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Coupling-length phase matching for efficient third-harmonic generation based on parallel-coupled waveguides Tianye Huang,1,2 Perry Ping Shum,1,2,* Xuguang Shao,1,2 Timothy Lee,3 Zhifang Wu,1,2 Huizi Li,1,2 Tingting Wu,1,2 Meng Zhang,1 Xuan Quyen Dinh,2,4 and Gilberto Brambilla3 1
Center for Optical Fiber Technology, School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 2 CNRS International-NTU-Thales Research Alliance (CINTRA), UMI 3288, 50 Nanyang Drive, Singapore 637553, Singapore 3
Optoelectronics Research Center, University of Southampton, Southampton, SO17 1BJ, UK 4
Thales Solutions Asia Pte Ltd, R&T, 28 Changi North Rise, Singapore 498755, Singapore *Corresponding author: [email protected] Received January 7, 2015; revised January 27, 2015; accepted January 28, 2015; posted January 28, 2015 (Doc. ID 231772); published March 3, 2015
We study third-harmonic generation (THG) in parallel-coupled waveguides where the spatial modulation of the mode intensity provides quasi-phase matching, called coupling-length phase matching (CLPM), for efficient nonlinear frequency conversion. Different types of CLPM are investigated for THG, and it is found that two sets of CLPM conditions can be practically implemented with traditional waveguides. These two CLPM conditions are further investigated by considering nonlinear phase modulations, which can degrade the CLPM-based THG conversion. However, up to 45% efficiency is still possible in this scheme. The greatest significance of this approach is that the requirement of perfect phase matching in a single waveguide is no longer necessary, leading to an alternative waveguide design for THG. © 2015 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.2620) Harmonic generation and mixing; (190.4975) Parametric processes; (230.1150) All-optical devices. http://dx.doi.org/10.1364/OL.40.000894
Recently, third-harmonic generation (THG) has become a topic of interest in various applications, such as the generation of new wavelengths [1–3], spectroscopy [4], and signal processing [5]. Compared to approaches that use nonlinear crystals for THG [2,3], waveguide-based schemes are attractive because of their capability for onchip integration without free-space alignment. Efficient THG in waveguides requires a large nonlinearity, large spatial overlap of the fundamental pump and harmonic mode fields, and the phase-matching condition (PMC). To satisfy the first key point, materials with inherently large χ 3 nonlinearity should be chosen. The necessary large spatial-mode overlap can be fulfilled by choosing two well-confined modes that have similar field distributions. The fulfillment of the PMC is a crucial requirement for efficient THG, as it ensures that the harmonic power can increase continuously. While several approaches have been investigated for THG [5–10], many suffer from either poor modal overlap or PMC issues. This is because the material dispersion prevents the pump, i.e., the fundamental frequency (FF), and the third harmonic frequency (THF) from experiencing the same material refractive index. This can be overcome by intermodal phase matching [5–8], which uses a waveguide designed to support the fundamental mode at the FF and a higher-order mode at the THF with nearly the same effective indices. For example, using a 14 mW pump, an average THG power of 1.3 pW was experimentally realized in a photonic crystal waveguide using a radiation THF mode [5]. A theoretical efficiency of 1.4% is predicted in plasmonic waveguides by employing a first-order THF mode [6]. By relying on the field enhancement effect in a microfiber-based ring resonator, a conversion efficiency of up to 50% is possible 0146-9592/15/060894-04$15.00/0
with the HE12 mode [7]. However, it remains complex and difficult to design and fabricate such waveguides, e.g., photonic crystal waveguides with detailed structures [5], plasmonic waveguides with nano-scale slot core regions [6], or ring resonators with uniform and accurate submicron fiber diameters [7]. Although efficient THG soliton-exploiting behavior has been predicted [10], this approach requires jointly engineering the phase mismatch and mode overlap, which also poses practical difficulties. Quasi-phase matching (QPM) by periodically modulating the nonlinear coefficient or modal refractive index is a promising solution to relax the strict requirement of a perfect PMC [11]. Practically, however, this again complicates fabrication by requiring additional steps to introduce the necessary waveguide modulations. On the other hand, the coupling-length phase matching (CLPM) method can be easily implemented with two parallelcoupled waveguides. It was previously investigated for second-order nonlinear processes [12,13]. However, the potential role of CLPM in THG has not been addressed. In this Letter, we theoretically study CLPM for THG waveguide design under the influence of different waveguide and coupling parameters. Furthermore, we consider the impact of third-order nonlinear phase modulations to ascertain their effect upon the expected THG conversion efficiency. We consider the structure of the CLPM in Fig. 1. Two identical parallel waveguides, a and b, are placed on the same substrate, close enough to permit the coupling of both the pump and harmonic modes during their propagation. The pump light is injected into waveguide a and subsequently generates the third harmonic in both waveguides. © 2015 Optical Society of America
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where A1;0 is the amplitude of the initial pump wave. Substituting Eqs. (3) and (4) into the coupled equations b for Aa 3 and A3 gives the following: iγ 3 A31;0 i3κ z ∂Aa 3 e 3 3ei3κ1 z 3e−i3κ1 z e−i3κ3 z e−iΔkz ∂z 8 iκ 3 Ab (5) 3 ; Fig. 1.
and
Schematic of coupled waveguides for THG.
