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We developed a method that can accurately calculate the theoretical quality factor (Q) of a two-dimensional photonic crystal slab nanocavity at a very high speed. Because our method is based on a direct calculation of the out-of-slab radiation loss rate, it does not suffer from in-plane loss, and this allows us to obtain the same Q with 0.18 times less calculation volume. In addition, we can obtain the Q immediately after finishing the cavity excitation, because our method uses only a snapshot of the wavevector space distribution of the resonant mode in contrast to the conventional method, where we need to fit the electro-magnetic field with an exponential decay that requires a relatively long data set. For a width-modulated line defect cavity that has a Q of 8.5 × 107 we obtained the same value as with a conventional method but with 94% less computation time.
© 2014 Optical Society of America
1. Introduction
A high quality factor (Q) photonic crystal (PhC) nanocavity [1–6] is a tiny μm-sized optical resonator that can trap light in a small volume V. The photon density in an optical cavity scales with Q/V. Hence, PhC nanocavities with a high Q/V value are attractive for use in applications that require a strong interaction between light and matter. Such applications include all-optical switches [7–9], optical bistable devices [10,11], four-wave mixing devices [12] and sensing [13]. Q/V, which is known as the Purcell factor [14], is an important index in terms of quantum optics for achieving weak and strong coupling [15,16].
The Q value of two-dimensional (2D) PhC slab nanocavities has increased rapidly as a result of improved fabrication techniques and better design. As regards good design, the breakthrough occurred when a design strategy was developed that could reduce out-of-slab radiation [1,5,6]. Since the light is confined by total internal reflection, it is not always easy to achieve a small out-of-slab radiation rate for a cavity mode with a small V. So, finding of a good design was a key for the improving the Q in a 2D-PhC slab nanocavity. We usually use three-dimensional finite-difference time-domain (3D-FDTD) calculations to determine a good 2D-PhC slab nanocavity design. We can rigidly simulate the electro-magnetic field and obtain important cavity parameters such as resonant wavelength, V and Q. However, 3D-FDTD consumes a lot of computational resources, particularly when Q is very high. Even when we parallelize and accelerate the calculation, for example by using graphics processing unit calculations, we need to calculate the total energy at every calculation step to obtain the energy decay, which is often required for obtaining accurate Q. This requires sequential computation, so calculating the ultrahigh Q of a 2D-PhC slab nanocavity by 3D-FDTD usually takes a lot of time. In order to reduce the computation time, various methods have been developed. Among them, a method that allows to obtain the intrinsic Q of a 2D PhC slab by calculating the propagating power through a plane placed on the top of the PhC slab has been widely accepted [17–19]. Another method calculates the Q from the coupling between the expanded guided modes of the PhC slab and radiative modes obtained by perturbation theory [20]. But direct calculation of the Q from FDTD with a short time is yet to be exploited.
In this study, we propose and demonstrate a method that allows us to obtain the Q of a 2D-PhC slab nanocavity from a snapshot of a 2D mode profile obtained at the center of the PhC slab, without the use of the temporal waveform data that we usually use to obtain the decay of the electro-magnetic field by an exponential fit. We first introduce the basic idea and describe the algorithm of our method in section 2. We call this method a Fresnel reflection approximation (FRA) method. We then show results for three different types of PhC nanocavities, namely L3, hexapole and width-modulated line defect PhC nanocavities in section 3. In section 4 we discuss the calculation time and show the advantage of the FRA method. We finish with a conclusion.
2. FRA method: Direct calculation of Q from wavevector space distribution
Out-of-slab radiation losses are the primary factor limiting the theoretical Q-factor of 2D-PhC slab nanocavities, because the light is confined by total internal reflection toward the vertical direction. Srinivasan and Painter described a strategy, called a momentum space design, with which they reduced the radiation of a resonant mode by designing its momentum distribution with small components in a light cone (LC) regime [21]. The amplitude of the wavevector k of the light in a slab with refractive index n is given as,
where k|| and k┴ are the in-plane (x and z axis) and vertical (y axis) wavevector components. ω0 is the angular frequency of the resonating light, and c is the speed of light in a vacuum. Snell’s law gives the critical angle for the total internal reflection condition, and we know that the light in an air-bridged 2D PhC nanocavity will not be perfectly confined by total internal reflection and this leads to out-of-slab radiation when |k||| is,This limit is known as a light line or an LC [21]. When we Fourier transform an in-plane optical mode [Fig. 1(a)] we obtain the in-plane wavevector distribution [Fig. 1(b)]. Equation (2) tells us that the cavity mode exhibits a smaller loss (i.e. higher Q) when its spatially Fourier transformed distribution has a smaller amplitude inside the LC [1]. It provides us with a qualitative analysis and enables us to understand whether a given mode has a high or low Q. However, the relation between the number of LC components and the Q-factor has not been shown quantitatively and we need to develop a different method to calculate the Q value of the optical mode directly.
