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Analysis of photonic crystal coupled resonator optical waveguide (CROW) structures with a highly dispersive background medium is presented. A finite-difference time-domain algorithm was employed which contains an exact representation of the permittivity of a three-level atomic system which exhibits electromagnetically induced transparency (EIT). We find that the coupling strength between nearest-neighbor cavities in the CROW decreases with increasing steepness of the background dispersion, which is continuously tunable as it is directly related to the control field Rabi frequency. The weaker coupling decreases the speed of pulse propagation through the waveguide. In addition, due to the dispersive nature of the EIT background, the CROW band shape is tuned around a fixed k-point. Thus, the EIT background enables dynamic tunability of the CROW band shape and the group velocity in the structure at a fixed operating point in momentum space.
©2007 Optical Society of America
1. Introduction
Signal routing on integrated optics platforms can be achieved through total internal reflection [1], Bragg reflection [2], and coupled optical resonators [3]. In the latter, a series of high-Q micro-resonators are aligned within proximity to one another such that photons can tunnel from one resonator to its nearest-neighbor and thus propagate as if in a waveguide. Figure 1 shows a schematic of such a coupled-resonator optical waveguide (CROW) formed by an infinite straight line of defect cavities in a two-dimensional (2D) photonic crystal (PhC) [4, 5, 6]. Yariv et al. showed through a tight-binding (TB) analysis that the dispersion relation and propagation velocity of a pulse in such a structure are determined solely by a coupling factor, κ :
and
respectively, where ωres is the mode frequency of a single, isolated resonator of the waveguide, R is the spacing between resonators, and k is the wave vector. Therefore, the group velocity, Eq. (2), of the pulse can be decreased by decreasing κ. However, κ is typically a fixed quantity based on the design of the structure, e.g. the intrinsic Q of a the individual resonators and/or the spacing R. Yet, for realizing active components in an optical circuit, a dynamic control over v g is desirable. The problem then becomes how one can manipulate κ in a seemingly fixed system. We propose that a tunable, highly dispersive process such as electromagnetically induced transparency (EIT) [7] will facilitate control over the propagation velocity in the CROW structure by modifying the coupling factor between cavities.
EIT has been demonstrated in multilevel atomic systems [7, 8]. In a three-level Λ system, three atomic energy levels are situated such that the transition probability amplitudes from the ground levels b and c to the upper level a show destructive quantum interference under the two-photon resonance condition. Phenomenologically, a transparency window opens between two absorption peaks in the susceptibility, which allows a probe pulse to propagate through the otherwise optically opaque medium. At the same time, this transparency window occurs at a highly dispersive region of the susceptibility. It has been reported that such a lossless, highly dispersive EIT medium within an optical resonator cavity has the effect of dramatically narrowing the cavity linewidth and increasing the cavity Q-factor [9, 10, 11]. In the following work, we consider a PhC CROW structure (Figure 1) which possesses a background exhibiting EIT. Thus, the system consists of a series of optical resonators embedded in a highly dispersive medium whose dispersive properties are linked to the Rabi frequency of the control field excitation. We show that control over the dispersion relation of the waveguide is gained through the control field Rabi frequency, which in turn enables control of v g in the waveguide.
Fig. 1. Cross-section schematic of a CROW structure formed in a 2D PhC square lattice of high index rods where single rods were removed to create the resonant cavities.
We studied the proposed CROW system through two-dimensional (2D) finite-difference time-domain (FDTD) calculations capable of modeling EIT phenomena [12]. The main advantage of our implementation is the direct representation of the EIT permittivity [13, 14] as a complex multipole expansion which is readily incorporated into the FDTD algorithm via the auxiliary differential equation (ADE) method [15, 16, 17]. The ADE method uses the link between the frequency dependent permittivity ε and the electric flux density, D⃗, via the polarization, P⃗l. This link introduces a phasor polarization current, J⃗l(t)=∂P⃗l/∂t, from which time-step update-equations can be derived. Since it adds only one extra step to the FDTD update algorithm, the ADE method does not increase the computation load significantly.