For waveguide a, the coupled-mode differential equations describing the evolution of the copropagating pump and harmonic amplitudes Aa and Aa can be 1 3 expressed as follows [7]: ∂Aa 2 a 2 a 1 iγ 1 jAa 1 j 2γ 2 jA3 j A1 ∂z 2 iΔkz iκ Ab ; γ 3 Aa 1 1 1 A3 e
(1)
and ∂Aa 2 a 2 a 3 i6γ 2 jAa 1 j 3γ 5 jA3 j A3 ∂z 3 −iΔkz γ 3 Aa iκ3 Ab 1 e 3 ;
iγ 3 A31;0 i3κ z ∂Ab 3 e 3 − 3ei3κ1 z 3e−i3κ1 z − e−i3κ3 z e−iΔkz ∂z 8 iκ3 Aa (6) 3 : From these two equations, Aa 3 z can be solved by taking the derivative with respect to z of Eq. (5) and substituting Eq. (6) into it, to give the analytical expression in Eq. (7): Aa 3
(2)
where γ 1 , γ 2 , and γ 5 correspond to the nonlinear coefficient for the pump’s self-phase modulation (SPM), pumpharmonic cross phase modulation, and harmonic SPM, respectively. The expressions for γ i (i 1, 2, 3, 5) can be found in [14]. Δk k3 − 3k1 is the phase mismatch. κ1 and κ3 are the coefficients describing the coupling of the pump and harmonic between the two waveguides. The equations for waveguide b are obtained by changing the superscripts (a) to (b) in Eqs. (1) and (2). To first determine the physical conditions under which CLPM can occur, we begin with a simplified analysis in which we neglect the phase modulation terms and utilize the “undepleted pump approximation.” With these assumptions, Eq. (1) and its counterpart for waveguide b can be integrated, thereby directly resulting in the following analytic solutions for the pump: Aa 1 z A1;0 cosκ 1 z;
(3)
Ab 1 z iA1;0 sinκ 1 z;
(4)
and
γ 3 A31;0 42κ 1 − κ 3 − Δk iκ z 42κ 1 − κ 3 Δk −iκ z − e 3 e 3 8 N 1N 3 N 2N 4 1 3 − eiκ3 −N 1 z e−iκ3 −N 2 z N1 N2 3 iκ3 −N 3 z 1 −iκ3 −N 4 z ; (7) − e e N3 N4
N 1 −3κ 1 κ3 Δk;
N 2 κ1 κ 3 − Δk;
(8)
N 4 −3κ 1 κ 3 − Δk:
(9)
and N 3 κ1 κ 3 Δk;
As Eq. (7) is composed of six different oscillatory terms, the harmonic amplitude Aa 3 cannot normally be built up continuously. However, when at least one of the coefficients N i are close to zero, no less than two of the oscillatory terms in Eq. (7) can interfere with each other constructively. However, their sum remains finite (the limit of Aa 3 ∕z exists with z approaching infinity), leading to the increment of Aa 3 . These are the conditions under which a CLPM can be achieved. To investigate THG performance under these different CLPM conditions, we first find the equivalent constraints represented by the waveguide parameters Δk, κ1 , and κ 3 ,
Table 1. 7 Conditions under Which CLPM Is Theoretically Possible in the Proposed Approach, with Their Corresponding Waveguide Constraints, Limits of Aa 3 ∕z, and Total Achievable Efficiency (Defined As jA3a j2 jA3b j2 ∕P0 ) Case 1 2 3 4 5 6 7
CLPM Condition
Waveguide Constraints
N 1 0 or N 4 0 N 2 0 or N 3 0 N 1 N 2 0 or N 3 N 4 0 N 1 N 3 0 or N 2 N 4 0 N1 N4 0 N2 N3 0 Any three N i 0 (i 1, 2, 3, 4)
Δk 3κ1 –κ3 or Δk −3κ1 κ3 Δk κ1 κ3 or Δk −κ 1 –κ3 Δk 2κ1 , κ1 κ3 or Δk −2κ 1 , κ1 κ3 Δk −κ3 , κ1 0 or Δk κ3 , κ 1 0 Δk 0, 3κ1 κ3 Δk 0, κ 1 −κ3 Δk κ1 κ3 0
limz→∞ A3a ∕z, T γ 3 A31;0 ∕8 Total Efficiency Tie iκ3 z T3ie∓iκ3 z Tie iκ3 z 3ie∓iκ3 z T4ie iκ3 z T2i cosκ3 z T6i cosκ3 z T8i
∼40% ∼75% ∼72.5% ∼98% ∼93% ∼100% ∼100%
