## Abstract

A frequency chirped continuous wave laser beam incident upon a resonant, two-level atomic absorber is seen to evolve into a Jacobi elliptic pulse-train solution to the Maxwell-Bloch equations. Experimental pulse-train envelopes are found in good agreement with numerical and analytical predictions.

©2001 Optical Society of America

Coherent propagation of trains of optical pulses through a two-level absorber has been investigated, from a theoretical viewpoint, by many researchers. These investigations have been based on the self-induced transparency (SIT) equations [1], both with [2–5], and without [6], the assumption of zero relaxation. Recently, theoretical predictions have been supported by experimental observations of the Jacobi elliptic dn solution [7]. In particular, experiments demonstrated the evolution of an arbitrarily shaped input optical pulse train into the analytic shape-preserving Jacobi elliptic pulse train solution to the Maxwell-Bloch equations [8]. A special feature of the observed solution was that the chirp in the optical frequency of the pulse train was zero. In this paper we report the experimental and numerical demonstration of the evolution of a continuous wave laser beam into an analytic shape-preserving pulse train solution. In this case, the analytic solution to the Maxwell-Bloch equations is a Jacobi elliptic function with a nonzero frequency chirp.

The interaction of a plane-wave optical field with an inhomogeneously broadened two-level absorber can be described by two sets of equations. The Bloch equations describe the effect of the optical field on the atom while the reduced Maxwell equations describe the effect of the atom on the optical field. For significant absorption, self-consistent solutions to both equations describe the propagation of the optical field through the two-level absorber. Together, the two sets of equations are known as the reduced Maxwell-Bloch equations.

For circularly polarized light traveling in the z-direction, the electric field at the position of the atom is:

where *ϕ*(*t*), the phase of the electromagnetic wave, is a function of time. The instantaneous field frequency may be defined as:

The Bloch equations are given by:

$$\frac{\partial v}{\partial t}=\left(\Delta \omega -\dot{\varphi}\right)u+{E}_{1}w-\frac{v}{{T}_{2}^{\prime}}$$

$$\frac{\partial w}{\partial t}=-{E}_{1}v+{E}_{2}u-\frac{\left(w+1\right)}{{T}_{1}}$$

while the reshaping of the input field is determined by the reduced Maxwell equations:

$$\frac{\partial {E}_{2}}{\partial z}=-\frac{\alpha}{2\pi g\left(0\right)}\underset{-\infty}{\overset{\infty}{\int}}u(t,z,\Delta \omega )g\left(\Delta \omega \right)d\Delta \omega $$

where *u* and *v* are the in-phase and in-quadrature components of the polarization, respectively, *w* is the atomic inversion, *E*_{1}
=*κE*_{0}
cos*ϕ*, *E*_{2}
=*κE*_{0}
sin*ϕ*,with *κ* defined as *2p*/*ħ* where *p* is the transition dipolemoment, ${T}_{\mathit{2}}^{\prime}$ is the relaxation rate of *u* and *v*, *T*_{1}
is the relaxation rate of *w*, *g*(*Δω*) is the line shape function where *Δω*=*ω*_{0}
-*ω* is the difference between the atomic resonance frequency and the frequency of the optical field. The field envelope pulse-train solution to these equations, in the limit of weak relaxation, is given by:

with the phase, *ϕ*(*t*), given by:

where *k*, *l*, and *τ* are constants with *k* being the modulus of the Jacobi elliptic sn. The full range of solutions can be found by allowing 0≤*k*
^{2}≤*l*
^{2}≤1.

In our earlier work [7], we observed that an input train of optical pulses, with rather arbitrary input envelope shape, evolves through the interaction with a two-level absorber to the analytical, shape-preserving, pulse-train Jacobi dn elliptic solution with *l*=*k*, or *dϕ*/*dt*=0, i.e., no frequency chirp. This was true, as predicted [6], even though the optical pulse-train had been allowed to come to equilibrium for a time much longer than ${T}_{\mathit{2}}^{\prime}$. Intuitively, this can be understood by observing the behavior of the Bloch equations in a pulse-train driving field. In this case, the Bloch equations, without relaxation, can be written as:

where **P** is the Bloch vector defined with components *u*, *v*, and *w*, while Ω is defined with components {-*E*_{1}
, -*E*_{2}
, (*Δω*-*dϕ*/*dt*)}. In Figure 1, the motion of the Bloch vector is depicted.

If the driving optical field is constant, the Bloch vector will simply precess about the torque vector, **Ω**. However, the situation is different with relaxation present since the Bloch vector will relax along the torque vector, **Ω**, and after several ${T}_{\mathit{2}}^{\prime}$, **P** will be aligned along **Ω** and no further oscillations will occur. With a pulse-train, **Ω** is a periodic function of time with a period shorter than ${T}_{\mathit{2}}^{\prime}$. As a result, **P** can never relax along **Ω**, since **Ω** is moving too fast. This understanding of the pulse train experiments suggested that similar results would be possible by making **Ω** time dependent by modulating (*Δω*-*dϕ*/*dt*) with time as opposed to the amplitude of the field, *E*_{0}
.

As a result, in this study we modulated the phase of the driving electric field, *ϕ*, with *dϕ*(*t*)/*dt* given by:

where *ϕ*_{0}
is a constant that determines the amplitude of the phase modulation, i.e., the frequency shift from the atomic resonance frequency, and *δ* is the frequency of the phase modulation.

