We investigate the dependence of quality factor Q of dipole modes in photonic crystal H1-defect nanocavity on the slab thickness and observe an increase of Q even after closing of the photonic bandgap both in numerical simulation and experimentation. This counter intuitive behavior results from the weak coupling between the cavity mode and the 2nd-guided mode in the photonic crystal slab. This is confirmed by computing the overlap between them in the momentum space.
©2008 Optical Society of America
Since the pioneering work of Purcell , who described the modification of the coupling between the electromagnetic field and an emitter placed inside a cavity and the discovery of photonic bandgap (PBG) material , the use of a photonic crystal (PC) with a defect as a cavity to control spontaneous emission has become an active field of research. Photonic crystal nanocavities are promising structures for nanophotonic devices, such as low-threshold lasers [3,4], and single photon sources (SPS) . This is because PC nanocavities can create resonant modes with high quality factor (Q-factor) with small mode volume, in other words, a large Purcell factor. Large enhancement of the spontaneous emission rate due to Purcell effect is a key phenomenon for reducing the threshold of nanolasers, increasing the repetition rate of SPS, and etc.
Especially, two-dimensional photonic crystal slab (2D PCS) defect cavities have a great potential to achieve that goal due to their simplicity in fabrication and their ability to achieve strong confinement of light in three dimensions. Several groups have already reported designs of high Q-factor photonic crystal slab cavity structure together with mode volume of the order of the cubic wavelength [6–13]. A designed Q in the order of 107–108 has been recently achieved [12,13]. With the maturation of nanometer-size photonic crystal fabrication technology in past ten years, PC nanocavities with a Q-factor of more than one million have been experimentally demonstrated [14,15]. However, these designs for high-Q cavity require precise controls of position and size of air holes in practical fabrication otherwise their Q-factors are significantly degraded [6,7]. For example, in Ref. , measured-Q of approximately 5,000 has dropped from its predicted value of 45,000 attributed to fabrication inaccuracies. This shows that these structures lack of robustness in practical systems. On the other hand, slab thickness, which is also a parameter of design, can be precisely controlled by using epitaxial growth techniques such as molecular beam epitaxy (MBE). So far, slab thickness of PCS is usually chosen in the order of half wavelength to ensure that PBG exists and to confine cavity modes strongly within the slab .
In this paper, we report an anomalous behavior in Q-factor of dipole modes of the H1-defect nanocavity with increased slab thickness. We numerically and experimentally investigated the dependence of Q-factor of dipole modes on slab thickness and found that high-Q can be obtained after closing of the photonic bandgap. These dipole modes in H1 cavity consist of two orthogonal polarized modes and are energetically degenerated, so the cavity possessing such modes can be applied to polarization entangled photon sources [17,18].
In Section 2, we briefly describe the computational methods used, which are mainly based on the three-dimensional (3D) finite-difference time-domain (FDTD) method . We then present a dependence of Q-factor of the dipole modes as a function of the slab thickness in Section 3. We show that high Q-cavities are achieved after closing of the PBG between 1st-and 2nd-guided bands. The origins of such an anomalous behavior are also discussed. In Section 4, we experimentally confirm our numerical prediction of high-Q cavities and show that the experimental results are in good agreement with the calculated results with maximum measured-Q of approximately 3,000.
2. Computational methods
In order to analyze the cavity characteristics, such as Q-factor, mode volume, and field distributions of the designed structure, we use the 3D FDTD method for our simulations. The calculation domain is 11a×11a in the x-y plane for all of our designed cavities, i.e. 5 periods of air holes surrounding the defect region. The size of the calculation domain in the z-direction is 10a. Perfectly matched layers (PML) are introduced to all outer boundaries as nonreflecting absorbers . The Q-factor is obtained by measuring the exponential decay of electromagnetic energy stored within the cavity after turning off the oscillation of the source:
where U(t) is the electromagnetic energy in the mode at time t, and ω is the frequency of the cavity mode. In addition, the total Q-factor (Q total) calculated by Eq. (1) is divided into two components, in-plane Q-factor (Q∥) and vertical Q-factor (Q⊥), by calculating the ratio of stored energy to radiating power:
where P is the total radiating power out of the cavity and is calculated by integrating the Poynting vectors over all the boundaries of the computational domain, in order to separately clarify the origins of the confinement mechanisms in each direction. Q∥ represents the propagation loss of the guided mode through the slab and Q⊥ corresponds to the vertical radiation loss due to the coupling of the cavity mode with leaky modes. Each component of Q-factor is calculated by spatial separation of the total radiating power into in-plane (P∥) and vertical components (P⊥) with boundaries of separating at about λ/2 from the surface of the slab .
The mode volume  is calculated using the following definition:
where ε(r→) is the dielectric constant at position r⃗, and E⃗(r⃗) is the total electric field at position r⃗. In addition, like the method in Ref. , the effective refractive index of the PCS structure is determined by taking the square root of the space-averaged dielectric constant:
In the next section, we will use this value to partly explain the origin of strong confinement of light in the cavity after the closing of the PBG.
