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Correlated Photon Radiometry at NIST
 
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,

omega   pump = omegasignal + omegaidler , (Eq. 1)

where omegapump is the frequency of the incident photon and omegasignal and omegaidler are the frequencies of the two downconverted photons, and the conservation of momentum,
  kpump = ksignal + kidler , (Eq. 2)
where kpump, ksignal, and kidler, are the pump, signal and idler wave vectors, respectively.
 
Figure 1. See caption below

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); (xprime, yprime, zprime) are rotated axes (rotation angle phi, about the axis z). (xprime, yprime, zprime) are laboratory frame axes (rotation angle phi, about the axis yprime).

Using spherical coordinates, the pump wave vector is expressed in the crystal principal dielectric axes x-hat, y-hat, z-hat with the polar and azimuthal angles thetapump and phipump 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, thetapump. Thus, for uniaxial crystals the result of the calculations will not depend on the azimuthal angle, phipump. 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 x-hatdouble prime, y-hatdouble prime, z-hatdouble prime, as shown in Fig. 1. In the lab frame, the signal and idler wave vectors are


  eq03 (Eq. 3)

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 s-hati. Here thetapump is the angle between s-hatpump and the z-hat axis, while phipump is the azimuthal angle (about z-hat) from the x-hat axis to s-hatpump in the x-y plane. For the down-conversion beams, the opening angles thetasignaland thetaidler are specified relative to s-hatpump, and the azimuthal angles phisignal and phiidler refer to rotations in the plane normal to s-hatpump (see Figure 2.)
Figure 2. See caption below

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 (phipump = 0 plane) is in the plane of the page. For uniaxial crystals, the choice phipump = 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 phisignal and phiidler are measured from the x-z plane. Dots indicate the points where the rays intersect the surface of the crystal.

The cosine vectors of the propagation direction s-hat are: sx = sin thetacosphi, sy = sinthetacosphi and sx = costheta. Note that the pump direction is specified with respect to the crystal axis (or axes) in the xyz (lab) frame via

  equation to calculate s-hat sub pump (Eq. 4)

while the signal and idler beams are specified relative to the pump beam via

  equation to calculate s-hat sub signal (Eq. 5a)

equation to calculate s-hat sub idler 		(Eq. 5b)

The transformation between coordinate systems is given by

  transformation equation (Eq. 6)

  transformation equation 		(Eq. 7)

where theta = thetapump and phi = phipump.

The problem to be solved has variables: thetapump, phipump, thetasignal, phisignal, thetaidler, phiidler, omegapump, omegasignal, and omegaidler. 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 thetapump, phipump, and omegapump) 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 omegasignal and phisignal 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,

   kpump(fast) = ksignal(slow) + kidler(slow)   , Type I  
 
kpump(fast) = ksignal(fast)   + kidler(slow)   ,   (Eq. 8)
  Type II
kpump(fast) = ksignal(slow) + kidler(fast)   .    


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:

  kpump(o) = ksignal(e) + kidler(e)   , 		 
 
kpump(o) = ksignal(o) + kidler(e)   , (Eq. 9)
 
kpump(o) = ksignal(e) + kidler(o)   .  

We find the index of refraction n(s-hat) in a given direction s-hat = (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]:
  Fresnel's equation of wave normals (Eq. 10)

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:

 Fresnel's equation of wave normals rewritten (Eq. 11)
where x = 1/[n2(s-hat)]. Solving for x, we obtain one solution for each possible polarization (fast or slow):
eq12 (Eq. 12)

with

Equations for B and C (Eq. 12a)

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, (omegapump, thetapump, phipump) and the signal frequency and azimuthal angle (omegasignal, phisignal). It is also clear from (Eq. 2) that the three wave vectors must lie in a plane so:

  phiidler = phisignal + pi (Eq. 13)
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 omegaidler to omegapump and omegasignal, leaving just two coupled equations and two unknowns.

