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Design of tapering one-dimensional photonic crystal ultrahigh-Q microcavities
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Photonics and Nanostructures – Fundamentals and Applications 7 (2009) 19–25 www.elsevier.com/locate/photonics
Design of tapering one-dimensional photonic crystal ultrahigh-Q microcavities Qin Chen, Duncan W.E. Allsopp * Department of Electronic and Electrical Engineering, University of Bath, Bath BA2 7AY, UK Received 18 June 2008; received in revised form 14 October 2008; accepted 19 November 2008 Available online 27 November 2008
Abstract One-dimensional (1D) photonic crystal (PC) microcavities can be readily embedded into silicon-on-insulator waveguides for photonic integration. Such structures are investigated by 2D Finite-Difference Time-Domain method to identify designs with high transmission which is essential for device integration. On-resonance transmission is found to decrease with the increasing mirror pairs, however, the quality factor (Q) increases to a saturated value. The addition to the Bragg mirrors of tapered periods optimized to produce a cavity mode with a near Gaussian shaped envelope results in a major reduction in vertical loss. Saturated Q up to 2.4 106 is feasible if the internal tapers are properly designed. The effect of increasing transmission is also demonstrated in a structure with the external tapers. # 2008 Elsevier B.V. All rights reserved. PACS : 42.55.Tv; 42.79.Ci; 42.79.Gn; 85.60.Bt Keywords: Photonic crystal; Microresonators; Filters; Finite difference methods; Q factor
1. Introduction Photonic crystals (PC) have attracted wide interest in the applications of low threshold lasers [1], high finesse filters [2], single photon devices [3], nonlinear optics [4], and slow light [5] in the last 20 years. A waveguide based one-dimensional (1D) PC is the simplest PC structure [6–12], in which the light confinement is realized by refractive index contrast in the two transverse directions and by photonic bandgap effect in the longitudinal direction. Joannopoulos and coworkers first proposed a novel air-bridge microcavity and predicted a quality factor (Q) of 104 in 1995 [6]. Two years later, they fabricated a 1D PC cavity in a
* Corresponding author. E-mail address: [email protected] (D.W.E. Allsopp). 1569-4410/$ – see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2008.11.002
silicon-on-insulator (SOI) optical waveguide with a measured Q of 265 [7]. One main loss source in this kind of 1D PC cavity is the etched air gaps, where there is no refractive index contrast in the vertical direction. The out-of-plane scattering loss, either into the air or into the substrate, causes a serious degeneration of Q. Krauss and De La Rue suggested that semiconductor-rich lattices with small air gaps could suppress the diffractive spreading loss [8]. Lalanne and co-workers concluded that the mode mismatch is the main cause of the loss and inserted both outside and inside tapers in their PC cavities with 1D gratings [9]. Jugessur et al. also applied the tapers in the structure discussed in Joannopoulos’s paper [10]. However, all the measured Qs were still several hundreds. Recently, significant improvements have been reported. Lalanne and co-workers demonstrated a 1D PC cavity with tapered reflectors on SOI wafers and
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obtained a Q of 8.9 103 [11]. Using a Fabry-Perot model, they estimated that an intrinsic Q of 3.8 105 could be obtained in a cavity with two tapered semiinfinite mirrors. Pruessner et al. observed a resonance with a Q of 2.7 104 in a 1D PC with a long cavity on SOI wafer by using a 4-micron-deep Si etch [12]. Although almost two-order improvement has been obtained, the Qs in 1D PC cavities are still much smaller than that in 2D PC cavities, such as the measured Q of 9.5 105 reported by Noda and co-workers in their double-hetero-PC cavity in which the lattice constant was changed at the interfaces [13], and the recorded experimental Q of 1.2 106 and theoretical Q of 7 107 realized in a photonic crystal nanocavity by the local width modulation of a line defect [5,14]. In this paper, the effect on Q and transmission of different kinds of microcavity formed by Bragg mirrors comprising aperiodic 1D PCs with tapering period etched into 2D SOI structures is investigated by the twodimensional Finite-Difference Time-Domain (FDTD) method. Momentum space analysis of the localized mode reveals that the improvement results from the tapers suppressing the amplitude of the wave vector components in the leaky region of the resonant mode, thereby suppressing the vertical radiation loss. 2. Method Fig. 1 shows the basic structure used in the simulation work reported here. It comprises a block of SOI material consisting of a Si substrate, SiO2 buffer layer of 1.5 mm, the top Si guide layer of 360 nm and the air cladding layer. The refractive indices of silicon and silicon dioxide are 3.48 and 1.46, respectively. It is assumed that a central cavity layer is then formed by etching Bragg mirrors, each comprising N pairs of Si and air gap layers. In the 2D FDTD simulations, the corrugated waveguide is assumed to be illuminated from the input waveguide by the transverse magnetic
(TM, Hy = 0) fundamental mode, which is a Gaussianmodulated cosine impulse covering a wide frequency band [15]. The ‘‘bootstrapping’’ technique is used to set the exciting source [15]. The perfectly matched layer (PML) absorbing boundary is used to terminate the FDTD calculation window, with the PML thickness of 0.5 and 1 mm in the x and z directions, respectively. The spatial cell size is 10 nm, and the time step is Courant limit [15]. The transmission, reflection and loss spectra are calculated from the power flux recorded at the detector planes, which are normalized by the source value [16]. For the reflection calculation, double FDTD runs are conducted. The output in the first run without gratings is subtracted from the output in the second run with the full structure to exclude the effect of the source in the detector plane. The resonance wavelength is found by fitting a Lorentzian to the transmission peak and Q is given by the ratio of the peak wavelength to its 3-dB bandwidth. The mode field distributions from the FDTD simulation are obtained by compressing the incident impulse spectral width into a range narrow enough to ensure that only on-resonance modes can be excited. The spatial Fourier transformation spectra, which represent the plane wave components of the cavity mode, are then calculated from these field distributions. The analysis is valid for 1D PC structures in 2D cross section geometry, which is an approximation to the actual 3D structures. In the case of air slots without transverse waveguide confinement, our 2D model neglects the scattering loss in the third dimension, i.e. Q reported here is the upper limit. In practice, the width of the access waveguide must be finite, even tapered [10], to ensure single mode behaviour in the PC cavity part and low insertion loss at each end of the device. 3. 1D PC cavities with non-tapered mirrors The basis of the simulations is a 1D PC microcavity with each reflector comprising 8 pairs of Bragg mirrors. Instead of quarter wave stacks, Si-rich mirrors (dH = 200 nm, dL = 90 nm) are used to suppress scattering losses at the interface between Si blocks and air slots [8]. The grating depth, h, is set to 650 nm, which covers the mode distribution in the vertical direction and provides a strong Bragg reflection. 3.1. Number of mirror periods
Fig. 1. Schematic of a 1D PC cavity on SOI material with N = 3 pairs of untapered Si/air mirrors and 3 pairs of internal tapers at each side the central cavity.
