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We demonstrate a slow light configuration that makes use of Bloch Surface Waves as an intermediate excitation in a double-prism tunneling configuration. This method is simple compared to the more usual technique for slowing light using the phenomenon of electromagnetically induced transparency in atomic gases or doped ionic crystals operated at temperatures below 4 K. Using a semi-numerical approach, we show that a 1D photonic crystal, a multilayer structure composed of alternating layers of TiO2 and SiO2, can be used to slow down light by a factor of up to 400. The results also show that better control of the speed of light can be achieved by changing the number of bilayers and the air-gap thickness appropriately.
© 2014 Optical Society of America
1. Introduction
Interest in slow light began after the research group led by Hau et al. [1] slowed down the speed of light to 17 meters per second using the technique of electromagnetically induced transparency (EIT) in a Bose-Einstein condensate. In a recent experiment [2], light has been stopped and used for image storage by EIT up to a time of one minute, setting a new benchmark for EIT-based light stopping. Although the recent experiment uses a doped Pr3+:Y2SiO5 crystal instead of atomic gas, it still requires temperatures lower than 4 K for its operation, which limits the adoption of the technique in practice. A more practical way to achieve slow light is through the use of photonic band gap materials (PBGM), which operate at room temperature [3–8]. Significant reduction in the speed of light in PBGMs is mainly attributed to photonic band-structure effects due to a spatially periodic dielectric function, rather than from material dispersion [9]. Theoretically and numerically, using loss-less photonic crystal waveguides, researchers have been able to completely stop light at the vicinity of band gap edges by dynamically tuning the properties of the material while the light is still in the material [10, 11]. With no dynamic tuning, the authors of [12] claim three orders of magnitude reduction in the speed of light (vg ∼ 0.0008c). However, due to leaky modes, out of plane radiation, material absorption, and structural distortion, even the three orders of magnitude reduction is not realizable in practice. These effects inherently impose limits on the maximum achievable slow down factor in PBGMs. Such imperfections can partially be accounted for in simulations by adding a small imaginary part to the otherwise real-valued refractive indices of the materials used in PBGMs.
Slowing down the speed of light is of significance for many practical applications. Slow light in optical devices can be used to make optical buffers which temporarily store light [13, 14]. Contrary to its name, it can actually increase the speed of telecommunications and data transfer in photonic crystal waveguides [15, 16]. Higher density of modes and enhanced light-matter interaction assisted by slow light in PBGMs can in turn enhance light amplification [17], nonlinear phase sensitivity [18], nonlinearities in the material response [19], and stimulated Raman scattering [20]. Recently, applications in sensing have also been proposed based on slow light in photonic crystal waveguides [21, 22].
In this paper, we introduce a new configuration that can reduce the speed of light by a factor of up to 400 in a prism tunneling configuration using Bloch Surface Waves (BSW) on a one dimensional photonic bandgap TiO2-SiO2 multilayer as an intermediate excitation. Bloch Surface Waves [23, 24] are propagating non-radiative, surface-bound electromagnetic waves that exist within the forbidden band gap of the multilayer. The frequency of the BSW can be located anywhere within the band gap by adjusting the thickness of the termination layer of the photonic multilayer. For the simulations presented here we chose an operating optical wavelength (632.8 nm) tuned to the BSW mode. To our knowledge, this technique of generating slow light assisted by BSW of a one-dimensional PBGM has not been reported in the literature.
