## Novel grating design approach by radiation modes coupling in nonlinear optical waveguides

Optics Express, Vol. 17, Issue 9, pp. 6982-6995 (2009)

http://dx.doi.org/10.1364/OE.17.006982

Acrobat PDF (915 KB)

### 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 λ_{FU} =1.55 μm and a codirectional second harmonic guided mode at λ_{SH} =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

^{(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.

^{(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]

## 2. Modes of the discontinuous periodic waveguide.

*E*(x,z) may be expressed as a modal expansion [5

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

*a*and

_{k}*b*(

*k*) 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

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

^{ω,2ω}and Δ

^{ω,2ω}are defined by the following equations

*ϕ*of the direction of each wave indicated in Fig. 3 is obtained from the equation

*β*indicates either,

*β*,

^{ω,2ω}*β*

^{(+)}, or

*β*

^{(-)},

*n*stands for

_{i}*n*and

_{1}*n*, and

_{2}*k*= 2π/λ

^{ω,2ω}_{FU,SH}. We observe that if the period Λ

_{2,3}satisfies the substrate mode condition

*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

*a*

^{+}

_{γi}and

*a*

^{-}

_{γi}of Fig. 3, and is represented by [7]

## 3. Coupled mode theory.

*a*denotes the duty ratio of the period.

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

## 4. Design and results

_{FU}=1.55 μm) is a polarized TE field. The GaAs core (

*n*(λ

_{1}_{FU}=1.55 μm)=3.374 and

*n*(λ

_{1}_{SH}=0.755 μm)=3.691) is characterized by a thickness d=0.22 μm and supports only a TE fundamental mode at λ

_{FU}=1.55 μm and only a SH TE mode at λ

_{FU}=0.775 μm. In this way all the TE power is matched with the two propagating modes and the modal dispersion is low. In order to minimize the reflections along the

*z*- propagating the direction and to conserve the single mode condition, the parameters h and t of Fig. 1 are fixed to the low value of 0.025 μm. The substrate material is Al

_{0.4}Ga

_{0.6}As (

*n*(λ

_{2}_{FU}=1.55 μm)=3.2 and

*n*(λ

_{2}_{SH}=0.755 μm)=3.4). The GaAs core, combined with the Al

_{0.4}Ga

_{0.6}As, characterizes the periodically switched nonlinearity (PSN) [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]

^{(2)}is modulated periodically along the direction of light propagation. We observe that in a generic case χ

^{(2)}is a tensor, however most semiconductor, which crystallize in zinc-blende structures, have a symmetry and their second-order susceptibility has a single nonzero independent component (for GaAs at the transparency region below the optical energy gap χ

_{x,y,z}

^{(2)}= 200 pm/V). The grating 1 couples the fundamental mode with the SH one. The period Λ

_{1}=2.31

^{*}10

^{-6}μm satisfies the QPM condition (phase mismatch δ=0):

*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

*k*versus

_{NL}*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]

*A*(z). In this case the set of equations to solve are (18) and (25) with the initial conditions

^{ω}*A*(0)=(P

^{ω}_{0})1/2 and

*A*(0)= 0 where P

^{2m}_{0}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

*A*(0)=

^{ω}*A*(L

^{ω}_{1}) and

*A*(0)=

^{2m}*A*(L

^{2m}_{1}). The period Λ

_{2}and Λ

_{3}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

*k*(strong coupling) are obtained with Λ

_{S}^{ω,2ω}_{2}=0.95

^{*}10

^{-6}μm and Λ

_{3}=1

^{*}10

^{-6}μm. The plots of Fig. 5(a) and Fig. 5(b) are obtained by evaluating the scattered fields

*E*(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

_{y}^{S}*k*and

_{S}^{ω}*k*versus

_{S}^{2ω}*z*-propagating direction. The SH conversion efficiency is given by η(z)=|

*A*(z)|

^{2ω}^{2}/|

*A*(0)

^{ω}^{2}| [8]. 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 L

_{1}=2mm. The input power used during the calculus is P

_{0}=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] L

_{eff}≅2.4(Λ

_{1}/2

*a*)

^{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 L

_{eff}. 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]

^{ω,2ω}=(β

_{d}

^{ω,2ω}- β

_{D}

^{ω,2ω})/(β

_{d}

^{ω,2ω}+ β

_{D}

^{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 Δ

*t*=1.66

^{*}10

^{-6}sec., dimension of spatial cell Δ

*z*=0.05 μm, and sinusoidal source at λ

_{0}= 1.55 μm given by

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]

## 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. |

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 |

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 |

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 |

5. | T. Rozzi and M. Mongiardo, |

6. | D. Marcuse, |

7. | D. Marcuse, “Hollow dielectric waveguides for distributed feedback lasers,” IEEE J. Quantum Electron. |

8. | T. Suhara and M. Fujimura, |

9. | T. Suhara and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped grating,” IEEE J. Quantum Electron. |

10. | S. Ura, S. Murata, Y. Awtsuji, and K. Kintaka, “Design of resonance grating coupler,” Opt. Express |

11. | A. Massaro and T. Rozzi, “Rigorous time-domain analysis of dielectric optical waveguides using Hertzian potentials formulation,” Opt. Express |

12. | A. Taflove and S. C. Hagness, |

13. | G. Mur, “Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic field equations,” IEEE Trans. Electromagn. Compat. |

**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

Sort: Year | Journal | Reset

### References

- 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]
- 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]
- 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]
- 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]
- 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).
- D. Marcuse, "Hollow dielectric waveguides for distributed feedback lasers," IEEE J. Quantum Electron. 26, 1265-1276 (1972).
- T. Suhara, and M. Fujimura, Waveguide Nonlinear-Optic Devices (Berlin: Springer, 2003).
- 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).
- S. Ura, S. Murata, Y. Awtsuji, and K. Kintaka, "Design of resonance grating coupler," Opt. Express 16, 12207-12213 (2008). [CrossRef] [PubMed]
- 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]
- 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.
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

### Figures

Fig. 1. |
Fig. 2. |
Fig. 3. |

Fig. 4. |
Fig. 5. |
Fig. 6. |

Fig. 7. |
Fig. 8. |
Fig. 9. |

Fig. 10. |
Fig. 11. |
Fig. 12. |

Fig. 13. |
||

« Previous Article | Next Article »

OSA is a member of CrossRef.