Dispersion of Interferometric Structures 73
should be affected. What is physically happening in the time domain is that the
various dielectric layers of the coating accumulate more or less energy at different
frequencies, resulting in a delay of some parts of the pulse. Therefore, significant
pulse reshaping with broadband coatings occurs only when the coherence length
of the pulse length is comparable to the coating thickness. Pulses of less than
30 fs duration were used in Weiner et al. [9, 10]. As shown previously, deter-
mination of the dispersion in the frequency domain can be made with a simple
Michelson interferometer. The latter being a linear measurement, yields the same
result with incoherent white light illumination or femtosecond pulses of the same
bandwidth.
An alternate method, advantageous for its sensitivity, but limited to the deter-
mination of the GVD, is to compare glass and coating dispersion inside a fs
laser cavity. As will be seen in Chapter 5, an adjustable thickness of glass is
generally incorporated in the cavity of mode-locked dye and solid state lasers, to
tune the amount of GVD for minimum pulse duration. The dispersion of mirrors
can be measured by substituting mirrors with different coatings in one cavity
position, and noting the change in the amount of glass required to compensate
for the additional dispersion [5, 11]. The method is sensitive, because the effect
of the sample mirror is multiplied by the mean number of cycles of the pulse in
the laser cavity. It is most useful for selecting mirrors for a particular fs laser
cavity.
2.3.2. Fabry–Perot and Gires–Tournois Interferometer
So far we have introduced (Michelson) interferometers only as a tool to split
a pulse and to generate a certain delay between the two partial pulses. In general,
however, the action of an interferometer is more complex. This is particularly
true for multiple-beam devices such as a Fabry–Perot interferometer. Let us con-
sider for instance a symmetric Fabry–Perot, with two identical parallel dielectric
reflectors spaced by a distance d. We will use the notations
˜
t
ij
for the field
transmission, and ˜r
ij
for the field reflection, as defined in Figure 2.5.
The complex field transmission function is:
˜
H() =
˜
t
12
˜
t
21
e
−ikd
+
˜
t
12
˜
t
21
e
−2ikd
·˜r
21
˜r
21
e
−ikd
+
˜
t
12
˜
t
21
e
−ikd
e
−2ikd
… ˜r
21
˜r
21
2
+···
=
˜
t
12
˜
t
21
e
−ikd
1
1 −˜r
2
21
e
−2ikd
(2.17)
where d
= d cos θ.
74 Femtosecond Optics
M
1
d
M
2
E
in
E
out
1
2
1
t
12
˜
t
12
˜
t
21
˜
t
21
˜
r
21
˜
r
12
˜
r
12
˜
r
21
˜
Figure 2.5 Schematic diagram of a Fabry–Perot interferometer.
˜
t
12
is the transmission from
outside (1) to inside (2);
˜
t
21
the transmission from inside (2) to outside (1); ˜r
12
the reflection from
outside (1) to inside (2) and ˜r
21
the reflection from inside (2) to outside (1).
Taking into account the interface properties derived in Appendix B,
˜
t
12
˜
t
21
−
˜r
12
˜r
21
= 1 and ˜r
12
=−˜r
∗
21
, the field transmission reduces to:
˜
H() =
(1 − R)e
−ikd
1 − Re
iδ
(2.18)
where,
δ() = 2ϕ
r
− 2k()d cos θ (2.19)
is the total phase shift of a roundtrip inside the Fabry–Perot, including the phase
shift ϕ
r
on reflection on each mirror, θ is the angle of incidence on the mirrors
(inside the Fabry–Perot), and R =|˜r
12
|
2
is the intensity reflection coefficient of
each mirror [12].
Similarly, one finds the complex reflection coefficient of the Fabry–Perot:
˜
R() =
√
R
e
iδ
− 1
1 − Re
iδ
. (2.20)
One can easily verify that, if—and only if—kd is real:
|
R
|
2
+
|
T
|
2
= 1. (2.21)
Dispersion of Interferometric Structures 75
Equations (2.18) and (2.20) are the transfer functions for the field spectrum.
The dependence on the frequency argument occurs through k = n()/c and
possibly ϕ
r
(). With n() complex, the medium inside the Fabry–Perot is either
an absorbing or an amplifying medium, depending on the sign of the imaginary
part of the index. We refer to a problem at the end of this chapter for a study of
the Fabry–Perot with gain.
The functions
˜
H() and
˜
R() are complex transfer functions, which implies
that, for instance, the transmitted field is:
˜
E
out
() = T ()
˜
E
in
() (2.22)
where
˜
E
in
is the incident field. Equation (2.22) takes into account all the dynamics
of the field and of the Fabry–Perot. In the case of a Fabry–Perot of thickness
d cτ
p
, close to resonance (δ() 1), the transmission function
˜
H()isa
Lorentzian, with a real and imaginary part connected by the Kramers Kronig
relation. We refer to a problem at the end of this chapter to show how dispersive
properties of a Fabry–Perot can be used to shape a chirped pulse.
In the case of a Fabry–Perot of thickness d cτ
p
, the pulse spectrum covers
a large number of Fabry–Perot modes. Hence the product (2.22) will represent a
frequency comb, of which the Fourier transform is a train of pulses. Intuitively
indeed, we expect the transmission and/or reflection of a Fabry–Perot interferom-
eter to consist of a train of pulses of decreasing intensity if the spacing d between
the two mirrors is larger than the geometrical pulse length, [Figure 2.6(a)].
M
1
M
2
(a)
M
1
(b)
M
2
Figure 2.6 Effect of a Fabry–Perot interferometer on a light pulse if the mirror spacing is larger (a)
and shorter (b) than the geometrical length of the incident pulse.
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