March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
1041
Designing photonic crystal waveguides for broadband four-wave mixing applications Panagiotis Kanakis,1,* Thomas Kamalakis,2 and Thomas Sphicopoulos1 1
2
Department of Informatics and Telecommunications, National and Kapodistrian University/Athens, Panepistimioupolis, Athens GR157 84, Greece
Department of Informatics and Telematics, Harokopio University/Athens, 9 Omirou Street, Athens GR17778, Greece *Corresponding author: [email protected] Received December 10, 2014; revised January 19, 2015; accepted January 27, 2015; posted February 4, 2015 (Doc. ID 229408); published March 11, 2015 We present photonic crystal waveguide designs which exhibit large four-wave mixing efficiencies over a wide wavelength region. These designs are identified using an optimization process taking into account sophisticated figure-ofmerits that depend on the pump bandwidth and the signal/pump tunability. The obtained designs achieve up to −18.9 dB conversion efficiency, tunable over a 10 nm tunability range. We also present alternative designs that are less efficient but have smaller power requirements and are far more compact. © 2015 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (190.4380) Nonlinear optics, four-wave mixing. http://dx.doi.org/10.1364/OL.40.001041
All-optical signal processing based on four-wave mixing (FWM) is currently attracting increased attention because of its potential applications [1,2]. In its degenerate form, FWM involves the interaction between a signal and an idler wave, located at different wavelengths, mediated through a strong pump wave located at a third wavelength [3]. Nano-photonic structures such as photonic crystals [4] may enable the realization of such functionalities in compact form. Perhaps the most commonly adopted figure-of-merit (FoM) for FWM is the conversion efficiency η defined as, η ≡ P i L∕P s 0;
(1)
where P i L is the power of the idler wave at the output of a waveguide (of length L) and P s 0 is the signal power at the input. For a given waveguide design, it can be shown that the efficiency is determined by the idler and signal wavelengths λi and λs , the length L, and the initial pump power P 0 , η ηλs ; λi ; L; P 0 . However, achieving a large peak value for η, obtained only near certain wavelength combinations, is not sufficient. One should rather seek a compromise between efficiency, bandwidth, power requirements, and device footprint. In our previous work [5], we developed new FoMs that include these important aspects. In the present work, we discuss how the FoMs in question can be used for optimizing a photonic crystal slab waveguide (PCSW). To apply these FoMs, we first need to define the available pump bandwidth Δλ and the signal/pump tunability δλ. Assuming that for a given P 0 and L the maximum efficiency is η0 P 0 ; L maxλi ;λs fηg, we define ΔλP 0 ; L as the pump wavelength range for which the conversion efficiency does not fall below mη0 where m is a specified tolerance level with 0 < m < 1. In addition, the parameter δλP 0 ; L is defined as the average wavelength separation jλp − λs j between the pump and the signal waves for which again η ≥ mη0 . To estimate the parameters Δλ and δλ for a given PCSW design, we obtain the values of η with respect to many possible wavelength combinations, which form a 2D grid λs ; λi assuming, for example, a 2 nm wavelength spacing. In our efficiency calculations 0146-9592/15/061041-04$15.00/0
we always exclude the diagonal λs λi elements since λs cannot be equal to λi in a practical experiment. Once ηλs ; λi is obtained, we readily obtain δλ as the average wavelength separation jλp − λs j for which η is higher than mη0 and Δλ as the wavelength range of the pump wavelengths in which η abides to the same condition. We define the efficiency-bandwidth-tunability (EBT) product FoM as EBT η0 × Δλ × δλ:
(2)
As EBT is increased, the design is expected to exhibit high-efficiency values over a wide pump wavelength range with a smooth efficiency variation with respect to λp and λs . To further underline the need for small device footprint and power requirements, we may also define an alternative FoM, EBTPL
η0 × Δλ × δλ : P0 × L
(3)
Optimizing the waveguide with respect to EBTPL is expected to yield shorter structures requiring less power at the expense of a smaller overall efficiency. From an application point of view, obtaining a smooth η is useful, since it guarantees that the achieved FWM performance will be maintained over a wide wavelength selection for the three waves. A photonic crystal slab is formed by the periodic modulation of the refractive index of a slab (e.g., by embedding holes of air with the dielectric constant εb 1 in a silicon layer with the dielectric constant εa ≅ 12). Forming a line defect (e.g., by removing an entire row of holes along the x-direction) introduces a guided defect mode inside the photonic band-gap. Figure 1(a) shows examples of the various geometric characteristics of the holes that can be altered to obtain a suitable waveguide design. The holes and their corresponding characteristics are labeled according to their proximity to the line defect. For example, Δy2 describes the shift of the centers of the holes which have the second smallest distance from the line defect (i.e., the second closest neighbors, labeled “2” in the © 2015 Optical Society of America
1042
OPTICS LETTERS / Vol. 40, No. 6 / March 15, 2015
η li
Fig. 1. (a) Parameters used in the optimization and (b) waveguide supercell considered in the photonic band calculations.
