1. Introduction

Dielectric periodic media can possess a complicated photonic band structure where allowed bands display strong dispersion and anisotropy [1, 2, 3]. The form-anisotropy is the origin of a number of interesting optical phenomena, usually referred to as superprism or ultrarefractive phenomena [4, 5, 6, 7]. The anisotropic energy flow of Bloch waves inside a photonic crystal can lead to extraordinary large and negative beam bending [6], self-collimation [8] or photon-focusing [9, 10]. In this paper we present a detailed theoretical study of self-collimation phenomena in two-dimensional photonic crystals.

A self-collimated beam of light does not spread when it propagates in photonic crystal. In contrast to spatial solitons, where the nonlinearity of homogeneous medium counteracts the natural spreading of the beam due to diffraction, the formation of self-collimated, or self-guided, beams in photonic crystal is a purely linear phenomenon. The spreading of the beam is counteracted by the crystal anysotropy, such that all wave vectors building the beam lead to the power flux in the same direction. This can be realized for the wave vectors ending at the flat regions on an iso-frequency surface of a photonic crystal. The self-collimation regime reported by Kosaka et al. [6] relies on the inflection points of an iso-frequency surface, where the Gaussian curvature of the surface tends to zero. Typically, a flat region spreads over very limited wave vector directions centered at the wave vector ending at the inflection point. As a consequence, self-collimation can occur only for very limited orientations of the beam with respect to the crystal lattice and for limited beam widths.

 

Fig. 1. Convex (red), concave (blue) and flat (black) iso-frequency contours. Iso-frequency contours on the left (right) panel correspond to the second band of a two-dimensional square (triangular) lattice photonic crystal made of dielectric rods in vacuum. Rods have the refractive index 2.9 and radius r=0.15d, where d is the lattice period. The wave vectors resulting in the Bloch eigenwaves with the group velocity pointing to the direction normal to the Brillouin zone boundary are depicted with arrows.

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In this paper, we show that for some frequencies the form of iso-frequency surface mimics the form of the Brillouin zone of the crystal. In Fig. 1, iso-frequency contours for realistic square (left) and triangular (right) lattice photonic crystals are depicted. With changing frequency, the anisotropy of a photonic crystal changes dramatically, its iso-frequency contours evolve from convex (red contours in Fig. 1) to concave (blue contours in Fig. 1). There exists a frequency range where an iso-frequency contour forms almost perfect square or hexagon. Then, a wide angular range of flat dispersion exists canceling out diffraction of a light beam with the corresponding range of wave vectors (Fig. 1).

 

Fig. 2. Photonic band structure of the square lattice photonic crystal made of dielectric rods in vacuum. Rods have the refractive index 2.9 and radius r=0.15d, where d is the lattice period. The band structure is given for TM polarization. The frequency is normalized to Ω=ωd/2πc=d/λ. c is the speed of light in the vacuum. Insets show the first Brillouin zone (left) and a part of the lattice (right).

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Further, without loss of generality we restrict our consideration to the case of a two-dimension photonic crystal based on a square lattice. In Section 2, we give a physical insight on the the self-guiding phenomenon. We analyze a radiation pattern of a point source in photonic crystal, taking advantage of an approximate analysis based on iso-frequency contours in the k-space. The direct rigorous numerical simulation of the beam propagation in a finite size photonic crystal is given in Section 3. Section 4 summarizes our results.

2. Fourier space analysis

In the present study we model an infinite size 2D photonic crystal (Fig. 2) and we restrict ourselves to the case of in-plane propagation. Consequently, the problem of electromagnetic wave interaction with a 2D photonic crystal is reduced to two independent problems, which we call TE (TM) when the magnetic (electric) field is parallel to the rods axis. In what follows, we limit ourselves to the case of TM modes. The photonic band structure of the square lattice photonic crystal made of dielectric rods in vacuum is shown in Fig. 2. The refractive index of the rods is 2.9 and their radius is r=0.15d, where d is the period of the lattice. A band structure calculations was done using the plane wave expansion method [11].

