ISSN 1054-660X, Laser Physics, 2007, Vol. 17, No. 2, pp. 148–151.
NONLINEAR OPTICS AND SPECTROSCOPY
© MAIK “Nauka /Interperiodica” (Russia), 2007. Original Text © Astro, Ltd., 2007.
Temporal Compression of Nanosecond Laser Pulses Using Coherent Control R. Buffaa, S. Cavalierib, L. Finib, E. Salib, *, and M. V. Tognettic a Dipartimento
di Fisica, Universita di Siena, Via Roma 56, I-53100 Siena, Italy di Fisica, and European Laboratory for Nonlinear Spectroscopy, Università di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy c CLOQ/Departamento de Física, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
b Dipartimento
*e-mail:
[email protected] Received September 22, 2006
Abstract—We present a study of the temporal compression of nanosecond laser pulses resulting from the coherent control peculiarities of the propagation dynamics in a regime of electromagnetically induced transparency. We describe the general theoretical framework and discuss the crucial conditions required in order to experimentally realize such a temporal compression scheme. A proof-of-concept experimental realization of a scheme of this type in a sample of hot sodium vapors is currently being implemented: we describe in detail the experimental setup designed for this purpose. PACS numbers: 42.50.Gy, 42.65.Re DOI: 10.1134/S1054660X0702017X
1. INTRODUCTION In the last decade, there has been a great deal of interest in the optical properties of coherently prepared atomic media. Particular attention has been given to electromagnetically induced transparency (EIT) and its use in the control of propagation of light [1–5]. More specifically, as far as this work is concerned, experimental evidence of the possibility to control the temporal shape of laser fields in the visible spectral region and in the microsecond temporal regime has been reported in cold atoms by Chien Liu et al. [6]. A theoretical study which discusses and explains how to exploit the peculiarities of EIT propagation dynamics to achieve coherent control of the temporal shaping and compression of laser pulses has also been recently published [7]. So far, this appears to be a unique technique in the far vacuum ultraviolet (VUV) or even extreme ultraviolet (XUV) spectral regions and may provide an important tool for nonlinear optics applications at very short wavelengths. In this paper, we present a study in which the main specific conditions required for an experimental realization of pulse compression are highlighted. The realization of a first proof-of-principle experiment in hot atoms and in the visible spectral range is under way in our laboratories at present. We describe in detail the experimental setup which has been designed and assembled in order to realize the experiment.
field envelopes Ep (probe) and Ec (coupling) and frequencies ωp = 2πc/ λp and ωc = 2πc/ λc, resonant with the atomic transitions 1–2 and 2–3, respectively. Figure 2 shows in a heuristic way the idea at the basis of the proposed experiment. The laser pulses, propagating along the z axis, enter the cell of length L containing the medium of three-level atoms with the temporal overlapping shown in (a). The probe pulse
2. THEORETICAL FRAMEWORK Figure 1 shows a schematic diagram of the physical system at the basis of our study: a three-level sodium atom in interaction with two laser pulses of electric148
|3〉
3d (2D) J = 3/2 λc = 818.5 nm 3p (2P) J = 1/2 |2〉
λp = 589.7 nm |1〉
3s (2S) J = 1/2
Fig. 1. Schematic diagram of the three-level ladder-scheme in sodium at the basis of the HIT scheme under investigation.
In a rigorous approach, the propagation equations of the electric-field envelopes Ep and Ec are written as ω p d 12 ∂ 1 ∂⎞ ⎛ ---- 〈 ρ ( v )〉 v , - + --- ----- E p = – iN -----------⎝ ∂z c ∂t⎠ 2ε 0 c 12 ω c d 23 ∂ 1 ∂⎞ ⎛ ---- 〈 ρ ( v )〉 v . - + --- ----- E = – iN -----------⎝ ∂z c ∂t⎠ c 2ε 0 c 23
2.5
2.5
2.0
2.0
Probe field Ep /Ep0
experiences EIT, and its propagation velocity vp slows down to a value vp c [1–3]. After some propagation in the cell, the coupling pulse overlaps the probe pulse as shown in (b). Under these conditions, different “points” of the probe pulse experience different values of Ec and “travel” with different “propagation velocity,” giving rise to a temporal reshaping of the probe pulse.
