1. Introduction

Scientists have long been seeking all-optical approaches to control and manipulate the light. In the past two decades, photonic crystals (PhCs) [1] have been used for e.g. high-Q nanocavities [2, 3], narrow waveguides [4] and microlasers [5] by confining the light on a wavelength scale. The major strategy for designing a PhC cavity is to introduce a defect and create a defect state in the photonic band gap so that it will trap the corresponding photons as a cavity. A periodic array of these cavities forms the coupled resonator optical waveguide (CROW)[6]. CROWs are used to manipulate the light flow, where the group velocity can be several orders of magnitude smaller than in bulk material (of the same refractive index)[6]. Another approach to trapping light is to consider a perfect PhC (without any defect) and enhance the local density of electromagnetic states at some critical points in the band dispersion diagram [7, 9, 8]. Near these points, the slope of the band is very small and the PhC supports slow light modes. Due to its low group velocity, a slow light mode has a stronger local density of electromagnetic states than a conventional mode with a large group velocity.

Very recently Tichit et al. [10] have shown an interesting phenomenon that a lamellar structure consisting of a conventional dielectric layer and a left-handed material (LHM) [11] layer can be used to confine light in the form of the so-called “light wheel”: light rotates locally inside the lamellar structure. Such a “light wheel” phenomenon is due to the backward wave (phase velocity and energy flow in opposite directions) in the LHM layer. When an LHM layer is coupled to a conventional layer under the phase matching condition, energy flows in the two layers are contra-directional and a “light wheel” can be formed. By using this “light wheel” phenomenon, a similar structure with a segment of LHM replaced by a phase-shift medium is proposed to act as an open cavity [12]. However, an LHM with low loss, especially in the optical frequency range, is difficult to realize experimentally. An easier approach to the realization of a “light wheel” is thus desirable.

In this letter, we show that a ”light wheel” phenomenon can be realized in a completely dielectric structure, namely, a composite dielectric waveguide [see Fig. 1(a)]. In some previous works on grating-assisted contra-directional couplers [13, 14], the coupling coefficient is very small, i.e., the modes which propagate contra-directionally in two waveguides are weakly coupled. While in the present properly designed composite waveguide structure, the contra-directional coupling is much stronger. Thus, these two contra-directionally propagating waveguide modes are strongly coupled with each other, and act together as a “light wheel” with near zero group velocity. By trapping this “light wheel” in truncated composite waveguides, we can achieve a high quality factor in a relatively short length without introducing any phase shift medium as used in Ref. [12].

 

Fig. 1. (a) Schematic diagram of the proposed composite dielectric waveguide structure. (b) Unit cell used in the calculation of the dispersion curves for the composite waveguide shown in (a). In all the numerical examples we choose the following geometric parameters: d ₁ = 0.5a, d ₂ = 0.5a, w ₁ = 1.4a, and w ₂ = 1.2a.

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Fig. 2. Extended band diagram of the 1D PhC waveguide (solid line) and the dispersion curve of a conventional dielectric waveguide (dashed line). A (in the second Brillouin zone) is the cross point between the two dispersion curves, while A’ is its corresponding point in the first Brillouin zone.

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2. Analysis

Figure 1(a) shows the schematic diagram of the present composite waveguide structure consisting of a conventional dielectric waveguide with index n ₂ = 2.7 (like the material SiC) and a one-dimensional (1D) photonic crystal waveguide. The 1D PhC waveguide is composed of air layers and high-index dielectric layers (n ₁ = 3.6;like the material Si) stacked alternately with a period of a along the x direction. The separation distance d between these two waveguides is adjustable. The optimized geometric parameters are shown in the caption of Fig. 1. In this letter, only TM-mode (the magnetic field is polarized along the y axis) is considered. Same conclusion can be drawn for TE-mode.

First, we consider the case that the 1D PhC waveguide and the conventional dielectric waveguide are separated. The band structure of the 1D PhC waveguide, calculated with the 2D finite-difference time-domain (FDTD) method [15], is shown by the solid line in Fig. 2. Here the dispersion curves are extended to the second Brillouin zone (BZ). The dashed line in Fig. 2 represents the dispersion curve of the fundamental mode [the only waveguide mode in the considered frequency range of 0.22(2πc/a) ~ 0.30(2πc/a)] in the conventional dielectric waveguide. As shown in Fig. 2, in the second BZ, the first band of the 1D PhC waveguide and the dispersion curve for the fundamental mode of the conventional waveguide intersect at point A, which means the modes of the two waveguides are perfectly matched there. However, the slopes of the dispersion curves at crosspoint A indicate that the group velocities v_g = ∂π/∂k_x (where π is the angular frequency and k_x is the propagation constant) of the two modes are opposite in the x direction. When these two waveguides are placed in close proximity, a contra-directional coupling (i.e., the coupling of two waveguide modes which propagate contra-directionally) may occur, like the case in Ref. [10], where an LHM layer is coupled to a conventional dielectric layer. Note that there is a big difference between the two cases: in the case of Ref. [10], the contra-directional coupling is due to the left-handed behavior of the LHM layer, i.e., the energy flow is in the opposite direction of the phase velocity. While in the present case, there is no real left-handed backward wave generated in the PhC waveguide. Bloch mode A in the PhC waveguide, can be expanded in a series [16]: $\sum _n{a}_n\exp \left[i\left(k_A+n2\pi /a\right)x\right],$, where a_n is a dimensionless Fourier coefficient of each plane wave. The phase velocity k_A is indicated in Fig. 2. The negative group velocity of this Bloch mode is caused by the dominant plane wave component A’ (see Fig. 2) in the first BZ, whose phase velocity in x direction k_A′ (k_A′ = k_A - 2π/a) is also negative. Such band folding effect has also been used to realize effective negative refraction in PhC [17].