Fig. 2 shows the transmission and Q as functions of the number, N, of mirror pairs for a structure of the
Q. Chen, D.W.E. Allsopp / Photonics and Nanostructures – Fundamentals and Applications 7 (2009) 19–25
Fig. 2. Transmission T, reflection R, loss L and Q versus mirror pairs N, where dH = 200 nm, dL = 90 nm, D = 400 nm and h = 650 nm.
generic type shown in Fig. 1. The variation of Q can be divided into three stages. Firstly, Q increases exponentially when N is less than 10. Secondly, it increases slowly from N = 10 to 16. Thirdly, it becomes saturated when N is larger than 16. There are two loss mechanisms in a 2D model of a 1D PC cavity in SOI slab waveguide of the type shown in Fig. 1. One is the longitudinal radiation loss, which depends on the degree of light confinement due to the Bragg mirrors, and the other is the vertical radiation loss caused by the mode coupling between the resonant mode in the cavity and the radiation mode in the cladding layer. With two such loss mechanisms, the Q of a 1D PC microcavity can be given by:
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Q can not be larger than QV irrespective of how many mirrors are located at each side of the central cavity. Obviously, this is much lower than those observed in 2D PC cavities [13]. The transmission T is found to decrease quickly with N, whereas the reflectivity R increases sharply as N > 8. Loss L defined as 1-T-R is also plotted in Fig. 2 (solid triangles). L initially increases with N, reaching a maximum at N = 10. Thereafter L decreases with N, finally saturating at 5% for N 14. This behaviour results from the increase in out-of-plane loss at low N with increasing number of mirror pairs due to the longer scattering region. For small N the out-of-plane loss derives from both mirrors. When the Bragg reflectors become longer at large N, the higher reflectivity of the first mirror decouples the cavity from the input light because of the reduced light tunnelling length into the whole structure. Consequently, the contribution from the second mirror to L falls with increasing N, tending to a constant level determined by the intrinsic impedance mismatch at the interface between the feeder waveguide and the first Bragg reflector. Therefore, the initial increase in loss with N and subsequent decoupling of the whole structure at large N values are the respective causes for the decrease of T with N. As shown in Fig. 2, less than 50% of incident power is transmitted through when N is larger than 8. If this cavity is used as a filter, a trade off between transmission and Q has to be considered. 3.2. Analysis by momentum space transformation
1 1 1 ¼ þ Q QL QV
(1)
where QL and QV are the values the quality factor would take if, respectively, longitudinal loss or vertical loss were the sole degradation mechanism. QL behaves in a similar way to the conventional loss mechanisms in a Fabry-Perot resonator, therefore it will increase exponentially with increasing number, N, of mirror periods. On the other hand the longitudinal confinement has only a slight effect on QV. Therefore, for small N the longitudinal loss dominates the total loss, i.e. QL QV and, from Eq. (1), Q QL. This corresponds to region I of the dependence of Q on N shown in Fig. 2. With an increase in N, the longitudinal and vertical loss become comparable, i.e. QL QV, and have almost the same effect on Q, region II of Fig. 2. With further increase in N, the vertical loss becomes dominant, i.e. QL QV, and Q saturates a value limited by QV, region III of Fig. 2. From Fig. 2, QV is found to be about 2.4 104 for SOI slab waveguides of the dimensions considered here.
1D PC cavities with periodic structures in one direction do not have a complete photonic band gap. The light confinement occurs via total internal reflection in the two other directions. The localized mode in the cavity can be seen as a combination of numerous plane wave components with wave vectors k, which may couple with the radiative modes in the cladding layer. The mode field in such a structure can be written as: Fðx; zÞ ¼ f ðx; zÞeikx x eikz z 2 2p kx2 þ kz2 ¼ k2 ¼ l
(2) (3)
where kx and kz are the vertical and tangential components of wave vector k, and l is the resonant wavelength. If kz lies within the range 0–2p/l, kx is a real number, which means the mode is not confined in the vertical direction. It is the plane waves with kz components in the range of 0–2p/l that give rise to the vertical loss that limits the Q of the resonant mode. The range 0 < kz < 2p/l is called the leaky region. The Spatial
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Fig. 3. Electric field (solid line) at the boundary between the top Si layer and the oxide buffer layer in a 1D PC cavity for (a) N = 4 and (c) N = 16, other parameters are the same as those in Fig. 2. The electric field in (a) is re-plotted in (c) as open circles for comparison. ‘B’ and ‘C’ separated by the dot line indicate Bragg mirrors and the central cavity. The modulus of SFT of electric field in (a) and (c) are shown in (b) and (d), respectively. The shadowed region is the leaky region. The vertical coordinate is broken for a detailed view.
