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The temperature dependence of the laser-induced damage threshold on optical coatings was studied in detail for laser pulses from 123 K to 473 K at different temperature. For pulses longer than a few picoseconds, the laser-induced damage threshold of coated substrates increased with decreasing temperature. This temperature dependence was reversed for pulses shorter than a few picoseconds. We describe the physics models to explain the observed scaling. The electron avalanche is essential to explain the differences in the temperature dependence.
© 2013 Optical Society of America
1. Introduction
In recent years, the progress of high power and energy laser systems has been outstanding and has accelerated the use in many scientific and industry fields. These advanced laser systems have many attractive applications such as debris removal, laser peening, laser rocket, neutron and gamma-ray sources, cancer therapy with laser driven electrons and ions, new material production and laser fusion. The output power and energy are determined in many cases by the laser induced damage threshold (LIDT) in these laser systems. The LIDT may be improved by understanding the laser damage mechanisms (LIDMs) at the conditions met in the laser systems.
LIDMs have been studied intensively in the last several decades [1]. According to a theoretical model [2–4], the laser-induced damage by short pulse laser (pulse width τ<2 ps) is initiated by multi-photon ionization. Further heating leads to collisional ionization and damage. An experimental examination of laser-induced damage in dielectrics by Stuart et al. demonstrated excellent agreement with the model [2,3]. In 2001, Schaffer et al. discussed the dominant process involving the electrons that trigger laser-induced damage at short pulse widths [5]. The study presented the laser intensity dependence of the multiphoton ionization and tunneling ionization rates for silica glass in terms of the Keldysh parameter [6]. The Drude, Mie theory, and thermal diffusion model revealed the τ1/2 scaling in more detail [7].
The role of temperature in the LIDM is extremely important for cryogenically cooled Yb:YAG laser systems that may be a promising candidate for the high power laser system in next generation. At the same time, it is very important to study the LIDM dependence on the pulse width from femtosecond to nanosecond regimes because the chirped pulse amplification (CPA) laser systems are widely in use in various fields. There were a few studies of temperature role more than 25 years ago [8,9]. However it is clear that detailed systematic study is called for obtaining high LIDT for the advanced laser systems with parameters both temperature and pulse width. We have demonstrated the temperature dependence of the LIDT under nanosecond infrared laser pulses for bulk silica glasses [10], a glass surface [11], single-layer optical coatings [11], and metal materials [12]. We have also confirmed the temperature dependence of nonlinear phenomena related to laser-induced damage [13].
In this paper, we elucidate physics models of the temperature dependence by evaluating the LIDT at different temperatures from 123 to 473 K and pulse widths 100 fs to 4 ns and suggest several physical mechanisms to explain the data.
2. Experiment
2.1 Experimental condition
We evaluated the temperature dependence of the LIDT of single-layer optical coatings using a Nd:YAG laser (wavelength 1064 nm, pulse width 4 ns) and a Ti:sapphire laser (wavelength 800 nm, pulse width 100 fs, 2 ps, and 200 ps). As experimental samples, six types of single-layer coatings (SiO2, Al2O3, HfO2, ZrO2, Ta2O5, and MgF2) were prepared on optically polished silica glass substrates (root mean square roughness <0.8 nm) by electron beam evaporation. The optical thickness of the single layers had a quarter wavelength (nd = λ/4) at 800 nm. Figure 1 shows an experimental layout for the evaluation. The laser pulse was focused on the sample by a lens with an 800-mm focal length at the 1064-nm wavelength or with a 200-mm focal length at the 800-nm wavelength. The focal spot sizes were defined by the 1/e2 peak intensity and were 80 μm (λ = 800 nm) and 160 μm (λ = 1064 nm) in diameter. The irradiation angle was 0°. The sample was placed in a copper holder and set in a vacuum chamber (~5 Pa) to avoid water condensation at low temperature. The temperature was controlled with a combination of liquid nitrogen and a heater, and was monitored with a platinum resistance temperature detector attached to the holder. We tested an N-on-1 irradiation scheme. A fixed site was irradiated by laser pulses. The fluence was gradually increased until damage occurred. The laser-induced damage was detected by plasma emission and by transferring the surface image with a co-aligned He–Ne laser [10–13]. The transferred image was monitored by a CCD camera with 3x magnification. We could detect a damage spot of a size as small as 2 μm in diameter. The irradiation of low fluence pulse under LIDT may cause defect treatment and/or defect generation in coating material. Thus, irradiation number of laser pulse is fixed to be 15 pulses at all N-on-1 testing. The electric field in the coatings for 800-nm and 1064-nm wavelengths is slightly different. The difference for Ta2O5 coating was estimated to be about 7%. The LIDT cannot be compared directly at different spot size. Since the LIDM is not affected by the spot size, the temperature dependence of LIDT is evaluated without comparison among the pulse widths.
