## Abstract

A novel high speed volumetric thickness profilometry based on a wavelength scanning full-field interferometer and its signal processing algorithm is described for a thin film deposited on pattern structures. A specially designed Michelson interferometer with a blocking plate in the reference path enables us to measure the volumetric thickness profile by decoupling two variables, thickness and profile, which affect the total phase function *ϕ(k)*. We show experimentally that the proposed method provides a much faster solution in obtaining the volumetric thickness profile data while maintaining the similar level of accurate measurement capability as that of the least square fitting method.

©2008 Optical Society of America

## 1. Introduction

There have been great interest and advances in interferometry over the past few decades [1–2]. The phase shifting method using monochromatic light has been a powerful and promising tool which is widely used for various applications, such as 3-D micro profilometry and digital holography [1–5]. However, the phase shifting method using a single wavelength inherently suffers from the 2*π* ambiguity problem. Likewise, as in piezo-electric transducer (PZT) based phase shifting in the *z*-scan domain, the spectral phase shifting approach also suffers from the 2*π* ambiguity problem. There have been various studies on solving the 2*π* ambiguity problem of the phase shifting method by use of multi-wavelength schemes [6–9]. Although multi-wavelength approaches can provide a feasible solution to some degree, a promising solution for a wide range of 3D measuring capability has been the use of wavelength scanning interferometric profilometry [10–13].

However, for patterned opaque structures upon which transparent thin films are deposited, a multi-reflection phenomenon occurs due to the thin film layers, making it difficult to obtain accurate 3-D surface profile data. Recently, some attempts were made to measure such 3-D volumetric thickness profile data of a patterned structure, upon which thin films were deposited [14–18]. Among them, the most accurate volumetric thickness profile measuring approach was in using the least square fitting approach. However, its inherently long calculation time made such an approach impractical. More recently, a one-dimensional line thickness profile measuring method using a grating as the dispersive device was proposed [19–20]. However, it cannot provide full volumetric thickness profile information which is usually used in various industrial practical applications.

In this paper, we propose a high speed volumetric thickness profile method based on a two-step operation. In order to decouple the two variables to be measured, thickness and upper surface profile data, a specially designed Michelson interferometer with a reference beam blocking mechanism was employed and the thickness and upper surface profile information were measured separately. We use the direct spectral phase function calculation method for enhancing the calculation time [16]. The proposed method enhances measurement speed dramatically while maintaining nanometer range accuracy.

## 2. Theory

Optical imaging profilometry can be divided into two alternative approaches: the white light scanning interferometer (z-scan) and the dispersive interferometric profilometer (k-scan) [10–11]. The former requires a precise moving actuator, such as PZT. On the other hand, the k-scan employs a wavelength scanning device instead of a moving actuator. Likewise, both approaches can be applied for measuring thickness profile information by compensating for the multi-reflection effect of thin films to analyze the phase data. This study is on a k-scan-based fast and accurate volumetric thickness profile measurement method. Figure 1(a) shows the system schematics, which consists of a white light source, a semi-collimating lens, an acousto-optic tunable filter (AOTF) with visible spectral scanning range, a 2-D CCD, and a Michelson interferometer with a specially designed switching plate that can block the optical wave traveling toward the reference mirror plane. The proposed system is a kind of 2-D spectral scanning interferometric system that provides a 3-D spectral imaging data set, i.e. a 2-D spatial axis and 1-D spectral axis data, as depicted in Fig. 1(b).

The proposed system has two measurement states depending on the position of the blocking plate in the reference beam path. The two separate measurement states are blocking plate ON and OFF, as depicted in Fig. 1(a). When the state is blocking plate ON, the reference beam blocker blocks the beam headed to the reference mirror surface such that the system acts like an imaging reflectometer. In this case, the AOTF acts as a wavelength scanning device that can measure the thin film thickness information over a certain range of a 2-dimensional region. In the next step, when the state is blocking plate OFF, another wavelength scanning was conducted to obtain the upper surface profile information through the direct phase function calculation method for high speed calculation [16].

Now, we describe a more detailed theory on the proposed volumetric high speed and accurate thickness profile measuring method. As illustrated in Fig. 1(a), the film thickness and upper surface profile information will hereafter be labeled as *d(x,y)* and *h(x,y)*, respectively. The upper surface profile *h(x,y)* indicates the directional distance from an imaginary reference mirror plane to the thin film upper surface.