Fig. 1 (a) In-plane spatial mode profile E(x, z) of a resonant mode of a 2D PhC nanocavity at the center of the slab. (b) Spatially Fourier tranformed wavevector (momentum) space distribution of the resonant mode shown in (a). The circle in the center is the LC. (c) Schematic illustration of a ray-optics view of the reflection and transmission at a 2D PhC slab.
In terms of ray-optics, the out-of-slab radiation occurs when light (that has a small k|| and is below the LC) reflects at the surface of the slab as shown in Fig. 1(c) at an angle smaller than the critical angle. Since the reflection and the transmittance per unit time are dependent on the angle φ1 we should be able to obtain the Q when we have the k|| space distribution of the resonant mode and calculate the loss rate for different k|| inside the LC.
Every time that light with a small k|| (i.e. small φ1) hits the surface of the slab, some of the light is transmitted and radiated. According to the Fresnel equations, the amplitude transmittance ts for a TE-like mode is given as,
where φ1 and φ2 are the angle of reflection and refraction as shown in Fig. 1(c). Then the energy transmittance Ts is given as,On the other hand, the number of reflections per unit time N is given as,
where d is the thickness of the slab. Thus, we can obtain the loss rate TsN of the field for a given k|| by substituting the following two equations into Eqs. (4) and (5), Finally, we can calculate the total loss rate of an optical mode by integrating the loss rate TsN over the LC area with k||, such as,where is the momentum space distribution of the resonant mode.Once we obtain L, it is straightforward to calculate the Q value. From the definition, Q is given as,
where U is the total energy stored in the cavity, which is the sum of the energies stored as Ex, Ez, and Hy. When we assume a TE-like mode, it is written as,But with the FRA method we use a 2D mode profile snapshot when all the energy is stored as an electrical field, so it is UHy = 0 and UEx + UEz = U (hence UHy is not used in [Eq. (9)]. Since the loss rates differ with propagation direction, we need to calculate them for each propagation direction. Indeed the loss rate for the z direction is higher than for the x direction for an L3 cavity, because the cross-sectional profile of the cavity mode along the x axis is similar to a lossless waveguide mode. If we use Hy for this calculation, we cannot take this effect into account. For this reason, we use a 2D mode profile when all the energy is confined as an electrical field. Note that LEx and LEz are the loss rates for Ex and Ez, respectively, calculated by using Eq. (8).In our calculations, we assume a silicon slab with air cladding (i.e. n0 = 1). To minimize the error caused by the ray-optics approximation, we use the effective refractive index neff instead of the refractive index n of the material in Eqs. (3)-(7). In addition we calculated neff separately for Ex and Ez to obtain better accuracy. neff is calculated by using,
where we use the electrical field or that are directly obtained from 3D-FDTD.To briefly summarize, although our model is based on ray-optics as shown in Fig. 1(c), the error caused by this approximation is kept at its minimum value, which is explained as follows:
- We use neff as n to take account of the effect of the filling factor (air hole versus silicon) as well as that of the evanescent wave. For instance the delay caused by the Goos-Hänchen shift is taken into account when we use neff in Eq. (5). We use the electrical field directly calculated by 3D-FDTD and this allows us to obtain neff with high accuracy.
- We calculate UExLEx and UEzLEz separately in order to model the Fabry-Pérot-like cavity resonance along the x and z axes separately. To accomplish this, neff is calculated separately for Ex, and Ey, and this allows us to take account of the different direction dependent penetration depths of the evanescent wave into the air.
