of 5 Hz and an amplitude of 400 nm, the residual after stabilization is only 8.5 nm. The maximum stable vibration amplitude and frequency of the system are mainly affected by the maximum stroke and response time of the PZT1.

3.3 Evaluation of vibration resistance for WSI

To test the performance of the WSI system, the S-303 was used to generate three sets of sinusoidal vibration with different amplitudes and frequencies (200 nm/1 Hz, 200 nm/5 Hz, and 400 nm/1 Hz). The sample to be tested was a 1.8 µm standard step (calibrated height value 1.806 ± 0.011 µm) manufactured by VLSI, and the plane surface and step height were measured. The wavelength-scanning range was 639.07 nm–549.13 nm, and a total of 200 frames of images were collected.

3.3.1 Plane-surface measurement

The PV value and root mean square height (Sq) are used as evaluation indicators, and their values are listed in Table 2. Before vibration resistance, the results are all distorted, while after vibration resistance, the measurement results are significantly improved.

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Table 2. Plane-surface measurement results under different vibrations.

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Figures 9 and 10 show the plane-surface measurement results and interference signal from a certain point before and after vibration resistance was switched on, respectively, in the same area when vibration with the amplitude of 200 nm and the frequency of 5 Hz was applied.

 figure: Fig. 9.

Fig. 9. Measurement results of plane surface before vibration resistance was turned on: (a) three-dimensional view of the plane, and (b) interference signal from a certain point.

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 figure: Fig. 10.

Fig. 10. Measurement results of plane after vibration resistance was turned on: (a) three-dimensional view of the plane; (b) interference signal from a certain point.

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When the sinusoidal vibration is applied, the effective information of the signal is overwhelmed by disturbance, and WSI cannot extract the correct surface topography. When the vibration resistance is turned on, the disturbance can be removed effectively.

3.3.2 Step-height measurement

The height of the step and its standard deviation (Std) are used as evaluation indicators [20], and their values are listed in Table 3. Before vibration resistance, the calculation results are all distorted, while after vibration resistance, the measured heights are within the range of the calibration value.

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Table 3. Step measurement results under different vibrations.

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Figures 11 and 12 show the results of step-height measurements before and after vibration resistance was switched on, respectively, when single-frequency vibration with the amplitude of 200 nm and the frequency of 5 Hz was applied.

 figure: Fig. 11.

Fig. 11. Step sample measured by WSI before vibration resistance was turned on: (a) three-dimensional view of the step, and (b) profile of a row of pixels.

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 figure: Fig. 12.

Fig. 12. Step sample measured by WSI after vibration resistance was turned on: (a) three-dimensional view of the step, and (b) profile of a row of pixels.

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3.4 Evaluation of vibration resistance for WTI

To test the performance of the wavelength-tuning interference system, the S-303 was used to generate three sets of sinusoidal interference signals with different amplitudes and frequencies (200 nm/1 Hz, 200 nm/5 Hz, and 400 nm/1 Hz). The sample to be tested was a 44 nm standard step (calibrated height value 43.2 ± 0.6 nm) manufactured by VLSI, and the plane surface and step height were measured. The initial wavelength was 632.72 nm, the initial position was about 10 µm, and the wavelength step was 5 nm. Four interferograms were collected in total.

3.4.1 Evaluation of vibration resistance for WTI

Figure 13 shows the results of plane-surface measurements before and after vibration resistance was turned on in the same area when single-frequency vibration is applied. The PV value and root mean square height (Sq) are regarded as evaluation indicators, and their values are listed in Table 4.

 figure: Fig. 13.

Fig. 13. Plane-surface measurement results under different vibrations: (a)-(c) before vibration resistance was turned on, and (d)-(f) after vibration resistance was turned on. (a) and (d) show 200 nm/1 Hz vibration, (b) and (e) show 200 nm/5 Hz vibration, (c) and (f) show 400 nm/1 Hz vibration.

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Table 4. Plane-surface measurement results under different vibrations.

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Table 4 shows that in a vibrating environment, the measurement results of the WTI are not distorted. This is because the iterative algorithm is not sensitive to phase-shift errors, but it still reduces the measurement accuracy. However, the measurement accuracy is improved by vibration resistance.

3.4.2 Step-height measurement

Figures 14 and 15 show the results of step-height measurement before and after vibration resistance was turned on, respectively, when single-frequency vibration is applied. The height of the step and its standard deviation are used as evaluation indicators.

 figure: Fig. 14.

Fig. 14. Step sample measured by WTI before vibration resistance was turned on: (a) three-dimensional view of the step, and (b) profile of a row of pixels.

