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Engineering of directional emission from photonic crystal waveguides

We analyze, by the finite-difference time-domain numerical methods, several ways to enhance the directional emission from photonic crystal waveguides through the beaming effect recently predicted by Moreno et al. [Phys. Rev. E 69, 121402(R) (2004)], by engineering the surface modes and corrugation of the photonic crystal surface. We demonstrate that the substantial enhancement of the light emission can be achieved by increasing the refractive index of the surface layer. We also measure power of surface modes and reflected power and confirm that the enhancement of the directional emission is related to the manipulation of the photonic crystal surface modes.

One of the recent advances in the physics of photonic crystals is the discovery of enhanced transmission and highly directional emission from photonic crystal waveguides predicted theoretically by Moreno et al. [1] and demonstrated independently in experiment by Kramper et al. [2]. These results provide a new twist in the study of surface modes in photonic crystals. Indeed, it is generally believed that surfaces and surface modes are a highly undesirable feature of photonic crystals, unlike point defects which are useful for creating efficient waveguides with mini-band gaps inside the photonic band gaps of a periodic structure. However, appropriate corrugation of the surface layer may lead to coherent enhancement of the radiating surface modes and highly directional emission of the light from a truncated waveguide [1,2].
As already mentioned by Moreno et al. [1], the major motivation for the discovery of highly directional emission from photonic crystal waveguides is largely provided by the physics of extraordinary optical transmission through subwavelength hole arrays in metallic thin films [3] and beaming of light from single nanoscopic apertures franked by periodic corrugations [4]. In both those cases, an incident light beam couples to the surface plasmon oscillations via corrugations in a metallic film, and is then emitted from the other side of the film being enhanced by its other corrugated surface. For photonic crystal waveguides, properties of the surface layer [1] or terminated surface [2] provide a key physical mechanism for the excitation of surface modes, their constructive interference, and subsequent highly directed emission.
In this paper, we study, by means of the finite-difference time-domain (FDTD) numerical method, the directional emission from a photonic crystal waveguide achieved by appropriate corrugation of the photonic crystal interface, following the original suggestion [1]. We analyze several strategies for enhancing the light beaming effect by varying the surface properties and by engineering the surface modes of a semi-infinite two-dimensional photonic crystal created by a square lattice of cylinders in vacuum. In particular, we optimize the corrugation at the surface, as well as vary the refractive index of the surface layer. We demonstrate that, in comparison with the previously published results [1], the substantial enhancement of the light emission and improved beaming effect can be achieved by increasing the refractive index of the surface layer while using a positive (i.e. opposite to that employed in Ref. [1]) corrugation displacement. We also measure the power of surface modes and reflected power and confirm that the enhancement of the directional emission through the beaming effect links closely to the manipulation of the surface modes supported by the photonic crystal interface.
We consider a photonic crystal slab created by a square lattice of cylinders with dielectric constant ε r = 11.56 (e.g. GaAs at a wavelength of 1.5 µm) and radius r = 0.18 a, where a is the lattice period. A row of cylinders removed along the plane x = 0 forms a single-mode waveguide (see Fig. 1) that supports a guided mode with frequencies between ω = 0.30×2πc/a and ω = 0.44 × 2πc/a propagating in the plane normal to the cylinders, with the electric field parallel to them.
When a source is placed in the waveguide at the point z = 0, it excites waves that propagate along the waveguide and are then emitted at the waveguide exit (at z = 9a). Since no surface modes are supported by a simple truncated slab, the light radiating from the waveguide undergoes uniform angular diffraction as demonstrated in Fig. 1(a) for the spatial distribution of the Poynting vector calculated for the source frequency ω = 0.408 × 2πc/a.
To characterize the transmission from the photonic crystal waveguide, we measure the directed power P D , normalized to the input power, incident upon a cross-sectional length of 2a centered at x = 0 and z = 45 a. A likewise normalized measure is taken of the reflected power P R incident upon a cross-sectional length of 20 a centered at the input to the waveguide, x = 0 and z = −a. This reflected power is considered a close measure of all reflected power. For the bulk photonic crystal with standard surface layer the directed power is P D = 0.0123, and the reflected power is P R = 0.0158.
Distribution of the Poynting vector for the directional emission from the photonic crystal waveguide demonstrated by Moreno et al. [1] is shown in Fig. 1(b). These results are produced by altering the surface layer geometry in two ways. Firstly, by reducing the radius of the surface cylinders to the value r s = 0.5r = 0.09 a, and thereby creating the conditions for a surface mode to exist at the truncated surface. And secondly, by displacing N = 9 even-numbered cylinders (numbered consecutively away from the waveguide) on both sides of the waveguide by ∆z = −0.3 a along the z−axis of the crystal, thus enhancing radiation of surface modes. Our calculations show that the directed power for such a structure is P D = 0.0723, while the reflected power is substantially large, P R = 0.2635. To further characterize the enhanced beaming effect, we measure one half of the total surface mode power, P S , incident upon a cross-sectional length 2 a positioned centrally at x = 24 a, z = 9 a; again normalized to the input power. Moreover, to characterize the containment of the directed power we measure the width of the central lobe of the directed emission w L between the first nulls at z = 45 a. For the geometry considered in Ref. [1], the surface mode power is P S = 0.0030, while the width of the central lobes is w L = 18.1 a.
