## Abstract

A snapshot multi-wavelength interference microscope is proposed for high-speed measurement of large vertical range discontinuous microstructures and surface roughness. A polarization CMOS camera with a linear micro-polarizer array and Bayer filter accomplishes snapshot multi-wavelength phase-shifting measurement. Four interferograms with 𝜋/2 phase shift are captured at each wavelength for phase measurement, the 2𝜋 ambiguities are removed by using two or three wavelengths.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Interference microscopes are commonly used to measure the surface shape and roughness of continuous and discontinuous surfaces [1]. One of the limitations is that the phase difference between two adjacent points must be less than 𝜋, limiting the measurement range. To overcome this limitation, vertical scanning is typically used [1–5]. Multi-wavelength interference microscope is the other approach [6–8]. However both methods have the same issue of slow speed and are not suitable for fast dynamic measurement as the measurement for each wavelength is obtained sequentially. To increase the measurement speed, several multi-wavelength interferometry have been developed. A phase shifting multi-wavelength dynamic interferometer with polarization masked camera was developed to obtain multi-wavelength data sequentially by a high speed polarization camera [9]. Parallel interferometry with multiple phase masked CCD arrays [10, 11] is another solution. Parallel interferometry approach can obtain fast measurement without sacrificing lateral resolution, but it increases the alignment difficulty and cost since multiply CCDs are used. Frequency-domain methods, such as digital holography microscopy [12, 13] and off-axis interferometry [14, 15], have also been developed, but they lack of the spatial information. A polarization masked camera has been developed to achieve a snapshot phase shift measurement in both interferometry [9, 16] and interference microscopy [17], but multi-wavelength measurement can only be achieved sequentially due to the lack of a color sensitive polarization camera.

In this paper, we propose a snapshot multi-wavelength interference microscope to achieve high-speed measurement of large vertical range discontinuous structure and surface roughness, using a custom CMOS camera with linear micro-polarizer array and Bayer filters [18]. Due to the combination of micro-polarizer array and Bayer filters, in each superpixel there are 4x4 pixels which have different polarization and color information, the spatial resolution is reduced but the imaging area is the same. This issue can be partially addressed by using the sensor with more and small pixels. Image interpolation can partially improve the system performance as well. Phase-shifted interferograms of different color channels are captured in a single shot. This system has the advantages of high speed and large vertical measurement range, particularly suitable for high dynamic measurement environment. In addition, our system can measure the surface roughness in a single shot.

## 2. System layout

The experimental setup is shown in Fig. 1. Three light emitting diodes (LEDs) in red (R), green (G) and blue (B) color channels provide multi-wavelength illumination; the central wavelengths of the bandpass filters in front of the RGB LEDs are 460 nm, 540 nm and 630 nm with a bandwidth of 10 nm. Light from LEDs is collimated by lens L1 and then goes through a polarizer (P) with the transmission axis at 45 degrees to optical axis. Linear polarized light is separated into *p* and *s* polarized light after passing though the polarized beam splitter (PBS). Two microscopy objectives (MO1 and MO2) with 0.28 numerical aperture (NA) are used in the Linnik interferometer configuration. Both beams pass through the achromatic quarter wave plate (QWP1 or QWP2) with fast axis at 45 degrees to the horizontal axis twice before returning to the PBS. QWP3 with its fast axis at 45 degrees is placed in front of the camera and transforms the two orthogonal linear polarized beams to opposite circular polarized beams. Lens L2 images the test surface onto the detector. A custom color polarization camera (PolarCam) with a wire-grid micro-polarizer array on the traditional RGB Bayer filter array is utilized to record phase-shifted interferograms in the RGB channels in a single shot.

## 3. Operating principle

#### 3.1 Micro-polarizer array based simultaneous phase shifting interferometer

The micro-polarizer array of the PolarCam in front of the CMOS focal plane contains four different linear polarizers at 0°, 45°, 90° and 135° in each superpixel. The Jones vectors of test and reference arms before PolarCam are

where ${E}_{i}=\frac{1}{\sqrt{2}}\left[\begin{array}{c}1\\ 1\end{array}\right]$ is the incident Jones vector after passing through the linear polarizer at 45 degrees.*Q*,

_{3}*Q*, and

_{2}*Q*are Jones matrix of QWP3, QWP2 and QWP1,

_{1}*Q*and

_{2}'*Q*are the Jones matrix of reversed QWP2 and QWP1, defined as $Q=\left[\begin{array}{cc}co{s}^{2}\theta +isi{n}^{2}\theta & (1-i)cos\theta sin\theta \\ (1-i)\mathrm{cos}\theta sin\theta & si{n}^{2}\theta +ico{s}^{2}\theta \end{array}\right]$where 𝜃 is the fast axis angle to the horizontal axis.

