1. Introduction

Tunable optical pulse delays have important applications in several areas, including phased-array antennas for radio frequency signals [1] and optical communications [2]. Recently the technique of slow light based on electromagnetically induced transparency (EIT) [3] in atomic media [4, 5, 6, 7] and later in solids [8, 9] has generated interest as a potential realization of fast, tunable optical delays [2]. In EIT, an otherwise opaque medium is rendered transparent to a “signal” laser pulse, in a small frequency window, by the application of a second continuous wave (c.w.) “pump” laser. Associated with this transparency window is a steep linear dispersion in the index of refraction, giving rise to extremely slow group velocities (down to ~10^-7 c) and time delays of pulses by several pulse widths [5]. The intensity of the pump laser controls the dispersion slope and thus the delay, satisfying the desire for a simple and fast method of delay tunability.

The very narrow frequency widths of EIT features in atomic media, while desirable for certain applications such as magnetometry [10], puts a strict limitation on the operational bandwidth of such a system (~1 MHz), negatively impacting the potential for devices applicable to radar and optical communications, which operate up to the gigahertz regime. The high frequency components of a wideband pulse are absorbed and distorted [11]. While in principle one could circumvent this limitation by increasing the laser power [4] it is desirable, from a cost and size perspective, to work at more modest powers. Furthermore, spurious effects due to additional levels in the atomic level structure can become more important at higher laser powers. Another approach has been to consider slow light in solid state materials [12, 13] due to their inherently faster light-matter interaction time-scales. One drawback of this approach is the faster decoherence time, which ultimately limits the pulse delay. Pulse delays in fibers, which are ultimately limited by distortion and gain, are also being actively explored [14, 15].

Yet another approach to achieving higher bandwidths is to use our proposed “channelization” architecture [16], in which the established EIT based slow light technique in ⁸⁷Rb atomic vapors is used with the addition of a spatially inhomogenous magnetic field. The scheme is diagrammed in Fig. 1(a). A high bandwidth input signal is spatially dispersed in the transverse direction (from point A to B in the figure), so different frequency components enter the EIT medium at different transverse positions x. By applying a longitudinal magnetic field, with varying magnitude along the transverse dimension, one can Zeeman shift the frequency position of the EIT transparency window to match the frequency component at each spatial location during propagation (B to C). By then recombining the frequency components after propagation (C to D), one can achieve delays for much wider bandwidth pulses.

 

Fig. 1. (a) Schematic of the channelization architecture. A wideband sigal pulse at Point A is spatially dispersed (e.g. via prisms) such that frequency components are displaced in the x direction by an amount proportional the frequency offset from the central carrier frequency (Point B), so the dispersed pulse covers a spatial width W. The dispersed signal then enters a Rb-87 cell illuminated by a monochromatic pump field and propagates with slow group velocity to Point C. A transversely varying longitudinal magnetic field is chosen to match the two-photon resonance at each position x. The dispersing process is reversed to recombine the frequency components of the pulse at Point D. (b) Schematic of particular energy levels used. Note the population is taken to be initially purely in a particular Zeeman sublevel (|1〉 indicated by the circle) so a nearly ideal 3-level Λ results. The pump field Ωp is on the bare resonance of |2〉↔|3〉, with the bare levels indicated by dotted lines. The green bars show the Zeeman shifts introduced to the sublevels for a positive magnetic field B_z . The Lande-g factors for each hyperfine level gF are indicated. The frequency of signal pulse Ω _s will depend on the position x, due to the transverse dispersion.

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In this paper we analyze this scheme in more detail, and present two new central results. First, we expand on an important subtlety noted in [16]. Because the slow group velocity arises from differential phase shifts across the pulse spectrum, one must analyze the relative phase shifts applied to pulse components at different spatial positions during the propagation. In [17], a model of the dispersion and propagation was developed which demonstrated delay of each frequency component. Here we use this model to consider in detail the recombination of the pulse and confirm our earlier conclusion [16] that no delay of the recombined pulse will occur if the EIT resonance exactly matches the central frequency at each location. We furthermore confirm how introducing a small mis-match between the central signal frequency and EIT resonance frequency at each position will recover the phase shifts necessary to recover a delay. The delays can be controlled with the pump power, just as in a conventional EIT slow light system. As will be shown in Section 2, this mis-match scheme contrasts with other proposals advocating the channelization concept [17, 18], in which additional methods, such as liquid crystals, must be employed to the induce the phase shifts necessary to achieve a delay. This requirement makes the delay time more difficult to tune. The analysis we present clarifies the central necessity of these additional phase shifters in such schemes.

