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ABSTRACT 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. SIMILAR CONTENT BEING VIEWED BY OTHERS QUANTUM-LIMITED DETERMINATION OF REFRACTIVE INDEX DIFFERENCE BY MEANS OF ENTANGLEMENT

Article Open access 16 May 2022 HIGH-PERFORMANCE QUANTUM ENTANGLEMENT GENERATION VIA CASCADED SECOND-ORDER NONLINEAR PROCESSES Article Open access 05 August 2021 INGAP _Χ_(2) INTEGRATED

PHOTONICS PLATFORM FOR BROADBAND, ULTRA-EFFICIENT NONLINEAR CONVERSION AND ENTANGLED PHOTON GENERATION Article Open access 15 October 2024 INTRODUCTION The Golden Ratio,\(\,\phi

=\frac{1+\surd 5}{2}\), is a sort of magic number, appearing in the most diverse fields of art1,2, biology3,4, mathematics5,6,7, physics and engineering8,9,10,11,12,13,14,15,16, often in

connection with fractal growth and self-similarity. This work shows for the first time its spontaneous occurrence in a nonlinear optical process, in the form of a growth-rate enhancement of

light generation. The relevant process is optical parametric generation (OPG)17,18, a paradigm for sources of classical and quantum-entangled light19,20,21,22,23,24. OPG implies the

spontaneous conversion of a photon from a high-energy pump laser (at frequency \({\omega }_{p}\)) into a pair of lower-energy photons, at frequencies \({\omega }_{a}\) and \({\omega

}_{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

(\({\omega }_{p}={\omega }_{a}+{\omega }_{b}\)) and momentum are preserved25. The latter condition is nowadays routinely enforced by quasi phase-matching (QPM) with nonlinear artificial

structuring26,27,28. 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. \({\overrightarrow{{\boldsymbol{k}}}}_{p},{\overrightarrow{{\boldsymbol{k}}}}_{{a}_{1}}\) and

\({\overrightarrow{{\boldsymbol{k}}}}_{{b}_{1}}\), respectively, is then typically achieved through the momentum (\({\overrightarrow{{\boldsymbol{G}}}}_{1}\)) provided by a 1D nonlinear

grating, so that:

\({\overrightarrow{{\boldsymbol{k}}}}_{p}={\overrightarrow{{\boldsymbol{k}}}}_{{a}_{1}}+{\overrightarrow{{\boldsymbol{k}}}}_{{b}_{1}}+{\overrightarrow{{\boldsymbol{G}}}}_{1}\), as

illustrated by the vectorial diagram of Fig. 1d. More advanced interaction schemes may be afforded by nonlinear photonic crystals27,28, 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 OPG29,30,31,32, 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 (\({\overrightarrow{{\boldsymbol{G}}}}_{1}\,\)and

\({\overrightarrow{{\boldsymbol{G}}}}_{2}\)), satisfying the following conditions29:

$$\{\begin{array}{c}\,{\overrightarrow{{\boldsymbol{k}}}}_{p}={\overrightarrow{{\boldsymbol{k}}}}_{{a}_{0}}+{\overrightarrow{{\boldsymbol{k}}}}_{{b}_{1}}+{\overrightarrow{{\boldsymbol{G}}}}_{1}\\

\,{\overrightarrow{{\boldsymbol{k}}}}_{p}={\overrightarrow{{\boldsymbol{k}}}}_{{a}_{0}}+{\overrightarrow{{\boldsymbol{k}}}}_{{b}_{2}}+{\overrightarrow{{\boldsymbol{G}}}}_{2}\end{array}$$

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

\({\overrightarrow{{\boldsymbol{G}}}}_{1}\,\)and \({\overrightarrow{{\boldsymbol{G}}}}_{2}\), respectively29,30,31. 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

context33, 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: (\({a}_{0},\,{b}_{2})\), \(({a}_{0},\,{b}_{0})\)

and \(({a}_{2},\,{b}_{0})\). This unique OPG configuration is underpinned by the following vectorial QPM conditions (Fig. 1f):

$$\{\begin{array}{c}{\overrightarrow{{\boldsymbol{k}}}}_{p}={\overrightarrow{{\boldsymbol{k}}}}_{{a}_{0}}+{\overrightarrow{{\boldsymbol{k}}}}_{{b}_{2}}+{\overrightarrow{{\boldsymbol{G}}}}_{2}\\

