Kerr effect in multilayer dielectric coatings

We report the utilization of the optical Kerr effect in multilayer dielectric coatings, previously discussed only theoretically. We present the design and realization of multilayer dielectric optical structures with layer-specific Kerr nonlinearities, which permit tailoring of the intensity-dependent effects. The modulation depth in reflectance reaches up to 6% for the demonstrated examples of dielectric nonlinear multilayer coatings. We show that the nonlinearity is based on the optical Kerr effect, with the recovery time faster than the laser pulse envelope of 1 ps. Due to high flexibility in design, the reported dielectric nonlinear multilayer coatings have the potential to open hitherto unprecedented possibilities in nonlinear optics and ultrafast laser applications. © 2016 Optical Society of America OCIS codes: (310.1620) Interference coatings; (320.7110) Ultrafast nonlinear optics; (310.6860) Thin films, optical properties; (310.4165) Multilayer design. References and links 1. 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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-tooptical 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][8][9][10][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.

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 10 9 W/cm 2 , 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]: 0 2 . n n n I = + ⋅ Here n 0 is the linear refractive index, n 2 [cm 2 /W] is the second-order nonlinear refractive index (often referred to as the Kerr coefficient), and I [W/cm 2 ] is the light intensity. For example, a light intensity of about 10 9 W/cm 2 can cause observable nonlinearities [14]. A typical response time for the nonlinearity induced by the electronic polarization is 1 fs (ref [1].). In order to enhance the impact of the optical Kerr effect, the optical characteristics of an NMC design need to be sensitive to slight changes in refractive index. An edge filter design, shown in Fig. 1, is extremely sensitive to small refractive index variations in the transition zone between high and low reflectance around 1030 nm ( Fig. 1(b)). The Abeles matrix method [15], derived directly from Maxwell's equations and commonly used for calculating the optical properties of multilayers, for a given wavelength λ determines the initial reflectance R i of an m-layer structure through the physical thicknesses of the layers d 1 , … d m , and the angle of incidence (AOI) ϑ: where n H and n L are the refractive indices of the high-index and low-index materials comprising the multilayer stack, respectively. In simplified form, the optical Kerr effect in multilayer coatings can be described as follows. The incident laser beam interferes with itself in the multilayer structure and, consequently, manifests a standing wave distributed along the coordinate perpendicular to the coating surface. Due to the presence of such a standing wave, a weak increase in layer refractive indices corresponding to the field intensity profile emerges, in accordance with Eq. (1). The latter can also be considered as an additional microstructure, which in turn leads to an increase in reflectance of the whole multilayer structure from R i at low intensities up to its final value R f at higher intensities: where ΔR(I) is the modulation depth of an NMC (see Fig. 1(b)). 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 Nb 2 O 5 (band gap E g ~3.5 eV) and SiO 2 (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 Ta 2 O 5 (E g ~3.8 eV) and SiO 2 , 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 fs 2 , 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 Nb 2 O 5 and Ta 2 O 5 in crystal form are on the order of 10 −15 cm 2 /W (refs [12,13].) and are, at times, used for reference purposes.

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 thindisk 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 laserinduced damage threshold of the samples.
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).  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.

Nonlinear pre-damage behavior
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 10 9 W/cm 2 , 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 Nb 2 O 5 /SiO 2 coatings is usually lower compared with that of Ta 2 O 5 /SiO 2 coatings [26], which is the case for the results presented here as well (Figs. 3 and  4, compare the last measurement points).

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: where ch temp R Δ and tot temp R Δ are the temperature-induced modulation depths with and without the chopper respectively. Analogous estimations were performed for F504 and yield similar results (Fig. 6). The negative values seen in Figs. 5 and 6 can be vindicated by an increased error budget of the powermeter at low intensities. Finally, the "pure" nonlinear changes estimated in this manner for both NMCs correspond to the measured values of modulation depths of reflectance in absence of temperature growth within the experimental uncertainty (Figs. 5 and 6).
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 cm 2 /W (refs [12,13].) and a typical change in refractive index is not higher than 0.01.   panels c and d). The error bars represent standard deviations. Parabola fits (solid black curves) serve as a guidance to the eye.

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.
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 1    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 intensitydependent 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.

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/cm 2 ) 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.

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. 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°. 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.

B. Pump-probe setup
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 pumpprobe 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 pumpprobe 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 pumpinduced 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.  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, Nb 2 O 5 is more susceptible to temperature than Ta 2 O 5 : 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 Ta 2 O 5 . 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: where d 1 , … 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. 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 .
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 Ta 2 O 5 thin films compared to those for Nb 2 O 5 as more studies have been performed for Ta 2 O 5 thin films than for films composed of Nb 2 O 5 (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.