1.1. Electromagnetically induced transparency
Figure 2(a) is a schematic of the three-level EIT system considered in this work (the Λ scheme). The susceptibility such a system can be expressed by [13, 14]:
where Δ=ω-ωab is the probe beam detuning from the resonance frequency ωab of the abtransition, γab is the loss due to the ab-transition, Ωc is the Rabi frequency of the control field, γbc is the decoherence rate, and C is a materials constant related to the density of excitation centers and the dipole moment. We express the permittivity, ε=ε 0(1+χ), of the EIT medium using an complex multipole expansion of Eq. (3):
where εb is the background dielectric constant and A 1,2 and B 1,2 are the coefficients:
and
Fig. 2. (a) Schematic of the energy levels in a three level Λ system. Representative plot of the real and imaginary parts of the permittivity for an EIT medium at two Ωc values of the control field: 0.08 (solid) and 0.03 (dashed)
Figure 2(b) is a representative plot of Eq. (4) assuming εb=1.0, ωab=0.35, γab=0.002, C=0.01s-1, γbc=0, and the two Rabi frequency values 0.08 (solid line) and 0.03 (dashed line) (all units are normalized by the lattice constant, a, of the PhC). Adjusting Ωc has two effects on the optical properties of the EIT medium. First, the frequency width of the transparency window increases with Ωc since the separation between the absorption peaks is equal to Ωc. Secondly, the slope of the dispersion, ℜ(ε), in the transparency region is inversely related to Ωc. Thus, vg of a pulse propagating in the EIT medium decreases as Ωc decreases since vg=dω/dk, ie. the steeper the dispersion, the slower the group velocity.
1.2. Coupled resonator optical waveguide structure
The CROW of this study consists of an infinitely long line of high-Q, point defect PhC cavities separated by a distance of R=3a (see Fig. 1). The cavities are formed in 2D square lattice of high-dielectric rods, ε=11.56, embedded in an EIT medium. The EIT medium has the same physical parameters as those given in section 1.1 except that ωab is chosen so that the EIT resonance frequency matches the mid-band frequency of the vacuum CROW, ω 0=0.3796, the subscript denoting the vacuum background structure. The size of the computational domain is 3a × 14a and contains and array of circular rods with diameter d=0.4a arranged in a square lattice. The point defect cavities are formed by removing a rod in the center of the array. Periodic boundaries are used along the long sides of the domain to form an infinitely long series of defect cavities separated by the distance R=3a. Perfectly matched layer [18] absorbing boundaries on the short sides of the domain truncate the structure in the transverse direction. A spatial resolution of Δx=Δy=0.025a is used in all of the calculations. A transverse-magnetic polarized field (magnetic field components normal to the rod axis) is used in the calculations since a photonic band gap exists for this polarization, and the fields are randomized throughout the domain at the beginning of the calculation. Data collected from a point detector positioned near the center of the point defect is used to calculate the resonant frequency of the CROW using the Padé approximation and Baker’s algorithm [19, 20].
2. Results and discussion
We investigate the effects of the EIT medium on the optical response of the CROW by calculating the dispersion relationship of the CROW for different values for Ωc. For the vacuum background CROW, the frequency width Δω of the waveguide band is 0.0109, and the maximum group velocity v g,0=0.1025vg/c 0 of the band occurs near the zone center. We can calculate the coupling factor from κ=Δω/(2ωres) if the resonant frequency of the isolated resonatorωres is known [6]. A separate calculation reveals that for the vacuum background single-defect cavity ωres=0.3793, thus giving a coupling factor of κ 0=0.0144. For the EIT background isolated resonator, the value of ωres changes for each value of Ωc because of a frequency pulling effect [21].
For convenience, we divide the response of the EIT CROW into two regimes: case 1) when the frequency width of the EIT transparency exceeds the Δω of the CROW and case 2) when the opposite is true.
Figures 3(a) and (b) show the CROW bands and group velocities as a function of wavevector for case 1. The vacuum background band is included as a comparison to the EIT-CROW bands under different values of Ωc. The points are the results from FDTD calculations while the solid lines are the results given by Eqs. (1) and (2), after calculating κ from the FDTD data, as done previously for the vacuum CROW. As shown, the bands become flatter as Ωc is decreased (e.g. as the dispersion of the EIT media increases). In addition, the bands share a common point at the zone center at ω 0=ωab around which the bands pivot as Ωc is tuned. When Ωc=0.01, slightly less than Δω of the vacuum CROW, the band is very flat with a maximum group velocity less than 2.0×10-3 vg/c 0 and a Δω<2.0×10-4 a/λ. This represents over a 50 times slowing of the group velocity in comparison to the vacuum background CROW.
