Highly dispersive photonic band-gap-edge optofluidic biosensors

Highly dispersive photonic band-gap-edge optoﬂuidic biosensors are studied theoretically. We demonstrate that these structures are strongly sensitive to the refractive index of the liquid, which is used to tune dispersion of the photonic crystal. The upper frequency band-gap edge shifts about 1.8 nm for δ n = 0.002 , which is quite sensitive. Results from transmission spectra agree well with those obtained from the band structure theory. [DOI: 10.2971/jeos.2006.06026]


I n t r o d u c t i o n
Photonic crystals (PhCs) are attractive optical materials for controlling and manipulating the flow of light [1]- [3]. One well-know property is the existence of photonic band gaps, i.e., regions of frequencies where electromagnetic wave propagation is prohibited. Another equally important aspect of a PhC is the existence of an unconventional ultra-strong dispersion. Such an ultra-strong dispersion was firstly reported by Lin et al. and demonstrated experimentally in the millimeterwave spectrum [4]. Kosaka et al. subsequently demonstrated the superprism effect in a highly dispersive photonic microstructure [5]. These unusual properties provide an exciting possibility for obtaining microphotonic and nanophotonic devices that can focus, disperse, switch, and steer light.
Optofluidics, the marriage of nano-photonics and microfluidics, refers to a class of optical systems that integrate optical and fluidic devices [6]. Due to unique properties of fluids, such integration provides a new way for dynamic manipulation of optical properties and shows many potential applications [7]- [14]. In particular, PhCs are interesting for optofluidics since they naturally have voids where fluids can be injected. Optical properties of the PhC can be easily reconfigured by selectively filling specific voids with liquid. Chow et al. demonstrated an ultra compact biosensor employing a two-dimensional (2D) photonic crystal microcavity [15]. Recently, we proposed simple biosensor structures based on highly dispersive PhC waveguides [16]. In this paper we will propose biosensor structures based on complete PhCs and the strong dispersion occuring near the Brillouin zone. In particular, the bandgap edges of the PhCs are strongly sensitive to the refractive index of the liquid which is used to tune the dispersion of the PhC. The suggested structures show a potential for biochemical sensing applications.