In order to obtain the volumetric thin film thickness profile, the effect of thin film on the interference intensity as expressed in Eq. (1) must be considered.

$$={i}_{0}(k,d)\left[1+\gamma (k,d)\mathrm{cos}\left\{2kh+\psi (k,d)\right\}\right]$$

Here, *E _{r}* and

*E*represent the reflected wave functions from the reference mirror and the measured sample, respectively, and

_{t}*I*is the interference intensity. As mentioned previously,

*d*and

*h*are the thickness and upper surface profile, respectively, and

*k*is the wavenumber defined by

*2π/λ. i*and

_{0}*γ*are the stationary part of the interference signal and visibility function, respectively. Furthermore, the phase change

*ψ(k,d)*can be represented as follows.

Here, *A* and *B* are the real and imaginary parts of the total reflection coefficient *R*, which can be described as follows. The thin film causes a multi-reflection phenomenon such that the total reflection coefficient *R* becomes the following.

Here, *θ* is the incidence angle and *r _{01}*,

*r*are the Fresnel reflection coefficients between mediums

_{12}*0*and

*1*and mediums

*1*and

*2*, respectively, where medium

*0*is air, medium

*1*is a thin film, and medium

*2*is a substrate or patterned metal. Here,

*N(k)*represents the complex refractive index of the deposited thin film. The spectral reflectance of the specimen is given by

*𝓢(k,d)*=

_{sample}*|R(k,d)*|

^{2}. Since the thickness is a constant at each measurement point, the phase change

*ψ (k,d)*can be reduced as a function of only wavenumber

*k*if the thickness is known.

With the assumption that the thin film is transparent and the incidence angle is zero, the thickness can be easily measured by the specially designed Michelson interferometer with a blocking plate ON. In the first place, *G(k,0) _{reference}*, the spectral density of reference specimen is measured. A bare crystalline silicon wafer without film coating was used as a standard specimen. Then the spectral density of a specimen,

*G(k,d)*, is measured to obtain the spectral reflectance

_{sample}*𝓢(k,d)*by Eq. (4).

_{sample}Here, the spectral reflectance of the standard specimen, *𝓢(k,0) _{reference}* is calculated using Eq. (3). The thickness information can be obtained by simply detecting the two wavenumbers

*k*and

_{1}*k*at which the first and the last peak of the spectral reflectance appears. By assuming that the distance between adjacent peaks is

_{2}*2π*, the thickness

*d*can be calculated by following Eq. (5).

Here, *n* is the number of maximum peaks and *N(k _{1})* and

*N(k*are the refractive indices of the thin film for

_{2})*k*and

_{1}*k*, respectively.

_{2}If the thickness of the thin film is too thin to measure the two consecutive peaks, we can apply the nonlinear least square fitting method to accurately measure the thickness information [20]. The measurement error of peak detection method was numerically simulated with thickness range from 700 nm to 4000 nm and the result is described in Fig. 2. The error starts to increase as the thickness of thin film decreases below 2000 nm. Specifically, the error abruptly increases with the thickness below 1000 nm. Therefore, the assumption is not valid for the thickness below 1000 nm.

Once the thickness data *d(x,y)* is obtained, the next step is to calculate the surface profile data *h(x,y)*. As mentioned, another wavelength scan without the blocking plate is conducted to measure the upper surface profile of thin film. First, *I(x,y,k)* must be measured throughout the entire wavenumber range. For each coordinate (*x,y*), we need to extract the spectral phase function *ϕ(k)* from *I(k)*. For this, both the Fourier transform and direct phase function calculation method based on the spectral phase shifting technique can be used. In this paper, however, we use the direct phase function calculation method since the latter can provide faster measurement capability. However, in order to apply the phase shifting technique for the spectral domain analysis, we must realize that the wavenumber *k* and the top surface profile data *h(x,y)* are coupled. At this stage, we redefine the surface profile as *h(x,y)*=*h0+h’(x,y)* and the wavenumber *k*=*kc+δk*. Here, *k _{c}* represents the central wavenumber for spectral phase shifting, and

