Coupled wave analysis of holographically induced transparency (HIT) generated by two multiplexed volume gratings

doi:10.1364/OE.19.007094

Coupled wave analysis of holographically induced transparency (HIT) generated by two multiplexed volume gratings

Luis Carretero, Salvador Blaya, Pablo Acebal, Antonio Fimia, Roque Madrigal, and Angel Murciano »View Author Affiliations

Optics Express, Vol. 19, Issue 8, pp. 7094-7105 (2011)
http://dx.doi.org/10.1364/OE.19.007094

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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.

1. Introduction

In recent years slow light phenomena have attracted great interest due to their potential applications [1

R. Boyd, O. Hess, C. Denz, and E. Paspalkalis, “Slow light,” J. Opt. 12(10), 100301 (2010). [CrossRef]

] such as optical processing, pulse buffering [2

J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005). [CrossRef]

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

] and interferometry [4

Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer,” Phys. Rev. Lett. 99(24), 240801 (2007). [CrossRef]

Z. M. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, “Enhancing the spectral sensitivity of interferometers using slow-light media,” Opt. Lett. 32(8), 915–917 (2007). [CrossRef] [PubMed]

]. 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

R. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in Progress in Optics , E. Wolf, ed., (Elsevier, 2002) vol. 43, chap. 6, pp. 497–530. [CrossRef]

]. After an initial demonstration of ultra slow group velocities of light pulses by Hau et al. [7

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]

], several methodologies have been used, such as electro-magnetically induced transparency (EIT) resonances [8

A. Kasapi, M. Jain, G. y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys. Rev. Lett. 74(13), 2447–2450 (1995). [CrossRef] [PubMed]

–10

L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics 2(8), 451–453 (2008). [CrossRef]

], coherent population oscillations [11

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]

, 12

M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90(11), 113903 (2003). [CrossRef] [PubMed]

], atomic double resonances [13

R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73(6), 063812 (2006). [CrossRef]

, 14

R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys. Rev. Lett. 98, 043902 (2007). [CrossRef] [PubMed]

], photonic crystal waveguides [15

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef] [PubMed]

–17

M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73(9), 096501 (2010). [CrossRef]

], coupled resonator optical waveguides (CROWs) [3

F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]

, 18

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]

, 19

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004). [CrossRef]

] and stimulated Brillouin scattering (SBS) in optical fibers [20

Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25(1), 201–206 (2007). [CrossRef]

, 21

L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2(8), 474–481 (2008). [CrossRef]

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

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]

]. 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

D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004). [CrossRef]

,23

M. F. Yanik, W. Suh, Z. Wang, and S. H. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). [CrossRef] [PubMed]

,24

K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency.” Phys. Rev. Lett. 98(21), 213904 (2007). [CrossRef] [PubMed]

]. 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

H. C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express 17(14), 11710–11718 (2009). [CrossRef] [PubMed]

]. 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

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2945 (1969).

], 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

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2945 (1969).

]. 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

R. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in Progress in Optics , E. Wolf, ed., (Elsevier, 2002) vol. 43, chap. 6, pp. 497–530. [CrossRef]

]. 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:

ε = ε_{0} + ε_{1} Cos (K_{1} . r) + ε_{2} Cos (K_{2} . r)

(1)

where r = (x,0,z), ε ₁ , ε ₂ are the dielectric constant modulations and:

K_{1} = \frac{n ω_{o}}{c} (Sin (θ_{r}) - Sin (θ_{s}), 0, Cos (θ_{r}) - Cos (θ_{s}))

(2)

K_{2} = \frac{n ω_{o}}{c} (Sin (θ_{s}) - Sin (θ_{w}), 0, Cos (θ_{s}) - Cos (θ_{w}))

(3)

where

n = \sqrt{ε_{0}}

is the average refractive index, ω_o the angular frequency used in the recording steps, c is the light velocity in the free space, and θ_r , θ_s , θ_w are the angles shown in Fig. 1. The wave propagation in the resulting multiplexed holographic structure is described by the Helmholtz equation:

\frac{\partial^{2} E}{\partial x^{2}} + \frac{\partial^{2} E}{\partial z^{2}} + {(\frac{ω}{c})}^{2} ε E = 0

(4)

Using the coupled wave theory methodology [26

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2945 (1969).

