## Abstract

Laser damage phenomena are governed by a number of different effects for the respective operation modes and pulse durations. In the ultra short pulse regime the electronic structure in the dielectric coating and the substrate material set the prerequisite for the achieved laser damage threshold of an optical component. Theoretical considerations have been done to assess the impact of contributing ionization phenomena in order to find a valid description for laser-induced damage in the femtosecond (fs) domain. Subsequently, a special set of sample has been designed to verify these considerations via ISO certified laser damage testing. Examining the theoretical and experimental data reveals the importance of multi-photon absorption for the optical breakdown. For titania, the influence of multi-photon absorption has been clearly shown by a quantized wavelength characteristic of the laser damage threshold.

©2009 Optical Society of America

## 1. Introduction

The femtosecond laser has come a long way from a complex, lab-filling and scientific tool to a tabletop machining device with applications in the medicine, commercial and industrial sector. With this development, an up-scaling of output power and pulse energy came along. Many applications of fs-pulses take advantage of the short interaction time of fs-radiation with matter resulting in a negligible thermal contribution to the involved interaction mechanisms. This special interaction type opens the way to a new area of laser material processing with outstanding precision and a significantly extended material range covering for example tungsten or molybdenum[1].

Additionally, these physical processes also play a vital role in numerous fundamental research experiments. For example, in the field of fundamental particle physics and fusion research high power, ultra short pulse laser systems are applied as energy source. Both, the pulse energy and the temporal pulse width are pushed towards the physical limits. Consequently, the power handling capability is of highest importance; amongst others, the involved optical components are exposed to extreme loads of power densities [2].

One of the major problems in the implementation of production cycles with fs-lasers is the power resistance of the optical components employed in the laser source and the beam guiding optics. As a consequence, from the beginning also for femtosecond laser technology the damage threshold and the corresponding mechanisms of a large variety of coatings and optical materials have been of high interest for scientists and manufacturers.

The present study reports on numerical calculations and the corresponding Laser-induced damage threshold measurements which were planned and designed to investigate in the correlation of femtosecond damage mechanisms to the electronic structure of dielectrics. Assuming that laser-induced damage is caused by Multi-Photon Ionization (MPI) and by the electronic structure of the solid, a characteristic damage behavior should be verifiable. In particular, a certain quantized damage behavior is expected which should be dependent either on the change in multi-photon absorption cross-section when continuously tuning the wavelength of the applied laser radiation or on the band gap of the optical material.

## 2. Theory and modeling

Currently, the damage of dielectrics in the fs-range is understood as a purely electronic damage process. Electrons are ionized during the laser-matter interaction to a specific critical electron density in the conduction band of the material. Usually authors have assumed the critical value is the critical density of the plasma when the respective plasma waves are resonant with the laser wavelength. For instance, for fused silica and the Ti:Sa laser irradiation at 0.8 micron this gives 10^{21} 1/cm^{3}. This has been reported in many publications, e.g. by Starke [8]. For the following calculations, the critical electron density is traced back to the plasma frequency. The mathematical description of the ionization process is available in a number of models. Precise predictions of the fs-damage threshold have been demonstrated by means of the following rate equation [3–5, 8]:

The growth of the electron density ρ(t) in the conduction band is given by the photo-ionization rate W* _{PI}* and the avalanche ionization rate W

*, respectively. To take into account relaxation processes from conduction band to other electronic states a relaxation rate Wrel is also introduced in Eq.1. The excitation rates can be described on the basis of the Keldysh theory for the photo-ionization and the Drude model of free carrier absorption. The corresponding expression for the photo-ionization rate W*

_{AV}*(Eq.2-5) contains a variety of material parameters as well as Dawson and elliptical integrals:*

