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. 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. SHG of a pump and self-phase modulation emulated by second-order processes of the signal and an optical parametric amplifier can be integrated monolithically. At high peak powers the Kerr effect can cause filamentation of light in air, in which the light travels without dispersion or divergence in a self-generated waveguide.

### 2nd Edition

When a noble gas atom is hit by an intense laser pulse, which has an electric field strength comparable to the Coulomb field of the atom, the outermost electron may be ionized from the atom. Once freed, the electron can be accelerated by the electric field of the light, first moving away from the ion, then back toward it as the field changes direction. The electron may then recombine with the ion, releasing its energy in the form of a photon. The light is emitted at every peak of the laser light field which is intense enough, producing a series of attosecond light flashes.

The photon energies generated by this process can extend past the th harmonic order up to a few K eV. This is called high-order harmonic generation. The laser must be linearly polarized, so that the electron returns to the vicinity of the parent ion. High-order harmonic generation has been observed in noble gas jets, cells, and gas-filled capillary waveguides.

One of the most commonly used frequency-mixing processes is frequency doubling , or second-harmonic generation. 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. 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 [17] [18] also called time reversal , wavefront reversal and is significantly different from retroreflection. One can interpret this nonlinear optical interaction as being analogous to a real-time holographic process.

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. The most common way of producing optical phase conjugation is to use a four-wave mixing technique, though it is also possible to use processes such as stimulated Brillouin scattering.

A device producing the phase-conjugation effect is known as a phase-conjugate mirror PCM.

As above, the phase-matching condition determines which of these waves is the dominant. This results in the retroreflecting property of the effect. Further, it can be shown that for a medium with refractive index n and a beam interaction length l , the electric field amplitude of the conjugate beam is approximated by. If the pump beams E 1 and E 2 are plane counterpropagating waves, then.

Since the imaginary part of the amplitude contains the phase of the beam, this results in the reversal of phase property of the effect. Note that the constant of proportionality between the signal and conjugate beams can be greater than 1. The power for this comes from the two pump beams, which are depleted by the process. The frequency of the conjugate wave can be different from that of the signal wave.

This is known as frequency flipping. Optical phase conjugation in the near field performs the reversal of classical rays, or retroreflection. In classical Maxwell electrodynamics a phase-conjugating mirror performs reversal of the Poynting vector :. The above identities are valid locally , i. In quantum electrodynamics the interpretation of phase conjugation is much simpler compared to classical electrodynamics. The photon reflected from phase conjugating-mirror out has opposite directions of linear and angular momenta with respect to incident photon in :.

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. The early studies of nonlinear optics and materials focused on the inorganic solids. With the development of nonlinear optics, molecular optical properties were investigated, forming molecular nonlinear optics [30]. Recently, many novel directions were proposed for enhanced nonlinearity and light manipulation, including twisted chromophores, combining rich density of states with bond alternation, microscopic cascading of second-order nonlinearity, etc.

Due to the distinguished advantages, molecular nonlinear optics have been widely used in the biophotonics field, including bioimaging [31] , phototherapy [32] ,biosensing [33] , etc. From Wikipedia, the free encyclopedia. See also: Second-harmonic generation. Physical Review Letters. Bibcode : PhRvL November A Study of the Phosphorescent State". Journal of the American Chemical Society. Nonlinear Optics.

Bibcode : Natur. Submitted manuscript. Bibcode : PhRvL.. Retrieved July 4, Nat Phys. Bibcode : NatPh Physica D: Nonlinear Phenomena. Bibcode : PhyD.. Kouzov, N. Egorova, M.

### Capabilities

Chrysos, F. Boyd 3rd ed. Boyd, Nonlinear optics, Third edition, Chapter 2. Bijlani; Amr S. Helmy The medium oscillators can be electronic transitions, molecular vibrations and rotations, and acoustic waves [ 2 ]. Typically, only a small number of linear and nonlinear oscillator modes are important that satisfy the resonance conditions [ 1 , 2 , 3 ]. Here we for the sake of definiteness consider the one-dimensional case. The evolution of the waves 1 is described by the system of the coupled equations in the so-called SVE approximation SVEA when the higher-order derivatives of the SVE can be neglected according to conditions 2 [ 1 , 2 , 3 ].

The typical nonlinear optical phenomena are self-focusing, self-trapping, sum- and difference-frequency generation, harmonic generation, parametric amplification and oscillation, stimulated light scattering SLS , and four-wave mixing FWM [ 1 ]. During the last decades, optical communications and optical signal processing have been rapidly developing [ 1 , 2 , 3 , 4 ]. In particular, the nonlinear optical effects in optical waveguides and fibers became especially important and attracted a wide interest [ 1 , 2 , 3 , 4 ]. The nonlinear optical interactions in the waveguide devices have been investigated in detail in Ref.

Nonlinear fiber optics as a separate field of nonlinear optics has been reviewed in Ref. Silicon photonics, i. The nonlinear optical phenomena in Si nanostructures such as quantum dots QD , quantum wells QW , and superlattices had been discussed [ 6 ]. It has been shown that the second harmonic generation SHG in silicon nanostructures is possible despite the centrosymmetric structure of Si crystals [ 6 ].

