## Surface plasmon polariton enhanced by optical parametric amplification in nonlinear hybrid waveguide |

Optics Express, Vol. 19, Issue 4, pp. 2858-2865 (2011)

http://dx.doi.org/10.1364/OE.19.002858

Acrobat PDF (1595 KB)

### Abstract

We theoretically studied nonlinear interactions between surface plasmon polariton (SPP) and conventional waveguide mode in nonlinear hybrid waveguide and proposed a possible method to enhance SPP wave via optical parametric amplification (OPA). The phase matching condition of this OPA process is fulfilled by carefully tailoring the dispersions of SPP and guided mode. The influences of incident intensity and phase of guided wave on the OPA process are comprehensively analyzed. It is found that not only a strong enhancement of SPP but also modulations on this enhancement can be achieved. This result indicates potential applications in nonlinear optical integration and modulations.

© 2011 OSA

## 1. Introduction

1. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter **33**(8), 5186–5201 (1986). [CrossRef] [PubMed]

2. I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B **78**, 161401 (2008). [CrossRef]

3. D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. **90**(2), 027402 (2003). [CrossRef] [PubMed]

4. R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature **461**(7264), 629–632 (2009). [CrossRef] [PubMed]

5. T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasman-polariton waveguides,” Phys. Rev. B **75**(24), 245405 (2007). [CrossRef]

6. T. Holmgaard, J. Gosciniak, and S. I. Bozhevolnyi, “Long-range dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express **18**(22), 23009–23015 (2010). [CrossRef] [PubMed]

10. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science **278**(5339), 843–846 (1997). [CrossRef]

11. H. J. Simon, D. E. Mitchell, and J. G. Watson, “Optical Second-Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. **33**(26), 1531–1534 (1974). [CrossRef]

13. S. Palomba and L. Novotny, “Nonlinear excitation of surface plasmon polaritons by four-wave mixing,” Phys. Rev. Lett. **101**(5), 056802 (2008). [CrossRef] [PubMed]

## 2. Theoretical model

14. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

*d*.

*k*is the in-plane wave vector (along

*x*direction) of corresponding modes. The permittivity components

*ε*

_{1}

*,*

_{x}*ε*

_{1}

*,*

_{z}*ε*

_{2}

*,*

_{x}*ε*

_{2}

*correspond to diagonal elements of permittivity tensor of NLD1 and NLD2. For TM polarization,*

_{z}*f*

_{1}=

*ε*

_{1}

*/ε*

_{x}*,*

_{m}*f*

_{2}=

*ε*

_{1}

*/*

_{x}*ε*

_{2}

*; while for TE polarization,*

_{x}*f*

_{1}=

*f*

_{2}= 1. By solving the dispersion relation of Eq. (1) with

*n*= 0 in TM polarization case, we can find in-plane wave vector

*k*always increases with the layer thickness

*d*. As the condition

*k*>

*k*

_{0}(

*ε*

_{1}

*)*

_{z}^{1/2}is satisfied by increase

*d*,

*k*

_{1}turns to be imaginary indicating an exponentially decay field from the interface. Thus we can make sure that the expected SPP in hybrid planar waveguide is just TM

_{0}mode supported by this structure. Of course, a higher guided TM

_{1}mode (n = 1) will be accommodated by further increasing

*d*to a proper value.

*ε*and

*μ*are the linear permittivity and permeability, and subscription

*i*= 1,2 refers to the considered SPP and guided mode (TM

_{1}) with frequencies of

*ω*and 2

*ω*, respectively.

*P*is the nonlinear polarization vector and

^{NL}*∂P*can be viewed as a source term that arises from the nonlinear interaction. For OPA process, a kind of second order nonlinear effect, we have

^{NL}/∂t**Tensor**

_{.}*χ*is the second order nonlinear susceptibility. According to coupled mode theory in waveguide,

*E*and

_{i}*H*can be expanded in terms of all canonical modes at the same frequency [15]:

_{i}*x*direction;

*A*(

_{l}*x*) is the amplitude, which evolves with

*x*due to coupling between different modes and propagation loss;

*E*(

_{l}*z*) and

*H*(

_{l}*z*) are mode profiles which have been normalized as

16. Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express **17**(16), 13502–13515 (2009). [CrossRef] [PubMed]

*β*and

_{i}*α*/2 (

_{i}*i*= 1,2) are real part and imaginary part of corresponding wave vectors, i.e.,

*k*=

_{i}*β*+ i

_{i}*α*/2;

_{i}*α*is defined as absorption coefficient; and

_{i}*κ*is the coupling coefficient that defined as

_{i}*Δβ = β*

_{2}

*-2β*

_{1}, also directly affects the conversion efficiency. Due to the dispersion caused by nonlinear material, it is usually difficult to satisfy phase matching conditions. Phase mismatching (

*Δβ≠*0) will lead to cycle flows of energy between these two modes and limit the one-way conversion efficiency, making the amplification of SPP impossible.

