Reducing radiation losses of one-dimensional photonic-crystal reflectors on a silica waveguide

Wei Ding; Rong-Juan Liu; Zhi-Yuan Li

doi:10.1364/OE.20.028641

1. Introduction

Optical cavities, which confine light by resonant recirculation [1], are essential for enhanced light-matter interactions and have a lot of applications in cavity quantum electrodynamics [2], enhancement/suppression of spontaneous emissions [3], filters [4], biochemical sensing [5], opto-mechanics [6], to name just a few. Based on non-dissipative dielectric materials, optical cavities can be implemented either by continuous total internal reflections in a whispering gallery geometry [7] or by photonic band gap (PBG) effects in a photonic crystal (PhC) structure [8]. Whereas the former enables ultrahigh cavity quality factor (Q-factor), which defines the light residence time inside the cavity in unit of the optical period, the latter has the merit of small cavity mode volume (V). In order to retain the high Q-factor property in the PhC version of optical cavities, the PhC reflectors, which form the boundaries of the cavities, must eliminate all the light escape out of the cavities (Fig. 1 ).

Fig. 1 Schematic of a PhC cavity which is surrounded by two PhC reflectors. The presence of the PhC reflectors not only leads to light recirculation inside the cavity but also results in light radiation out of the waveguide.

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Efforts of searching ideal PhC reflectors accompanies with the development of high-Q PhC cavities, from three-dimension (3D) [8] to two-dimension (2D) [9] and then to one-dimension (1D) [10]. The first high-Q (~10⁶) PhC cavity was demonstrated in a 2D slab [11,12] with the out-of-plane light confinement implemented by total internal reflections and the in-plane light confinements by PBGs. Similarly, in 1D PhC cavities, also known as nanobeam structures [13], it is also the total internal reflection that takes charge of the light confinements in two perpendicular directions. When PhC reflectors evolve from 3D to 2D, the improved fabrication quality leads to a rise of Q-factors of PhC cavities. When PhC reflectors evolve from 2D to 1D, device footprints shrink dramatically, PhC forbidden bands become attainable in low-index waveguides. However, besides these beneficial consequences, with the decrease of the PhC dimension, additional light leakages of PhC reflectors appear via radiation, which is unavoidable for total internal reflection light confinements (see Fig. 1). Extra efforts must be paid to lower this loss in low-dimensional PhC reflectors [11–13].

From the viewpoint of applications, further advances of 1D PhC cavities are driven by three goals: higher Q-factors for enhanced light-matter interactions [14,15], higher on-line transmissions at resonance for good coupling from feeding waveguides to cavities [16,17], and lower refractive index of the waveguide material for expanded field profiles of guided modes [18,19]. For all these goals, the common challenge is how to reduce the radiation loss of the 1D PhC reflectors, which form the boundaries of the cavity in Fig. 1. This paper focuses on studying the radiation generation process in a 1D PhC reflector on a low-index (silica) waveguide. In order to minimize geometrical complexities and highlight physics, we rule out the cavity structures in this work and concentrate on a slab waveguide with one dimension infinitely long. The study to more practical devices with cavity geometry and 3D dimension will be demonstrated elsewhere. The general conclusion is same with this paper.

Although the radiation process has been intensively studied in the context of PhC cavities, most investigations either are pure mathematical [20,21] or rely on a posteriori information of global cavity mode profiles [22,23]. The physics-inspired Fabry-Perot picture sometimes leads to incorrect predictions, for example to the abnormal Q-factor variation in Ref [24]. The geometry complexity of the PhC cavity may obscure the essence of the radiation process. We believe it is deserved to go back to the simpler structure of 1D PhC reflector, especially in the context of low-index waveguides, to clarify the radiation generation process.

Below, in section 2, we first give the design rules of our 1D PhC reflectors, and then give the calculations of the eigen-modes of the PhC waveguide segments. In section 3, numerical simulations to the radiation losses of periodic 1D PhC reflectors are followed by discussions of their oscillatory natures. In section 4, an analytical model reflecting the interferometric characters of the radiation process is proposed. We corroborate and apply this model in a variety of 1D PhC reflectors, including periodic and non-periodic ones. The limitation of our model is also discussed. Section 5 summarizes the conclusions.

