We propose a novel photonic crystal waveguide design with defects at the termination to provoke directional emission. Because the resulting optical field distribution at the waveguide exit resembles a triple point source, the light can emit with low angular divergence. This phenomenon appears both in photonic crystal structures formed by dielectric rods as well as by air holes.
© 2006 Optical Society of America
Photonic crystal (PC) structures have been intensively studied for several applications such as waveguides in purely 2D systems , waveguides in 2D slab PCs , waveguides in novel materials [3,4], microsized light sources , reflectors , etc. Some of these devices were designed based on peculiarities of the dispersion relation of the pertinent Bloch waves which are difficult to meet in conventional optics. Prominent examples are negative refraction , self-collimation [4,8], and directional emission [9–11]. This directional emission has been attributed to the excitation of a surface mode at the PC endface, which couples via a surface corrugation to the continuum of air modes. It has been shown  that any photonic crystal – dielectric medium interface may support a surface mode provided that an appropriate PC termination has been selected. By appropriately selecting the corrugation period, directional emission can be achieved at a predefined angle , in particular, normal to the interface. However, a 2D square lattice PC consisting of dielectric rods requires a different interface monolayer for surface mode excitation. To improve the directional emission characteristics, the configuration can be optimized in decreasing the input wavelength, increasing the refractive index of the surface layer and using a positive corrugation displacement for the periodic modulation of the rods position at the PC surface . Experimental evidence of the phenomenon of collimation and enhanced transmission has been presented in Ref. 13 by measuring the field distribution of microwave emitted by PC waveguide formed by dielectric rods. Directional emission has also been experimentally proven in the reversed configuration, namely a 2D silicon PC with a hexagonal lattice of air holes . From the application point of view, directional emission is attractive for enhancing the coupling efficiency of light from photonic crystal waveguides into conventional dielectric waveguides or to control the emission direction of PC edge-emitting lasers . However, the involved engineering of the interface monolayer may prevent the easy implementation of this concept.
Thus, in this paper we propose an alternative PC structure to obtain directional emission, even for a homogeneous square lattice. Two point defects are added near the PC termination producing a kind of triple point source, which evokes directional emission regardless of the details of the very PC termination. We demonstrate that this concept holds for PCs formed by dielectric rods, but also by air-holes.
2. Simulation and results
We consider a square lattice of dielectric rods in air. We choose the dielectric constant and the radius of the rods to be εr=9 and r=0.2a, respectively, where a is the lattice constant. For TE-polarization (electric field pointing along the rod axis), the photonic bandgap of the PC structure is calculated by the plane wave expansion (PWE) method and can be found in the normalized frequency range between 0.322 and 0.441ωa/2πc where ω and c are the angular frequency and the light velocity in vacuum, respectively. A so-called W1 PC waveguide is built in the structure by removing one row of rods along the Γ-X direction. Two point defects are added by removing two rods near the end of the PC waveguide as shown in Fig. 1(a). [Detail of the structure is shown in Fig. 2(a)] A Gaussian TE-polarized light with the wavelength of 2.5a is launched into the PC waveguide. The width of the launched Gaussian beam is 1a. The finite-difference time-domain method is used to calculate the propagation of the light. Figures 1(a) and (b) show the output field distribution of the PC waveguides with and without the two point defects, respectively. By comparing Fig. 1(a) and (b), we can observe that the angular divergence of the output beam is significantly narrowed by adding the two point defects.
To study the sensitivity of the beam directionality to the wavelength, we launch a pulse into the PC waveguide with two point defects [Fig. 1(a)]. The temporal change of the field intensity from the detectors as shown in Fig. 1(a) is recorded. The distance between the detectors and the output of the waveguides is 15a. Fast Fourier transform is used to obtain the spectral information. Figure 1(c) shows the spectrum for each corresponding detector. The intensity of the optical field for different output angles can be analyzed. We can observe that the intensity difference between the detectors changes significantly as the wavelength of the light is 2.5a. The ripples at the wavelength from 2.8a to 3.1a in these spectra might be due to a difference in the coupling to the waveguide for different frequencies or to a Fabry-Pérot effect in the finite length waveguide. The ratio of the field amplitude received by the detector at 0° and 30° is also shown in Fig. 1(c). At the wavelength of 2.5a, we can observe that the ratio is maximum indicating that the divergence of the emitting beam is minimum. We plot the angular distribution of the output intensity in Fig. 1(d). The full-width at half-maximum is around 30°. By using this design procedure, the wavelength for minimum beam divergence can be found. Inspecting the curve, which displays the ratio, the full-width at half-maximum of the peak at the wavelength of 2.5a is around 0.0515a. If the normalized wavelength of 2.5a corresponds to 1.55μm, the 3dB bandwidth of the device covers a range from 1531nm-1568nm, which corresponds to the entire C-band for fiber optics communication systems.