The thermo-optic effect (dn/dT) and thermal expansion affect the electric field distribution and the optical spectrum in the coating. Prior to measure the LIDT, the optical spectrum at different temperature has been evaluated with a spectral photometer and a cryostat. Figure 2 shows the spectrum for (a) SiO2 and (b) HfO2 coating at 123 K, 298 K (room temperature), and 473 K. The variation was less than accuracy of the wavelength resolution (~1%), and spectrum shift was not appeared. Thus the variation of the optical thickness is estimated to be less than 1% and the effect of thermo-optic effect and thermal expansion were neglected in this study.
2.2 Experimental result
Figure 3 shows the temperature dependence of the LIDT and morphologies of laser-induced damage site. The damage site was observed with Nomarski differential interference microscope. The damage thresholds were normalized by the LIDT at room temperature. Under 100-fs [Fig. 3(a)] laser pulses, the LIDT decreased with decreasing temperature. On the other hand, a 2-ps laser pulse [Fig. 3(b)], the dependence for some materials exhibited increase of LIDT with decreasing temperature. Additionally, under 200-ps [Fig. 3(c)] and 4-ns [Fig. 3(d)] laser pulses, the LIDT exhibited the opposite trend. The slope of the LIDT for each coating material was changed for pulse widths around a few picosecond. Note that this region roughly agrees with the region of deviation from τ1/2 scaling in the pulse width dependence of the LIDT. Figures 3(e)–3(h) show the morphologies of laser-induced damage and the doted circle means the irradiated spot size. The laser-induced damage is caused by LIDT fluence at room temperature as shown in Fig. 3. The geometries in 100-fs and 2-ps were a dot at center of irradiated laser as shown in Figs. 3(e) and 3(f). It means that the LIDM may have started with multiphoton ionization because the laser-induced damage caused at center of irradiated laser where the highest intensity has. On the other hands, the geometries in 200-ps and 4-ns testing were different as shown in Figs. 3(g) and 3(h). The substrate contains impurities consisting of residues from the abrasive compounds used in optical polishing. The geometry of the damage sites exhibited dots in the center, where the intensity was strongest in the irradiated area, indicating that precursors for photoionization existed. In addition, the damage sites produced by the 4-ns laser pulse were distributed in a wide area in which the refractive index of the coatings changed. The change in the refractive index was triggered by the composition change caused by laser-induced damage during a long pulse. The damage site produced by the 4-ns laser pulse was the largest among those produced by all the pulses. The temperature and coating material did not affect the geometries of the laser damage sites.
Fig. 3 Temperature dependence of LIDT under (a) 100-fs, (b) 2-ps, (c) 200-ps, and (d) 4-ns laser pulses. Photos for SiO2 coatings at room temperature under (e) 100-fs, (f) 2-ps, (g) 200-ps, and (h) 4-ns laser pulses. Doted circle shows focal spot.
3. Quantitative analysis
3.1 Theoretical model
Laser-induced damage occurs when the relation
is satisfied, where n is the electron number density generated by photoionization, phonon–electron interaction, or multiphoton ionization; β(Τ) is the multiplication rate for impact ionization; τ is the pulse width; and V is the volume in which free electrons are generated. The critical density ncr can be estimated using Bose–Einstein statistical mechanics as [14] Here, p = (2mεe)1/2 is the momentum of free electrons, m is the mass of an electron, εe is the energy of an electron, ω(Τ) is the optical phonon frequency, e*≈e(1/ε)1/2 is the effective electron charge, e is the electron charge, ε is the dielectric constant, τs(ε) = 30 fs is the mean collision time, Nw(T) is the average phonon number, k0 is the Boltzmann constant, and T is the temperature. The plasma density at which laser-induced damage began to appear was 1018–1019 cm−3 [2,15–19]. For calculation, Godmanis et al. used ηω(Τ) = 58 meV as a parameter for silica glass [20], and Worlock and Fleury reported the temperature dependence of the optical phonon frequency for oxide materials [21]. The Fig. 4(a) shows the calculated temperature dependence of the critical density ncr(T) for silica glass, which decreased with decreasing temperature. The number of free electrons that caused laser-induced damage at low temperature was lower than that at high temperature.Fig. 4 Calculated temperature dependence of (a) critical density and (b) multiple rate for silica glass.
