1. Introduction

Optofluidics opens a new research field in photonics by combining optics and microfluidics. The compact optofluidic devices have been widely researched for a number of applications such as biochemical sensors [1-8], fluidic-tuning systems [9,10], and platforms of interaction of gaseous molecules with light [11]. In the biochemical sensors, the resonant frequency shift induced by the change of the index of refraction (IOR) is used to analyze samples. Owing to the compactness and the relatively high spectral resolution of the microcavities, the sensors have been realized by using various microcavities such as PhC cavities [2-4], microtoroids [5,6], microring [7], etc. Very recently, White et al. analyzed the performances of various sensors by comparing the detection limit of the devices in the existence of system noises and sample absorptions [8]. They showed that it is desirable for the sensors to have smaller sensor resolution and larger sensitivity. Such two factors enable sensors to measure the small frequency shift precisely and hence the small amount of the corresponding IOR change can be measured. The resolution and the sensitivity depend on the design of the sensor as well as various noises and sample absorptions.

On the other hand, there has been increasing interests in measuring the change of IOR of the environmental material by using photonic crystals (PhC) due to their ability to confine photons in wavelength-dimension by photonic bandgap (PBG). Subsequently PhC cavities can have small mode volumes on the order of 0.1 μm³, which permits to sense the index change within very small volumes (~ 0.1 fL) of sample material. Such ultra-small devices are crucial for biochemical sensors where the amount of sample is often limited. Moreover, the miniaturization of the sensing devices enables very compact photonic integrated chips, in which various analyses of chemical species can be performed. However, there have been few researches on the optimizations of the PhC cavities for the index sensing. In this paper, we focus on the design of the PhC cavity to have smaller resolution and larger sensitivity for index sensing. Here we describe two factors as a quality factor and a response factor.

First we define a response factor, R, which relates the IOR change of the background material with the shift of the resonant wavelength:

where n is the IOR of the background material, λ and ω are the resonant wavelength and the frequency, Δn, Δλ, and Δω are the IOR change, the wavelength shift, and the frequency shift, respectively. Equation (1) is valid only if Δn, Δλ, and Δω are much smaller than n, λ, and ω. R shows the sensitivity of the device to environmental IOR change. As the amount of the electric field in the background increases, R becomes larger. Since the R is normalized by the resonant wavelength, R enables to compare the performances of various sensors operating at the different wavelengths.

Second, the Q (=λ/Δλ, Δλ is the full-width half maximum of the resonance) of the cavity should be high enough to resolve the wavelength shift caused by the IOR change. Since the resonance linewidth can be considered as the amount of the smallest resonance shift which is clearly measurable, there is a lower limit of Q to measure the change of the IOR for the certain material. The smallest change of the IOR was measured in the ambient gas materials, which was as small as 10⁻⁴ [4,7]. R for the normal PhC cavity is on the order of 0.1, as will be shown later. Therefore the normalized frequency shift (Δω/ω) corresponding to the gas index of 10⁻⁴ is estimated as 10⁻⁵ and hence the Q larger than 10⁵ is required to sense the gas.

In this paper, we optimize PhC cavities for larger R by increasing the overlapping of electric field with the background material while high Q is maintained. The optimizations are performed by three ways such as thinning slab thickness, introducing central holes, and using slot waveguides. By tuning three structure parameters, we theoretically investigate R and Q for the cavities to optimize through three-dimensional (3D) finite-difference time-domain (FDTD) simulations. The cavity for sensing is finally optimized by combining two effects of the optimized structure parameters. Having in mind that the lower limit of Q is 10⁵ for gas sensing, we design the finally optimized PhC cavity to have Q larger than 10⁵. On the other hand, the cavities with larger R tend to have smaller Q for each optimization here. Therefore we set a lower limit of Q to 10⁶ for each optimization and hence the final cavity with two optimization parameters has Q of 10⁵ at least. In Section 2, the structure of the heterostructure cavity, S3, is briefly explained and R and Q are estimated. In Section 3 and 4, R is enhanced by reducing the slab thickness and drilling air holes in the waveguide region of the S3 cavity. In Section 5, we propose a new heterostructure cavity, T3, built in the slotted waveguide and R and Q are investigated for various slot widths. In Section 6, the optimized cavity structures for the slab thickness, the central holes, and the slot widths are summarized and the finally optimized cavity structures are discussed. In addition, the behavior of R and Q are briefly explained in the waveguide coupled system.

