Peterson 
Using the paraxial approximations, Z>> )F-x | , and the Fraunhofer 
(i.e., far field) approximation klé |? ~ 0, then it can be shown 
that 
2Z 
; - oe ee eS Ss 
ve) =Giy tte oF se g() ez **aF (44) 
The boundary conditions, e(£) , selected to represent the bubble 
are 
eH =O +g + 5,08 
sagpye! 2p CRY le BEEP Heer ae 
(5A) 
nes 1; Pel?! 
where Dy) 
o, Ele 
The first 2 terms of (5A) represent an opaque object andthe3rd term 
is an equivalent to a lens with a negative focal length f, attenuation 
factor a , and Gaussian transmittance distribution. The use of a 
negative lens is an approximation to ray 2 of Figure 1A, It is assu- 
med here that a uniform plane wave is incident on the object plane 
(€). 
Substituting g, ) in equation 4A gives 
i.e., a plane wave. 
. . 7—* —? . . 
The second integral gives W2 (x) when g,(&) is substitut- 
ed into 4A. Fora circular disc of radius [f|=fo 
ifo _ikz ikp* 
¥, &) = Sues ela ved [225 4 pbo/Z) (7A) 
where p= |x| 
—, 
The thirdintegral W, (x) can be shown to be 
s ee Des eae As ree 
v, @) z Ee ic elke JtnZ a7 715 git,P? (ga) 
1150 