2. Simulation approach
The optical configuration for the generation of slow light using BSW on a one-dimensional PBGM is shown in Fig. 1. The set up consists of two prisms separated by an air gap. One prism has a PBGM multilayer structure deposited on its reflecting face such that when light is incident at the appropriate phase matching angle, BSWs are excited at the multilayer-air interface. Excitation of BSWs is indicated by a sharp drop in the reflected light. The BSW is a surface bound excitation with evanescent fields that penetrate both into the multilayer and the air. The second prism permits the evanescent BSW field in the air to become a radiative wave that is transmitted out of the prism as shown. The process is similar to the frustrated total internal reflection, a configuration much explored in previous work on group velocity manipulation [25]. For the simulation, a collimated optical beam with a Gaussian temporal pulse profile is incident through the prism at θres, the resonant angle for BSW excitation. The Gaussian pulse is described by
where A is the amplitude of the pulse, t is time, t0 is the center of the pulse, and σ is the pulse envelope width.
Fig. 1 A schematic diagram of the configuration for slow light generation using BSW as an intermediate excitation.
Using the 2 × 2 matrix formulation for a layered medium [23], transmittance coefficients of each frequency present in the pulse are computed. These transmittance coefficients are complex numbers, containing both the amplitude and phase information of transmitted light as a function of frequency. To determine the effect of this transmittance function on our Gaussian pulse, we form the Fast Fourier Transform (FFT) of the pulse and multiply each frequency component by the corresponding complex transmittance value. The result is the spectrum of the transmitted pulse in the frequency domain. An inverse FFT is then used to recover the transmitted pulse as a function of time. The difference between the peaks of the input and transmitted pulses is called the group-delay (τg), which can be computed using the phase information of the transmittance function and is given by
where ϕ is the phase of the pulse at a given frequency ω. The group velocity (vg) is consequently given by where z is the total propagation distance of the pulse. The group index (ng), which is a slowdown factor from the velocity c [14] is For structural slow light devices, a figure of merit is frequently employed [26] as group-index bandwidth product also known as normalized Delay Bandwidth Product (nDBP), where the working bandwidth is the range over which ng does not vary by more than 10%.3. Simulation results and discussions
The results represented here are based upon a one-dimensional PBGM which consists of a multilayer composed of eight bilayers of TiO2 (refractive index = (4.84 + i0.0007)1/2) and SiO2 (refractive index = (2.1316 + i0.0001)1/2), with thicknesses of 82.8 nm and 154.9 nm respectively. The termination layer has a thickness of 185 nm which results in a BSW mode near the center of the bandgap. As indicated in [27] the imaginary values of the dielectric constants of SiO2 and TiO2 in this wavelength range are difficult to find in the literature because they are so small and because, in the case of TiO2, are very dependent on the mode of preparation. The values used here are based on published sources [28,29] and selected to match with our groups long history of experimental investigations with BSWs [24, 27, 30]. The band structure of the multilayer (Fig. 1(b)) is shown in Fig. 2. The vertical axis represents normalized frequency and the horizontal axis the wave vector component parallel to the plane of the multilayer. The BSW, as a surface bound wave, has its wave vector entirely in this plane. The dispersion diagram shows regions in which light is radiative into the multilayer (green shaded) and regions in which it is non-radiative into the multilayer (blue shaded). The band of blue that rises from left to right in the figure represents the photonic band gap in the multilayer. The plot also shows the limiting light line as the red dashed line. This line is the dispersion for light incident at grazing incidence along the multilayer surface. For light at other angles of incidence in air the parallel component will be smaller, thus the entire region to the left of the limiting light line is radiative into the air side of the multilayer. BSWs exist in a region in which the mode is not radiative into either the air or the multilayer. This region is shown as the narrow strip to the right of the light line and within the photonic band gap. Given this dispersion relation we set the wavelength of light for our simulation to be 632.8 nm (ω = 0.47 [2πc/Λ]) and θres = 44.3950 (β = 0.68 [2π/Λ]), corresponding to the narrow defect mode region in Fig. 2.
Fig. 2 Surface dispersion diagram for an 8 bilayer multilayer TiO2-SiO2 PBGM used in the simulations. The red dashed line represents the limiting lightline in air. The green and blue shaded areas represent radiative and non-radiative regions respectively into the multilayer. The plot units are in reduced angular frequency (2πc/Λ) and wavevector (2π/Λ) where Λ is the periodicity of the multilayer.
