Abstract

Design and reconstruction of two-dimensional (2D) and three-dimensional photonic structures are usually carried out by forward simulations combined with optimization or intuition. Reconstruction by means of layer stripping has been applied in seismic processing as well as in design and characterization of one-dimensional photonic structures such as fiber Bragg gratings. Layer stripping is based on causality, where the earliest scattered light is used to recover the structure layer by layer. Our setup is a 2D layered nonmagnetic structure probed by plane-polarized harmonic waves entering normal to the layers. It is assumed that the dielectric permittivity in each layer only varies orthogonal to the polarization. Based on obtained reflectance data covering a suitable frequency interval, time-localized pulse data are synthesized and applied to reconstruct the refractive index profile in the leftmost layer by identifying the local, time-domain Fresnel reflection at each point. Once the first layer is known, its impact on the reflectance data is stripped off and the procedure repeated for the next layer. Through numerical simulations it will be demonstrated that it is possible to reconstruct structures consisting of several layers. The impact of evanescent modes and limited bandwidth is discussed.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  3. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).
  4. A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
    [CrossRef]
  5. A. E. Yagle and B. C. Levy, “Layer-stripping solutions of multidimensional scattering problems,” J. Math. Phys. 27, 1701–1710 (1986).
    [CrossRef]
  6. J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
    [CrossRef]
  7. O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
    [CrossRef]
  8. A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Statist. Comput. 7, 1331–1349(1986).
    [CrossRef]
  9. J. Skaar and R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).
    [CrossRef]

2007 (1)

O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
[CrossRef]

2002 (1)

2001 (1)

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

1986 (2)

A. E. Yagle and B. C. Levy, “Layer-stripping solutions of multidimensional scattering problems,” J. Math. Phys. 27, 1701–1710 (1986).
[CrossRef]

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Statist. Comput. 7, 1331–1349(1986).
[CrossRef]

1985 (1)

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Bruckstein, A. M.

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Statist. Comput. 7, 1331–1349(1986).
[CrossRef]

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Erdogan, T.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Feced, R.

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Kailath, T.

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Statist. Comput. 7, 1331–1349(1986).
[CrossRef]

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Koltracht, I.

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Statist. Comput. 7, 1331–1349(1986).
[CrossRef]

Levy, B. C.

A. E. Yagle and B. C. Levy, “Layer-stripping solutions of multidimensional scattering problems,” J. Math. Phys. 27, 1701–1710 (1986).
[CrossRef]

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Skaar, J.

O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
[CrossRef]

J. Skaar and R. Feced, “Reconstruction of gratings from noisy reflection data,” J. Opt. Soc. Am. A 19, 2229–2237 (2002).
[CrossRef]

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Waagaard, O. H.

O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
[CrossRef]

Wang, L.

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yagle, A. E.

A. E. Yagle and B. C. Levy, “Layer-stripping solutions of multidimensional scattering problems,” J. Math. Phys. 27, 1701–1710 (1986).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Skaar, L. Wang, and T. Erdogan, “On the synthesis of fiber Bragg gratings by layer peeling,” IEEE J. Quantum Electron. 37, 165–173 (2001).
[CrossRef]

J. Math. Phys. (1)

A. E. Yagle and B. C. Levy, “Layer-stripping solutions of multidimensional scattering problems,” J. Math. Phys. 27, 1701–1710 (1986).
[CrossRef]

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

Phys. Rev. Lett. (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

SIAM J. Appl. Math. (2)

A. M. Bruckstein, B. C. Levy, and T. Kailath, “Differential methods in inverse scattering,” SIAM J. Appl. Math. 45, 312–335 (1985).
[CrossRef]

O. H. Waagaard and J. Skaar, “Inverse scattering in multimode structures,” SIAM J. Appl. Math. 68, 311–333 (2007).
[CrossRef]

SIAM J. Sci. Statist. Comput. (1)

A. M. Bruckstein, I. Koltracht, and T. Kailath, “Inverse scattering with noisy data,” SIAM J. Sci. Statist. Comput. 7, 1331–1349(1986).
[CrossRef]

Other (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(a) A plane wave pulse is incident to a layered 2D structure. For the structure, different colors indicate different refractive indices. (b) Immediately after t = 0 , the field has only been affected by the first layer; thus, we may identify the first layer from the first part of the reflected field in the time domain.