 

Fig. 3. The band structures for the composite waveguide with different separation distance d. The star (^*) points B and C indicate the band edge modes for the cases of separation distances d = 0 and d = 0.6a, respectively.

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The band diagrams of the composite waveguide for different separation distance d are calculated and shown in Fig. 3. The unit-cell (see Fig. 1(b)) for the 2D FDTD method has a length of a in the x direction and a width of 8a in the z direction. In Fig. 3, one sees that a band gap is induced by the contra-directional coupling between the 1D PhC waveguide and the conventional dielectric waveguide. When the separation distance d ≥ 0.2a , the relationship between the band gap width ∆ω and the contra-directional coupling coefficient κ can be derived from the coupled mode theory (CMT) and can be approximately expressed as [18],

where λ₀ is the wavelength corresponding to the central frequency of the band gap and ∆λ is the band width (in terms of wavelength) for the contra-directional coupling. c is the light velocity in free space. v _g+ and v _g- are the group velocities of the conventional dielectric waveguide and the 1D PhC waveguide, respectively. From Fig. 2, one can obtain $\frac{c}{v_{g+}}\approx \frac{c}{v_{g-}}\approx 3.3.$ When the two waveguides get closer, the coupling becomes stronger and the band gap width, according to Eq. (1), obviously increases (see Fig. 3). As for the case when separation distance d = 0, CMT is no longer valid because of too strong coupling and Eq. (1) is not reliable. The dispersion curves of the composite waveguide, as shown in Fig. 3, are greatly distorted by the large band gap caused by the contra-directional coupling.

 

Fig. 4. (a) The modulus of H_y field in a domain of 15a height and 70a length. White arrows show the propagation direction of light. A Gauss beam is coupled into (a) the composite waveguide; and (b) a single conventional waveguide.

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Inside the band gap caused by the contra-directional coupling, a localized “light wheel” can be formed as the case in Ref. [10]. The composite waveguide with separation distance d = 0.2a is chosen as an example. To excite the “light wheel”, we put a high refractive index (n = 3) material with a gap of 0.3a above the top interface of the composite waveguide. A Gaussian beam, at normalized frequency ω = 0.25(2πc/a) (i.e., the central frequency of the band gap, see Fig. 3), is launched with an incident angle θ = 47° in the high refractive index material. In the FDTD simulation, PML (perfectly matched layer) boundary treatment is used. Fig. 4(a) shows the steady-state pattern of H-field modulus. The fringes above the structure are due to interference between the incident and reflected beams. As illustrated in Fig. 4(a) with the propagation directions of light energy (denoted by the white arrows), a clockwisely rotating “light wheel” is formed. This phenomenon can be explained as follows. Since inside the band gap no real propagation mode exists in this composite waveguide, the right-propagating light is excited (by the evanescent coupling of the incident Gaussian beam) in the upper conventional dielectric waveguide. After some propagation distance L in the upper conventional dielectric waveguide, the right-propagating light may be totally coupled into the PhC waveguide, where the energy propagates contra-directionally (see Fig. 4(a)). In a similar way, the left-propagating light in the PhC waveguide will also be totally coupled back into the conventional dielectric waveguide. Consequently, a rotating “light wheel” is formed locally. The propagation distance L, which is related to the transmissivity, can be estimated approximately by [19]:

where T is the transmissivity through the composite waveguides after a propagation distance of L. If one wants 90% contra-directional coupling (enough for illustrating the “light wheel” phenomenon), i.e., the transmissivity would be below 10%, the length L of the composite waveguide should be longer than 35a for κ = 0.052a ^-1, which is the coupling coefficient estimated from the band gap width ∆ω = 0.01 (2πc/a) according to Eq. (1). Thus, we set the length of the composite waveguide to 70a (i.e., 2L) in our simulation. For comparison, the case of a single conventional waveguide below the high refractive index material is shown in Fig. 4(b). One sees that different from the light wheel of the composite waveguide shown in Fig. 4(a), the mode excited in the single conventional waveguide just keeps propagating toward the right in Fig. 4(b).

