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In this letter, we show that the Q factors of the latest high-Q cavities in two dimensional photonic crystals, measured experimentally to be ~1000000, are determined by losses due to imperfections in the fabricated structures, and not by the cavity design. Quantitative analysis shows that the dominant sources of loss include the tilt of air-holes within the cavity, the roughness of the inner walls of the air-holes, variation in the radii of the air-holes, and optical absorption by adsorbed material. We believe that cavities with experimental Q factors of the order of several millions will be obtained in the future by reducing the losses due to imperfections through improved fabrication techniques.
©2006 Optical Society of America
1. Introduction
High Q factor photonic nanocavities are structures which can strongly confine light within a very small volume, of the order of the cubic wavelength. Several interesting phenomena are expected to be exhibited by high-Q photonic nanocavities, including an increase in light-matter interaction, and long-term preservation of photons. Among the many approaches to the construction of high-Q nanocavities, engineering photonic crystal cavities is considered to be one of the most promising. Recently, a very high Q factor of 600,000 was reported for a cavity with a double-heterostructure, in a two-dimensional (2D) photonic crystal (PC) slab [1]. A Q factor of this order means that light is confined within a μm-scale region for the length of time in which light can travel several tens of cm in a vacuum. Cavities with Q factors one or more orders of magnitude greater than this may ultimately have applications in fields such as photonic memory and quantum computation. In principle, three-dimensional (3D) photonic crystals are a very attractive prospect for the realization of ultra-high-Q nanocavities, as the propagation of light can be blocked in all directions. At present, however, the fabrication technology of 3D photonic crystals is not sufficiently advanced to have surpassed the latest high-Q nanocavities in 2D-PC slabs. In this letter, we analyze the factors which determine the Q factors of the latest high-Q nanocavities, and suggest how higher Q factors may be obtained.
2. Additional Q factor
First, we examine the status of the latest high-Q nanocavity reported in Ref. 1. A three-dimensional finite-difference time-domain (3D-FDTD) calculation showed that the Q factor determined by the cavity design (Q design) is 2,000,000.[1] By contrast, the experimentally determined Q factor of the fabricated cavity (Q exp) was 600,000. This mismatch can be explained by introducing an additional Q factor due to imperfections of the fabricated cavity (Q imperfect), such as roughness and contamination. The relationship between these three Q factors is given by:
Q imperfect was evaluated to be 850,000 for this cavity, indicating that the loss due to imperfections (1/Q imperfect = 1/850,000) is larger than the loss determined by design (1/Q design = 1/2,000,000). Therefore, we propose that Q exp will saturate at the level of Q imperfect despite further increases in Q design. In early studies on PC cavities, Q design of the cavities experimentally investigated was 250-5,000 [2–5]. In such cavities, Q exp is considered to have been determined by Q design, as 1/Q imperfect was negligible compared to 1/Q design. In 2003, experimental results on a Gaussian confinement cavity [6], which had Q design ~ 100,000, were reported. For this cavity, 1/Q imperfect ~ 1/Q design. Most recently, Q design was increased to – 2,000,000 by the invention of the (single step) double-heterostructure cavity [1], in which Gaussian confinement is realized more exactly. In this case, Q exp is determined by 1/Q imperfect and not by 1/Q design as discussed above, despite the reduction in 1/Q imperfect due to improvements in fabrication techniques.
To confirm that Q exp will saturate at the level of Q imperfect, despite further increases in Q design, we fabricated a cavity in a 2D-PC slab, with a very large Q design, and measured the Q factor. If Q design ≫ Q exp, then we expect Q exp ≈ Q imperfect . The cavity had a two-step heterostructure [1], illustrated in Fig. 1. The PC consisted of a triangular lattice of circular airholes with radii of 118 nm, in a 250-nm thick Si slab. The cavity consisted of a line-defect waveguide, along which the lattice constant of the PC increased in two-steps towards the center, as shown in Fig. 1. The lattice constants of the central, intermediate and outer regions were 420 nm, 415 nm, and 410 nm, respectively. Light was mainly confined to the central region due to the differences between the mode-gap frequencies along the line defect [1]. A second, wider, line-defect waveguide was formed near the cavity, and was used to introduce light to the cavity. The cavity structure was designed so that the envelope of the cavity electric-field was very similar to a Gaussian function. As a result, radiation loss is strongly suppressed and the calculated Q factor (Q design) was 16,000,000.
Fig. 1. SEM image of the heterostructure photonic crystal cavity designed to have a Q factor of 16,000,000.
