1. Introduction

All materials have a nonzero nonlinear susceptibility of third order, χ⁽³⁾, and are thus able to produce the third harmonic (TH) of laser radiation. For efficient conversion materials with large χ⁽³⁾ are desirable and in addition the phase mismatch of fundamental and third harmonic waves must be kept small. Both demands are difficult to be satisfied simultaneously in isotropic materials.

Several techniques have been developed to mitigate the phase mismatch problem. Optical birefringence in crystals is the most widely used method in particular for high power lasers. Both direct TH generation (THG) in one crystal [1] and cascade schemas comprising of second-harmonic generation (SHG) followed by sum-frequency generation of fundamental and second-harmonic resulted in efficiencies of 10% to 15% for femtosecond pulses. Perfect phase mismatch provided, the conversion increases with the square of the interaction length L². For the conversion of ultrashort laser pulses certain bandwidth requirements have to be met, which limits the useable length of the nonlinear optical component.

Periodic modulation of the linear and / or nonlinear susceptibility was suggested early on to mediate the phase mismatch [2]. Known as quasi-phase matching considerable SHG conversion efficiencies have been reported, for example in periodically poled materials [3] where the sign of the nonlinearity changes, for a review see [4].

More recently photonic crystal metamaterials have gained interest for harmonic, in particular SH, generation, for a review see [5]. Larger lengths of these periodic structures lead to larger conversion usually at the expense of conversion bandwidth. For sufficiently long structures, the SH signal was predicted to scale as L⁶ [6, 7] and a power coefficient of about 5 was observed [7]. This considerably larger L-dependence compared to bulk materials stems from field enhancement in addition to the phase mismatch compensation. Femtosecond pulse, direct TH generation with efficiencies on the order of 6 * 10⁻⁵ were observed in a 3D photonic crystal structure comprised of polysterene nanoparticles at intensities before the onset of continuum generation [8]. Enhanced third-order optical nonlinearities were also observed in ZnO/Al₂O₃ nanolaminates [9] and in silicon nanoparticles driven by magnetic response [10].

Thin film coatings of metal oxides and fluorides deposited, for example, by ion-beam sputtering or e-beam evaporation are widely used in optical components such as mirrors and filters. Stacks of such films can be designed and fabricated with tailored reflection, transmission and dispersion characteristics. Laser induced damage thresholds of these structures are among the highest of any optical material, which allows the handling of high power and high intensity laser radiation. This suggests that specially designed stacks of such films can also be used as nonlinear optical components such as harmonic generators with predefined spectral and dispersive behaviors. Considerable conversion (> 20%) to TH was predicted in optimized designs due to the interplay of an effective phase mismatch correction and local field enhancements in aperiodic structures [11]. Due to the mostly amorphous (isotropic) structure of these coatings the nonlinear optical susceptibility of second order χ⁽²⁾ and hence SHG are negligibly small.

We describe here a frequency tripling mirror (FTM) based on a stack of dielectric films designed for maximum THG, see Fig. 1. While the laminar structure has some similarities with 1D photonic crystals and metamaterials the aperiodic architecture and individual film thicknesses approaching a wavelength cause fundamental differences in behavior and require different analysis. With a 25-layer stack with a total thickness of only about 3 μm conversion efficiencies (in reflection) approaching the percent level have been demonstrated at 790 nm.

 

Fig. 1 Photo of an experimental setup demonstrating the operation of a frequency tripling mirror (FTM) and schematic diagram of the beam propagation. A train of femtosecond pulses (791 nm, 55 fs, 100 MHz) was focused onto the FTM with a parabolic mirror. The TH in reflection produces a fluorescence signal from a white card. The experimental data presented later were obtained with a focusing lens replacing the parabolic mirror.

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2. Frequency tripling mirror design and fabrication

The mirrors consist of alternating layers of a low (SiO₂) and a high (HfO₂ or ternary Hf_xAl_yO) index material, which were ion-beam sputtered on a fused silica substrate. The deposition chamber was equipped with a load lock and an oil-free vacuum pumping system consisting of a turbo molecular and a scroll pump. Starting at a base pressure of 10⁻⁶ mbar, coating materials were sputtered from metal targets under a reactive atmosphere with a typical partial oxygen pressure in the range of 3 × 10⁻⁴ mbar using a three-grid rf ion source (RIM-source). For some samples, in order to achieve low absorption in the UV spectral region and improved structural properties, a few (less than five) percent of Alumina was mixed to Hafnia by a co-sputtering process from a zone target of Hafnium and Aluminum [12]. The refractive index of the Hf_xAl_yO ternary compound is about 2.5% smaller than that for pure HfO₂ in the near IR. Because of their superior mechanical and optical properties the ternary oxides were used for the stacks with large number of films.

The deposition rates were adjusted to 0.14 nm/s and 0.3 nm/s for the high-index and low-index material, respectively. For precise control of the individual layer thicknesses an optical broad band monitoring device [13, 14] (BBM) was employed. The BBM-system monitored the transmission of the layer stack during the deposition in a spectral range between 400 nm to 1000 nm. This allowed us to control the thickness of individual layers in the stack to within one percent.

