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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 8 — Apr. 9, 2012
  • pp: 8920–8928
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Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides

Zhaolu Wang, Hongjun Liu, Nan Huang, Qibing Sun, and Jin Wen  »View Author Affiliations


Optics Express, Vol. 20, Issue 8, pp. 8920-8928 (2012)
http://dx.doi.org/10.1364/OE.20.008920


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Abstract

Terahertz (THz) wave generation via four-wave mixing (FWM) in silicon membrane waveguides is theoretically investigated with mid-infrared laser pulses. Compared with the conventional parametric amplification or wavelength conversion based on FWM in silicon waveguides, which needs a pump wavelength located in the anomalous group-velocity dispersion (GVD) regime to realize broad phase matching, the pump wavelength located in the normal GVD regime is required to realize collinear phase matching for the THz-wave generation via FWM. The pump wavelength and rib height of the silicon membrane waveguide can be tuned to obtain a broadband phase matching. Moreover, the conversion efficiency of the THz-wave generation is studied with different pump wavelengths and rib heights of the silicon membrane waveguides, and broadband THz-wave can be obtained with high efficiency exceeding 1%.

© 2012 OSA

1. Introduction

The development of efficient and compact sources of THz-wave is of great interest for applications in various fields such as applied physics, communications, sensing, and life sciences [1

1. Y. Takushima, S. Y. Shin, and Y. C. Chung, “Design of a LiNbO(3) ribbon waveguide for efficient difference-frequency generation of terahertz wave in the collinear configuration,” Opt. Express 15(22), 14783–14792 (2007). [CrossRef] [PubMed]

]. The difference-frequency generation (DFG) in nonlinear optical crystals is an important technique for coherent THz-wave generation [2

2. K. Kawase, H. Minamide, K. Imai, J. Shikata, and H. Ito, “Injection-seeded terahertz-wave parametric generator with wide tenability,” Appl. Phys. Lett. 80(2), 195–198 (2002). [CrossRef]

7

7. Y. J. Ding, “Efficient generation of high-frequency terahertz waves from highly lossy second-order nonlinear medium at polariton resonance under transverse-pumping geometry,” Opt. Lett. 35(2), 262–264 (2010). [CrossRef] [PubMed]

]. However, it is difficult to increase the conversion efficiency for DFG based THz-wave generation, because most of nonlinear optical crystals have a large absorption in the THz–wave region. Surface-emitting THz-wave generation can be used to overcome the high absorption loss [8

8. Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference-frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett. 30(21), 2927–2929 (2005). [CrossRef] [PubMed]

11

11. Y. H. Avetisyan, “Terahertz-wave surface-emitted difference-frequency generation without quasi-phase-matching technique,” Opt. Lett. 35(15), 2508–2510 (2010). [CrossRef] [PubMed]

]. Unfortunately, this method requires a specially designed crystal, and the interaction length is limited by the size of the base material [12

12. K. Suizu and K. Kawase, “Terahertz-wave generation in a conventional optical fiber,” Opt. Lett. 32(20), 2990–2992 (2007). [CrossRef] [PubMed]

].

To cope with these difficulties, Suizu et al. proposed a way to generate THz-waves in an optical fiber via FWM process, which is a promising method for realizing a reasonable THz-wave source [12

12. K. Suizu and K. Kawase, “Terahertz-wave generation in a conventional optical fiber,” Opt. Lett. 32(20), 2990–2992 (2007). [CrossRef] [PubMed]

]. FWM in silicon waveguide had been studied not only theoretically but also experimentally [13

13. H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13(12), 4629–4637 (2005). [CrossRef] [PubMed]

, 14

14. R. L. Espinola, J. I. Dadap, R. M. Osgood Jr, S. J. McNab, and Y. A. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13(11), 4341–4349 (2005). [CrossRef]

]. Compared with conventional fiber, the silicon rib membrane waveguide will be a more viable structure for THz-wave generation via FWM. There are five major inherent advantages. First, the silicon membrane has an absorption loss below 0.23 cm−1 over 1.2-6.9 μm and 25-200 μm [15

15. R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840 (2006). [CrossRef]

], while the absorption coefficient of the optical fiber in the THz-wave region is about 5 cm−1. Second, the nonlinear refractive index n2 of silicon is about 200 times larger than that of silica [16

16. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). [CrossRef] [PubMed]

]. Third, the refractive index of silicon (around 3.5) is much larger than that of air, which implies a much stronger light confinement. Fourth, the crystalline nature of silicon that makes stimulated Raman scattering (SRS) depend strongly on the waveguide geometry and mode polarization, and SRS cannot occur when an input pulse excites the TM mode [16

16. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). [CrossRef] [PubMed]

]. Fifth, the silicon membrane waveguide is also CMOS compatible and enable low-cost large-scale integration [15

15. R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840 (2006). [CrossRef]

]. Moreover, the silicon waveguide also have been modified to show second-order nonlinearity at technically relevant levels [17

17. R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441(7090), 199–202 (2006). [CrossRef] [PubMed]

, 18

18. B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Nagel, F. Merget, and H. Kurz, “Pockels effect based fully integrated, strained silicon electro-optic modulator,” Opt. Express 19(18), 17212–17219 (2011). [CrossRef] [PubMed]

]. Even the THz-wave generation based on DFG has already been experimentally demonstrated in a silicon waveguide by Waechter et al [19

19. M. Wächter, C. Matheisen, M. Waldow, T. Wahlbrink, J. Bolten, M. Nagel, and H. Kurz, “Optical generation of terahertz and second-harmonic light in plasma-activated silicon nanophotonic structures,” Appl. Phys. Lett. 97(16), 161107 (2010). [CrossRef]

]. Despite this progress, there is still a strong motivation to investigate the THz-wave generation based on FWM due to the high third-order nonlinearity of silicon waveguide.

In this paper, we investigate efficient THz-wave generation via FWM in silicon membrane waveguides using Mid-infrared pump and signal waves. The organization of the paper is as follows. In Section 2, we analyze the collinear phase matching condition and phase matching bandwidth with the dispersion relation of silicon membrane waveguides. In section 3, we numerically investigate the conversion efficiency of the THz-wave generation for different pump wavelengths and rib heights of the waveguides. Finally, we summarize this paper in Section 4.

2. Phase matching condition and phase matching bandwidth for THz-wave generation

We use degenerate FWM to generate THz-wave, which typically involves two pump photons at angular frequency ωp passing their energy to a signal wave at angular frequency ωs and a THz-wave at angular frequency ωTHz. Figure 1
Fig. 1 Schematic of (a) energy-conservation diagram and (b) the phase-matching condition for collinear phase matching.
shows the energy conservation diagrams and phase-matching condition for collinear configuration, which ensures that the THz-wave is generated through FWM and grows while copropagating with the pump and signal beam. These relationships can be written as the following equations [20

20. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]:
2ωpωsωTHz=0,
(1)
ks+kTHz2kp+kNL=0,
(2)
where kp, ks and kTHz represent the propagation wave number of pump, signal and THz-wave, respectively. kNL is the nonlinear phase mismatch, which induced by self phase modulation (SPM) and cross phase modulation (XPM) [21

21. M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441(7096), 960–963 (2006). [CrossRef] [PubMed]

]. We can also define kL = ks + kTHz-2kp as the linear phase mismatch due to dispersion [22

22. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14(11), 4786–4799 (2006). [CrossRef] [PubMed]

]. Since the signal and THz-wave are located symmetrically around the pump frequency, the linear phase mismatch only depends on even-order dispersion parameters as [20

20. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]
kL=β2pΩsp2+2m=2β2mp(2m)!Ωsp2m,
(3)
where β2p is the group-velocity dispersion, and β2mp is the even-order dispersion at the pump frequency. Ωsp = ωs-ωp = ωp-ωTHz is the signal-pump (or pump-THz) frequency detuning, which is a large value due to ωTHz<<ωp. Since Ωsp is so large, the higher-order dispersions become important for the phase-matching [20

20. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

]. However, the higher-order dispersions cannot be accurately calculated using numerical method. Therefore, Eq. (3) can’t be used to calculate the linear phase mismatch, and we use it only to explain the influence of the higher-order dispersions on the linear phase mismatch in the following part of the paper. The linear phase mismatch kL can be calculated using the Equation: kL = ks + kTHz-2kp = (nsωs + nTHzωTHz-2npωp)/c, where ns, nTHz, and np represent the fundamental TM mode effective indices of the signal, THz-wave, and pump. c is the speed of light in vacuum.