{\overrightarrow{{\boldsymbol{k}}}}_{p}={\overrightarrow{{\boldsymbol{k}}}}_{{a}_{0}}+{\overrightarrow{{\boldsymbol{k}}}}_{{b}_{0}}+{\overrightarrow{{\boldsymbol{G}}}}_{1}\\

{\overrightarrow{{\boldsymbol{k}}}}_{p}={\overrightarrow{{\boldsymbol{k}}}}_{{a}_{2}}+{\overrightarrow{{\boldsymbol{k}}}}_{{b}_{0}}+{\overrightarrow{{\boldsymbol{G}}}}_{2}\end{array}\,$$ (1)

where the subscript ‘0’ denotes the shared (signal and idler) modes, coupled via \({\overrightarrow{{\boldsymbol{G}}}}_{1}\), and the subscript ‘2’ denotes the remaining output modes,

coupled via \({\overrightarrow{{\boldsymbol{G}}}}_{2}\), 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: $$\{\begin{array}{l}\begin{array}{l}{\partial

}_{z}{a}_{0}=\,g\,({b}_{0}^{\dagger }+{b}_{2}^{\dagger })\,\\ \begin{array}{l}{\partial }_{z}{a}_{2}=\,g\,{b}_{0}^{\dagger }\\ {\partial }_{z}{b}_{0}=\,g\,({a}_{0}^{\dagger

}+{a}_{2}^{\dagger })\end{array}\end{array}\\ {\partial }_{z}{b}_{2}=\,g\,{a}_{0}^{\dagger }\end{array}\,$$ (2) where _z_ is the propagation variable and \({\partial }_{z}\) the

_z-_derivative.\(\,{a}_{m},\,\,{b}_{m}\) denote photon annihilation operators (or, equivalently, complex field amplitudes in a classical description)29 for the signal and idler modes at

frequencies \({\omega }_{a}\) and \({\omega }_{b}\), 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:\(\,{N}_{{a}_{1}},{N}_{{b}_{1}}\sim {e}^{2gz}\), featuring an exponential growth factor equal to 2_g_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: \({N}_{{a}_{0}},{N}_{{b}_{1}},\,{N}_{{b}_{2}}\sim

\,{e}^{2\sqrt{2}gz}\), with an exponential gain enhancement of \(\sqrt{2}\) with respect to standard OPG32. The 4-mode parametric coupling of Fig. 1c, described by equations (2), instead,

gives rise to an asymptotic exponential growth of the form \({N}_{{a}_{0}},{N}_{{a}_{2}},{N}_{{b}_{0}},\,{N}_{{b}_{2}}\sim {e}^{2\phi gz}\), where \(\phi \) 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 (\({\lambda }_{p}\)), which propagates at a small angle (\({\theta }_{p}\)) 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 LiTaO3 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 poling35. 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 LiTaO3, i.e. _d_33 = 16 pm/V35.

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 mm2. 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 \({\overrightarrow{{\boldsymbol{G}}}}_{1}\,\)and \({\overrightarrow{{\boldsymbol{G}}}}_{2}\) (inset of Fig. 2), for which:

\(|{\overrightarrow{{\boldsymbol{G}}}}_{1}|=|{\overrightarrow{{\boldsymbol{G}}}}_{2}|=4\pi /({\rm{\Lambda }}\sqrt{3})\). Each of them supports, for a given value of \({\theta }_{p}\),

infinite sets of QPM solutions, defined in terms of the wavelength (\(\lambda \)) and transverse propagation angles \(({\theta }_{x},{\theta }_{y}\)) of the signal and idler modes. In the 3D