figure) along the y axis. A positive Δy2 indicates that the holes move away from the defect. In the same way Δr 1 indicates the radius increase of the closest neighbors. We assume that the changes are made so that the waveguide supercell retains its mirror symmetry with respect to the x axis. Varying the waveguide characteristics, it is possible to obtain different dispersion mode relations k kω which can be tailored to correspond to a flat group index variation ng ng ω. A sufficiently flat ng ng ω can, in principle, favor nonlinearities over a large bandwidth [6]. A large ng also leads to increased losses which may eventually limit the waveguide performance. A compromise must therefore be sought. We define the flat band region as the wavelength range in which the group index (ng ) varies no more that 10% from the ng value ng0 obtained when the group velocity dispersion (GVD) is minimum [7]. We expect that this region favors nonlinear interactions; however, in our calculations, the wavelengths of the three waves were not only considered in the region above, but also in all adjacent regions for which ng varies less than 10% of ng0 and the losses will be similar. We also vary the wavelengths inside the portions of the dispersion relation between these regions as well. Using exhaustive search, we have previously identified optimal PCSW designs [5] with respect to the FoMs in Eqs. (2) and (3). Although exhaustive search methods are frequently adopted in the literature [8] with alternative FoMs, they usually limit the number of design parameters that can be considered concurrently and, as their number increases, one must resort to alternative optimization methods [9]. In this work, we maximize the above FoMs based on an interior-point optimization method that combines a direct method for solving the constrained maximization problem, along with conjugate gradient steps using trust regions performed by MATLAB’s fmincon function [10]. In this context, EBT can be written as EBT f a; k; r a ; h; εa ; εb ; Δx1 ; Δy1 ; Δr 1 ; …; ΔxN ; ΔyN ; Δr N ; P 0 ; L, where f is a function, a is the lattice constant, k is the wavenumber, h is the slab height, N is the number of hole groups assumed in the optimization process and r a are the radii of all holes that are not considered in the optimization. The function f is not known in closed, form but can be computed using Eq. (2) and a model for the efficiency η. The treatment of EBTPL is quite similar. The estimation of η can be carried out by numerically solving the coupled ordinary differential equations (ODEs) for the three propagating waves [11]. In our case, it turns out that this is impractical, since η must be numerically estimated for many combinations λs ; λi for each waveguide candidate design. Alternatively, one can approximate η as
λs κ2 1 tot2 sinh2 gL: λi 4g
(4)
Equation (4) is analytical and includes the influence of free-carriers (FCs) generation. It is generally in very good agreement with the numerical ODE solution [1,5] and requires much less computational time. The parameter li is the loss experienced by the idler wave, li P i L∕P i 0 exp−ai L 2L RefF i gP¯ 2p ;
(5)
where ai is the linear loss coefficient for the idler wave. In Eq. (4), κ tot and g are the total phase mismatch and the parametric gain given by [3] κ tot κ ImfF s F i − 2F p gP¯ 2p ;
(6)
1∕2 2 : g n22 S 2p S i S s ωi ωs P¯ 2p c−2 A−2 psi − κ ∕4
(7)
In Eqs. (5) and (6), F μ is the FC coefficient [11]: Fμ ≅
N C 2π C 2 λμ 2 C − ; j λ0 λμ 1 2 P 2p
(8)
where the index μ p, s, i corresponds to the pump, signal, and idler waves, respectively. The FC density N C is given by N C βTPA S 3p τC P 2p ∕2ℏωp A2ppp [12], S μ is the slow-down factor for μth wave [12], λ0 1550 nm, ωμ 2πc∕λμ , τC is the FC lifetime and for silicon waveguides C 1 1.35 × 10−27 m3 , and C 2 1.45 × 10−21 m3 [13]. The FC lifetime τC is assumed to be 600 ps [3], while the effective modal areas Aρκψ are given by [11] R Aρκψ
V
R R jE ρ j2 dV V jE κ j2 dV V jE ψ j2 dV 1∕2 R : α V E ρ E ρ E κ E ψ dV
(9)
In the equation above, E μ is the electric field component of the μth wave along the y direction, and V is the volume of the waveguide supercell used in the calculations. In our calculations, the modal fields are obtained by a 3D plane wave expansion (PWE) [14] mode solver which takes into account the geometric parameters of the waveguide. The number of plane waves used for the x, y, and z directions are 14, 84, and 28, respectively. The supercell assumed in the PWE calculations is shown in Fig. 1(b). In Eq. (6), κ is the phase mismatch in the presence of linear loss only, given by [3] −1 −1 −1 −1 −1 κ Δk 4πn2 P¯ p A−1 pss λs Apii λi − Appp λp ;
