Abstract

In integrated optics the radiation modes represent a negative aspect regarding the propagation of guided modes. They characterize the losses of the substrate region but can contribute to enhance the guided modes by considering the coupling through properly designed gratings arranged at the core/substrate interface. By tailored gratings, the radiation modes become propagating modes and increase the guided power inside the waveguide guiding region. This enhancement is useful especially in low intensity processes such as second harmonic χ(2) conversion process. For this purpose, we analyze accurately the radiation modes contribution in a χ(2) GaAs/AlGaAs nonlinear waveguide where second harmonic signal is characterized by a low power intensity. This analysis considers a new design approach of multiple grating which enhances a fundamental guided mode at λFU =1.55 μm and a codirectional second harmonic guided mode at λSH =0.775 μm. In particular we analyze the second harmonic conversion efficiency by studying the coupling effect of three gratings. The combined effects of the gratings provide an efficient second harmonic field conversion. Design considerations, based on the coupled mode equations analysis, are theoretically discussed. A good agreement between analytical and numerical results is observed.

© 2009 Optical Society of America

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References

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  1. E. U. Rafailov, P. L. Alvarez, C. T. A. Brown, W. Sibbett, R. M. De la Rue, P. Millar, D. A. Yanson, J. S. Roberts, and P. A. Houston, "Second-harmonic generation from a first-order quasi-phase-matched GaAs/AlGaAs waveguide crystal," Opt. Lett. 26, 1984-1986 (2001)..
    [CrossRef]
  2. X. Yu, L. Scaccabarozzi, J. S. Harris, P. S. Kuo, and M. M. Fejer, "Efficient continuous wave second harmonic generation pumped at 1.55 ?m in quasi-phase-matched AlGaAs waveguides," Opt. Express 13, 10742-10748 (2005).
    [CrossRef] [PubMed]
  3. X. Yu, L. Scaccabarozzi, A. C. Lin, M. M. Fejer, and J. S. Harris, "Growth of GaAs with orientation-patterned structures for nonlinear optics," J. Cryst. Growth 301, 163-167 (2007).
    [CrossRef]
  4. A. Massaro, V. Tasco, M. T. Todaro, R. Cingolani, M. De Vittorio, and A. Passaseo, "Scalar time domain modeling and coupling of second harmonic generation process in GaAs discontinuous optical waveguide," Opt. Express 16, 14496-14511 (2008).
    [CrossRef] [PubMed]
  5. T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides, (IEE Electromagnetic Waves Series 43, London 1997).
    [CrossRef]
  6. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York 1974).
  7. D. Marcuse, "Hollow dielectric waveguides for distributed feedback lasers," IEEE J. Quantum Electron. 26, 1265-1276 (1972).
  8. T. Suhara, and M. Fujimura, Waveguide Nonlinear-Optic Devices (Berlin: Springer, 2003).
  9. T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 8, 661-669 (1972).
  10. S. Ura, S. Murata, Y. Awtsuji, and K. Kintaka, "Design of resonance grating coupler," Opt. Express 16, 12207-12213 (2008).
    [CrossRef] [PubMed]
  11. A. Massaro, and T. Rozzi, "Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation," Opt. Express 14, 2027-2036 (2006).
    [CrossRef] [PubMed]
  12. A. Taflove, S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London 2000), ch. 2,3,4,7.
  13. G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
    [CrossRef]

2008 (2)

2007 (1)

X. Yu, L. Scaccabarozzi, A. C. Lin, M. M. Fejer, and J. S. Harris, "Growth of GaAs with orientation-patterned structures for nonlinear optics," J. Cryst. Growth 301, 163-167 (2007).
[CrossRef]

2006 (1)

2005 (1)

2001 (1)

1981 (1)

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

1972 (2)

D. Marcuse, "Hollow dielectric waveguides for distributed feedback lasers," IEEE J. Quantum Electron. 26, 1265-1276 (1972).

