Abstract

In this paper we present an efficient method for designing discrete, nearly-uniform Bragg gratings in generic planar waveguides. Various schemes have already been proposed to design continuous Bragg gratings in optical fibers, but a general scheme for creating their discrete counterpart is still lacking. Taking a continuous Bragg grating as our starting point, we show that the same grating functionalities can also be realized in any planar waveguide by discretizing it into a series of air holes. The relationship between the two gratings is established in terms of grating strength and local grating period.

© 2005 Optical Society of America

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References

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  1. J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558–1560 (2003).
    [Crossref]
  2. J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
    [Crossref]
  3. H. P. Li, Y. L. Sheng, and Y. Li, et al. “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21, 2074–2083 (2003).
    [Crossref]
  4. L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939–941 (2003).
    [Crossref]
  5. M. Ibsen and R. Feced, “Fiber Bragg gratings for pure dispersion-slope compensation,” Opt. Lett. 28, 980–982 (2003).
    [Crossref] [PubMed]
  6. A. Yariv, “Coupled-wave formalism for optical waveguiding by transverse Bragg reflection,” Opt. Lett. 27, 936–938 (2002).
    [Crossref]
  7. W. Kuang and J.D. O’Brien, “Reducing the out-of-plane radiation loss of photonic crystal waveguides on high-index substrates,” Opt. Lett. 29, 860–862 (2004).
    [Crossref] [PubMed]
  8. M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. 22, 223–226 (1999).
    [Crossref]
  9. R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
    [Crossref]
  10. A. Yariv, Optics Electronics (4th edition, Saunders College Publishing, 1991).

2004 (1)

2003 (5)

J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558–1560 (2003).
[Crossref]

J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[Crossref]

H. P. Li, Y. L. Sheng, and Y. Li, et al. “Phased-only sampled fiber Bragg gratings for high-channel-count chromatic dispersion compensation,” J. Lightwave Technol. 21, 2074–2083 (2003).
[Crossref]

L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939–941 (2003).
[Crossref]

M. Ibsen and R. Feced, “Fiber Bragg gratings for pure dispersion-slope compensation,” Opt. Lett. 28, 980–982 (2003).
[Crossref] [PubMed]

2002 (1)

1999 (2)

M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. 22, 223–226 (1999).
[Crossref]

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[Crossref]

Chuang, K. P.

L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939–941 (2003).
[Crossref]

Feced, R.

M. Ibsen and R. Feced, “Fiber Bragg gratings for pure dispersion-slope compensation,” Opt. Lett. 28, 980–982 (2003).
[Crossref] [PubMed]

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[Crossref]

Ibsen, M.

Kuang, W.

Lai, Y.C.

L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939–941 (2003).
[Crossref]

Laso, M. A. G.

M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. 22, 223–226 (1999).
[Crossref]

Li, H. P.

Li, S. Y.

J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558–1560 (2003).
[Crossref]

Li, Y.

Muriel, M. A.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[Crossref]

O’Brien, J.D.

Sheng, Y. L.

Sheu, L. G.

L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939–941 (2003).
[Crossref]

Shum, P.

J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558–1560 (2003).
[Crossref]

Skaar, J.

J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[Crossref]

Waagaard, O. H.

J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[Crossref]

Yariv, A.

Zervas, M. N.

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[Crossref]

Zhang, J.

