Slow light of dark pulses in a photorefractive crystal

Slowdown of Gaussian “dark pulses” in photorefractive crystal using a two-wave mixing method is shown at room temperature. In agreement with our theoretical model, the experiment shows the dependence of the dark pulse delay and the group velocity values on the photorefractive gain (cid:2) and the input pulse width t 0 . The use of dark pulses allows us to improve the time-delay values and achieve maximal fractional of about 1 for a pulse duration close to the crystal response time.

Slow light in dispersive nonlinear media was actively studied theoretically and experimentally [1][2][3][4][5], including in photorefractive (PR) materials [6][7][8][9][10][11].The two-wave mixing (TWM) process in PR crystal yields amplification of the output pulse during the recording of the photorefractive grating.Simultaneously this modifies the light dispersion inside the crystal, hence reducing the group velocity of the output amplified pulse down to 1 cm/s in BaTiO 3 [12] and Sn 2 P 2 S 6 [10] at room temperature.The four-wave mixing effect can also be used to decelerate the light velocity both of the transmitted and the phase conjugate waves [13].The output pulse delays vary as a function of the PR nonlinearity [14], the crystal response time, the input pulse duration [7], and the photorefractive gain [10].Small values of group velocity (large delays) are achieved for large values.However, the dispersion inside the crystal limits the delay bandwidth product because of the broadening of the output pulse.
Previous studies of slow light used Gaussian pulses because it is better suited to numerical calculations and easier to process than other pulse shapes.However, experimental and numerical studies have shown that the dark pulse or "Gaussian holes" can be more stable and less sensitive to noise in optical fibers than the Gaussian light pulses.For example, the dark soliton can be interesting for long-distance optical communications [15] due to its stable propagation in optical fibers under the influence of noise.Since their first observation [16], the dark pulses have attracted great interest, particularly for studying the propagation of solitons in the optical fiber [17,18] and in Bose Einstein condensate [19].Dark pulses arise in many distinct areas of physics: Coherent quantum control of two-photon transitions in caesium (Cs) [20], timeresolved x-ray spectroscopy [21], generation of Kerr combs * nacera.bouldja@centralesupelec.fr in normal-dispersion microresonators with mode-interactionassisted excitation, and mode-locking transitions [22].The use of dark pulses has been recently suggested in slow-light systems including in the plasmas [23] and insulator-insulatormetal plasmonic waveguides [24].It has been analytically and numerically demonstrated that the dispersion of the output signal can be compensated by using Kerr dielectrics and dark solitons, and hence it of interest for high-bit-rate telecommunications [24].

