## Diffraction management of focused light beams in optical lattices with a quadratic frequency modulation

Optics Express, Vol. 13, Issue 11, pp. 4244-4249 (2005)

http://dx.doi.org/10.1364/OPEX.13.004244

Acrobat PDF (598 KB)

### Abstract

We reveal that the effective diffraction experienced by light beams launched along the central guiding channel of optical lattice with a quadratic frequency modulation can be tuned in strength and sign. Complete suppression of the linear diffraction in the broad band of spatial frequencies is shown to be possible, thus profoundly affecting properties of nonlinear self-sustained beams as well. In particular, we report on the properties of a new class of stable solitons supported by such lattices in defocusing media.

© 2005 Optical Society of America

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides” Opt. Lett. **13**, 794 (1988). [CrossRef] [PubMed]

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**, 1863 (2000). [CrossRef] [PubMed]

3. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices” Nature **424**, 817 (2003). [CrossRef] [PubMed]

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**, 1863 (2000). [CrossRef] [PubMed]

4. J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. **29**, 268 (2004). [CrossRef] [PubMed]

5. T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. **29**, 468 (2004). [CrossRef] [PubMed]

6. A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. **92**, 093901 (2004). [CrossRef] [PubMed]

7. O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Grating-mediated waveguiding,” Phys. Rev. Lett. **93**, 103902 (2004). [CrossRef] [PubMed]

8. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices” Phys. Rev. E. **66**, 046602 (2002). [CrossRef]

13. O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature **433**, 500 (2005). [CrossRef] [PubMed]

8. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices” Phys. Rev. E. **66**, 046602 (2002). [CrossRef]

15. Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express **12**, 2831 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2831 [CrossRef] [PubMed]

*η*axis, described by the nonlinear Schrödinger equation for dimensionless complex field amplitude

*q*:

*ξ*and transverse η coordinates are scaled to the diffraction length and the input beam width, respectively. The parameter

*p*is proportional to the depth of refractive index modulation, while the function

*R*(

*η*)=cos[Ωη(1+

*αη*

^{2})] stands for the transverse profile of refractive index, where Ω is the carrying spatial frequency and parameter 0≤

*α*<1 characterizes the frequency modulation rate. Parameter

*σ*=±1 defines the nonlinearity sign (focusing/defocusing). Further we assume that the refractive index modulation depth is small compared to the unperturbed index and is of the order of the nonlinear contribution. It should be pointed out that one can use either technique of Fourier-transform synthesis or computer generated holograms for optical induction of the desired lattice profiles as well as their direct technological fabrication. Notice that Eq. (1) conserves the total energy flow

*U*and Hamiltonian

*H*

*σ*=0) Eq. (1) can be reduced to the following equation

*η*=0. The possibility of broadband diffraction control of such beams in the lattice with a quadratic frequency modulation (FM) follows from analysis of dispersion relations. The dispersion relations can be obtained explicitly upon substitution of spectrum for the symmetric pairs of plane waves

*q*(

*η*,ξ)=[exp(

*ikη*)+exp(-

*ikη*)]exp(

*ibξ*) having frequencies ±

*k*and propagation constants

*b*into Eq. (3):

*k*, and the last one describes the impact of the lattice due to the Bragg-type scattering. The sign of diffraction and its strength is dictated by the quantity

*d*

^{2}

*b/dk*

^{2}[2

2. H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. **85**, 1863 (2000). [CrossRef] [PubMed]

6. A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. **92**, 093901 (2004). [CrossRef] [PubMed]

16. E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E **62**, 2882 (2000). [CrossRef]

17. R. E. Slusher, B. J. Eggleton, T. A. Strasser, and C. M. de Sterke, “Nonlinear pulse reflection from chirped fiber gratings,” Opt. Express **3**, 465 (1998),
http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-11-465 [CrossRef] [PubMed]

*(0)+2*

_{p}R_{k}*(2*

_{p}R_{k}*k*) in Eq. (4).

*α*≪1 a first maximum of

*R*(

_{k}*k*) is achieved at

*k*≈Ω, and it shifts to the high-frequency region with increase of FM rate α. Notice almost parabolic behavior of

*R*(

_{k}*k*) dependence in a frequency interval Ω>

*k*>-Ω. The dispersion curves

*b*(

*k*) are depicted in Fig. 1(b) for different values of the refractive index modulation depth

*p*.

