Abstract

An optical kicked system with free-space light propagation along a sequence of equally spaced thin phase gratings is presented and investigated. We show, to our knowledge for the first time in optics, the occurrence of the localization effect in the spatial frequency domain, which suppresses the spreading of diffraction orders formed by the repeated modulation by the gratings of the propagating wave. Resonances and antiresonances of the optical system are described and are shown to be related to the Talbot effect. The system is similar in some aspects to the quantum kicked rotor, which is the standard system in the theoretical studies of the suppression of classical (corresponding to Newtonian mechanics) chaos by interference effects. Our experimental verification was done in a specific regime, where the grating spacing was near odd multiples of half the Talbot length. It is shown that this corresponds to the vicinity of antiresonance in the kicked system. The crucial alignment of the gratings in-phase positioning in the experiment was based on a diffraction elimination property at antiresonance. In the present study we obtain new theoretical and experimental results concerning the localization behavior in the vicinity of antiresonance. The behavior in this regime is related to that of electronic motion in incommensurate potentials, a subject that was extensively studied in condensed matter physics.

© 2000 Optical Society of America

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2000 (1)

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, “Experimental observation of localization in the spatial frequency domain of a kicked optical system,” Phys. Rev. E 61, 4694R–4697R (2000).
[CrossRef]

1999 (1)

1997 (2)

M. V. Berry and S. Klein, “Transparent mirrors: rays, waves, and localization,” Eur. J. Phys. 18, 222–228 (1997).
[CrossRef]

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

1995 (1)

F. L. Moore, J. C. Robinson, C. F. Barucha, B. Sundaram, and M. G. Raizen, “Atom optics realization of the quantum δ-kicked rotor,” Phys. Rev. Lett. 75, 4598–4601 (1995).
[CrossRef] [PubMed]

1991 (1)

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

1989 (1)

1987 (1)

J. Krug, “Optical analog of a kicked quantum oscillator,” Phys. Rev. Lett. 59, 2133–2136 (1987).
[CrossRef] [PubMed]

1986 (1)

R. Blumel, S. Fishman, and U. Smilansky, “Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization,” J. Chem. Phys. 84, 2604–2614 (1986).
[CrossRef]

1984 (1)

M. Wilkinson, “Critical properties of electron eigenstates in incommensurate systems,” Proc. R. Soc. London, Ser. A 391, 305–350 (1984).
[CrossRef]

1983 (2)

L. A. Pastur and A. L. Figotin, “Localization in an incommensurate potential: exactly solvable multidimensional model,” JETP Lett. 37, 686–688 (1983).

D. J. Thouless and Q. Niu, “Wavefunction scaling in a quasi-periodic potential,” J. Phys. A 16, 1911–1919 (1983).
[CrossRef]

1982 (3)

D. R. Grempel, S. Fishman, and R. E. Prange, “Localization in an incommensurate potential: an exactly solvable model,” Phys. Rev. Lett. 49, 833–836 (1982).
[CrossRef]

B. Simon, “Almost periodic Schrödinger operators: a review,” Adv. Appl. Math. 3, 463–490 (1982).
[CrossRef]

P. Sarnak, “Spectral behavior of quasi periodic potentials,” Commun. Math. Phys. 84, 377–401 (1982).
[CrossRef]

1981 (1)

B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, “Dynamical stochasticity in classical and quantum mechanics,” Sov. Sci. Rev. Sect. C 2, 209–267 (1981).

1979 (1)

S. Aubry and G. Andre, “Analyticity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1979).

1964 (1)

M. Ya. Azbel, “Energy spectrum of a conduction electron in a magnetic field,” Sov. Phys. JETP 19, 634–645 (1964).

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Andre, G.

S. Aubry and G. Andre, “Analyticity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1979).

Aubry, S.

S. Aubry and G. Andre, “Analyticity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1979).

Azbel, M. Ya.

M. Ya. Azbel, “Energy spectrum of a conduction electron in a magnetic field,” Sov. Phys. JETP 19, 634–645 (1964).

Bartolini, P.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Barucha, C. F.

F. L. Moore, J. C. Robinson, C. F. Barucha, B. Sundaram, and M. G. Raizen, “Atom optics realization of the quantum δ-kicked rotor,” Phys. Rev. Lett. 75, 4598–4601 (1995).
[CrossRef] [PubMed]

Bekker, A.