T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 8, 661-669 (1972).

Alvarez, P. L.

Awtsuji, Y.

Brown, C. T. A.

Cingolani, R.

De la Rue, R. M.

De Vittorio, M.

Fejer, M. M.

X. Yu, L. Scaccabarozzi, A. C. Lin, M. M. Fejer, and J. S. Harris, "Growth of GaAs with orientation-patterned structures for nonlinear optics," J. Cryst. Growth 301, 163-167 (2007).
[CrossRef]

X. Yu, L. Scaccabarozzi, J. S. Harris, P. S. Kuo, and M. M. Fejer, "Efficient continuous wave second harmonic generation pumped at 1.55 ?m in quasi-phase-matched AlGaAs waveguides," Opt. Express 13, 10742-10748 (2005).
[CrossRef] [PubMed]

Harris, J. S.

X. Yu, L. Scaccabarozzi, A. C. Lin, M. M. Fejer, and J. S. Harris, "Growth of GaAs with orientation-patterned structures for nonlinear optics," J. Cryst. Growth 301, 163-167 (2007).
[CrossRef]

X. Yu, L. Scaccabarozzi, J. S. Harris, P. S. Kuo, and M. M. Fejer, "Efficient continuous wave second harmonic generation pumped at 1.55 ?m in quasi-phase-matched AlGaAs waveguides," Opt. Express 13, 10742-10748 (2005).
[CrossRef] [PubMed]

Houston, P. A.

Kintaka, K.

Kuo, P. S.

Lin, A. C.

X. Yu, L. Scaccabarozzi, A. C. Lin, M. M. Fejer, and J. S. Harris, "Growth of GaAs with orientation-patterned structures for nonlinear optics," J. Cryst. Growth 301, 163-167 (2007).
[CrossRef]

Marcuse, D.

D. Marcuse, "Hollow dielectric waveguides for distributed feedback lasers," IEEE J. Quantum Electron. 26, 1265-1276 (1972).

Massaro, A.

Millar, P.

Mur, G.

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

Murata, S.

Nishihara, H.

T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 8, 661-669 (1972).

Passaseo, A.

Rafailov, E. U.

Roberts, J. S.

Rozzi, T.

Scaccabarozzi, L.

X. Yu, L. Scaccabarozzi, A. C. Lin, M. M. Fejer, and J. S. Harris, "Growth of GaAs with orientation-patterned structures for nonlinear optics," J. Cryst. Growth 301, 163-167 (2007).
[CrossRef]

X. Yu, L. Scaccabarozzi, J. S. Harris, P. S. Kuo, and M. M. Fejer, "Efficient continuous wave second harmonic generation pumped at 1.55 ?m in quasi-phase-matched AlGaAs waveguides," Opt. Express 13, 10742-10748 (2005).
[CrossRef] [PubMed]

Sibbett, W.

Suhara, T.

T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 8, 661-669 (1972).

Tasco, V.

Todaro, M. T.

Ura, S.

Yanson, D. A.

Yu, X.

X. Yu, L. Scaccabarozzi, A. C. Lin, M. M. Fejer, and J. S. Harris, "Growth of GaAs with orientation-patterned structures for nonlinear optics," J. Cryst. Growth 301, 163-167 (2007).
[CrossRef]

X. Yu, L. Scaccabarozzi, J. S. Harris, P. S. Kuo, and M. M. Fejer, "Efficient continuous wave second harmonic generation pumped at 1.55 ?m in quasi-phase-matched AlGaAs waveguides," Opt. Express 13, 10742-10748 (2005).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (2)

D. Marcuse, "Hollow dielectric waveguides for distributed feedback lasers," IEEE J. Quantum Electron. 26, 1265-1276 (1972).

T. Suhara, and H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating," IEEE J. Quantum Electron. 8, 661-669 (1972).