The sample was fabricated as described in Ref 1. A silicon-on-insulator (SOI) substrate was prepared, and the designed pattern was drawn on top by electron-beam lithography. The patterns were transferred to the Si slab by SF6-based dry-etching. The insulator layer (SiO2) underneath the patterned region was removed by selective wet-etching to form an air-bridge structure. The cavity properties were measured as described in Ref 7. Light from a tunable-wavelength semiconductor laser (line-width < 1 MHz) was introduced into the excitation waveguide; the intensity of the light emitted from the cavity to free space was measured as a function of the wavelength. The wavelength was varied in ~0.3 pm steps, and was monitored by a wavelength meter to a differential accuracy of ± 0.15 pm [7]. The measured emission spectrum of the cavity is plotted in Fig. 2 (filled circles); a sharp resonant peak with a width of the order of a few pm was seen. The solid lines in Fig 2. are Lorentzian fits to the spectrum with full width at half maximum (FWHM) values of 1.8 pm (red) and 2.1 pm (blue) [7]. The Q factor of the cavity loaded by the excitation waveguide (Qloaded) was calculated from the FWHM values, to be in the range 750,000 – 880,000. Q exp is related to Q loaded by [8]
where T is the transmittance through the excitation waveguide at the resonant wavelength. T was evaluated to be ~ 0.85, from the transmittance spectrum of the excitation waveguide (open circles, Fig. 2). Thus, from Eq. 2, we obtain Qexp = 820,000–950,000. (In addition to the results shown in Fig. 2, more than 30 cavities having Q design > 15,000,000 were fabricated and measured, and the maximum value of Q exp was found to be ~1,000,000.)
As Q design (= 16,000,000) is more than two orders of magnitude greater than Q exp, it is clear that Q exp ≈ Q imperfect, and so for this cavity, Q imperfect is evaluated to be in the range 860,000 – 1,010,000. This range coincides with Q imperfect of the cavity reported in Ref 1. These results clearly indicate that values of Q exp of the latest PC nanocavities are limited to ~1,000,000 by Q imperfect. We conclude that the key to further increasing the experimental Q factor is to reduce the loss due to imperfections.
Fig. 2. Measured cavity characteristics: Emission spectrum of the cavity (filled circles), fitted by Lorentzian functions with full width half maximum values of 1.8 pm (red line) and 2.1 pm (blue line); and transmission spectrum of the excitation waveguide (open circles).
4. Analysis of losses due to imperfections
As a first step towards this aim, the details of the losses due to imperfections were analyzed. The origins of such losses were categorized as either (A) imperfections of the cavity shape, or (B) imperfections of the cavity material. We considered the following five imperfections of the cavity shape: (A1) surface roughness of the Si slab, (A2) surface roughness of the inner walls of the air-holes, (A3) variation in the radii of the air-holes, (A4) variation in the positions of the air-holes, and (A5) tilt of the inner walls of the air-holes. Other factors such as major deviation in air hole shapes, bending of the slab, or a lack of uniformity of slab thickness might in principle give rise to additional losses, but using the present fabrication techniques and materials, these imperfections were not observed in scanning electron microscopy measurement and were expected to be minor considerations compared to factors A1-A5. We also investigated the following two imperfections of the cavity material: (B1) optical absorption by residual free carriers in the Si slab, and (B2) optical absorption by material, such as water, adsorbed to the surfaces of the cavity. Optical scattering by refractive index fluctuations in the slab, and optical absorption and scattering due to the air surrounding the cavity might potentially induce additional losses, but we considered that these losses would be negligible compared to those caused by factors B1 and B2. (The former is negligible since the slab is made of single crystal silicon, and the latter is negligible since the attenuation coefficient the air is much smaller than the value of the water discussed below.)
The losses due to each of the above factors were quantitatively evaluated. Here, we present the results without showing the details of the evaluation methods due to space limitations. (The details will be reported in a separate paper.) Table 1 contains the measured quantities on which the calculations were based, and the calculated loss Q factors, for each of the factors A1-A5, B1, and B2. The imperfections of the cavity shape were measured by atomic force microscopy (A1) and scanning electron microscopy (A2-A5). The measured quantities are shown in the second column of Table 1. The roughness values (A1 and A2) are quantified by standard deviations (s) and correlation lengths (Lc). The variations in the radii and positions of the air-holes (A3 and A4) are quantified by standard deviations only.
Deviations of the dielectric material distribution from the designed structure can be treated as additional electric polarizations from which propagation modes are excited to induce radiation losses. No transverse-electric-like (TE-like) slab modes exist at the resonant frequency of this cavity, due to the photonic band gap effect, so radiation losses are expected to free space modes and transverse-magnetic-like (TM-like) slab modes only [9]. An important consideration is that radiation losses will only occur to modes which have the same symmetry about the center plane of the slab as the additional electric polarizations due to imperfections. The electric field of the cavity modes is TE-like and is even about the center plane, whereas TM-like slab modes are odd, and both odd and even free space modes occur. A3 and A4 preserve the mirror symmetry about the center plane of the slab and so may induce even free space modes. A1, A2, and A5 break the mirror symmetry, and so may induce both even and odd free space modes and TM-like slab modes.
