## Observation of the optical components inherent in multi-wave non-collinear acousto-optical coupled states

Optics Express, Vol. 10, Issue 24, pp. 1398-1403 (2002)

http://dx.doi.org/10.1364/OE.10.001398

Acrobat PDF (322 KB)

### Abstract

Three- and four-wave spatial Bragg solitons in the form of weakly coupled states, originating with one- and two-phonon non-collinear scattering of light in anisotropic medium, are uncovered. The spatial-frequency distributions of their optical components are investigated both theoretically and experimentally.

© 2002 Optical Society of America

## 1. Introduction

## 2. One- and two-phonon scattering of light in a uniaxial crystal

^{5}times less than the number of phonons injected into a medium, where the powers of the incident light and ultrasound are close to each other. In this case, the amplitude of the acoustic wave is governed by a homogeneous wave equation, and it is agreed that the regime of so-called weak coupling takes place. Let us assume that the area of propagation for the acoustic wave, traveling almost perpendicularly to the light beams, is bounded by two planes x = 0 and x = L in a uniaxial crystal, and take into account both angular and frequency mismatches in the wave vectors. Usually, the Bragg acousto-optical process includes three waves, incident and scattered light modes as well as an acoustic mode, and incorporates conserving both the energy and the momentum for each partial act of a one-phonon scattering. However, under certain conditions, i.e. at a set of the angles of light incidence on selected crystal cuts and at fixed frequency of acoustic wave, one can observe Bragg scattering of the light caused by participating two phonons. The conservation laws are given by (ω

_{1}= ω

_{0}+ Ω , k→

_{1}= k→

_{0}+ K→ , ω

_{2}= ω

_{0}+ 2Ω , and k→

_{2}= k→

_{0}+ 2K→ simultaneously (ω

_{m}, k→

_{m}and Ω, K→ are the frequencies and wave vectors of light and acoustic waves, m = 0,1,2). Such a four-wave process occurs at the frequency of the acoustic wave, peculiar to just a two-phonon scattering, which can be determined from Ω

_{0}= 2πλ

^{-1}v ∣

^{1/2}, here n ≠ n

_{1}are the refractive indices of uniaxial crystal, v is the ultrasound velocity, λ is the incident light wavelength. The polarization of light in the zero-th and the second orders is orthogonal to the polarization in the first order, whereas the frequencies of light beams in the first and second orders are shifted by Ω

_{0}and 2Ω

_{0}, respectively, with respect to the zero-th order.

## 3. Three-wave non-collinear acousto-optical weakly coupled states

_{0}(x) and C

_{1}(x) of light waves, describing a one-phonon Bragg non-collinear acousto-optical interaction, is governed by the well-known set of combined equations [6,7]

_{0,1}as q

_{0}≈ q

_{1}≈q = 2πΔn/(λcosθ). The amplitude Δn of varying the refractive indices due to the action of a continuous-wave ultrasound reflects the photo-elastic properties of the crystal and includes the amplitude of acoustic wave; θ is the angle of incidence for the wave C

_{0}. The parameter η = k

_{0,x}-k

_{1,x}represents the joint angular-frequency mismatch. Using the boundary conditions ∣ C

_{0}(x = 0) ∣

^{2}= 1, C

_{1}(x = 0) = 0, and the conservation law ∣ C

_{0}∣

^{2}+ ∣C

_{1}∣ = 1, resulting from Eqs. (1), we write the solutions to Eqs. (1) in terms of the light intensities

_{0}∣

^{2}is independent of the coordinate x and exhibits the contribution of some background, while the second one, representing the oscillating portion of the solution, describes localizing the incident light field. The oscillating portion of the field is imposed on the background, whereby the existence of the joint angular-frequency mismatch η and its absolute value determine the level of that background. The intensity of the scattered light contains the only oscillating portion of a field. Applying the additional condition of localizing the scattered light within the spatial interval L of interaction, we find from Eqs. (2) that

