1. Introduction

Dielectric multilayer coatings are fundamental to almost all kinds of laser systems and have therefore significantly contributed to progress in optics. High-power lasers, for example, heavily rely on highly-reflective mirrors. Control of pulse phase properties by dispersive multilayer mirrors (also known as chirped mirrors) is commonly used in ultrafast lasers and applications. The design of the dielectric coatings enables a highly flexible control of almost the entire range of linear optical properties, such as reflectance, bandwidth, phase and dispersion.

Nonlinear optics, however, is still mostly based on bulk materials with mm-sized dimensions. A prominent mechanism is the optical Kerr effect, which can be viewed as an intensity-dependent change of refractive index. Two technologically important consequences of the Kerr effect are self-focusing, used for laser mode-locking, and self-phase modulation, used for compressing laser pulses to few-cycle duration. The quasi-instantaneous response time of several femtoseconds of the Kerr effect is essential for both applications [1]. Adopting the optical Kerr effect for laser technology has enabled a plethora of ultrafast applications, ranging from the ultrafast light gate [2] to the revolutionary Kerr-lens mode-locking technique [3], from near-at-hand light modulators, presented here, to potential optical-to-optical computers. Previous implementations of nonlinear multilayer mirrors comprised of dielectric materials have not been based on the Kerr effect, but rather relied on nonlinear absorption [4], or frequency conversion [5]. However, a lossless element is essential for many applications, in particular within lasers or enhancement cavities. To this end, dielectric oxide films with low losses (in contrast to metals) come into sight as promising sources of nonlinearity. Moreover, dielectric materials are appealing as they are highly resistant to laser damage, can excellently withstand abrasion and environmental influence, as well as due to the maturity of available deposition technologies.

In this work, we realize dielectric optical coatings with nonlinear behavior dominated by the optical Kerr effect, in order to produce a device with almost instantaneous nonlinearity and, preferably, low losses. This idea has not, to date, transcended theoretical investigations [6]. In many simulations foreshowing a strong nonlinear response of dielectric multilayer structures, authors have used materials (often not specified) with overestimated values for Kerr coefficients [7–11]. In reality, the fundamental challenges emerge from the comparatively small nonlinear indices of dielectric materials [1,12,13] and the necessity to work at high intensities. Here, we present the design guidelines for creating multilayer structures in which the overall optical response is made extremely sensitive to slight changes of the layer refractive indices. The result is a Kerr-effect-driven dielectric multilayer element with intensity-dependent reflectance and transmittance. Specifically, reflectance increases at higher intensities, making the device potentially useful for mode-locking applications. This result holds the prospect for devices purely based on dielectric materials and the optical Kerr effect—nonlinear multilayer coatings (NMCs)—allowing nearly-lossless operation for future applications.

2. Materials and methods

2.1 Design considerations

In contrast to all previous works, the present paper is concerned primarily with multilayer dielectric coatings for the near-infrared spectral region (namely 1030 nm) operating at high intensities on the order of 10⁹ W/cm², leaving semiconductor, organic, metal and exotic materials outside the scope of this study.

The optical Kerr effect in a medium results in an almost instantaneously occurring intensity-induced change in the refractive index [14]:

 

Fig. 1 The optical Kerr effect illustration: (a) NMC design F502 over a wide wavelength range; (b) NMC design F502 in the vicinity of 1030 nm: Reflectance at the edge increases from R_i at low intensities to R_f at higher intensities.

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The first realization of an NMC (F502) shown in Fig. 1 (low intensity) with a central wavelength of 1030 nm consisted of 69 alternating layers of the dielectric materials Nb₂O₅ (band gap E_g ~3.5 eV) and SiO₂ (E_g ~8.3 eV), resulting in a total thickness of 8.7 μm. The working range of the edge filter design is the edge itself (Fig. 1(b)), the position of which can be easily adjusted by varying the AOI of the incident light [16]. The design of F502 was therefore tailored for a non-zero AOI = 20° for p-polarized light. The next realization of an NMC, F504 was obtained with the dielectric materials Ta₂O₅ (E_g ~3.8 eV) and SiO₂, with a total thickness of 9.3 μm. The multilayer structure of F504 was designed to have an optimized flat group delay dispersion of −5000 fs², essential for potential laser applications, at an AOI = 20° over a wavelength range 1024–1033 nm. A commercial software package OptiLayer [17] was used to obtain these designs.

