In nonlinear optics, the superposition principle no longer holds. The first nonlinear optical effect to be predicted was two-photon absorption , by Maria Goeppert Mayer for her PhD in , but it remained an unexplored theoretical curiosity until and the almost simultaneous observation of two-photon absorption at Bell Labs [1] and the discovery of second-harmonic generation by Peter Franken et al. Nonlinear optics explains nonlinear response of properties such as frequency , polarization, phase or path of incident light.

In these processes, the medium has a linear response to the light, but the properties of the medium are affected by other causes:. Nonlinear effects fall into two qualitatively different categories, parametric and non-parametric effects. A parametric non-linearity is 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.

Parametric and "instantaneous" i. Note that the polarization density P t and electrical field E t are considered as scalar for simplicity. Central to the study of electromagnetic waves is the wave equation.

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Starting with Maxwell's equations in an isotropic space, containing no free charge, it can be shown that. For nonlinear medium, Gauss's law does not imply that the identity. 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:. 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".

As an example, if we consider only a second-order nonlinearity three-wave mixing , then the polarization P takes the form. Plugging this into the expression for P gives. These three-wave mixing processes correspond to the nonlinear effects known as second-harmonic generation , sum-frequency generation , difference-frequency generation and optical rectification respectively. Note: Parametric generation and amplification is a variation of difference-frequency generation, where the lower frequency of one of the two generating fields is much weaker parametric amplification or completely absent parametric generation.

In the latter case, the fundamental quantum-mechanical uncertainty in the electric field initiates the process. The above ignores the position dependence of the electrical fields. In a typical situation, the electrical fields are traveling waves described by. The above equation is known as the phase-matching condition. Typically, three-wave mixing is done in a birefringent crystalline material, where the refractive index depends on the polarization and direction of the light that passes through.

## Photonics: Linear and Nonlinear Interactions of Laser Light and Matter

The polarizations of the fields and the orientation of the crystal are chosen such that the phase-matching condition is fulfilled. This phase-matching technique is called angle tuning. Typically a crystal has three axes, one or two of which have a different refractive index than the other one s. Uniaxial crystals, for example, have a single preferred axis, called the extraordinary e axis, while the other two are ordinary axes o see crystal optics.

There are several schemes of choosing the polarizations for this crystal type.

## ISBN 13: 9783540670742

If the signal and idler have the same polarization, it is called "type-I phase matching", and if their polarizations are perpendicular, it is called "type-II phase matching". However, other conventions exist that specify further which frequency has what polarization relative to the crystal axis.

These types are listed below, with the convention that the signal wavelength is shorter than the idler wavelength. Most common nonlinear crystals are negative uniaxial, which means that the e axis has a smaller refractive index than the o axes.

In those crystals, type-I and -II phase matching are usually the most suitable schemes. Types II and III are essentially equivalent, except that the names of signal and idler are swapped when the signal has a longer wavelength than the idler. 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. These methods are called temperature tuning and quasi-phase-matching.

Temperature tuning is used when the pump laser frequency polarization is orthogonal to the signal and idler frequency polarization. The birefringence in some crystals, in particular lithium niobate is highly temperature-dependent. The crystal temperature is controlled to achieve phase-matching conditions.

## Nonlinear optics - Wikipedia

The other method is quasi-phase-matching. Hence, these crystals are called periodically poled. This results in the polarization response of the crystal to be shifted back in phase with the pump beam by reversing the nonlinear susceptibility. This allows net positive energy flow from the pump into the signal and idler frequencies.

Quasi-phase-matching can be expanded to chirped gratings to get more bandwidth and to shape an SHG pulse like it is done in a dazzler. Light from the new lasers may become even cheaper than that from light bulbs. Thus, laser optic devices will in? Many of these processes will be based on nonlinear optical interactions of the laser light with suitable optical material. In these interactions the - terial is modi?

The light is then in turn modi? Finally, the nonlinear modi? Light is modi? To use laser light in this sense in science, technology and medicine, kn- ledge from di? From the reviews of the second edition: "Menzel covers a narrower subject area in greater depth. We are treated to a comprehensive discussion of linear and nonlinear interactions between light and matter with more of the mathematics than some others offer.

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