Golden Ratio Gain Enhancement in Coherently Coupled Parametric Processes

Nonlinear optical processes are an essential tool in modern optics, with a broad spectrum of applications, including signal processing, frequency conversion, spectroscopy and quantum optics. Ordinary parametric devices nevertheless still suffer from relatively low gains and wide spectral emission. Here we demonstrate a unique configuration for phase-matching multiple nonlinear processes in a monolithic 2D nonlinear photonic crystal, resulting in the coherent parametric emission of four signal and idler modes, featuring an exponential gain enhancement equal to the Golden Ratio. The results indicate a new route towards compact high-brightness and coherent sources for multi-photon generation, manipulation and entanglement, overcoming limitations of conventional parametric devices.

OPG implies the spontaneous conversion of a photon from a high-energy pump laser (at frequency ω p ) into a pair of lower-energy photons, at frequencies ω a and ω b 17,18 . This occurs in a medium possessing a quadratic optical nonlinearity and can lead to an exponential build-up of the signal and idler fields, provided that photon energy (ω ω ω = + p a b ) and momentum are preserved 25 . The latter condition is nowadays routinely enforced by quasi phase-matching (QPM) with nonlinear artificial structuring [26][27][28] .
Standard twin-beam (2-mode) OPG couples one signal and one idler mode, a m and b m , where the subscript m designates the mode pair, as illustrated in Fig. 1a (with m = 1). QPM among the wave-vectors of the pump, signal and idler, i.e. → → k k , p a 1 and → k b 1 , respectively, is then typically achieved through the momentum ( → G 1 ) provided by a 1D nonlinear grating, so that: 1 1 , as illustrated by the vectorial diagram of Fig. 1d. More advanced interaction schemes may be afforded by nonlinear photonic crystals 27,28 , where a two-dimensional patterning of the quadratic nonlinearity provides multiple reciprocal lattice vectors, sustaining several concurrent QPM processes. Figure 1b illustrates the case of 3-mode OPG [29][30][31][32] , in which two idler modes, b 1 and b 2 , are coherently coupled to a shared signal, a 0 . The index m = 0 designates here the shared mode in the overall OPG output. In this case, as shown in Fig. 1e, QPM of two concurrent OPG processes is achieved via two reciprocal lattice vectors of the nonlinear photonic crystal ( → G 1 and → G 2 ), satisfying the following conditions 29 : Similarly, 3-mode OPG can also be sustained by a shared idler, b 0 , coherently coupled to two signal modes, a 1 and a 2 , quasi phase-matched via → G 1 and → G 2 , respectively 29-31 . Here we use the QPM degrees of freedom of a 2D nonlinear photonic crystal to achieve OPG with four coupled output modes (two signals and two idlers), accessing a super-resonant regime, which may have some analogy with coherent effects observed in a different context 33 , whereby the parametric emission rates are enhanced as a result the coherent interplay of multiple parametric interactions. The waves in the nonlinear crystal interact according to the diagram of Fig. 1c, where a single pump excites three concurrent OPG processes, simultaneously generating three signal-idler mode pairs: . This unique OPG configuration is underpinned by the following vectorial QPM conditions (Fig. 1f): where the subscript '0' denotes the shared (signal and idler) modes, coupled via → G 1 , and the subscript '2' denotes the remaining output modes, coupled via → G 2 , as depicted in Fig. 1f.

Theoretical predictions
In the limit of perfect QPM, described by equations (1), plane-wave, monochromatic and undepleted pump, the evolution of the signal and idler modes in the crystal is approximately described by the following set of equations: where z is the propagation variable and ∂ z the z-derivative. a b , m m denote photon annihilation operators (or, equivalently, complex field amplitudes in a classical description) 29 for the signal and idler modes at frequencies ω a and ω b , respectively.
For standard OPG (Fig. 1a), in the same parametric limit under which equations (2) are derived, the mean photon-numbers in the high gain regime (gz ≫ 1) evolve in propagation as: ∼ N N e , a b gz 2 1 1 , featuring an exponential growth factor equal to 2g 25 .
For the 3-mode OPG case (Fig. 1b), implemented with a shared signal (or, equivalently, with a shared idler), the mean photon-numbers evolve as: , with an exponential gain enhancement of 2 with respect to standard OPG 32 .
The 4-mode parametric coupling of Fig. 1c, described by equations (2), instead, gives rise to an asymptotic exponential growth of the form , where ϕ is the Golden Ratio (see also Methods). The appearance of this number can be traced back to the peculiar asymmetric form of the coupling, where the populations of modes a 0 and b 0 originate from two possible frequency down-conversion processes, while modes a 2 and b 2 are populated by a single OPG process (Fig. 1c).