*d*

^{2}

*b/dk*

^{2}, including reversal of its sign, for a broad frequency band

*k*∈ [-Ω,Ω]. The critical lattice depth that corresponds to the case of zero diffraction, i.e.,

*d*

^{2}

*b/dk*

^{2}=0, is given by

*p*

_{cr}={8Ω(3

*α*Ω)

^{-3/2}Ai[Ω(3

*α*Ω)

^{-1/3}]}

^{-1}and is depicted in Fig. 1(c) for different

*α*. For

*p*>

*p*

_{cr}the “positive” diffraction is replaced by the “negative” one. In the spatial domain this is accompanied by a qualitative change of the phase front shape from convex to concave. Notice, that the possibility to control diffraction sign in the broad frequency band [-Ω,Ω] with FM lattices is advantageous in comparison with the diffraction management in harmonic lattices, where zero diffraction can be achieved only for one particular frequency

*k*and, therefore, diffraction can be suppressed only for broad beams with the correspondingly narrow spectrum.

*S*(

*k*) can be characterized by the integral factor

*q*(

*η*,ξ=0)=

*q*

_{0}sech(

*η*), with

*S*(

*k*)=

*S*

_{0}sech

^{2}(

*πk*/2). The existence of parameter range where diffraction is remarkably suppressed (

*C*≪1) is readily apparent. These analytical results are confirmed by the direct numerical integration of Eq. (1). The propagation of such beam in the harmonic lattice is accompanied by strong diffraction (Fig. 2(a)), while in the lattice with a suitable quadratic FM the similar beam experiences only small reshaping. Thus, quadratic modulation of the lattice frequency offers a powerful tool for diffraction control of focused light beams.

*η*=0. It is especially important in the case of focusing media where reduced diffraction requires smaller light intensities for its compensation. We have found soliton profiles from Eq. (1) in the form

*q*(

*η,ξ*)=

*w*(

*η*)exp(

*ibξ*), where

*w*(

*η*) is the real function and

*b*is the propagation constant. The properties of the ground-state soliton solutions supported by FM lattice in focusing media (

*σ*=1) are summarized in Fig. 3. With growth of the propagation constant, the soliton gets narrower, while its peak amplitude increases (Fig. 3(a)). There exists a lower cutoff on propagation constant for soliton existence. Close to this cutoff, the soliton transforms into a linear guided mode. The presence of linear localized guided modes even for

*α*≪1 clearly distinguishes lattices with quadratic FM from their harmonic counterparts that can support in the linear regime only periodic Bloch waves. Consequently, the width

*D*of soliton supported by FM lattice does not diverge when its energy flow

*U*→0, as it occurs in the harmonic lattices (Fig. 3(b)). The propagation constant cutoff

*b*

_{co}increases monotonically with growth of lattice depth

*p*, and drops off with increase of lattice frequency Ω (Fig. 3(c)). Growth of α causes only slight diminishing of cutoff value. A comprehensive linear stability analysis revealed that solitons supported by FM lattices with focusing nonlinearity are stable in the entire domain of their existence.

*σ*=-1), we discovered that lattices with quadratic FM can support a specific type of solitons that have no analogs in harmonic lattices. Their properties are summarized in Fig. 4. In contrast to the usual gap solitons (Fig. 4(d)) the field of soliton supported by FM lattice does not change its sign (Fig. 4(a)). Such solitons exist in the propagation constants interval

*b*∈ (0,

*b*

_{co}), where upper cutoff

*b*

_{co}for existence coincides with lower cutoff for solitons in focusing media. At

*b*→

*b*

_{co}such solitons also transform into linear guided modes of FM lattice, thus their width remains finite in contrast to gap solitons that become delocalized in both lower and upper cutoffs. At b→0 the soliton width diverges as shown in Fig. 4(b) and it acquires the form of a bright peak superimposed on the pedestal. The energy flow is a monotonically decreasing function of propagation constant (Fig. 4(c)). Linear stability analysis showed that solitons supported by defocusing FM lattices are stable in the entire domain of their existence. This result emphasizes the potential of the concept put forward with the quadratic FM lattices – the competition between a negative diffraction and defocusing nonlinearity can result in formation of new types of stable solitons.

## Acknowledgments

## References and links

1. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides” Opt. Lett. |

2. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Diffraction management,” Phys. Rev. Lett. |

3. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behavior in linear and nonlinear waveguide lattices” Nature |

4. | J. Hudock, N. K. Efremidis, and D. N. Christodoulides, “Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,” Opt. Lett. |

5. | T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, “Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,” Opt. Lett. |

6. | A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. |

7. | O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Grating-mediated waveguiding,” Phys. Rev. Lett. |

8. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive optically induced photonic lattices” Phys. Rev. E. |

9. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in optically induced real time waveguide arrays,” Phys. Rev. Lett. |

10. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices” Nature |

11. | D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically induced gratings” Opt. Lett. |

12. | H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, “Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices” Phys. Rev. Lett. |