17. R. H. Stolen, M. A. Bösch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. **6**(5), 213–215 (1981). [CrossRef] [PubMed]

9. J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. **127**(6), 1918–1939 (1962). [CrossRef]

10. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science **278**(5339), 843–846 (1997). [CrossRef]

18. T. Sugita, K. Mizuuchi, Y. Kitaoka, and K. Yamamoto, “31%-efficient blue second-harmonic generation in a periodically poled MgO:LiNbO3 waveguide by frequency doubling of an AlGaAs laser diode,” Opt. Lett. **24**(22), 1590–1592 (1999). [CrossRef]

21. A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Quadratic phase matching in nonlinear plasmonic nanoscale waveguides,” Opt. Express **17**(22), 20063–20068 (2009). [CrossRef] [PubMed]

22. Z. J. Wu, X. K. Hu, Z. Y. Yu, W. Hu, F. Xu, and Y. Q. Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B **82**(15), 155107 (2010). [CrossRef]

*β*

_{1}(

*ω*) and

*β*

_{2}(2

*ω*) as the wave vectors of SPP and guided mode respectively, the phase matching condition 2

*β*(

*ω*) =

*β*(2

*ω*) will be appropriately obtained by carefully adjusting the modes dispersions. In this regard, it is reasonable to using a guided wave of

*β*

_{2}(2

*ω*) as an auxiliary light to compensate the propagating loss of SPP and even amplify it.

## 3. Example and analysis

_{3}for the anisotropic nonlinear dielectric that forms a conventional dielectric planar waveguide [20

20. Y. L. Lee, T. J. Eom, W. Shin, B.-A. Yu, D.-K. Ko, W.-K. Kim, and H.-Y. Lee, “Characteristics of a multi-mode interference device based on Ti:LiNbO3 channel waveguide,” Opt. Express **17**(13), 10718–10724 (2009). [CrossRef] [PubMed]

*ε*

_{1}

*=*

_{i}*ε*

_{2}

*+ 0.04 (*

_{i}*i = x,y,z*), corresponding to a numerical aperture of about 0.2 for the conventional dielectric waveguide, where

*ε*

_{2}

*is permittivity tensor elements of common LiNbO*

_{i}_{3}[23

23. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. **16**(4), 373–375 (1984). [CrossRef]

*χ*,

*d*

_{33}for LiNbO

_{3}[23

23. G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. **16**(4), 373–375 (1984). [CrossRef]

*z*, and use TM

_{1}mode as the auxiliary pumping wave, the coupling coefficients can be simplified by only considering the transverse fields assince the integration over the longitudinal components (

*d*

_{31},

*d*

_{22}and

*E*

_{i}_{,}

*) is too small and can be neglected as well as Ref. 20*

_{x}20. Y. L. Lee, T. J. Eom, W. Shin, B.-A. Yu, D.-K. Ko, W.-K. Kim, and H.-Y. Lee, “Characteristics of a multi-mode interference device based on Ti:LiNbO3 channel waveguide,” Opt. Express **17**(13), 10718–10724 (2009). [CrossRef] [PubMed]

*μ*m. To obtain phase matching, the wavelength of SPP and TM

_{1}mode is selected carefully. Dispersion relations of SPP and TM

_{1}mode are plotted in Fig. 2(a) . Since phase matching condition can also be described as

*n*

_{TM1}

*= n*

_{SPP}, we plot frequency versus effective index in the inset of Fig. 2(a) to make it more obvious. The intersection point of the two curves, 168.4THz for SPP and 336.8THz for TM

_{1}, indicates the satisfaction of phase matching. The corresponding effective indices are

*n*

_{SPP}= 2.1731 +

*i*0.001 and

*n*

_{TM1}= 2.1731 +

*i*2.593 × 10

^{−6}. Figure 1(b) shows the mode profiles of SPP and TM

_{1}used in this work. The field of SPP is tightly localized at the metal surface, with over 93% power confined in a 1

*μ*m thin dielectric layer, while only ~90% power of the TM

_{1}mode in the 3μm waveguide layer though it has a doubled frequency. The priority of SPP as the sub-wavelength waveguide is clearly demonstrated.

