1. Introduction

Four wave mixing (FWM) is a well-known nonlinear process[1, 2, 3, 4, 5], in which a dielectric medium with a significant nonlinear susceptibility χ³ is simultaneously pumped by two coherent waves E ₁ and E ₂ and probed by a third wave E ₃, producing a fourth wave E ₄ radiated in the opposite direction to, and phase conjugate with, E ₃. The particular case when the frequencies of all four waves E _i=1,2,3,4 are the same, is referred to as degenerate four-wave mixing (DFWM). The production of a fourth wave is essentially a result of the interaction between the pump 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 considering the reversal of the propagation direction of the generated wave with respect to the probe beam. To date, several different nonlinear media have been used to demonstrate FWM, such as metal vapors[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 momentum (OAM) between two waves via optical pumping of an atomic sample using a vortex beam. The observation of a conjugated-phase replica of the beam demonstrated the transfer of OAM into the system and subsequently into the backgenerated beam. Barreiro et al have also made use of an atomic coherence grating to generate the phase conjugated counterparts of beams carrying OAM [12, 13], further demonstrating mechanisms for the transfer and the conservation of OAM in FWM experiments.

One particular instance of nonlinear medium can be artificially produced by means of ordered arrays of microscopic dielectric particles suspended in a transparent medium. The particles can be spatially arranged in crystal-like structures making use of an optical gradient force exerted on them by a spatially varying optical field [14, 15, 16]. Similar effects can also be achieved in non-isotropic media[17]. The seminal work by Smith et al[14, 15] clearly demonstrated that the backward generated beam from such a colloidal crystal was due to FWM, but did not look into its phase conjugation.

In this paper, we present the direct observation of phase conjugation of a coherent optical wavefield via DFWM following the work of Tabosa and Barreiro[11, 12, 13], using in our case a colloidal crystal as the nonlinear medium. Making use of a Laguerre-Gauss (LG) beam carrying OAM[18] allows us to probe the phase conjugation in a very simple manner making use of the clearly observable spatial phase structure of such class of vortex beams. This is, to the best of our knowledge, the first experiment in which direct verification of phase conjugation using a nonlinear colloidal medium is made. We also probe the spatial structure of the crystal via small angle scattering (SAS) and measure the reflectivity of the artificial crystal as a function of 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 the nonlinearity can be arbitrarily controled.

2. Nonlinear colloidal crystal media

Microscopic spherical particles set in suspension are known to form ordered structures by means of laser-induced freezing[19]. Typically, two crossed laser beams in a monodisperse colloidal liquid suspension of spherical particles produce a standing wave that stimulates a number density grating within the suspension due to the optical gradient force associated to this spatially periodic potential. The structure of the resulting spatial distribution of particles can be inferred by observing the diffraction pattern of an additional laser beam incident on the resulting grating. When subject to the radiation of the pump fields E _i=1,2, each particle in the suspension is affected by the optical gradient force of the resulting interference beams such that two orthogonal spatial gratings are formed in the suspension. A coarse and a fine grating periods are given by

respetively, where θ is the angle between E ₁ and E ₃. One of the pump waves and the probe E ₃ drive the particles into the spatial grating, which scatters the second pump wave so as to form the conjugated-phase wave while this and the second pump create another grating, which in turn scatters the first pump wave back into the probe. The ordered medium created under the influence of the standing wave, shows a strong nolinearity as a result of the of refractive index contrast between the particles and the liquid, which modulates the collective refractive index of the suspension spatially[14].

