1. Introduction

Nonlinear optics (NLO) is the interaction of electromagnetic (EM) waves oscillating at different frequencies. Since EM waves cannot interact directly, a material system must be involved. Two broad categories of NLO exist, the parametric and the non-parametric. Non-parametric processes involve charge excitation and re-emission, as in photoluminescence, while parametric interactions involve virtual states and, thus, can, in principle, become instantaneous lossless. The topic of this paper is parametric NLO.

It is common to expand the induced polarization as a power series in the applied EM field,

Given the applied significance of NLO and the smallness of the material susceptibilities [2], an effort has been made to enhance the electric field at the NLO interaction site using methods such as gap-plasmons, bowtie-antennas, non-centrosymmetric metallic structures, plasmonic gratings, nanoslits, and other plasmonic/nano-optics approaches [3–28]. Despite the impressive progress in that direction, appreciable NLO still requires high-intensity lasers or long interaction lengths with a material system. A complementary approach would seek to strengthen the susceptibility elements beyond that of a conventional material.

Recently, we have explored the possibility of NLO that does not emerge from the nonlinear susceptibility of some material but the interaction between the particles composing a nano-optical system, each with a strictly linear response [29,30]. To demonstrate this effect, we have composed a system that should have produced undetectable SHG according to the convention but produced a noticeable one in practice. In addition, we proposed a toy model whose sole source of nonlinearity comes from the interaction between linear oscillators to illustrate the source of the nonlinearity we have observed. The model reproduced the measured SHG spectrum, including unique features beyond the conventional approach’s reach. Accordingly, we call this kind of NLO interaction-based nonlinear optics (INLO). Xu et al. used a similar model to explain the emergence of SHG, sum, and difference frequency generation from a collection of small interacting metal nanoparticles [31,32].

This paper broadens our study of INLO to parametric optical rectification—a 2nd-order NLO process where a static polarization, V, emerges from an oscillating EM field, E, i.e., $V \propto {\chi ^{(2 )}}EE$. Note that the above relation admits a quadratic dependence between the rectified potential, V, and the deriving electric field, E, or a linear dependence on its intensity, which is ${\propto} EE$ [33–35]. Namely, within the convention, irrespective of the sample geometry or the type of NLO material, a quadric dependence of the signal over the field strength must always emerge. Alternatively, a linear dependence has to emerge concerning the intensity of the oscillating driving force.

Optical rectification measurements are usually conducted by filling the space between a capacitor’s conductive plates with a nonlinear optical material, i.e., one with an appreciable nonlinear susceptibility [35–37]. As a capacitor, we use a slab waveguide with two nanometer-thin buried metal layers separated by a nanometer-thin dielectric gap. However, unlike the standard approach, our sample has no nonlinear material. In addition, the nano-thin metallic layers suppress the ability of the metal to produce second-order nonlinear plasmonic effects due to the homogenization of the electric field within the metal layers. At the same time, such small thicknesses are prerequisites for an INLO to emerge.

Three experiments were conducted, each testing a different aspect of an NLO process; the results agree with what is expected, i.e., facilitating the necessary conditions for an INLO process. Sufficient proof is hard to come by purely based on experimental measures. Therefore, we devised a toy model whose sole source of nonlinearity is the interaction between linear oscillators. The interaction follows what is expected from confined charges driven to oscillate by an external field while experiencing a quasi-static near-field interaction with an additional lump of externally driven nearby charges. The model shows a good qualitative agreement with what was observed. Thus, based on the experimental results and the model, we conclude that our samples have an active INLO mechanism.

This study is not the first report we believe to result from interaction-based optical nonlinearity. Indeed, bi-metal structures have long been studied and often produce abnormal 2nd-order nonlinear activity [38–45]. However, they are not about rectification, the explanation of unexpected outcomes is often heuristic, and the possibility of beyond material-based nonlinearity is overlooked. Another closely related topic is the plasmoelectric effect, the origin of which is rooted in the thermodynamics of free energy [46,47]. While such reasoning has its virtue, it leaves the microscopic origin of the said effect obscure. Contrary to these studies, we conducted experiments designed to test the possibility of INLO in our samples supplemented with a dynamic model highlighting the role of interactions over all other nonlinearity sources. We hope the combination of different approaches will contribute to understanding the, so far, elusive INLO effect.