3. High Q-factor dipole modes in photonic crystal H1-defect nanocavities after closing of photonic crystal bandgap
3.1 Structural parameters and defect mode
The calculation model is based on air-bridge type photonic crystal slab (n=3.4) with a triangular lattice of air holes. A defect, called a H1-defect, is formed by simply removing one of the air holes at the center. The defect cavity contains no modification in shape and position of the surrounding holes. The defect mode of interest is the x-dipole mode , which is one of the doubly-degenerated dipole modes, and its field distributions of Ey component is shown in Fig. 1(a). For discussion in a later section, we also show an Ex-field distribution of the y-dipole mode in Fig. 1(b). In the FDTD calculation, we used a cubic lattice, in which size of grid cells in three dimensions are identical. Therefore, the structure is not a completely perfect triangular lattice. As a result, anisotropy occurs in the x and y directions, which leads to splitting of the dipole degeneracy. The full-width at half-maximum (FWHM) of the emitter inside the cavity is ensured to be narrow enough to excite only the mode of interest when we calculate cavity Q and mode volume.
3.2 Slab thickness dependence of cavity Q of H1 dipole mode
Figure 2 shows the dependence of Q-factor of x-dipole mode on slab thickness d with radius of air holes r=0.40a, where a is the lattice periodicity. The Q-factor of an x-dipole mode is significantly improved by slightly increasing the slab thickness. The Q total reaches the highest value up to 16,200 at the slab thickness d=1.35a with small mode volume Veff=0.44(λ/n)3. Interestingly, such a high Q-factor is obtained after the photonic bandgap for TE-like modes between the lowest two TE-like bands, denoted as 1st- and 2nd-guided band, is closed at d=1.20a. The Q total was divided into two components, Q∥ and Q⊥, in order to separately observe a behavior of Q-factor in each direction. Firstly, let us explain a tendency of Q-factor in a conventional range of slab thickness, before closing of PBG. The in-plane Q can be increased by increasing the number of air holes surrounding the cavity, however, the Q total of dipole mode is limited by decay to the free space, thus the Q total is only a few hundreds. It gradually drops as the slab thickness increases resulted from the reduction of Q∥ due to the decrease in PBG size . It should be clarified that the peak of Q⊥ at the minimum value of Q total and Q∥ has no physical significance. In that region, because of the coupling between the cavity mode and the 2nd-guided mode, the cavity mode mainly leaks into the slab guided mode. Therefore, the component of light that can radiate toward the vertical direction becomes very small, which unavoidably results in a large value of Q⊥ without any evidence of strong confinement of light in this direction due to the cavity. After the gap is closed, both Q⊥ and Q∥ (and thus Q total), increase together to their peaks at d=1.35a and then decrease when the slab thickness exceeds that value. Only Q⊥ significantly rises up again when the slab thickness keeps increasing and another maximum peak occurs at d=2.70a. The increase of Qtotal after closing of the PBG is unexpected because Q∥ is supposed to greatly drops limiting Q total. We separately consider the origins of the strong light confinement mechanisms at the optimized slab thickness in each direction. In the vertical direction, there are two maximum peaks of Q⊥ at d=1.35a and 2.70a. These values of slab thickness correspond to ~1λ and 2λ, where the effective refractive indices of the structures were calculated to be 2.608 and 2.619, respectively, using Eq. (4). So we conclude that the strong confinement mechanism in the vertical direction is due to the resonance of the cavity mode when the slab thickness is equal to a multiple of wavelength.
If the slab is thick enough, by more than 1.20a, the 2nd-guided band falls below the lowest-order band edge and is responsible for destroying the photonic bandgap. As a result, the PBG effect cannot be used to explain the strong light confinement mechanism in the in-plane direction. Figure 3 shows a band diagram for the structure with d=1.35a calculated by using the plane-wave expansion (PWE) method. The cavity mode in this structure has a normalized frequency of a/λ=0.292, which is spectrally matched with the 2nd-guided mode. In addition, the cavity mode also has the same symmetry as that of the guided mode, even symmetry about the center of the slab, so they can be coupled with each other. On the other hand, these modes hardly overlap in the momentum space. We calculate the equifrequency contour (EFC) of the 2nd-guided mode at the cavity resonant frequency. Figure 4(a) shows an Ey-field distribution in momentum space for the cavity mode with d=1.35a, including the light line and the EFC of the 2nd-guided mode. The EFC is almost a perfect circle and only overlaps with faint components of the cavity mode field, which leads to weak coupling between the cavity mode and the 2nd-guided mode and thus high Q∥. In contrast, shown in Fig. 4(b), in the structure with d=1.75a with Q∥ of about 1,000, there are strong components of the field distribution that overlap with the EFC of the 2nd-guided mode, therefore light is not strongly confined in the cavity but well guided through the slab. We summarize amounts of overlap between the cavity mode and the EFC of the 2nd-guided mode of each structure with the slab thickness from 1.20 to 1.75a in Fig. 4(c). These amounts of overlap were calculated by firstly normalizing the field distributions in momentum space and then taking them a line integral over the EFC. As we expected, the amounts of overlap show the opposite behavior to Q∥, while the minimum amount of overlap occurs at the slab thickness where the Q∥ is maximum. Our conclusion of the origin of high Q∥ resulted from the mode mismatching between the cavity mode and the guided mode in the momentum space is similar to that in Ref. , where the authors avoid momentum space couplings of the cavity mode to slab guided mode through the dielectric perturbation by tailoring the defect geometry.