*δk*is the amount by which the wavenumber is shifted. Then, for each coordinate (

*x,y*), Eq. (1) can be re-written as follows:

$$\approx {i}_{0}(x,y,k)\left\{1+\gamma (x,y,k)\mathrm{cos}\left(2{k}_{c}h(x,y\right)+2{h}_{0}\delta k+\psi (k,d)\right)\}.$$

With the condition that *h _{0}*≫

*h*’(

*x,y*), the last term 2

*h’δk*can be omitted. Also,

*δk*is redefined as (3-

*m*)

*Δk*. Here,

*Δk*indicates the minimum wavenumber variation induced by the spectral scanning device. The interfered intensity

*I*can be re-written with the subscript of

*m*as follows.

$${I}_{2}(x,y)={i}_{0}(x,y)\left\{1+\gamma (x,y)\mathrm{cos}\left[2{k}_{c}h(x,y)-2{h}_{0}2\Delta k+\psi ({k}_{c},d)\right]\right\}$$

$${I}_{3}(x,y)={i}_{0}(x,y)\left\{1+\gamma (x,y)\mathrm{cos}\left[2{k}_{c}h(x,y)+\psi ({k}_{c},d)\right]\right\}$$

$${I}_{4}(x,y)={i}_{0}(x,y)\left\{1+\gamma (x,y)\mathrm{cos}\left[2{k}_{c}h(x,y)+2{h}_{0}\Delta k+\psi ({k}_{c},d)\right]\right\}$$

$${I}_{5}(x,y)={i}_{0}(x,y)\left\{1+\gamma (x,y)\mathrm{cos}\left[2{k}_{c}h(x,y)+2{h}_{0}\left(2\Delta k\right)+\psi ({k}_{c},d)\right]\right\}$$

When the appropriate spectral carrier frequency *h _{0}* is applied,

*i*and

_{0}(k)*γ(k)*can be treated as slowly varying functions, which means that they can be considered to be constants regardless of the variation of wavenumber

*k*. In order to obtain the phase value

*ϕ(k*at the central wavenumber

_{c})*k*, five intensity values

_{c}*I*~

_{1}*I*are used as follows:

_{5}The above equation explains how to obtain a phase value at the specific wavenumber *k _{c}*. By sweeping the central wavenumber

*k*throughout the entire wavenumber scanning range, we can obtain the spectral phase function

_{c}*ϕ(k)*. With an accurate total phase function, one can expect to obtain accurate upper surface profile information

*h*. Once the total phase function

*ϕ(k)*is obtained, the profile information

*h*can be found using the following equation.

Here, *k _{1}* and

*k*can be arbitrary wavenumbers since

_{0}*ψ(k)*and

*ϕ(k)*are both obtained as fully defined functions.

## 3. Experimental results

Experiments were carried out in order to examine the effectiveness of the AOTF-based fast and accurate volumetric thickness profile measurement method. The optical setup has a magnification of 4.4. The sample is located at the focal position of the objective, therefore, the light from the sample is collimated and passes through the AOTF and finally imaged on the CCD. The whole image is captured by 330×250 pixel window. Finally, the region we are interested is determined by the window size of 150×150 pixels. The AOTF used in the experiment is made of TeO2 crystal and has a visible diffraction region of 400nm to 650nm. The diffraction in the AOTF occurs by applying radio frequency (RF) signal ranging from 120MHz to 220MHz and the diffracted wavelength can be tuned within microseconds.

For an accurate measurement, the image shift induced by scanning the wavelength of the AOTF should be compensated. The image shift is inevitable while using an AOTF. The diffraction angle is measured and plotted against the wavelength as described in Fig. 3(a). For calibrating the image shift accurately, a USAF resolution target was used and the correlation between adjacent images was calculated. More specifically, the total image shift on the CCD was 163 pixels and 2000 images were captured as the AOTF scanned linearly from 120 MHz to 170 MHz for calibration. That is, approximately 12 images were recorded per pixel separation on CCD. A rectangular binary mask was multiplied to the first image to define the region of interest (ROI) and the ROI was chosen as a reference. A single pixel shifted mask was multiplied to the next 11 images and the correlation between the reference and these images were calculated. The accurate frequency to operate the AOTF to shift the reference image by a single pixel was d6ecided by the maximum correlation value. With this processing method and the polynomial fitting, the image shift can be calibrated to sub-pixel deviation as described in Fig. 3(b).