] 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):

E = R (z) exp (- j K_{R} . r) + S (z) exp (- j K_{S} . r) + W (z) exp (- j K_{W} . r)

(5)

The two superposed gratings described by the grating vectors K₁ and K₂ connect R with S and S with W respectively, so the Bragg conditions are satisfied:

K_{S} = K_{R} - K_{1}, K_{W} = K_{S} - K_{2}

(6)

Assuming that K_R is given by:

K_{R} = \frac{n ω}{c} (Sin (θ), 0, Cos (θ))

(7)

where ω is the angular frequency used in the reconstruction step. From Eqs. (2), (3), and (6) it follows that:

K_{S} = \frac{n}{c} (ω Sin (θ) - ω_{o} (Sin (θ_{r}) - Sin (θ_{s})), 0, ω Cos (θ) - ω_{o} (Cos (θ_{r}) - Cos (θ_{s})))

(8)

K_{W} = \frac{n}{c} (ω Sin (θ) - ω_{o} (Sin (θ_{r}) - Sin (θ_{w})), 0, ω Cos (θ) - ω_{o} (Cos (θ_{r}) - Cos (θ_{w})))

(9)

Introducing the electric field E given by Eq. (5) in the Helmholtz equation (4) and using the coupled wave theory [26

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2945 (1969).

] approximations, we obtain the differential equation system:

\frac{d}{d z} (\begin{matrix} R (z) \\ S (z) \\ W (z) \end{matrix}) = - j (\begin{matrix} 0 & \frac{κ_{1}}{c_{r}} & 0 \\ \frac{κ_{1}}{c_{s}} & \frac{ϑ_{s}}{c_{s}} & \frac{κ_{2}}{c_{s}} \\ 0 & \frac{κ_{2}}{c_{w}} & \frac{ϑ_{w}}{c_{w}} \end{matrix}) (\begin{matrix} R (z) \\ S (z) \\ W (z) \end{matrix})

(10)

where:

\begin{matrix} κ_{i} = \frac{ε_{i} ω}{4 c n} & with & i = (1, 2) \\ ϑ_{p} = \frac{β^{2} - | K_{p} |^{2}}{2 β} & with & p = (s, w) \\ c_{q} = \frac{K_{Q z}}{2 β} & with & q = (r, s, w) \end{matrix}

(11)

where 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.

Fig. 1 Scheme of the multiplexing of volume holographic gratings.

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

\begin{matrix} {\bar{κ}}_{i} = \frac{ε_{i} ω_{o}}{4 c n} & with & i = (1, 2) \\ {\bar{ϑ}}_{p} = \frac{n δ ω}{c} (1 - Cos (θ_{p})) & with & p = (s, w) \\ {\bar{c}}_{q} = Cos (θ_{q}) & with & q = (r, s, w); \end{matrix}

(12)

where all of them are functions of recording parameters (θ_r , θ_w , θ_s , ω_o ) material properties (ε ₁, ε ₂ , n) and only

\bar{ϑ}

_p depends on frequency detuning (δω). Introducing approximations (12) into Eq. (10) we obtain the differential equation system given by:

\frac{d}{d z} (\begin{matrix} R (z) \\ S (z) \\ W (z) \end{matrix}) = A (\begin{matrix} R (z) \\ S (z) \\ W (z) \end{matrix}) = - j (\begin{matrix} 0 & \frac{\sqrt{c_{r} c_{s} ν_{1}}}{c_{r}} & 0 \\ \frac{\sqrt{c_{r} c_{s} ν_{1}}}{c_{s}} & ξ_{s} & \frac{\sqrt{c_{w} c_{s} ν_{2}}}{c_{s}} \\ 0 & \frac{\sqrt{c_{w} c_{s} ν_{2}}}{c_{w}} & ξ_{w} \end{matrix}) (\begin{matrix} R (z) \\ S (z) \\ W (z) \end{matrix})