2. Design of a heterostructure cavity using air hole modulation

Since Song et al. proposed a two-dimensional (2D) PhC heterostructures cavity by using PhCs with two different lattice constants [12,13], various heterostructure cavities have been demonstrated by modulating the structure parameters of the waveguides such as the width [14] and the refractive index [10]. Owing to the slightly different parameters in the heterostructure, the confined mode profile is gently varying, which suppress the vertical loss out of the slab. Therefore high Q PhC cavities with Q over 10⁶ have been demonstrated in the photonic heterostructures by exploiting the mode-gap [12-14].

On the other hand, Srinivasan et al. demonstrated the cavity with graded air hole radius [15] and Kwon et al. demonstrated that the PhC surrounded by the PhC with the different air hole radii can confine photons [16]. Very recently, we proposed a high-Q PhC cavity created by the local modulation of the air hole radius along the waveguide [17]. The air holes modulated cavity extends the degrees of freedom in the fabrication of the heterostructure cavities. In addition, the structural modifications in an air hole modulated cavity are limited to several air holes so that the cavity has a great advantage for integrating photonic devices in a small region. We successfully fabricated the proposed cavity and measured the change of the IOR for the different gas environments [4]. In this paper, we present the optimizations of the structure parameters in the air-hole modulated heterostructure cavity for index sensing. Here it should be noted that the optimizations based on the slab thickness, the central air holes, and the slot width can be applied to any types of the heterostructure cavities because of the similar mode shape.

Figure 1(a) shows the structure of the proposed cavity. In a single line defect waveguide, the radius of the innermost air holes, r_wg, vary along the waveguide direction. In comparison with the air hole radius of the background PhC, r_pc, of 0.25a, the r_wg of the mirror regions is enlarged to 0.28a, as shown in the yellow boxes of Fig. 1(a). Since the dispersion curve shifts to higher frequency for larger r_wg in Fig. 1(d), the frequency near the bandedge of the waveguide with the r_wg~0.25a is placed at the mode-gap of the waveguide mode with the r_wg~0.28a. Mode-gap of the waveguide and PBG of PhC makes photons to be confined horizontally, as shown in Fig. 1(b). Photons are vertically confined in the slab with the slab thickness T~0.625a by total internal reflection (TIR). The refractive index of a slab is 3.4, which corresponds to the indexes of typical semiconductor at λ=1.55μm. The parameters used in the FDTD simulations are Δx=Δy=Δz=1/20a for the spatial grids and Δt=0.5Δx/c for the time interval, where c is the speed of light. In this paper, the frequencies and the Q of the modes are calculated by applying filter-diagonalization method (FDM) to the detected fields in the FDTD simulations, which provides much better accuracy than conventional Fourier transform method for the special form of the signals such as the exponentially decreasing sinusoidal signal [18]. In order to find the optimal grid size, we perform the simulations for different spatial grids, 1/10a, 1/20a, and 1/40a to confirm the validation of the simulations. In contrast to 30% difference of the Q in the case of 1/10a, the frequency and the Q are same within 1% differences for the 1/20a and 1/40a simulations and hence we use 1/20a for the following simulations. In addition, the Q obtained by FDM was confirmed by the Q obtained from the decay rate of the total energy of the mode.

The tapered regions with linearly increasing air holes are introduced between the center and the mirror regions to further increase Q by changing mode-gap gently, as shown in the red boxes of Fig. 1(a). In the tapered regions indicated by the red boxes, the radii of the innermost air holes are 0.26a, 0.27a from the hole close to the center. Figure 1(c) presents the frequency shift of the dispersion curve along the waveguide schematically. In the tapered region, the mode-gap shifts gradually so that the cavity mode has an ultrahigh-Q of 7.83×10⁶ [17]. The mode volume of the cavity is 1.29 (λ/n)³. In this cavity, we calculated the R by obtaining the resonant frequencies for different IOR of the background materials from n=1.0 to n=1.001. The change of the resonant frequency is proportional to the change of IOR in the case that the change is much smaller than the IOR. The relation is given by Eq. (1). The estimated R is 0.047. This value corresponds to 73nm/RIU (refractive index unit) for the resonant wavelength of 1.55 μm.