Figure 2(a) shows the optical field distribution at the end of the PC waveguide of Fig. 1(a). The two point defects produce two strong resonant modes, which radiate off the PC structure. The resonance frequency of the single defect (only one rod missing without waveguide) is calculated by PWE to be 0.395, which corresponds to the frequency for the directional emission to be 0.4 (=1/2.5) as shown in Fig.1(c). The two resonant modes and the optical field emitted by the PC waveguide can be regarded as three radiating sources. The interference between these three sources produces a directional emitting beam and also two weak lateral lobes of radiation as shown in Fig. 2(a). This phenomenon can be simulated by three point sources in free space. The resulting optical field distribution is illustrated in Fig. 2(b). The distance between the point sources is 2a, i.e., the position of the point sources is (-2a, -2a), (0, -2a) and (2a, -2a), respectively, and they radiate at a wavelength of 2.5a. We can also observe the directional emitting beam and two weak lateral lobes of radiation. In Fig. 2(a), the wavefront at the waveguide output (region encircled by dashed lines) is similar to that in Fig. 2(b). These phenomena confirm that the directional beam emitted from the PC waveguide with two defects originates from the interference between the waveguide and defect modes.
By tuning the properties (position and phase, etc.) of the defect modes at the output of the PC waveguide, the diffraction efficiency of the directional emitting beam can be modified. As an example, we add two additional rods at the position of (-4a, 3a) and (4a, 3a) as shown in Fig. 2(c). The same spectral analysis as in Fig. 1(c), now performed for Fig. 2(c). The ratio of the field amplitude received by the detector at 0° and 30° is shown in Fig. 2(d). At the wavelength of 2.442a, we can observe that the ratio is maximum indicating that the divergence of the emitting beam is minimum. We plot the angular distribution of output intensity in Fig. 2(e). The full-width at half-maximum is around 16°. The intensity of the lateral lobes of radiation is also reduced. The results imply that the number of the point defects may have influence on the directionality of the beam and the diffraction efficiency of the lateral lobes of radiation. However the maximum intensity of the directional beam in Fig. 2(e) is around 0.052 which is lower than that in Fig. 1(d) to be 0.16. This behavior may be due to the fact that as increasing the number of point defects, the power stored in the point defects is increased. More power of the light should flow into the defects. Since the defects are near the surface of PC structure, a part of light radiates into the air region. The rest of power reflected back into the waveguides may be increased. The result implies that there may be a tradeoff between the output efficiency and the directionality. In Fig. 2(f), we move the point defects a rod away from the waveguide. The same spectral analysis is performed. The results are shown in Fig. 2(g) and (h). The maximum intensity difference between 0° and 30° is found at the wavelength of 2.527a, which is similar to the resonant wavelength of the point defect. The results obtained these two structures above imply that the frequency for directional emission is tightly correlated with the resonant frequency of the point defect. In Fig. 2(f), light at the normalized frequency of 0.396 is launched into the PC waveguide. The two resonant modes and two lateral lobes of radiation can still be observed in the point defects. The angle between the two lateral lobes of radiation is 52°. The diffraction efficiency for the two lateral lobes of radiation is much higher than that for the directional emission. The result shows that by modifying the structure of the point defects, the directional emission properties can be changed. The proposed structures producing directional emission may be used to enhance the coupling efficiency between conventional waveguides and PC waveguides .
Another application of the directional emission is to narrow down the beam divergence of edge-emitting lasers. Usually, this is achieved by adding some optical lenses  at the output of the laser. This increases the size of the devices and the optical loss. Therefore, we propose to use the air-hole defects adjacent to the PC waveguides in dielectric material to obtain a directionally emitting beam.
The second structure has been designed in a low-index-contrast PC  due to its potential application in the visible and infrared regions. In order to obtain vertical light confinement such PCs are often experimentally realized as patterned slab waveguides. Because we focus on the inplane dynamics of the light propagation we restrict the analysis to purely 2D calculations. However, we account for the slab waveguide by using the effective index of the corresponding fundamental waveguide mode instead of the (larger) material index in these calculations. The air-holes are hexagonally arranged in a Nb2O5 substrate with an effective refractive index of 1.73. The radius of the air-holes is 0.4a. For TM-polarization, the photonic bandgap of the PC structure can be found in the normalized frequency range between 0.455 and 0.508ωa/2πc. A W1 PC waveguide is built in the structure by removing one row of airholes along Γ-K direction. The output of waveguide is closed by three air holes to obtain a higher reflection for the application of edge-emitting lasers. Two defects are added by removing four air-holes near the end of the PC waveguide as shown in Fig. 3. A TM-polarized light is launched into the PC waveguides. Using the same design procedure mentioned above, the wavelength of 2a is chosen to obtain the minimum angular divergence. Figure 3(b) shows the optical field distribution at the end of the PC waveguide. We can observe strong resonant modes at the defects as those in Fig. 2(a) and the fact that the output beam is directionally emitted. This directional emission in low index materials could find a potential application in an erbium-doped PC edge-emitting laser. For the emission lines of erbium are at 537 and 1550nm , the corresponding diameter of the air-hole is 215 and 620nm, respectively. These feature sizes of the PC structure can be easily fabricated by contemporary e-beam photolithography technique.
In this work, we have proposed novel PC structures to obtain directional emission of light. Modes located at properly designed defects near the waveguide exit interfere with the waveguide mode leading to directional emission. The corresponding frequency is closely related to the resonant frequency of the point defect. We demonstrated that this structure can be applied in PCs formed by dielectric rods but also by air-holes. The former can be used to enhance the light coupling efficiency from PC waveguides to conventional waveguides. The latter can be used for narrowing the beam divergence of the edge emitting PC lasers.
The authors are grateful for the financial support of the National Science Council (Grant number 43080F) and the Deutsche Forschungsgemeinschaft, Priority Program ‘Photonic crystals’.
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