Second-harmonic-generation rings and refractive-index measurement in uniaxial crystals

Rick Trebino

doi:10.1364/AO.20.002090

Applied Optics
Vol. 20,
Issue 12,
pp. 2090-2096
(1981)
•https://doi.org/10.1364/AO.20.002090

Second-harmonic-generation rings and refractive-index measurement in uniaxial crystals

Rick Trebino

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Rick Trebino, "Second-harmonic-generation rings and refractive-index measurement in uniaxial crystals," Appl. Opt. 20, 2090-2096 (1981)

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Abstract

Noncollinear second-harmonic-generation (SHG) processes involving an unfocused laser beam and its scattered radiation can act to produce one or more cones of phase-matched second harmonic, resulting in rings on an observation screen. We calculate expressions for the ring parameters (center and radius) for types I and II uniaxial crystals in terms of the crystal indices of refraction and geometry. Measurement of the ring parameters for several crystal tilt angles can yield very accurate relative values for the indices of refraction of a uniaxial SHG crystal. A method for the automatic maintenance of optimum SHG which does not require the placement of optics in the beam is also suggested.

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Equations (45)

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	Type of Phase–matching	Polarization			Formulas for Ring Centers (α_C) and Ring Radii (α_R) (to first order)

		Main Beam	Scattered Fundamental	Second Harmonic

	I	0	0	E	$α_{C} = \frac{- A_{2}}{n}; α_{R} = \sqrt{{(\frac{A_{2}}{n})}^{2} + \frac{2 Δ n}{n}}$
^(a)	II	0	E	E	$α_{C} = \frac{A_{1} - A_{2}}{n}; α_{R} = \sqrt{{(\frac{A_{1} - A_{2}}{n})}^{2} + \frac{2 Δ n}{n}}$
	II	E	0	E	$α_{C} = \frac{- A_{2}}{n}; α_{R} = \sqrt{{(\frac{A_{2}}{n})}^{2} + \frac{2 Δ n}{n}}$

	Type of Phase–matching	Polarization			Formulas for Ring Centers (α_C) and Ring Radii (α_R) (to first order)

		Main Beam	Scattered Fundamental	Second Harmonic

	I	E	E	0	$α_{C} = \frac{A_{1}}{n}; α_{R} = \sqrt{{(\frac{A_{1}}{n})}^{2} + \frac{2 Δ n}{n}}$
^(b)	II	0	E	0	$α_{C} = \frac{A_{1}}{n}; α_{R} = \sqrt{{(\frac{A_{1}}{n})}^{2} + \frac{2 Δ n}{n}}$
	II	E	0	0	α_C = 0; $α_{R} = \sqrt{\frac{2 Δ n}{n}}$

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Applied Optics