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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 15 — Jul. 28, 2014
  • pp: 18612–18624
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Efficient phase-matched third harmonic generation in an asymmetric plasmonic slot waveguide

Tingting Wu, Yunxu Sun, Xuguang Shao, Perry Ping Shum, and Tianye Huang  »View Author Affiliations


Optics Express, Vol. 22, Issue 15, pp. 18612-18624 (2014)
http://dx.doi.org/10.1364/OE.22.018612


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Abstract

An asymmetric plasmonic slot waveguide (APSW) for efficient phase-matched third harmonic generation (THG) is proposed and demonstrated theoretically. Nonlinear organic material DDMEBT polymer is integrated into the bottom of the metallic slot, while silicon is used to fill the top of the slot. We introduce the rigorous coupled-mode equations of THG in the lossy APSW and apply them to optimize the waveguide geometry. Taking advantage of the surface plasmon polaritons (SPPs), the electric fields can be tightly confined in the metallic slot region and the nonlinear effect is greatly enhanced accordingly. Then, we investigate the relationships between THG efficiency and parameters such as slot width and height, phase matching condition (PMC), modal overlap related nonlinear parameter, figure-of-merit, pump power and detuning. With the proposed asymmetric waveguide, we demonstrate a high THG conversion efficiency of 4.88 × 10−6 with a pump power of 1 W and a detuning constant of −36 m−1 at a waveguide length of 10.65 𝜇m.

© 2014 Optical Society of America

1. Introduction

During the past few years, mid-infrared (mid-IR) photonics have been attracting increasing attentions [1

1. J. Ma and S. Fathpour, “Pump-to-Stokes relative intensity noise transfer and analytical modeling of mid-infrared silicon Raman lasers,” Opt. Express 20(16), 17962–17972 (2012). [CrossRef] [PubMed]

, 2

2. Z. Wang, H. Liu, N. Huang, Q. Sun, and X. Li, “Mid-infrared Raman amplification and wavlength conversion in dispersion engineered silicon-on-sapphire waveguides,” J. Opt. 16(1), 015206(2014). [CrossRef]

]. Accompany with the development of this special waveband, various all-optical integrated functional devices in the mid-IR region, such as amplifiers and wavelength converters, have been studied for potential applications in free space communications, chemical and biological sensing, and medical procedures [3

3. R. K. Lau, M. Ménard, Y. Okawachi, M. A. Foster, A. C. Turner-Foster, R. Salem, M. Lipson, and A. L. Gaeta, “Continuous-wave mid-infrared frequency conversion in silicon nanowaveguides,” Opt. Lett. 36(7), 1263–1265 (2011). [CrossRef] [PubMed]

, 4

4. C. J. Fredricksen, J. W. Cleary, W. R. Buchwald, P. Figueiredo, F. Khalilzadeh-Rezaie, G. Medhi, I. Rezadad, M. Shahzad, M. Yesiltas, J. Nath, J. Boroumand, E. Smith, and R. E. Peale, “Planar integrated plasmonic mid-IR spectrometer,” Proc. SPIE 8353, 835321 (2012). [CrossRef]

]. However, there is still little investigation on passive photonics devices based in mid-IR waveband to realize the highly desired signal processing functions such as dispersion monitoring, switching and wavelength conversion. In 1.55 μm region, third harmonic generation (THG) has been demonstrated to be a promising way to realize high speed optical performance monitoring of optical signal to noise ratio (OSNR) and residual dispersion [5

5. B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Optical signal processing on a silicon chip at 640Gb/s using slow-light,” Opt. Express 18(8), 7770–7781 (2010). [CrossRef] [PubMed]

]. Due to the development of mid-IR communication [6

6. A. Soibel, M. Wright, W. Farr, S. Keo, C. Hill, R. Q. Yang, and H. C. Liu, “Free space optical communication utilizing mid-infrared interband cascade laser,” Proc. SPIE 7587, 75870S (2010). [CrossRef]

, 7

7. N. S. Prasad, D. D. Smith, and J. R. Magee, “Data communication in mid-IR using a solid-state laser pumped optical parametric oscillator,” Proc. SPIE 4821, 214–224 (2002). [CrossRef]

], THG devices working in this special waveband can find applications in the area of signal processing. Therefore, design for efficient THG devices working in the mid-IR waveband is interesting and valuable.