2. Design rules and eigen-mode calculations

Our task is to construct a 1D PhC reflector in a low-index slab waveguide with compact sizes. Size compactness distinguishes a micron-scaled PhC reflector from a millimeter-scaled fiber Bragg grating based on UV-photosensitivity [25]. Device compactness is also a requirement of practical nanofabrication techniques, which cannot keep high resolutions over a large number of periods. For our 1D PhC reflectors, the size compactness, however, constitutes a challenge because of the low refractive index of the waveguide material and the narrow PBG arising therefrom. Figure 2 plots the photonic band structure of a silica/air quarter wave plate stack and that of a silicon/air quarter wave plate stack. The PBG of the former is much narrower and shallower, implying that a PhC reflector based on a silica waveguide requires more number of periods to overcome light evanescent couplings through it (see Fig. 1). In contrast, a PhC reflector based on a silicon waveguide can inhibit light evanescent couplings very easily even setting working wavelengths close to the edges of PBG [17]. The first design rule of our 1D PhC reflectors is, therefore, to set the working wavelength (λ = 1.55μm in this work) to the center of the PBGs for all the PhC segments, which form the PhC reflector, to keep the device compact. Note that, in 1D situation, low refractive index of the waveguide material still allows a complete PBG to open.

Fig. 2 Photonic band structures of (a) a silica/air quarter wave plate stack and (b) a silicon/air quarter wave plate stack. The real [Re(k)] and the imaginary (α) parts of the wave vectors are plotted in different colors (black and red respectively). The gray shadow areas represent the PBGs. The inserts are the schematics of the wave plate stacks. n_Silica = 1.46, n_Silicon = 3.46.

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The second design rule arises from people’s quests for both high Q-factors and high transmissions at resonance in cavity applications. In the Fabry-Perot picture [24], when two PhC reflectors form a cavity, the Q-factor of the cavity is determined by two factors: the light evanescent coupling through the reflectors and the light radiative scattering at the ends of the cavity waveguide (i.e. the radiation loss on the side of the PhC reflector facing the cavity, Fig. 1). In the language of Q-factor, the reciprocal of the total Q-factor equals to the sum of the reciprocals of two Q-factor components ( $Q^{- 1} = Q_{C}^{- 1} + Q_{R}^{- 1}$ , with $Q_{C}$ and $Q_{R}$ being coupling-induced and radiation-induced Q-factors respectively). Here, the Q-factor represents the ratio of the stored energy inside the cavity to the loss per cycle. Reducing radiation losses of the 1D PhC reflector helps improve the radiation-induced Q-factor and benefits the total Q-factors of the cavity. On the other hand, when light is fed from the outer waveguide, effective light excitation inside the cavity occurs only when the radiation loss on the side of the PhC reflector facing the outer waveguide (see Fig. 1) is smaller than the evanescent coupling efficiency from the feeding waveguide to the cavity. Reducing radiation losses of 1D PhC reflectors gives rise to a high transmission at resonance of the PhC cavity as Ref [16]. points out. Altogether, in the context of the PhC cavity application, a high Q-factor requires a low-radiation design of the PhC reflector facing the cavity, and a high transmission at resonance requires a low-radiation design facing the feeding waveguide. Our second design rule of the PhC reflector is, therefore, the mirror symmetry of the PhC reflector with both sides having low radiation properties.

Figure 3(a) plots the schematic of a 1D PhC reflector in a silica slab waveguide (with the y-direction infinitely long). The working wavelength λ = 1.55μm, the refractive index of silica n = 1.46, the environment is air, the excitation mode is transverse magnetic (TM, with E_x, E_z, and H_y nonzero), and the waveguide width 2a = 1.4μm < 1.457μm is determined by the single even mode criterion $2 a \sqrt{n^{2} - 1} < λ$ [26] for removing multi-mode plagues. The 1D PhC reflector consists of one row of elliptical holes with the period to be Λ and the transverse (longitudinal) diameter to be w (t).

Fig. 3 (a) Schematic of a 1D PhC reflector in a silica slab waveguide. The geometrical parameters, the coordinate, and the polarization direction are defined in the picture. (b) Wave vector diagram of a periodic PhC segment with w = 1.3μm, t = 0.4μm, and Λ = 0.664μm. The gray area is the air light cone, and the dashed line represents the working wavelength. (c) The upper (red squares) and the lower (blue squares) edges of the PBGs as a function of the parameters w and t. The values of w are 0.1/0.3/0.5/0.7/0.9/1.1/1.3μm respectively.

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Applying plane-wave expansion methods and Bloch boundary conditions [27,28] to one PhC segment, defined by the dashed lines in Fig. 3(a), Fig. 3(b) plots the wave vector diagram. All the geometrical parameters are listed in the caption. Note that the elliptical hole locates in the center of the PhC waveguide segment, and adjacent PhC segments are in connection with each other via the unaltered waveguide part, not via the air hole. In Fig. 3(b), a PBG (~17% for the ratio of the gap width to the midgap) appears at the edge of the Brillouin zone under the air light cone (the gray area). Inside this PBG, the wave vector of the Bloch mode is a complex number with the real part to be $\frac{π}{Λ}$ and the imaginary part to be α. When we adjust the period of the PhC segment, Λ, to 0.664μm, the center of the PBG moves to the working wavelength as the design rule 1 requires. We do this Λ-adjustment work for a number of parameters of (w, t) and plot all the upper/lower edges of the PBG (the red/blue squares) in Fig. 3(c). It is seen that the widest PBG occurs when the elliptical hole is the widest and the longitudinal diameter of the hole, t, roughly equals to one quarter of the wavelength, which coincides with the fact that quarter wave plate stacks have the widest PBGs. Following our preceding analysis, the wider the PBG, the greater the imaginary part of the wave vector, α, in the center of the PBG, the faster the evanescent decay of the Bloch mode, and the more compact the 1D PhC reflector. In Fig. 3(c), it is also seen that, for each value of w, there are two t’s having the same band gap width (one in the green area, the other in the yellow area). A detailed study shows that the one in the green area has a smaller period Λ, corresponding to a longer distance from the PBG to the air light cone in the wave vector diagram of Fig. 3(b), which is favorable for reducing radiation losses. In the following calculations, we restrict the parameter t within 0 to 0.4 μm. In this region, t, Λ, PBG width, and α vary monotonically.

