Calculating Characteristics of Noncollinear Phase Matching
in Uniaxial and Biaxial Crystals
3. Practice: Computational Results
We have implemented the second of the above methods in a FORTRAN program
designed to solve the phase-matching problem for a wide variety of pumping
conditions and crystal materials. The program may be freely downloaded from the
link at the top of this page and is capable of generating data for the
following kinds of plots (f indicates a function of the variables in
parentheses):
- 2-D Plot, nx, ny,
nz = f(
 pump)
- 3-D Plot, nslow -
nfast = f(
 signal,  signal)
- 3-D Plot,
 kminimum =
f(signal,
signal)
- 3-D Plot, Phase-Matching Function
 = f(ktransverse, kz)
- 2-D Plot, signal versus idler (signal fixed) for a Chosen Value of the
Phase-Matching Function
- Polar Plot, (Optimum signal, Optimum idler) = f(signal)
- 2-D Plot, Optimum signal = f(signal) at Chosen signal with Spreading in signal and signal, idler and idler Fixed
- 2-D Plot, Optimum signal = f(signal) at Chosen signal with Spreading in signal and signal
- 3-D Plot, Phase-Matching Function
=
f(signal,
signal)
- 3-D Plot, Phase-Matching Function
= f(signal,
signal)
We now proceed to give examples and discussion of the results for each of these
options.
3.1 Option 1: 2-D Plot, nx, ny,
nz = f(pump)
This plots the most basic information available for a specific crystal material,
namely, the variation of the indices of refraction with wavelength. The program
includes the coefficients in the Sellmeier-type index dispersion relations
[9] for a number of common nonlinear optical
materials, drawn from references that are cited in comment lines in the code.
Plots of this kind provide the first clue as to whether any phase matching will
be possible for a particular combination of pump, signal, and idler wavelengths.
Such a plot for  -barium borate (BBO) is shown in Fig. 4.
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Figure 4. Indices of refraction versus wavelength for BBO, a negative
uniaxial crystal. The graph indicates that "extraordinary" waves (polarized
parallel to the optical axis z) travel faster than "ordinary" waves (polarized
transverse to this axis).
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3.2 Option 2: 3-D Plot, nslow -
nfast = f(signal, signal)
Although it is fairly simple to determine crystal configurations that produce
phase matching in a uniaxial crystal (because there is only a single variable,
pump), determining
phase-matching regions in a biaxial crystal, such as KNbO3 is more
complex [10-12]. To determine the effect of both
variables (pump and
pump), graphs of
nslow - nfast versus
pump and
pump can be produced
(Fig. 5). As will be seen in the next option, phase matching usually
occurs where nslow - nfast differs
significantly from zero.
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Figure 5. Plot for nslow -
nfast versus
pump and
pump in
KNbO3 crystal with pump = 0.633 µm.
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3.3 Option 3: 3-D Plot, kminimum =
f(signal,
signal)
Graphs of the minimum value of
|k| versus
pump and
pump can also be produced
(see Fig. 6.) One can notice the similarity between Fig. 5 and
Fig. 6. If nslow - nfast
 0, then
kminimum
is large, while if nslow -
nfast is large, then
|kminimum|
0. It is clear that some difference between
nslow and nfast is required for
phase matching. These plots can be then used to quickly determine which crystal
configurations lead to phase matching.
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Figure 6. Graph of
|k| versus
pump and
pump in KNbO3
crystal with pump =
0.633 µm and signal = 0.950 µm.
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3.4 Option 4: 3-D Plot, Phase-Matching Function
= f(ktransverse, kz)
For crystals of finite length, the signal and idler vectors need not sum
exactly to the pump vector for some down-conversion to occur (see
Fig. 3). For these cases, the
down-conversion intensity will be weighted by the phase-matching function
, as defined in
(Eq. 20). This option generates data for
plots of (see Fig. 7), indicating
the regions of momentum-space around the pump vector into which the sum of
the signal and idler vectors must fall for down-conversion to occur. The
down-conversion intensity will be highest for the central regions where
1 (i.e., |k| 0)
and lowest for the outer regions where 0. Note
that the longer the crystal, the more constricted the phase-matching region
becomes in the  direction. Similarly, a
wider pump beam would restrict the phase-matching region, but in the transverse
direction.
