Say you have a slit with light passing through it. You will get a diffraction
pattern, lets call it
E
⃗
s
E
⃗
s
.
If you cover the slit with a piece of material that fits just inside the slit,
then there is no
E
⃗
E
⃗
field in the Fraunhofer limit. The is means that the
E
⃗
E
⃗
field of the blocker, lets call it
v
e
c
E
b
v
e
c
E
b
,
must exactly cancel
E
⃗
s
E
⃗
s
.
The only way this can happen is if
E
⃗
b
=
−
E
⃗
s
E
⃗
b
=
−
E
⃗
s
. Now if you take the slit away the
E
⃗
E
⃗
field of the blocker must still remain and then irradiance must be
I
b
=
(
−
E
s
)
2
=
I
s
I
b
=
(
−
E
s
)
2
=
I
s
The interference pattern looks the same. You can verify this yourself by
taking a strand of your hair and a laser pointer. Shine the laser pointer at a
wall and then put a strand of your hair in front of the light beam. The
resulting interference pattern is the exact same as one would obtain from a
slit with the same width as your hair.

We can use Babinet's Principle to solve complex problems. For example, say you
have square aperture with sides of length
L
L
.
The the diffraction pattern for light passing through it is
E
⃗
=
ɛ
⃗
A
e
i
(
k
r
−
ω
t
)
L
2
R
[
sin
β
L
β
L
]
[
sin
α
L
α
L
]
E
⃗
=
ɛ
⃗
A
e
i
(
k
r
−
ω
t
)
L
2
R
[
sin
β
L
β
L
]
[
sin
α
L
α
L
]
where
(assuming the aperture lies in the
y
−
z
y
−
z
plane)
β
L
=
k
L
Y
/
2
R
β
L
=
k
L
Y
/
2
R
α
=
k
L
Z
/
2
R
.
α
=
k
L
Z
/
2
R
.
Now
put an opaque square of length
d
d
in the middle of the aperture. Now the resulting
E
⃗
E
⃗
field is
E
⃗
=
ɛ
⃗
A
R
e
i
(
k
r
−
ω
t
)
[
L
2
sin
β
L
β
L
sin
α
L
α
L
−
d
2
sin
β
d
β
d
sin
α
d
α
d
]
E
⃗
=
ɛ
⃗
A
R
e
i
(
k
r
−
ω
t
)
[
L
2
sin
β
L
β
L
sin
α
L
α
L
−
d
2
sin
β
d
β
d
sin
α
d
α
d
]
or
I
=
I
0
[
L
2
sin
β
L
β
L
sin
α
L
α
L
−
d
2
sin
β
d
β
d
sin
α
d
α
d
]
2
I
=
I
0
[
L
2
sin
β
L
β
L
sin
α
L
α
L
−
d
2
sin
β
d
β
d
sin
α
d
α
d
]
2
β
L
=
k
L
Y
/
2
R
β
L
=
k
L
Y
/
2
R
α
L
=
k
L
Z
/
2
R
.
α
L
=
k
L
Z
/
2
R
.
β
d
=
k
d
Y
/
2
R
β
d
=
k
d
Y
/
2
R
α
d
=
k
d
Z
/
2
R
.
α
d
=
k
d
Z
/
2
R
.

Comments:"This book covers second year Physics at Rice University."