Besides PBG widths, eigen-mode calculations of this section also provide information about the period of the PhC segment (Λ) and the decay rate of the Bloch mode (α). All these information will be used in the following calculations to the radiation loss.

3. Radiation-loss simulations

Radiation loss of a 1D PhC reflector can be accurately obtained by numerical methods. We carry out finite-difference time-domain (FDTD) simulations by sending a modulated guided wave towards the tested PhC reflector and monitoring its transmission and reflection. Before and after the wave arrives at the PhC reflector, it respectively travels through one piece of uniform waveguide with the length greater than 30 microns to guarantee true single mode in the interested region.

The blue (green) curve in Fig. 4(c) gives the simulated transmission (reflection) of a 1D PhC reflector with w = 1.3μm, t = 0.2μm, Λ = 0.614μm, N = 9, whose schematic is depicted in Fig. 4(a). Based on these results, the radiation loss is derived (Rad = 1 – Tran – Refl) and plotted in black. The increase of the radiation loss with the decrease of the wavelength coincides with the fact that the upper branch in Fig. 3(b) crosses the air light cone. In Fig. 4(c), we also plot the radiation loss (in red) of another 9-period PhC reflector [depicted in Fig. 4(b)] with w = 0.7μm, t = 0.4μm, Λ = 0.626μm. The radiation loss of the second PhC reflector is greater than that of the first one. Since the eigen-mode calculations in section 2 have proven these two PhC reflectors have the same PBG width [see the two black circles in Fig. 3(c)], so that the similar photonic band structure, the discrepancy of their radiation losses should be ascribed to their different hole shapes. In a light ray picture, the input light rays in Fig. 3(a) and 3(b) are reflected to different directions with the former falling inside and the latter outside the total internal reflection angle range. A prolate elliptical hole looks more preferred for resisting radiation losses, especially when the total internal reflection angle is small in a low-index waveguide. Below, we set all the elliptical hole widths to be 1.3μm [the bold lines in Fig. 3(c)] to ensure the prolate hole shape as much as possible. This unified choice of w makes our PhC reflectors more like the ladder structure [6,14,18,19], rather than the circular hole structure [13,15–17] where w varies.

Fig. 4 Schematics of two 9-period PhC reflectors having (a) prolate (1.3μm × 0.2μm) and (b) oblate (0.7μm × 0.4μm) elliptical holes respectively. These two PhC reflectors have the same PBG width as the black circles in Fig. 3(c) indicate. The red arrows represent the input and the reflected light rays. (c) Simulated transmission (blue) and reflection (green) curves of the PhC reflector in (a). The black and the red curves are the derived radiation losses of the two PhC reflectors. The vertical gray line is the working wavelength.

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We apply FDTD simulations to a range of periodic 1D PhC reflectors and plot their radiation losses at the working wavelength [the values denoted by the black circles in Fig. 4(c)] into one diagram of Fig. 5(a) . As expected, when the thickness of the elliptical hole, t, increases from 0.05μm to 0.4μm, light is radiated out of the waveguide more intensely due to the decrease of the hole ellipticity. However, the striking feature of Fig. 5(a) is that with the increase of the period number of PhC reflectors, the radiation losses oscillate periodically, which means adding PhC segments sometimes does not give rise to more radiations and some route exists for coupling radiated-out lights back to the waveguide. The periodic oscillations also imply an interferometric character in the radiation generation process. To the best of our knowledge, this oscillatory feature of the radiation loss at one fixed working wavelength has not been systematically studied in the context of 1D PhC reflectors. Analyses to it will help understand the radiation generation process and help design low-radiation PhC structures, especially on a low-index waveguide. Setting t = 0.1μm, Fig. 5(c) further plots the radiation losses of one PhC reflector at three different working wavelengths, all lying inside the PBG. The oscillatory features appear again. With the increase of the working wavelength, the oscillation period shortens.

