Abstract

In this article it is shown that for specific initial conditions an input beam injected in a multimode periodic segmented waveguide does not diffract and remains collimated all along the waveguide, whereas a speckle-like pattern is expected at the output of a multimode structure. This nonintuitive behavior can be explained with the help of ray and wave chaos properties. A modal analysis developed in this article reveals that this nondiffractive beam regime is due to a specific superposition of modes with regularly spaced propagation constants. A discrepancy with the commonly used equivalent continuous waveguide model is also identified.

© 2013 Optical Society of America

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2011

2010

P. Aschiéri and V. Doya, “Ray dispersion strongly modified by a periodic index segmentation,” Opt. Commun. 283, 3673–3677 (2010).
[CrossRef]

2009

D. Castaldini, P. Bassi, P. Aschiéri, S. Tascu, M. D. Micheli, and P. A. Baldi, “High performance mode adapters based on segmented SPE:LiNbO3 waveguides,” Opt. Express 17, 17868–17873 (2009).
[CrossRef]

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

2007

S. Shinohara and T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216 (2007).
[CrossRef]

D. Castaldini, P. Bassi, S. Tascu, P. Aschiéri, M. P. D. Micheli, and P. Baldi, “Soft-proton-exchange tapers for low insertion-loss LiNbO3 devices,” J. Lightwave Technol. 25, 1588–1593 (2007).
[CrossRef]

E. D. Leonel, “Corrugated waveguide under scaling investigation,” Phys. Rev. Lett. 98, 114102 (2007).
[CrossRef]

2006

P. Aschiéri, V. Doya, and A. Picozzi, “Complex behaviour of a ray in a Gaussian index profile periodically segmented waveguide,” J. Opt. A 8, 386–390 (2006).
[CrossRef]

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

2005

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

O. Bendix, J. Méndez-Bermúdez, G. Luna-Acosta, U. Kuhl, and H.-J. Stöckmann, “Design of beam splitters and microlasers using chaotic waveguides,” Microelectron. J. 36, 285–288 (2005).
[CrossRef]

2004

F. Fogli, G. Bellanca, P. Aschieri, M. D. Micheli, and P. Bassi, “Spatial harmonics analysis of deep waveguide Bragg gratings,” J. Opt. A 6, 433–438 (2004).
[CrossRef]

P. Aschiéri, F. Fogli, P. Aumont, M. D. Micheli, G. Bellanca, and P. Bassi, “Optical power management using second harmonic generation in periodic segmented waveguides,” Opt. Commun. 235, 55–61 (2004).
[CrossRef]

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef]

2002

M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

J. Dingjan, E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef]

2001

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Light scarring in an optical fiber,” Phys. Rev. Lett. 88, 014102 (2001).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872–874 (2001).
[CrossRef]

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[CrossRef]

1999

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4 channel waveguides,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

1998

1997

J. U. Nockel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[CrossRef]

1996

1995

1994

1993

Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11, 1831–1838 (1993).
[CrossRef]

1992

Z. Weissman and A. Hardy, “2-D mode tapering via tapered channel waveguide segmentation,” Electron. Lett. 28, 1514–1516 (1992).
[CrossRef]

L. Li and J. Burke, “Linear propagation characteristics of periodically segmented waveguides,” Opt. Lett. 17, 1195–1197 (1992).
[CrossRef]

1990

J. D. Bierlein, D. B. Laubacher, J. B. Brown, and C. J. van der Poel, “Balanced phase matching in KTiOP4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

1983

M. De Micheli, J. Botineau, S. Neveu, P. Sibillot, and D. B. Ostrowsky, “Independent control of index and profile in proton-exchanged lithium niobate guide,” Opt. Lett. 8, 114–115 (1983).
[CrossRef]

J. Vollmer, J. P. Nisius, P. Hertel, and E. Krtzig, “Refractive index profiles of LiNbO3: Ti waveguides,” Appl. Phys. A 32, 125–127 (1983).
[CrossRef]

1980

1940

K. Husimi, “Some formal properties of the density matrix,” Math. Soc. Jpn. 22, 264–314 (1940).

Aitchison, J. S.