Experiments
The 4-mode OPG concept was implemented in a hexagonally poled crystal, with a single pump beam at a fixed wavelength (λ p ), which propagates at a small angle (θ p ) with respect to the symmetry axis of the nonlinear lattice (z), as sketched in Fig. 2.
The nonlinear medium was a nearly stoichiometric Z-cut periodically poled LiTaO 3 crystal, doped with 1 mol% MgO (Oxide Corp) 34 . The pattern consisted of a hexagonal lattice of ferroelectric domains, with total inverted area ~33% and period Λ = 8.3 μm, fabricated by room temperature electric field poling 35 . The nonlinear chip was 2 cm-long, 4 mm-wide and 500 µm-thick along its X, Y and Z crystallographic directions, corresponding to the z, x and y axes, respectively, in the reference frame adopted in what follows. The crystal input and output facets were polished to optical grade, parallel to the (x, y) plane. The sample was mounted on a rotation stage and held at a constant temperature (T = 80 ± 0.1 °C) throughout the experiments.
The pump for the experiments was an 11 ps, 527.5 nm wavelength source, obtained from the second harmonic of a 1055 nm pulsed Nd:Glass laser, having a 10 Hz repetition rate. The input field was polarized along the y-axis (Z in the crystal frame) to make use of the largest nonlinear coefficient of LiTaO 3 , i.e. d 33 = 16 pm/V 35 . The pump beam was shaped into an elliptical transverse profile and loosely focused along the vertical (y) direction. The beam size at the entrance of the crystal was 2 × 0.2 mm 2 . Twin beams in the near-infrared and telecom ranges (λ a ~ 800 nm for the signal and λ b ~ 1550 nm for the idler) were generated during the high-gain OPG process. During the experiment a complete characterization of the far-field signal radiation was performed, including a spectral and angular mapping of the OPG response in the near infrared for different pulse energies (10-100 μJ) and incidence angles (−2.5° to 2.5°) of the pump beam. The idler radiation was also monitored by means of an infrared camera (Xeva, Xenics) to verify the emission in the correct wavelength range, further confirming the expected OPG distributions (see Methods for details).

