Non-linear Optical Materials 3.1 Introduction
3.2 Considerations for material selection
3.2.2 Tuning range
Even if a material has a large non-linearity it may not be of use unless the crystal is optically transparent in the wavelength region of interest and there is sufficient birefringence available to facilitate phase matching*. As mentioned previously in chapter 2, phase matching requires the three interacting waves to satisfy the vector
relation given in eqn 2.4.8 This can be achieved by all three beams having parallel wavevectors, collinear phase matching, or their being small angles between the wavevectors, non-collinear phase matching. Non-collinear phase matching has the experimental advantage of eliminating feedback on the resonator axis for one or both of the generated waves [14,15], but it does so at the expense of a beam walk-off effect analogous to Poynting vector walk-off.
Phase matching calculations
To simplify calculation of tuning curves it was decided to limit the operation of the OPO to tuning in the principal dielectric planes. This may result in a small reduction in the maximum non-linear coefficient available for two of the materials considered, but does so in favour of simplified calculations and experimental operation.
When the wavevectors of the interacting beams are restricted to operation in one of the principal dielectric planes, then biaxial crystals may be treated with the same, and much simpler, formalism which can be applied to uniaxial crystals. That is, beams which have their polarisation vector in the chosen principal dielectric plane produce elliptical wavefronts and have a refractive index which depends on the angle of propagation to the dielectric axes. These beams are termed extra-ordinary waves, or e-waves. If the polarisation vector is normal to the plane, then the refractive index is independent of the angle of propagation in the plane. These beams are termed ordinary waves, or o-waves. Crystals can be classified by their birefringence, n^ - no, where n^ and no are the refractive indices of the e and o waves respectively. Thus, a crystal which has ne less than no is termed negative, and the case ne larger than no is termed positive. This gives rise to four types of phase matching. In a negative crystal, the possible phase matching interactions are, type I - two o waves produce (or are produced by) an e wave, and type II - one o wave and one e wave produce (or are produced by) an e-wave. The corresponding cases for positive crystals are, type I - ee <-> o, and type II - oe <-> o. The phase matching conditions for second harmonic generation in uniaxial crystals [16]
Note here that although phase tnatching is usually achieved by use o f the birefringence of the material, natural or induced, quasi-phase matching, as originally proposed by Bloembergen et al [12], where the sign of the phase mismatch or non-linear coefficient is periodically reversed is now proving to be a viable alternative [14], This technique also allows use o f non-linear coefficients which are not accessible via
Ch. 3 : Non-linear Optical Materials
and biaxial crystals [17,18,19] have already been published in the literature. However these results are cumbersome and not simple to apply. By restricting our approach to consideration of propagation in the principal planes, each plane can be treated in an analogous way to a uniaxial crystal, as previously discussed.
% n
fig. 3.1 - axial convention for phase match calculations
If we consider the coordinate system shown in fig. 3.1, then the two refractive indices of interest for the case 0 = 90° (i.e. propagation in the ij plane) will be
Hj sin 0 + «,• cos 0
(3.2.2.1) (3.2.2.2) These equation apply to any of the three principal planes of the xyz dielectric axial system if the following index relations are applied
xy plane : n,- = n^,rij = rty^n^ = xz plane :
yz plane : n, = n^yiij = riy^rif. =
(3.2.2.3a) (3.2.2.3b) (3.2.2.3c) The way in which the equations for phase matching have been derived makes it only necessary to refer to one angle. The angle of interest is (J> for the xy plane, and 0 for the xz and yz planes. The general equations for phase matching in the principal planes were obtained by inserting the refractive index equations (3.2.2.1 and 2) into the phase match equation 2.5.1.1 to obtain the following
e-oo cos d =
y >12
1 1
0-ee f{6 ) =
Ch. 3 : Non-linear Optical Materials
+(rt,(A,j)cos0y
+
\V2
o-oe cos^ Q =
(rt^(A,)sin0) +(«,(A ,)cos0)
1 -»jk(^p) = 0 (3.2.2.4b) e-eo f{9 ) = \^jn,{X,)nj{Xs) sy2 (3.2.2.4c) (ny(Ap)sine) +(n,(A^)cos0y _______ >h{Xj)nj{Xi)
&
■\‘A (n^(A;)sinô) +(/ij(Ai)cos0)‘N
"k{X,) = 0 (3.2.2.4d)The appropriate equations for the uniaxial case require only the additional relation nj = nic which gives
rig = rii and = rij - n^. (3.2.2.5) Equations (3.2.2.4b and d) are written in the form f(9) = 0. This is because they are not analytically solvable but can be solved easily by numerical iterative techniques. The method of solution chosen was 'the method of false position'. Despite the fact that some other methods converge faster, this method was chosen as the initial estimates are not required to be accurate and no derivatives are involved.