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Figure 7. Phase-matching function for KDP crystals of three different
lengths and constant pump beam width of 2 mm (FWHM).
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3.5 Option 5: 2-D Plot, signal versus idler (signal fixed) for a Chosen Value of the Phase-Matching
Function
For a crystal of finite length and pump beam of finite width, there are many
combinations of signal and idler opening angles that can lead to
down-conversion at a given pair of signal and idler wavelengths. This option
generates a plot of all possible combinations of signal versus idler which result in the phase-matching function
falling to some specific value, say, = 0.5, for a particular pair of fixed down-conversion
wavelengths (see Fig. 8).
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Figure 8. Graph of signal versus idler ( = 0.5) for a KDP crystal (5 mm crystal length and
2 mm pump width) with pump = 0.351 µm,
pump = 0 deg,
pump = 52 deg,
signal =
0.633 µm, signal = 0 deg.
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3.6 Option 6: Polar Plot, (Optimum
signal,
Optimum idler) =
f(signal)
To map the down-conversion output, this option produces 2-D graphs
of the signal and idler output directions for a given signal frequency
(Fig. 9). This graph corresponds to a single crystal configuration
(pump and
 pump are fixed with
pump arbitrary because BBO
is uniaxial) and a single signal [which can be used with
(Eq. 1) to calculate
idler]. The configuration
in Fig. 9 was chosen because it shows both the collinear
(idler =
signal = 0) and
noncollinear cases. Both the internal and external angles for the emission are
calculated, although only the internal results are shown below. Multiple plots
of this kind with different signal and idler frequencies can be examined if
more complete results of the down-conversion are desired.
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Figure 9. Polar plot of the direction of signal and idler output photons
(signal, idler =
0.702 µm) for = 49.2 deg and pump = 0.351 µm from a BBO crystal. The
pump beam propagates out of the page at the origin.
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3.7 Option 7: 2-D Plot, Optimum
signal =
f(signal)
at Chosen signal
with Spreading in signal
and signal,
idler and
idler Fixed.
For any given pair of conjugate signal and idler wavelengths, there may exist
an optimum pair of emission angles
signal and
idler producing perfect
phase matching (i.e., satisfying (Eq. 2) and
yielding = 1).
Down-conversion will be strongest for these optimum combinations of wavelengths
and angles. This option provides data for plotting the optimum signal angle as
a function of signal wavelength, as shown in Fig. 10. For type I
down-conversion, the names "signal" and "idler" are
completely arbitrary, so that this is in fact a graph of both the signal and
idler emission angles. For type II down-conversion, one may find the idler
angles by running the option again and choosing the "signal" (now
really the idler) to be the slow wave instead of the fast wave, or vice versa.
Both the internal and external angles are reported (Fig. 10 displays
internal angles). The opening angles can be plotted for any choice of emission
plane, such as signal
= 0 deg.
If the crystal were infinitely long, down-conversion would occur only at these
optimal combinations of wavelength and angle. For crystals of finite length,
however, some emission will occur in a range of angles about the optimum for
each wavelength. The broader the phase-matching function, the larger this range
of angles becomes, as one might guess from examining
Fig. 3 and Fig. 7.
Therefore, option 7 also provides a first-order estimate of this spreading
in both signal and
signal as a function of
wavelength. For each signal wavelength, the spreading in the signal angles is
calculated assuming that the conjugate idler photon is emitted at precisely the
optimum opening angle for its wavelength, so that only
signal and
signal are allowed to vary.