2. Basic THG theory in the APSW

It should be noted that, the nonlinear response of silver in the waveguide is neglected during the THG process, since the electric field does not penetrate deeply inside the metal and decays exponentially with distance from the metal-dielectric interface. Therefore, to study the THG process in the considered APSW, the nonlinear coupled mode theory for waveguides with substantial loss is outlined. To correctly model the nonlinear optical interactions, all components of the excited electric and magnetic fields in the APSW should be taken into consideration. From Maxwell’s curl equations,
×E(r,t)=μH(r,t)t
(1)
×H(r,t)=εE(r,t)t+PNLt
(2)
where ε and μ are the linear permittivity and permeability. PNLis the nonlinear polarization vector and PNL/t can be viewed as a source term that arises from the nonlinear interaction. Assuming only two different propagating modes in the APSW, i.e. pump at frequency ω1 and its third harmonics at ω3, and neglecting their linear losses at first, the total excited electric fields E and magnetic fields H at any location of the waveguide can be described as below [21

21. A. A. Sukhorukov, A. S. Solntsev, S. S. Kruk, D. N. Neshev, and Y. S. Kivshar, “Nonlinear coupled-mode theory for periodic plasmonic waveguides and meramaterials with loss and gain,” Opt. Lett. 39(3), 462–465 (2014).

, 22

22. 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]

]:
E(r,t)=12jA˜j(z)Z01/2Fj(r)exp[i(βjzωjt)]+c.c.
(3)
H(r,t)=12jA˜j(z)Z0-1/2Gj(r)exp[i(βjzωjt)]+c.c.
(4)
where βj=ωjcneff(ωj) is the propagation constant, A˜j(z)is the slowly-varying mode amplitude, Fj(r)and Gj(r)are mode profiles which have been normalized with 14ANL(Fj×Gj*+Fj*×Gj)z^dxdy=1, where j=1refers to the case of FF, and j=3refers to the case of THF. Z0=μ0/ε0 here is used to simplify the numerical calculation. r=(x,y,z) andr=(x,y). With the normalization, the corresponding field power can be expressed asPj(z)=|Aj(z)|2.

PNL=ε0χ(3)(r)(E(r,t)E(r,t))E(r,t)
(6)

For lossy waveguides case, the complex propagation constant can be written as: β=βj+iαj/2, where βjandαj are both real and positive. They represent the phase propagation constant and linear propagation loss coefficient, respectively. We define
Aj=A˜jexp(αjz2)
(7)
and substitute it into Eq. (6), the nonlinear polarization can be rewritten as:
PNL'=ε0χ(3)(r)(E'(r,t)E'(r,t))E'(r,t)
(8)
where E'(r,t)=12jAj(z)Z01/2Fj(r)exp[i(βjzωjt)]+c.c.. We then can obtain the following nonlinear coupled-wave equation for the lossy APSW:

dA˜jdz=Z01/22exp(αjz2)ANLexp[i(βjzωjt)]Fj*PNL'ttdxdy
(9)

dP1dz=α1P12I3P132P312sinΨ
(20)
dP3dz=α3P3+2I6P132P312sinΨ
(21)
dΨdz=δβ+3(I4I1)P1+(I53I2)P3+(I6P132P3123I3P112P312)cosΨ
(22)

It is found that P3 can be neglected in Eq. (22) because of its low value as compared to pump power, i.e. P1. Then Eq. (22) can be rewritten as dΨdz=δβ+3(I4I1)P1. This new equation depicts that Ψ is determined mainly by detuning term and pump power term. If the right hand side of Eq. (22) is kept approximately to be 0, power can be transferred from pump (P1) to third harmonic (P3) efficiently. We will use the Runge-Kutta method in MATLAB to numerically calculate Eqs. (18) and (19) in the section 4 to investigate the THG efficiency.

