Abstract

We introduce a Fourier analysis of the waveform of periodic light-irradiance variation to capture Fourier coefficients for multichannel rotating-element ellipsometers. In this analysis, the Fourier coefficients for a sample are obtained using a discrete Fourier transform on the exposures. The analysis gives a generic function that encompasses the discrete Fourier transform or the Hadamard transform, depending on the specific conditions. Unlike the Hadamard transform, a well-known data acquisition method that is used only for conventional multichannel rotating-element ellipsometers with line arrays with specific readout-mode timing, this Fourier analysis is applicable to various line arrays with either nonoverlap or overlap readout-mode timing. To assess the effects of the novel Fourier analysis, the Fourier coefficients for a sample were measured with a custom-built rotating-polarizer ellipsometer, using this Fourier analysis with various numbers of scans, integration times, and rotational speeds of the polarizer.

© 2011 Optical Society of America

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References

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  1. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).
  2. R. W. Collins, I. An, J. Lee, and J. A. Zapien, in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew, 2005), pp. 481–566.
    [CrossRef]
  3. D. E. Aspnes, Thin Solid Films 455–456, 3 (2004).
    [CrossRef]
  4. I. An and R. W. Collins, Rev. Sci. Instrum. 62, 1904 (1991).
    [CrossRef]
  5. N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, J. Opt. Soc. Am. A 8, 919 (1991).
    [CrossRef]
  6. W. Chegal, Y. J. Cho, and H. M. Cho, Department of Convergence Technology, Korea Research Institute of Standards and Science, 1 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea, are preparing a manuscript to be entitled “Source-polarization-free rotating-polarizer spectroscopic ellipsometer based on novel Fourier analysis.”
  7. B. Johs, Thin Solid Films 234, 395 (1993).
    [CrossRef]

2004 (1)

D. E. Aspnes, Thin Solid Films 455–456, 3 (2004).
[CrossRef]

1993 (1)

B. Johs, Thin Solid Films 234, 395 (1993).
[CrossRef]

1991 (2)

An, I.

I. An and R. W. Collins, Rev. Sci. Instrum. 62, 1904 (1991).
[CrossRef]

N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, J. Opt. Soc. Am. A 8, 919 (1991).
[CrossRef]

R. W. Collins, I. An, J. Lee, and J. A. Zapien, in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew, 2005), pp. 481–566.
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, Thin Solid Films 455–456, 3 (2004).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

Chegal, W.

W. Chegal, Y. J. Cho, and H. M. Cho, Department of Convergence Technology, Korea Research Institute of Standards and Science, 1 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea, are preparing a manuscript to be entitled “Source-polarization-free rotating-polarizer spectroscopic ellipsometer based on novel Fourier analysis.”

Cho, H. M.

W. Chegal, Y. J. Cho, and H. M. Cho, Department of Convergence Technology, Korea Research Institute of Standards and Science, 1 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea, are preparing a manuscript to be entitled “Source-polarization-free rotating-polarizer spectroscopic ellipsometer based on novel Fourier analysis.”

Cho, Y. J.

W. Chegal, Y. J. Cho, and H. M. Cho, Department of Convergence Technology, Korea Research Institute of Standards and Science, 1 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea, are preparing a manuscript to be entitled “Source-polarization-free rotating-polarizer spectroscopic ellipsometer based on novel Fourier analysis.”

Collins, R. W.

N. V. Nguyen, B. S. Pudliner, I. An, and R. W. Collins, J. Opt. Soc. Am. A 8, 919 (1991).
[CrossRef]

I. An and R. W. Collins, Rev. Sci. Instrum. 62, 1904 (1991).
[CrossRef]

R. W. Collins, I. An, J. Lee, and J. A. Zapien, in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew, 2005), pp. 481–566.
[CrossRef]

Johs, B.

B. Johs, Thin Solid Films 234, 395 (1993).
[CrossRef]

Lee, J.

