Abstract

Nondestructive reconstruction of the location and width of a thin-strip defect in a strongly confined planar waveguide is considered. Explicit reconstruction formulas are given for a quick reconstruction of a thin-strip defect whose width is small by measuring the scattered fields at the two end faces of the planar waveguide for two frequencies. Numerical results are given, and the analytical reconstruction method is shown to be reliable regardless of the location of the defect.

© 2006 Optical Society of America

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  1. L. B. Soldano and E.C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
    [CrossRef]
  2. M. K. Smit and C. V. Dam, "PHASAR-based WDM devices: principles, design and applications," IEE J. Sel. Top. Quantum Electron. 2, 236-250 (1996).
    [CrossRef]
  3. W. K. Kim, W. Yang, and H. Lee, "Effects of parasitic modes in high-speed LiNbO3 optical modulators," Opt. Express 12, 2568-2573 (2004).
    [CrossRef] [PubMed]
  4. S. He, R. Liao, and V. Romanov, "Explicit formulas for the identification of a small defect in a planar waveguide," J. Opt. Soc. Am. A 22, 1414-1419 (2005).
    [CrossRef]
  5. M. Huang and X. Yan, "Analytical solutions to estimate the stress induced polarization shift and the temperature sensitivity of optical waveguides," J. Appl. Phys. 95, 2820-2826 (2004).
    [CrossRef]
  6. A. Sharma, "Analysis of integrated optical waveguides: variational method and effective-index method with built-in perturbation correction," J. Opt. Soc. Am. A 18, 1383-1387 (2001).
    [CrossRef]
  7. S. He, X. Y. Ao, and V. Romanov, "General properties of N×M self-images in a strongly confined rectangular waveguide," Appl. Opt. 42, 4855-4859 (2003).
    [CrossRef] [PubMed]
  8. R. F. Harrington, Field Computation by Moment Methods (IEEE, New York, 1993).
    [CrossRef]

2005 (1)

2004 (2)

W. K. Kim, W. Yang, and H. Lee, "Effects of parasitic modes in high-speed LiNbO3 optical modulators," Opt. Express 12, 2568-2573 (2004).
[CrossRef] [PubMed]

M. Huang and X. Yan, "Analytical solutions to estimate the stress induced polarization shift and the temperature sensitivity of optical waveguides," J. Appl. Phys. 95, 2820-2826 (2004).
[CrossRef]

2003 (1)

2001 (1)

1996 (1)

M. K. Smit and C. V. Dam, "PHASAR-based WDM devices: principles, design and applications," IEE J. Sel. Top. Quantum Electron. 2, 236-250 (1996).
[CrossRef]

1995 (1)

L. B. Soldano and E.C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Ao, X. Y.

Dam, C. V.

M. K. Smit and C. V. Dam, "PHASAR-based WDM devices: principles, design and applications," IEE J. Sel. Top. Quantum Electron. 2, 236-250 (1996).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (IEEE, New York, 1993).
[CrossRef]

He, S.

Huang, M.

M. Huang and X. Yan, "Analytical solutions to estimate the stress induced polarization shift and the temperature sensitivity of optical waveguides," J. Appl. Phys. 95, 2820-2826 (2004).
[CrossRef]

Kim, W. K.

Lee, H.

Liao, R.

Pennings, E.C. M.

L. B. Soldano and E.C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Romanov, V.

Sharma, A.

Smit, M. K.

M. K. Smit and C. V. Dam, "PHASAR-based WDM devices: principles, design and applications," IEE J. Sel. Top. Quantum Electron. 2, 236-250 (1996).
[CrossRef]

Soldano, L. B.

L. B. Soldano and E.C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

Yan, X.

M. Huang and X. Yan, "Analytical solutions to estimate the stress induced polarization shift and the temperature sensitivity of optical waveguides," J. Appl. Phys. 95, 2820-2826 (2004).
[CrossRef]

Yang, W.

