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A dynamically reconfigurable nested photonic crystal cavity suitable for frequency conversion applications is proposed. Dynamic switching between two distinct cavities allows intermodal transition via spatially-uniform tuning of the refractive index. Exclusion of the initial resonant mode from the Eigen modes of the tuned cavity precludes the adiabatic wavelength conversion process. Multiple intermodal transitions are suppressed by the symmetry of the mode profiles of the two cavities. Over 90nm wavelength shift (from L-band to the S-band) is shown numerically.
©2010 Optical Society of America
1. Introduction
Converting the wavelength of optical pulses is an important functionality for the wavelength division multiplexing scheme used in optical communication and signal processing applications [1]. In silicon platform, wavelength conversion is conventionally realized using a nonlinear optical process with relatively high input power, e.g. four-wave mixing in centimeter-scale silicon wires [2, 3]. While resonant enhancement of the nonlinear effect considerably reduces the required on-chip footprint [4], the alternative approaches based on dynamic tuning of a wavelength-scale optical resonator have the potential to deliver ultra-compact devices with low-power operation [5].
Wavelength conversion by adiabatic tuning of the resonator within the photon life-time, whereby trapped photons follow the instantaneous resonance frequency of the cavity, has been proposed [6–8] and demonstrated [9]. However, because of the mutual orthogonality of the Eigen states in a single cavity, stored energy remains in its original resonant mode throughout the process. Hence the refractive index change (Δn) results in a wavelength shift (Δλ) which is proportionally less than that of the index change (i.e. Δλ/λ<Δn/n). This is because of the scaling properties of the Maxwell equations and the presence of the field outside the index modulated region. The commonly adopted free-carrier plasma dispersion effect [10] in a silicon platform yields about 10nm wavelength shift around the telecom wavelength [11].
Dynamically induced coupling between distinct modes of a resonator may extend the achievable frequency shift beyond that possible due to the simple scaling of the refractive index. Coupling is possible by tailoring both the temporal and spatial profiles of the perturbation, based on the spectral separation and the field profile of the modes respectively [12, 13]. Within the optical C-band a frequency comb generator coupling 15 resonances of a ring resonator over an 8nm range has been demonstrated [14]. Over 200nm wavelength shift using a heterostructure photonic crystal (PhC) cavity was proposed [15]. However, the required spatially non-uniform index modulation profile and the inevitable presence of the slightly frequency-shifted adiabatic transition of the initially excited modes are generally not desired in a typical wavelength conversion application. The latter issue is more pronounced considering the much higher conversion efficiency of the adiabatic process compared to the intermodal transition in a single resonator. The relative strength of the transitions from an initial mode to the available Eigen modes is approximately proportional to the overlap of the modes weighted by the perturbation profile (assuming a small number of modes with a narrow free spectral range). Simple modulation profiles, e.g. spatially uniform index change at an offset relative to the cavity center, yield a higher overlap for the adiabatic conversion than the intermodal transition. The overlap in the former case is between the initial and the perturbed profiles of an identical mode whereas the latter involves distinct resonances with orthogonal field profiles.
As an alternative approach we propose intermodal transition in a dynamically reconfigurable nested PhC cavity realized along a line defect waveguide. The resonator consists of a statically formed positive-defect heterostructure cavity embedded in an outer, partially defined negative-defect cavity. Tuning the refractive index with a uniform spatial profile, transforms the resonator from a positive-defect to a dynamically formed negative-defect cavity. Each heterostructure cavity has a distinct set of resonances originating from different regions on the dispersion bands of the waveguide. In this scheme, elimination of the initially excited mode from the available Eigen modes of the cavity formed after the index tuning, suppresses the adiabatic transition; whereas possible overlap between the modes of the initial and tuned cavities allows mode-mixing with a uniform spatial perturbation. A large wavelength shift with only a limited number of interacting resonances is possible by tailoring the structure and using the symmetry of the mode profiles. Over 90nm of conversion range around the telecom wavelength (i.e. 6.19%) is achieved with 0.7% tuning of the refractive index.
2. Nested cavity design
2.1 Structure
We consider the two generic configurations of a heterostructure PhC cavity [16], namely the positive-defect and negative-defect type cavities [17], and design a nested resonator which allows dynamic reconfiguration from one type to the other without a significant loss of the stored energy (see section 3).
