We describe a photonic crystal nanocavity with multiple spatially overlapping resonances that can serve as a platform for nonlinear frequency conversion. We show nonlinear characterization of structures with two resonances nearly degenerate in frequency. We also demonstrate structures with resonances separated by up to 523 nm.
© 2011 OSA
Photonic band gap nanocavities confine light into subwavelength volumes with high quality factor, facilitating ultracompact, low-power optoelectronic devices such as lasers [1, 2], as well as studies of fundamental physics such as cavity quantum electrodynamics [3,4]. State of the art photonic crystal nanocavity designs [5–8] can have quality factors exceeding one million for a single cavity resonance. For nonlinear optical interactions such as frequency conversion or stimulated Raman scattering, however, it is desirable to have nanostructures that support multiple resonances having arbitrary frequency separation and good spatial field overlap which can be coupled through the nonlinearity of the cavity material. In high quality factor microcavities, the phase matching condition is satisfied by the spatial overlap of the field patterns for each frequency [9–11], allowing high conversion efficiency in a small volume, on-chip device.
In a nanocavity with a single photonic band gap, it is difficult to independently control the frequencies of each mode. Additionally, the field patterns of different modes typically have minimal spatial overlap, and the absolute frequency separation between resonances is limited by the band gap size. Band gaps for different polarizations (e.g. transverse magnetic (TM) and transverse electric (TE)) [12–15] can generate additional resonant modes; however, it is difficult to independently tune their frequencies, and TM resonances require relatively thick membranes that are more difficult to fabricate.
To enable nonlinear coupling of cavity resonances, structures can be fabricated in a III–V semiconductor such as GaP or GaAs (for second or third order optical nonlinearity) or a group IV semiconductor such as silicon (for third order nonlinearities, such as Raman scattering or four wave mixing). Previous experimental work in three and four-wave mixing in semiconductor photonic crystal nanocavities has been implemented using designs featuring a single photonic band gap [16–20]. Recently , we proposed a crossed beam photonic crystal cavity suitable for nonlinear frequency conversion that allows at least two individually tunable resonances with a frequency separation larger than the size of the photonic bandgap in a single nanobeam. Here, we extend that work by describing the design as well as experimental linear and nonlinear characterization of structures with resonances that are nearly degenerate with orthogonal polarization, as well as the design and linear characterization of structures with resonances separated by more than 500 nm.
2. Cavity design
An illustration of the design in shown in Fig. 1(a). The basic element of our design is the nanobeam [6, 7, 22–25], a 1D periodic photonic crystal waveguide clad in the other two directions by air, shown in Fig. 1(b). A single unit cell of a single nanobeam, comprising a rectangular airhole with dimensions hx and hy in a high index dielectric slab of width w, is shown inside the red box (thickness t of the air-bridged membrane is out of the plane). The structure has periodicity a in the x̃ direction; this periodicity produces a 1D TE photonic band gap with size determined by the difference in frequency between the first two bands (dielectric band and air band) at the wavevector kx = π/a. We first investigate the periodicities required in a nanobeam structure to achieve different wavelength resonances using 3D simulations performed using plane wave expansion with supercell approach (MIT Photonic Bands (MPB)) . Figure 1(c) plots the wavelengths of the dielectric and air bands obtained from simulations as a function of lattice constant a. Solid lines show wavelengths for fixed absolute thickness of the membrane and are calculated using the refractive index n of GaAs at each wavelength; dotted lines are calculated by scaling the result for wavelength 1550 nm (i.e. fixed t/a, fixed n). Because a real structure has a fixed membrane thickness, decreasing the periodicity in one beam relative to the other leads to larger relative thickness t/a, redshifting the wavelengths of the bands; therefore achieving resonances with relative frequency f 2/f 1 requires superlinear scaling of the feature size, e.g. a 2/a 1 > f 2 /f 1, as shown by the divergence between solid and dotted lines. (There is a second less important contribution because refractive index increases for higher frequencies, from 3.37 to 3.53 for range plotted in Fig. 1(c)). Figure 1(d) shows the frequencies of dielectric band and air band, as well the relative size of the band gap, as the width of the beam is changed.