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and calculate the limit of Aa 3 ∕z with z approaching infinity. These are listed in Table 1 for each of the seven different possible conditions satisfying CLPM, which shall each be analyzed in the proceeding analysis. We define also the following parameters: normalized initial pump power P 0 1, conversion efficiency in one of the wave2 guides jAa 3 j ∕P 0 , and normalized nonlinearity γ 3 LP 0 , where L 1 is the normalized waveguide length. For the first case of N 1 0, assuming ΔkL 500, κ1 L 200, κ3 L 100 and normalized nonlinearity γ 3 LP 0 2, the solutions obtained from both the undepleted pump approximation and numerically from Eqs. (1) and (2) are demonstrated in Fig. 2(a). The analytical approximation and numerical solution show good agreement for THG in waveguide a within the normalized waveguide length of 0.8, with an efficiency close to 5%. However, since the pump depletion becomes significant when the normalized nonlinearities are higher, we must resort to the numerical solution. For example, increasing the normalized nonlinearity to γ 3 LP 0 12 provides a peak conversion efficiency of 22.2%, as presented by the red curve in Fig. 2(a). Note that both waveguides act as the nonlinear medium, and therefore waveguide b carries nearly the same power of third harmonic, as is verified in Fig. 2(b). Thus, the total achievable conver2 b 2 sion is jAa 3 j jA3 j ∕P 0 40%. Next, we consider the second case N 2 0, and assume ΔkL 500, κ 1 L 400, and κ 3 L 100. The analytically and numerically calculated conversion efficiency with different normalized nonlinearities, illustrated in Fig. 3(a), are initially similar but start to depart from each other once the conversion becomes significant (above 5%). In contrast with the previous CLMP case N 1 0, with a normalized nonlinearity of 12, the conversion efficiency not only grows more rapidly with distance, but attains a higher maximum, exceeding 35% over a much shorter normalized length in both waveguides a and b, as shown in Figs. 3(a) and 3(b). The total efficiency is about 75%.
Fig. 2. THG conversion in the CLPM condition of N 1 0. (a) Conversion efficiency in waveguide a with normalized nonlinearity of 2 and 12. (b) Conversion efficiency in waveguide b with normalized nonlinearity of 12.
Fig. 3. THG conversion in the CLPM condition of N 2 0. (a) Conversion efficiency in waveguide a with normalized nonlinearity of 2 and 12. (b) Conversion efficiency in waveguide b with normalized nonlinearity of 12.
Finally, with a normalized nonlinearity of 12, we calculate the conversion in waveguide a under other CLPM conditions. For N 1 N 2 0, N 1 N 3 0, N 1 N 4 0, and N 2 N 3 0 (rows 3, 4, 5, and 6 in Table 1), Figs. 4(a)–4(d) indicate total conversion efficiencies of 72.5%, 98%, 93%, and 100%, respectively. Also, it can be found that the higher the magnitude of the limit of Aa 3 ∕z, the faster the harmonic amplitude grows. For the condition of three or more N i 0 (row 7 in Table 1), the situation reduces to the perfect traditional phase-matching condition in a single waveguide, with no coupling between waveguides. Note that in conventional evanescently coupled waveguides, the coupling coefficient for longer wavelength modes is larger than that at shorter wavelengths (κ1 > κ 3 > 0). Therefore, the CLPM conditions listed in rows 3 and 4 of Table 1 would be difficult to implement. Since perfect phase matching Δk 0 is a necessary requirement for the conditions in rows 5, 6, and 7, these are of no interest in our discussion. The most meaningful CLPM conditions are thus the first two, namely Δk 3κ1 –κ 3 and Δk κ 1 κ3 . These conditions represent a flexible waveguide design process, since Δk ≈ 0 does not need to be fulfilled in either
Fig. 4. THG conversion in waveguide a for the CLPM condition of (a) N 1 N 2 0 (κ1 L 100, κ 3 L 100, ΔkL 200), (κ 1 L 0, κ3 L 100 ΔkL 100), (b) N1 N3 0 (c) N 1 N 4 0 (κ 1 L 50, κ3 L 150, ΔkL 0), and (d) N 2 N 3 0 (κ1 L 100, κ3 L 100, ΔkL 0).
