1. Introduction

Light velocity control by light has been a hot research topic for the last decade in the area of nonlinear quantum optics [1–5], all-optical information processing [6,7], and quantum information processing [8–11]. In the area of nonlinear optics, the group velocity of light propagating through an optical medium is normally understood to have a fixed value determined by the refractive index of the medium at the specific wavelength. However, the group velocity of one pulse (a probe pulse) can be controlled by applying another light beam (the coupling or pump beam) to induce an abrupt change in the absorption spectrum [1–11]. For the group velocity control by using absorption spectrum modification, the Kramers-Kronig relation contains the fundamental physics [12]. The absorption spectrum is coupled with the dispersion spectrum in an integral form. By narrowing the absorption spectrum of the medium, one can obtain very high value of derivative of refractive index resulting in an ultraslow group velocity of light.

Electromagnetically induced transparency (EIT) [13] has been intentionally studied for the quantum coherent control of light for absorption modification resulting in a group velocity change. The EIT spectral width of a transparency window for a probe field, however, is typically quite narrow and determined by the coupling Rabi frequency applied. Thus, EIT-based slow light in general has limitations in potential applications of ultrahigh speed information processing due to its narrow transparency spectrum. In the early 2000’s a modified spectral hole-burning technique, the so-called coherent population oscillation (CPO), was introduced as an additional method to achieve an ultraslow group velocity of light in solids [14,15]. In CPO, however, signal field must be spectrally closer to the pump field to get efficient slow-down effects. In this case, pump field induced spectral noise and/or population relaxation time-limited bandwidth limits potential applications of CPO [15], even though there are some advantages over EIT such as in a relatively simple scheme and a room-temperature operation.

As mentioned above, the present method uses a simple absorption modification in an inhomogeneously broadened medium using a spectral hole-burning phenomenon [16]. Here, we present observations of self-induced ultraslow group velocity of light in a rare-earth doped crystal using persistent spectral hole-burning phenomenon, where the group velocity or group delay is comparable with the EIT or CPO based. Unlike EIT or CPO, the present method uses a single signal pulse with and without repump fields, where the repump serves to control atom numbers interacting with the signal. The lifetime of the spectral hole burnt by the repump fields or the signal pulse is as long as the spin population relaxation time T^SPIN ₁, where T^SPIN ₁ ranges from minutes to hours in rare-earth doped solids [16,17]. The spectral-hole width is of course controllable simply by modifying the intensity of the signal field using power broadening effect [17]. Lower bound of the persistent hole-width is similar to that by CPO or EIT, but the physics is fundamentally different from CPO in using optical phase relaxation time and from EIT in using hole burning itself. Benefits of the present method over EIT or CPO are an inherently simple scheme of controlling the group delay and elimination of spectral noise. First, we use repump fields (R1 and R2 in Fig. 1), which are far-detuned, nonoverlapping in time, and different in propagation directions. Second, the group velocity of light can be easily controlled by modifying the atom population interacting with the probe. Third, the antihole by the repump remains persistent as long as several hours, so that consecutive signal pulses do not need the repump field at each time.

The persistent spectral hole burning is a simple spectroscopic phenomenon observed in an inhomogeneously broadened three-level optical medium whose population relaxation rates are different [16]. In most rare-earth doped solids such as Pr³⁺ doped Y₂SiO₅ (Pr:YSO) or ion-doped solids such as Cr³⁺ doped Al₂O₃, the population relaxation time for spin transitions is very slow compared with that for optical transitions. In such an optical medium interacting with a narrow bandwidth light, the absorption spectrum is modified due to the persistent spectral hole burning. The Kramers-Kronig relation predicts a frequency dependent dispersion modification as a result, and this in turn causes a change in the group velocity of the light pulse. The narrower the width of the spectral hole, the slower the group velocity [1–5]. The spectral hole-width, in general, depends on laser jitter, where the laser jitter is normally much wider than the optical homogeneous width. Moreover, the hole-width can be widened by increasing the laser intensity to obtain power broadened spectral hole [16,17]. Thus, the group velocity control range is wide if the light (either repump or signal) is spectrally narrow. Here, we present self induced slow light and how to control the group velocity of a signal pulse by controlling the repump field intensity.

