Pis'ma v ZhETF, vol. 102, iss. 4, pp. 281-285 © 2015 August 25
Optical storage based on coupling of one-way edge modes and cavity
modes
Y. Fang+, Zh. 1Vi+, H. Q. He+, T. Jiang* ^
+ School of Computer Science and Telecommunication Engineering, Jiangsu University, 212013 Zhenjiang, China * School of Statistics and Mathematics, Zhejiang Gongshang University, 310018 Hangzhou, China
Submitted 5 June 2015 Resubmitted 7 July 2015
We design a new kind of optical storage composed of a ring resonator that is based on both the one-way property of the edge modes of magneto-optical photonic crystals and the coupling effect of cavities. The ring resonator can be served as an optical storage through a close field circulation. Through another edge waveguide and coupling cavity, the electromagnetic signals can be either written into the storage or be taken out from it.
DOI: 10.7868/S0370274X15160122
I. Introduction. For many years scientists believed that "optical data cannot be stored statically and must be processed and switched on the fly". The reason for this conclusion was that the speed of light is so fast that stopping and storing an optical signal is thought to be impossible. However, recent progresses of science and technology annulled such assertions and prove that the group velocity of light can be decelerated to zero, effectively trapping or "stopping" the light pulse [1]. In addition, the ring resonator has been widely used in the development of microwave bandpass filters, couplers, mixers, oscillators, and antennas. The ring resonators combined with other waveguides make light circulating along a single direction, which has useful roles in modulating optical circuits [2-4]. However, the early ring resonator systems have many ports and the optical flow cannot be formed in a close path. Thus, the conventional ring resonator can not be used as a storage system. If the optical flow is circulated in an isolated ring resonator in a close path, the optical data can be saved. Thus, besides "trapping" or "stopping" the light pulse, another way for dynamically storing optical data may be with the help of the new ring resonator. To design such new ring resonator is the subject of this study. In addition, basing on the new ring resonator we will design an optical storage.
Our idea comes from the existence of one-way electromagnetic modes in 2D magneto-optical (MO) photonic crystals similar to the chiral edge states found in the integer quantum Hall (QH) effect [5,6]. Also, Zheng Wang et al. demonstrated the unidirectional edge modes
e-mail: jtao@263.net
through experimental and theoretical study [7-9]. These modes are confined at the edge of certain 2D MO photonic crystals and possess group velocities pointing in only one direction, determined by the direction of an applied dc magnetic field. Backscattering in the unidirectional edge modes is completely suppressed. As is well known, the coupling among different modes may excite many new and interesting phenomena. In this study we try to make the coupling of the unidirectional edge modes and cavity modes in order to achieve a new kind of ring resonator and find some new transmission mechanism.
II. Structure and method. We use a truncated 2D MO PC with its two edges combined with two regular PCs. All the dielectric rods extend along the z direction and the cross plane is the xy plane. The structure may excite two unidirectional edge modes on the two edges of the 2D MO PCs, respectively. Two regular PCs are used as gapped layers to prevent light scattering into air. The two edge modes may be independent if the interval of the two edges is large enough. The structure parameters for the 2D MO PC are the same as those in Ref. [7]. A square lattice of YIG rods (e = 15eo) of radius O.ffa in air is considered, where a is the lattice constant. The regular PCs consist of a square lattice of alumina rods with dielectric constant e = 10eo in air background, and its lattice direction is with 45° to the YIG lattice. The regular PC lattice constant is a/y/2 and the alumina rod radius is 0.106a. An external direct current (dc) magnetic field applied in the out-of-plane (z) direction induces strong gyromag-netic anisotropy, with the permeability tensor taking the form [7]
282
Y. Fang, Zh. Ni, H. Q. He, T. Jiang
Mi 3№ 0 where K
-3V 2 Mi 0 (1)
0 0 M3 e(G -
/x(r)
where m = 14/xo, M2 = 12.4/xo- The two edge modes of the 2D MO PC can be obtained from the modified plane wave expansion method. For E polarization (the electric field is along the 2 axis), we eliminate the magnetic field from Maxwell's equations to obtain the wave equation
f UJ
V x —V x E(r) = uj2e{r)eoMoE(r) = e(r) — E(r) Mr ) c
(2)
where ^
M 3V
and £(r) are the
0
■JV>" M 0
0 0 (J,'"
position-dependent periodic structure in the xy plane. Taking advantage of the periodic nature of the problem, the ¿7-field may be expanded into a sum of plane waves using Bloch's theorem as
Ez{r) = ]TE(k + G) exp[«(k + G) • r],
(3)
