## Spectrum broadening of high-efficiency second harmonic generation in cascaded photonic crystal microcavities |

Optics Express, Vol. 21, Issue 1, pp. 756-763 (2013)

http://dx.doi.org/10.1364/OE.21.000756

Acrobat PDF (1307 KB)

### Abstract

An effective approach is proposed to broaden the spectrum of high-efficiency second harmonic generation in a one-dimensional photonic crystal based on the cascaded structure. By controlling the thickness of the joint layer, it is possible to realize a flat-top or quasiflat-top impurity band centered at the fundamental wavelength due to mode splitting effect in coupled cavities. Simulation results reveal that the spectrum of generated second harmonic exhibits a hump-like or multi-peak profile with wavelength tuning. It is a salient feature that the spectral stability of efficiency enhancement could be greatly improved compared to the conventional Lorentzian profile while maintaining an ultrahigh *Q* factor. Such merit of spectra re-shaping can significantly relax the stringent requirements for ultrahigh-*Q* microcavities in practical applications, e.g. fabrication inaccuracy, thermal variation, and wavelength detuning.

© 2013 OSA

## 1. Introduction

1. H. Thyrrestrup, S. Smolka, L. Sapienza, and P. Lodahl, “Statistical theory of a quantum emitter strongly coupled to Anderson-Localized modes,” Phys. Rev. Lett. **108**(11), 113901 (2012). [CrossRef] [PubMed]

8. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science **284**(5421), 1819–1821 (1999). [CrossRef] [PubMed]

*Q*and ultra-small modal volume

*V*

_{m}. These two conditions pave a way to achieve dramatic enhancement of light-matter interaction over a compact space that is critical for optical nonlinear effects such as second harmonic generation (SHG). It is therefore expected to significantly enhance the frequency conversion efficiency, for instance, if spectrally tuning the fundamental frequency to a cavity/defect mode in a one-dimensional (1D) photonic crystal (PC) [9

9. B. Shi, Z. M. Jiang, and X. Wang, “Defective photonic crystals with greatly enhanced second-harmonic generation,” Opt. Lett. **26**(15), 1194–1196 (2001). [CrossRef] [PubMed]

6. F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. Qiu, J. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B **70**(24), 245109 (2004). [CrossRef]

11. Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express **14**(26), 12670–12678 (2006). [CrossRef] [PubMed]

12. S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B **65**(16), 165208 (2002). [CrossRef]

12. S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B **65**(16), 165208 (2002). [CrossRef]

16. E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett. **106**(20), 203902 (2011). [CrossRef] [PubMed]

12. S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B **65**(16), 165208 (2002). [CrossRef]

*cascaded*PCMC structure. The spectra re-shaping of cavity modes is simply controlled by the thickness of joint layers, instead of the size of each cavity, which offers great freedom to increase the transmission bandwidth and maintain a high-efficiency SHG process simultaneously. The improved spectral stability for the SHG process also has great potential applications in relaxing the critical or rigorous requirement for ultrahigh-

*Q*microcavities as it allows quite large tolerance in fabrication, design scale inaccuracy, thermal variation, and wavelength detuning.

## 2. Method and design

*D(LH)*

^{PN}*(Fig. 1(a) ) originates from a matrix of (HL)*

^{PN}*H(LH)*

^{PN}*, provided that the central layer H is changed into the cavity layer D (the geometric length is thus changed from*

^{PN}*d*

_{H}to

*d*

_{D}), where H and L stand for the high- and low-refractive-index layers, respectively. The structure of (HL)

*or (LH)*

^{PN}*is a distributed feedback Bragg reflector and has a period number of*

^{PN}*PN*. They are composed of alternately stacked H and L layers, satisfying the condition of a quarter-wavelength optical thickness of

*n*

_{H}

*d*

_{H}=

*n*

_{L}

*d*

_{L}=

*λ*

_{0}/4, where

*n*

_{H(L)}represents the refractive index of H(L) layer and

*λ*

_{0}is the central wavelength of the photonic bandgap of the Bragg reflectors. To generate a cavity mode at the wavelength of

*λ*

_{0}, the cavity layer D can be designed with a half-wavelength optical thickness (i.e.