The parameters for the crossed beam are shown in Figs. 2(a) and 2(b). A cavity is formed in each beam by introducing a central region with no holes (cavity length l) and tapering the lattice constant and hole size near the cavity region [7,24]. In each beam, confinement along the periodic direction is provided by distributed Bragg reflection; confinement out-of-plane is provided by total internal reflection. Confinement in the in-plane direction orthogonal to the beam axis is provided by total internal reflection, and in the case of beams with overlapping photonic band gaps, also by distributed Bragg reflection . This structure allows nearly independent tuning of each resonant frequency by tuning the parameters (e.g. width, lattice constant, cavity length) of each beam and has natural channels for coupling through each beam to an access waveguide at each wavelength . A cavity with orthogonally polarized resonances degenerate in frequency can be formed by using the same parameters for each beam. This could be used for applications such as coupling to spin states of embedded quantum emitters  or for building polarizaton entangled photon sources based on a bi-excitonic cascade from a single quantum dot [30, 31]. Figures 2(c) and 2(d) show the field patterns from 3D finite difference time domain (FDTD) simulations for a doubly degenerate structure. The cavities also have several additional higher order modes the number of which is determined primarily by cavity length (two per beam for the structures shown in Fig. 2) with additional field pattern nodes.
3. Linear and nonlinear spectroscopy of cavities with nearly degenerate resonant frequencies
We now present linear nonlinear characterization of cavities with nearly degenerate resonant frequencies. A scanning electron microscope (SEM) image of structure fabricated in GaAs is shown in Fig. 3(a). The structures are defined by e-beam lithography and dry etching, as well as wet etching of a sacrificial AlGaAs layer underneath the 164 nm thick GaAs membrane. To characterize the resonant frequencies of the structures, we measure reflectivity, using vertical incidence through an objective lens and free space coupling, in the cross-polarized configuration using a tungsten halogen lamp as a broadband light source . The principle of the measurement is illustrated in Fig. 3(b). The input and measurement polarizations are orthogonal; the cavity polarization is oriented 45 degrees to both output and input. The cavity acts as a frequency-selective polarization rotator, filtering light at the frequencies of the cavity resonances, resulting in a high-signal to noise measurement of the cavity reflectivity from which the quality factor can be extracted. The result of this measurement for the structure in Fig. 3(a) is shown in Fig. 3(c). There are two resonances visible at 1571.2 nm (Q=4700) and 1573.9 nm (Q=7200). The difference between the frequencies of the resonances is due to fabrication imperfections; improving fabrication or employing local tuning [31,33,34] could assist in yielding perfectly degenerate resonances. Quality factors are extracted by fitting the measured data to the sum of two Fano lineshapes , plus linear background (to account for the measured background from the white light source).
This cross-polarized configuration, however, cannot easily distinguish between cavities with vertical and horizontal polarizations, and also cannot resolve resonances with frequencies separated by less than a linewidth. To more sensitively measure cavity resonances, we use intra-cavity second harmonic generation [16, 19]. This allows us to absolutely determine the far-field polarization of the cavity, since second harmonic generation is maximized when incident light is aligned to the polarization of the cavity. Additionally, because the measured lineshape from the second harmonic is proportional to the square of the lineshape from the cavity, we can more easily distinguish cavities separated by small differences in frequency. A second harmonic measurement of a different structure with nearly degenerate resonances in shown in Fig. 4. Figure 4(a) shows the second harmonic counts measured for three different incident laser polarizations as the laser wavelength is varied. The angle of alignment between input polarization and the polarization of each cavity mode determines the fraction of power coupled into the cavity, and accordingly affects the magnitude of second harmonic generated. The intensity of second harmonic generation for each mode also depends on the free space coupling efficiency, quality factor of the mode, spatial overlap of fundamental and second harmonic fields, and detuning of laser wavelength from the cavity resonance . Two modes with orthogonal polarization (with resonant wavelengths 1541.75 nm and 1542.1 nm), as expected, are clearly visible. To further illustrate this, we scan additional incident polarizations and wavelengths, as shown in Figs. 4(b) and 4(c). Figure 4(c) shows 2D slices at three incident laser wavelengths. The red lines show fits for two resonances with polarization separated by exactly 90 degrees, in good agreement with the measured data.
We also perform nonlinear mixing with two resonant modes detuned in frequency. Figure 5 shows sum-frequency generation performed in a structure with resonances at 1552.8 nm and 1558.9 nm (selected to have two resonances overlapping with the high power range of our tunable lasers and large enough frequency separation to observe the SFG peak). The central sum-frequency peak is smaller due to the smaller spatial overlap between the two modes than with a single mode. There is no cavity resonance at either the second harmonic or sum frequency.