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Fig. 5. THG conversion in waveguide a with different normalized nonlinearity in the CLPM condition of (a) N 1 0, and (b) N 2 0. (c) Total conversion in the CLPM condition of N 2 0.
waveguide and the de-tuning can instead be compensated by the field coupling between two identical waveguides, as κ1 and κ3 are adjustable by tuning the waveguide separation. Furthermore, efficient THG can take place between two fundamental modes at the FF and THF even when Δk ≠ 0. In the above, nonlinear phase modulation terms were neglected; however, in real THG devices, these terms can have impact on THG performance. In the following, we investigate the CLPM conditions N 1 0 and N 2 0, while considering these nonlinear phase shifts. We assume that both the FF and THF waves are propagating as fundamental modes. As the THF mode is more tightly confined than the FF in a conventional waveguide, γ 5 (which represents the harmonic SPM) would be larger than the other γ i and so we treat γ 5 2γ 1 2γ 2 2γ 3 . By numerically solving the coupling mode equations, we obtain the result for N 1 0 with different γ 3 LP 0 values, which is shown in Fig. 5(a). The maximum achievable conversion falls to 6% in a single waveguide, which means the phase modulation degrades the performance significantly in this particular CLPM condition. The total efficiency from both waveguides is 10%, only a quarter of that without phase modulations. However, when N 2 0, the conversion process demonstrates better tolerance to the nonlinear phase modulation, as the maximum conversion remains ∼25% per waveguide, as shown in Fig. 5(b). This is regardless of whether the normalized nonlinearity is 2, 8, or 12. Additionally, for a given normalized nonlinearity, the length required to reach maximum conversion is shorter
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than the case of N 1 0. Figure 5(c) depicts the total efficiency against the waveguide length. Although the normalized length required to reach maximum conversion decreases with the increasing nonlinearity, the achievable peak efficiency of ∼45% is nearly unchanged, indicating a promising approach for efficient THG. For practical implementation, the waveguide parameters (e.g., width, height, and index distribution) can be engineered to achieve an optimized phase mismatch that can be compensated by tuning the separation. For example, the flexible Si/Ge platform offers great opportunities to design CLPM waveguides for THG operation in the mid-IR band [15]. In summary, we have discussed how to realize efficient THG in coupled waveguides based on CLPM. Among the different possible types of CLPM conditions, there are two, namely Δk 3κ1 –κ 3 , and Δk κ 1 κ3 , that are relatively easy to implement, using conventional evanescently coupled waveguides. Studying these two conditions further reveals that nonlinear phase modulation can degrade the conversion in both CLPM schemes. However, when Δk κ1 κ 3 , a conversion efficiency of up to 45% is still possible in the proposed structure. The key advantage of this approach is that there is no need to satisfy a perfect PMC (Δk ≈ 0) in either waveguide, and Δk can be compensated by the spatial coupling between the waveguides, hence offering an alternative method in designing devices for THG. The authors would like to thank Professor Ivan Biaggio for his insightful discussion and guidance. This work was partially supported by the Ministry of Education of Singapore Tier 2 and the Singapore A*STAR SERC Grant “Advanced Optics in Engineering” Program. References 1. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photonics 3, 206 (2009). 2. J. Mes, E. J. Duijn, R. Zinkstok, S. Witte, and W. Hogervorst, Appl. Phys. Lett. 82, 4423 (2003). 3. H. Zhong, P. Yuan, S. Wen, and L. Qian, Opt. Express 22, 4267 (2014). 4. I. Appelbaum, Appl. Phys. Lett. 103, 122604 (2013). 5. B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. OFaolain, T. Krauss, B. J. Eggleton, and D. J. Moss, Opt. Express 18, 7770 (2010). 6. T. Huang, X. Shao, Z. Wu, T. Lee, T. Wu, J. Zhang, H. Q. Lam, and G. Brambilla, IEEE Photon. J. 6, 4800607 (2014). 7. T. Lee, N. G. R. Broderick, and G. Brambilla, J. Opt. Soc. Am. B 30, 505 (2013). 8. K. Bencheikh, S. Richard, G. Melin, G. Krabshuis, F. Gooijer, and J. A. Levenson, Opt. Lett. 37, 289 (2012). 9. T. Cheng, W. Gao, M. Liao, Z. Duan, D. Deng, M. Matsumoto, T. Misumi, T. Suzuki, and Y. Ohishi, Opt. Lett. 39, 1005 (2014). 10. V. Shahraam Afshar, M. A. Lohe, T. Lee, T. M. Monro, and N. G. R. Broderick, Opt. Lett. 38, 329 (2013). 11. S. Zhu, Y. Zhu, and N. Ming, Science 278, 843 (1997). 12. P. Dong and A. G. Kirk, Phys. Rev. Lett. 93, 100901 (2004). 13. I. Biaggio, V. Coda, and G. Montemezzani, Phys. Rev. A 90, 043816 (2014). 14. V. Grubsky and A. Savchenko, Opt. Express 13, 6798 (2005). 15. M. Brun, P. Labeye, G. Grand, J. M. Hartmann, F. Boulila, M. Carras, and S. Nicoletti, Opt. Express 22, 508 (2014).
