70 Femtosecond Optics
Equation (2.12) is not limited to dielectric samples. Instead, any optical trans-
fer function
˜
H which can be described by an equation similar to Eq. (1.164),
can be determined from such a procedure. For instance, the preceding discussion
remains valid for absorbing materials, in which case the wave vector is complex,
and Eq. (2.12) leads to a complete determination of the real and imaginary part
of the index of refraction of the sample versus frequency. Another example is the
response of an optical mirror, as we will see in the following subsection.
2.3. DISPERSION OF INTERFEROMETRIC
STRUCTURES
2.3.1. Mirror Dispersion
In optical experiments, mirrors are used for different purposes and are usually
characterized only in terms of their reflectivity at a certain wavelength. The
latter gives a measure about the percentage of incident light intensity that is
reflected. In dealing with femtosecond light pulses, one has, however, to consider
the dispersive properties of the mirror [4, 5]. This can be done by analyzing the
optical transfer function which, for a mirror, is given by
˜
H() = R()e
−i()
. (2.13)
It relates the spectral amplitude of the reflected field
˜
E
r
() to the incident
field
˜
E
0
()
˜
E
r
() = R()e
−i()
˜
E
0
(). (2.14)
Here R()
2
is the reflection coefficient and () is the phase response of
the mirror. As mentioned earlier a nonzero () in a certain spectral range
is unavoidable if R() is frequency dependent. Depending on the functional
behavior of () (cf. Section 1.3.1), reflection at a mirror not only introduces
a certain intensity loss but may also lead to a change in the pulse shape and to
chirp generation or compensation. These effects are usually more critical if the
corresponding mirror is to be used in a laser. This is because its action is mul-
tiplied by the number of effective cavity round trips of the pulse. Such mirrors
are mostly fabricated as dielectric multilayers on a substrate. By changing the
number of layers and layer thickness, a desired transfer function, i.e., reflectivity
and phase response, in a certain spectral range can be realized. As an example,
Figure 2.4 shows the amplitude and phase response of a broadband high-reflection
Dispersion of Interferometric Structures 71
1.10.9 1.0
2
3
2
2
–()
0.8 1.2
/
0
0
50
100
|
R
|
2
(%)
|
R
|
2
–
–
–
Figure 2.4 Amplitude and phase response for a high reflection multilayer mirror (dashed line)
and a weak output coupler (solid line) as a function of the wavelength (Adapted from Dietel
et al. [5]).
mirror and a weak output coupler. Note that, although both mirrors have similar
reflection coefficients around a center wavelength λ
0
, the phase response dif-
fers greatly. The physical explanation of this difference is that R() [or R(λ)]
far from ω
0
= 2πc/λ
0
(not shown) influences the behavior of () [or (λ)]
near ω
0
.
Before dealing with the influence of other optical components on fs pulses, let
us discuss some methods to determine experimentally the mirror characteristics.
In this respect the Michelson interferometer is not only a powerful instrument
to analyze a sample in transmission, but it can also be used to determine the
dispersion and reflection spectrum of a mirror. The interferogram from which
the reference spectrum can be obtained is shown on the left of Fig. 2.3. Such
a symmetric interference pattern can only be achieved in a well compensated
Michelson interferometer (left part of Fig. 2.2) with identical (for symmetry)
mirrors in both arms, which are also broadband (to obtain a narrow correlation
pattern). For a most accurate measurement, the mirror to be measured should be
inserted in one arm of the interferometer rather than substituted to one of the
reference mirrors. Otherwise, the dispersive properties of that reference mirror
cannot be canceled. In Fig. 2.2 (left), a sample mirror is indicated as the dotted
line, deflecting the beam (dashed lines) towards a displaced end mirror. As in
the example of the transmissive sample, insertion of the reflective sample can in
general not be done without losing the relative phase and delay references. The
cross-correlation measured after substitution of the sample mirror in one arm of
the interferometer (right-hand side of Fig. 2.3) is
˜
A
+
12
(τ + τ
f
) exp(iϕ
f
), which is
the function
˜
A
+
12
(τ) with an unknown phase (ϕ
f
) and delay (τ
f
) error. The ratio
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