After calibration of the image shift, the measurement accuracy was evaluated. Three samples with different thickness (1µm, 2µm, 3µm) of SiO2 deposited on the Si wafer substrate were prepared and we compared the measurement results by using the commercial equipments and the proposed method as listed in table 1. The fitting method applied to the raw data provides accurate results for the three samples which implies that the system is well calibrated. Similar accurate results were obtained for the thickness of 2µm and 3µm by peak detection method which is approximately 30 times faster. The thickness analyzed by peak detection method for the thin film of 1µm includes error as expected from the numerical simulation of Fig. 2. The measurement error at 1um thickness which is larger than the theoretical expectation can be attributed to the limited spectral resolution of an AOTF.

Experiments to measure the thickness of a thin film were carried out to compensate for the phase change effect due to the thin film. The silicone pattern structures, over which a SiO_{2} thin film is deposited, were used for the experiments. The sample is fabricated by following multiple steps. First, the SiO_{2} is deposited on the silicon wafer and the photo resistance is coated and selectively etched. Finally, the SiO_{2} is etched to make patterns at the top surface. Figure 4 shows a photograph of the rectangular patterned structure used in the experiment.

In the first step, wavelength scanning with the blocking plate ON is carried out to obtain the thin film thickness information. Figure 5 shows the spectral reflectance of the sample at two different positions A and B in the Fig. 4. The moving average algorithm was applied to the raw spectrum two times serially with subset of 10 data points. The thickness distribution can be obtained by Eq. (5) detecting the first and the last peak at each measurement point.

Secondly, the same wavelength scanning is carried out with the blocking plate OFF to obtain the *h(x,y)* information. In order to apply the spectral carrier frequency concept, the reference mirror plane is positioned such that the distance between the two arms is around 30 µm. Figure 6(a) is the measurement result at the central point of the pattern sample. With this interference intensity data, spectral domain signal processing is conducted for all measurement points using Eq. (9) to calculate the total phase function *ϕ(k)*, as described in Fig. 6(b).

Figure 7(a) is the obtained phase function *ψ(k)* at the central point of the measured sample. In this way, the phase change effect due to thin films can be compensated by obtaining the phase function *ψ(k)* at all measurement points. Since the phase function is obtained for all the measurement points, the surface profile *h(x,y)* can be calculated using Eq. (9), as described by a solid line in Fig. 7(b).

Finally, by using the obtained thickness *d(x,y)* and upper surface information *h(x,y)*, we can reconstruct the thickness profile, as shown in Fig. 8. As seen, some error peaks exist near the region of the abrupt change in height gap, which is expected to be tackled by appropriate signal processing.

The thickness d_{1} and d_{2} and step height (h_{1}–h_{2}) were measured to be 1478 nm, 2007 nm and 530 nm, respectively, by using commercial reflectometer and surface profiler. The measurement results by the proposed method are plotted in Fig. 9. The thickness measurement error for d_{1} and d_{2} is 19 nm and 1 nm, respectively, and the error of surface step height is 14 nm. The measurement result by peak detection method was accurate for the thin film when the thickness is larger than 2µm. But for the thickness below 2µm, the error increases as the film becomes thinner. An AOTF or a spectrometer with higher spectral resolution would improve the accuracy of the peak detection method. The fitting method can be applied to obtain high accuracy for the thin film below 2µm at the expense of ~30 times longer signal processing time.

## 3. Conclusion

A novel high speed volumetric thickness profilometry based on wavelength scanning full-field interferometer using a two-step operation has been described for a thin film deposited on pattern structures. In order to decouple the two variables to be measured, thickness and upper surface profile data, a specially designed Michelson interferometer with a reference beam blocking mechanism has been employed and the thickness and upper surface profile information have been measured separately. In this paper, we have used the direct spectral phase function calculation method for enhancing the calculation time. Experimental results for a SiO_{2} thin film pattern structure showed that the thickness and surface profile of the sample can be measured around 30 times faster than the least square fitting method while maintaining the same level of accurate measurement capability.