An ordinary heterostructure cavity is formed by embedding a core section with locally modified band structure within an otherwise uniform line defect waveguide which acts as the reflectors on either side of the cavity. Nano-cavity modes [18] are pinned to the portion of dispersion bands of the core that are shifted within the mode-gap of the corresponding bands in the surrounding reflectors and hence confined within the core. In a triangular lattice PhC, for a positive-defect cavity with a red-shifted core, mode-gap confined resonances are only possible near the band-edge portion of the upward bending bands of the core (P1 to P3 in Fig. 1 ), these being in the mode-gap of the corresponding reflector bands. In contrast, the blue-shifted bands in the core of a negative-defect cavity support locally confined modes only near the downward bending stationary points on the dispersion bands of the core (N1 and N2 in Fig. 1) placed in the mode gap of the reflector bands.
Fig. 1 Dispersion bands of TE-like modes of the single line-defect waveguide (red) standard and (blue) blue-shifted bands for the waveguide with enlarged holes close to the defect. Positive (negative) defect heterostructure cavity with red (blue)-shifted bands at the center supports mode-gap-confined resonances close to the up (down)-concave stationary points of the dispersion relation.
The proposed nested cavity, assuming a triangular lattice of air-holes in silicon, and the band structure along the structure are schematically shown in Fig. 2 . The positive-defect inner cavity is statically defined by enlarging the holes close to the line defect from the bulk radius at the center to form a pair of reflectors with blue-shifted dispersion bands on either side of the core. After a sufficient number of reflector layers to maintain the in-plane confinement and hence the high Q of the inner cavity, the enlarged holes are down-tapered back to the bulk radius which partially defines the outer cavity. The added standard waveguide sections act as the reflectors of the cavity following the index tuning. As in the case of the inner cavity, the second mirror is subsequently terminated by another reflector. The third mirror at the tailing edges of the cavity has a higher band edge than that of the first reflector and is formed by up-tapering the radius from the bulk size to a larger radius than that of the first reflector.
Fig. 2 Schematic of the nested cavity and band structure; open circles represent the bulk PhC radius. Red, blue and violet colored sections have increasingly higher band edge frequency. The slab index in the solid box is dynamically tuned. Stored energy is calculated within the dotted box. The band-edge and the down-bending stationary point frequency of the first even mode (see Fig. 1) along the line-defect before (after) the tuning are schematically shown along the upper (lower) half of the cavity.
In this scheme lowering the refractive index at the center of the structure to eliminate the slight band edge mismatch that forms the inner cavity, reconfigures the structure to effectively that of a dynamically formed negative-type cavity. The latter has a core section consisting of the former inner cavity and its mirror regions with blue-shifted dispersion bands compared to the second pair of statically defined reflectors. Based on our simulation, the third mirror has negligible effect on the resonance wavelength and field profiles of the cavity before and after the tuning. It is devised to minimize the possible in-plane energy leakage from the inner resonator while reconfiguring the nested cavity by removing the mode-gap barrier that forms the negative-defect cavity.
For a smooth modulation of the field profile a linear tapering of the radius over two lattice periods to define the reflectors of the inner cavity is used (tapering over four lattice periods is correspondingly selected for the outer cavity and the third mirror). This choice along with a minimum number of reflector layers for the inner cavity, by minimizing the effective size of the tuned resonator, increases its free spectral range and hence limits the number of intermodal transitions. It also partially offsets the lower Q of the negative-defect cavity because of the proximity of the resonant mode to the light cone of air-cladding leaky modes in the slab platform.
2.2 Mode selection
Switching from a positive-defect to a negative-defect cavity eliminates the unwanted adiabatic transition due to the absence of the initial resonant modes in the available Eigen states after tuning. While a sufficient change of the refractive index to reverse the relative displacement of the bands between the core and the reflectors of an ordinary heterostructure cavity may also give the same end result, the nested cavity requires half the index change. More importantly because of the continuous presence of the confined resonance states throughout the transition in a nested cavity, it is possible to convert the initially stored energy to the available photonic states throughout the tuning process.
