## Second-harmonic generation in photonic crystals with a pair of epsilon-negative and mu-negative defects

Optics Express, Vol. 17, Issue 8, pp. 6682-6687 (2009)

http://dx.doi.org/10.1364/OE.17.006682

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### Abstract

A conventional one-dimensional photonic crystal with a conjugated pair of *ε*-negative and *μ*-negative defects has been presented, and only the defects are presumed to possess quadratic nonlinearity. Large enhancement of second-harmonic generation is predicted in numerical simulation. Interface and volume nonlinearity are both utilized in the process of second-harmonic generation due to the strong localization of the fundamental wave.

© 2009 Optical Society of America

1. H. Kogelnik and C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett. **18**, 152–154 (1971). [CrossRef]

3. Y. Fink, J. N. Winn, S Fan, C Chen, J Michel, J. D. Joannopoulos, and E. L. Thomas, “A Dielectric Omnidirectional Reflector,” Science **282**, 1679–1682 (1998). [CrossRef] [PubMed]

4. A.J. Ward, J. B. Pendry, and W. J. Stewart, “Photonic dispersion surfaces,” J. Phys.: Condens.Matter **7**, 2217–2224 (1995). [CrossRef]

5. M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF_{2} Photonic Band Gaps,” Appl. Phys. Lett. **72**, 1676–1678 (1998). [CrossRef]

6. M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A **56**, 3166–3174 (1997). [CrossRef]

7. 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**, 245109 (2004). [CrossRef]

8. P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, and P. N. Prasad, “Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,” Appl. Phys. Lett. **87**, 051102 (2005). [CrossRef]

9. 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**, 033104 (2006). [CrossRef]

*Ren*that simultaneous localization of the fundamental wave (FW) and second harmonic wave (SHW) in the same defective layer could bring giant enhancement of the conversion efficiency by more than eight orders. [7

7. 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**, 245109 (2004). [CrossRef]

10. G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and M. Scalora, “Large enhancement of second harmonic generation near the zero-n gap of a negative index Bragg grating,” Phys. Rev. E **73**, 036603 (2006). [CrossRef]

11. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. **76**, 4773 (1996). [CrossRef] [PubMed]

*ε*< 0 and

*μ*> 0 and mu-negative (MNG) materials with

*μ*< 0 and

*ε*> 0. The conventional PC with a pair of ENG-MNG defects have higher localization at the interface of ENG and MNG than the PC made of layered ENG-MNG materials,[13

13. H. T. Jiang, H. Chen, and S. Y. Zhu, “Localized gap-edge fields of one-dimensional photonic crystals with an e-negative and a m-negative defect,” Phys. Rev. E **73**, 046601 (2006). [CrossRef]

_{PN}CD(AB)

_{PN-1}A. Here, A and B are PIMs, C and D represent mu-negative (MNG) material and epsilon-negative (ENG) material respectively. PN represents the period number of AB layers. Here we choose PN = 9.

*ε*

_{A,B,C,D}and

*μ*

_{A,B,C,D}are the permittivity and permeability of A, B, C, D layers respectively.

*d*

_{A,B,C,D}is the thickness of the corresponding layers respectively. In this paper,

*ε*= 9,

_{A}*μ*= 1,

_{A}*ε*= 2,

_{B}*μ*= 1, the structure is surrounded with air. μ

_{B}_{C}and

*ε*are described by lossy Drude model, and the parameters are chosen as

_{D}*ω*= 2

*πf*is angular frequency in unit of gigahertz,

*γ*(

_{m}*γ*) is the damping coefficient. In the calculation, the normally incident electromagnetic wave is a transverse magnetic (TM) wave (the magnetic field

_{e}**H**is in the y direction) propagating along the z direction as shown in Fig. 1. The transverse electric (TE) wave can be treated in a similar way. The parameters of the pair of SNG materials should be chosen specially. Only if the pair of the SNG materials are conjugatedly matched pair of MNG-ENG layers,

*i.e*. the parameters of MNG-ENG layers must satisfy the relationship described as

*ε*= −

_{C}*ε*,

_{D}*μ*= −

_{C}*μ*, and

_{D}*d*=

_{C}*d*, so that the defects will not influence the transmission property of the structure. It is actually a particular case of zero-reflection condition. Using transmission-line model, the zero-reflection condition can be written as

_{D}*Z*= −

_{C}*Z*,

_{D}*k*=

_{MNG}d_{C}*k*, [14

_{ENG}d_{D}14. A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. **51**, 2558–2571 (2003). [CrossRef]

*Z*(

_{C}*D*) and

*k*

_{MNG(ENG)}indicate the impedance and wave number in MNG and ENG layers respectively.

