Introduction

Photonic crystals (PhCs) [1–3] are periodic dielectric patterns with feature sizes comparable to the wavelength of light. Such a strong modulation of the dielectric constant within the medium results in banded features much like what a real crystal does to an electron. Moreover, the dispersion relation of light can differ significantly from that of a homogeneous medium especially near the photonic band edge. Such an effect can lead to many unconventional optical properties, e.g., negative refraction [4], self-collimation [5], superprism effects [6–9], etc.

The term “superprism” refers to the effect that light beams with slightly different frequencies propagate in very different directions within the PhC region. Such a strong angular dispersion is promising for constructing compact on-chip demultiplexers for spectroscopic or wavelength-division-multiplexing communication applications [10–13].

Many previous studies of the superprism effect concentrated primarily on achieving an angular group dispersion (AGD) as large as possible. However, large AGD typically occurs only over a very narrow range of frequencies, and is highly frequency dependent. Using such a su-perprism structure to construct a spectrometer would lead to highly nonuniform channel spacing over a narrow working bandwidth. On the contrary, an ideal spectrometer would possess uniform spectral resolution over a large working bandwidth, i.e., would possess a flat-top dispersion profile. To incorporate such an idea into the design of a superprism structure, we here propose a new figure of merit (FOM), namely the angular-group-dispersion–bandwidth product (AGDBP) to quantitatively describe the spectroscopic performance of PhC superprism structures. Using this new metric, we perform an optimization of the geometry of a two-dimensional parallelogram-lattice PhC structure to achieve flat-band operation. The spectral performance of one example of a superprism-based on-chip spectrometer is further analyzed numerically using finite-difference time-domain simulations.

Analysis of PhC superprism structures

We first briefly summarize the mathematical formula used to quantitatively describe the angular dispersion performance of a given superprism PhC structure. We start with a 2-D square-lattice PhC structure as shown in the inset of Fig. 1, where a₁ and a₂ are the basis vectors and r is the radius of each air hole drilled into the host material. We use a plane wave expansion analysis algorithm [14] to calculate the bandstructure of such a PhC, and the calculated equi-frequency contours (EFC) in the first Brillouin zone are plotted in Fig. 1.

 

Fig. 1 Equi-frequency contours (EFCs) of a 2-D square lattice PhC in the first Brillouin zone. The light beam enters the PhC region at an incidence angle α_inc and the “refraction” angle inside the PhC region is labeled as α_pc. Solid arrows are the Bloch k vectors indicating the direction of phase velocities of the beams. Dashed arrows normal to the EFC contour indicates the directions of group velocities. Near the band edge in k-space, a small change in frequency leads to a large directional variation in group velocity. Inset: the PhC structure in real space. a₁ and a₂ are the two basis vectors. The radius of the air holes is r = 0.3a where a = |a₁| is the lattice constant.

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We assume that the interface between the PhC and the incident slab medium is along one lattice direction, denoted by a₁. To quantify the superprim effect, we define the angular group dispersion (AGD), also sometimes referred to as the q factor [15] at a frequency ω as follows:

Since one can obtain the numerical values of the partial derivatives ∂ω/∂k_x and ∂ω/∂k_y by analyzing the calculated band structure, expression (1) can be rewritten as follows:

For light incident from a fixed direction, a frequency change δω leads to a change in the Bloch wave vector k inside the PhC as follows:

Lattice structure optimization

One major limitation of a square-lattice PhC superprism structure is that its AGD is highly frequency dependent, which leads to very narrow working bandwidth over which the super-prism effect is appreciable and to very unevenly-spaced channel spacing.

On the other hand, spectroscopic applications typically desire a spectrometer to distinguish many uniformly-distributed spectral lines over the bandwidth of interest. Such concern indicates that an ideal superprism spectrometer should have uniform AGD over an extended working bandwidth. To take into account such considerations, we here propose a new figure of merit, namely the angular-group-dispersion–bandwidth product (AGDBP) defined as follows:

Here q(ω₀) is the AGD at a central frequency ω₀, and Δω is the frequency bandwidth in the vicinity of ω₀ within which the AGD varies less than 5% from q(ω₀).

To optimize the PhC structure to achieve better spectral performance, we generalize the square PhC lattice into a parallelogram lattice as shown in Figs. 2(a) and 2(b). A parallelogram lattice can be described using several free parameters: the angle θ between the two basis vectors a₁ and a₂, and the distance d between neighboring rows of air holes, and the radius of air holes r. For illustrative purposes, we here choose to fix r at a typical value r = 0.3a and to optimize only the two parameters θ and d. Note that |a₂| ≡ a₂ = d/cosθ.

 

Fig. 2 Geometric analysis of beam propagation in a general 2-D PhC. (a), the PhC structure in coordinate space; θ and d are the two adjustable parameters. (b), corresponding reciprocal lattice; k₁ is perpendicular to a₁ and k₂ to a₂. (c), EFC plot of the structure. The k_x axis is chosen to be along the k₂ direction.

