Page 312 - High Power Laser Handbook

280 So l i d - S t at e La s e r s Heat-Capacity Lasers 281

reduction of the gain coefficient and, consequently, of the stored
energy density. Physically, the gain coefficient “clamps” at a given
value; at this point, the pump energy goes into generating more ASE
as opposed to increasing the gain coefficient.
Under ideal conditions, there would be no mismatch in refractive
index between the slab and the edge cladding. These ideal conditions
can be achieved by diffusion bonding the edge cladding to the slab or
by co-sintering the edge cladding to the slab in the case of ceramic
media. However, both of these approaches have proved to be time
consuming and not very repeatable in terms of yield. Another approach
is to use a bonding agent (such as epoxy) between the slab and the
edge cladding. Unfortunately, most epoxies have a refractive index
significantly lower than that of YAG. Thus, the Fresnel reflection of
spontaneous emission at the slab/epoxy interface lowers the thresh-
old for parasitic oscillations within the slab. If, however, the slab edge
is roughened before bonding, the diffuse scattering that results raises
the threshold and acts to inhibit the formation of parasitics.
The roughened surface in the ASE model can be treated as fol-
lows: The surface is characterized by z(x, y), which represents the dif-
ference in height (in the z-direction) at any point (x, y) from the mean
z-value of the surface. We assume z is a normally distributed, station-
ary, random variable with 0 mean and variance s . The random dis-
2
tribution is further described by correlation distance.
In the limit where one assumes that the surface is quite rough, so
that s/l >> 1, where l is the wavelength of the light, the probability
density for normally incident light to be scattered into angle q may be
written as 9

cξ  ξ 2 sin q 2 
qξ
p(; ) = exp −   (11.13)
( +
1 + cos q  81 cos ) q 
2
where
  ξ  −1
c =  2π erf  
  22  
and ξ = T/s characterizes the surface. A rough surface is given by
ξ→ 0; a smooth surface, by ξ→ ∞. Each time a ray hits the slab edge,
its new (reflected) direction is randomized according to the probabil-
ity distribution shown in Eq. (11.13). If U represents a uniformly dis-
tributed random number on (0, 1), the scattering angle q, as given by
Eq. (11.13), may be generated from 10
  22   ξ 

q U ξ(; ) = 2 tan   erf −1 1 U  erf   (11.14)
−
 
  ξ   22 

where erf is the inverse error function.
–1