The strength of the intermodal transition is determined by the coupling matrix elements proportional to the overlap of the field profile of the two modes weighted by the index perturbation. Additionally the temporal variation of the tuning profile must have a nonzero Fourier component at the transition frequency (i.e. the frequency difference of the coupled resonances) for a non-vanishing time-averaged power transfer. Theoretically, a perfect temporal/spatial phase matching (resonant transition) allows the complete energy transfer between the modes in the absence of radiation and absorption (specially the free-carrier absorption) losses. For instance complete indirect interband transition in the PhC and ring resonator structures has been shown numerically [12, 13].
Coupling between the non-orthogonal modes of the distinct cavities in the nested structure is possible even with a spatially uniform perturbation profile. For a uniform tuning, the possible intermodal transitions are readily determined from the symmetry of the Eigen states which are the same as the underlying line defect modes close to the corresponding stationary point of the dispersion bands as depicted in Fig. 3 . From all the available states of the unperturbed positive-defect cavity (labeled as P1 to P3 in Fig. 1) transition to the N1 mode of the dynamically formed negative-defect cavity is effectively zero due to the orthogonality of the distinct waveguide modes with the same wave vector. Considering the N2 mode, coupling from P1 is suppressed due to the opposite symmetry of the field profiles about the ΓK axis. Likewise the opposite parity along the ΓM direction eliminates the coupling from the P3 mode of the positive defect cavity. The latter is more evident from the even symmetry of the N2 profile about the ΓM axis as shown in Fig. 4 (b) . Finally the P2 and N2 modes with the same symmetry about both axes have a nonzero overlap (see Fig. 4) and are selected for the frequency conversion application with uniform tuning as the initial and final resonance states respectively.
Fig. 3 (a)-(e) Magnetic-field profile in the symmetry plane of the slab membrane at the stationary point of the dispersion bands indicated as P1, P2, P3, N1 and N2 in Fig. 1 respectively. (f) Schematic of the waveguide structure.
Fig. 4 Magnetic field profile of the resonant modes of the nested cavity shown in Fig. 1; (a) the initial P2 mode and (b) the N2 mode after the index tuning.
3. Numerical results
For a concrete example consider the nested cavity of Fig. 2, with the normalized radii of the bulk PhC and the first, second and third reflectors as 0.31, 0.34, 0.31 and 0.36 respectively. Two, four and four periods of linear tapering of the hole radius proceeds the first, second and third reflectors respectively. The first mirror is truncated after five periods which ensure a total Q in the order of 106, limited by the in-plane confinement, for the fundamental mode of the unperturbed cavity [19].
In this section we employ a 2D analysis to numerically investigate the wavelength conversion by tuning the proposed nested cavity. It should be noted that capturing the impact of the limited out-of-plane confinement in the slab PhC platform would require 3D analysis. This is particularly important for quantitative assessment of the implication of the close proximity of the N2 resonance to the light-line. Nevertheless, using a modified effective index to match the actual band edges of the slab line defect modes, which are the origin of the cavity modes, yields the resonant wavelengths with reasonable accuracy [20, 21]. For a 250nm silicon membrane (refractive index of 3.4), lattice constant of a=460nm and TE-like polarization the effective index is chosen as 2.78.
This is justified by noting both the already demonstrated high Q (in the order of 106 experimentally [22]) of the fundamental (P1) mode of the single cavities with similar structure [19, 22] and a relatively high expected Q (an order of magnitude lower than the P1 resonance) for the P2 mode based on 3D FDTD simulation [17]. The feasibility of the adiabatic and interband transitions in the slab platform despite the limited out-of-plane confinement has been already shown [7, 13]. Furthermore the necessarily ultra-fast tuning for the transition within the expected wavelength shift range significantly relaxes the required photon life time of the cavity. Both adiabatic and intermodal transitions have been experimentally demonstrated based in ring resonators with Q’s in the range of 104 [9, 14]. In particular for the considered cavity, the tuning interval is not longer than 100fs which is less than 20 optical cycles of the initial resonance.
We first analyzed the selected modes of the resonator before (P2) and after the tuning (N2) process to verify dynamic reconfiguration of the cavity by changing the slab index within the region indicated in Fig. 2. As shown in Fig. 4, initially the resonance at λ=1566nm, confined in the center of the cavity, has the same symmetry as the P2 mode of a positive-defect heterostructure cavity. Its resonance wavelength is also close to the band-edge of the second order mode of the standard line defect forming the core of the inner cavity. The presence of the third reflector further enhances the in-plane confinement with a Q in the order of 108 for the P2 mode extracted from the 2D model.
