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A photonic crystal defect consisting of several subwavelength holes was investigated as a means to increase the surface area of the defect region without compromising the quality factor of the structure. Finite-difference time-domain calculations were performed to determine the relationships between the size of the multi-hole defect (MHD) region, resonance frequency, quality factor, and refractive index of the defect holes. The advantage of using the MHD for sensing applications is demonstrated through a comparison with a single hole defect (SHD) photonic crystal structure. Assuming the same monolayer thickness of biomaterial coats the defect hole walls of the MHD and SHD, the MHD has a three times larger change in resonance frequency and two times larger quality factor.
©2008 Optical Society of America
1. Introduction
Photonic crystals, periodic dielectric structures that have the capability to manipulate light propagation, have been studied extensively for several applications, including optical waveguides and interconnects [1–4], modulators and switches [5,6], lasers [7,8], and sensors [9–11]. Defects in photonic crystals, which break the periodicity of the dielectric function and localize light, are necessary for these applications. Designing photonic crystals with strong field confinement, small mode volumes, and low extinction losses enables lower loss interconnects, lower threshold lasers, and higher sensitivity sensors. Over the past several years, research to optimize defect location, size, and shape has led to photonic crystal structures with quality factors exceeding 106 [12]. However, this research has not specifically examined the available surface area of the defect, which is a critical parameter for biosensing and other applications that rely on the attachment of material in the photonic crystal defect.
In this paper, we present the design and analysis of multi-hole photonic crystal defects that allow substantial design freedom with the photonic crystal defect surface area. Multi-hole defects (MHD) are photonic crystal defects consisting of several subwavelength holes that are significantly smaller than the surrounding photonic crystal lattice holes. By replacing the traditional single hole defect (SHD) previously reported for biosensing applications [11] with a MHD, a significant increase in surface area for small molecule attachment, along with a quality factor improvement, can be achieved. While linear defects such as the L3 configuration provide a large top-side surface area and high quality factor [13], the MHD configuration can provide up to 25 times greater overall surface area (top-side + internal) for each defect site without significantly compromising quality factor [14]. We emphasize that available surface area in the volume of the defect is the essential parameter; the highest sensitivity detection occurs when there is the greatest overlap of field with molecules. Solid defects only enable interaction of molecules with an evanescently decaying field at the photonic crystal slab surface, while MHDs enable strong interaction between the resonant mode and attached molecules in the defect holes. In the following sections, we explore the relationships between surface area, sensitivity of the defect state frequency to small changes in refractive index, and quality factor for MHDs containing different numbers of defect holes.
2. Design of multi-hole defects
2.1 Multi-hole defect structures
An example of a MHD photonic crystal is shown in Fig. 1(a). The small holes constituting the MHD are more than an order of magnitude smaller than wavelengths falling within the photonic band gap of the surrounding photonic crystal. Since a mixed dielectric media with features smaller than the wavelength can be considered optically as a single effective medium [15–16], the MHD can be treated optically as a single unit. The effective dielectric constant of mixed media depends on the dielectric constant of each constituent material and the fill fraction of each material. In this work, we consider mixed dielectric regions of air and silicon.
The positions of the multiple defect holes were generated using a simple MATLAB program, which assumed a hexagonal lattice as its basis in order to facilitate the most efficient filling of the defect region. The maximum number of complete defect holes was positioned accordingly within the defect region, starting with a hole in the center. Partial holes were not considered. Different effective defect radii were evaluated, as shown in Fig. 1(b–d). The size and spacing of the defect holes were fixed in this analysis; a study examining the effect of changing the defect hole size and spacing was reported elsewhere [14]. Due to the scalable nature of Maxwell’s Equations, a relative unit system was used in which all quantities are denoted as multiples of a spatial unit called ‘a’. Here, the photonic crystal lattice spacing is designated to be ‘a’, and all parameters of the MHD simulations were done relative to ‘a’.
Fig. 1. (a) Dielectric constant plot of MHD simulation space, where black indicates ε=12, white indicates ε=1. Detailed MHD regions with effect radius (b) 0.2a, (c) 0.3a, and (d) 0.4a are also shown. The defect hole radius in all cases is 0.04a, with defect hole spacing 0.12a.
