Optically-induced potentials, which are generated by propagating beams that create a refractive index change in their host medium, are becoming increasingly important for photonic science and material characterization. Examples include plasma filamentation and self-focusing of femtosecond pulses [1–3], holographic optical trapping [4–6], and photonic lattice generation [7–9]. Great progress at reconstructing these potentials has been made in the temporal domain, where digital holography has been used to measure nonlinear index changes [10,11]. In these experiments, the refractive index field is optically induced by a pump beam and then probed by another (scattered) beam oriented perpendicular to it. Typically, though, the methods use Gabor, or in-line, holography [10] and therefore are limited to axially-thin objects whose reconstructions may suffer degradation from the twin image. Off-axis digital holography has avoided this limitation [11], but still provides only projected, not volumetric, data. While such information may be obtained using variants of optical tomography, these setups would require complicated geometries, such as multi-angle shots or a rotating object, or computationally burdensome post-processing algorithms for asymmetric or strongly refracting objects [12]. In the spatial domain, z-scan techniques have been used to characterize both thin [13] and thick [14,15] nonlinear samples; however, they cannot measure the full transverse index profile and usually are limited by assuming cylindrically symmetric input beams. Here, we propose a digital holography technique for reconstructing optically-induced, axially-thick refractive index fields in the spatial domain and demonstrate it in a defocusing photorefractive crystal. The method, which does not make any assumptions about the generating beam, is a two-step procedure: (1) holographically reconstructing the beam that generates the potential [16,17] and (2) using a physical model to calculate the corresponding index change. As an example, we experimentally generate and numerically reconstruct a cylindrical test potential. The reconstructed index profile is then verified by comparing experimental and numerical scattering of a plane-wave probe beam.

As in free-space digital holography [18], the reconstruction method requires knowledge of the full complex field exiting the sample. The signal amplitude can be recorded directly in a camera, while the output phase requires an interference (holographic) measurement. In the experiments, the object phase was measured by incrementing the phase of a reference beam with a piezo-actuated mirror (in λ/4 increments), as shown in Fig. 1 , and using a standard phase-shifting algorithm [19]. For a known propagator, this complex field then can be integrated numerically forwards or backwards. For example, in the linear homogeneous regime, the field may be reconstructed simply by implementing the Fresnel transformation integral. For nonlinear media, this kernel is modified by intensity-induced changes in the phase. Recently, we demonstrated this numerical back-propagation for scalar wave propagation in a self-defocusing medium [16]. Here, we generalize these results to vector propagation by including electro-optic index differences and spatial inhomogeneities due to different polarizations of light.

 

Fig. 1 Experimental setup. A plano-convex lens focuses an ordinarily polarized Gaussian beam onto the photorefractive SBN:75 crystal to induce an axially-thick potential. The phase is captured by a phase-shifting technique with a piezo-actuated mirror. The plane wave probe beam is created by removing the lens and rotating the polarization so that it is parallel to the c-axis (extraordinary polarization).

Download Full Size | PDF

In the paraxial approximation, the beam dynamics of the slowly varying wavefunction ψ(x,y,z) is well described by the nonlinear Schrödinger equation (NLSE),

The algorithm, Eq. (2), assumes that the field is a scalar one and is not generally applicable to propagation in anisotropic media, e.g. crystals, in which polarization dependence becomes important. In this case, reconstruction of both components requires a pair of coupled NLSEs, as the different polarizations can interact through and potentially be converted by Δn and V. Here, we consider a much simpler setup in which one beam optically induces an index change which then serves as the potential for a separate probe beam. This would be the normal situation for index characterization of a single beam and is the typical configuration for most optical induction experiments, e.g. the all-optical creation of spatial photonic lattices [7–9]. In these cases, there is no dynamical coupling between the components, and the propagation may be considered sequentially: induction beam first, then scattered probe. As a minimal system, we consider nonlinear induction but linear scattering, giving the following pair of Schrödinger equations:

Our particular experimental system uses a 2 × 5 × 10 mm photorefractive strontium barium niobate (SBN:75) crystal with a self-defocusing nonlinearity produced by applying an electric field of −600 V/cm along the crystalline axis. For ordinary polarization, the refractive index is n′_o² = n_o² − n_o⁴r₁₃E_sc, where E_sc is the space-charge field resulting from charge transport and r_ij is the ij-component of the electro-optic tensor. For extraordinary beams, the index n′_e² = n_e² − n_e⁴r₃₃E_sc. For SBN, r₁₃ << r₃₃, so it is assumed typically that the ordinary refractive index change is negligible, i.e., that ordinary beams produce an index change but do not experience it. While this is usually a good approximation for many experiments, it is not wholly accurate or generally applicable. Indeed, other photorefractive crystals have comparable electro-optic coefficients for different polarizations, so we include it in this analysis.