_{1}∣

^{2}will be nonzero only in the spatial interval occupied by elastic wave and, therefore, the envelope of scattered light wave will be localized; i.e. the distribution of ∣C

_{1}∣

^{2}over the transverse extent of elastic wave has n partial peaks in its envelope, while the intensity ∣C

_{1}∣

^{0}has n holes. Here, we have assumed that the transverse distribution of acoustic power density is rather uniform in behavior. When all these phenomena take place, we may say that a n -pulse weakly coupled state is shaped with a three-wave non-collinear acousto-optical interaction. The efficiency ξ, of localization can be obtained from Eqs. (2): ξ, = q

^{2}(q

^{2}+η

^{2})

^{-1}. It depends on the acoustic power density P ~q

^{2}and the joint mismatch η. With η ≠ 0, one can shape the non-collinear weakly coupled acousto-optical states at a low level of the power density P, but the efficiency I of localization will be less than unity. The two-dimensional plots for the spatial-frequency distributions of the optical components in a two-pulse three-wave coupled state are presented in Fig. 1.

## 4. Shaping four-wave non-collinear acousto-optical weakly coupled states via two-phonon light scattering

_{m}(x) of light waves (m = 0,1,2), with stationary two-phonon light scattering in Bragg regime is given by [6,7]

_{m}=k

_{m,x}-k

_{m+1,x}, explained in terms of x-components for the light wave vectors, represent the joint angular-frequency mismatches. The factor q describes both the material properties and the acoustic power density and it is set equal to a constant. We analyze Eqs. (3) with the simplest boundary conditions ∣ C

_{0}(x = 0)∣

^{2}= I

^{2}, C

_{l,2}(x = 0) = 0 and exploit the conservation law ∣C

_{0}∣

^{2}+∣C

_{1}∣

^{2}+∣C

_{2}∣

^{2}=I

^{2}, resulting from Eqs. (3), where I

^{2}is the intensity of continuous-wave incident light. The exact solutions to Eqs. (3) in this regime can be written as

_{m}are real roots of the cubic algebraic equation a

^{3}-(η

_{0}+η)a

^{2}- (2q

^{2}-ηη

_{0})a + q

^{2}η = 0 and η = η

_{0}+η

_{1}. As follows from Eqs. (4–6), the intensities ∣C

_{m}(x)∣

^{2}are periodic in x, so such values x

_{n}exist that ∣C

_{0}(x

_{n}=0)∣

^{2}=I

^{2}, C

_{l,2}(x

_{n}= 0) = 0. Thus, the intensities of scattered waves are zero outside the area occupied by the acoustic wave, i.e. the effect of localization occurs for the scattered light. Inside this area, the spatial distributions of the scattered waves contain a number of peaks, and simultaneously the distribution of incident light has holes at the same positions. If η

_{0}= η

_{1}= 0, we find

_{n}) has the form of x

_{n}q = πn√2 . Now we assume the precise angular alignment and expend η

_{0}and η

_{1}into a power series only in terms of the frequency detuning Δf = ∣f - f

_{0}∣ for the current frequency f relative to the frequency f

_{0}. In the second order approximation the diagram of wave vectors gives us η

_{0}≈ πλ

^{-2}(Δf)

^{2}and η

_{1}≈ πλ,

^{-2}(4f

_{0}(Δf) + 7 (Δf)

^{2}). Therefore, in the first order approximation, we may put η

_{0}≈ 0 and η ≈ η

_{1}∝ Δf. A graphic example of the spatial distributions for light intensities inside the rectangular acoustic pulse for this case is presented in Fig. 2. It is seen that well localized and uniform distribution of light with η

_{1}= 0 becomes to be broken due to the increasing mismatch η

_{1}, i.e. the detuning Δf, or converted into other states of localization. These plots demonstrate various opportunities for shaping multi-pulse four-wave weakly coupled states via two-phonon light scattering and for observing their optical components.