In the end, the definition of the conversion efficiency of THG is given as η=P3(Lp)/P1(0), whereP1(0)stands for the pump power at FF, Lp stands for the waveguide length when THF reaches its maximum output power P3(Lp), respectively. Note that, the value of δβ = β3-3β1 can directly influence the conversion efficiency. Phase-mismatch (δβ ≠ 0) always results in cycle flowing of the energy between FF and THF and limits the one-way conversion efficiency which makes the THG impossible. Recently, the quasi-phase matching (QPM) technique was theoretically proposed to realize PMC by adopting index grating [26

26. K. Tarnwski, B. Kibler, C. Finot, and W. Urbanczyk, “Quasi-phase-matched third harmonic generation in optical fibers using refractive-index gratings,” IEEE. J. of Quant. Electro. 47(5), 622–629 (2011). [CrossRef]

], but it leads to large mode sizes and complexity in structure fabrications. Fortunately, the APSW supports several modes and the effective index of the fundamental mode is larger than the higher-order mode at any specific frequency. PMC then can be appropriately obtained if the FF propagates at a lower-order mode. In our proposed configuration, inter-modal phase-matching technique as described above is adopted [17

17. T. Lee, Y. Jung, C. A. Codemard, M. Ding, N. G. R. Broderick, and G. Brambilla, “Broadband third harmonic generation in tapered silica fibres,” Opt. Express 20(8), 8503–8511 (2012). [CrossRef] [PubMed]

, 18

18. K. Bencheikh, S. Richard, G. Mélin, G. Krabshuis, F. Gooijer, and J. A. Levenson, “Phase-matched third-harmonic generation in highly germanium-doped fiber,” Opt. Lett. 37(3), 289–291 (2012). [CrossRef] [PubMed]

].

3. Key factors to increase the THG efficiency

We first fixed the slot width to be w = 40 nm and silicon slot height to be hs = 100 nm, respectively. The effective indices of the 0-th mode at FF (λ1 = 3600 nm) and 2-nd mode at THF (λ1 = 1200) are calculated by using finite-element-based commercial COMSOL software, with an element size of 4 nm for slot and 80 nm for other region. Figure 2(a)
Fig. 2 Design of the waveguide geometry satisfying the PMC. Effective indices of the guided modes versus (a) the DDMEBT slot height ho with certain slot width w = 40 nm and silicon slot height hs = 100 nm, (b) the silicon slot height hs with certain DDMEBT slot height ho = 500 nm. Note that ‘0’ and ‘2’ stand for the 0-th and 2-nd waveguide modes, respectively.
shows the effective indices as a function of DDMEBT polymer slot height ho. We found that, with a wide range of the DDMEBT slot height from 460 nm to 540 nm, the indices of the two guided modes do not change too much. For simplicity, we consider a DDMEBT polymer slot height of 500 nm and study the possibility of achieving PMC by adjusting the height of the silicon slot. Figure 2(b) plots the effective indices of the two guided modes as a function of the silicon slot height for w = 40 nm and ho = 500 nm. We can see that the PMC occurs when the silicon slot height is set to be hs = 207 nm. At this point, the effective indices of FF and THF are 2.5663 + 0.0325i and 2.5661 + 0.0089i, respectively. Here, for FF wave, the supported 0-th mode results from the coupling between the fundamental mode in the DDMEBT polymer slot and fundamental mode in the silicon slot. On the other hand, for THF wave, the 2-nd mode results from the coupling between the fundamental mode in the DDMEBT polymer slot and 1-st mode in the silicon slot.