(10)
where n2 is the nonlinear Kerr coefficient and Δk ks ki − 2kp is the linear phase mismatch (where kμ kωμ are the propagation constants of the three waves). The power of the pump wave, its average value, and its average square value are given by [3] P p z P 0 e−ap z 1 δ1 − e−2ap z −1∕2 ;
(11)
March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
1043
Table 1. PCSW Design Optimization with Respect to EBT Parameter (m 1∕2) Design Parameters Δy1 ∕a
Δy2 ∕a
Δy3 ∕a
Δr 1 · a
Δr 2 · a
Δr 3 · a
EBT (nm2 )
η0 (dB)
Δλ (nm)
δλ (nm)
P 0 (W)
L(μm)
ng0
0.08303 0.144 0.15071 0.1484 0.14957 0.14968
0 0.0981 0.10834 0.10743 0.09808 0.09919
0 0 0.1090 0.0193 0.0131 0.0124
0 0 0 −0.0112 −0.0406 −0.0407
0 0 0 0 −0.0396 −0.0397
0 0 0 0 0 −0.0034
1.6 2.5 3.6 5.6 6.4 7.9
−22.3 −19.5 −19.8 −18.7 −19.9 −18.9
39.8 36 43 48 58 58
6.9 6.3 7.9 8.7 10.8 10.7
0.81 0.93 1.96 1.28 1.51 1.66
509.6 500 477.8 508.7 506.2 507.4
6.1 13.5 10.3 11.7 9.2 9.3
P¯ p
−a L P0 1 e p p sin−1 p − sin−1 p ; ap L δ δ−1 1 δ−1 1 (12) P2 P¯ 2P 0 ln1 δ1 − e−2ap L ; 2ap δ
(13)
2 where δ 2a−1 p RefF p gP 0 . The linear loss coefficient aμ of the μth wave can be calculated using the loss model proposed in [13,15], assuming a fast light loss level equal to 2 dB∕cm [13]. In our calculations, we assume a silicon PCSW with fixed a 412 nm, h 0.5a while the radii of holes not included in the optimization are also held fixed at r a 0.27a. We also assume that the range of the design parameters is −0.04a ≤ Δr i ≤ 0, 0 ≤ Δyi ≤ 0.15a and 0.1 W ≤ P 0 ≤ 2 W, 25 μm ≤ L ≤ 510 μm. The PCSW designs obtained by maximizing EBT and EBTPL are described in Tables 1 and 2, for a gradually increasing number of design parameters. We designate the two designs quoted in the last rows of Tables 1 and 2 as designs A and B, respectively. We also quote the values obtained for Δλ, δλ, and ng0 , as well as the required P 0 and L. In the optimization process, we include the parameters of the three closest neighbors (N 3). This brings the total number of parameters considered to eight (three for the horizontal shifts Δyi , three for the radii Δr i , plus two for P 0 and L). Because Δxi in the optimization process yield only a marginal improvement on the presented FoMs, these parameters are not considered in the tables. Because of the tight confinement of the mode, varying holes situated farther away also had negligible bearing. Table 1 shows the optimum design parameters when EBT is maximized as more parameters are included in the optimization. The optimum waveguides yield progressively larger η0 , Δλ, and δλ values. In all optimizations except the first, we used the parameters of the previous optimum design as a starting point. For the first
optimization, we chose the initial point randomly. We deduce that EBT is optimized for relatively long waveguides and large pump powers. It is worth noticing that the optimum length is always near the maximum allowable length (≅ 500 μm). For this optimization, we have chosen m 1∕2, i.e., a −3 dB tolerance level. We have found that this value generally produced high-efficiency designs with smooth wavelength dependence. The results obtained when optimizing the design of PCSW with respect to the EBTPL are shown in Table 2. In this case, one obtains more compact and lower power designs at the expense of a smaller η0 , compared to Table 1. As the number of parameters included is increased, P 0 and L are reduced. We have observed that, for these optimizations it is better to choose a much smaller value for m 10−2 (−20 dB level). This is because EBTPL favors small L and P 0 values at the expense of lower efficiencies and poorer wavelength dependence and, from an application point of view, one must be prepared to tolerate larger variations of η. Figure 2 illustrates the conversion efficiencies ηλi ; λs achieved by designs A and B. As shown in Fig. 2(a), design A yields a cross-like variation centered near the maximum η value (η0 −18.9 dB). Because of this cross-like behavior, it is more indicative to consider δλ as the average