(\(\lambda ,{\theta }_{x},{\theta }_{y})\) parameter space, the signal (idler) solutions lie on two surfaces, corresponding to QPM via \({\overrightarrow{{\boldsymbol{G}}}}_{1}\) or

\({\overrightarrow{{\boldsymbol{G}}}}_{2}\), respectively. Their intersection with the plane of the lattice (\({\theta }_{y}=0\)) yields two (\(\lambda ,{\theta }_{x}\)) branches were OPG

emission is concentrated, as illustrated by Fig. 3. Figure 3a and b, show the numerical results of a realistic modelling of the experiments (see also Methods) for 3-mode OPG with symmetric

(\({\theta }_{p}=0\)) 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 findings29,30,31. 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 (\({\theta }_{p}=1.47\)°, Fig. 3b), the shared

signal (idler) mode gets closer to one of the coupled signal (idler) modes, i.e.: \({a}_{0}\to {a}_{1}\) (\({b}_{0}\to {b}_{1}\)) until, for a given pump incidence angle (\({\theta

}_{p}=2.1\)°, 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 \({\overrightarrow{{\boldsymbol{G}}}}_{1}\) or \({\overrightarrow{{\boldsymbol{G}}}}_{2}\). 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 phenomena33. 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 _μ_m) 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 \({L}_{p-i}\sim 14\) mm (shorter than the overall propagation length: _L_ = 20 mm). 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), corresponding to \({\theta }_{p}={0}^{{\rm{o}}},\,\,{1.47}^{{\rm{o}}}\) and \({2.1}^{{\rm{o}}}\), respectively, i.e. the three configurations analyzed in 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 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 field25,32, an exponential growth of the

output photon number with the square root of the input pump energy (\({E}_{p})\,\,\)is expected, i.e.: \(N\propto {e}^{\gamma \sigma \sqrt{{E}_{p}}}\), where \(\sigma \,\,\)is a constant,

while the pre-factor \(\gamma \) takes the value of \(1,\,\sqrt{2}\,\,\)and \(\phi =1.618\ldots \) 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 \(\sqrt{{E}_{p}}\). 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 \(\mathrm{ln}\,N\) are apparent in the 15–50 \(\mu J\) 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: \({\gamma }^{(exp)}\) = 1.51 ± 0.38 and \(\gamma \)(exp) = 2.07 ± 0.71 for 3-mode and 4-mode OPG,

respectively, which compare well with numerical simulations, yielding: \(\gamma \)(sim) = 1.47 ± 0.17 and \(\gamma \)(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 radiation36. 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:\(\,d(x,z)\approx {d}_{01}\,{e}^{i{G}_{z}z}({e}^{i{G}_{x}x}+\,{e}^{-i{G}_{x}x})\), where

\({\overrightarrow{{\boldsymbol{G}}}}_{1,2}=\,{G}_{z}{\overrightarrow{{\boldsymbol{e}}}}_{z}\pm \,{G}_{x}{\overrightarrow{{\boldsymbol{e}}}}_{x}\) are the two fundamental vectors of the

reciprocal lattice (Fig. 2)27. For the hexagonally poled crystal used in the experiments: \({G}_{z}=\frac{2\pi }{{\rm{\Lambda }}}\), \({G}_{x}=\frac{2\pi }{{\rm{\Lambda }}\sqrt{3}}\), and

\({d}_{01}=0.29\,{d}_{33}\)30. Propagation equations are written in the Fourier domain spanned by the 3D vector \({\boldsymbol{w}}=({k}_{x},{k}_{y},\,{\rm{\Omega }})\), where

\({k}_{x},\,{k}_{y}\) are the transverse components of the field wave-vectors and \({\rm{\Omega }}\) is the frequency shift from the carrier frequencies37,38. They account for pump

depletion, spatial and temporal walk-off, diffraction and dispersion at any order, and take the following form: $$\begin{array}{c}{\partial }_{z}{A}_{s}({{\boldsymbol{w}}}_{s},z)=\chi \int

