Cascaded Second Harmonic Generation with a Half-Order Periodical Orientation
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INEAR OPTICS
Cascaded Second Harmonic Generation with a Half-Order Periodical Orientation V. Yu. Mylnikova, N. S. Averkieva, and G. S. Sokolovskiia, * a
Ioffe Institute, Russian Academy of Sciences, St. Petersburg, 194021 Russia *е-mail: [email protected] Received June 28, 2020; revised June 28, 2020; accepted July 16, 2020
Abstract—We demonstrate theoretically cascaded fourth and second harmonics generation in a periodically oriented nonlinear crystal with a half-order of the periodical orientation. We consider a cascaded process in which four photons of the fundamental harmonic are initially converted into an intermediate photon of the fourth harmonic, which parametrically decays at the second stage into two photons of the second harmonic. We show that fractional phase matching can be achieved by using an asymmetric periodically poled grating. To describe the propagation and transformation of light inside a nonlinear crystal, the quantum spatial Heisenberg equations have been applied, using which the average numbers of photons of the fourth and second harmonics have been calculated as functions of the nonlinear crystal length. Keywords: second harmonic generation, nonlinear optics, phase matching DOI: 10.1134/S0030400X2011017X
INTRODUCTION Designing a visible laser with a wide tuning range is motivated by demands of the biomedical community, biophotonics, fluorescence microscopy, and spectroscopy. However, because the tuning is usually achieved by the use of several radiation sources, these systems are cumbersome, expensive, and difficult to manufacture, adjust, and operate. The use of the second harmonic generation (SHG) effect makes it possible to obviate these difficulties and appears to be a promising way to achieve tunable generation of radiation in the visible range. In the near future, a laser source based on an infrared semiconductor laser and frequency doubling in a nonlinear crystal [1–3] should replace gas and solid-state lasers, which are currently used for a wide class of biomedical applications. Due to the dispersion of the refractive index in a nonlinear crystal, the momentum conservation law is not satisfied in the general case, and the SHG is efficient only on the scale of the so-called coherence length: Lc = 1/2λ/|n(λ) – n(2λ)| [4], where λ is the wavelength of the second harmonic, while n(λ) and n(2λ) are the refractive indices of the crystal at the wavelengths of the second and fundamental harmonics, respectively. At the present time, the most used method to surmount the problem of violation of the law of conservation of momentum or the “phasematching” is the periodic orientation of a nonlinear crystal under the action of a high voltage with a period equal to two coherence lengths, Λ = 2Lc. As a result, the mismatch between the wave vectors for the SHG,
k(λ) – 2k(2λ), becomes exactly compensated by the grating wave vector of the periodic orientation, q0 = 2π/Λ. It was shown in [5] by Fejer et al. that the phase mismatch can be compensated not only by one, but also by several grating wave vec
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