During the last two decades, the different pulse duration domains have been connected to a number of fundamental mechanisms that govern laser-induced damage in optical materials and thin films. On the time scale of nanosecond pulses, breakdown usually occurs because of inclusions within the material. Because of higher absorption in these inclusions, they are heated by the laser pulse, generate a plasma, and cause a collapse of the surrounding matrix [1]. For shorter pulses in the subpicosecond regime, damage can be traced back to the generation of free electrons in the conduction band triggered by multiphoton absorption (MPA) [2]. MPA-excited electrons will generate additional electrons by avalanche ionization, and when a certain electron density is exceeded, the film will be damaged [3].

In this Letter we report on nonlinear absorption measured in tantala/silica single layers at a wavelength of 532 nm. With a specially designed set of single layers, it was possible to demonstrate the influence of MPA on laser-induced damage behavior also in the nanosecond time domain. MPA is most likely to be observed when a resonance condition is created. This occurs when either the test wavelength is matched to the bandgap of the material of interest or, if possible, the bandgap is chosen according to the available wavelength. Since laser damage testing needs a certain pulse energy, the latter option is preferable. With a certain hardware configuration, ion beam sputtering (IBS) has been set up to promote the option of a tailored bandgap. By shifting two directly connected sputter targets relative to the ion beam, any mixture of the pure materials can be achieved. This also results in bandgap energies in between the values of the two pure components [4,5]. With this technique, any index of refraction can be set, and even gradual index profiles and Rugate filters can be deposited.

For this work, ${\mathrm{SiO}}_2$ and ${\mathrm{Ta}}_2{\mathrm{O}}_5$ have been utilized to form a set of single layers ($2\lambda /4$ at 1064 nm) showing bandgap values ranging from 4.5 (pure ${\mathrm{Ta}}_2{\mathrm{O}}_5$) to 7.4 eV (pure ${\mathrm{SiO}}_2$). In Fig. 1 the transmission spectra of these single layers in the UV are displayed. According to the Tauc algorithm, these are used together with the reflectance data to derive the bandgap energies for each mixing ratio [6].

 

Fig. 1. Transmission spectra of single-layer mixtures of tantala and silica ranging from pure silica to pure tantala. Optical thickness: $2\lambda /4$ at 1064 nm.

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Besides the well established $z$-scan method of measuring the nonlinear absorption indirectly via the transmission, one can also test for the absorptance directly. In this case of single layers, the interaction length within the coating is so small that the direct absorptance test is preferable. These samples have been analyzed concerning their absorptance in a laser calorimetric measurement. This technique offers the absolute absorptance of the entire sample in a calibrated measurement [7,8]. Consequently, the substrate (bulk and surface) and thin film contribute to the measured value. As these tests are performed at 532 nm and the substrate consists of synthetic fused silica, the MPA contribution can be neglected. The photon energy is 2.33 eV, and the bandgap of the substrate is 7.5 eV, making a three-photon process very unlikely for the applied intensities. All samples have been tested for absorption several times with a ramped-up input power for each consecutive test. The sample with the bandgap meeting the resonance condition was found to show an increased absorption as a function of the applied intensity. This is illustrated in Fig. 2. For comparison, the measurement of the pure silica layer is also plotted with a constant absorption for different intensities. This result suggests that within resonance, and also at low intensities, two-photon absorption (2PA) can be observed in tantala-based films.

 

Fig. 2. Absorptance of a tantala/silica single layer with a bandgap energy of 4.51 eV and a pure silica single layer with a bandgap of 7.4 eV. These are selected samples from the set displayed in Fig. 1.

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Since the damage threshold under nanosecond irradiation is in the order of ${10}^9\mathrm{W}/{\mathrm{cm}}^2$, the influence of 2PA on the breakdown of the material is also investigated. This resonance is slightly off the calculated required bandgap, especially because there was also a sample with a gap energy of 4.64 eV, which is almost identical to the energy of two photons of 532 nm radiation. The exact origin of this shift is not yet understood, but possible reasons can be mentioned. First, the determination of the bandgap energy from the transmission measurements is accompanied by an error of 3%–5%. This shift alone could account for this offset. Second, there might be some intermediate states located close beneath the conduction band, which might show a lifetime long enough to promote a transition into the conduction band by absorbing another photon. However, this assumption still needs to be validated.

One consequence of MPA-governed laser-induced breakdown is a change in the damage threshold when the bandgap changes to a value requiring a higher order of MPA. For example, when the resonance is shifted from 2PA to three-photon absorption (3PA). Since a 3PA process is by orders of magnitude less probable, the damage threshold should increase when this range is entered. To illustrate the extent of the decreased probability of MPA processes at higher orders, the multiphoton ionization rate calculated according to Keldysh [9] in fused silica (bandgap 8.3 eV) is shown in Fig. 3 as a function of intensity. Each of the displayed curves represents one MPA order associated with a common laser wavelength. For this illustration, the ionization rate was chosen, because an MPA cross section or an absorption coefficient will change its dimension for higher orders. Simply because these quantities are subject to the nonlinear character of the MPA process, they always scale with the intensity and higher powers thereof. The rate, however, is a quantity of identical dimension and can be compared for the different orders of MPA.