{d}^{3}{{\boldsymbol{w}}}_{p}\,{A}_{p}({{\boldsymbol{w}}}_{p},z)[{A}_{i}^{\dagger

}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{s}-{{\bf{G}}}_{{\boldsymbol{x}}},z){e}^{-i{{\mathfrak{D}}}_{1}({{\boldsymbol{w}}}_{s},{{\boldsymbol{w}}}_{p})z}\\ \,\,\,\,\,+\,{A}_{i}^{\dagger

}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{s}+{{\bf{G}}}_{{\boldsymbol{x}}},z){e}^{-i{{\mathfrak{D}}}_{2}({{\boldsymbol{w}}}_{s},{{\boldsymbol{w}}}_{p})z}]\end{array}$$ (3)

$$\begin{array}{c}{\partial }_{z}{A}_{i}({{\boldsymbol{w}}}_{i},z)=\chi \int {d}^{3}{{\boldsymbol{w}}}_{p}\,{A}_{p}({{\boldsymbol{w}}}_{p},z)[{A}_{s}^{\dagger

}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{i}-{{\bf{G}}}_{{\boldsymbol{x}}},z){e}^{-i{{\mathfrak{D}}}_{1}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{i}-{{\bf{G}}}_{x},{{\boldsymbol{w}}}_{p})z}\,\\

\,\,\,\,\,+\,\,{A}_{s}^{\dagger

}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{i}+{{\bf{G}}}_{{\boldsymbol{x}}},z){e}^{-i{{\mathfrak{D}}}_{2}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{i}+{{\bf{G}}}_{{\boldsymbol{x}}},{{\boldsymbol{w}}}_{p})z}]\,\end{array}$$

(4) $$\begin{array}{c}{\partial }_{z}{A}_{p}({{\boldsymbol{w}}}_{p},z)=-\,\chi \int {d}^{3}{{\boldsymbol{w}}}_{s}\,{A}_{s}({{\boldsymbol{w}}}_{s},z)[{A}_{i}^{\dagger

}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{s}-{{\bf{G}}}_{{\boldsymbol{x}}},z){e}^{i{\mathfrak{D}}{}_{1}({{\boldsymbol{w}}}_{s},{{\boldsymbol{w}}}_{p})z}\\ \,\,\,\,\,\,+\,{A}_{i}^{\dagger

}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{s}+{{\bf{G}}}_{{\boldsymbol{x}}},z){e}^{i{{\mathfrak{D}}}_{2}({{\boldsymbol{w}}}_{s},{{\boldsymbol{w}}}_{p})z}]\end{array}$$ (5) where

\({{\bf{G}}}_{{\boldsymbol{x}}}\) is a short-hand notation for the vector \(({G}_{x},\,0,0)\) in _W_-space, and \({A}_{j}({\boldsymbol{w}},z)\) are the positive frequency parts of the field

operators for the signal \((j=s)\), idler \((j=i)\,\,\)and pump \((j=p)\) beams, with dimensions such that \(\langle {{A}_{j}}^{\dagger }{A}_{j}\rangle \) is a number of photons per unit

frequency and wavevector-squared. The first and second term at the R.H.S. of equations (3–5) describe all the possible OPG processes mediated by the lattice vectors

\({\overrightarrow{{\boldsymbol{G}}}}_{1}\,\,\)and \({\overrightarrow{{\boldsymbol{G}}}}_{2}\), respectively. Their phase-matching functions

\({{\mathfrak{D}}}_{1,2}({{\boldsymbol{w}}}_{s},{{\boldsymbol{w}}}_{p})={k}_{sz}({{\boldsymbol{w}}}_{s})+{k}_{iz}({{\boldsymbol{w}}}_{p}-{{\boldsymbol{w}}}_{s}\mp

{{\bf{G}}}_{{\bf{x}}})-{k}_{pz}({{\boldsymbol{w}}}_{p})+{G}_{z}\) account for the efficiency by which each of these processes occurs, where

