Light propagation and localization in modulated photonic lattices and waveguides

We review both theoretical and experimental advances in the recently emerged physics of modulated photonic lattices. Artificial periodic dielectric media, such as photonic crystals and photonic lattices, provide a powerful tool for the control of the fundamental properties of light propagation in photonic structures. Photonic lattices are arrays of coupled optical waveguides, where the light propagation becomes effectively discretized. Such photonic structures allow one to study many useful optical analogies with other fields, such as the physics of solid state and electron theory. In particular, the light propagation in periodic photonic structures resembles the motion of electrons in a crystalline lattice of semiconductor materials. The discretized nature of light propagation gives rise to many new phenomena which are not possible in homogeneous bulk media, such as discrete diffraction and diffraction management, discrete and gap solitons, and discrete surface waves. Recently, it was discovered that applying periodic modulation to a photonic lattice by varying its geometry or refractive index is very much similar to applying a bias to control the motion of electrons in a crystalline lattice. An interplay between periodicity and modulation in photonic lattices opens up unique opportunities for tailoring diffraction and dispersion properties of light as well as controlling nonlinear interactions.

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Diffraction-free optical beams propagate freely without change in shape and scale. Monochromatic beams that avoid diffractive spreading require two-dimensional transverse profiles and there are no corresponding solutions for profiles restricted to one transverse dimension. Here, we demonstrate that the temporal degree of freedom can be exploited to efficiently synthesize one-dimensional pulsed light sheets that propagate self-similarly in free space, with no need for nonlinearity or dispersion. By introducing programmable conical (hyperbolic, parabolic or elliptical) spectral correlations between the beam’s spatiotemporal degrees of freedom, a continuum of families of propagation-invariant light sheets is generated. The spectral loci of such beams are the reduced-dimensionality trajectories at the intersection of the light-cone with spatiotemporal spectral planes. Far from being exceptional, self-similar axial-propagation in free space is a generic feature of fields whose spatial and temporal degrees of freedom are tightly correlated. These ‘space–time’ light sheets can be useful in microscopy, nonlinear spectroscopy, and non-contact measurements. One-dimensional non-diffracting sheets of light are achieved without exploiting nonlinearity. Such light sheets may be exploited in microscopy and sensing applications.

Diffraction-free space–time light sheets

Diffraction-free optical beams propagate freely without change in shape and scale. Monochromatic beams that avoid diffractive spreading require two-dimensional transverse profiles, and there are no corresponding solutions for profiles restricted to one transverse dimension. Here, we demonstrate that the temporal degree of freedom can be exploited to efficiently synthesize one-dimensional pulsed optical sheets that propagate self-similarly in free space. By introducing programmable conical (hyperbolic, parabolic, or elliptical) spectral correlations between the beam's spatio-temporal degrees of freedom, a continuum of families of axially invariant pulsed localized beams is generated. The spectral loci of such beams are the reduced-dimensionality trajectories at the intersection of the light-cone with spatio-temporal spectral planes. Far from being exceptional, self-similar axial propagation is a generic feature of fields whose spatial and temporal degrees of freedom are tightly correlated. These one-dimensional `space-time' beams can be useful in optical sheet microscopy, nonlinear spectroscopy, and non-contact measurements.

Diffraction-free space-time beams

We have proposed an ultracompact all-optical photonic crystal AND gate based on nonlinear ring resonators, consisting of two Kerr nonlinear photonic crystal ring resonators inserted between three parallel line defects. We have employed a Si nanocrystal as the nonlinear material for its appropriate nonlinear properties. The gate has been simulated and analyzed by finite difference time domain and plane wave expansion methods. The proposed logic gate can operate with a bit rate of about 120 Gbits/s.

All-optical ultracompact photonic crystal AND gate based on nonlinear ring resonators

The authors demonstrate experimentally that interaction between nonlocal solitons in nematic liquid crystals (NLCs) can be controlled by the degree of nonlocality. For a given beam width, the degree of nonlocality can be modulated by changing the pretilt angle θ0 of NLC molecules through bias voltage V. As V increases (so does θ0), the degree of nonlocality decreases. When the degree of nonlocality is below a critical value, the solitons behave in the way like their local counterpart, i.e., in-phase solitons attract while out-of-phase solitons repulse each other. Such a voltage-controlled interaction between the solitons can be readily implemented in experiments.

Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals

In analogy to a perturbed harmonic oscillator, we calculate the fundamental and some other higher order soliton solutions of the nonlocal nonlinear Sch\"odinger equation (NNLSE) in the second approximation in the generally nonlocal case. Comparing with numerical simulations we show that soliton solutions in the second approximation can describe the generally nonlocal soliton states of the NNLSE more exactly than that in the zeroth approximation. We show that for the nonlocal case of an exponential-decay type nonlocal response the Gaussian-function-like soliton solutions cannot describe the nonlocal soliton states exactly even in the strongly nonlocal case. The properties of such nonlocal solitons are investigated. In the strongly nonlocal limit, the soliton's power and phase constant are both in inverse proportion to the fourth power of its beam width for the nonlocal case of a Gaussian function type nonlocal response, and are both in inverse proportion to the third power of its beam width for the nonlocal case of an exponential-decay type nonlocal response.

Perturbative analysis of generally nonlocal spatial optical solitons.

Propagation properties of elegant Hermite-cosh-Gaussian laser beams

We have introduced a novel (to our knowledge) class of complex-variable-function (CVF)—Gaussian solitons that constitute the exact soliton solutions of the Snyder-Mitchell model. The CVF-Gaussian solitons whose transverse structure has an inherent rotation are the product of an arbitrary analytic CVF and a Gaussian function. A distribution factor that is the parameter for the description of the transverse distribution of the CVF–Gaussian solitons is discussed.

Complex-variable-function-Gaussian solitons.

We find a new family of solutions of the nonparaxial wave equation that represents ultrashort pulsed light beam propagation in free space. The spatial and temporal parts of these pulsed beams are separable; the spatial transverse part is described by a Bessel function and remains unchanged during propagation, but the temporal profile can be arbitrary. Therefore the pulsed beam exhibits diffraction-free behavior with no transverse spreading, but the temporal part changes as if in a dispensive medium; the change is dominated by what we call spatially induced group-velocity dispersion. The analytical and numerical investigations show that the even- and odd-order spatially induced dispersions partially compensate for each other so as to give rise to pulse spreading, weakening, asymmetry, and center shift.

Wei Hu

Papers

Nonlocality-controlled interaction of spatial solitons in nematic liquid crystals

Perturbative analysis of generally nonlocal spatial optical solitons.

Propagation properties of elegant Hermite-cosh-Gaussian laser beams

Complex-variable-function-Gaussian solitons.

Ultrashort pulsed Bessel beams and spatially induced group-velocity dispersion