Altewischer, E.

J. Dingjan, E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef]

Arbore, M. A.

Aschieri, P.

F. Fogli, G. Bellanca, P. Aschieri, M. D. Micheli, and P. Bassi, “Spatial harmonics analysis of deep waveguide Bragg gratings,” J. Opt. A 6, 433–438 (2004).
[CrossRef]

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[CrossRef]

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4 channel waveguides,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Aschiéri, P.

P. Aschiéri and M. P. de Micheli, “Highly efficient coupling in lithium niobate photonic wires by the use of a segmented waveguide coupler,” Appl. Opt. 50, 3885–3888 (2011).
[CrossRef]

P. Aschiéri and V. Doya, “Unexpected light behaviour in periodic segmented waveguides,” Chaos 21, 043118 (2011).
[CrossRef]

P. Aschiéri and V. Doya, “Ray dispersion strongly modified by a periodic index segmentation,” Opt. Commun. 283, 3673–3677 (2010).
[CrossRef]

D. Castaldini, P. Bassi, P. Aschiéri, S. Tascu, M. D. Micheli, and P. A. Baldi, “High performance mode adapters based on segmented SPE:LiNbO3 waveguides,” Opt. Express 17, 17868–17873 (2009).
[CrossRef]

D. Castaldini, P. Bassi, S. Tascu, P. Aschiéri, M. P. D. Micheli, and P. Baldi, “Soft-proton-exchange tapers for low insertion-loss LiNbO3 devices,” J. Lightwave Technol. 25, 1588–1593 (2007).
[CrossRef]

P. Aschiéri, V. Doya, and A. Picozzi, “Complex behaviour of a ray in a Gaussian index profile periodically segmented waveguide,” J. Opt. A 8, 386–390 (2006).
[CrossRef]

P. Aschiéri, F. Fogli, P. Aumont, M. D. Micheli, G. Bellanca, and P. Bassi, “Optical power management using second harmonic generation in periodic segmented waveguides,” Opt. Commun. 235, 55–61 (2004).
[CrossRef]

Aumont, P.

P. Aschiéri, F. Fogli, P. Aumont, M. D. Micheli, G. Bellanca, and P. Bassi, “Optical power management using second harmonic generation in periodic segmented waveguides,” Opt. Commun. 235, 55–61 (2004).
[CrossRef]

Baldi, P.

D. Castaldini, P. Bassi, S. Tascu, P. Aschiéri, M. P. D. Micheli, and P. Baldi, “Soft-proton-exchange tapers for low insertion-loss LiNbO3 devices,” J. Lightwave Technol. 25, 1588–1593 (2007).
[CrossRef]

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[CrossRef]

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4 channel waveguides,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Baldi, P. A.

Bassi, P.

D. Castaldini, P. Bassi, P. Aschiéri, S. Tascu, M. D. Micheli, and P. A. Baldi, “High performance mode adapters based on segmented SPE:LiNbO3 waveguides,” Opt. Express 17, 17868–17873 (2009).
[CrossRef]

D. Castaldini, P. Bassi, S. Tascu, P. Aschiéri, M. P. D. Micheli, and P. Baldi, “Soft-proton-exchange tapers for low insertion-loss LiNbO3 devices,” J. Lightwave Technol. 25, 1588–1593 (2007).
[CrossRef]

P. Aschiéri, F. Fogli, P. Aumont, M. D. Micheli, G. Bellanca, and P. Bassi, “Optical power management using second harmonic generation in periodic segmented waveguides,” Opt. Commun. 235, 55–61 (2004).
[CrossRef]

F. Fogli, G. Bellanca, P. Aschieri, M. D. Micheli, and P. Bassi, “Spatial harmonics analysis of deep waveguide Bragg gratings,” J. Opt. A 6, 433–438 (2004).
[CrossRef]

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[CrossRef]

Bava, G. P.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

Becker, K.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

Bellanca, G.