2.2. Solving the Equations

The remaining variables, thetasignal and thetaidler 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 thetasignal and thetaidler 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

  |Deltak| = 0 , (Eq. 14)

where

  Deltak = kpump - ksignal - kidler , (Eq. 15)

and find its minimum as a function of thetasignal and thetaidler. A final method is to apply a 1-D minimization algorithm after obtaining a relation between thetasignal and thetaidler.

The first method finds the Deltak minimum by resolving (Eq. 14) into the three following equations:

   Deltakx = 0 , (Eq. 16)
  Deltaky = 0 , (Eq. 17)
  Deltakz = 0 . (Eq. 18)

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: Deltakx = 0, Deltaky = 0, and Deltakz = 0 can be solved independently, but the resulting |Deltak| may not necessarily equal zero. This can happen because the thetasignal and thetaidler values required for Deltakx = 0 can be different from those required for Deltaky = 0 and Deltakz = 0. Therefore, although this method is faster than the other methods, it requires an independent check of |Deltak| = 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 |Deltak| not equal 0.

The second method, treats Deltak as a vector quantity and finds the minimum of |Deltak| = f(thetasignal, thetaidler). For the idealized case of an infinitely long crystal and infinitely wide pump beam, |Deltak| = 0. is required for phase matching, because the interaction Hamiltonian contains an integral over all space [8] producing a delta function:

  a delta function (Eq. 19)

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 Deltak not equal 0, that is, with imperfect phase matching. In this case, the interaction Hamiltonian integral yields the phase-matching function:

  a phase-matching function (Eq. 20)

This function is a weighting function for the intensity of the emitted down-conversion that has a maximum value of 1 for |Deltak| = 0, and falls to zero as the phase mismatch |Deltak| increases. We may then arbitrarily say that phase matching occurs for values of |Deltak| that yield Phi greater or equal to 1/2 (see Fig. 3). This corresponds to |Deltakz| less than or equal to 2.783/L in the direction of pump propagation (Deltaktransverse = 0) or |Deltaktransverse| less than or equal to 1.177/W in the plane orthogonal to pump direction (Deltakz = 0). For this situation, the goal of our method is still to find the minimum of |Deltak| as a function of two variables, thetasignal and thetaidler, but we now must also evaluate the resulting value of Phi and determine whether Phi greater or equal to 1/2 or not.

 
Imperfect phase matching of the pump, signal, and idler propagation vectors.

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 |Deltak| 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 :
  1. Set the value of lambdapump, thetapump, phipump, lambdasignal, and phisignal.


  2. Calculate kpump.


  3. Calculate lambdaidler and phiidler (cf. [Eq. 1) and (Eq. 13)].


  4. Initialize both the unknowns thetasignal and thetaidler 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.


  5. Call UNCMND, a 2-D minimization routine which returns the minimum value of |Deltak| and the optimum phase-matching values of thetasignal and thetaidler which correspond to this minimum. UNCMND computes |Deltak| 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.]


  6. Write these values to an output file.


  7. If lambdasignal or phisignal is final value, then end; otherwise, increment lambdasignal or phisignal and go to step 3.


The third method for solving (Eq. 2) begins by rewriting it as follows:

  eq21 (Eq. 21)
  eq22 (Eq. 22)
By adding the squares of these two equations, one obtains:

 eq23 (Eq. 23)

Then, using (Eq. 23), (Eq. 21) can be rewritten as:

 eq24 (Eq. 24)

to provide a relation between the two unknowns. We can then use a 1-D minimization function for Deltak. although it can save calculation time, this method was not implemented because it assumes thetasignal is given by a definite relation to thetaidler (i.e., perfect phase matching) and so it does not lend itself to finding output spreading where Deltak not equal to 0.

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For technical information or questions, contact:

Alan Migdall
Phone: (301) 975-2331
Fax: (301) 869-5700
Email: amigdall@nist.gov

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Online: August 2002   -   Last updated: May 2003