chp42_3 - PHY2061 R D Field Single-Slit Diffraction...

PHY2061 R. D. Field
Department of Physics chp42_3.doc University of Florida
Single-Slit Diffraction Intensity Pattern (2)
The overall wave function at the point P on the screen is given by
) sin(
) sin(
2
1
2
1
2 / ) 1 (
1
1 1
φ
φ
φ ω ω
∆
∆
= = Ψ = Ψ
∆ − − −
=
∑
N
e e Ae S e Ae
N i ikr t
N
ikr t i
N
n
n N
.
The distance where r from the center of the slit to the point P is given by,
φ θ ∆ − + = + = ) 1 ( sin
2
1
1 2
1
1
N r W r r
k
.
Now as N becomes large
φ φ ∆   →  ∆ − = ∆Φ
>>
N N
N 1
) 1 (
and
∆Φ
∆Φ
≈
∆Φ
∆Φ
≈
∆
∆
= Ψ
−
− −
2
1
2
1
) (
2
1
2
1
) (
2
1
2
1
) (
) sin(
) / sin(
) sin(
) sin(
) sin(
t kr i
t kr i t kr i
N
NAe
N
Ae
N
Ae
ω
ω ω
φ
φ
where I used
N N N 2 2
sin
1
∆Φ
  → 






∆Φ
>>
.
Thus for large N
∆Φ
∆Φ
≈ Ψ
−
2
1
2
1
) (
) sin(
)