Figure 2
Fig. 2 Left: rib silicon membrane waveguide dimension; Right: the TM mode profiles of the silicon membrane waveguide at the wavelength of 35 μm for different rib heights.
shows the dimension of silicon membrane waveguide, in which the rib is suspended over an air filled cavity comprising the lower cladding, whilst the upper cladding is also air. This waveguide is constructed by etching away the buried-oxide insulating material in silicon on insulator (SOI) locally under the rib [15

15. R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840 (2006). [CrossRef]

]. The waveguide should be designed not only confining the THz-wave, but also satisfying the single-mode condition [23

23. G. Z. Mashanovich, M. Milosevic, P. Matavulj, S. Stankovic, B. Timotijevic, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol, and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23(6), 064002 (2008). [CrossRef]

]:
WH0.3+r1r2,
(4)
where W/H is the ratio of the rib width to overall rib height, and r is the ratio of the slab height (H-h) to overall rib height. Here we set h = H/2 and r = 0.5.

The dimension of the waveguide is optimally designed to realize the collinear phase matching. The rib waveguides with width of 12 μm and rib heights varied from 14 μm up to 17 μm can satisfy the single-mode condition. To determine the performance of these waveguides, we need to simulate the mode profiles at the THz-band using a finite-difference mode solver [24

24. T. E. Murphy, software available at http://www.photonics.umd.edu.

]. There are two assumptions before simulation. The first is that the pump and signal waves are both fundamental TM mode, thus the SRS can be neglected [16

16. L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). [CrossRef] [PubMed]

]. The second is that the pump and signal waves are Mid-infrared waves and exceed 2.2 μm, leading to a negligible of two-photon absorption (TPA) and free-carrier absorption (FCA) and thus enabling efficient parametric generation [25

25. Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express 16(14), 10596–10610 (2008). [CrossRef] [PubMed]

, 26

26. X. Liu, R. M. Osgood Jr, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophtonic waveguides,” Nat. Photonics 4(8), 557–560 (2010). [CrossRef]

]. The fundamental TM mode profiles of the silicon membrane waveguide at the wavelength of 35 μm for several rib heights are shown in Fig. 2. It is clear that the designed waveguides can confine THz-wave.

To realize phase matching, we first simulate the dispersion of the waveguides. For the silicon membrane waveguides mentioned above, the fundamental TM mode effective indices neff as a function of pump wavelength are numerically determined using the finite-difference mode solver. Note that in the calculations, the material dispersion of the silicon is determined by a Sellmeier equation mentioned in [27

27. R. M. Osgood Jr, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon. 1(1), 162–235 (2009). [CrossRef]

].The dispersion relation is then calculated from β(ω) = neff(ω)ω/c. Higher-order dispersion is finally calculated via numerical differentiation from βn = dnβ/dωn, and the results of the neff and even-order dispersion are shown in Fig. 3
Fig. 3 Plots of computed effective index of refraction (a), and (b) GVD dispersion (c) forth order dispersion (d) sixth order dispersion as a function of pump wavelength for different rib heights.
. The zero dispersion wavelengths of the four waveguides are 6 μm, 6.1 μm, 6.2 μm, and 6.3 μm, respectively. The fourth-order dispersion β4 and sixth-order dispersion β6 are all negative for the pump wavelength ranging from 4 μm to 7 μm. In the calculation process, the initial errors would be amplified when taking so many derivatives of discrete date and the curves of higher-order dispersion will be distorted for a high spectral resolution. As we only concern the sign of higher-order dispersions other than precision, we can decrease spectral resolution to obtain smooth curves as shown in Fig. 3. Despite the value of higher-order dispersions can’t be accurately calculated, we can distinguish the sign of higher-order dispersions, which will be used to explain the change of linear phase mismatch with different pump wavelengths for a large signal-pump frequency detuning Ωsp in the following part.