Lowering the index by 0.7% to align the band-edge of the second order line defect mode in the middle of the first tapered region with the corresponding band of the first reflector effectively reconfigures the structure to a negative-defect cavity bounded by the second set of mirrors. This is evident from the relatively slower decay of the field over the entire inner cavity region and the much stronger field intensity beyond the first mirror (see Fig. 4). The extracted resonance with N2-like symmetry is at λ=1469nm. The resonance is close to the down-concave stationary point on the second order mode of the line defect with enlarged holes, which now forms the core of the heterostructure resonator. This further supports dynamic reconfiguration of the cavity. The third mirror has negligible effect on the negative defect resonances despite being blue-shifted (i.e. acting as a loss channel) compared to the core of the tuned cavity. This is because having a sufficient number of reflector periods, consisting of the second taper and mirror sections and partially the third taper region, effectively buffers the resonance modes from the third mirror. The 2D model yields a Q in the order of 107 for the N2 resonance.
Tuning the refractive index of silicon within the same order, based on the plasma dispersion of the free carriers induced by strong linear absorption of silicon in the visible spectrum, has been experimentally demonstrated using ultra-short pump pulses. Dynamic control of the cavity Q by an index change in the range of 10−3-10−2 along a PhC waveguide was demonstrated in [23]. Index changes of 0.6% and higher were estimated based on the free carrier concentration in a wavelength conversion application in the same platform [24].
We next investigated the wavelength conversion process. To this end after numerically exciting the initial structure with the previously calculated P2 mode, the slab refractive index of the inner cavity (see Fig. 2) is linearly lowered by 0.7%. The tuning is assumed in a 100fs time slot which is in the same order as the inverse of the frequency difference between the considered resonances to ensure power transfer between the modes. The temporal evolution of the field is recorded over the subsequent 218 finite difference time domain (FDTD) time steps. The spectrum obtained by Fourier transforming this time signal, as depicted in Fig. 5 , clearly shows a peak at the previously calculated N2 resonance of the switched cavity also with the same field profile. As a reference the same spectrum obtained with no index tuning is also shown in the figure (with a peak at the initial resonance), indicating a dynamic conversion range of 6.19% or 97nm around the telecom wavelength band for the chosen lattice constant.
Fig. 5 Spectra of the field inside the nested cavity (solid) with (dashed) without tuning obtained by 2D FDTD. Tuning is performed in 100fs by 0.7% lowering of refractive index in the solid box of Fig. 2.
In order to further investigate the confinement of the initially stored energy within the 2D model, we also calculated the temporal variation of the field energy inside the cavity throughout the tuning process. As depicted in Fig. 6 , following the index tuning, a 4.6% increase in the stored energy is observed. The energy conservation in the photon level, or alternatively, the invariance of the energy-frequency product requires an increase of 6.6% (i.e. 1566nm/1469nm≈1.066). The deviation explains the initial decrease of the energy in Fig. 6, possibly due to an in-plane leakage while recapturing the photons from the initial cavity into the dynamically formed cavity. The blue-shift eventually outweighs the loss and the energy increases following the tuning. The initial and final temporal oscillation periods of the field energy, as clarified in the insets, corresponds to the P2 and N2 resonance wavelengths of the initial and tuned cavities, which is also in agreement with the spectrum given in Fig. 5.
Fig. 6 Temporal variation of the field energy inside the cavity (solid) with and (dotted) without index modulation. Cavity is numerically excited with P2 mode at t = 0, energy density integrated within the dotted box in Fig. 2. Insets show the magnified views of the energy plot before and after the tuning. Time is multiplied by the light speed in vacuum.
4. Conclusion
Wavelength conversion beyond the relative index change by dynamic tuning of a proposed nested PhC cavity was numerically demonstrated within the limit of the 2D analysis. Dynamic reconfiguration of the cavity allowed intermodal transition of the initially stored energy with suppressed adiabatic conversion through a spatially uniform change of the refractive index. Other applications of the nested resonator in dynamic changing of the Q of the cavity and reconfigurable optical frequency comb generation are under investigation.
Acknowledgements
This work was in part supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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