Electron avalanche, which dominated the LIDM at long pulse widths (τ > 20 ps), were also studied. The process is described using the electron multiplication rate β(Τ) [20,22],
where A is an arbitrary constant, E is the electric field, Eg is the band gap energy, and ρ(Τ) is the electric resistivity. The product of β(Τ)τ was calculated with an arbitrary constant A = 17 [15] for silica glass, as shown in Fig. 4(b). Roy and Chakravorty reported the temperature dependence of the electric resistivity for silica glasses [23]. We assumed that the electric resistivity of silica glass and a SiO2 thin film at room temperature is ρ(298 Κ) = 1015 Ωm. The calculated electron multiplication rates decreased with temperature. Thus, free electrons are multiplied less at low temperature than at high temperature. For an irradiated pulse width nearly equal to the mean collision time [τ ~τs(ε) = 30 fs], the impact ionization is calculated as [24]The temperature dependence of the LIDT will vary because the dominant process in the LIDM varies with the laser pulse width. The critical density increased only by 10% when the temperature of the sample changed from 123 K to 473 K, as shown in Fig. 4(a). On the other hand, the multiplication rates increased by two times over the same temperature range, as shown in Fig. 4(b). Thus, the temperature dependence of the LIDT may be affected by the variations in the multiplication rate and critical density. Electron avalanche will play an important role in the LIDM in determining the trend and direction of the temperature dependence of the LIDT. In particular, the electric resistivity in Eq. (4) is a function of temperature, which reflects the electron mobility in materials.The electron number density n is given by
where np, ni(T), and nm are the contributions of photoionization, phonon–electron interaction, and multiphoton absorption, respectively. Photoionization, which is an important process at long pulse widths, has been defined as the product of the transition rate and the number density of electrons at precursor and/or defect levels [25]. On the other hands, multiphoton ionization transfers electrons in valance band via absorption of two or more photons (multiphoton absorption). The transition rate in photoionization does not depend on the temperature along with multiphoton ionization [26]. The number density of electrons is a function of the temperature known as the Boltzmann distribution; however, an increase in temperature of 350 K is represents an energy change of only 0.035 eV. Because this energy is much lower than the band gap energy of dielectric materials, the variation in the number of electrons is negligible. The number density was assumed to be np = 1013–1015 cm−3, assuming an impurity concentration on the order of 0.01 ppm to 1 ppm. Phonon–electron interactions are also an important process at long pulse widths, and were studied using the energy gain rate G and average energy loss rate L of an electron and the average-electron model [15]. The number density of electrons ni(T) can be estimated asOn the other hand, multiphoton ionization, the main process in the short pulse region (τ < 20 ps) can be estimated by [2]where σ4 = 2 × 10−114 cm8s3 is the cross section of four-photon absorption, Ns is the solid atom density, and ν is the laser frequency. In this estimation, we defined four-photon absorption as a dominant process in multiphoton ionization. The four-photon absorption of 800-nm light yields an energy of approximately 6.2 eV. The band gap of silica glass is 9 eV, but non-negligible defect levels from the E′ center [27–29] and non-bridging oxygen hole center [30,31] exist at around 6 eV. Thus, the transition from valence band to 6 eV state with four-photon absorption will be induced. Then it is straightforward for electrons to move to the conduction band from the defect levels around 6 eV. On the other hands, possibility of six-photon absorption of 1064-nm light may have to be considered. Since the intensity of 1064-nm laser pulse was lower than that of 800-nm laser pulse due to the pulse width difference, the number of density from six-photon absorption with 1064-nm laser pulse has not been calculated.3.2 Calculation result and Discussion
The above discussion enables us to model the temperature dependence of the LIDT. Figure 5(a) shows the measured and calculated temperature dependence of the LIDT for SiO2 coatings. The theoretical LIDT for SiO2 was calculated with reported parameters as presented in section 3.1. The same calculation for other materials may become possible with further investigation about required parameters for each materials. The electric fields that trigger the laser-induced damage were calculated using Eqs. (1)–(8). Next, the LIDT was then calculated using the electric field. The solid lines in Fig. 5(a) show the results of calculations of the linearphenomena: electron avalanche, electron–phonon interaction, and photoionization (excluding nonlinear processes, e.g., multiphoton ionization). For the calculation, we assumed ni(T) = 1013cm−3. In contrast, the dashed and dotted lines in Fig. 5(a) indicate the results with only the nonlinear processes of multiphoton ionization (β(Τ)τV = 1) and multiphoton ionization including impact ionization (β(Τ)τV = 2τ/τs(ε)), respectively. The measured results for 100-fs, 200-ps, and 4-ns pulses are in excellent agreement with the solid line or the dotted line. The model suggests the following:
Fig. 5 (a) Measured and calculated temperature dependence of LIDT of SiO2 coating. 5(b) Measured and calculated temperature dependence of LIDT under 2-ps pulses for SiO2 and HfO2 coatings.