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 figure: Fig. 15.

Fig. 15. Step sample measured by WTI after vibration resistance was turned on: (a) three-dimensional view of the step, and (b) profile of a row of pixels.

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Table 5 shows that without vibration resistance, the measured average step height was 43.97 nm, and its standard deviation was 1.81 nm. After vibration resistance was turned on, the measured average step height was 43.44 nm, and its standard deviation was 1.27 nm. It is clear that the surfaces of the steps appear smoother after turning on the vibration resistance.

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Table 5. Step measurement results under different vibrations.

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3.4.3 Performance comparison

To evaluate the performance of the system, the WTI with active vibration resistance was compared to a PSI placed on the active vibration-isolation table. The tested sample was the 44 nm standard step, and the PSI was solved by the five-step phase-shift algorithm. Figure 16 shows the measurement results from the two methods.

 figure: Fig. 16.

Fig. 16. Results of the two interferometric measurement methods for (a)/(b) a plane surface and (c)/(d) a step. (a) and (c) show the WTI results, and (b) and (d) show the PSI results.

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Both WTI and PSI can accurately restore the surface topography of the sample. The accuracy of WTI is slightly lower than that of PSI. However, PSI requires an accurate phase shift and is very sensitive to external interference (such as environmental vibration, mechanical errors, etc.), while WTI is insensitive to phase-shift errors. Therefore, WTI has lower stability requirements for experiments and a wider range of applications.

4. Conclusion

A multi-mode interferometric measurement system based on wavelength modulation and active vibration resistance has been introduced in this paper. It supports two measurement modes: WSI and WTI, and these modes are suitable for examining structured and smooth surfaces without mechanical scanning. In this system, AOTF is applied to achieve wavelength modulation, and a common optical-path design is adopted with a feedback control system to actively suppress the influence of external environmental vibrations. The control performance is mainly affected by the PZT response speed and its maximum stroke.

In addition, a method for rough acquisition of the measurement position of the system is proposed here. The current measurement location information can be obtained through this method in real time, which facilitates the adjustment of wavelength scanning and the calculation of the phase-shift value of the wavelength tuning.

In order to verify the effect of the vibration resistance, wavelength modulation experiment was carried out with step heights and plane surfaces as the tested samples, and vibrations of different frequencies and different amplitudes were added. The experimental results show that compared to the vibration resistance turned off, the WSI signal can be greatly improved, and the surface topography of the tested samples can be restored accurately; under varying degrees of vibration, the system can restore the surface topography of the step height with an accuracy of nanometers by WTI, which is similar with the result of PSI placed on the active vibration-isolation table. There is no mechanical scanning in the system, its scanning speed is fast, and it can use active vibration resistance. This means this system can meet a variety of test requirements in industrial applications.

Funding

National Key Research and Development Program of China (2018YFB1107600); 111 Project (B0714).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (16)
Fig. 1.
Fig. 1. Schematic diagram of interferometric measurement system based on wavelength modulation. LDLS: laser-driven light source. SLD: super-luminescent diode. BS: beam splitter. COL: collimator. OL: 5 ${\times} $ objective lens. OF: multimode optical fiber. FT1/FT2: hot mirrors. FL: focusing lens. PZT1: piezoelectric ceramic actuator for vibration resistance. PZT2: piezoelectric ceramic actuator for scanning. PD: photodetector.

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Fig. 2.
Fig. 2. WSI signal in (a) the time domain, and (b) the frequency domain.

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Fig. 3.
Fig. 3. Extracted envelopes with different light-source powers and sample tilt angles: (b) before normalization, and (b) after normalization.

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Fig. 4.
Fig. 4. Schematic diagram of closed-loop control system.

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Fig. 5.
Fig. 5. Process of obtaining the rough height value: (a) SLD interference signal and its upper envelope, and (b) comparison before and after upper envelope fitting.

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Fig. 6.
Fig. 6. Interference envelopes of SLD extracted under different conditions.

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Fig. 7.
Fig. 7. Output of the optical detector before and after the system is stabilized when a single-frequency sinusoidal vibration is applied: (a) overall output, and (b) zoomed-in section.

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Fig. 8.
Fig. 8. Residual change after the engagement of the vibration-resistance system when single-frequency sinusoidal vibration is applied: (a) constant frequency and varying amplitude, and (b) constant amplitude and varying frequency.

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Fig. 9.
Fig. 9. Measurement results of plane surface before vibration resistance was turned on: (a) three-dimensional view of the plane, and (b) interference signal from a certain point.