*d*

_{D}= 2

*d*

_{H}). As shown in Fig. 1(b), a double-cascaded PCMC structure is formed by connecting two identical single PCMCs using a joint layer (layer J). If we cascade

*N*identical S-PCMCs through

*N*-1 joint layers with the same thickness of

*d*

_{J}, a multiple-cascaded PCMC can be constructed with the arrangement of (HL)

*D(LH)*

^{PN}*[J(HL)*

^{PN}*D(LH)*

^{PN}*]*

^{PN}*as depicted in Fig. 1(c). When*

^{N-1}*N*= 1, the proposed structure degenerates into a typical single PCMC. In this paper, we consider the cascaded PCMC structures formed by high-index dielectric layers LiNbO

_{3}embedded in a low-index background of the air, i.e. the materials of H/L/D/J layer are LiNbO

_{3}/air/LiNbO

_{3}/air, respectively. This kind of multilayer structure is able to be fabricated using the top-down lithography. The dispersion of LiNbO

_{3}is temperature dependent and the effective nonlinear coefficient

*d*

_{eff}is 43.9 pm/V as described in Ref. 6

6. F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. Qiu, J. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B **70**(24), 245109 (2004). [CrossRef]

*λ*

_{0}here is 1064 nm, a typical lasing generation from Nd:YAG and Nd:YVO

_{4}which are extensively used in many fields.

## 3. Results and discussion

*PN*= 5 and a normal incidence is considered for simplicity. When

*d*

_{J}= 0, the cavity modes appear in the transmission spectrum as separated Lorentzian profiles. As the air gap

*d*

_{J}varies from 0 to

*d*

_{L}, the antibonding mode located at the shorter resonance wavelength exhibits a red-shift from 1058.61 nm to

*λ*

_{0}, while the bonding mode is almost insensitive to the variation of

*d*

_{J}and keeps its resonance peak at the wavelength of

*λ*

_{0}. When

*d*

_{J}keeps increasing and deviates from

*d*

_{L}to 2

*d*

_{L}, the antibonding mode is standing at the wavelength of

*λ*

_{0}, while the bonding mode moves from

*λ*

_{0}to 1067.72 nm. In the special case of

*d*

_{J}=

*d*

_{L}, two cavity modes are almost degenerate at

*λ*

_{0}= 1064 nm and difficult to be distinguished. Under this condition, the original Lorentzian transmission profile becomes a flat-top window. The red and black dotted lines in Fig. 2 show the evolution of the two resonance modes with different thickness of joint layer (

*d*

_{J}), where two modes are continuously modulated. It is worth noting that one of the resonance peaks is always centered at

*λ*

_{0}= 1064 nm for any value of

*d*

_{J}, provided that the refractive index difference between layer J and background is rather small, which is crucial to observe flat-top unit transmission spectra. Otherwise, the antibonding and bonding modes can be clearly distinguished even in the case of

*d*

_{J}=

*d*

_{L}. Further simulations with

*d*

_{J}> 2

*d*

_{L}indicate that the mode splitting have periodicity of

*d*

_{J}with a period of 2

*d*

_{L}, implying the phase correlation of two cavity modes.

*d*

_{J}=

*d*

_{L}are further studied for different

*N*as shown in Fig. 3 , where the phase and transmission spectra of coupled cavity modes for three cascaded structures with

*N*= 2, 3, and 4, as well as a single PCMC (

*N*= 1) for reference are calculated. The phase of the modes in Fig. 3(a) is relative to that of the incident field, i.e. a phase of zero is assumed for the incident field. It can be seen that the phase near the antibonding mode drastically shifts from -π to + π, while it always remains zero for the bonding state. As mentioned above, in a double-cascaded PCMC with

*d*

_{J}=

*d*

_{L}, a flat-top window can be obtained in the transmission spectrum (red dotted line in Fig. 3(b)) due to cavity modes degeneration. The flat-top profile can be well analyzed by a differentiator model, which is synthesized as a linear superposition of original Lorentzian profiles and their successive derivatives [11

11. Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express **14**(26), 12670–12678 (2006). [CrossRef] [PubMed]

*N*= 3, a quasiflat-top resonance window appears around

*λ*

_{0}and three coupled modes can be distinguished by the two shallow valleys near the band edges (green solid line in Fig. 3(b)). The two side modes are antibonding states with quick phase shifts, and the middle resonance at

*λ*

_{0}belongs to bonding state with a phase close to zero. For a quadruple-cascaded PCMC with

*N*= 4, there are four modes in the spectrum (blue dashed line in Fig. 3(b)), corresponding to antibonding/bonding/antibonding/bonding states from shorter to longer wavelengths in the frequency range of our study. It is clear that when the value of

*N*increases, the coupled resonance peak gradually approaches to a rectangle shape but with deeper valleys near the impurity band edges, which may introduce non-negligible variation to the optical nonlinear process.

