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Calculating Characteristics of Noncollinear Phase Matching
in Uniaxial and Biaxial Crystals
1. Introduction
The process of spontaneous parametric down conversion, in which a
"pump" photon is effectively split into a pair of lower energy
"signal" and "idler" photons in a nonlinear optical medium,
has proved abundantly useful in the last decade. The twin photons, which are
entangled in energy, momentum, and emission time, have been used in a variety
of striking demonstrations of the most nonclassical aspects of quantum theory
[1,2]. In addition, the downconverted photons have found
applications in the field of metrology, where they can be used to determine the
quantum efficiency of photon-counting detectors, and also to determine the
spectral radiance of an infrared source. The photon correlations of
down-converted light allow these measurement applications to be performed in a
fundamentally absolute manner as opposed to conventional methods which rely on
previously calibrated standards [3,4].
Calculation of the three-wave down-conversion interaction requires the use of
conservation of energy and conservation of momentum, commonly referred to as
phase matching. Because the process is nonresonant, a downconverted photon may
be emitted over a wide range of wavelengths, so long as the energy and momentum
conservation conditions for the pair of photons are met. The individual photons
of a pair may also propagate along different directions; this is referred to as
noncollinear phase matching. Collinear phase matching, where the
incident photon and the output pair of photons propagate in the same direction
inside the crystal, is generally well understood, while the noncollinear
geometry is more difficult to calculate and thus is poorly documented. One of
the advantages of noncollinear phase matching over the collinear case is that
it allows easy discrimination between each of the two downconverted photons and
the pump beam.
In this paper, we will describe a broadly applicable method of finding
noncollinear phase-matching configurations. We also provide examples obtained
from a computer program we have developed that implements our method and is
freely available on the Internet. We hope that the broad pool of calculable
crystal data included with this program (both uniaxial and biaxial crystals are
included) and wide spectral ranges that can now be calculationally investigated
will aid other researchers in designing their parametric down-conversion
experiments.
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