Direct detection of optical phase conjugation in a colloidal medium

Direct detection of optical phaseconjugation in a colloidal medium

Carlos Lopez-Mariscal1∗, Julio C. Guti errez-Vega1,David McGloin2 and Kishan Dholakia2

1Photonics and Mathematical Optics Group, Tecnologico de Monterrey, Mexico 648492School of Physics and Astronomy, The University of St Andrews, KY16 9SS, Scotland

[email protected]

Abstract: Degenerate four-wave mixing is demonstrated using an artificialKerr medium and is evidenced by directly observing the phaseconjugationof a vortex signal beam. The nonlinear susceptibility is produced by arefractive index grating created in a suspension of dielectric microscopicparticles optically confined in the intensity grating distribution of twointerfering laser beams.

© 2007 Optical Society of America

OCIS codes: (140.7010) Trapping; (170.4520) Optical confinement and manipulation;(190.3970) Microparticle nonlinear optics; (190.5040) Phase conjugation.

References and links1. R. W. Hellwarth, “Generation of time-reversed wave frontsby nonlinear refraction,” J. Opt. Soc. Am.67, 1-3

(1977).2. D. M. Bloom and G. C. Bjorklund, “Conjugate wave-front generation and image reconstruction by four-wave

mixing,” Appl. Phys. Lett.31, 592-594 (1977).3. A. Yariv and D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave

mixing,” Opt. Lett.1, 16-18 (1977).4. S. M. Jensen and R. W. Hellwarth, “Observation of the time-reversed replica of a monochromatic optical wave,”

Appl. Phys. Lett.32, 166-168 (1978).5. M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave

mixing and phase conjugation,” Opt. Lett.7, 313-315 (1982).6. P. F. Liao, D. M. Bloom, and N. P. Economou, “CW optical wave-front conjugation by saturated absorption in

atomic sodium vapor,” App. Phys. Lett.32, 813-815 (1978).7. T. Mikropoulos, S. Cohen, M. Kompitsas, S. Goutis, and C. Baharis, “Phase conjugation by degenerate four-wave

mixing in barium vapor,” Opt. Lett.15, 1270-1272 (1990).8. P. F. Liao and D. M. Bloom, “Continuous-wave backward-wavegeneration by degenerate four-wave mixing in

ruby,” Opt. Lett.3, 4-6 (1978).9. E. Freysz, E. Laffon and A. Ducasse, “Phase conjugation used as a test of the local and nonlocal characteristics

of optical nonlinearities in microemulsions,” Opt. Lett.16, 1644-1646 (1991).10. D. Fekete, J. C. Auyeung, and A. Yariv, “Phase-conjugatereflection by degenerate four-wave mixing in a nematic

liquid crystal in the isotropic phase,” Opt. Lett.5, 51-53 (1980).11. J. W. R. Tabosa and D. V. Petrov, “Optical pumping of orbital angular momentum of light in cold cesium atoms,”

Phys. Rev. Lett.83, 4967-4970 (1999).12. S. Barreiro and J. W. R. Tabosa, “Generation of Light Carrying Orbital Angular Momentum via Induced Coher-

ence Grating in Cold Atoms,” Phys. Rev. Lett.90, 133001 (2003).13. S. Barreiro, J. W. R. Tabosa, J. P. Torres, Y. Deyanova, and L. Torner, “Four-wave mixing of light beams with

engineered orbital angular momentum in cold cesium atoms,” Opt. Lett.29, 1515-1517 (2004).14. P. W. Smith, A. Ashkin, and W. J. Tomlinson, “Four-wave mixing in an artificial Kerr medium,” Opt. Lett.6,

284-286 (1981).15. P. W. Smith, P. J. Maloney, and A. Ashkin, “Use of a liquid suspension of dielectric spheres as an artificial Kerr

medium,” Opt. Lett.7, 347-349 (1982).16. A. E. Neeves, M. H. Birnboim, “Polarization selective optical phase conjugation in a Kerr-like medium,” J. Opt.

Soc. Am. B.5, 701-708 (1988).

#81676 - $15.00 USD Received 30 Mar 2007; revised 3 May 2007; accepted 4 May 2007; published 7 May 2007

(C) 2007 OSA 14 May 2007 / Vol. 15, No. 10 / OPTICS EXPRESS 6330

17. R. Pizzoferrato, M. De Spirito, U. Zammit, M. Marinelli, F.Scudieri, and S. Martelluci, “Optical phase conjuga-tion through translational and rotational diffusive rearrangements of liquid-dispersed microparticles,” Phys. Rev.A 41, 2882-2885 (1990).