The largest nonoptimal values of
signal and
signal that result in
falling to some specific value, say,
= 0.5, are found, and the
difference between these nonoptimal angles and the optimal angles are reported
in the data set as "spreads." They may be used to construct
error-bars or plotted independently as in Fig. 10.
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Figure 10. Upper curve shows optimum signal = f(signal), while the lower two curves show the
spreads in signal and
signal, with
idler and
idler fixed,
= 0.5. All curves for a
5 mm long KDP crystal and 2 mm pump beam width,
pump =
0.351 µm, pump = 52 deg,
pump =
0 deg, and signal = 0 deg.
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3.8 Option 8: 2-D Plot, Optimum
signal =
f(signal)
at Chosen signal with
Spreading in signal and
signal.
This option is the same as option 7, but the spreads in
signal at each wavelength
are computed in an iterative fashion that allows both the signal and the idler
to be emitted at a nonoptimal opening angle (Fig. 11). This provides a
more realistic estimate for the spreads than that given by the previous option,
but also requires more computing time. However, the spread in
signal is computed exactly
as in the previous option. For if the idler were not constrained to be emitted
in the plane chosen by the user (say,
idler = 180 deg
corresponding to the choice of optimum
signal = 0 deg)
then the sequence of iterations would simply map out the entire circle of
emission for both the signal and the idler. As in option 7, the spreads
that result in falling to some
user-defined "target" value like
= 0.5 are computed.
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Figure 11. Upper curve shows optimum signal = f(signal), while the lower two curves show the
spreads in signal and
signal, with
idler fixed,
= 0.5, for a 5 mm
crystal length and 2 mm pump width. Here idler is allowed to vary, as opposed to Fig. 10
where idler was fixed. This
difference produces a larger spread in signal. All curves for a 5 mm long KDP crystal and
2 mm pump beam width, pump = 0.351 µm,
pump = 52 deg,
pump = 0 deg, and
signal = 0 deg.
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3.9 Option 9: 3-D Plot, Phase-Matching Function
= f(signal, signal)
In this option, the value of the phase-matching function is computed for the
entire range of signal wavelength and angle combinations, within the domain of
validity of the Sellmeier coefficients for the chosen crystal. This is done by
repetition of option 8, with the "target" value of
incremented from 0.1 to 1. Because the
phase-matching function is a weight function for the emission of down-converted
pairs, a 3-D plot of (signal,
signal) can serve as a
crude picture of the relative intensity of the down-conversion as a function of
wavelength and angle (see Fig. 12). The intensity will be highest for
the optimum phase-matching combinations that result in
= 1. It is important to note
that such plots cannot provide completely accurate pictures of the
down-conversion intensity, since the probability of down-conversion is also
affected by the strength of the nonlinear electric susceptibility - another
frequency dependent quantity. However, if the values of
are compared over a range of frequencies
with nearly constant susceptibility, then their interpretation as relative
intensities for the down-conversion should be valid over that range.
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Figure 12. Plot for =
f(signal,
signal) for a KDP
crystal (5mm crystal length and 2 mm pump width) with
pump =
0.351 µm, pump = 0 deg, pump = 52 deg,
signal = 0 deg.
(Here signal is an external
angle.)
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3.10 Option 10: 3-D Plot, Phase-Matching Function
=
f(signal,
signal)
This option shows the variation of as a
function of signal wavelength and signal azimuthal angle (rather than opening
angle as in the previous option), assuming that the azimuthal angle of the
idler is fixed (as in option 7 and
option 8.) A 3-D plot of the results (shown in
Fig. 13) can be interpreted as plots of relative down-conversion
intensity versus wavelength and azimuthal angle, with the same caveats as
listed for option 9.
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Figure 13. Graph of =
f(signal,
signal) for a KDP
crystal (5 mm crystal length and 2 mm pump width) with
pump =
0.351 µm, pump = 0 deg, pump = 52 deg, and
signal = 0 deg
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