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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 9 — Apr. 27, 2009
  • pp: 6982–6995
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Novel grating design approach by radiation modes coupling in nonlinear optical waveguides

A. Massaro, R. Cingolani, M. De Vittorio, and A. Passaseo  »View Author Affiliations


Optics Express, Vol. 17, Issue 9, pp. 6982-6995 (2009)
http://dx.doi.org/10.1364/OE.17.006982


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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

1. Introduction

Studies of χ(2) and the quasi-phase matching (QPM) technique promote research works on nonlinear optic (NLO) devices for ultrafast signal processing based on second-order nonlinearity. After the development of high-efficiency QPM- second harmonic generation (SHG) devices using ferroelectric crystal waveguides, the research was directed to implementation of a compact and efficient coherent light source by combining the SHG device with a semiconductor laser. Moreover a strong motivation of research on NLO devices has been the need for the development of all-optical wavelength converters for dense wavelength division multiplexing (DWDM) optical communication systems. In particular the second harmonic generation in χ(2) nonlinear process represents a good solution for DWDM applications in telecommunication systems. But this process requires high intensity input power and large interaction lengths for a good second harmonic (SH) conversion. For this reasons the introduction of tailored gratings which enhance the SH field is important.

A second grating (grating 2 of Fig. 1), placed at second the interface between the core and the substrate region, increases the fundamental power through the coupling between the substrate modes and the fundamental guided mode. On the same second interface, after the grating 2, we consider a third grating (grating 3 of Fig. 1) which enhances the power transferred to the SH mode through the coupling between the substrate modes and the SH guided mode by optimizing the conversion efficiency.

We resume the analysis of this work in the following steps: i) we analyze the modes involved in the designed structure including substrate modes and diffraction theory; ii) we study the coupled mode equations systems by combining the effect of each grating; iii) we provide the design criteria steps of the periodic structure; iv) finally we model the tailored waveguide through the time domain Hertzian potential formulation (HPF) used to solve nonlinear χ(2) processes [4

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]

], and by the finite difference time domain (FDTD) modelling. A good agreement between numerical and analytical results is checked.

Fig. 1. Multiple gratings design in nonlinear optical symmetrical waveguide.

2. Modes of the discontinuous periodic waveguide.

Hx(x,z)=1jωμzEy(x,z)
Hz(x,z)=1jωμxEy(x,z)
(1)

Ey(x,z)={kakψk(x)+0b(kx)φ(x;kx)dkx}ejβz
(2)

and ak and b(kx) are the amplitudes of the guided and continuum modes respectively. In the case of two TE guided modes (fundamental and SH modes) the guided field are represented by [5

5. T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides, (IEE Electromagnetic Waves Series 43, London1997). [CrossRef]

]

Eyω=aωψω(x)ejβωz
Ey2ω=a2ωψ2ω(x)ejβ2ωz
(3)

with

ψω,2ω(x)={cos(σω,2ωx)corecos(σω,2ωd/2)exp[Δω,2ω(xd/2)]substrate
(4)

and

aω,2ω=d1+1Δω,2ω
(5)

where the parameters σω,2ω and Δω,2ω are defined by the following equations

Δω,2ω=(n22(kω,2ω)2(βω,2ω)2)1/2
(6)
σω,2ω=(n12(kω,2ω)2(βω,2ω)2)1/2
(7)
Fig. 2. Modes in symmetrical planar waveguide.

f(z)=tsin(2πΛ2,3z)
(8)

As reported in Fig. 3 the sinusoidally deformed dielectric interface scatters the incident radiation. The arrows shown in the Fig. 3 indicate the propagation vector of the incident, reflected, and transmitted plane waves labeled i, r, and t. The shorter arrows labeled 1 and -1 indicate scattered plane waves. Two scattered waves emerge on either side of the interface. One set of scattered waves accompanies the reflected wave in the core region and another set accompanies the transmitted wave in the substrate region. The angle ϕ of the direction of each wave indicated in Fig. 3 is obtained from the equation

cos(ϕ)=β/nikω,2ω
(9)

where the propagation constant β indicates either, βω,2ω, β (+), or β (-), ni stands for n1 and n2, and kω,2ω= 2π/λFU,SH. We observe that if the period Λ2,3 satisfies the substrate mode condition

2πΛ2,3±βω,2ω<nikω,2ω
(10)

for i=1, and i=2, propagating scattered waves will exist. Far from the surface only the zero-order reflected and transmitted waves can be observed. Only in the immediate vicinity of the distorted surface is there any field distortion. Though the grating, this distorted field exchanges power with the zero-order field by increasing the guided power. The field of zero-order are the incident i, reflected r, and transmitted t plane waves corresponding to the fundamental and SH fields. In particular in the TE case the zero-order field is given by

eyω,2ω=Aiexp[j(σω,2ωx+βω,2ωz)]+Arexp[j(σω,2ωxβω,2ωz)]core region
(11)
eyω,2ω=Atexp[j(Δω,2ωx+βω,2ωz)]substrateregion
(12)

The total scattered field includes in each region the contributions of the scattered field a + γi and a - γi of Fig. 3, and is represented by [7

7. D. Marcuse, “Hollow dielectric waveguides for distributed feedback lasers,” IEEE J. Quantum Electron. 26, 1265–1276 (1972).