 

Fig. 1. (a). Design of a S3 cavity. Dotted purple lines indicate the center of the cavity. The mirror regions are represented by yellow boxes, the tapered regions by red boxes. The radii of the air holes of the PhC and the innermost air hole are represented by r_pc, r_wg, respectively. (b). Electric field intensity pattern. The black circles indicate air holes. (c). Schematic of the band diagram along the waveguide direction. (d). Dispersion curves of the line defect waveguides with r_wg~0.25a (red lines) and 0.28a (blue lines). The gray area indicates the leaky mode region.

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3. Effect of slab thickness on R and Q in the S3 cavity

Figure 2(a) shows the waveguide-direction side view of the electric field intensity profile of the S3 cavity with T~0.625a. The mode is mostly confined inside the slab and the evanescent field of the mode experiences the background material. In the electric field intensity distribution along the z-axis, very small amounts of the electric field are placed in the background, which are described by the components in the IOR~1.0 in Fig. 2(b). In Fig. 2(c), the field distribution in the cavity with T~0.125a shows two significant differences comparing with the distribution in the cavity with T~0.625a. First, the mode size along the z-axis becomes smaller in the thinner slab cavity. The full-width half-maximums (FWHM) of the distributions for the cavities with T~0.125a and 0.625a are 0.25a and 0.55a, respectively. Second, the fraction of the mode energy in the air increases in the thinner slab cavity. In the Figs. 2(b) and 2(c), 59% (8.2%) of the field in the cavity with T~0.125a (0.625a) exists in the air. The larger evanescent field components and the smaller mode size in the cavity with thinner slab affect R and Q.

 

Fig. 2. (a). Side view of the electric field intensity profile in the cavity with T~0.625a. Black solid lines indicate PhC slab surface. Along the z-axis, refractive index (black dashed line) and the electric field intensity distributions (blue solid line) of the modes in the cavities with (b) T~0.625a and (c) T~0.125a. (d) Response factor (black line) and quality factor (blue line) vs. slab thickness.

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We investigate R and Q of the S3 cavities for various T from 0.725a to 0.125a in Fig. 2(d). In the thinner cavity, R increases rapidly from 0.0403 (T~0.725a) to 0.206 (T~0.125a) due to larger overlapping of the field with the air, as shown in Figs. 2(b) and 2(c). However, Q decreases for the thinner cavity because of the weaker vertical confinement. For example, when T is 0.125a, Q becomes 1.1×10⁵ which is one order smaller than the Q for T~0.225a.

As the PhC cavity with thinner slab has larger R and smaller Q because of the larger evanescent field and the smaller mode size, a trade-off between R and Q is needed. In addition, since the modifications of the cavity structure in the following sections tend to decrease Q and the final cavity with a Q higher than 10⁵ is required to measure the change of the IOR of the ambient gas, we select the cavity with the Q larger than 10⁶ for the optimization of the slab thickness. In the range of the slab thickness where Q is larger than 10⁶, the largest R~0.119 is obtained in the cavity with the slab thickness of T~0.225a while Q is 1.58×10⁶. This R value corresponds to 184nm/RIU for the λ of 1.55 μm, which is 2.5 times larger than the R of the cavity with T~0.625a. T~0.625a is generally used for the each optimization in Section 4 and 5.

4. Effect of central air holes on R and Q in the S3 cavity

Having in mind the electric field pattern of the S3 cavity in which maximums of the field are placed in the dielectric region in Fig. 1(b), we introduce the air holes with the radius r_c at the center of the waveguide in order to increase the overlap of the electric field with the background material. In Fig. 3(a), the electric intensity pattern is shown for the S3 cavity with central air holes with radii r_c~0.20a and T~0.625a. Since the field pattern is conserved even though air holes are drilled, most of electric field overlaps the central air holes. We investigate R and Q for a range of r_c, as shown in Fig. 3(b). As r_c increases, R also increases rapidly from 0.047 (r_c~0.0a) to 0.135 (r_c~0.225a) due to higher field overlap. In contrast to R, Q decreases slowly for larger rc and drops one order of magnitude at r_c~0.225a.

