## All-optical Landau-Zener tunneling in waveguide arrays

Optics Express, Vol. 14, Issue 5, pp. 2021-2026 (2006)

http://dx.doi.org/10.1364/OE.14.002021

Acrobat PDF (125 KB)

### Abstract

We investigate Landau-Zener all-optical tunneling in a voltage-controlled waveguide array realized in undoped nematic liquid crystals. From the material governing equations we derive the original Zener model and demonstrate a novel approach to Floquet-band tunneling.

© 2006 Optical Society of America

## 1. Introduction

1. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear arrays of coupled waveguides,” Opt. Lett. **13**, 794–796 (1988). [CrossRef] [PubMed]

5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. **83**, 4756–4760 (1999). [CrossRef]

11. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-Gap Structure of Waveguide Arrays and Excitation of Floquet-Bloch Solitons,” Phys. Rev. Lett. **90**, 053902–(4) (2003). [CrossRef]

12. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Optical Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” in *Proc. Top. Meet. On Nonlinear Guided Waves and their Applications*, ThD1 (Opt. Soc. Am., Dresden, Germany, 2005).

5. R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. **83**, 4756–4760 (1999). [CrossRef]

11. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-Gap Structure of Waveguide Arrays and Excitation of Floquet-Bloch Solitons,” Phys. Rev. Lett. **90**, 053902–(4) (2003). [CrossRef]

6. B. Wu and Q. Niu, “Nonlinear Landau-Zener tunneling,” Phys. Rev. A **61**, 023402–(5) (2000). [CrossRef]

12. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Optical Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” in *Proc. Top. Meet. On Nonlinear Guided Waves and their Applications*, ThD1 (Opt. Soc. Am., Dresden, Germany, 2005).

7. R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index,” Phys. Rev. Lett. **94**, 113904–(4) (2005). [CrossRef]

8. V. V. Konotop, P. G. Kevrekidis, and M. Salerno, “Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice,” Phys. Rev. A **72**, 023611–(5) (2005). [CrossRef]

12. H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Optical Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” in *Proc. Top. Meet. On Nonlinear Guided Waves and their Applications*, ThD1 (Opt. Soc. Am., Dresden, Germany, 2005).

*transition region*). [13

13. C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. London Ser. A **137**, 696–702 (1932). [CrossRef]

*connect*, in spite of their distinct properties and features. [10] In optical lattices, Landau-Zener tunneling can occur between diverse FB bands, provided that a non-adiabatic

*acceleration*is available (e.g. a refractive index gradient in the transition region). Light waves initially coupled to a specific band can therefore be transmitted to another band, [7

7. R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index,” Phys. Rev. Lett. **94**, 113904–(4) (2005). [CrossRef]

8. V. V. Konotop, P. G. Kevrekidis, and M. Salerno, “Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice,” Phys. Rev. A **72**, 023611–(5) (2005). [CrossRef]

*director*field. [14] In the presence of an applied electric field (either static or high-frequency or optical), NLC molecules can alter their mean angular orientation through the (induced) dipole-field reaction and tune their refractive index. This field-driven

*reorientational*process is at the basis of the electro-optic and all-optical response of NLC and entails fully tunable architectures. [14,15] Linear and nonlinear phenomena in periodic geometries, such as discrete diffraction, self-localized waves [16–18

16. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete light propagation and self-trapping in liquid crystals,” Opt. Express **13**, 1808–1815 (2005), http://www.opticsexpress.org/abstract.cfm?id=82980. [CrossRef] [PubMed]

19. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. **86**, 051112–(3) (2005). [CrossRef]

*pump*) is injected along the waveguides (Fig. 1), producing a nonlinear refractive index decrease and defining two transition regions (Fig. 1 right). Such all-optical acceleration can act on a second beam (the

*probe*) (Fig. 1 left) which, initially consisting of light coupled to a specific FB spectral band, crosses FB levels and exchanges energy with a lower FB band. In order to transfer light from an upper to a lower band, a negative index shift (Fig. 1 right) hence a self-defocusing nonlinearity need be exploited. Interestingly enough, after the transfer to a lower band, probe light is able to propagate through the lattice at a transverse velocity higher than the maximum defined by its initial FB band, undergoing angular steering and spatial switching (see Sec. 3).

