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arXiv:2209.06145 · Symmetry

Exact Solutions for Vector Phase-Matching Conditions in Nonlinear Uniaxial Crystals

Juan López-Durán, Óscar Rosas-Ortiz

Published Oct 29, 2022• 4 citations
arXivHTMLPDF

Abstract

The transcendental equations of vector phase matching are transformed into a fourth-order polynomial equation that admits an analytical solution. The real roots of this equation provide the optical axis orientations that are useful for efficient down-conversion in nonlinear uniaxial crystals. The production of entangled photon pairs is discussed in both collinear and non-collinear configurations of the spontaneous parametric down-conversion (SPDC) process. Degenerate and non-degenerate cases are also distinguished. As a practical example, SPDC processes of type-I and type-II are studied for beta-barium borate (BBO) crystals. The predictions are in very good agreement with experimental measurements already reported in the literature and include theoretical results of other authors as particular cases. Some properties that seem to be exclusive to BBO crystals are reported; the experimental verification of the latter would allow a better characterization of these crystals.

DOI
10.3390/sym14112272

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The transcendental equations of vector phase matching are transformed into a fourth-order polynomial equation that admits an analytical solution. The real roots of this equation provide the optical axis orientations that are useful for efficient down-conversion in nonlinear uniaxial crystals. The production of entangled photon pairs is discussed in both collinear and non-collinear configurations of the spontaneous parametric down-conversion (SPDC) process. Degenerate and non-degenerate cases are also distinguished. As a practical example, SPDC processes of type-I and type-II are studied for beta-barium borate (BBO) crystals. The predictions are in very good agreement with experimental measurements already reported in the literature and include theoretical results of other authors as particular cases. Some properties that seem to be exclusive to BBO crystals are reported; the experimental verification of the latter would allow a better characterization of these crystals.
Abstract Spontaneous parametric down-conversion (SPDC) is the most widely used process for generating photon pairs entangled in various degrees of freedom such as polarization, time-energy, position-transverse momentum, and angle-orbital angular momentum (OAM). In SPDC, a pump photon interacts with a non-linear optical crystal and splits into two entangled photons called the signal and the idler photons. The SPDC process has been studied extensively in the last few decades for various pump and crystal configurations, and the entangled photon pairs produced by SPDC have been used in numerous experimental studies on quantum entanglement and entanglement-based real-world quantum-information applications. In this tutorial article, we present a thorough study of phase matching in BBO crystals for spontaneous parametric down-conversion and thereby also investigate the generation of entangled photons in such crystals. First, we present a theoretical derivation of two-photon wavefunction produced by SPDC in the frequency and transverse momentum bases. We then discuss in detail the effects due to various crystal and pump parameters including the length of the crystal, the angle between the optic axis and the pump propagation direction, the pump incidence angle on the crystal surface, the refraction at the crystal surfaces, and the pump propagation direction inside the crystal. These effects are extremely relevant in experimental situations. We then present our numerical and experimental results in order to illustrate how various experimental parameters affect the phase matching and thus the generation of entangled photons. Finally, using the two-photon wavefunction in the transverse wave-vector basis, we show how to derive the two-photon wavefunction in the OAM basis and thereby calculate the two-photon angular Schmidt spectrum. We expect this article to be useful for researchers working in various capacities with entangled photons generated by SPDC in BBO crystals.
Spontaneous parametric upconversion (SPUC) and spontaneous parametric downconversion (SPDC) have been observed in the β-barium borate (BBO) crystal, and their mechanisms are experimentally and theoretically investigated. SPUC, tuned from 530 to 600 nm with a FWHM of about 25 nm, can be attributed to the sum frequency between the quantum noise and the fundamental laser pulse in the type II phase matching condition (e+o→e). SPDC, tuned from 480 to 520 nm with a FWHM of about 15 nm, can be attributed to the difference frequency between the quantum noise and the fundamental laser pulse in the type I phase matching condition (e→o+o).
We theoretically investigate the effective nonlinear coefficient d eff of spontaneous parametric down-conversion (SPDC) process in biaxial nonlinear crystal BiB3O6 (BIBO). We solve the Fresnel equations in the collinear degenerate down-converted configuration, and get the relations of phase-matching angles. Considering the double phase-matching, we compute the phase-matching angles with different wavelength. Furthermore, we simplify the expression of effective nonlinear coefficient out of the principal planes, and numerically calculate it varying with the angle and wavelength. These results indicate that there are three effective phase-matching schemes for SPDC which can be used to construct a system to generate entangled photon pairs.
We investigate the high-purity entangled photon pair generation in a recently developed borate crystal, Ba3Mg3(BO3)3F3. The technique is based on the spontaneous parametric down-conversion under the extended phase matching (EPM), where the phase matching and the group velocity matching between the interacting photons are satisfied simultaneously in bulk crystals with point symmetry of orthorhombic mm2 (thus showing biaxial birefringence). We will discuss all the theoretical aspects required for the generation of photon pairs in mm2 biaxial crystals, which are much more complex than the cases of uniaxial crystals (e.g., β-BaB2O4 and LiNbO3) and periodically poled crystals that are widely used in the field. Our study includes theoretical and numerical investigations of two types of EPM and their corresponding effective nonlinearities and spatial walk-offs. The results show that two types of EPM are satisfied over the specific range in the direction of pump wave vector, corresponding to its spectral ranges of 876.15–1052.77 nm for Type I and 883.92–914.33 nm for Type II. The joint spectral analyses show that photon-pairs can be generated with high purities of 0.997 with a proper pump filtering (for Type II), and 0.833 even without pump filtering (for Type I).
Nonlinear crystals are typically used when interaction of different frequencies of light is requested. In classical optics these nonlinear phenomena are used for second-harmonic generation, sum-frequency generation, optical parametric amplification and many other effects. In quantum optics, dealing with optical interaction on the level of individual photons, the most prominent process is spontaneous parametric down-conversion (SPDC),1 where the crystal is pumped by intensive laser light and the crystal can mediate the splitting of a pump photon to a photon pair. The two generated photons are typically called signal and idler. Influence of magnetic field on these nonlinear processes was not thoroughly tested yet. This topic deserves intensive study both from theoretical and experimental point of view, because the magnetic field can decrease the symmetry of the nonlinear crystal and so it may allow to use new types of phase-matching conditions. We started to test the SPDC process in BBO crystals. Nonlinear magneto-optic tensor of this material is not known and we can hardly predict it. According to our first theoretical derivations the efficiency of the nonlinear processes has to oscillate when rotating the magnetic-field orientation.