 

Fig. 3. Ionization rate as a function of intensity calculated according to Keldysh [9] in fused silica with a bandgap energy of 8.3 eV. Multi photon ionization rate of the orders 2 to 8 (top curve to bottom) are displayed associated with common laser wavelengths.

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At the intensity of interest for nanosecond damage testing (approx. ${10}^9\mathrm{W}/{\mathrm{cm}}^2$), the ionization rates differ in the range of five to six orders of magnitude in between two orders of MPA. In fact, the transition from 2PA to 3PA is the largest one in these calculations. Smith [10] already discussed an abrupt change of MPA when changing from 2PA to 3PA in 1978. This hints at an abrupt behavior change in damage threshold also, if laser-induced damage is MPA driven. Since this is clearly expected for the high intensities in the subpicosecond range, an experimental effort has been shown before to prove this assumption, and Jupé et al. reported on this topic [11]. In their publication, ${\mathrm{TiO}}_2$ was investigated with a tunable laser source with a pulse duration of 130 fs, and a step was observed in the damage threshold as a function of wavelength. The increase was evident, and the laser induced damage threshold was four times higher in the region of the higher order.

The set of samples discussed here was tested for damage thresholds at 532 nm and a pulse duration of 8 ns. By applying 10.000 pulses on each test site at 100 Hz repetition frequency with an ISO21254-2 protocol, the damage threshold of each of these samples was experimentally determined. While Jupé et al. tuned the test wavelength and kept the bandgap of the specimen fixed, in the present experiment the tuned and fixed parameters were interchanged. With a fixed wavelength, samples of changing bandgap were investigated. The result is shown in Fig. 4. The plotted threshold values are the 0.2% damage probability of a power-law fit [12]. In this case, the anticipated step is located at the energy corresponding to the step from 2PA to 3PA. With a photon energy of 2.33 eV, any gap beyond approximately 4.7 eV will require three photons for electrons to be excited into the conduction band. The center of this transition is fairly well located at this value. With a transition range of 0.5 eV, there is a certain width to this effect. The laser-induced damage threshold is increased by a factor of 5 because of this transition. Compared to the increase with ${\mathrm{TiO}}_2$ in the subpicosecond tests [11], this is a rather high and very clear effect to observe with these lower intensities. Another abrupt transition toward four-photon absorption (4PA) is not expected because of the low probability of 4PA. Here, pure defect-induced damage is very likely.

 

Fig. 4. Laser-induced damage threshold of tantala/silica mixtures as a function of bandgap energy. Test parameters: 532 nm, 8 ns, 100 Hz, 10.000-on-1 protocol.

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To support these results, the error budget of the tests has to be discussed. The absolute calibration of the test setup is only of minor importance, because this empirical approach was aiming for a relative increase of the damage threshold. So far there has been no model involved that could suggest a certain value as the damage threshold of tantala in the nanosecond regime. Therefore, only the relative error is relevant and is defined by the stability of the laser. The beam diameter was set to approximately 270 μm ($1/e^2$), and the pulse-to-pulse stability is 3%. The laser system is injection seeded, and spiking of the pulse energy beyond the 3% cannot be observed. For these reasons and with the given pointing stability, the fluence error can be estimated to 10%. The absolute bandgap error is set to 3% because of the empirical fit algorithm.

Morphology investigations of the damaged test sites support a change in the damage mechanism. When entering the 3PA range, defect-induced damage is clearly dominant. In Fig. 5 one image of each range is displayed: in 5(a), 2PA; in 5(b) and 5(c), transition; in 5(d), 3PA—defect-induced damage. In the 2PA region all samples show the same type of damage morphology. As in Fig. 5(a), separated damage sites are distributed under the beam diameter. For each pit, the surface of the substrate is open and the coating in the immediate vicinity is affected by absorbed energy, showing a changed index of refraction. The first sample in the transition slope can also be assigned to these kinds of damage. Starting from the second sample within in the slope [Fig. 5(c)] and for all other samples [Fig. 5(d) as another example of the 3PA plateau], the fully removed film and the index changes in the coating right by the pits are not observed anymore. There are a number of black pinholes that remain after a defect has been removed from the film. This distinct change of damage morphology coincides with the position of the step and therefore supports the influence of 2PA within the damage initiation in tantala film.

 

Fig. 5. Damaged sites in (a) the 2PA region, (b), (c) the transition, and (d) the 3PA or defect-induced damage region.

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