PLAN, AEROPUERTO DE CHINCHERO CUSCO

crystal itself, such as thermal lensing. The second consideration is that the additional optical elements within the laser cause additional losses, both from

absorption and scattering and also from the conversion of fundamental power to second harmonic. This can have a serious effect on the intracavity power, particularly if the laser is operating near threshold.

2-1-6-1 Focusing effects

A detailed discussion of most aspects of SHG in the

presence of focused beams is given by Boyd and Kleinmann [15] and only a brief summary is therefore given here.

Gaussian intensity distribution with a beam waist radius of Wq. Two parameters are then used to describe the

interaction of the beam with the crystals

^ b w^k-j 2 n-|w2 (31)

where

^

fundamental beam within the crystal, and:

B = (32)

where B is termed the double refraction parameter and p is the walkoff angle.

Boyd and Kleinmann collected all the optimi sable parameters of the SHG interaction within a single

function, "h", which they then proceeded to optimise for the case of zero absorption and the beam focused at the centre of the crystal. This process reduced the

dependence of h to two variables, B and The variation of h^(B,^), where the subscript "m" represents

optimisation, was computed for various values of B and ^ , and their graphs are shown in Fig 2-2a. It is seen that for any value of B , the curves of h^^B,^) against Ç each have a single maximum, corresponding to an optimum value of J , J ._{J Jm}

The value of Ç has its maximum of 2.84 when B - 0, and falls sharply to 1.92 at B = 1, and then reaches a limit of 1.392 when B > 6 . Since the crystal length is

proportional to g , the optimum length reduces as B rises, This of course is expected, since B rises due mainly to increased walkoff, which means that the length of the crystal over which phasematching can occur is

correspondingly reduced. The variation of over the range of B = O to B = 6 is shown in Fig 2-2b, from Boyd

and Kleinmann. This curve is defined by the maxima of Fig 2-2a.

The value of h^ at the peaks of the curves in Fig 2-2a i now considered. This then involves the double

optimisation of both ^ and the absorption and position already considered. The variation of , here termed h <B) is given in Fig 2-2c._{mm ^}

Given the above assumptions of the Boyd and Kleinmann analysis regarding the optimisation, the earlier

expression for the SH power, Eq«<20) eventually reduces to:

^ (33)

where the possibility of a small absorption has been

considered outwith the h^^(B) term by the expression e“^'-, where (% is the absorption coefficient, discussed more

fully in a later section.

To illustrated the effect of walkoff and defocusing on the h|^<B,i^) term, we consider the case of a 1° walkoff in a

1.5cm crystal having a refractivity of 1.5 at 4BBnm. In such a case, B = 5 approximately, and even with optimum

focusing, “ 0.13 (from Fig 2-2c), and thus the SH power is only 13% of its maximum value in the absence of walkoff. The walkoff angle is of course determined by the dispersion and wavelength chosen, but a value of i® is not untypical for many materials where critical

phasematching is present. Alteration of the focus in such a case only serves to reduce the SH power still further, as has assumed optimum focusing in the first place.

2 - 1 -6—2 (Alai kof f

When critical phasematching is used, the SH and

fundamental beams do not overlap over the full length of the crystal 5 due to walkoff. Thus the length over which

SHG takes place is not the full length of the crystal, but the aperture length, 1q , defined as the length of the

crystal over which the two beams overlap. This is given in terms of the beam waist Wq and the walkoff angle

p

Ig = (34)

Where there is significant walkoff, it is not correct to use the length of the crystal L, in the expression for SH power (2 0), but to use a more complex expression which

depends on the relative sizes of L , 1q and Ij; , the

effective length of focus in the crystal. A detailed analysis requires the use of numerical integration,

discussed fully in [153 and C163. However, where there is considerable walkoff, and 1q << L, it is appropriate to

replace

i?

is, to regard SHG to take place only within the aperture , length. As L rises, however, the value of the double refraction parameter B also rises, and thus does the optimum beam waist which increases 1q providing that

optimum focusing is maintained.

By combining (31) with (34), we obtains

^ / *'

P

N/2n,|

which illustrates the point thuss as L rises, so does B , in proportion to /L. This also increases the optimum value of ? , but once B is larger than about 2, F varies very little. Thus in the presence of significant

walkoff, ^ can be regarded as more or less constant in the above expression, and an increase in crystal length

results in a rather smaller increase in SH power, which is then influenced by further absorption.

Hence, in these circumstances, it is the aperture length* 1q , which is the limiting factor in SHG, and not the size

of the crystal itself.

For a 1.5cm crystal with a refractivity of 1.5 at 48Snm, a walkoff of 1° yields an aperture length of about 2mm, and

for the case of LFM, where at this wavlength the walkoff is about 8°, it is about 0.3mm. Clearly, therefore, in

the absence of noncritical phasematching, or a close approximation to it, the choice of crystal length is determined by the optical requirements of the laser as a whole, for example in terms of the dimensions of the

Z-fold, rather than by any SHG considerations, unless the