]

EyS=t2Aiσω,2ω(Δω,2ωσω,2ω)[exp[j(σ()xβ()z)]σ()+Δ()exp[j(σ(+)xβ(+)z)]σ(+)+Δ(+)]
coreregion
(13)
EyS=t2Aiσω,2ω(Δω,2ωσω,2ω)[exp[j(Δ()x+β()z)]σ()+Δ()exp[j(Δ(+)x+β(+)z)]σ(+)+Δ(+)]
substrateregion
(14)

with the definitions [7

7. D. Marcuse, “Hollow dielectric waveguides for distributed feedback lasers,” IEEE J. Quantum Electron. 26, 1265–1276 (1972).

]

β(±)=βω,2ω±2πΛ2,3
(15)
σ(±)=(n12(kω,2ω)2(β(±))2)1/2
(16)
Δ(±)=(n22(kω,2ω)2(β(±))2)1/2
(17)

where (15) indicates the phase matching between the zero order field and the scattered waves.

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.

3. Coupled mode theory.

{dAω(z)dz+j(2kLωcosKz)Aω(z)=j[kNLexp(j2δz)]*[Aω(z)]*A2ω(z)dA2ω(z)dz+j(2kL2ωcosKz)Aω(z)=jkNLexp(j2δz)[Aω(z)]2
(18)

with

kLω=ωε04[Eyω(x)]*ΔεEyω(x)dx
(19)
kL2ω=2ωε04[Ey2ω(x)]*ΔεEy2ω(x)dx
(20)
kNL=2ωε04[Ey2ω(x)]*χ˜NL[Eyω(x)]2dx
(21)

where

2δ=β2ω(2βω+K),K=2π/Λ1
(22)

is the phase mismatch between the fundamental and SH field,

Δε=(ε1ε2)sin(aπ)πexp[jKz]
(23)

is the first term of the permittivity Fourier series, and finally

χ˜NL=(χ1(2)χ2(2))sin(aπ)πexp[jKz]
(24)

is the first term of the nonlinear coefficient Fourier series, and a denotes the duty ratio of the period.

The grating 2 of Fig. 1 enhances the low pump fundamental field coupled into the waveguide. In this case coupling between the fundamental field and the scattered fields (generated by grating 2) is considered. In particular the equations to solve are the following [10

10. S. Ura, S. Murata, Y. Awtsuji, and K. Kintaka, “Design of resonance grating coupler,” Opt. Express 16, 12207–12213 (2008). [CrossRef] [PubMed]

]

{dAω(z)dz=jiaγi±(z)kγi,Aω*exp[j(β±(βω2π/Λ2)z]dβ±daγi±(z)dz=jAω(z)kγi,Aω*exp[j(β±(βω2π/Λ2)z]
(25)

with the coupling coefficient defined as

kγi,Aω=kSω=Eyω(x)ΔεEyS(x)dx
(26)

Finally the grating 3 optimize the SH conversion by coupling the SH field with the scattered field. In particular we take into account the system

{dA2ω(z)dz=jiaγi±(z)kγi,A2ω*exp[j(β±(βω2π/Λ3)z]dβ±daγi±(z)dz=jA2ω(z)kγi,A2ω*exp[j(β±(βω2π/Λ3)z]
(27)

where

kγi,A2ω=kS2ω=Ey2ω(x)ΔεEyS(x)dx
(28)

4. Design and results

2βω+2πΛ1=β2ωorΛ1=(λFU)/(N2ωNω)
(29)

where N ω,2ω are the effective refractive indices of both modes. The duty ratio a is assumed equal to ½. In Fig. 4 we show the normalized coupling coefficient kNL versus z obtained by applying the QPM condition. Usually the input power is of the order of 1–100 mW [1

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]