18. L. Allen, M. W. Bijersbergen, R. J. C. Spreeuw and J. P. Woerdman, “Orbital angular momentum of light and thetransformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).

19. A. Chowdhury, B. J. Ackerson and N. A. Clark, “Laser-induced freezing,” Phys. Rev. Lett.55, 833-836 (1985).20. N. R. Heckenberg, R. McDuff, C. P Smith, H. Rubinsztein-Dunlop and M. J. Wegener, “Laser beams with phase

singularities,” Opt. Quantum Elect.24, S951-S962 (1992)21. M. Harris, C. A. Hill, and J. M. Vaughan, “Laser modes with helical wave fronts, ” Phys. Rev. A49, 3119-3122

(1994).22. I. G. Marienko, M. S. Soskin, and M. V. Vasnetsov, “Phase conjugation of wavefronts containing phase singular-

ities,” Proc. SPIE 3487, 39-41 (1998).23. W. Jiang, Q. Chen, Y. Zhang and G.-C. Guo, “Computation of topological charges of optical vortices via nonde-

generate four-wave mixing,” Phys. Rev. A74, 043811 (2006).24. W. Loose, B.J Ackerson, “Model calculations for the analysis of scattering data from layered structures,” J. Chem.

Phys.101, 7211-7220 (1994).

1. Introduction

Four wave mixing (FWM) is a well-known nonlinear process[1, 2, 3, 4, 5], in which a dielec-tric medium with a significant nonlinear susceptibilityχ3 is simultaneously pumped by twocoherent wavesE1 andE2 and probed by a third waveE3, producing a fourth waveE4 radiatedin the opposite direction to, and phase conjugate with,E3. The particular case when the fre-quencies of all four wavesEi=1,2,3,4 are the same, is referred to asdegeneratefour-wave mixing(DFWM). The production of a fourth wave is essentially a result of the interaction between thepump electric fields and individual dipoles in the medium. The phase conjugation of the back-generated wave is in turn a consequence of the conservation of momentum when consideringthe reversal of the propagation direction of the generated wave with respect to the probe beam.To date, several different nonlinear media have been used todemonstrate FWM, such as metalvapors[6, 7], a ruby crystal[8], microemulsions[9] and liquid crystals[10].

In a more recent work, Tabosa and Petrov[11] reported the transfer of orbital angular mo-mentum (OAM) between two waves via optical pumping of an atomic sample using a vortexbeam. The observation of a conjugated-phase replica of the beam demonstrated the transferof OAM into the system and subsequently into the backgenerated beam. Barreiroet al havealso made use of an atomic coherence grating to generate the phase conjugated counterpartsof beams carrying OAM [12, 13], further demonstrating mechanisms for the transfer and theconservation of OAM in FWM experiments.

One particular instance of nonlinear medium can be artificially produced by means of orderedarrays of microscopic dielectric particles suspended in a transparent medium. The particles canbe spatially arranged in crystal-like structures making use of an optical gradient force exertedon them by a spatially varying optical field [14, 15, 16]. Similar effects can also be achievedin non-isotropic media[17]. The seminal work by Smithet al[14, 15] clearly demonstrated thatthe backward generated beam from such a colloidal crystal was due to FWM, but did not lookinto its phase conjugation.

In this paper, we present the direct observation of phase conjugation of a coherent opticalwavefield via DFWM following the work of Tabosa and Barreiro[11, 12, 13], using in ourcase a colloidal crystal as the nonlinear medium. Making useof a Laguerre-Gauss (LG) beamcarrying OAM[18] allows us to probe the phase conjugation ina very simple manner makinguse of the clearly observable spatial phase structure of such class of vortex beams. This is, tothe best of our knowledge, the first experiment in which direct verification of phase conjugationusing a nonlinear colloidal medium is made. We also probe thespatial structure of the crystal viasmall angle scattering (SAS) and measure the reflectivity ofthe artificial crystal as a functionof pump power. Using a colloidal medium as the source of nonlinearities sets the basis for



further understanding of the conservation of OAM in microscopic nonlinear systems where thenonlinearity can be arbitrarily controled.