Fig. 5 (a) Simulated and (b) calculated total radiation losses of periodic 1D PhC reflectors as a function of the number of periods. w = 1.3μm, t = 0.05/0.1/0.15/0.2/0.25/0.3/0.35/0.4μm, and λ = 1.55μm. The y-axes in (a) and (b) are both logarithmic and span 22.2dB. (c) Simulated radiation losses at three different working wavelengths for the 1D PhC reflector with w = 1.3μm and t = 0.1μm. The gray vertical lines in all the plots denote the periods of the oscillations.

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4. Modeling

For a 1D PhC reflector consisting of one row of elliptical holes, it is hypothesized that the total radiation is composed of component radiated waves from individual holes. Our analysis starts from considering the perturbation of an infinitely thin elliptical hole to a uniform waveguide as depicted in the inset of Fig. 6 . The presence of this perturbation gives rise to energy coupling from guided waves to radiated waves. Omitting the coupling to the backward radiated waves, which has been verified in a lot of literatures [29], Fig. 6 plots the transverse coupling coefficient [30] from the guided wave ( $e^{(G)}$ ) to the forward radiated waves ( $e^{(R)}$ ) in a silica slab waveguide,

K^{t} = \frac{ω}{4} \int_{- \frac{w}{2}}^{\frac{w}{2}} Δ ε (x) \cdot e_{x}^{(G)} (x) \cdot {[e_{x}^{(R)} (x)]}^{*} d x,

where the permittivity perturbation

Δ ε = ε_{0} (1 - {1.46}^{2})

is assumed to be constant in the region

x \in [- \frac{w}{2}, \frac{w}{2}]

, and the eigen-modes of the guided and the radiated waves are normalized by their z-directional Poynting flux (see Appendix). Figure 6 shows that most radiations distribute in one angular range around one tilting direction θ. This characteristic will be utilized in the following.

Fig. 6 Transverse coupling coefficients from the guided wave to the forward-radiated waves in a slab silica waveguide (2a = 1.4μm). The coupling is caused by an infinitely thin elliptical hole with w = 1.3μm. The inset shows the schematic.

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In the schematic of Fig. 7(a) , for a periodic 1D PhC reflector, each component radiated wave coming from one elliptical hole has the same titling angle. Their sum should take into account their relative phases and amplitudes. The phase differences originate from the different wave vectors between the radiated waves (k₀, the wave vector of free space) and the guided Bloch wave ( $π / Λ$ ). And the amplitude differences are due to the evanescent decay of the guided Bloch wave (α) inside the PhC waveguide. It seems the former gives rise to the oscillation of the total radiation loss shown in Fig. 5, and the latter explains the damping of these oscillation evolutions. In order to test the first statement, the oscillation periods are derived as $2 π / [Λ (\frac{π}{Λ} - k_{0} \cdot \cos θ)] = 2 / [1 - (2 Λ \cdot \cos θ / λ)]$ and denoted in Figs. 5(a) and 5(c) with the vertical gray lines. Here, the values Λ obtained in section 2 are used, and the calculation agrees well with the simulated curves. As the thickness of the elliptical hole, t, increases, the period of the PhC segment, Λ, increases, leading to the increase of the oscillation period in Fig. 5(a). Similarly, as the working wavelength blue-shifts, the oscillation period increases as well [see Fig. 5(c)]. In all the calculations, the tilting angle is assumed to be constant.

Fig. 7 Schematics of radiation generation processes occurring in a periodic 1D PhC reflector.

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In above derivation of the oscillation period formula, we assume the guided Bloch wave decays exponentially inside the PhC waveguide. Although this assumption is questionable, it brings a mathematical convenience that the amplitudes of the component radiated waves constitute a geometric sequence and the total radiation can be added up to two terms. In physics, this implies that all the radiated waves actually only occur at two end interfaces of the PhC waveguide, and the evolution formation of the guided Bloch wave inside the PhC waveguide does not matter indeed [24]. This understanding makes our treatments easier and enables us to focus attentions to the interfaces of PhC waveguides, as Fig. 7(b) shows.

Actually, inside the PhC waveguide, the guided Bloch wave experiences an energy exchange process with its counter-directional Bloch wave [31]. The coupling equations of the forward (A_f) and the backward (A_b) guided waves

{\begin{matrix} \frac{d A_{f}}{d z} = κ \cdot A_{b} \\ \frac{d A_{b}}{d z} = κ^{*} \cdot A_{f} \end{matrix},

and their solutions

{\begin{matrix} A_{f} (z) = \frac{A_{f} (0) \cdot | κ | \cos h [| κ | (z - L)] + A_{b} (L) \cdot κ^{*} \sin h (| κ | z)}{| κ | \cos h (| κ | L)} \\ A_{b} (z) = \frac{A_{f} (0) \cdot κ \sin h [| κ | (z - L)] + A_{b} (L) \cdot | κ | \cos h (| κ | z)}{| κ | \cos h (| κ | L)} \end{matrix},

show that the coupling coefficient

κ

can be used to solve the amplitudes of the forward and the backward guided waves in any position. Inside the PBG,

κ

is a negative real number with the absolute value to be the imaginary part of the wave vector of the Bloch mode, α, (see section 2). When a forward guided Bloch wave impinges upon the interfaces of z = 0 and z = L, a backward guided Bloch wave impinges upon the interface of z = 0. All these impinging’s lead to radiated waves in their forward tilting directions, as depicted in Fig. 7(c).

