The Fresnel–Arago laws are three laws which summarise some of the more important properties of interference between light of different states of polarization. Augustin-Jean Fresnel and François Arago, both discovered the laws, which bear their name.
The laws are as follows:
- Two orthogonal, coherent linearly polarized waves cannot interfere.
- Two parallel coherent linearly polarized waves will interfere in the same way as natural light.
- The two constituent orthogonal linearly polarized states of natural light cannot interfere to form a readily observable interference pattern, even if rotated into alignment (because they are incoherent).
One may understand this more clearly when considering two waves, given by the form
E
1
(
r
,
t
)
=
E
01
cos
(
k
1
⋅
r
−
ω
t
+
ϵ
1
)
and
E
2
(
r
,
t
)
=
E
02
cos
(
k
2
⋅
r
−
ω
t
+
ϵ
2
)
, where the boldface indicates that the relevant quantity is a vector, interfering. We know that the intensity of light goes as the electric field squared (in fact,
I
=
ϵ
v
⟨
E
2
⟩
T
, where the angled brackets denote a time average), and so we just add the fields before squaring them. Extensive algebra yields an interference term in the intensity of the resultant wave, namely:
I
12
=
ϵ
v
E
01
⋅
E
02
cos
δ
, where
δ
=
(
k
1
⋅
r
−
k
2
⋅
r
+
ϵ
1
−
ϵ
2
)
represents the phase difference arising from a combined path length and initial phase-angle difference.
Now it can be seen that if
E
01
is perpendicular to
E
02
(as in the case of the first Fresnel–Arago law),
I
12
=
0
and there is no interference. On the other hand, if
E
01
is parallel to
E
02
(as in the case of the second Fresnel–Arago law), the interference term produces a variation in the light intensity corresponding to
cos
δ
. Finally, if natural light is decomposed into orthogonal linear polarizations (as in the third Fresnel–Arago law), these states are incoherent, meaning that the phase difference
δ
will be fluctuating so quickly and randomly that after time-averaging we have
⟨
cos
δ
⟩
T
=
0
, so again
I
12
=
0
and there is no interference (even if
E
01
is rotated so that it is parallel to
E
02
).
