The measurement of surface roughness of optical thin

Thin Solid Films 517 (2009) 5110–5115

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Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / t s f

The measurement of surface roughness of optical thin films based on fast Fourier transform Chuen-Lin Tien a,⁎, Huei-Min Yang a, Ming-Chung Liu b a b

Department of Electrical Engineering, Feng Chia University, Taichung, Taiwan, ROC Energy and Resources Laboratories, Industrial Technology Research Institute, Hsinchu, Taiwan, ROC

a r t i c l e

i n f o

Available online 1 April 2009 Keywords: Thin film Surface roughness Fast Fourier transform

A

b s t r a c t

The measurement of surface roughness of optical thin films based on fast Fourier transform (FFT) associated with a Gaussian filter was presented. The measurement of the surface roughness is performed by a Fizeautype microscopic interferometer combined with the Matlab program to analyse the captured interferograms. The surface profile can be obtained by the fringe pattern analysis program using FFT method. In order to improve the accuracy, we normalize the fringe pattern to eliminate the background variation before using the FFT. The roughness profile is filtered by the Gaussian filter after the phase change is converted to surface height distribution. The root-mean-square value of surface roughness of optical thin films was determined by our proposed method. The results were verified by atomic force microscopy (AFM). © 2009 Elsevier B.V. All rights reserved.

1. Introduction Surface roughness is a critical parameter in various fields, especially for high-quality optics and optical coatings. Hence the surface roughness of optical thin films is an important factor in determining the performance of optical devices. An optical surface can be imagined as a sum of an infinite number of sinusoidally varying surfaces of different spatial frequencies. The surface roughness is a characteristic of high spatial frequency. Detailed knowledge of the roughness characteristics of optical thin films will aid in the optimization of the process parameters of the optical coatings. A variety of suitable methods for surface characteristics inspection are available to measure the surface roughnesses such as atomic force microscopy (AFM) [1], scanning electron microscopy (SEM) [2], total integrated scattering (TIS) [3,4], stylus profilometry [5], and interferometry methods. However, the above instruments used in general are fairly expensive and the analysis of thin films in highly complex and time-consuming. For example, one kind of roughness measurements is normally done through the use of stylus type instruments. The major disadvantage of using stylus profilometry for such measurements is that it requires direct physical contact, which limits the measuring speed. In addition, the instrument readings are based on a limited number of line samplings, which may not represent the real characteristics of the surface. Because of these drawbacks, contact

⁎ Corresponding author. Tel.: +886 4 24517250x3809; fax: +886 4 24516842. E-mail address: [email protected] (C.-L. Tien). 0040-6090/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2009.03.193

type instruments are not suitable for high-speed automated inspection. For non-contact surface roughness measurement, white-light interferometry [6–8], interference microscopy [9,10], phase-shifting interferometry [11], coherence scanning microscopy [12], and Fourier transform method (FTM) [13] are used to measure surface roughness. FTM was first published by Takeda et al. [13] in 1982, and it has some advantages such as analysing using only one interferogram, high accuracy and near real-time measurement. In this study, we present a Fizeau-type microscopic interferometer associated with FTM to reconstruct surface topography of thin film. After reconstructing the film's surface, we make use of a Gaussian filter [14] to filter out the high-frequency signal and to obtain roughness profile. This non-contact method was used to measure different rough TiO2 thin films by the fast Fourier transform algorithm and digital filter process. Atomic force microscopy (AFM) was also used to confirm the results obtained by means of the proposed method. 2. Method This paper presents a new approach for surface roughness measurement using optical interferometry, fast Fourier transform and digital filtering process. It has an advantage over the traditional method where the surface geometry is not touched and line to line scanning is not required. The general interference equation for the intensity measured in a two-beam interferometer is as follows: iðx; yÞ = aðx; yÞ + bðx; yÞ cos ½2π f0 x + /ðx; yÞ

ð1Þ

C.-L. Tien et al. / Thin Solid Films 517 (2009) 5110–5115

where a(x,y)and b(x,y) are the bias of the fringe pattern intensity and the contrast of fringes coefficient, respectively. ϕ(x,y) is the phase of the wave front, and f0 is the spatial-carrier frequency. The intensity distribution can be rewritten in the following form

root-mean-square value of surface roughness is determined by the following formula.

Rrms 4

iðx; yÞ = aðx; yÞ + cðx; yÞ expð2πif0 xÞ + c ðx; yÞ expð − 2πif0 xÞ

ð2Þ

where the asterisk indicates the complex conjugate and cðx; yÞ = 1 2 bðx; yÞ exp ½i/ðx; yÞ is a complex fringe pattern. Then Eq. (2) takes the Fourier transform with respect to x using the fast Fourier transform (FFT), which gives 4