Fourier Transform (SFT) of the longitudinal field distribution provides the spectral distribution of its plane wave components and enables analysis of the vertical loss [13,17]. Fig. 3 shows the electric field distribution of the resonant mode in the longitudinal direction in 1D PC microcavities for N = 4 and N = 16 and their SFT’s. The location of the central cavity relative to the mode field profiles is indicated by dotted lines. As Figs. 3(a) and (c) reveal, the longitudinal field of the cavity mode is antisymmetric with respect to a vertical symmetry plane through the centre of the cavity. Values of wave vector corresponding to leaky modes (i.e. kz = 0–2p/l) are indicated by the shaded regions in Fig. 3(b) and (d). The value of the SFT at each kz indicates the relative amplitude of the plane wave with longitudinal wave vector with that value of kz. Plane waves in the leaky region will couple with the radiation modes in the cladding layer and introduce the vertical loss. For example, the Q of the microcavity formed by N = 16 mirrors, at 2.4 104, is 100 times higher than that with N = 4 mirrors, however the corresponding values of the normalized integral of the spectral intensity in the leaky regions of the SFTs are
1.5 103 and 3.5 103, respectively (Fig. 3(b) and (d)). However, the difference in the vertical loss is only of 2. This is reasonable because the electric field profile around the central cavity in the longitudinal direction shows no obvious change with the increase in the number of mirror periods, as can be seen in Fig. 3(c). The slightly higher integrated spectral intensity in the leaky region for the N = 4 structure, compared to that of the N = 16 device, derives from the differences in the electric field distribution at the outer edge of the reflectors. It is obvious that in the former the electric field is still strong where the reflector interfaces with the surrounding Si feeder slab waveguides. The abrupt change in the electric field distribution at the boundary of the whole structure introduces an additional vertical radiation loss. 4. Cavities with mirrors of tapering period Based on the behaviour of 2D PC microcavity technology, the theoretically optimum shape for the field envelope of the resonant mode is a sinc function as its SFT is rectangular and can be engineered not to overlap with the leaky range of kz components. A
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envelope for the cavity mode in 1D PC microcavities as optimal holes positioning have in 2D structures. 4.1. Internal tapers with linearly varying air gaps
Fig. 4. Transmission and Q versus the number of mirror pairs in a 1D PC microcavity with linearly (square symbols, four air slots close to the central Si cavity at each side are 50, 60, 70, 80 nm wide) and nonlinearly (circular symbols, three air slots close to the central Si cavity at each side are 50, 70, 80 nm wide) tapered mirrors. Other parameters are the same as for Fig. 2.
Gaussian shaped envelope has been shown to be a practical alternative, requiring less rigorous optimization [13]. It shown here that mirror pairs with tapering variations in their periodicity fulfill the same role in increasing Q by creating a nearly Gaussian shaped
The relative merits of linearly varying the air gap widths as against a nonlinear variation are now considered. First, tapers are added to the Bragg reflectors by reshaping the four air slots in each mirror closest to the central cavity layer so that their widths increase from 50 to 80 nm, in steps of 10 nm, moving away from the cavity while the Si section lengths remain constant. Fig. 4 shows the variations in Q (open squares) and transmission (solid squares) with the total number of mirror periods, N. The variations in Q and transmission with N follows the trends shown in Fig. 2, except that the saturated value of Q, i.e. QV, has increased significantly from 2.4 104 in Fig. 2 to 5.0 105 with the introduction of tapers. The electric field variation in the longitudinal direction and its SFT are shown in Fig. 5 for two tapered 1D PC cavities, N = 8 (Fig. 5(b)) and N =14 (Fig. 5(d)). Only the field distribution in a half region
Fig. 5. The analogous figures to Fig. 3 for a 1D PC cavity with linear tapers in Fig. 4 at (a) N = 8 and (c) N = 14. The Gaussian function fit is plotted in (a) as a thick dashed line. The electric field profile in Fig. 3(c) is re-plotted in (c) as a thick dashed line for comparison. The regions labelled ‘B’, ‘T’ and ‘C’ and delineated by the dotted lines indicate the positions of the Bragg mirrors, tapers and the central cavity relative to the mode field (only distribution at the half region z > 0 are plotted). The modulus of SFT of electric field (solid lines) in (a) and (c) are shown in (b) and (d), respectively. The shadowed region is the leaky region. The vertical coordinate is broken for a detailed view.