With this mis-match scheme, bandwidth is in effect gained at the expense of delay, so no increase in the maximum delay-bandwidth product is achieved. However, importantly we find that, given a desired delay, one can increase the bandwidth. The second central result of this paper is the systematic optimization of the available bandwidths, given constraints on desired delay, laser power, and magnetic field gradients, and we find substantial improvements in parameter ranges of interest. For example, for a 10 ns delay, we expect bandwidths of ~100 MHz (as opposed to 20 MHz with conventional slow light) for modest pump powers ~10 mW. Importantly, the channelization architecture allows one to enter the regime of delay-bandwidth products exceeding unity for delays of interest (~ns). This is only possible with conventional EIT for much longer pulse widths. We focus on ⁸⁷Rb due to the successful implementations of slow light in this medium, but our analysis can easily be applied to other atomic media. The general concept of channelization should also be applicable to solids, semiconductors and other candidate systems.

In Section 2 we consider carefully the subtleties involved in spatially separating the frequency components of a pulse, then recombining them after propagation through a slow light medium. We obtain a complete expression for the recombined pulse which shows how the above mentioned mis-match and the dispersive resolution enter the problem. In Section 3 we outline our model which accounts for atomic diffusion in the inhomogenous magnetic field, an important limiting effect which prevents one from using arbitrarily steep magnetic field gradients. Our novel approach of including diffusion directly in the Optical Bloch Equations is also applicable to other inhomogenous effects in EIT systems. We then present numerical analysis of this model and compare it with analytic expressions. The analytic expressions are then used in Section 5 to learn the maximum possible bandwidths as a function of desired delay, given constraints on laser power and magnetic fields. Our results are summarized in Section 6.

2. Slow light with channelization and recombination

We now analyze the channelization scheme. By analyzing the splitting, propagation, and recombination in frequency space, we see clearly the need for introducing a slight frequency mis-match to achieve a delay.

The channelization scheme is diagrammed in Fig. 1(a). Assume a broadband, circularly-polarized signal pulse with an envelope of transverse spatial width σ, temporal length τ, and central frequency ω _s input at location A (see Fig. 1(a)): ${\Omega }_s^{\left(A\right)}(x,t)={\Omega }_{s0}\exp \left(-\frac{x^2}{2{\sigma }^2}-\frac{t^2}{2{\tau }^2}\right)$. Fourier transforming to frequency this pulse can be written:

where δ _s is the frequency deviation from ω_s . Sending this pulse through a spatial dispersing device, such as a prism system [17], displaces each frequency component δ _s by a distance δs/S_d . For applications in phased-array radar, such dispersion can be achieved by mixing a wide-band radio signal onto an optical carrier with an Acousto-Optic device [19]. For optical signals, achieving the necessary dispersive resolution will be difficult, however, available wavelength division multiplexers are now approaching the desired resolution, or one could also use an etalon. The dispersive slope S_d is determined by the power of the disperser. After this step, at point B, the pulse is written

If transformed back into time, ${\mathrm{\Omega }}_s^{\left(\mathrm{B}\right)}$ (x,t) represents a pulse whose central frequency is a strong function of x: ω_s +Sd^x . When the dispersion slope S_d is sufficiently small, the first term in the exponential dominates and the pulse bandwidth at each x is significantly smaller (and thus temporally longer) than the original input pulse (τ _disp=1/S_d σ≫τ) [17].

 

Fig. 2. (a) Imaginary part of the susceptibility (Eq. (9) calculated according to the method outlined in Section 3) versus signal frequency with B_z =0 G (black) and B_z =30 G. The insets zoom in on the EIT resonance at each value of B_z . The solid curves show the homogeneous field case, while the dark and light dots show the cases with gradients S_B /µB=3 G/mm and S_B /µ_B =10 G/mm, respectively. The pump is chosen to be 50 mW/cm². The buffer gas pressure is p=30 Torr, for which D_g =5 cm²/s. (b) Real part of the susceptibility.