_{1}power (blue in linear scale). The calculation is performed with incident intensity of 1kW/cm for SPP mode as a seed signal and 50MW/cm and for TM

_{1}guided mode as the pumping wave. The highest conversion efficiency is 1.69% and the peak of SPP appears at the position of 5.8 mm. To make a comparison, we also plot the trend of SPP attenuation without the TM

_{1}pumping wave (the dashed curve). It is clearly found that the intensity of this pure SPP drops to 1/

*e*after only about 140

*μ*m, while SPP interacted with pumping wave gets amplification by about 845 times at the peak position.

_{1}mode. In Fig. 3(a) , OPA efficiency (left label) and the SPP amplification peak position (right label) with respect to the pumping intensity are depicted. It is apparently shown that the OPA process has a threshold pumping power (

*P*

_{TM1}

^{(0)}~35MW/cm), below which the pumping TM

_{1}is unable to amplify the seed SPP and no SPP peak in propagation can be observed as a result. When the pumping intensity exceed the threshold, the efficiency increases rapidly (from 0% to 27% as

*P*

_{TM1}

^{(0)}ranges from 35 to 150MW/cm), while the SPP peak position experiences a maximum value (at

*P*

_{TM1}

^{(0)}~43MW/cm) and tends to descend with the increase of

*P*

_{TM1}

^{(0)}. In addition, we plot another OPA dependence of the intensity of seed SPP for a comparison as shown in Fig. 3(b), where the intensity of seed SPP ranges from 0.1kW/cm to 10kW/cm while the efficiency varies within a narrow range (1% ~2.5%). This influence of the amplification effect is rather weaker than

*P*

_{TM1}

^{(0)}. This phenomenon is reasonably explained by deducing coupled wave equation from Eq. (3) to where

*ψ*(

*x*) =

*Φ*+ 2

*φ*

_{1}(

*x*)-

*φ*

_{2}(

*x*),

*A*

_{1}= |

*A*

_{1}|exp[

*iφ*

_{1}(

*x*)],

*A*

_{2}= |

*A*

_{2}|exp[

*iφ*

_{2}(

*x*)] and

*κ*= |

*κ*|exp(i

*Φ*). To get an amplification of SPP,

*d*|

*A*

_{1}|/

*dx*>0 should be satisfied. But the absorption coefficient (

*α*

_{1}) of SPP, obviously acts as the obstacle for the energy transfer from TM

_{1}to SPP. Then, we can get a critical intensity of TM

_{1}by solving

*d*|

*A*

_{1}|/

*dx*= 0, and obtain |

*A*

_{2}| = 2

*α*

_{1}[

*ωε*

_{0}|

*κ*|sin

*ψ*(

*x*)]

^{−1}, which corresponds to a balance value for a “lossless” SPP. On one hand, this critical balance value of

*P*

_{TM1}determines the threshold of

*P*

_{TM1}

^{(0)}; on the other hand, it results in an SPP peak in propagation [see Fig. 2(b)]. In the peak position, the pumping energy decreases to the balance value, after which the pumping gain is overcome by the loss and leads to attenuation of SPP wave. Commonly, with stronger incident intensity SPP signal will be amplified at a shorter distanced due to the higher enhancement rate indicated in Eq. (5)a), which coincides with the major tendency. However, if the pumping intensity is merely above the threshold, the amplification will stop soon since the energy may consume to below the balance value quickly within a short distance. With the pumping energy increases, this distance will extend and a maximum peak position emerges at a proper value of

*P*

_{TM1}

^{(0)}. It well explains the non-monotonous evolution of the SPP peak position in Fig. 3(a). Besides, Eq. (5)a) and (5b) also indicate the OPA efficiency is more relevant to |

*A*

_{2}| than |

*A*

_{1}| revealing different dependences on the intensity of pumping TM

_{1}and seed SPP in Fig. 3(a) and 3(b), respectively.