OPTICS LETTERS / Vol. 40, No. 6 / March 15, 2015
Coupling-length phase matching for efficient third-harmonic generation based on parallel-coupled waveguides Tianye Huang,1,2 Perry Ping Shum,1,2,* Xuguang Shao,1,2 Timothy Lee,3 Zhifang Wu,1,2 Huizi Li,1,2 Tingting Wu,1,2 Meng Zhang,1 Xuan Quyen Dinh,2,4 and Gilberto Brambilla3 1
Center for Optical Fiber Technology, School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 2 CNRS International-NTU-Thales Research Alliance (CINTRA), UMI 3288, 50 Nanyang Drive, Singapore 637553, Singapore 3
Optoelectronics Research Center, University of Southampton, Southampton, SO17 1BJ, UK 4
Thales Solutions Asia Pte Ltd, R&T, 28 Changi North Rise, Singapore 498755, Singapore *Corresponding author: [email protected] Received January 7, 2015; revised January 27, 2015; accepted January 28, 2015; posted January 28, 2015 (Doc. ID 231772); published March 3, 2015
We study third-harmonic generation (THG) in parallel-coupled waveguides where the spatial modulation of the mode intensity provides quasi-phase matching, called coupling-length phase matching (CLPM), for efficient nonlinear frequency conversion. Different types of CLPM are investigated for THG, and it is found that two sets of CLPM conditions can be practically implemented with traditional waveguides. These two CLPM conditions are further investigated by considering nonlinear phase modulations, which can degrade the CLPM-based THG conversion. However, up to 45% efficiency is still possible in this scheme. The greatest significance of this approach is that the requirement of perfect phase matching in a single waveguide is no longer necessary, leading to an alternative waveguide design for THG. © 2015 Optical Society of America OCIS codes: (190.0190) Nonlinear optics; (190.2620) Harmonic generation and mixing; (190.4975) Parametric processes; (230.1150) All-optical devices. http://dx.doi.org/10.1364/OL.40.000894
Recently, third-harmonic generation (THG) has become a topic of interest in various applications, such as the generation of new wavelengths [1–3], spectroscopy [4], and signal processing [5]. Compared to approaches that use nonlinear crystals for THG [2,3], waveguide-based schemes are attractive because of their capability for onchip integration without free-space alignment. Efficient THG in waveguides requires a large nonlinearity, large spatial overlap of the fundamental pump and harmonic mode fields, and the phase-matching condition (PMC). To satisfy the first key point, materials with inherently large χ 3 nonlinearity should be chosen. The necessary large spatial-mode overlap can be fulfilled by choosing two well-confined modes that have similar field distributions. The fulfillment of the PMC is a crucial requirement for efficient THG, as it ensures that the harmonic power can increase continuously. While several approaches have been investigated for THG [5–10], many suffer from either poor modal overlap or PMC issues. This is because the material dispersion prevents the pump, i.e., the fundamental frequency (FF), and the third harmonic frequency (THF) from experiencing the same material refractive index. This can be overcome by intermodal phase matching [5–8], which uses a waveguide designed to support the fundamental mode at the FF and a higher-order mode at the THF with nearly the same effective indices. For example, using a 14 mW pump, an average THG power of 1.3 pW was experimentally realized in a photonic crystal waveguide using a radiation THF mode [5]. A theoretical efficiency of 1.4% is predicted in plasmonic waveguides by employing a first-order THF mode [6]. By relying on the field enhancement effect in a microfiber-based ring resonator, a conversion efficiency of up to 50% is possible 0146-9592/15/060894-04$15.00/0
with the HE12 mode [7]. However, it remains complex and difficult to design and fabricate such waveguides, e.g., photonic crystal waveguides with detailed structures [5], plasmonic waveguides with nano-scale slot core regions [6], or ring resonators with uniform and accurate submicron fiber diameters [7]. Although efficient THG soliton-exploiting behavior has been predicted [10], this approach requires jointly engineering the phase mismatch and mode overlap, which also poses practical difficulties. Quasi-phase matching (QPM) by periodically modulating the nonlinear coefficient or modal refractive index is a promising solution to relax the strict requirement of a perfect PMC [11]. Practically, however, this again complicates fabrication by requiring additional steps to introduce the necessary waveguide modulations. On the other hand, the coupling-length phase matching (CLPM) method can be easily implemented with two parallelcoupled waveguides. It was previously investigated for second-order nonlinear processes [12,13]. However, the potential role of CLPM in THG has not been addressed. In this Letter, we theoretically study CLPM for THG waveguide design under the influence of different waveguide and coupling parameters. Furthermore, we consider the impact of third-order nonlinear phase modulations to ascertain their effect upon the expected THG conversion efficiency. We consider the structure of the CLPM in Fig. 1. Two identical parallel waveguides, a and b, are placed on the same substrate, close enough to permit the coupling of both the pump and harmonic modes during their propagation. The pump light is injected into waveguide a and subsequently generates the third harmonic in both waveguides. © 2015 Optical Society of America
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where A1;0 is the amplitude of the initial pump wave. Substituting Eqs. (3) and (4) into the coupled equations b for Aa 3 and A3 gives the following: iγ 3 A31;0 i3κ z ∂Aa 3 e 3 3ei3κ1 z 3e−i3κ1 z e−i3κ3 z e−iΔkz ∂z 8 iκ 3 Ab (5) 3 ; Fig. 1.
and
Schematic of coupled waveguides for THG.