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The dispersed signal is then sent into a ⁸⁷Rb cell containing a buffer gas to reduce diffusion of the ⁸⁷Rb [20]. We propose initially spin-polarizing the atoms into a single magnetic sublevel |1〉≡|3S _1/2,F=1, _mF =+1〉. The central signal laser frequency ω_s is chosen resonant with |1〉↔|3〉≡|3P _1/2,F′=2,m′_F =+2〉 (the D ₁ line), the strong, circularly-polarized pump field is resonant with |2〉≡|3S _1/2,F=2, _mF =+1〉↔|3〉, and we form a nearly ideal Λ system to achieve EIT and slow light (see Fig. 1(b)). Previous experimental work [6, 11] has implemented this scheme without the initialization into a particular magnetic sub-level. Such preparation is necessary here due to our utilization of linear Zeeman shifts of the EIT resonance. A magnetic field B_z oriented along the propagation direction z differentially shifts the two ground states |1〉, |2〉, by ($g_F^{\left(2\right)}$ $m_F^{\left(2\right)}$ -$g_F^{\left(1\right)}$$m_F^{\left(1\right)}$ )µ_B =(2π) 1.4 MHz/G in our configuration. Fig. 2 the shows the real and imaginary parts of the susceptibility ${\chi }_R^{\left(D\right)}$ , ${\chi }_I^{\left(D\right)}$ as a function of signal frequency detuning from the resonance δ _S at two different values of B_z (Section 3 describes the model used for these calculations). The larger scale plots show the overall Doppler broadened absorption resonance. Far from two-photon (Raman) resonance (the point where the difference of the signal and pump frequencies match the energy splitting of |1〉 and |2〉), the pump field has little effect on the χ ^(D). However, as the insets show, the pump introduces an EIT feature, with reduced absorption, at the two-photon resonance frequency δ_s =($g_F^{\left(2\right)}$ $m_F^{\left(2\right)}$ -$g_F^{\left(1\right)}$ $m_F^{\left(1\right)}$ )µ_BB_z . We choose a transverse linear gradient for the magnetic field such that the two-photon resonance frequency varies according to S_Bx across the cell. This could be achieved, for example, via a single wire to the side of the cell [17], where here we would need ~100 A to create the desired resonance shift (~100 G) across the width the cell (~cm). More sophisticated designs with several wires would improve the homogeneity of the field and reduce the current requirements. In this vicinity of the resonance, the susceptibility can be expanded in the form:

In Section 4 we will see that this form is applicable and calculate the coefficients a, s and w. Including only these few terms will give us accurate results for our purposes because the absorption profile acts to damp out any frequency components far from the resonance. Assuming this susceptibility for the EIT medium, the pulse at Point C can be expressed as

where D is the optical density of the medium. The linear slope of ${\chi }_R^{\left(D\right)}$ with δ_s gives differential phase shifts across the frequency spectrum at each x, which gives rise to slow group velocity and a pulse delay τ_d = _sD. To prevent attenuation, the residual absorption at the EIT resonance aD must be sufficiently small. The Zeeman shift of the EIT resonance S_Bx appears directly in the expression and the frequency components at each x must fit within the local bandwidth window (S_Bx-w/2√D, S_Bx+w/2√D) to prevent distortion and absorption. An important feature to note in Fig. 2 is that, to a good approximation, the susceptibility curves at the two different fields B_z are identical except for the shift S_Bx, so a, s and w do not vary with x. This will occur so long as the shift S_Bx is comparable to or smaller than the Doppler width of the atoms (~(2π) 300 MHz in room temperature rubidium) [16]. The wide Doppler width of atomic vapors is helpful in this sense.

After propagation through the EIT medium, the various delayed pulses are recombined back into a single, wide-band pulse, using the converse of the process used in the dispersion:

Inverting the Fourier transform will then reveal the temporal characteristics of the recombined pulse ${\mathrm{\Omega }}_s^{\left(D\right)}$ (x,t). The final expression Eq. (5) reveals several important results. First, in the limit of no magnetic field gradient S_B =0 it reduces to the conventional slow light result, where the term τ_d =sDδ_s gives rise to the pulse delay _sD and the last term enforces the finite bandwidth of the transmission window w/√D. A wideband (temporally short) input pulse τ≪√D/w is absorbed and broadened.

Second, in the channelization case, the signal is dispersed with some power S_d , and choosing a field gradient S_B =S_d corresponds to exact matching of the two-photon resonance at all x. In this case, one sees from Eq. (5) that the absorption of high frequency components is eliminated as desired (the fifth term in the exponential). However, the delay inducing term (the fourth in the exponential) likewise vanishes, resulting in an undelayed pulse upon recombination. Physically, the frequency component at each x is exactly centered on the EIT window (see Fig. 2(b)) and thus, does not interact with the atomic medium and receives no phase shift. One can only recover a delay if one chooses a slight mismatch between the resonance slope and dispersive slope S_B =S_d (1-β), for β≡0. Then there is a δ_s -dependent detuning from the EIT resonance, of the same form as conventional slow light. So long as all the dispersed frequency components are within their local EIT bandwidth, the total bandwidth of the channelization system is B _ch=S_BW, where W is the physical spatial width of the dispersive system (see Fig. 1(a)). To keep frequency components at all x within their local bandwidth one must choose a small mis-match, constrained by βS_BW<w/√D. This results in a bandwidth increased by a factor β^-1 over conventional slow light. The delays are correspondingly smaller sDβ.

