Short pulse lasers are used in numerous applications such as time resolved spectroscopy, precision material
processing and large bandwidth telecommunication. Driven by these applications, recent developments in this
field are directed to lasers generating higher output power and shorter pulses. Nowadays most of the work
in short pulse physics is done with Ti:Sapphire lasers, but also dye lasers and solid state lasers on the basis
of other transition metal or rare earth metal doped crystals such as Yb:KGW are used for the generation of
femtosecond pulses. The reproducible generation of sub100fspulses is closely connected with the development
of broadband low loss dispersive delay lines consisting of prism or grating pairs or of dispersive multilayer
reflectors.
The spectral bandwidth of a pulse is related to the pulse duration by a well known theorem of Fourier Analysis.
For instance, the bandwidth (FWHM) of a 100fs gaussian pulse at 800nm is 11nm. For shorter pulses, the
wavelength spectrum becomes significantly broader. A 10fs pulse has a bandwidth of 107nm.
If such a broad pulse passes through an optical medium, the spectral components of this pulse propagate with
different speeds. Dispersive media such as glass impose a so called “positive chirp” on the pulse, meaning
that the short wavelength (“blue”) components are delayed with respect to the long wavelength (“red”) components
(see schematic drawing in figure 1).
Figure 1: Broadening of a pulse by propagation through an optical medium (schematic drawing)
A similar broadening can be observed if a pulse is reflected by a dielectric mirror and the bandwidth of the
pulse is larger or equal to the width of the reflection band of the mirror. Also broadband mirrors consisting
of a double stack system cause pulse broadening, because the path lengths of the spectral components of
the pulse are extremely different in these coatings.
In the sub100fsregime it is essential to control the phase properties of each optical element over the extremely
wide bandwidth of the fslaser. This holds not only for the stretcher and compressor units, but also for the
cavity mirrors, output couplers and the beam propagation system. In addition to the power spectrum, i.e.
reflectance or transmittance, also the phase relationship among the Fourier components of the pulse must be
preserved in order to avoid broadening or distortion of the pulse.
A mathematical analysis of the phase shift which is applied to a pulse passing through a medium or being
reflected by a mirror (see insert on page 53 of our catalog) shows that the main physical properties which describe this
phenomenon are the group delay dispersion (GDD) and the third order dispersion (TOD). These properties
are defined as the second and third derivative of the reflected phase with respect to the frequency.
Especially designed dielectric mirrors offer the possibility to impose a “negative chirp” on a pulse. Thus,
the positive chirp which results from crystals, windows etc. can be compensated. The schematic drawing
in figure 2 explains this effect in terms of different optical path lengths of blue, green and red light in such
a negative dispersion mirror.
Figure 2: Optical path lengths of blue green and red light in a negative dispersion mirror
(schematic drawing)
LAYERTEC offers femtosecond laser optics with different bandwidths. This catalog shows e.g. optics for the
wavelengths range of the Ti:Sapphire laser in three chapters, each representing a characteristic bandwidth of
the optics: standard components with a bandwidth of about 120nm, broadband components (bandwidth about
300nm) and ultra broadband components (bandwidth of one octave or more).
Each of these chapters shows low dispersion laser and turning mirrors, negative dispersion mirrors or mirror pairs,
output couplers and beam splitters of the corresponding bandwidth. Moreover, we want to present silver mirrors
for fs applications which are the components with the broadest lowGDD wavelength region available.
Please note that the GDD spectrum of a dielectric negative dispersion mirror is not a continuous flat graph.
All types of negative dispersion mirrors exhibit oscillations in the GDD spectrum. These oscillations are small
for standard bandwidths. However, broadband and ultra broadband negative dispersion mirrors exhibit strong GDD
oscillations. A considerable flattening of these oscillations can be achieved by using mirror pairs consisting of
mirrors with slightly shifted GDD oscillations which compensate each other. These mirror pairs are especially
designed for this compensation behaviour. Figure 3 shows a schematic drawing of such a mirror pair and the
corresponding GDD spectra.