2. Methods

When designing an optical rectification experiment, it is necessary to consider the prerequisites: First, the charge oscillations should be homogeneous (lumped) in each part of the system, and second, the lumped bodies of charges should be close enough for electrostatic interaction to commence. We chose to work with silver and gold, metals that served us well in our past studies of INLO-based SHG [29,30]. The first condition is met once the particles are smaller than the EM penetration depth into the respective metal, which is ∼30 nm in our case. When this happens, phase shifts are minor, and charge oscillation becomes uniform within each particle, i.e., they are lumped. The second condition is met once the particles are placed at a distance smaller than the free-space wavelength apart, a few hundred nanometers in our case. However, this is the maximally allowed separation; smaller distances are advantageous due to the inverse square decay of near-field electrostatic forces. If these two conditions are not met, then the surface nonlinearity of the metals would become the dominant nonlinear mechanism [48,49]. We emphasize, however, that particles should be smaller than the EM penetration only in the direction of charge oscillations, whereas in any other dimension, the sample may extend well beyond this range.

Given the above considerations, we have conceived a sample made from two glass slides, each covered with a thin 20-nm metal layer and an additional 20-nm dielectric layer over one of them. A slab waveguide with buried nanometer-thin metal layers is thus formed by placing these slides over each other with the metal layers facing inward, as shown in Fig. 1(a), (b), and (c). An electromagnetic wave with polarization shown by the two-sided black arrow traveling down this waveguide causes charge oscillations in the metal layers, as illustrated by the lines of wavey red dots in Fig. 1(a). Due to the confinement of the nanometer-thin layers, charge oscillations are lumped but not along the light’s propagation direction. Rectification emerges from a static deflection of these oscillating lumped charges caused by the interaction they experience across the gap layer—deflections that are indicated by ${x_0}$ and ${y_0}$ in Fig. 1(a).

 

Fig. 1. (a) The slab arrangement with a red arrow indicating the incoming illumination and a two-sided black arrow showing its polarization and direction of charge oscillations. The oscillating charges are illustrated as red dots, and their static deflections are ${x_0}$ and ${y_0}$. (b) The partially overlapping slab waveguide sample. (c) Front view of the sample with red ellipse illustrating the laser illumination. (d) Simulation of a guided mode in the partially overlapping slab waveguide.

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The slab layout of our sample confines the light to the volume of the glass slides, which enhances the interaction with the buried metal layers. Further confinement can be achieved if the overlap between the two glass slides is reduced, as shown in Fig. 1(b). The result is a unique sample arrangement explicitly designed for this experiment. The proposed sample confines light in two dimensions while guiding it along the remaining one. In addition, it allows easy access to the buried metal layers, as seen in Fig. 1(c). Figure 1(d) shows the mode of this waveguide obtained from a commercial solver (COMSOL). In this case, the ability to guide light is attributed to a combined effect of slab and ridge waveguides [50,51]. However, this arrangement’s precise linear-optics guiding properties are not yet known. We will mitigate this shortcoming by exploiting the convention that material-based 2nd-order NLO processes must yield a quadratic dependence over the EM field strength. Therefore, any deviation from this rule indicates a non-conventional NLO process, such as INLO.

The proposed waveguide provides ample experimental tunability (metal composition, separation, overlap) while negating competing effects: Having layers thinner than the EM penetration depth means that field gradients responsible for the surface nonlinearity of metals cannot be supported [48,49]. Also, the dielectric separation between metal layers prevents the development of a thermocouple (Seebeck) effect. In addition, the thin dielectric separation, combined with the relatively thick dielectric substrates, guarantees that negligible temperature differences develop in the thin metal layers, yet another potential source of an erroneous signal [35,52]. In addition, unlike SHG, the static optical rectification considered here has no phase associated with it. As a result, there are no phase-matching restrictions, which allows us to collect the rectification signal over the entire sample length—a fact that we will use later to prove that INLO-based rectification resides in our samples. Finally, it is well-known that 2nd-order NLO processes require inversion symmetry breaking, and the proposed sample presents us with two: The internal one comes from using different metals, and an external one from aligning the laser to interact more strongly with one of the layers. The external mechanism is expected to yield rectification from samples with identical metal layers. In the following, we will test both of these mechanisms.