_{1}mode, which were both set to zero in previous calculations, i.e.,

*φ*

_{1}(0) =

*φ*

_{2}(0) = 0. In fact, this phase is another important factor that affects the optical parametric process, according to Eq. (5)a) and (5b). We can derive the changes of phase from coupled wave equations as

*ψ*= 0.5π), which is clearly revealed in the evolutions of the phases with different initial value (0.5π, 1.0π, 1.2π and 1.4π) in Fig. 4(a) . According to Eq. (6)a) and (6b), cos

*ψ*(

*x*) = 0 [or

*ψ*(

*x*) = (1/2 +

*n*)π] is stable values for the phase. However, for even and odd number

*n*, sin

*ψ*(

*x*) is 1 and −1 respectively, which directly decides the sign of

*d*|

*A*

_{1}|/

*dx*according to Eq. (5)a). Thus, for some improper incident phases, OPA efficiency may be much lower and even get a negative contribution at the beginning. In this case, the SPP wave has to propagate much longer and experience extra losses until the phases get a stable value beneficial to SPP amplification. To prove our prediction, we change

*ψ*(0) from 0 to 2π to see the influence of incident phase. Figure 4(b) shows the OPA efficiency and peak position of different

*ψ*(0), where the incident intensities of TM

_{1}and SPP are 50MW/cm and 1kW/cm respectively. When

*ψ*(0) = 1.5π (corresponding to sin

*ψ*(0) = −1), a dip of efficiency appears together with a longest SPP peak position as predicted, while the highest efficiency is achieved at

*ψ*(0) = 0.5π. In Fig. 4(c), the intensity evolution of SPP with respect to different initial phase

*ψ*(0) are shown in different color curves, respectively. Combined with the phase evolution [Fig. 4(a)], we may find when incident phase at desirable value, such as 0.5π,

*ψ*(x) reach the stable value immediately and SPP get amplified once the process started. For some improper incident phases near to undesirable value, for example,

*ψ*(0) = 1.4π, a long distance of propagation is necessary to get a desirable stable phase value for amplification of the seed SPP wave. For such incident phases indeed, SPP will firstly decay to an extremely low level before getting amplified.

## 4. Discussions

_{1}mode is acceptable [see Fig. 1(b)], which fortunately provides us the possibility to realize the direct phase matching condition without any additional structural design (like Ref. 22

22. Z. J. Wu, X. K. Hu, Z. Y. Yu, W. Hu, F. Xu, and Y. Q. Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B **82**(15), 155107 (2010). [CrossRef]

## 5. Conclusion

## Acknowledgement

## References and Links

1. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter |

2. | I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B |

3. | D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. |

4. | R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature |

5. | T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasman-polariton waveguides,” Phys. Rev. B |

6. | T. Holmgaard, J. Gosciniak, and S. I. Bozhevolnyi, “Long-range dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express |

7. | R. W. Boyd, |

8. | R. A. Baumgartner and R. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. |

9. | J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. |

10. | S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science |

11. | H. J. Simon, D. E. Mitchell, and J. G. Watson, “Optical Second-Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. |

12. | H. J. Simon, R. E. Benner, and J. G. Rako, “Optical second harmonic generation with surface plasmons in piezoelectric crystals,” Opt. Commun. |

13. | S. Palomba and L. Novotny, “Nonlinear excitation of surface plasmon polaritons by four-wave mixing,” Phys. Rev. Lett. |

14. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

15. | G. Lifante, Integrated Photonics: Fundamentals (Wiley, England, 2003). |

16. | Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express |

17. | R. H. Stolen, M. A. Bösch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. |

18. | T. Sugita, K. Mizuuchi, Y. Kitaoka, and K. Yamamoto, “31%-efficient blue second-harmonic generation in a periodically poled MgO:LiNbO3 waveguide by frequency doubling of an AlGaAs laser diode,” Opt. Lett. |

19. | H. Jiang, G. H. Li, and X. Y. Xu, “Highly efficient single-pass second harmonic generation in a periodically poled MgO:LiNbO3 waveguide pumped by a fiber laser at 1111.6 nm,” Opt. Express |

20. | Y. L. Lee, T. J. Eom, W. Shin, B.-A. Yu, D.-K. Ko, W.-K. Kim, and H.-Y. Lee, “Characteristics of a multi-mode interference device based on Ti:LiNbO3 channel waveguide,” Opt. Express |

21. | A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Quadratic phase matching in nonlinear plasmonic nanoscale waveguides,” Opt. Express |

22. | Z. J. Wu, X. K. Hu, Z. Y. Yu, W. Hu, F. Xu, and Y. Q. Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B |

23. | G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(230.7390) Optical devices : Waveguides, planar