GDD and TOD
If a pulse is reflected by a dielectric mirror, i. e. a stack of alternating high and low refractive index
layers, there will be a phase shift between the original and the reflected pulse resulting from the time
which it takes the different Fourier components of the pulse to pass through the layer system of the
mirror. In general, the phase shift Φ(ω) near the centre frequency ω_{0} may be expanded
in a Taylor series for frequencies near ω_{0}:
Φ (ω) = Φ (ω_{0}) + Φ′ (ω_{0}) (ω – ω_{0})
+ 1/2 Φ′′(ω_{0}) (ω – ω_{0})^{2} + 1/6 Φ′′′
(ω_{0}) (ω – ω_{0})^{3} + …
The derivatives are, respectively, the Group Delay (GD) Φ´(ω_{0}), the
Group Delay Dispersion (GDD) Φ´´(ω_{0}) and the Third
Order Dispersion (TOD) Φ′′′(ω_{0}). More strictly speaking,
this expansion is only useful in an exactly soluble model, for the propagation of a transform limited Gaussian
pulse and for pure phase dispersion. For extremely short pulses and combinations of amplitude and phase
dispersion numerical calculations may be necessary. Nevertheless, this expansion shows clearly the physical
meaning of the single terms:
Assuming the phase shift is linear in frequency (i. e. GD ≠ 0, GDD = 0 and TOD = 0 over the pulse
bandwidth), the reflected pulse is delayed in time by the constant group delay and, of course, scaled
by the amplitude of reflectance R. The pulse spectrum will remain undistorted.
If GDD ≠ 0, two important effect are observed:

The reflected pulse is temporarily broadened. This broadening effect depends only on the
absolute value of the GDD. LAYERTEC offers “low GDD mirrors”, i. e. mirrors with GDD< 20
fs^{2} over a given wavelength range, which are needed to preserve the pulse shape when
the pulse is reflected by these mirrors.

Moreover, the pulse becomes “chirped”, i. e. it changes its momentary frequency during the pulse
time. This effect depends on the sign of the GDD, so that the momentary frequency may become
higher (up chirp, GDD > 0) or lower (down  chirp, GDD < 0). This allows to compensate positive
GDD effects of nonlinear optical elements by using negative GDD mirrors.
The TOD determines also pulse length and pulse shape (distortion of the pulse) and becomes a very
important factor at pulse lengths of 20fs and below.
Figure 3: Schematic drawing of a negative dispersion mirror pair
It is also possible to use negative dispersion mirrors with high values of negative GDD for pulse compression.
These so called GiresTournoisInterferometer (GTI)mirrors (see pages 72–73 of our catalog) are successfully used in Ti:Sapphire
lasers, Yb:YAG and Yb:KGWoscillators and Er:Fibre lasers. Pulse compression in Yb:YAG and Yb:KGWoscillators
provides pulses of some hundred femtoseconds pulse length. For each wavelength components
with different amounts of negative GDD are presented.
Besides these optics for the spectral range of the Ti:Sapphire ground wave and for the very promising Yb:YAGand
Yb:KGWlasers we also offer optics for the harmonics of this radiation down to the VUV wavelength range,
optics for femtosecond lasers in the 1500nmrange and especially designed optics for high power ultra short
pulse lasers.
LAYERTEC has own capabilities for design calculation and also for GDD measurements in the wavelength range
from 250–1100nm. A GDD measurement setup for wavelengths up to 1700nm is under construction.
References:
H.Holzwarth, M.Zimmermann, Th.Udem, T.W.Hänsch, P.Russbüldt, K.Gäbel, R.Poprawe, J.C.Knight, W.J.Wadsworth and P.St.J. Russell; Optics Letters
Vol. 26, No.17 (2001), p.1376–1378
Y.S.Lim, H.S.Jeon, Y.C.Noh, K.J.Yee, D.S.Kim, J.H. Lee, J.S.Chang, J.D.Park; Journal of the Korean Physical Society Vol.40, No.5 (2002), p. 837–843
G. Tempea, V.Yakovlev and F. Krausz; “Interference coatings for Ultrafast Optics” in N.Kaiser, H.K.Pulker (eds.) “Optical Interference Coatings “,
SpringerVerlag Berlin Heidelberg 2003, p. 393–422 and the references therein