Abstract

We demonstrate a technique for a single shot mapping of nonlinear phase shift profiles in spatial solitons that are formed during short pulse propagation through one-dimensional slab AlGaAs waveguides, in the presence of a focusing Kerr nonlinearity. The technique uses a single beam and relies on the introduction of a lithographically etched reflective planar mirror surface positioned in proximity to the beam’s input position. Using this setup we demonstrate nonlinearity-induced sharp lateral phase variations for certain initial conditions, and creation of higher spatial harmonics when the beam is in close proximity to the mirror.

© 2007 Optical Society of America

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References

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  1. See, for example: T. Dauxois and M. Peyrard, Physics of Solitons (Cambridge University Press, London, 2006).
  2. See, for example: J. R. Taylor, Optical Solitons - Theory and Experiment (Cambridge University Press, New York, 1999).
  3. See, for example: S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).
  4. O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68,026610 (2003).
    [CrossRef]
  5. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
    [CrossRef] [PubMed]
  6. J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo,M. Sorel, and J. S. Aitchison, "Beam interactions with a blocker soliton in one-dimensional arrays," Opt. Lett. 30,1027-1029 (2005).
    [CrossRef] [PubMed]
  7. R. Jin, C. L. Chuang, H. M. Gibbs, S.W. Koch, J. N. Polky, and G. A. Pubanz, "Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers," Appl. Phys. Lett. 53,1792-1793 (1988).
    [CrossRef]
  8. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81,3383-3386 (1988).
    [CrossRef]
  9. J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, "Discrete Vector Solitons in Kerr Nonlinear Waveguide Arrays," Phys. Rev. Lett. 91,143907 (2003).
    [CrossRef] [PubMed]
  10. See, for example: J. D. Jackson, Classical Electrodynamics, 3rd edition (JohnWiley and Sons, New York, 1998), pp. 352-366, 378-389.
  11. The BPM code is available for free at http://www.freeBPM.com.
  12. H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87,043902 (2001).
    [CrossRef] [PubMed]
  13. Y. Linzon, I. Ilsar, D. Cheskis, R. Morandotti, J. S. Aitchison, and S. Bar-Ad, "Near-field imaging of nonlinear pulse propagation in planar silica waveguides," Phys. Rev. E 72,066607 (2005).
    [CrossRef]
  14. J. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11,662-664 (1986).
    [CrossRef] [PubMed]
  15. J. A. Giordmaine, "Mixing of light beams in crystals," Phys. Rev. Lett. 8,19-20 (1962).
    [CrossRef]
  16. G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97,073906 (2006).
    [CrossRef] [PubMed]
  17. D. V. Skryabin and A. V. Yulin, "Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers," Phys. Rev. E 72,016619 (2005).
    [CrossRef]
  18. A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, and P. Russell, "Interaction of an Optical Soliton with a Dispersive Wave," Phys. Rev. Lett. 95,213902 (2005).
    [CrossRef] [PubMed]
  19. S. Flach, V. Fleurov, A. V. Gorbach, and A. E. Miroshnichenko, "Resonant Light Scattering by Optical Solitons," Phys. Rev. Lett. 95,023901 (2005).
    [CrossRef] [PubMed]
  20. D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly Induced Relaxation to the Ground State in a Two-Level System," Phys. Rev. Lett. 95,073902 (2005).
    [CrossRef] [PubMed]
  21. A. Avidan, Y. Lahini, D. Mandelik, and Y. Silberberg, "Ground-state selection as a four-wave-mixing process," Phys. Rev. A 73,063811 (2006).
    [CrossRef]

2006 (2)

G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97,073906 (2006).
[CrossRef] [PubMed]

A. Avidan, Y. Lahini, D. Mandelik, and Y. Silberberg, "Ground-state selection as a four-wave-mixing process," Phys. Rev. A 73,063811 (2006).
[CrossRef]

2005 (6)

Y. Linzon, I. Ilsar, D. Cheskis, R. Morandotti, J. S. Aitchison, and S. Bar-Ad, "Near-field imaging of nonlinear pulse propagation in planar silica waveguides," Phys. Rev. E 72,066607 (2005).
[CrossRef]

