Abstract

This paper presents a computationally efficient model to simulate the interference signal of vertical scanning interferometry. Existing models are either oversimplified or computationally intensive. Our model incorporates the geometric and spectral effects on vertical scanning interferometry, but removes the computationally intensive numerical integration process by modeling the light spectrum as a sum of piecewise cosine functions. Compared to direct numerical integration of the generalized model, the computational time (for an interference signal) of the proposed model is 256,800 times faster. To verify the accuracy of the proposed model, we simulate the interference signal of a phosphor-based LED, and verify our result with experimental data and a computationally intensive counterpart. Other than reduced computational time, the elementary form of an interference signal derived in this paper will facilitate future work.

© 2010 Optical Society of America

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References

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    [CrossRef]

2005

M. Li, C. Quan, C. J. Tay, I. Reading, and S. Wang, “Measurement of transparent coating thickness by the use of white light interferometry,” Proc. SPIE 5852, 401–406 (2005).
[CrossRef]

2004

2002

2001

A. Hirabayashi, H. Ogawa, and K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” Proc. SPIE 4451, 356–367 (2001).
[CrossRef]

1999

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

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput. 21, 1–23 (1999).
[CrossRef]

1996

1995

Branch, M. A.

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput. 21, 1–23 (1999).
[CrossRef]

Coleman, T. F.

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput. 21, 1–23 (1999).
[CrossRef]

Colonna de Lega, X.

de Groot, P.

Ermolaeva, E.

Gurov, I.

Hirabayashi, A.

A. Hirabayashi, H. Ogawa, and K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” Proc. SPIE 4451, 356–367 (2001).
[CrossRef]

Kim, G.

Kim, S.

Kitagawa, K.

A. Hirabayashi, H. Ogawa, and K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” Proc. SPIE 4451, 356–367 (2001).
[CrossRef]

Larkin, K. G.

Li, M.

M. Li, C. Quan, C. J. Tay, I. Reading, and S. Wang, “Measurement of transparent coating thickness by the use of white light interferometry,” Proc. SPIE 5852, 401–406 (2005).
[CrossRef]

Li, Y.

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput. 21, 1–23 (1999).
[CrossRef]

Ogawa, H.

A. Hirabayashi, H. Ogawa, and K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” Proc. SPIE 4451, 356–367 (2001).
[CrossRef]

Olszak, A.

Quan, C.

M. Li, C. Quan, C. J. Tay, I. Reading, and S. Wang, “Measurement of transparent coating thickness by the use of white light interferometry,” Proc. SPIE 5852, 401–406 (2005).
[CrossRef]

Reading, I.

M. Li, C. Quan, C. J. Tay, I. Reading, and S. Wang, “Measurement of transparent coating thickness by the use of white light interferometry,” Proc. SPIE 5852, 401–406 (2005).
[CrossRef]

Schmit, J.

Sheppard, C. J. R.

Tay, C. J.

M. Li, C. Quan, C. J. Tay, I. Reading, and S. Wang, “Measurement of transparent coating thickness by the use of white light interferometry,” Proc. SPIE 5852, 401–406 (2005).
[CrossRef]

Wang, S.

M. Li, C. Quan, C. J. Tay, I. Reading, and S. Wang, “Measurement of transparent coating thickness by the use of white light interferometry,” Proc. SPIE 5852, 401–406 (2005).
[CrossRef]

Zakharov, A.

Appl. Opt.

J. Opt. Soc. Am. A

Proc. SPIE

A. Hirabayashi, H. Ogawa, and K. Kitagawa, “Fast surface profiler by white-light interferometry using a new algorithm, the SEST algorithm,” Proc. SPIE 4451, 356–367 (2001).
[CrossRef]

M. Li, C. Quan, C. J. Tay, I. Reading, and S. Wang, “Measurement of transparent coating thickness by the use of white light interferometry,” Proc. SPIE 5852, 401–406 (2005).
[CrossRef]

SIAM J. Sci. Comput.

M. A. Branch, T. F. Coleman, and Y. Li, “A subspace, interior, and conjugate gradient method for large-scale bound-constrained minimization problems,” SIAM J. Sci. Comput. 21, 1–23 (1999).
[CrossRef]

Other

A. Olszak and J. Schmit, “Scanning interferometry with reference signal,” U.S. patent 6,624,894 (23 September 2003).

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Figures (3)

Fig. 1
Fig. 1

Analyzing the number of piecewise cosine functions required to represent a Gaussian function. The closer R 2 equals to 1, the better it fits.