In order to investigate why Q drops seriously at r_c~0.225a, we separate the optical loss of the cavity to the vertical loss, the loss along the waveguide, and the loss through the PhC, which are related to the Qz, Qx, and Qy, respectively. The total Q (Qtot) is mostly limited by Qz except for r_c~0.225a, i.e. vertical loss is the dominant loss channel in the cavity, as shown in Fig. 3(c). However, at r_c~0.225a, Qy limits Qtot. This means that most of the photons leak out of the cavity through the PhC region. As the r_c increases, the frequency of the cavity mode (blue line in Fig. 3(d)) approaches the bandedge of the air band of the PhC and hence the gap between the cavity mode and the air bandedge is getting smaller. Here the frequency (black line in Fig. 3(d)) of the bandedge mode of the waveguide with the r_wg~0.25a changes similarly, which implies that the cavity mode originates from the bandedge mode of the waveguide. Since the cavity mode is placed near the edge of the PBG, the cavity mode with r_c~0.225a becomes leaky in the surrounding PhC. Therefore enlarging the central air holes guarantees higher sensitivity (larger R), however, as r_c is close to r_pc, the Q of the mode drops seriously.

 

Fig. 3. (a). Electric field intensity pattern of the S3 cavity with air holes at the center of a waveguide. r_c and T are 0.20a and 0.625a, respectively. (b) Response factor and quality factors vs. r_c (c) Quality factors as a function of r_c (d) The normalized frequencies of the waveguide bandedge mode and the cavity mode as a function of r_c. Yellow boxes represent air band and dielectric band regions of the surrounding PhC with r_pc~0.25a.

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In our cavity structure, the cavity with r_c~0.20a and T~0.625a shows R~0.110 while Q is 3.82×10⁶ still larger than 10⁶. This R value corresponds to 171nm/RIU for the λ of 1.55 μm, which is 2.3 times larger than the R of the cavity without r_c.

5. Design of a heterostructure cavity in a slotted waveguide

 

Fig. 4. (a). Structure of a slotted waveguide with r_wg~0.28a and the slot width, v~0.20a. (b). The dispersion curves of the waveguide modes with r_wg~0.28a (red line) and 0.25a (blue line), and v~0.20a. (c). Design of a T3 cavity in a slotted waveguide with v~0.20a. The purple lines, the black dashed lines indicate the center of the cavity and the boundaries between the mirror and the tapered regions. Mirror, tapered, center regions are described by yellow, red, and blue boxes. (d). Electric field intensity profile of the T3 cavity (e). Schematic of the band diagram along the waveguide direction. (f). Refractive index and electric field intensity distribution along y1-axis. (g). Response factor and quality factor vs. slot width (h). Quality factors vs. slot width

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Recently, Almeida et al. proposed a new waveguide structure where two ridge waveguide are closely located with nanometer size gap of low index so that strong electric field is confined within the gap because of the large discontinuity of the field [19]. Owing to the strong light-matter interaction, the slotted waveguide can be advantageously utilized in optical sensors [7]. On the other hand, a slotted waveguide has been realized to control dispersion in PhCs, which also allows confining strong fields in the low-index gap [20].

Here we propose a heterostructure cavity, T3, in a slotted waveguide to use the field concentration in low-index material for sensors. Figure 4(a) shows the dispersion curves of the slotted waveguide with r_wg~0.28a and 0.25a. The simulated structure with r_wg~0.28a is described by Fig. 4(b). The slot width is v~0.20a. As r_wg increases, the frequency of the dispersion curve shifts to higher frequencies like in the normal waveguide in Fig. 1(d). However, since the dispersion curve is convex in contrary to the concave curve of the normal PhC waveguide, the mirror region should have the smaller r_wg to build a heterostructure cavity, as shown in Fig. 4(c). This heterostructure cavity with slotted waveguide is named a T3 cavity. In the T3 cavity, the center region (blue box) with r_wg~0.28a is sandwiched by the mirror regions (yellow boxes) with r_wg~0.25a and the tapered regions (red boxes) are introduced between them. The radii of the air holes in the tapered regions are 0.27a, 0.26a from the hole close to the center. The slab thickness is 0.625a. As is expected from the slotted waveguide characteristics, the electric field is strongly confined in the slot in Fig. 4(d). In Fig. 4(e), the schematic diagram describes the mode-gap shift along the waveguide in the T3 cavity. The transmission range linearly shifts to lower frequency range from the cavity center to the mirror regions according to the change of the radius of the air holes. It is clearly visible that most of electric field is confined within the slot of low-index material in the electric field distribution along the y1 axis, as shown in Fig. 4(f).