_{17}A and (AB)

_{9}CD(AB)

_{8}A respectively, using transfer-matrix method.[12

12. L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional resonance modes in photonic crystal heterostructures containing single-negative materials,” Phys. Rev. B **70**, 245102 (2004). [CrossRef]

15. N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E **65**, 046607 (2002). [CrossRef]

*ε*

_{1}=

*μ*

_{1}= 1,

*ε*

_{2}=

*μ*

_{2}= 1,

*α*=

*β*= 400,

*γ*=

_{m}*γ*= 2

_{e}*π*×3×10

^{-6}. The values of the damping chosen here are small enough, so that their influence on

*μ*,

_{C}*ε*can be neglected.

_{D}*d*=

_{C}*d*= 48

_{D}*mm*,

*d*=

_{C}*d*= 18

_{D}*mm*, respectively, while

*d*=

_{A}*d*= 18 mm are invariable in these three cases. It is noted that the transmission function within the bands is periodically modulated. This periodic modulation is mainly due to the reflections from end facets of the structure yielding Fabry-Pérot resonances. We can see from Fig. 2(a) that the higher frequency edge of band gap is always at 2.25 GHz, which means the pair of the defects do not change the transmittance at FW because the zero-reflection condition is satisfied. However, the transmittance of SHW changes slightly. This change is owing to the consideration of the lossy Drude model, and does not have much effect on the conversion efficiency of SHG. We will illustrate this phenomenon clearly in the following calculation.

_{B}*d*=

_{A}*d*= 18

_{B}*mm*,

*d*=

_{C}*d*= 48

_{D}*mm*. A high localization of the electric field is found in the MNG and ENG layers, and the intensity of relative electric field is amplified by 600 times. Obviously, the amplitude of intensity of localized electric field is improved greatly in the MNG-ENG defects due to the confinement of the two conventional PCs outside of the defects. In this defective PC, the dramatically localized field will greatly contribute to the SHG.

*d*(=

_{C}*d*) on the maximum of the relative electric field intensity distribution which was called

_{D}*localization peak*here. The thickness of the MNG and ENG materials have to be changed simultaneously, as the thickness of the MNG and ENG materials affects the localization peak obviously. In Fig. 3(a), we plot the relationship between the localization peak and

*d*by red solid line. The localization peak increases exponentially when

_{C}*d*becomes thicker, until to a maximum value about 2250 a.u. with

_{C}*d*= 55

_{C}*mm*. Then the localization peak decreases if

*d*grows continuously , because the zero-reflection condition does not exactly satisfied at this frequency (2.25 GHz), otherwise, the localization peak would keep increasing when dC strides over 55 mm.[16

_{C}16. A. Alu, N. Engheta, and R. W. Ziolkowski, “Finite-difference time-domain analysis of the tunneling and growing exponential in a pair of epsilon-negative and mu-negative slabs,” Phys. Rev. E **74**, 016604 (2006). [CrossRef]

*ω*and 2

*ω*can be written as:[17

17. N. Mattiucci, G. D’Aguanno, M. J. Bloemer, and M. Scalora, “Second harmonic generation from a positive-negative index material heterostructure,” Phys. Rev. E **72**, 066612 (2005). [CrossRef]

*ε*

_{ω,2ω,μω,2ω}are susceptibility and permeability respectively of the FW or SHW.

*d*(

*z*) is the quadratic coupling coefficient.

17. N. Mattiucci, G. D’Aguanno, M. J. Bloemer, and M. Scalora, “Second harmonic generation from a positive-negative index material heterostructure,” Phys. Rev. E **72**, 066612 (2005). [CrossRef]

^{±}

_{ω,2ω}is the right-to-left (RTL) (+) or left-to-right (LTR) (-) linear modes of the structure at

*ω*or 2

*ω*as described in Ref. 18

18. G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis,” Phys. Rev. E **70**, 016612 (2004). [CrossRef]

12. L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional resonance modes in photonic crystal heterostructures containing single-negative materials,” Phys. Rev. B **70**, 245102 (2004). [CrossRef]

15. N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E **65**, 046607 (2002). [CrossRef]

*d*(

*z*) as a Dirac

*δ*function. But in this paper, we consider both of the interface and volume nonlinearity contribution, and for convenience we chose the nonlinear coefficients of them as the same.