R. W. Collins, I. An, J. Lee, and J. A. Zapien, in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew, 2005), pp. 481–566.
[CrossRef]

Nguyen, N. V.

Pudliner, B. S.

Zapien, J. A.

R. W. Collins, I. An, J. Lee, and J. A. Zapien, in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew, 2005), pp. 481–566.
[CrossRef]

J. Opt. Soc. Am. A (1)

Rev. Sci. Instrum. (1)

I. An and R. W. Collins, Rev. Sci. Instrum. 62, 1904 (1991).
[CrossRef]

Thin Solid Films (2)

D. E. Aspnes, Thin Solid Films 455–456, 3 (2004).
[CrossRef]

B. Johs, Thin Solid Films 234, 395 (1993).
[CrossRef]

Other (3)

W. Chegal, Y. J. Cho, and H. M. Cho, Department of Convergence Technology, Korea Research Institute of Standards and Science, 1 Doryong-Dong, Yuseong-Gu, Daejeon 305-340, South Korea, are preparing a manuscript to be entitled “Source-polarization-free rotating-polarizer spectroscopic ellipsometer based on novel Fourier analysis.”

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1987).

R. W. Collins, I. An, J. Lee, and J. A. Zapien, in Handbook of Ellipsometry, H.G.Tompkins and E.A.Irene, eds. (William Andrew, 2005), pp. 481–566.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Fourier coefficients for a 34-nm-thick SiO 2 film on c-Si measured using the custom-built ellipsometer with various numbers of scans and integration times.

Fig. 2
Fig. 2

Fourier coefficients for a 34-nm-thick SiO 2 film on c-Si measured using the custom-built ellipsometer when the rotational speed of the polarizer was varied.

Fig. 3
Fig. 3

Dotted lines denote the ellipsometric spectra for a 34-nm-thick SiO 2 film on c-Si measured using the custom-built ellipsometer. Solid lines represent the best-fit one-parameter simulation, yielding the oxide thickness of 33.93 nm ± 0.01 nm .

Equations (7)

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I ( t ) = I 0 { 1 + n = 1 N [ α 2 n cos ( 4 π n t / T ) + β 2 n sin ( 4 π n t / T ) ] } , ( N 1 ) .
S j = ( j 1 ) T / M j T / M I ( t ) d t , ( j = 1 , , M ) = I 0 T M + n = 1 N I 0 T 2 n π sin ( 2 n π M ) { α 2 n cos [ 2 n π ( 2 j 1 ) M ] + β 2 n sin [ 2 n π ( 2 j 1 ) M ] } .
S j = ( j 1 ) T / M + T d ( j 1 ) T / M + T d + T i I ( t ) d t , ( j = 1 , , M ) .
S j = I 0 T i + n = 1 N I 0 T i ξ 2 n sin ξ 2 n ( cos [ 4 n π ( j 1 ) M ] { α 2 n cos [ ξ 2 n ( 1 + 2 T d T i ) ] + β 2 n sin [ ξ 2 n ( 1 + 2 T d T i ) ] } sin [ 4 n π ( j 1 ) M ] { α 2 n sin [ ξ 2 n ( 1 + 2 T d T i ) ] β 2 n cos [ ξ 2 n ( 1 + 2 T d T i ) ] } ) ,
F 2 n 2 M d 0 j = 1 M S j exp [ i 4 n π ( j 1 ) M ] ,
α 2 n = ξ 2 n sin ξ 2 n { Re ( F 2 n ) cos [ ξ 2 n ( 1 + 2 T d T i ) ] Im ( F 2 n ) sin [ ξ 2 n ( 1 + 2 T d T i ) ] } ,
β 2 n = ξ 2 n sin ξ 2 n { Re ( F 2 n ) sin [ ξ 2 n ( 1 + 2 T d T i ) ] + Im ( F 2 n ) cos [ ξ 2 n ( 1 + 2 T d T i ) ] } ,

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