## 2. Sample and NLC model

*d*) of the nematic liquid crystal PCB (

*n*

_{⊥}= 1.516,

*n*

_{∥}= 1.681 at λ = 1.064

*μ*m) sandwiched between two BK7 (n=1.507) glass plates; the top plate is coated with an array of parallel strips of Indium-Tin-Oxide (ITO) electrodes, the bottom one is uniformly covered by a grounded ITO film. Such electrodes, via the application of an external low-frequency (1kHz) voltage, can define a periodic set of identical channel waveguides evanescently coupled to one another, hence make a one dimensional dielectric lattice of constant Λ. When an electric field distribution is applied thru a bias

*V*, in fact, molecular reorientation takes place in the principal plane (

*x*,

*z*), thereby increasing the refractive index

^{2}

*θ*

_{0}(with

*x*–polarized guided mode and originating a periodic index modulation across the sample. [17

17. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. **29**, 1530–1532 (2004). [CrossRef] [PubMed]

*probe*, is ruled by:

*A*

_{probe}its electric field envelope. The steady-state distribution of the molecular director

*θ*

_{0}(

*x*,

*y*) is described by: [14,15]

*K*being the NLC mean elastic constant, Δ

*ε*

_{RF}=

*ε*

_{0}(

*ε*

_{2225}-

*ε*

_{⊥}) the low-frequency birefringence and

*E*

_{x}the

*x*-component of the external field. A self-defocusing nonlinearity is provided by the thermo-optic response of the liquid crystal. The temperature distribution

*T*=

*T*

_{0}+ Δ

*T*, with

*T*

_{0}the temperature of the bulk and Δ

*T*its pump-induced variation, is described by:

*I*

_{pump}the heat generated per unit time and space by the pump-intensity

*I*

_{pump}- By working near the NLC transition temperature [14] and acting on the bias

*V*in order to operate at a large director angle, [15] we could maximize the thermal response with respect to reorientation, making the former the dominant all-optical effect and the latter entirely negligible in our experiments (and model). The index change can be expressed as

**r**;

*V*;

*T*) =

*n*

_{0}(

**r**;

*V*;

*T*

_{0})+ Δ

*n*

_{e}(

**r**;

*V*; Δ

*T*), with

## 3. Theory

*x*,

*y*) = ε

_{0}(

*x*)+ Δε(

*x*,

*y*), with a periodic modulation Δε(

*x*,

*y*) = Δε(

*x*,

*y*+ Δ), and perform a factorization

*A*

_{probe}(

*x*,

*y*,

*z*) = Δ(

*y*,

*z*)

*B*(

*x*)exp(

*i*β

_{x}

*z*) with

*B*(

*x*)exp(

*i*β

_{x}

*z*) the solution of (1) for ε = ε

_{0}, [4] thus reducing Eq. (1) to the one-dimensional Schrödinger equation:

*θ*

_{0}(

*x*,

*y*) =

*θ*

_{r}+ γ(

*x*,

*y*), i.e. a mean value

*θ*

_{r}(≈

*π*/4) and a small periodic modulation γ(

*x*,

*y*) = γ(

*x*,

*y*+ Λ). By applying the method of strained parameters, assuming an applied field of the form

*F*(

*Y*) =

*F*(

*Y*+ Λ), [20

20. A. Fratalocchi and G. Assanto, “Discrete light localization in one dimensional nonlinear lattices with arbitrary non locality,” Phys. Rev. E **72**, 066608–(6) (2005). [CrossRef]

*x*than on

*y*, due to the strong asymmetry of the cell, much wider than thick. An input beam with an

*x*-waist comparable with the cell thickness

*d*does not undergo a strong non locality owing to the boundaries at a fixed temperature. Conversely, no boundaries are present along

*y*and the temperature distribution is free to widen. Therefore, we can conveniently take a local response along x (∂

^{2}/∂

*x*

^{2}≈ 0), factorize

*I*(

*x*,

*y*) =

*I*(

*x*)

*I*(

*y*) and expand

*I*(

*y*) in the eigenfunctions ϕ

_{v}= exp(

*i*v

*y*) of the homogeneous kernel ∂

^{2}ϕ

_{v}/∂

*y*

^{2}= -v

^{2}ϕ

_{v}:

*I*(

*y*) = ∫

*Ĩ*ϕ

_{v}dv. In the case of a small transition region (≈ Λ), we can further expand the thermal shift Δ

*T*at first-order. By substituting Eq. (5)–(6) into (4), after some lengthy but otherwise straightforward calculations we obtain the following dimension-less Schrödinger-like equation:

*W*=

*θ*

_{r}∫γ(

*x*)∥

*B*∣

^{2}dx,

*u*

_{-1}(

*Y*) the Heavyside function. We can then Fourier expand the periodic term

*V*(

*Y*) in a series

*V*= Σ

_{n}v

_{n}cos(2

*nY*) and the field ψ on a plane-wave basis ψ =

*a*

_{1}(

*Z*) exp(

*iKY*) +

*a*

_{2}(

*Z*) exp(-

*iKY*). After substituting and projecting on Eq. (7), we finally obtain the original Zener model: [13

