Classical optics second-harmonic generation

Second-harmonic generation, which was observed in the early days of lasers [18] is probably the best known nonlinear optical process. Because of its simplicity and variety of practical applications, it is a starting point for presentation of nonlinear optical processes in the textbooks on nonlinear optics [1,2]. Classically, the second-harmonic generation means the appearance of the field at frequency 2co (second harmonic) when the optical field of frequency co (fundamental mode) propagates through a nonlinear crystal. In the quantum picture of the process, we deal with a nonlinear process in which two photons of the fundamental mode are annihilated and one photon of the second harmonic is created. The classical treatment of the problem allows for closed-form solutions with the possibility of energy being transferred completely into the second-harmonic mode. For quantum fields, the closed-form analytical solution of the... 

After an overview of the main papers devoted to chaos in lasers (Section I.A) and in nonlinear optical processes (Section I.B), we present a more detailed analysis of dynamics in a process of second-harmonic generation of light (Section II) as well as in Kerr oscillators (Section III). The last case we consider particularly in the context of coupled nonlinear systems. Finally, we present a cumulant approach to the problem of quantum corrections to the classical dynamics in second-harmonic generation and Kerr processes (Section IV). 

Nonlinear optics deals with physical systems described by Maxwell equations with an nonlinear polarization vector. One of the best known nonlinear optical processes is the second-harmonic generation (SHG) of light. In this section we consider a well-known set of equations describing generation of the second harmonic of light in a medium with second-order nonlinear susceptibility %(2 The classical approach of this section is extended to a quantum case in Section IV. 

Let us consider an optical system with two modes at the frequencies oo and 2oo interacting through a nonlinear crystal with second-order susceptibility placed within a Fabry-Perot interferometer. In a general case, both modes are damped and driven with external phase-locked driving fields. The input external fields have the frequencies (0/, and 2(0/,. The classical equations describing second-harmonic generation are [104,105] ... 

The phase mismatch which impedes the second-harmonic generation leads also to the cascading of quadratic nonlinearities and the induced phase shift. This effect has been used in continuous-wave optical devices and much effort has been devoted to the classical description of light propagation. We address the quantum theory of steady-state propagation of light and compare this formalism... 

Equations (9.21) arise in various physical contexts. In particular, they describe the process of second harmonic generation in nonlinear optics (see, e.g. (8)). In fact, they can be applied to any process in which two classical oscillating modes (waves) transform into one and other under the resonance condition 2uja wb. 

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