Nonlinear Optical Systems

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Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity is typically observed only at very high light intensities (when the electric field of the light is >108 V/m and thus comparable to the atomic electric field of 1011 V/m) such as those provided by lasers. Above the Schwinger limit, the vacuum itself is expected to become nonlinear. In nonlinear optics, the superposition principle no longer holds.[1][2][3]

The first nonlinear optical effect to be predicted was two-photon absorption, by Maria Goeppert Mayer for her PhD in 1931, but it remained an unexplored theoretical curiosity until 1961 and the almost simultaneous observation of two-photon absorption at Bell Labs[4]and the discovery of second-harmonic generation by Peter Franken et al. at University of Michigan, both shortly after the construction of the first laser by Theodore Maiman.[5] However, some nonlinear effects were discovered before the development of the laser.[6] The theoretical basis for many nonlinear processes were first described in Bloembergen's monograph \"Nonlinear Optics\".[7]

Nonlinear effects fall into two qualitatively different categories, parametric and non-parametric effects. A parametric non-linearityis an interaction in which the quantum state of the nonlinear material is not changed by the interaction with the optical field. As a consequence of this, the process is \"instantaneous\". Energy and momentum are conserved in the optical field, making phase matching important and polarization-dependent.[15][16]

where the coefficients χ(n) are the n-th-order susceptibilities of the medium, and the presence of such a term is generally referred to as an n-th-order nonlinearity. Note that the polarization density P(t) and electrical field E(t) are considered as scalar for simplicity. In general, χ(n) is an (n + 1)-th-rank tensor representing both the polarization-dependent nature of the parametric interaction and the symmetries (or lack) of the nonlinear material.

is true in general, even for an isotropic medium. However, even when this term is not identically 0, it is often negligibly small and thus in practice is usually ignored, giving us the standard nonlinear wave equation:

The nonlinear wave equation is an inhomogeneous differential equation. The general solution comes from the study of ordinary differential equations and can be obtained by the use of a Green's function. Physically one gets the normal electromagnetic wave solutions to the homogeneous part of the wave equation:

acts as a driver/source of the electromagnetic waves. One of the consequences of this is a nonlinear interaction that results in energy being mixed or coupled between different frequencies, which is often called a \"wave mixing\".

In general, an n-th order nonlinearity will lead to (n + 1)-wave mixing. As an example, if we consider only a second-order nonlinearity (three-wave mixing), then the polarization P takes the form

One undesirable effect of angle tuning is that the optical frequencies involved do not propagate collinearly with each other. This is due to the fact that the extraordinary wave propagating through a birefringent crystal possesses a Poynting vector that is not parallel to the propagation vector. This would lead to beam walk-off, which limits the nonlinear optical conversion efficiency. Two other methods of phase matching avoid beam walk-off by forcing all frequencies to propagate at a 90 with respect to the optical axis of the crystal. These methods are called temperature tuning and quasi-phase-matching.

Practically, frequency doubling is carried out by placing a nonlinear medium in a laser beam. While there are many types of nonlinear media, the most common media are crystals. Commonly used crystals are BBO (β-barium borate), KDP (potassium dihydrogen phosphate), KTP (potassium titanyl phosphate), and lithium niobate. These crystals have the necessary properties of being strongly birefringent (necessary to obtain phase matching, see below), having a specific crystal symmetry, being transparent for both the impinging laser light and the frequency-doubled wavelength, and having high damage thresholds, which makes them resistant against the high-intensity laser light.

It is possible, using nonlinear optical processes, to exactly reverse the propagation direction and phase variation of a beam of light. The reversed beam is called a conjugate beam, and thus the technique is known as optical phase conjugation[21][22] (also called time reversal, wavefront reversal and is significantly different from retroreflection).