13. C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. London Ser. A **137**, 696–702 (1932). [CrossRef]

*Z*) = - δ

*Z*. Eqs (8–9) predict tunneling between bands at the characteristic exponential rate exp(-

*π*

7. R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index,” Phys. Rev. Lett. **94**, 113904–(4) (2005). [CrossRef]

*z*) than initially imposed by excitation. To elucidate this concept, we numerically integrated Eq. (7) for

*V*(

*Y*) = sin

^{2}(

*Y*),

*V*

_{0}= 1 and δ = 0.5. A linear superposition of FB modes belonging to band 1, launched with the maximum transverse velocity (Fig. 3(b)), LZ-tunnels to band 2 as it travels through the accelerated region (Fig. 3(c), dotted line). Clearly, the angle of propagation increases beyond the maximum dictated by band 1 (Fig. 3(b)–(c)), unambiguously witnessing an LZ tunneling.

## 4. Experimental results

*μ*m and

*d*= 6

*μ*m were designed and realized with the nematic PCB (5CB). [21

21. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Optically-induced Zener tunneling in one dimensional lattices,” Opt. Lett. , to be published. [PubMed]

*μ*m and acquired images of the light scattered from the (

*y*,

*z*) plane with a microscope and a high resolution CCD camera. The pump was mechanically modulated and the CCD synchronized in order to acquire images of the (cw) probe only when

*I*

_{pump}= 0. The response of NLC is slow enough to permit the use of a standard chopper to implement this temporal separation. To characterize the nonlinear response, we performed a first series of experiments injecting just the probe in a single channel of the array. As its power was raised from

*P*

_{probe}= 1

*mW*to 6

*mW*, the refractive change reduced the transverse modulation causing a wider spreading of the beam in the plane (

*y*,

*z*) (Fig. 4(a)–(b)), demonstrating self-defocusing due to the dominant thermal response. Landau-Zener tunneling was then implemented by launching an intense pump in order to lower the NLC refractive index and induce transition regions around the accelerated portion of the array [21

21. A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Optically-induced Zener tunneling in one dimensional lattices,” Opt. Lett. , to be published. [PubMed]

_{y}= 15

*μ*m, with a diffraction length of about 900

*μ*m. The latter condition prevents any overlap with the probe after tunneling. A clear demonstration of all-optical LZ tunneling is visible in the photo sequence displayed in Fig. 5, showing the linear propagation of a signal beam (

*y*-waist ω

_{y}= 1.5 Λ, power

*P*

_{probe}= 1

*mW*) in the presence of the pump (dotted line) with ω

_{y}= 3Λ, 0 ≤

*P*

_{pump}≤ 30

*mW*). Light, initially coupled to band 1 at the maximum transverse velocity (for

*P*

_{pump}= 0), discretely diffracts owing to evanescent coupling. Once the pump is turned on (Fig. 5 dotted green line) no changes are appreciable until its power reaches

*P*

_{pump}= 25

*mW*(Fig. 5 dotted red line). Beyond this value, the nonlinear acceleration causes the signal to LZ-tunnel to band 2 and propagate at a larger angle in the observation plane. Such visible increment over the maximum imposed by the initial band (Fig. 5) unambiguously demonstrates that the probe has changed state, tunneling to a higher-order band in the spectrum. The LC transition region is smoother than the employed first order potential step (Fig. 3(b)), hence it reduces reflections. As apparent in Fig. 5, the tunneling rate is quite high as the residual light in band 1 can be hardly distinguished from the noisy background.

## 5. Conclusions

## References and links

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

2. | S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, “Channel optical waveguide directional couplers,” Appl. Phys. Lett. |

3. | A. A. Sukhorukov and Y. S. Kivshar, “Spatial optical solitons in waveguide arrays,” IEEE J. Quantum Electron. |

4. | Y. S. Kivshar and G. P. Agrawal, |

5. | R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, “Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,” Phys. Rev. Lett. |

6. | B. Wu and Q. Niu, “Nonlinear Landau-Zener tunneling,” Phys. Rev. A |

7. | R. Khomeriki and S. Ruffo, “Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index,” Phys. Rev. Lett. |

8. | V. V. Konotop, P. G. Kevrekidis, and M. Salerno, “Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice,” Phys. Rev. A |

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

10. | K. Sakoda, |

11. | D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-Gap Structure of Waveguide Arrays and Excitation of Floquet-Bloch Solitons,” Phys. Rev. Lett. |

12. | H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, “Optical Bloch oscillations and Zener tunneling in two-dimensional photonic lattices,” in |

13. | C. Zener, “Non-adiabatic crossing of energy levels,” Proc. R. Soc. London Ser. A |

14. | I. C. Khoo, |

15. | F. Simoni, |

16. | A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete light propagation and self-trapping in liquid crystals,” Opt. Express |