], but the SH conversion requires high intensity fundamental power. For this purpose the grating 2 is designed in order to enhance the fundamental power inside the waveguide through the coupling with the scattered field generated by the fundamental mode Aω(z). In this case the set of equations to solve are (18) and (25) with the initial conditions Aω(0)=(P0)1/2 and A2m(0)= 0 where P0 is the input pump power. Moreover the grating 3 is added in order to optimize the SH conversion efficiency by increasing the SH coupled power. Concerning the grating 3 the set of equations to solve are (18) and (27) with the initial conditions Aω(0)= Aω(L1) and A2m(0)= A2m(L1). The period Λ2 and Λ3 satisfy the propagation condition (10) and provide a constructive interference given by the phase matching condition (15) between the incident and the scattered field. In particular we choose a strong coupling condition by analyzing the coupling coefficient versus the period and versus the z-direction. Figure 5 shows that regular coupling coefficients kSω,2ω (strong coupling) are obtained with Λ2=0.95*10-6 μm and Λ3=1*10-6 μm. The plots of Fig. 5(a) and Fig. 5(b) are obtained by evaluating the scattered fields EyS (see Fig. 6(a) and Fig. 6(b)) generated by the fundamental and SH modes. Figures 7 and 8 illustrates the normalized substrate coupling coefficients kSω and kS versus z-propagating direction. The SH conversion efficiency is given by η(z)=|A(z)|2/|Aω(0)2| [8

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

]. Figure 9 shows the analytical solution of the conversion efficiency η by considering the grating 1, the grating 1 with the effect of the grating 2, and finally the effect of the three grating together by fixing a reference length of L1=2mm. The input power used during the calculus is P0 =100 mW. A conversion efficiency of about 70 % is obtained after a length of z=10 mm. It is clear from the Fig. 9 that the grating 3 increases the efficiency of about 20 % (optimization) respect to the case of grating 1 combined with the grating 2. In this way is possible to reduce the total length of the device by overcoming the problems of losses and reflections along the propagating direction. Concerning the total length of the structure, we observe that the theoretical effective interaction length [8

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

] Leff≅2.4(Λ1/2a)1/2 =3.6 mm not considers the variation of the fundamental power along the longitudinal direction, and not takes into account the grating parameters h and t which characterize the power reflected on each step discontinuity, thus, in our practical case, a high interaction efficiency is obtained with grating lengths larger than Leff. Moreover the choice of the parameter h=t=0.025 μm is justified by the following considerations. As reported in Fig. 10, low reflection coefficients [5

5. T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides, (IEE Electromagnetic Waves Series 43, London1997). [CrossRef]

] Γω,2ω=(βd ω,2ω - βD ω,2ω)/(βd ω,2ω + βD ω,2ω) at each step discontinuity are obtained in the single SH mode region by decreasing the values of h. But a strong coupling is obtained in the single mode SH region by increasing the h values (see Fig. 11 where are reported the coupling coefficients versus h). Figure 12 justifies the final choice of h=t=0.025 μm: the figure analyzes the numerical and analytical efficiency η versus z for different h values by showing a good convergence between numerical HPF, numerical two dimensional (2D) FDTD, and analytical results. A good matching between the HPF and 2D FDTD spectra is observed in Fig. 13 where we calculate the discrete Fourier transform (DFT). The convergence of the solutions confirms the accuracy of the presented theoretical model. The numerical simulations are performed by parallel calculus with time steps Δt=1.66*10-6 sec., dimension of spatial cell Δz=0.05 μm, and sinusoidal source at λ0= 1.55 μm given by

E0fundamental=sin(ωfundamental·t·Δt)
(30)

The used numerical boundary conditions are the absorbing boundary conditions (ABCs) [13

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]

].

Fig. 4. Grating 1: normalized nonlinear coupling coefficient versus z.
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. (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. Grating 2: substrate coupling coefficient versus L1.
Fig. 8. Grating 3: substrate coupling coefficient versus L2.
Fig. 9. Analytical and HPF numerical results of the efficiency SH conversion versus z.
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. Coupling coefficients versus h calculated by assuming z=0.
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. 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.

5. Conclusion

References and links

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 μm 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, London1997). [CrossRef]

6.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic Press, New York1974).

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 and S. C. Hagness, Computational Electrodynamic: the Finite-difference Time-domain Method, (Arthec House Publishers, sec. ed., London2000), 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]

OCIS Codes
(000.4430) General : Numerical approximation and analysis
(310.0310) Thin films : Thin films

ToC Category:
Integrated Optics

History
Original Manuscript: January 7, 2009
Revised Manuscript: February 24, 2009
Manuscript Accepted: March 9, 2009
Published: April 13, 2009

Citation
A. Massaro, R. Congolani, M. De Vittorio, and A. Passaseo, "Novel grating design approach by radiation modes coupling in nonlinear optical waveguides," Opt. Express 17, 6982-6995 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-6982


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References

  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]

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