Phase-Matched Second-Harmonic Generation from Metasurfaces Inside Multipass Cells

We demonstrate a simple and scalable approach to increase conversion efficiencies of nonlinear metasurfaces by incorporating them into multipass cells and by letting the pump beam to interact with the metasurfaces multiple times. We experimentally show that by metasurface design, the associated phase-matching criteria can be fulfilled. As a proof of principle, we achieve phase matching of second-harmonic generation (SHG) using a metasurface consisting of aluminium nanoparticles deposited on a glass substrate. The phase-matching condition is verified to be achieved by measuring superlinear dependence of the detected SHG as a function of number of passes. We measure an order of magnitude enhancement in the SHG signal when the incident pump traverses the metasurface up to 9 passes. Results are found to agree well with a simple model developed to estimate the generated SHG signals. We also discuss strategies to further scale-up the nonlinear signal generation. Our approach provides a clear pathway to enhance nonlinear optical responses of metasurface-based devices. The generic nature of our approach holds promise for diverse applications in nonlinear optics and photonics.


Introduction
Nonlinear optical responses of materials are of paramount importance in a wide spectrum of modern photonic applications ranging from the development of ultrafast high-power laser sources [1][2][3] , and optical metrology 4 to recent realizations of optical neural networks 5 .The main challenge in nonlinear optics is often to realize material systems where the inherently weak nonlinearly generated signal fields are, upon propagation through the system, coherently built-up to strengths of practical relevance.In other words, the challenge is to phase match the system 6 , which can be achieved e.g. using special nonlinear crystals 7 , fibers 8 , waveguides 9 , and/or resonators made of such materials 10,11 .
Metamaterials are artificial structures made of sub-wavelength building blocks 12 .Interestingly, metamaterials can exhibit properties not (easily) achieved using natural materials 13,14 , and have recently emerged as a powerful technology for realizing novel flat photonic components, such as metalenses and metaholograms 15,16 .Nonlinear optical metamaterials have also been recently suggested as a solution for the phase-matching problem, because optically thin nonlinear metamaterials are vir-tually free of phase-matching issues 17 .
Despite advantages and steady progress of nonlinear metasurfaces/metamaterials, their usefulness for applications is still mostly limited by their poor conversion efficiencies 18 .Approaches to enhance their nonlinear responses have included various works utilizing resonance enhancement 19,20 , mode-overlap optimization 21 and/or index-near-zero behavior 22 .Recently, nonlinear responses have also been enhanced by phase-engineered bulk nonlinear metamaterials, consisting of several layers of nonlinear metasurfaces 23 .Despite steady progress, it seems necessary to come up with new approaches to further enhance the nonlinear responses of metasurfaces/materials.
Here, we experimentally demonstrate an easy approach to enhance nonlinear responses of metamaterials, by incorporating them inside multipass cells and allowing the pump beam to pass through the metamaterial several times.As a proof-of-principle demonstration, we measure second-harmonic generation (SHG) emission from a plasmonic metasurface and demonstrate an order of magnitude enhancement in the measured SHG signal at the wavelengths where the metasurface was phase-matched.Notably, we systematically investigate the dependence of SHG emission on the number of passes, distinctly demonstrating clear signatures of successful phase matching.Importantly, this demonstrated approach is quite generic, offering compatibility with various existing enhancement techniques.