(13)

where we defined the parameters:

\begin{matrix} n_{p} = n (S e c (θ_{p}) - 1) & with p = (s, w) \\ ν_{1} = {\bar{κ}}_{1}^{2} (\frac{n_{s}}{n} + 1) \\ ν_{2} = {\bar{κ}}_{2}^{2} (\frac{n_{s}}{n} + 1) (\frac{n_{w}}{n} + 1) \\ ξ_{p} = \frac{ϑ_{p}}{c_{p}} = \frac{δ ω}{c} n_{p} & w i t h p = (s, w) \end{matrix}

(14)

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

Fig. 2 Index group (n_p , p = s, w) normalized to refractive index n as a function of recording angle θ_p .

The differential equations system (13) is similar to that obtained for the grating induced transparency waveguide (GIT) in reference [25

H. C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express 17(14), 11710–11718 (2009). [CrossRef] [PubMed]

]. 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:

R (z) = \sum_{i = 0}^{2} C_{i} exp (x_{i} z)

(15)

and the amplitude transmittance t:

t = R (L) = \sum_{i = 0}^{2} C_{i} exp (x_{i} L)

(16)

where x_i are the three roots of the characteristic polynomial (P) of matrix A in Eq. (13):

P = - x^{3} - j (ξ_{s} + ξ_{w}) x^{2} + (ξ_{s} ξ_{w} - ν_{1} - ν_{2}) x - j ν_{1} ξ_{w}

(17)

and C_i are integration constants given by:

\begin{array}{r} C_{0} = \frac{Γ_{12}}{Γ_{01} - Γ_{02} + Γ_{12}} \\ C_{1} = - \frac{Γ_{02}}{Γ_{01} - Γ_{02} + Γ_{12}} \\ C_{2} = \frac{Γ_{01}}{Γ_{01} - Γ_{02} + Γ_{12}} \end{array}

(18)

where:

Γ_{ip} = exp (L (x_{i} + x_{p})) Ω_{ip} w i t h Ω_{ip} = (x_{i} - x_{p}) (- ν_{1} + x_{i} x_{p})

(19)

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_w = 0 and in this case the roots of P given by Eq. (17) are: x ₀ = 0,

x_{1} = j \sqrt{ν_{1} + ν_{2}}

x_{2} = - j \sqrt{ν_{1} + ν_{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 = \frac{ν_{1} + ν_{2}}{ν_{2} + ν_{1} Cos (L \sqrt{ν_{1} + ν_{2}})}

(20)

It may be deduced from Eq. (20) that the set of values

L = L_{m} = 2 m π / \sqrt{ν_{1} + ν_{2}}

(where m is a natural number), means that t = 1 at frequency design ω_o . Therefore, if the HRS has a thickness L = L ₁, in the forbidden band of the reflection grating, a permitted band appears, centered at ω_o . 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 band

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

t = \frac{exp (j ψ)}{\sqrt{ρ_{01}^{2} + ρ_{02}^{2} + ρ_{12}^{2} - 2 (ρ_{01} ρ_{02} Cos (ϕ_{21}) - ρ_{01} ρ_{12} Cos (ϕ_{20}) + ρ_{02} ρ_{12} Cos (ϕ_{01}))}}

(21)

where we introduced:

\begin{array}{l} ρ_{ip} = \frac{Ω_{ip}}{Ω_{01} - Ω_{02} + Ω_{12}} \\ ϕ_{ip} = (x_{i} - x_{p}) L / j \\ ψ = arctan (\frac{ρ_{01} Sin (x_{2} L / j) - ρ_{02} Sin (x_{1} L / j) + ρ_{12} Sin (x_{0} L / j)}{ρ_{01} Cos (x_{2} L / j) - ρ_{02} Cos (x_{1} L / j) + ρ_{12} Cos (x_{0} L / j)}) \end{array}