_{Pi}$$x=\genfrac{}{}{0.1ex}{}{2{U}_{i}}{\pi \stackrel{\u0304}{h}{\omega}_{0}\sqrt{\mathrm{\Gamma}}}E\left(\sqrt{\zeta}\right)\phantom{\rule[-0ex]{.9em}{0ex}}\mu =]x+1[-x\phantom{\rule[-0ex]{.9em}{0ex}}\eta =\genfrac{}{}{0.1ex}{}{{\pi}^{2}}{2K\left(\sqrt{\zeta}\right)E\left(\sqrt{\zeta}\right)}$$

$$\mathrm{\Phi}={\int}_{0}^{z}\mathrm{exp}\left({y}^{2}-{z}^{2}\right)dyK\left(k\right)={\int}_{0}^{\pi \u20442}\genfrac{}{}{0.1ex}{}{1}{\sqrt{1-{k}^{2}{\mathrm{sin}}^{2}}\left(\varphi \right)}d\varphi \phantom{\rule[-0ex]{.2em}{0ex}}E\left(k\right)={\int}_{0}^{\pi \u20442}\sqrt{1-{k}^{2}{\mathrm{sin}}^{2}\varphi}d\varphi $$

*γ*: Keldysh parameter, *ω*
_{0}: central frequency of the Laser irradiation, *e*: elementary charge, *m*: effective electron mass, *U _{i}*: intrinsic material gap,

*c*: speed of light (vacuum),

*ε*

_{0}: ermittivity,

*n*

_{0}: index of refraction,

*I*: intensity of laser irradiation,

*h*̄: Planck constant, ]

*Z*[: integer part of Z, Φ: Dawson integral, and K, E: elliptic integrals.

A general analytical solution of Eq.2 is not available, and therefore, the Keldysh ionization rate has to be determined numerically. Keldysh’s model supports two approximations of Eq.2 governed by the parameter for *γ*≫1 multi-photon ionization (MPI) is assumed as dominant contribution (MPI-approximation), and for *γ*≪1 tunnel ionization (TI) is the primary mechanism (TI-approximation). Consequently, the Keldysh formula can be employed to describe two very different ionization mechanisms. In TI-approximation, the band gap is deformed by the influence of the extremely high field strength acting during the pulse. The Keldysh theory predicts a continuous variation of the ionization probability with photon energy for TI. In MPI-approximation, the electron is transferred to the conduction band by an instantaneous absorption of the necessary number of photons. In the case of dominating MPI an abrupt change in the ionization probability and the ionization rate is assumed if the ionization switches from the n- to the n+1-photon absorption. This effect should be observed for appropriate combinations of photon and band gap energies. Numerous publications are focused on quartz investigated with radiation of a Ti-Sapphire laser operating around 780nm. In this constellation the Keldysh parameter is approximately 1 for the peak intensity of the femtosecond pulse. Therefore, it is not possible two distinguish between the TI- and MPI effect. In the present study, titanium oxide (TiO_{2}, titania) is investigated for two reasons: First, *γ* is approximately three at the peak fluence and therefore, a small tendency towards MPI can be assumed for TiO_{2}. Second, the LIDT of dielectric stacks is often dominated by the high refractive index material suggesting detailed investigations in the corresponding materials to improve the power handling capability. In Fig.1, the computed ionization rates for quartz and TiO_{2} are compared. The rates were calculated on the basis of the complete solution of the Keldysh equation (solid line) as well as in MPI-approximation (dotted line) and in TI-approximation (dashed line). For *γ*≫1 the complete Keldysh solution is fitted very well to the MPI approximation. Therefore, a realistic probability can be assumed that damage can located in the MPI - range for TiO_{2}. The second dominant contribution to ionization can be described by the avalanche process: electrons in the conduction band absorb photons until the electron energy is exceeding the band gap energy. By relaxation of these electrons additional electrons can be excited from the valance band to the conduction band via energy transfer. The avalanche ionization rate can be calculated from the Drude model [8, 9].

with the absorption cross section σ, and the resulting collision time *τ _{c}*. In contrast to the photo ionization, the Avalanche effect postulates a significant electron density in the conduction band. Typically, more then 10

^{19}to 10

^{20}electrons/cm

^{3}are necessary for a significant effect of the avalanche ionization process. Nevertheless, the aspect of avalanche ionization is of fundamental importance for a quantitative analysis.