rather than the maximum wavelength separation jλp − λs j, and this is the reason we adopted the former. In Fig. 2(b), we observe a large efficiency value for design B, occurring at opposing edges of the grid and an overall smooth variation of η at lower values (η ≅ −45 dB). The rather small efficiencies obtained by design B are the result of the small waveguide length and the low pump power. They can be increased by adjusting any of these two parameters. This increase is illustrated in Fig. 3(a), where we plot η with respect to λi assuming constant λs 1.602 μm and L 74.6 μm, while varying the input pump power P 0 . It is deduced that η remains practically constant inside a 40 nm range,
Table 2. PCSW Design Optimization with Respect to EBTPL Parameter (m 10−2 ) Design Parameters Δy1 ∕a
Δy2 ∕a
Δy3 ∕a
Δr 1 · a
Δr 2 · a
Δr 3 · a
EBTPL (fm/W)
η0 (dB)
Δλ(nm)
δλ(nm)
P 0 (W)
L(μm)
ng0
0.14434 0.13656 0.14969 0.15099 0.14882 0.15094
0 0.09193 0.10946 0.10999 0.10974 0.10998
0 0 0.10619 0.01779 0.03162 0.00013
0 0 0 −0.041 −0.0405 −0.04
0 0 0 0 −0.006 −0.0097
0 0 0 0 0 −0.0013
6.3 6.9 20.5 63.4 63.7 73.9
−25.2 −27.5 −25.5 −28.1 −28.1 −28
16 28 43 56 56 61.3
2.8 4.9 7.8 10.5 10.5 11.6
0.16 0.29 0.26 0.15 0.15 0.2
132.3 121.9 177.6 91.8 91.8 74.6
25.2 14.3 10.4 9.4 9.5 9.5
1044
OPTICS LETTERS / Vol. 40, No. 6 / March 15, 2015
Furthermore, it is interesting to analyze the influence of optical loss in our designs. The total losses of the pump wave amount to −27.9 and −6.17 dB for designs A and B, respectively. The linear losses experienced by the pump wave are ≅ −1.5 and ≅ −0.4 dB, respectively, implying that the designs are limited by FC-induced loss. Finally, it is worth noting that the nonlinear losses can be mitigated through the application of an external DC field, driving the FCs away from the center of the waveguide [16] which may potentially lead to higher efficiencies because of the corresponding decrease in the loss level. In conclusion, we have discussed how PCSWs can be designed for broadband FWM applications. Two alternative FoMs were used in this design process which involved many waveguide parameters simultaneously. The obtained designs are characterized by a smooth efficiency variation with respect to the pump and signal wavelengths and should be useful in signal processing applications.
Fig. 2. FWM conversion efficiencies with respect to the wavelengths of the idler (λi ) and the signal (λs ) waves for designs A and B.
Fig. 3. Efficiency variation of the design B at λs 1.602 nm with respect to λi , assuming different (a) P 0 and (b) L values.
regardless of P 0 . Figure 3(b) shows the efficiency obtained by varying the waveguide length L while keeping a constant P 0 0.2 W. In this case, the flat nature of η is maintained, but the corresponding wavelength region is slowly reduced with increasing L, implying a trade-off between the waveguide length and the pump/signal tunability δλ.
This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund. References 1. M. Ebnali-Heidari, C. Monat, C. Grillet, and M. K. MoravvejFarshi, Opt. Express 17, 18340 (2009). 2. C. Husko, T. D. Vo, B. Corcoran, J. Li, T. F. Krauss, and B. J. Eggleton, Opt. Express 19, 20681 (2011). 3. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, J. Opt. Soc. Am. B 31, 366 (2014). 4. J. Li, L. O’Faolain, and T. F. Krauss, Opt. Express 20, 17474 (2012). 5. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, Opt. Lett. 39, 884 (2014). 6. C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Opt. Express 17, 2944 (2009). 7. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, Opt. Express 14, 9444 (2006). 8. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, Opt. Express 16, 6227 (2008). 9. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, Opt. Lett. 37, 4585 (2012). 10. R. H. Byrd, M. E. Hribar, and J. Nocedal, SIAM J. Optim. 9, 877 (1999). 11. T. Chen, J. Sun, and L. Li, Opt. Express 20, 20043 (2012). 12. S. Rawal, R. K. Sinha, and R. M. De La Rue, J. Nanophoton. 6, 063504 (2012). 13. L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, Opt. Express 18, 27627 (2010). 14. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001). 15. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, J. Opt. Soc. Am. B 29, 2787 (2012). 16. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, Opt. Express 18, 3582 (2010).