2. Results and discussions

 

Fig. 1. Self-induced ultraslow light of P in a rare-earth Pr³⁺ doped Y₂SiO₅. (a). Schematics of energy level diagram, (b). pulse sequence, and (c). propagation. (d). Ultraslow group velocity of the probe P versus the probe detuning: R1=24mW; R2=32mW; P=2.6mW. The frequency detuning of the probe is made by using an acousto-optic modulator.

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Figure 1 shows observations of self induced ultraslow group velocity in Pr:YSO (0.05 at. %) at T=5K. Figs. 1(a), 1(b), and 1(c) show schematics of energy level diagram, pulse sequence, and propagation direction, respectively. The repetition rate of the pulse sequence is 20 Hz. Using appropriate beam splitters and acousto-optic modulators all three laser beams are generated by the same laser output (Technoscan ring-dye laser) in a planar configuration and focused into the Pr:YSO sample through a 40 cm focal length lens with a crossing angle of ~25 mrad. The beam spot diameter (FWHM) of R1, R2, and P are 210, 310, and 100 µm, at the focal point, respectively. The repump pulses R1 and R2 are used for reinitialization of state |2> to have identical atom (ion) distribution before each probe pulse, P. Thus, the spectral distribution of atoms in state |2> should depend on the laser jitter (~300 kHz), where the laser jitter is time dependent. The laser jitter in a dye laser system is mostly caused by a dye jet noise, where the noise is timely accumulative to some extent in a short time scale of ≤10 µs (will be discussed in Fig. 3). Initially all three ground states |1>, |2> and |3> are equally distributed within a 4 GHz inhomogeneous line shape. The repump R1 and R2, respectively, excite atoms from states |1> and |3> populating state |2> by optical pumping generating an anti-hole [18]. Thus, the anti-hole spectral line shape should be symmetric and is expected to be quasi-Gaussian. The maximum atom density in state |2> is expected to be three times higher than atoms at thermal equilibrium.

The 3-mm-long Pr:YSO medium is made optically dense using the repump fields as described above. As the probe pulse P enters into the medium, part of the pulse is absorbed by the resonant atoms, creating a spectral hole in the atoms along the propagation path. Thus, the refractive index variation created in both frequency and space causes a group velocity change to the rest of P pulse (the space refractive index-dependent nonlinear effect will be discussed elsewhere). The group velocity of the probe P in the medium is inversely proportional to the number of atoms [1]:

where N is the number of atoms. Among the optical fields, R1, R2, and P, no coherence is induced to the interacting atoms because R1 and R2 interact with different groups of atoms, and the delay time of P is much longer than the inverse of laser jitter (~300 kHz). Figure 1(d) shows 30 sample averaged results of the self-induced ultraslow light as a function of the probe detuning. The positive detuning stands for a blue shift, and has the same effect as the red shift does (not shown). As shown in Fig. 1(d), the ultraslow group velocity of the P pulse depends on its frequency detuning. More frequency detuning causes faster group velocity. This means that the redistributed population in state |2> is maximum at line center and decreases as it is detuned. From the result of the group velocity versus the detuning we conclude that the line shape of the redistributed atom population in state |2> should be quasi-Gaussian caused by the repump spectral lineshape. The spectral width of the atom redistribution depends on the timedependent laser jitter.

 

Fig. 2. Atom number dependent ultraslow light. (a) Group velocity versus repump power R2, and (b) Group delay versus repump power R2. The number of atoms is controlled by the repump R2 using persistent spectral hole burning.