13. | O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, “Observation of random-phase lattice solitons,” Nature |

14. | Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, “Spatial soliton switching in quasi-continuous optical arrays” Opt. Lett. |

15. | Y. V. Kartashov, V. A. Vysloukh, and L. Torner, “Soliton trains in photonic lattices,” Opt. Express |

16. | E. N. Tsoy and C. M. de Sterke, “Propagation of nonlinear pulses in chirped fiber gratings,” Phys. Rev. E |

17. | R. E. Slusher, B. J. Eggleton, T. A. Strasser, and C. M. de Sterke, “Nonlinear pulse reflection from chirped fiber gratings,” Opt. Express |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: May 2, 2005

Revised Manuscript: May 20, 2005

Published: May 30, 2005

**Citation**

Yaroslav Kartashov, Lluis Torner, and Victor Vysloukh, "Diffraction management of focused light beams in optical lattices with a quadratic frequency modulation," Opt. Express **13**, 4244-4249 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4244

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### References

- D. N. Christodoulides and R. I. Joseph, �??Discrete self-focusing in nonlinear arrays of coupled waveguides�?? Opt. Lett. 13, 794 (1988). [CrossRef] [PubMed]
- H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, �??Diffraction management,�?? Phys. Rev. Lett. 85, 1863 (2000). [CrossRef] [PubMed]
- D. N. Christodoulides, F. Lederer, and Y. Silberberg, �??Discretizing light behavior in linear and nonlinear waveguide lattices�?? Nature 424, 817 (2003). [CrossRef] [PubMed]
- J. Hudock, N. K. Efremidis, and D. N. Christodoulides, �??Anisotropic diffraction and elliptic discrete solitons in two-dimensional waveguide arrays,�?? Opt. Lett. 29, 268 (2004). [CrossRef] [PubMed]
- T. Pertsch, U. Peschel, F. Lederer, J. Burghoff, M. Will, S. Nolte, and A. Tunnermann, �??Discrete diffraction in two-dimensional arrays of coupled waveguides in silica,�?? Opt. Lett. 29, 468 (2004). [CrossRef] [PubMed]
- A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, �??Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,�?? Phys. Rev. Lett. 92, 093901 (2004). [CrossRef] [PubMed]
- O. Cohen, B. Freedman, J. W. Fleischer, M. Segev, and D. N. Christodoulides, �??Grating-mediated waveguiding,�?? Phys. Rev. Lett. 93, 103902 (2004). [CrossRef] [PubMed]
- N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, �??Discrete solitons in photorefractive optically induced photonic lattices�?? Phys. Rev. E. 66, 046602 (2002). [CrossRef]
- J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, �??Observation of discrete solitons in optically induced real time waveguide arrays,�?? Phys. Rev. Lett. 90, 023902 (2003). [CrossRef] [PubMed]
- J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, �??Observation of two-dimensional discrete solitons in optically induced nonlinear photonic lattices�?? Nature 422, 147 (2003). [CrossRef] [PubMed]
- D. Neshev, E. Ostrovskaya, Y. Kivshar, and W. Krolikowski, �??Spatial solitons in optically induced gratings�?? Opt. Lett. 28, 710 (2003). [CrossRef] [PubMed]
- H. Martin, E. D. Eugenieva, Z. Chen, and D. N. Christodoulides, �??Discrete solitons and soliton-induced dislocations in partially coherent photonic lattices�?? Phys. Rev. Lett. 92, 123902 (2004). [CrossRef] [PubMed]
- O. Cohen, G. Bartal, H. Buljan, T. Carmon, J. W. Fleischer, M. Segev, and D. N. Christodoulides, �??Observation of random-phase lattice solitons,�?? Nature 433, 500 (2005). [CrossRef] [PubMed]
- Y. V. Kartashov, A. S. Zelenina, L. Torner, and V. A. Vysloukh, �??Spatial soliton switching in quasi-continuous optical arrays�?? Opt. Lett. 29, 766 (2004). [CrossRef] [PubMed]
- Y. V. Kartashov, V. A. Vysloukh, and L. Torner, �??Soliton trains in photonic lattices,�?? Opt. Express 12, 2831 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2831">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-2831</a> [CrossRef] [PubMed]
- E. N. Tsoy and C. M. de Sterke, �??Propagation of nonlinear pulses in chirped fiber gratings,�?? Phys. Rev. E 62, 2882 (2000). [CrossRef]
- R. E. Slusher, B. J. Eggleton, T. A. Strasser, and C. M. de Sterke, �??Nonlinear pulse reflection from chirped fiber gratings,�?? Opt. Express 3, 465 (1998), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-11-465">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-3-11-465</a> [CrossRef] [PubMed]

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