For a complete picture of the rate model, the relaxation process described by the parameter W* _{rel}* in Eq.1 has to be discussed. Mero [5] has shown that the contribution of relaxation of the electrons in dielectric materials is relatively small compared to the excitation processes, and that the relaxation time is within the time scale of the pulse duration. A generic expression for the relaxation process is given in Eq.7. Typical values of

*τ*are in the range of 200fs [6].

_{rel}In the performed numerical solution of Eq.1, the temporal intensity profile of the pulse is assumed to be Gaussian. Fig.2 displays the electron density calculated for the time period of the laser pulse. The different contribution to the generation of the electron densities is indicated separately. At high fluences, the avalanche ionization increases the electron density by more than one order of magnitude. In contrast to this, the influence of the relaxation is nearly negligible. The electron densities of Fig.2 were calculated for a wavelength of 670nm, a pulse duration of 130fs, and a fluence of 0,29J/cm^{2}. For the material gap the experimental value of 3.65eV for the applied material is used, and the effective mass of the electron is approximated to 0.3 times the electron mass. Obviously, the electron density exceeds the critical threshold of the material, inducing damage.

## 3. Consequences of multi-photon ionization on laser-induced damage

As discussed in the previous section, MPI provides a substantial contribution to the damage mechanism for ultra short laser pulses. Considering the large differences in cross-sections of n and n+1 photon absorption, a quantized behavior of the damage threshold is expected. The presented theoretical approach indicates clear steps in the ionization characteristic behavior of the material with photon energies varying over orders of the multi-photon process. For instance, if one assumes a certain photon energy this results in exactly n photons to pass the absorption gap of the material. Furthermore, decreasing marginally the photon energy so that n+1 photons are necessary to exceed the material gap should lead in an abrupt decrease of the ionization rate.

Therefore, the LIDT should also show a respective change at the transition point between the n and n+1 photon orders. Translated into terms of damage thresholds and wavelengths, the theory postulates a cascaded increase of the damage threshold with increasing wavelength, where steps are always located at the transition points between the multi-photon orders. To detect an expected step, two different experimental approaches can be considered. Firstly, a threshold measurement can be performed on a material series which contains species with adjusted material gap energies. Recent investigations in material mixtures indicate that the material gap of a ternary compound can be tuned by the mixing ratio of oxides[7]. For example, a transition point from the two-photon to a three-photon process may be reached by mixtures of TiO_{2} with SiO_{2} for the wavelength of the Ti-Sapphire laser. However, since the detailed electronic material structure of the mixed ternary compounds is not precisely known, the mixture approach may suffer from additional electronic effects which are not integrated in the present theory. In contrast to this, the second method based on a direct tuning of the laser wavelength for the threshold measurement is a straight forward concept. Varying the photon energy of the test laser over a transition point for a dielectric material with well defined band structure should reveal the expected quantization effect of the LIDT-value.

In the present study, a wavelength tunable source with an optical parametric amplifier system is employed to detect a transition point in TiO_{2}-coatings. The material gap of the TiO2 layer deposited by ion beam sputtering was determined by a spectrophotometric method to a value of E=3.61eV, corresponding to a wavelength of 680nm. The expected orders of multi-photon processes are indicated in Fig.3 for a wavelength range from 400nm to 2000nm. For a selected wavelength range (insert Fig.3), which is well accessible by the experimental set-up and covers the transition between the second and third multi photon order, the ionization rates as a function of power density were calculated. Below an energy density of 10^{11}W/cm^{2} the graphs reveal a clear difference in the ionization rates for the two-photon transition in contrast to the three-photon transition (Fig.4).