(22)

If transmittance is studied near the recording frequency ω ₀ (δω << 1), it can be assumed that the parameters ξ_w << 1 and ξ_s << 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 ξ_w and ξ_s , at ξ_w = 0 and ξ_s = 0) by:

\begin{array}{l} x_{0} = - j \frac{ξ_{w} ν_{1}}{ν_{1} + ν_{2}} \\ x_{1} = - \frac{j}{4} (\frac{2 ξ_{w} ν_{2}}{ν_{1} + ν_{2}} + 4 \sqrt{ν_{1} + ν_{2}} + ξ_{s} (2 - \frac{ξ_{w} ν_{2}}{{(ν_{1} + ν_{2}))}^{3 / 2}})) \\ x_{2} = \frac{j}{4} (- \frac{2 ξ_{w} ν_{2}}{ν_{1} + ν_{2}} + 4 \sqrt{ν_{1} + ν_{2}} + ξ_{s} (- 2 - \frac{ξ_{w} ν_{2}}{{(ν_{1} + ν_{2}))}^{3 / 2}})) \end{array}

(23)

The transmittance can be expressed according to Eq. (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 ξ_w and ξ_s ) and finally a second order Taylor series expansion on ξ_w and first order on ξ_s on the argument of the root square in the denominator of Eq. (21).

t = \frac{{(ν_{1} + ν_{2})}^{5 / 2} exp (\frac{- j π ν_{1} (3 ξ_{w} ν_{2} + ξ_{s} (ν_{1} + ν_{2}))}{{(ν_{1} + ν_{2})}^{5 / 2}})}{\sqrt{{(ν_{1} + ν_{2})}^{5} + π^{2} (2 ξ_{s} ξ_{w} (2 ν_{1} - ν_{2}) ν_{2} (ν_{1} + ν_{2}) - ξ_{w}^{2} ν_{1} ν_{2} {(ν_{2} - 2 ν_{1})}^{2})}}

(24)

Finally if we introduce 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:

t (ω) = \frac{exp (j τ_{d} δ ω)}{\sqrt{1 + {(δ ω / γ)}^{2}}} = \frac{exp (j τ_{d} (ω - ω_{o}))}{\sqrt{1 + {((ω - ω_{o}) / γ)}^{2}}}

(25)

where τ_d is the time-delay induced by the device and 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:

τ_{d} = \frac{4 n^{3} π (n_{s} n + (n_{w} + n) (3 n_{w} + n_{s}) α^{2})}{ε_{1} ω_{o} \sqrt{n_{s} + n} {(α^{2} (n_{w} + n) + n)}^{3 / 2}}

(26)

γ = \frac{ε_{1} ω_{o} \sqrt{n_{s} + n} {(α^{2} (n_{w} + n) + n)}^{5 / 2}}{4 α π n^{5 / 2} \sqrt{n_{w} (n + n_{w}) (2 n - (n + n_{w}) α^{2}) (2 n (n_{s} - n_{w}) + (n + n_{w}) (2 n_{s} + n_{w}) α^{2})}}

(27)

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:

α = η \sqrt{\frac{- n}{n_{w} + n}}

(28)

Introducing Eq. (28) in Eqs. (26) and (27), it follows that time delay and scale factor can be written as:

τ_{d} = \frac{4 n^{2} π (- n_{s} + (n_{s} + 3 n_{w}) η^{2})}{ε_{1} ω_{o} \sqrt{- n (n_{s} + n) (η^{2} - 1) {(η^{2} - 1)}^{2}}}

(29)