Finally, our result Eq. (5) gives a requirement for the resolution of the initial dispersion. The bandwidth of the channelized pulse at each x is τ^-1_disp. If this is too large (due to a large input beam σ) then a wide range of frequencies will be present at each location during the EIT propagation, and the last term DS²_Bx²/w² will result in a large absorption and spreading of the pulse. A spatially dependent phase shift DsS_Bx will also appear in the recombined pulse in this limit.

In the case of perfect matching (β=0), as proposed in [17, 18], one could recover a delay only by the application of an additional spatially dependent phase shifts (with, e.g., liquid crystal spatial light modulators) after the EIT medium (Point C in Fig. 1). A delayed, wideband pulse would require application of a phase shift pattern ϕ(x)=S_dxτ_d , where τ_d is the desired delay. Putting this additional phase shift in Eq. (4) leads to a delay in Eq. (5), even for β=0. In this case, one is not using the EIT to induce the delay of the wideband pulse, but rather the additional phase shifters. Indeed, delay purely via spatially dependent phase shifts of a dispersed pulse has been explored in the context of radar true time delay [19]. One disadvantage of this approach is the phase shift magnitudes must be adjusted to match the desired delay, so the tunability rate is limited by the tunability of the phase shifters. This is in contrast to our mis-match scheme which, like conventional slow light, has the desirable characteristic of fast, continuous, and easily tunable delays via the pump power.

3. Microscopic diffusion model

We have derived expressions for the dispersion, propagation and subsequent recombination for our channelization scheme and can derive quantitative results given the correct susceptibility of the EIT medium ${\chi }_R^{\left(D\right)}$ +${i\chi }_I^{\left(D\right)}$ . We now turn toward a microscopic calculation of this susceptibility and, in particular, analytic expressions for the parameters a, s, and w of Eq. (3), which we find are indeed independent of the magnetic field (and therefore x) over a wide range. The important physical mechanism which we add to previous work on slow light propagation in warm atomic vapors [6] is the effect of atomic diffusion within a highly inhomogeneous field. For parameters of interest here, this mechanism will be the dominant mechanism in determining the minimum absorption coefficient a while having little effect on the EIT feature’s slope s and width w. This model was first developed by us in [16], and here we briefly review it. Later it will be used for numerical and analytic calculations for a, which are used to optimize the channelization scheme bandwidths.

Our starting point is to consider the three relevant internal states of the atom and compute evolution of the 3×3 density matrix in the presence of light fields (the Optical Bloch Equations [21]). We consider the weak signal regime, in which we can linearize in the signal field amplitude. We then integrate over the velocity distribution (Doppler profile). The susceptibility seen by the signal can then be written in terms of this calculated density matrix.

Consider the three level model diagrammed in Fig. 1(b), where Ω _{s (p)}=-e r ₁₃₍₂₃₎· $\widehat{\epsilon }$ _s(p) E_{s (p)}/h̄ are the Rabi frequencies of the signal and pump field, respectively, r _ij are the dipole matrix elements, $\widehat{\epsilon }$ _s(p) are unit polarization vectors (chosen to be σ+ for both fields, see Fig. 1(b)), and E _s(p) are the slowly varying envelopes of the electric fields $E_{s\left(p\right)}(e^{i\left(k_{s\left(p\right)}z-{\omega }_{s\left(p\right)}t\right)}+c.c.)⁄2$. Here the wavenumbers k_s ,k_p ≈2π/λ, with λ=795 nm the wavelength, are taken to be equal. In the weak probe regime, the population of state |1〉 is ρ ₁₁≈1, while ρ ₂₂,ρ ₃₃=𝒪(${\mathrm{\Omega }}_p^2$ )≈0. All coherences except ρ ₂₁,ρ ₃₁=𝒪(Ω _p ) can be similarly neglected, so the relevant dynamics terms in the density matrix can be written as a two-component vector ρ=(ρ ₂₁,ρ ₃₁) ^T . The density matrix evolution is given by [16]:

where ${\mathrm{\Delta }}_Z^{\left(i\right)}$=µ_B$g_F^{\left(i\right)}$ $m_F^{\left(i\right)}$ Bd are the Zeeman shifts of level |i〉, with the Bohr magneton µ_B =(2π)1.4 MHz/G, and the Lande-g factors $g_F^{\left(i\right)}$ given in Fig. 1(a) [23]. The EIT resonance shift at x is S_Bx, where $S_B={\mu }_B\left(g_F^{\left(2\right)}{m}_F^{\left(2\right)}-g_F^{\left(1\right)}{m}_F^{\left(1\right)}\right)\left(\frac{d{B}_z}{dx}\right)=\mu B\left(\frac{d{B}_z}{dx}\right)$. The time-of-flight broadening and atomic diffusion in the magnetic field is reduced by the presence of a polarization preserving neon buffer gas, which has been used successfully to preserve ground state coherences in atomic systems [20, 24]. The buffer gas pressure shifts and broadens the excited state. For ⁸⁷Rb with a neon buffer gas, the shift is $S_p^{\left(\mathrm{e}\right)}$ p, where p is the pressure, and $S_p^{\left(\mathrm{e}\right)}$ =-(2π)0.9 MHz/Torr (the ground state pressure shifts are much smaller and negligible for our parameters). The pressure broadening is B_pp where B_p (2π) 5 MHz/Torr [25]. The total dephasing rate of the optical transition is γ_e =Γ _r /2+B_pp, where Γ _r =(2π) 6 MHz is the radiative decay rate. The bare detunings are Δ _{s (p)}=ω _s(p)-(ω₃-ω₁₍₂₎), are chosen to be zero throughout this paper. We have also introduced a Doppler shift δ_D =(2π)v_z /λ, where v_z is the velocity of a particular atom along the light propagation direction z.

The effect of the drift in the inhomogeneous magnetic field is captured with the last term, where D_g is the drift coefficient (150 cm²/s)(Torr/p)(T/273K)^3/2 [26]. The factor 1/3 in the first equation of Eq. (6) accounts for the inhomogeneity being in only the x direction. This is the leading order effect in determining the absorption at the EIT resonance a. We take the ground state decoherence (the decay of ρ ₂₁), due to effects such as diffusion out of the pump laser, to be γ _coh=(2π) 1 kHz in this paper (slightly larger than measured in [6]). We will see the absorption from diffusion in the magnetic field dominates this for typical parameters.

Our strategy is to solve Eq. (6) in Fourier space, in which the time derivative is replaced by Fourier component iδ_s (see Eq. (1)). (We will use tildes to indicate the Fourier transformed quantities.) Without the diffusion term, solution can be solved by simply inverting $\widehat{ℳ}$ ̃. Assuming the drift term is a small perturbation, we first solve the equation without it to get a zeroeth order solution $\tilde{\rho }$ ⁽⁰⁾=$\widehat{ℳ}$ ̃ ^-1 S̃, then plug this back into the diffusion term and solve for a first-order solution. The result is:

The final step is to then average over the thermal distribution of Doppler shifts δ_D , which is an exponential distribution with a width ${\Delta }_D=\sqrt{2⁄3}\left(2\pi \right){\nu }_{\mathrm{Th}}⁄\lambda$. Unfortunately, this last step complicates the iterative solution Eq. (7) as the first order correction is not sufficiently small to assure convergence for δ_D in the wings of the Doppler profile. To combat this problem, we implement an alternate calculation method, whereby we instead integrate the zeroeth order solution $\overline{\rho }$ ⁽⁰⁾ over a small range of magnetic fields which, in the small diffusion limit, will reproduce the iterative solution Eq. (7):

The {λ _j } and {v _j } are, respectively, the eigenvalues and eigenvectors of $\widehat{ℳ}$ ̃ ^-1 and the a_j are the coefficients a_j =v _j ·(∂² $\tilde{\rho }$ ⁽⁰⁾/∂$B_z^2$ ). For cases where the iterative correction is too large to allow convergence, the integration merely causes the susceptibility to return to the background absorption value (in the absence of the pump field). For all numerical results presented in Section 4, we calculate Eq. (8), Doppler integrate, then calculate derivatives to extract a, s, w. For analytic results, we use the iterative solution Eq. (7) and Doppler integrate.