Two material constants are required for the calculation of these values: The dispersion of refractive index which is determined by spectrophotometric measurements and a subsequent fitting, and the effective electron mass (m=0.3) which can be only integrated as a fitting parameter. The pulse duration is selected to 130fs, and the critical electron density ρ* _{c}* is calculated by:

As expected, the modeled LIDT-values growth dramatically at the transition wavelength around 680nm. As a consequence of the mentioned wavelength shift of the material gap according to Keldysh, the step is shifted 10nm to shorter wavelength. This shift is reasoned by the high field strength. Obviously, the change does not occur as a sharp but rather as a continuous change in a range of 10nm. Obviously, the properties of the photo-ionization dominate the wavelength behavior of the calculated LIDT and accordingly, the minimum LIDT is observed for the resonance of the photon energy with multiples of the band gap energy. Appropriately, a discrepancy of the applied photon energy to the material gap will set the process out of resonance and will cause a higher damage threshold. However, the calculation predicts the LIDT as periodic sequence of the multi-photon ionization orders. This behavior is displayed in Fig.5. Such a periodical structure of higher and lower damage threshold ranges can be explained with a resonant or non-resonant excitation of the electronic states and the quantized photon energy. It is surprising that the damage threshold doesn’t grow significantly for titania in the investigated wavelength range. This behavior results from the decreasing critical electron density and an increasing cross section of the avalanche ionization towards longer wavelengths (see Fig.6). This effect compensates the decreasing of the photo-ionization rate. Consequently, the damage threshold will be growing for longer wavelengths.

## 4. Experimental set-up

The experimental set-up is based on a tuneable radiation with a commercial fs-optical parametric amplifier (OPA, light conversion, ”Topas 4-800”) coupled to a Ti:Sapphire laser system (Spectra Physics, ”spit fire”) with a pulse duration of 130fs, and a repetition rate of 1kHz. In the OPA the light is converted to the required wavelength in a wide tuning range. The power of the combined radiation source is dependent on the wavelength with a maximum of 200mW in the visible spectral range. The OPA emits the signal and the idler wavelength as well as a fraction of the pump radiation. In the present paper, all LIDT-measurements are performed with the second harmonic of the signal wavelength which is separated from the other wavelengths emitted by the source using adapted dichroic mirrors. The resulting wavelength spectrum is monitored by a fiber spectrometer. Behind the separator, the beam is switched by a fast shutter and attenuated by a rotatable broadband polarizer. A calibrated photodiode is employed to determine the pulse energy of each pulse, and the beam is focused by a fused silica lens onto the sample. The condition of the sample is monitored by measuring the scattered light. All samples are optically inspected by Nomarski microscopy after the test and the results are compared to the observations of the in situ damage detection system. The LIDT measurement protocol described in ISO11254 [10] was applied for the damage testing.

## 5. Sample preparation and experimental results

A set of TiO_{2} single layers deposited by ion beam sputtering with a physical thickness of 440nm on B270 substrate was selected for the experiment. The band gap was determined by the second derivation of the transmission measurement, and amounts to 3.61eV for the titania layers. With this material gap the step from the 2 to the 3 photon ionization order is calculated to be located at a wavelength of 680nm, as illustrated in Fig.3. In Fig.7, the 0% and the 50%-LIDT measured with 1000 pulses per test site are displayed for the wavelength range from 590nm up to 750nm. The behavior of the measured LIDT-values indicates a significant growth in the transition range from the two-photon to the three-photon excitation. The LIDT remains almost constant at a level of 0.15J/cm^{2} below the transition range and steps up to a constant value of around 0.40J/cm^{2}. The exact wavelength position of the step is determined to 682nm, which is in good agreement with the theoretical prediction. In comparison to the measured and calculated damage thresholds, the wavelength of the multi-photon order transition is red-shifted by 10nm. With respect to the measurement accuracy of the material gap determination, this difference is negligible. There are different methods to determine the material gap from total loss measurement data. For typical coating materials the value of the material gap can show variations of 15%. Additionally, the measured LIDT increase is significantly higher than the calculated step. It is assumed that avalanche is overestimated within the rate equation (Eq.6) applied in the calculation.