_{1}'*T*and

_{pbs}*R*are transmittance and reflectance Jones matrix of PBS, defined as ${T}_{pbs}=\left[\begin{array}{cc}{T}_{p}& 0\\ 0& {T}_{s}\end{array}\right]$, ${R}_{pbs}=\left[\begin{array}{cc}{R}_{p}& 0\\ 0& {R}_{s}\end{array}\right]$,

_{pbs}*T*and

_{p}*T*are the transmittance of the PBS for

_{s}*p*and

*s*polarized light,

*R*and

_{p}*R*are the reflectance for

_{s}*p*and

*s*polarized light. For ideal PBS,

*T*=

_{p}*R*= 1 and

_{s}*T*=

_{s}*R*= 0.

_{p}*M*and

_{T}*M*are Jones matrices of optical surface under test and the reference mirror, ${M}_{T}={M}_{R}=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$. For ideal components, test and reference beams after QWP3 should be opposite circular polarized. After passing through the micro-polarizer array, the Jones vector is ${E}_{i}={P}_{i}*({E}_{T}+{E}_{R})$, where $P=\left[\begin{array}{cc}{\mathrm{cos}}^{2}\theta & \mathrm{sin}\theta \mathrm{cos}\theta \\ \mathrm{sin}\theta \mathrm{cos}\theta & {\mathrm{sin}}^{2}\theta \end{array}\right]$ is the Jones matrix for linear polarizer with the transmission axis at 𝜃.

_{R}After the light passes through the linear polarizer array, the two incident orthogonally circular test and reference beams interfere at the focal plane. The irradiance at each pixel is

where*I*is the total irradiance behind a linear polarizer with transmission axis at 𝜃,

*I*and

_{T}*I*are the irradiance of the test and reference arms. For the linear polarizers in each superpixel at 0°, 45°, 90° and 135°, phase shifts of 0°, 90°, 180° and 270° between test and reference beams are introduced. A basic four step phase shift algorithm can be used to calculate the surface phase information:

_{R}In the experimental setup, the retardance of the achromatic QWP is not exactly 𝜋/4. As shown in Fig. 2(a), the retardance are 0.2527, 0.2455 and 0.2460 for B (460 nm), G (540 nm) and R (630 nm), respectively provided by the manufacturer Bolder Vision Optik, Inc. Figure 2(b) shows the phase errors caused by the QWP retardance error for the working wavelengths, they are very small and will be ignored in this study.

In Eq. (4), phase calculation is made with the assumption that the phase is constant over the region covered by a 4 x 4 super pixel as shown in Fig. 3. This assumption introduces a phase related error which was studied previously using a monochrome PolarCam [19]. In a monochrome PolarCam, one super pixel contains 2 x 2 pixels with linear polarization state in 0°, 45°, 90° and 135°. Instead of using a 2 x 2 grid, the phase is calculated by a weighted average in a 3 x 3 grid. In the current RGB PolarCam configuration, the phase is calculated by a weighted average in a 6 x 6 grid, as shown in Fig. 4. The irradiance *I _{0°}*,

*I*,

_{45°}*I*and

_{90°}*I*of pixel (i, j) in Eq. (4) are replaced by:

_{135°}The weighted average is a linear combination of neighboring pixels, which has the same format of linear interpolation of *I _{0°}*,

*I*,

_{45°}*I*and

_{90°}*I*polarization state in pixel (i, j). Thus a linear-spline interpolation can also be used to recover the missing intensity of each pixel as $I={\left[\begin{array}{cccccccccccc}{I}_{R0}& {I}_{R1}& {I}_{R2}& {I}_{R3}& {I}_{G0}& {I}_{G1}& {I}_{G2}& {I}_{G3}& {I}_{B0}& {I}_{B1}& {I}_{B2}& {I}_{B3}\end{array}\right]}^{T}$, where R, G and B are the input light at red, green and blue channels, 0, 1, 2 and 3 are corresponding to 0°, 45°, 90° and 135° linear polarization states. Probably due to a small misalignment between the wire-grid micro-polarizer array and the RGB Bayer filter array, one G channel has larger polarization crosstalk error than the other in each sub-super pixel [18]. Thus only half of the G channel is used in the interpolation. The surface phase information in RGB channel can be calculated as:

_{135°}#### 3.2 Multi-wavelength metrology

One of the limitations in single wavelength interference microscopy is that the optical path difference (OPD) between two adjacent measured points must be less than 𝜆/2. If the OPD is larger than 𝜆/2, a 2𝜋 ambiguity is introduced. In order to get an ambiguity-free phase measurement, multiple wavelengths are often used. Using the phase values at multiple wavelengths, the 2𝜋 ambiguity can be avoided.