3. Experiment

In order to verify and measure the nonlinear refraction from the formed colloidal crystal, we have built a standard experimental setup (see Fig. 1) for observing the FWM backgenerated signal. In our experiment, the probe is a special instance of LG beams, which are a family of vortex beams characterized by two parameters l and p. LG beams posses a coaxial phase dislocation with a topological charge given by the azimuthal integer index l. This parameter represents the number of intertwined phase helices embedded in the beam, which originate from the helical phase term exp(ilφ), where φ is the azimuthal coordinate, and thus also determines the OAM content of the beam. A second index p relates to the radial structure of the beam. For p = 0 and ∣l∣ > 0, the transverse intensity of LG beams consists of one very bright thin ring, and thus this particular subset of LG beams is often referred to as annular LG beams. For this experiment, we have used an annular LG beam (l = 4, p = 0) generated by a computer-generated phase hologram as probe beam [20]. When the annular LG beam is interfered with a plane, mutually coherent reference wave, its azimuthally dependent term allows for the observation of the rotating phase of the beam in the intensity pattern of the resulting interference[21,22]. This feature can be taken advantage of to easily observe the inversion of the spatial phase l → -l of the backgenerated 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 laser operating at 532 nm as both pump and probe beams in a DFWM configuration. The pumps are two counterpropagating beams focused and set to interfere in a region within the sample volume to form the standing interference pattern. Both pump beams were focused to a diameter of aproximately 26μm and the interaction region is set to a fraction of the Rayleigh range of the beams, so that the pumps can 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 interference pattern at the sample cell. A neutral density filter (ND₂) was placed in the path of one 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 in the path of one of the pump beams were used to ensure that both pump beams were linearly polarized in the same plane when they reached the sample cell. A lock-in amplifier in conjunction with a photodiode and a beam chopper were used in order to discriminate the FWM signal from scattered light and parasitic reflections. Colloidal samples were made using 100 nm diameter monodisperse polystyrene microspheres suspended in deionized water for preparing the medium. The probe beam made an angle of approximately 4 degrees with respect to the optical axis of the pumps and formed fine and coarse gratings with spatial periods of 265nm and 7.6 μm respectively.

 

Fig. 1. Setup used for FWM experiments. The probe is shaped into an LG beam using an offline blazed-phasehologram. The length of the delay line is varied for the reference beam 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-sampling mirror, ND₁ and ND₂ are neutral density filters. BGW and PW denote the backgenerated wave and the reference plane wave for the interferograms.

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4. Results and discussion

From OAM conservation, it follows that the angular momenta per photon in the beams must comply with l _F + l _B - l _p = l_bg, where the first two terms account for the forward and backwards pump beams and l_p and l_bg are the probe and backgenerated signal beams. In our experiments, the Gaussian pumps have l_F = l_B = 0, hence -l_p = l_bg. However, in principle, OAM can also be 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 with l = 4, p = 0. Colored lines represent phase shifts, while circles represent the limits of the extent of the probe beam alone.

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Upon the onset of FWM, the backgenerated beam was set to interfere with a reference plane wave 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 centre of the LG beam. The interference pattern was isolated and registered using a CCD camera and compared to that of a beam reflected on a mirror in place of the nonlinear medium to determine the occurrence of phase conjugation. We used a variable delay line to keep the optical path difference of the reference beam to a minimum with respect to the backgenerated beam. An additional neutral density filter (ND ₁) is placed in the path of the reference beam for enhanced contrast of the interferogram.

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

In order to assess the long-range order of the microspheres that make up the nonlinear medium, we probed the structure with a He-Ne laser (1 mW) in order to look at the small angle scattering pattern in the far field. The structure of scattering patterns observed on a distant screen suggests that the particles in the medium are packed locally in a lattice with residual hexagonal symmetry[24]. Samples exhibited severe scattering losses at this wavelength as high as 18 cm^-1 for a particle radius of 100 nm. The measured backgenerated power (Fig. 3) relates to the total pump power by a cubic fit, indicating that the signal is indeed a result of the DFWM process[14]. The maximum power value measured for the backgenerated signal was less than 4% that of the probe signal, suggesting an upper bound for the efficiency of the nonlinear effect that can be attributed to absorption and multiple scattering within the colloidal suspension.

The efficient formation of the colloidal crystal grating was observed only above a threshold value P ₀ of pump powers, below which the intensity of the probe had no influence on the occurrence of the backgeneration of the signal. In the absence of absorption, the efficiency of the medium is limited by scattering losses, thus the colloidal crystal has, in principle, an effective bandwidth that spans the whole visible and near infrared spectra. At pump powers above P ₀ = 150 mW, the backgenerated signal was discerned and observed to increase with signal 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 considering Brownian motion alone. This effect can be attributed to damping caused by the viscosity of the solution.

 

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

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5. Conclusions

We have demonstrated the optical phase conjugation of a coherent wavefield via DFWM using a colloidal crystal as the nonlinear medium. Phase conjugation was verified by directly observing the 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 a signature of FWM and hence indirectly demonstrated the OAM exchange between the probe beam and the backgenerated beam, as expected from this parametric process. We have also measured the backgeneration efficiency of the crystal as a function of pump power and probed the structure of the phase transition of the colloidal crystal. Finally, we have investigated the dynamics of the spatial index grating formation in real time and found unusually large diffusion times possibly due to viscosity of the suspension medium.