1.5
Here, N is the density of the atomic sample, d12 and d23 are the electric-dipole moments of the transitions 1–2 and 2–3, respectively, and the coherences ρnm(v ) are averaged over the velocity distribution fD(v ):
1.5
b
1.0
1.0 0.5
(1)
149
0
0.5
a –4
–2 0 2 Time (t – z/c), arb. units
4
Coupling field Ec /Ec0
TEMPORAL COMPRESSION OF NANOSECOND LASER PULSES
0
Fig. 2. The idea at the basis of the temporal shape control. In the framework moving at velocity c, the probe pulse (continuous line) “slips” under the coupling pulse (dashed line) inside the cell containing the EIT modified medium. Different “points” of the probe pulse “travel” at different “propagation velocity” resulting in a reshaping of the probe pulse temporal profile.
+∞
∫ρ
〈 ρ nm ( v )〉 v =
nm ( v )
f D ( v ) dv .
(2)
–∞
For a weak probe laser pulse (ρ11 ≈ 1), the coherences ρnm(v ) that appear in Eqs. (1) and (2) satisfy the following Liouville equation: ρ˙ 12 = – iΩ p – iΩ c ρ 13 – ( i∆ p + γ 12 )ρ 12 , ρ˙ 13 = – iΩ c ρ 12 – [ i ( ∆ p + ∆ c ) + γ 13 ]ρ 13 ,
(3)
ρ˙ 23 = 0, where Ωp = d12Ep /2 and Ωc = d23Ec /2 are the Rabi couplings, ∆p and ∆c are detunings from resonance, γnm represent all kind of dephasing rates, and the explicit dependence on the atomic velocity v has been dropped. In a sample of hot atoms, the detunings depend on the v velocity v of the atom as ∆p = (ω2 – ω1) ---- and ∆c = c v (ω3 – ω2) ---- . Equations (1)–(3) can be numerically c solved to provide Ep at the cell output (z = L). In [7] it is shown that once the conditions γ 12 /Ω c ,
γ 13 /Ω c ,
δω D /Ω c ,
δω p /Ω c ,
δω c /Ω c 1
(4)
are fulfilled (δωp and δωc are the spectral widths of Ωp and Ωc, and δωD is the generic Doppler broadening), then a straightforward analytical expression for Ep can be provided. In particular, if the probe field Ep is overlapped, at cell input and output, by flat regions of the LASER PHYSICS
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coupling field of value Ec0 at z = 0 and nEc0 at z = L (see Fig. 2), then 2
E p ( L, t ) = nE p [ 0, n ( t – t L ) ] 2
× exp { – γ 13 [ n ( t L – t ) + t – L/c ] },
(5)
where tL is the arrival time of the probe field [8]. For a negligible dephasing rate γ13, Eq. (5) shows how the probe field at the cell output Ep(L, t), while preserving its functional shape, is temporally compressed by a factor n2 and its amplitude amplified by a factor n. However, for a nonnegligible dephasing rate γ13, Eq. (5) shows an absorption of the probe peak intensity given by I p ( L, t L ) 1 – ----------------------- = 1 – exp ( – 2γ 13 ∆t ), 2 n I p ( 0, 0 )
(6)
where ∆t = tL – L/c is the temporal delay accumulated by the probe pulse with respect to the coupling pulse. It is important to note that, once approximations (4) are fulfilled, the total absorption of the probe pulse is dependent on γ13 through the temporal characteristics of the laser pulses, and it turns out to be negligible only when γ13∆t 1. Numerical integration of Eqs. (1)–(3) has been performed in the case of the sodium ladder scheme discussed in the next section and represented in Fig. 1 by using the temporal injection scheme shown in Fig. 3. The calculation was carried out using the following values for the relevant parameters: NL = 2 × 1015 cm–2; coupling field intensity at the initial time Ic0 = 8.3 KW/cm2 (Rabi frequency ΩC0 = 3.52 × 1010 rad/s); (2γ12)–1 = 16 ns; (2γ13)–1 = 22 ns; Tp = 4 ns; Tc = 4 ns.