From Eq. (2), we further argue that, different with the real part of the wave vector, the imaginary part of the wave vector of the guided Bloch wave should not be looked upon as an intrinsic quantity. The decay of the Bloch wave is actually a composite process and can be split to one energy exchange process, described by Eq. (2), and one phase lagging process, described by the propagation constant. The rate of the energy exchange is actually determined by the amplitude of another guided Bloch wave, not of itself. And, the energy exchange process occurs over one distance of waveguide, not in one specific point. While, as Figs. 7(b) and 7(c) claim, any radiation is generated at some interface. In our following analyses to the radiation loss, we only consider the propagation constant of the guided wave, no matter it is propagating or evanescent.

Till now, our description of a radiated wave is still according to its angular direction, which is inconvenient. Exploiting the characteristic that most radiation distributes in one narrow angular range, we convert the finite angular spread Δθ (full angle) of a radiated wave to the finite beam width Δx’ of a non-divergent radiated beam [see Fig. 7(d)]. The relationship of Δθ and Δx’ can be attained via the formula, $\frac{1}{2} \approx \bar{Δ k_{x'}} \cdot \bar{Δ x'} \approx \frac{k_{0} Δ θ}{2} \cdot \frac{Δ x'}{2 \sqrt{3}}$ (uncertainty principle), with the bar symbol standing for the standard deviation and the prime symbol representing the direction perpendicular to the tilting angle. The inversely proportional relationship of Δθ and Δx’ can also be understood in the context of a Gaussian beam having a finite angular spread and a finite beam waist. This is a critical simplification of our model and enables us to deal with the interferences of component radiated beams by more intuitive spatial overlaps. Figure 7(d) illustrates that the oscillations of the total radiation losses exhibited in Fig. 5 are determined by two factors: the relative phases and amplitudes of beam 1 and 2, and the extent of their spatial overlapping. Beam 3 in the backward tilting direction contributes to the total radiation non-interferometrically. Inspecting the details of the oscillation evolutions in Fig. 5 is a test to the verification of our model.

Before applying the model, an accurate estimate to the radiation angular spread, Δθ, is required. We launch a FDTD simulation to a 1D PhC reflector and plot the radiation pattern in Fig. 8 . All the geometrical parameters are given out in the caption. The estimated value of $Δ θ \approx {8.4}^{\circ}$ gives rise to the beam width $Δ x' \approx 5.8 μ m$ and the beam width in the z-direction [defined in Fig. 7(d)] $Δ x = \frac{Δ x'}{\cos θ} \approx 6 μ m$ , with $θ \approx 15^{\circ}$ . Comparing with the result in Fig. 6, which is from general coupled-mode theory, Fig. 8 provides more precise information about the characteristics of the radiation pattern.

Fig. 8 FDTD simulated radiation pattern of a 1D PhC reflector with w = 1.3μm, t = 0.1μm, and N = 4. The inset shows the schematic.

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Making use of the quantities obtained in section 2 (Λ and α of PhC segments) and the parameters $θ \approx 15^{\circ}$ , $Δ x \approx 6 μ m$ , we carry out calculations to the total radiation losses of the periodic PhC reflectors. First, the amplitudes of the forward and the backward guided Bloch waves at both ends of one PhC reflector are obtained by using Eq. (3). The amplitudes of the component radiated beams are proportional to these values. Second, the relative phases of the component radiated beams are solved by using the parameters Λ and θ. The phase difference between beam 1 and beam 2 need add one additional π because of the reverse changes of the propagation constant at two interfaces (one is from a high-index wave to a low-index wave and the other reverses). Third, the spatial overlap between beam 1 and beam 2 is solved by using θ, Δx and Λ. Outside the overlap region, the lights in two beams contribute to the total radiation loss non-interferometrically; inside the overlap, the lights from two beams interfere with each other. When beam 1 and beam 2 are coherently out of phase, the light radiated out from the input interface can be coupled back into the waveguide via the route of the spatial overlap. Finally, beam 3 is added to the total radiation loss non-interferometrically. Figure 5(b) plots the calculated radiation losses of a variety of periodic 1D PhC reflectors. The results agree well with the simulation in Fig. 5(a) in terms of the oscillation period, the damping speed of the oscillation, the initial phase of the oscillation evolution, and the relative intensity of different curves. Forward tilted radiations are only produced at the input and the output interfaces. As the distance of these two interfaces increases, the amplitude difference of beam 1 and beam 2 increases, and the spatial overlap between them decreases. Both these two factors determine the damping speed of the oscillation curves in Fig. 5. Simultaneously, as the PhC reflector becomes longer, the backward radiation component strengthens, which causes the base of the oscillation curves raise. All these features are clearly presented in our calculated curves, matching well with the simulations. This undoubtedly proves that our analytical model correctly embodies the interferometric nature in the radiation process. In calculations, we use the same θ and Δx for all the PhC reflectors because of their same elliptical hole widths (w = 1.3μm). The insensitivity of these two parameters, θ and Δx, with regard to the elliptical hole thickness (t) also implies that our model may be applicable in more complicated structures, for example non-periodic 1D PhC reflectors.