Thorough studies of the nonlinear refractive indices of the dielectric thin-films used in this work have, to date, not been presented in the available literature. However, the nonlinear indices of Nb₂O₅ and Ta₂O₅ in crystal form are on the order of 10⁻¹⁵ cm²/W (refs [12,13].) and are, at times, used for reference purposes.

2.2 Sample fabrication and characterization

The designed multilayer structures were deposited on fused silica substrates with thicknesses of 145 μm and 6.35 mm by use of the magnetron sputtering technique [18] (see Appendix A). The pre-damage behavior of the NMCs F502 and F504 coated on 145 μm-thick substrates was studied experimentally with the setup shown in Fig. 2. The laser was an Yb:YAG thin-disk regenerative amplifier with a repetition rate of 50 kHz (ref [19].). The pulse intensity was varied with a half-wave plate followed by a polarization cube placed in front of the focusing lens. A strong escalation of the scattered light from the sample surface, monitored with a charge-coupled device camera, served as an indication of damage occurrence [20]. The laser beam was focused on a sample in order to provide intensities up to 5% below the laser-induced damage threshold of the samples.

 

Fig. 2 Schematic of the setup for investigating the pre-damage behavior of the samples. Yb:YAG thin-disk regenerative amplifier: 1030 nm central wavelength, 50 kHz repletion rate, 1 ps pulse duration.

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Both the reflected and transmitted average power values were measured for each incident power value with a powermeter (PM-10, COHERENT). The relative temperature change of the sample was monitored during the measurements via an IR camera (FLIR SC 300 Series camera A325) with a measurement accuracy of ± 2°C. The IR camera realized an additional control in monitoring damage occurrence, as the relative temperature increased rapidly at the onset of damage. The IR camera was set to measure the temperature difference ΔTemp between the spot where the laser hit the NMC sample surface and the ambient medium. The sample was placed on a rotational stage in order to adjust the AOI and achieve different initial reflectance R_i to initial transmittance T_i ratios (R_i/T_i).

Pump-probe measurements were carried out in order to investigate the time response of the NMCs. The experiments were performed with a conventional degenerative pump-probe setup utilizing the aforementioned laser source (see Appendix B).

3. Results and discussion

3.1 Nonlinear pre-damage behavior

Figures 3 and 4 illustrate the pre-damage performance of the NMCs F502 (Nb₂O₅/SiO₂) and F504 (Ta₂O₅/SiO₂) carried out using the setup shown in Fig. 2. The plots reveal evidence that the reflectance of both samples increases, the transmittance decreases and the relative temperature grows with the incident intensity; the total losses (100%–R–T) are on the scale of measurement error.

 

Fig. 3 Intensity-dependent change of reflectance, transmittance, relative temperature and total losses of NMC F502 for R_i/T_i of 50/50 (a) and 80/20 (b). The error bars represent standard deviations.

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Fig. 4 Intensity-dependent change of reflectance, transmittance, relative temperature and total losses of NMC F504 for R_i/T_i of 50/50 (a) and 80/20 (b). The error bars represent standard deviations.

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The observed temperature growth could be justified by absorption, inevitable even in multilayers of high quality [21]. The absorption origin lies outside the present discussion, however, among the possible causes the following might be important: multiphoton absorption, dust particles and impurities, native defects [22] (shallow traps), self-trapped excitons (deep traps) subsequently leading to long-lived color center formations [23–25], or a complex combination thereof. At intensities on the order of 10⁹ W/cm², absorption processes become relevant and result in the temperature increase. However, the extent to which the absorption contributes to the total losses is not quantifiable as the latter lie within the measurement error (Figs. 3 and 4), which allows one to conclude that absorption is not the dominant process in the present case. On the other hand, the modulation depths for both F502 and F504, both in reflectance and in transmittance, are well-defined and are certainly larger than the errors (Figs. 3 and 4).