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, “Experimental observation of localization in the spatial frequency domain of a kicked optical system,” Phys. Rev. E 61, 4694R–4697R (2000).
[CrossRef]

Berry, M. V.

M. V. Berry and S. Klein, “Transparent mirrors: rays, waves, and localization,” Eur. J. Phys. 18, 222–228 (1997).
[CrossRef]

Blumel, R.

R. Blumel, S. Fishman, and U. Smilansky, “Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization,” J. Chem. Phys. 84, 2604–2614 (1986).
[CrossRef]

Chirikov, B. V.

B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, “Dynamical stochasticity in classical and quantum mechanics,” Sov. Sci. Rev. Sect. C 2, 209–267 (1981).

Dalichaouch, R.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Figotin, A. L.

L. A. Pastur and A. L. Figotin, “Localization in an incommensurate potential: exactly solvable multidimensional model,” JETP Lett. 37, 686–688 (1983).

Fischer, B.

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, “Experimental observation of localization in the spatial frequency domain of a kicked optical system,” Phys. Rev. E 61, 4694R–4697R (2000).
[CrossRef]

B. Fischer, A. Rosen, and S. Fishman, “Localization in frequency for periodically kicked light propagation in a dispersive single-mode fiber,” Opt. Lett. 24, 1463–1465 (1999).
[CrossRef]

Fishman, S.

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, “Experimental observation of localization in the spatial frequency domain of a kicked optical system,” Phys. Rev. E 61, 4694R–4697R (2000).
[CrossRef]

B. Fischer, A. Rosen, and S. Fishman, “Localization in frequency for periodically kicked light propagation in a dispersive single-mode fiber,” Opt. Lett. 24, 1463–1465 (1999).
[CrossRef]

R. Blumel, S. Fishman, and U. Smilansky, “Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization,” J. Chem. Phys. 84, 2604–2614 (1986).
[CrossRef]

D. R. Grempel, S. Fishman, and R. E. Prange, “Localization in an incommensurate potential: an exactly solvable model,” Phys. Rev. Lett. 49, 833–836 (1982).
[CrossRef]

Grempel, D. R.

D. R. Grempel, S. Fishman, and R. E. Prange, “Localization in an incommensurate potential: an exactly solvable model,” Phys. Rev. Lett. 49, 833–836 (1982).
[CrossRef]

Izrailev, F. M.

B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, “Dynamical stochasticity in classical and quantum mechanics,” Sov. Sci. Rev. Sect. C 2, 209–267 (1981).

Klein, S.

M. V. Berry and S. Klein, “Transparent mirrors: rays, waves, and localization,” Eur. J. Phys. 18, 222–228 (1997).
[CrossRef]

Krug, J.

J. Krug, “Optical analog of a kicked quantum oscillator,” Phys. Rev. Lett. 59, 2133–2136 (1987).
[CrossRef] [PubMed]

Lagendijk, A.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Liu, L.

McCall, S. L.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Moore, F. L.

F. L. Moore, J. C. Robinson, C. F. Barucha, B. Sundaram, and M. G. Raizen, “Atom optics realization of the quantum δ-kicked rotor,” Phys. Rev. Lett. 75, 4598–4601 (1995).
[CrossRef] [PubMed]

Niu, Q.

D. J. Thouless and Q. Niu, “Wavefunction scaling in a quasi-periodic potential,” J. Phys. A 16, 1911–1919 (1983).
[CrossRef]

Pastur, L. A.

L. A. Pastur and A. L. Figotin, “Localization in an incommensurate potential: exactly solvable multidimensional model,” JETP Lett. 37, 686–688 (1983).

Platzman, P. M.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Prange, R. E.

D. R. Grempel, S. Fishman, and R. E. Prange, “Localization in an incommensurate potential: an exactly solvable model,” Phys. Rev. Lett. 49, 833–836 (1982).
[CrossRef]

Raizen, M. G.

F. L. Moore, J. C. Robinson, C. F. Barucha, B. Sundaram, and M. G. Raizen, “Atom optics realization of the quantum δ-kicked rotor,” Phys. Rev. Lett. 75, 4598–4601 (1995).
[CrossRef] [PubMed]

Righini, R.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Robinson, J. C.

F. L. Moore, J. C. Robinson, C. F. Barucha, B. Sundaram, and M. G. Raizen, “Atom optics realization of the quantum δ-kicked rotor,” Phys. Rev. Lett. 75, 4598–4601 (1995).
[CrossRef] [PubMed]

Rosen, A.