A significant drawback of a surface layer design suggested in Ref. [1] is a large amount of the reflected power. We find that the reflected power can be reduced by trapping the electrical field mostly within the surface layer, as occurs for the uncorrugated surface. Increasing the applied wavelength by 4.4% from λ = 2.45 a to λ = 2.55 a to account for the proportionally increased distance resulting from the corrugated surface cylinders allows us to decrease the reflected power to P R = 0.048, while increasing the directed power and surface mode power marginally to P D = 0.0768 and P S = 0.0484, respectively. A measure of the average wave impedance in the vicinity of the waveguide shows that the increased wavelength reduces the impedance from ∼ 1000Ω to ∼ 320Ω.
In order to increase the directional power, we alter the surface layer structure by shifting the even-numbered cylinders forward by the distance of ∆z = 0.4 a, while leaving the oddnumbered cylinders on the lattice sites (i.e. no displacement). As the increased distance due to this corrugation over that of the uncorrugated distance is 7.7%, the applied wavelength is increased proportionally to λ = 2.63865 a. This new surface produces the directed power of P D = 0.15418 and decreased reflected and surface mode powers of P R = 0.0318 and P S = 0.0100, respectively. Furthermore, the central lobe of the directed emission is now contained within w L 7.79 a. Our analysis shows that substantial improvement to the directed power can be achieved by increasing the refractive index of the surface layer from n s = 3.4 to n s = 3.6. This results in the directed power increasing to P D = 0.1689, while decreasing the reflected and surface-mode power to P R = 0.0295 and P S = 0.0023, respectively. The width of the directed beam's central lobe resulting from the increased surface layer's refractive index is w L = 9.553 a. The increased power is achieved by decreasing the light-line slope, thus placing the surface mode closer to the continuum of radiative modes.
Additional improvement of the directed power can be achieved by decreasing the radius of the cylinders one layer prior to the surface layer, z = 8 a to r s−1 = 0.135 a. This change induces a near-surface defect mode that leaks coherently into the surface layer before being radiated, increasing the directed power to P D = 0.2104, the reflected power, to P R = 0.1028, and decreasing the surface power to P S = 0.0078. The width of the central lobe of the directed emission becomes w L = 8.642 a. The spatial distribution of the Poynting vector for this optimal design is shown in Fig. 1(c). A comparison of the significantly enhanced beaming over the standard interface and that of Ref. [1] is provided in Fig. 2, for a cross-section of the power density measured at z = 45 a.
Control of the directed emission is achieved through the manipulation of the refractive index of the surface layer cylinders. This is illustrated in the attenuation of the directed power shown in Fig. 1(d), where the refractive index of the surface cylinders is increased to the value n = 4.5. In this case, the outgoing beam splits, the directed power vanishes, and the surface-mode is in cut-off with a localized state formed within the first two surface cylinders next to the waveguide exit. Figure 2(d) shows a cross-section of the power density measured at z = 45 a for the beam splitting depicted in Fig. 1(d).
The effect produced by a change of the surface refractive index is demonstrated is Fig. 3 where the index is varied from n = 2.4 to n = 4.4. As already mentioned, the refractive index of the surface layer has a profound effect on both the directed and reflected powers, suggesting that it could be used not only for achieving a control over the beaming effect but also for matching Top: surface layer used in Ref. [1]. Bottom: with the surface cylinders' radius reduced to r s = 0.09 a and the refractive index n s = 3.6, with N = 9 odd-numbered cylinders displaced ∆z = +0.4 a, and the radius of the cylinders in the layer prior to the surface layer reduced to r s−1 = 0.135 a. Fig. 4. Normalized power density incident upon a cross-section of the length 2 a centered at x = 0 and z = 45 a as the radius of the surface cylinders varies. Top: surface layer from Ref. [1] but for different values of r s . Bottom: with the radius of the surface cylinders reduced to=20 r s = 0.09 a and their refractive index increased to=20 n s = 3.6, with N = 9 odd-numbered cylinders displaced by ∆z = +0.4 a.
the waveguide to the surrounding media.
The influence of the radius of the surface cylinders on the beaming effect is summarized in Fig. 4, where the radius is varied from r s = 0.045 a to r s = 0.2 a. The radius is the key parameter in the inducement of the surface mode and these results illustrate clearly that the optimum radius is indeed close to r s = 0.9 a used in Ref. [1].
In conclusion, we have implemented different strategies for the enhancement of the light beaming effect by engineering the surface modes of photonic crystals. In particular, we have revealed that, in comparison with the previous studies, the substantial enhancement of the light emission and improved light beaming can be achieved by increasing the refractive index of the surface layer. We have provided a link of the observed enhancement of the directional emission with the properties of the surface modes supported by the photonic crystal interface.
We acknowledge a partial support of the Australian Research Council and useful discussions with Sergei Mingaleev and Costas Soukoulis.