Ið f ; yÞ = Aðf0 ; yÞ + C ð f − f0 ; yÞ + C ð f + f0 ; yÞ

ð3Þ

where the capital letters indicate the Fourier spectra and f is the spatial frequency in the x-direction. The spatial variations of a(x,y), b(x,y) and ϕ(x,y) are slow compared with the carrier frequency f0. The Fourier spectra in Eq. (3) are separated by the carrier frequency f0. The function C(f − f0,y) can be isolated by using a filter centered at f0 and the carrier frequency can be removed by shifting C(f − f0,y) to the origin to give C(f0,y). We compute the inverse Fourier transform of C(f0,y) with respect to f and obtain c(x,y). Thus the phase distribution ϕ(x,y) can be calculated by

/ðx; yÞ = arctan

Im½cðx; yÞ Re½cðx; yÞ

ð4Þ

where Re½cðx; yÞ and Im½cðx; yÞ are the real part and imaginary part of c(x,y), respectively. Eq. (4) gives the phase modulo 2π, the arctangent function is defined over a range from − π/2 to π/2, and there may be discontinuities present in the calculated phase. A phase unwrapping algorithm is needed to remove the discontinuities by adding or subtracting multiples of 2π to a pixel until the difference between it and its adjacent pixel is less than π. Since the sampling theorem requires that there must be at least two pixels per fringe, so the phase difference between two adjacent pixels must be less than π. In this study, we use the phase unwrapping algorithm which was described by Macy [15]. Once the value of ϕ(x,y) is known, the surface profile of the sample can be reconstructed. A Gaussian filter is used to separate the roughness profile from the original profile. The defined wavelength value is referred to as the cutoff of the Gaussian filter. Thus the Gaussian filter acts as a low-pass filter. An important property of the Gaussian filter is its linear phase. The filter is designed to have 50% transmission at the cutoff. The weighting function and transmission characteristic of Gaussian filter are given by

SðxÞ =

1 x 2 exp −π αλc αλc

Aoutput λ 2 = exp −π α c Ainput λ

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91 = 2 8 = < 1 X M X N 2 = ½zði; jÞ− hzði; jÞi ; :MN j = 1 i = 1

ð7Þ

where z(i,j) is the height of the surface profile, and hzði; jÞi is the mean value. For comparison, an atomic force microscope (Digital Instruments Nanoscope II) was used to scan surface terrain in a typical area of 5 µm × 5 µm on a vibration-free platform. The parameter commonly used for describing the roughness characteristics of a coating surface is the root-mean-square (rms) roughness. It very rapidly conveys to the user an impression of the quality of the surface under study and presents a suitable tool for roughness description. 3 Experimental setup 3.1. Thin film preparation Titanium oxide films were deposited by conventional electronbeam evaporation. The coatings under study have to be on a flat, well polished substrate. The substrates used in this experiment were B270 glass plate and silicon wafer. Prior to deposition, the substrates had undergone ultrasonic cleaning in acetone and ethanol and had been dried in a vacuum dryer. Ti3O5 granule was chosen as a starting material to make titanium dioxide films. A vacuum thin film deposition system was used for this study. The system consisted of a chamber equipped with dual e-beam sources for evaporation, a substrate holder carried with different substrates (circular glass plate

ð5Þ

ð6Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi where α = ln2 = π = 0.4697, x is position from the origin of the weighting function, and λc is the long wavelength roughness cutoff. The mean phase is obtained by convolving the surface profile with the weighing function in Eq. (5). The mean phase is then subtracted from the surface height profile to obtain the roughness profile. Finally, the

Fig. 1. A Fizeau microscope interferometer used for measuring surface roughness.

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Fig. 2. Interference pattern obtained from the TiO2 thin film.

and silicon wafer), and an optical monitor to control evaporation. TiO2 films were deposited by electron-beam evaporation. The distance between the starting material and the substrates was 120 cm. Samples were mounted onto the rotation of a substrate holder with a 900 mm diameter that rotated at a speed of 12 rpm. A thermocouple was placed near the sample holder to monitor the chamber temperature. An e-beam gun was used to evaporate Ti3O5 granules. The film thickness and the rate of deposition were controlled by both optical and quartz crystal monitors. The deposition rate was 0.592 nm/s. The vacuum chamber was initially pumped down by a mechanical pump and cryopump to a base pressure of less than 3 × 10− 5 Torr. Oxygen, the active gas, was fed near the material source at a flow rate regulated by a needle valve. During deposition, the total chamber pressure was maintained at 1.4 × 10− 4 Torr by adjusting the oxygen flow. The oxygen gas flow was controlled at 75 sccm. The thin films are deposited with a physical thickness of about 200–400 nm as monitored by an optical monitor.

Fig. 4. Cross-section profile of surface roughness (a) in the x-direction; (b) in the y-direction.