P. Aschiéri, F. Fogli, P. Aumont, M. D. Micheli, G. Bellanca, and P. Bassi, “Optical power management using second harmonic generation in periodic segmented waveguides,” Opt. Commun. 235, 55–61 (2004).
[CrossRef]

F. Fogli, G. Bellanca, P. Aschieri, M. D. Micheli, and P. Bassi, “Spatial harmonics analysis of deep waveguide Bragg gratings,” J. Opt. A 6, 433–438 (2004).
[CrossRef]

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[CrossRef]

Bendix, O.

O. Bendix, J. Méndez-Bermúdez, G. Luna-Acosta, U. Kuhl, and H.-J. Stöckmann, “Design of beam splitters and microlasers using chaotic waveguides,” Microelectron. J. 36, 285–288 (2005).
[CrossRef]

Bierlein, J.

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4 channel waveguides,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Bierlein, J. D.

J. D. Bierlein, D. B. Laubacher, J. B. Brown, and C. J. van der Poel, “Balanced phase matching in KTiOP4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Bona, G.

Botineau, J.

Brown, J. B.

J. D. Bierlein, D. B. Laubacher, J. B. Brown, and C. J. van der Poel, “Balanced phase matching in KTiOP4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Burke, J.

Cao, H.

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

Castaldini, D.

Choi, M.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef]

Chou, M. H.

De Micheli, M.

de Micheli, M. P.

Debernardi, P.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

Degen, C.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

Dingjan, J.

J. Dingjan, E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef]

Djellali, N.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

Dorgeuille, F.

Doya, V.

P. Aschiéri and V. Doya, “Unexpected light behaviour in periodic segmented waveguides,” Chaos 21, 043118 (2011).
[CrossRef]

P. Aschiéri and V. Doya, “Ray dispersion strongly modified by a periodic index segmentation,” Opt. Commun. 283, 3673–3677 (2010).
[CrossRef]

P. Aschiéri, V. Doya, and A. Picozzi, “Complex behaviour of a ray in a Gaussian index profile periodically segmented waveguide,” J. Opt. A 8, 386–390 (2006).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872–874 (2001).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Light scarring in an optical fiber,” Phys. Rev. Lett. 88, 014102 (2001).
[CrossRef]

Eger, D.

Elsäßer, W.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

Ern, D.

Fang, W.

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

Feit, M. D.

Fejer, M. M.

Filoche, M.

Fischer, I.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

Fleck, J. J. A.

Fogli, F.

F. Fogli, G. Bellanca, P. Aschieri, M. D. Micheli, and P. Bassi, “Spatial harmonics analysis of deep waveguide Bragg gratings,” J. Opt. A 6, 433–438 (2004).
[CrossRef]

P. Aschiéri, F. Fogli, P. Aumont, M. D. Micheli, G. Bellanca, and P. Bassi, “Optical power management using second harmonic generation in periodic segmented waveguides,” Opt. Commun. 235, 55–61 (2004).
[CrossRef]

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[CrossRef]

Francois, S.

Gensty, T.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

Germann, R.

Ghatak, A.

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University, 1998).

Ghatak, A. K.

Gozhyk, I.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

Greco, N.

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[CrossRef]

Harayama, T.

S. Shinohara and T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216 (2007).
[CrossRef]

Hardy, A.

Z. Weissman, A. Hardy, M. Katz, M. Oron, and D. Eger, “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguides,” Opt. Lett. 20, 674–676 (1995).
[CrossRef]

Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11, 1831–1838 (1993).
[CrossRef]

Z. Weissman and A. Hardy, “2-D mode tapering via tapered channel waveguide segmentation,” Electron. Lett. 28, 1514–1516 (1992).
[CrossRef]

Hendel, I.

Z. Weissman and I. Hendel, “Analysis of periodically segmented waveguide mode expanders,” J. Lightwave Technol. 13, 2053–2058 (1995).
[CrossRef]

Hentschel, M.