- It should be noted that the interference effect of the multipath reflection at the surface is automatically taken into account since we use the result of the Fourier transformation of a 2D electrical field distribution profile instead of the intensity field profile.
3. Applying FRA method to 2D PhC nanocavities
3.1. L3 resonator
First, we apply the FRA method to an L3 resonator [1]. The resonator is a PhC nanocavity formed with three missing air holes, and the design is shown in Fig. 2. The air-bridged PhC nanocavity is made of a material with a refractive index of 3.45 (silicon). The lattice constant a is 420 nm, the air-hole radius r is 115.5 nm, and the slab thickness d is 210 nm. The nearby air holes on either side of the resonator are 32-nm shifted toward the outside from their original position, and their radius is reduced to 63 nm. This enables us to realize a high Q. For the FDTD calculation, we set a yz symmetry plane at the center of the resonator. The time step is set at 0.05 fs/step and the grid size is 30 nm square. We excite the cavity for 1.25 ps with a TE polarized Gaussian pulse whose center wavelength is 1572 nm.
Fig. 2 Schematic illustration of the design of an L3 PhC nanocavity. The number of rows R is defined as shown in this picture.
The spatial and momentum mode profiles are shown in Fig. 3. The resonant wavelength of this cavity mode is 1572 nm. The Q-factor of this cavity mode was obtained as 8.5 × 104 by exponentially fitting the temporal decay of the electro-magnetic field. (We call this method the conventional method.) Obviously, with this method, we cannot obtain an accurate Q when the calculation area is too small. This is because the Q of a 2D-PhC slab nanocavity is given as Q−1 = QH−1 + Qv−1, where QH and Qv are the Qs determined by the loss towards the horizontal and vertical directions, respectively. It should be noted that Q = Qv for an isolated 2D-PhC slab nanocavity. The light will leak outside the structure in a horizontal direction and penetrates into the perfect matching layer (PML) when there are too few barrier photonic crystal layers (i.e. QH has a finite value). To investigate this, we show the dependence of the calculated Q as a function of the number of the rows R. R is defined as shown in Fig. 2 and corresponds to the distance between the cavity and the PML in horizontal plane. The result is summarized in Fig. 4 with black squares. We see that Q decreases rapidly when R is less than 9, which is due to the penetration of the cavity mode into the PML. This shows that we need to set the calculation area sufficiently large in order to obtain the Q accurately. As a result of this, we need large computation cost for a conventional method when Q is high. On the other hand, FRA method calculates the Q directly from the out-of-slab radiation loss. Therefore it does not suffer from horizontal scattering loss, and we can reduce the R.
Fig. 3 The spatial and wavevector space profile of an L3 cavity. (a) Ex(r) spatial mode distribution of the cavity resonance at the center of the slab. (b) Intensity profile of the spatially Fourier transformed wavevector distribution of (a). (c) Ez(r) spatial mode distribution of the cavity resonance at the center of the slab. (d) Intensity profile of the spatially Fourier transformed wavevector distribution of (c).
Fig. 4 Q calculated with different methods as a function of the number of rows. The details are provided in the text.
Before showing the result we obtained with the FRA method, we explain the calculation procedure in more detail. We excite and then take a snapshot of the cavity mode at a moment when all the energy is in the electrical field. The Ex and Ez values are shown in Figs. 3(a) and 3(c), respectively. These two mode profiles are spatially Fourier transformed and their momentum space distributions are shown in Figs. 3(b) and 3(d). The loss rates in the LC of Figs. 3(b) and 3(d) are separately calculated with Eq. (8) and then Q is obtained by using Eq. (9). The results of the obtained Q for different R values are shown as red dots in Fig. 4. Even though we employ a ray-optics assumption for the calculation of Eq. (8), it shows that the result obtained with the FRA method agrees well with that obtained with the conventional method, which uses no assumption. As discussed before, we replaced n with neff in Eqs. (4)-(7) to take the effect of the evanescent field into account, and this ensures high accuracy. Indeed we obtain a larger error and a Q of 1.3 × 105 when we use n = 3.46. In addition, we calculate LEx and LEz separately, which also contributes to the higher accuracy, because we have a Q of 1.7 × 102 when ignore the difference in the propagation direction. These contrivances make it possible to obtain an accurate Q.