\({k}_{jz}({\boldsymbol{w}})=\sqrt{{k}_{j}^{2}({\rm{\Omega }})-({k}_{x}^{2}+{k}_{y}^{2})}\) is the _z_-component of the wave-vector of the mode, with wavenumber \({k}_{j}({\rm{\Omega

}})=\frac{{\omega }_{j}+{\rm{\Omega }}}{{\boldsymbol{c}}}n({{\rm{\omega }}}_{{\rm{j}}}+{\rm{\Omega }})\), _n_ being the extraordinary refractive index of the nonlinear crystal35. Numerical

simulations were performed in the framework of the Wigner representation, where field operators are replaced by _c_-number fields39. The input signal and idler fields, initially in the

vacuum state, are simulated stochastically with Gaussian white noise, while the injected pump field is a high-intensity coherent pulse. The equations are integrated with a pseudo-spectral

(split-step) method38. The dimensions of the numerical grid used for the simulations (512 × 256 × 512 points along _x, y_ and the temporal axis, respectively) were carefully chosen to

accommodate the spectral and angular bandwidths of the pump (with Gaussian spatial and temporal profiles) as well as the generated signal and idler distributions, under realistic modelling

conditions. ANALYTICAL MODEL Analytical results can be derived in the parametric limit, where the pump is approximated by a classical monochromatic plane-wave, undepleted by the OPG process.

We assume that the plane-wave pump propagates in the \((x,z)\)-plane at a small angle \({\theta }_{p}={q}_{p}/{k}_{p}\) with the _z_-axis. As will be detailed in a further theoretical

publication, the shared signal modes are then characterized by the same _x_-component of the wave-vector as the pump, \({k}_{x}={q}_{p}.\) In these conditions, when the pump is not tilted to

super-resonance (\({q}_{p}\ne \pm \,{{G}}_{x}\)), the 3-mode OPG condition is realized. With a shared signal and for perfect phase-matching

\(({{\mathfrak{D}}}_{1}={{\mathfrak{D}}}_{2}=0)\), the configuration is described by the following set of equations: $$\{\begin{array}{c}{\partial }_{z}{a}_{0}=g[{b}_{1}^{\dagger

}+{b}_{2}^{\dagger }]\\ {\partial }_{z}{b}_{1}=g{a}_{0}^{\dagger }\\ {\partial }_{z}{b}_{2}=g{a}_{0}^{\dagger }\,\end{array}$$ (6) where the coupling strength is \(\,g=\chi {\alpha

}_{p}=\frac{\sigma }{2L}\sqrt{{E}_{p}}\), \({\alpha }_{p}\) 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: $$\,{N}_{{a}_{0}}(L)\sim

{N}_{{b}_{1,2}}(L)\propto {\sinh }^{2}(\sqrt{2}gL)\simeq \frac{1}{4}{e}^{2\sqrt{2}gL}\,$$ (7) where the last line holds in the high-gain regime \(gL\gg 1\). The above expression is to be

compared with the standard behavior of 2-mode OPG, where \(\,N\propto {\sinh }^{2}(gL)\simeq \frac{1}{4}\,{e}^{2gL}\), Analogous results apply to the dual case of 3-mode OPG with a shared

idler, with the substitutions: \({a}_{0}\to {b}_{0},\,{b}_{1}\to {a}_{1}\) and \({b}_{2}\to {a}_{2}\). When the pump is tilted to super-resonance, i.e. for \({q}_{p}=\pm \,{{G}}_{x}\), 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 \(\pm g\phi \) and \(\pm g/\phi \), where _φ_ is the _Golden Ratio_. Remarkably, \(g\phi \) and \(g/\phi \) correspond to two independent parametric processes, one showing an

enhancement of the growth rate of light \(g\,\to \,g\times 1.618\,\mathrm{..}\) and the other a decrease \(g\,\to g\times \mathrm{0.618...}\) 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:

$${N}_{{a}_{0}}(L)={N}_{{b}_{0}}(L)\propto \frac{{\phi }^{2}}{1+{\phi }^{2}}{\sinh }^{2}(g\phi L)\,\simeq \frac{{\phi }^{2}}{4(1+{\phi }^{2})}{e}^{2\phi gL}$$ (8)

$${N}_{{a}_{2}}(L)={N}_{{b}_{2}}(L)\propto \frac{1}{1+{\phi }^{2}}{\sinh }^{2}(g\phi L)\,\simeq \frac{1}{4(1+{\phi }^{2})}{e}^{2\phi gL}$$ (9) Therefore, when the chain of stimulated

processes becomes long enough, the intensity of the hot-spots exhibits a power-law increase with respect to the surrounding 2-mode fluorescence