Once two phases ${\varphi}_{1}$ and ${\varphi}_{2}$ are obtained for two wavelengths 𝜆_{1} and 𝜆_{2}, the measurable step height *H* can be expanded from 𝜆_{1}/4 to 𝛬_{12}/4 where ${\Lambda}_{12}={\lambda}_{1}{\lambda}_{2}/\left|{\lambda}_{1}-{\lambda}_{2}\right|$is the equivalent wavelength. The equivalent phase $\Phi ={\varphi}_{1}-{\varphi}_{2}$is related to the step height as [20]

_{12}/𝜆

_{1}in this method. To avoid this induced phase error, the fringe order method is used [21]. Using the step height measured by the equivalent wavelength, the fringe order

*N*of 𝜆

_{1}_{1}is determined. Then the step height

*H*is calculated: where

_{1}*round*is the nearest integer. By using the fringe order method, a high precision and long measurement range can be achieved. In order to get correct fringe order

*N*, the phase noise in 𝜙

_{1}_{1}must be smaller than 𝜆

_{1}/(4𝛬

_{12}) [22]. A longer equivalent wavelength has a tighter noise limit.

For the case of three wavelengths, three phases 𝜙_{1}, 𝜙_{2} and 𝜙_{3} are obtained. The equivalent wavelength for 𝜆_{1} and 𝜆_{2} is ${\Lambda}_{12}={\lambda}_{1}{\lambda}_{2}/\left|{\lambda}_{1}-{\lambda}_{2}\right|$, for 𝜆_{2} and 𝜆_{3} is ${\Lambda}_{23}={\lambda}_{2}{\lambda}_{3}/\left|{\lambda}_{2}-{\lambda}_{3}\right|$. Then a longer equivalent wavelength is obtained by ${\Lambda}_{12-23}={\Lambda}_{12}{\Lambda}_{23}/\left|{\Lambda}_{12}-{\Lambda}_{23}\right|$, with step height:

Instead of calculating the fringe order of 𝜆_{1} directly, we first calculate the fringe order of 𝛬_{12} based on H_{12-23} and then calculate *H _{12}*, from which the final step height

*H*is obtained. The maximum phase noise limit is significantly increased from 𝜆

_{1}_{1}/(4𝛬

_{12-23}) to 𝜆

_{1}/(4𝛬

_{12}) or 𝛬

_{12}/(4𝛬

_{12-23}).

## 4. Camera calibration

As a key component of the snapshot multi-wavelength interference microscope, the RGB PolarCam must be calibrated to ensure accurate measurement. This RGB PolarCam contains a Bayer color filter array with a wire-grid micro-polarizer array, forming a 4 x 4 superpixel with RGB in 0°, 45°, 90° and 135° linear polarization states, as shown in Fig. 3. The pitch for each Bayer color filter superpixel is 4.5𝜇m, and the pitch for micro-polarizer array superpixel is 9𝜇m. Since this RGB PolarCam contains both Bayer color filter array and micro-polarizer array, all related errors, such as color crosstalk, polarization crosstalk, and micropolarizer misalignment, should be calibrated to increase measurement accuracy.

In camera calibration, missing intensity measurements due to the use of superpixel are recovered at each pixel by linear-spline interpolation. The input Stokes vectors for R, G and B are $\overrightarrow{{S}_{R}}={[{S}_{R0}{S}_{R1}{S}_{R2}{S}_{R3}]}^{T}$, $\overrightarrow{{S}_{G}}={[{S}_{G0}{S}_{G1}{S}_{G2}{S}_{G3}]}^{T}$, $\overrightarrow{{S}_{B}}={[{S}_{B0}{S}_{B1}{S}_{B2}{S}_{B3}]}^{T}$, *S _{R3}*,