In order to determine the effective mode area Aeff, we calculate the mode profiles at wavelength from 2 μm to 7 μm. Figure 4
Fig. 4 The fundamental TM mode profiles of the waveguides with different rib heights.
shows the fundamental TM mode profiles of the waveguides with different heights. Because of the little variation of the mode profiles for the Mid-infrared waves, we use 4.3 μm as the operation wavelength to simulate the mode profiles and calculate the effective mode areas. The effective mode areas Aeff of the waveguides are 42 μm2, 48 μm2, 54 μm2 and 58 μm2, respectively.

When the linear phase mismatch kL is compensated by the nonlinear phase mismatch kNL, the phase matching is realized according to Eq. (2). The nonlinear phase mismatch is defined as kNL = 2γPP, where γ = ωn2/cAeff is the effective nonlinearity coefficient of the waveguide, and PP represents the pump peak power. If we assumed the pump wavelength is 4.3 μm and the rib height is 15 μm, the nonlinearity coefficient γ = 152.2 W−1km−1 with n2 = 5 × 10−18 m2W−1, which is assumed according to [29

29. E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express 18(21), 21981–21989 (2010). [CrossRef] [PubMed]

31

31. N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011). [CrossRef]

]. The nonlinear phase mismatch kNL = 6.08 cm−1 when the pump peak power is set to be 2000 W. Thus, the phase matching is realized when the linear phase mismatch is about −6 cm−1 as shown in Fig. 6(a), and the points of intersection are the phase matching point. The corresponding relationship of pump, signal and THz-wave wavelength for the phase matching point is described in Fig. 6(b). Therefore, phase matching for a widely tunable THz-wave ranging from 8.57 THz to 10 THz (or from 30 μm to 35 μm) can be realized by tuning the pump wavelength from 4.2 μm to 4.4 μm in the silicon membrane waveguide with rib height of 15 μm. If the pump wavelength is less than 4.2 μm, much broader THz-wave bandwidth can be achieved as the trend shown in Fig. 6. However, the corresponding signal wavelength satisfying the phase matching will be reduced less than 2.2 μm and the efficiency for THz-wave generation will be decreased due to TPA.

3. The efficiency of THz-wave generation

The FWM process can be described by the following coupling equations [29

29. E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express 18(21), 21981–21989 (2010). [CrossRef] [PubMed]

]:
dApdz=αp2Ap+iγp|Ap|2Ap+2iγps|As|2Ap+2iγpt|At|2Ap+2iγpAsAtAp*exp(ikLz),
(5)
dAsdz=αs2As+iγs|As|2As+2iγsp|Ap|2As+2iγst|At|2As+iγsAp2At*exp(ikLz),
(6)
dAtdz=αt2At+iγt|At|2At+2iγtp|Ap|2At+2iγts|As|2At+iγtAp2As*exp(ikLz),
(7)
where Ap, As and At represent the slowly varying amplitude of the pump, signal and THz-waves, and z is the propagation distance. The parameters αp, αs and αt represent the linear propagation losses of the pump, signal and THz-wave, which are assumed as 0.138 cm−1, 0.092 cm−1 and 0.23 cm−1, respectively [15

15. R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840 (2006). [CrossRef]

, 32

32. G. Z. Mashanovich, M. M. Milošević, M. Nedeljkovic, N. Owens, B. Xiong, E. J. Teo, and Y. Hu, “Low loss silicon waveguides for the mid-infrared,” Opt. Express 19(8), 7112–7119 (2011). [CrossRef] [PubMed]

]. The nonlinearity coefficient γxy (xy = ps, st, pt) can be calculated with the averaged frequency, ωxy = (ωx + ωy)/2 [29

29. E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express 18(21), 21981–21989 (2010). [CrossRef] [PubMed]

]. Here, we assume n2=5×10−18 m2W−1 for the pump and THz-wave [29

29. E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express 18(21), 21981–21989 (2010). [CrossRef] [PubMed]

31

31. N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011). [CrossRef]

]. Despite the n2 for the signal wave is predicted to be about 8×10−18 m2W−1 [30

30. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007). [CrossRef]

], for simplicity, we also use n2=5×10−18 m2W−1 for the signal in the simulation. Moreover, the value of n2 used to calculate γxy is also assumed as 5×10−18 m2W−1.