The FRA method demonstrates its power when we reduce the calculation volume; namely when we reduce R. Although Q decreases with the conventional method due to the horizontal scattering loss, the Q obtained with the FRA method remains at its original value. This is because the FRA method calculates the “true” Q (i.e. Q = Qv), which is determined by the out-of-slab radiation and does not suffer from horizontal scattering, as has been discussed in [17] that also calculates the vertical and horizontal Qs separately. Indeed, we obtain a Q of 8.8 × 104 when R = 2 with the FRA method. It is surprising that the Q maintains even with such a small structure. In addition to the small volume, we can also reduce the required calculation step with the FRA method. Since we gradually excite the cavity for 1.25 ps, we obtain the resonating cavity mode immediately after we end the excitation (t = 0). So we can obtain the Q by the FRA method from a 2D snapshot at t = 0, without exponentially fitting the temporal decay of the electromagnetic field. When we compare the Q values at t = 0, 250 fs, and 1 ps, we do not observe any significant differences. It should be noted that the conventional method requires us to run the calculation step at least for 1 ps after the excitation for an L3 cavity, because we must fit the electromagnetic field with the exponential decay. (The required calculation time is even longer for a cavity with a higher Q because the decay is slower.)
Thanks to the smaller calculation volume and shorter step, we can significantly reduce the computation time needed to obtain an accurate Q. Indeed, we can reduce the computation time by 94% by using R = 2.
3.2. Hexapole resonator
Next, we apply our method to a hexapole PhC nanocavity [6,22,23]. The lattice constant a is 400 nm, the air-hole radius r is 88 nm, and the slab thickness d is 188 nm. The air holes around the resonator are shifted 96 nm from their original position. The design is detailed in [22]. Again we set a yz symmetry plane at the center of the resonator. The grid size is 40 nm, and the time step is 0.067 fs. The resonance wavelength of this resonator is 1467 nm. We excite the cavity for 3.75 ps using a Gaussian shaped light source with a spectrum width (FWHM) of ~1.2 nm at a center wavelength of 1467 nm. A hexapole cavity mode is a hexagonal symmetric intensity profile mode, where the Ex and Ez are shown in Figs. 5(a) and 5(c), respectively. The momentum space distributions that we use for the FRA method calculation are shown in Figs. 5(b) and 5(d). First we calculate the Q value obtained with the conventional decay method, in which we fit the decaying electromagnetic field with an exponential curve after we calculate the electro-magnetic field for t = 1 ps. In contrast, we calculate the Q value with the FRA method from a snapshot taken immediately after we ended the excitation of the cavity (t = 0). The results are summarized in Fig. 5(e). Again we obtain good agreement between the FRA and conventional methods when R is large. Furthermore, the FRA method is more advantageous than the conventional decay method when R is small.
Fig. 5 Results for a hexapole PhC nanocavity. (a) Ex(r) spatial mode distribution of the cavity resonance at the center of the slab. (b) Intensity profile of the spatially Fourier transformed k-wavevector distribution of (a). (c) Ez(r) spatial mode distribution of the cavity resonance at the center of the slab. (d) Intensity profile of the spatially Fourier transformed k-wavevector distribution of (c). (e) Q calculated with the conventional decay and FRA methods when t = 0 ps.
Although Fig. 5(e) confirms the advantage of the FRA method, we observe that Q decreases when R is smaller than 11 even with the FRA method. The reason is clear when we observe the mode profile carefully. Figure 6 compares the mode profile (|Hy|2) for an L3 and a hexapole PhC nanocavity. Since the spatial mode penetrates deep into the PhC and decays gradually, the mode reaches the PML. As a result, the Fourier transformed momentum space distribution is strongly modified by the convolution of the sinc function. This results in a decrease in the Q value even with our FRA method. However, we would like to emphasize that the FRA method still maintains a much higher Q even with a smaller R.
Fig. 6 Spatial mode profile |Hy|2 for an L3 (a) and a hexapole (b) PhC nanocavity. The mode profile for a hexapole PhC nanocavity reaches the boundary of the picture, which is the boundary of the PML.