Nonlinear dynamics in complex optical systems such as solid-state lasers, CO 2 lasers, and semiconductor lasers is caused by the light-matter interaction [ 7 ]. Under certain conditions, the nonlinear optical processes in such optical complex systems result in instabilities and transition to chaos [ 7 ].

In this chapter we briefly describe the basic nonlinear optical phenomena. The detailed analysis of these phenomena may be found in [ 1 , 2 , 3 , 4 , 5 , 6 , 7 ] and references therein. The chapter is constructed as follows. The mechanisms and peculiarities of the basic nonlinear effects mentioned above are discussed in Section 3. Conclusions are presented in Section 4. They have the form [ 4 ]. Equations 3 — 6 describe the vectors averaged over the volumes which contain many atoms but have linear dimensions smaller than substantial variations of the applied electric field [ 8 ].

## Fundamentals of Nonlinear Optics - CRC Press Book

Combining Eqs. Here c is the free space light velocity. Suppose that the electric field is a group of monochromatic plane waves given by [ 1 ].

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Then, the Fourier transform of the nonlinear polarization 1 yields [ 1 ]. We do not present here the analytical properties of the nonlinear susceptibilities which are discussed in detail in Ref. In some simple cases, the nonlinear susceptibilities can be evaluated by using the anharmonic oscillator model [ 1 , 8 ]. It is assumed that a medium consists of N classical anharmonic oscillators per unit volume [ 1 ]. Such an oscillator may describe an electron bound to a core or an infrared-active molecular vibration [ 1 ]. The nonlinear terms become essential when the electromagnetic power is large enough in such a way that a medium response cannot be considered linear anymore [ 8 ].

We limit our analysis with quadratic and cubic nonlinearities proportional to x 2 and x 3 , respectively [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 ]. In general case, the microscopic expressions for nonlinear susceptibilities of a medium are calculated by using the quantum mechanical approach. In particular, the density matrix formalism is a powerful and convenient tool for such calculations [ 1 , 2 , 7 , 8 ].

Electromagnetic waves in a medium interact through the nonlinear polarization 8 [ 1 ]. Typically, a nonlinear optical effect that occurs due to such an interaction is described by the coupled wave equations of the type 7 with the nonlinear susceptibilities 12 as the coupling coefficients [ 1 ]. In general case, the coupled wave method can also include waves other than electromagnetic [ 1 ].

For instance, in the case of SBS process, the acoustic waves are taken into account, and in the case of SRS process, the molecular vibrations are typically considered [ 1 , 2 , 4 ]. The coupled wave equations are usually solved by using SVEA 2 [ 1 ]. We start with the sum-frequency, difference-frequency, and second harmonic generation.

Similarly, in the case of the difference-frequency generation, we obtain [ 1 ]. The efficient nonlinear wave mixing can occur only under the phase-matching conditions. The detailed analysis of the sum-frequency generation, difference-frequency generation, and SHG in different configurations may be found in [ 1 , 3 , 6 ].

## Study Guide 12222-2020

It can be shown that the efficient sum-frequency generation can be realized under the following conditions [ 1 ]. The length of the nonlinear crystal must provide the required conversion efficiency. The efficient SHG can be realized with the single-mode laser beams focused into the nonlinear optical crystal [ 1 ].

Sum-frequency generation, difference-frequency generation, and SHG can be also carried out in the waveguide nonlinear optical devices [ 3 ]. Waveguide SHG devices can be used in optical signal processing such as laser printer, laser display, optical memory, short pulse, multicolor, and ultraviolet light generation [ 3 ].

These phenomena are much weaker than the second-order ones. Self-focusing is an induced lens effects caused by the self-induced wavefront distortion of the optical beam propagating in the nonlinear medium [ 1 ]. In such a medium, a refractive index n has the form [ 1 ]. Consequently, the central part of the beam travels at a smaller velocity than the beam edge.

As a result, the gradual distortion of the original plane wavefront of the beam occurs, and the beam appears to focus by itself [ 1 ]. The self-focusing results in the local increase of the optical power in the central part of the beam and possible optical damage of transparent materials limiting the high-power laser performance [ 1 ]. SPM is also caused by the positive refractive index change It is the temporal analog of self-focusing which leads to the spectral broadening of optical pulses [ 4 ].

In optical fibers, for short pulses and sufficiently large fiber length L f , the combined effect of the group velocity dispersion GVD and SPM should be taken into account [ 4 ]. In such a case, the pulse propagates in the optical fiber as an optical soliton, i. The optical solitons can propagate undistorted over long distances, and they can be applied in fiber-optic communications [ 4 ]. Consider now THG. Unlike SHG, it is always allowed [ 1 ].

For this reason, the laser intensity required for the efficient THG is limited by the optical damage in crystals [ 1 ]. THG can be realized in highly nonlinear optical fibers where the phase matching can be accomplished [ 4 ]. SBS is a nonlinear optical effect related to parametric coupling between light and acoustic waves [ 1 ]. The acoustic wave enhanced by the interacting pump and signal Stokes wave modulates the mass density of the medium which in turn modulates the refractive index [ 1 , 3 , 4 ].