Ellipticized rubber acoustical lens providing balanced astigmatism

An ellipticized singlet azimuth versus elevation optimized and aperture extremized nonspherical acoustic lens made out of rubber with surfaces S and S' specified by the partial differential equations ##EQU9## the symmetry conditions ##EQU10## and the boundary conditions

• z(x,y)=z'(x,y)=0,
• on the ellipse
• .GAMMA.:
• (x.sup.2 /b.sub.o.sup.2 cos.sup.2 .psi..sub.o)+(y.sup.2 /b.sub.o.sup.2)=1
• where
• S and S' are the lens surfaces having functional representations of the forms z=z(x,y) and z'=z' (x',y') wherein x' and y' are themselves functions x'=x'(x,y) and y'=y'(x,y) of the independent variables x and y, ##EQU11## are the partial derivatives of z with respect to the independent variables x and y respectively, ##EQU12## are the Jacobian of z' and y' with respect to the independent variables x and y, the Jacobian of x' and y' with respect to the independent variables x and y and the Jacobian of x' and z' with respect to the independent variables x and y respectively, F(A), G(A), F'(A) and G'(A) are the functions of the arguments A=(x,y,z,x',y',z') defined as ##EQU13## and where p and p' denote the path length elements defined by the expressions
• p.sup.2 =x.sup.2 +(y-y.sub.o).sup.2 +(z-z.sub.o).sup.2, p'.sup.2 =(x'-x).sup.2 +(y'-y).sup.2 +(z'-z).sup.2,
• and n.sub.o, y.sub.o, z.sub.o and .psi..sub.o are respectively, the index of refraction of the lens material, the y and z coordinates of the finite focal point F and the off-axis angle to the infinite focal point F.sub..infin., .GAMMA. is the ellipse bounding the lens formed by S and S' and is defined by the equation shown in which x and y are the independent variables and b.sub.o is the maximum radius of the lens formed by S and S' and the semi-major axis of the ellipse .GAMMA..