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z > 0 is plotted for a clear view. The electric field profile (solid line) around the central cavity in the tapered structure matches closely that of a Gaussian function (thick dashed line) as shown in Fig. 5(a). From Fig. 5(c), it can be seen that the electric field profile now changes more gently around the central cavity than it does in the non-tapered structure to reduce the vertical loss. The SFT’s in Fig. 5(b) and (d) provide supporting evidence of greatly reduced loss. The wave vector components in the leaky region are greatly reduced compared to those in Fig. 3(b) and (d). The normalized integral of the spectral intensity in the leaky region are 1.3 104 and 8.9 105 in Fig. 5(b) and (d) for N = 8 and N = 14, respectively, a 20-times suppression of the integrated spectral intensity of wave vector components in the leaky region. Fig. 5(b) and (d) show that the formation of more nearly Gaussian shaped longitudinal mode envelopes as result of inserting the tapers is the cause of the suppression of the leaky components of the cavity modes. Although the Q (1.3 104) of the N = 8 microcavity is less than that of the N = 16 cavity without tapers considered in Fig. 3(c), it is now limited by longitudinal loss and hence by QL rather by QV as in the case of the latter. The reduction of the intensity of the wave vector components in the leaky region as shown in Fig. 5(b), indicates a method to obtain an ultrahigh Q via increases in QV by the use of tapers. For example, Q values as high as 7.4 105 result from incorporating linear tapers with 50, 70 and 90 nm air gaps increasing from the cavity edge towards the unperturbed mirrors. For both linearly varying air gap tapers considered, the transmission is significantly improved compared to the non-tapered structures with the same number of mirror pairs. However, transmission still drops rapidly with increasing number of mirror pairs. 4.2. Internal tapers with nonlinearly varying air gaps It is now shown that nonlinear tapers provide a flexible means of optimizing the field profile around the boundary of the central cavity to match the desired Gaussian profile. Even higher Q, around 2.4 106, is obtained in a structure with three pairs of tapered mirrors at each side, i.e. dL1–3 = 50, 70, 80 nm, where dLi is the width of the i’ th air gap numbered from the central cavity, together with 16 pairs of periodic mirrors. A 100 times improvement is obtained compared to the non-tapered structure. Transmission and Q versus mirror pairs are shown as circular symbols
in Fig. 4. The transmission also improves significantly, with the nonlinear tapers making theoretically possible a 1D PC microcavity with Q > 3 105 with 75% transmission. 4.3. External tapers Lalanne et al. designed a 1D PC cavity based on a 2D SOI geometry using the Fourier-expansion method. By considering the impedance match and radiation recycling, these researchers inserted external tapers between the normal mirrors and the feeder waveguides and obtained a single resonance with a theoretical Q as high as 3.4 104 together with 89% transmission [18]. Here an initial external design is included as a complement of the taper design. The external tapers consists of 4 Si blocks of 300 nm, 260 nm, 230 nm and 210 nm separated by three air slots with constant thickness of 90 nm. The thickness of Si block generally decreases from the feeder waveguide to the normal mirrors. Compared to a structure as in Fig. 2 at N = 8, a structure with four pairs of normal mirrors plus the external tapers at each side of the cavity gives a 11% increase of the transmission and keeps the same Q at the same wavelength. When the external tapers are applied together with the internal tapers, for example, in the case of internal nonlinear tapers consisting of 50 nm, 70 nm, 80 nm thick air slots as shown in Fig. 4, the transmission in the structure with six pairs of normal mirrors and both tapers reaches 71.5% compared to 20.5% in a structure with 10 pairs normal mirrors and only internal tapers. And the Q decreases from 1.3 106 to 3.8 105. However, the almost the same simultaneously high values of transmission and Q can be simply obtained by taking out two normal mirrors at each side of the central cavity in the structure with only internal tapers, i.e. at N = 11 in Fig. 4. Based on the simulation work presented here, tapering the period of the 1D PC over the two-to-three periods closest to the cavity is the main factor in the simultaneous improvement Q and transmission via the effect on the shape of the cavity mode profile. 5. Conclusion The FDTD method has been used to demonstrate how the structure of a SOI 1D PC microcavity can be optimized to support an ultra-high Q resonance yet retain acceptably high transmission. Using a 2D model, it is shown how tapers inserted between the central cavity section and the Bragg reflectors can be used to
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shape the envelope of the resonant mode to minimize the vertical radiation loss. Analysis of the Spatial Fourier Transform of the cavity field has revealed that the tapers have much the same effect as optimizing the positions of the holes in 2D PC SOI microcavities in reducing the intensity of leaky spectral components of the resonant mode. In particular, the advantage of nonlinearly varying the air gaps in the internal tapers results in predicted Q of greater than 2 106. Significantly, the transmission also improves without the addition of external tapers to match the Bloch modes of the Bragg mirrors to the guided modes in the access waveguides. However, the addition of mode matching external tapers does give rise to an improvement in transmission, but at the cost of some reduction in Q. Acknowledgement The authors wish to acknowledge support from the European Union under Framework 6 contract number 017481, STREP ‘‘N2T2’’. References [1] O. Painter, R.K. Lee, A. Scherer, A. Yariv, J.D. O’Brien, P.D. Dapkus, I. Kim, Science 284 (1999) 1819. [2] S. Noda, A. Chutinan, M. Imada, Nature 407 (2000) 608.
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[3] P. Michler, A. Kiraz, C. Becher, W.V. Schoenfeld, P.M. Petroff, L. Zhang, E. Hu, A. Imamog˘lu, Science 290 (2000) 2282. [4] L. Fekete, F. Kadlec, P. Kuzˇel, H. Neˇmec, Opt. Lett. 32 (2007) 680. [5] T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, H. Taniyama, Nat. Photon. 1 (2007) 49. [6] P.R. Villeneuve, S. Fan, J.D. Joannopoulos, K.-Y. Lim, G.S. Petrich, L.A. Kolodziejski, R. Reif, Appl. Phys. Lett. 67 (1995) 167. [7] J.S. Foresi, P.R. Villeneuve, J. Ferrera, E.R. Thoen, G. Steinmeyer, S. Fan, J.D. Joannopoulos, L.C. Kimerling, H.I. Smith, E.P. Ippen, Nature 390 (1997) 143. [8] T.E. Krauss, R.M. De La Rue, Appl. Phys. Lett. 68 (1996) 1613. [9] D. Peyrade, E. Silberstein, P. Lalanne, A. Talneau, Y. Chen, Appl. Phys. Lett. 81 (2002) 829. [10] A.S. Jugessur, P. Pottier, R.M. De La Rue, Electron. Lett. 39 (2003) 367. [11] P. Velha, J.C. Rodier, P. Lalanne, J.P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, N. J. Phys. 8 (2006) 204. [12] M.W. Pruessner, T.H. Stievater, W.S. Rabinovich, Opt. Lett. 32 (2007) 533. [13] T. Asano, B. Song, Y. Akahane, S. Noda, IEEE J. Sel. Topics Quantum Electron. 12 (2006) 1123. [14] E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, Appl. Phys. Lett. 88 (2006) 041112. [15] A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, vol. 13, Artech House, Boston, MA, 1998, pp. 561–612. [16] Q. Chen, Y.-D. Yang, Y.-Z. Huang, Appl. Phys. Lett. 89 (2006) 061118. [17] K. Srinivasan, O. Painter, Opt. Exp. 10 (2002) 670. [18] P. Lalanne, J.P. Hugonin, IEEE J. Quantum Electron. 39 (2003) 1430.