The layer thicknesses were chosen for maximum THG of the stack. This was done using optical matrices for nonlinear frequency conversion taking into account the layered structure and associated interfaces for one fundamental input wavelength (monochromatic input) and a genetic search algorithm [11]. The search space was constrained to individual layer thicknesses between 5 nm and 350 nm. The relevant nonlinear susceptibility of hafnia is about 4.5 times larger than for silica [15] and can be estimated with 9 × 10⁻²² m²/V². Therefore the silica layers act mainly through their phase response. The nonlinear properties (χ⁽³⁾) of the ternary compound that was used in some of the samples were within measurement errors the same as those for hafnia. The dispersive properties of the high and low-index material were determined from a fit of the measured spectral characteristics of corresponding single layers using commercially available software (SPEKTRUM [16]).

The general layout of a 25-layer FTM is shown in Fig. 2. The overall layer thickness is 3.2 μm with a total of 1.5 μm of Hf_xAl_yO. For comparison the coherence length of the high-index material is about 630 nm. The transmission spectrum of the design shows a local maximum around the expected wavelength for the TH (262.3 nm) and the fundamental (787 nm). The actual (measured) spectrum is Stokes shifted by about 5 nm in the near IR region, which is likely a results of deposition tolerances and uncertainties in the absolute values of the refractive indices of the film materials. Local intensity maxima are near, but not necessarily exactly at, the film interfaces.

 

Fig. 2 (a) Schematic diagram of the film stack of a frequency tripling mirror with 25 layers, and intensity distribution of a monochromatic incident wave (λ_F = 787 nm). The layer stack consists of alternating films of Hf_xAl_yO (dark stripes) and SiO₂ deposited on a substrate (fused silica). Transmission spectra in the (b) near IR and (c) UV of the design (dashed line) and actual sample (solid line). Linear absorption in the stack is negligibly small. The relative, measured THG conversion efficiency (data points) is also shown as function of the fundamental peak wavelength in (b), see text. The vertical arrows indicate the peak wavelength of the fundamental and TH spectrum at maximum conversion according to the design.

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The mirrors were tested using the pulses from a femtosecond Ti:sapphire oscillator (55 fs) focused with an aspheric lens (f ≈ 15 mm). Figure 3 shows the experimental layout. The generated UV radiation in reflection was coupled out with a dichroic beam splitter after passing through the lens. For some experiments, a Pockels cell selected single or bursts of pulses from the train.

 

Fig. 3 Experimental setup to demonstrate the frequency tripling mirror (FTM). The Ti:sapphire laser delivered 55-fs pulses at 100 MHz. PD - avalanche photodiode.

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3. Experimental results and discussion

The TH power was measured as a function of the input fluence, see Fig. 4. In the low-fluence region the expected cubic conversion law was reproduced independent of whether single pulses or the 100-MHz pulse train were used. In the high-fluence region, where we illuminated with single pulses to avoid incubation [17], the slope decreased. Possible reasons for this are the presence of other nonlinear optical processes like Kerr effect and simultaneous (two-photon) absorption of TH and fundamental. Nonlinear absorption in dielectric multilayers with tailored field enhancement has been observed recently [18]. An intensity dependent refractive index changes the optical properties of the stack near the pulse center including the phase mismatch and thus it can reduce the conversion efficiency. In the single pulse regime we measured the conversion for both increasing and decreasing pulse energy without changing the illuminated spot and could not detect any permanent sample changes that manifest themselves in a reduced conversion efficiency.

 

Fig. 4 Third-harmonic signal as a function of the input pulse energy for an FTM with 25 layers [(Hf_xAl_yO/SiO₂)¹²Hf_xAl_yO] deposited on a fused silica substrate. In the low-(high) energy region, data were taken with pulse trains (single pulses). The solid line shows a cubic dependence.

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We measured the TH signal as a function of center wavelength of the Ti:sapphire oscillator and found that it peaked at a fundamental wavelength λ_F of about 791 nm, which is somewhat off the design wavelength of 787 nm, cf. Fig. 2. A similar Stokes shift of the measured transmission spectrum was already mentioned. It is interesting to note that maximum TH conversion coincides with a ”resonance” of the layer stack. On the short-wavelength side (λ_F ≈ 785 nm) the spectral width (FWHM) of the fundamental was Δλ_F ≈ 15 nm. The FWHM of the TH spectrum of Δλ₃ ≈ 2.9 nm agrees with the expected $\mathrm{\Delta }{\lambda }_F/3\sqrt{3}$ for full conversion assuming bandwidth limited Gaussian pulses. The UV peak wavelength λ₃ was shifted upward by about 1 nm relative to λ_F/3. At maximum conversion efficiency λ₃ = λ_F/3 but there were signs of bandwidth limitations for 55-fs fundamental pulses; the UV spectrum was about 20% narrower than the theoretical limit. At the maximum output power available from the oscillator corresponding to a fluence at the mirror of about 0.15 J/cm² the single pulse conversion reached values close to 1%. We also designed stacks where we forced periodicity. The predicted conversion efficiencies were considerably lower compared to the aperiodic designs.