Fig. 2
Fig. 2

Reconstruction of the second layer for different window functions, in (a)  n 2 2 = 1.2 and in (b)  n 2 2 = 2.0 . Other parameters: M = 300 , L = 100 , N ω = 100 , ω [ 9 , 19 ] , Δ z = π / 2 , n 1 2 = 1.0 . Note that only parts of the computational domain L are shown in the plots.

Fig. 3
Fig. 3

Reconstruction of the second layer for different ω 1 and ω 2 , but with constant bandwidth. In (a)  n 2 2 = 1.05 , and in (b)  n 2 2 = 2.0 . Other parameters: M = 300 , L = 100 , N ω = 100 , Δ z = π / 2 , n 1 2 = 1.0 .

Fig. 4
Fig. 4

Reconstruction of (a) the first layer, and (b) the second layer, for different choices of n 2 . Other parameters: M = 300 , L = 100 , ω [ 9 , 19 ] , N ω = 100 , Δ z = π / 2 , n 1 2 = 1.0 . The solid black lines represent the exact permittivity.

Fig. 5
Fig. 5

Error in each layer, given as | ϵ comp ϵ | . Parameters: M = 300 , L = 100 , ω [ 9 , 19 ] , N ω = 100 , Δ z = π / 2 , n 1 2 = 1.0 , n 2 2 = 1.05 .

Fig. 6
Fig. 6

Reconstruction of the square function (a) for the first layer, and (b) for the second layer. The computations were done for different ω 1 and ω 2 , but with constant bandwidth. Other parameters: M = 300 , L = 100 , N ω = 100 , Δ z = π / 2 , n 1 2 = 1.0 , n 2 2 = 1.05 .

Equations (57)

Equations on this page are rendered with MathJax. Learn more.

