TuA4.5 (Contributed) 17:00 – 17:15
Optimization of Compact Photonic Crystal Cavity Using the Gentle Confinement Method Jon Øyvind Kjellman∗ , Takuo Tanemura, and Yoshiaki Nakano
Research Center for Advanced Science and Technology, the University of Tokyo, Japan ∗ Email:
[email protected] II. O PTIMIZATION STRATEGY
Abstract—We demonstrate that the gentle confinement method is effective for optimizing area-limited photonic crystal cavities. We also show the importance of the boundary and that optimal boundary parameters improves the Q-factor significantly.
For passive cavities, Q is maximized by minimizing the radiative loss. Q can generally be decomposed into two parts −1 Q−1 = Q−1 ⊥ + Qk , where the first term represents the outof-plane loss and the second term represents the in-plane loss. Our strategy is to first consider out-of-plane loss and then inplane loss. To minimize the out-of-plane loss, we apply the gentle confinement method [5], [6], which is an established technique for designing high-Q line-defect cavities (e.g. PhC hetero structures [5]). In this method, out-of-plane loss is minimize by ensuring that the electric field at the cavity surface has a Gaussian envelope. Recognizing that the cavity shown in Fig. 1a can be decomposed into three crossed line-defects, we apply this method of optimization on these line defects. Significant in-plane losses are expected due to the limited structure of the island. The importance of the boundary has already been demonstrated in optically thick, area-limited, PhC cavities without photonic band-gap confinement [7]. We demonstrate that the boundary is also important for our bandgap confined cavity in an optically thin slab.
I. I NTRODUCTION Photonic crystal (PhC) cavities are attractive for highly integrated laser sources due to their high Q-factor, subwavelength-scale mode volumes and the possibility of achieving ultra-low lasing thresholds. PhC technology is maturing and electrically pumped PhC lasers [1]–[3] have been demonstrated. Despite the small mode volumes, however, high-Q PhC cavities generally require a large number of PhC periods to provide strong confinement. This makes PhC cavities less suited for applications requiring dense integration. In addition, for many applications, including electrically pumped lasers, non-conventional cavity designs are necessary to tailor the field to avoid excessive losses near absorbing regions such as metal electrodes. We have previously proposed the PhC cavity illustrated in Fig. 1a [4]. It was designed to provide confinement within a limited area while reducing the field at the cavity center. Here, we computationally explore an optimization strategy aiming to reduce the number of design parameters for this cavity and to be able to predict its resonance wavelength. The optimization problem is divided into minimization of outof-plane losses and in-plane losses. To minimize out-of-plane losses we employ the gentle confinement method [5]. In-plane losses are minimized by changing the boundary of the cavity. Using this strategy we demonstrate a twofold improvement in Q-factor over the previously reported value.
III. R ESULTS Full-field, 3D finite-difference time-domain (FDTD) and plane wave expansion method (PWEM) calculations are performed. We choose the PhC lattice constant a = 433.2 nm and set the hole radius to 150 nm and slab thickness to 260 nm based on fabrication considerations; the cladding/hole and slab refractive indices are set to 1.45 and 3.48, respectively. The cavity has a corner-to-corner diameter of less than 8 µm. A. Out-of-plane loss In the gentle confinement method the electric field along a line defect is tailored to a Gaussian envelope in order to reduce out-of-plane losses [5], [6]. This is done by engineering the line-defect dispersion relation such that the imaginary component of the complex wave vector, the attenuation constant q, satisfies q = B|x|. Here, B is a constant which determines 2 the sharpness of the Gaussian envelope Eenv ∝ e−Bx . First, using the PWEM on the unit cell shown in Fig. 1b, we calculate the dispersion relation fw (k) for several line defects of different widths w. From this, the method of Tanaka et al. [5] lets us derive the relationship between frequency and attenuation fw (q), as plotted in Fig. 2a.
a w a/2
b w4 w3 w2 w1 (a)
w0 x=0 ~ x
(b)
Fig. 1. (a) Basic layout of hexagonal cavity. The line-defect widths w0 ..w4 can be changed by changing the spacing between the diagonal arrays of holes (black, dashed lines.) (b) Line-defect unit-cell for PWEM calculation of dispersion relation.
978-1-4673-5060-0/$31.00 ©2013 IEEE
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0.30
w = 0.475 0.488 0.500
√ 3a
0.525
0.29 0.28
0.550 0.575
0
0.1
0.2
Attenuation, q [2π/a]
Frequency [a/λ]
0.31
0.2 0.1
Attenuation, q [2π/a] (a)
(a) b = 0.25a, Q = 5490
0 0.48 0.50 0.52 0.54 0.56
Fig. 4. log of the optical intensity at the surface of the cavity at resonance.