It is important to note that due to the pulsed laser excitation used in our experiments, there is likely at least some THz generation in addition to the static rectification produced by our samples. SHG is another NLO process that will likely be produced in our experiment, albeit in small efficiency, due to improper phase matching. Nonetheless, the THz generation, SHG, and any other non-static effect are not detected in our experiments and, as a result, do not affect the claims of this paper which are all based on a relative comparison between static OR signals from different experimental configurations.

Samples were formed by depositing a 10-nm-thin adhesive Ti layer on standard 100-µm-thick 1.5 × 1.5 cm microscopy cover slides. The slides were then covered with 20 nm of silver or gold as the active material for our experiments. The silver-coated slides were additionally coated with 20 nm of SiO₂, acting as a dielectric separation layer and preventing silver oxidation. Ellipsometry and STEM characterization of our samples indicated layer thicknesses within 3 nm of their nominal values. In addition, an AFM scanning shows surface roughness of ∼1 nm. The Supplement 1 gives more details about the sample characterization and its results.

3. Experimental results

Two samples were used, one with silver and gold-coated glass slides (silver-gold) and a second with two silver-coated slides (silver-silver). As a result, the silver-gold invokes both inversion symmetry violating mechanisms from above—internal and external—while the silver-silver sample relies solely on the external one. The rectification signal was excited with a pulsed Ti:Sapphire laser (Coherent Chameleon) with 150 fs pulses at an 80 MHz repetition rate. The laser was focused on the edge of the sample using a 5-cm focal length lens that produced a Gaussian beam with a ∼50 µm wide waist extending ∼2 mm along the beam propagation direction (the Rayleigh range). We adopted an experimental procedure where the beam’s lateral position was tuned to maximize the received signal. The ∼200 µm wide slab sample gave us ample tunability. As a result, the beam is not necessarily impinging on the center of the slab. Waveguiding was confirmed by observing light emerging on the opposite end of the sample, as seen in the Supplement 1.

The laser intensity was controlled by placing a half-wave plate in front of a polarizing beam splitter. Detection was achieved by inserting an optical chopper into the beamline and connecting the metal layers to a lock-in amplifier (SR830) through a 30-dB pre-amplifier. The optical chopper’s modulation frequency and the lock-in amplifier’s detection frequency were always identical. A schematic depiction of the experimental setup is shown in the Supplement 1. Generally, the signal increased for shorter wavelengths. Accordingly, we chose to work with a 700 nm excitation wavelength. The optimal size of the overlap region between the two metal-coated glass slides was 1 mm—more reduced the signal due to the loss of confinement while less degraded the signal due to extensive scattering. It is important to note that the pulsed excitation of our sample is likely to produce higher harmonics and THz radiation. However, such fast oscillating signals are not registered by our detection circuit and, therefore, do not affect the results of this paper, even if they exist.

Sequences of on/off cycles were captured at different laser intensities. A typical sequence of laser on/off cycles is shown in the Supplement 1. The rectification is the average difference between what is measured over the ‘on’ and ‘off’ stages. Not all the injected light interacts with the metal layers. However, above 1 kW of average laser power, the signal from the silver-gold sample gradually degraded after initiation, indicating a temperature rise. Accordingly, the laser intensity was kept below this critical value for the silver-gold sample. No such effect was observed for the silver-silver sample.

In the first experiment, we tested the rectification as a function of the laser power. Figure 2(a) and (b) show the resulting rectification from the silver-silver and silver-gold samples with error bars showing the standard deviation from different experiments. The source of the progressively more significant errors in the silver-silver sample is not entirely clear but consistent. Perhaps it can be attributed to the lower silver losses that make this sample more sensitive to power fluctuations of the laser excitation. The OR signal responds instantly to the excitation laser pulse [37]. Therefore, OR signal from our pulse laser (150fs, 80 MHz repetition) is not zero only for a ∼1/8333 of the time. The lock-in amplifier cannot respond that fast. As a result, the registered OR is smaller by the same factor than the actual one due to temporal averaging by the lock-in amplifier. The depicted rectification values in Fig. 2 consider this fact.