Results for the ordinary beam propagation are shown in Fig. 2 . Because of both the relatively wide input beam (FWHM ~50 μm, Fig. 2(a),(b)) and the small ordinary electro-optic coefficient, beam spreading in the nonlinear case (Figs. 2(e),(f)) is slightly more than that in the linear case (Figs. 2(c),(d)). Note that this slightly defocused nonlinear output is markedly different from the optical shock waves generated by extraordinarily polarized beams [20]. The output field was then measured and back-propagated digitally to recover the input using Eq. (3a). Technically, the photorefractive screening nonlinearity of the crystal is saturable [21], with a response ∆n_ordinary ∝ r_ijÎ_ordinary /(1 + Î_ordinary), where Î_ordinary is the beam intensity

 

Fig. 2 Numerical reconstruction of ordinarily polarized beam. (a,b) Intensity and phase of measured input. (c,d) Intensity and phase of linear output. (e,f) Intensity and phase of measured nonlinear output. Note enhanced defocusing compared to the linear case. (g,h) Reconstructed input using Eq. (2). i) Intensity and j) phase cross sections through the beam center of reconstructed and measured inputs.

Download Full Size | PDF

I_ordinary = |ψ_ordinary|², normalized to a background (dark current) intensity. However, recent experiments have shown that for the defocusing parameters considered here, the simpler Kerr nonlinearity ∆n_ordinary = −|γ|I_ordinary proves sufficient for modeling [20]. The algorithm was iterated with these parameters and the value of γ was adjusted until the input field was successfully reconstructed, as shown in Figs. 2(g)-(j). The calibrated Kerr coefficient ratio is γ_e/γ_o = 18.8, in good agreement with the nominal value, n_e³r₃₃/n_o³r₁₃ = 19.7, for SBN:75. In effect, the reconstruction method integrates the nonlinearity over the entire length of propagation, thus providing a thick-sample z-scan technique [14,15] to characterize the optical properties of a material. Details of the transverse response, such as nonlocality [22,23] are still retained.

Once the beam was successfully reconstructed at the input, as shown in Fig. 2, the calibrated algorithm can then be used to calculate the field at an arbitrary location within the crystal simply by propagating the code to different values of z [16]. Illustrated in Fig. 3 is this

 

Fig. 3 Reconstruction of axial index perturbation along its propagation direction. Note only slight defocusing of beam in both intensity and phase.

Download Full Size | PDF

evolution of the induced potential along the propagation axis. Because of the optical induction phenomenon, the ordinary beam produces a spatially-varying index change (Δn/n₀ ~10⁻⁴) that can be written as V( r ) = −|v₀|·I_ordinary( r ), where v₀ is a constant. This induced index change will act as an axially-thick scattering object for extraordinary light. We stress that this method does not rely on assumptions about the object profile, i.e. the procedure is general for all optically-induced index changes. Thus, to within the resolution limits of the setup, it allows for the full volumetric reconstruction of arbitrary optically-induced index changes with a single-surface field measurement and calibrated algorithm. Such information can be stored for future reconstructions and experiments.

To test the accuracy of the method for measuring the potential profile, we treat the induced potential as a fixed object and perform a linear scattering experiment. Experimentally, we launched as a probe beam a low-intensity, extraordinarily-polarized plane wave (constant phase and amplitude) at normal incidence to the induced cylinder. Both the input (plane wave) and output (scattered wave) fields were measured. As a control, we substituted the measured input field into the beam propagation code, with the reconstructed potential from Fig. 2 in Eq. (3b), and numerically simulated the output. This semi-numerical output (Figs. 4(a),(b) ) compares favorably with experimental measurement (Figs. 4(c),(d)), in both amplitude and phase. Cross-sections of these outputs (Figs. 4(e),(f)) show how accurately the holographic reconstruction captures the potential itself. Slight variations in the field profiles can be accounted for by noting that the bulk crystal has natural defects and inhomogeneities that are not modeled in the reconstruction algorithm.

 

Fig. 4 Semi-empirical simulation and experimental measurement of plane-wave scattering from potential. (a,b) Intensity and phase of simulated output, using measured input and reconstructed potential. (c,d) Experimental output of scattered wave. e) Intensity cross sections. f) Phase cross sections.

Download Full Size | PDF

Finally, it is important to consider limitations of the technique. Experimentally, measurement of the output field is limited by the spatial resolution and quantization error of the CCD camera and by the numerical aperture of the imaging system. These standard effects preclude measurement of the high frequency content, i.e., the fine details, of the object, but can be minimized by using high resolution digital devices and high NA lenses, respectively. Numerically, the propagation Eqs. (1) and (3) break down when the paraxial and envelope approximations are violated. On the other hand, the reconstruction algorithm (2) is independent of the particular evolution equation and can be generalized to include non-paraxial behavior [24–26]. Note, though, that these vectorial algorithms are still one-way and do not include counter-propagating effects such as back-reflections. Regardless, near-field holography, e.g. through total internal reflection, has been demonstrated experimentally [27,28]. Nonlinear behavior at the sub-wavelength scale has been demonstrated as well [29,30]. The ultimate resolution limits therefore arise from noise effects, both as competition to the signal and as a source for instability. An examination of this will be the subject of future work.

In conclusion, we have demonstrated the volumetric reconstruction of optically-induced index changes via digital holography. While some a priori knowledge of the material response proved useful for developing a correct algorithm, the technique is independent of the index profile itself, provided that the generating beam can be captured holographically. The example discussed here was the simplest case for investigating the role of nonlinearity and spatial non-uniformity in the propagating medium. More complex examples, using both the nonlinear and potential terms in Eq. (2) concurrently, can be accommodated easily. That is, the method can be generalized to spatially-inhomogeneous media, such as optical lattices, as well as multi-component signal beams. The method thus provides a platform on which to develop more complex imaging and characterization systems.