*I*is the power density of the pump beam.

^{Input}_{ω}*d*of the MNG and ENG is chosen to be 43.9 pm/V, the same as in Ref. 7

_{eff}7. 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**, 245109 (2004). [CrossRef]

^{2}. The conversion efficiency of the conventional PC with a conjugated pair of MNG-ENG defects (represented by blue solid line) is largely improved by more than five orders of magnitude comparing with that of the conventional PCs (represented by red dashed line, which has been amplified by a factor of 10

^{5}for clarity). We should summarize that there are two reasons resulting in the large improvement of the conversion efficiency: 1) FW is strongly localized in the MNG-ENG layers and the amplification is much higher than that in the conventional PC, although the FW lies in the band edge similarly. In conventional PC, the field distribution corresponding to the edge of band gap is very small, which is ascribed to the low quality factor of F-P cavity. While in our SNG defective PC, the EM is decaying from the center of the defect to both sides, which represents a localized state. 2) The maximum of the field distribution of FW is exactly at the interface of MNG-ENG defects, which makes the surface nonlinearity to be usable in SHG. It is necessary to point out that the improvement of the conversion efficiency of second harmonic generation is mainly due to the introduction of the meta-material defects and the contribution of the surrounded PC is negligible. Our calculation shows the conversion efficiency due to the surrounded PC is only in the order of 10

^{-7}, which is much smaller than the total conversion efficiency. The main reason of the conversion efficiency enhancement is due to field localization in the defects of the meta-material as we discussed above.

*d*by blue dashed line in Fig. 3(a) with the input power equaling to 1MW/cm

_{C}^{2}, and the parameters used here are the same as Fig. 2(b). The conversion efficiency also depends on

*d*and nearly coincides with the curve of localization peak, which implies that the localization peak of FW of the structure mainly determines the conversion efficiency. So, we can conclude that the SHW does not influence the conversion efficiency significantly. When

_{C}*d*= 55 mm, the conversion efficiency reaches the maximum (3×10

_{C}^{-2}).

^{2}in the calculation of conversion efficiency as in Fig. 3(a). Since the transmittance and the localization of the FW are affected by PN periodically, correspondently, the conversion efficiencies will also be periodically influenced by PN similarly. Each localization peak means the FW is one of transmission resonance modes. When the FW is close to the second transmission resonance, the conversion efficiency is larger than the others.[18

18. G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis,” Phys. Rev. E **70**, 016612 (2004). [CrossRef]

## References and links

1. | H. Kogelnik and C. V. Shank, “Stimulated emission in a periodic structure,” Appl. Phys. Lett. |

2. | V. P. Bykov, “Spontaneous emission in a periodic structure,” Zh. Expr. Teor. Fiz. |

3. | Y. Fink, J. N. Winn, S Fan, C Chen, J Michel, J. D. Joannopoulos, and E. L. Thomas, “A Dielectric Omnidirectional Reflector,” Science |

4. | A.J. Ward, J. B. Pendry, and W. J. Stewart, “Photonic dispersion surfaces,” J. Phys.: Condens.Matter |

5. | M. J. Bloemer and M. Scalora, “Transmissive properties of Ag/MgF |

6. | M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J. W. Haus, “Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures,” Phys. Rev. A |

7. | 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 |

8. | P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, and P. N. Prasad, “Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating,” Appl. Phys. Lett. |

9. | 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 |

10. | G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and M. Scalora, “Large enhancement of second harmonic generation near the zero-n gap of a negative index Bragg grating,” Phys. Rev. E |

11. | J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. |

12. | L. G. Wang, H. Chen, and S. Y. Zhu, “Omnidirectional resonance modes in photonic crystal heterostructures containing single-negative materials,” Phys. Rev. B |

