Acousto-Optics, Optical Computing, and Signal Processing - Applied Optics Fundamentals and Device Applications - page 132

( )

r exp − 2 r λ

⎡
⎢

⎤
⎥

π z

L

⎡
⎢

z
L

1
2

⎤
⎥

2 h

h

h
h

X cos exp − 2 β + X d d d d

e

θ r z z (3.32)

h h

2 2 1 2

] /

[( z − z ) + r

e h

The integral over θ is trivial and the integrals over z e and z h must be done
using numerical methods. We can calculate the integral over r using the fol-
lowing equation:

2 r

⎛
⎜

⎞
⎟

r exp d r

∞

λ ⎡ ⎤

⎡
⎢

⎤
⎥

2 2 | | t π ⎛

⎜

2 | | t ⎞

⎟ −

⎛
⎜

2 | t |

λ

⎞
⎟ −

∫

G t ( ) = = H N 1

(3.33)

⎢ ⎥

1 1

2

λ λ λ

t

2

+ r

2

⎣ ⎦

r = 0

where
H 1 ( u ) is the first-order Struve function
N 1 ( s ) is the first-order Neumann function or Bessel function of the second
kind

The Struve function is defined by [44]

⎡
⎢

2 4 6

⎤
⎥

2 u u u

…

H u ( ) = − + −

1

1 3

2

1 3 5

2 2

1 3 5 7

2 2 2

π ⋅ ⋅ ⋅ ⋅ ⋅ ⋅

while the Neumann function is defined by [45]

π π

N s ( ) = Y s ( ) + (ln 2 − ϕ ) J s ( )

1 1 1

2

where φ = Euler's constant,

⎡

∞
∑

⎤

2 k

= −

2 2 ⎛

⎜

s ⎞

⎟

s

)} (

− s /

1

4 )

Y s ( ) + ln J s ( ) − ⎢

⎢

{ ( ξ k + 1 ) + ξ ( k + 2 ⎥

⎥

(3.34)

1 1

s k k

π π 2 2 π !( + )

⎣ ⎦

k = 0

( )

k

2

∞
∑

− s / 4

J s ( ) = ( s / 2 )

1

k !( 1 k )

+

k = 0

ξ ( 1 = − ϕ

−
∑ 1
1

n 1

−

ξ ( ) n = − + ϕ k n ≥ 2

k =