We derive explicit expressions in the frame of the optical indicatrix for the second-order effective nonlinearity in biaxial crystals with point groups 2, m, and 1, governing the conversion efficiency in three-wave nonlinear optical interactions. The tabulated expressions for the monoclinic symmetry classes 2 and m are valid for all possible orientations of the optical indicatrix relative to the crystallographic frame and for propagation along an arbitrary direction outside the principal planes. They can be used for direct estimation of the effective nonlinearity in the same frame where the phase-matching loci are calculated. The relevant properties and conventions used for the newly emerging acentric monoclinic crystals belonging to the borate family are summarized and tabulated. The derivations are expected to help establish adherence to uniform nomenclature and conventions for these novel inorganic nonlinear crystals, and to eliminate ambiguity and increasing confusion in the literature and in the industrial specifications. The general expressions for the effective nonlinearity are reduced for triclinic crystals of point group 1 to simplified forms in the principal planes.
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Forms of the dil Tensor in the Dielectric Frame xyza
b≡
Point group 2
Point group m
±x
±y
±z
The two orientations of the xyz frame relative to the crystallographic frame abc correspond to the reversal of two of the principal optical axes that preserves the chirality. Assuming that none of the dielectric axes lies with its positive direction in the sector spanned by the −a and −c crystallographic axes, the reversed axes in the three cases (rows in the table) are xz, yx, and zy, respectively.
Table 2
Effective Nonlinearity for Symmetry Class 2 Crystals and Arbitrary Propagation Direction
b
x
− (d11 cos2 φ + 3d12 sin2 φ)cos3θ cos φ sin δ cos2 δ
+ (d11 cos2 φ + 3d12 sin2 φ)cos3θ cos φ sin2 δ cos δ
The SHG results and effective nonlinearities used for rescaling refer to 1064 nm. The convention from Table 5 required a change in the signs of d11, d12, and d13 in comparison with the literature cited.
Phase-matched SHG, relative to β -barium borate (BBO) rescaled here assuming deff(BBO) = 2.16 pm/V.4
Absolute SHG measurements.
Phase-matched SHG, relative to KTP, rescaled here for comparison using deff(KTP) = 2.45 pm/V.
Phase-matched SHG and Maker fringes relative to KTP with deff(KTP) = 2.45 pm/V assumed.
Separated-beams method.
Maker fringes relative to LBO, signs of d24 and d33 determined from complete neglect of differential overlap (CNDO) theory, rescaled here to d31(LBO) = 0.67 pm/V.4
Gaussian 92 theory.
CNDO theory.
Phillips–Van Vechten dielecric theory and bond charge model, transformed here from a XYZ frame, where Y ≡ b and Z ≡ c, to the xyz frame.
Table 8
Expressions for deff in the Principal Planes of the Dielectric Frame xyz for Triclinic Acentric Crystals of Point Group 1
Forms of the dil Tensor in the Dielectric Frame xyza
b≡
Point group 2
Point group m
±x
±y
±z
The two orientations of the xyz frame relative to the crystallographic frame abc correspond to the reversal of two of the principal optical axes that preserves the chirality. Assuming that none of the dielectric axes lies with its positive direction in the sector spanned by the −a and −c crystallographic axes, the reversed axes in the three cases (rows in the table) are xz, yx, and zy, respectively.
Table 2
Effective Nonlinearity for Symmetry Class 2 Crystals and Arbitrary Propagation Direction
b
x
− (d11 cos2 φ + 3d12 sin2 φ)cos3θ cos φ sin δ cos2 δ
+ (d11 cos2 φ + 3d12 sin2 φ)cos3θ cos φ sin2 δ cos δ
The SHG results and effective nonlinearities used for rescaling refer to 1064 nm. The convention from Table 5 required a change in the signs of d11, d12, and d13 in comparison with the literature cited.
Phase-matched SHG, relative to β -barium borate (BBO) rescaled here assuming deff(BBO) = 2.16 pm/V.4
Absolute SHG measurements.
Phase-matched SHG, relative to KTP, rescaled here for comparison using deff(KTP) = 2.45 pm/V.
Phase-matched SHG and Maker fringes relative to KTP with deff(KTP) = 2.45 pm/V assumed.
Separated-beams method.
Maker fringes relative to LBO, signs of d24 and d33 determined from complete neglect of differential overlap (CNDO) theory, rescaled here to d31(LBO) = 0.67 pm/V.4
Gaussian 92 theory.
CNDO theory.
Phillips–Van Vechten dielecric theory and bond charge model, transformed here from a XYZ frame, where Y ≡ b and Z ≡ c, to the xyz frame.
Table 8
Expressions for deff in the Principal Planes of the Dielectric Frame xyz for Triclinic Acentric Crystals of Point Group 1