# Introduction to Optics for Ultrafast Lasers

## Introduction

Ultrafast lasers are used in numerous applications such as time resolved spectroscopy, precision material processing and non-linear optics. Driven by these applications, recent developments in this field are directed to lasers generating higher output power and shorter pulses. Currently, most short pulse physics is done with Ti:Sapphire lasers. Solid-state lasers based on other transition metal or rare earth metal doped crystals (Nd:YAG, Nd:YVO

_{4}, Yb:KGW) are also used for the generation of ultrashort pulses. The reproducible generation of sub-100 fs-pulses is closely connected to 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 Fourier analysis. For instance, the bandwidth (FWHM) of a 100 fs Gaussian pulse at 800 nm is 11 nm. For shorter pulses, the wavelength spectrum becomes significantly broader. A 10 fs pulse has a bandwidth of 107 nm.

If such a broad pulse passes through an optical medium, the spectral components of this pulse propagate with different speeds. Dispersive media, like 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 Fig. 1).

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. Consequently, 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 sub-100 fs-regime it is essential to control the phase properties of each optical element over the extremely wide bandwidth of the fs-laser. 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, the phase relationship among the Fourier components of the pulse must be preserved in order to avoid broadening or distortion of the pulse.

Mathematical analysis of the phase shift, which is applied to a pulse passing through a medium or being reflected by a mirror (see part GDD and TOD), 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 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 Fig. 2 explains this effect in terms of different optical path lengths of blue, green and red light in a negative dispersion mirror.

LAYERTEC offers ultrafast laser optics with different bandwidths. This catalog shows 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 120 nm, broadband components (bandwidth about 300 nm) and octave spanning components.

Moreover, silver mirrors for fs applications are presented which offer the broadest low-GDD bandwidth available.

Please note that the GDD spectrum of a dielectric negative dispersion mirror is not a 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. Considerable reduction of these oscillations can be achieved by using mirror pairs consisting of mirrors with carefully shifted GDD oscillations. Fig. 3 shows a schematic drawing of said mirror pair and the corresponding GDD spectra.

It is also possible to use negative dispersion mirrors with high values of negative GDD for pulse compression. These so-called Gires-Tournois-Interferometer (GTI) mirrors (see page Gires-Tournois-Interferometer (GTI) Mirrors (600 – 1600 nm)) are successfully used in Ti:Sapphire lasers, Yb:YAG and Yb:KGW oscillators and Er:Fiber lasers. Pulse compression in Yb:YAG and Yb:KGW oscillators provides pulses of some hundred femto-seconds pulse length. For each wavelength, components with different amounts of negative GDD are presented on the following pages. Besides these optics for the spectral range of the Ti:Sapphire fundamental and for the very promising Yb:YAG and Yb:KGW lasers, LAYERTEC also offers optics for the harmonics of this radiation down to the VUV wavelength range, optics for ultrafast lasers in the 1500 nm-range and especially designed optics for high power ultra-short pulse lasers. LAYERTEC has its own capabilities for design calculation and also for GDD-measurements in the wavelength range from 250 to 1700 nm.

## 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 center 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}) + Φ´´(ω_{0}) (ω – ω_{0})²/2 + Φ´´´(ω_{0}) (ω – ω_{0})^{3}/6 + …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 clearly shows 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 temporally broadened. This broadening effect depends only on the absolute value of the GDD. LAYERTEC offers “low GDD mirrors”, i.e. mirrors with |GDD| < 20 fs² over a given wavelength range, which guarantee the preservation of the pulse shape when the pulse is reflected by these mirrors.
- Moreover, the pulse becomes “chirped”, i.e. it changes its momentary frequency during 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 20 fs and below.

## References

R. 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: “White-light frequency comb generation with a diode-pumped Cr:LiSAF laser”; 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“; Springer-Verlag Berlin Heidelberg 2003, p. 393 – 422 and the references therein

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