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Volumn 59, Issue 5, 1999, Pages 4021-4040

Transverse patterns in nondegenerate intracavity second-harmonic generation

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EID: 0001431259     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.59.4021     Document Type: Article
Times cited : (21)

References (44)
  • 1
    • 85037252683 scopus 로고    scopus 로고
    • For a recent review on pattern formation in nonlinear optics see, for instance, L. A. Lugiato, M. Brambilla, and A. Gatti, in Advances in Atomic, Molecular and Optical Physics, edited by B. Bederson and H. Walther (Academic, Boston, 1998), Vol. 40, and references quoted therein
    • For a recent review on pattern formation in nonlinear optics see, for instance, L. A. Lugiato, M. Brambilla, and A. Gatti, in Advances in Atomic, Molecular and Optical Physics, edited by B. Bederson and H. Walther (Academic, Boston, 1998), Vol. 40, and references quoted therein.
  • 10
    • 0000991772 scopus 로고    scopus 로고
    • Phys. Rev. AS. Longhi and A. Geraci, 54, 4581 (1996)
    • (1996) , vol.54 , pp. 4581
    • Longhi, S.1    Geraci, A.2
  • 21
    • 85037189597 scopus 로고    scopus 로고
    • (a) C. Etrich, D. Michaelis, U. Peschel, and F. Lederer, Chaos Solitons Fractals (to be published)
    • (a) C. Etrich, D. Michaelis, U. Peschel, and F. Lederer, Chaos Solitons Fractals (to be published)
  • 22
    • 0013176490 scopus 로고    scopus 로고
    • (b) Phys. Rev. E 58, 4005 (1998)
    • (1998) Phys. Rev. E , vol.58 , pp. 4005
  • 24
    • 85037243919 scopus 로고    scopus 로고
    • The directions of polarization correspond to the two principal axes of the ellipse obtained by intersection of the Fresnel ellipsoid with the plane transverse to the propagation direction (collinear) of the fields
    • The directions of polarization correspond to the two principal axes of the ellipse obtained by intersection of the Fresnel ellipsoid with the plane transverse to the propagation direction (collinear) of the fields.
  • 29
    • 85037218122 scopus 로고    scopus 로고
    • The form of amplitude equations given by Eq. (36) is valid for an arbitrary number M of interacting modes on the critical circle provided that the angle between two arbitrary wave vectors entering in the expansion (27) is much larger than the angular width of modes (of order (Formula presented) that become active when considering finite bandwidth effects (see, for instance, Ref
    • The form of amplitude equations given by Eq. (36) is valid for an arbitrary number M of interacting modes on the critical circle provided that the angle between two arbitrary wave vectors entering in the expansion (27) is much larger than the angular width of modes (of order (Formula presented) that become active when considering finite bandwidth effects (see, for instance, Ref. 14). Finite bandwidth effects could be taken into account by introducing slow space variables in the multiple-scale analysis, and the final amplitude equations would have the form of Newell-Whitehead-Segel equations (see, for instance, Refs. 28(b)). The nonlocal terms in the resulting equations are important to reveal the existence of long-wavelength instabilities of patterns (such as Eckhaus and zigzag instabilities), but this analysis goes beyond the scope of the present work.
  • 34
    • 85037218919 scopus 로고    scopus 로고
    • We solved numerically the amplitude equations (38) considering a large number of competing modes (typically (Formula presented) for (Formula presented) and for different values of detuning parameters. A variable-step fourth-order Runge-Kutta method was used, and small random values for mode amplitudes were assumed as initial conditions. Steady-state solutions corresponding to squeezed structures can be found indeed for some range of detuning parameters. As an example, squeezed hexagons with wave vectors oriented at relative angles (Formula presented) and (Formula presented) are found as stable states for (Formula presented) and (Formula presented) The free energy of the squeezed hexagons is (Formula presented) which should be compared to the value (Formula presented) for regular hexagons and (Formula presented) for square patterns
    • We solved numerically the amplitude equations (38) considering a large number of competing modes (typically (Formula presented) for (Formula presented) and for different values of detuning parameters. A variable-step fourth-order Runge-Kutta method was used, and small random values for mode amplitudes were assumed as initial conditions. Steady-state solutions corresponding to squeezed structures can be found indeed for some range of detuning parameters. As an example, squeezed hexagons with wave vectors oriented at relative angles (Formula presented) and (Formula presented) are found as stable states for (Formula presented) and (Formula presented) The free energy of the squeezed hexagons is (Formula presented) which should be compared to the value (Formula presented) for regular hexagons and (Formula presented) for square patterns.
  • 43
    • 85037233734 scopus 로고    scopus 로고
    • For a narrower computing window, for instance, for a box size of (Formula presented) with 128 Fourier modes, we typically observed spontaneous appearance of patterns with a decagonal or octagonal symmetry solely in the first stage of the instability, the final stable states usually being squares. It should be noticed that the prevalence of square patterns at low aspect ratios and transition to a quasicrystalline order at larger aspect ratios is a phenomenon analogous to that observed in Faraday wave experiments at low viscosity and due to boundary constraints (see
    • For a narrower computing window, for instance, for a box size of (Formula presented) with 128 Fourier modes, we typically observed spontaneous appearance of patterns with a decagonal or octagonal symmetry solely in the first stage of the instability, the final stable states usually being squares. It should be noticed that the prevalence of square patterns at low aspect ratios and transition to a quasicrystalline order at larger aspect ratios is a phenomenon analogous to that observed in Faraday wave experiments at low viscosity and due to boundary constraints (see 21).


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