Photonics and Nanostructures – Fundamentals and Applications 7 (2009) 19–25 www.elsevier.com/locate/photonics
Design of tapering one-dimensional photonic crystal ultrahigh-Q microcavities Qin Chen, Duncan W.E. Allsopp * Department of Electronic and Electrical Engineering, University of Bath, Bath BA2 7AY, UK Received 18 June 2008; received in revised form 14 October 2008; accepted 19 November 2008 Available online 27 November 2008
Abstract One-dimensional (1D) photonic crystal (PC) microcavities can be readily embedded into silicon-on-insulator waveguides for photonic integration. Such structures are investigated by 2D Finite-Difference Time-Domain method to identify designs with high transmission which is essential for device integration. On-resonance transmission is found to decrease with the increasing mirror pairs, however, the quality factor (Q) increases to a saturated value. The addition to the Bragg mirrors of tapered periods optimized to produce a cavity mode with a near Gaussian shaped envelope results in a major reduction in vertical loss. Saturated Q up to 2.4 106 is feasible if the internal tapers are properly designed. The effect of increasing transmission is also demonstrated in a structure with the external tapers. # 2008 Elsevier B.V. All rights reserved. PACS : 42.55.Tv; 42.79.Ci; 42.79.Gn; 85.60.Bt Keywords: Photonic crystal; Microresonators; Filters; Finite difference methods; Q factor
1. Introduction Photonic crystals (PC) have attracted wide interest in the applications of low threshold lasers [1], high finesse filters [2], single photon devices [3], nonlinear optics [4], and slow light [5] in the last 20 years. A waveguide based one-dimensional (1D) PC is the simplest PC structure [6–12], in which the light confinement is realized by refractive index contrast in the two transverse directions and by photonic bandgap effect in the longitudinal direction. Joannopoulos and coworkers first proposed a novel air-bridge microcavity and predicted a quality factor (Q) of 104 in 1995 [6]. Two years later, they fabricated a 1D PC cavity in a
* Corresponding author. E-mail address: [email protected] (D.W.E. Allsopp). 1569-4410/$ – see front matter # 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2008.11.002
silicon-on-insulator (SOI) optical waveguide with a measured Q of 265 [7]. One main loss source in this kind of 1D PC cavity is the etched air gaps, where there is no refractive index contrast in the vertical direction. The out-of-plane scattering loss, either into the air or into the substrate, causes a serious degeneration of Q. Krauss and De La Rue suggested that semiconductor-rich lattices with small air gaps could suppress the diffractive spreading loss [8]. Lalanne and co-workers concluded that the mode mismatch is the main cause of the loss and inserted both outside and inside tapers in their PC cavities with 1D gratings [9]. Jugessur et al. also applied the tapers in the structure discussed in Joannopoulos’s paper [10]. However, all the measured Qs were still several hundreds. Recently, significant improvements have been reported. Lalanne and co-workers demonstrated a 1D PC cavity with tapered reflectors on SOI wafers and
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obtained a Q of 8.9 103 [11]. Using a Fabry-Perot model, they estimated that an intrinsic Q of 3.8 105 could be obtained in a cavity with two tapered semiinfinite mirrors. Pruessner et al. observed a resonance with a Q of 2.7 104 in a 1D PC with a long cavity on SOI wafer by using a 4-micron-deep Si etch [12]. Although almost two-order improvement has been obtained, the Qs in 1D PC cavities are still much smaller than that in 2D PC cavities, such as the measured Q of 9.5 105 reported by Noda and co-workers in their double-hetero-PC cavity in which the lattice constant was changed at the interfaces [13], and the recorded experimental Q of 1.2 106 and theoretical Q of 7 107 realized in a photonic crystal nanocavity by the local width modulation of a line defect [5,14]. In this paper, the effect on Q and transmission of different kinds of microcavity formed by Bragg mirrors comprising aperiodic 1D PCs with tapering period etched into 2D SOI structures is investigated by the twodimensional Finite-Difference Time-Domain (FDTD) method. Momentum space analysis of the localized mode reveals that the improvement results from the tapers suppressing the amplitude of the wave vector components in the leaky region of the resonant mode, thereby suppressing the vertical radiation loss. 2. Method Fig. 1 shows the basic structure used in the simulation work reported here. It comprises a block of SOI material consisting of a Si substrate, SiO2 buffer layer of 1.5 mm, the top Si guide layer of 360 nm and the air cladding layer. The refractive indices of silicon and silicon dioxide are 3.48 and 1.46, respectively. It is assumed that a central cavity layer is then formed by etching Bragg mirrors, each comprising N pairs of Si and air gap layers. In the 2D FDTD simulations, the corrugated waveguide is assumed to be illuminated from the input waveguide by the transverse magnetic
(TM, Hy = 0) fundamental mode, which is a Gaussianmodulated cosine impulse covering a wide frequency band [15]. The ‘‘bootstrapping’’ technique is used to set the exciting source [15]. The perfectly matched layer (PML) absorbing boundary is used to terminate the FDTD calculation window, with the PML thickness of 0.5 and 1 mm in the x and z directions, respectively. The spatial cell size is 10 nm, and the time step is Courant limit [15]. The transmission, reflection and loss spectra are calculated from the power flux recorded at the detector planes, which are normalized by the source value [16]. For the reflection calculation, double FDTD runs are conducted. The output in the first run without gratings is subtracted from the output in the second run with the full structure to exclude the effect of the source in the detector plane. The resonance wavelength is found by fitting a Lorentzian to the transmission peak and Q is given by the ratio of the peak wavelength to its 3-dB bandwidth. The mode field distributions from the FDTD simulation are obtained by compressing the incident impulse spectral width into a range narrow enough to ensure that only on-resonance modes can be excited. The spatial Fourier transformation spectra, which represent the plane wave components of the cavity mode, are then calculated from these field distributions. The analysis is valid for 1D PC structures in 2D cross section geometry, which is an approximation to the actual 3D structures. In the case of air slots without transverse waveguide confinement, our 2D model neglects the scattering loss in the third dimension, i.e. Q reported here is the upper limit. In practice, the width of the access waveguide must be finite, even tapered [10], to ensure single mode behaviour in the PC cavity part and low insertion loss at each end of the device. 3. 1D PC cavities with non-tapered mirrors The basis of the simulations is a 1D PC microcavity with each reflector comprising 8 pairs of Bragg mirrors. Instead of quarter wave stacks, Si-rich mirrors (dH = 200 nm, dL = 90 nm) are used to suppress scattering losses at the interface between Si blocks and air slots [8]. The grating depth, h, is set to 650 nm, which covers the mode distribution in the vertical direction and provides a strong Bragg reflection. 3.1. Number of mirror periods
Fig. 1. Schematic of a 1D PC cavity on SOI material with N = 3 pairs of untapered Si/air mirrors and 3 pairs of internal tapers at each side the central cavity.