R is calculated as 0.215~0.240 for the range of v from 0.10a to 0.30a, as shown in Fig. 4(g). Since the amount of the optical energy inside slot is almost constant even for larger slot width [19], the change of R is less than 10% for various v. In the T3 cavity, Q drops seriously at v~0.30a. For v smaller than 0.10a or larger than 0.30a, there exists no confined mode. When v is smaller than 0.10a, the dispersion curve disappears below the dielectric band. When v is larger than 0.30a, the curve touches the air band. This is why Q drops to 20000 at v~0.30a. When v is 0.30a, Qtot is limited by Qy, as shown in Fig. 4(h). This means that the frequency of the cavity mode is close to the air band and hence the mode is leaky in the surrounding PhC like in the case of the S3 cavity with r_c~0.225a.

We choose the T3 cavity of T~0.625a and v~0.20a with large R~0.232 while Q is 1.01×10⁶ still larger than 10⁶. This R value corresponds to 360nm/RIU for the λ of 1.55 μm, which is 4.9 times larger than the R of the S3 cavity with T~0.625a.

6. Discussion

Table 1. R and Q of the optimized cavities.

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In Table 1, we summarize the PhC heterostructure cavities optimized by tuning three structural parameters, the slab thickness, the radius of the central air holes, and the slot width of the waveguide. The reference cavity (A) with T~0.625a has R~0.047 and 7.83×10⁶.

First, we investigated the thickness effect on R and Q of the cavity. When the cavity is built on the thinner slab, R increases because of larger evanescent field experiencing the background material. However, for the cavity with T~0.125a, Q drops seriously due to weaker TIR. As a result of the trade-off between R and Q, the cavity (B) with the slab thickness of 0.225a is selected to have R~0.119 and Q~1.58×10⁶. Second, the central air holes are introduced at the center of the waveguide, which results in more overlapping of the electric field and the background material. The cavity with larger central holes tends to have larger R and smaller Q due to larger overlapping and weaker horizontal confinement. The cavity (C) with the central holes of r_c=0.20a is selected to have R~110 and Q~3.82×10⁶. Third, the heterostructure cavity (D) is designed in the slotted waveguide. Owing to the strong electric field confinement inside the slot, the R is strongly enhanced. However, Q is degraded in the large slot width due to the weak horizontal confinement. Therefore the slot width, v=0.20a is selected to have R~0.232 and Q~1.01×10⁶.

When the cavity with the central air holes or the cavity with the slot waveguide is built in the thin slab with T~0.225a, the R is further enhanced and the Q decreases more. The S3 cavity (E) with the central air hole (r_c=0.20a) has R~0.200 and Q~7.30×10⁵. On the other hand, the T3 cavity (F) with the slot (v=0.20a) has R~0.330 and Q~1.03×10⁵. In this T3 cavity, R corresponds to 512nm/RIU while Q is on the order of 10⁵. Therefore this T3 cavity has largest R value and is still able to measure the change of the IOR in the ambient gas material.

In this paper, we assume that the considered cavities are isolated from the input/output coupled waveguides. Since the mode shape of the heterostructure cavity is very similar with the waveguide mode, the coupling efficiency is relatively high enough to get good transmission ratio while additional vertical loss is not introduced by the existence of the input/output waveguides with r_wg~0.25a [17]. When the S3 cavity with T~0.625a (R~0.047, Q~7.83×10⁶) is connected to the input/output waveguides with 5 mirror layers, the system Q is kept to be larger than 10⁶ and 60% of the input signal is transmitted to the output waveguides [17]. On the other hand, since it is negligible the effect of the input/output waveguides on the cavity mode shape, the changes of the R are not observed for the cavity connected with the waveguides in our simulations. In the case of the cavity with the thinner slab, the central holes, or the slot, the transmission efficiency and the Q should be further studied. However, the relatively high transmission and the large Q are expected while the enhanced R is kept to be same.

7. Conclusion

In this paper, we optimize the heterostructure cavities for high response factor, R, to measure the change of the IOR in the background material by 3D FDTD method. The R is enhanced by increasing the field overlapping with the background material through three methods such as thinning the slab thickness, drilling the air holes at the center of the waveguide, and making the slot in the waveguide. To keep enough spectral resolution for gas sensing, the Q for the final cavity is kept to be larger than 10⁵. For the thinner slab, we analyzed the enhancement of the R through the mode patterns in the real spaces. The behaviors of the R and the Q in the cavities with the central holes or the slot were explained by the electric field patterns, the channels of the optical losses, and the position of the frequencies of the modes within the PBG. In order to increase R and analyze the behavior of the Q, the analysis used for the optimizations of the cavities in this paper are generally applied to any PhC heterostructure cavities created by modulation of the lattice constants, the width, the refractive index, and the radius of the air holes.