## Transverse power flow reversing of guided waves in extreme nonlinear metamaterials

Optics Express, Vol. 18, Issue 11, pp. 11911-11916 (2010)

http://dx.doi.org/10.1364/OE.18.011911

Acrobat PDF (525 KB)

### Abstract

We theoretically prove that electromagnetic beams propagating through a nonlinear cubic metamaterial can exhibit a power flow whose direction reverses its sign along the transverse profile. This effect is peculiar of the hitherto unexplored extreme nonlinear regime where the nonlinear response is comparable or even greater than the linear contribution, a condition achievable even at relatively small intensities. We propose a possible metamaterial structure able to support the extreme conditions where the polarization cubic nonlinear contribution does not act as a mere perturbation of the linear part.

© 2010 Optical Society of America

## 1. Introduction

*ε*and permeability

*μ*has triggered an intense research effort aimed at investigating novel regimes of linear electromagnetic propagation and suitable configurations have been devised for observing remarkable effects such as, for example, superlensing [1

1. J. B. Pendry, “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. **85**, 3966 (2000). [CrossRef] [PubMed]

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science **312**, 1780 (2006). [CrossRef] [PubMed]

3. J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter”, Opt. Lett. **22**, 475 (1997). [CrossRef] [PubMed]

4. N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials”, Science **317**, 1698 (2007). [CrossRef] [PubMed]

5. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials”, Phys. Rev. Lett. **91**, 037401 (2003). [CrossRef] [PubMed]

6. I. V. Shadrivov and Y. S. Kivshar, “Spatial solitons in nonlinear left-handed metamaterials”, J. Opt. A: Pure Appl. Opt. **7**, 68 (2005). [CrossRef]

7. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. **99**, 153901 (2007). [CrossRef] [PubMed]

8. M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized Nonlinear Schrdinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials”, Phys. Rev. Lett. **95**, 013902 (2005). [CrossRef] [PubMed]

## 2. Nonlinear guided waves

*e*

^{-iωt}, where

*ω*is the angular frequency) propagating through a nonlinear metamaterial characterized by the constitutive relations (holding for the field complex amplitudes)

*ε*> 0 and

*μ*> 0 are the linear permittivity and permeability, respectively, whereas

*χ*> 0 and 0 <

*γ*< 1 are the parameters characterizing the cubic defocusing nonlinear response. We focus our attention on transverse magnetic (TM) nonlinear guided waves propagating along the

*z*- axis of the form

*c*is the speed of light in vacuum) are dimensionless spatial coordinates,

*β*is a real dimensionless propagation constant and

*u*and

_{x}*u*are dimensionless electric field components. Substituting the fields of Eqs. (2) into Maxwell equations ∇ ×

_{z}**E**=

*iω*

**B**and ∇ ×

**H**= -

*iω*

**D**and using the constitutive relations of Eqs. (1) we get

*u*and

_{x}*u*are spatially even (

_{z}*u*(

_{x}*ξ*) =

*u*(-

_{x}*ξ*)) and odd (

*u*(

_{z}*ξ*) = -

*u*(-

_{z}*ξ*)), respectively and, as a consequence, we adopt the boundary conditions

*u*(0) =

_{x}*u*

_{x0},

*u*(0) = 0 and

_{z}*u*(+∞) =

_{x}*u*

_{x∞},

*u*(+∞) =

_{z}*u*

_{z∞}. Since

*u*(

_{x}*ξ*) and

*u*(

_{z}*ξ*) have to asymptotically approach two constant values, their first and second derivative vanish for

*ξ*→ +∞ so that, exploiting the boundary conditions, we require the right hand sides of Eqs. (3) to vanish at

*u*=

_{x}*u*

_{x∞}and

*u*=

_{z}*u*

_{z∞}. Therefore we obtain

*u*

^{2}

_{x∞}< 1/2 is a necessary condition for the existence of the considered nonlinear waves. In order to prove their existence, we exploit the fact that the system of Eqs. (3) is integrable [9

9. A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwells equations”, J. Opt. Soc. Am. B **22**, 1384 (2005). [CrossRef]