2.2 Simulation methods
Two-dimensional photonic crystal lattices were first analyzed using freely available software for solving fully-vectorial eigenmodes of Maxwell’s equations with periodic boundary conditions (mpb) [17, 18]. After extracting the photonic bandgap for an air hole radius of 0.4a (shown in Fig. 2(a)), the MHD structure data was generated using MATLAB. The photonic crystal lattice hole and defect hole positions were then integrated to define the complete photonic crystal structure. Finite-difference time-domain (FDTD) analysis [19] of the photonic crystal structure was carried out using software with subpixel averaging for increased accuracy (meep) [20]. Due to the relatively small size of the defect holes in the MHD, the resolution of the simulation was increased to 150 grid lines per unit a in order to resolve the smaller holes. All simulations, including those of SHD sensors, were carried out at this higher resolution in order to obtain consistent comparison results.
A Gaussian source with center frequency and width matching that of the photonic band gap was positioned in the middle of the MHD region and run for several periods. The resulting fields were then analyzed with freely available harmonic inversion software based on the filter diagonalization method (harminv) [21], which extracted the decay rates and frequencies of the MHD cavity modes.
The simulation area was bordered by a 1-spatial unit thick perfectly matched layer (PML) [22], which absorbed the fields leaving the simulated region in order to prevent reflections. Such reflections could interfere with the measureable quantities of interest, namely cavity quality factor and resonance frequency. The simulation grid covered approximately 10 periods of the photonic crystal lattice, which increased the maximum achievable cavity quality factor for the device to that of comparable devices [9].
3. Analysis of multi-hole defect photonic crystals
The application of MHD photonic crystals as sensors requires a study of the device sensitivity. Sensitivity can be evaluated by determining the magnitude of change in resonance frequency and quality factor for small changes in defect hole refractive index (e.g., due to the addition of biomolecules). We assume a change of refractive index that is isolated to the MHD region.
3.1 Resonance frequency
The dielectric constant of the individual defect holes was varied between 1 and 12 (silicon) for three different MHD region effective radii. The defect hole radius and spacing were held constant at 0.04a and 0.12a, respectively. Figure 2(b) shows the relationship between the MHD photonic crystal resonance frequency and dielectric constant of the defect holes.
Fig. 2. (Color online) (a) Photonic bands for the photonic crystal with hole radii of 0.4a. A photonic bandgap exists for only for the TE polarization between 0.2462 and 0.4052. (b) Resonance frequency for varied defect hole dielectric constant and MHD effective radius.
The slope of each curve shown in Fig. 2(b) represents the sensitivity of the specified MHD photonic crystal. For sensing applications, it is desirable to design a device that has a large resonance frequency change for small dielectric constant changes (i.e., steep slope). In general, the largest slope and maximum sensitivity occurs for larger effective MHD radii. However, below a defect hole dielectric constant of approximately 5, the mode corresponding to a MHD with effective radius 0.4a is no longer supported due to the state being pushed into the air band. For the surrounding photonic crystal with air holes of radius 0.4a, the bandgap occurs between the frequencies of 0.2462 and 0.4052, as shown in Fig. 2(a). Upon close inspection of Fig. 2(b), the slopes of the MHD resonance plots are not linear, but change with increasing slope for lower defect hole dielectric constant. The maximum achievable sensitivity thus increases for defect hole dielectric constants near that of air, or ε=1.
3.2 Cavity quality factor
Fig. 3(a) shows the cavity quality factor for MHD photonic crystals with different defect hole dielectric constants and different effective radii. The quality factor increases with decreasing effective MHD radius. The field distribution for a 0.2a effective radius MHD shows the mode is monopole in Fig. 3(b). Larger effective radii contribute a greater perturbation to the field and dielectric constant within the defect region, causing the quality factor to drop by nearly a factor of 5. For all MHDs, the quality factor reaches the same maximum value when the defect area is completely filled with high-index material (silicon); in this case, the MHD is essentially replaced with a solid defect (i.e., missing lattice hole).