B I O S E N S O R S T R U C T U R E S A N D R E S U L T S
Let us first consider a 2D triangular PhC with air holes extending through a high index ε = 10.5 dielectric material, shown in the inset of Figure 1. The holes have a radius of 0.36a, where a is the lattice constant. Here, we will focus our study on transmission spectra of the PhC with air holes being filled with different liquids. It was shown in our previous work that both the surface termination and surface direction of the PhC are critical for high transmission (i.e., coupling) at an interface between air and the PhC [17,18]. Consider the TE-polarized (magnetic field parallel to the air holes) light normally incident into the PhC. To enhance the coupling at the interface, we choose to couple light to the PhC along the ΓM direction, i.e., the surface of the PhC slab is along ΓK direction. The symmetric PhC slab is composed of 11 layers along the ΓM direction and the distance (surface termination) between right boundary and the centers of the first right holes is 0.5a. Transmission spectra for the PhC are obtained using the 2D finite-difference timedomain (FDTD) method [19]. For this case, we use the periodic condition in ΓK direction and perfectly matched layers in the ΓM direction [20] as the numerical boundary treatment. For the present application we are not interested in the details of the Fabry-Perot pattern in Figure 1, but rather the spectral position of the band-gap edge which is a highly sensitive measure for changes in the refractive index of the liquid. To see it more clearly, the change of the band-gap edge as a function of the refractive index of the liquid is shown in Figure 2. As seen, the low-frequency mode-gap edges slightly change with the refractive index of the liquid. However, the highfrequency mode-gap edge is strongly dependent on the refractive index of the liquid, as shown by squares in Figure 2. As an example, the high-frequency band-gap edge shifts δ(a/λ) = 0.012987 when the air holes (with index n = 1) are filled by a liquid of index n = 1.1. For comparison, it is only δ(a/λ) = 0.001278 for the low-frequency band-gap shift. Now, consider a commercial silicone fluid with a calibrated refractive-index accuracy of δn = 0.002, as mentioned in Ref. [15], where the refractive index of the liquid varies from n = 1.446 to 1.454 in increments of 0.002. For the working wavelength around 1.55µm (here we choose a = 450nm), the high-frequency bandgap edge shifts up to 1.17 nm for δn = 0.002, while 0.33 nm for the low-frequency band-gap edge. For comparison, we note that the shift in resonant wavelength for the high-qualityfactor PhC cavity is about 0.4 nm for δn = 0.002 [15]. The above results demonstrate that even such a simple PhC has potential applications as a sensitive biosensor.
To further elucidate the physics behind the strong sensitivity, we next support the picture by dispersion calculations for the PhC. For this purpose we use a Block-iterative frequencydomain method [21]. The dispersion of the PhC, in absence or presence of a fluid, is shown in Figure 3.  One can clearly see that these PhCs have photonic band gaps for the TE polarization, which are related to the band gaps in Figure 1, though the band-gap regions appear slightly different with those obtained from the transmission spectra. Note that the band gaps in Figure 1 are larger than those in Figure 3. This is because the band gaps in Figure 3 are for all incident directions while the band gaps in Figure 1 are only for propagation along the normal direction. From Figures 1 and 3, we also observe that the position of the gap in the transmission spectra, which are obtained for plane electromagnetic waves incident normally on the PhC, agree very well with the position of the gaps in the frequency band structure of the corresponding infinite crystal along the ΓM direction (denoted by red regions). When increasing the refractive index of the liquid [going from panel (a) toward panel (f)], the high-frequency band-gap edge is significantly downward shifted, while the low-frequency band-gap edge slightly decreases. We emphasize that all results obtained from band structures are consistent with those from the transmission spectra. The sensitivity of this structure is mainly attributed to the strong dispersion of the PhC mode. Figures 4a and 4b  As seen, the low-frequency band-edge Bloch mode hardly changes with varying refractive index of the liquid. However, for the high-frequency band-edge mode, the ratio of the energy in holes becomes lager as the refractive index of the liquid increases, i.e., this mode distribution is strongly dependent on the refractive index of the liquid, which is in agreement with the results in Figures 1-3. A somewhat similar structure has been realized experimentally by Okamoto et al. [22], but in this paper we have studied the shift for the band-gap edges in details and offered a physical explanation for the shifts. Besides, compared to the device in Ref. [22], our proposed device offers a better resolution.
Let us next consider a square PhC with dielectric rods in air, as shown in the inset of Figure 5. The permittivity of the rods is ε = 10.5, and the radius of the rods is r=0.2a. Transmission spectra for TM polarization are shown in Figure 5, where the background of the PhC is filled by different liquids. The band-gap edge as a function of the refractive index of the liquid is shown in Figure 6. Similar to the result shown in Figure 2, the low-frequency band-gap edge hardly changes as the refractive index of the liquid increases, while the high-frequency band-gap edge is strongly dependent on the liquid. Compared with the result for the high-frequency band-gap edge shown in Figure 2, the results in Figure 6 illustrate a higher sensitivity. The highfrequency band-gap edge shifts δ(a/λ) = 0.025939 when the air holes are filled by a liquid of index n = 1.1. For comparison we have δ(a/λ) = 0.012987 for the structure shown in the inset of Figure 1.
Band structures are shown in Figure 7, where red regions represent the band gap for ΓX direction. From Figure 5 and Figure 7, we find that the position of the gap in the transmission spectra agree very well with the position of the gaps for the PhC along the ΓM direction. When increasing the refractive index of the liquid [going from panel (a) toward panel (f)], the high-frequency band-gap edge is significantly downward shifted, while the low-frequency bandgap edge slightly decreases. Again we consider a commercial silicone fluid with a calibrated refractive-index accuracy of δn = 0.002. For the working wavelength around 1.55µm (a = 450nm), the mode-gap edge shifts up to 1.60 nm for δn = 0.002. The proposed biosensor relies strongly on the dispersion of the PhC band-edge mode and the presence of a band gap. To further improve the sensitivity, we optimize the PhC structure by varying the radius of the rods. By a careful design of the structure shown in the inset of Figure 5, we have been able to improve the design further. For the working wavelength around 1.55µm (a = 450nm), the band-gap edge shifts about 1.8 nm for δn = 0.002, when r is tuned to 0.1a. Compared to the biosensor we proposed before [16], this structure not only shows much better sensitivity but it also seems relatively easy to realize experimentally, since the design involves no cavities or waveguide structures. Finally, this device, with a size of 5µm × 5µm, is sufficiently compact for most applications.

C O N C L U S I O N S
To conclude, we have theoretically studied optofluidic biosensors based on highly dispersive 2D photonic crystal. Our study shows that these structures are strongly sensitive to the refractive index of the liquid, which is used to tune dispersion of photonic crystal. For the working wavelength around 1.55µm, we predict shifts in the band-gap edge up to 1.8 nm for δn = 0.002. Although our study is based on 2D photonic crystals, it can be extended to the case of a 2D photonic crystal slab. For a 2D photonic crystal slab, the field will attenuate due to out-of-plane loss, but the shift of the high-frequency mode-gap edge at X/M point (when tuning by liquid) is unaffected by the out-of-plane radiation. The high sensitivity makes such devices interesting for biochemical sensing applications.

A C K N O W L E D G M E N T S
This work is financially supported by the Danish Council for Strategic Research through the Strategic Program for Young Researchers (grant no: 2117-05-0037).