γ = \frac{ε_{1} ω_{o} \sqrt{n_{s} + n} {(η^{2} - 1)}^{5 / 2}}{4 π n^{3 / 2} \sqrt{- n_{w} η^{2} (η^{2} + 2) (2 n s (η^{2} - 1)) + n_{w} (η^{2} + 2)}}

(30)

In order to obtain a Lorentzian function when we introduce 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

L_{1} (L_{1} = 2 π / \sqrt{ν_{1} + ν_{2}})

, it follows that the device thickness must be:

L_{1} = \frac{8 π c n}{ω_{o} ε_{1} \sqrt{- (1 + n_{s} / n) (η^{2} - 1)}}

(31)

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:

v_{g} = \frac{L_{1}}{τ_{d}} = \frac{2 c {(η^{2} - 1)}^{2}}{- n_{s} + (n_{s} + 3 n_{w}) η^{2}}

(32)

In the following section we will analyze the propagation of pulses in this kind of system.

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:

Φ_{inc} (t) = exp (- \frac{{(c t)}^{2}}{2 {(W_{0} n)}^{2}}) Cos (ω_{o} t)

(33)

where 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

\hat{Φ}

_inc (ω) as the direct Fourier transform of the incident field (33):

{\hat{Φ}}_{inc} (ω) = \int_{- \infty}^{\infty} Φ_{inc} (t) exp (j ω t) d ω = \sqrt{A π} (exp (- A {(ω - ω_{o})}^{2}) + exp (- A {(ω + ω_{o})}^{2}))

(34)

where

A = n^{2} T_{0}^{2} / 2

The transmitted field Φ _tr (t) will then be given by the inverse Fourier transform:

Φ_{tr} (t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} {\hat{ϕ}}_{inc} (ω) t (ω) exp (- j ω t) d ω

(35)

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

Φ_{tr} (t) = \frac{1}{\sqrt{2 π}} \int_{- \infty}^{\infty} {\hat{ϕ}}_{inc} (ω) \frac{exp (j τ_{d} (ω - ω_{o}))}{\sqrt{1 + {((ω - ω_{o}) / γ)}^{2}}} exp (- j ω t) d ω

(36)

Replacing

\hat{Φ}

_inc (ω) by the expression given in Eq. (34) in Eq. (36), and approximating t(ω) to:

t (ω) = \frac{exp (j τ_{d} (ω - ω_{o}))}{\sqrt{1 + {((ω - ω_{o}) / γ)}^{2}}} \approx exp (j τ_{d} (ω - ω_{o})) (1 - \frac{1}{2} {((ω - ω_{o}) / γ)}^{2})

(37)

it follows that the transmitted field Φ _tr (t) is given by:

Φ_{tr} (t) = \frac{exp (- \frac{c^{2} {(τ_{d} - t)}^{2}}{2 {(W_{0} n)}^{2}} Cos (ω_{o} t) (c^{4} {(τ_{d} - t)}^{2} - c^{2} n^{2} W_{0}^{2} + 2 γ^{2} n^{4} W_{0}^{4}))}{2 γ^{2} n^{4} W_{0}^{4}}

(38)

where we have used that 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} {(τ_{d} - t)}^{2} - c^{2} n^{2} W_{0}^{2} + 2 γ^{2} n^{4} W_{0}^{4} / (2 γ^{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_rms given by [27

B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003). [CrossRef]

D_{rms} = \sqrt{\frac{\int_{- \infty}^{\infty} {(Φ_{tr}^{e} (t) - Φ_{inc}^{e} (t - τ_{d}))}^{2} d t}{\int_{- \infty}^{\infty} Φ_{inc}^{e} {(t)}^{2} d t}}

(39)

where the function

Φ_{inc, tr}^{e} (t) = Φ_{inc, tr} (t) / Cos (ω_{0} t)

are the incident Gaussian and transmitted Gaussian distorted envelopes. Introducing Eqs. (38) and (33) into Eq. (39), and integrating gives:

D_{rms} = \frac{\sqrt{3} c^{2}}{4 γ^{2} n^{2} W_{0}^{2}}

(40)

5. Numerical simulations

In our numerical calculations, we take ω_o = 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 n_s (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 n_s , n_w .