J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558–1560 (2003).
[Crossref]

IEEE J. Quantum Electron. (2)

J. Skaar and O. H. Waagaard, “Design and characterization of finite-length fiber gratings,” IEEE J. Quantum Electron. 39, 1238–1245 (2003).
[Crossref]

R. Feced, M. N. Zervas, and M. A. Muriel, “An efficient inverse scattering algorithm for the design of nonuniform fiber Bragg gratings,” IEEE J. Quantum Electron. 35, 1105–1115 (1999).
[Crossref]

IEEE Photonics Technol. Lett. (2)

J. Zhang, P. Shum, and S. Y. Li, et al., “Design and fabrication of flat-band long-period grating,” IEEE Photonics Technol. Lett. 15, 1558–1560 (2003).
[Crossref]

L. G. Sheu, K. P. Chuang, and Y.C. Lai, “Fiber bragg grating dispersion compensator by single-period overlap-step-scan exposure,” IEEE Photonics Technol. Lett. 15, 939–941 (2003).
[Crossref]

J. Lightwave Technol. (1)

Microwave Opt. Tech. Lett. (1)

M. A. G. Laso et al., “Analysis and design of 1-D photonic bandgap microstrip structures using a fiber grating model,” Microwave Opt. Tech. Lett. 22, 223–226 (1999).
[Crossref]

Opt. Lett. (3)

Other (1)

A. Yariv, Optics Electronics (4th edition, Saunders College Publishing, 1991).

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Figures (4)

Fig. 1.
Fig. 1.

Top view of a 3-D planar waveguide (the y-dimension is not shown). A series of air holes (with spacing ~Λ) drilled at the center of the waveguide serve as the perturbation. Note even though we have plotted here a conventional, index-confined rectangular waveguide, our analysis can also be extended to more exotic structures such as the multilayer Bragg waveguide or the 2-D photonic crystal slab waveguide.

Fig. 2.
Fig. 2.

Fourier spectrum of Δ(n 2). Valid when Δn(z) and θ(z) are slowly varying.

Fig. 3.
Fig. 3.

Fourier spectrum of Δ(ñ 2). Valid when am ’s and bm ’s are slowly varying. Note only the two resonant components at K 0 and -K 0 can provide coupling between the forward and backward propagating waves.

Fig. 4.
Fig. 4.

Local period at the mth air hole. The local period at a given air hole is defined to be the average of the distances to the two adjacent air holes, i.e., (Λ12)/2.

Equations (48)