M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Hertel, P.

J. Vollmer, J. P. Nisius, P. Hertel, and E. Krtzig, “Refractive index profiles of LiNbO3: Ti waveguides,” Appl. Phys. A 32, 125–127 (1983).
[CrossRef]

Herve-Gruyer, G.

Hierle, R.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

Husimi, K.

K. Husimi, “Some formal properties of the density matrix,” Math. Soc. Jpn. 22, 264–314 (1940).

Katz, M.

Kim, C.-M.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef]

Kippelen, B.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

Krtzig, E.

J. Vollmer, J. P. Nisius, P. Hertel, and E. Krtzig, “Refractive index profiles of LiNbO3: Ti waveguides,” Appl. Phys. A 32, 125–127 (1983).
[CrossRef]

Kuhl, U.

O. Bendix, J. Méndez-Bermúdez, G. Luna-Acosta, U. Kuhl, and H.-J. Stöckmann, “Design of beam splitters and microlasers using chaotic waveguides,” Microelectron. J. 36, 285–288 (2005).
[CrossRef]

Kwon, T.-Y.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef]

Laubacher, D. B.

J. D. Bierlein, D. B. Laubacher, J. B. Brown, and C. J. van der Poel, “Balanced phase matching in KTiOP4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

Lauret, J. S.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

Lautru, J.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

Lebental, M.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

Lee, S.-Y.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef]

Legrand, O.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Light scarring in an optical fiber,” Phys. Rev. Lett. 88, 014102 (2001).
[CrossRef]

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872–874 (2001).
[CrossRef]

Leonel, E. D.

E. D. Leonel, “Corrugated waveguide under scaling investigation,” Phys. Rev. Lett. 98, 114102 (2007).
[CrossRef]

Li, L.

Lozenko, S.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

Luna-Acosta, G.

O. Bendix, J. Méndez-Bermúdez, G. Luna-Acosta, U. Kuhl, and H.-J. Stöckmann, “Design of beam splitters and microlasers using chaotic waveguides,” Microelectron. J. 36, 285–288 (2005).
[CrossRef]

Mahalakshmi, V.

Massarek, I.

Méndez-Bermúdez, J.

O. Bendix, J. Méndez-Bermúdez, G. Luna-Acosta, U. Kuhl, and H.-J. Stöckmann, “Design of beam splitters and microlasers using chaotic waveguides,” Microelectron. J. 36, 285–288 (2005).
[CrossRef]

Mersali, B.

Micheli, M. D.

D. Castaldini, P. Bassi, P. Aschiéri, S. Tascu, M. D. Micheli, and P. A. Baldi, “High performance mode adapters based on segmented SPE:LiNbO3 waveguides,” Opt. Express 17, 17868–17873 (2009).
[CrossRef]

P. Aschiéri, F. Fogli, P. Aumont, M. D. Micheli, G. Bellanca, and P. Bassi, “Optical power management using second harmonic generation in periodic segmented waveguides,” Opt. Commun. 235, 55–61 (2004).
[CrossRef]

F. Fogli, G. Bellanca, P. Aschieri, M. D. Micheli, and P. Bassi, “Spatial harmonics analysis of deep waveguide Bragg gratings,” J. Opt. A 6, 433–438 (2004).
[CrossRef]

Micheli, M. P. D.

Miniatura, C.

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Light scarring in an optical fiber,” Phys. Rev. Lett. 88, 014102 (2001).
[CrossRef]

Mortessagne, F.

V. Doya, O. Legrand, and F. Mortessagne, “Optimized absorption in a chaotic double-clad fiber amplifier,” Opt. Lett. 26, 872–874 (2001).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Light scarring in an optical fiber,” Phys. Rev. Lett. 88, 014102 (2001).
[CrossRef]

Neveu, S.

Nisius, J. P.