Fig. 3 L1 as a function of group index ns.

Equation (31) and Fig. 3 show that a value of L1 = 2mm 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_w = −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

\hat{Φ}

_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_d = 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 D_rms << 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 D_rms 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.

Fig. 4 T as a function of detuning δω. Inset curves (blue) shows the relative error of our approximated function (25) in comparison with exact result given by Eq. (16).

Fig. 5 Phase as a function of detuning δω. Inset curve shows the relative error of our approximated function (25) in comparison with exact result given by Eq. (16).

Fig. 6 Intensity of incident pulse (brown) and time-delayed pulses obtained using equation 35 and approximations (25) (blue) or (37)–(38) (pink) as a function of time. Inset curve shows the relative error in comparison with exact solution.

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.

References and links

1.	R. Boyd, O. Hess, C. Denz, and E. Paspalkalis, “Slow light,” J. Opt. 12(10), 100301 (2010). [CrossRef]
2.	J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005). [CrossRef]
3.	F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]
4.	Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer,” Phys. Rev. Lett. 99(24), 240801 (2007). [CrossRef]
5.	Z. M. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, “Enhancing the spectral sensitivity of interferometers using slow-light media,” Opt. Lett. 32(8), 915–917 (2007). [CrossRef] [PubMed]
6.	R. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in Progress in Optics , E. Wolf, ed., (Elsevier, 2002) vol. 43, chap. 6, pp. 497–530. [CrossRef]
7.	L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
8.	A. Kasapi, M. Jain, G. y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys. Rev. Lett. 74(13), 2447–2450 (1995). [CrossRef] [PubMed]
9.	M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75(2), 457–472 (2003). [CrossRef]
10.	L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics 2(8), 451–453 (2008). [CrossRef]
11.	M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]
12.	M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90(11), 113903 (2003). [CrossRef] [PubMed]
13.	R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73(6), 063812 (2006). [CrossRef]
14.	R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys. Rev. Lett. 98, 043902 (2007). [CrossRef] [PubMed]
15.	Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef] [PubMed]
16.	T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]
17.	M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73(9), 096501 (2010). [CrossRef]
18.	A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]
19.	D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004). [CrossRef]
20.	Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25(1), 201–206 (2007). [CrossRef]
21.	L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2(8), 474–481 (2008). [CrossRef]
22.	S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]
23.	M. F. Yanik, W. Suh, Z. Wang, and S. H. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). [CrossRef] [PubMed]
24.	K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency.” Phys. Rev. Lett. 98(21), 213904 (2007). [CrossRef] [PubMed]
25.	H. C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express 17(14), 11710–11718 (2009). [CrossRef] [PubMed]
26.	H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2945 (1969).
27.	B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003). [CrossRef]

OCIS Codes
(050.7330) Diffraction and gratings : Volume gratings
(090.0090) Holography : Holography
(090.4220) Holography : Multiplex holography

ToC Category:
Holography

History
Original Manuscript: February 1, 2011
Revised Manuscript: March 1, 2011
Manuscript Accepted: March 5, 2011
Published: March 29, 2011

Citation
Luis Carretero, Salvador Blaya, Pablo Acebal, Antonio Fimia, Roque Madrigal, and Angel Murciano, "Coupled wave analysis of holographically induced transparency (HIT) generated by two multiplexed volume gratings," Opt. Express 19, 7094-7105 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7094