We analyse the first two bands of the crystal, the iso-frequency contours of which evolve from convex to concave topology. In Fig. 3 transition iso-frequency contours corresponding to the normalized frequencies Ω=d/λ=0.3567 (the first band) and Ω=d/λ=0.5765 (the second band) are depicted.

The iso-frequency contour for the normalized frequency Ω=d/λ=0.3567 forms a square with rounded corners, rotated by 45± with respect to the Brillouin zone and centered at M-point of the first Brillouin zone (Fig. 3, left). The iso-frequency contour for the normalized frequency Ω=d/λ=0.5765 consists of two branches, which are plotted in red and blue in Fig. 3, right. Both branches mimic the form of the first Brillouin zone of the crystal being squares with rounded corners. The “red” branch is centered at the Γ-point and the “blue” branch is centered at the M-point of the Brillouin zone (Fig. 3, right).

 

Fig. 3. Iso-frequency diagram for the normalized frequencies Ω=d/λ=0.3567 (left) and Ω=d/λ=0.5765 (right). The shaded regions show the wave vectors resulting in the Bloch eigenwaves with the group velocity vectors pointing to the same direction. The crystal parameters are given in Fig. 2 caption.

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To analyze how this special kind of anisotropy affects light propagation in a photonic crystal, we calculate a radiation pattern of a point light source radiating into the crystal.

The field of an arbitrary light source embedded in the periodic medium can be constructed by a suitable superposition of the eigenwaves of the medium, ${\mathbf{A}}_{n\mathbf{k}}\left(\mathbf{r}\right)={\mathbf{a}}_{n\mathbf{k}}\left(\mathbf{r}\right)e^{i{\mathbf{k}}_n\mathbf{r}}$ . The eigenwave A _nk(r) is dimensionless and satisfies orthogonalization and normalization relations [12]. Here, A denotes the vector potential. We assume that a light source is a harmonically oscillating dipole with a frequency ω₀ and a real dipole moment d, located at the position r ₀ inside a photonic crystal. Then the field at the position r in the crystal reads [12]

where ω_nk is the Bloch eigenfrequency, V is the volume of the unit cell of the crystal, * denotes the complex conjugate and c is the speed of light in vacuum. Integration is performed over the first Brillouin zone and summation is over different photonic bands, where n is a band index.

For large x=r-r₀ the exponential function in the integral (1) will oscillate very rapidly. To evaluate this integral we use the fact that in a typical experiment |x|≫λ, where λ is the wavelength of the electromagnetic wave. Then, by the method of stationary phase, the main contribution to the value of the integral (1) comes from those regions of the iso-frequency surface ω=ω₀, where product k _n x is stationary. We denote the stationary phase points by ${\mathbf{k}}_n^{\nu }$ . Following the evaluation presented by Camley and Maradudin [13] for the asymptotic analysis of the Green’s function of the phonon scattering problem, one can show [14] that the asymptotic form of the vector potential (1) at the position r far from the point dipole is given by

 

Fig. 4. Radiation pattern of a point source inside the photonic crystal with parameters given in Fig. 2 caption. The normalized frequencies are Ω=d/λ=0.3567 (left) and Ω=d/λ=0.5765.

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where V _nk=∇_kω_nk is the group velocity of the eigenwave (n,k), $K_{n\mathbf{k}}^{\mathrm{\nu }}$=${\mathrm{\alpha }}_1^{\mathrm{\nu }}$${\mathrm{\alpha }}_2^{\mathrm{\nu }}$ determines the Gaussian curvature of the iso-frequency surface at the stationary point k _n =${\mathbf{k}}_n^{\mathrm{\nu }}$ and summation is over all stationary points with x·${\mathbf{V}}_{n\mathbf{k}}^{\mathrm{\nu }}$>0.