√ Line defect width, w [a 3] (b)
√ case of w4 = 0.52 3a. The plot shows strong dependence of Q on b with maximum Q = 5490 at b = 0.25a. This is a 30% improvement over the case of b = 0 and a 110% improvement over the initially reported value for this cavity design [4]. We also note that the resonance wavelength stays within a range of 0.5 nm around 1.537 µm for all investigated values of b. To elucidate the mechanism behind the effect of b on Q, we investigate the field distribution in the cavity. Fig. 4a and b shows the logarithm of the optical intensity for the cases of the highest (b = 0.25a) and lowest (b = 0.50a) Q-factor, respectively. The low-Q case clearly shows stronger fields along the boundary compared with the optimized case. Based on this, we believe that the dominating mechanism for inplane losses is radiation from the cavity side walls due to suboptimal phase matching at the boundary interface. Tailoring of this region may yield further improvements in Q.
Quality factor
Fig. 2. √ (a) Attenuation at selected line-defect widths between √ w = 0.475 3a (top) and w = 0.575 3a (bottom). The dashed line indicates the design resonance wavelength λres . From the intersection points, we derive (b) attenuation as a function of line-defect width.
5000 4000 3000 2000 0.475
0.500
0.525
0.550 −0.5
√ Outer defect width, w4 [a 3] (a)
0.5
0
1
Boundary width, b/a (b)
Fig. 3. (a) Q-factor as a function of the line-defect width at the outermost holes w4 with √b = 0. (b) Q-factor as a function of the boundary width b with w4 = 0.52a 3.
IV. C ONCLUSION
For the design shown in Fig. 1a, the variables w0 ..w4 √ must be determined. We first fix w0 = 0.575 3a to allow a propagating mode (q = 0) at λres = 1.532 µm. The corresponding resonance frequency (fres = 0.2828) is indicated with a horizontal line in Fig. 2a and the indicated intersection points give q as a function of w at the resonance frequency. This relationship, q(w), is plotted in Fig. 2b. Next, to determine w1 ..w4 , we determine a value for the constant B by choosing a value for w4 that satisfies q(w4 ) > 0. When B has been determined, w1 , w2 and w3 follows by satisfying q(wn ) = B|x|. Using FDTD, the Q-factor is calculated for various cavities with different w4 and Q(w4 ) is plotted Fig. 3a. We observe that there exists an optimum value of w4 . This seems reasonable as a small value should result in a sharp envelope which according to gentle confinement theory should have significant losses. On the other hand, for large values of w4 , the field would not be sufficiently confined and there would be significant scattering loss √ at the corners. Maximum Q = 4240 is found at w4 = 0.52 3a with a resonance wavelength of 1537 nm—only 5 nm larger than the design value. Fig. 1a shows how the parameter b can be used to change the boundary width of the cavity. To verify that the PhC structure itself confines the mode, we set b = ∞. Although Q is reduced to 3620, the cavity mode persists and we conclude that the PhC structure does effectively confine the mode.
We have shown that the gentle confinement method is effective for realizing PhC cavities with a deterministic resonance wavelength within a strictly limited area. We have also demonstrated that Q is strongly dependent on the boundary width. The gentle confinement method followed by boundary optimization reduced the cavity optimization problem to two dimensions (w4 and b.) Optimization of the investigated cavity yielded a theoretical Q-factor of 5490 in a hexagonal island with a corner-to-corner diameter of 7.6 µm. This is more than two times the previously reported Q-factor. The resonance wavelength was only 5 nm larger than the design wavelength. R EFERENCES [1] M. Seo et al., “Low threshold current single-cell hexapole mode photonic crystal laser,” Applied physics letters, vol. 90, no. 17, p. 171122, 2007. [2] B. Ellis et al., “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nature Photonics, vol. 5, no. 5, pp. 297–300, May 2011. [3] S. Matsuo et al., “Room-temperature continuous-wave operation of lateral current injection wavelength-scale embedded active-region photonic-crystal laser,” Opt. Express, vol. 20, no. 4, pp. 3773–3780, Feb 2012. [4] J. Kjellman, A. Higo, and Y. Nakano, “Design of photonic crystal cavity for hexagonal islands,” in IEEE Photonics Conference, sept. 2012, pp. 272–273. [5] Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with q-factor of ∼109 ,” J. Lightwave Technol., vol. 26, no. 11, pp. 1532–1539, Jun 2008. [6] K. Welna et al., “Novel dispersion-adapted photonic crystal cavity with improved disorder stability,” IEEE J. Quantum Electron., vol. 48, no. 9, pp. 1177–1183, sept. 2012. [7] S.-H. Kim, A. Homyk, S. Walavalkar, and A. Scherer, “High-q impurity photon states bounded by a photonic band pseudogap in an optically thick photonic crystal slab,” Phys. Rev. B, vol. 86, p. 245114, Dec 2012.
B. In-plane loss To minimize the in-plane loss we examine the effect of b on the Q-factor. Fig. 3b shows how Q depends on b for the
978-1-4673-5060-0/$31.00 ©2013 IEEE
(b) b = 0.50a, Q = 2400
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