## Acknowledgments

This research was supported by a grant (04-K14-01-013-00) from Center for Nanoscale Mechatronics & Manufacturing, one of the 21st Century Frontier Research Programs, which are supported by Ministry of Education, Science and Technology, KOREA.

## References and links

**1. **K. Creath, “Temperal phase measuring methods,” in *Interferogram Analysis: Digital Fringe Pattern Measurement Techniques*, D. W. Robinson and G. T. Reid, eds., (Institute of Physics, Bristol, UK, 1993).

**2. **Y. Ishii, J. Chen, and K. Murata, “Digital phase-measuring interferometry with a tunable laser diode,” Opt. Lett. **12**, 233–235 (1988). [CrossRef]

**3. **I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. **22**, 1268–1270 (1997). [CrossRef] [PubMed]

**4. **B. Javidi and D. Kim, “Three-dimensional object recognition by use of single-exposure on-axis digital holography,” Opt. Lett. **30**, 236–238 (2005). [CrossRef] [PubMed]

**5. **D. Kim, J. W. You, and S. Kim, “White light on-axis digital holographic microscopy based on spectral phase shifting,” Opt. Express **14**, 229–234 (2006). [CrossRef] [PubMed]

**6. **C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. **12**, 2071–2074 (1973). [CrossRef] [PubMed]

**7. **Y. Cheng and J. C. Wyant, “Two-wavelength phase shifting interferometry,” Appl. Opt. **23**, 4539–4543 (1984). [CrossRef] [PubMed]

**8. **A. Pfortner and J. Schwider, “Red-green-blue interferometer for the metrology of discontinuous structures,” Appl. Opt. **42**, 667–673 (2003). [CrossRef] [PubMed]

**9. **J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. **28**, 1141–1143 (2003). [CrossRef] [PubMed]

**10. **P. de Groot and L. Deck, “Three-dimensional imaging by sub-Nyquist sampling of white-light interferomgrams,” Opt. Lett. **18**, 1462–1464 (1993). [CrossRef] [PubMed]

**11. **J. Schwider and L. Zhou, “Dispersive interferometric profilometer,” Opt. Lett. **19**, 995–997 (1994). [CrossRef] [PubMed]

**12. **M. Kinoshita, M. Takeda, H. Yago, Y. Watanabe, and T. Kurokawa, “Optical frequency-domain microprofilometry with a frequency-tunable liquid-crystal Fabry-Perot etalon device,” Appl. Opt. **38**, 7063–7068 (1999). [CrossRef]

**13. **I. Yamaguchi, “Surface tomography by wavelength scanning interferometry,” Opt. Eng. **39**, 40–46 (2000). [CrossRef]

**14. **S. W. Kim and G. H. Kim, “Thickness profile measurement of transparent thin-film layers by white-light scanning interferometry,” Appl. Opt. **38**, 5968–5973 (1999). [CrossRef]

**15. **D. Kim, S. Kim, H. Kong, and Y. Lee, “Measurement of the thickness profile of a transparent thin film deposited upon a pattern structure with an acousto-optic tunable filter,” Opt. Lett. **27**, 1893–1895 (2002). [CrossRef]

**16. **D. Kim and S. Kim, “Direct spectral phase calculation for dispersive interferometric thickness profilometry,” Opt. Express **12**, 5117–5124 (2004). [CrossRef] [PubMed]

**17. **H. Akiyama, O. Sasaki, and T. Suzuki, “Sinusoidal wavelength-scanning interferometer using an acousto-optic tunable filter for measurement of thickness and surface profile of a thin film,” Opt. Express **13**, 10066–10074 (2005). [CrossRef] [PubMed]

**18. **K. Kitagawa, “Simultaneous measurement of film surface topography and thickness variation using white-light interferometry,” Proc. SPIE **6375**, 637507 (2006). [CrossRef]

**19. **Y. S. Ghim and S. W. Kim, “Thin-film thickness profile and its refractive index measurements by dispersive white-light interferometry,” Opt. Express **14**, 11885–11891 (2006). [CrossRef] [PubMed]

**20. **Y. S. Ghim and S. W. Kim, “Fast, precise, tomograpic measurements of thin films,” Appl. Phys. Lett. **91**, 091903 (2007). [CrossRef]