*S*, and

_{G3}*S*are the circular parts in Stokes vector of RGB light, which are 0 in our experiment. Thus $\overrightarrow{{S}_{R}}$ can be simplified as ${[{S}_{R0}{S}_{R1}{S}_{R2}]}^{T}$, same with $\overrightarrow{{S}_{G}}$ and $\overrightarrow{{S}_{B}}$. The relationship between intensity

_{B3}*I*, analyzer matrix

*A*, and input Stokes vector

*S*of a pixel is:

The setup for camera calibration is shown in Fig. 5. RGB LEDs with bandpass filters are used as light sources. Only one LED is turned on at each time for the color crosstalk calibration. The integration sphere is used to produce uniform unpolarized light illumination, resulting in a constant ${S}_{0}$ in Stokes vector. The input Stokes vector for the given wavelength is $\overrightarrow{S}={[{S}_{0}{S}_{1}{S}_{2}]}^{T}={[{S}_{0}{S}_{0}\text{cos}\theta {S}_{0}\mathrm{sin}\theta ]}^{T}$, while the other two wavelengths are 0. For each wavelength, 18 measurements are taken with linear polarizer angle 𝜃 at 0°, 10°, 20°,…170°. There are a total of 54 measurements used to calculate analyzer matrix *A*:

*S*is the pseudo inverse of input Stokers vector

^{+}*S*. The ideal analyzer matrix for each wavelength that transforms Stokes vector into linear polarization state at 0°, 45°, 90° and 135° is

The corrected intensity $\tilde{I}$ contains the information for weighted averaged intensities of 0°, 45°, 90° and 135° polarization states at R, G and B wavelengths, which could be used in Eq. (6) for the phase calculation of the RGB channels. This calibration process contains the correction of color crosstalk, nonuniformity in micropolarizer extinction ratio and micropolarizer orientation misalignment based on each pixel of the RGB PolarCam.

## 5. Experimental result

#### 5.1 Measurement with RGB LED light source

The VLSI step height standard (SHS-1.8 QC) and a diamond turned copper surface are measured to demonstrate the system performance. The step height of the VLSI standard is 1.8 𝜇m. Figure 6 shows the spectrum of the RGB light source used for the measurement. Three LEDs with central wavelengths at 460 nm, 540 nm and 630 nm with a 10 nm FWHM bandwidth is used in the experiment. The spectrum bandwidth of the light source is an important factor that will affect the measurement accuracy, especially for the proposed system because the Bayer filter in the sensor has a wide transmission bandwidth. In order to obtain more accurate measurement, bandpass filter with 10nm bandwidth are used in front of each LED. LED spectrum is not exactly a symmetry Gaussian distribution, the mean wavelength must be calibrated before measurement. Generally speaking, a narrower bandwidth will have a more accurate extraction of the mean wavelength.

Since a small shift in single wavelength will cause a large difference in effective wavelength, we will calibrate the mean wavelength of each LED before the measurement. For example, the effective wavelength 𝛬_{12-23} for 460 nm, 540 nm and 630 nm is 17388 nm, the effective wavelength 𝛬_{12-23} becomes 15538 nm when the central wavelength of the Green LED is 541 nm, and an 1850 nm difference is introduced by this 1 nm shift in wavelength. The mean wavelengths are calibrated by measuring a VLSI step height standard (SHS 4606 Å). The calibrated wavelengths are 459.8 nm for the B channel, 540 nm for the G channel, and 629.7 nm for the R channel. The effective wavelength is 𝛬_{12-23} = 16888 nm, which means the measurement range for this RGB light source is 𝛬_{12-23}/4 = 4222 nm.

The surface roughness of the diamond turned copper surface is first measured and compared in RGB channels, as shown in Fig. 7. The measure results of surface root mean square (RMS) are 8.62 nm from the R channel shown in Fig. 7(a), 8.78 nm from the G channel shown in Fig. 7(b), and 8.74 nm from the B channel shown in Fig. 7(c), compared to 8.36 nm measured by Zygo white light interference microscope Newview 8300. The difference in RMS surface roughness between RGB channel and Zygo is caused by the different bandwidth of light source, residual color crosstalk, and polarization crosstalk.

The measurement results of the VLSI height standard are shown in Fig. 8. The step height standard is measured 10 times, calculated and averaged in accordance to the definition in ISO 5436-1 standard. Figures 8(a)-8(d) are the interferograms in RGB, R channel, G channel and B channel, Fig. 8(e) is the 3D surface profile and Fig. 8(f) is the line profile. As shown in Table 1, the averaged step height is 1720.51 nm with a standard deviation of 0.54 nm. The same step height standard is also measured by Zygo NewView 8300 white light interference microscope for comparison, as shown in Fig. 9. The measured step height with the Zygo Newview 8300 is 1722 nm. The corresponding difference between our system and the Zygo interferometer is 1.49 nm.

