1. Introduction

The optical engineering of the speed of light attracted a large interest during the last decade. Two main nonlinear interactions are exploited: Electromagnetically Induced Transparency [1,2] and Coherent Population Oscillation (CPO) [3]. However other nonlinear interactions such as Raman [4] and Brillouin [5,6] scatterings are also considered. In most of the studies, the slow light propagation is demonstrated in atomic vapors and Bose Einstein Condensates, where impressive low group velocities (few tens of m/s) were demonstrated implementing very low optical powers (in the range of few tens of mW/cm²). More recently, the field extends to solid state systems that are attractive due to the elimination of effect associated to atomic motion, the stability, the compactness and the compatibility with usual optical devices and systems. Starting from the initial demonstration in ruby [7], through alexandrite [8] and semiconductor quantum wells [9], slow light in solid state systems becomes a field of growing interest. This is specially the case for nonlinear mechanisms that, like CPO, require only a two level system to be implemented. Among solid state systems, a particular mention should be made to Er-doped materials for two main reasons. Firstly, as many rare earth, Er allows slow and ultra slow light propagation due to the very long excited level lifetime available. Secondly, such long lived states can be addressed in the important wavelength range around 1.5 µm, fitting the transparency region of silica fibers at the heart of many telecommunication applications and for which stable commercial laser sources are available.

Ultraslow light propagation with a group velocity of 3 m/s was reported via CPO effect in an Er³⁺ doped Y₂SiO₅ crystal [10]. As usual the fluctuation of the local Er environment induces an inhomogeneous broadening often considered as a limiting factor. Baldit et al. [10] demonstrated that this inhomogeneous broadening can be a resource in order to tune both the group velocity and the optical transmission. These results are also attractive from the point of view of the very low optical power involved, comparable to those implemented in experiments with atomic vapors or cold atoms. In turn, they are performed at low temperature, a serious limitation for most applications. Room temperature slow light propagation was demonstrated in erbium doped fibers (EDF) [11–13], taking benefit both from the mode confinement and the increased interaction length. In addition, as a key element of optical communication, EDF offer the compatibility with applications close to the telecommunication industry. Slow light was demonstrated via CPO in this context using EDF, with group velocities at 1.5 µm of the order of 1200 m/s, both for modulated beams, in a pump-probe configuration and for self induced effect in the pulsed propagation. In addition by injecting a pump beam at 980 nm, either delay or advancement can be achieved, depending on the pumping level [14]. Group velocities between 35210 m/s and –40000 m/s where reported [14]. In all of these demonstrations, the absorbing/gain Er medium is located in the fiber core, i.e. at the center of the guided mode.

A new generation of EDF is nowadays developed with the aim of optimizing the pumping conditions while keeping the outstanding performance of regular telecommunication fibers used in transmission applications [15–17]. In these fibers, the doped, Er or Yb, medium has a ring shape surrounding a single-mode core. The core is undoped and its diameter is of the order of those of regular telecommunication fibers for transmission applications, to allow low losses and single mode propagation, as well as ease integration to regular transmission systems. The distribution of the absorbing/gain medium allows to increase the pumping power injected in the, doped, second section that acts as a multimode waveguide for the pump wavelength. The amplification is obtained thanks to the partial overlap between the tail of the light propagating at 1,5 µm, mostly in the single mode core and the doped ring region. In this paper we demonstrate slow light propagation in a Ring Erbium-Doped Fiber (REDF). In this configuration the strong dispersion induced by the CPO effect is obtained through the interaction of the light mode wings extending in the ring doped with erbium ions. As predicted by [18], in the context of EIT, this condition is sufficient to modify the effective index of refraction seen by the light propagating in the single mode core.

2. Experiment

The REDF used in the experiment was pulled from a solution-doped preform fabricated by modified chemical-vapor deposition. The spatial distribution of the index of refraction of the fiber is represented in Fig. 1 . Two main regions appear corresponding to the single mode core with a diameter of 5.6 µm and the multimode core having a diameter of 38 µm. The Er³⁺ ions, shown in blue in Fig. 1, are distributed in a ring located in the multimode core in the vicinity of the single mode core. The concentration of the Er³⁺ ions is 600 ppm and the overlap with the optical mode propagating in the fiber is about 9%, calculated by implementing finite-element method and using the index of refraction profile depicted in Fig. 1. The fiber used in our experiment is 1.5 meter long and its absorption per unit length is represented in Fig. 2 as a function of the wavelength.