The modulation depths of the NMCs are determined by the AOI (according position of the edge), which also defines the initial proportion R_i/T_i. As one can see (Figs. 3 and 4), the modulation depths in reflectance and transmittance are much higher for the case R_i/T_i = 50/50 (left panels) compared to those at R_i/T_i = 80/20 (right panels), for both F502 and F504. It is evident, that the optical properties of the edge filter design are highly responsive to an intensity increase, and ΔR can reach 16% under particular conditions (for R_i/T_i = 50/50). Note, that the damage threshold of Nb₂O₅/SiO₂ coatings is usually lower compared with that of Ta₂O₅/SiO₂ coatings [26], which is the case for the results presented here as well (Figs. 3 and 4, compare the last measurement points).

3.2 Decoupling of the temperature influence

As a temperature rise of the NMCs irradiated by a laser was observed (see Figs. 3 and 4), studies were conducted in order to analyze the influence of temperature on the NMC performance. For the presented NMC design, due to thermal expansion and the thermo-optic effect, a temperature rise unavoidably causes a shift of the edge position to longer wavelengths, i.e. in the same direction as the optical Kerr effect (see Fig. 1). Essentially, the steeper the edge of the NMC (enhancing the Kerr effect), the more susceptible it is to temperature shifts. The dependence of ΔR on ΔTemp was estimated by simultaneously recording reflectance spectra and measuring the surface temperature of the NMCs (coated on 6.35 mm substrates) via an IR camera. This dependence was found to be linear and allows one to separate the temperature impact resulting in the increase in reflectance of the NMCs (for more details see Appendix C).

In order to experimentally exclude the impact of temperature on the NMCs, a chopper wheel was introduced into the setup described above (behind the lens, Fig. 2). When the laser beam is modulated by the chopper wheel (10% duty cycle), the incident average power drops by approximately a factor of ten (as well as the induced temperature modulation) while the incident intensity remains the same.

Figure 5 shows the decoupling of the temperature-induced and nonlinear behavior for F502 obtained using the setup shown in Fig. 2 with the incorporated chopper wheel. The temperature-induced modulation depths ΔR_temp for F502 (red triangles in Fig. 5) were computed with the help of the experimentally determined linear slope of the dependence of ΔR on ΔTemp, mentioned above. Next, these ΔR_temp estimations were subtracted from both the modulation depths measured with the chopper ΔR_ch (blue diamonds in Fig. 5(b) and 5(d)) and the total modulation depths measured without the chopper ΔR_tot (violet diamonds in Fig. 5(a) and 5(c)). Thus, the nonlinear change in reflectance ΔR_nl was obtained as follows:

 

Fig. 5 Estimation of the “pure” nonlinear effect for NMC F502: Estimated nonlinear ΔR_nl (yellow diamonds) is obtained as a result of subtraction of the temperature-induced ΔR_temp (red triangles) from the measured ΔR_tot (a, c: violet diamonds) without chopper and ΔR_ch (b, d: blue diamonds) with chopper for two cases of R_i/T_i of 50/50 (upper panels a and b) and 80/20 (lower panels c and d). The error bars represent standard deviations. Parabola fits (solid black curves) serve as a guidance to the eye.

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Fig. 6 Estimation of the “pure” nonlinear effect for NMC F504: Estimated nonlinear ΔR_nl (yellow circles) is obtained as a result of subtraction of the temperature-induced ΔR_temp (red triangles) from the measured ΔR_tot (a, c: green circles) without chopper and ΔR_ch (b, d: blue circles) with chopper for two cases of R_i/T_i of 50/50 (upper panels a and b) and 80/20 (lower panels c and d). The error bars represent standard deviations. Parabola fits (solid black curves) serve as a guidance to the eye.