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References

R. Boyd, O. Hess, C. Denz, and E. Paspalkalis, “Slow light,” J. Opt. 12(10), 100301 (2010). [CrossRef]
J. B. Khurgin, “Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis,” J. Opt. Soc. Am. B 22, 1062–1074 (2005). [CrossRef]
F. N. Xia, L. Sekaric, and Y. Vlasov, “Ultracompact optical buffers on a silicon chip,” Nat. Photonics 1(1), 65–71 (2007). [CrossRef]
Z. Shi, R. W. Boyd, R. M. Camacho, P. K. Vudyasetu, and J. C. Howell, “Slow-light fourier transform interferometer,” Phys. Rev. Lett. 99(24), 240801 (2007). [CrossRef]
Z. M. Shi, R. W. Boyd, D. J. Gauthier, and C. C. Dudley, “Enhancing the spectral sensitivity of interferometers using slow-light media,” Opt. Lett. 32(8), 915–917 (2007). [CrossRef] [PubMed]
R. Boyd and D. J. Gauthier, ““Slow” and “fast” light,” in Progress in Optics , E. Wolf, ed., (Elsevier, 2002) vol. 43, chap. 6, pp. 497–530. [CrossRef]
L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397(6720), 594–598 (1999). [CrossRef]
A. Kasapi, M. Jain, G. y. Yin, and S. E. Harris, “Electromagnetically induced transparency - propagation dynamics,” Phys. Rev. Lett. 74(13), 2447–2450 (1995). [CrossRef] [PubMed]
M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75(2), 457–472 (2003). [CrossRef]
L. V. Hau, “Optical information processing in Bose-Einstein condensates,” Nat. Photonics 2(8), 451–453 (2008). [CrossRef]
M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Superluminal and slow light propagation in a room-temperature solid,” Science 301(5630), 200–202 (2003). [CrossRef] [PubMed]
M. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, “Observation of ultraslow light propagation in a ruby crystal at room temperature,” Phys. Rev. Lett. 90(11), 113903 (2003). [CrossRef] [PubMed]
R. M. Camacho, M. V. Pack, and J. C. Howell, “Low-distortion slow light using two absorption resonances,” Phys. Rev. A 73(6), 063812 (2006). [CrossRef]
R. M. Camacho, C. J. Broadbent, I. Ali-Khan, and J. C. Howell, “All-optical delay of images using slow light,” Phys. Rev. Lett. 98, 043902 (2007). [CrossRef] [PubMed]
Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438(7064), 65–69 (2005). [CrossRef] [PubMed]
T. Baba, “Slow light in photonic crystals,” Nat. Photonics 2(8), 465–473 (2008). [CrossRef]
M. Notomi, “Manipulating light with strongly modulated photonic crystals,” Rep. Prog. Phys. 73(9), 096501 (2010). [CrossRef]
A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24(11), 711–713 (1999). [CrossRef]
D. D. Smith, H. Chang, K. A. Fuller, A. T. Rosenberger, and R. W. Boyd, “Coupled-resonator-induced transparency,” Phys. Rev. A 69(6), 063804 (2004). [CrossRef]
Z. M. Zhu, A. M. C. Dawes, D. J. Gauthier, L. Zhang, and A. E. Willner, “Broadband SBS slow light in an optical fiber,” J. Lightwave Technol. 25(1), 201–206 (2007). [CrossRef]
L. Thevenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2(8), 474–481 (2008). [CrossRef]
S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997). [CrossRef]
M. F. Yanik, W. Suh, Z. Wang, and S. H. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. Rev. Lett. 93(23), 233903 (2004). [CrossRef] [PubMed]
K. Totsuka, N. Kobayashi, and M. Tomita, “Slow light in coupled-resonator-induced transparency.” Phys. Rev. Lett. 98(21), 213904 (2007). [CrossRef] [PubMed]
H. C. Liu and A. Yariv, “Grating induced transparency (GIT) and the dark mode in optical waveguides,” Opt. Express 17(14), 11710–11718 (2009). [CrossRef] [PubMed]
H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48(9), 2909–2945 (1969).
B. Macke and B. Segard, “Propagation of light-pulses at a negative group-velocity,” Eur. Phys. J. D 23, 125–141 (2003). [CrossRef]

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