*N*) in the cascaded structure affects the conversion efficiency at the SH wavelength of 532 nm (i.e.

*λ*

_{0}/2). Figure 5(a) shows the dependence of the conversion efficiency on the incident FW intensity

*I*

_{0FW}for

*N*= 1, 2, 3, and 4 with

*d*

_{J}=

*d*

_{L}. As concluded in Refs. 6

6. F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. Qiu, J. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B **70**(24), 245109 (2004). [CrossRef]

19. R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. **31**(22), 3327–3329 (2006). [CrossRef] [PubMed]

21. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structures with deep gratings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **67**(1), 016606 (2003). [CrossRef] [PubMed]

19. R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. **31**(22), 3327–3329 (2006). [CrossRef] [PubMed]

*PN*= 5 is not strong enough to ensure a quite small coherent length; thus the dispersive propagation through the multilayer has obvious influence on the efficiency of SHG. Compared to the single PCMC, the efficiency of the cascaded PCMC structure is greatly enhanced mainly due to the added cavities for nonlinear interaction. For instance, under the same incident FW intensity of 460 MW/cm

^{2}, the quadruple-cascaded PCMC can reach its maximum efficiency up to 59%, while the efficiency of structures with

*N*= 1, 2, and 3 are 37%, 40%, and 57%, respectively. However, the enhancement factor is not simply proportional to the number of cavities. As shown in Fig. 5(a), the efficiency of the triple-cascaded PCMC exhibits more evident enhancement than the structures with

*N*= 2 and 4. It originates from the fact that in the odd-cascaded structures, one of cavity modes exactly takes place at the wavelength of

*λ*

_{0}and the FW is thus more strongly confined in the cavity layer, resulting in the prominent enhancement of nonlinear process.

*I*

_{0FW}of 200 MW/cm

^{2}, where PCMC structures with

*N*= 1 to 4 are simulated. It is evident that the generated SH energy varies with the wavelength tuning. For the single PCMC, the SHG curve exhibits a Lorentzian profile with a maximum of 30% around

*λ*

_{0}/2. With

*N*increasing from 1 to 4, the SHG efficiency at

*λ*

_{0}/2 monotonically increases as described above in Fig. 5(a). In the double-cascaded PCMC structure (

*N*= 2), the SHG efficiency curve exhibits a typical lump-like profile with two peaks occurring at 532 ± 0.03 nm. The corresponding spectral linewidth at 30% (the maximum SHG efficiency from the single PCMC) is 0.095 nm. For the case of triple-cascaded structure (

*N*= 3), it is found that the points corresponding to the maximum efficiencies are located at 532 and 532 ± 0.04 nm in the spectra. The spectral linewidth at 30% is broadened to 0.105 nm. Such a wider SH linewidth is consistent with the wider high-transmission window at the fundamental wavelength in the triple-cascaded PCMC. However, if one wants to get a linewidth of 0.105 nm in the single PCMC structure, only 7% of SHG efficiency could be obtained. These phenomena confirm that the introduction of cascaded structures not only broadens the spectral range of high-efficient frequency conversion but also significantly enhances the efficiency of the nonlinear optical process. For the case of

*N*= 4, the shape of SHG efficiency curve becomes not regular but still exhibits four peaks at the wavelengths of 531.962, 531.968, 532.006, and 532.045 nm, respectively. The spectral linewidth of SHG efficiency at 30% is still wide but drops down to 0.085 nm due to the deeper valleys close to the impurity band edges. These valleys correspond to the peaks of group velocities that obviously are undesirable for the nonlinear interaction [12

**65**(16), 165208 (2002). [CrossRef]

*N*, the broadening effectiveness is unusually reduced due to the stronger fluctuation in the spectra. Moreover, we notice that the third-order optical nonlinearity may affect the overall conversion efficiency under strong pumping conditions. To clarify this issue, we have taken the third harmonic (TH) generation into account in a single PCMC structure with

*PN*= 5 as an example. The output power intensity of TH is six orders of magnitude less than that of SH with the incident fundamental power intensity up to 1000 MW/cm

^{2}. Therefore, the influence can be ignored when we numerically simulate the SHG process in the single and cascaded PCMC structures.