The amplitude of the vector potential in (2) is proportional to |$K_{n\mathbf{k}}^{\mathrm{\nu }}$|^-1/2 and |x|^-1 steady-state emission intensity far from a point dipole is given by the time averaged Poynting’s vector S(r)=(c/8π)Re[E(r)×H*(r)] and it is proportional to the inverse Gaussian curvature of the iso-frequency surface, ~|$K_{n\mathbf{k}}^{\mathrm{\nu }}$|^-1 and to the inverse square of the distance between a source and an observation point, ~|x|^-2. The asymptotic energy flux shows the necessary amount of decrease with distance (~|x|^-2, providing a finite value of the energy flux in any finite interval of a solid angle, assuming non vanishing Gaussian curvature. A vanishing curvature formally implies an infinite flux along the corresponding observation direction. This phenomenon is known as photon focusing [9, 10, 14].

Using the method described above, we have modeled radiation patterns due to a point source inside the photonic crystal (Fig. 4) for the normalized frequencies Ω=d/λ=0.3567 and Ω=d/λ=0.5765 (Fig. 3). We assume that a point source produces an isotropic and uniform distribution of wave vectors k _n . Then after an averaging of (2) over the dipole moment orientation, the main contribution to the relative radiation intensity far from the point source is given by an inverse Gaussian curvature of the iso-frequency surface, ~|$K_{n\mathbf{K}}^{\mathrm{\nu }}$|^-1.

In Fig. 4, an angular distribution of radiated intensity is presented for the normalized frequencies Ω=d/λ=0.3567 (left) and Ω=d/λ=0.5765 (right) for the crystal parameters given in the caption of Fig. 2. The important feature of the wave propagation inside a photonic crystal at the considered normalized frequencies is that even a cylindrical initial wave results in an energy flow strongly focused along some specific directions, while it remains negligible for all other directions. To emphasize this special type of self-collimated electromagnetic wave propagation inside a photonic crystal we will refer to it as self-guiding.

For the first band (Ω=d/λ=0.3567, Fig. 4, left), the radiation pattern is strongly focused along the [11] direction of the lattice. The radiation pattern for the second band (Ω=d/λ=0.5765, Fig. 4, right) is more complex, due to the presence of two branches of the iso-frequencies contour. While the “red” branch displays very strong focusing along the [10] direction of the square lattice, the “blue” branch, having some finite curvature, shows a more complex focusing pattern with two rays focused in the directions separated from the [10] direction by approximately 1° (Fig. 4, right).

 

Fig. 5. Modulus of the electric field map for a 30×30 rod photonic crystal excited by a point isotropic source. Left, Ω=d/λ=0.3567; right, Ω=d/λ=0.5765. The axis scales are in units of d and the colorscale is from 0 (blue) to 1 (red).

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In the self-guiding regime the radiation pattern due to a point source implies that the strongly collimated light propagation is insensitive to the divergence of the initial beam and almost insensitive to the orientations of the beam with respect to the crystal lattice. This property can lead to useful applications of self-guiding for waveguide design.

A mechanism of spatial light confinement in a self-guiding regime differs significantly, form the wave-guiding mechanisms in both conventional dielectric waveguides [15] and in more recent photonic bandgap waveguides [16, 17]. In the former cases, total reflection at the boundary of a waveguide, due to total internal reflection in a dielectric waveguide [15] and due to a full photonic band gap in a photonic crystal waveguide [16, 17], is the reason of the light confinement. Neither a full photonic bandgap nor a physically defined wave-guiding region are necessary in the case of self-guiding. The spreading of the beam in a photonic crystal is counteracted then by the crystal anisotropy, like in the case of spatial solitons nonlinearity of the medium counteracts the natural spreading of the beam due to diffraction.

3. Real space analysis

Several questions remain to be answered: are the iso-frequency considerations valid for an actual finite size crystal? What is the coupling at the interfaces of the crystal? And what is the coupling between two crossing beams in the crystal?

In this section, we use a rigorous method based on a Fourier-Bessel development of the electromagnetic field around each rod of the photonic crystal [18]. We model a 2D finite size photonic crystal excited by a point isotropic source or illuminated by a Gaussian beam expressed as a plane wave packet with a Gaussian amplitude distribution:

with α=ksin(θ), β²=k ²-α², k=2π/λ. Note that the mean angle of incidence of the beam is α0=ksin(θ0).