3.3. Width-modulated line defect nanocavity
Finally, we show that the FRA method works even for a PhC nanocavity with the highest Q. We apply our method to a width-modulated line defect nanocavity [2,5,11,24]. The schematic image of the cavity design is shown in Fig. 7(a). The lattice constant a is 420 nm, the air-hole radius r is 108 nm, and the slab thickness d is 205 nm. The air holes on either side of the resonator are shifted 9, 6, and 3 nm from their original positions away from the line defect waveguide whose width is . We set a yz symmetry plane at the center of the resonator. The grid size is 28 nm, and the time step is 0.047 fs. We excite the cavity for 5.88 ps using a Gaussian pulse with a spectrum width (FWHM) of ~0.8 nm. The resonance wavelength of this resonator is 1568 nm. The spatial mode and wavevector profiles are shown in Figs. 7(b)-7(e).
Fig. 7 (a) Schematic illustration of a width-modulated line defect PhC nanocavity. (b) Ex(r) spatial mode distribution of the cavity at the resonance. (c) Intensity profile of the spatially Fourier transformed k-wavevector distribution of (b). (d) Ez(r) spatial mode distribution of the cavity at the resonance. (e) Intensity profile of the spatially Fourier transformed k-wavevector distribution of (d).
We summarize the Q factors with different R values in Fig. 8, which shows that the FRA method works even for a width-modulated line defect nanocavity with a Q close to 108. Again, the obtained Q is accurate even when R = 2.
Fig. 8 The Q value as a function of the number of air-hole rows R for a width-modulated line defect PhC nanocavity.
4. Calculation time
In this section, we discuss the calculation time required for the FRA method and describe its benefit. As shown above, we can reduce the calculation volume and shorten the calculation step. Hence we can significantly reduce the calculation time required to obtain the Q value. Here we show two examples, namely L3 and width-modulated line defect PhC nanocavities. In all cases we gradually excite the cavity (1.25 ps for L3 and 5.88 ps for a width-modulated line defect) to obtain the cavity mode immediately after ending the excitation.
With the conventional decay method, we need to continue calculating the electromagnetic field for 1 ps for the L3 PhC nanocavity to accurately fit the field decay with the exponential decay. Since the Q of a width-modulated line defect PhC nanocavity is even higher we need 6 ps, which results in a very long calculation time before we can obtain Q. In addition, Figs. 4 and 8 show that we need R = 11.
On the other hand, with the FRA method we can use the 2D snapshot obtained immediately after we finish the excitation. In addition, the required calculation volume is much smaller, and R = 2 is sufficient both for L3 and width-modulated line defect PhC nanocavities.
Figure 9 summarizes the calculation time (time is converted for a single core) required to obtain the Q value for L3 and width-modulated line defect PhC nanocavities. It shows a significant improvement. We can reduce the original calculation time by 94% for an L3 cavity. The reduction is greater when the Q of the cavity is high. The time reduction is 81% for a width-modulated line defect PhC nanocavity. Whereas the conventional decay method takes almost a week to finish the calculation, we can obtain the Q after a day with the FRA method. Usually we need a trial-and-error calculation when searching for an optimized structure, hence having a short calculation time is important. Since FRA method uses 2D data, there is even a possibility to compute the Q from pure 2D FDTD by combining the FRA method.
Fig. 9 Computation time required to obtain the Q values for L3 and width modulated line defect 2D PhC nanocavities. The calculation times needed with the decay and FRA methods are shown.
5. Summary
We proposed and formulize the FRA method, which computes the Q of a 2D PhC nanocavity from a snapshot of a 2D mode profile at the center of a PhC slab. We showed that we can use the FRA method to accurately calculate Q for L3, hexapole and width-modulated line defect 2D PhC nanocavities. Since the FRA method employs only a snapshot, we can use the mode profile immediately after we excite the cavity mode. We confirmed that the Q value is accurate in all cases. In addition, we can use a smaller calculation volume because there is no Q reduction caused by horizontal scattering loss to the PML. As a result, we were able to greatly shorten the calculation time. We can use computer resources more efficiently, and it may be able to combine this approach with an optimization algorithm to search for an even higher Q.
Acknowledgment
Part of this work was supported by a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology, Japan for the Photon Frontier Network Program.
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