2. Nonlinear colloidal crystal media

Microscopic spherical particles set in suspension are known to form ordered structures bymeans of laser-induced freezing[19]. Typically, two crossed laser beams in a monodispersecolloidal liquid suspension of spherical particles produce a standing wave that stimulates anumber density grating within the suspension due to the optical gradient force associated tothis spatially periodic potential. The structure of the resulting spatial distribution of particlescan be inferred by observing the diffraction pattern of an additional laser beam incident on theresulting grating. When subject to the radiation of the pump fieldsEi=1,2, each particle in thesuspension is affected by the optical gradient force of the resulting interference beams suchthat two orthogonal spatial gratings are formed in the suspension. A coarse and a fine gratingperiods are given by

Λ+ =λ

2cos(θ/2), and Λ− =

λ2sin(θ/2)

, (1)

respetively, whereθ is the angle betweenE1 andE3. One of the pump waves and the probeE3

drive the particles into the spatial grating, which scatters the second pump wave so as to formthe conjugated-phase wave while this and the second pump create another grating, which inturn scatters the first pump wave back into the probe. The ordered medium created under theinfluence of the standing wave, shows a strong nolinearity asa result of the of refractive indexcontrast between the particles and the liquid, which modulates the collective refractive index ofthe suspension spatially[14].

3. Experiment

In order to verify and measure the nonlinear refraction fromthe formed colloidal crystal, wehave built a standard experimental setup (see Fig. 1) for observing the FWM backgenerated sig-nal. In our experiment, the probe is a special instance of LG beams, which are a family of vortexbeams characterized by two parametersl andp. LG beams posses a coaxial phase dislocationwith a topological charge given by the azimuthal integer index l . This parameter represents thenumber of intertwined phase helices embedded in the beam, which originate from the helicalphase termexp(il ϕ), whereϕ is the azimuthal coordinate, and thus also determines the OAMcontent of the beam. A second indexp relates to the radial structure of the beam. Forp= 0 and|l | > 0, the transverse intensity of LG beams consists of one very bright thin ring, and thus thisparticular subset of LG beams is often referred to asannular LG beams. For this experiment,we have used an annular LG beam (l = 4, p = 0) generated by a computer-generated phasehologram as probe beam [20]. When the annular LG beam is interfered with a plane, mutuallycoherent reference wave, its azimuthally dependent term allows for the observation of the rotat-ing phase of the beam in the intensity pattern of the resulting interference[21, 22]. This featurecan be taken advantage of to easily observe the inversion of the spatial phasel → −l of thebackgenerated beam compared to the phase of a specular reflection as a result of FWM.

We have used the cw output of a diode-pumped solid state laseroperating at 532 nm as bothpump and probe beams in a DFWM configuration. The pumps are two counterpropagatingbeams focused and set to interfere in a region within the sample volume to form the standinginterference pattern. Both pump beams were focused to a diameter of aproximately 26µm andthe interaction region is set to a fraction of the Rayleigh range of the beams, so that the pumpscan be locally approximated by plane waves. Two glass windows served as compensating plates(CP) to adjust for the relative optical paths of the pumps producing small displacements of the



Fig. 1. Setup used for FWM experiments. The probe is shaped into an LG beam usingan offline blazed-phasehologram. The length of the delay line is varied for the referencebeam to match the optical path of the backgenerated wave. CP-compensating plate pair,CH-chopper, LP-linear polarizer, QW-quarter-wave plate, HOL-hologram, SM-samplingmirror, ND1 and ND2 are neutral density filters. BGW and PW denote the backgeneratedwave and the reference plane wave for the interferograms.

interference pattern at the sample cell. A neutral density filter (ND2) was placed in the path ofone of the pump beams to equalize the optical power in the pumps so that the fringe contrast,and thus the intensity gradient, were maximized. A half-wave plate and a linear polarizer inthe path of one of the pump beams were used to ensure that both pump beams were linearlypolarized in the same plane when they reached the sample cell. A lock-in amplifier in conjunc-tion with a photodiode and a beam chopper were used in order todiscriminate the FWM signalfrom scattered light and parasitic reflections. Colloidal samples were made using 100 nm di-ameter monodisperse polystyrene microspheres suspended in deionized water for preparing themedium. The probe beam made an angle of approximately 4 degrees with respect to the opticalaxis of the pumps and formed fine and coarse gratings with spatial periods of 265nm and 7.6µm respectively.