*u*(

_{x}*ξ*),

*u*(

_{z}*ξ*) of Eqs. (3). Evidently, after substituting the obtained

*β*into Eq. (4),

*F*has a stationary point at (

*u*

_{x∞},

*u*

_{z∞}) and the guided waves are represented by curves of constat

*F*in the plane (

*u*,

_{x}*u*) joining (

_{z}*u*

_{x0}, 0) to the stationary point. Therefore, requiring that

*F*has a saddle point at (

*u*

_{x∞},

*u*

_{z∞}) and exploiting the above necessary condition, we conclude that the considered nonlinear waves exist in the range

*γ*> 0. In addition the relation

*F*(

*u*

_{x0},0) =

*F*(

*u*

_{x∞},

*u*

_{z∞}) yields the possible values of

*u*(0) =

_{x}*u*

_{x0}corresponding to the asymptotical value

*u*(+∞) =

_{x}*u*

_{x∞}. In Fig. 1(a) and 1(b) we plot the profiles of

*u*and

_{x}*u*corresponding to different values of

_{z}*u*

_{x∞}, spanning the range of Eq. (5), for

*γ*= 0.5 obtained by numerically integrating Eqs. (3) with the above boundary conditions. The power flow carried by these waves is described by the time-average Poynting vector

*z*- axis. In Fig. 1(c) we plot the profiles of

*S*evaluated for the fields reported in Fig. 1(a) and 1(b), and we note that the sign of

_{z}*S*is not constant along the transverse profiles, a region where

_{z}*S*< 0 (red portion of the curves) existing around

_{z}*ξ*= 0. This reversing of the power flow along the transverse profile of the nonlinear guided waves is particularly evident from Fig. 1(d) where we draw the vector field

**S**on the plane (

*ξ*,

*ζ*) for one of the fields of Fig. 1(a) and 1(b). In order to physically grasp and discuss this unusual effect we recast Eq. (6) in the form

*D*along the wave profile while

_{x}*E*does not change its sign. This implies that, regardless the absolute sign of the fields, the overall effective dielectric response undergoes a sign reversing due to the fact that, in the first of Eq. (1), the nonlinear cubic term can be both smaller and greater than the linear part, depending on the local field strengths. It is worth noting that, although we have discussed this effect using the considered nonlinear guided waves admitting analytical treatment, the phenomenon is more general and holds for the wider class of TM fields of the form

_{x}**E**(

*x*,

*z*) =

*e*[

^{ikz}*E*(

_{x}*x*,

*z*)

**e**̂

_{x}+

*iE*(

_{z}*x*,

*z*)

**e**̂

_{z}] since, if ∣∂

_{z}

*E*∣ ≪

_{x}*k*∣

*E*∣ and ∣∂

_{x}_{z}

*E*∣ ≪

_{z}*k*∣

*E*∣ (i.e. the field manly propagates along the

_{z}*z*- axis) it is simple to obtain from Maxwell equations that

*ε*< 0 and

*μ*< 0 (with

*n*< 0). On the other hand, in our case,

*μ*> 0 and the sign of the power flow is not uniform being controlled through the field intensity. Note that such an extreme condition can be achieved when the field intensity ∣

*E*∣

^{2}is comparable or greater than

*ε*/

*χ*, so that, in standard materials where

*ε*is generally of the order of unity and

*χ*is very small (of the order of 10

^{-20}m

^{2}/V

^{2}in semiconductors [10]), the required intensity is so large to rule out the whole discussed phenomenology. However, if a metamaterial is employed where

*ε*can be chosen to be much smaller than unity, the intensity threshold can be reduced to the point of making the extreme nonlinear regime accessible even for intensities much smaller than those employed in standard nonlinear optics experiments.

## 3. Nonlinear layered medium supporting the extreme nonlinear regime

*ε*≪ 1) makes feasible intensities able to trigger the above linear-nonlinear competition, the main issue remains of finding a medium whose dielectric response is, in the considered intensity range, purely cubic. In fact, the first of Eqs. (1) is a power series expansion of the constitutive relation

*D*=

*D*(

*E*) in the field strength

*E*and therefore, if the third order is comparable with the first one, one generally has to consider higher order terms. In order to show that the discussed extreme nonlinear regime can effectively be achieved, consider the metamaterial structure reported in Fig. 2 consisting of alternate linear metamaterial and nonlinear medium layers, along the

