## Focusing properties of Gaussian beams by a slab of Kerr-type left-handed metamaterial

Optics Express, Vol. 16, Issue 7, pp. 4774-4784 (2008)

http://dx.doi.org/10.1364/OE.16.004774

Acrobat PDF (514 KB)

### Abstract

Kerr-type left-handed metamaterial (LHM) slab is proved to have an effect of focusing paraxial Gaussian beams and changing their waist radius, as conventional lens can do. The expressions for the focusing distance and the spot radius at the focal point are derived by the variational approach. We show that the incident Gaussian beams can be compressed or expanded by a single Kerr LHM slab, according to the sign of the Kerr nonlinearity and the divergence of the incident beam. Especially, it is demonstrated the focusing properties are significantly tuned by the slab thickness, the beam power and the divergence of the incident Gaussian beam.

© 2008 Optical Society of America

## 1. Introduction

1. J. B. Pendry, “Negative Refraction Makes a Perfect Lens,” Phys. Rev. Lett. **85**, 3966–3969 (2000). [CrossRef] [PubMed]

2. J. B. Pendry and D. R. Smith, “Reversing light with negative refraction,” Phys. Today **57**(6), 37–44 (2004). [CrossRef]

3. N. Garcia and M. Nieto-Vesperinas, “Left-handed materials do not make a perfect lens” Phys. Rev. Lett. **88**, 207403(2002). [CrossRef] [PubMed]

9. M. W. Feise and Y. S. Kivshar, “Sub-wavelength imaging with a left-handed material flat lens,” Phys. Lett. A **334**, 326 (2005). [CrossRef]

10. J. J. Chen, T. M. Grzegorczyk, B.-I. Wu, and J. A. Kong, “Limitation of FDTD in simulation of a perfect lens imaging system,” Opt. Express **13**, 10840 (2005). [CrossRef] [PubMed]

11. A. N. Lagarkov and V. N. Kissel, “Near-Perfect Imaging in a Focusing System Based on a Left-Handed-Material Plate,” Phys. Rev. Lett. **92**, 077401 (2004). [CrossRef] [PubMed]

12. V. N. Kissel and A. N. Lagarkov, “Superresolution in left-handed composite structures: From homogenization to a detailed electrodynamic description,” Phys. Rev. B **72**, 085111 (2005). [CrossRef]

13. H. L. Luo, W. Hu, Z. Z. Ren, W. X. Shu, and F. Li, “Focusing and phase compensation of paraxial beams by a left-handed material slab,” Opt. Commun. **266**, 327–331 (2006). [CrossRef]

14. R. Ziolkowski, “Pulsed and CW Gaussian beam interactions with double negative metamaterial slabs,” Opt. Express **11**, 662–681 (2003). [CrossRef] [PubMed]

18. P. P. Banerjee and G. Nehmetallah, “Linear and nonlinear propagation in negative index materials,” J. Opt. Soc. Am. B. **23**, 2348–2355 (2006). [CrossRef]

22. V. Yannopapas, “Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies,” Appl. Phys. A **87**, 259–264 (2007). [CrossRef]

24. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E **69**, 016617 (2004). [CrossRef]

## 2. Models

*L*(region D2), and then in another free space D3. In Fig. 1,

_{LHM}*Z*

_{w}_{,1}and

*Z*

_{w,3}are the locations of waist of the Gaussian beam in free space D1 and D3, respectively, while

*Z*

_{0,2}and

*Z*

_{0,3}denote the on-axis coordinates of the front and back surfaces of the Kerr LHM slab, respectively. The Kerr LHM slab is assumed to be lossless and is transversely infinite, and the Gaussian beam is assumed to pass through the interfaces without reflection for simplicity.