The THz-wave generation via FWM is numerically studied by simultaneously injecting pump pulses and signal pulses in the Mid-infrared band [33]. The pump and signal pulses are taken to be hyperbolic-secant pulses with same pulse width of 12 ns and same repetition rate [33]. Here, we assume the signal peak power is half of the pump peak power. The multi-photon absorption can be neglected in this paper, which does not presents a significant obstacle for practical applications [26

26. X. Liu, R. M. Osgood Jr, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophtonic waveguides,” Nat. Photonics 4(8), 557–560 (2010). [CrossRef]

].

η=PTHzout/Ppumpin
(8)

For simplicity, we assume the waveguide length is 6 mm and the pump peak power is 2000 W in the following part.

THz-wave generation based on FWM is also a discrete wavelength conversion. Figure 9
Fig. 9 The conversion efficiency as a function of THz-wavelength when the signal wavelength is tuned from 2299.8 nm to 2304.9 nm for a fixed pump located at 4.3 μm.
shows the conversion efficiency of the generated THz-wave when the signal wavelength is tuned from 2299.8 nm to 2304.9 nm for a fixed pump located at 4.3 μm. It is shown that the maximum conversion efficiency occurs at the THz-wavelength of 32.5μm when the signal is tuned to satisfy the phase-matching condition. For this discrete wavelength conversion, the bandwidth of the generated THz-wave is about 1 μm, while the corresponding bandwidth of the signal is only 5.1 nm.

4. Conclusion

We have presented a theoretical study of THz-wave generation using Mid-infrared pump and signal waves in silicon membrane waveguides. The simulation model allows us to show the importance of the pump wavelength to the collinear phase matching, which can be realized only when the pump wavelength locates in the normal GVD regime. Moreover, broadband phase matching can be achieved by tuning the pump wavelength and the rib height. Finally, we numerically discuss the conversion efficiency of the THz-wave generation in the silicon membrane waveguides. This method and results show a promising way to realize an efficient and compact THz-wave source.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant 61078029.

References and links

1.

Y. Takushima, S. Y. Shin, and Y. C. Chung, “Design of a LiNbO(3) ribbon waveguide for efficient difference-frequency generation of terahertz wave in the collinear configuration,” Opt. Express 15(22), 14783–14792 (2007). [CrossRef] [PubMed]

2.

K. Kawase, H. Minamide, K. Imai, J. Shikata, and H. Ito, “Injection-seeded terahertz-wave parametric generator with wide tenability,” Appl. Phys. Lett. 80(2), 195–198 (2002). [CrossRef]

3.

A. C. Chiang, T. D. Wang, Y. Y. Lin, S. T. Lin, H. H. Lee, Y. C. Huang, and Y. H. Chen, “Enhanced terahertz-wave parametric generation and oscillation in lithium niobate waveguides at terahertz frequencies,” Opt. Lett. 30(24), 3392–3394 (2005). [CrossRef] [PubMed]

4.

X. Xie, J. Xu, and X.-C. Zhang, “Terahertz wave generation and detection from a cdte crystal characterized by different excitation wavelengths,” Opt. Lett. 31(7), 978–980 (2006). [CrossRef] [PubMed]

5.

T. D. Wang, S. T. Lin, Y. Y. Lin, A. C. Chiang, and Y. C. Huang, “Forward and backward terahertz-wave difference-frequency generations from periodically poled lithium niobate,” Opt. Express 16(9), 6471–6478 (2008). [CrossRef] [PubMed]

6.

K. L. Vodopyanov and Y. H. Avetisyan, “Optical terahertz wave generation in a planar GaAs waveguide,” Opt. Lett. 33(20), 2314–2316 (2008). [CrossRef] [PubMed]

7.

Y. J. Ding, “Efficient generation of high-frequency terahertz waves from highly lossy second-order nonlinear medium at polariton resonance under transverse-pumping geometry,” Opt. Lett. 35(2), 262–264 (2010). [CrossRef] [PubMed]

8.

Y. Sasaki, Y. Avetisyan, H. Yokoyama, and H. Ito, “Surface-emitted terahertz-wave difference-frequency generation in two-dimensional periodically poled lithium niobate,” Opt. Lett. 30(21), 2927–2929 (2005). [CrossRef] [PubMed]

9.

K. Suizu, Y. Suzuki, Y. Sasaki, H. Ito, and Y. Avetisyan, “Surface-emitted terahertz-wave generation by ridged periodically poled lithium niobate and enhancement by mixing of two terahertz waves,” Opt. Lett. 31(7), 957–959 (2006). [CrossRef] [PubMed]

10.

