1. Introduction

In the past decade, there has been significant effort in the development of photonic crystals (PCs) due to their potential applications in ultra compact photonic components [1], such as waveguides [2], filters [3], lasers [4], cavities [5], and sensors [6]. Many of these applications take advantage of the photonic bandgap imposed by the periodic structure to the propagating electromagnetic fields. On the other hand, there has been a growing interest for PC applications based on the behavior of modes outside the bandgap such as negative refraction index [7] and self-collimation [8, 9].

In recent years, the application of PCs to obtain both enhanced transmission and strong beaming of light coming out of PC waveguides has increased significantly. It is shown both theoretically [10] and experimentally [11] that the use of surface leaky modes, appearing in PCs under proper conditions, can be engineered to collimate electromagnetic radiation leaving a PC waveguide. To improve the beaming effect achieved by using the properties of surface modes of a PC interface, different ways, such as increasing the input source frequency and surface layer refractive index, as well as using a positive surface corrugation, have been proposed [12]. In addition, the beaming effect of light by using increased-index PC waveguides was studied in Ref. [13]. The authors have numerically analyzed the conditions required to achieve optimal beaming from the increased-index waveguide by determining the sources of losses and inefficiencies using FDTD calculations.

Furthermore, it has been proposed to obtain directional emission by using a PC waveguide with two point defects added near its termination. It was shown theoretically that this structure can be applied for PCs not only with dielectric rods, but also with air holes [14]. All these proposed structures rely on the photonic bandgap phenomenon and use the confinement properties of PCs.

Alternatively, the negative refraction index, the outside-gap phenomenon, which can occur in PCs under certain conditions, has been used to realize focusing devices [15–17]. In addition, self-collimation has been exploited to collimate the propagating waves without considerable broadening and changing in the beam profile [18–22].

In Ref. [23] a combination of self-collimation and directional emission has been suggested to realize the coupling of free space terahertz waves into a silicon slab. The authors have used FDTD calculations to theoretically show that highly-efficient directional emission is provided by a PC structure with two parallel line-defect waveguides.

In this work, we experimentally confirm the theoretical predictions of Ref. [23]. To back up our experiments theoretically we employ the Finite-Element-Method (FEM). We study a PC waveguide with two adjacent line defects. As a reference we also investigate a PC slab without any waveguide structure. The latter structure is discussed first.

2. Theory and design

Figure 1(a) depicts the PC slab, a structure consisting of a square array of cylindrical air holes in a block of high density polyethylene (HDPE). The refractive index of HDPE in the microwave range is n _HDPE=1.53. The radius of each hole is r=0.185a, where a=1.89 mm is the lattice periodicity. The PC dimensions are 20a (width) and 11a (length) and 6a (thickness). The photonic crystal structure starts at Y=0 and ends at Y=11a and the dielectric slab starts at Y=11a and ends at Y=32a. Air/dielectric interfaces are at Y=0 and Y=32a.

 

Fig. 1. (a). PC slab. (b). Electric field distribution in the PC slab.

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Light propagation through a PC structure is controlled by its dispersion surface. The dispersion surface presents the spatial variation of the spectral properties of a certain band or eigenmode supported by a PC structure in k-space. To evaluate the information content within a certain dispersion surface, a section of this surface is selected at a constant frequency. Such a cross section is identified as an equi-frequency contour (EFC). Incident light waves propagate in direction normal to the dispersion surface. The importance of the EFCs comes from the fact that the group velocity, or the direction of the light propagation, corresponds to the direction of the steepest ascent of the dispersion surface and is perpendicular to the EFCs [22].

The equifrequency contours (EFCs) of the second band of the PC for TM polarization (electric field parallel to the hole axis) are shown in Fig. 2. They are obtained with a semi-analytical method, in which the full-wave analysis of multilayer PCs can be reduced to the analysis of an equivalent multi-conductor transmission line (TL) by using a matrix formulation. In the particular case of single-layer two-dimensional PCs the inductance and capacitance matrices of this equivalent TL provide the propagation characteristics of the crystal under investigation [24].

 

Fig. 2. The equifrequency contours of the second band of the PC for TM polarization.

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For self-collimation square EFCs where the wave is only allowed to propagate in directions normal to the sides of the squares are required. As a result, it is possible to change the incident wave vector over a wide range of angles and retain a narrow range of propagation angles within a PC structure. As can be seen from Fig. 2 the EFCs can be approximated by a square for the normalized frequencies 0.5≤a/λ≤0.62, which corresponds to the frequency range 79–98 GHz. We chose this particular frequency range since the PC structure can easily be fabricated in a mechanical machine shop.

The propagation of the electromagnetic waves in the PC structures is calculated using the FEM based software Comsol [25]. To obtain the electric field distribution of the TM polarization a Gaussian wave is used as an excitation source. The source of the sub-terahertz waves is placed outside of the photonic crystal structure. From the left side of the photonic crystal a normal incident Gaussian wave with 6.5 mm of width is launched into the structure. The simulated electrical field distribution for 95 GHz is shown in Fig. 1(b). It can be seen that the waves are efficiently coupled to the PC structure by self-collimation. The advantage of the self-collimation effect is that it does not require a precise alignment and specific physical boundaries for coupling the free space electromagnetic waves into the PC slab.