\({N}_{{\rm{4}}-{\rm{mode}}}=\,{({N}_{{\rm{2}}-{\rm{mode}}})}^{\phi }\). 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. REFERENCES * Hemenway, P. _Divine proportion: phi in art,

nature, and science_. (Sterling Publishing, New York, USA, 2005). Google Scholar  * Divine golden ingenious: The Golden Ratio as a theory of everything? O. Götze and L. Kugler Eds, Hirmer

Publishers, Munich, Germany (2016). * Mitchison, G. J. Phyllotaxis and the Fibonacci series. _Science_ 196, 270–275 (1977). Article  ADS  PubMed  CAS  Google Scholar  * Marinković, S.,

Stanković, P., Štrbac, M., Tomić, I. & Ćetković, M. Cochlea and other spiral forms in nature and art. _Am. J. Otolaryngol._ 33, 80–87 (2012). Article  PubMed  Google Scholar  * Dunlap,

R. A. The Golden Ratio and Fibonacci numbers, World Scientific Publishing (1997). * Lefebvre, V. A. The golden section and an algebraic model of ethical cognition. _J. Math. Psychol._ 29,

289–310 (1986). Article  MATH  Google Scholar  * Schuster, S. A new solution concept for the ultimatum game leading to the Golden Ratio. _Sci. Rep._ 7, 5642 (2017). Article  ADS  PubMed 

PubMed Central  CAS  Google Scholar  * Thawornwong, S. & Enke, D. The adaptive selection of financial and economic variables for use with artificial neural networks. _Neurocomputing_ 56,

205–232 (2004). Article  Google Scholar  * Sen, S. & Agarwal, R. Golden ratio in science, as random sequence source, its computation and beyond. _Comp. Math. Appl._ 56, 469–498 (2008).

Article  MathSciNet  MATH  Google Scholar  * Hubsch, T. & Katona, G. A. Golden-ratio-controlled chaos in supersymmetric dynamics. _Int. J. Mod. Phys. A_ 28, 1350156 (2013). Article  ADS

  MathSciNet  MATH  CAS  Google Scholar  * Coldea, R. _et al_. Quantum criticality in an Ising chain: experimental evidence for emergent E8 symmetry. _Science_ 327, 177–180 (2010). Article 

ADS  PubMed  CAS  Google Scholar  * Willard, B. C. The Golden Ratio in optics. _Opt. Photon. News_ 4, 22–25 (1993). Article  ADS  Google Scholar  * Coelho, F. S. & Herdeiro, C. A. R.

Relativistic Euler’s three-body problem, optical geometry, and the golden ratio. _Phys. Rev. D_ 80, 104036 (2009). Article  ADS  CAS  Google Scholar  * Cruz, N., Olivares, M. &

Villanueva, J. R. The golden ratio in Schwarzschild–Kottler black holes. _Eur. Phys. J. C_ 77, 123 (2017). Article  ADS  CAS  Google Scholar  * Sigalotti, L. & Mejias, A. The golden mean

in special relativity. _Chaos Solitons Fract._ 30, 521–524 (2006). Article  ADS  MATH  Google Scholar  * Heyrovska, R. The golden ratio ionic and atomic radii and bond lengths. _Mol. Phys._

103, 877–882 (2005). Article  ADS  CAS  Google Scholar  * Harris, S. E., Oshman, M. K. & Byer, R. L. Observation of tunable optical parametric fluorescence. _Phys. Rev. Lett._ 18,