Semi-analytical model
The phase-matching condition of the secondharmonic response within metamaterials can be expressed as where m is an integer.The terms ϕ ω = k ω h and ϕ 2ω = k 2ω h denote the phase accumulations of the fundamental and second-harmonic fields, respectively, due to their propagation through a distance h.Terms δ ω and δ 2ω denote the phase shifts incurred due to the interaction between the fundamental and second-harmonic fields with the metasurface, respectively.These phase shifts can be determined from the optical responses of the metamaterials (here from their transmittance spectra), enabling to fulfill the phase-matching condition through appropriate metamaterial design.
By measuring the linear transmission T of the samples, we can determine their phase responses and subsequently estimate the strength of the phase-matched second-harmonic signal upon propagation of the sample metasurfaces multiple times.The transmission T is linked to the polarizability α, dictated by the localized surface plasmon resonances (LSPRs) via where k = 2π/λ represents the wavevenumber.Assuming α to be a complex-valued Lorentzian function, the full complex-valued α can be found after solving the imaginary part of α from the above equation.The phase shifts δ associated with the sample can then be found by using α = |α| exp(iδ).
In the case when the pump beam passes many times through the metasurface, the SHG response can be estimated using the approach already introduced for the stacked metasurfaces 23 .The phase-matched SHG signal is given by where χ (2) ms is the nonlinear susceptibility of the metasurface and N is the number of passes through the metasurface.The transmittance T near the pump and SHG frequencies is experimentally measured, while the phase mismatch ∆k is calculated using Eq. ( 1).
The sample used in our experiments consisted of V-shaped aluminum nanoparticles (Fig. 1a) deposited on a 0.5 mm-thick SiO 2 substrate using standard electron-beam lithography tech- The transmission spectrum measured from the sample and the calculated phase changes for x-and y-polarizations.(c) The calculated phase term ∆k for the SHG signal emitted from the sample in the wavelength range of the pump (900-1000 nm).Dashed lines highlight the wavelengths (920 nm, 940 nm, 964 nm, and 989 nm) at which the phase-matching condition is fulfilled.
niques.The nanoparticles had a thickness of 30 nm, arm lengths of 140 nm, and arm widths of 70 nm.The nanoparticles were positioned randomly (being however similarly oriented) with a particle density of 11.11 particles/µm 2 .This particle density corresponded to density of a periodic square nanoparticle array with periodicity of 300 nm.The reason to investigate randomly positioned nanoparticle arrays was that such arrays do not exhibit collective responses, such as surface lattice resonances 24,25 , that would unnecessarily complicate the phase extraction protocol and subsequent data analysis.Based on numerical simulations (Ansys/Lumerical FDTD), these particle dimensions resulted in LSPRs near 420 nm for x-polarized incident light.We confirmed this by measuring transmission spectra shown in Fig. 1b (see the Supplemental material for description of the transmission setup [LINK HERE BY THE PUBLISHER]).Due to the presence of LSPRs, the nanoparticles induce phase-changes (red curve in Fig 1b ) both at pump and SHG wavelengths, which we then used, with the substrate thickness, to calculate ∆k shown in Fig. 1c.At the wavelengths where ∆k is an integer of 2π, we expect phase matching and coherent build-up of the SHG signal.

Nonlinear characterization
The experimental setup, illustrated in Fig. 2, was used to characterize the SHG response of the sample as a function of the number of passes N of the pump beam through the sample.A tunable titanium sapphire femtosecond laser (Chameleon Vision II), operating at an 80 MHz repetition rate and offering a spectral range of 680-1080 nm with pulse duration of 140 fs (near 800 nm), was used as the pump source.The FWHM value for the bandwidth of the pump laser at 950 nm was measured to be 5 nm, and consequently a pulse duration of 200 fs was used in our calculations.The combination of the half-wave plate HWP1 and the linear polarizer LP was used to control the power incident on the sample.Furthermore, the half-wave plate HWP2 was used to adjust the polarization of the incident beam to x-polarization.Importantly, the use of a low-pass filter LPF before the sample ensured no SHG signal possibly originating from optical components preceding the sample were detected.Three lenses, denoted as L1 (f 1 = 75 mm), L2 (f 2 = 25.4 mm), and L3 (f 3 = 250 mm), were used to control the beam size on the sample as well as to adjust the Rayleigh range associated with the Gaussian beam propagation.Using a camera, the beam size before the first lens L1 was measured to be 2.6 mm (FWHM).The lens L2 was positioned to reduce the beam size by a factor of 0.34, resulting in divergence angle of 1.8 mrad for the beam after lens L3.Subsequently, the calculated FWHM value for the beam waist was 343 µm for pump at 950 nm, which enabled to maintain a fairly focused pump beam during its passes through the metasurface (Rayleigh range of 9.75 cm).We note that a considerably larger Rayleigh range would have resulted in a wider beam profile and correspondingly reduced beam intensity, which subsequently would have resulted in decreased SHG signal.
The number of passes N of the pump beam through the sample was adjusted by displacing mirror M4 along the trajectory indicated by the arrow shown in Fig. 2. Changes in N resulted detectable offsets for both the pump and the SHG beams, which were readily recorded with the two cameras used to detect both the pump and the SHG beams.The beam offsets have been highlighted in Fig. 2 as three lines of varying color intensity.
The lens L4 (f 4 = 50 mm) was used to collect the generated SH signal emitted from the sample.The dichroic mirror (DM) effectively filtered out the pump beam while reflecting it towards the camera used to image the sample.The SHG beam passed through the DM and was focused by the lens L5 (f 5 = 100 mm) to the SHG camera (ZWO-ASI1600MM).The short-pass filter (SPF) removed any residual pump and made it possible to measure the generated SHG signal.