One can interpret optical phase conjugation as being analogous to a real-time holographic process.[23] In this case, the interacting beams simultaneously interact in a nonlinear optical material to form a dynamic hologram (two of the three input beams), or real-time diffraction pattern, in the material. The third incident beam diffracts at this dynamic hologram, and, in the process, reads out the phase-conjugate wave. In effect, all three incident beams interact (essentially) simultaneously to form several real-time holograms, resulting in a set of diffracted output waves that phase up as the \"time-reversed\" beam. In the language of nonlinear optics, the interacting beams result in a nonlinear polarization within the material, which coherently radiates to form the phase-conjugate wave.

Reversal of wavefront means a perfect reversal of photons' linear momentum and angular momentum. The reversal of angular momentum means reversal of both polarization state and orbital angular momentum.[24] Reversal of orbital angular momentum of optical vortex is due to the perfect match of helical phase profiles of the incident and reflected beams. Optical phase conjugation is implemented via stimulated Brillouin scattering,[25] four-wave mixing, three-wave mixing, static linear holograms and some other tools.

Optical fields transmitted through nonlinear Kerr media can also display pattern formation owing to the nonlinear medium amplifying spatial and temporal noise. The effect is referred to as optical modulation instability.[13] This has been observed both in photo-refractive,[27] photonic lattices,[28] as well as photo-reactive systems.[29][30][31][32] In the latter case, optical nonlinearity is afforded by reaction-induced increases in refractive index.[33] Examples of pattern formation are spatial solitons and vortex lattices in framework of nonlinear Schrödinger equation.[34][35]

In this Research Letter it is shown how certain optical wave amplification phenomena, which do not obey standard phase-matching conditions and are generally classified as non-Hermitian phase matching or gain-through-loss(-filtering) processes, can be considered as particular cases of the more general phenomenon of converse symmetry breaking predicted and observed in various networks of coupled oscillators. It is shown that the abovementioned optical amplification processes are possible thanks to a phase-locking dynamics enabled by asymmetry or heterogeneity in the coupled-modes equations.

Schematic of the synchronization-through-diversity concept: (i) In a set of coupled heterogeneous oscillators, CSB consists in the existence of a stable synchronized state which would be otherwise impossible for a set in which the oscillators are identical; and (ii) two optical modes synchronize and are amplified during propagation in a nonlinear medium with an asymmetric spectral response, while for a symmetric response function this does not occur.

Researchers at the Cockrell School of Engineering at The University of Texas at Austin have created a new nonlinear metasurface, or meta mirror, that could one day enable the miniaturization of laser systems. googletag.cmd.push(function() { googletag.display('div-gpt-ad-1449240174198-2'); }); The invention, called a \"nonlinear mirror\" by the researchers, could help advance nonlinear laser systems that are used for chemical sensing, explosives detection, biomedical research and potentially many other applications. The researchers' study will be published in the July 3 issue of Nature.

The metamaterials were created with nonlinear optical response a million times as strong as traditional nonlinear materials and demonstrated frequency conversion in films 100 times as thin as human hair using light intensity comparable with that of a laser pointer.

Nonlinear optical effects are widely used by engineers and scientists to generate new light frequencies, perform laser diagnostics and advance quantum computing. Due to the small extent of optical nonlinearity in naturally occurring materials, high light intensities and long propagation distances in nonlinear crystals are typically required to produce detectable nonlinear optical effects.

The research team led by UT Austin's Department of Electrical and Computer Engineering professors Mikhail Belkin and Andrea Alu, in collaboration with colleagues from the Technical University of Munich, has created thin-film nonlinear metamaterials with optical response many orders of magnitude larger than that of traditional nonlinear materials. The scientists demonstrated this functionality by realizing a 400-nanometer-thick nonlinear mirror that reflects radiation at twice the input light frequency. For the given input intensity and structure thickness, the new nonlinear metamaterial produces approximately 1 million times larger frequency-doubled output, compared with similar structures based on conventional materials.

\"This work opens a new paradigm in nonlinear optics by exploiting the unique combination of exotic wave interaction in metamaterials and of quantum engineering in semiconductors,\" said Professor Andrea Alu. 781b155fdc