For waveguide a, the coupled-mode differential equations describing the evolution of the copropagating pump and harmonic amplitudes Aa and Aa can be 1 3 expressed as follows [7]: ∂Aa 2 a 2 a 1 iγ 1 jAa 1 j 2γ 2 jA3 j A1 ∂z 2 iΔkz iκ Ab ; γ 3 Aa 1 1 1 A3 e
(1)
and ∂Aa 2 a 2 a 3 i6γ 2 jAa 1 j 3γ 5 jA3 j A3 ∂z 3 −iΔkz γ 3 Aa iκ3 Ab 1 e 3 ;
iγ 3 A31;0 i3κ z ∂Ab 3 e 3 − 3ei3κ1 z 3e−i3κ1 z − e−i3κ3 z e−iΔkz ∂z 8 iκ3 Aa (6) 3 : From these two equations, Aa 3 z can be solved by taking the derivative with respect to z of Eq. (5) and substituting Eq. (6) into it, to give the analytical expression in Eq. (7): Aa 3
(2)
where γ 1 , γ 2 , and γ 5 correspond to the nonlinear coefficient for the pump’s self-phase modulation (SPM), pumpharmonic cross phase modulation, and harmonic SPM, respectively. The expressions for γ i (i 1, 2, 3, 5) can be found in [14]. Δk k3 − 3k1 is the phase mismatch. κ1 and κ3 are the coefficients describing the coupling of the pump and harmonic between the two waveguides. The equations for waveguide b are obtained by changing the superscripts (a) to (b) in Eqs. (1) and (2). To first determine the physical conditions under which CLPM can occur, we begin with a simplified analysis in which we neglect the phase modulation terms and utilize the “undepleted pump approximation.” With these assumptions, Eq. (1) and its counterpart for waveguide b can be integrated, thereby directly resulting in the following analytic solutions for the pump: Aa 1 z A1;0 cosκ 1 z;
(3)
Ab 1 z iA1;0 sinκ 1 z;
(4)
and
γ 3 A31;0 42κ 1 − κ 3 − Δk iκ z 42κ 1 − κ 3 Δk −iκ z − e 3 e 3 8 N 1N 3 N 2N 4 1 3 − eiκ3 −N 1 z e−iκ3 −N 2 z N1 N2 3 iκ3 −N 3 z 1 −iκ3 −N 4 z ; (7) − e e N3 N4
N 1 −3κ 1 κ3 Δk;
N 2 κ1 κ 3 − Δk;
(8)
N 4 −3κ 1 κ 3 − Δk:
(9)
and N 3 κ1 κ 3 Δk;
As Eq. (7) is composed of six different oscillatory terms, the harmonic amplitude Aa 3 cannot normally be built up continuously. However, when at least one of the coefficients N i are close to zero, no less than two of the oscillatory terms in Eq. (7) can interfere with each other constructively. However, their sum remains finite (the limit of Aa 3 ∕z exists with z approaching infinity), leading to the increment of Aa 3 . These are the conditions under which a CLPM can be achieved. To investigate THG performance under these different CLPM conditions, we first find the equivalent constraints represented by the waveguide parameters Δk, κ1 , and κ 3 ,
Table 1. 7 Conditions under Which CLPM Is Theoretically Possible in the Proposed Approach, with Their Corresponding Waveguide Constraints, Limits of Aa 3 ∕z, and Total Achievable Efficiency (Defined As jA3a j2 jA3b j2 ∕P0 ) Case 1 2 3 4 5 6 7
CLPM Condition
Waveguide Constraints
N 1 0 or N 4 0 N 2 0 or N 3 0 N 1 N 2 0 or N 3 N 4 0 N 1 N 3 0 or N 2 N 4 0 N1 N4 0 N2 N3 0 Any three N i 0 (i 1, 2, 3, 4)
Δk 3κ1 –κ3 or Δk −3κ1 κ3 Δk κ1 κ3 or Δk −κ 1 –κ3 Δk 2κ1 , κ1 κ3 or Δk −2κ 1 , κ1 κ3 Δk −κ3 , κ1 0 or Δk κ3 , κ 1 0 Δk 0, 3κ1 κ3 Δk 0, κ 1 −κ3 Δk κ1 κ3 0
limz→∞ A3a ∕z, T γ 3 A31;0 ∕8 Total Efficiency Tie iκ3 z T3ie∓iκ3 z Tie iκ3 z 3ie∓iκ3 z T4ie iκ3 z T2i cosκ3 z T6i cosκ3 z T8i
∼40% ∼75% ∼72.5% ∼98% ∼93% ∼100% ∼100%