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The temperature impact appears to be almost 50% (or even more depending on the value of R_i/T_i) of the observed modulation depths in reflectance and transmittance for the unmodulated laser radiation case (chopper off in Figs. 5 and 6). Basically, the experimentally observed pre-damage behavior of the studied elements is an interplay between the temperature-induced and nonlinear effects in these particular NMC realizations. Nevertheless, it is possible to circumvent the temperature influence by introducing a chopper wheel. Thus, the modulation depths measured in such a way can be regarded as “pure” nonlinear effect estimations. As one can notice, the effect of about 6% occurs at 50% edge of the filter F504 (Fig. 6), and the effect of about 3% – at 80% of the same edge filter. These values can be concluded to be rather large if to take into consideration that the nonlinear coefficients of dielectrics are estimated to be on the order of 10⁻¹⁵ cm²/W (refs [12,13].) and a typical change in refractive index is not higher than 0.01.

3.3 Pump-probe measurements

Pump-probe measurements were performed in order to fully verify the Kerr-based nature of the nonlinear response of the new elements. Time-resolved recovery curves of the NMCs F502 and F504 for different values of R_i/T_i, obtained with a conventional pump-probe setup, are presented in Fig. 7, panels a and b respectively. Any possible coherence spike, typically observed near zero delay time at about 5–20 fs (see for example ref [27].), cannot be resolved due to the much longer source pulse duration of 1 ps. The detected relative changes of R are presented normalized as these values are most probably affected by the presence of the coherence spike, and thus are not of interest. Note also, that pump-probe technique was used in order to measure the time response of the NMCs, not the absolute values of the modulation depth in reflectance. Figures 7 and 8 provide evidence that the reflectance change disappears very quickly after the pump pulse for both initial conditions. Over the course of the measurements, the recovery of F504 was consistently below the 1 ps sensitivity of the detection system. The response time of F502, however, contained some features observed after the pump pulse is gone (Fig. 7). These undetermined features are under ongoing investigation.

 

Fig. 7 Normalized time response of NMC F502, hit by a 1-ps pulse at zero time delay for two cases of R_i/T_i of 50/50 (upper panels a and b) and 80/20 (lower panels c and d). Left panels a and c illustrate the long time behavior, whereas right panels b and d illustrate the short time behavior. Solid black curves are Gaussian fitted ACF.

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Fig. 8 Normalized time response of NMC F504, hit by a 1-ps pulse at zero time delay for two cases of R_i/T_i of 50/50 (upper panels a and b) and 80/20 (lower panels c and d). Left panels a and c illustrate the long time behavior, whereas right panels b and d illustrate the short time behavior. Solid black curves are Gaussian fitted ACF.

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The autocorrelation function (ACF) of the laser source used in the experiments was measured with the help of a commercial autocorrelator (A·P·E pulseCheck 15). The measured ACF was then fitted by the expected Gaussian function with a full width at half maximum (FWHM) of 1.46 ps, from which the pulse duration τ_p of 1 ps can be calculated via ${\tau }_p=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}$}\right.\cdot FWHM$. The normalized time response of both NMCs is compared to the Gaussian fitted ACF (solid black curves in Figs. 7 and 8). Panels b and d in Fig. 8 make it obvious that the recovery of F504 entirely follows the 1-ps pulse. Much slower temperature effects, usually occurring within seconds, cannot contribute to response of NMCs on 1 ps time scale. Thus, at least for F504, the “pure” change in reflectance measured previously (Fig. 6) is fully related to the nonlinear time response with an upper limit of 1 ps. For comparison, the nonlinear response of a metallodielectric mirror [28] does not fully recover after a delay of 16 ps.

