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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 1 — Jan. 14, 2013
  • pp: 756–763
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Spectrum broadening of high-efficiency second harmonic generation in cascaded photonic crystal microcavities

Fang-Fang Ren, Jiandong Ye, Hai Lu, Rong Zhang, and Youdou Zheng  »View Author Affiliations


Optics Express, Vol. 21, Issue 1, pp. 756-763 (2013)
http://dx.doi.org/10.1364/OE.21.000756


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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

2. Method and design

A single PCMC with the structure of (HL)PND(LH)PN (Fig. 1(a)
Fig. 1 Schematic diagram of PCMC structures. (a) Single PCMC, (b) double-cascaded PCMC, and (c) multiple-cascaded PCMC.
) originates from a matrix of (HL)PNH(LH)PN, provided that the central layer H is changed into the cavity layer D (the geometric length is thus changed from dH to dD), where H and L stand for the high- and low-refractive-index layers, respectively. The structure of (HL)PN or (LH)PN is a distributed feedback Bragg reflector and has a period number of PN. They are composed of alternately stacked H and L layers, satisfying the condition of a quarter-wavelength optical thickness of nHdH = nLdL = λ0/4, where nH(L) represents the refractive index of H(L) layer and λ0 is the central wavelength of the photonic bandgap of the Bragg reflectors. To generate a cavity mode at the wavelength of λ0, the cavity layer D can be designed with a half-wavelength optical thickness (i.e. dD = 2dH). 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 dJ, a multiple-cascaded PCMC can be constructed with the arrangement of (HL)PND(LH)PN[J(HL)PND(LH)PN]N-1 as depicted in Fig. 1(c). When 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 LiNbO3 embedded in a low-index background of the air, i.e. the materials of H/L/D/J layer are LiNbO3/air/LiNbO3/air, respectively. This kind of multilayer structure is able to be fabricated using the top-down lithography. The dispersion of LiNbO3 is temperature dependent and the effective nonlinear coefficient deff 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]

. The interested fundamental wavelength λ0 here is 1064 nm, a typical lasing generation from Nd:YAG and Nd:YVO4 which are extensively used in many fields.

3. Results and discussion

We have firstly done numerical investigations on the properties of the cavity modes in linear transmission spectra of a double-cascaded PCMC as shown in Fig. 2
Fig. 2 The dependence of the resonance peaks on the air gap distance dJ in a double-cascaded PCMC with PN = 5. The transmission spectra are shifted upwards for clarity with offset = 1.
. The period number is taken as PN = 5 and a normal incidence is considered for simplicity. When dJ = 0, the cavity modes appear in the transmission spectrum as separated Lorentzian profiles. As the air gap dJ varies from 0 to dL, the antibonding mode located at the shorter resonance wavelength exhibits a red-shift from 1058.61 nm to λ0, while the bonding mode is almost insensitive to the variation of dJ and keeps its resonance peak at the wavelength of λ0. When dJ keeps increasing and deviates from dL to 2dL, the antibonding mode is standing at the wavelength of λ0, while the bonding mode moves from λ0 to 1067.72 nm. In the special case of dJ = dL, two cavity modes are almost degenerate at λ0 = 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 (dJ), where two modes are continuously modulated. It is worth noting that one of the resonance peaks is always centered at λ0 = 1064 nm for any value of dJ, 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 dJ = dL. Further simulations with dJ > 2dL indicate that the mode splitting have periodicity of dJ with a period of 2dL, implying the phase correlation of two cavity modes.

The flat-top transmission characteristics of cascaded PCMCs with dJ = dL are further studied for different N as shown in Fig. 3
Fig. 3 (a) The simulated phase and (b) transmission spectra of the coupled cavity modes in single and cascaded PCMCs with N = 1 to 4. The structures are with PN = 5 and dJ = dL.
, 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 dJ = dL, 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]

, 17

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. 2012, 895281 (2012).

]. For a triple-cascaded PCMC with N = 3, a quasiflat-top resonance window appears around λ0 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 λ0 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.