J. Vollmer, J. P. Nisius, P. Hertel, and E. Krtzig, “Refractive index profiles of LiNbO3: Ti waveguides,” Appl. Phys. A 32, 125–127 (1983).
[CrossRef]

Nockel, J. U.

J. U. Nockel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[CrossRef]

Offrein, B.

Oron, M.

Ortega, D.

Ostrowsky, D. B.

Owens, D.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

Picozzi, A.

P. Aschiéri, V. Doya, and A. Picozzi, “Complex behaviour of a ray in a Gaussian index profile periodically segmented waveguide,” J. Opt. A 8, 386–390 (2006).
[CrossRef]

Rastogi, V.

Richter, K.

M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Rim, S.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef]

Rue, R. M. D. L.

Ryu, J.-W.

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef]

Shenoy, M. R.

Shinohara, S.

S. Shinohara and T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216 (2007).
[CrossRef]

Sibillot, P.

Spuhler, M. M.

Stancil, D.

Stöckmann, H.-J.

O. Bendix, J. Méndez-Bermúdez, G. Luna-Acosta, U. Kuhl, and H.-J. Stöckmann, “Design of beam splitters and microlasers using chaotic waveguides,” Microelectron. J. 36, 285–288 (2005).
[CrossRef]

Stone, A. D.

J. U. Nockel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[CrossRef]

Sundheimer, M.

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4 channel waveguides,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Tascu, S.

Thyagarajan, K.

Ulysse, C.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

van der Poel, C. J.

J. D. Bierlein, D. B. Laubacher, J. B. Brown, and C. J. van der Poel, “Balanced phase matching in KTiOP4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

van Exter, M. P.

J. Dingjan, E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef]

Vollmer, J.

J. Vollmer, J. P. Nisius, P. Hertel, and E. Krtzig, “Refractive index profiles of LiNbO3: Ti waveguides,” Appl. Phys. A 32, 125–127 (1983).
[CrossRef]

Weissman, Z.

Z. Weissman, A. Hardy, M. Katz, M. Oron, and D. Eger, “Second-harmonic generation in Bragg-resonant quasi-phase-matched periodically segmented waveguides,” Opt. Lett. 20, 674–676 (1995).
[CrossRef]

Z. Weissman and I. Hendel, “Analysis of periodically segmented waveguide mode expanders,” J. Lightwave Technol. 13, 2053–2058 (1995).
[CrossRef]

Z. Weissman and A. Hardy, “Modes of periodically segmented waveguides,” J. Lightwave Technol. 11, 1831–1838 (1993).
[CrossRef]

Z. Weissman and A. Hardy, “2-D mode tapering via tapered channel waveguide segmentation,” Electron. Lett. 28, 1514–1516 (1992).
[CrossRef]

Woerdman, J. P.

J. Dingjan, E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef]

Yamilov, A.

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

Zyss, J.

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

Appl. Opt.

Appl. Phys. A

J. Vollmer, J. P. Nisius, P. Hertel, and E. Krtzig, “Refractive index profiles of LiNbO3: Ti waveguides,” Appl. Phys. A 32, 125–127 (1983).
[CrossRef]

Appl. Phys. Lett.

M. Lebental, J. S. Lauret, R. Hierle, and J. Zyss, “Highly directional stadium-shaped polymer microlasers,” Appl. Phys. Lett. 88, 031108 (2006).
[CrossRef]

N. Djellali, I. Gozhyk, D. Owens, S. Lozenko, M. Lebental, J. Lautru, C. Ulysse, B. Kippelen, and J. Zyss, “Controlling the directional emission of holey organic microlasers,” Appl. Phys. Lett. 95, 101108 (2009).
[CrossRef]

J. D. Bierlein, D. B. Laubacher, J. B. Brown, and C. J. van der Poel, “Balanced phase matching in KTiOP4 waveguides,” Appl. Phys. Lett. 56, 1725–1727 (1990).
[CrossRef]

M. Sundheimer, P. Aschieri, P. Baldi, and J. Bierlein, “Modeling and experimental observation of parametric processes in segmented KTiOPO4 channel waveguides,” Appl. Phys. Lett. 74, 1660–1662 (1999).
[CrossRef]

Chaos

P. Aschiéri and V. Doya, “Unexpected light behaviour in periodic segmented waveguides,” Chaos 21, 043118 (2011).
[CrossRef]

Electron. Lett.

