Parametric gain and pump threshold

We now consider parametric gain and pump powers necessary to reach oscillation threshold. Generally, obtaining low thresholds enable high conversion efficiencies to be reached.

In a singly resonant optical parametric oscillator, when a signal frequency at cOg is incident on the parametric gain medium, the single pass power gain is[l 1]

where

— 2c0iC0sdeff^Ip / aingUpEgC^ , 2-34

g2 = r 2 - ( ^ ) 2 . 2-35

Ip is the pump power intensity, Eq is the permittivity of free space, and c is the speed of

light in vacuum. For the small gain and perfect phase matching (AK = 0) case, the gain is thus sinh2(TL), and can be approximated as F^L^. As in a laser oscillator, threshold is reached when the parametric gain equals the loss, namely

r2L2 = 6g , 2-36

where ôg is the round-trip power loss of the signal wave. The pump power required to reach threshold is then

| ^ = 7uKoL2 , 2-37

Where Kqis defined as

TtEoningnpC^

However, this is a simple analysis for the plane wave approximation. In practice, there are many options and conditions, for instance, considering the nonlinear medium geometry, there are both type I and type II phase matching, and both CPM and NCPM etc. An optimisable parameter for the three wave interaction process is the focusing. An optimised focusing parameter can be selected, which significantly reduces the pump threshold and increases the conversion efficiency. But, an accompanying problem is crystal damage. It is clear that there is a watershed in the theoretical analysis. If the available pump energy or power is high enough then optical parametric oscillation does

CHAPTER 2 Theoretical background

not require tight focusing, in this case it is appropriate to use the plane wave approximation; if the available pump energy or power is not adequate, and optical parametric oscillation requires resort to tight focusing of the pump wave to reach the

pump threshold, then we use the Boyd and Kleinman theory. However, here the

limiting condition is crystal damage.

Boyd and Kleinman (1968)[12] have shown that the pump power threshold of second harmonic generation, sum-frequency mixing and optical parametric oscillation can be accurately described as a function of two parameters. One is the focusing parameter which is defined as the radio of the crystal length L to the confocal beam parameter b in the crystal, namely

Ç = ç 2-39

For Gaussian beam profiles the confocal parameter is given in terms of the beam waist Wo as follows

b = 2zo = kwo^ = Wq^ 2-40

K

The other parameter is the walkoff parameter B. In the optical parametric oscillator, this is defined by

B = § (L k )'/2 (^ )'" 2-41

where

tto “ 2 tt{) 2-42

and p is defined as the double refraction walkoff angle. For a given pump beam waist Wq, this walkoff limits the effective nonlinear interaction length in the crystal.

The original Boyd and Kleinman theory is valid in situations involving: CW pumping, doubly resonant oscillation of signal and idler wavelengths, and operation near degeneracy. Another assumption in the theory was that pump, signal and idler beam shared the same confocal parameter, and the consequences of them having different confocal parameters was not investigated. Following Boyd and Kleinman,

Fischer et al (1977)[19] first theoretically derived the threshold conditions for singly resonant OPO's with tightly focused pumps. However in their derivations the other assumptions are exactly the same as the Boyd and Kleinman's, including the condition of pump and signal sharing the same confocal parameter. In 1982, Guha, Wu and Falk theoretically extended the analysis to express the consequences of unequal confocal parameters[20]. Their conclusion was that lower thresholds can be generally achieved with unequal confocal parameters. They developed Boyd and Kleinman's theory, and derived singly resonant and doubly resonant OPO threshold formulae with unequal confocal parameters. Their results can be used to describe steady-state operation of pulsed, as well as CW optical parametric oscillators.

Consider an interaction under conditions of type I phase matching, where the pump is an extraordinary ray and signal and idler are ordinary rays then the reciprocal pump threshold can be written as

^ = KLh(Ç,B) 2-43

where

K = KoLkps , (W-i) 2-44

and h(4,B) is a function containing all of the dependence of the generated signal upon

the optimisable parameters[ 12] [20]. The h(^,B) function can be numerically calculated using a computer. In the general case, Guha's results shows that h(%,B) is maximised with unequal confocal parameters and that for certain values of and Çg, h(^,B) can be increased appreciably over its value for = %g. For example if ^p = 0.1, h(%,B) is increased by nearly a factor of two if the signal focusing parameter is increased from 0.1 to 0.5.

A theoretical derivation of the OPO threshold conditions for a type II phase matching geometry with a tightly focused pump has not appeared in the literature. However, a few papers have discussed type II second harmonic generation using Boyd

and Kleinman's theory[21][22]. Deriving the optimum threshold condition for a type II

phase matching geometry is more complicated than the type I situation. Since now in the general case two double refraction angles result. Fortunately in most OPO cases, the

CHAPTER 2 Theoretical background