We consider diffraction from apertures other than a slit. For example consider
a rectangular aperture as shown below. If
ɛ
A
⃗
ɛ
A
⃗
is the source strength per unit area (assumed to be constant over the entire
area in this example) and
d
S
=
ⅆ
y
ⅆ
z
d
S
=
ⅆ
y
ⅆ
z
is an infinitesmal area at a point in the aperture then we have:
d
E
⃗
=
ɛ
A
⃗
r
e
i
(
k
r
−
ω
t
)
ⅆ
y
ⅆ
z
d
E
⃗
=
ɛ
A
⃗
r
e
i
(
k
r
−
ω
t
)
ⅆ
y
ⅆ
z
We see from the figure that
r
=
[
x
2
+
(
Y
−
y
)
2
+
(
Z
−
z
)
2
]
r
=
[
x
2
+
(
Y
−
y
)
2
+
(
Z
−
z
)
2
]
and that
R
2
=
x
2
+
Y
2
+
Z
2
.
R
2
=
x
2
+
Y
2
+
Z
2
.
Thus we use
x
2
=
R
2
−
Y
2
−
Z
2
x
2
=
R
2
−
Y
2
−
Z
2
to write
r
=
[
R
2
−
Y
2
−
Z
2
+
Y
2
+
y
2
−
2
y
Y
+
Z
2
+
z
2
−
2
z
Z
]
1
/
2
r
=
[
R
2
−
Y
2
−
Z
2
+
Y
2
+
y
2
−
2
y
Y
+
Z
2
+
z
2
−
2
z
Z
]
1
/
2
or
r
=
[
R
2
−
2
Y
y
−
2
Z
z
+
y
2
+
z
2
]
1
/
2
r
=
[
R
2
−
2
Y
y
−
2
Z
z
+
y
2
+
z
2
]
1
/
2
r
=
R
[
1
−
2
Y
y
/
R
2
−
2
Z
z
/
R
2
+
(
y
2
+
z
2
)
/
R
2
]
1
/
2
.
r
=
R
[
1
−
2
Y
y
/
R
2
−
2
Z
z
/
R
2
+
(
y
2
+
z
2
)
/
R
2
]
1
/
2
.
We are only considering Fraunhofer diffraction so
R
,
Z
,
Y
R
,
Z
,
Y
are much larger than
y
y
and
z
z
and we can rewrite
r
≈
R
[
1
−
2
(
Y
y
+
Z
z
)
/
R
2
]
1
/
2
r
≈
R
[
1
−
2
(
Y
y
+
Z
z
)
/
R
2
]
1
/
2
and then finally expanding using the binomial theorem and taking only the most
significant terms
r
≈
R
[
1
−
(
Y
y
+
Z
z
)
/
R
2
]
.
r
≈
R
[
1
−
(
Y
y
+
Z
z
)
/
R
2
]
.
So we have
d
E
⃗
=
ɛ
A
⃗
r
e
i
(
k
r
−
ω
t
)
ⅆ
y
ⅆ
z
d
E
⃗
=
ɛ
A
⃗
r
e
i
(
k
r
−
ω
t
)
ⅆ
y
ⅆ
z
or using the fact that
R
R
is large
d
E
⃗
=
ɛ
A
⃗
R
e
i
(
k
r
−
ω
t
)
ⅆ
y
ⅆ
z
d
E
⃗
=
ɛ
A
⃗
R
e
i
(
k
r
−
ω
t
)
ⅆ
y
ⅆ
z
which we integrate to get the field
E
⃗
=
∫
−
b
/
2
+
b
/
2
∫
−
a
/
2
+
a
/
2
ɛ
A
⃗
R
e
i
(
k
r
−
ω
t
)
ⅆ
z
ⅆ
y
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
∫
−
a
/
2
+
a
/
2
e
−
i
k
y
Y
/
R
ⅆ
y
∫
−
b
/
2
+
b
/
2
e
−
i
k
z
Z
/
R
ⅆ
z
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
[
e
−
i
k
y
Y
/
R
−
i
k
Y
/
R

−
a
/
2
+
a
/
2
[
e
−
i
k
z
Z
/
R
−
i
k
Z
/
R

−
b
/
2
+
b
/
2
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
[
e
−
i
k
a
Y
/
2
R
−
e
i
k
a
Y
/
2
R
−
i
k
Y
/
R
]
[
e
−
i
k
b
Z
/
2
R
−
e
i
k
b
Z
/
2
R
−
i
k
Z
/
R
]
E
⃗
=
∫
−
b
/
2
+
b
/
2
∫
−
a
/
2
+
a
/
2
ɛ
A
⃗
R
e
i
(
k
r
−
ω
t
)
ⅆ
z
ⅆ
y
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
∫
−
a
/
2
+
a
/
2
e
−
i
k
y
Y
/
R
ⅆ
y
∫
−
b
/
2
+
b
/
2
e
−
i
k
z
Z
/
R
ⅆ
z
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
[
e
−
i
k
y
Y
/
R
−
i
k
Y
/
R

−
a
/
2
+
a
/
2
[
e
−
i
k
z
Z
/
R
−
i
k
Z
/
R

−
b
/
2
+
b
/
2
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
[
e
−
i
k
a
Y
/
2
R
−
e
i
k
a
Y
/
2
R
−
i
k
Y
/
R
]
[
e
−
i
k
b
Z
/
2
R
−
e
i
k
b
Z
/
2
R
−
i
k
Z
/
R
]
Now lets define
α
=
k
a
Y
/
2
R
α
=
k
a
Y
/
2
R
β
=
k
b
Z
/
2
R
β
=
k
b
Z
/
2
R
and
β
=
k
b
Z
/
2
R
β
=
k
b
Z
/
2
R
and we see that
E
⃗
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
[
e
i
α
−
e
−
i
k
α
i
k
Y
/
R
]
[
e
i
β
−
e
−
i
β
i
k
Z
/
R
]
E
⃗
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
[
e
i
α
−
e
−
i
k
α
i
k
Y
/
R
]
[
e
i
β
−
e
−
i
β
i
k
Z
/
R
]
or rearranging
E
⃗
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
[
e
i
α
−
e
−
i
α
2
i
α
/
a
]
[
e
i
β
−
e
−
i
β
2
i
β
/
b
]
E
⃗
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
R
[
e
i
α
−
e
−
i
α
2
i
α
/
a
]
[
e
i
β
−
e
−
i
β
2
i
β
/
b
]
E
⃗
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
a
b
R
[
e
i
α
−
e
−
i
α
2
i
α
]
[
e
i
β
−
e
−
i
β
2
i
β
]
E
⃗
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
a
b
R
[
e
i
α
−
e
−
i
α
2
i
α
]
[
e
i
β
−
e
−
i
β
2
i
β
]
E
⃗
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
a
b
R
[
sin
α
α
]
[
sin
β
β
]
E
⃗
=
ɛ
A
⃗
e
i
(
k
R
−
ω
t
)
a
b
R
[
sin
α
α
]
[
sin
β
β
]
So finally we can write the intensity as
I
(
Y
,
Z
)
=
I
(
0
,
0
)
[
sin
α
α
]
2
[
sin
β
β
]
2
I
(
Y
,
Z
)
=
I
(
0
,
0
)
[
sin
α
α
]
2
[
sin
β
β
]
2
Below is a plot of
[
sin
β
β
]
2
[
sin
α
α
]
2
[
sin
β
β
]
2
[
sin
α
α
]
2
Below is a plot of
[
sin
β
β
]
[
sin
α
α
]
[
sin
β
β
]
[
sin
α
α
]
"This book covers second year Physics at Rice University."