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For a given PhC structure with a set of values for θ and d, we first calculate the value of $\mathcal{P}$ for all possible incidence angle α_inc as shown in Fig. 2(a) using the procedure described in the previous section. We then select the maximum value ${\mathcal{P}}_{max}$ as the optimum AGDBP of the structure with the current set of θ and d. Considering practical fabrication conditions, we here limit the value of d to be between 0.9a₁ and 1.5a₁, and the value of θ between 90° and 90° − tan⁻¹(0.5a₁/d). Note that for a given value of d, our scanning range of θ covers all possible geometries for all incidence angles due to the symmetry and periodicity of the PhC structures. The optimum AGDBP ${\mathcal{P}}_{max}$ is plotted in Fig. 3(a) as a function of θ and d. Here the effective refractive index of the slab is 2.83, and only TE polarization is considered. Figure 3(b) shows the optimized AGD as a function of frequency for three different sets of optimized parameters as circled in Fig. 3(a). All three designs exhibit flat-band AGD over extended frequency ranges, and therefore can satisfy various differing needs for the working bandwidth.

 

Fig. 3 Optimization results for obtaining a flat-band PhC superprism. (a), ${\mathcal{P}}_{max}$ as a function of position in the two dimensional parameter space. (b) Superprism factor q as a function of the normalized frequency ω plotted for 3 structures: ① ${\mathcal{P}}_{max}$ = 0.839 rad, d = 0.9a₁,θ = 89°; ② P_max = 0.802 rad,d = 0.9a₁,θ = 83°; ③ ${\mathcal{P}}_{max}$ = 0.809 rad,d = 1.1a₁,θ = 88°.

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Numerical demonstration of an optimized superprism spectrometer

To further demonstrate the performance of our flat-band PhC superprism structure obtained through the optimization procedure, we numerically simulate spectral performance of an optimized superprism spectrometer using finite-difference time-domain (FDTD) algorithms [16]. The lattice parameter of the PhC structure is chosen to be d = 0.9a₁, θ = 89°, i.e., the structure ① marked in Fig. 3(b). We further choose the lattice constant a₁ to be 310 nm, which would set the center wavelength at 1550 nm with a working bandwidth of approximately 80 nm.

The schematic diagram of this spectrometer is shown in Fig. 4(a). The device is composed of an adiabatically tapered input waveguide, a preconditioning free-propagation region (a slab waveguide) [11], a PhC superprism region, a second free-propagation slab region, and a series of single-mode output waveguides corresponding to individual output channels. Light is launched into the structure from the input waveguide. This input waveguide is adiabatically widened to an exit-width of 3.72 μm in order to reduce the angular spatial frequency components of the input beam. The preconditioning region is designed to be 140 μm long and has opposite sign in the second-order diffraction coefficient [17] as compared to that in the PhC region. As a result, the beam first diverges in the preconditioning region and then re-focuses as it propagates within the PhC region. The total PhC area is approximately 135 by 42 μm². The spacing between neighboring output waveguides is approximately 3.5 μm. The output signals are coupled into 8 single-mode waveguides. Figure 4(b) shows the normalized spectral transmittance of the 8 channels. As predicted by the design, this device shows a quite flat spectral response over its designed working bandwidth, and the number of channels it can support is approximately doubled as compared to previous designs [10–13]. The peak transmission varies by approximately 4 dB over the eight channels, and the spectral response slightly widens from channel 8 to channel 1. Such non-uniformity in transmission and the crosstalk between the channels are primarily caused by the aberrations, e.g., spherical and coma, of the spectrometer geometry that involves a rectangular PhC region. These aberrations can be potentially reduced by optimizing the shape and local lattice geometries of the PhC region [18]. The on-chip loss of the spectrometer is simulated to be approximated 11-15 dB, which is mainly caused by the undesired reflection at the slab-PhC interfaces. Such reflections can be mitigated by adiabatically optimizing the PhC structures at the vicinity of the slab-PhC interfaces [19,20].

 

Fig. 4 (a) Schematic diagram of a flat-band superprism spectrometer; (b) Spectral transmission of the 8 channels that are evenly spaced over the working bandwidth.

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Conclusion

In this work, we have proposed a new figure of merit, namely the angular-group-dispersion–bandwidth-product to quantify the spectroscopic performance of a PhC superprism structure. Using this FOM, we have performed optimization of a parallelogram-lattice PhC structure for building a superprism spectrometer. We have shown that a flat angular group dispersion can be achieved for a wide range of working bandwidths. Furthermore, the performance of an 8-channel spectrometer with 10 nm channel spacing at the center wavelength of 1550 nm has been determined using numerical simulation. Our method provides a systematic procedure to design flat-band on-chip miniaturized spectrometers and demultiplexers based on photonic crystals, and can be extended straightforwardly to other dispersive mechanisms.