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, “Experimental observation of localization in the spatial frequency domain of a kicked optical system,” Phys. Rev. E 61, 4694R–4697R (2000).
[CrossRef]

B. Fischer, A. Rosen, and S. Fishman, “Localization in frequency for periodically kicked light propagation in a dispersive single-mode fiber,” Opt. Lett. 24, 1463–1465 (1999).
[CrossRef]

Sarnak, P.

P. Sarnak, “Spectral behavior of quasi periodic potentials,” Commun. Math. Phys. 84, 377–401 (1982).
[CrossRef]

Schultz, S.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Shepelyansky, D. L.

B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, “Dynamical stochasticity in classical and quantum mechanics,” Sov. Sci. Rev. Sect. C 2, 209–267 (1981).

Simon, B.

B. Simon, “Almost periodic Schrödinger operators: a review,” Adv. Appl. Math. 3, 463–490 (1982).
[CrossRef]

Smilansky, U.

R. Blumel, S. Fishman, and U. Smilansky, “Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization,” J. Chem. Phys. 84, 2604–2614 (1986).
[CrossRef]

Smith, D.

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Sundaram, B.

F. L. Moore, J. C. Robinson, C. F. Barucha, B. Sundaram, and M. G. Raizen, “Atom optics realization of the quantum δ-kicked rotor,” Phys. Rev. Lett. 75, 4598–4601 (1995).
[CrossRef] [PubMed]

Thouless, D. J.

D. J. Thouless and Q. Niu, “Wavefunction scaling in a quasi-periodic potential,” J. Phys. A 16, 1911–1919 (1983).
[CrossRef]

Wiersma, D. S.

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Wilkinson, M.

M. Wilkinson, “Critical properties of electron eigenstates in incommensurate systems,” Proc. R. Soc. London, Ser. A 391, 305–350 (1984).
[CrossRef]

Adv. Appl. Math. (1)

B. Simon, “Almost periodic Schrödinger operators: a review,” Adv. Appl. Math. 3, 463–490 (1982).
[CrossRef]

Ann. Isr. Phys. Soc. (1)

S. Aubry and G. Andre, “Analyticity breaking and Anderson localization in incommensurate lattices,” Ann. Isr. Phys. Soc. 3, 133–164 (1979).

Appl. Opt. (1)

Commun. Math. Phys. (1)

P. Sarnak, “Spectral behavior of quasi periodic potentials,” Commun. Math. Phys. 84, 377–401 (1982).
[CrossRef]

Eur. J. Phys. (1)

M. V. Berry and S. Klein, “Transparent mirrors: rays, waves, and localization,” Eur. J. Phys. 18, 222–228 (1997).
[CrossRef]

J. Chem. Phys. (1)

R. Blumel, S. Fishman, and U. Smilansky, “Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization,” J. Chem. Phys. 84, 2604–2614 (1986).
[CrossRef]

J. Phys. A (1)

D. J. Thouless and Q. Niu, “Wavefunction scaling in a quasi-periodic potential,” J. Phys. A 16, 1911–1919 (1983).
[CrossRef]

JETP Lett. (1)

L. A. Pastur and A. L. Figotin, “Localization in an incommensurate potential: exactly solvable multidimensional model,” JETP Lett. 37, 686–688 (1983).

Nature (1)

D. S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini, “Localization of light in a disordered medium,” Nature 390, 671–673 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[CrossRef]

Phys. Rev. E (1)

B. Fischer, A. Rosen, A. Bekker, and S. Fishman, “Experimental observation of localization in the spatial frequency domain of a kicked optical system,” Phys. Rev. E 61, 4694R–4697R (2000).
[CrossRef]

Phys. Rev. Lett. (4)

D. R. Grempel, S. Fishman, and R. E. Prange, “Localization in an incommensurate potential: an exactly solvable model,” Phys. Rev. Lett. 49, 833–836 (1982).
[CrossRef]

F. L. Moore, J. C. Robinson, C. F. Barucha, B. Sundaram, and M. G. Raizen, “Atom optics realization of the quantum δ-kicked rotor,” Phys. Rev. Lett. 75, 4598–4601 (1995).
[CrossRef] [PubMed]

