## Abstract

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

## 1. Introduction

In recent years slow light phenomena have attracted great interest due to their potential applications [1] such as optical processing, pulse buffering [2,3] and interferometry [4,5]. The ability to slow down the group velocity (*ν _{G}*) of a propagating pulse requires control over the frequency dependence of the refractive index by using strong material dispersion [6]. After an initial demonstration of ultra slow group velocities of light pulses by Hau

*et al.*[7], 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 [22]. 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 [25]. 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 [26], 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 [26]. 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 [6]. 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:

**r**= (

*x,*0

*,z*),

*ε*

_{1}

*, ε*

_{2}are the dielectric constant modulations and:

*ω*the angular frequency used in the recording steps, c is the light velocity in the free space, and

_{o}*θ*,

_{r}*θ*,

_{s}*θ*are the angles shown in Fig. 1. The wave propagation in the resulting multiplexed holographic structure is described by the Helmholtz equation:

_{w}**K**and

_{1}**K**connect R with S and S with W respectively, so the Bragg conditions are satisfied:

_{2}**K**is given by:

_{R}*ω*is the angular frequency used in the reconstruction step. From Eqs. (2), (3), and (6) it follows that:

*β*=

*nω/c*. The boundary conditions for 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

*ω*(

_{o}*ω*=

*ω*+

_{o}*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 *θ _{r}* = 0 and at the Bragg angle in the reconstruction step

*θ*=

*θ*= 0. Assuming normal incidence parameters given by Eq. (11) can be approximated to:

_{r}*θ*,

_{r}*θ*,

_{w}*θ*,

_{s}*ω*) material properties (

_{o}*ε*

_{1},

*ε*

_{2}

*, n*) and only $\overline{\vartheta}$

*depends on frequency detuning (*

_{p}*δω*). Introducing approximations (12) into Eq. (10) we obtain the differential equation system given by:

*n*is the group index of waves S and W. It is important to note that Eq. (14) shows that the normalized group index values

_{p}*n*/

_{s}*n*and

*n*/

_{w}*n*are negatives in the range of recording angles for transmission gratings (see Fig. 2) where |

*n*|

_{w}*> n*and |

*n*|

_{s}*> 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 [25]. 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:

*x*are the three roots of the characteristic polynomial (P) of matrix

_{i}**A**in Eq. (13):

*C*are integration constants given by:

_{i}In the particular case that the HRS is reconstructed at the Bragg angle (*θ* = *θ _{r}*) and at frequency

*ω*(

_{o}*δω*= 0), according to Eqs. (14),

*ξ*=

_{s}*ξ*= 0 and in this case the roots of P given by Eq. (17) are:

_{w}*x*

_{0}= 0, ${x}_{1}=j\sqrt{{\nu}_{1}+{\nu}_{2}}$, ${x}_{2}=-j\sqrt{{\nu}_{1}+{\nu}_{2}}$ (where we have assumed that |

*ν*

_{2}| > |

*ν*

_{1}|), therefore, taking into account Eqs. (16), (18), and (19), transmittance t can be expressed as:

*t*= 1 at frequency design

*ω*. Therefore, if the HRS has a thickness

_{o}*L*=

*L*

_{1}, in the forbidden band of the reflection grating, a permitted band appears, centered at

*ω*. In the next section we will describe the process for obtaining a practicable analytical expression of transmittance near the recording frequency

_{o}*ω*

_{0}that permits us to analytically study the pulse propagation inside the HRS.

## 3. Analytical expression for the permitted band

Introducing Eqs. (18) and (19) into Eq. (16), and performing some algebraic manipulations, transmittance t can be expressed as:

*ω*

_{0}(

*δω <<*1), it can be assumed that the parameters

*ξ*<< 1 and

_{w}*ξ*<< 1 under the conditions that the resulting permitted band is narrow, so the roots of polynomial P can be approximated (using a first order Taylor series expansion on variables

_{s}*ξ*and

_{w}*ξ*, at

_{s}*ξ*= 0 and

_{w}*ξ*= 0) by:

_{s}*ψ*(on variables

*ξ*and

_{w}*ξ*) and finally a second order Taylor series expansion on

_{s}*ξ*and first order on

_{w}*ξ*on the argument of the root square in the denominator of Eq. (21).

_{s}*δω*variable, and a phase that is linear on

*δω*:

*τ*is the time-delay induced by the device and

_{d}*γ*is the scale parameter which specifies the half-width at half-maximum of the Lorentzian function (the accuracy of this analytical approximation may be seen in the numerical section). Time delay and scale parameter only depend on the material properties and the recording geometry of HRS. Assuming that

*ε*

_{2}=

*αε*

_{1}, we obtain that their values are:

From Eq. (26) it is easy to deduce that *τ _{d}* → ∞ if

*α*approaches $\sqrt{\frac{-n}{{n}_{w}+n}}$, 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:

*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 ${L}_{1}({L}_{1}=2\pi /\sqrt{{\nu}_{1}+{\nu}_{2}})$, it follows that the device thickness must be: The analytical 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:

*W*

_{0}is the free space pulse spatial width, defined as the length from the maximum at which the pulse amplitude decreases a factor

*e*

^{−1/2}. If we introduce the free-space pulse time length as

*T*

_{0}=

*W*

_{0}

*/c*, we can obtain the frequency domain of the incident field $\widehat{\mathrm{\Phi}}$

*(*

_{inc}*ω*) as the direct Fourier transform of the incident field (33):

The transmitted field Φ* _{tr}*(

*t*) will then be given by the inverse Fourier transform:

Introducing Eq. (25) in Eq. (35) it follows that:

*(*

_{inc}*ω*) by the expression given in Eq. (34) in Eq. (36), and approximating t(

*ω*) to:

*(*

_{tr}*t*) is given by:

*t*(–

*ω*) =

*t*

^{*}(

*ω*). Equation (38) shows that the transmitted field is equal to a time-delayed incident field (33), distorted by a factor $({c}^{4}{({\tau}_{d}-t)}^{2}-{c}^{2}{n}^{2}{W}_{0}^{2}+2{\gamma}^{2}{n}^{4}{W}_{0}^{4}/(2{\gamma}^{2}{n}^{4}{W}_{0}^{4})$. In order to characterize the distortion of the transmitted field we are going to use the root-mean-square

*D*given by [27]:

_{rms}## 5. Numerical simulations

In our numerical calculations, we take *ω _{o}* = 3.5431 × 10

^{15}

*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

*n*(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

_{s}*n*,

_{s}*n*.

_{w}Equation (31) and Fig. 3 show that a value of *L*1 = 2*mm* corresponds to *n _{s}* = −4.8793, which is the value that we are going to take in order to analyze an example of our system,together with

*n*= −4.8793 (which implies that the transmission grating was recorded in symmetrical geometry). For these parameters Fig. 4 shows the values of transmissivity (

_{w}*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 $\widehat{\mathrm{\Phi}}$

*(*

_{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

*τ*= 124

_{d}*ns*and

*γ*= 80.8

*MHz*. 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

*D*<< 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

_{rms}*D*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.

_{rms}## 6. Conclusions

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.

## Acknowledgments

The authors acknowledge support from project FIS2009-11065 of Ministerio de Ciencia e Innovación of Spain.

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