Results and Discussion
QPM in the nonlinear photonic crystal was achieved through the fundamental reciprocal lattice vectors → G 1 and → G 2 (inset of Fig. 2), for which: . Each of them supports, for a given value of θ p , infinite sets of QPM solutions, defined in terms of the wavelength (λ) and transverse propagation angles θ θ ( , x y ) of the signal and idler modes. In the 3D (λ θ θ , , ) x y parameter space, the signal (idler) solutions lie on two surfaces, corresponding to QPM via → G 1 or → G 2 , respectively. Their intersection with the plane of the lattice (θ = 0 y ) yields two (λ θ , x ) branches were OPG emission is concentrated, as illustrated by Fig. 3. ) and asymmetric pumping, respectively. Both cases make apparent the existence of OPG hot-spots at the intersections of the two QPM branches, where output intensities are significantly enhanced. This is ascribed to 3-mode OPG, consistently with previous findings [29][30][31] . The labels highlight the coupling of a shared signal (a 0 ) to two symmetric idler modes (b 1 , b 2 ) and the dual case, i.e. a shared idler (b 0 ) coupled to two signal modes (a 1 , a 2 ), according to the previously introduced notation. The diagrams on the R.H.S. illustrate the corresponding momentum conservation laws.
As the pump is tilted away from the symmetry axis (θ = .
1 47 p°, Fig. 3b), the shared signal (idler) mode gets closer to one of the coupled signal (idler) modes, i.e.: → a a 0 1 ( → b b 0 1 ) until, for a given pump incidence angle (θ = . 2 1 p°, Fig. 3c), a peculiar super-resonant condition is met, where the x-components of all the optical wave-vectors (pump, signal and idler) are individually matched to the x-component of either → G 1 or → G 2 . The interacting optical fields then experience the same spatial transverse modulation as the crystal lattice, resulting in constructive interference of the parametric emission in the specific directions satisfying equations (1). The coherence of the pump in the (x, z) plane is also expected to play a role, in analogy with other superradiance emission  33 . Specifically, the longitudinal and transverse coherence lengths of the pump, here coinciding with its extensions in the z (~1.5 mm) and x (2 mm) directions, must be much larger than the poling period (Λ ~ 8.3 μm) of the pattern. Accordingly, the expected nonlinear interference effect is clearly observed even if the pump pulse length is sub-optimal with respect to temporal walk-off between the pump and idler pulses (duly accounted for in the simulations), occurring over a distance When the super-resonant condition is met, the two triplets of OPG modes, originally uncoupled (Fig. 3b), coalesce into 4 coupled modes (Fig. 3c). This enables the interaction scenario of Fig. 1c, whereby 2 modes of the signal (a 0 and a 2 ) and 2 of the idler (b 0 and b 2 ), pairwise frequency-degenerate, realize the 4-mode coupling of equations (2).
To access in the experiments the super-resonant regime, the pump propagation direction was adjusted by rotating the crystal. For each pump incidence angle, a full 3D characterization of the signal fluorescence was performed at the output, by measuring the intensity angular distributions in the far-field and their spectral-angular distributions in the x and y directions. The key experimental results are illustrated by Fig. 4(a-i) Fig. 3. The hot spots of 3 and 4-mode OPG were order of magnitudes brighter than the surrounding 2-mode OPG background (barely visible in the plots of Fig. 4a,d and g). For 3-mode OPG, three nearly straight lines were observed in the far-field plane (Fig. 4b and e): a central line featuring all the hot-spots associated with a shared signal (a 0 ), and two 1 47 p 0 , and (g-i) θ = .
2 1 p 0 . By tilting the pump incidence angle from the symmetric configuration to the superresonance angle, the three hot-spots seen in the θ λ ( , ) x -maps (a,d) coalesce into two (g). The effect is accompanied by a marked enhancement in OPG brightness and by the merging of the three lines seen in the far field (b,c,e,f) into two (h,i). The experimental data were obtained by recording the signal radiation over 20 laser shots. The numerical simulations show single-shot results.
outer lines featuring those of the signals coupled to a shared idler (a 1 , a 2 ), in full agreement with simulations ( Fig. 4c and f). The wavelength of the radiation emitted along the hot-spot lines varied over a broad range (Fig. s3,  Supplementary Information).
As the crystal was rotated away from the symmetric pumping condition (Fig. 4b,c), the central (shared signal) line approached one of the outer lines (Fig. 4e,f). Their merging at the super-resonance condition (Fig. 4h,i) was accompanied by a sudden and dramatic increase of the hot-spot intensity (Fig. 4g). In these conditions, modes a 0 and a 1 become fully degenerate (in both wavelength and angle) as confirmed by the measurements of Fig. 4g.
To confirm the predictions for the Golden Ratio gain enhancement, the evolution of the output signal intensities in 2-, 3-and 4-mode OPG was systematically investigated as a function of the pump energy in the experiments (Fig. 5). Based on a simplified analytical model [equations (2) and Methods] and on the proportionality of the parameter g therein to the amplitude of the pump field 25,32 , an exponential growth of the output photon number with the square root of the input pump energy (E ) p is expected, i.e.: ∝ γσ N e E p , where σ is a constant, while the pre-factor γ takes the value of 1, 2 and ϕ = . … 1 618 for the case of 2-, 3-and 4-mode OPG, respectively. Accordingly, the OPG response is analyzed by plotting the values of N in logarithmic scale as a function of E p . Figure 5 shows the experimental data for 2-, 3-and 4-mode OPG as: black circles, blue squares and red triangles, respectively. Open symbols connected by solid lines show the results of numerical simulations, based on a realistic modelling of the experiments (Methods).
Three distinct linear trends in the evolution of N ln are apparent in the 15-50 μJ energy range, with a marked increase in the slopes when moving from 2-, to 3-, and finally to 4-mode OPG. Linear fits on the experimental data yield gain enhancement factors: γ exp ( ) = 1.51 ± 0.38 and γ (exp) = 2.07 ± 0.71 for 3-mode and 4-mode OPG, respectively, which compare well with numerical simulations, yielding: γ (sim) = 1.47 ± 0.17 and γ (sim) = 1.83 ± 0.17 in the two cases (details of the data analysis and fit procedure provided in Supplementary Information).
Overall the experimental results summarized by Fig. 5 agree, within the errors, with analytical predictions based on equations (2), even though the experimental conditions satisfy only very roughly the approximations implied by the analytical model. This is a further indicator of the robustness of the predicted physics. It is also noteworthy that the experimental data consistently deviate from the analytical predictions in the same direction as the results of the numerical simulations.