Obeying the design rules 1 and 2 in section 2, we construct a non-periodic 1D PhC reflector depicted in the schematic of Fig. 9 . The whole reflector consists of eleven PhC segments with each having a PBG centered at the working wavelength λ = 1.55μm. The PhC segments in the middle region compose one uniform PhC waveguide section with t = 0.4μm, whose relatively wide PBG (see Fig. 3) enables the whole PhC reflector to well resist light evanescent couplings. At two ends of the PhC reflector, one or two PhC segments are tapered in order to reduce the radiation loss of the whole reflector.

Fig. 9 (a) Calculated (solid line) and simulated (square symbols) radiation losses of 11-layer 1D PhC reflectors with the variation of t₁. The other PhC segments have t = 0.4μm. The minimums of the two curves are labeled by the vertical red lines. (b) Calculated and (c) Simulated radiation losses as a function of $(t_{1}, t_{2}) \in [0, 0.4] μ m \times [0, 0.4] μ m$ . Same color bars are used in (b) and (c). The non-periodic PhC reflector, the five uniform PhC waveguide sections, and the definitions of $t_{1, 2}$ are denoted in the schematic. The white circles in (b) and (c) label the minimum-radiation points. The red crosses represent the progressly-tapered point.

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In case 1, we let one PhC segment tapered with the thickness of the elliptical hole varied within $t_{1} \in [0, 0.4] μ m$ . We model this problem by splitting the whole PhC reflector to three uniform PhC waveguide sections. At the interfaces of these waveguide sections, the directions and the relative phases of the component radiated beams are derived by using Λ and $θ \approx 15^{\circ}$ . The amplitudes of the forward and the backward guided Bloch waves at these interfaces are also calculated. Equation (3) is converted to its matrix form

[\begin{matrix} A_{f} (z) \\ A_{b} (z) \end{matrix}] = [\begin{matrix} \cos h (α z) & - \sin h (α z) \\ - \sin h (α z) & \cos h (α z) \end{matrix}] \cdot [\begin{matrix} A_{f} (0) \\ A_{b} (0) \end{matrix}] = M \cdot [\begin{matrix} A_{f} (0) \\ A_{b} (0) \end{matrix}],

and is used in series through different waveguide sections. The amplitudes of the component radiated beams are estimated by assuming they are proportional to the amplitudes of the guided Bloch waves and are proportional to the Fresnel coefficient

(\frac{π}{Λ_{i n}} - \frac{π}{Λ_{o u t}}) / (\frac{π}{Λ_{i n}} + \frac{π}{Λ_{o u t}})

. Here,

Λ_{i n}

(

Λ_{o u t}

) is the period of the PhC segment before (after) the interface, and the Fresnel coefficient reflects the influence of the change of the propagation constant across the interface. Estimating radiation amplitudes with Fresnel coefficient has been widely used in propagating wave cases [32]. We extend this approach to evanescent Bloch waves based on our preceding arguments to the propagating and the evanescent guided Bloch waves. Next, the spatial overlaps between component radiated beams are solved by using θ, Δx and Λ. The total radiation losses are summed up by counting interferences inside these overlap regions. Note that, interferences occur in both forward and backward tilting directions. Finally, Fig. 9(a) plots the calculated radiation losses (solid line) with

t_{1} \in [0, 0.4] μ m

, the minimum is found at

t_{1} = 0.165 μ m

. We compare with the simulated result (square symbols), whose minimum is at

t_{1} = 0.18 μ m

. The minimums of two curves appear in nearly same places. In contrast to the calculation, the FDTD simulation takes much longer time to find this optimal t₁. Note that the discrepancy of the calculation and the simulation with regard to their line shapes may attribute to the simplification in our model, which omits the radiation in directions outside the tilting angular ranges.