1041
Designing photonic crystal waveguides for broadband four-wave mixing applications Panagiotis Kanakis,1,* Thomas Kamalakis,2 and Thomas Sphicopoulos1 1
2
Department of Informatics and Telecommunications, National and Kapodistrian University/Athens, Panepistimioupolis, Athens GR157 84, Greece
Department of Informatics and Telematics, Harokopio University/Athens, 9 Omirou Street, Athens GR17778, Greece *Corresponding author: [email protected] Received December 10, 2014; revised January 19, 2015; accepted January 27, 2015; posted February 4, 2015 (Doc. ID 229408); published March 11, 2015 We present photonic crystal waveguide designs which exhibit large four-wave mixing efficiencies over a wide wavelength region. These designs are identified using an optimization process taking into account sophisticated figure-ofmerits that depend on the pump bandwidth and the signal/pump tunability. The obtained designs achieve up to −18.9 dB conversion efficiency, tunable over a 10 nm tunability range. We also present alternative designs that are less efficient but have smaller power requirements and are far more compact. © 2015 Optical Society of America OCIS codes: (230.5298) Photonic crystals; (190.4380) Nonlinear optics, four-wave mixing. http://dx.doi.org/10.1364/OL.40.001041
All-optical signal processing based on four-wave mixing (FWM) is currently attracting increased attention because of its potential applications [1,2]. In its degenerate form, FWM involves the interaction between a signal and an idler wave, located at different wavelengths, mediated through a strong pump wave located at a third wavelength [3]. Nano-photonic structures such as photonic crystals [4] may enable the realization of such functionalities in compact form. Perhaps the most commonly adopted figure-of-merit (FoM) for FWM is the conversion efficiency η defined as, η ≡ P i L∕P s 0;
(1)
where P i L is the power of the idler wave at the output of a waveguide (of length L) and P s 0 is the signal power at the input. For a given waveguide design, it can be shown that the efficiency is determined by the idler and signal wavelengths λi and λs , the length L, and the initial pump power P 0 , η ηλs ; λi ; L; P 0 . However, achieving a large peak value for η, obtained only near certain wavelength combinations, is not sufficient. One should rather seek a compromise between efficiency, bandwidth, power requirements, and device footprint. In our previous work [5], we developed new FoMs that include these important aspects. In the present work, we discuss how the FoMs in question can be used for optimizing a photonic crystal slab waveguide (PCSW). To apply these FoMs, we first need to define the available pump bandwidth Δλ and the signal/pump tunability δλ. Assuming that for a given P 0 and L the maximum efficiency is η0 P 0 ; L maxλi ;λs fηg, we define ΔλP 0 ; L as the pump wavelength range for which the conversion efficiency does not fall below mη0 where m is a specified tolerance level with 0 < m < 1. In addition, the parameter δλP 0 ; L is defined as the average wavelength separation jλp − λs j between the pump and the signal waves for which again η ≥ mη0 . To estimate the parameters Δλ and δλ for a given PCSW design, we obtain the values of η with respect to many possible wavelength combinations, which form a 2D grid λs ; λi assuming, for example, a 2 nm wavelength spacing. In our efficiency calculations 0146-9592/15/061041-04$15.00/0
we always exclude the diagonal λs λi elements since λs cannot be equal to λi in a practical experiment. Once ηλs ; λi is obtained, we readily obtain δλ as the average wavelength separation jλp − λs j for which η is higher than mη0 and Δλ as the wavelength range of the pump wavelengths in which η abides to the same condition. We define the efficiency-bandwidth-tunability (EBT) product FoM as EBT η0 × Δλ × δλ:
(2)
As EBT is increased, the design is expected to exhibit high-efficiency values over a wide pump wavelength range with a smooth efficiency variation with respect to λp and λs . To further underline the need for small device footprint and power requirements, we may also define an alternative FoM, EBTPL
η0 × Δλ × δλ : P0 × L
(3)
Optimizing the waveguide with respect to EBTPL is expected to yield shorter structures requiring less power at the expense of a smaller overall efficiency. From an application point of view, obtaining a smooth η is useful, since it guarantees that the achieved FWM performance will be maintained over a wide wavelength selection for the three waves. A photonic crystal slab is formed by the periodic modulation of the refractive index of a slab (e.g., by embedding holes of air with the dielectric constant εb 1 in a silicon layer with the dielectric constant εa ≅ 12). Forming a line defect (e.g., by removing an entire row of holes along the x-direction) introduces a guided defect mode inside the photonic band-gap. Figure 1(a) shows examples of the various geometric characteristics of the holes that can be altered to obtain a suitable waveguide design. The holes and their corresponding characteristics are labeled according to their proximity to the line defect. For example, Δy2 describes the shift of the centers of the holes which have the second smallest distance from the line defect (i.e., the second closest neighbors, labeled “2” in the © 2015 Optical Society of America
1042
OPTICS LETTERS / Vol. 40, No. 6 / March 15, 2015
η li
Fig. 1. (a) Parameters used in the optimization and (b) waveguide supercell considered in the photonic band calculations.