*N*= 1 to 4, the forward FW energy is not only converting into SH, but also strongly reflected or coupling to the backward FW due to the high index contrast between layers. It leads to a consequence that the SH conversion efficiency tends to saturate in both directions. (ii) In the single PCMC structure, the SH outputs in both directions are approximately the same at all times, while in the cascaded PCMC structures the SH output is backward-dominant due to the symmetry breaking. As discussed in our previous work [19

19. R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. **31**(22), 3327–3329 (2006). [CrossRef] [PubMed]

## 4. Summary

## Acknowledgments

## References and links

1. | H. Thyrrestrup, S. Smolka, L. Sapienza, and P. Lodahl, “Statistical theory of a quantum emitter strongly coupled to Anderson-Localized modes,” Phys. Rev. Lett. |

2. | D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vucković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett. |

3. | J. Pan, Y. Huo, S. Sandhu, N. Stuhrmann, M. L. Povinelli, J. S. Harris, M. M. Fejer, and S. Fan, “Tuning the coherent interaction in an on-chip photonic-crystal waveguide-resonator system,” Appl. Phys. Lett. |

4. | T. W. Lu and P. T. Lee, “Ultra-high sensitivity optical stress sensor based on double-layered photonic crystal microcavity,” Opt. Express |

5. | S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett. |

6. | F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. Qiu, J. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B |

7. | F. F. Ren, R. Li, C. Cheng, J. Chen, Y. X. Fan, J. P. Ding, and H. T. Wang, “Low-threshold and high-efficiency optical parametric oscillator using a one-dimensional single-defect photonic crystal with quadratic nonlinearity,” Phys. Rev. B |

8. | O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science |

9. | B. Shi, Z. M. Jiang, and X. Wang, “Defective photonic crystals with greatly enhanced second-harmonic generation,” Opt. Lett. |

10. | J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, |

11. | Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express |

12. | S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B |

13. | P. Royo, R. P. Stanley, and M. Ilegems, “Coupling of impurity modes in one-dimensional periodic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

14. | M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett. |

15. | M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett. |

16. | E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett. |

17. | J. Azña, L. K. Oxenløwe, E. Palushani, R. Slavík, M. Galili, H. C. H. Mulvad, H. Hu, Y. Park, A. T. Clausen, and P. Jeppesen, “In-fiber subpicosecond pulse shaping for nonlinear optical telecommunication data processing at 640 Gbit/s,” Int. J. Opt. |

18. | Y. H. Ja, “Using the shooting methos to solve boundary-value problems involving nonlinear coupled-wave equations,” Opt. Quantum Electron. |

19. | R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett. |

20. | M. L. Ren and Z. Y. Li, “High conversion efficiency of second harmonic generation in a short nonlinear photonic crystal with distributed Bragg reflector mirrors,” Appl. Phys., A Mater. Sci. Process. |

21. | G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structures with deep gratings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(140.3945) Lasers and laser optics : Microcavities

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 10, 2012

Revised Manuscript: December 8, 2012

Manuscript Accepted: December 11, 2012

Published: January 7, 2013

**Citation**

Fang-Fang Ren, Jiandong Ye, Hai Lu, Rong Zhang, and Youdou Zheng, "Spectrum broadening of high-efficiency second harmonic generation in cascaded photonic crystal microcavities," Opt. Express **21**, 756-763 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-756