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n ( z ) = n 0 + Δ n ( z ) cos ( K 0 z + θ ( z ) )
n ˜ ( x , y , z ) = n ˜ 0 + Δ n ˜ ( x , y , z )
Δ n ˜ ( x , y , z ) = δ ( x ) W ( y ) m a m δ ( z m Λ + b m )
Size of the mth air hole at z m Λ Δ n ( m Λ )
Local period of the air holes 2 π K 0 + θ ' ( z )
2 E x μ ε ( r ) 2 E x t 2 = μ 2 t 2 [ P pert ( r , t ) ]
P pert ( r , t ) Δ ( n 2 )
P pert Δ ( n 2 ) 2 n 0 Δ n ( z ) cos ( K 0 z + θ ( z ) ) .
Δ ( n 2 ) K 0 = n 0 π [ m Δ n ( m Λ ) e i θ ( m Λ ) sin [ ( z m Λ ) Γ ] z m Λ ] e i K 0 z
P pert Δ ( n ˜ 2 ) 2 n ˜ 0 Δ n ˜ ( x , y , z ) = 2 n ˜ 0 δ ( x ) W ( y ) m a m δ ( z m Λ + b m )
Δ ( n ˜ 2 ) K 0 = 2 n ˜ 0 π [ m a m e i b m K 0 sin [ ( z m Λ + b m ) Γ ] z m Λ + b m ] e i K 0 z .
a m = n 0 2 n ˜ 0 Δ n ( m Λ )
b m = θ ( m Λ ) K 0 .
Local period at the mth air hole = Λ 1 + Λ 2 2
= 2 Λ b m + 1 + b m 1 2
= Λ [ 1 θ [ ( m + 1 ) Λ ] θ [ ( m 1 ) Λ ] 4 π ]
Λ [ 1 2 Λ θ ' ( m Λ ) 4 π ]
= 2 π K 0 [ 1 θ ' K 0 ]
2 π K 0 + θ '
n ˜ ( x , y , z ) = n ˜ 0 + Δ n ˜ ( x , y , z ) = n ˜ 0 + γ δ ( x ) W ( y ) m a m δ ( z m Λ + b m )
Δ ( n 2 ) continuous = 2 n 0 Δ n ( z ) cos ( K 0 z + θ ( z ) )
Δ ( n ˜ 2 ) discrete = γ δ ( x ) W ( y ) [ 2 n 0 Δ n ( z ) cos ( K 0 z + θ ( z ) ) ]
e i ω t 2 [ b B E B K 0 β e i K 0 z 2 + b F E F K 0 β e i K 0 z 2 i K 0 b B E B e i K 0 z 2 + i K 0 b F E F e i K 0 z 2 ]
= μ 2 t 2 { γ δ ( x ) W ( y ) n 0 ε 0 Δ n ( z ) cos ( K 0 z + θ ( z ) ) [ ( b B E B e i K 0 z 2 e i ω t + b F E F e i K 0 z 2 e i ω t ) ] }
E m E n * d x d y = 2 ω μ k m δ m n
2 ω μ k m = γ t + t E m ( 0 , y ) 2 d y
γ = 2 ω μ k m t + t E m ( 0 , y ) 2 d y 2 ω 0 μ K 0 2 t + t E m ( 0 , y ) 2 d y = 2 μ c n ˜ 0 t + t E m ( 0 , y ) 2 d y
d b B ( z , β ) d z + i β b B ( z , β ) = q ( z ) b F ( z , β )
d b F ( z , β ) d z i β b F ( z , β ) = q * ( z ) b B ( z , β )
β = 2 k n 0 K 0 2
q ( z ) = i K 0 2 n 0 Δ n ( z ) 2 e j θ ( z )
E x = 1 2 e i ω t [ b B ( z , β ) E B ( x , y ) e i K 0 z 2 + b F ( z , β ) E F ( x , y ) e i K 0 z 2 ]
2 E x μ ε ( r ) 2 E x t 2 = μ 2 t 2 [ P pert ( r , t ) ]
e i ω t 2 [ b B e i K 0 z 2 ( K 0 2 4 E B + 2 E B x 2 + 2 E B y 2 + ω 2 μ ε E B ) +
b F e i K 0 z 2 ( K 0 2 4 E F + 2 E F x 2 + 2 E F y 2 + ω 2 μ ε E F )
i K 0 b B E B e i K 0 z 2 + i K 0 b F E F e i K 0 z 2 ]
= μ 2 t 2 P pert ( r , t )
b B K 0 2 b B ; b F K 0 2 b F .
( k n 0 ) 2 E B + 2 E B x 2 + 2 E B y 2 + ω 2 μ ε E B = 0
( k n 0 ) 2 = ( K 0 2 + β ) 2 K 0 2 4 + K 0 β
e i ω t 2 [ b B E B K 0 β e i K 0 z 2 + b F E F K 0 β e i K 0 z 2 i K 0 b B E B e i K 0 z 2 + i K 0 b F E F e i K 0 z 2 ]
= μ 2 t 2 P pert
= μ 2 t 2 [ Δ ( n 2 ) ε 0 E y ]
= μ 2 t 2 { 2 n 0 Δ n ( z ) cos ( K 0 z + θ ( z ) ) ε 0 × 1 2 [ ( b B E B e i K 0 z 2 e i ω t + b F E F e i K 0 z 2 e i ω t ) ] }
b B + i β b B = i n 0 k 0 ( ω 2 μ ε 0 n 0 2 ) Δ n ( z ) e i θ ( z ) b F .
ω 2 μ ε 0 n 0 2 = n 0 2 ω 2 c 2 = ( k n 0 ) 2 = ( K 0 2 + β ) 2 K 0 2 4 .
b B + i β b B = i K 0 4 n 0 Δ n ( z ) e i θ ( z ) b F = q ( z ) b F

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