Results and discussions
First, we measured power dependence of the collected SHG signal for the case of a single pass (Fig. 3).The power dependence was verified to be close to quadratic, confirming that the collected signal was of nonlinear origin and that no apparent sample damage occurred at the used power levels.This was not surprising, because we used a relatively large beam waist, and our samples exhibited low absorption at the pump wavelenghts (900-1000 nm).Consequently, the pump power incident on the metasurface was set to be 350 mW in the following experiments.The y-polarized SHG response of the sample was measured as a function of the number of passes for x-polarized incident light.
Our semi-analytical model predicted an enhancement of the SHG response at specific wavelengths as the number of passes N increases, as depicted in Fig. 4a.These wavelengths corresponded to the phase-matched wavelengths [see Fig. 1c], where the SHG response reached its local maximum.This trend was confirmed by experiments [Fig.4b], where we measured the SHG response as a function of the pump wavelength (900-1000 nm).These measurements were repeated for different number of passes (up to N = 9 passes).The observed phase-matched wavelengths were found to be 904 nm, 920 nm, 940 nm, 960 nm, and 985 nm, agreeing well with the calculated phase-matched wavelengths [Fig.1c].Notably, the SHG response exhibited a superlinear dependence (SHG ∝ N βexp = N 1.12 ) on the number of passes N of the pump beam, confirming that the SHG signal was successfully phasematched.
In the ideal scenario of a lossless and perfectly phase-matched sample, the SHG signal would exhibit a quadratic dependence on the number of passes N (SHG ∝ N β ideal , where β ideal = 2) 23 .However, our sample was neither perfectly lossless nor perfectly phase-matched.The sample transmittance at the pump and the SHG signals was measured to be approximately 94% and 89%, respectively [Fig.1b].Taking into account these transmission losses in our semianalytical model, the loss-corrected scaling factor was expected to be β loss = 1.60.
In addition to losses, also the bandwidth of the input pump (∆λ FWHM = 9.5 nm near 950 nm) affects the SHG power dependence because the sample should be phase-matched for all the wavelengths within the pump bandwidth.As this was not the case with our sample, we also took into account this partial phase mismatching in our model, resulting in a further reduced scaling factor of β pulsed = 1.25.Although the used model was simple, e.g.changes in the pump beam intensity were neglected, the calculated scaling factor value agrees quite well with the measured value of β exp = 1.12.
In future, the strength of the SHG signal could be straightforwardly enhanced by further increasing the number of passes N and/or the scaling factor β. Based on our calculations and experiments, it seems advantageous to focus first on increasing β.A straightforward approach to increase β would be to reduce the transmission losses of the relevant wavelengths by incorporating anti-reflection coatings into the metasurface devices.Furthermore, the scaling factor could be increased (simultaneously with N) also by moving from monolayer metasurfaces to using stacked metasurfaces 23 , which would reduce the overall transmission losses by minimizing the number of interfaces.
Once transmission losses are properly remedied, it becomes relevant to tackle the reduction of β due to partial phase mismatch when spectrally broad fs-pulses are used as the pump source.One could for example reduce the amount of normal dispersion of the system by reducing the thickness of the used glass substrate, on top of which the metasurfaces have been fabricated.In addition, one could design the metasurfaces to be anomalously dispersive, flattening the overall dispersion profile of the system.Instead of tackling the problems associated with spectrally broad pumps, in future one could also focus on nonlinear applications where spectrally narrow pump beams, i.e. continuouswave lasers, would be used.
We note that it would be challenging to further increase N using our current experimental setup.However, finding ways to increase N would be in future of paramount interest.The current challenges were attributed to the finite length of the Rayleigh range of the laser beam and to the physical dimensions of the sample.Together, these issues necessitated us to separate the two mirrors by several millimeters.We think that one solution to above would be to effectively increase the Rayleigh range by using slightly curved (convex) mirrors [26][27][28] .
As other potential approaches to further increase N, we are also considering to fabricate metasurfaces directly on top of the mirrors, or completely replacing the mirrors with highlyreflecting metasurfaces 29,30 .This replacement would allow the metasurfaces to serve both as sources of harmonic signals and as substitutes of the mirrors.This way the propagation distance between each pass could be considerably reduced, leading to higher N and improved efficiency.In addition, this approach would reduce the amount of normal dispersion, simultaneously increasing the achievable β, sounding therefore as an effective approach to considerably increase achievable conversion efficiencies.