It is appropriate to also present, for the case of propagation within a principal dielectric plane, the equation for the Poynting vector walk-off angle p, which in this formalism is
[
20]
p = tan-1 i^nf -? ij)sin2 0
2{^nj sin^ 9 + nf cos^ (S.2.2.6) The walk off angle in the case of uniaxial ciystals is defined in terms of He and Uq. As it is not solely uniaxial crystals being considered here, the walk off equation has been defined to apply to any principal plane with indices n\ and nj, where 9 is the angle measured from the i-axis. The relations between ijk and xyz defined previously in
Ch. 3 : Non-linear Optical Materials (3.2.2.3) still apply.
Equations (3.2.2.4) were used to investigate a number of materials which looked promising for use in an OPO. Some of the phase match curves so generated will be presented later when individual materials are discussed with respect to their potential for construction of an OPO.
The tuning range of an OPO is ultimately limited by the region of the spectrum over which the material is transparent, assuming that the gain is sufficient and the process can be phase matched over the region of transparency. However what must also be borne in ntind is that the transparency range quoted in the literature for a crystal is likely to be between the two wavelengths at which the transmission goes to zero. This does not allow for transmission roll-offs where the transmission gradually falls to zero, which then acts as an additional loss mechanism for the OPO and will eventually prevent oscillation. Transmission roll-offs are particularly relevant at the infrared end of the transparency, as is the case in KTP [21,22].
3.2.3 Calculation of deff for the principal planes
As briefly discussed in section 3.2.1, the non-linear coefficient tensor is determined by the symmetry of the crystal. This can be described, in the reduced notation [23], by a 6
X 3 matrix which relates the vector components of the induced non-linear polarisation to those of the electric fields which generate them. In the previous section, the use of different electric field polarisation vectors to obtain biréfringent phase matching was discussed. It is therefore desirable to be able to calculate the magnitude of the non- linearity that will be available for different phase matching geometries. In this section we will discuss the calculation of an effective non-linear coefficient, deff, which may be applied to individual phase match geometries. We will also see that although certain wavelength combinations can be phase matched, the phase match geometry required can result in a deff of zero, which results in the exchange of energy between waves being disallowed.
The effective non-linear coefficient can be defined for each crystal point group as [18] (3.2.3.1) where Ui (i =1,2,3) are the unit vectors of the three interacting fields, d is the non-linear coefficient tensor, and summation over indices is assumed. The unit vectors can be expressed in terms of the direction cosines, which in turn are expressed in terms of the polar coordinates 0 and (j) [18,19]. This approach gives general results applicable to propagation in an arbitrary direction through the crystal. However, the results so produced are complicated expressions which do not give any immediate insight and are often solved numerically.
Ch. 3 : Non-linear Optical Materials
The adoption of operation in the principal planes explained earlier with respect to phase match calculations also allows a simplified derivation of deff which gives a more immediate insight into the behaviour of the non-linear coefficient with propagation direction.
The second order polarisation vector components induced by the interaction of two applied electric fields Ei and E2 can be expressed as
(3.2.3.2) where i = x,y,z are the components along the dielectric axes, are direction cosines, dijk is the non-linear coefficient tensor, and summation over the indices jk is implied. The easiest way to demonstrate this with respect to its simple application to operation in the principal planes is with an example, e- 0 0 phase matching in the yz-plane of KNbOs,
a crystal with mm2 symmetry.
Pc
a
fig. 3.2 - calculation of deff for e- 0 0 phase matching in the yz plane of an mm2 crystal.
For an interaction in the yz plane, the effective o-wave will be polarised in the x direction. Thus, the components of two waves E% and E%, magnitude Eq, will be
ax-=l a i- 0 (3.2.3.3a) (3.2.3.3b) (3.2.3.3c) (3.2.3.4a) (3.2.3.4b) (3.2.3.4c)
For mm2 symmetry the only nonzero coefficients are dsi, ds2, dg], d2 4, and d\^, where
the specific relations for this crystal class between the dielectric and crystallographic axes have been applied. The induced non-linear polarisation components are then
Px = ^xxz^i^oaiEo + dx^aiEoülEo = 0 (3.2.3.5a)
Py ~ dyy^UyEoüiEo + dy,yaiEoüyE(j = 0 (3.2.3.5b)
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Phase matching requires an e-wave from the induced polarisation of the two o-waves. Py and Pz both contribute to Pg, the e-wave driving polarisation. From fig.3.2 and with Py = 0, the effective polarisation producing the e-wave is
P, = P^sm6 (3.2.3.6)
Thus the effective non-linear coefficient is
deff ~ djj sin 0 (3.2.3.7)
Note that in the example calculation above the formalism is of two o-waves creating an e-wave. In parametric oscillation the case of interest is of one high frequency component splitting into two lower frequency components. However these processes are physically identical as they have been shown to satisfy permutation symmetry [1 2].