## 6. Summary

In this paper, Keldysh’s theory for electron ionization was employed to investigate the laser-induced damage threshold of titania single layers. For the exemplary optical system of a titania single layer most necessary material parameters were experimentally determined. With these, the laser-induced damage threshold in dependence of the wavelength was calculated assuming photo-ionization as governingmechanism. ExpectingMPI, characteristic steps in a wavelengths scaling should be observed. Based on Keldysh, these steps were found at the wavelengths position of the MPI-order transition. To follow up on these results experimentally, a laser damage experiment was conducted with a tunable fs-laser system to test the MPI transition from two-photon absorption to three-photon absorption. An OPA laser source was tuned between 590nm and 750nm, and, for the first time, measurements confirmed the theoretical description in a thin film dielectric sample. The predicted step was observed at 680 nm introducing an increase of the LIDT by a factor of three. As a consequence, photo-ionization is verified to be the dominating mechanism for laser-induced damage in titania for ultra-short laser pulses. There is no simple way to prove the electronic nature of the damage process in complex electronic systems. The applied titania single layers could be used as a simple model system providing comparable simple electronic structure allowing for this effect to be observed. In systems of higher electronic complexity, it is assumed that the observability of this effect would have been superpositioned by additional electronic transition channels.

## Acknowledgments

The authors would like to thank the Deutsche Forschungsgemeinschaft (DFG) for the financial support within the Exzellenzcluster 201 ”Quest - Centre for Quantum Engineering and Space-Time Research”.

## References and links

**1. **G. Kamlage, T. Bauer, A. Ostendorf, and B. N. Chichkov, “Deep drilling of metal by femtosecond laser pulses” Appl. Phys. A ,**A77**, 307–310 (2003).

**2. **T. Ditmire, J. Zweiback, V. P. Yanovsky, T. E. Cowan, G. Hays, K. B. Wharton, and K. B, “Nuclear fusion from explosions of femtosecond laser-heated deuterium clusters,” Nature **383**, 489–492 (**1999**).

**3. **B. C. Stuart, M. D. Feit, S. Herman, A. M. Rubenchik, B. W. Shore, and M. D. Perry, “Nanosecond-to femtosecond laser induced breakdown in dielectrics” Phys. Rev. B **53**, 1749–1761 (1996). [CrossRef]

**4. **L. Sudrie, A. Couairon, M. Franco, B. Lamouroux, B. Prade, S. Tzortzakis, and A. Mysyrowicz, “Femtosecond Laser-Induced Damage and Filamentary Propagation in Fused Silica,” Phys. Rev. Lett. **89,(18) **186601-1-4 (2002).

**5. **M. Mero, A. J. Sabbah, J. Liu, B. Clapp, J. Zeller, W. Rudolph, K. Starke, and D. Ristau, “Femtosecond pulse damage behavior of oxide and fluoride dielectric thin films,” Proc. SPIE 5273 (2004). [CrossRef]

**6. **M. Mero, J. Liu, W. Rudolph, D. Ristau, and K. Starke, “Scaling laws of femtosecond laser pulse induced breakdown in oxide films,” Phys. Rev. B **71**, 115109 (2005). [CrossRef]

**7. **M. Jup, L. Jensen, M. Lappschies, K. Starke, and D. Ristau, “Improvement in laser irradiation resistance of fs-dielectric optics using silica mixtures,” Proc. SPIE **6403 **64031A (2006). [CrossRef]

**8. **K. Starke, D. Ristau, H. Welling, T. V. Amotchkina, M. Trubetskov, A. A. Tikhonravov, and A. S. Chirkin, “Investigations in the non-linear behavior of dielectric coatings by using ultrashort pulses,” Proc. SPIE **5273 **(2004). [CrossRef]

**9. **M. D. Feit and J. J. A. Fleck, “Effect of refraction on spot-size dependence of laser-induced breakdown,” Appl. Phys. Lett. **24(4)**169–172 (1974). [CrossRef]

**10. **“ISO 11254: Optics and optical instruments. Lasers and laser related equipment. Test methods for laser induced damage threshold of optical surfaces. Part 1: 1 on 1-test, 2000, Part 2: S on 1 test, 2001, Part 3: Assurance of laser power handling capabilities 2006” International Organization of Standardisation

**11. **M. D. Feit, A. M. Rubenchik, B. W. Shore, B. C. Stuart, and M. D. Perry, “Laser-induced damage in dielectrics with nanosecond to subpicosecond pulses: II. theory,” Proc. SPIE **2428**, 469–478 (1995). [CrossRef]