A second PC structure is shown in Fig. 3(a). It includes two adjacent parallel line defects that are introduced in the PC by removing some of the air holes of the two rows.

 

Fig. 3. (a). PC waveguide with two line defects. (b). Electric field distribution in the PC waveguide.

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It has been shown theoretically that a higher coupling efficiency can be achieved if two waveguides are introduced to confine the waves [23]. This effect adds to the self-collimation effect discussed above. In fact, the free space sub-terahertz waves are coupled efficiently to the PC structure via self-collimation in a first step. In a second step they are guided via the two waveguides. At the interface of the PC structure and the dielectric slab, the outputs of these waveguides act analogously to a two-point source [23]. The constructive interference of these outputs results in an efficient directional emission.

The simulated electrical field distribution of the PC waveguide at 95 GHz is shown in Fig. 3(b). As can be seen, the beam focusing is more efficient with two line defects than in the case of the PC slab without line defects.

As a measure for the coupling efficiency of the two structures shown in Figs. 1(a) and 3(a) we numerically integrate the total power which is transmitted through the spatial window ranging from -1.85a≤X≤1.85a at Y=32a. A Gaussian beam profile is assumed at the PC entrance. For comparison we also investigate a bare dielectric slab (not shown), i.e. a slab without any PC structure. Figure 4 shows the simulated normalized transmitted power of the three structures versus the frequency. The transmitted power directly in front of the photonic crystal (Y=0) has been normalized to the power directly after the photonic crystal (Y=32a). For 95 GHz a coupling efficiency of 70% is reached for the structure with two adjacent waveguides. Over the entire frequency range the PC slab has a lower efficiency than the PC structure with waveguides. Hence, we can conclude that an increased coupling of TM modes to a dielectric slab can be achieved by a combination of the self-collimation effect and a guiding structure such as two line defects. However, an enhanced coupling efficiency is obtained only in a certain frequency window. From Fig. 4 it can be seen that the coupling efficiency of the PC waveguide is higher than that of the bare slab only for frequencies below 98 GHz. For the frequencies above this value the EFCs are no longer square-like and the electromagnetic waves can not be spatially concentrated via self-collimation. Consequently, the PC waveguide has a lower coupling efficiency than the bare slab for frequencies above 98 GHz.

 

Fig. 4. Simulated normalized transmission versus frequency for the three structures studied.

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3. Experimental results

To validate these theoretical findings, the structures of Figs. 1(a) and 3(a) as well as a bare slab are fabricated from HDPE using a standard machining process. To measure their transmission properties we use an Agilent E8361A network analyzer. The network analyzer is equipped with two transmitter/receiver mixers (Agilent N5260-60004) enabling the S-matrix parameters in the W band (75–110 GHz) and in part of the V band (67–75 GHz) to be characterized.

The sub-terahertz radiation is focused and guided by two polyethylene plano-convex lenses with focal length of 100 mm. The samples are placed in an intermediate focus where we have a Gaussian beam profile. To ensure that the microwaves propagate exclusively through the structures, two metal pinholes are used at the input. The first metal pinhole has a circular shape with the diameter of 24 mm. The second which is attached to the input side, i.e., to the left side of the photonic crystal structure has rectangular shape with the dimension of the photonic crystal cross section in x-z plane. Moreover, an additional pinhole ranging from -1.85a≤X≤1.85a and 0≤Z≤6a was attached to the output of the structures at Y=32a. The setup of the experiment is shown in Fig. 5. Admittedly, there are some diffractions of the radiation through the pinholes that deteriorate the performance of the experimental work in comparison with simulation result. In this paper, we do not consider the diffraction quantitatively; however, we attribute some differences between the experiment and simulation to the wave diffraction.

 

Fig. 5. The experimental setup.

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The simulations shown above are carried out only for a single propagation through the structure. Reflections which might occur at the ends of the structure, i.e. at Y=0 and Y=32a, are not considered. Yet, experimentally one sees the effect of those reflections as Fabry-Perot resonances that produce a fine-scale structure on the transmission spectra. Hence, we smooth the raw data using a fast Fourier transform filter. The resulting spectra are shown in Fig. 6. They clearly confirm that the highest coupling efficiency can be obtained when using a PC waveguide structure in the self-collimation frequency range. The maximum coupling efficiency obtained experimentally is 63%.

 

Fig. 6. Measured normalized transmission versus frequency for the three structures studied.

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The results of the experiment are in good agreement with those of the simulation. We note that the bare slab has a bigger efficiency than a PC slab in the experimental work. We attribute this to diffraction caused by the input pinhole. The diffraction is less pronounced for the bare slab, as it has a lower refractive index than the PC slab. In addition, some discrepancies in the magnitude of the normalized transmission are related to the Fabry-Perot resonances, scattering losses, and the mismatch at the interface between the structure output and the free space part before the receiver antenna. However, in real applications of this structure for the integrated circuits these problems do not exist and an improved coupling efficiency can be achieved.

4. Conclusion

We have studied the coupling of free space waves to dielectric slabs using two-dimensional PC waveguides by simulations and experiments. Introducing two line defects into the PC slab has increased the coupling efficiency significantly. The results show that the PC waveguide can be applied in photonic integrated circuits to serve as an efficient coupling device.