Z. Weissman and A. Hardy, “2-D mode tapering via tapered channel waveguide segmentation,” Electron. Lett. 28, 1514–1516 (1992).
[CrossRef]

J. Lightwave Technol.

J. Opt. A

F. Fogli, G. Bellanca, P. Aschieri, M. D. Micheli, and P. Bassi, “Spatial harmonics analysis of deep waveguide Bragg gratings,” J. Opt. A 6, 433–438 (2004).
[CrossRef]

P. Aschiéri, V. Doya, and A. Picozzi, “Complex behaviour of a ray in a Gaussian index profile periodically segmented waveguide,” J. Opt. A 8, 386–390 (2006).
[CrossRef]

Math. Soc. Jpn.

K. Husimi, “Some formal properties of the density matrix,” Math. Soc. Jpn. 22, 264–314 (1940).

Microelectron. J.

O. Bendix, J. Méndez-Bermúdez, G. Luna-Acosta, U. Kuhl, and H.-J. Stöckmann, “Design of beam splitters and microlasers using chaotic waveguides,” Microelectron. J. 36, 285–288 (2005).
[CrossRef]

Nature

J. U. Nockel and A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature 385, 45–47 (1997).
[CrossRef]

Opt. Commun.

P. Aschiéri and V. Doya, “Ray dispersion strongly modified by a periodic index segmentation,” Opt. Commun. 283, 3673–3677 (2010).
[CrossRef]

P. Aschiéri, F. Fogli, P. Aumont, M. D. Micheli, G. Bellanca, and P. Bassi, “Optical power management using second harmonic generation in periodic segmented waveguides,” Opt. Commun. 235, 55–61 (2004).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

F. Fogli, N. Greco, P. Bassi, G. Bellanca, P. Aschieri, and P. Baldi, “Spatial harmonics modelling of planar periodic segmented waveguides,” Opt. Quantum Electron. 33, 485–498 (2001).
[CrossRef]

Phys. Rev. A

W. Fang, A. Yamilov, and H. Cao, “Analysis of high-quality modes in open chaotic microcavities,” Phys. Rev. A 72, 023815 (2005).
[CrossRef]

Phys. Rev. E

S. Shinohara and T. Harayama, “Signature of ray chaos in quasibound wave functions for a stadium-shaped dielectric cavity,” Phys. Rev. E 75, 036216 (2007).
[CrossRef]

M. Hentschel and K. Richter, “Quantum chaos in optical systems: the annular billiard,” Phys. Rev. E 66, 056207 (2002).
[CrossRef]

Phys. Rev. Lett.

T. Gensty, K. Becker, I. Fischer, W. Elsäßer, C. Degen, P. Debernardi, and G. P. Bava, “Wave chaos in real-world vertical-cavity surface-emitting lasers,” Phys. Rev. Lett. 94, 233901 (2005).
[CrossRef]

J. Dingjan, E. Altewischer, M. P. van Exter, and J. P. Woerdman, “Experimental observation of wave chaos in a conventional optical resonator,” Phys. Rev. Lett. 88, 064101 (2002).
[CrossRef]

S.-Y. Lee, S. Rim, J.-W. Ryu, T.-Y. Kwon, M. Choi, and C.-M. Kim, “Quasiscarred resonances in a spiral-shaped microcavity,” Phys. Rev. Lett. 93, 164102 (2004).
[CrossRef]

V. Doya, O. Legrand, F. Mortessagne, and C. Miniatura, “Light scarring in an optical fiber,” Phys. Rev. Lett. 88, 014102 (2001).
[CrossRef]

E. D. Leonel, “Corrugated waveguide under scaling investigation,” Phys. Rev. Lett. 98, 114102 (2007).
[CrossRef]

Other

A. Ghatak and K. Thyagarajan, Introduction to Fiber Optics (Cambridge University, 1998).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic of the investigated longitudinally periodic waveguide with a period Λ and the transverse Gaussian index profile of the high index segments. Everywhere else, the index is constant and is equal to the substrate index n2 and an oscillating ray path with a period of 2×Λ.