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and calculate the limit of Aa 3 ∕z with z approaching infinity. These are listed in Table 1 for each of the seven different possible conditions satisfying CLPM, which shall each be analyzed in the proceeding analysis. We define also the following parameters: normalized initial pump power P 0 1, conversion efficiency in one of the wave2 guides jAa 3 j ∕P 0 , and normalized nonlinearity γ 3 LP 0 , where L 1 is the normalized waveguide length. For the first case of N 1 0, assuming ΔkL 500, κ1 L 200, κ3 L 100 and normalized nonlinearity γ 3 LP 0 2, the solutions obtained from both the undepleted pump approximation and numerically from Eqs. (1) and (2) are demonstrated in Fig. 2(a). The analytical approximation and numerical solution show good agreement for THG in waveguide a within the normalized waveguide length of 0.8, with an efficiency close to 5%. However, since the pump depletion becomes significant when the normalized nonlinearities are higher, we must resort to the numerical solution. For example, increasing the normalized nonlinearity to γ 3 LP 0 12 provides a peak conversion efficiency of 22.2%, as presented by the red curve in Fig. 2(a). Note that both waveguides act as the nonlinear medium, and therefore waveguide b carries nearly the same power of third harmonic, as is verified in Fig. 2(b). Thus, the total achievable conver2 b 2 sion is jAa 3 j jA3 j ∕P 0 40%. Next, we consider the second case N 2 0, and assume ΔkL 500, κ 1 L 400, and κ 3 L 100. The analytically and numerically calculated conversion efficiency with different normalized nonlinearities, illustrated in Fig. 3(a), are initially similar but start to depart from each other once the conversion becomes significant (above 5%). In contrast with the previous CLMP case N 1 0, with a normalized nonlinearity of 12, the conversion efficiency not only grows more rapidly with distance, but attains a higher maximum, exceeding 35% over a much shorter normalized length in both waveguides a and b, as shown in Figs. 3(a) and 3(b). The total efficiency is about 75%.
Fig. 2. THG conversion in the CLPM condition of N 1 0. (a) Conversion efficiency in waveguide a with normalized nonlinearity of 2 and 12. (b) Conversion efficiency in waveguide b with normalized nonlinearity of 12.
Fig. 3. THG conversion in the CLPM condition of N 2 0. (a) Conversion efficiency in waveguide a with normalized nonlinearity of 2 and 12. (b) Conversion efficiency in waveguide b with normalized nonlinearity of 12.
Finally, with a normalized nonlinearity of 12, we calculate the conversion in waveguide a under other CLPM conditions. For N 1 N 2 0, N 1 N 3 0, N 1 N 4 0, and N 2 N 3 0 (rows 3, 4, 5, and 6 in Table 1), Figs. 4(a)–4(d) indicate total conversion efficiencies of 72.5%, 98%, 93%, and 100%, respectively. Also, it can be found that the higher the magnitude of the limit of Aa 3 ∕z, the faster the harmonic amplitude grows. For the condition of three or more N i 0 (row 7 in Table 1), the situation reduces to the perfect traditional phase-matching condition in a single waveguide, with no coupling between waveguides. Note that in conventional evanescently coupled waveguides, the coupling coefficient for longer wavelength modes is larger than that at shorter wavelengths (κ1 > κ 3 > 0). Therefore, the CLPM conditions listed in rows 3 and 4 of Table 1 would be difficult to implement. Since perfect phase matching Δk 0 is a necessary requirement for the conditions in rows 5, 6, and 7, these are of no interest in our discussion. The most meaningful CLPM conditions are thus the first two, namely Δk 3κ1 –κ 3 and Δk κ 1 κ3 . These conditions represent a flexible waveguide design process, since Δk ≈ 0 does not need to be fulfilled in either
Fig. 4. THG conversion in waveguide a for the CLPM condition of (a) N 1 N 2 0 (κ1 L 100, κ 3 L 100, ΔkL 200), (κ 1 L 0, κ3 L 100 ΔkL 100), (b) N1 N3 0 (c) N 1 N 4 0 (κ 1 L 50, κ3 L 150, ΔkL 0), and (d) N 2 N 3 0 (κ1 L 100, κ3 L 100, ΔkL 0).