E ( x , t ) = 1 2 π e ( x , z , ω ) e i ω t d ω ,
× e ( x , z , ω ) i ω μ 0 h ( x , z , ω ) = 0 ,
× h ( x , z , ω ) + i ω ϵ 0 ϵ ( x , z ) e ( x , z , ω ) = 0 ,
· ( ϵ ( x , z ) e ( x , z , ω ) ) = 0 ,
· h ( x , z , ω ) = 0 .
ϵ 1 e 1 = ϵ 2 e 2 ,
e 1 = e 2 ,
h 1 = h 2 .
e ( x , z ) = e ( x , z ) y ^ ,
h ( x , z ) = 1 i ω μ 0 ( e ( x , z ) z x ^ + e ( x , z ) x z ^ ) ,
2 e x 2 + 2 e z 2 + ϵ ( x , z ) k 2 e = 0 ,
e ( x , z ) = e ( x , z ) y ^ = m E ( m ) ( z ) exp ( i k x ( m ) x ) y ^ ,
ϵ ( x , z ) = m ϵ ( m ) ( z ) exp ( i k x ( m ) x ) ,
ϵ ( m ) ( z ) = 1 L 0 L ϵ ( x , z ) exp ( i k x ( m ) x ) d x .
d 2 E ( m ) ( z ) d z 2 ( k x ( m ) ) 2 E ( m ) ( z ) + k 2 m ϵ ( m m ) ( z ) E ( m ) ( z ) = 0 ,
d 2 E ( z ) d z 2 + ( k z 2 + V ( z ) ) E ( z ) = 0 ,
V ( z ) = k 2 I + k 2 [ ϵ ( 0 ) ϵ ( 1 ) ϵ ( 2 ) ϵ ( 1 ) ϵ ( 0 ) ϵ ( 1 ) ϵ ( 2 ) ϵ ( 1 ) ϵ ( 0 ) ] .
d E + ( z ) d z = i k z E + ( z ) + i ( 2 k z ) 1 V ( z ) ( E + ( z ) + E ( z ) ) ,
d E ( z ) d z = i k z E ( z ) i ( 2 k z ) 1 V ( z ) ( E + ( z ) + E ( z ) ) .
Ψ ( z ) = [ E + ( z ) E ( z ) ] , C ( z ) = [ i k z + i ( 2 k z ) 1 V ( z ) i ( 2 k z ) 1 V ( z ) i ( 2 k z ) 1 V ( z ) i k z i ( 2 k z ) 1 V ( z ) ] ,
d Ψ ( z ) d z = C ( z ) Ψ ( z ) .
Ψ ( z b ) = exp [ ( z b z a ) C ] Ψ ( z a )
M i = exp ( Δ i C i ) ,
Ψ ( z i ) = M i Ψ ( z i 1 ) .
Ψ ( z N ) = i = N 0 M i Ψ ( z 0 ) = M Ψ ( z 0 ) .
Ψ ( z 0 ) = [ E + ( z 0 ) E ( z 0 ) ] ,
Ψ ( z N ) = [ E + ( z N ) 0 ] ,
M = [ M 11 M 12 M 21 M 22 ] ,
[ E + ( z N ) 0 ] = [ M 11 M 12 M 21 M 22 ] [ E + ( z 0 ) E ( z 0 ) ] ,
E ( z 0 ) = R E + ( z 0 ) ,
E + ( z N ) = T E + ( z 0 ) ,
R = M 22 1 M 21 ,
T = M 11 M 12 M 22 1 M 21 .
[ T 0 ] = M [ I R ] .
F ( t z c 0 ) + R 1 ( x ) F ( t + z c 0 ) = T 1 ( x ) F ( t z c 1 ( x ) ) ,
1 c 0 F ( t z c 0 ) + R 1 ( x ) c 0 F ( t + z c 0 ) = T 1 ( x ) c 1 ( x ) F ( t z c 1 ( x ) ) ,
1 + R 1 ( x ) = T 1 ( x ) ,
1 + R 1 ( x ) = T 1 ( x ) c 0 c 1 ( x ) ,
ϵ 1 ( x ) = ϵ 0 ( 1 R 1 ( x ) 1 + R 1 ( x ) ) 2 .
[ E + ( z 1 ) E ( z 1 ) ] = M 1 [ E + ( z 0 ) E ( z 0 ) ] = M 1 [ I R ] .
E + ( z N ) = T ˜ E + ( z 1 ) ,
E ( z 1 ) = R ˜ E + ( z 1 ) ,
R ˜ = E ( z 1 ) E + ( z 1 ) 1 .
r ( x , ω ) = m R ( k x ( m ) = 0 , k x ( m ) , ω ) e i k x ( m ) x ,
k x ( m ) = 2 π m L , m Z , and ω [ ω 1 , ω 2 ] .
R ( x , t ) = 1 2 π r ( x , ω ) e i ω t d ω .
R w ( x , t ) = 1 2 π ω 1 ω 2 r ( x , ω ) W ( ω ) e i ω t d ω .
R 1 ( x ) = R w ( x , 0 ) w ( 0 ) = ω 1 ω 2 r ( x , ω ) W ( ω ) d ω W ( ω ) d ω .
2 π ω 2 ω 1 2 Δ 1 c 1 .
Δ x = L M ,
max k x = π M L .
k z = k 2 k x 2 = ω 2 k x 2 ,
ϵ ( x ) = η ± γ cos ( x 2 ) ,
min ϵ = n 1 2 = 1.0 ,
max ϵ = n 2 2 ,
ϵ 1 ( x ) = { n 2 2 , 0 x < π n 1 2 , π x < 2 π ,
ϵ 2 ( x ) = { n 1 2 , 0 x < π n 2 2 , π x < 2 π .

Metrics