Fig. 2.
Fig. 2.

(a) Poincaré section of the waveguide that exhibits two resonances surrounded by a chaotic sea. Waveguide parameters are δn=0.024, Λ=300μm; the waveguide width is w=12.3μm, and DC=0.8. (b) Poincaré section for a different waveguide configuration that exhibits four peripheral resonances surrounded by closed curves corresponding to stable trajectories. Waveguide parameters are δn=0.05, Λ=146μm; the waveguide width is w=15μm; and DC=0.8.

Fig. 3.
Fig. 3.

Field distribution with a schematic of the waveguide segments for an incident Gaussian beam launched in a resonance of the Poincaré section of the MPSW. (a) High index segments are schematically represented by rectangles. (b) Husimi representation. (c) modal spectrum distribution, which shows that a set of modes regularly spaced are excited. (d) length spectrum of the modal spectrum in the case of the collimated beam.

Fig. 4.
Fig. 4.

(a) Field distribution, (b) Husimi representation, and (c) modal spectrum for an incident Gaussian beam launched outside of the resonance of the Poincaré section. The field is spread on many waveguide modes, which is confirmed by the Husimi distribution and the modal spectrum. (d) is the corresponding length spectrum.

Fig. 5.
Fig. 5.

(a) Field distribution for an incident Gaussian beam launched in a resonance [noted 1 in Fig. 2(a)] of the Poincaré section of the MPSW. Because of the large number of periods for this waveguide configuration, the high index segments are not superimposed on the field distribution in order to keep a clear picture. (b) Husimi representation. (c) represents the modal spectrum distribution, which shows a large set of modes compared to Fig. 3(c), but they are still regularly spaced, which is confirmed by the length spectrum shown in (d) that only exhibits the period of 584 μm and its harmonics.

Fig. 6.
Fig. 6.

(a) Field distribution for an incident Gaussian beam launched outside of a resonance [noted 2 in Figure 2(a)] of the Poincaré section of the MPSW. (b) Husimi representation. (c) represents the modal spectrum distribution, which shows a large set of modes but, unlike the previous case, modes are not regularly spaced, as can be seen with (d) where the Fourier transform of the spectrum exhibits a nonharmonic distribution of period length.

Fig. 7.
Fig. 7.

(a) Total modal spectrum of the ECWG (in blue) and the modes excited when an incident Gaussian beam is launched in a resonance of the Poincaré section for the first waveguide configuration. The distribution of the excited modes of the MPSW does not match with the modal distribution of the ECWG. In (b), due to a larger number of modes and in order to have a clear picture, it is compared to only one part of the spectrum of the ECWG and modes excited when an incident Gaussian beam is launched in a resonance. The mismatch is still present.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

n(x)=n2+δnex2w2,
d2xdz2={2n2δnβ2w2xex2w2;(p1)Λz(DC+p1)Λ0;(DC+p1)Λ<z<pΛ,
ΔΨ+k0n2(x,z)Ψ=0,
2jk0n0Ez2Ex2k02(n2(x,z)n22)E=0.
c(z)=+E*(x,0)E(x,z)dx.
C(β)=0Lc(z)h(z)exp(iβz)dz,
H(x,k)=n=1NH(x,k,z=nΛ).
H(x,k,z=nΛ)=|1(πσ2)14+exp(ikx)exp((xx)22σ2)E(x,z)dx|2,
δneq=DC×δn.
E(x,z)=nAnun(x)exp(iβnz),

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