Reduction in the speed of light in the defect mode of the multilayer can be attributed to the generation of BSWs at the resonance frequency. The light is stored temporarily on the surface of the multilayer in the form of BSWs before being transmitted to the second prism through the air gap. The thickness of the air gap determines the strength of coupling between the BSW and the second prism and hence establishes the lifetime of BSW before it becomes radiative. The air gap can be adjusted appropriately to obtain the maximum time delay possible. Figure 3(a) shows the maximum time delays gained at different air gap thickness for a eight bilayer TiO2-SiO2 multilayer. An optimum time delay of 4.209 ps (ng ∼ 350) was achieved with air gap thickness of 1600 nm, which can also be clearly seen in Fig. 3(b). The transmitted (red) pulse is shifted to the right compared to the incident (blue) pulse, meaning that the pulse takes a longer time to travel the same distance when the multilayer is present on its way compared to the situation when there is a glass-slab of equal thickness in place of the multilayer. To observe the optimum group-delay, however, a sufficiently long-enough pulse is required. This is important because the frequency bandwidth of the pulse has to lie within the narrow transmission bandwidth of the multilayer to observe slow light. In our simulation, this was ensured by requiring the FFT spectrum of the incident pulse to completely lie within the transmittance region. Moreover, the transmission bandwidth of the multilayer becomes narrower with the increase in the number of bilayers. So, an even longer pulse is required for multilayers with a higher number of bilayers. We obtain nDBP of 0.02 for the eight bilayered TiO2-SiO2 multilayer, which is comparatively smaller than the values reported in [26, 31]. However, the large nDBP in [26, 31] are realized by sending the light pulse through much longer (∼ 100 μm) photonic crystal waveguides, as opposed to a thin (∼ 2 μm) PBGM in our study.
Fig. 3 (a) Delay time as a function of wavelength for four different air gap values. (b) Incident (top) and transmitted (bottom) pulses showing 4.209ps delay.
To exhibit the behavior of the group index (slow-down factor) and transmittance with respect to increasing air-gap thickness, a plot is presented in Fig. 4(a) with the multilayer of 8 bilayers. At smaller air-gap thickness, higher transmittance is observed but the group index is low, and vice-versa. Adjusting these properties, a required group delay with some acceptable transmittance can be easily acquired. The plot also shows that increasing the air-gap thickness after a certain level does not further increase the group index. In fact, the group index starts decreasing, owing to the fact that increasing the air-gap thickness decreases the transmittance significantly. The transmitted pulse tunnels through the air-gap in the form of an evanescent wave which is a very short-range wave. Thus, with increasing air-gap thickness, only a small amount of input light is tunneled across the air-gap which eventually starts decreasing the group index.
Fig. 4 (a) Group Index (blue) and corresponding transmittance (green) as a function of air-gap thickness with the number of TiO2-SiO2 bilayers = 8. (b) Group Index (blue) and Air Gap thickness (green) as a function of number of bilayers.
A similar behavior is also seen with the increase in the number of bilayers in the multilayer as in Fig. 4(b). Increasing the number of bilayers and setting an appropriate air gap thickness initially increases the group index, but after reaching a certain limiting number of bilayers, improvement in the group index is not seen. Adding more bilayers increases internal reflections, absorption, and scattering which limits the amount of transmitted light resulting in the low transmittance and group index.
4. Conclusion
Our study shows that a simple one-dimensional PBG multilayer structure in a prism-coupled BSW configuration can reduce the speed of light by about a factor of up to 400 when operated at its narrow defect mode transmission region. This finding is remarkable, keeping in mind that losses due to material absorption are not ignored. The slow-down factor depends greatly on the number of bilayers in the multilayer and on the air gap thickness. Further improvement in the slow down factor is possible by wisely choosing materials with higher refractive index contrast for the multilayers.
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