March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
Fig. 5. THG conversion in waveguide a with different normalized nonlinearity in the CLPM condition of (a) N 1 0, and (b) N 2 0. (c) Total conversion in the CLPM condition of N 2 0.
waveguide and the de-tuning can instead be compensated by the field coupling between two identical waveguides, as κ1 and κ3 are adjustable by tuning the waveguide separation. Furthermore, efficient THG can take place between two fundamental modes at the FF and THF even when Δk ≠ 0. In the above, nonlinear phase modulation terms were neglected; however, in real THG devices, these terms can have impact on THG performance. In the following, we investigate the CLPM conditions N 1 0 and N 2 0, while considering these nonlinear phase shifts. We assume that both the FF and THF waves are propagating as fundamental modes. As the THF mode is more tightly confined than the FF in a conventional waveguide, γ 5 (which represents the harmonic SPM) would be larger than the other γ i and so we treat γ 5 2γ 1 2γ 2 2γ 3 . By numerically solving the coupling mode equations, we obtain the result for N 1 0 with different γ 3 LP 0 values, which is shown in Fig. 5(a). The maximum achievable conversion falls to 6% in a single waveguide, which means the phase modulation degrades the performance significantly in this particular CLPM condition. The total efficiency from both waveguides is 10%, only a quarter of that without phase modulations. However, when N 2 0, the conversion process demonstrates better tolerance to the nonlinear phase modulation, as the maximum conversion remains ∼25% per waveguide, as shown in Fig. 5(b). This is regardless of whether the normalized nonlinearity is 2, 8, or 12. Additionally, for a given normalized nonlinearity, the length required to reach maximum conversion is shorter
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than the case of N 1 0. Figure 5(c) depicts the total efficiency against the waveguide length. Although the normalized length required to reach maximum conversion decreases with the increasing nonlinearity, the achievable peak efficiency of ∼45% is nearly unchanged, indicating a promising approach for efficient THG. For practical implementation, the waveguide parameters (e.g., width, height, and index distribution) can be engineered to achieve an optimized phase mismatch that can be compensated by tuning the separation. For example, the flexible Si/Ge platform offers great opportunities to design CLPM waveguides for THG operation in the mid-IR band [15]. In summary, we have discussed how to realize efficient THG in coupled waveguides based on CLPM. Among the different possible types of CLPM conditions, there are two, namely Δk 3κ1 –κ 3 , and Δk κ 1 κ3 , that are relatively easy to implement, using conventional evanescently coupled waveguides. Studying these two conditions further reveals that nonlinear phase modulation can degrade the conversion in both CLPM schemes. However, when Δk κ1 κ 3 , a conversion efficiency of up to 45% is still possible in the proposed structure. The key advantage of this approach is that there is no need to satisfy a perfect PMC (Δk ≈ 0) in either waveguide, and Δk can be compensated by the spatial coupling between the waveguides, hence offering an alternative method in designing devices for THG. The authors would like to thank Professor Ivan Biaggio for his insightful discussion and guidance. This work was partially supported by the Ministry of Education of Singapore Tier 2 and the Singapore A*STAR SERC Grant “Advanced Optics in Engineering” Program. References 1. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photonics 3, 206 (2009). 2. J. Mes, E. J. Duijn, R. Zinkstok, S. Witte, and W. Hogervorst, Appl. Phys. Lett. 82, 4423 (2003). 3. H. Zhong, P. Yuan, S. Wen, and L. Qian, Opt. Express 22, 4267 (2014). 4. I. Appelbaum, Appl. Phys. Lett. 103, 122604 (2013). 5. B. Corcoran, C. Monat, M. Pelusi, C. Grillet, T. P. White, L. OFaolain, T. Krauss, B. J. Eggleton, and D. J. Moss, Opt. Express 18, 7770 (2010). 6. T. Huang, X. Shao, Z. Wu, T. Lee, T. Wu, J. Zhang, H. Q. Lam, and G. Brambilla, IEEE Photon. J. 6, 4800607 (2014). 7. T. Lee, N. G. R. Broderick, and G. Brambilla, J. Opt. Soc. Am. B 30, 505 (2013). 8. K. Bencheikh, S. Richard, G. Melin, G. Krabshuis, F. Gooijer, and J. A. Levenson, Opt. Lett. 37, 289 (2012). 9. T. Cheng, W. Gao, M. Liao, Z. Duan, D. Deng, M. Matsumoto, T. Misumi, T. Suzuki, and Y. Ohishi, Opt. Lett. 39, 1005 (2014). 10. V. Shahraam Afshar, M. A. Lohe, T. Lee, T. M. Monro, and N. G. R. Broderick, Opt. Lett. 38, 329 (2013). 11. S. Zhu, Y. Zhu, and N. Ming, Science 278, 843 (1997). 12. P. Dong and A. G. Kirk, Phys. Rev. Lett. 93, 100901 (2004). 13. I. Biaggio, V. Coda, and G. Montemezzani, Phys. Rev. A 90, 043816 (2014). 14. V. Grubsky and A. Savchenko, Opt. Express 13, 6798 (2005). 15. M. Brun, P. Labeye, G. Grand, J. M. Hartmann, F. Boulila, M. Carras, and S. Nicoletti, Opt. Express 22, 508 (2014).