4. Simulation results

Now we numerically study the nonlinear THG performance in the proposed APSW. Assuming a pump power of 1 W, the conversion efficiency and the corresponding waveguide length Lp at different PMCs are illustrated in Fig. 5
Fig. 5 Conversion efficiency of THG and the corresponding waveguide length Lp as a function of the slot width at different PMCs for a fixed pump power of 1 W.
. The conversion efficiency η increases together with shorter waveguide length with the reduction of the slot width. These results are consistent with the FOM variation as discussed above. Note that the pump power can be obtained in practice by typical mid-IR lasers, such as CW optical parametric oscillators [28

28. W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-W continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. 21(17), 1336–1338 (1996). [CrossRef] [PubMed]

34

34. M. M. J. W. van Herpen, S. Li, S. E. Bisson, S. te Lintel Hekkert, and F. J. M. Harren, “Tuning and stability of a continuous-wave mid-infrared high-power single resonant optical parametric oscillator,” Appl. Phys. B 75(2–3), 329–333 (2002). [CrossRef]

]. In Fig. 6(a)
Fig. 6 (a) Contour map of conversion efficiency in MPSW with different δβ and (b) optical powers of FF and THF along the propagation distance with δβ = −36 m−1 with a fixed pump power of 1 W; (c) conversion and the corresponding detuning versus the pump power.
, we give the efficiency contour map with different detuning constants for the waveguide geometry as shown in Fig. 3. A maximum conversion efficiency of 4.88 × 10−6 happens with an optimized detuning of δβ = −36 m−1. The value is negatively offset from zero to compensate for the nonlinear phase shift during the nonlinear process, although a smaller but nonetheless significant third harmonic signal would still be achieved if δβ = 0 m−1. Figure 6(b) shows the evolution simulation results of the FF and THF powers along the propagation distance with a fixed pump power of P1 = 1 W and a detuning of δβ = −36 m−1. One can see that the power of the fundamental mode at FF decreases monotonously due to the nonlinear conversion process and its linear propagation loss. The nonlinear conversion mainly contributed to the THF, which increases first before decreases and reaches its maximum power of 4.88 × 10−6 W at a 10.65 μm waveguide length. This short waveguide length is also crucial in overcoming other competing nonlinear effects (such as stimulated Raman and Brillouin scattering), which increase exponentially with propagation distance. We also examine the conversion efficiency and the corresponding optimized detuning as a function of pump power, as shown in Fig. 6(c). The optimized detuning decreases from −6 m−1 to −36 m−1 as pump power increases from 0.2 to 1 W, which conforms to the Eq. (22). The conversion efficiency increases with increasing pump power and can be further increased with pump power higher than 1 W.

According to our calculation, the value of Re(I6), FOM_FF, FOM_THF, and conversion η all increase with the reduction the slot width as shown in Figs. 7(a)
Fig. 7 (a) Re(I6), (b) FOMs, and (c) conversion efficiency as a function of the slot width.
-7(c). It can be summarized that the THG performance will be further improved with smaller slot width. However, slots that are too narrow should not be considered due to the difficulty in fabrication.

5. Conclusion

Acknowledgments

This work was supported by the Singapore A*STAR SERC Grant: “Advanced Optics in Engineering” Program (Grant No. 1223600001).

References and links

1.

J. Ma and S. Fathpour, “Pump-to-Stokes relative intensity noise transfer and analytical modeling of mid-infrared silicon Raman lasers,” Opt. Express 20(16), 17962–17972 (2012). [CrossRef] [PubMed]

2.

Z. Wang, H. Liu, N. Huang, Q. Sun, and X. Li, “Mid-infrared Raman amplification and wavlength conversion in dispersion engineered silicon-on-sapphire waveguides,” J. Opt. 16(1), 015206(2014). [CrossRef]

3.

R. K. Lau, M. Ménard, Y. Okawachi, M. A. Foster, A. C. Turner-Foster, R. Salem, M. Lipson, and A. L. Gaeta, “Continuous-wave mid-infrared frequency conversion in silicon nanowaveguides,” Opt. Lett. 36(7), 1263–1265 (2011). [CrossRef] [PubMed]

4.

C. J. Fredricksen, J. W. Cleary, W. R. Buchwald, P. Figueiredo, F. Khalilzadeh-Rezaie, G. Medhi, I. Rezadad, M. Shahzad, M. Yesiltas, J. Nath, J. Boroumand, E. Smith, and R. E. Peale, “Planar integrated plasmonic mid-IR spectrometer,” Proc. SPIE 8353, 835321 (2012). [CrossRef]

5.

