Ultrashort Laser Pulse Phenomena, 2nd Edition

Appendix B

Phase Shifts on Transmission

and Reflection

B.1. THE SYMMETRICAL INTERFACE

Let us consider first the simple situation sketched in Figure B.1. The interface

can be a mirror with a reflecting coating on the front face and an antireflection

coating on the back face. We are only interested in fields propagating outside the

mirror. The energy conservation relation between the reflected (field reflection

coefficient ˜r) and transmitted (field transmission coefficient

˜

t) waves implies:

|˜r|

2

+|

˜

t|

2

= 1, (B.1)

where we assumed a unity field amplitude.

Another relation can be found by adding another incident field of amplitude 1

(beam 2 in the figure), and taking advantage of the symmetry. Summing the

intensities:

|˜r +

˜

t|

2

+|˜r +

˜

t|

2

= 2. (B.2)

Combination of Eqs. (B.1) and (B.2) leads to

2[˜r

˜

t

∗

+˜r

∗

˜

t]=0, (B.3)

which implies that the phase shifts on transmission and reflection are com-

plementary:

ϕ

r

− ϕ

t

=

π

2

. (B.4)

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