 

Fig. 2. The mean net rectification of (a) the silver-silver and (b) the silver-gold samples as a function of the laser excitation power. Error bars correspond to the standard deviation. (c) The rectification for TM (orange) and TE (blue) 700-nm polarized laser excitation. The inset shows the TE and TM polarization relative to the slab layout. (d) Rectification from a full-length (blue) and half-length (orange) sample for 700-nm TM excitation.

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The linear dependence of the rectification on the excitation power is typical of a parametric process since it corresponds to a quadric dependence over the field amplitude [35]. However, the deviation from this trend, mainly for the silver-gold sample at high laser excitation power, indicates that the active parametric mechanisms may be beyond the simple material-based NLO that is described by the ${P^2} \propto {\chi ^2}EE$ rule. Since laser power was kept below the threshold of thermal degradation, we view this observation as a genuine non-material NLO-base effect. Note that this observation is irrespective of the guiding properties of our samples or the possible existence of NLO active material therein.

In a second experiment, we tested the rectification for different polarizations of the excitation laser. INLO-based rectification is supposed to emerge from the electrostatic interaction between oscillating charges in our sample’s two metal layers. As shown in Fig. 1(a), oscillations must be transverse to the layers to facilitate the required interaction. Accordingly, we expect stronger rectification to emerge once the laser is polarized transversely to the metal layers—the so-called TM polarization. Figure 2(c) shows that an order-of-magnitude stronger rectification emerges for TM polarization relative to the orthogonal (TE) one, as expected for an INLO-based process. The short time-scale fluctuation in the signal reflects detection noise, and the long time-scale ones are attributed to laser power instability. However, while a TM excitation is necessary for an INLO-based rectification to emerge, it is insufficient since rectification from surface plasmon hybridization in the metal layers may also be enhanced [48,53]. Therefore, we need an additional, sufficient condition to rule out the possibility of surface plasmon enhancement of a conventional nonlinearity, be that of the metal surface or any other material-based effect.

Since some nonlinearity of metal surfaces might have resided in our sample despite the ultra-thin metal layers (below the EM penetration length), and since this weak nonlinearity might have been intensified by surface plasmon to a detectable level, a third experiment was conducted to rule out this possibility. To do so, we take advantage of the wavenumber of surface plasmons that is always larger than that of a wave in free space (we assume that the low confinement has a negligible effect on the modal wavenumber) [50,54]. As a result, some spatial discontinuity must be present to bridge the resulting momentum mismatch. Due to the smoothness of our sample (<1 nm), the only available spatial discontinuities are those from our sample’s edges. Thus, it is expected that cleaving the sample to half its original length would not significantly alter the detected rectification signal if it had originated from surface plasmons.

On the other hand, the INLO process emerges from local interactions between oscillating charges in the two metal layers. Also, since the rectification has no associated phase, it accumulates along the waveguiding sample. Therefore, an INLO-based rectification strength is expected to correlate to the sample’s length. Figure 2(d) compares the rectification signal for a whole and a half length sample. Here, for illustrative purposes, we show the raw data that emerges from manually operating the laser shutter at ∼20 sec on and off cycles. It is seen that cleaving the sample to half its length halved the signal, ruling out the possibility of surface plasmon enhancement. Accordingly, based on the three experiments, we conclude that our samples have a dominant non-material-based NLO mechanism responsible for the detected rectification signal.

4. Model fitting

So far, we have proven that a non-material-based nonlinear optical mechanism is responsible for our detected rectification. However, it would be premature to conclude that it is necessarily an INLO-based effect. In the past, we have used a toy model to illustrate the source of INLO-based SHG interaction in our measured data [29,30]. Here, we use a similar approach to show that the interaction between oscillating charges can lead to the observed rectification. The model’s purpose, in this case, is to provide sufficient evidence for an INLO-based rectification.