 

Fig. 1 Defraction index profile of the REDF.

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Fig. 2 Absorption of the REDF.

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In order to excite the extremely narrow CPO hole, the pump-probe relative frequency stability should be better than the CPO hole linewidth which is limited by the Er³⁺ ions lifetime T ₁ = 10 ms. These long lifetimes impose a serious constraint to the mutual pump-probe stability, when two lasers are used. To overcome the problems associated to the use of two synchronized laser sources we adopt the approach introduced by Boyd et al [7]. It consists in implementing an intensity-modulation of a continuous-wave (cw) laser source, which produces in the Fourier space new spectral components playing the role of the probe field, whereas the carrier frequency plays the role of the pump.

The scheme of the experimental setup is depicted in Fig. 3 . The light source is a cw 2-kHz linewidth fiber laser (KOHERAS) emitting at 1536.1 nm. At this wavelength, the absorption in the REDF is 0.01 cm⁻¹. The laser beam goes through an acousto-optic modulator (AOM) in order to generate the pump and probe fields for the CPO effect. This is achieved by modulating the intensity of the first order beam diffracted by the AOM. In order to obtain a good and constant optical coupling, the REDF is spliced to a standard single mode 2-meter-long SMF28 fiber having a core diameter of 9 µm. The optical losses induced by the splicing, measured at a wavelength of 1.31 µm at which the Er³⁺ ions do not absorb, are 0.8 dB. The laser beam is injected into the SMF 28 fiber using a 5 × microscope objective having a numerical aperture NA = 0.12, equivalent to the NA of the SMF 28 fiber. For 10 mW incident on the fiber, 1 mW is measured at the output. Considering the splicing losses and the absorption of the fiber, a coupling efficiency of about 60% in the SMF28 fiber is deduced. A fraction of the laser beam, picked up before the objective with a polarizing beam splitter, is used as a reference to measure the delay achieved in REDF. The main beam propagating in the REDF and the reference beam are detected using InGaAs detectors (NEWFOCUS 1181) and recorded with lock-in Amplifiers. The group delay is then computed from the phases measured by the lock-in amplifiers and is simply given by τ_g = Δϕ/(2πδ) where Δϕ is the signal – reference phase difference and δ the modulation frequency.

 

Fig. 3 Scheme of the experimental setup.

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3. Slow light propagation

The group delay is measured as a function of the input intensity for a set of modulation frequencies (Fig. 4 ). Firstly, the group delay τ_g increases as the pump power is increased. This is due to the increased number of Er³⁺ ions contributing to the CPO process. Secondly, as the pump power is further increased, the delay reaches a maximum value and then declines. This is mainly due to two effects. When the saturation level is attained the number of Er³⁺ ions whose populations are oscillating in the CPO process reaches a stationary level. A further increase on the pump power induces a power broadening of the CPO hole. This in turn is associated to a smoothed dispersion profile at the origin of weaker delays. The maximum delay at 10 Hz is reached for a pump power of about 4 mW, corresponding to an intensity of about 6.4 W/cm². As the frequency modulation gets higher, the maximum delay tends to zero since the modulation is faster than 1/T ₁ and the population can no longer follow the pump-probe beating. The behavior obtained in REDF shown in Fig. 4 is similar to the one of CPO-based slow light propagation in a crystal doped with Er³⁺ ions [10]. It is also consistent with the results shown in Fig. 5 obtained in a regular EDF using the same experimental setup of Fig. 3. The EDF is 3 meter long, with an index of refraction contrast between the 4-µm-diameter core and the cladding of 18 10⁻³. The absorption in the EDF is 4 dB/m at 1532 nm.

 

Fig. 4 Group delay obtained in REDFA for different frequencies δ as a function of the pump power.

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Fig. 5 Group delay obtained in a regular EDFA.