Appl. Opt. (1)

IEE J. Sel. Top. Quantum Electron. (1)

M. K. Smit and C. V. Dam, "PHASAR-based WDM devices: principles, design and applications," IEE J. Sel. Top. Quantum Electron. 2, 236-250 (1996).
[CrossRef]

J. Appl. Phys. (1)

M. Huang and X. Yan, "Analytical solutions to estimate the stress induced polarization shift and the temperature sensitivity of optical waveguides," J. Appl. Phys. 95, 2820-2826 (2004).
[CrossRef]

J. Lightwave Technol. (1)

L. B. Soldano and E.C. M. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Other (1)

R. F. Harrington, Field Computation by Moment Methods (IEEE, New York, 1993).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry and coordinates for the waveguide and the thin-strip defect (the thick vertical bar).

Fig. 2
Fig. 2

Fields (a) V s ( x , L , K ) and (b) V s ( x , L , K ) at the two end surfaces with K = 13.9851 μ m 1 (corresponding to λ = 1.55 μ m ).

Fig. 3
Fig. 3

Relative reconstruction error of defect product δ n c as the width δ of the thin-strip defect increases.

Fig. 4
Fig. 4

Relative reconstruction error of defect location z 0 as the width δ of the thin-strip defect increases.

Fig. 5
Fig. 5

Reconstruction errors for product δ n c and location z 0 of the thin-strip defect.

Fig. 6
Fig. 6

Relative reconstruction errors for product δ n c as the width δ of the thin-strip defect increases.

Fig. 7
Fig. 7

Relative reconstruction errors for product δ n c as the location z 0 of the thin-strip defect varies.

Equations (62)