figure) along the y axis. A positive Δy2 indicates that the holes move away from the defect. In the same way Δr 1 indicates the radius increase of the closest neighbors. We assume that the changes are made so that the waveguide supercell retains its mirror symmetry with respect to the x axis. Varying the waveguide characteristics, it is possible to obtain different dispersion mode relations k kω which can be tailored to correspond to a flat group index variation ng ng ω. A sufficiently flat ng ng ω can, in principle, favor nonlinearities over a large bandwidth [6]. A large ng also leads to increased losses which may eventually limit the waveguide performance. A compromise must therefore be sought. We define the flat band region as the wavelength range in which the group index (ng ) varies no more that 10% from the ng value ng0 obtained when the group velocity dispersion (GVD) is minimum [7]. We expect that this region favors nonlinear interactions; however, in our calculations, the wavelengths of the three waves were not only considered in the region above, but also in all adjacent regions for which ng varies less than 10% of ng0 and the losses will be similar. We also vary the wavelengths inside the portions of the dispersion relation between these regions as well. Using exhaustive search, we have previously identified optimal PCSW designs [5] with respect to the FoMs in Eqs. (2) and (3). Although exhaustive search methods are frequently adopted in the literature [8] with alternative FoMs, they usually limit the number of design parameters that can be considered concurrently and, as their number increases, one must resort to alternative optimization methods [9]. In this work, we maximize the above FoMs based on an interior-point optimization method that combines a direct method for solving the constrained maximization problem, along with conjugate gradient steps using trust regions performed by MATLAB’s fmincon function [10]. In this context, EBT can be written as EBT f a; k; r a ; h; εa ; εb ; Δx1 ; Δy1 ; Δr 1 ; …; ΔxN ; ΔyN ; Δr N ; P 0 ; L, where f is a function, a is the lattice constant, k is the wavenumber, h is the slab height, N is the number of hole groups assumed in the optimization process and r a are the radii of all holes that are not considered in the optimization. The function f is not known in closed, form but can be computed using Eq. (2) and a model for the efficiency η. The treatment of EBTPL is quite similar. The estimation of η can be carried out by numerically solving the coupled ordinary differential equations (ODEs) for the three propagating waves [11]. In our case, it turns out that this is impractical, since η must be numerically estimated for many combinations λs ; λi for each waveguide candidate design. Alternatively, one can approximate η as
λs κ2 1 tot2 sinh2 gL: λi 4g
(4)
Equation (4) is analytical and includes the influence of free-carriers (FCs) generation. It is generally in very good agreement with the numerical ODE solution [1,5] and requires much less computational time. The parameter li is the loss experienced by the idler wave, li P i L∕P i 0 exp−ai L 2L RefF i gP¯ 2p ;
(5)
where ai is the linear loss coefficient for the idler wave. In Eq. (4), κ tot and g are the total phase mismatch and the parametric gain given by [3] κ tot κ ImfF s F i − 2F p gP¯ 2p ;
(6)
1∕2 2 : g n22 S 2p S i S s ωi ωs P¯ 2p c−2 A−2 psi − κ ∕4
(7)
In Eqs. (5) and (6), F μ is the FC coefficient [11]: Fμ ≅
N C 2π C 2 λμ 2 C − ; j λ0 λμ 1 2 P 2p
(8)
where the index μ p, s, i corresponds to the pump, signal, and idler waves, respectively. The FC density N C is given by N C βTPA S 3p τC P 2p ∕2ℏωp A2ppp [12], S μ is the slow-down factor for μth wave [12], λ0 1550 nm, ωμ 2πc∕λμ , τC is the FC lifetime and for silicon waveguides C 1 1.35 × 10−27 m3 , and C 2 1.45 × 10−21 m3 [13]. The FC lifetime τC is assumed to be 600 ps [3], while the effective modal areas Aρκψ are given by [11] R Aρκψ
V
R R jE ρ j2 dV V jE κ j2 dV V jE ψ j2 dV 1∕2 R : α V E ρ E ρ E κ E ψ dV
(9)
In the equation above, E μ is the electric field component of the μth wave along the y direction, and V is the volume of the waveguide supercell used in the calculations. In our calculations, the modal fields are obtained by a 3D plane wave expansion (PWE) [14] mode solver which takes into account the geometric parameters of the waveguide. The number of plane waves used for the x, y, and z directions are 14, 84, and 28, respectively. The supercell assumed in the PWE calculations is shown in Fig. 1(b). In Eq. (6), κ is the phase mismatch in the presence of linear loss only, given by [3] −1 −1 −1 −1 −1 κ Δk 4πn2 P¯ p A−1 pss λs Apii λi − Appp λp ;
