Discrete quadratic solitons with competing second-harmonic components

Setzpfandt, Frank; Sukhorukov, Andrey A.; Pertsch, Thomas

doi:10.1103/physreva.84.053843

Cited by 5 publications

(5 citation statements)

References 45 publications

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“…This system conserves the total power |E 1 | 2 + |E 2 | 2 = 3 2 P . (28) is, up to a scaling of the fields and the length variable, equivalent to the easily recognisable system describing the nonlinear interaction between two modes via the Kerr nonlinearity; for example E 1 and E 2 might describe continuous wave propagation in the two linearly polarised, phase velocity mismatched components of the electric field of an optical fibre [12] (see A for how to transform from a conventional set of equations with material parameters in physical units to the normalised form in (28)). For the first time it is now evident that the general solution of this classic system is simply expressible via (26).…”

Section: Cubic Casementioning

confidence: 99%

“…In the quadratic case, the form of the solution presented herein may prove particularly useful when trying to extend the algebraic manipulations to solitonic like or continuous wave solutions of the relevant time varying equations with dispersion [24,25]. The methods presented herein may also generalize to higher mode systems such as waveguide arrays with quadratic nonlinearity [26,27,28]. In the cubic case, the solution method may prove particularly useful when trying to extend the algebraic manipulations to spatial or temporal solitonic like solutions of the relevant partial differential equations with diffraction or dispersion [30,29] and it may also generalize to higher mode systems such as three-wave mixing [36] and four-wave mixing [31,32].…”

Section: Wider Applicationsmentioning

confidence: 99%

“…followed by the introduction of a new length variable z = 2∆βZ and function E j (z) = a j (Z) yields (28).…”

Section: B Periods As Elliptic Integralsmentioning

confidence: 99%

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General complex envelope solutions of coupled-mode optics with quadratic or cubic nonlinearity

Hesketh¹

2015

J. Opt. Soc. Am. B

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The analytic general solutions for the complex field envelopes are derived using Weierstrass elliptic functions for two and three mode systems of differential equations coupled via quadratic χ2 type nonlinearity as well as two mode systems coupled via cubic χ3 type nonlinearity. For the first time, a compact form of the solutions is given involving simple ratios of Weierstrass sigma functions (or equivalently Jacobi theta functions). A Fourier series is also given. All possible launch states are considered. The models describe sum and difference frequency generation, polarization dynamics, parity-time dynamics and optical processing applications.Coupled mode nonlinear optical systems form a fundamental building block in optical processing functionality. Herein, we provide an extensive analytical study of both the coupled mode system for quadratic nonlinearity, found in materials such as periodically poled lithium niobate (PPLN) crystals and commonly referred to as χ 2 in optics, as well as the coupled mode system for cubic nonlinearity commonly referred to as χ 3 and typically found in silica waveguides such as optical fibers. The functionality enabled by the quadratic nonlinearity includes, but is not limited to, sum and difference frequency generation [1,2], modulation format conversion [3], optical logic [4,5], and all-optical switching [6,7,8,9]. Respectively, the cubic nonlinear system in optics can describe nonlinear polarization mode dynamics; which include instabilities [10] and has enabled such functionality as the optical Kerr shutter [11,12] and all-optical regeneration [13,14], adjacent waveguides coupled via evanescent optical fields; which are utilized in all-optical switching and optical logic [15], and most recently, nonlocal non-Hermitian parity-time (PT) symmetric coupled mode systems; which may provide a test bed within optics for theoretical physics and possibly enable novel unique optical properties such as non-reciprocity between modes and power threshold behaviour (see [16] and references therein). Outside of optics, the coupled mode equations with cubic nonlinearity also describe Bose-Einstein condensates (BECs) in atomic physics [17].The equations describing light interactions in a dielectric with quadratic or cubic nonlinearity were first derived by Armstrong et. al. in 1962 [1]. Therein, the amplitudes of two or three mixing frequencies were solved for in terms of Jacobi elliptic functions. In a Hamiltonian analysis, bifurcations and instabilities were examined in the two mode quadratic nonlinear system in [18] and the phase of the fundamental mode was solved for as an elliptic integral of the third kind in [6] under the special condition that the second mode (or second harmonic) was zero at input. It was proposed therein that the intensity dependent refractive index could be used to induce phase shifts in a Mach-Zehnder waveguide to produce an all-optical switch. A similar technique was explored in a 3 mode case in [7] where in addition to optical switching, wavelength demultiplexing a...

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Section: Cubic Casementioning

confidence: 99%

Section: Wider Applicationsmentioning

confidence: 99%

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General complex envelope solutions of coupled-mode optics with quadratic or cubic nonlinearity

Hesketh¹

2015

J. Opt. Soc. Am. B

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“…1(d) shows a scheme of the three bands governing the propagation of the modes taken into account here. The nonlinear propagation of cw-waves in the lithium niobate WGA, taking into account one FW and two SH modes, can be described with the following coupled mode equations for the mode amplitudes in the waveguides [29,31]:…”

Section: Spatial Cw Soliton Phase Transition With Two Sh Modesmentioning

confidence: 99%

Spectral pulse transformations and phase transitions in quadratic nonlinear waveguide arrays

et al. 2011

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We study experimentally and numerically the dynamics of a recently found topological phase transition for discrete quadratic solitons with linearly coupled SH waves. We find that, although no stationary states are excited in the experimental situation, the generic feature of the phase transition of the SH is preserved. By utilizing simulations of the coupled mode equations we identify the complex processes leading to the phase transition involving spatial focusing and the generation of new frequency components. These distinct signatures of the dynamic phase transition are also demonstrated experimentally.

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“…In contrast to previous works [11] we analyze the influence of coupling to multiple TH modes [14] of the gas filed Kagome fiber, with individual dispersion. We assume filling with Xe gas, being more nonlinear with a reduced relative impact of n 4 than Ar.…”

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confidence: 97%