732–735 (1967). Article  ADS  CAS  Google Scholar  * Klyshko, D. N. Scattering of light in a medium with nonlinear polarizability. _Sov Phys JETP_ 28, 522–526 (1969). ADS  Google Scholar  *

Dunn, M. H. & Ebrahimzadeh, M. Parametric generation of tunable light from continuous-wave to femtosecond pulses. _Science_ 286, 1513–1518 (1999). Article  PubMed  CAS  Google Scholar  *

Huang, S.-W. _et al_. High-energy pulse synthesis with sub-cycle waveform control for strong-field physics. _Nature Photon._ 5, 475–479 (2011). Article  ADS  CAS  Google Scholar  * Berger,

V. & Rosencher, E. Optical parametric sources for the infrared. _Compt. Rend. Phys._ 8, 1099–1226 (2007). Article  ADS  CAS  Google Scholar  * Burnham, D. C. & Weinberg, D. L.

Observation of simultaneity in parametric production of optical photon pairs. _Phys. Rev. Lett._ 25, 84–86 (1970). Article  ADS  CAS  Google Scholar  * Pan, J.-W., Gasparoni, S., Aspelmeyer,

M., Jennewein, T. & Zeilinger, A. Experimental realization of freely propagating teleported qubits. _Nature_ 421, 721–725 (2003). Article  ADS  PubMed  CAS  Google Scholar  * Ren, J.-G.

_et al_. Ground-to-satellite quantum teleportation. _Nature_ 549, 70–73 (2017). Article  ADS  PubMed  CAS  Google Scholar  * Boyd, R. W. Nonlinear Optics, Academic press, Elsevier (2003),

Chapter 2. * Myers, L. E. _et al_. Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3. _J. Opt. Soc. Am. B_ 12, 2102–2116 (1995). Article  ADS  CAS  Google

Scholar  * Berger, V. Nonlinear photonic crystals. _Phys. Rev. Lett._ 81, 4136 (1998). Article  ADS  CAS  Google Scholar  * Gallo, K., Gawith, C. B. E. & Smith, P. G. R. Bidimensional

hexagonal poling of LiNbO3 for nonlinear photonic crystals and quasi-crystals. _Ferroelectrics_ 340, 69–74 (2006). Article  CAS  Google Scholar  * Gallo, K., Levenius, M., Laurell, F. &

Pasiskevicius, V. Twin-beam optical parametric generation in χ(2) nonlinear photonic crystals. _Appl. Phys. Lett._ 98, 161113 (2011). Article  ADS  CAS  Google Scholar  * Conforti, M.,

Baronio, F., Levenius, M. & Gallo, K. Broadband parametric processes in χ(2) nonlinear photonic crystal. _Opt. Lett._ 39, 3457–3460 (2014). Article  ADS  PubMed  CAS  Google Scholar  *

Levenius, M., Pasiskevicius, V. & Gallo, K. Angular degrees of freedom in twin-beam parametric down-conversion. _Appl. Phys. Lett._ 101, 121114 (2012). Article  ADS  CAS  Google Scholar

  * Liu, H.-C. & Kung, A. H. Substantial gain enhancement for optical parametric amplification and oscillation in two-dimensional χ(2) nonlinear photonic crystals. _Opt. Expr._ 16,

9714–9725 (2008). Article  ADS  Google Scholar  * Onodera, T., Liscidini, M., Sipe, J. E. & Helt, L. G. Parametric fluorescence in a sequence of resonators: an analogy with Dicke

supperadiance. _Phys. Rev. A_ 93, 043837 (2016). Article  ADS  CAS  Google Scholar  * Lim, H. H., Kurimura, S., Katagai, T. & Shoji, I. Temperature-dependent Sellmeier equation for

refractive index of 1.0 mol% Mg-doped stoichiometric lithium tantalate. _Jpn. J. Appl. Phys._ 52, 032601 (2013). Article  ADS  CAS  Google Scholar  * Yu, N. E., Kurimura, S., Nomura, Y.