Conclusion
In summary, we revealed the possibility of a unique four-mode coupling in optical parametric generation, resulting in a gain enhancement equal to the Golden Ratio, which stems from the peculiar symmetries of the nonlinear coupling. We realized it in a monolithic format by exploiting the quasi-phase-matching degrees of freedom of a hexagonally poled crystal. Experiments performed in the high-gain regime provided evidence for enhanced OPG emission with localized intensity hot-spots, in very good agreement with theoretical predictions. The results provide a new paradigm for tailoring gain and coherent emission in optical parametric amplifiers, oscillators and entangled photon sources, by engineering the coherent coupling of multiphoton processes. Further generalizations to even higher number of modes may be envisaged through the extra degrees of freedom afforded by nonlinear quasi-crystals, higher dimensionality lattices and excitation with structured pump beams.

Methods
Experiments. The emission from the crystal around the signal wavelength was first spatially characterized in the far field by means of a f = 50 mm focal length lens, placed at a distance f from the output face of the crystal, and the far field images were recorded by a high efficiency, 16 bit, charged-coupled-device (CCD) camera (Andor) placed in the lens focal plane ( Supplementary Information Fig. s1). A razor edge filter with 99% transmission for wavelengths >530 nm was used to cut the pump. Neutral density filters were used to control the intensity level onto the CCD chip. Single shot and multiple shot images were recorded by controlling the exposure time of the CCD camera. The spectrally resolved angular spectrum of the signal radiation was then recorded by means of an imaging spectrometer (Lot Oriel) with a 150 μm-wide slit. This spectrometer, constituted by a grating placed in the center of a telescopic system made of two spherical mirrors, allows the reconstruction of the angular spectrum of the injected far-field radiation along the slit direction, and of the wavelength spectrum along the orthogonal direction (diffraction plane of the grating). The result is a 2D image recorded by a CCD camera that gives an angular and wavelength mapping of the OPG radiation 36 .
The signal emission was characterized both in the θ x = 0 plane and in the θ y = 0 plane ( Supplementary  Information Fig. s3). For the characterization of the emission in the horizontal plane, the far-field signal beam underwent a 90° rotation by means of two-beam steering mirrors and was then focused by the f = 50 mm lens onto the vertical slit of the imaging spectrometer. The spectra were recorded at the very output of the spectrometer (imaging plane of the slit plane), by the CCD camera ( Supplementary Information Fig. s2).
Theory and simulations. The OPG process is modelled in terms of coupled propagation equations for the three (pump, signal and idler) interacting field operators, described as wave-packets centered around their carrier frequencies. The periodic nonlinear modulation in the (x, z)-plane is described by keeping only the leading order terms in the Fourier expansion of the nonlinear susceptibility: are the two fundamental vectors of the reciprocal lattice (Fig. 2)  , where k k , x y are the transverse components of the field wave-vectors and Ω is the frequency shift from the carrier frequencies 37,38 . They account for pump depletion, spatial and temporal walk-off, diffraction and dispersion at any order, and take the following form: super-resonance ( ≠ ± q G p x ), the 3-mode OPG condition is realized. With a shared signal and for perfect phase-matching = = D D ( 0 ) 1 2 , the configuration is described by the following set of equations: where the coupling strength is , α p being the classical pump amplitude and L the crystal length. These equations are analogous to those first derived and analyzed in a classical framework in ref. 32 . When applied to our specific lattice configuration, the mean photon numbers at the crystal output ( = z L) are given by: where the last line holds in the high-gain regime  gL 1. The above expression is to be compared with the standard behavior of 2-mode OPG, where ∝  N g L e sinh ( ) . When the pump is tilted to super-resonance, i.e. for = ± q G p x , two sets of 3-modes coalesce into a single set of 4 coupled modes, evolving in the sample according to the model described by equations (2) and equally valid for classical field amplitudes, and 4-mode OPG can be realized. Eigenvalues and eigenvectors of this 4-mode dynamical coupling can be found with some algebraic manipulations. In particular, for perfect phase matching the eigenvalues are ϕ ±g and ϕ ±g/ , where ϕ is the Golden Ratio. Remarkably, ϕ g and ϕ g/ correspond to two independent parametric processes, one showing an enhancement of the growth rate of light → × .
.. g g 1 618 and the other a decrease → × . ... g g 0 618 with respect to standard 2-mode OPG. At high parametric gains the first process clearly dominates. Accordingly, in this regime the mean photon numbers exhibit the following asymptotic behaviors: . These analytical predictions agree quite well with the experimental findings shown in Fig. 5, confirming the robustness of the predicted phenomena, despite the deviation of the experimental conditions from the approximations of the model, in particular due to the onset of temporal walk-off effects in this relatively long crystal. Data availability. Supplementary information is available in the online version of the paper.