J. Meier, G. I. Stegeman, D. N. Christodoulides, Y. Silberberg, R. Morandotti, H. Yang, G. Salamo,M. Sorel, and J. S. Aitchison, "Beam interactions with a blocker soliton in one-dimensional arrays," Opt. Lett. 30,1027-1029 (2005).
[CrossRef] [PubMed]

D. V. Skryabin and A. V. Yulin, "Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers," Phys. Rev. E 72,016619 (2005).
[CrossRef]

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, and P. Russell, "Interaction of an Optical Soliton with a Dispersive Wave," Phys. Rev. Lett. 95,213902 (2005).
[CrossRef] [PubMed]

S. Flach, V. Fleurov, A. V. Gorbach, and A. E. Miroshnichenko, "Resonant Light Scattering by Optical Solitons," Phys. Rev. Lett. 95,023901 (2005).
[CrossRef] [PubMed]

D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly Induced Relaxation to the Ground State in a Two-Level System," Phys. Rev. Lett. 95,073902 (2005).
[CrossRef] [PubMed]

2003 (3)

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, "Discrete Vector Solitons in Kerr Nonlinear Waveguide Arrays," Phys. Rev. Lett. 91,143907 (2003).
[CrossRef] [PubMed]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68,026610 (2003).
[CrossRef]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

2001 (1)

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87,043902 (2001).
[CrossRef] [PubMed]

1988 (2)

R. Jin, C. L. Chuang, H. M. Gibbs, S.W. Koch, J. N. Polky, and G. A. Pubanz, "Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers," Appl. Phys. Lett. 53,1792-1793 (1988).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81,3383-3386 (1988).
[CrossRef]

1986 (1)

1962 (1)

J. A. Giordmaine, "Mixing of light beams in crystals," Phys. Rev. Lett. 8,19-20 (1962).
[CrossRef]

Appl. Phys. Lett. (1)

R. Jin, C. L. Chuang, H. M. Gibbs, S.W. Koch, J. N. Polky, and G. A. Pubanz, "Picosecond all-optical switching in single-mode GaAs/AlGaAs strip-loaded nonlinear directional couplers," Appl. Phys. Lett. 53,1792-1793 (1988).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

A. Avidan, Y. Lahini, D. Mandelik, and Y. Silberberg, "Ground-state selection as a four-wave-mixing process," Phys. Rev. A 73,063811 (2006).
[CrossRef]

Phys. Rev. E (3)

Y. Linzon, I. Ilsar, D. Cheskis, R. Morandotti, J. S. Aitchison, and S. Bar-Ad, "Near-field imaging of nonlinear pulse propagation in planar silica waveguides," Phys. Rev. E 72,066607 (2005).
[CrossRef]

O. Jedrkiewicz, J. Trull, G. Valiulis, A. Piskarskas, C. Conti, S. Trillo, and P. Di Trapani, "Nonlinear X waves in second-harmonic generation: Experimental results," Phys. Rev. E 68,026610 (2003).
[CrossRef]

D. V. Skryabin and A. V. Yulin, "Theory of generation of new frequencies by mixing of solitons and dispersive waves in optical fibers," Phys. Rev. E 72,016619 (2005).
[CrossRef]

Phys. Rev. Lett. (9)

A. Efimov, A. V. Yulin, D. V. Skryabin, J. C. Knight, N. Joly, F. G. Omenetto, A. J. Taylor, and P. Russell, "Interaction of an Optical Soliton with a Dispersive Wave," Phys. Rev. Lett. 95,213902 (2005).
[CrossRef] [PubMed]

S. Flach, V. Fleurov, A. V. Gorbach, and A. E. Miroshnichenko, "Resonant Light Scattering by Optical Solitons," Phys. Rev. Lett. 95,023901 (2005).
[CrossRef] [PubMed]

D. Mandelik, Y. Lahini, and Y. Silberberg, "Nonlinearly Induced Relaxation to the Ground State in a Two-Level System," Phys. Rev. Lett. 95,073902 (2005).
[CrossRef] [PubMed]

P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, "Spontaneously generated X-shaped light bullets," Phys. Rev. Lett. 91,093904 (2003).
[CrossRef] [PubMed]

J. A. Giordmaine, "Mixing of light beams in crystals," Phys. Rev. Lett. 8,19-20 (1962).
[CrossRef]

G. Bartal, O. Manela, and M. Segev, "Spatial four wave mixing in nonlinear periodic structures," Phys. Rev. Lett. 97,073906 (2006).
[CrossRef] [PubMed]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81,3383-3386 (1988).
[CrossRef]