4. Multiple resonances with large frequency separation
To create nonlinear optical sources at arbitrary wavelengths, it is also desirable to design the cavity to have large separation between cavity resonances. Figures 6(a) and 6(b) show simulated cavity fields for structures with resonances at 1.55 μm and 1.103 μm. The maximum separation in frequency that can be achieved is related to the ratio of the two beam widths, as the quality factor of the higher frequency mode is limited by diffraction into the orthogonal beam (Fig. 6(c)) as the difference between the beam widths grows (arbitrarily increasing the beam width for a fixed lattice constant would decrease the size of the photonic band gap and reduce the mode quality factor, as shown in Fig. 1(d)). The quality factor is limited for both resonances by out-of-plane scattering. An SEM of a fabricated structure with resonances separated by a large frequency difference is shown in Fig. 6(d).
For nonlinear frequency conversion applications, it is important for cavity field patterns to have large spatial overlap. Defining the nonlinear overlap, normalized to 1, as Fig. 6. This number could be increased to 0.07 by decreasing the number of taper periods from 5 to 3; the quality factors are reduced to 1440 and 1077 respectively, although this could likely be increased by reoptimization of other parameters. The value of γ for the degenerate cavity presented earlier is also 0.02. The value of γ for a single resonance of the degenerate structure overlapped with itself is 0.89 (i.e. γ is determined by the fraction of field in the nonlinear material), indicating the size of γ is limited by the amount of field concentrated in the central region. This overlap is smaller than that for TE00 and TM00 modes in a single nanobeam ; however, our design uses a thinner membrane which is easier to fabricate and can support larger frequency separations.
Figure 7 shows cross-polarized reflectivity measurements using a broadband source of structures with resonances at 1546.6 nm (Q=1600) and 1023 nm (Q=500), a frequency separation of 523.6 nm. We believe the experimental Q factors are limited by fabrication inaccuracies caused by the small feature size of the central holes in the vertical beam, which are located in regions of high field for both resonances. We could not perform nonlinear characterization of these structures due to lack of availability of a laser at 1023 nm.
5. Improving the crossbeam frequency conversion platform
For highly efficient three wave mixing, it would be desirable to have all three frequencies confined by cavity modes. This could be achieved by additionally using a higher order mode in one beam. However, all cavity resonances in our design have electric field primarily in plane (TE), while nonlinear frequency conversion in III–V semiconductors grown on (100)-oriented wafers requires at least one frequency to have out of plane (TM) polarization (because the only non-zero elements of the bulk have i ≠ j ≠ k ). This can be circumvented by using a wafer with different crystal orientation such as (111) [36, 37] so that the effective χ (2) tensor is rotated to have more non-zero elements. Figure 8 shows a concept illustration of highly efficient doubly resonant second harmonic generation using the crossbeam structure.
Our design is also interesting for integration with single semiconductor emitters, which is important for creating quantum photonic interfaces between different quantum emitters and telecommunications wavelength. Recently , we demonstrated second harmonic excitation of a InAs single quantum dot emitting at 900 nm, creating a fast single photon source triggered at telecommunication wavelengths (triggered single photons at 100 MHz, with nonclassical statistics visible at 300 MHz). The speed of that source could be improved by using a second cavity resonance, as in our crossed beam design, to enhance the spontaneous emission rate of the quantum dot through the Purcell effect , with possible speeds exceeding 1 GHz. Finally, our design is also promising for intracavity frequency conversion of single photons from an integrated quantum emitter .
In summary, we describe a crossed nanobeam platform for nonlinear frequency conversion that allows small mode volume cavities with spatial overlap and large frequency separation. We demonstrate cavities with nearly-degenerate frequencies, as well as frequencies separated by more than 500 nm, which we characterize by linear and nonlinear spectroscopy. Finally, we propose extensions of this work to allow high efficiency frequency conversion, which might be used in conjunction with integrated single emitters.
Financial support was provided by the National Science Foundation (NSF Grant ECCS-10 25811). KR and SB supported by Stanford Graduate Fellowships and the NSF GRFP (SB). This work was performed in part at the Stanford Nanofabrication Facility of NNIN supported by the National Science Foundation under Grant No. ECS-9731293.
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