Calculating Characteristics of Noncollinear Phase Matching
in Uniaxial and Biaxial Crystals
2. Theory: Phase-Matching Conditions in Uniaxial and Biaxial Crystals
2.1. Coordinate System, Equations and Variables
Consider a three-wave mixing process, where one photon incident on the crystal
interacts to produce a pair of lower-energy correlated photons by parametric
down-conversion. This study is carried out for the most general case, including
biaxial and uniaxial crystals, for noncollinear or collinear geometries and
for pairs of downconverted photons with or without equal frequencies. The two
main constraints are the conservation of energy,
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pump =
signal +
idler ,
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(Eq. 1)
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where pump is the frequency
of the incident photon and signal and idler are the frequencies of the two downconverted
photons, and the conservation of momentum,
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kpump = ksignal + kidler ,
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(Eq. 2)
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where kpump, ksignal, and
kidler, are the pump, signal and idler wave vectors,
respectively.
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Figure 1. The crystal axes and the laboratory frame axes.
(x, y, z) are crystal dielectric axes (the
optical plane is the x-z plane and
nz > ny >
nx);
(x,
y,
z) are rotated axes
(rotation angle , about
the axis z). (x,
y,
z) are laboratory frame
axes (rotation angle ,
about the axis y).
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Using spherical coordinates, the pump wave vector is expressed in the crystal
principal dielectric axes ,
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with the polar and azimuthal angles
pump and
pump
defined as shown in Fig. 1. In uniaxial crystals there is only one axis
allowing symmetry of revolution, so the direction of the pump can be specified
by a single angle, pump.
Thus, for uniaxial crystals the result of the calculations will not depend on
the azimuthal angle, pump.
However, for biaxial crystals, which lack that symmetry, two angles are
required. The angles are defined here according to the positive nonlinear
optics frame convention of Roberts [5].
Since the crystal dielectric axes are not convenient for calculating the
resulting output, we express the signal and the idler wave vectors in the lab
frame defined by the rotated axes ,
, , as shown
in Fig. 1. In the lab frame, the signal and idler wave vectors are
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where ni (i = pump, signal, idler) are the
refractive indices for the photons (for their individual states of polarization)
in the given direction of propagation i. Here pump is the angle between pump and the axis,
while pump is the azimuthal
angle (about ) from the
axis to pump in the x-y plane. For the down-conversion
beams, the opening angles signaland idler are specified relative to
pump, and the azimuthal angles
signal and
idler refer to rotations in
the plane normal to pump (see
Figure 2.)
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Figure 2. Another view of the crystal and laboratory frame
coordinates, showing a typical experimental arrangement for parametric
down-conversion within a crystal. In this figure, the x-z plane
(pump = 0 plane)
is in the plane of the page. For uniaxial crystals, the choice
pump = 0 can
always be made, but for biaxial crystals, this drawing represents a special
case in which the crystal axes C1 and C2,
and the pump beam all lie in the plane of the page. The signal beam is emerging
low and towards the viewer, while the idler beam is propagating high and away
from the viewer. The azimuthal angles signal and idler are measured from the x-z plane. Dots
indicate the points where the rays intersect the surface of the crystal.
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The cosine vectors of the propagation direction are: sx = sin cos,
sy = sincos and
sx = cos.
Note that the pump direction is specified with respect to the crystal axis (or
axes) in the xyz (lab) frame via
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while the signal and idler beams are specified relative to the pump beam via
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The transformation between coordinate systems is given by
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where =
pump and
=
pump.
The problem to be solved has variables:
pump,
pump,
signal,
signal,
idler,
idler,
pump,
signal, and
idler. These are related by
equations (Eq. 1) and (Eq. 2)
which yield one and three equations, respectively. Thus, we have nine variables
related by four equations. Five variables can therefore be chosen as
parameters to reduce the number of unknowns to equal the number of equations.
The pump direction and frequency (as given by
pump,
pump, and
pump) can clearly be chosen
as parameters. In addition, one of the downconverted photon frequencies can be
chosen, as well as its azimuthal angle. (In our analysis
signal and
signal are selected.)
In general (for uniaxial and biaxial crystals), there are two different
indices of refraction for a single direction of propagation. For uniaxial
crystals, those are the "ordinary" and the "extraordinary"
indices of refraction. For biaxial crystals, they are referred to as the
"fast" and the "slow," where the fast index is the smaller
of the two indices. Having two possible indices for each wavelength enables the
phase matching of kpump, ksignal and
kidler to be achieved in several ways, for example,
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kpump(fast) = ksignal(slow)
+ kidler(slow) ,
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Type I
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kpump(fast) = ksignal(fast)
+ kidler(slow) ,
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(Eq. 8)
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Type II
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kpump(fast) = ksignal(slow)
+ kidler(fast) .
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These are the most common phase-matching configurations, and are usually
classified by type [6]. The first line of (Eq. 8)
where the signal and idler beams have similar polarizations is referred to as
type I phase matching. The second and third lines are examples of type II
phase matching, in which the signal and idler polarizations are orthogonal; the
names "signal" and "idler" are arbitrary, and can be
assigned to either the fast or the slow wave. While it is theoretically
possible for the pump to be the slow ray, this does not usually lead to
phase matching in most materials.
Phase-matching in uniaxial crystals is often described in terms of the ordinary
and extraordinary indices. For example, in a "positive uniaxial"
crystal - one for which the extraordinary ray travels slower than the ordinary
ray - phase matching is achieved with the following combinations of the
ordinary and the extraordinary light:
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kpump(o)
= ksignal(e)
+ kidler(e) ,
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kpump(o)
= ksignal(o)
+ kidler(e) ,
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(Eq. 9)
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kpump(o)
= ksignal(e)
+ kidler(o) .