J. Krug, “Optical analog of a kicked quantum oscillator,” Phys. Rev. Lett. 59, 2133–2136 (1987).
[CrossRef] [PubMed]

S. L. McCall, P. M. Platzman, R. Dalichaouch, D. Smith, and S. Schultz, “Microwave propagation in two-dimensional dielectric lattices,” Phys. Rev. Lett. 67, 2017–2020 (1991).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

M. Wilkinson, “Critical properties of electron eigenstates in incommensurate systems,” Proc. R. Soc. London, Ser. A 391, 305–350 (1984).
[CrossRef]

Sov. Phys. JETP (1)

M. Ya. Azbel, “Energy spectrum of a conduction electron in a magnetic field,” Sov. Phys. JETP 19, 634–645 (1964).

Sov. Sci. Rev. Sect. C (1)

B. V. Chirikov, F. M. Izrailev, and D. L. Shepelyansky, “Dynamical stochasticity in classical and quantum mechanics,” Sov. Sci. Rev. Sect. C 2, 209–267 (1981).

Other (22)

D. L. Shepelyansky, “Localization of quasienergy eigenfunctions in action space,” Phys. Rev. Lett. 56, 677–680 (1986); “Localization of diffusive excitation in multi-level systems,” Physica D 28, 103–114 (1987).
[CrossRef] [PubMed]

R. Graham, M. Schlautmann, and P. Zoller, “Dynamical localization of atomic-beam deflection by a modulated standing light wave,” Phys. Rev. A 45, R19–R22 (1992); see also R. Graham, M. Schlautmann, and D. L. Shepelyansky, “Dynamical localization in Josephson junctions,” Phys. Rev. Lett. 67, 255–258 (1991), and references therein.
[CrossRef] [PubMed]

G. Casati, I. Guarneri, and D. L. Shepelyansky, “Hydrogen atom in monochromatic field: chaos and dynamical photonic localization,” IEEE J. Quantum Electron. 24, 1420–1444 (1988), and references therein; R. Blümel and U. Smilansky, “Microwave ionization of highly excited hydrogen atoms,” Z. Phys. D 6, 83–105 (1987), and references therein; E. J. Galvez, B. E. Sauer, L. Moorman, P. M. Koch, and D. Richards, “Microwave ionization of H atoms: breakdown of classical dynamics for high frequencies,” Phys. Rev. Lett. PRLTAO 61, 2011–2014 (1988); J. E. Bayfield, G. Casati, I. Guarneri, and D. W. Sokol, “Localization of classically chaotic diffusion for hydrogen atoms in microwave fields,” Phys. Rev. Lett. PRLTAO 63, 364–367 (1989), and references therein; R. Blümel, R. Graham, L. Sirko, U. Smilansky, H. Walther, and K. Yamada, “Microwave excitation of Rydberg atoms in the presence of noise,” Phys. Rev. Lett. PRLTAO 62, 341–344 (1989); R. Blümel, A. Buchleitner, R. Graham, L. Sirko, U. Smilansky, and H. Walther, “Dynamical localization in the microwave interaction of Rydberg atoms: the influence of noise,” Phys. Rev. A PLRAAN 44, 4521–4540 (1991), and references therein.
[CrossRef] [PubMed]

B. G. Klappauf, W. H. Oskay, D. A. Steck, and M. G. Raizen, “Observation of noise and dissipation effects on dynamical localization,” Phys. Rev. Lett. 81, 1203–1206 (1998); H. Ammann, R. Gray, I. Shvarchuck, and N. Christensen, “Quantum delta-kicked rotor: experimental observation of decoherence,” Phys. Rev. Lett. 80, 4111–4115 (1998).
[CrossRef]

For reviews, see D. J. Thouless, “Critical phenomena, random systems, gauge theories,” in Proceedings of the Les-Houches Summer School, K. Osterwalder and R. Stora, eds. (North-Holland, Amsterdam, 1986), p. 681; I. M. Lifshits, S. A. Gredeskul, and L. A. Pastur, Introduction to the Theory of Disordered Systems (Wiley, New York, 1988).

See, for example, A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 7.