The model consists of two harmonic oscillators with a strictly linear response to the external excitation and coulomb-like interaction between them. Significantly, these interactions are made for a fixed amount of charge, expressing the homogeneous lumped oscillations expected to emerge in layers thinner than the EM penetration depth into the respective metal. Accordingly, these Coulomb-like interaction terms consider a fixed amount of charges in their numerators and dynamic separation between them in their denominators; these denominators give our model its characteristic nonlinear behavior. We, therefore, adopted the following coupled oscillators toy model to describe only the most fundamental aspects of our samples’ nonlinear response:

The model consists of two harmonic oscillators on the left of the equation sign. The unitless dynamic parameters of these oscillators are x and y. Accordingly, all terms in the equations are expressed in units of ${s^{ - 2}}$. The natural frequency and damping of the x oscillators are given by ${\omega _x}$ and by ${\mathrm{\Gamma }_x}$, respectively, and similarly also for the y oscillator.

The first term on the right of the equation signs is the driving forces that are proportional to the product of electric field amplitudes, ${E_x}$ and ${E_y}$, by the elementary charge e. In the denominators of these force terms, we find the interparticle separation D, which is the normalization factor of x and y. Therefore, the actual oscillation amplitude of the x oscillator, for example, is given by $xD$. Also, ${m_e}$ is the mass of an electron and ${N_x}$ and ${N_y}$ are the number of oscillating charges in each piece of metal. Normalization by the ${N_x}$ and ${N_y}$ distinguishes this model from the previous ones that considered a 3-dimensional particle [29–32]. The number of charges, ${N_x}$ and ${N_y}$, are calculated based on the layer volume times the charge density of the respective metal taken as $6 \times {10^{22}}\; c{m^{ - 3}}$ and $5 \times {10^{22}}\; c{m^{ - 3}}$ for silver and gold, respectively. Therefore, this normalization resolves the issue of dealing with the enormous number of charges in our relatively broad and elongated samples.

The coupling terms are the second to appear on the right of the equation signs. Lumped bodies of charges are assumed to interact only with those closest to them in the opposite metal layer. This assumption is supported by (i) the prevalence of electrostatic forces over radiative ones at a close distance and (ii) the cancelation of long-distance interactions due to the phase accumulation between far-apart oscillations. The fixed amount of homogeneously oscillating charges are expressed in the coefficients of each straight bracket, and ${\varepsilon _0}$ is the vacuum permittivity. The unitless expressions in the square brackets are the electrostatic forces between two bodies of oscillating charges placed within a deep sub-wavelength distance. The first term in each bracket is the interaction with the adjacent particle’s positive static charges, while the second is the interaction with the oscillating negative charges. Note that this unique characteristic of having fixed amounts of charges with a dynamic separation between them is the sole source of nonlinearity in our model. This implementation of a Coulomb coupling differs from the more common one that favors a dynamic charge-related property in its numerator and fixed separation in its denominator as in the well-known dipole approximation or plasmonic hybridization models [51].

The proposed model is sensitive to the laser intensity and illumination misalignment on the sample edge through ${E_x}$ and ${E_y}$, the resonances and damping of the metallic structures through the natural frequency and damping parameters ${\omega _x}$, ${\omega _y}$, ${\mathrm{\Gamma }_x}$, and ${\mathrm{\Gamma }_y}$, the thickness of each metallic layer through the number of charges ${N_x}$ and ${N_y}$, and, finally, the separation between them through D. However, some aspects of our samples’ waveguiding properties and coupling are EM in nature and, thus, must phenomenologically be introduced to our model.

We wish to determine whether the toy model from Eq. (1) could reproduce the measured rectification. One way of doing this would be to expand the nonlinear coupling in powers of the charge displacements, x and y, and find closed-form expressions by inserting a harmonic series as a trial solution. However, this approach gives a quadratic response by construction and, therefore, could not reproduce our experimental results. A much better fit emerges from numerically calculating a time series for each oscillator based on Eq. (1). The rectified electric potential can then be found from:

The model is stiff; we used an implicit Runge-Kutta algorithm to achieve convergence with a relatively small amplitude of the oscillating excitation force. The model is not a simulation—the calculated and the experimental rectifications had to be normalized to be viewed on the same scale. Accordingly, we use the model only to find if the Coulomb-like nonlinear interaction terms in Eq. (1) could qualitatively reproduce the measured rectification.

