Verify 10.2 normalization in Gaussian Beams handouts.

Clear[u, c, a, k, b]
u[r_] = (c/( - I b)) E^(I k r^2/(2 ( - I b))) ;
absUsq[r_] = Abs[u[r]]^2 ;
usq[r_] = FullSimplify[ absUsq[r] , r > 0 && k > 0 && b > 0 && c > 0]
i = Integrate[ 2 Pi r usq[r], {r, 0, Infinity}]  ;
FullSimplify[ i, r > 0 && k > 0 && b > 0 && c > 0]

Express the Gaussian beam function u_00 (with initial intention of plotting it)

Clear[u]
u[z_, r_, zNought_, wNought_, k_] =
Module[{w, varR, phi, x, y, q}, 
w[z_, zNo_, wNo_] = Sqrt[wNo^2 ( 1 + z^2/zNo^2) ];
varR[zz_, zNo_] = (zz^2 + zNo^2)/zz ;
phi[zz_, zNo_] = ArcTan[ zz/zNo ] ;
q[zz_, zNo_] = zz - I zNo ;

Sqrt[2/Pi]/w[z, zNought, wNought] 
Exp[-I phi[z, zNought]]
Exp[-(r/w[z, zNought, wNought])^2]
Exp[ I k r^2/(2 varR[z, zNought])]
]
ur[z_, r_, zNought_, wNought_, k_] = Real[u[z, r, zNought, wNought, k]]

Real[(((E^(-((r^2)/((wNought^2) (1+((z^2)/(zNought^2)))))+((I k (r^2) z)/(2 ((z^2)+(zNought^2))))-I ArcTan[(z/zNought)])) (Sqrt[(2/π)]))/(Sqrt[(wNought^2) (1+((z^2)/(zNought^2)))]))]

Manipulate[
ContourPlot3D[
b^2(x^2/a^2 + y^2/a^2- 1) == z^2
, {x, -r,r}
, {y, -r,r}
, {z, -r,r}
, AxesLabel -> {x, y, z}
]
,{{r, 5}, 0.1, 10}
,{{a, 1}, 0.1, 10}
,{{b, 1}, 0.1, 10}
]

Module[
 {r, a, b},
r = 5. ;
a = 1 ;
b = 1 ;
ContourPlot3D[
b^2(x^2/a^2 + y^2/a^2- 1) == z^2
, {x, -r,r}
, {y, -r,r}
, {z, -r,r}
, AxesLabel -> {x, y, z}
]
]

(*Export["C:\\Users\\Peeter\\cygwin_home\\physicsplay\\notes\\blogit\\q.pdf",First[ImportString[ExportString[%,"PDF"],"PDF"]]]*)