Overall, in multilayer thin films, nonlinear absorption has been shown to only lead to decreasing reflectance (e.g., ref [4].), which is not the case in the present study (Figs. 3 and 4). The temperature-induced effects were isolated, enabling the observation of the “pure” nonlinear effect (Figs. 5 and 6). Recovery of the NMCs with an upper limit of 1 ps was shown (Fig. 8). Hence, from a physical point of view, the nonlinear pre-damage performance of the NMCs is dominated by the optical Kerr effect. Previously reported calculations [6,9,11] and experiments with various nonlinear optical devices (e.g., refs [5,28].) have predominantly been performed for the wavelengths equal to or shorter than 800 nm, where multiphoton absorption processes arise in dielectrics with a high probability. For the present investigation, any multiphoton absorption processes are considered detrimental. For the same bandgap, the longer the wavelength, the less the probability of multiphoton absorption and other intensity-dependent absorption effects, hindering the beneficial optical Kerr effect. A step towards longer wavelengths and, consequently, to a nonlinear optical response dominated solely by the optical Kerr effect, was undertaken herein. Operating at longer wavelengths such as 1.5 μm or 2 μm is expected to bypass the limitations originating from intensity-dependent absorption.

3.4 Possible applications

Here, one of the possible applications is demonstrated, namely performance of an NMC as a passive, intensity-stabilizing element for laser pulse trains. Figure 9 shows a plot of average transmitted power versus incident laser peak intensity, illustrating that the transmitted power practically remains constant in the nonlinear regime (starting from about 100 GW/cm²) despite the increasing incident intensity. This result shows that NMC F504 exhibits “limiting performance”, i.e., limits the transmitted power to a certain value which depends on the R_i/T_i ratio. The instantaneous Kerr effect thereby ensures fast stabilization of the transmitted radiation, whereas the temperature effect additionally provides slow stabilization at a time scale of 1 s.

 

Fig. 9 Limiting performance of NMC F504: Dependence of transmitted power on the peak intensity for different R_i/T_i values enumerated by the legend.

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4. Conclusions

Nonlinear multilayer coatings (NMCs) for the near-infrared spectral range have been designed, manufactured and characterized in detail within the frame of this work. The observed behavior of the NMCs has been found to be the result of two separate manifestations: a quasi-instantaneous nonlinear response driven by the optical Kerr effect, and temperature-induced effects arising through thermal expansion and the thermo-optic effect. The detrimental temperature influence has been studied independently. The modulation depth in reflectance of the NMCs due to the optical Kerr effect has been shown to be as high as 6%, which is fairly significant considering the small Kerr coefficients of the composing dielectric materials. Pump-probe measurements have shown that the nonlinearity of the NMCs has a rapid recovery time shorter than the laser envelope width of 1 ps.

It can be realistically conceived that dielectric NMC elements, which can be fabricated using standard coating techniques available in many facilities around the world, could become a substantial alternative for conventional nonlinear materials in laser applications. NMCs can potentially be integrated in laser oscillators as intensity-limiting, spatial or temporal filters. With further optimization, NMCs may serve as multilayer amplitude modulators for initiating or assisting laser mode-locking.

Appendix

A. Deposition technique and post-production characterization

The designed nonlinear multilayer coatings (NMCs) were produced by means of the coating plant (Helios, Leybold Optics) equipped with two proprietary TwinMags magnetrons and a plasma source for plasma/ion-assisted reactive middle-frequency dual-magnetron sputtering. It exploits precise time-controlled monitoring of layer thicknesses, based on accurate calibration of the deposition process parameters affecting deposition rates. Additionally, built-in broadband in-situ monitoring [29] with a B270 test glass as a witness sample was used.

After production, the samples were characterized with the help of a commercial spectrophotometer (Perkin Elmer Lambda 950) and a home-built white-light interferometer [30,31]. A comparison between the designed and measured data is presented in Fig. 10. The agreement between theoretical and experimental data is quite good, given that the design is rather sensitive to manufacturing errors. However, some deviation, that can be noticed in Fig. 10(b), is not important as it can be easily compensated for by tuning the angle of incidence (AOI) of the incident light at an NMC.

 

Fig. 10 Comparison of the designed and measured data for NMC F504: (a) The designed reflectance (blue curve) and the measured data (red curve) at AOI = 0°; (b) The designed group delay dispersion (blue curve) and the measurement performed with a white-light interferometer (red circles) for p-polarized light at AOI = 20°.