Fig. 2 shows the transmission and Q as functions of the number, N, of mirror pairs for a structure of the
Q. Chen, D.W.E. Allsopp / Photonics and Nanostructures – Fundamentals and Applications 7 (2009) 19–25
Fig. 2. Transmission T, reflection R, loss L and Q versus mirror pairs N, where dH = 200 nm, dL = 90 nm, D = 400 nm and h = 650 nm.
generic type shown in Fig. 1. The variation of Q can be divided into three stages. Firstly, Q increases exponentially when N is less than 10. Secondly, it increases slowly from N = 10 to 16. Thirdly, it becomes saturated when N is larger than 16. There are two loss mechanisms in a 2D model of a 1D PC cavity in SOI slab waveguide of the type shown in Fig. 1. One is the longitudinal radiation loss, which depends on the degree of light confinement due to the Bragg mirrors, and the other is the vertical radiation loss caused by the mode coupling between the resonant mode in the cavity and the radiation mode in the cladding layer. With two such loss mechanisms, the Q of a 1D PC microcavity can be given by:
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Q can not be larger than QV irrespective of how many mirrors are located at each side of the central cavity. Obviously, this is much lower than those observed in 2D PC cavities [13]. The transmission T is found to decrease quickly with N, whereas the reflectivity R increases sharply as N > 8. Loss L defined as 1-T-R is also plotted in Fig. 2 (solid triangles). L initially increases with N, reaching a maximum at N = 10. Thereafter L decreases with N, finally saturating at 5% for N 14. This behaviour results from the increase in out-of-plane loss at low N with increasing number of mirror pairs due to the longer scattering region. For small N the out-of-plane loss derives from both mirrors. When the Bragg reflectors become longer at large N, the higher reflectivity of the first mirror decouples the cavity from the input light because of the reduced light tunnelling length into the whole structure. Consequently, the contribution from the second mirror to L falls with increasing N, tending to a constant level determined by the intrinsic impedance mismatch at the interface between the feeder waveguide and the first Bragg reflector. Therefore, the initial increase in loss with N and subsequent decoupling of the whole structure at large N values are the respective causes for the decrease of T with N. As shown in Fig. 2, less than 50% of incident power is transmitted through when N is larger than 8. If this cavity is used as a filter, a trade off between transmission and Q has to be considered. 3.2. Analysis by momentum space transformation
1 1 1 ¼ þ Q QL QV
(1)
where QL and QV are the values the quality factor would take if, respectively, longitudinal loss or vertical loss were the sole degradation mechanism. QL behaves in a similar way to the conventional loss mechanisms in a Fabry-Perot resonator, therefore it will increase exponentially with increasing number, N, of mirror periods. On the other hand the longitudinal confinement has only a slight effect on QV. Therefore, for small N the longitudinal loss dominates the total loss, i.e. QL QV and, from Eq. (1), Q QL. This corresponds to region I of the dependence of Q on N shown in Fig. 2. With an increase in N, the longitudinal and vertical loss become comparable, i.e. QL QV, and have almost the same effect on Q, region II of Fig. 2. With further increase in N, the vertical loss becomes dominant, i.e. QL QV, and Q saturates a value limited by QV, region III of Fig. 2. From Fig. 2, QV is found to be about 2.4 104 for SOI slab waveguides of the dimensions considered here.
1D PC cavities with periodic structures in one direction do not have a complete photonic band gap. The light confinement occurs via total internal reflection in the two other directions. The localized mode in the cavity can be seen as a combination of numerous plane wave components with wave vectors k, which may couple with the radiative modes in the cladding layer. The mode field in such a structure can be written as: Fðx; zÞ ¼ f ðx; zÞeikx x eikz z 2 2p kx2 þ kz2 ¼ k2 ¼ l
(2) (3)
where kx and kz are the vertical and tangential components of wave vector k, and l is the resonant wavelength. If kz lies within the range 0–2p/l, kx is a real number, which means the mode is not confined in the vertical direction. It is the plane waves with kz components in the range of 0–2p/l that give rise to the vertical loss that limits the Q of the resonant mode. The range 0 < kz < 2p/l is called the leaky region. The Spatial
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Fig. 3. Electric field (solid line) at the boundary between the top Si layer and the oxide buffer layer in a 1D PC cavity for (a) N = 4 and (c) N = 16, other parameters are the same as those in Fig. 2. The electric field in (a) is re-plotted in (c) as open circles for comparison. ‘B’ and ‘C’ separated by the dot line indicate Bragg mirrors and the central cavity. The modulus of SFT of electric field in (a) and (c) are shown in (b) and (d), respectively. The shadowed region is the leaky region. The vertical coordinate is broken for a detailed view.