& Kitamura, K. Stable-high-power green light generation with thermally conductive periodically poled stoichiometric lithium tantalate. _Jpn. J. Appl. Phys._ 43(10A), L1265–L1267 (2007).

Article  ADS  CAS  Google Scholar  * Jedrkiewicz, O., Picozzi, A., Clerici, M., Faccio, D. & Di Trapani, P. Emergence of X-shaped spatiotemporal coherence in optical waves. _Phys. Rev.

Lett._ 97, 243903 (2006). Article  ADS  PubMed  CAS  Google Scholar  * Gatti, A., Zambrini, B. & Lugiato, L. & San Miguel, M. Multiphoton, multimode polarization entanglement in

parametric downconversion. _Phys. Rev. A_ 68, 053807 (2003). Article  ADS  CAS  Google Scholar  * Brambilla, E., Jedrkiewicz, O., Di Trapani, P. & Gatti, A. Space-time coupling in

upconversion of broadband down-converted light. _J. Opt. Soc. Am. B_ 31, 1383 (2014). Article  ADS  CAS  Google Scholar  * Gatti, A. _et al_. Langevin treatment of quantum fluctuations and

optical patterns in optical parametric oscillators below threshold. _Phys. Rev A_ 56, 877 (1997). Article  ADS  CAS  Google Scholar  Download references ACKNOWLEDGEMENTS This work was

supported by the Swedish Research Council - Vetenskapsrådet (grants VR 622-2010-526 and 621-2014-5407), by the ADOPT Linneaus Center for Optics and Photonics in Stockholm and by a short-term

mobility mission of the Italian CNR (2016AMMCNT38916). Helpful discussions with F. Prati and P. Di Trapani (Insubria University) and J. Hirohashi (Oxide Corp.) are also gratefully

acknowledged. AUTHOR INFORMATION AUTHORS AND AFFILIATIONS * Istituto di Fotonica e Nanotecnologie del CNR, Udr Como, Via Valleggio 11, 22100, Como, Italy Ottavia Jedrkiewicz & Alessandra

Gatti * Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100, Como, Italy Alessandra Gatti & Enrico Brambilla * Department of Applied Physics,

KTH – Royal Institute of Technology, Roslagstullsbacken 21, SE-10691, Stockholm, Sweden Martin Levenius & Katia Gallo * Laser Research Centre, Vilnius University, Saulėtekio 10,

LT-10223, Vilnius, Lithuania Gintaras Tamošauskas Authors * Ottavia Jedrkiewicz View author publications You can also search for this author inPubMed Google Scholar * Alessandra Gatti View

author publications You can also search for this author inPubMed Google Scholar * Enrico Brambilla View author publications You can also search for this author inPubMed Google Scholar *

Martin Levenius View author publications You can also search for this author inPubMed Google Scholar * Gintaras Tamošauskas View author publications You can also search for this author

inPubMed Google Scholar * Katia Gallo View author publications You can also search for this author inPubMed Google Scholar CONTRIBUTIONS The samples and experiments were devised by K.G.,

with essential contributions by M.L. K.G. and O.J. performed the experiments, with essential contributions from G.T. to mounting the setup. A.G. developed the theory and derived the

analytical model and its solutions. E.B. developed and performed the numerical simulations. The gain data analysis was performed mainly by E.B. and O.J., with relevant contributions from

K.G. and A.G. The manuscript was written by K.G., A.G., O.J. and E.B. All authors discussed the results, reviewed the manuscript, and agreed on its final version. CORRESPONDING AUTHOR

Correspondence to Katia Gallo. ETHICS DECLARATIONS COMPETING INTERESTS The authors declare no competing interests. ADDITIONAL INFORMATION PUBLISHER'S NOTE: Springer Nature remains

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Brambilla, E. _et al._ Golden Ratio Gain Enhancement in Coherently Coupled Parametric Processes. _Sci Rep_ 8, 11616 (2018). https://doi.org/10.1038/s41598-018-30014-7 Download citation *

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