(10)
where n2 is the nonlinear Kerr coefficient and Δk ks ki − 2kp is the linear phase mismatch (where kμ kωμ are the propagation constants of the three waves). The power of the pump wave, its average value, and its average square value are given by [3] P p z P 0 e−ap z 1 δ1 − e−2ap z −1∕2 ;
(11)
March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
1043
Table 1. PCSW Design Optimization with Respect to EBT Parameter (m 1∕2) Design Parameters Δy1 ∕a
Δy2 ∕a
Δy3 ∕a
Δr 1 · a
Δr 2 · a
Δr 3 · a
EBT (nm2 )
η0 (dB)
Δλ (nm)
δλ (nm)
P 0 (W)
L(μm)
ng0
0.08303 0.144 0.15071 0.1484 0.14957 0.14968
0 0.0981 0.10834 0.10743 0.09808 0.09919
0 0 0.1090 0.0193 0.0131 0.0124
0 0 0 −0.0112 −0.0406 −0.0407
0 0 0 0 −0.0396 −0.0397
0 0 0 0 0 −0.0034
1.6 2.5 3.6 5.6 6.4 7.9
−22.3 −19.5 −19.8 −18.7 −19.9 −18.9
39.8 36 43 48 58 58
6.9 6.3 7.9 8.7 10.8 10.7
0.81 0.93 1.96 1.28 1.51 1.66
509.6 500 477.8 508.7 506.2 507.4
6.1 13.5 10.3 11.7 9.2 9.3
P¯ p
−a L P0 1 e p p sin−1 p − sin−1 p ; ap L δ δ−1 1 δ−1 1 (12) P2 P¯ 2P 0 ln1 δ1 − e−2ap L ; 2ap δ
(13)
2 where δ 2a−1 p RefF p gP 0 . The linear loss coefficient aμ of the μth wave can be calculated using the loss model proposed in [13,15], assuming a fast light loss level equal to 2 dB∕cm [13]. In our calculations, we assume a silicon PCSW with fixed a 412 nm, h 0.5a while the radii of holes not included in the optimization are also held fixed at r a 0.27a. We also assume that the range of the design parameters is −0.04a ≤ Δr i ≤ 0, 0 ≤ Δyi ≤ 0.15a and 0.1 W ≤ P 0 ≤ 2 W, 25 μm ≤ L ≤ 510 μm. The PCSW designs obtained by maximizing EBT and EBTPL are described in Tables 1 and 2, for a gradually increasing number of design parameters. We designate the two designs quoted in the last rows of Tables 1 and 2 as designs A and B, respectively. We also quote the values obtained for Δλ, δλ, and ng0 , as well as the required P 0 and L. In the optimization process, we include the parameters of the three closest neighbors (N 3). This brings the total number of parameters considered to eight (three for the horizontal shifts Δyi , three for the radii Δr i , plus two for P 0 and L). Because Δxi in the optimization process yield only a marginal improvement on the presented FoMs, these parameters are not considered in the tables. Because of the tight confinement of the mode, varying holes situated farther away also had negligible bearing. Table 1 shows the optimum design parameters when EBT is maximized as more parameters are included in the optimization. The optimum waveguides yield progressively larger η0 , Δλ, and δλ values. In all optimizations except the first, we used the parameters of the previous optimum design as a starting point. For the first
optimization, we chose the initial point randomly. We deduce that EBT is optimized for relatively long waveguides and large pump powers. It is worth noticing that the optimum length is always near the maximum allowable length (≅ 500 μm). For this optimization, we have chosen m 1∕2, i.e., a −3 dB tolerance level. We have found that this value generally produced high-efficiency designs with smooth wavelength dependence. The results obtained when optimizing the design of PCSW with respect to the EBTPL are shown in Table 2. In this case, one obtains more compact and lower power designs at the expense of a smaller η0 , compared to Table 1. As the number of parameters included is increased, P 0 and L are reduced. We have observed that, for these optimizations it is better to choose a much smaller value for m 10−2 (−20 dB level). This is because EBTPL favors small L and P 0 values at the expense of lower efficiencies and poorer wavelength dependence and, from an application point of view, one must be prepared to tolerate larger variations of η. Figure 2 illustrates the conversion efficiencies ηλi ; λs achieved by designs A and B. As shown in Fig. 2(a), design A yields a cross-like variation centered near the maximum η value (η0 −18.9 dB). Because of this cross-like behavior, it is more indicative to consider δλ as the average rather than the maximum wavelength separation jλp − λs j, and this is the reason we adopted the former. In Fig. 2(b), we observe a large efficiency value for design B, occurring at opposing edges of the grid and an overall smooth variation of η at lower values (η ≅ −45 dB). The rather small efficiencies obtained by design B are the result of the small waveguide length and the low pump power. They can be increased by adjusting any of these two parameters. This increase is illustrated in Fig. 3(a), where we plot η with respect to λi assuming constant λs 1.602 μm and L 74.6 μm, while varying the input pump power P 0 . It is deduced that η remains practically constant inside a 40 nm range,