Fourier Transform (SFT) of the longitudinal field distribution provides the spectral distribution of its plane wave components and enables analysis of the vertical loss [13,17]. Fig. 3 shows the electric field distribution of the resonant mode in the longitudinal direction in 1D PC microcavities for N = 4 and N = 16 and their SFT’s. The location of the central cavity relative to the mode field profiles is indicated by dotted lines. As Figs. 3(a) and (c) reveal, the longitudinal field of the cavity mode is antisymmetric with respect to a vertical symmetry plane through the centre of the cavity. Values of wave vector corresponding to leaky modes (i.e. kz = 0–2p/l) are indicated by the shaded regions in Fig. 3(b) and (d). The value of the SFT at each kz indicates the relative amplitude of the plane wave with longitudinal wave vector with that value of kz. Plane waves in the leaky region will couple with the radiation modes in the cladding layer and introduce the vertical loss. For example, the Q of the microcavity formed by N = 16 mirrors, at 2.4 104, is 100 times higher than that with N = 4 mirrors, however the corresponding values of the normalized integral of the spectral intensity in the leaky regions of the SFTs are
1.5 103 and 3.5 103, respectively (Fig. 3(b) and (d)). However, the difference in the vertical loss is only of 2. This is reasonable because the electric field profile around the central cavity in the longitudinal direction shows no obvious change with the increase in the number of mirror periods, as can be seen in Fig. 3(c). The slightly higher integrated spectral intensity in the leaky region for the N = 4 structure, compared to that of the N = 16 device, derives from the differences in the electric field distribution at the outer edge of the reflectors. It is obvious that in the former the electric field is still strong where the reflector interfaces with the surrounding Si feeder slab waveguides. The abrupt change in the electric field distribution at the boundary of the whole structure introduces an additional vertical radiation loss. 4. Cavities with mirrors of tapering period Based on the behaviour of 2D PC microcavity technology, the theoretically optimum shape for the field envelope of the resonant mode is a sinc function as its SFT is rectangular and can be engineered not to overlap with the leaky range of kz components. A
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envelope for the cavity mode in 1D PC microcavities as optimal holes positioning have in 2D structures. 4.1. Internal tapers with linearly varying air gaps
Fig. 4. Transmission and Q versus the number of mirror pairs in a 1D PC microcavity with linearly (square symbols, four air slots close to the central Si cavity at each side are 50, 60, 70, 80 nm wide) and nonlinearly (circular symbols, three air slots close to the central Si cavity at each side are 50, 70, 80 nm wide) tapered mirrors. Other parameters are the same as for Fig. 2.
Gaussian shaped envelope has been shown to be a practical alternative, requiring less rigorous optimization [13]. It shown here that mirror pairs with tapering variations in their periodicity fulfill the same role in increasing Q by creating a nearly Gaussian shaped
The relative merits of linearly varying the air gap widths as against a nonlinear variation are now considered. First, tapers are added to the Bragg reflectors by reshaping the four air slots in each mirror closest to the central cavity layer so that their widths increase from 50 to 80 nm, in steps of 10 nm, moving away from the cavity while the Si section lengths remain constant. Fig. 4 shows the variations in Q (open squares) and transmission (solid squares) with the total number of mirror periods, N. The variations in Q and transmission with N follows the trends shown in Fig. 2, except that the saturated value of Q, i.e. QV, has increased significantly from 2.4 104 in Fig. 2 to 5.0 105 with the introduction of tapers. The electric field variation in the longitudinal direction and its SFT are shown in Fig. 5 for two tapered 1D PC cavities, N = 8 (Fig. 5(b)) and N =14 (Fig. 5(d)). Only the field distribution in a half region
Fig. 5. The analogous figures to Fig. 3 for a 1D PC cavity with linear tapers in Fig. 4 at (a) N = 8 and (c) N = 14. The Gaussian function fit is plotted in (a) as a thick dashed line. The electric field profile in Fig. 3(c) is re-plotted in (c) as a thick dashed line for comparison. The regions labelled ‘B’, ‘T’ and ‘C’ and delineated by the dotted lines indicate the positions of the Bragg mirrors, tapers and the central cavity relative to the mode field (only distribution at the half region z > 0 are plotted). The modulus of SFT of electric field (solid lines) in (a) and (c) are shown in (b) and (d), respectively. The shadowed region is the leaky region. The vertical coordinate is broken for a detailed view.
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z > 0 is plotted for a clear view. The electric field profile (solid line) around the central cavity in the tapered structure matches closely that of a Gaussian function (thick dashed line) as shown in Fig. 5(a). From Fig. 5(c), it can be seen that the electric field profile now changes more gently around the central cavity than it does in the non-tapered structure to reduce the vertical loss. The SFT’s in Fig. 5(b) and (d) provide supporting evidence of greatly reduced loss. The wave vector components in the leaky region are greatly reduced compared to those in Fig. 3(b) and (d). The normalized integral of the spectral intensity in the leaky region are 1.3 104 and 8.9 105 in Fig. 5(b) and (d) for N = 8 and N = 14, respectively, a 20-times suppression of the integrated spectral intensity of wave vector components in the leaky region. Fig. 5(b) and (d) show that the formation of more nearly Gaussian shaped longitudinal mode envelopes as result of inserting the tapers is the cause of the suppression of the leaky components of the cavity modes. Although the Q (1.3 104) of the N = 8 microcavity is less than that of the N = 16 cavity without tapers considered in Fig. 3(c), it is now limited by longitudinal loss and hence by QL rather by QV as in the case of the latter. The reduction of the intensity of the wave vector components in the leaky region as shown in Fig. 5(b), indicates a method to obtain an ultrahigh Q via increases in QV by the use of tapers. For example, Q values as high as 7.4 105 result from incorporating linear tapers with 50, 70 and 90 nm air gaps increasing from the cavity edge towards the unperturbed mirrors. For both linearly varying air gap tapers considered, the transmission is significantly improved compared to the non-tapered structures with the same number of mirror pairs. However, transmission still drops rapidly with increasing number of mirror pairs. 