*y*-axis, of thickness

*d*

_{1}and

*d*

_{2}respectively. The metamaterial is a negative dielectric (ND) whose constitutive relation is

**D**=

*ε*

_{0}

*ε*

_{1}

**E**(where

*Re*(

*ε*

_{1}) < 0) whereas the nonlinear medium (NL) is characterized by the constitutive relation

**D**=

*ε*

_{0}

*ε*

_{2}

**E**-

*ε*

_{0}

*χ*

_{2}[(

**E**·

**E**

^{*})

**E**+

*γ*(

**E**·

**E**)

**E**

^{*}], i.e. it is an isotropic defocusing (

*R*(

_{e}*ε*

_{2}) > 0,

*χ*

_{2}> 0) Kerr medium. The ND medium is generally a metal so that, in order to compensate losses, we suppose that the NL medium is also an active medium and that

*Im*(

*ε*

_{2}) can be tuned by adjusting the pumping efficiency (see example below) [11]. The media relative permeability are [

*μ*

_{1}and

*μ*

_{2}, respectively. If the spatial period

*d*=

*d*

_{1}+

*d*

_{2}is much smaller than the field vacuum wavelength 2

*πc*/

*ω*, the TM field propagating through the structure experiences the effective response described by Eqs. (1) and characterized by the spatially averaged parameters

*f*=

*d*

_{1}/(

*d*

_{1}+

*d*

_{2}) is the fraction of negative dielectric. From these relations, it is evident that suitable values of

*ε*

_{1},

*f*and

*Im*(

*ε*

_{2}) can be chosen so that 0 <

*Re*(

*ε*) ≪ 1 and ∣

*Im*(

*ε*)∣ ≪

*Re*(

*ε*) i.e. the overall medium effective response coincides with the one considered in present Letter. Most importantly, the medium is able to support the extreme nonlinear regime since if the field is such that ∣

*E*∣

^{2}~

*ε*

_{2}/

*χ*, at the same time one has that ∣

*E*∣

^{2}≪

*ε*

_{2}/

*χ*

_{2}. Therefore the nonlinear medium layers (NL) are in the presence of a field for which their response is purely cubic and, as a consequence, the overall averaged structure response is purely cubic as well.

*λ*= 0.810

*μm*propagating through a layered metamaterial structure for which

*ε*

_{1}= -28.79800 + 1.55000

*i*,

*μ*

_{1}= 1 and

*ε*

_{2}= 10.90000 - 0.56750

*i*,

*μ*

_{2}= 1,

*χ*

_{2}= 6.56 × 10

^{-18}

*m*

^{2}/

*V*

^{2},

*γ*= 0.5. For the considered wavelength

*λ*, the chosen

*ε*

_{1}coincides with the silver permittivity [12] whereas

*Re*(

*ε*

_{2}) and

*χ*

_{2}are the linear and nonlinear parameters characterizing the AlGaAs [13

13. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. **27**, 1296 (1991). [CrossRef]

*Im*(

*ε*

_{2}) can be attained simply by adjusting the pump laser intensity [7

7. Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. **99**, 153901 (2007). [CrossRef] [PubMed]

*f*= 0.2754, from Eqs. (8) we obtain the effective medium parameters

*ε*= 0.00235 + 0.00003

*i*,

*μ*= 1 and

*χ*= 4.76 × 10

^{-18}m

^{2}/

*V*

^{2}. The absorption coefficient of the considered effective medium is

*α*= (4

*π*/

*λ*)

*Im*(∞

*ε*) ≃ 4×10

^{-3}

*μm*

^{-1}so that the above power flow reversing effect can be observed for propagation distances up to the decay length 1/

*α*≃ 208

*μm*(note that this decay length can be made larger by improving the balance between losses and gain). Consider now the nonlinear guided wave whose power flow is reported in Fig. (2)b which is characterized by

*u*

_{x∞}= 0.65 and a transverse dimensionless width ∆

*ξ*≃ 2 (∆

*ξ*also coincides with the width of the transverse portion of the field where the Poynting vector is antiparallel to the propagation direction). The physical width of the considered wave is

*S*/

_{z}*S*

_{0}≃ 0.1 (see Fig. 2) so that the wave is characterized by the intensity

*S*= 0.1

_{z}*S*

_{0}≃ 0.3

*MW*/

*cm*

^{2}. It is worth stressing that the considered micron-sized confined wave is observable with an intensity (~

*MW*/

*cm*

^{2}) much smaller than that (~

*GW*/

*cm*

^{2}) required for exciting a spatial soliton (of the same width and at the same wavelength) propagating through a AlGaAs sample [15].