Fig. 2
Fig. 2

Intensity spectrum of phosphor-based LED, LXHL-LW6C by LumiLEDs.

Fig. 3
Fig. 3

Comparison of the correlogram by experiment, the proposed model, and an existing computationally intensive model: the proposed model agrees with the computationally intensive prior art and experimental data well.

Equations (9)

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I interference ( z ) = C 1 bandwidth 0 θ 0 { k 2 × cos [ 2 k ( z z 0 ) cos θ + ϕ ] × sin θ cos θ d θ } F ( k ) d k ,
I ( z ) = I dc ( z ) + I amplitude ( z ) exp ( ( z z 0 ) 2 σ 2 ) cos ( 2 π ( z z 0 ) λ m + ϕ ) ,
a b f ( x ) d x ( b a ) n ( f ( a ) + f ( b ) 2 k = 1 n 1 f ( a + k b a n ) ) ,
a exp ( ( x x m ) 2 σ 2 ) = { a 1 cos ( 2 π ( x x m ) b 1 ) + a 2 cos ( 2 π ( x x m ) b 2 ) for ( x m c ) x ( x m + c ) 0 else ,
[ a 1 a 2 x m b 1 b 2 ] = [ A 1 0 0 0 A 2 0 0 0 0 1 0 0 0 0 B 11 B 12 0 0 B 21 B 22 ] [ a x m σ 1 ] , c = C range min ( | b 1 | , | b 2 | ) ,
{ a 1 = 0.7888 a a 2 = 0.2049 a x m = x m b 1 = 7.6777 σ 0.0078 b 2 = 2.4769 σ + 0.0024 c = 0.852 min ( | b 1 | , | b 2 | ) .
I interference ( z ) = C bandwidth k 2 F ( k ) 0 θ 0 cos [ 2 k ( z z 0 ) cos θ + ϕ ] sin θ cos θ d θ d k = C bandwidth k 2 F ( k ) [ ( 2 k z cos θ 2 k z 0 cos θ ) sin ( 2 k z cos θ 2 k z 0 cos θ + ϕ ) + cos ( 2 k z cos θ 2 k z 0 cos θ + ϕ ) 4 k 2 z 2 8 k 2 z 0 z + 4 k 2 z 0 2 ] 0 θ 0 d k = C k l l k u l a cos ( 2 π k k m b ) 8 z 0 z 4 z 2 4 z 0 2 [ ( 2 k z cos θ 0 2 k z 0 cos θ 0 ) sin ( 2 k z cos θ 0 2 k z 0 cos θ 0 + ϕ ) + cos ( 2 k z cos θ 0 2 k z 0 cos θ 0 + ϕ ) ( 2 k z 2 k z 0 ) sin ( 2 k z 2 k z 0 + ϕ ) cos ( 2 k z 2 k z 0 + ϕ ) ] d k = C k l l k u l a cos ( 2 π k k m b ) 8 z 0 z 4 z 2 4 z 0 2 [ 2 k cos θ 0 ( z z 0 ) sin ( 2 k cos θ 0 ( z z 0 ) + ϕ ) + cos ( 2 k cos θ 0 ( z z 0 ) + ϕ ) cos ϕ ] d k = C [ ( g ( z , z 0 , θ 0 , k u l , b ) g ( z , z 0 , 0 , k u l , b ) ) ( g ( z , z 0 , θ 0 , k l l , b ) g ( z , z 0 , 0 , k l l , b ) ) ] .
g ( z , z 0 , θ , k , b ) = D 4 ( b U ( ( 2 ( π b U ) k cos ( ( b ϕ + 2 k m π 2 π k + 2 b U k ) b ) + b sin ( b ϕ + 2 k m π 2 π k + 2 b U k b ) ) / ( π b U ) 2 + ( 2 ( π + b U ) k cos ( b ϕ 2 k m π + 2 π k + 2 b U k b ) + b sin ( b ϕ 2 k m π + 2 π k + 2 b U k b ) ) / ( π + b U ) 2 ) + ( b ( sin ( b ϕ + 2 k m π 2 π k + 2 b U k b ) / ( π + b U ) + sin ( b ϕ 2 k m π + 2 π k + 2 b U k b ) / ( π + b U ) ) ) ) ,
D = 1 / ( 8 z z 0 4 z 2 4 z 0 2 ) , U = ( z z 0 ) cos θ .

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