Figures 4 (a) and (b) present the bands and group velocities for the structure under case 2, respectively. A single band from just above the transition between case 1 and 2 is included for comparison (Ωc=0.02). As shown, the bands in case 2 are significantly flatter than in case 1. For Ωc=0.003, the group velocity is below 1.8×10-4 vg/c 0, approximately 570 times slower than that of the vacuum CROW. For values of Ωc ≤ 0.002, the absorption of the EIT media (recall γab=0.002 in our calculations) would reduce the Q-factor of the cavities, thus setting a lower limit for the group velocity. (However, we note that this limit occurs only for a steady-state control field. If Ωc were ramped down adiabatically to zero as the pulse propagates in a finite CROW structure, the pulse would be halted and stored in the EIT [22].) In addition to the slower group velocity, the rate of change in the slope of the bands with change in Ωc decreases in case 2 in comparison to case 1. This can be seen more clearly in a plot of the group velocity in the EIT CROW as a function of Ωc. Since this quantity is directly related to the coupling factor by Eq. (2), we present κ (Ωc) in Figure 5.
Fig. 5. Coupling coefficient (normalized by the vacuum CROWκ 0) as a function of control field Rabi frequency. The points and crosses are taken from the FDTD and TB analysis data, respectively, and the solid line is a plot of the EIT group velocity relationship (right axis) with a fitted α parameter.
On the right axis in Figure 5, we plot the vg vs. Ωc relationship of the EIT background [23]. Working from the group velocity at the ab-transition frequency we can derive an expression for vg(Ωc):
where α is (2Cωab)-1. From the values given earlier, we calculate α=1.317×102 for the EIT background medium and plot Eq. (7) in Figure 5. We find that the relationship of the EIT CROW κ vs. Ωc follows that of the EIT background. We used a minimization algorithm to fit the FDTD data points to an equation of the form of Eq. (7) to find α=1.940×10 2. The shape of κ (Ωc) can be explained as follows: Since the coupling is dependent upon the overlap of the field between adjacent resonators, κ ∝1/Q. At the same time, as Soljačić et al. have shown, Q increases as the steepness of the dispersion increases, thus vg ∝1/Q. It follows that κ ∝vg, meaning that in the EIT CROW system, manipulation of the coupling coefficient is directly coupled to a controllable parameter, namely Ωc.
3. Conclusion
In conclusion, we have successfully modeled a PC CROW structure embedded in a three-level EIT medium with the FDTD method. The ADE method was employed to incorporate the frequency dependence of ε of the highly dispersive EIT background into the time-stepping algorithm. We found that by adjusting the control field strength, dynamic control the shape of the CROW band was enabled, whereas the shape of the CROW band is typically fixed since it is controlled by the coupling strength of the cavities, eg. distance R between the defects. We classified two regimes of the dynamic system by the relative frequency band width of the vacuum CROW structure verses the band width of the EIT transparency window. In the two regimes, the rate of change in the coupling factor (or the group velocity) as a function of the control field Rabi frequency differed, being larger in the case where Ωc ≥ Δω. We also found that the relationship of the EIT-CROW coupling factor, κ (Ωc) follows that of the EIT group velocity, vg(Ωc). In the spirit of the TB analysis, one can draw a parallel between the atomic separation distance and the resonator cavity separation in the CROW. Therefore, one can also think of the dispersion of the media or Ωc in the cavity as the ‘atomic potential’. As Ωc decreases, the atomic potential increases, reducing the coupling between modes. To achieve a usable slow light effect, a large bandwidth is required to accept the input pulse into the system before the band is flattened and the pulse stopped [24]. Through our investigation, we have shown an implementation of the CROW which would allow dynamic control of the propagation speed of a pulse through the system. We chose the 2D square lattice PhC system because it has a monopole point defect mode and simple band structure, making it a ‘clean’ system for this study; however, these effects are not limited to this configuration. More importantly, the unique dispersion characteristics of the EIT in the CROW fixes a pivot point in the CROW bands, around which the bands change shape. This pivot offers a fixed operation point in momentum space which could useful in coupling in and out of the structure. The EIT and CROWimplementation could be applied in optical buffer or optical memory schemes.
Acknowledgments
Funding for this work was through the Swedish Foundation for Strategic Research (SSF) under the INGVAR program, the SSF Strategic Research Center in Photonics, and the Swedish Research Council (VR).
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