#### 5.2 Measurement with RGB laser and red LED

The upper limit of the discontinuity in step height measurement is limited by the coherence length of the light source and the effective wavelength. By replacing RGB LEDs with RGB lasers at 442.8 nm, 533.3 nm, and 659.8 nm, the effective wavelength is increased to 42 𝜇m, which increases our measurement range to about 10.5 𝜇m. One of the problems of using a laser as light source is that the measurement results suffer from coherent noise [23], limiting the accuracy of surface roughness measurement. To reduce the spatial coherence, the laser sources are focused onto a rotating diffuser and then coupled into a multi-mode fiber with a 600 𝜇m core and 0.2 NA. The spatial coherence of the laser beam is now limited by the spatial size of the multi-mode fiber.

In order to increase the measurement range while maintaining high quality surface roughness measurement, a light source combing three lasers at ${\lambda}_{1}$ = 442.8 nm, ${\lambda}_{2}$ = 533.3 nm, and ${\lambda}_{4}$ = 659.8 nm with a red light LED at ${\lambda}_{3}$ = 629.7 nm is used in the experiment shown in Fig. 10. The spectrum is shown in Fig. 11. By adding a Red LED into the fringe order calculation, we now have four wavelengths as 𝛬_{12-24} = 42𝜇m, 𝛬_{12-23} = 10.4𝜇m, 𝛬_{12} = 2.6𝜇m and 𝜆_{3} = 0.6297𝜇m. With one more step in the fringe order calculation, the maximum phase noise limit is significantly increased from 𝛬_{12}/(4𝛬_{12-24}) = 0.015 waves to 𝜆_{3}/(4𝛬_{12}) = 0.0603 waves, which makes our system more robust for measuring large step heights. Thus we can achieve a long measurement range without speckle noise by combining a RGB laser and a Red LED. One problem of this configuration is that the camera response to the Red LED and the red laser are similar, and the two signals cannot be distinguished in a single shot. This problem can be solved by using two RGB PolarCams with color filters to separate the two wavelengths, or by capturing the two images sequentially, one with only the Red LED on and the other with only the RGB lasers on. In our experiment, we choose the second option to demonstrate the concept.

With this RGB laser and Red LED light source, we take measurements for a diamond turned copper surface with multiple step heights from 3𝜇m to 10𝜇m. The surface roughness of the diamond turned cupper surface is measured and compared with a Zygo NewView 8300 white light interference microscope, as shown in Fig. 12. The measure results are 8.87 nm from the R channel in our system (Fig. 12(a)) vs. 8.49 nm from the Zygo NewView 8300 (Fig. 12(b)). The difference may be caused by the different objective numerical apertures and the longer coherence length of the red LED in our system, which has a bandwidth of 10nm, while Zygo system uses a white LED. The measurement for each step height is repeated 10 times. Figure 13 shows the measurement results for 3 𝜇m, 4 𝜇m and 5 𝜇m step heights, Figs. 13(a)-13(d) are the interferograms for the Blue laser, Green laser, Red LED and Red laser respectively, Figs. 13(e) and 13(f) show the 3D surface profile and line profile. Figure. 14 plots the measurement errors for the 3𝜇m to 10𝜇m step heights, there is a trend of increasing measurement error as the step height becomes larger. Table 2 shows the measurement results of Zygo NewView 8300 and snapshot multi-wavelength interference microscope. The measurement results for snapshot multi-wavelength interference microscope are the averaged result of 10 measurements. The corresponding difference between Zygo interferometer and snapshot multi-wavelength interference microscope is within 0.26% of the step height.

## 6. Conclusion

We propose a snapshot multi-wavelength interference microscope for optical metrology. A custom RGB PolarCam captures the phase shifted images for red, green and blue wavelengths. A multi-wavelength method is used to remove the 2𝜋 ambiguities and extend the vertical measurement range. We demonstrate the system performance with a VLSI step height standard and a diamond turned copper surface. This system is capable of measuring optical components at high speed and with a large vertical range.

## Funding

National Science Foundation (NSF) (1455630, 1607358); National Institutes of Health (NIH) (S10OD018061); and TRIF Space Exploration & Optical Sciences (SEOS).

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