T. Ikari, X. Zhang, H. Minamide, and H. Ito, “THz-wave parametric oscillator with a surface-emitted configuration,” Opt. Express 14(4), 1604–1610 (2006). [CrossRef] [PubMed]

11.

Y. H. Avetisyan, “Terahertz-wave surface-emitted difference-frequency generation without quasi-phase-matching technique,” Opt. Lett. 35(15), 2508–2510 (2010). [CrossRef] [PubMed]

12.

K. Suizu and K. Kawase, “Terahertz-wave generation in a conventional optical fiber,” Opt. Lett. 32(20), 2990–2992 (2007). [CrossRef] [PubMed]

13.

H. Fukuda, K. Yamada, T. Shoji, M. Takahashi, T. Tsuchizawa, T. Watanabe, J. Takahashi, and S. Itabashi, “Four-wave mixing in silicon wire waveguides,” Opt. Express 13(12), 4629–4637 (2005). [CrossRef] [PubMed]

14.

R. L. Espinola, J. I. Dadap, R. M. Osgood Jr, S. J. McNab, and Y. A. Vlasov, “C-band wavelength conversion in silicon photonic wire waveguides,” Opt. Express 13(11), 4341–4349 (2005). [CrossRef]

15.

R. A. Soref, S. J. Emelett, and W. R. Buchwald, “Silicon waveguided components for the long-wave infrared region,” J. Opt. A, Pure Appl. Opt. 8(10), 840 (2006). [CrossRef]

16.

L. Yin, Q. Lin, and G. P. Agrawal, “Soliton fission and supercontinuum generation in silicon waveguides,” Opt. Lett. 32(4), 391–393 (2007). [CrossRef] [PubMed]

17.

R. S. Jacobsen, K. N. Andersen, P. I. Borel, J. Fage-Pedersen, L. H. Frandsen, O. Hansen, M. Kristensen, A. V. Lavrinenko, G. Moulin, H. Ou, C. Peucheret, B. Zsigri, and A. Bjarklev, “Strained silicon as a new electro-optic material,” Nature 441(7090), 199–202 (2006). [CrossRef] [PubMed]

18.

B. Chmielak, M. Waldow, C. Matheisen, C. Ripperda, J. Bolten, T. Wahlbrink, M. Nagel, F. Merget, and H. Kurz, “Pockels effect based fully integrated, strained silicon electro-optic modulator,” Opt. Express 19(18), 17212–17219 (2011). [CrossRef] [PubMed]

19.

M. Wächter, C. Matheisen, M. Waldow, T. Wahlbrink, J. Bolten, M. Nagel, and H. Kurz, “Optical generation of terahertz and second-harmonic light in plasma-activated silicon nanophotonic structures,” Appl. Phys. Lett. 97(16), 161107 (2010). [CrossRef]

20.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).

21.

M. A. Foster, A. C. Turner, J. E. Sharping, B. S. Schmidt, M. Lipson, and A. L. Gaeta, “Broad-band optical parametric gain on a silicon photonic chip,” Nature 441(7096), 960–963 (2006). [CrossRef] [PubMed]

22.

Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express 14(11), 4786–4799 (2006). [CrossRef] [PubMed]

23.

G. Z. Mashanovich, M. Milosevic, P. Matavulj, S. Stankovic, B. Timotijevic, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol, and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol. 23(6), 064002 (2008). [CrossRef]

24.

T. E. Murphy, software available at http://www.photonics.umd.edu.

25.

Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express 16(14), 10596–10610 (2008). [CrossRef] [PubMed]

26.

X. Liu, R. M. Osgood Jr, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophtonic waveguides,” Nat. Photonics 4(8), 557–560 (2010). [CrossRef]

27.

R. M. Osgood Jr, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon. 1(1), 162–235 (2009). [CrossRef]

28.

Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Impact of dispersion profiles of silicon waveguides on optical parametric amplification in the femtosecond regime,” Opt. Express 19(24), 24730–24737 (2011). [CrossRef] [PubMed]

29.

E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express 18(21), 21981–21989 (2010). [CrossRef] [PubMed]

30.