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Figure 2 shows the group velocity of pulse P versus repump R2 power (intensity). Below the saturation limit, the number of atoms transferred into state |2> by the repump R2 from state |3> should be proportional to the repump power. Figure 2 demonstrates that the group delay of P is linearly proportional to the repump (R2) power (intensity). It should be noted that the oscillator strength (or transition coupling efficiency) for the transition |3>-|5> in Fig. 1(a) is very weak [18]. Thus, the group delay of P from Eq. (1) is proportional to the number of atoms, where the number of atoms is proportional to the repump power (intensity) of R2:

where L is medium length, and Ω_R2 is Rabi frequency of the repump R2. For the repump power R2=32mW and the group delay τ_g=20 µs, the group velocity of P is v_g=L/τ_g=3 (mm)/20 (µs)=150 m/s=c/2,000,000. Combining Eqs. (1) and (2), the results are very similar to EIT based slow light in terms of pump power broadening [5].

When a transition is resonantly driven, the optical homogeneous width is broadened as the laser power increases [17]. This is so called power broadening. Figure 3 demonstrate probe power dependent group delay for different probe pulse length ranging from 2 µs to 10 µs. As shown in Fig. 3(a), the group delay is inversely proportional to the probe power. According to the Kramers-Kronig relation, a narrow absorption spectrum induces a stiffer dispersion slope $\left(\frac{\partial \chi }{\partial {\omega }_{,}}\gg 1\right)$, and the stiffer dispersion slope becomes a dominating factor determining the group velocity [1],

where ω is the probe frequency, and χ is the susceptibility of the medium related with the absorption spectrum. Thus, Fig. 3(a) demonstrates that the probe group delay is inversely proportional to the probe power. For a fixed probe power, the group delay increases as the pulse length decreases if the pulse length is less than 4 µs (see Fig. 3(b)). A possible explanation for this could be that the laser jitter is time (pulse length) dependent, and shorter pulse length has less jitter showing that narrower absorption line in Eq. (3). The laser jitter dependent group delay, however, gives less effect as the P power increases (see the results of Fig. 3(b) for t>4 µs). This is due to competition between the laser jitter and power broadening. Figure 3(c) shows maximum group delay of ~40 µs of the red-dotted circle in Fig. 3(a). The pulse broadening is due to increased interaction time in the dispersive solid medium. The longer interaction time, the longer group delay. The delay-bandwidth product in Fig. 3(c) is ~4, which is similar to those results obtained in EIT [1,2,5] or CPO [14,15].

 

Fig. 3. (a). Group delay of P as functions of power and pulse length, (b). Pulse length dependent group delay for a fixed power, (c). Ultralong (40 µs) group delay for the red circle of (a). R1=17 mW; R2=26 mW.

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Figure 4 shows a continuous group velocity control of the pulse P by adjusting the repump power of R2. The power of P is a little bit less than that used in Fig. 2. The R1 power remains the same as in Fig. 2. The R2 power is manually adjusted from maximum (~40 mW) to minimum (15 mW) by rotating a disk type of a continuously varying neutral density filter. As shown in Fig. 4, the group velocity of P can be controlled continuously by adjusting the repump power. As already discussed in Fig. 2, the group velocity change of P is due to the interacting atom number control by the repump R2.

 

Fig. 4. (Media 1) Continuous control of ultraslow group velocity of P by adjusting the repump power R2. (973 KB)

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3. Conclusion

In summary we have presented observations of self-induced ultraslow light in a rare earth Pr³⁺ doped Y₂SiO₅ crystal. The velocities obtained are comparable with those observed in EIT and CPO. We have discussed the group delay as functions of detuning, pulse length, power, and atom population or optical density. Utilizing advantages of a simple scheme for spectrally tailoring the absorbing materials and active control of the group delay by using either a separate light beam or the probe light itself, the present demonstrations have potential applications of using persistent spectral hole-burning medium to all-optical information processing such as actively controllable all-optical buffer memory. To increase the delay-bandwidth product, atom population control may be a good technique for potential applications of buffer memories, where a precursor of light can be added to burn a narrow spectral hole before input pulses to avoid the input power dependent group velocity.