 

Fig. 6. Self-guiding in a limit-size photonic crystal. Left: Modulus of the electric field map for a 20×40 rod photonic crystal illuminated by a Gaussian beam with W/d=2.5, Ω=d/λ=0.5765. Right: Modulus of the electric field of the incident Gaussian beam. The axis scales are in units of d and the colorscale is from 0 (blue) to 1 (red).

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In Fig. 5 the electric modulus field map is shown, when a 30×30 rod crystal is excited by a point isotropic source at the normalized frequencies Ω=d/λ=0.3567 (left) and Ω=d/λ=0.5765 (right). The point source is placed in the middle of the crystal, at x ₀=0 and y ₀=0. The emitted light is guided in channels in the [11] and [10] directions, correspondingly, as it was predicted in the previous section.

In Fig. 6, a rigorous numerical simulation of self-guided Gaussian beam is presented for a normalized frequency Ω=d/λ=0.5765. Figure 6, right shows the field map of the TM polarized incident beam, where the width of the beam is W/d=2.5 and the waist is located at x ₀=0 and y ₀/d=19.5. The self-guiding in the [10] lattice direction is shown in Fig. 6 left. The crystal is made of 20×40 rods and is illuminated from the top by an incident Gaussian beam. The guiding of the beam is obvious in the photonic crystal, as seeing by comparing the width of the beam at the bottom of the crystal and at the same ordinate in free space. The width of the incident beam fully defines the width of the self-guided beam in the crystal. In the same time, the presence of the self-guiding phenomenon does not depend on the incident beam width. It is important to note that in this study we did not make any attempt to optimize the coupling of the incident beam. The transmitted energy below the crystal is about 33% of the incident one. We can expect that it is possible to increase transmission largely using, for example, the method proposed by Lalanne et al. [19].

Figure 7 illustrates the influence of the mean angle of incidence on the self-guiding. We consider the 20 rod width and 40 rod high crystal (identical to Fig. 6, left). It is illuminated by a gaussian beam with width W/d=2.5. The upper panel shows the modulus of the incident field when the angle of incidence θ₀=0, the panel second from the top shows the modulus of the transmitted field above the crystal for the same angle of incidence. To see a clear difference on the transmitted field the angle of incidence must be increased up to 5° degrees (see Fig. 7 third curve from the top). For the angle of incidence equal to 10° degrees the shape of the transmitted beam is strongly modified as shown in Fig. 7, bottom.

Figure 8 shows a 30×30 rod photonic crystal illuminated by two Gaussian beams when the normalized frequency is Ω=d/λ=0.5765. The first beam impinges from the top of the crystal and the second impinges from the right. The first beam is identical to the incident beam of Fig. 6 but with y ₀/d=14.5 and the second one propagates from the right with W/d=2.5. The waist is now located at x ₀/d=14.5 and y ₀=0. The key feature of the self-guiding illustrated in Fig. 8 is that two beams can cross each other without cross-talk, in a contrast to the case of narrow dielectric waveguides. This effect offers an advantage for applications as it is not trivial to design crossed waveguides with no cross-talk in a comparable size scale.

 

Fig. 7. Modulus of the field as a function of x. The crystal is the one depicted in Fig. 6, left. The normalized frequency is Ω=d/λ=0.5765. From the top to the bottom: modulus of the incident field at y=19.5d for θ₀=0°, modulus of the total field at y=-21.5d for θ₀=0°, modulus of the total field at y=-21.5d for θ₀=5°, modulus of the total field at y=-21.5d for θ₀=10°

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Fig. 8. Crossing of two self-guided beams.Modulus of the electric field map for a 30×30 rod photonic crystal illuminated two perpendicular Gaussian beams with W/d=2.5, d/λ=0.5714 and the colorscale is from 0 (blue) to 1 (red).

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4. Conclusion

In conclusion, we have shown unique features of self-guiding: the tolerance to misalignment and no cross-talk between two self-guided beams. Moreover, it should be noticed that there is no need to obtain a full bandgap as required by the usual photonic bandgap guides and that there is obviously no need to align the emitters and the receptors with respect to the guide (only the emitter and the receptor must be aligned with each other).