**E**

_{1}=

*x̂A*

_{1}(

*x*,

*y*,

*z*)

*e*

^{i(kz-ωt)}, where

*ω*denotes the circular frequency of the electric field,

*A*

_{1}(

*x*,

*y*,

*z*) is the envelope of the complex amplitude which can be written as

*A*

_{1}/∂

*z*|≪

*k*

_{1}

*A*

_{1}, and the scalar approximation, where

*w*

_{1}(

*z*)=

*w*

_{0,1}{1+[(

*z*-

*Z*

_{w,1})/

*L*

_{R,1}]

^{2}}

^{1/2}is the beam spot radius,

*A*

_{0,1}and

*w*

_{0,1}are the peak amplitude and the spot radius at the beam waist,

*R*

_{1}(

*z*)=(

*z*-

*Z*

_{w,1}){1+[

*L*

_{R,1}/(

*z*-

*Z*

_{w,1})]

^{2}} is the curvature radius of the beam phase front,

*L*

_{R,1}=

*k*

_{1}

*w*

_{0,1}

^{2}/2,

*k*

_{1}=

*n*

_{L,1}2

*π*/

*λ*

_{0},

*n*

_{L,1}=1 is the refractive index of free space and

*λ*

_{0}is the vacuum wavelength of the incident electromagnetic wave. Thus the

*q*parameter of the incident beam is 1/

*q*

_{1}(

*z*)=[1/

*R*

_{1}(

*z*)]-[2/

*k*

_{1}

*w*

_{1}(

*z*)]. We should note here that we assume the paraxial approximation holds in the following theory and calculation.

27. 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]

30. S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E **73**, 036617 (2006). [CrossRef]

32. S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A **75**, 033815 (2007). [CrossRef]

*k*

_{2}=

*n*

_{L,2}2

*π*/

*λ*

_{0},

*n*

_{L,2}is the linear refractive index of the LHM slab,

*A*

_{2}is the complex envelope of the electric field,

*C*

_{NL,2}=

*χ*

_{p,2}/(2

*n*

_{L,2}) is the nonlinear index coefficient,

*χ*

_{p,2}is the cubic susceptibility,

*µ*

_{rl,2}is the relative magnetic permeability of the LHM slab.

## 3. Theory analysis

*q*parameter of Gaussian beam in Ref. [18

18. P. P. Banerjee and G. Nehmetallah, “Linear and nonlinear propagation in negative index materials,” J. Opt. Soc. Am. B. **23**, 2348–2355 (2006). [CrossRef]

*w*

_{0,2i}and the location of the waist

*Z*

_{w,2i}of the transformed Gaussian beam as

*w*

_{2i}(

*z*)=

*w*

_{0,2i}{1+[(

*z-Z*

_{w,2i})/

*L*

_{R,2i}]

^{2}}

^{1/2},

*R*

_{2i}(

*z*)=(

*z-Z*

_{w,2i}){1+[

*L*

_{R,2i}/(

*z-Z*

_{w,2i})]

^{2}},

*L*

_{R,2i}=

*k*

_{2}

*w*

_{0,2i}

^{2}/2, and

*k*

_{2}=

*n*

_{L,2}2

*π*/

*λ*

_{0}.

33. D. Anderson, M. Bonneal, and M. Lisak, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids **22(1)**, 105–109 (1979). [CrossRef]

*C*

_{0}=

*w*

_{2i}(

*Z*

_{0,2})=

*w*

_{1}(

*Z*

_{0,2}),

*K*=4-

*M a*

_{2}(

*z*)

^{2}

*w*

_{2}(

*z*)

^{2},

*M*=

*k*

^{2}

_{2}

*µ*

_{rl,2}

*C*

_{NL,2}/

*n*

_{L,2},

*a*

_{2}(

*z*)

*w*

_{2}(

*z*)=

*a*

_{2}(

*Z*

_{0,2})

*w*

_{2}(

*Z*

_{0,2}) is a constant due to the energy conservation law. Defining the power of the incident beam as

*P*=(1/2)

*ε*

_{0}|

*n*

_{L,2}|

*πa*

_{2}(

*Z*

_{0,2})