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It should be mentioned, that peak powers not higher than 75 MW (average power 4 W) were used for sample characterization, and the beam waste w at the sample was rather large (about 180 μm). On the other hand, many systems operate at lower intracavity peak powers of 5–40 MW (ref [32].) requiring much tighter focusing, which may cause self-focusing over the 6.35-mm propagation distance of a standard fused silica substrate. In such thick substrates, self-focusing processes may alter the properties of the transmitted beam, which is detrimental for the potential laser applications. In order to characterize the potentially applicable laser components, the NMCs deposited on the particular thin (145 μm) substrates were used in the course of measurements with the laser involved. Notice, that the thin (145 μm) substrates cannot be polished as good as standard (6.35 mm) ones, which might justify the reduced damage threshold values.

The peak intensity I_p for a Gaussian pulse, both temporally and spatially in transverse direction, was calculated as follows [33]:

(5)$$I_p={\left(\frac{\pi }{2}\right)}^{-\frac{3}{2}}\frac{P}{f_{rep}{\tau }_p{w}^2},$$

where P is the average incident power, f_rep is the laser repetition rate (50 kHz), τ_p is the pulse duration (1 ps), and w is Gaussian beam radius – the distance from the beam axis, at which the intensity drops to 1/e² of its maximum value (about 180 μm in this study). Notice also that the intensity values presented in this work should only be used as a guide due to imperfect beam radius estimation.

B. Pump-probe setup

Figure 11 displays the pump-probe setup used. The incoming pulse was divided by a beam splitter into pump and probe pulses, and the latter was sent through a variable delay line. The probe beam was attenuated by a factor of 100 with respect to the pump in order to measure only the pump-induced reflectance changes.

 

Fig. 11 Schematic of a degenerate pump-probe setup. BS, beam splitter; AF, attenuating filter.

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Both pulse replicas were focused on the sample by lenses, and the reflected probe pulse was detected by a photodiode. To examine an approximately uniformly excited region of the sample, the probe was focused onto a spot significantly smaller than the area of the pump. The probe beam remained s-polarized whereas the polarization of the pump was rotated through 90° after propagating across a half-wave plate and a polarization cube, which decreased the possibility of a pronounced so-called “coherence spike” appearing [34]. Many scientific groups have reported on observing a peak around zero delay time, termed as coherence spike (known as coherent interaction or coherent nonlinearity as well) in pump-probe femtosecond measurements [27, 35–37]. The possible presence of a coherence spike needs to be taken into account, because the measurement results achieved with the pump-probe method may contain artefacts which can falsely be attributed to the sample response.

The photodiode was set to detect s-polarization by employing another half-wave plate and a cube immediately before the detector, hence reducing the influence of unwanted scattered light (Fig. 11). The pump pulse was mechanically modulated by a chopper and the pump-induced reflectance modulation of the probe was detected by a lock-in amplifier in order to increase the signal-to-noise ratio. Typically, this kind of setup is capable of a sensitivity up to ΔR/R_i = 10⁻⁴ at 1 kHz (ref [38]).

The linear reflectance R_i (initial reflectance at low intensities) of the samples was measured separately by a powermeter in order to set the desired initial reflectance R_i to initial transmittance T_i ratios (R_i/T_i). The time response measurements of the NMCs were performed at the intensities of approximately 15% below the damage threshold.

C. Temperature influence estimations

First, NMCs F502 and F504 coated on 6.35 mm fused silica substrates were placed into an oven (for 2-2.5 hours at 100°C and 150°C respectively) and were characterized with the help of the industrial spectrophotometer Perkin Elmer Lambda 950 and a portable FLIR E5 camera. Figure 12 illustrates representative temperature measurements for the NMCs. Both the reflectance spectra and the absolute temperature of the samples were recorded starting from the maximal temperature (red curves in Fig. 12) down to the ambient air temperature (blue curves in Fig. 12). Figure 12 experimentally confirms that a temperature rise leads to a shift of the edge position to longer wavelengths.