The steady state solution of atomic density matrix is then used to calculate the linear susceptibility. In the linear signal regime the pump field Rabi frequency Ω _p is constant in space and time. The signal field Ω _s propagates according to the slowly varying envelope Maxwell equation [21], with the polarization written in terms of the atomic density matrix. In frequency space, the Maxwell equation is

where σ=3λ²/(2π) is the resonant cross section for unity oscillator strength and f ₁₃ is the transition oscillator strength. This gives us a dimensionless susceptibility in terms of the Doppler averaged atomic coherence $\tilde{\rho }$ ^(D) ₃₁. We neglect the term accounting for free space propagation at c, which can be ignored for time scales of interest.

4. Results

Figure 2 presents the calculated susceptibilities at two magnetic fields both in the absence and presence of an magnetic field gradients. First, we note that the features are nearly identical in shape at the two values. We found numerically that this behavior persists over a wide range of magnetic fields (beyond ±200 G [16]) due to the one-photon detuning being well within the Doppler width, as seen on the larger scale plots. When one-photon detuning is well within the Doppler width Δ _D , the susceptibility will depend only on the two-photon detuning and the EIT features thus look nearly identical except for the shift of the EIT resonance. In [22] it was experimentally confirmed that pulse delays were only a weak function of the one-photon detuning over a range of several hundred megahertz. Quantitatively, we see from the Zeeman shifts (see Fig. 1(b)) that in the presence of a magnetic field, the EIT resonance is shifted ${\mathrm{\Delta }}_Z^{\left(2\right)}$ -${\mathrm{\Delta }}_Z^{\left(1\right)}$ =µ_BB_z , while the the pump field (on the bare resonance) will be detuned by from the one-photon resonance by only ${\mathrm{\Delta }}_Z^{\left(3\right)}$ -${\mathrm{\Delta }}_Z^{\left(2\right)}$ =(1/3)µ_BB_z .

 

Fig. 3. (a) The relative transparency parameter R _EIT≡a/a|Ω_p0=0 versus the magnetic field gradient S_B . The red (blue) dots show the case for pump power 120 mW/cm² (60 mW/cm²) and p=30 Torr buffer gas pressure. The solid curves show the prediction Eq. (10). (b) R _EIT versus buffer gas pressure p for pump intensity 120 mW/cm² and S_B =4 G/mm (red), 40 mW/cm² and S_B =2 G/mm (green), 120 mW/cm² and S_B =8 G/mm (blue). (c) The slope at the EIT resonance s versus S_B for the same powers as in (a). The solid curves show the analytic estimate described in the text. (d) The curvature in the absorption profile at the resonance w for the same cases.

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Second, we see the diffusion in the inhomogeneous field leads to a slight washing out of the EIT features, as each atom samples a region with a finite range of EIT resonance frequencies while it is being pumped into the dark state. When the field gradient is smaller (3 G/mm; black dots in the insets of Figure 2), this leads primarily to an increase in the minimum absorption, while leaving the slope and width mostly unaffected. In the small diffusion limit, where the iterative result Eq. (7) is valid, one can evaluate the imaginary part of this expression at EIT resonance δ_s =S_Bx, Doppler average and obtain the analytic results:

As we mentioned above, the first term, due to the diffusion, will typically dominate the absorption due to decoherence of the ground state. Thus we expect a quadratic dependence of the minimum absorption on the field gradient S_B . In Fig. 3(a) we compare this estimate and numerical calculations of the absorption versus S_B , for two different pump powers. To isolate the diffusion effects clearly, we set γ _coh=0 in these plots. We plot the relative transparency R _EIT=a/a|Ω_p0=0, where a|Ω_p0=0=Γ _r /2Δ _D ) is the background absorption away from the EIT resonance (1 due to Doppler broadening). We see fairly good agreement until the relative transparency becomes significant (~0.1), at which point saturation effects play a role. While for smaller gradients, the dominant effect is an increase in the minimum absorption a, at higher gradients (see 10 G/mm in Fig. 2) we see the diffusion begins to actually broaden the EIT feature as well.

In Fig. 3(b) we show the comparison versus buffer gas pressure at different gradients and powers. Again we see fairly good agreement with Eq. (10). There is a non-trivial dependence on the buffer gas pressure. At lower pressures, the most important aspect is the reduction in the diffusion constant D_g ∝1/p, which is essential to reduce high absorption. However, when B_pp>Δ _D the behavior turns over, and begins to grow due to the ${\gamma }_r^2$ ∝p ² in the numerator, showing that pressure broadening can be detrimental. We find that choosing p≈30 Torr, for which B_pp~Δ _D is optimal regardless of other parameters. Finally, we note the rather strong dependence of the absorption on the pump intensity ∝|Ω _p |^-6.