## 5. Experimental data

_{0}=63.3 MHz, v=0.6 mm/μs) on a light wavelength of λ= 633 nm and λ = 488 nm, respectively. The Bragg scattering of circularly polarized light, which gives the maximum efficiency of interaction, was performed without any effect on the acoustic wave. The incident light power was about 1 W, while the acoustic pulse power was in excess of 3.3 W. Acoustic pulses had the rectangular shape, whose width was varied to control stability of the states formed. The intensities of the optical components in four-wave coupled states have been measured as functions of the product qx and the detuning Δf on the first interval of localization (n = 1). The corresponding oscilloscope traces are presented in Figs. 4 and 5. Because L = 1.3 cm was constant, the acoustic power was varied to control the product qx .

## 6. Conclusion

9. D. E. Pelinovsky and Yu. S. Kivshar, “Stability criterion for multicomponent solitary waves,” Phys. Rev. E **62**, 8668–8676 (2000). [CrossRef]

_{1}decreases to 3.25 at Δf = 482 kHz) in agreement with Refs. [4–6].

## References and links

1. | A. P. Sukhorukov, |

2. | A. S. Shcherbakov, |

3. | A. I. Maimistov, |

4. | A. S. Shcherbakov, “Properties of solitary three-wave coupled states in a two-mode optical waveguide,” in Nonlinear Guided Waves and Their applications, OSA Technical Digest (Optical Society of America, Washington DC, 2001), pp. 100–102. |

5. | A. S. Shcherbakov, “Shaping the optical components of solitary three-wave weakly coupled states in a two-mode crystalline waveguide,” in Nonlinear Guided Waves and Their Applications, OSA Technical Digest (Optical Society of America, Washington DC, 2002), NLMD7, pp.1–3. |

6. | V. I. Balakshy, V. N. Parygin, and L. I. Chirkov, |

7. | A. Korpel, |

8. | F. Yu, |

9. | D. E. Pelinovsky and Yu. S. Kivshar, “Stability criterion for multicomponent solitary waves,” Phys. Rev. E |

**OCIS Codes**

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: October 21, 2002

Revised Manuscript: November 19, 2002

Published: December 2, 2002

**Citation**

Alexandre Shcherbakov and A. Lopez, "Observation of the optical components inherent in multi-wave non-collinear acousto-optical coupled states," Opt. Express **10**, 1398-1403 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-24-1398

Sort: Journal | Reset

### References

- A. P. Sukhorukov, Nonlinear Wave Interactions in Optics and Radiophysics, (Nauka, Moscow, 1988).
- A. S. Shcherbakov, A three-wave interaction. Stationary coupled states, (Saint-Petersburg Technical University, St. Petersburg, 1998).
- A. I. Maimistov, Nonlinear Optical Waves, (Kluwer, Boston, 1999).
- A. S. Shcherbakov, �??Properties of solitary three-wave coupled states in a two-mode optical waveguide,�?? in Nonlinear Guided Waves and Their applications, OSA Technical Digest (Optical Society of America, Washington DC, 2001), pp. 100-102.
- A. S. Shcherbakov, �??Shaping the optical components of solitary three-wave weakly coupled states in a two-mode crystalline waveguide,�?? in Nonlinear Guided Waves and Their Applications, OSA Technical Digest (Optical Society of America, Washington DC, 2002), NLMD7, pp.1-3.
- V. I. Balakshy, V. N. Parygin, and L. I. Chirkov, Physical Principles of Acousto-Optics, (Radio i Svyaz, Moscow, 1985).
- A. Korpel, Acousto-Optics, (Marcel Dekker, New-York, 1988).
- F. Yu, Introduction to Information Optics, (Academic Press, San Diego, 2001).
- D. E. Pelinovsky and Yu. S. Kivshar, �??Stability criterion for multicomponent solitary waves,�?? Phys. Rev. E 62, 8668-8676 (2000). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.