IEEE Trans. Electromagn. Compat. (1)

G. Mur, "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations," IEEE Trans. Electromagn. Compat. 23, 377-382 (1981).
[CrossRef]

J. Cryst. Growth (1)

X. Yu, L. Scaccabarozzi, A. C. Lin, M. M. Fejer, and J. S. Harris, "Growth of GaAs with orientation-patterned structures for nonlinear optics," J. Cryst. Growth 301, 163-167 (2007).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Other (4)

A. Taflove, S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London 2000), ch. 2,3,4,7.

T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides, (IEE Electromagnetic Waves Series 43, London 1997).
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York 1974).

T. Suhara, and M. Fujimura, Waveguide Nonlinear-Optic Devices (Berlin: Springer, 2003).

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Figures (13)

Fig. 1.
Fig. 1.

Multiple gratings design in nonlinear optical symmetrical waveguide.

Fig. 2.
Fig. 2.

Modes in symmetrical planar waveguide.

Fig. 3.
Fig. 3.

The sinusoidally distorted core-substrate interface acts like a phase grating. The incident i, reflected r, and diffracted rays are shown. The sinusoidal approximation is applied to the grating 2 and 3 of Fig. 1.

Fig. 4.
Fig. 4.

Grating 1: normalized nonlinear coupling coefficient versus z.

Fig. 5.
Fig. 5.

Substrate coupling coefficient: (a) coupling coefficient versus Λ2; (b) coupling coefficient versus Λ3. Strong couplings are obtained with Λ2 = 0.95 μm and Λ3 = 1 μm, where the coupling coefficients are regular respect to the z- direction.

Fig. 6.
Fig. 6.

(a) Scattered field generated by the fundamental mode. (b) Scattered field generated by the SH field. The coordinate system is the same of Fig. 1.

Fig. 7.
Fig. 7.

Grating 2: substrate coupling coefficient versus L1.

Fig. 8.
Fig. 8.

Grating 3: substrate coupling coefficient versus L2.

Fig. 9.
Fig. 9.

Analytical and HPF numerical results of the efficiency SH conversion versus z.

Fig. 10.
Fig. 10.

Reflection coefficients Γω,2ω=(βd ω,2ω - βD ω,2ω)/(βd ω,2ω + βD ω,2ω) of the propagating modes at each step discontinuity. Inset: schematic diagram of a step discontinuity. The propagation constants βω,2ω d and βω,2ω D are related to region with core thickness d and D, respectively.

Fig. 11.
Fig. 11.

Coupling coefficients versus h calculated by assuming z=0.

Fig. 12.
Fig. 12.

Structure with grating 1, 2 and 3: analytical, 2D-FDTD and HPF results of the efficiency SH conversion versus z for different h values. The analytical results are averaged values.

Fig. 13.
Fig. 13.

Comparison between HPF and 2D-FDTD normalized spectra of the fundamental and second harmonic modes. Inset: time evolution of the fundamental and second harmonic normalized fields generated inside the waveguide of Fig. 1.

Equations (34)