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In this work, we investigate theoretically and experimentally how dark pulses may be decelerating in a PR crystal at room temperature.First, we developed an analytical model describing the temporal envelope of the dark pulse propagating through the PR crystal.As for the Gaussian light pulses, the model shows that the delay and shape of the output dark pulse change as a function of the TWM gain and the input pulse duration.Using the TWM between the dark pulse and a continuous pump beam, delays greater than the input pulse duration can be reached.For ms pulse duration, the corresponding delays are larger than those we have reported in Ref. [10] for Gaussian pulses and the fractional delay is also larger close to 1.To analyze the delay performances of slow dark pulse in the PR material, the numerical calculations are implemented using the experimental parameters for a Tedoped Sn 2 P 2 S 6 crystal.Let us consider the input dark pulse A(0, t ) as the following: Here A 0 and t 1 are respectively the amplitude and 1/e halfwidth of the input pulse intensity.Using the Taylor expansion, Eq. (1) becomes where a n is given by ( The propagation of the dark pulse in the photorefractive crystal with thickness d after the interference with a continuous pump wave is expressed as follows: where A ω (0) is the Fourier transform of A(0, t ), g = d is the dimensionless parameter conventionally referred to as the coupling strength with being the PR gain, d being the crystal thickness, and f ω [25] being the characteristic response function.By using the inverse Fourier transform, the output dark pulse amplitude A(d, t ) transmitted through the PR crystal is given by We use the expression of the output signal (Eq.( 7) in Ref. [25]) and Eq. ( 5) to plot respectively the temporal envelope of bright [Fig.1(a)] and dark [Fig.1(b)] pulses calculated for | d| = 6.62 and full input width at half maximum t 0 = 60 ms.By analyzing the profiles of the normalized pulse intensities, we can retrieve the output pulse delay τ , which is the time shift between the maximum or (minimum) between the input and output light pulses or (dark pulses).The result shows that for the same t 0 the minimum of the output dark pulse [Fig.1(b)] is more delayed than to the output bright pulse maximum [Fig.1(a)].For t 0 = 60 ms, the bright and the dark pulses are respectively delayed by τ = 67 ms and τ = 77 ms for d = |6.62|.To illustrate the dependence of the time delay versus the TWM gain , we plot in Fig. 2 the temporal profile of the input and output intensities for t 0 = 60 ms for three values of | d| = 5, 8, 10.As for the bright pulses, it is clearly seen that the dark pulse delay can be controlled by the gain.Both delay and output pulse width increase with increasing d values.For | d| = 5, 8, 10, the transmitted pulse delays are τ = 50, 80, 100 ms respectively.Also, Figs. 1 and 2 show that the transmitted pulse broadens when propagating through the PR crystal.This broadening increases with the increase of the PR gain value.For | d| = 5, 8, and 10, the durations of the output pulse become 120, 130, and 150 ms respectively.Below we present the results of experimental studies of the slow dark pulses using the setup presented in Fig. 3.The setup is similar to that used in Ref. [10].The measurements are performed with a laser at λ = 638 nm with intensity of 3.8 W/cm 2 and a 5-mm-thick Sn 2 P 2 S 6 :Te PR crystal.The electro-optic modulator which is controlled by a signal generator generates bright or dark pulses by a change of the electric signal phase (the phase difference between a dark and bright pulse is 180 • ).
With this setup, we study first the experimental performances of the slow dark pulses in Sn 2 P 2 S 6 for the full input pulse duration at half maximum t 0 = 10 ms as a function of the TWM gain.To measure , we change the pump and signal ratio and we measure the amplification of the output signal which gives the value of the gain.The results are shown in Fig. 4. Figure 4(a) presents the normalized input and output pulse intensities for two experimental values of the gain d = 2 and d = 2.9.As is shown, the delay τ between the minimum of the input and output pulse changes with d.The transmitted pulse propagates in the crystal without any broadening for d = 2 and when d increases to 4.6, the output pulse duration t out (FHW M) changes from 10 to 28 ms.As is presented in Fig. 4(b), beyond d = 2.4, the measured delay τ is larger than the dark pulse duration t 0 and a maximum value of 16.5 ms is measured for d = 4.6, which is larger than the delay reported in Ref. [10] for the same input pulse duration.Figure 4(c Now, as reported in Ref. [10], we analyze how the delay varies with the input duration for large TWM d = 5.This d value is achieved for a very weak dark pulse intensity, i.e., the ratio between the pump and the signal is of order 7 × 10 3 .The curves (ac) in Fig. 5 show the profiles of the normalized input and output pulse intensities for three input pulse durations.For example, at t 0 = 10.3 ms close to the crystal response time τ = 10 ms, the output pulse of duration t out = 15.7 ms is delayed by τ = 7.9 ms. Figure 5(b) presents a case where the delay τ = 28 ms is greater than the input pulse duration t 0 = 23.5 ms.If we increase the input pulse duration, the delay increases with t 0 , and a delay of the order of 146 ms is measured for a pulse duration of order of 480 ms [Fig.5(c)].Therefore, there is a range of input pulse durations for which the delay reaches a maximum value relatively to t 0 .
The variation of the delay, the output pulse duration, and the corresponding FD with respect to the input pulse durations are plotted in Fig. 6. Figure 6(a) shows that delay values increase for the short input pulses durations t 0 < 250 ms decreasing to zero.A maximum delay value of order of 156 ms corresponding to the envelop group velocity v g = 3.2 cm/s is measured for input pulse duration t 0 = 240 ms.Also, for pulse durations higher than 880 ms, the delay values become negative which cause the acceleration of the output dark pulse.This case is possible because of the Sn 2 P 2 S 6 selfcompensation in PR response [9].The light acceleration can be observed by rotating the c axis of the crystal by 180 • or by increasing the input pulse duration t 0 > τ.In our case, the advance time is measured for a pulse duration greater than 800 ms.The output dark pulse time width versus the input pulses durations are plotted in Fig. 6(b).We note the enlargement of the output pulse duration for higher d values.For example, the input pulse of t 0 = 19 ms is slowed down by τ = ms and its duration is almost doubled at the output of the crystal.Knowing the output pulse duration and the pulse delay, we can compute in Fig. 6(c) the fractional delay.Figure 6(c) shows that the FD is maximum for a given input pulse duration t 0 which is close to the response time of the crystal τ .When t 0 τ and increasing t 0 , the delay remains quite constant while the broadening of the output pulse increases, hence decreasing FD.When t 0 τ and increasing t 0 , the delay increases much slowly than the increase of the broadening of the pulse, hence also decreasing FD.An optimal value of FD is therefore achieved for an intermediate value of t 0 close to τ .
In summary, we have studied the slowdown of dark pulses using TWM in the Sn 2 P 2 S 6 :Te photorefractive crystal at room temperature.In addition to the experiments, the numerical calculations illustrate the performances of pulses deceleration including the time delay, the resulting distortion, and the fractional delay.The results show that the delay increases with the input duration and also with the nonlinear gain d.The experimental results confirm the validity of the theoretical model.It is shown that the dark pulse is more delayed than bright pulse [10] for input pulse duration in the range 0 < t 0 < 100 ms.A maximum time delay of 156 ms is achieved for input dark duration t 0 = 240 ms, corresponding to a group velocity of v g = 3.2 cm/s.Increasing d leads to longer delay but the dark output pulse also propagates in the PR crystal with a larger distortion.Finally, the fractional delays 1 have been obtained for input pulse duration t 0 around the crystal PR response time t r = 10 ms, a situation for which the delay τ is larger than t 0 .
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FIG. 1 .
FIG. 1. Temporal envelopes of the normalized input (black line) and output (red line) pulses for input full pulse width at half maximum t 0 = 60 ms and | d| = 6.62,(a) time delay τ = 67 ms for the Gaussian light pulse and (b) τ = 77 ms for the dark Gaussian pulse.

FIG. 6 .
FIG.3.Experimental setup: BS is the beam splitter, A is the attenuator, EOM is the electrooptic modulator, D 1,2 are the detectors, M is the mirror, θ is the incidence angle, and d is the crystal thickness.