Handbook of Biomedical Optics

38 Handbook of Biomedical Optics

From geometry,  = D sin  (Figure 3.6b), so

 

 

E E

E E

j ks t jkD

j ks t

=

=

−

( )

⎡

⎣

⎤

⎦

+

⎡

⎣

⎤

⎦

{ }

−

( )

⎡

0

0

1exp exp sin ,

exp

ω θ

ω

⎣⎣

⎤

⎦

+

⎡

⎣

⎤

⎦

{ }

1 2exp sin .π θ λjD /

e intensity (see Section 3.2.4.3) is given by

I E E E∝ ⋅=

  

2

*

.

erefore,

I E jD jD∝ +

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨

⎩

⎫

⎬

⎭

+ −

⎡

⎣

⎢

⎤

⎦

⎥

⎧

⎨



0

2

1 1 2 1 2( ) exp sin exp sinπ

θ

λ

π

θ

λ

⎩⎩

⎫

⎬

⎭

∝

⎛

⎝

⎜

⎞

⎠

⎟

+

⎡

⎣

⎢

⎤

⎦

⎥

,

cos sin ,I E D



0

2

2 2

2π

λ

θ

making use of the Euler relation cos(x) = (1/2)(exp(jx) + exp(−jx)).

Via the identity 2[1 + cos(2x)] = cos

2

(x), the intensity becomes

I E

D

∝

⎛

⎝

⎜

⎞

⎠

⎟

4

0

2

2



cos

sin

π

θ

λ

Note that without interference—just considering the incident

plane wave, for example, I E∝



0

2

, wit ...