## Single beam mapping of nonlinear phase shift profiles in planar waveguides with an embedded mirror

Optics Express, Vol. 15, Issue 19, pp. 12068-12075 (2007)

http://dx.doi.org/10.1364/OE.15.012068

Acrobat PDF (449 KB)

### 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

## 1. Introduction, theoretical and numerical background

### 1.1. Introduction

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]

*i.e*. in the study of the interaction of spatial solitons with micro-structured inhomogeneities [7

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, and J. S. Aitchison, “Discrete Vector Solitons in Kerr Nonlinear
Waveguide Arrays,” Phys. Rev. Lett. **91**, 143907 (2003). [CrossRef] [PubMed]

### 1.2. Theoretical and numerical background

*ϕ*and

*β*in spatial soliton formation, Figs. 1 and 2 show solutions of the NLSE obtained from beam propagation method (BPM) simulations [11

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

*z*direction (Fig. 1(a)). The accumulated phase at the center of the beam (Fig. 1(b)) is linear with a slope of approximately unity in the dimensionless units of Fig. 1(b). Note that the slope

*difference*relative to Fig. 1(b) is shown in Fig. 1(d), and is indeed nonlinear. The positive slope corresponds to a self-focusing nonlinearity

*n*

_{2}>0, with a local increase of the propagation constant, associated with its penetration into the semi-infinite gap. Lateral Profiles of the output phases and propagation constants obtained from the above simulations are shown in Figs. 2(a),(b) for different input powers. Beyond a certain threshold power, a spatial breakup occurs that is accompanied by a breakup of the phase profile (Fig. 2(a)) and by the recovery of the linear propagation constant (Fig. 2(b)).

*β*is uniform and larger than

*β*

_{0}(Fig. 2(d)).

## 2. Optical setup and sample composition

### 2.1. Optical setup

*ϕ(x)*is then directly measured as it results in phase shifts of the imaging interference pattern. We stress that the interference fringe shift is exactly equal to the nonlinear phase only in the case of

*spatial*solitons (which are dispersive in the temporal domain) [2, 3], such as in AlGaAs waveguides. When

*spatiotemporal*compression is considered (

*i.e*. as self-focusing is observed in both space and time dimensions) [12

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]

### 2.2. Sample geometry

_{x}GaAs

_{1-x}waveguides have been used extensively for nonlinear optics applications [7

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, and J. S. Aitchison, “Discrete Vector Solitons in Kerr Nonlinear
Waveguide Arrays,” Phys. Rev. Lett. **91**, 143907 (2003). [CrossRef] [PubMed]

*x*=0.18 in the core layer and

*x*=0.24 in the clad layer give rise to a physical vertical refractive index difference of Δ

*n*=0.03 at

*λ*

_{0}=1.5

*µm*, with a core index of

*n*

_{0}=3.34. A 25

*µm*-wide shallow etching applied to the top clad (Fig. 4(a)) effectively decreases the mode area, and therefore the effective refractive index, of a beam confined to the core [8

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, and J. S. Aitchison, “Discrete Vector Solitons in Kerr Nonlinear
Waveguide Arrays,” Phys. Rev. Lett. **91**, 143907 (2003). [CrossRef] [PubMed]

## 3. Experimental results

### 3.1. Phase shift profile mapping

*µm*(to enable efficient coupling to the core layer, see Fig. 4(c)) and an input width of ≃100

*µm*at the beam’s waist. The formation of a spatial soliton as a function of the input peak power is shown in Fig. 5(a) for a coupling location that is far away from the mirror,

*i.e*. in an homogeneous region.

### 3.2. Observation of sharp phase gradients

*i.e*. with initial conditions in which the input phase front is not flat. An interesting phenomenon that we have observed in such cases was the occurrence of a series of nonlinearity-induced sharp phase shifts (larger than 2

*π*) at

*x*separations that are smaller than the fringe period, even when the excitation is in the weak perturbation regime. An example is shown in Fig. 6(a). Sharp nonlinear phase gradients, which are extended in

*x*, were observed as local “blurring” of the interference pattern around these regions, as shown in Fig. 6(b).

### 3.3. Strong perturbation regime

*p*>800

*µm*), the power-dependent characteristics of the interference pattern are substantially different.

*k*appear in the spectrum of the interference pattern in the intermediate and high power cases.

_{x}15. J. A. Giordmaine, “Mixing of light beams in
crystals,” Phys. Rev. Lett. **8**, 19–20
(1962). [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]

## 4. Conclusion

## Acknowledgments

## References and links

1. | T. Dauxois and M. Peyrard, |

2. | J. R. Taylor, |

3. | S. Trillo and W. E. Torruellas, |

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 |

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. |

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. |

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. |

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. |

9. | J. Meier, J. Hudock, D. Christodoulides, G. Stegeman, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Discrete Vector Solitons in Kerr Nonlinear
Waveguide Arrays,” Phys. Rev. Lett. |

10. | J. D. Jackson, |

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. |

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 |

14. | J. P. Gordon, “Theory of the soliton self-frequency
shift,” Opt. Lett. |

15. | J. A. Giordmaine, “Mixing of light beams in
crystals,” Phys. Rev. Lett. |

16. | G. Bartal, O. Manela, and M. Segev, “Spatial four wave mixing in nonlinear
periodic structures,” Phys. Rev. Lett. |

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 |

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. |

19. | S. Flach, V. Fleurov, A. V. Gorbach, and A. E. Miroshnichenko, “Resonant Light Scattering by Optical
Solitons,” Phys. Rev. Lett. |

20. | D. Mandelik, Y. Lahini, and Y. Silberberg, “Nonlinearly Induced Relaxation to the Ground
State in a Two-Level System,” Phys. Rev.
Lett. |

21. | A. Avidan, Y. Lahini, D. Mandelik, and Y. Silberberg, “Ground-state selection as a four-wave-mixing
process,” Phys. Rev. A |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(190.0190) Nonlinear optics : Nonlinear optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.4000) Optical devices : Microstructure fabrication

(230.7390) Optical devices : Waveguides, planar

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 2, 2007

Revised Manuscript: August 30, 2007

Manuscript Accepted: September 2, 2007

Published: September 6, 2007

**Citation**

Yoav Linzon, Yaron Shavit, Moshe Elazar, Roberto Morandotti, Maite Volatier-Ravat, Vincent Aimez, Richard Arès, and Shimshon Bar-Ad, "Single beam mapping of nonlinear phase shift profiles in planar waveguides with an embedded mirror," Opt. Express **15**, 12068-12075 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12068

Sort: Year | Journal | Reset

### References

- 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).
- 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]
- 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]
- 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]
- 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]
- 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.
- 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]
- 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. P. Gordon, "Theory of the soliton self-frequency shift," Opt. Lett. 11,662-664 (1986). [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]
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.