Conclusions
In this study, we have demonstrated an order of magnitude enhancement of the SHG signal in nonlinear metasurfaces by utilizing a simple multipass configuration, where the pump beam traverses the metasurface multiple times.We designed and fabricated a nonlinear metasurface composed of random array of V-shaped aluminum nanoparticles.In our multipass configuration, the pump beam undergoes up to 9 passes through the sample, resulting in an enhancement of the SHG signal at specific wavelengths that satisfy the phase-matching condition.Our findings, validated by the agreement between our semi-analytical model and experimental observations, underscore the robustness of our approach.Notably, this study introduces an innovative strategy for achieving phase matching of the SHG signal within nonlinear metamaterials, eliminating the need for complex fabrication procedures required to realize phase-matched stacked metasurfaces.Furthermore, our work demonstrates a clear path-way to systematically increase the efficiencies of metasurface-based nonlinear devices.

Funding
We acknowledge the support of the Academy of Finland (Grant No. 308596), the Flagship of Photonics Research and Innovation (PREIN) funded by the Academy of Finland (Grants No. 320165 and 320166).TS acknowledges also Jenny and Arttu Wihuri Foundation for doctoral research grant.TKH acknowledges Academy of Finland project number (322002).RF acknowledges the support of the Academy of Finland through the Academy Research Fellowship (332399).

Figure 1 :
Figure 1: (a) A schematic representing a single V-shaped nanoparticle from the metasurface.(b)The transmission spectrum measured from the sample and the calculated phase changes for x-and y-polarizations.(c) The calculated phase term ∆k for the SHG signal emitted from the sample in the wavelength range of the pump (900-1000 nm).Dashed lines highlight the wavelengths (920 nm, 940 nm, 964 nm, and 989 nm) at which the phase-matching condition is fulfilled.

Figure 2 :
Figure 2: A schematic of the setup used to measure SHG response of the sample at different number of passes N. The number of passes is controlled by moving the mirror M4 laterally along the direction shown by the arrow, and observing the shifting of the pump and SHG beams on the cameras.The inset on the right shows the phase-matched SHG emission in case of 3 passes of the pump beam through the metasurface device.

Figure 3 :
Figure 3: Quadratic dependence of the SHG signal intensity on the average power and peak power density of the pump laser for one pass through the sample.

FundamentalFigure 4 :
Figure 4: (a) Calculated and (b) measured SHG signal for different number of passes N. (c) The enhancement factor of the SHG signal as a function of the number of passes N. The calculated (β pulsed = 1.25) and the experimentally measured data (β exp = 1.12) are in good agreement and show a superlinear dependence.