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: December 10, 2010

Revised Manuscript: January 18, 2011

Manuscript Accepted: January 21, 2011

Published: January 31, 2011

**Citation**

F. F. Lu, T. Li, J. Xu, Z. D. Xie, L. Li, S. N. Zhu, and Y. Y. Zhu, "Surface plasmon polariton enhanced by optical parametric amplification in nonlinear hybrid waveguide," Opt. Express **19**, 2858-2865 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-2858

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### References

- J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter 33(8), 5186–5201 (1986). [CrossRef] [PubMed]
- I. De Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B 78, 161401 (2008). [CrossRef]
- D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90(2), 027402 (2003). [CrossRef] [PubMed]
- R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009). [CrossRef] [PubMed]
- T. Holmgaard and S. I. Bozhevolnyi, “Theoretical analysis of dielectric-loaded surface plasman-polariton waveguides,” Phys. Rev. B 75(24), 245405 (2007). [CrossRef]
- T. Holmgaard, J. Gosciniak, and S. I. Bozhevolnyi, “Long-range dielectric-loaded surface plasmon-polariton waveguides,” Opt. Express 18(22), 23009–23015 (2010). [CrossRef] [PubMed]
- R. W. Boyd, Nonlinear Optics (Elsevier Science, 2003).
- R. A. Baumgartner and R. Byer, “Optical parametric amplification,” IEEE J. Quantum Electron. 15(6), 432–444 (1979). [CrossRef]
- J. Armstrong, N. Bloembergen, J. Ducuing, and P. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev. 127(6), 1918–1939 (1962). [CrossRef]
- S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278(5339), 843–846 (1997). [CrossRef]
- H. J. Simon, D. E. Mitchell, and J. G. Watson, “Optical Second-Harmonic Generation with Surface Plasmons in Silver Films,” Phys. Rev. Lett. 33(26), 1531–1534 (1974). [CrossRef]
- H. J. Simon, R. E. Benner, and J. G. Rako, “Optical second harmonic generation with surface plasmons in piezoelectric crystals,” Opt. Commun. 23(2), 245–248 (1977). [CrossRef]
- S. Palomba and L. Novotny, “Nonlinear excitation of surface plasmon polaritons by four-wave mixing,” Phys. Rev. Lett. 101(5), 056802 (2008). [CrossRef] [PubMed]
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]
- G. Lifante, Integrated Photonics: Fundamentals (Wiley, England, 2003).
- Z. Ruan, G. Veronis, K. L. Vodopyanov, M. M. Fejer, and S. Fan, “Enhancement of optics-to-THz conversion efficiency by metallic slot waveguides,” Opt. Express 17(16), 13502–13515 (2009). [CrossRef] [PubMed]
- R. H. Stolen, M. A. Bösch, and C. Lin, “Phase matching in birefringent fibers,” Opt. Lett. 6(5), 213–215 (1981). [CrossRef] [PubMed]
- T. Sugita, K. Mizuuchi, Y. Kitaoka, and K. Yamamoto, “31%-efficient blue second-harmonic generation in a periodically poled MgO:LiNbO3 waveguide by frequency doubling of an AlGaAs laser diode,” Opt. Lett. 24(22), 1590–1592 (1999). [CrossRef]
- H. Jiang, G. H. Li, and X. Y. Xu, “Highly efficient single-pass second harmonic generation in a periodically poled MgO:LiNbO3 waveguide pumped by a fiber laser at 1111.6 nm,” Opt. Express 17(18), 16073–16080 (2009). [CrossRef] [PubMed]
- Y. L. Lee, T. J. Eom, W. Shin, B.-A. Yu, D.-K. Ko, W.-K. Kim, and H.-Y. Lee, “Characteristics of a multi-mode interference device based on Ti:LiNbO3 channel waveguide,” Opt. Express 17(13), 10718–10724 (2009). [CrossRef] [PubMed]
- A. R. Davoyan, I. V. Shadrivov, and Y. S. Kivshar, “Quadratic phase matching in nonlinear plasmonic nanoscale waveguides,” Opt. Express 17(22), 20063–20068 (2009). [CrossRef] [PubMed]
- Z. J. Wu, X. K. Hu, Z. Y. Yu, W. Hu, F. Xu, and Y. Q. Lu, “Nonlinear plasmonic frequency conversion through quasiphase matching,” Phys. Rev. B 82(15), 155107 (2010). [CrossRef]
- G. J. Edwards and M. Lawrence, “A temperature-dependent dispersion equation for congruently grown lithium niobate,” Opt. Quantum Electron. 16(4), 373–375 (1984). [CrossRef]

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