Table 2. PCSW Design Optimization with Respect to EBTPL Parameter (m 10−2 ) Design Parameters Δy1 ∕a
Δy2 ∕a
Δy3 ∕a
Δr 1 · a
Δr 2 · a
Δr 3 · a
EBTPL (fm/W)
η0 (dB)
Δλ(nm)
δλ(nm)
P 0 (W)
L(μm)
ng0
0.14434 0.13656 0.14969 0.15099 0.14882 0.15094
0 0.09193 0.10946 0.10999 0.10974 0.10998
0 0 0.10619 0.01779 0.03162 0.00013
0 0 0 −0.041 −0.0405 −0.04
0 0 0 0 −0.006 −0.0097
0 0 0 0 0 −0.0013
6.3 6.9 20.5 63.4 63.7 73.9
−25.2 −27.5 −25.5 −28.1 −28.1 −28
16 28 43 56 56 61.3
2.8 4.9 7.8 10.5 10.5 11.6
0.16 0.29 0.26 0.15 0.15 0.2
132.3 121.9 177.6 91.8 91.8 74.6
25.2 14.3 10.4 9.4 9.5 9.5
1044
OPTICS LETTERS / Vol. 40, No. 6 / March 15, 2015
Furthermore, it is interesting to analyze the influence of optical loss in our designs. The total losses of the pump wave amount to −27.9 and −6.17 dB for designs A and B, respectively. The linear losses experienced by the pump wave are ≅ −1.5 and ≅ −0.4 dB, respectively, implying that the designs are limited by FC-induced loss. Finally, it is worth noting that the nonlinear losses can be mitigated through the application of an external DC field, driving the FCs away from the center of the waveguide [16] which may potentially lead to higher efficiencies because of the corresponding decrease in the loss level. In conclusion, we have discussed how PCSWs can be designed for broadband FWM applications. Two alternative FoMs were used in this design process which involved many waveguide parameters simultaneously. The obtained designs are characterized by a smooth efficiency variation with respect to the pump and signal wavelengths and should be useful in signal processing applications.
Fig. 2. FWM conversion efficiencies with respect to the wavelengths of the idler (λi ) and the signal (λs ) waves for designs A and B.
Fig. 3. Efficiency variation of the design B at λs 1.602 nm with respect to λi , assuming different (a) P 0 and (b) L values.
regardless of P 0 . Figure 3(b) shows the efficiency obtained by varying the waveguide length L while keeping a constant P 0 0.2 W. In this case, the flat nature of η is maintained, but the corresponding wavelength region is slowly reduced with increasing L, implying a trade-off between the waveguide length and the pump/signal tunability δλ.
This research has been co-financed by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: Heracleitus II. Investing in knowledge society through the European Social Fund. References 1. M. Ebnali-Heidari, C. Monat, C. Grillet, and M. K. MoravvejFarshi, Opt. Express 17, 18340 (2009). 2. C. Husko, T. D. Vo, B. Corcoran, J. Li, T. F. Krauss, and B. J. Eggleton, Opt. Express 19, 20681 (2011). 3. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, J. Opt. Soc. Am. B 31, 366 (2014). 4. J. Li, L. O’Faolain, and T. F. Krauss, Opt. Express 20, 17474 (2012). 5. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, Opt. Lett. 39, 884 (2014). 6. C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Opt. Express 17, 2944 (2009). 7. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, Opt. Express 14, 9444 (2006). 8. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, Opt. Express 16, 6227 (2008). 9. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, Opt. Lett. 37, 4585 (2012). 10. R. H. Byrd, M. E. Hribar, and J. Nocedal, SIAM J. Optim. 9, 877 (1999). 11. T. Chen, J. Sun, and L. Li, Opt. Express 20, 20043 (2012). 12. S. Rawal, R. K. Sinha, and R. M. De La Rue, J. Nanophoton. 6, 063504 (2012). 13. L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenovic, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, Opt. Express 18, 27627 (2010). 14. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001). 15. P. Kanakis, T. Kamalakis, and T. Sphicopoulos, J. Opt. Soc. Am. B 29, 2787 (2012). 16. A. C. Turner-Foster, M. A. Foster, J. S. Levy, C. B. Poitras, R. Salem, A. L. Gaeta, and M. Lipson, Opt. Express 18, 3582 (2010).