Most dielectric materials used for optical coatings are known to change their optical properties when illuminated with high intensity (fs) lasers - a process known as laser conditioning and incubation [17, 19]. While for linear optical applications these changes are often subtle and do not compromise the functionality of the optical component we find that incubation becomes important for nonlinear optical applications. If we illuminate the FTM with a burst of fs pulses of high fluence the conversion drops after the first pulse. The conversion showed partial recovery when a second burst was incident on the same sample spot after a few hundred milliseconds. The transients of this incubation and recovery behavior are consistent with observations made in fs double pulse damage experiments with hafnia coatings [20].

Several samples were designed with different number of layers from N = 3 to N = 35 and some of them tested experimentally. Figure 5 shows the predicted relative conversion efficiencies as a function of the number N of layers and corresponding measurements. Note that there is no guarantee that the search found the solution with the absolute maximum THG for a given N. The results suggest that the expected efficiency increases approximately according to a power law with an exponent between 5 and 6 for N > 3, the bandwidth decreases with N (not shown). These trends are similar to SHG scaling with respect to length of photonic crystals [6, 7]. The predicted TH signal for a 35-layer stack is six orders of magnitude larger than for a 3-layer stack. The sensitivity of THG with respect to random layer thickness fluctuations, which are unavoidably introduced by the deposition process, was found to increase with N.

 

Fig. 5 Predicted TH signal of frequency tripling mirrors as a function of the number N of layers. The sample with 10 layers contains a 30-nm Al film between the dielectric layer stack and the substrate, see text. The measurement points refer to pulse trains with a single pulse fluence of about 0.01 J/cm² for which incubation was not observed. The high-index material of the 25- and 35-layer sample was Hf_xAl_yO, it was HfO₂ for all other samples. The two lines, anchored arbitrarily at N = 3, show the N⁵ and N⁶ dependencies as guide for the eye. The data point for a single layer refers to a 377-nm thick hafnia film.

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In the designs involving only dielectric films the optimized layer stack compromises between an effective compensation of the overall phase mismatch of fundamental and TH, a large field enhancement in the high-index layers producing most of the TH, high penetration and reflection of the fundamental wave leading to two passes through the stack, and high reflection of the TH produced in forward direction. High reflection of both fundamental and TH wave can also be accomplished by a single metal layer between dielectric stack and substrate. In this location the fundamental field is typically considerably smaller than the input mediating potential problems arising from absorption and subsequent damage of the metal film. We designed and tested a 9-layer sample with an additional 30-nm Aluminum film and the predicted (measured) TH output in reflection is more than 100 (10) times larger than the corresponding signals from a 9-layer sample without Al, cf. Fig. 5.

The ultimate limit of the single pulse conversion efficiency is given by the maximum intrafilm fluence below optical damage. In the sub ps regime this critical fluence scales as F_c ≈ (a + bE_g)τ^κ, where E_g is the material bandgap, τ is the pulse duration, κ ≈ 0.3 and a and b are constants [21]. The nonlinear susceptibility of dielectric film materials was measured to follow ${\chi }^{(3)}\propto {\left(E_g^2-E_3^2-i\gamma E_3\right)}^{-1}$, where γ is a damping parameter and E₃ is the TH photon energy [15]. Thus, provided a large enough bandwidth, the maximum possible conversion efficiency for sub ps pulses is expected to scale as ${\left[{\chi }^{(3)}{F}_c\right]}^2\propto {\tau }^{2\kappa -2}/{\left(E_g^2-E_3^2-i\gamma E_3\right)}^2$, which favors short pulses and materials where the UV photon energy is close to the bandgap as long as multi-photon absorption and the Kerr effect can be neglected. In principle, the Kerr effect, if important, can be taken into account in the design process to maximize conversion in the pulse center.

4. Summary

We designed, fabricated and demonstrated a frequency tripling mirror (FTM) based on an aperiodic sequence of high-index layers (Hf_xAl_yO or HfO₂) and low-index layers (SiO₂). Dielectric oxide films are attractive because of their high laser damage resistance, excellent environmental robustness, high transmission from the IR to the UV spectral region and the availability of well-developed deposition techniques. Despite the monochromatic design, bandwidths supporting the conversion of 55 fs pulses were observed and single pulse conversion efficiencies approaching 1% demonstrated with a 25-layer sample. Efficiencies of a few percent seem possible for fluences approaching 50% of the critical fluence for laser damage with the current design. Both bandwidth and conversion efficiencies can be improved by using a larger number of layers including a metal film between the substrate and the dielectric stack, and by making bandwidth and thickness sensitivity an additional optimization parameter. Materials with low laser incubation are desirable if the laser induced modification of relevant material properties cannot be taken into account in the design. The FTM can also be configured with linear optical functions for use in multi-reflection geometries, laser cavities, and external cavities. The demonstration of an FTM based on aperiodic dielectric layers opens the way for functional optical elements utilizing interference coatings for nonlinear optical processes like harmonic generation and nonlinear wave mixing.