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Fig. 10.
Fig. 10. Measurement results of plane after vibration resistance was turned on: (a) three-dimensional view of the plane; (b) interference signal from a certain point.

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Fig. 11.
Fig. 11. Step sample measured by WSI before vibration resistance was turned on: (a) three-dimensional view of the step, and (b) profile of a row of pixels.

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Fig. 12.
Fig. 12. Step sample measured by WSI after vibration resistance was turned on: (a) three-dimensional view of the step, and (b) profile of a row of pixels.

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Fig. 13.
Fig. 13. Plane-surface measurement results under different vibrations: (a)-(c) before vibration resistance was turned on, and (d)-(f) after vibration resistance was turned on. (a) and (d) show 200 nm/1 Hz vibration, (b) and (e) show 200 nm/5 Hz vibration, (c) and (f) show 400 nm/1 Hz vibration.

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Fig. 14.
Fig. 14. Step sample measured by WTI before vibration resistance was turned on: (a) three-dimensional view of the step, and (b) profile of a row of pixels.

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Fig. 15.
Fig. 15. Step sample measured by WTI after vibration resistance was turned on: (a) three-dimensional view of the step, and (b) profile of a row of pixels.

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Fig. 16.
Fig. 16. Results of the two interferometric measurement methods for (a)/(b) a plane surface and (c)/(d) a step. (a) and (c) show the WTI results, and (b) and (d) show the PSI results.

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Tables (5)

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Table 1. Comparison of rough acquisition method and WSI calculation position.

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Table 2. Plane-surface measurement results under different vibrations.

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Table 3. Step measurement results under different vibrations.

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Table 4. Plane-surface measurement results under different vibrations.

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Table 5. Step measurement results under different vibrations.

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Equations (19)

Equations on this page are rendered with MathJax. Learn more.

I x y ( k i ) = a x y ( k i ) + b x y ( k i ) cos ⁡ ( 4 π k i h x y ) = a x y ( k i ) + b x y ( k i ) 2 ( e j φ x y ( i ) + e − j φ x y ( i ) ) ,
F F T ( I x y ( k i ) ) = A ( f ) + B ( f − O P D ) + B ( f + O P D ) ,
I F T ( B ( f − O P D ) ) = b x y ( k i ) e j φ x y ( i ) / 2.
φ x y ( i ) = i m a g ( l n ( b x y ( k i ) 2 e j φ x y ( i ) ) ) = 4 π k i h x y .
h x y = Δ φ 4 π Δ k .
h x y ′ = ϕ x y k i = φ x y + 2 m π k i = φ x y − 2 π i n t ( ϕ x y − 2 h x y k i 2 π ) k i ,
I t ( x , y ) = a t ( x , y ) + b t ( x , y ) cos ⁡ ( 4 π h ( x , y ) λ t ) ,
ϕ t ( x , y ) = 4 π h ( x , y ) λ t = 4 π h ( x , y ) λ 0 + t Δ λ .
ϕ t ( x , y ) ≈ 4 π h ( x , y ) λ 0 − 4 π h ( x , y ) λ 0 2 t Δ λ = ϕ 0 ( x , y ) + δ t ,
I xy ( k t ) = a xy ( k t ) + b xy ( k t ) cos ⁡ ( φ 0 ( x,y ) + δ t ) ,
I x y ( k t ) = a x y ( k t ) + b x y ( k t ) c o s ϕ 0 ( x , y ) c o s δ t − b x y ( k t ) s i n ϕ 0 ( x , y ) s i n δ t .
E = ∑ t M ⁡ ∑ x = 1 m ⁡ ∑ y = 1 n ⁡ [ I x y ′ ( k t ) − I x y ( k t ) ] 2 = e r r o r m i n ,
I x y ( k t ) = a x y ( k t ) + B x y ( k t ) c o s δ t + C x y ( k t ) s i n δ t ,
ϕ 0 ( x , y ) = a r c t a n − C x y ( k t ) B x y ( k t ) .
I x y ( k t ) = a x y ( k t ) + B x y ′ ( k t ) c o s ϕ 0 ( x , y ) + C x y ′ ( k t ) s i n ϕ 0 ( x , y ) ,
δ t = a r c t a n − C x y ′ ( k t ) B x y ′ ( k t ) .
I ( z ) = e x p [ − 2 π 2 ( z L c ) 2 ] c o s ( 4 π z λ 0 + ψ ) ,
I e n v ( z ) = e x p [ − 2 π 2 ( z L c ) 2 ] .
h e r r o r Δ V = h p v V p v = λ π 4 π V p v = λ 4 V p v ,
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