17. | A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Discrete propagation and spatial solitons in nematic liquid crystals,” Opt. Lett. |

18. | A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Optical multiband vector breathers in tunable waveguide arrays,” Opt. Lett. |

19. | A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “All-optical switching and beam steering in tunable waveguide arrays,” Appl. Phys. Lett. |

20. | A. Fratalocchi and G. Assanto, “Discrete light localization in one dimensional nonlinear lattices with arbitrary non locality,” Phys. Rev. E |

21. | A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, “Optically-induced Zener tunneling in one dimensional lattices,” Opt. Lett. , to be published. [PubMed] |

**OCIS Codes**

(160.3710) Materials : Liquid crystals

(260.0260) Physical optics : Physical optics

**ToC Category:**

Physical Optics

**History**

Original Manuscript: November 15, 2005

Revised Manuscript: February 15, 2006

Manuscript Accepted: February 18, 2006

Published: March 6, 2006

**Citation**

Andrea Fratalocchi and Gaetano Assanto, "All-optical Landau-Zener tunneling in waveguide arrays," Opt. Express **14**, 2021-2026 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-5-2021

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### References

- D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988). [CrossRef] [PubMed]
- S. Somekh, E. Garmire, A. Yariv, H. Garvin, and R. Hunsperger, "Channel optical waveguide directional couplers," Appl. Phys. Lett. 22, 46-48 (1972). [CrossRef]
- A. A. Sukhorukov and Y. S. Kivshar, "Spatial optical solitons in waveguide arrays," IEEE J. Quantum Electron. 39, 31-50 (2003). [CrossRef]
- Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from fibers to photonic crystals (Academic Press, San Diego, 2003).
- R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg, and Y. Silberberg, "Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations," Phys. Rev. Lett. 83, 4756-4760 (1999). [CrossRef]
- B. Wu and Q. Niu, "Nonlinear Landau-Zener tunneling," Phys. Rev. A 61, 023402 (2000). [CrossRef]
- R. Khomeriki and S. Ruffo, "Nonadiabatic Landau-Zener Tunneling in Waveguide Arrays with a Step in the Refractive Index," Phys. Rev. Lett. 94, 113904 (2005). [CrossRef]
- V. V. Konotop, P. G. Kevrekidis, and M. Salerno, "Landau-Zener tunneling of Bose-Einstein condensates in an optical lattice," Phys. Rev. A 72, 023611 (2005). [CrossRef]
- S. Trillo and W. E. Torruellas, Spatial Solitons (Springer-Verlag, Berlin, 2001).
- K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Berlin, 2001).
- D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, "Band-Gap Structure ofWaveguide Arrays and Excitation of Floquet-Bloch Solitons," Phys. Rev. Lett. 90, 053902 (2003). [CrossRef]
- H. Trompeter, W. Krolikowski, D. N. Neshev, A. S. Desyatnikov, A. A. Sukhorukov, Y. S. Kivshar, T. Pertsch, U. Peschel, and F. Lederer, "Optical Bloch oscillations and Zener tunneling in two-dimensional photonic lattices," in Proc. Top. Meet. On Nonlinear Guided Waves and their Applications, ThD1 (Opt. Soc. Am., Dresden, Germany, 2005).
- C. Zener, "Non-adiabatic crossing of energy levels," Proc. R. Soc. London Ser. A 137, 696-702 (1932). [CrossRef]
- I. C. Khoo, Liquid Crystals: Physical Properties and Nonlinear Optical Phenomena (Wiley, New York, 1995).
- F. Simoni, Nonlinear Optical Properties of Liquid Crystals (World Scientific, Singapore, 1997).
- A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete light propagation and self-trapping in liquid crystals," Opt. Express 13, 1808-1815 (2005), http://www.opticsexpress.org/abstract.cfm?id=82980. [CrossRef] [PubMed]
- A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Discrete propagation and spatial solitons in nematic liquid crystals," Opt. Lett. 29, 1530-1532 (2004). [CrossRef] [PubMed]
- A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optical multiband vector breathers in tunable waveguide arrays," Opt. Lett. 30, 174-176 (2005). [CrossRef] [PubMed]
- A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "All-optical switching and beam steering in tunable waveguide arrays," Appl. Phys. Lett. 86, 051112 (2005). [CrossRef]
- A. Fratalocchi and G. Assanto, "Discrete light localization in one dimensional nonlinear lattices with arbitrary non locality," Phys. Rev. E 72, 066608 (2005). [CrossRef]
- A. Fratalocchi, G. Assanto, K. A. Brzdakiewicz, and M. A. Karpierz, "Optically-induced Zener tunneling in one dimensional lattices," Opt. Lett., to be published. [PubMed]

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