13. | H. T. Jiang, H. Chen, and S. Y. Zhu, “Localized gap-edge fields of one-dimensional photonic crystals with an e-negative and a m-negative defect,” Phys. Rev. E |

14. | A. Alu and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. |

15. | N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, “Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect,” Phys. Rev. E |

16. | A. Alu, N. Engheta, and R. W. Ziolkowski, “Finite-difference time-domain analysis of the tunneling and growing exponential in a pair of epsilon-negative and mu-negative slabs,” Phys. Rev. E |

17. | N. Mattiucci, G. D’Aguanno, M. J. Bloemer, and M. Scalora, “Second harmonic generation from a positive-negative index material heterostructure,” Phys. Rev. E |

18. | G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, “Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis,” Phys. Rev. E |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: January 6, 2009

Revised Manuscript: March 18, 2009

Manuscript Accepted: March 25, 2009

Published: April 8, 2009

**Citation**

Qing G. Du, Fang-Fang Ren, Chan H. Kam, and Xiao W. Sun, "Second-harmonic generation in photonic crystals with a pair of epsilon-negative and mu-negative defects," Opt. Express **17**, 6682-6687 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-8-6682

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### References

- H. Kogelnik and C. V. Shank, "Stimulated emission in a periodic structure," Appl. Phys. Lett. 18, 152-154 (1971). [CrossRef]
- V. P. Bykov, "Spontaneous emission in a periodic structure," Zh. Expr. Teor. Fiz. 62, 505-513 (1972).
- Y. Fink, J. N. Winn, S Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, "A Dielectric Omnidirectional Reflector," Science 282,1679-1682 (1998). [CrossRef] [PubMed]
- A. J. Ward, J. B. Pendry, and W. J. Stewart, "Photonic dispersion surfaces," J. Phys.: Condens.Matter 7,2217-2224 (1995). [CrossRef]
- M. J. Bloemer and M. Scalora, "Transmissive properties of Ag/MgF2 Photonic Band Gaps," Appl. Phys. Lett. 72,1676-1678 (1998). [CrossRef]
- M. Scalora, M. J. Bloemer, A. S. Manka, J. P. Dowling, C. M. Bowden, R. Viswanathan, and J.W. Haus, "Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures," Phys. Rev. A 56,3166-3174 (1997). [CrossRef]
- 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 70,245109 (2004). [CrossRef]
- P. P. Markowicz, V. K. S. Hsiao, H. Tiryaki, A. N. Cartwright, and P. N. Prasad, "Enhancement of third-harmonic generation in a polymer-dispersed liquid-crystal grating," Appl. Phys. Lett. 87,051102 (2005). [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. B 73,033104 (2006). [CrossRef]
- G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and M. Scalora, "Large enhancement of second harmonic generation near the zero-n gap of a negative index Bragg grating," Phys. Rev. E 73,036603 (2006). [CrossRef]
- J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76,4773 (1996). [CrossRef] [PubMed]
- L. G. Wang, H. Chen, and S. Y. Zhu, "Omnidirectional resonance modes in photonic crystal heterostructures containing single-negative materials," Phys. Rev. B 70,245102 (2004). [CrossRef]
- H. T. Jiang, H. Chen, and S. Y. Zhu, "Localized gap-edge fields of one-dimensional photonic crystals with an e-negative and a m-negative defect," Phys. Rev. E 73,046601 (2006). [CrossRef]
- A. Alu and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51,2558-2571 (2003). [CrossRef]
- N. H. Liu, S. Y. Zhu, H. Chen, and X. Wu, "Superluminal pulse propagation through one-dimensional photonic crystals with a dispersive defect," Phys. Rev. E 65,046607 (2002). [CrossRef]
- A. Alu, N. Engheta, and R. W. Ziolkowski, "Finite-difference time-domain analysis of the tunneling and growing exponential in a pair of epsilon-negative and mu-negative slabs," Phys. Rev. E 74,016604 (2006). [CrossRef]
- N. Mattiucci, G. D’Aguanno, M. J. Bloemer, and M. Scalora, "Second harmonic generation from a positivenegative index material heterostructure," Phys. Rev. E 72,066612 (2005). [CrossRef]
- G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, "Density of modes and tunneling times in finite, one-dimensional, photonic crystals: a comprehensive analysis," Phys. Rev. E 70,016612 (2004). [CrossRef]

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