4.2. Internal tapers with nonlinearly varying air gaps It is now shown that nonlinear tapers provide a flexible means of optimizing the field profile around the boundary of the central cavity to match the desired Gaussian profile. Even higher Q, around 2.4 106, is obtained in a structure with three pairs of tapered mirrors at each side, i.e. dL1–3 = 50, 70, 80 nm, where dLi is the width of the i’ th air gap numbered from the central cavity, together with 16 pairs of periodic mirrors. A 100 times improvement is obtained compared to the non-tapered structure. Transmission and Q versus mirror pairs are shown as circular symbols
in Fig. 4. The transmission also improves significantly, with the nonlinear tapers making theoretically possible a 1D PC microcavity with Q > 3 105 with 75% transmission. 4.3. External tapers Lalanne et al. designed a 1D PC cavity based on a 2D SOI geometry using the Fourier-expansion method. By considering the impedance match and radiation recycling, these researchers inserted external tapers between the normal mirrors and the feeder waveguides and obtained a single resonance with a theoretical Q as high as 3.4 104 together with 89% transmission [18]. Here an initial external design is included as a complement of the taper design. The external tapers consists of 4 Si blocks of 300 nm, 260 nm, 230 nm and 210 nm separated by three air slots with constant thickness of 90 nm. The thickness of Si block generally decreases from the feeder waveguide to the normal mirrors. Compared to a structure as in Fig. 2 at N = 8, a structure with four pairs of normal mirrors plus the external tapers at each side of the cavity gives a 11% increase of the transmission and keeps the same Q at the same wavelength. When the external tapers are applied together with the internal tapers, for example, in the case of internal nonlinear tapers consisting of 50 nm, 70 nm, 80 nm thick air slots as shown in Fig. 4, the transmission in the structure with six pairs of normal mirrors and both tapers reaches 71.5% compared to 20.5% in a structure with 10 pairs normal mirrors and only internal tapers. And the Q decreases from 1.3 106 to 3.8 105. However, the almost the same simultaneously high values of transmission and Q can be simply obtained by taking out two normal mirrors at each side of the central cavity in the structure with only internal tapers, i.e. at N = 11 in Fig. 4. Based on the simulation work presented here, tapering the period of the 1D PC over the two-to-three periods closest to the cavity is the main factor in the simultaneous improvement Q and transmission via the effect on the shape of the cavity mode profile. 5. Conclusion The FDTD method has been used to demonstrate how the structure of a SOI 1D PC microcavity can be optimized to support an ultra-high Q resonance yet retain acceptably high transmission. Using a 2D model, it is shown how tapers inserted between the central cavity section and the Bragg reflectors can be used to
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shape the envelope of the resonant mode to minimize the vertical radiation loss. Analysis of the Spatial Fourier Transform of the cavity field has revealed that the tapers have much the same effect as optimizing the positions of the holes in 2D PC SOI microcavities in reducing the intensity of leaky spectral components of the resonant mode. In particular, the advantage of nonlinearly varying the air gaps in the internal tapers results in predicted Q of greater than 2 106. Significantly, the transmission also improves without the addition of external tapers to match the Bloch modes of the Bragg mirrors to the guided modes in the access waveguides. However, the addition of mode matching external tapers does give rise to an improvement in transmission, but at the cost of some reduction in Q. Acknowledgement The authors wish to acknowledge support from the European Union under Framework 6 contract number 017481, STREP ‘‘N2T2’’. References [1] O. Painter, R.K. Lee, A. Scherer, A. Yariv, J.D. O’Brien, P.D. Dapkus, I. Kim, Science 284 (1999) 1819. [2] S. Noda, A. Chutinan, M. Imada, Nature 407 (2000) 608.
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[3] P. Michler, A. Kiraz, C. Becher, W.V. Schoenfeld, P.M. Petroff, L. Zhang, E. Hu, A. Imamog˘lu, Science 290 (2000) 2282. [4] L. Fekete, F. Kadlec, P. Kuzˇel, H. Neˇmec, Opt. Lett. 32 (2007) 680. [5] T. Tanabe, M. Notomi, E. Kuramochi, A. Shinya, H. Taniyama, Nat. Photon. 1 (2007) 49. [6] P.R. Villeneuve, S. Fan, J.D. Joannopoulos, K.-Y. Lim, G.S. Petrich, L.A. Kolodziejski, R. Reif, Appl. Phys. Lett. 67 (1995) 167. [7] J.S. Foresi, P.R. Villeneuve, J. Ferrera, E.R. Thoen, G. Steinmeyer, S. Fan, J.D. Joannopoulos, L.C. Kimerling, H.I. Smith, E.P. Ippen, Nature 390 (1997) 143. [8] T.E. Krauss, R.M. De La Rue, Appl. Phys. Lett. 68 (1996) 1613. [9] D. Peyrade, E. Silberstein, P. Lalanne, A. Talneau, Y. Chen, Appl. Phys. Lett. 81 (2002) 829. [10] A.S. Jugessur, P. Pottier, R.M. De La Rue, Electron. Lett. 39 (2003) 367. [11] P. Velha, J.C. Rodier, P. Lalanne, J.P. Hugonin, D. Peyrade, E. Picard, T. Charvolin, E. Hadji, N. J. Phys. 8 (2006) 204. [12] M.W. Pruessner, T.H. Stievater, W.S. Rabinovich, Opt. Lett. 32 (2007) 533. [13] T. Asano, B. Song, Y. Akahane, S. Noda, IEEE J. Sel. Topics Quantum Electron. 12 (2006) 1123. [14] E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, Appl. Phys. Lett. 88 (2006) 041112. [15] A. Taflove, Advances in Computational Electrodynamics: The Finite-Difference Time-Domain Method, vol. 13, Artech House, Boston, MA, 1998, pp. 561–612. [16] Q. Chen, Y.-D. Yang, Y.-Z. Huang, Appl. Phys. Lett. 89 (2006) 061118. [17] K. Srinivasan, O. Painter, Opt. Exp. 10 (2002) 670. [18] P. Lalanne, J.P. Hugonin, IEEE J. Quantum Electron. 39 (2003) 1430.