^{2}

*w*

_{2}(

*Z*

_{0,2})

^{2}and the reference power as

*P*

_{r,2}=

*ε*

_{0}

*cλ*

^{2}/(2

*π*|

*C*

_{NL,2}|), we have

*R*=

_{P}*P*/

*P*

_{r,2}. Further, to reflect the influence of the divergence of the incident beam, we define parameters

*θ*

_{o,1}=(

*Z*

_{w,1}-

*Z*

_{0,2})/

*L*

_{R,1}and

*θ*

_{i,2}=(

*Z*

_{w,2i}-

*Z*

_{0,2})/

*L*

_{R,2i}. Thus, we obtain

*θ*

_{i,2}=

*θ*

_{o,1},

*C*

_{0}=

*w*

_{0,1}(1+

*θ*

_{o,1}

^{2})

^{1/2}and

*C*

_{1}=

*w*

_{0,1}(1+

*θ*

_{o,1})

^{-1/2}(-

*θ*

_{o,1})/

*L*

_{R,2i}. By these relations, Eq. (6) becomes

*Z*

_{w,1}=

*Z*

_{0,2}=0, indicating

*C*

_{0}=

*w*

_{0,1}and

*θ*

_{o,1}=0. Under these conditions,

*b*

_{2}(

*z*) and

*φ*

_{2}(

*z*) can be obtained by the variational approach and

*b*

_{2}(

*z*) has the form

*Z*

_{0,3}, we have

*K*is a crucial parameter for the output field of the Kerr LHM slab. According to Eq. (7),

*K*can be significantly influenced by the sign of the nonlinear polarization of the LHM slab and the beam power, so we discuss the following three cases: (i),

*χ*

_{p,2}>0; (ii),

*χ*

_{p,2}<0 and

*R*<1/(-

_{P}*µ*

_{rl,2}); and (iii),

*χ*

_{p,2}<0 and

*R*>1/(-

_{P}*µ*

_{rl,2}). For the former two cases, we have

*K*>0 and

*w*

_{2}(

*Z*

_{0,3})>

*C*

_{0}, indicating the beam is self-defocusing in the Kerr LHM slab. For the last case, we obtain that

*K*is smaller than zero and

*w*

_{2}(

*Z*

_{0,3}) decreases from

*C*

_{0}as

*L*increases, indicating the beam is self-focusing in the Kerr LHM slab, and so the reference power

_{LHM}*P*

_{r,2}can be viewed as the critical power for self-focusing. Thus, the conditions for selffocusing and self-defocusing effects in the Kerr LHM case obtained here are in sharp contrast with their counterparts in the conventional Kerr medium (CKM) case. These results provide arguments for the prediction in Refs. [24

24. I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E **69**, 016617 (2004). [CrossRef]

25. P. P. Banerjee and G. Nehmetallah, “Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials,” J. Opt. Soc. Am. B **24**, A69–A76 (2007). [CrossRef]

*L*=[(

_{LHM}*k*

_{2}

*C*

_{0}

^{2})

^{2}/(

*-K*)]

^{1/2},

*w*

_{2}(

*Z*

_{0,3}) becomes approximately zero. If the slab thickness is close to or bigger than such a value, the beam will be totally self-focused in the Kerr LHM slab and Eq. (10) is no longer valid under such a condition.

*Z*

_{0,3}similar to those at

*Z*

_{0,2}, we obtain the expressions for the waist location and the spot radius of the waist of the Gaussian beam as

*χ*

^{p,2}<0 and

*R*>1/(-

_{P}*µ*

_{rl,2}) and

*L*<[(

_{LHM}*k*

_{2}

*C*

_{0}

^{2})

^{2}/(-

*K*)]

^{1/2}, (ii)

*χ*

_{p,2}<0 and

*R*<1/(-

_{P}*µ*

_{rl,2}), and (iii)