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Δ x , z V ( x , z , K ) + K 2 [ 1 + n ( z ) ] V ( x , z , K ) = 0 ,
V ( 0 , z , K ) = V ( h , z , K ) = 0 ,
n ( z ) = { n c z [ z 0 , z 0 + δ ] 0 otherwise } ,
V inc ( x , z , K ) = sin ( π x h ) exp ( i γ 1 z ) ,
Δ x , z V inc ( x , z , K ) + K 2 V inc ( x , z , K ) = 0 ,
V inc ( 0 , z , K ) = V inc ( h , z , K ) = 0 .
Δ x , z V s ( x , z , K ) + K 2 V s ( x , z , K ) + K 2 n ( z ) V ( x , z , K ) = 0 ,
V s ( 0 , z , K ) = V s ( h , z , K ) = 0 .
V s ( x , z , K ) = V inc ( x , z , K ) ψ ( z , K ) .
[ ψ ( z , K ) ] + 2 i γ 1 [ ψ ( z , K ) ] + K 2 n ( z ) ψ ( z , K ) + K 2 n ( z ) = 0 .
ψ ( z , K ) = { C 1 ( K ) exp ( 2 i γ 1 z ) z < z 0 1 + C 2 ( K ) exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z } + C 3 ( K ) exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z } z [ z 0 , z 0 + δ ] C 4 ( K ) z > z 0 + δ } .
C 1 ( K ) exp ( 2 i γ 1 z 0 ) = 1 + C 2 ( K ) exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z 0 } + C 3 ( K ) exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z 0 } ,
2 γ 1 C 1 ( K ) exp ( 2 i γ 1 z 0 ) = C 2 ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z 0 } + C 3 ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z 0 } ,
C 4 ( K ) = 1 + C 2 ( K ) exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] ( z 0 + δ ) } + C 3 ( K ) exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] ( z 0 + δ ) } ,
0 = C 2 ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] ( z 0 + δ ) } + C 3 ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] ( z 0 + δ ) } .
C 2 ( K ) = C 3 ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] 1 × exp [ 2 i ( γ 1 2 + K 2 n c ) 1 2 ( z 0 + δ ) ] .
2 γ 1 = C 2 ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z 0 } + C 3 ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z 0 } .
C 3 ( K ) = C 2 ( 0 ) ( K ) exp { i [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] z 0 } ,
C 3 ( 0 ) ( K ) = 2 γ 1 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] { [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] 2 × exp [ 2 i δ ( γ 1 2 + K 2 n c ) 1 2 ] + [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] 2 } 1 .
C 2 ( K ) = C 2 ( 0 ) ( K ) exp { i [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] z 0 } ,
C 1 ( K ) = C 1 ( 0 ) ( K ) exp ( 2 i γ 1 z 0 ) ,
C 4 ( K ) = 1 + C 2 ( 0 ) ( K ) exp { i δ [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] } + C 3 ( 0 ) ( K ) exp { i δ [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] } ,
C 2 ( 0 ) ( K ) = C 3 ( 0 ) ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] 1 exp [ 2 i δ ( γ 1 2 + K 2 n c ) 1 2 ] ,
C 1 ( 0 ) ( K ) = 1 + C 2 ( 0 ) ( K ) + C 3 ( 0 ) ( K ) .
exp [ 2 i δ ( γ 1 2 + K 2 n c ) 1 2 ] 1 + 2 i δ ( γ 1 2 + K 2 n c ) 1 2 .
C 3 ( 0 ) ( K ) γ 1 ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × { 2 γ 1 i δ [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] 2 } 1 ( 1 2 ) ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × { 1 + [ i δ ( 2 γ 1 ) ] [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] 2 } ( 1 2 ) ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] + i δ α 3 ,
α 3 = ( γ 1 2 + K 2 n c ) 1 2 [ K 2 n c ( 4 γ 1 ) ] [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] .
C 2 ( 0 ) ( K ) C 3 ( 0 ) ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] 1 [ 1 + 2 i δ ( γ 1 2 + K 2 n c ) 1 2 ] ( 1 2 ) ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] + i δ α 2 ,