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Note that in all cases the delay does not rapidly vanish as the power is further increased. This is a signature of the inhomogeneous broadening as evidenced in the following theoretical analysis.

The behavior of the group delay is theoretically very well accounted for using a simple analysis based on the dynamical evolution of the ions population when excited by a modulated light. In this approach, the ground state population evolution is given by

Solving Eq. (1) in the approximation of a weak modulation, , we can deduce the complex absorption coefficient of the modulated intensity. It is given by

The dashed line in Fig. 4 is the theoretical plot of expression (4) obtained for T ₁ = 10 ms, δ = 10 Hz and by adjusting the saturation parameter S. When the powers are below the optimum value for which the maximum delay is reached, theory agrees with the observation. However as the pump power is increased above the optimum value of 4 mW, theory does not follow anymore the experimental data. This disagreement is due to the inhomogeneous broadening of the optical transition that cannot be considered in the model based on the evolution of the population and thus by expression (4). A more rigorous theoretical approach based on the evolution of the matrix density should be used. This formalism takes account of the different relative detunings (pump – probe and pump – atomic resonance) and more importantly the inhomogeneous broadening. In this approach, the dielectric susceptibility is calculated. The detailed analysis can be found in reference [19]. The dielectric susceptibility of the probe is given by

In order to take account of the inhomogeneous broadening, one has to integrate the susceptibility ${\chi }_s\left(\delta ,\mathrm{\Delta }\right)$ over the inhomogeneous line. It came out from the calculations, that knowing the exact value of the inhomogeneous broadening is not relevant when it is much larger than homogeneous line since the important contribution to the integral is around few homogeneous lines. This is the case for the REDF. In addition T ₁ = 10 ms and the absorption profile are independently measured, and the only unknown parameter is the saturation intensity that can be inferred from the fit of one of the curves of Fig. 4. The other fits are thus obtained without any adjustable parameter. The continuous lines in Fig. 4 are the theoretical group delays obtained using the density matrix formalism. The agreement is now excellent and points to the crucial role of the inhomogeneous broadening. We would like to stress that, from a more practical point of view, the inhomogeneous broadening does not kill the slowing down of the light. It constitutes instead an interesting resource to tune the group delay, by taking benefit from the change of the absorption level among the inhomogeneous profile.

The group delay is then measured as a function of the modulation frequency in order to clearly highlight its CPO origin. The results, presented in Fig. 6 for a pump power of 5 mW, clearly show a decline of the optical delay as the modulation frequency is increased, with a full width at half maximum of 25 Hz. This value is consistent with the natural linewidth (1 + S)^1/2/πT ₁ = 25 Hz of the Er³⁺ transition, where S = 1.5 for a power of 5 mW. The continuous line is the theoretical group delay calculated using the matrix density formalism and taking into account the inhomogeneous broadening. The maximum delay achieved of 1.1 ms corresponds to a group velocity as low as 1360 m/s. Since the Er³⁺ ions ring introduces a strong localized absorption a question arises on the eventual distortion of the spatial profile or more subtle an eventual variation of the delay along the spatial profile. We perform measurements where the output beam was screened by a movable razor blade. They confirm that both the spatial intensity profile is undistorted and that the optical delay is uniform along the whole spatial profile. This is expected for an optical mode undergoing, as a whole, CPO effect wherever the absorption associated to the two-level system is located.

 

Fig. 6 Group delay as function of the frequency δ for a pump power of 5 mW. The continuous line is the theoretical curve.

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4. Conclusion

In conclusion, we have demonstrated slow light propagation in an REDF via CPO effect. Although the doping is annularly distributed in a second core surrounding the undoped central core, the whole optical mode experience a group delay. The measured group velocity is as low as 1360 m/s, lower than the speed of sound in silica. It can be tuned by taking benefit from the inhomogeneous broadening, inherent to the dispersion of the local environment of erbium ions in the silica. Finally, we would like to stress that such REDF are particularly suitable for implementing tunable group delays (positive or negative) by pumping in the secondary, doped, core at a wavelength of 980 nm. Indeed, the pumping effect will change the absorption of the fiber turning it into gain, modifying the slop of the index of refraction dispersion.