We present a holographic system that can be used to manipulate the group velocity of light pulses. The proposed structure is based on the multiplexing of two sequential holographic volume gratings, one in transmission and the other in reflection geometry, where one of the recording beams must be the same for both structures. As in other systems such as grating induced transparency (GIT) or coupled-resonator-induced transparency (CRIT), by using the coupled wave theory it is shown that this holographic structure represents a classical analogue of the electromagnetically induced transparency (EIT). Analytical expressions were obtained for the transmittance induced at the forbidden band (spectral hole) and conditions where the group velocity was slowed down were analyzed. Moreover, the propagation of Gaussian pulses is analyzed for this system by obtaining, after further approximations, analytical expressions for the distortion of the transmitted field. As a result, we demonstrate the conditions where the transmitted pulse is slowed down and its shape is only slightly distorted. Finally, by comparing with the exact solutions obtained, the range of validity of all the analytical formulae was verified, demonstrating that the error is very low.
© 2011 OSA
In recent years slow light phenomena have attracted great interest due to their potential applications  such as optical processing, pulse buffering [2,3] and interferometry [4,5]. The ability to slow down the group velocity (νG) of a propagating pulse requires control over the frequency dependence of the refractive index by using strong material dispersion . After an initial demonstration of ultra slow group velocities of light pulses by Hau et al. , several methodologies have been used, such as electro-magnetically induced transparency (EIT) resonances [8–10], coherent population oscillations [11, 12], atomic double resonances [13, 14], photonic crystal waveguides [15–17], coupled resonator optical waveguides (CROWs) [3, 18, 19] and stimulated Brillouin scattering (SBS) in optical fibers [20, 21].
The pioneering EIT phenomenon produces a slow down of the light group velocity by means of the destructive quantum interference that creates a narrow transparency window in the absorption line of an atomic media . Similar effects have been theoretically and experimentally demonstrated in classical systems such as coupled optical resonators, where coupled resonator induced transparency (CRIT) is given by mode splitting and classical destructive interference [19,23,24]. Moreover, as an optical analog of EIT, grating induced transparency (GIT), based on a three-mode waveguide modulated by two co-spatial gratings has recently been proposed . In this system the three waveguide modes are equivalent to the three quantum states of EIT while the gratings are counterparts of the electromagnetic waves. So, the coupled mode equations describe the system, (as does the Hamiltonian in the EIT), due to the continuous coupling along the waveguide.
Similarly, in this paper we propose multiplexing two sequential holographic volume gratings, one in transmission and the other in reflection geometry, where one of the recording beams must be the same for both structures. Reconstruction of the reflection grating at the Bragg angle will give three waves inside the material which can be described by the coupled wave theory , resulting in a transmission spectrum with a unity transmission at the forbidden band of the reflection grating (spectral hole) similar to that of EIT. A theoretical analysis of the coupled wave theory for this system was performed to obtain an analytical formulae for the transmittance at the generated permitted band (spectral hole), thus making it possible to analyze the slow down of the light group velocity as a function of the parameters of the gratings. Finally, analytical expressions were also obtained to analyze of the propagation of Gaussian pulses and their distortion of the transmitted field.
2. Theoretical background
The theory of coupled waves describes the diffraction of volume holographic gratings . For this, it is assumed that only two waves are present in the material. So, if two holographic volume gratings are multiplexed with a specific geometry, it is reasonable to assume that three waves are propagating inside the material. Therefore, it is expected to obtain for the propagating waves an analog mathematical description analogous to the Hamiltonian of a three level atomic system in the Λ configuration with two fields . We propose a holographic recording structure (HRS) obtained by using sequential multiplexing of two holographic gratings (in a material with no real time response), a reflection grating and a transmission grating, with one of the recording beams used being the same for both gratings (see Fig. 1) The device obtained can be represented by a spatial modulation of the relative dielectric constant ε (assuming zero conductivity in the medium) given by:Fig. 1. The wave propagation in the resulting multiplexed holographic structure is described by the Helmholtz equation: 26] to solve Eq. (4), only three waves will be present in the material, so the total electric field inside the device will be given by the superposition of the three waves with complex amplitudes R(z), S(z) and W(z): Eqs. (2), (3), and (6) it follows that: Eq. (5) in the Helmholtz equation (4) and using the coupled wave theory  approximations, we obtain the differential equation system: Eqs. (10) are R(0) = 1, S(L) = 0 and W(L) = 0, where we have assumed that the thickness of the HRS is L. In order to obtain an analytical expression for the HRS amplitude transmittance t = R(L) near the recording frequency ωo (ω = ωo +δ ω, δ ω << 1) and for Bragg angle reconstruction.