150
BUFFA et al.
Intensity, a.u. 5 4 3 2 1 0
5
10
15
20
25 Time, ns
Fig. 3. Numerical simulation for the case of the sodium scheme represented in Fig. 1 and described in the text. Dashed curve: coupling field. Solid curves: probe field at the entrance of the cell (left-hand side) and at the cell output (right-hand side). Calculation parameters: NL = 2 × 1015 cm–2; coupling field intensity at the initial time Ic0 = 8.3 KW/cm2 (Rabi frequency ΩC0 = 3.52 × 1010 rad/s); (2γ12)–1 = 16 ns; (2γ13)–1 = 22 ns; Tp = 4 ns; Tc = 4 ns.
The numerical integration of Eqs. (1)–(3) in the presence of Doppler broadening was a prohibitively timeconsuming task for “ordinary” computing capabilities; therefore, it was performed using a parallel computing facility [9]. The numerical results, shown in Fig. 3, exactly reproduce the analytical solution given by (5), predicting in this case a transmission near 65%. 3. THE EXPERIMENT In this section, we describe the experimental scheme whose implementation is under way in our laboratories at the present, with the aim of demonstrating the theoretical scheme presented in the previous sections. Although the most interesting application of our theoretical scheme would be in the VUV, we envisage a first proof-of-principle experimental realization in the visible–near infrared (VIS–NIR) spectral region, due to the fact that this solution is easier to implement. We identify sodium as a convenient sample, due to the relative simplicity in handling it and to the fact that it presents convenient transitions that we can exploit with our experimental capabilities. The atomic energy levels involved and the corresponding laser wavelengths are shown in Fig. 1. As the 1–2 transition, we will use the 3S–3P transition at λp = 589.7 nm (probe field), while the 3P–3D transition at λc = 818.5 nm (coupling field) will be the 2–3 transition. A lambdatype schema allows the use of a metastable final state. This may lead to a coherence ρ13 with a small dephasing rate γ13, which turns out to be a crucial parameter
when the goal is to attain ultraslow light propagation. However, when the goal is to control the shape of VUV probe pulses by using a coupling field which is still in the visible spectral region, the use of a metastable final state may not be possible, and the use of a ladder scheme could be necessary. Having the final level as the highest energy level implies a larger value of γ13, which is less convenient for the realization of the experiment (as explained in the previous section). On the other hand, this scheme is closer to the configuration that one would use in a pulse compression experiment in the VUV spectral region and hence is also particularly interesting for us to investigate. The complete experimental setup envisaged for the proposed experiment is shown in Fig. 4. The probe field is provided by a Quantel Dye laser (pumped by a frequency-doubled Nd:YAG laser at λ = 532 nm), which delivers pulses with energy of approximately 10 mJ, wavelength tunable from 550 to 740 nm, 10 Hz repetition rate, and a bandwidth of δωp/2π ≈ 2.5 GHz (FWHM). As far as the coupling laser is concerned, the solution that was implemented is a single-mode injection-seeded titanium-doped sapphire (Ti:S) ring laser [10]. The Ti:S laser can be conveniently pumped by a fraction of the same Nd:YAG laser that pumps the Dye laser. It also offers a wide tuning range in the NIR, given by the very broad transition of the titanium-doped sapphire. In particular, it can be efficiently operated at λc = 818.5 nm, which is the wavelength we need for the coupling field. The temporal duration of the generated pulses essentially corresponds to the photon lifetime in the cavity if the pump duration is short enough to be considered instantaneous (this is our case, as the pump pulse duration is 5 ns). The output pulse duration is therefore controllable to a certain extent, for example, by changing the length of the cavity or the reflectivity of the output coupler. This system was assembled; with a cavity length of 85 cm and an output coupler with a reflectivity R = 85%, it delivers pulses with an energy exceeding 10 mJ and a duration of approximately 