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### References

- H. Thyrrestrup, S. Smolka, L. Sapienza, and P. Lodahl, “Statistical theory of a quantum emitter strongly coupled to Anderson-Localized modes,” Phys. Rev. Lett.108(11), 113901 (2012). [CrossRef] [PubMed]
- D. Englund, A. Majumdar, A. Faraon, M. Toishi, N. Stoltz, P. Petroff, and J. Vucković, “Resonant excitation of a quantum dot strongly coupled to a photonic crystal nanocavity,” Phys. Rev. Lett.104(7), 073904 (2010). [CrossRef] [PubMed]
- J. Pan, Y. Huo, S. Sandhu, N. Stuhrmann, M. L. Povinelli, J. S. Harris, M. M. Fejer, and S. Fan, “Tuning the coherent interaction in an on-chip photonic-crystal waveguide-resonator system,” Appl. Phys. Lett.97(10), 101102 (2010). [CrossRef]
- T. W. Lu and P. T. Lee, “Ultra-high sensitivity optical stress sensor based on double-layered photonic crystal microcavity,” Opt. Express17(3), 1518–1526 (2009). [CrossRef] [PubMed]
- S. V. Zhukovsky, D. N. Chigrin, A. V. Lavrinenko, and J. Kroha, “Switchable lasing in multimode microcavities,” Phys. Rev. Lett.99(7), 073902 (2007). [CrossRef] [PubMed]
- F. F. Ren, R. Li, C. Cheng, H. T. Wang, J. Qiu, J. Si, and K. Hirao, “Giant enhancement of second harmonic generation in a finite photonic crystal with a single defect and dual-localized modes,” Phys. Rev. B70(24), 245109 (2004). [CrossRef]
- F. F. Ren, R. Li, C. Cheng, J. Chen, Y. X. Fan, J. P. Ding, and H. T. Wang, “Low-threshold and high-efficiency optical parametric oscillator using a one-dimensional single-defect photonic crystal with quadratic nonlinearity,” Phys. Rev. B73(3), 033104 (2006). [CrossRef]
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science284(5421), 1819–1821 (1999). [CrossRef] [PubMed]
- B. Shi, Z. M. Jiang, and X. Wang, “Defective photonic crystals with greatly enhanced second-harmonic generation,” Opt. Lett.26(15), 1194–1196 (2001). [CrossRef] [PubMed]
- J. M. Lourtioz, H. Benisty, V. Berger, J. M. Gerard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices, 2nd ed. (Springer-Verlag 2008).
- Y. Park, M. Kulishov, R. Slavík, and J. Azaña, “Picosecond and sub-picosecond flat-top pulse generation using uniform long-period fiber gratings,” Opt. Express14(26), 12670–12678 (2006). [CrossRef] [PubMed]
- S. Lan, S. Nishikawa, Y. Sugimoto, N. Ikeda, K. Asakawa, and H. Ishikawa, “Analysis of defect coupling in one- and two-dimensional photonic crystals,” Phys. Rev. B65(16), 165208 (2002). [CrossRef]
- P. Royo, R. P. Stanley, and M. Ilegems, “Coupling of impurity modes in one-dimensional periodic systems,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.64(1), 016604 (2001). [CrossRef] [PubMed]
- M. Bayer, T. Gutbrod, A. Forchel, T. L. Reinecke, P. A. Knipp, R. Werner, and J. P. Reithmaier, “Optical demonstration of a crystal band structure formation,” Phys. Rev. Lett.83(25), 5374–5377 (1999). [CrossRef]
- M. Bayindir, B. Temelkuran, and E. Ozbay, “Tight-binding description of the coupled defect modes in three-dimensional photonic crystals,” Phys. Rev. Lett.84(10), 2140–2143 (2000). [CrossRef] [PubMed]
- E. Gavartin, R. Braive, I. Sagnes, O. Arcizet, A. Beveratos, T. J. Kippenberg, and I. Robert-Philip, “Optomechanical coupling in a two-dimensional photonic crystal defect cavity,” Phys. Rev. Lett.106(20), 203902 (2011). [CrossRef] [PubMed]
- J. Azña, L. K. Oxenløwe, E. Palushani, R. Slavík, M. Galili, H. C. H. Mulvad, H. Hu, Y. Park, A. T. Clausen, and P. Jeppesen, “In-fiber subpicosecond pulse shaping for nonlinear optical telecommunication data processing at 640 Gbit/s,” Int. J. Opt.2012, 895281 (2012).
- Y. H. Ja, “Using the shooting methos to solve boundary-value problems involving nonlinear coupled-wave equations,” Opt. Quantum Electron.15(6), 529–538 (1983). [CrossRef]
- R. Li, J. Chen, Q. Xu, F. F. Ren, Y. X. Fan, J. Ding, and H. T. Wang, “Saturation effect and forward-dominant second-harmonic generation in single-defect photonic crystals with dual localizations,” Opt. Lett.31(22), 3327–3329 (2006). [CrossRef] [PubMed]
- M. L. Ren and Z. Y. Li, “High conversion efficiency of second harmonic generation in a short nonlinear photonic crystal with distributed Bragg reflector mirrors,” Appl. Phys., A Mater. Sci. Process.107(1), 71–76 (2012). [CrossRef]
- G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. Bertolotti, M. J. Bloemer, and C. M. Bowden, “Energy exchange properties during second-harmonic generation in finite one-dimensional photonic band-gap structures with deep gratings,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.67(1), 016606 (2003). [CrossRef] [PubMed]

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