2.1. Normal incidence in reconstruction step
A particular case of special interest is that in which we work at normal incidence θr = 0 and at the Bragg angle in the reconstruction step θ = θr = 0. Assuming normal incidence parameters given by Eq. (11) can be approximated to:Eq. (10) we obtain the differential equation system given by: Eq. (14) shows that the normalized group index values ns/n and nw/n are negatives in the range of recording angles for transmission gratings (see Fig. 2) where |nw| > n and |ns| > n as can be seen in Fig. 2, so the values of the group index range from −2n to −4n. Negative values of the group index show us that S and W are backward propagating waves.
The differential equations system (13) is similar to that obtained for the grating induced transparency waveguide (GIT) in reference . However there are some differences since:
- The A matrix coefficients given by Eq. (13), are all purely imaginary. However, in the GIT matrix A 12 and A 21 are reals.
- Due to the holographic recording process A 12 ≠ A 21 and A 23 = A 32 will be only be fulfilled if the transmission grating is recorded with symmetrical geometry.
- According to Eqs. (13) all the coupling coefficients depend on group index parameters, and in the GIT matrix only diagonal elements depend on the group index.
- According to Eq. (14), in the HIT model, the group index depends on the geometry of the recoding process.
Solving the differential equation system (13), it follows that R(z) can be expressed as:Eq. (13):
In the particular case that the HRS is reconstructed at the Bragg angle (θ = θr) and at frequency ωo (δω = 0), according to Eqs. (14), ξs = ξw = 0 and in this case the roots of P given by Eq. (17) are: x 0 = 0, , (where we have assumed that |ν 2| > |ν 1|), therefore, taking into account Eqs. (16), (18), and (19), transmittance t can be expressed as:Eq. (20) that the set of values (where m is a natural number), means that t = 1 at frequency design ωo. Therefore, if the HRS has a thickness L = L 1, in the forbidden band of the reflection grating, a permitted band appears, centered at ωo. In the next section we will describe the process for obtaining a practicable analytical expression of transmittance near the recording frequency ω 0 that permits us to analytically study the pulse propagation inside the HRS.
3. Analytical expression for the permitted bandEq. (24) by introducing Eq. (23) in Eqs. (21) and (22), taking into account Eq. (19) and after making a first order Taylor series expansion for ψ (on variables ξw and ξs) and finally a second order Taylor series expansion on ξw and first order on ξs on the argument of the root square in the denominator of Eq. (21). Eqs. (14) and (12) in Eq. (24), we deduce that transmittance is given by a complex function whose module is the square root of a Lorentzian function on δω variable, and a phase that is linear on δω:
From Eq. (26) it is easy to deduce that τd → ∞ if α approaches , which implies that group velocity of the transmitted field is reduced to 0, but, as can be seen in Eq. (27), γ will also approach 0. Therefore, in order to obtain the maximum time delay with non-null transmittance we are going to analyze the case in which:Eq. (28) in Eqs. (26) and (27), it follows that time delay and scale factor can be written as: Eqs. (29) and (30) in Eq. (25), the condition γ 2 > 1 must be fulfilled.Thus it follows from Eq. (30) that |η| > 1, but obviously as demonstrated before, values closer to 1 provides higher time delays and a narrower Lorentzian transmittance function. If we introduce Eqs. (12), (14), and (28) in , it follows that the device thickness must be: Eqs. (25), (29), (30), and (31) completely determine the transmittance behavior of the permitted band generated by the HRS, which makes it possible to control the group velocity of a propagating pulse given by:
4. Propagation through the permitted band
In this section we are going to analyze the delay and deformation of temporal signals that propagate through the permitted band previously analyzed. Let us consider an incident Gaussian electric field given by:
The transmitted field Φtr(t) will then be given by the inverse Fourier transform:Eq. (34) in Eq. (36), and approximating t(ω) to: Equation (38) shows that the transmitted field is equal to a time-delayed incident field (33), distorted by a factor . In order to characterize the distortion of the transmitted field we are going to use the root-mean-square Drms given by : Eqs. (38) and (33) into Eq. (39), and integrating gives:
5. Numerical simulations
In our numerical calculations, we take ωo = 3.5431 × 1015 s −1, n= 1.5, ε 1 = 0.0075, and η = 1.01. Introducing these parameters into Eqs. (29), (30), and (31), it is easy to observe that the thickness L 1, which is given by equation 31 depends on group index ns (see Fig. 3). So it can be deduced that the reflection grating determines the HRS thickness. In the case of the time delay and scale factor both of them depend on group index ns, nw.
Equation (31) and Fig. 3 show that a value of L1 = 2mm corresponds to ns = −4.8793, which is the value that we are going to take in order to analyze an example of our system,together with nw = −4.8793 (which implies that the transmission grating was recorded in symmetrical geometry). For these parameters Fig. 4 shows the values of transmissivity (T = |t|2) obtained using the exact solution (16) and the analytical expression given by Eq. (25). As can be seen, a narrow permitted band is shown in the center of the forbidden band of the reflection grating. In the center of the figure the permitted band is zoomed, and the analytical solution is identical to the exact one, with a relative error of our approximated function lower than 0.012 %, as shown in the inset curve of Fig. 4. Figure 5 shows the phase values obtained by using exact solution (16) and the analytical expression given by Eq. (25). It may be seen that both solutions are identical in a region δω lower than the region where T functions coincides (compare Figs. 4 and 5), so if we want to use our analytical approximated solution, and the results shown in Eqs. (38) and (39), we must restrict the width of the incident field in the frequency domain inc(ω) to a region where the exact and approximated phase will be in good agreement. For the parameters previously mentioned we find, using Eqs. (29) and (30), that time delay τd = 124 ns and γ = 80.8MHz. Taking into account these values, we introduced in our system a Gaussian pulse (see Eqs. (33), (34)) with parameters W 0 = 108 μm,T 0 = 333 ns in order to ensure that Drms << 1 (see Eq. (40)). Figure 6 shows the incident pulse (brown color) and the time-delayed pulses obtained using Eq. (35) taking into account approximations (25) (blue) and (37)–(38) (pink). As can be observed both results are very similar (it is important that the pulses were not normalized for comparison), where the relative error with respect to the exact value is lower than 3% in the region of interest, as can be observed in the inset curve. The values of Drms obtained by using Eqs. (39) and (40) are 0.024 and 0.026 respectively, which implies that there is no significant distortion, and that the analytical expression obtained in Eq. (38) can be used.
We have described a holographic system that can be used to delay light pulses. The HRS proposed is based on the multiplexing of two holographic volume gratings, one in transmission and the other in reflection geometry. Using coupled wave theory, we obtained an analytical expression for the transmittance induced at the forbidden band (spectral hole) and analyzed the conditions where group velocity is slowed down. We have shown that the exact and approximated solutions are in good agreement. Moreover, the propagation of Gaussian pulses is analyzed for this system by obtaining, after further approximations, analytical expressions for the transmitted field. We have demonstrated that the transmitted pulse is slowed down and its shape is only slightly distorted. Finally, by comparing with the exact solutions obtained, the range of validity of all the analytical formulae was verified, demonstrating that the error is very low.
The authors acknowledge support from project FIS2009-11065 of Ministerio de Ciencia e Innovación of Spain.
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