30 ns. The theory described above requires a smooth coupling field and hence single-mode operation for the coupling laser. In our Ti:S laser, this condition will be satisfied by employing the injection-seeded ring geometry of the laser cavity. Proper matching of the cavity length to the wavelength of the injected seed is ensured by a feedback-control system employing a piezoelectric [11]. The theory presented in the above section requires, for an analytical solution, the condition that the coupling field overlapped with the probe pulse upon entering the cell be constant and different from the coupling field overlapped upon exiting the cell. Essentially, the temporal behavior of the coupling field should feature two flat regions at different amplitudes and an interval of (ideally fast) rise time, as presented in Figs. 2 and 3. Also, the rise time between the two flat regions should LASER PHYSICS
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TEMPORAL COMPRESSION OF NANOSECOND LASER PULSES
Nd:YAG Pump 532 nm 10 Hz, 5 ns 200 mJ
Controllable delay
Beam splitter Tunable Dye laser
589 nm, 3 ns (multi mode)
Ti:Sapphire Ring cavity ECDL (CW) 818 nm, 100 mW Sindle-mode See ding
818 nm 1 mJ, 50 ns Single-mode
Dichroic mirror
t
151
ns pulse shaper
Dichroic mirror
Sodium vapours cell
Detection (fast photodiode and oscilloscope) Fig. 4. Complete experimental layout.
be as short as possible in order to minimize the necessary propagation distance and, as a consequence, the absorption factor (see discussion above). In order to achieve this result, we use a combination of Pockels cells and polarizers as an “electronic pulse shaper.” Conditions (4)—required by the discussion of the previous section—impose a lower limit on the coupling pulse intensity. In the presented scheme with sodium, the most stringent conditions are those on δωp /Ωc (with our laser, δωp /2π ≈ 2.5 GHz) and δωD /Ωc (δωD/2π ≈ 3.1 GHz for a sample at a temperature of T = 300°C). The limit imposed by the Doppler broadening can be reduced by employing a counter-propagating geometry. However, this is not strictly necessary, as a value of Ωc /2π of the order of 10 GHz will be sufficient to fulfill conditions (4) and an energy of less than 1 mJ in the coupling pulse will be enough for this purpose. 4. CONCLUSIONS In this paper, we presented a study in which the use of a time-dependent coupling pulse, modifying the complex optical properties of an atomic medium, allows us to coherently control the temporal duration of a probe laser pulse. The effect of a nonnegligible dephasing rate γ13 on the two-photon coherence ρ13 was discussed in detail, and straightforward analytical results have been presented. We showed that the total absorption of the probe pulse upon propagation along the cell is dependent on γ13 through the temporal characteristics of the laser pulses. Numerical calculations confirm the validity of this analysis.
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The main specific conditions required for an experimental realization of pulse compression were highlighted, and a first proof-of-principle experiment— under way in our laboratories now—in a sample of hot Na atoms and in the visible spectral region was discussed. The details of the experimental setup were described and discussed. ACKNOWLEDGMENTS M.V.T. acknowledges the HPC Europe Visitor Program for financial and technical support and thanks Dr. Andrea Tarsi for kind technical assistance. REFERENCES 1. S. E. Harris, Phys. Today 50 (7), 36 (1997). 2. J. P. Marangos, J. Mod. Opt. 45, 471 (1998). 3. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, Rev. Mod. Phys. 77, 633 (2005). 4. P. W. Milonni, J. Phys. B: At. Mol. Opt. Phys. 35, R31 (2002) 5. D. F. Phillips, A. Fleischhauer, A. Mair, et al., Phys. Rev. Lett. 86, 783 (2001). 6. Chien Liu, Z. Dutton, C.H. Behroozi, and M. V. Hau, Nature 409, 490 (2001). 7. R. Buffa, S. Cavalieri and M. V. Tognetti, Phys. Rev. A 69, 033815 (2004). 8. R. Buffa, S. Cavalieri, L. Fini, et al., Opt. Comm. 264, 471 (2006). 9. http://www.hpc-europa.org/. 10. T. D. Raymond and A.V. Smith, Opt. Lett. 16, 33 (1991). 11. C. Hamilton, Opt. Lett. 17, 728 (1992).