B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Optical signal processing on a silicon chip at 640Gb/s using slow-light,” Opt. Express 18(8), 7770–7781 (2010). [CrossRef] [PubMed]

6.

A. Soibel, M. Wright, W. Farr, S. Keo, C. Hill, R. Q. Yang, and H. C. Liu, “Free space optical communication utilizing mid-infrared interband cascade laser,” Proc. SPIE 7587, 75870S (2010). [CrossRef]

7.

N. S. Prasad, D. D. Smith, and J. R. Magee, “Data communication in mid-IR using a solid-state laser pumped optical parametric oscillator,” Proc. SPIE 4821, 214–224 (2002). [CrossRef]

8.

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

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H. Junginger, H. Puell, H. Scheingraber, and C. Vidal, “Resonant third-harmonic generation in a low-loss medium,” IEEE J. Quantum Electron. 16(10), 1132–1137 (1980). [CrossRef]

10.

S. B. Hasan, C. Rockstuhl, T. Pertsch, and F. Lederer, “Second-order nonlinear frequency conversion processes in plasmonic slot waveguides,” J. Opt. Soc. Am. B 29(7), 1606–1611 (2012). [CrossRef]

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M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics 6(11), 737–748 (2012). [CrossRef]

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M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011). [CrossRef] [PubMed]

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S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95(4), 046802 (2005). [CrossRef] [PubMed]

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D. F. P. Pile, T. Ogawa, D. K. Gramotnev, Y. Matsuzaki, K. C. Vernon, K. Yamaguchi, T. Okamoto, M. Haraguchi, and M. Fukui, “Two-dimensionally localized modes of a nanoscale gap plasmon waveguide,” Appl. Phys. Lett. 87(26), 261114 (2005). [CrossRef]

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T. Lee, Y. Jung, C. A. Codemard, M. Ding, N. G. R. Broderick, and G. Brambilla, “Broadband third harmonic generation in tapered silica fibres,” Opt. Express 20(8), 8503–8511 (2012). [CrossRef] [PubMed]

18.

K. Bencheikh, S. Richard, G. Mélin, G. Krabshuis, F. Gooijer, and J. A. Levenson, “Phase-matched third-harmonic generation in highly germanium-doped fiber,” Opt. Lett. 37(3), 289–291 (2012). [CrossRef] [PubMed]

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A. A. Sukhorukov, A. S. Solntsev, S. S. Kruk, D. N. Neshev, and Y. S. Kivshar, “Nonlinear coupled-mode theory for periodic plasmonic waveguides and meramaterials with loss and gain,” Opt. Lett. 39(3), 462–465 (2014).

22.

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]

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27.

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28.

W. R. Bosenberg, A. Drobshoff, J. I. Alexander, L. E. Myers, and R. L. Byer, “93% pump depletion, 3.5-W continuous-wave, singly resonant optical parametric oscillator,” Opt. Lett. 21(17), 1336–1338 (1996). [CrossRef] [PubMed]

29.

A. J. Henderson, P. M. Roper, L. A. Borschowa, and R. D. Mead, “Stable, continuously tunable operation of a diode-pumped doubly resonant optical parametric oscillator,” Opt. Lett. 25(17), 1264–1266 (2000). [CrossRef] [PubMed]

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F. Kühnemann, K. Scheider, A. Hecker, A. A. E. Martis, W. Urban, S. Schiller, and J. Mlynek, “Photoacoustic trace-gas detection using a cw single-frequency parametric oscillator,” Appl. Phys. B 66(6), 741–745 (1998). [CrossRef]

33.

M. M. J. W. van Herpen, S. Te Lintel Hekkert, S. E. Bisson, and F. J. M. Harren, “Wide single-mode tuning of a 3.0- 3.8-mum, 700-mW, continuous-wave Nd:YAG-pumped optical parametric oscillator based on periodically poled lithium niobate,” Opt. Lett. 27(8), 640–642 (2002). [CrossRef] [PubMed]

34.