I. Dana, E. Eisenberg, and N. Shnerb, “Dynamical localization near quantum antiresonance: exact results and a solvable case,” Phys. Rev. Lett. 74, 686–689 (1995); “Antiresonance and localization in quantum dynamics,” Phys. Rev. E 54, 5948–5963 (1996); E. Eisenberg and I. Dana, “Limited sensitivity to analyticity: a manifestation of quantum chaos,” Found. Phys. FNDPA4 27, 153–170 (1997).
[CrossRef] [PubMed]

R. E. Prange, D. R. Grempel, and S. Fishman, “Wave functions at a mobility edge: an example of a singular continuous spectrum,” Phys. Rev. B 28, 7370–7372 (1983); R. E. Prange, D. R. Grempel, and S. Fishman, “Solvable model of quantum motion in an incommensurate potential,” Phys. Rev. B 29, 6500–6512 (1984); R. E. Prange, D. R. Grempel, and S. Fishman, “Long-range resonance in Anderson insulators: finite-frequency conductivity of random and incommensurate systems,” Phys. Rev. Lett. PRLTAO 53, 1582–1585 (1984).
[CrossRef]

R. E. Prange and S. Fishman, “Experimental realizations of kicked quantum chaotic systems,” Phys. Rev. Lett. 63, 704–707 (1989); O. Agam, S. Fishman, and R. E. Prange, “Experimental realizations of quantum chaos in dielectric waveguides,” Phys. Rev. A 45, 6773–6802 (1992).
[CrossRef] [PubMed]

A. W. Snyder and S. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

S. Fishman, D. R. Grempel, and R. E. Prange, “Chaos, quantum recurrences, and Anderson localization,” Phys. Rev. Lett. 49, 509–512 (1982); D. R. Grempel, R. E. Prange, and S. Fishman, “Quantum dynamics of a nonintegrable system,” Phys. Rev. A 29, 1639–1647 (1984).
[CrossRef]

F. Haake, Quantum Signatures of Chaos (Springer, New York, 1991).

E. Ott, Chaos in Dynamical Systems (Cambridge U. Cambridge, UK, 1993).

P. Bergé, Y. Pomeau, and C. Vidal, Order Within Chaos (Hermann, Paris, 1984).

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983).

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion (Springer, New York, 1983).

M. C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, New York, 1990).

M. J. Giannoni, A. Voros, and J. Zinn-Justin, eds., “Chaos and quantum physics,” in Proceedings of the Les-Houches Summer School, Session LII, 1989 (North-Holland, Amsterdam, 1991).

G. L. Oppo, S. M. Barnett, E. Riis, and M. Wilkinson, eds., “Quantum dynamics of simple systems,” in Proceedings of the 44th Scottish Universities Summer School in Physics (Scottish Universities Summer School in Physics Publications and Institute of Physics, Bristol, UK, 1996).

G. Casati, B. V. Chirikov, F. M. Izrailev, and J. Ford, in Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Vol. 93 of Lecture Notes in Physics, G. Casati and J. Ford, eds. (Springer-Verlag, Berlin, 1979), p. 334.

J. Avron and B. Simon, “Singular continuous spectrum for a class of almost periodic Jacobi matrices,” Bull. Am. Math. Soc. 6, 81–85 (1982); “Almost periodic Schrödinger operators. II. The integrated density of states,” Duke Math. J. 50, 369–391 (1983).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Schematic description of the free-space kicked optical system with the array of phase gratings at equidistant locations. Some of the light paths are shown.

Fig. 2
Fig. 2

Numerical simulation of the evolution of the spatial frequency width σ as a function of the number of kicks N, for a classical system (without interference) that exhibits diffusive behavior (curve a) and for phase-disordered gratings (where we added for each grating N a random phase, φN, such that γn2 is replaced by γn2+φNn; curve b). A similar diffusive behavior was obtained for random γ. Confinement behavior is obtained for ordered gratings, far from resonance or antiresonance (with γ=0.74π; curve c) and near antiresonance (with γ=2.97π, or δ=-0.03; curve d). For all graphs, κ=5.94, which matches the experiment, and the starting state is n=0, resulting in n=0.

Fig. 3
Fig. 3

Experimental confined spatial frequency spectrum intensity after the eighth and ninth gratings for the ordered (filled points) and disordered (filled triangles) gratings. The initial direction is n=0, within experimental accuracy.

Fig. 4
Fig. 4

Experimental spatial frequency intensity width σ after each of the nine kicks (the symbols are the same as in Fig. 3).

Fig. 5
Fig. 5

(a) Typical numerically calculated confined spatial frequency spectrum |ψ|2, with a fir-tree shape after N=200 kicks, starting from a wave with n=0. We used κ=15 and δ=0.011. Zooms of tree head and one of the plateaus are given in (b) and (c), respectively.