While the diffusion can easily be the dominant mechanism in determining the absorption a, the slope s and width w of the feature tend to be much less sensitive. In the absence of a magnetic gradient (S_B =0), we find they are [16] s=Γ _r /|Ω _p |², $w={\mid {\Omega }_p\mid }^2⁄2\sqrt{8{\Gamma }_r{\gamma }_e}$. In Figs. 3(c–d) we see they maintain approximately these values until the diffusion causes to the atoms to sample frequency widths comparable to the EIT feature size w, which is the same point at which saturation begins to cause disagreement in our estimate of the absorption, Eq. (10).

5. Bandwidth optimization

Armed with analytic expressions for a, s,w we will now use them to systematically optimize the available bandwidths given sets of constraints on the desired delays, pump powers and focusing, and available magnetic field gradients and transverse frequency dispersion. We will first derive the relationship between bandwidth and delay for conventional slow light, and see how the delay-bandwidth product can only be increased by via increased delay and decreased bandwidth. We will then find the corresponding relationship for the channelization architecture and see how the dispersion and mis-match provides an additional degree of freedom, allowing large delay-bandwidth products even for small delays. We conclude that for many parameters of interest, channelization gives us a substantial improvement.

Suppose we desire some delay ${\tau }_d^{\left(0\right)}$ . Using a conventional slow light configuration, our delay would be ${\tau }_d^{\left(0\right)}$ =sD=DΓ _r /|Ω _p |², which can be inverted to give the optical density D necessary for a given pump Rabi frequency Ω _p . The bandwidth (defined as the signal frequency component δs where the transmission attenuation is 1/e) is $B_{\mathrm{conv}}={\mid {\Omega }_p\mid }^2⁄\sqrt{8{\gamma }_e{\Gamma }_{r}D}=\mid {\Omega }_p\mid ⁄\sqrt{8{\gamma }_e{\tau }_d^{\left(0\right)}}$. We can also write this in terms of the power P≡|Ω _p |² WH where W and H are the width and height of the focused laser beam in the medium (the units of P are not true units of power but allow us to scale properly the laser power with intensity and beam area):

This equation shows the direct relationship between bandwidth and delay, given finite power and focusing constraints. It is plotted as the black curve in Fig. 5, and we see that one can get arbitrarily large bandwidths only by lowering ${\tau }_d^{\left(0\right)}$ (via lower optical density D). The inset of Fig. 5 shows the delay-bandwidth product B _conv ${\tau }_d^{\left(0\right)}$ . One can only get arbitrarily large delay bandwidth products by increasing ${\tau }_d^{\left(0\right)}$ via D. This product will ultimately be limited by the maximum allowable D, determined by the absorption a due to ground state decoherence γ _coh (see Eq. (10)).

 

Fig. 4. The maximum bandwidth B _ch Eq. (12) versus the chosen delay ${\tau }_D^{\left(0\right)}$ . In the blue curves, we hold the the slope S_B =1 G/mm constant while adjusting W (both for γ _coh=0 (dotted) and γ _coh=1 kHz (solid)). In the red curves, we hold W=4 mm while adjusting S_B . In both cases, the beam height H=0.5 mm and the total pump power is 10 mW. For comparison, the black curve shows the bandwidth for a conventional slow light system, with a pump of the same power focused to an area W=H=0.5 mm. Inset: The delay-bandwidth product versus desired delay for the same parameters (note the larger range of delays).

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When we use the channelization to separate frequency components, the relationship between bandwidth and delay is altered. Using the channelization configuration, with mismatch parameter β≪1, we have a delay ${\tau }_d^{\left(0\right)}$ =sβD=βDΓ _r /|Ω _p |². Since a frequency component δ_s enters the medium at a position where it is detuned βS_Bx, the highest frequency components, at ±W/2, are the mis-matched from the EIT resonance ±βS_B (W/2). Setting this mis-match equal to the local bandwidth B _conv puts a constraint on the parameter β=B _conv/B _ch, where B _ch=S_BW is the bandwidth in the channelization architecture. Plugging this into our above expression for ${\tau }_d^{\left(0\right)}$ and solving for B _ch yields $B_{\mathrm{ch}}=\left(\sqrt{D{\Gamma }_r⁄8{\gamma }_e}\right)⁄{\tau }_d^{\left(0\right)}$. We use our expression Eq. (10) to constrain our maximum optical density D<D _max=1/(4a _anal) and substituting this result finally yields:

where C=64/3.