The experimental apparatus is shown in Figure 2. An argon ion laser was used to pump a single-longitudinal-mode, tunable, 1MHz linewidth, continuous-wave dye laser using Rhodamine 590 as the gain medium. The output of the continuous wave dye laser was chirped by using a fast electro-optic crystal to introduce a *ϕ*(*t*) onto the optical beam. The chirped continuous wave laser light was directed through a quarter-wave plate to produce (σ^{+}) circularly polarized light and focused by a lens (*L*_{1}
) into the sodium cell, which was housed in an oven between the pole faces of a magnet. The magnetic field could be varied from 0 to 10 kG and was used to resolve and tune the sodium mj transitions. For our experiment we excited the ^{2}S_{1/2} (m_{j}=1/2) to ^{2}P_{3/2} (m_{j}=3/2) transition. This transition has the advantage that the ^{2}P_{3/2} (m_{j}=3/2) excited state can only decay back to the ^{2}S_{1/2} (m_{j}=1/2) ground state thereby avoiding any complications due to optical pumping. For this transition ${T}_{\mathit{2}}^{\prime}$ is 32 ns, while *T*_{1}
is 16 ns. Another lens (*L*_{2}
) was used between the sodium cell and a high speed detector to image the output laser beam from the sodium cell onto an aperture (*A*_{1}
) with a magnification of 2.45. This was done in order to select only the uniform plane-wave region of the output signal for observation. This is important since comparison between theory and experiment will be limited to the uniform plane wave region. A third lens (*L*_{3}
) was used to focus the output from the aperture onto the fast detector. The mode structure of the dye laser was continuously monitored by two Fabry-Perot interferometers to insure single mode operation and to allow the laser to be tuned on or off the sodium absorption line.

In thiswork, the frequency of the power amplifier which was used to drive the phase modulator was 31.25 MHz. This is, therefore, the frequency of the phase modulation of our laser light. It dictates the cycle time of the phase modulation to be 1/f=32 nanoseconds. The applied voltage on the phase modulator determines the amplitude of the phase modulation. Then, by applying a magnetic field the sodium absorption line is brought on resonance with the laser beam. The sodium cell was kept at a temperature of 190° C which yields a Beer’s absorption length, *αL*=5. That *αL*=5 was confirmed by low intensity absorption measurements of the non-chirped continuous wave laser beam.

Figure 3 shows the input continuous wave laser light signal incident on the sodium cell and the intensity of the laser beam coming through the sodium cell while a phase modulation with a frequency of 31.25 MHz and amplitude of only 20 MHz is applied to the laser beam. These curves show a modulation in the intensity of the laser beam corresponding to a period of 16 ns, or a frequency of 62.5MHz, which is twice the frequency of modulation of the laser beam and equal to *T*_{1}
. For this case, the amplitude of the phase modulation was low so that the Bloch vector was not significantly perturbed from its otherwise equilibrium position reached if *dϕ*/*dt* is zero. As a result, even with an *αL*=5, not much reshaping takes place. The envelope shape has not yet evolved to a Jacobi elliptic function and attempts to fit the data to a Jacobi elliptic function were unsuccessful. However, the modulated solid line shows that the experimental observation is nevertheless in good agreement with the numerical solution of Eqns. (3) and (4).

Figure 4 shows the output when the amplitude of phase modulation is increased to 30 MHz. In this case *ϕ*_{0}
is the order of *κE*_{0}
so that the movement of the Bloch vector and the corresponding reshaping is sufficient to forma Jacobi elliptic envelope. The experimental observation is in good agreement with both the Jacobi elliptic analytic solution (modulated solid line) and the numerical prediction (slightly better fit to the data but not shown here) of Eqns. (3) and (4).

The analogy between varying the phase and amplitude of the optical field also suggested that by increasing the magnitude of the phase variation it should be possible to drive the atoms through two Rabi oscillations in one period.

Figure 5 shows the intensity of the laser beam coming through the sodium cell (dotted line), while a phase modulation with frequency of 31.25 MHz and a higher amplitude of 50 MHz is applied to the laser beam. In this case, we observe the expected pulse break-up behavior characteristic of 4π or higher area self-induced transparency.

In conclusion, we report the observation of the evolution of a chirped continuous wave laser beam, interacting with a two-level absorber, into a Jacobi elliptic pulse-train solution to the Maxwell-Bloch equations. The evolved envelope is shown to be in good agreement with both analytic Jacobi elliptic functions and with numerical solutions of the Maxwell-Bloch equations.

## References and links

**1. **S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. **183**, 457–485 (1969). [CrossRef]

**2. **J. H. Eberly, “Optical pulse and pulse-train propagation in a resonant medium,” Phys. Rev. Lett. **22**, 760–762 (1969). [CrossRef]

**3. **M. D. Crisp, “Distortionless propagation of light through an optical medium,” Phys. Rev. Lett. **22**, 820–823 (1969). [CrossRef]

**4. **D. Dialetis, “Propagation of electromagnetic radiation through a resonant medium,” Phys. Rev. A **2**, 1065–1075 (1970). [CrossRef]

**5. **L. Matulic and J. H. Eberly, “Analytic study of pulse chirping in self-induced transparency,” Phys. Rev. A **6**, 822–836 (1972). [CrossRef]

**6. **M. A. Newbold and G. J. Salamo, “Effects of relaxation on coherent continuous-pulse-train propagation,” Phys. Rev. Lett. **42**, 887–890 (1979). [CrossRef]

**7. **J. L. Shultz and G. J. Salamo, “Experimental observation of the continuous pulse-train soliton solution to the Maxwell-Bloch equations,” Phys. Rev. Lett. **78**, 855–858 (1997). [CrossRef]

**8. **N. Akhmediev and J. M. Soto-Crespo, “Dynamics of solitonlike pulse propagation in birefringent optical fibers,” Phys. Rev. E **49**, 5742–5754 (1994). [CrossRef]