3.2. Roughness measurements A PC-based Fizeau-type microscopic interferometer for evaluating the surface roughness of optical thin films is shown in Fig. 1. This

system contains a Fizeau-type microscopic interferometer, a 20× microscope objective and a CCD camera combined with a frame grabber to capture images of the specimen surface to be characterized,

Fig. 3. 3-D roughness profile without using the Gaussian filter.


and a specially developed Matlab program to analyze the captured interference images. A He–Ne laser with the wavelength of 632.8 nm is used as the light source. The light beam falls on a cube beam-splitter which splits the beam into a reference and a sample beam that emerge in perpendicular directions. One beam is used to direct onto a test surface through a cube beam-splitter and a microscope objective (20×), another beam is reflected by a reference mirror (flatness λ/20) and a cube beam-splitter. The test surface within the field of view of the objective is thus uniformly illuminated and the area of interest on the test surface can be adjusted through alignment of the iris diaphragm. The beams reflected off the reference and the test surface onto a CCD sensor through a lens. The resulting interference fringe pattern is recorded by a CCD camera (AVT PIKE, Model: F-032B) and the corresponding signal is digitized by a frame grabber for further processing with a computer. The process of image acquisition with spatial-carrier phase is carried out with the support of the frame grabber. Only the central 240 × 240 pixels, representing an actual dimension of 100 µm × 100 µm, were chosen for analysis in order to prevent any edge effects. A fringe pattern with a Gaussian distribution background can be seen on the LCD monitor as shown in Fig. 2. The surface profile can be obtained by fringe pattern analysis and FTM. Fig. 3 shows the 3-D plot

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of the roughness profile before digital filter processing. In order to improve the accuracy, we normalize the fringe pattern to eliminate the background variation before using FTM. The roughness profile is filtered by the Gaussian filter with a cutoff wavelength of 3.5 µm, and the phase is translated to height scale by multiplying λ/4π. Fig. 4 shows the plot of cross-section profile of surface roughness in the xdirection and y-direction, respectively. The rms value obtained from the roughness profile was determined with all the low frequency components removed from the filter. Often it is necessary to fully characterize surface texture with direct visualization of a film's surface. Fig. 5 represents the surface roughness of TiO2 thin film by means of a 3-D surface terrain and 2-D contour map to get quantitative surface texture information. 4. Results and discussion When TiO2 thin-film specimens were tested by use of a Fizeautype microscopic interferometer, they were compared to a reference plate with a surface flatness of λ/20. The interference fringes of the TiO2 film specimen were obtained by the Fizeau microscopic interferometer. The roughness evaluation is conducted by the MATLAB program.

Fig. 5. The surface roughness of the TiO2 thin film by using the Gaussian filter (a) 3-D surface terrain; (b) 2-D contour map.

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Fig. 6. AFM images of TiO2 thin films with different rms surface roughnesses of (a) 1.201 nm; (b) 1.501 nm; (c) 1.551 nm; (d) 1.577 nm.

The surface roughness of TiO2 films deposited on silicon wafers was observed by AFM. In order to find the relationship between the AFM observation and the proposed method, rms surface roughness measurements of TiO2 films have been carried out by the above measuring procedure. The surface morphology pictures of TiO2 thin films obtained from AFM are shown in Fig. 6. Fig. 6(a) to (d) show the surface morphology images of TiO2 specimens with the rms surface roughness of 1.201–1.577 nm corresponding to the film thickness of 200–400 nm, respectively. It can be seen that the rms surface roughness increases with increasing the variation of the TiO2 film's thickness. In this paper, four different roughness surfaces of TiO2 thin

films with different thicknesses are measured by our proposed method. The proposed method can be verified by atomic force microscopy (AFM). The comparison between the proposed methods associated with FFT, without FFT, and the AFM approach for the surface roughness measurement of TiO2 thin films are listed in Table 1. A good agreement between the results of an AFM and the results of our proposed method is demonstrated on different thicknesses of TiO2 thin films as shown in Fig. 7. It indicates clearly the possibility of successful implementation of the proposed approach in practice.

5. Conclusions Table 1 Comparison of rms surface roughness measurements between the proposed methods with FFT, without FFT, and an atomic force microscope. TiO2 thin films (thickness in nm)

AFM method

Microscopic interferometer for Rrms (nm)

Rrms (nm)

With FFT

Without FFT

Sample Sample Sample Sample

1.201 1.517 1.551 1.577

1.209 ± 0.0424 1.478 ± 0.0094 1.554 ± 0.0078 1.669 ± 0.0321

7.943 ± 0.0860 8.260 ± 0.0169 8.294 ± 0.0305 8.365 ± 0.0330

1 (200 2 (300 3 (350 4 (400

nm) nm) nm) nm)

An experimental investigation has been performed to study the fast Fourier transform method and digital Gaussian filter for measuring the surface roughness of thin films. In this paper a MATLAB program was developed to determine the surface roughness of thin films by using a PC-based Fizeau-type microscopic interferometer. Our measuring results are in good agreement with the results of AFM measurements. The deviation between both methods is less than 5.9%. The proposed technique should be useful in measuring the surface roughness of optical thin films. It offers


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Acknowledgement The authors gratefully appreciate the support of the National Science Council of R.O.C with Contract No. NSC 96-2221-E-035-067MY3. References

Fig. 7. Plot of surface roughness of TiO2 thin films by AFM and proposed method.

many advantages compared to the mechanical stylus method such as simplicity, low cost, non-destructive interaction and high speed of measurement.

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