Fig. 6
Fig. 6

Same as Fig. 5, but for κ=5.94 and δ=-3×(5/501)-0.03, corresponding to the experimental parameters.

Fig. 7
Fig. 7

Widths of the plateaus Δn0 for various values of κ and δ, determined by visual inspection of figures like Figs. 5 and 6 (circles), as compared with relation (8), with a=3/π (solid lines), for (a) δ=(5-1)/320 and variable κ and for (b) κ = 10 and variable δ.

Fig. 8
Fig. 8

Eigenstates of model (1) for κ=10 and δ=51/4096 (solid curves): (a) eigenstate of Eq. (1), centered near the origin, and eigenstates (5) of Eq. (4), with ν=0 and s=1 (dashed curve) and ν=54 and s=1/2 (dotted curve); (b) eigenstate of Eq. (1), centered near n=20, and eigenstates (5) of Eq. (4), with ν=22 and s=1 (dashed curve) and ν=54 and s=1/2 (dotted curve).

Fig. 9
Fig. 9

Experimental spatial frequency spectrum intensity after the eighth [(a), (b)] and ninth [(c), (d)] gratings for ordered [filled circles in (a) and (c)] and phase-disordered [filled triangles in (b) and (d)], gratings, as compared with the theoretical results from Eq. (1), near antiresonance, where δ=-0.03, showing localization [plus sign in (a) and (c)] or a spread spectrum away from antiresonance [minus sign in (b) and (d)], where γ=5-1, and when intensities, rather than field amplitudes, are used [diamonds in (b) and (d)], which gives results similar to the case in which the phase of the gratings is random. The results of the linear model (4), for τ=-0.18π (δ=-0.03, n0=3; squares), are presented as well.

Fig. 10
Fig. 10

Experimental spatial frequency intensity width σ after each of the nine gratings (filled circles), as compared with the theoretical results of Eq. (1) near antiresonance (plus signs), with γ=2.97π, and away from antiresonance (minus signs), where γ=5-1. Also shown is the result (squares) in the linear model with δ=-0.03, n0=3, which give s=1.92.

Equations (31)

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i ψz=-λ4π2ψx2+κ cos(kgx)Nδ(z-Nz0)ψ,
H=γ n^2+V(x, z),
U^KR=(exp-iγ n^2)exp[-iκ cos(kgx)].
U^LR=exp[-iκ cos(kg x)]exp[-i(τ+π)nˆ].
ψν(n)=exp[i(τ/2)(n-ν)]
×(-i)|n-ν|J|n-ν|κ2 sin[(τ+π)/2],
dAκ2πδdn,
Δn0κ2πδΔn0
Δn0aκ/δ,
U^TS=exp(-iγ2n^2)exp[-iκ cos(kgx)]×exp(-iγ2n^2)exp[+iκ cos(kgx)].
exp(-iγ n^2)=exp(-iπnˆ)exp(-iγ2n^2),
exp(-iπnˆ)exp[-iκ cos(kg x)]
=exp[-iκ cos(kgx-π)]exp(-iπnˆ)
U^KR2=exp(-iπnˆ)U^TSexp(-iπnˆ),
ψ(x, z=0)=nanexp(inkg x).
ψ(x, z)=nanexp(inkg x-iγn2).
ψ(x, z=NzT)=ψ(x, z=0),
ψx, z=(2N+1) zT2=ψx+λg2, z=0,
i ψt=τnˆ+κ cos yNδ(t-N)ψ,
t=z/z0,y=kg x.
Uˆ=exp(-iκ cos y)exp(-iτnˆ).
ψν(y, t)=exp(-iωνt)ϕν(y, t),
Uˆψν=exp(-iων)ψν.
ϕν(y)=1/2πexp[iφν(y)],
φν(y)=νy+lClνexp(ily).
-ων+φν(y)=-κ cos y+φν(y-τ),
ων=ντ mod 2π,
C±1ν=κ4iexp(±iτ/2)sin(τ/2),
φν(y)=νy-κ2sin(y+τ/2)sin(τ/2).
ϕν(y)=exp(iνy)mimJm-κ2 sin(τ/2)×exp{-im[(π-τ)/2-y]}
ϕν(n)=Jn-νκ2 sin(τ/2)(-1)n-νexp[i(n-ν)τ/2].

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