*χ*

_{p,2}>0 and

*R*<1/(-

_{P}*µ*

_{rl,2}); and (iv),

*χ*

_{p,2}>0 and

*R*>1/(-

_{P}*µ*

_{rl,2}). For the first case, we have

*Z*

_{w,3}<

*Z*

_{0,3}, indicating the output beam of the Kerr LHM slab is divergent. For the rest three cases, we have

*b*

_{2}(

*Z*

_{0,3})>0 and

*Z*

_{w,3}<

*Z*

_{0,3}, indicating the output beam of the Kerr LHM slab is convergent. In the rest three cases, the effect of the Kerr LHM slab is similar to the effect of a lens. So, in the following part of this paper, we call a Gaussian beam in D3 a focused beam if

*Z*

_{w,3}<

*Z*

_{0,3}and call this effect as the focusing (or lensing) effect of Kerr LHM slabs on Gaussian beams. Thus, it can be concluded that: (i) both Kerr LHM slabs with a positive nonlinear polarization and those with a negative nonlinear polarization can act as lenses to focus the incident Gaussian beam to a new waist; (ii) for the positive Kerr nonlinearity case, beam focusing happens for both 0<

*R*<1/(-

_{P}*µ*

_{rl,2}) and

*R*>1/(-

_{P}*µ*

_{rl,2}) conditions; and (iii) for the negative Kerr nonlinearity case, beam focusing happens for

*R*<1/(-

_{P}*µ*

_{rl,2}) only. For such a focusing effect, the location and the radius of the focal spot are determined by Eq. (11). For convenience, we define the corresponding focusing distance as

*L*=

_{f}*Z*

_{w,3}-

*Z*

_{0,3}and especially, define the focusing distance for the case

*θ*

_{o,1}=0 as

### 3.1 Focusing properties of Gaussian beams by LHM slabs with positive or negative Kerr nonlinearities

*χ*

_{p,2}>0, and (ii)

*χ*

_{p,2}<0 and

*R*<1/(-

_{P}*µ*

_{rl,2}). Unless specially pointed out, in the following calculations and simulations, we use the following parameters:

*λ*

_{0}=1053 nm,

*w*

_{2}(

*Z*

_{0,2})=0.4 mm,

*R*=10,

_{P}*Z*

_{w,1}=

*Z*

_{0,2}=0,

*L*=10 cm,

_{LHM}*n*

_{L,2}=-1 and

*µ*

_{rl,2}=-1.

*χ*

_{p,2}>0, the theoretical predictions for the focusing distance and the waist radius in D3 are shown in Figs. 2(a) and 2(b), respectively. To show the influence of

*L*on

_{LHM}*L*

_{f0}, we define the slope of a curve corresponds to certain beam power as

*s*=∂

*L*

_{f0}/∂

*L*. By

_{LHM}*s*, we divide the results in Fig. 2(a) into two cases: (i) beam power is some times as high as

*P*

_{r,2}, e.g.

*R*is about 5, and (ii) power ratio is large, e.g.

_{p}*R*is about 18. For case (i),

_{p}*s*is always positive and

*L*

_{f0}keeps increasing as

*L*increases. For case (ii), though generally positive,

_{LHM}*s*can be negative in a certain value range of

*L*where

_{LHM}*L*

_{f0}decreases as

*L*increases. It should also be noted that

_{LHM}*s*approaches one finally in both cases. Figure 2(b) shows that the waist radius in D3 is inversely proportional to both the slab thickness and the beam power. Thus, the same focal spot radius can be obtained by changing the incident beam power or the thickness of the LHM slab. Besides, for Gaussian beams with very high power, evidently focusing effect can be obtained by a thin LHM slab.