In case 2, two PhC segments at the ends of the 1D PhC reflector are tapered. The whole reflector is split to five uniform waveguide sections (see the schematic in Fig. 9). Following the same procedure described above, Fig. 9(b) plots the calculated radiation losses with the variations of t₁ and t₂ ( $\in [0, 0.4] μ m$ ). The minimum radiation loss occurs at ( $t_{1} = 0.26 μ m$ , $t_{2} = 0.04 μ m$ ). Similarly, FDTD simulations [Fig. 9(c)] find the minimum radiation loss at ( $t_{1} = 0.28 μ m$ , $t_{2} = 0.05 μ m$ ), very close to the calculated result. Again, much less time is taken for our analytical model to find this minimum-radiation point. A striking feature in Fig. 9(b) is that the minimum radiation appears at $t_{1} > t_{2}$ . This is against the conventional wisdom that a gradually tapered transition from a uniform waveguide to a PhC waveguide ( $t_{1} < t_{2}$ ) yields the lower radiation according to the impedance matching of the local guided modal profile. If we define a progressive tapering ( $t_{1} = \frac{0.4 μ m}{3}$ , $t_{2} = 2 \times \frac{0.4 μ m}{3}$ ) by linearly varying t from the uniform waveguide section (t = 0) to the PhC waveguide section (t = 0.4μm), the monotonic relationship between t and the PBG width (Fig. 3) ensures the local guided modal profile changes gradually in the transition region. We label this geometry by a red cross in Fig. 9 and verify with FDTD simulations that the progressively tapered PhC reflector ( $t_{1} < t_{2}$ ) produces 2.3dB more radiation loss than the geometry found by our model ( $t_{1} > t_{2}$ ). This result means that, by shrinking the second elliptical hole, the relative phases of the radiation components generated at different interfaces acquire some kind of distribution, which causes most radiation components to cancel out with each other. On the other hand, with a progressively tapered transition, although the amplitude of every radiation component reduces, their relative phases make them interfere constructively. In a PhC reflector on a low-index waveguide, the interferometric effect dominates the radiation process because of comparable amplitudes of radiation components. Exploiting this interferometric effect, we can lower the total radiation loss. Altogether, the success of finding the minimum-radiation non-periodic PhC reflector is another verification to our analytical model. We merely consider the radiation components in two narrow angular ranges, around two tilting directions, the underneath interferometric mechanism determines the relative strength of the total radiation.

Actually, impedance matching of guided modal profile enables a good energy transfer from one local guided mode to its neighbor. However, from our viewpoint, the impedance matching picture omits the interplays between component radiated waves after these waves radiate out of the waveguide. The implicit assumption that component radiated waves non-interferometrically leave away from the waveguide is valid only when a PhC reflector is constructed on a high-index waveguide where the quick decay of the guided Bloch wave diminishes the interference between component radiated waves. For a PhC reflector on a low-index waveguide, our model stresses the interferometric effects. The slow decay of the guided Bloch wave leads to comparable amplitudes of the radiation components, and the small propagation constant of the guided Bloch wave results in slow evolutions of the phase differences. As a matter of fact, even in a high-index waveguide, the interferometric character has also been reported in 1D PhC cavity structures, where this character is expressed as the variation of the Q-factor and has been explained as the radiation recycling effect inside cavities [33].

Though exhibiting usefulness in a variety of 1D PhC reflectors, our analytical model has some limitations. First, based on the characteristic that most radiations distribute around one tilting direction, our model assumes the radiations in other directions can be ignored. This assumption fails when the elliptical hole of the PhC reflector becomes more oblate. In that case, the radiation in the vertical direction is strong. Second, our model works well when the angular spread of the radiation, Δθ, is small. This can be understood in the context of a Gaussian beam. The smaller the angular spread, the greater the Rayleigh range, and the conversion from a divergent radiated wave having a finite angular spread to a non-divergent radiated beam having a finite width becomes more reasonable. We have examined a 1D PhC reflector on a single-mode silicon slab waveguide (n = 3.46, 2a = 0.45μm, w = 0.4μm, λ = 1.55μm). The deep sub-wavelength size in the transverse direction (2a, w < λ) results in a large radiation angular spread and invalidates our analytical model. Third, in this paper, we focus on the TM polarization (E_x, E_z, and H_y nonzero). The transverse electric mode excitation (E_y, H_x, and H_z nonzero) not only causes more radiations but also makes the radiation pattern more complicated.