## 4. Conclusions

## References and links

1. | J. B. Pendry, “Negative Refraction Makes a Perfect Lens”, Phys. Rev. Lett. |

2. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields”, Science |

3. | J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter”, Opt. Lett. |

4. | N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials”, Science |

5. | A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials”, Phys. Rev. Lett. |

6. | I. V. Shadrivov and Y. S. Kivshar, “Spatial solitons in nonlinear left-handed metamaterials”, J. Opt. A: Pure Appl. Opt. |

7. | Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials”, Phys. Rev. Lett. |

8. | M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized Nonlinear Schrdinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials”, Phys. Rev. Lett. |

9. | A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwells equations”, J. Opt. Soc. Am. B |

10. | R. W. Boyd, |

11. | Note that the presented scheme can be improved by considering more than two basic layers constituents. This can simplify the identification of suitable active media (not coinciding with the nonlinear medium) to steer loss compensation. |

12. | E. D. Palik, |

13. | M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids”, IEEE J. Quantum Electron. |

14. | S. A. Ramakrishna and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain”, Phys. Rev. B |

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

**OCIS Codes**

(190.3270) Nonlinear optics : Kerr effect

(160.3918) Materials : Metamaterials

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 4, 2010

Revised Manuscript: April 8, 2010

Manuscript Accepted: April 13, 2010

Published: May 21, 2010

**Citation**

A. Ciattoni, C. Rizza, and E. Palange, "Transverse power flow reversing of guided waves in extreme nonlinear metamaterials," Opt. Express **18**, 11911-11916 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-11-11911

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### References

- J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. 85, 3966 (2000). [CrossRef] [PubMed]
- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling Electromagnetic Fields,” Science 312, 1780 (2006). [CrossRef] [PubMed]
- J. Takahara, S. Yamagishi, H. Taki, A. Morimoto, and T. Kobayashi, “Guiding of a one-dimensional optical beam with nanometer diameter,” Opt. Lett. 22, 475 (1997). [CrossRef] [PubMed]
- N. Engheta, “Circuits with Light at Nanoscales: Optical Nanocircuits Inspired by Metamaterials,” Science 317, 1698 (2007). [CrossRef] [PubMed]
- A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear Properties of Left-Handed Metamaterials,” Phys. Rev. Lett. 91, 037401 (2003). [CrossRef] [PubMed]
- I. V. Shadrivov, and Y. S. Kivshar, “Spatial solitons in nonlinear left-handed metamaterials,” J. Opt. A, Pure Appl. Opt. 7, 68 (2005). [CrossRef]
- Y. Liu, G. Bartal, D. A. Genov, and X. Zhang, “Subwavelength Discrete Solitons in Nonlinear Metamaterials,” Phys. Rev. Lett. 99, 153901 (2007). [CrossRef] [PubMed]
- M. Scalora, M. S. Syrchin, N. Akozbek, E. Y. Poliakov, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, and A. M. Zheltikov, “Generalized Nonlinear Schrödinger Equation for Dispersive Susceptibility and Permeability: Application to Negative Index Materials,” Phys. Rev. Lett. 95, 013902 (2005). [CrossRef] [PubMed]
- A. Ciattoni, B. Crosignani, P. Di Porto, and A. Yariv, “Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwell’s equations,” J. Opt. Soc. Am. B 22, 1384 (2005). [CrossRef]
- R. W. Boyd, Nonlinear Optics (Academic Press, New York, 1994).
- Note that the presented scheme can be improved by considering more than two basic layers constituents. This can simplify the identification of suitable active media (not coinciding with the nonlinear medium) to steer loss compensation.
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, San Diego, 1998).
- M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of Bound Electronic Nonlinear Refraction in Solids,” IEEE J. Quantum Electron. 27, 1296 (1991). [CrossRef]
- S. A. Ramakrishna, and J. B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
- Y. S. Kivshar, and G. P. Agrawal, Optical Solitons (Academic Press, San Diego, 2003).

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