*χ*

_{p,2}<0 and

*R*<1/(-

_{P}*µ*

_{rl,2}), the theoretical predictions for the focusing distance and the waist radius in D3 are shown in Fig. 3. In this case, for each slab thickness larger than zero, the focusing distance

*L*

_{f0}decreases as the beam power increases, and the variation of the waist radius

*w*

_{0,3}in D3 with the beam power looks like a parabola. These are different from those shown by Fig. 2. Moreover, for Fig. 3(b), it should be noted that the waist radius in D3 is larger than the incident beam waist, which is contrary to that shown by Fig. 2(b), and that the variation in absolute waist radius is smaller when compared to that in Fig. 2(b).

*Z*

_{0,3}is

*G*=

*-K L*/(

_{LHM}*k*

_{2}

*C*

_{0}

^{2}). Considering the half-width (at 1/

*e*-intensity point) of the spatial spectrum of Gaussian beam satisfies ΔΩ=(1+

*G*

^{2})

^{1/2}/

*w*, where

*w*is the half-width (at 1/

*e*-intensity point) of the beam spot, we obtain from Eqs. (4), (5) and (10) the half-width of the spatial spectrum at the back surface of the slab:

### 3.2 Comparisons with the linear LHM slab case and the conventional Kerr medium slab case

*µ*

_{rl,2}=1 and

*n*

_{L,2}>1. For incident Gaussian beams with

*θ*

_{o,1}=0, there are two cases: (i) they will be self-defocused for

*χ*

_{p,2}<0 or for

*χ*

_{p,2}>0 and

*R*<1, and (ii) they will be self-focused for

_{P}*χ*

_{p,2}>0 and

*R*>1. Then, it can be inferred from Eq. (10) that: (i) for the former case,

_{P}*b*

_{2}(

*Z*

_{0,3})<0 and the beam in D3 is divergent; (ii) for the latter case, if the conditions for

*w*

_{2}(

*Z*

_{0,3})>0 are satisfied,

*b*

_{2}(

*Z*

_{0,3})>0 and the output beam of the CKM slab is convergent, indicating the beam can also be focused in D3. However, for the latter case, the beam suffers from modulation instability and thus filamentation may occur when there is small-scale modulation. For convenience, we assume that the waists of the incident beams are identical. Then, we find the main differences between the focusing properties of Gaussian beams by a Kerr LHM slab and those by a CKM slab case to be the following:

*R*<1/(-

_{P}*µ*

_{rl,2}), both LHM slabs with positive nonlinear polarization and those with negative nonlinear polarization has the lensing effect on them;

*χ*

_{p,2}>0, beams don’t suffer from modulation instability or filament formation in Kerr LHMs, as shown in Ref. [30

30. S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E **73**, 036617 (2006). [CrossRef]

*µ*

_{rl,2}=-1 is not as strong as that of the CKM slab. The reason is that the power density in the Kerr LHM slab is smaller than that in the CKM slab, which weakens the nonlinear effect, and thus the focal spot size in D3 from the Kerr LHM slab is larger than that from the CKM slab.

## 4. Numerical simulations

*χ*

_{p,2}>0 and

*R*>1/(-

_{P}*µ*

_{rl,2}).

### 4.1 *θ*_{o,1}=0

### 4.2 *θ*_{o,1}≠0

*θ*

_{o,1}depends on the convergence/divergence of incident Gaussian beams, i.e. for convergent incident beams,

*θ*

_{o,1}>0, and for divergent incident beams,

*θ*

_{o,1}<0. On the other hand, it is easy to infer from Eq. (4) that

*θ*

_{o,1}represents the linear spatial chirp of incident Gaussian beams of the Kerr LHM slab. To see the influence of

*θ*

_{o,1}, we keep the spot radius at

*Z*

_{0,2}fixed for simplicity. In addition, in the Kerr LHM slab, the thickness of the slab also plays an important role in this case according to Eq. (8). Thus, we discuss the following two sub-cases: one is for a thick slab (

*L*=10 cm); the other is for a thin slab (

_{LHM}*L*=4 cm).