where k is a wave vector in the Brillouin zone, G represents a lattice vector in reciprocal space (also a square lattice), describing the periodic structure, and E(k + G) is the expansion coefficient corresponding to G. The tensor elements of f//x(r) can be expressed clS cl Fourier series expansion [fO, f f]
M' = ]TM'(G)ex p(iGT), (4)
G
M" = ]TM"(G)ex p(iGT), (5)
G
M'(G) = J M'(r) exp(-iG ■ r)dr, (6)
M"(G) = ±- J M"(r) exp(-iG ■ r)dr. (7)
where
In (6), (7) Au indicates the area of Wigner-Seitz (WS) unit cell that may be used to represent the periodic structure. By taking Eqs. (3)-(5) into (2), we finally obtain
^[M'(G-G')K'.K-№"(G-G')(K'xK-ez)]JB(k+G') =
G'
= 4Ee(G-G')£(k+G'), (8)
i
J e(r) exp[—«(G - G') • r]dr, (9)
Eq. (8) includes the sum of infinite number of reciprocal vectors G' and we select finite reciprocal vectors instead in the allowed precision range. Then the equation becomes a matrix eigenvalue equation. For a fixed wave vector k, the frequencies uj of the allowed modes in the periodic structure are found through solving Eq. (8). In the band calculations for the edge modes, a supercell must be used. Theoretically, the supercell must contain infinite rods. In practical calculations a finite number of rods can satisfy the needed accuracy.
III. Results and analysis. According to Eqs. (1)-(8), we calculate the projected band diagram of the structure system. The 2D MO PC has a width of 6a. The band diagram for the compound structure is shown in Fig. fa. There are two symmetric dispersion curves in the common gap of the MO PC and the regular PC. The red dashed curve and the black solid curve correspond to the upper edge modes and the lower edge modes, respectively. Each dispersion curve has only one group velocity direction, but the two group velocity directions corresponding to the two dispersion curves are opposite. Because the two edges have enough interval and the two edge modes are not coupled, the black dispersion curve is close to the single edge modes in Ref. [7]. To give a direct illustration of the one-way edge modes, we place two point sources with frequency uj = 0.54 1 7(27rc/a) at the centers of the two edges, respectively. The calculated steady-state Ez field distributions in Fig. lb show that the lower source excites edge mode only propagating to the right, whereas the upper source excites edge mode only propagating to the left. The results are calculated by the finite-element frequency-domain method and the scatter boundary conditions are used.
If we set up a cavity in the MO PC, the cavity resonance may be coupled with the two edge modes, and the two edge channels can tunnel through the coupling. The cavity can be formed by increasing one rod radius denoted as rc. The coupling only occurs when the frequency of the cavity mode is close to that of the edge mode. One of the cavity mode frequencies with rc = 0.33a is found at w = 0.5522(27rc/a). Fig. 2 shows that the one-way wave from a point source with uj = 0.55 25(27tc/o) at the lower edge channel is completely tunneled to the upper edge channel through the coupling of cavity. Because of the one-way property, the propagating directions of the two channels are opposite.
The above results provide us the design basis. It is clear that a single cavity cannot form a close circulation.
Fig. 1. (Color online) (a) - The projected band diagram of the MO PC combined with two regular PCs. The red dashed curve and the black solid curve correspond to the upper edge modes and the lower edge modes, respectively, (b) - The Er field distributions from two point sources (denoted as stars) with ui = 0.541 7(27tc/a) at the edges, respectively. The two point sources both excite one-way transmission modes with opposite directions
To form a close ring resonator, we try to set up another cavity to the left of the existed one in Fig. 2. But we find that a close circulation has not been formed with the two cavities because the coupling intensity among the cavities and the edge channels is too small. Thus, we set up four cavities in the MO PC by changing the radius of four rods as 0.33a. The four cavities form the four vertexes of a rectangle shape. Such a configuration can form a close wave circulation if a source with frequency ui from 0.5437(27tc/o) to 0.5537(27tc/o) is excited in one of the two edges. Fig. 3 shows the calculated result for a wave frequency u> = 0.55f7(27rc/a). The field forms a clear rectangle loop in an anticlockwise direction. The loop leads to a resonance effect and concentrates the
Fig. 2. (Color online) The Er field distributions correspond to the tunnelling from the lower channel to the upper channel. The white arrows denote the propagation directions
Fig. 3. (Color online) A close one-way field circulation from the coupling of two unidirectional edge modes and four cavity modes. The white arrows denote the propagation directions
electromagnetic energy. Thus the whole path of the loop can be view
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