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We find the index of refraction n()
in a given direction =
(sx, sy, sz) using
the indicatrix equation given by Fresnel's equation of wave normals,
expressed in terms of the crystal principal dielectric axes
[7]:
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Here nx , ny , and
nz are the crystal principal refractive indices at a given
wavelength. For a biaxial crystal, nx <
ny < nz , while for a
uniaxial crystal, nx = ny =
no (ordinary) and nz =
ne (extraordinary). (Eq. 10) can be rewritten
as:
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where x = 1/[n2()]. Solving for x, we obtain one solution for each
possible polarization (fast or slow):
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with
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To solve the phase-matching problem, we choose a crystal and type of
phase matching. The only data needed are the indices of refraction of the
crystal. As already mentioned, we can select the pump frequency and direction,
(pump,
pump,
pump) and the signal
frequency and azimuthal angle
(signal,
signal). It is also clear
from (Eq. 2) that the three wave vectors must lie in a
plane so:
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This relation makes one of the three component equations represented by
(Eq. 2) redundant. So now we have three equations and
three unknowns remaining. Of these (Eq. 1) simply
relates idler to
pump and
signal, leaving just two
coupled equations and two unknowns.
2.2. Solving the Equations
The remaining variables,
signal and
idler must be found
simultaneously using (Eq. 2). This problem is complex
because the index of refraction depends on the wave vector direction, so in the
general biaxial case, we must solve (Eq. 10) to find
an index. This affects the magnitude of the wave vector as shown in
(Eq. 3) requiring that we solve (Eq. 2) using both
(Eq. 3) and (Eq. 10). Because this problem has no analytic solution,
it requires an iterative search routine. We can deal with this situation three
different ways. First, we may use two equations of (Eq. 2) to find a
relation between signal and
idler and then use the
remaining equation of (Eq. 2) to find its root with a root finding
subroutine (one equation and one unknown). Second, we may rewrite (Eq. 2)
as
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where
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k
= kpump -
ksignal -
kidler ,
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(Eq. 15)
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and find its minimum as a function of
signal and
idler. A final method is to
apply a 1-D minimization algorithm after obtaining a relation between
signal and
idler.
The first method finds the
k
minimum by resolving (Eq. 14) into the three following
equations:
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Then a root-finding subroutine is needed to solve these equations. This method
works well for uniaxial crystals, but produces erroneous results for some
biaxial crystals:
kx = 0,
ky = 0,
and
kz = 0
can be solved independently, but the resulting |k|
may not necessarily equal zero. This can happen because the
signal and
idler values required for
kx = 0
can be different from those required for
ky = 0
and
kz = 0.
Therefore, although this method is faster than the other methods, it requires
an independent check of |k| = 0. Furthermore,
in the case of a finite length crystal it is difficult to determine whether
phase matching is allowed, because in practice one can have phase matching even
when |k|
0.
The second method, treats k as
a vector quantity and finds the minimum
of |k| =
f(signal,
idler). For the idealized
case of an infinitely long crystal and infinitely wide pump beam,
|k| = 0. is
required for phase matching, because the interaction Hamiltonian contains an
integral over all space [8] producing a delta
function:
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However, for a finite crystal length L and a Gaussian transverse pump
intensity profile of finite width W, it is possible for down-conversion
to occur even when k
0, that is, with
imperfect phase matching. In this case, the interaction Hamiltonian integral
yields the phase-matching function:
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This function is a weighting function for the intensity of the emitted
down-conversion that has a maximum value of 1 for
|k| = 0, and
falls to zero as the phase mismatch |k| increases. We may then arbitrarily say that phase
matching occurs for values of |k| that yield 1/2
(see Fig. 3). This corresponds to
|kz|
2.783/L
in the direction of pump propagation (ktransverse = 0) or
|ktransverse|
1.177/W
in the plane orthogonal to pump direction
(kz = 0).
For this situation, the goal of our method is still to find the minimum of
|k| as a function of two variables,
signal and
idler, but we now must also
evaluate the resulting value of and
determine whether
1/2 or not.
Figure 3. Imperfect phase matching of the pump, signal, and idler
propagation vectors.
Because there is no general analytical method to find the minimum value of
|k| for each
possible signal angle under a given set of pumping conditions, we search for
this minimum iteratively, via a computer algorithm. This method is slower than
the first, but produces more reliable results for both uniaxial and biaxial
crystals. It is implemented in our computer program (see
Section 3) as follows :
- Set the value of pump,
pump,
pump,
signal, and
signal.
- Calculate kpump.
- Calculate idler and
idler (cf.
[Eq. 1) and (Eq. 13)].
- Initialize both the unknowns
signal and
idler to the value S
times 0.03 rad, where S is a scale factor chosen by the user.
alternatively, after the first iteration the user may choose to initialize
these variables with the optimum values found in the previous iteration.
- Call UNCMND, a 2-D minimization routine which returns
the minimum value of |k| and the optimum phase-matching values of
signal and
idler which correspond to
this minimum. UNCMND computes |k| and its first derivative, and uses Newton's method
to find the zero of the first derivative.
[UNCMND is
a public-domain FORTRAN routine available from a web site maintained by
the Information Technology Laboratory at NIST.]
- Write these values to an output file.
- If signal or
signal is final value, then
end; otherwise, increment signal or signal and go to step 3.
The third method for solving (Eq. 2) begins
by rewriting it as follows:
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By adding the squares of these two equations, one obtains:
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Then, using (Eq. 23), (Eq. 21)
can be rewritten as:
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to provide a relation between the two unknowns. We can then use a 1-D
minimization function for k.
although it can save calculation time, this method was not implemented because
it assumes signal is given
by a definite relation to idler (i.e., perfect phase matching) and so it does not
lend itself to finding output spreading where
k
0.
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