## Experimental observation of chirped continuous pulse-train soliton solutions to the Maxwell-Bloch equations

Optics Express, Vol. 8, Issue 2, pp. 153-158 (2001)

http://dx.doi.org/10.1364/OE.8.000153

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### 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.

© Optical Society of America

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]

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]

*ϕ*(

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

*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,

*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:

*ϕ*(

*t*), given by:

*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.

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]

*l*=

*k*, or

*dϕ*/

*dt*=0, i.e., no frequency chirp. This was true, as predicted [6

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]

**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.

**Ω**. However, the situation is different with relaxation present since the Bloch vector will relax along the torque vector,

**Ω**, and after several

**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

**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}.

*ϕ*, with

*dϕ*(

*t*)/

*dt*given by:

*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.

*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*

_{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.

*αL*=5. That

*αL*=5 was confirmed by low intensity absorption measurements of the non-chirped continuous wave laser beam.

*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).

*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).

## References and links

1. | S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. |

2. | J. H. Eberly, “Optical pulse and pulse-train propagation in a resonant medium,” Phys. Rev. Lett. |

3. | M. D. Crisp, “Distortionless propagation of light through an optical medium,” Phys. Rev. Lett. |

4. | D. Dialetis, “Propagation of electromagnetic radiation through a resonant medium,” Phys. Rev. A |

5. | L. Matulic and J. H. Eberly, “Analytic study of pulse chirping in self-induced transparency,” Phys. Rev. A |

6. | M. A. Newbold and G. J. Salamo, “Effects of relaxation on coherent continuous-pulse-train propagation,” Phys. Rev. Lett. |

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. |

8. | N. Akhmediev and J. M. Soto-Crespo, “Dynamics of solitonlike pulse propagation in birefringent optical fibers,” Phys. Rev. E |

**OCIS Codes**

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(190.5940) Nonlinear optics : Self-action effects

**ToC Category:**

Focus Issue: Quantum control of photons and matter

**History**

Original Manuscript: November 22, 2000

Published: January 15, 2001

**Citation**

Shihadeh Saadeh, John Shultz, and Gregory Salamo, "Experimental observation of chirped continuous pulse-train soliton solutions to the Maxwell-Bloch equations," Opt. Express **8**, 153-158 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-2-153

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### References

- S. L. McCall and E. L. Hahn, "Self-induced transparency," Phys. Rev. 183, 457-485 (1969). [CrossRef]
- J. H. Eberly, "Optical pulse and pulse-train propagation in a resonant medium," Phys. Rev. Lett. 22, 760-762 (1969). [CrossRef]
- M. D. Crisp, "Distortionless propagation of light through an optical medium," Phys. Rev. Lett. 22, 820-823 (1969). [CrossRef]
- D. Dialetis, "Propagation of electromagnetic radiation through a resonant medium," Phys. Rev. A 2, 1065-1075 (1970). [CrossRef]
- L. Matulic and J. H. Eberly, "Analytic study of pulse chirping in self-induced transparency," Phys. Rev. A 6, 822-836 (1972). [CrossRef]
- M. A. Newbold and G. J. Salamo, "Effects of relaxation on coherent continuous-pulse-train propagation," Phys. Rev. Lett. 42, 887-890 (1979). [CrossRef]
- 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]
- N. Akhmediev and J. M. Soto-Crespo, "Dynamics of solitonlike pulse propagation in birefringent optical fibers," Phys.Rev.E 49, 5742-5754 (1994). [CrossRef]

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