We now find this bandwidth for a given laser power P and given desired delay. In principle, both the slope S_B and system width W are free parameters, but in practice there will be determined by technical considerations in producing the dispersion and magnetic field gradients. To make a meaningful comparison between conventional and channelization bandwidths, we assume the some pump power P. For the conventional set-up, which benefits from tight focusing of the pump, we use W=H=0.5 mm, while for the channelization scheme, we set H=0.5 mm but adjust W to be wider as desired. We also set Δ _D =330 MHz (T=60 Celcius) and γe=(2π) 150 MHz and D_g =5.0 cm²/s, their values at the optimum pressure p=30 Torr (see Fig. 3(b)).

We first find B _ch, holding the slope at a constant value S_B =1 G/mm and adjusting W. The procedure is to solve Eq. (12) for W, given ${\tau }_d^{\left(0\right)}$ and S_B . The resulting maximum bandwidths B _ch are plotted as the blue curves in Fig. 4, with the dotted curve representing a case with no ground state decoherence γ _coh=0 and solid curve representing γ _coh=(2π) 1 kHz. For smaller ${\tau }_D^{\left(0\right)}$ we see substantially larger bandwidths for the channelization case over the conventional case. Decoherence γ _coh has the effect of slightly lowering the advantage of channelization, but not substantially in the regime plotted. In the inset of Fig. 4 we plot the delay-bandwidth products and plot over a much larger range of delays. Here we clearly see a large relative advantage over conventional slow light for smaller delays but a much smaller advantage for longer delays ~µs and finite γ _coh. In this sense, if one was concerned only with obtaining the largest delay-bandwidth product and was unconcerned with the magnitude of the delays, there is little advantage to using channelization. As an alternative constraint, the red curves show the results of holding W=4 mm and letting S_B vary and one sees a similar trend, though here there is an even larger advantage of channelization at lower delays, and more disadvantage at longer delays.

We emphasize that the results presented in Fig. 4 are not fundamental limits. Rather they depend on available freedom in choosing S_B and W, which is governed by technical implementation issues. They also depend on the decoherence γ _coh, available pump power P, focusing ability H,W, and other system parameters. Equations (11–12) provide a systematic method for comparing the bandwidth advantage of channelization to conventional slow light for different desired delays. Generally speaking, there is a larger advantage for smaller delays and weaker focusing. At larger delays, the extra technical requirements to implement channelization becomes an undesirable burden for the small advantage gained.

6. Conclusion

We have expanded upon the analysis of our proposed channelization architecture in two important respects. First, we have analyzed in more detail how the pulse acts upon recombination of the different frequency components, including the effect of finite resolution of the transverse dispersion procedure. We concluded that indeed perfect matching of the EIT resonance at all locations lead to no delay of the recombined wideband pulse, and that a slight mismatch must be introduced to recover a delay [16]. We emphasize that, using this procedure, the delays can still be continuously and quickly adjusted by varying pump power as in conventional EIT. Our result Eq. (5) also includes how the finite resolution of the dispersion will play a role, and could lead to additional attenuation and phase shifts in the recombined pulse.

Second, doing a systematic analysis of how the delays and bandwidths scale with various physical constraints, we found that, using our results Eqs. (11–12) that one can obtain substantially larger bandwidths for shorter delays. In particular, for typical parameters we found bandwidth increases of ~5 (to about 100 MHz) for 10 ns delays. Thus one can achieve significant improvements for particular applications, such as radar true-time delay and optical buffers, which require significantly shorter delays but larger bandwidths than those available with conventional EIT. This is true despite the fact that one cannot increase the optimal delay-bandwidth product. Instead, one is using channelization to trade-off delay for bandwidth, without having to use prohibitively large pump powers to broaden the EIT feature. The physical limitations on the bandwidth are related to the magnetic field gradients and light dispersion available, and the diffusion rates of atoms in inhomogeneous magnetic fields.

Looking to future developments, we want to emphasize the general concept of channelization, both continuous (analyzed here) and discrete, can be applied to systems other than atomic EIT. Quite generally, it can be applied to systems in which one can shift the resonance frequency spatially in a controlled way. When one has this ability, it can be exploited to increase the available bandwidth. Also, we note that the mis-match between the two-photon resonance can be chosen to have either sign, raising the possibility of exploring both superluminal and slow light propagation with this technique.

The authors thank Irina Novikova and Fredrik Fatemi for useful discussions. This work was supported by the Office of Naval Research and the DARPA-DSO Slow Light Program.