5. Conclusions

In summary, we theoretically investigate the radiation generation process in a simple PhC structure: a 1D PhC reflector on a silica slab waveguide. Emphases are put on the component radiated waves, rather than the guided Bloch wave. An analytical model is proposed after the eigen-mode property (Λ, α) of the PhC segments and the radiation pattern characteristic (θ, Δx) of the waveguide are obtained. Our model points out that all the radiation components are generated at the interfaces between different waveguide sections, they radiate outward with some tilting angle and with some beam width in the forward and the backward directions respectively, and the interferences between these radiation components determine the total radiation loss. Taking into account the relative phases/amplitudes of the radiation components and the extent of their spatial overlap, our model analytically calculates the total radiation loss produced in the whole PhC reflector. By comparing with FDTD simulations, the applicability of our model is verified in periodic and non-periodic 1D PhC reflectors. In one example, we demonstrate how to use this interferometric effect to reduce the total radiation loss by 2.3dB, comparing to the gradually tapered PhC reflector. Besides making design easier, our model provides a physical insight into the interferometric nature of the radiation generation process, which can also be extended to more practical and more important PhC devices, like PhC nanocavities on 3D dielectric waveguides. Our explorations to radiation mechanisms in PhC waveguide structures call for more efforts to physics and applications of these promising nanophotonic devices, especially in the low-index waveguide context.

Appendix

The TM eigen-modes of the guided and the radiated waves in a slab wave-guide can be expressed as follows [26].

Guided wave (G):

\begin{array}{l} \frac{H_{y} (x)}{e^{i (β z - ω t)}} \equiv h_{y} (x) = {\begin{matrix} A \cos (k_{1} a) e^{- k_{2} (x - a)}, x > a \\ A \cos (k_{1} x), - a < x < a \\ A \cos (k_{1} a) e^{k_{2} (x + a)}, x < - a \end{matrix}, \\ e_{x} (x) = \frac{β}{ω ε_{0} ε (x)} h_{y} (x), e_{z} (x) = \frac{i}{ω ε_{0} ε (x)} \cdot \frac{d h_{y} (x)}{d x} . \end{array}

Radiated wave (R):

\begin{array}{l} h_{y} (x) = {\begin{matrix} B {\cos (k_{1}' a) \cos [k_{2}' (x - a)] - \frac{k_{1}' ε_{2}}{k_{2}' ε_{1}} \sin (k_{1}' a) \sin [k_{2}' (x - a)]}, x > a \\ B \cos (k_{1}' x), - a < x < a \\ B {\cos (k_{1}' a) \cos [k_{2}' (x + a)] + \frac{k_{1}' ε_{2}}{k_{2}' ε_{1}} \sin (k_{1}' a) \sin [k_{2}' (x + a)]}, x < - a \end{matrix}, \\ e_{x} (x) = \frac{β'}{ω ε_{0} ε (x)} h_{y} (x), e_{z} (x) = \frac{i}{ω ε_{0} ε (x)} \cdot \frac{d h_{y} (x)}{d x} . \end{array}

where the coordinate in Fig. 3(a) is used,

k_{0}

is the wavenumber of free space, the electric fields (e_x, e_z) and the magnetic fields (h_y) are given in the core (ε₁) and in the cladding (ε₂), respectively. Different propagation constants of the guided modes (β) and the radiated mode (β’) lead to different transverse wave vectors in the core k₁(k₁’) and in the cladding k₂(k₂’) with

β^{2} + k_{1}^{2} = β'^{2} + k_{1}'^{2} = k_{0}^{2} ε_{1}

and

β^{2} - k_{2}^{2} = β'^{2} + k_{2}'^{2} = k_{0}^{2} ε_{2}

, respectively. The normalization coefficients A and B are determined by the z-directional Poynting flux.

\frac{β {| A |}^{2}}{ω ε_{0}} [\frac{a}{2 ε_{1}} + \frac{\sin (2 k_{1} a)}{4 ε_{1} k_{1}} + \frac{1 + \cos (2 k_{1} a)}{4 ε_{2} k_{2}}] = 1,

\frac{β' {| B |}^{2}}{ω ε_{0} ε_{2}} [\cos^{2} (k_{1}' a) + {(\frac{k_{1}' ε_{2}}{k_{2}' ε_{1}})}^{2} \sin^{2} (k_{1}' a)] = \frac{2}{π} .

To obtain the transverse intermodal coupling coefficient $K^{t}$ , Eq. (1) is integrated from $x = - \frac{w}{2}$ to $\frac{w}{2}$ ,

K^{t} = \frac{Δ ε β β' A B^{*}}{4 ω ε_{0}^{2} ε_{1}^{2}} {\frac{\sin [(k_{1} + k_{1}') w / 2]}{k_{1} + k_{1}'} + \frac{\sin [(k_{1} - k_{1}') w / 2]}{k_{1} - k_{1}'}} .

Acknowledgment

This work was supported by the National Basic Research Foundation of China under Grant No. 2011CB922002, the National Natural Science Foundation of China (No. 61275044 & No. 11204366), and 100 Talents Programme of The Chinese Academy of Sciences (No. Y1K501DL11).

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Reducing radiation losses of one-dimensional photonic-crystal reflectors on a silica waveguide

Abstract

1. Introduction

2. Design rules and eigen-mode calculations

3. Radiation-loss simulations

4. Modeling

5. Conclusions

Appendix

Acknowledgment

References and links

Cited By

Figures (9)

Equations (9)

Optics Express