J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, J. S. Aitchison, "Discrete Vector Solitons in Kerr Nonlinear Waveguide Arrays," Phys. Rev. Lett. 91,143907 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, S. Bar-Ad, D. Ross, and J. S. Aitchison, "Kerr spatiotemporal self-focusing in a planar glass waveguide," Phys. Rev. Lett. 87,043902 (2001).
[CrossRef] [PubMed]

Other (5)

See, for example: J. D. Jackson, Classical Electrodynamics, 3rd edition (JohnWiley and Sons, New York, 1998), pp. 352-366, 378-389.

The BPM code is available for free at http://www.freeBPM.com.

See, for example: T. Dauxois and M. Peyrard, Physics of Solitons (Cambridge University Press, London, 2006).

See, for example: J. R. Taylor, Optical Solitons - Theory and Experiment (Cambridge University Press, New York, 1999).

See, for example: S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).

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Figures (7)

Fig. 1.
Fig. 1.

(a), (b) Low power (20 W) and (c), (d) high power (5.2 kW) simulations of Eq. (1) at λ0=1.5 µm for a 100 µm-wide input Gaussian wave packet, with flat phase-front initial conditions, over a propagation length of z=6.5 mm in a planar AlGaAs waveguide. (a), (c): Intensity evolution (|E(x,z)|2). (b) Phase evolution ϕ(z) (estimated at a central x position indicated by the dashed lines in (a),(c)). (d) Phase difference Δϕ(z) between the nonlinear propagation (c) and the linear propagation (a),(b), at the same x position.

Fig. 2.
Fig. 2.

Simulated phase profiles (left panels, (a) and (c)) and propagation constant profiles (right panels, (b) and (d)) at the output following linear (green lines) and nonlinear (blue/red lines) propagation over a 6.5 mm-long 2D AlGaAs waveguide. The input beam intensity profile is Gaussian with a width of 100 µm in the top panels ((a) and (b)), while it has the form a square Hyperbolic Secant with a width of 13 µm in the bottom panels ((c) and (d)). In all cases, the excitation wavelength is λ 0=1.5 µm.

Fig. 3.
Fig. 3.

Possible nonlinear phase shift measurement setups. (a) A two-beam setup involving remote coupling with different lenses, following the two beams overlapping on a nonlinear crystal. (b) A two-beam setup involving close coupling with the same lens following output facet imaging. (c) and (d): A single beam setup with a plane mirror surface embedded in the waveguide and output facet imaging: (c) Low power and (d) high power (soliton) beams.

Fig. 4.
Fig. 4.

(a),(c): Vertical sample cross-sections with local barriers formed by (a) a shallow 0.6 µm etching, and (c) a deep 1.6 µm etching. (b),(d): Output facet images as a function of the sample position p (along the x direction, with respect to the input beam) in samples with a barrier that is (b) perturbative (as in (a)) and (d) reflective (as in (b)). Both samples have the same length (6.5 mm), and are excited by a low power (100 W) 100 µm-wide Gaussian input beam. The left boundaries of the barriers are indicated by the dashed white lines in (b),(d). The same sample position scale p is used in subsequent figures.

Fig. 5.
Fig. 5.

Experimental results of phase shift profile estimation in the weak perturbation regime: (a), (b) Formation of a spatial soliton as a function of input peak power in (a) an homogeneous region (p≃0 in Fig. 4(d)), and (b) in proximity to the mirror (p≃400 µm). (c) Low power (green) and high power (orange) interference fringe patterns of the output beam. (d) The soliton’s lateral phase shift profile as extracted from (c).

Fig. 6.
Fig. 6.

Experimental results of sharp phase gradients with a non-flat input phase front in the weak perturbation regime: (a) Interference fringe patterns exhibiting sharp phase gradients (indicated by the red arrows). (b) An example of local phase “blurring” as a function of the power, induced by sharp nonlinear phase gradients.

Fig. 7.
Fig. 7.

Experimental results in the strong perturbation regime (p≃1200 µm). (a) Low power (green), intermediate power (orange) and high power (cyan) interference fringe patterns of the output beam. (b) Discrete Fourier transforms of the data in (a) showing the transverse spatial frequency content of the light emerging from the waveguide.

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