626 Phase Shifts on Transmission and Reflection
1
Figure B.1 Reflection and transmission by a plane mirror between two identical media.
It is because of the latter phase relation that the antiresonant ring reflects back
all the incident radiation and has zero losses if |˜r|
2
=|
˜
t|
2
= 0. 5.
B.2. COATED INTERFACE BETWEEN TWO
DIFFERENT DIELECTRICS
Let us consider—as in Figure B.2—a partially reflecting coating at an interface
between air (index 1) and a medium of index n. A light beam of amplitude E
1
=
1/
√
cos θ
1
is incident from the air, at an angle of incidence θ
1
. The transmitted
beam is refracted at the angle θ
2
and has an amplitude
˜
t
1
/
√
cos θ
1
. The reflected
beam has an amplitude ˜r
1
/
√
cos θ
1
. Energy conservation leads to the relation:
|˜r
1
|
2
+|
˜
t
1
|
2
n cos θ
2
cos θ
1
= 1, (B.5)
where we took into account the change in beam cross section on refraction.
We have a similar energy conservation equation for a beam of amplitude
E
2
= 1/
√
n cos θ
2
incident at an angle θ
2
on the dielectric–air interface:
|˜r
2
|
2
+|
˜
t
2
|
2
cos θ
1
n cos θ
2
= 1. (B.6)
The amplitude of the reflection coefficient is equal on both sides of the interface.
For the phase, the only sign relation consistent with energy conservation in a
Coated Interface Between Two Different Dielectrics 627
Figure B.2 Reflection and transmission at an interface.
Gires–Tournois interferometer and with the known phase shift on pure dielectric
interfaces, is:
˜r
1
=−˜r
∗
2
, (B.7)
or, r
1
= r
2
, with the relation between phase angles:
ϕ
r,1
=−ϕ
r,2
− π. (B.8)
To find a relation between the phase shift on transmission and reflection, we
consider the energy conservation for light incident from the upper half of the
figure (the axis of symmetry being the dashed normal to the interface):
1 + 1 = cos θ
1
˜r
1
√
cos θ
1
+
˜
t
2
√
n cos θ
2
2
+ n cos θ
2
˜r
2
√
n cos θ
2
+
˜
t
1
√
cos θ
1
2
. (B.9)
Taking into account the energy conservation relations (B.5) and (B.6) leads to
the following trigonometric relations between phase shifts on transmission and
628 Phase Shifts on Transmission and Reflection
reflection:
cos(ϕ
r,1
− ϕ
t,2
)
cos(ϕ
r,2
− ϕ
t,1
)
=−1, (B.10)
which leads to the relation between phase angles:
ϕ
t,2
− ϕ
r,2
= ϕ
r,2
− ϕ
t,1
+ (2n + 1)π. (B.11)
A direct consequence of this phase relation is:
˜
t
1
˜
t
2
−˜r
1
˜r
2
= 1. (B.12)
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