- (i)When the calculation result does not include an electron avalanche, the LIDT slightly decreases with temperature in the short pulse region.
- (ii)When the calculation results include an electron avalanche, the LIDT increases with decreasing temperature.
- (iii)The electron avalanche will be important for determining the trend and direction of the temperature dependence of the LIDT.
The proposed model may also suggest a way to obtain extremely high LIDT optics because the calculated LIDT indicated by solid lines in Fig. 5(a) exponentially increases with decreasing temperature. It is important to evaluate the temperature dependence of the electric resistivity in particular.
However, the measured results for 2-ps pulses show an irregular temperature dependence of the LIDT Fig. 3(b), which is not explained by the simple model. The plots for 2-ps pulses lie between those for linear processes (solid lines) and nonlinear processes (dashed lines) in Fig. 5(a). Because the LIDM includes both linear and nonlinear process, the LIDT should be considered as a combination of both types. Figure 4(b) shows the measured and estimated results for 2-ps pulses for SiO2 and HfO2 coatings. In the estimation of the LIDT of HfO2, the electric resistivity reported by Pereira et al. [32] was used, and the dielectric constant was calculated from the refractive index. For the other parameters, we used the same parameters as those for SiO2. The lines in Fig. 5(b) were obtained using a value corrected by γ, which is defined as
The corrected value γ means a balance between linear and nonlinear processes in LIDM, i.e. the γ gives the solid line (γ = 1) and the dashed line (γ = 0) in Fig. 5(a). The experimental LIDT values are in excellent agreement with Eq. (9) with γ = 0.9815 for SiO2 and γ = 0.97 for HfO2. The corrected values explain the irregular temperature dependence of the LIDT under 2-ps pulses. The dot-dashed lines indicate a moderate difference in the temperature dependence of the LIDT under a 2-ps pulse, as shown in Fig. 5(b). Figure 6 shows calculation results of the temperature dependence of LIDT for SiO2 and HfO2 at 2-ps testing with corrected value γ in Eq. (9). Our theory indicates that the bottom of the dot-dashed lines is easily influenced by the electric resistivity and band gap. In the 2-ps pulse measurement, the temperature dependence of the LIDT in which the LIDT increases with decreasing temperature appears to be the result of a low electric resistivity and a wide band gap. Thus, the corrected value will be described as function of electric resistively and band gap.
Fig. 6 Calculation result for (a) SiO2 and (b) HfO2 material at 2-ps pulse testing with different corrected value γ in Eq. (9).
The model for temperature dependence of LIDT may suggest a method of producing extremely high LIDT optics (at least several times higher than the conventional) because the calculated LIDT [solid line in Fig. 5(a)] exponentially increases with decreasing temperature. LIDT at a low temperature can be improved mainly by controlling the electric resistivity. If the temperature dependence of each parameter (especially electric restively) is fixed at cryogenic condition, we might get the extremely high LIDT optics.
4. Conclusion
In conclusion, the temperature dependence of the LIDT of optical coatings was different when a laser pulse width of a few picoseconds was used compared to the LIDT under other pulses. The geometries of the laser damage site also varied with the laser pulse width. The results of calculations using a proposed model agreed well with the measured results. The electric resistivity, which is related to electron avalanche, i.e., the electron mobility, is an essential parameter affecting the temperature dependence. The irregular temperature dependence of the LIDT under a 2-ps pulse is explained in terms of the proposed model and a corrected value. The proposed model may also suggest a method of producing extremely high LIDT optics because the calculated LIDT [solid line in Fig. 5(a)] exponentially increases with decreasing temperature. To improve the LIDT of optical materials, it is important to evaluate the temperature dependence of the electric resistivity in particular.
We may also propose a method of producing extremely high LIDT optics (at least several times higher than the conventional) because the calculated LIDT [solid line in Fig. 5(a)] exponentially increases with decreasing temperature. LIDT at a low temperature can be improved mainly by controlling the electric resistivity. The new type optics could be realized just by the combination of existing optics and a cryostat where the mirror substrate can be cooled down to either 77 K or 2 K degrees with liquid N2 or He. The new optics will have a potential to demonstrate a high-power that has never been achieved.
Acknowledgments
This study was funded by a Grant-in-Aid for Research Fellow from the Japan Society for the Promotion of Science (to K. Mikami, Grant No. 25-275). The part of K.A Tanaka contribution is supported by JSPS PIRE project (Partnership International for Research and Education).
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