A good fit emerged for natural frequencies corresponding to a wavelength of 280 nm for silver and 468 nm for gold. These are not the free plasma resonance nor those of bound charges, but the effective material resonances as they appear in the literature [55,56]. Likewise, these are not the resonances of the slab waveguide, which are shifted due to optical confinement. These facts agree with the electrostatic nature of the nonlinear activity in our samples.

The damping rates were calculated from the apparent ∼30 nm and ∼100 nm full-width half maximum of the prominent absorption peaks of the respective metals [55,56]. In addition, we have assumed that losses are small, so constant excitations can be considered. Thicknesses of the silver, gold and SiO₂ layers were taken to be 20 nm, in agreement with the ellipsometry and STEM data presented in the Supplement 1.

Fitting was performed by manually adjusting each oscillator’s driving force and interlayer coupling strengths that correspond to our model’s phenomenological EM aspects and the arbitrariness of the beam alignment procedure. Agreement with the normalized experimental results for the silver-silver sample emerged once the unit value of the driving force and the coupling terms were multiplied by 3 and 3.4, respectively. For the silver-gold case, a good agreement emerged for 1.1 and 8.5 multiplication of the force and coupling terms, respectively. The beam was misaligned by 40% in both cases (towards the gold layer in the silver-gold case). The fitting parameters can be considered acceptable given how the true EM nature of the sample affects this naïve toy model. Fitting was performed manually, and the results are not necessarily unique.

Figure 3 compares the fitted results in green to the experimental ones in blue. The graphs are now plotted as a function of the driving force, which is proportional to the electric field, so that a primarily quadratic trend emerges. Recall that a purely quadratic trend is the only one possible for a material-based NLO, irrespective of our sample’s material or guiding properties. Accordingly, we plot the best-fitted quadratic trends in orange.

 

Fig. 3. The measured rectification in blue, the model fit in green, and the best fit of a quadratic trend in orange for (a) the silver-silver and (b) the silver-gold samples. Note that the model’s root-mean-square error (RMS) is smaller than the quadratic trend one.

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Root-mean-square (RMS) values for the deviation relative to the experimental data were calculated for each case presented and are shown in the respective figure legends. The better agreement of the toy model we proposed, whose nonlinearity comes from its interaction, as seen by the smaller RMS it produces compared to what a material-based mechanism yields, is a further confirmation of the type of INLO our samples possess. The better agreement of our model with the experimental data relative to a poor agreement with the analytical quadratic trend of the silver-gold sample is attributed to the combined effect of the internal and external inversion symmetry-breaking mechanisms that this type of sample possesses. Notably, the model captures the high-excitation non-quadratic trend of this case. The silver-silver sample has only an external inversion symmetry mechanism. Therefore, its experimental data and our model’s results for this sample resemble an analytical quadratic trendline.

5. Summary and conclusions

This paper extends our study of interaction-based nonlinear optics to parametric optical rectification. Accordingly, we have devised samples that minimize competing effects such as material-based nonlinear optics, thermocouple effects, and others. By observing the rectified signal dependence of the excitation strength, polarization, and sample length, we conclude that a nonlinear mechanism beyond the conventional material-based one is active in our sample. Fitting the results with a toy model whose sole source of nonlinearity comes from a near-field Coulomb-like interaction suggests that this is the source of the observed rectification in our sample. Significantly, the model reproduced the non-quadratic trend that is entirely beyond a material-based nonlinearity.

This work aims to prove the existence of an interaction-based rectification in the simplest, most straightforward manner. As a result, this qualitative study is formulated as a set of experiments with a clear true or false outcome and comparing a carefully crafted toy model to a best-case scenario of conventional nonlinearity. The wavelength-dependent spectral response and close-to-resonance excitation are left to future studies. Another challenge is having a more accurate model of our proposed interaction.