When the working frequency is getting close to band gap edge, the light wheel gradually becomes delocalized. At band gap edge, propagation modes of small group velocity appear. For the case of separation distance d = 0, slow light mode B, at normalized frequency ω = 0.2500(2πa/c), is obtained (cf. Fig. 3). The corresponding H-field pattern is shown in Fig. 5(a) for a length of 6a in x direction. The nearly zero energy flow of this slow light is caused by the overall effect of contra-directional energy flows in the two waveguides. In each waveguide, the light flow is non-zero. Thus we call it as “light wheel slow light”. The distribution of time averaged Poynting vector of mode B is shown in Fig. 5(a), from which one can see that the whole light wheel can be regarded as a composite of many small light wheels caused by contra-directional coupling. Each unit cell (see Fig. 1(b)) possesses a small light wheel and acts like a small cavity. It seems like the case in CROWs [6]. But the difference is that the sole unit cell in the present structure has no resonant modes with high Q-factor. The light energy in the each unit can be classified into two parts: one part locally rotates in the unit cell, and the other part flows in and out of the unit cell from both sides. The ratio between the localized energy and the total energy, which determines the ability of this composite waveguide to trap light, mainly depends on the contra-directional coupling coefficient. To illustrate the point, the Poynting wave vector of mode C (see Fig. 3), which is the band edge mode for the case of separation distance d = 0.6a, is given in Fig. 5(b) for comparison. Because of the large separation distance, the contra-directional coupling is small and little part of light energy confinedly rotates in each unit cell. Thus, this mode acts more likely two separated waveguide modes propagating contra-directionally, which can be obtained if the composite waveguide is long enough. While in a truncated composite waveguide with relatively short length, this mode will not be achieved as shown in the following simulation.

 

Fig. 5. FDTD-simulated H-field pattern for (a) band edge mode B and (b) band edge mode C. Arrows shows the time averaged Poynting vector.

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Based on the light wheel slow light, the composite waveguide can be used to trap light as a cavity. Here a composite waveguide with separation distance d = 0 is used as an example. A Gauss point pulse with center frequency ω = 0.2500(2π/a) and pulse width dw = 0.0008(2π/a) is excited in the center of the conventional waveguide. By using the 2D FDTD method combined with fast harmonic analysis [20], the Q-factor of the resonant mode is calculated (with same discretization and same alignment of the epsilon inside the Yee cell [15] as that for the bandstructure calculation) and shown in Fig. 6 as the length of the composite waveguide varies. Here the finite Q-factor is due to the vertical radiation loss and the diffraction loss at the edges of the composite waveguide, both of which is caused by the finite length of the composite waveguide. From Fig. 6 one sees that Q-factor of the resonant mode is in the range of 8,000-10,000, not so sensitive to the length change of the composite waveguide (this is because the Fabry-Perot resonance caused by reflection at the two edges has little effect on the Q-factor in this case) To illustrate the enhancement of the Q-factor by the light wheel effect, we choose a composite waveguide consisting of 27 unit cells as an example. A resonant mode with Q-factor of 9.3×10³ has been obtained at normalized frequency ω = 0.2501(2π/a), which is very close to the band edge mode B. As shown in Fig. 6, when the cavity length is increased, we can always get a high Q resonant mode close to the band edge. The H-field pattern (as well as the profile of the structure) of the resonant mode is shown in the inset of Fig. 6. Since the light wheel slow light mode confined in the composite waveguide is far below the light line, the vertical radiation loss is relatively small. Thus the main loss in this structure is caused by the part of energy diffracted at the edges of both waveguides. Because of the strong contra-directional coupling of the waveguides when separation distance d = 0, a large part of energy rotates confinedly in each unit cell, and a high Q-factor is obtained. When separation distance changes to 0.6a, contra-directional coupling between them becomes really small. In this case, no resonant mode is found with Q-factor larger than 100 in the frequency range of 0.23(2πc/a) ~ 0.27(2πc/a).

 

Fig. 6. The Q-factor of the resonant mode near the band gap edge as the composite waveguide length varies. The inset shows H-field pattern of the resonant mode in the composite waveguide consisting of 27 unit cells. The numbers denote the resonant frequencies in the unit of 2πc/a

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3. Conclusion

To summarize, we have numerically demonstrated that a localized “light wheel” can be realized with only conventional dielectric materials, namely, a composite waveguide consisting of a 1D dielectric PhC waveguide and a conventional dielectric waveguide. The physical mechanism of this localized ”light wheel” has been explained in detail. Moreover, due to the strong contra-directional coupling, the light wheel slow light mode at the edge of the band gap is achieved in the composite waveguide with relatively short length. Consequently, a quality factor of 9.3 × 10³, which is not sensitive to the length of the composite waveguide, has been obtained. To the best of our knowledge, such a high quality factor in such a short length has not been reported in previous structures of grating-assisted contra-directional couplers. The proposed “light wheel” structure is purely dielectric and thus easier to realize in the optical frequency range as compared with the ones proposed in Ref. [10, 12]. In comparison with the line-defect PhC waveguide [21] which is broadly used to slow light, the present structure of the composite waveguide is simpler (two waveguides without PhC surrounding) and more compact in z direction. Importantly, it can provide a low loss alternative to novel slow light and even light trapping.