The losses from imperfections in the air-hole radii, positions, and tilts (A3, A4, and A5 respectively) were evaluated from 3D-FDTD calculations. To evaluate the loss due to A3 (A4), the radius (position) of a certain air-hole nearest to the cavity was changed, and the calculated change in the photon lifetime of the cavity mode were used to evaluate the additional loss Q factor. If the variations of the radii (positions) of different air-holes did not correlate, then interference between the radiation from different the air-holes was neglected. There were effectively about 10 air-holes in the main part of the cavity, so the total losses due to A3 and A4 were roughly evaluated by multiplying the losses due to a single air-hole by 10. To evaluate the loss Q factor due to A5, the changes in the photon lifetimes of the cavity mode were calculated by gradually tilting the inner walls of all the air-holes from 1° to 10°.
A 3D-FDTD calculation method could not be used directly to evaluate the losses due to roughness (A1 and A2), as the scale of the roughness was smaller than the cell size used to discretize the electro-magnetic field. The evaluation was carried out as follows: First, additional electric polarizations due to roughness [10] were calculated by multiplying the calculated cavity electric field and the deviation of the dielectric material distribution from the designed structure. Second, the radiation powers from the additional electric polarizations were calculated, taking into account the random nature of roughness and the fact that the correlation lengths were shorter than the wavelength of light in the material. The loss Q factors were obtained from the ratio of the radiation powers to the electro-magnetic energy in the cavity.
The loss Q factors due to imperfections of the cavity material (B1 and B2) were evaluated using the equation
where α, n 0, and λ. are the absorption coefficient, refractive index of the material, and resonant wavelength of the cavity, respectively. This equation was derived by assuming that the light confined in the cavity travels at the phase velocity and is absorbed according to the absorption coefficient. For loss due to optical absorption by residual free carriers in the Si slab (B1), the absorption coefficient was estimated, using the Drude model and the specifications of the SOI substrate, to be 2.3 × 10-2 cm-1. For optical absorption by material adsorbed to the surfaces of the cavity (B2), we considered all materials abundant in air and with a large absorption coefficient at the resonant wavelength of the cavity. We propose that water is the candidate most likely to give rise to significant loss, as the absorption coefficient at 1500 nm is greater than 10 cm-1 [11]. We assumed that one monolayer of water molecules were adsorbed on the surfaces of the cavity, including the inner walls of the air-holes. (Surfaces of Si are naturally oxidized and hydroxilated in ambient conditions, and covered with several monolayers of water molecules [12].) The effective absorption coefficient was calculated by multiplying the volume ratio of adsorbed water to Si by the absorption coefficient of pure water. The loss Q factor was then calculated using Eq. 3. The results of all the calculations are summarized in Table 1. The Q factors are divided into those associated with radiation to free space modes, radiation to TM-like slab modes, and absorption.
Table 1. Summary of the origins, measured quantities, and calculated Q factors for each loss. Lc: Correlation length, σ: Standard deviation, nc: Carrier density , ρ: Resistivity, σ: Absorption coefficient.
The inverse of the total theoretical additional loss Q-factor (1/Q imperfect) was obtained by summing the inverses of all the calculated Q factors in Table 1, giving Q imperfect = ~ 900,000. This theoretical value is of the same order as the experimental value (~ 1,000,000), indicating that the results in Table 1 are a measure of the real situation, despite the assumptions used. The dominant losses were associated with the tilt of the inner walls of air-holes (Q ~ 3,000,000 for a tilt of 3°), variation in radii of air-holes (Q ~ 3,000,000), absorption by surface water (Q ~ 4,000,000), and surface roughness of the inner walls of air-holes (Q ~ 5,500,000, for radiation to free space and TM-like slab modes). The tilt and surface roughness of the inner walls of air-holes, and variation in air-hole radii are likely to be decreased by reviewing the fabrication processes e.g. dry-etching and EB lithography. Adsorption of water is poorly understood, and needs to be investigated further: measurements in vacuo may clarify the extent of the problem. Another surface absorption mechanism such as absorption by surface states [13] might also have some contribution.
5. Summary
In summary, we have shown that the experimental Q factors of the latest point-defect cavities in two dimensional photonic crystals, which are of the order of 1,000,000, are determined by imperfections of the fabricated structures and not by the cavity design. The imperfections of the fabricated structure have been categorized, and the associated losses quantified. It has been shown that the dominant sources of loss are the tilt and roughness of the inner walls of the air-holes, variation in air-hole radii, and optical absorption by material (water) adsorbed on the cavity surfaces. We believe that experimental Q factors of the order of a few to several millions will be obtained in the future by solving these problems.
Acknowledgments
This work is partly supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and also by CREST, Japan Science and Technology Agency.
References and links
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