M. M. J. W. van Herpen, S. Li, S. E. Bisson, S. te Lintel Hekkert, and F. J. M. Harren, “Tuning and stability of a continuous-wave mid-infrared high-power single resonant optical parametric oscillator,” Appl. Phys. B 75(2–3), 329–333 (2002). [CrossRef]

OCIS Codes
(070.4340) Fourier optics and signal processing : Nonlinear optical signal processing
(190.2620) Nonlinear optics : Harmonic generation and mixing
(250.5403) Optoelectronics : Plasmonics
(250.4390) Optoelectronics : Nonlinear optics, integrated optics

ToC Category:
Nonlinear Optics

History
Original Manuscript: May 20, 2014
Revised Manuscript: July 5, 2014
Manuscript Accepted: July 6, 2014
Published: July 24, 2014

Citation
Tingting Wu, Yunxu Sun, Xuguang Shao, Perry Ping Shum, and Tianye Huang, "Efficient phase-matched third harmonic generation in an asymmetric plasmonic slot waveguide," Opt. Express 22, 18612-18624 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-15-18612


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References

  1. J. Ma and S. Fathpour, “Pump-to-Stokes relative intensity noise transfer and analytical modeling of mid-infrared silicon Raman lasers,” Opt. Express20(16), 17962–17972 (2012). [CrossRef] [PubMed]
  2. Z. Wang, H. Liu, N. Huang, Q. Sun, and X. Li, “Mid-infrared Raman amplification and wavlength conversion in dispersion engineered silicon-on-sapphire waveguides,” J. Opt.16(1), 015206(2014). [CrossRef]
  3. R. K. Lau, M. Ménard, Y. Okawachi, M. A. Foster, A. C. Turner-Foster, R. Salem, M. Lipson, and A. L. Gaeta, “Continuous-wave mid-infrared frequency conversion in silicon nanowaveguides,” Opt. Lett.36(7), 1263–1265 (2011). [CrossRef] [PubMed]
  4. C. J. Fredricksen, J. W. Cleary, W. R. Buchwald, P. Figueiredo, F. Khalilzadeh-Rezaie, G. Medhi, I. Rezadad, M. Shahzad, M. Yesiltas, J. Nath, J. Boroumand, E. Smith, and R. E. Peale, “Planar integrated plasmonic mid-IR spectrometer,” Proc. SPIE8353, 835321 (2012). [CrossRef]
  5. B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. O’Faolain, T. F. Krauss, B. J. Eggleton, and D. J. Moss, “Optical signal processing on a silicon chip at 640Gb/s using slow-light,” Opt. Express18(8), 7770–7781 (2010). [CrossRef] [PubMed]
  6. A. Soibel, M. Wright, W. Farr, S. Keo, C. Hill, R. Q. Yang, and H. C. Liu, “Free space optical communication utilizing mid-infrared interband cascade laser,” Proc. SPIE7587, 75870S (2010). [CrossRef]
  7. N. S. Prasad, D. D. Smith, and J. R. Magee, “Data communication in mid-IR using a solid-state laser pumped optical parametric oscillator,” Proc. SPIE4821, 214–224 (2002). [CrossRef]
  8. J. A. Armstrong, N. Bloembergen, J. Ducuing, and P. S. Pershan, “Interactions between light waves in a nonlinear dielectric,” Phys. Rev.127(6), 1918–1939 (1962). [CrossRef]
  9. H. Junginger, H. Puell, H. Scheingraber, and C. Vidal, “Resonant third-harmonic generation in a low-loss medium,” IEEE J. Quantum Electron.16(10), 1132–1137 (1980). [CrossRef]
  10. S. B. Hasan, C. Rockstuhl, T. Pertsch, and F. Lederer, “Second-order nonlinear frequency conversion processes in plasmonic slot waveguides,” J. Opt. Soc. Am. B29(7), 1606–1611 (2012). [CrossRef]
  11. M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photonics6(11), 737–748 (2012). [CrossRef]
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