A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett. 90(19), 191104 (2007). [CrossRef]

31.

N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” J. Appl. Phys. 110(1), 011301 (2011). [CrossRef]

32.

G. Z. Mashanovich, M. M. Milošević, M. Nedeljkovic, N. Owens, B. Xiong, E. J. Teo, and Y. Hu, “Low loss silicon waveguides for the mid-infrared,” Opt. Express 19(8), 7112–7119 (2011). [CrossRef] [PubMed]

33.

http://www.nature.com/nphoton/journal/v4/n8/full/nphoton.2010.173.html.

OCIS Codes
(190.4380) Nonlinear optics : Nonlinear optics, four-wave mixing
(230.7370) Optical devices : Waveguides
(310.2790) Thin films : Guided waves

ToC Category:
Nonlinear Optics

History
Original Manuscript: February 16, 2012
Revised Manuscript: March 17, 2012
Manuscript Accepted: March 20, 2012
Published: April 2, 2012

Citation
Zhaolu Wang, Hongjun Liu, Nan Huang, Qibing Sun, and Jin Wen, "Efficient terahertz-wave generation via four-wave mixing in silicon membrane waveguides," Opt. Express 20, 8920-8928 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8920


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References

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  22. Q. Lin, J. Zhang, P. M. Fauchet, and G. P. Agrawal, “Ultrabroadband parametric generation and wavelength conversion in silicon waveguides,” Opt. Express14(11), 4786–4799 (2006). [CrossRef] [PubMed]
  23. G. Z. Mashanovich, M. Milosevic, P. Matavulj, S. Stankovic, B. Timotijevic, P. Y. Yang, E. J. Teo, M. B. H. Breese, A. A. Bettiol, and G. T. Reed, “Silicon photonic waveguides for different wavelength regions,” Semicond. Sci. Technol.23(6), 064002 (2008). [CrossRef]
  24. T. E. Murphy, software available at http://www.photonics.umd.edu .
  25. Q. Lin, T. J. Johnson, R. Perahia, C. P. Michael, and O. J. Painter, “A proposal for highly tunable optical parametric oscillation in silicon micro-resonators,” Opt. Express16(14), 10596–10610 (2008). [CrossRef] [PubMed]
  26. X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, “Mid-infrared optical parametric amplifier using silicon nanophtonic waveguides,” Nat. Photonics4(8), 557–560 (2010). [CrossRef]
  27. R. M. Osgood, N. C. Panoiu, J. I. Dadap, X. Liu, X. Chen, I. Hsieh, E. Dulkeith, W. M. J. Green, and Y. A. Vlasov, “Engineering nonlinearities in nanoscale optical systems: physics and applications in dispersion-engineered silicon nanophotonic wires,” Adv. Opt. Photon.1(1), 162–235 (2009). [CrossRef]
  28. Z. Wang, H. Liu, N. Huang, Q. Sun, and J. Wen, “Impact of dispersion profiles of silicon waveguides on optical parametric amplification in the femtosecond regime,” Opt. Express19(24), 24730–24737 (2011). [CrossRef] [PubMed]
  29. E. K. Tien, Y. Huang, S. Gao, Q. Song, F. Qian, S. K. Kalyoncu, and O. Boyraz, “Discrete parametric band conversion in silicon for mid-infrared applications,” Opt. Express18(21), 21981–21989 (2010). [CrossRef] [PubMed]
  30. A. D. Bristow, N. Rotenberg, and H. M. van Driel, “Two-photon absorption and Kerr coefficients of silicon for 850-2200 nm,” Appl. Phys. Lett.90(19), 191104 (2007). [CrossRef]
  31. N. K. Hon, R. Soref, and B. Jalali, “The third-order nonlinear optical coefficients of Si, Ge, and Si1-xGex in the midwave and longwave infrared,” J. Appl. Phys.110(1), 011301 (2011). [CrossRef]
  32. G. Z. Mashanovich, M. M. Milošević, M. Nedeljkovic, N. Owens, B. Xiong, E. J. Teo, and Y. Hu, “Low loss silicon waveguides for the mid-infrared,” Opt. Express19(8), 7112–7119 (2011). [CrossRef] [PubMed]
  33. http://www.nature.com/nphoton/journal/v4/n8/full/nphoton.2010.173.html .

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