4. Results and discussion

From OAM conservation, it follows that the angular momenta per photon in the beams mustcomply withlF + lB− lp = lbg, where the first two terms account for the forward and backwardspump beams andlp andlbg are the probe and backgenerated signal beams. In our experiments,the Gaussian pumps havelF = lB = 0, hence−lp = lbg. However, in principle, OAM can alsobe transferred from the pump beams[23].



Fig. 2. Interferograms of (a) the specularly reflected signal and a reference plane wave, (b)the backgenerated signal and the reference plane wave. The probe beam is a LG beam withl = 4, p = 0. Colored lines represent phase shifts, while circles represent the limitsof theextent of the probe beam alone.

Upon the onset of FWM, the backgenerated beam was set to interfere with a reference planewave so that its transverse phase profile could be directly inferred from the interferogram[22] -we expect a classic forked interference pattern arising from the phase discontinuity at the centreof the LG beam. The interference pattern was isolated and registered using a CCD camera andcompared to that of a beam reflected on a mirror in place of the nonlinear medium to determinethe occurrence of phase conjugation. We used a variable delay line to keep the optical pathdifference of the reference beam to a minimum with respect tothe backgenerated beam. Anadditional neutral density filter(ND1) is placed in the path of the reference beam for enhancedcontrast of the interferogram.

The OAM content of the probe beam is transferred into the colloidal crystal via scatteringand subsequently to the backgenerated wave by the collective superposition of the oscillationsof the nonlinear polarization driven by the probe beam. The phase profile of the backgeneratedsignal shows an inversion of the sign of the phase with respect to the beam reflected by a mirroras expected for phase conjugation (see Fig. 2). The density grating here is thus analogous tothe atomic coherence gratings induced in Refs. [12] and [13], with the pump beams in ourexperiment confining the particles due to the gradient forceas well as feeding the amplificationand backgeneration within the DFWM process.

In order to assess the long-range order of the microspheres that make up the nonlinearmedium, we probed the structure with a He-Ne laser (1 mW) in order to look at the smallangle scattering pattern in the far field. The structure of scattering patterns observed on a dis-tant screen suggests that the particles in the medium are packed locally in a lattice with residualhexagonal symmetry[24]. Samples exhibited severe scattering losses at this wavelength as highas 18 cm−1 for a particle radius of 100 nm. The measured backgenerated power (Fig. 3) relatesto the total pump power by a cubic fit, indicating that the signal is indeed a result of the DFWMprocess[14]. The maximum power value measured for the backgenerated signal was less than4% that of the probe signal, suggesting an upper bound for theefficiency of the nonlinear effectthat can be attributed to absorption and multiple scattering within the colloidal suspension.

The efficient formation of the colloidal crystal grating wasobserved only above a thresholdvalue P0 of pump powers, below which the intensity of the probe had no influence on theoccurrence of the backgeneration of the signal. In the absence of absorption, the efficiencyof the medium is limited by scattering losses, thus the colloidal crystal has, in principle, aneffective bandwidth that spans the whole visible and near infrared spectra. At pump powers



Fig. 3. Backgenerated signal power as a function of input power. Thesolid line is the cubicfit. The inset shows the diffraction pattern for SAS.

aboveP0 = 150 mW, the backgenerated signal was discerned and observedto increase withsignal power. Long-range order of the particles occured faster with increasing pump power.The structures were seen to decay difussively in a longer time than expected, by consideringBrownian motion alone. This effect can be attributed to damping caused by the viscosity of thesolution.

5. Conclusions

We have demonstrated the optical phase conjugation of a coherent wavefield via DFWM using acolloidal crystal as the nonlinear medium. Phase conjugation was verified by directly observingthe transverse phase profile of the backgenerated beam. By using a vortex beam carrying OAM,we have been able to directly observe the inversion of the topological charge of the beam as asignature of FWM and hence indirectly demonstrated the OAM exchange between the probebeam and the backgenerated beam, as expected from this parametric process. We have alsomeasured the backgeneration efficiency of the crystal as a function of pump power and probedthe structure of the phase transition of the colloidal crystal. Finally, we have investigated thedynamics of the spatial index grating formation in real timeand found unusually large diffusiontimes possibly due to viscosity of the suspension medium.

Acknowledgments

This research was supported by the Consejo Nacional de Ciencia y Tecnologıa grant 42808 andthe Research Chair in Optics at Tecnologico de Monterrey grant number CAT-007. DM is aRoyal Society University Research Fellow. We thank the European Science Foundation SONSproject.



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Direct detection of optical phase conjugation in a colloidal medium