There is a clear trade-off that occurs between sensitivity and cavity quality factor, which affects the design of the MHD photonic crystal for specific applications. For example, at low defect hole dielectric constants, sensitivity is high for effective radii of 0.3a; however, in this regime of dielectric constant, the cavity quality factor is the lowest. For silicon-based photonic crystal sensors, depending on the refractive index of the material to be sensed and the accuracy of available measurement equipment, a MHD pattern can be generated to appropriately balance quality factor and sensitivity by changing the MHD effective radius. It is also possible to change the defect hole radius and spacing to modify the sensitivity and quality factor [14]: smaller defect holes spaced closer together offer larger surface area and greater sensitivity, but also lower quality factor.
Fig. 3. (Color online) (a) Change in cavity quality factor for varied defect hole dielectric constant and effective MHD radius, (b) Field distribution for a MHD with effective radius 0.2a, with defect hole radius and dielectric constant 0.04a and 1.05, respectively.
4. Sensor applications
The enhanced surface area afforded by MHD photonic crystals that allows strong interaction between the resonant photonic crystal mode and biomolecules attached inside the MHD holes is extremely favorable for small molecule detection. To quantitatively demonstrate the advantage of MHD compared to SHD biosensors, we simulate the effect of adding a variable thickness of biomolecules on the defect hole walls of MHD and SHD photonic crystals. We note that for sensing applications, chemical linkers are used to attach biomolecules to inorganic surfaces [23]. Thus, our simulations can accurately model the addition of a uniform monolayer of a chemical linker, such as silane, or approximate the addition of discrete molecules, such as DNA and proteins.
For the MHD, we choose an effective radius of 0.2a, defect hole radius of 0.04a, and defect hole spacing of 0.12a. The SHD is specified with a radius of 0.2a, such that the defect region of the SHD has the same footprint as the MHD. Figure 4 shows the resonance frequency shift as a function of monolayer optical thickness in the MHD and SHD. The Maxwell-Garnett approximation was used to calculate the effective refractive index for each defect hole containing the added monolayer of biomaterial [15]. Both defect configurations were assumed to be initially filled with air (n=1), and all resonance shifts were measured relative to the calculated resonance of an empty, as-fabricated structure.
Fig. 4. (Color online) Plot of resonance shift as a function of monolayer optical thickness. The monolayer optical thickness is specified as a fraction of the photonic crystal lattice constant, a.
For monolayers thinner than the defect hole radius, the resonance frequency shift of MHD photonic crystals is approximately 3 times larger than that of SHD photonic crystals. Hence, the sensitivity of detection is clearly superior for the MHD structures. We note that while the resonance shift is approximately linear with increasing monolayer thickness for thinner monolayers, the resonance shift of the MHD sensor tapers off as it approaches the condition of completely filling the defect holes. As the defect holes in the MHD become nearly filled with the monolayer, the change in effective index becomes smaller since there is increasingly less available surface area. The SHD sensor experiences a similar phenomenon near its complete filling condition when the monolayer optical thickness approaches 0.2a.
Also notable is the difference in quality factor of the MHD and SHD sensors. The quality factor of the SHD sensor is 2.8×103, whereas the MHD sensor has a quality factor of 6.3×103 when the defect holes are filled with air. Higher quality factors correspond to sharper resonances in measured transmission spectra. Since narrower resonances facilitate the detection of small resonance frequency shifts, the larger cavity quality factor of the MHD sensors further enhances the detection capabilities of the MHD sensor compared to the SHD sensor. The higher quality factor of the MHD can be explained in part due to the larger effective refractive index of the MHD, which results in stronger field confinement. For the same radius defect region, the MHD has index contributions from both air and silicon, while the SHD has an index contribution only from air. Additionally, the more abrupt index change at the boundary of the SHD contributes to scattering and loss, which reduce the quality factor. While we anticipate challenges in fabricating MHD structures with slab thicknesses of 0.5a to 1.0a, we believe the device dimensions to be within realistic fabrication tolerances for e-beam lithography and reactive-ion etching [24].
5. Conclusion
We have presented a multi-hole defect in a photonic crystal, which consists of several holes that are significantly smaller than the photonic crystal lattice holes. The advantages of the MHD compared to single-hole photonic crystal defects are larger surface area and increased cavity quality factor for equivalent defect effective radius. The increased surface area of the MHD contributes to a factor of 3 increase of the resonance frequency shift when material is attached inside the defect. The properties of the MHD can be tailored to a particular figure of merit such as quality factor, sensitivity, or surface area, depending on the application, by changing the effective defect radius, as well as the defect hole radius and spacing.
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