136 Femtosecond Optics
which explicitly varies as
exp
−
iπ
λ
˜
Q
r
xx
x
2
in
+ 2
˜
Q
r
xt
x
in
t
in
−
˜
Q
r
tt
t
2
in
×exp
π
λ
˜
Q
i
xx
x
2
in
+ 2
˜
Q
i
xt
x
in
t
in
−
˜
Q
i
tt
t
2
in
, (2.164)
where
˜
Q
r
ij
,
˜
Q
i
ij
are the real and imaginary coordinates of the matrix (
˜
Q
in
)
−1
and
˜
Q
xt
=−
˜
Q
tx
. The first factor in Eq. (2.164) expresses the phase behavior and
accounts for the wave front curvature and chirp. The second term describes the
spatial and temporal beam (pulse) profile. Note that unless
˜
Q
r,i
xt
= 0 the diagonal
elements of (
˜
Q
in
) do not give directly such quantities as pulse duration, beam
width, chirp parameter, and wave front curvature. One can show that the field at
the output of an optical system is again a generalized Gaussian beam where in
analogy to (2.162) the generalized beam parameter (
˜
Q
out
) can be written as
˜
Q
out
=
A 0
G 1
˜
Q
in
+
BE/λ
HI/λ
C 0
00
˜
Q
in
+
DF/λ
01
. (2.165)
The evaluation of such matrix equations is quite complex since it generally
gives rather large expressions. However, the use of advanced algebraic formula
manipulation computer codes makes this approach practicable.
2.8. NUMERICAL APPROACHES
The analytical and quasi-analytical methods to trace pulses give much phys-
ical insight but fail if the optical systems become too demanding and/or many
dispersion orders have to be considered.
There are commercial wave and ray tracing programs available that allow
one to calculate not only the geometrical path through the system but also the
associated phase. Thus complete information on the complex field distribution
(amplitude and phase) in any desired plane is retrievable.
2.9. PROBLEMS
1. Dispersion affects the bandwidth of wave plates. Calculate the maximum
pulse duration for which a 10
th
order quarter wave plate can be made of
crystalline quartz, at 266 nm, using the parameters given with Eq. (2.2).
Problems 137
We require that the quarter-wave condition still be met with 5% accu-
racy at ± (1/τ
p
) of the central frequency. What is the thickness of the
wave plate?
2. We consider here a Fabry–Perot cavity containing a gain medium. To sim-
plify, we assume the gain to be linear and uniform in the frequency range
around a Fabry–Perot resonance of interest. Consider this system to be
irradiated by a tunable probe laser of frequency v
p
.
(a) Find an expression for the transmission and reflection of this Fabry–
Perot with gain as a function of the frequency of the probe laser.
(b) Find the gain for which the expression for the transmission tends to
infinity. What does it mean?
(c) Describe how the gain modifies the transmission function of the
Fabry–Perot (linewidth, peak transmission, peak reflection). Sketch
the transmission versus frequency for low and high gain.
(d) With the probe optical frequency tuned to the frequency for which
the empty (no gain) Fabry–Perot has a transmission of 50%, find its
transmission factor for the value of the gain corresponding to lasing
threshold.
3. Calculate the transmission of pulse propagating through a Fabry–Perot
interferometer. The electric field of the pulse is given by E(t) = E(t)e
iω
t
,
where E(t) = exp(−|t|/τ) and τ = 10 ns. The Fabry–Perot cavity is 1 mm
long, filled with a material of index n
0
= 1. 5, and both mirrors have a
reflectance of 99.9%. The wavelength is 1 µm. What is the transmission
linewidth (FWHM) of this Fabry–Perot? Find analytically the shape (and
duration) of the pulse transmitted by this Fabry–Perot, assuming exact
resonance.
4. Consider the same Fabry–Perot as in the previous problem, on which a
Gaussian pulse (plane wave) is incident. The frequency of the Gaussian
pulse is 0.1 ns
−1
below resonance. Calculate (numerically) the shape of
the pulse transmitted by this Fabry–Perot, for various values of the pulse
chirp a. The pulse envelope is:
˜
E(t) = e
−(1 +ia)(
t
τ
G
)
2
.
Is there a value of a for which the pulse transmitted has a minimum
duration?
5. Consider the Gires–Tournois interferometer. (a) As explained in the text,
the reflectivity is R = constant = 1, while the phase shows a strong
variation with frequency. Does this violate the Kramers–Kronig relation?
Explain your answer. (b) Derive the transfer function [Eq. (2.30)].
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