Another important aspect is how the number of coupled cavities (N) in the cascaded structure affects the conversion efficiency at the SH wavelength of 532 nm (i.e. λ0/2). Figure 5(a)
Fig. 5 (a) The dependence of SHG efficiency on the incident FW intensity for the examples of N = 1, 2, 3, and 4 with dJ = dL. (b) The comparison of high-efficiency spectral linewidth in PCMC structures with N = 1 to 4 under I0FM = 200 MW/cm2.
shows the dependence of the conversion efficiency on the incident FW intensity I0FW for N = 1, 2, 3, and 4 with dJ = dL. 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]

and 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]

-21

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]

, with an increase of incident FW power, the total conversion efficiency including both forward and backward SH is rapidly enhanced till a saturation point and then gradually drops down when the backward FW intensity goes over the forward FW intensity. It is due to that the FW energy mainly flows to backward FW rather than forward and backward SH. In a symmetric single PCMC structure that satisfies the conditions of pump depletion and global phase matching, the SH outputs are balanced in the forward and backward directions, and the maximum of total conversion efficiency can reach 50%. By breaking the symmetry of the structure, the saturation efficiency may increase or decrease depending on the reflectivity or transmittance of the left and right PC mirrors at the both sides of an individual cavity layer [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]

]. In Fig. 5(a), the saturation efficiency of the symmetric single PCMC is nearly 40%, which is lower than 50% because the field confinement in the case of 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/cm2, 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 λ0 and the FW is thus more strongly confined in the cavity layer, resulting in the prominent enhancement of nonlinear process.

Figure 5(b) shows the SHG efficiency with detuned wavelengths, i.e. the dependence of the SH outputs on the SH wavelength with constant input FW intensity I0FW of 200 MW/cm2, 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 λ0/2. With N increasing from 1 to 4, the SHG efficiency at λ0/2 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

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]

]. Therefore, in the cascaded structures with even lager 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/cm2. Therefore, the influence can be ignored when we numerically simulate the SHG process in the single and cascaded PCMC structures.

4. Summary

In summary, we proposed an effective approach to broaden the resonance peak of cavity modes by introducing the cascaded photonic crystal microcavities based on the concept of spectra re-shaping. The numerical simulations clearly confirm its potential application in achieving a wider spectral range with high SHG conversion efficiency and other nonlinear optical processes. Such improvement of spectral stability is practically important for high-performance nonlinear optical devices.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 11104130, 60825401, and 60936004, the Natural Science Foundation of Jiangsu Province under Grant Nos. BK2011556 and BK2011437, and the State Key Program for Basic Research of China under Grant Nos. 2010CB327504 and 2011CB301900.

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. 108(11), 113901 (2012). [CrossRef] [PubMed]

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. 104(7), 073904 (2010). [CrossRef] [PubMed]

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. 97(10), 101102 (2010). [CrossRef]

4.

T. W. Lu and P. T. Lee, “Ultra-high sensitivity optical stress sensor based on double-layered photonic crystal microcavity,” Opt. Express 17(3), 1518–1526 (2009). [CrossRef] [PubMed]

5.

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]

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]

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 73(3), 033104 (2006). [CrossRef]

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]

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]

10.

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).

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]

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. 64(1), 016604 (2001). [CrossRef] [PubMed]

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. 83(25), 5374–5377 (1999). [CrossRef]

15.

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]

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]

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. 2012, 895281 (2012).

18.

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]

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]

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. 107(1), 71–76 (2012). [CrossRef]

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]

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

  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]
  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.104(7), 073904 (2010). [CrossRef] [PubMed]
  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.97(10), 101102 (2010). [CrossRef]
  4. 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]
  5. 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]
  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. B70(24), 245109 (2004). [CrossRef]
  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. B73(3), 033104 (2006). [CrossRef]
  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,” Science284(5421), 1819–1821 (1999). [CrossRef] [PubMed]
  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]
  10. 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).
  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. Express14(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. B65(16), 165208 (2002). [CrossRef]
  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.64(1), 016604 (2001). [CrossRef] [PubMed]
  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.83(25), 5374–5377 (1999). [CrossRef]
  15. 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]
  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]
  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.2012, 895281 (2012).
  18. 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]
  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]
  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.107(1), 71–76 (2012). [CrossRef]
  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]

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