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H x ( x , z ) = 1 jωμ z E y ( x , z )
H z ( x , z ) = 1 jωμ x E y ( x , z )
E y ( x , z ) = { k a k ψ k ( x ) + 0 b ( k x ) φ ( x ; k x ) d k x } e jβz
E y ω = a ω ψ ω ( x ) e j β ω z
E y 2 ω = a 2 ω ψ 2 ω ( x ) e j β 2 ω z
ψ ω , 2 ω ( x ) = { cos ( σ ω , 2 ω x ) core cos ( σ ω , 2 ω d / 2 ) exp [ Δ ω , 2 ω ( x d / 2 ) ] substrate
a ω , 2 ω = d 1 + 1 Δ ω , 2 ω
Δ ω , 2 ω = ( n 2 2 ( k ω , 2 ω ) 2 ( β ω , 2 ω ) 2 ) 1 / 2
σ ω , 2 ω = ( n 1 2 ( k ω , 2 ω ) 2 ( β ω , 2 ω ) 2 ) 1 / 2
f ( z ) = t sin ( 2 π Λ 2,3 z )
cos ( ϕ ) = β / n i k ω , 2 ω
2 π Λ 2,3 ± β ω , 2 ω < n i k ω , 2 ω
e y ω , 2 ω = A i exp [ j ( σ ω , 2 ω x + β ω , 2 ω z ) ] + A r exp [ j ( σ ω , 2 ω x β ω , 2 ω z ) ] core region
e y ω , 2 ω = A t exp [ j ( Δ ω , 2 ω x + β ω , 2 ω z ) ] substrate region
E y S = t 2 A i σ ω , 2 ω ( Δ ω , 2 ω σ ω , 2 ω ) [ exp [ j ( σ ( ) x β ( ) z ) ] σ ( ) + Δ ( ) exp [ j ( σ ( + ) x β ( + ) z ) ] σ ( + ) + Δ ( + ) ]
core region
E y S = t 2 A i σ ω , 2 ω ( Δ ω , 2 ω σ ω , 2 ω ) [ exp [ j ( Δ ( ) x + β ( ) z ) ] σ ( ) + Δ ( ) exp [ j ( Δ ( + ) x + β ( + ) z ) ] σ ( + ) + Δ ( + ) ]
substrate region
β ( ± ) = β ω , 2 ω ± 2 π Λ 2,3
σ ( ± ) = ( n 1 2 ( k ω , 2 ω ) 2 ( β ( ± ) ) 2 ) 1 / 2
Δ ( ± ) = ( n 2 2 ( k ω , 2 ω ) 2 ( β ( ± ) ) 2 ) 1 / 2
{ d A ω ( z ) dz + j ( 2 k L ω cos Kz ) A ω ( z ) = j [ k NL exp ( j 2 δz ) ] * [ A ω ( z ) ] * A 2 ω ( z ) d A 2 ω ( z ) dz + j ( 2 k L 2 ω cos Kz ) A ω ( z ) = j k NL exp ( j 2 δz ) [ A ω ( z ) ] 2
k L ω = ω ε 0 4 [ E y ω ( x ) ] * Δ ε E y ω ( x ) dx
k L 2 ω = 2 ω ε 0 4 [ E y 2 ω ( x ) ] * Δ ε E y 2 ω ( x ) dx
k NL = 2 ω ε 0 4 [ E y 2 ω ( x ) ] * χ ˜ NL [ E y ω ( x ) ] 2 dx
2 δ = β 2 ω ( 2 β ω + K ) , K = 2 π / Λ 1
Δ ε = ( ε 1 ε 2 ) sin ( a π ) π exp [ jKz ]
χ ˜ NL = ( χ 1 ( 2 ) χ 2 ( 2 ) ) sin ( a π ) π exp [ jKz ]
{ d A ω ( z ) dz = j i a γi ± ( z ) k γi , A ω * exp [ j ( β ± ( β ω 2 π / Λ 2 ) z ] d β ± d a γi ± ( z ) dz = j A ω ( z ) k γi , A ω * exp [ j ( β ± ( β ω 2 π / Λ 2 ) z ]
k γi , A ω = k S ω = E y ω ( x ) Δ ε E y S ( x ) dx
{ d A 2 ω ( z ) dz = j i a γi ± ( z ) k γi , A 2 ω * exp [ j ( β ± ( β ω 2 π / Λ 3 ) z ] d β ± d a γi ± ( z ) dz = j A 2 ω ( z ) k γi , A 2 ω * exp [ j ( β ± ( β ω 2 π / Λ 3 ) z ]
k γi , A 2 ω = k S 2 ω = E y 2 ω ( x ) Δε E y S ( x ) dx
2 β ω + 2 π Λ 1 = β 2 ω or Λ 1 = ( λ FU ) / ( N 2 ω N ω )
E 0 fundamental = sin ( ω fundamental · t · Δ t )

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