α 2 = ( γ 1 2 + K 2 n c ) 1 2 [ K 2 n c ( 4 γ 1 ) ] [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] .
C 4 ( K ) 1 + C 2 ( 0 ) ( K ) + C 3 ( 0 ) ( K ) + i δ { γ 1 [ C 2 ( 0 ) ( K ) + C 3 ( 0 ) ( K ) ] + [ C 3 ( 0 ) ( K ) C 2 ( 0 ) ( K ) ] ( γ 1 2 + K 2 n c ) 1 2 } = i δ K 2 n c [ 1 ( 2 γ 1 ) i δ ] .
C 4 ( K ) i δ n c K 2 ( 2 γ 1 ) ,
C 1 ( 0 ) ( K ) i δ ( α 2 + α 3 ) = i δ n c K 2 ( 2 γ 1 ) .
C 4 ( K ) C 1 ( 0 ) ( K ) ,
C 1 ( K ) C 4 ( K ) exp ( 2 i γ 1 z 0 ) .
δ n c 2 γ 1 C 4 ( K ) ( i K 2 ) .
z 0 1 2 [ γ 1 ( K ) γ 1 ( K ) ] arg [ C 1 ( K ) C 4 ( K ) C 4 ( K ) C 1 ( K ) ] .
exp [ 2 i δ ( γ 1 2 + K 2 n c ) 1 2 ] 1 + 2 i δ ( γ 1 2 + K 2 n c ) 1 2 2 δ 2 ( γ 1 2 + K 2 n c ) .
C 3 ( 0 ) ( K ) 1 2 ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] × { 1 ( 2 γ 1 ) 1 [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] 2 × [ i δ δ 2 ( γ 1 2 + K 2 n c ) 1 2 ] } 1 1 2 ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] + i δ α 3 ( K ) + δ 2 β 3 ( K ) ,
α 3 ( K ) = K 2 n c ( 4 γ 1 ) 1 ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] ,
β 3 ( K ) = K 2 n c ( 8 γ 1 2 ) 1 ( 2 γ 1 2 + K 2 n c ) ( γ 1 2 + K 2 n c ) 1 2 × [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] .
C 2 ( 0 ) ( K ) C 3 ( 0 ) ( K ) [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] 1 × [ 1 + 2 i δ ( γ 1 2 + K 2 n c ) 1 2 2 δ 2 ( γ 1 2 + K 2 n c ) ] 1 2 ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] + i δ α 2 ( K ) + δ 2 β 2 ( K ) ,
α 2 ( K ) = K 2 n c ( 4 γ 1 ) 1 ( γ 1 2 + K 2 n c ) 1 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] ,
β 2 ( K ) = ( 8 γ 1 2 ) 1 ( γ 1 2 + K 2 n c ) 1 2 × [ ( K 2 n c ) 2 + 8 γ 1 2 ( γ 1 2 + K 2 n c ) ] [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] .
C 4 ( K ) 1 + C 2 ( 0 ) ( K ) { 1 i δ [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] 1 2 δ 2 [ γ 1 + ( γ 1 2 + K 2 n c ) 1 2 ] 2 } + C 3 ( 0 ) ( K ) { 1 i δ [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] 1 2 δ 2 [ γ 1 ( γ 1 2 + K 2 n c ) 1 2 ] 2 } i δ α 4 ( K ) + δ 2 β 4 ( K ) ,
α 4 ( K ) = K 2 n c ( 2 γ 1 ) 1 ,
β 4 ( K ) = β 2 ( K ) + β 3 ( K ) + K 2 n c 2 = ( 4 γ 1 2 ) 1 ( K 2 n c ) 2 .
C 1 ( 0 ) ( K ) i δ α 1 ( K ) + δ 2 β 1 ( K ) ,
α 1 ( K ) = K 2 n c ( 2 γ 1 ) 1 ,
β 1 ( K ) = β 2 ( K ) + β 3 ( K ) = ( 4 γ 1 2 ) 1 ( K 2 n c ) 2 K 2 n c 2 .
V ( x , z , K ) V inc ( x , z , K ) = 1 + ψ ( z , K ) .
V ( x , z , K ) V inc ( x , z , K ) { 1 + [ i δ α 1 ( K ) + δ 2 β 1 ( K ) ] exp [ 2 i γ 1 ( z 0 z ) ] z < z 0 1 + i δ α 4 ( K ) + δ 2 β 4 ( K ) z > z 0 + δ } { 1 δ α 1 ( K ) sin [ 2 γ 1 ( z 0 z ) ] z < z 0 1 + δ 2 [ 2 β 4 ( K ) + α 4 2 ( K ) ] 2 z > z 0 + δ } .
A ( K ) V ( x , L , K ) sin ( π x h ) 1 δ α 1 ( K ) sin φ ( K ) ,
B ( K ) V ( x , L , K ) sin ( π x h ) 1 + δ 2 [ 2 β 4 ( K ) + α 4 2 ( K ) ] 2 ,
2 β 4 ( K ) + α 4 2 ( K ) = ( K 2 n c ) 2 [ 4 γ 1 2 ( K ) ] 1 .
δ n c K 2 γ 1 ( K ) { 8 [ 1 B ( K ) ] } 1 2 .
sin φ ( K ) A ( K ) 1 δ α 1 ( K ) 1 A ( K ) 1 { 2 [ 1 B ( K ) ] } 1 2 .
φ ( K ) φ ( K ) = 2 ( z 0 + L ) [ γ 1 ( K ) γ 1 ( K ) ] .
sin 2 φ ( K ) sin 2 φ ( K ) 2 sin φ ( K ) cos φ ( K ) [ φ ( K ) φ ( K ) ] ,
φ ( K ) φ ( K ) sin 2 φ ( K ) sin 2 φ ( K ) × 2 sin φ ( K ) 1 { 1 [ sin φ ( K ) ] 2 } 1 2 .
z 0 = L + φ ( K ) φ ( K ) 2 [ γ 1 ( K ) γ 1 ( K ) ] 1 ,
C 1 C 4 = exp ( 2 i γ 1 z rec ) ,
C 1 C 1 ( 0 ) = exp ( 2 i γ 1 z 0 ) .

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