Finally, let us note that such a counter intuitive behavior of Q-factor after closing of the PBG is also attainable in the y-dipole mode of the same cavity structure. The results will be shown in the next section.
4. Fabrication and Experimental Results
4.1 Crystal growth and photonic crystal nanocavity fabrication
The numerical prediction described above was experimentally confirmed. We investigated PC nanocavities fabricated in four samples, each grown on an undoped (100)-oriented GaAs substrate by molecular beam epitaxy. First, a 300-nm-thick GaAs buffer layer was deposited on the substrate at 600°C followed by a 700-nm-thick Al0.7Ga0.3As sacrificial layer. Finally, GaAs slab layers with the slab thickness of 190, 315, 390, and 450 nm including a self-assembled InGaAs quantum dot (QD) layer as an active material at the center of the slab were grown on each sample. The quantum dot density was ~ 1010 cm-2. Patterns of each sample were designed to have a radius of air holes r=0.40a and contain a wide range of periodicity between a=245–360 nm to cover a range of the slab thickness of interest. One air hole at the center of each pattern was omitted to form the H1-defect cavity. The PC structures were patterned using an electron-beam lithography system and then transferred through the GaAs layer by inductive coupled plasma reactive ion etching (ICP-RIE) using a Cl2/Ar mixture. Finally, the sacrificial layer was removed by dipping samples in an HF:H2O (1:9) acid solution to form suspending air-bridge structures. The fabricated air-bridge H1-defect nanocavity with d=390 nm is shown in the scanning electron micrographs of Fig. 5.
4.2 Experimental setup
Micro-photoluminescence (µ-PL) measurements were performed in a temperature-controlled liquid-helium cryostat at 4 K. The samples were optically pumped by a continuous-wave Ti:sapphire laser operated at 780 nm. The pump laser beam was focused to a 4 µm-diameter spot on the sample surface by a microscope objective [50×, numerical aperture=0.42], and was positioned on the PCs using piezo-electric nanopositioners. The PL from the QDs was collected by the same microscope objective and analyzed with both a monochromator equipped with an InGaAs multichannel detector array and a triple grating monochromator equipped with a Si CCD.
4.3 Experimental results and discussions
The emission from a QD ensemble can be used to prove the cavity characteristics because of its broad PL across a wide spectrum of 920–1050 nm. Figure 6 shows the PL spectra for the structures with d=1.345a with different polarization. The polarization dependence confirms the dipole-nature of these kinds of cavity modes. Sharp peaks of both x-dipole and y-dipole modes reflect their high-Q. Figure 7 shows the measured-Q and calculated-Q over a wide range of the slab thicknesses for both the x-dipole mode and y-dipole mode. The measured-Q has the tendencies very close to that of the calculated-Q with the highest measured-Q being ~ 3,000 at d ~ 1.345a for the x-dipole mode and ~ 2,000 at d=1.393a for the y-dipole mode. The calculated-Q for the x-dipole and y-dipole mode are 16,200 at d=1.35a and 7,500 at d=1.40a, respectively. These differences between the measured and calculated values can be attributed to the fabrication errors causing the roughness of the sidewall and the fluctuation of shape and size of the etched air holes. The results clearly show that the structural parameters of the best fabricated cavity and those of the predicted one are almost exactly the same (the slab thickness d=1.345a of the experimental results is close to d=1.35a of the calculated ones), because the only parameter that needs to be adjusted is the slab thickness which can be precisely controlled by using epitaxial growth techniques such as MBE. The cavity modes in the samples with the slab thickness around 1a cannot be recognized due to their very low Q.
In summary, we have numerically and experimentally demonstrated a significant increase of Q-factor of dipole modes in photonic crystal H1-defect nanocavity after closing of the photonic bandgap by tuning the slab thickness. The optimal slab thickness is equal to a wavelength of light confined in the cavity. The maximum calculated-Q and measured-Q are ~ 16,200 at d=1.35a and ~ 3,000 at d=1.345a, respectively. They were obtained after closing of the photonic bandgap. In this cavity, the strong light confinement of the cavity in the in-plane direction is not caused by the photonic bandgap effect due to a lack of the photonic bandgap but resulted from the mode mismatching between the cavity mode and the guided mode in the momentum space yielding only weak coupling between these two modes. This finding will contribute to extending the freedom of cavity design, such as that for the application to polarization entangled photon source, where it is required to form cavity modes with prescribed Q-factor and polarization.
The authors would like to thank Katsuyuki Watanabe and Satomi Ishida for their useful technical support. This work was supported by the Specially Appointed Funds for Promoting Science and Technology.
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