 

Fig. 12 Reflectance spectra of NMC F502 (a) and NMC F504 (b): Spectra shift to longer wavelengths with increasing temperature. Black circles positioned at dashed lines set for two cases of R_i/T_i of 50/50 and 80/20 show the increase of reflectance due to the temperature rise.

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For an accurate characterization, it is necessary to determine the evolution of the optical properties of each layer material composing the multilayer stacks in response to a change in temperature. Therefore, knowledge of the physical parameters describing each dielectric material, such as the coefficient of thermal expansion α and the thermo-optic coefficient β is required. Franc et al. have summarized the thermo-mechanical and optical parameters for several optical coating materials (ref [39]. and references therein). The coefficients α and β for the dielectric materials used in this study are shown in Table 1. However, these parameters have mainly been determined for single layers and are in general different for multilayer stacks. Moreover, the coefficients α and β of a single layer always depend on the deposition process. Thus, the coefficients α and β for the NMCs presented here most probably differ from the literature ones. As evident from Table 1, Nb₂O₅ is more susceptible to temperature than Ta₂O₅: thermal expansion coefficient for the former is 1.6 times larger, and the thermo-optic coefficient is one order of magnitude higher than that of Ta₂O₅. One can estimate the change in reflectance due to thermal effects ΔR_temp, for a certain temperature change ΔTemp, based on the coefficients of thermal expansion α and the thermo-optic coefficient β for each layer of the m-layer structure:

(6)$$\begin{array}{l}\Delta R_{temp}=R(d_1(1+{\alpha }_1\cdot \Delta Temp),\dots ,d_m(1+{\alpha }_m\cdot \Delta Temp);\\ \begin{array}{cccc}& & & \end{array}\begin{array}{cccc}\begin{array}{ccc}& & \end{array}& & & \end{array}{n}_H+{\beta }_H\cdot \Delta Temp,n_L+{\beta }_L\cdot \Delta Temp;\lambda ;\vartheta )-R_i,\end{array}$$

where d₁, … d_m are the physical thicknesses of the layers, n_H and n_L are the refractive indices of the high-index and low-index materials comprising the multilayer stack, respectively, and ϑ is the AOI.

Table 1. Thermo-mechanical and optical parameters for the layer materials at room temperature^a

View Table

Further, the measured data in Fig. 13 allows one to estimate the dependence of ΔR on the temperature for both NMC samples (violet diamonds for F502 and green circles for F504 in Fig. 13) at two levels of R_i/T_i (50/50 and 80/20). Theoretical estimations for ΔR (black diamonds for F502 and black circles for F504 in Fig. 13) based on the literature coefficients (Table 1) were computed by virtue of Eq. (6) for the temperature values measured in the experiment at same two levels of R_i.

 

Fig. 13 Comparison of the temperature influence for NMCs F502 and F504 for two cases of R_i/T_i of 50/50 (a) and 80/20 (b) based on experimental (violet diamonds and green circles, accordingly) and theoretical estimations (black diamonds and circles, accordingly). Solid black lines are linear fits of both experimental and theoretical estimations.

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As indicated in Fig. 13, the theoretical and experimental estimations correlate quite well for NMC F504, while the data for NMC F502 does not quite agree with each other. This discrepancy may be the result of more reliably defined coefficients for Ta₂O₅ thin films compared to those for Nb₂O₅ as more studies have been performed for Ta₂O₅ thin films than for films composed of Nb₂O₅ (see, e.g., ref [39]. and references therein). Furthermore, one can notice, that the sample F504 is less prone to temperature effects than the F502, as it was expected based on the thermal parameters of the comprised materials (Table 1).

Funding

Deutsche Forschungsgemeinschaft (DFG) Cluster of Excellence, “Munich Centre for Advanced Photonics,” (http://www.munich-photonics.de). TA receives the funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie agreement No 657596.

Acknowledgments

The authors gratefully acknowledge Prof. Dr. Ferenc Krausz for his support and fruitful discussions. We also thank W. Schneider, A. Ryabov, D. Ehberger and Bo-Han Chen for providing and maintaining the laser source, and M. Walbran for his assistance in styling the text.

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