_{LHM}*θ*

_{o,1}|=0.5. In this figure, it should be noted that the solid curve also represents the transform limited waist radius of the incident beam. Figure 5(a) shows the results for the thick-slab case. For

*θ*

_{o,1}>0, the beam waist radius decreases as

*θ*

_{o,1}increases. For

*θ*

_{o,1}<0, as |

*θ*

_{o,1}| increases, the beam waist radius increases at first and then decreases after reaching the maximum. This shows that irrespective of the initial beam divergence/convergence the incident beam can be focused by thick Kerr LHM slabs under certain conditions. When compared to the values shown by the solid curve, the numerical results for

*θ*

_{i,2}>0 are always smaller, while those for

*θ*

_{i,2}<0 can also be bigger when |

*θ*

_{o,1}| is big enough. For the thin-slab case shown by Fig. 5(b), for the case

*θ*

_{o,1}>0, the variation trend of

*w*

_{0,3}is the same as that in Fig. 5(a), but for the case

*θ*

_{i,2}<0,

*w*

_{0,3}keeps increasing as |

*θ*

_{i,2}| increases and finally the beam becomes divergent in D3.

*χ*

_{p,2}<0 and 0<

*R*<1 case, however, because the absolute increment of beam waist radius is relatively small (see Fig. 3(b)), we set

_{P}*R*=0.5 and

*P**L*=40 cm to obtain evident results. The influence of

_{LHM}*θ*

_{o,1}on the new beam waist is presented in Fig. 6. It is clear that because of

*θ*

_{o,1}, the difference between

*w*

_{0,3}and

*w*

_{0,1}can be larger than the

*θ*

_{o,1}=0 case. For the case

*θ*

_{o,1}>0,

*w*

_{0,3}increases at first and then decreases as |

*θ*

_{o,1}| increases and it always bigger than

*w*

_{0,1}; for the case

*θ*

_{o,1}<0,

*w*

_{0,3}decreases as |

*θ*

_{o,1}| increases and basically smaller than

*w*

_{0,1}. Comparing Fig. 6 to Fig. 5(a), it is easy to find that the effects of the convergence/divergence of incident Gaussian beams in these two cases are opposite to each other.

## 5. Conclusion

## Acknowledgments

## References and links

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20. | V. Yannopapas and N. V. Vitanov, “Photoexcitation-induced magnetism in arrays of semiconductor nanoparticles with a strong excitonic oscillator strength,” Phys. Rev. B |

21. | V. Yannopapas, “Negative refractive index in the near-UV from Au-coated CuCl nanoparticle superlattices,” Phys. Stat. Sol. (RRL) |

22. | V. Yannopapas, “Artificial magnetism and negative refractive index in three-dimensional metamaterials of spherical particles at near-infrared and visible frequencies,” Appl. Phys. A |

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

24. | I. V. Shadrivov, A. A. Sukhorukov, Y. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E |

25. | P. P. Banerjee and G. Nehmetallah, “Spatial and spatiotemporal solitary waves and their stabilization in nonlinear negative index materials,” J. Opt. Soc. Am. B |

26. | V. M. Agranovich, Y. R. Shen, R. H. Baughman, and A. A. Zakhidov, “Linear and nonlinear wave propagation in negative refraction metamaterials” Phys. Rev. B |

27. | 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. |

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

29. | I. Kourakis and P. K. Shukla, “Nonlinear propagation of electromagnetic waves in negative-refraction-index composite materials,” Phys. Rev. E |

30. | S. C. Wen, Y. W. Wang, W. H. Su, Y. J. Xiang, X. Q. Fu, and D. Y. Fan, “Modulation instability in nonlinear negative-index material,” Phys. Rev. E |

31. | S Wen, Y. Xiang, W. Su, Y. Hu, X. Fu, and D. Fan, “Role of the anomalous self-steepening effect in modulation instability in negative-index material,” Opt. Express |

32. | S. Wen, Y. Xiang, X. Dai, Z. Tang, W. Su, and D. Fan, “Theoretical models for ultrashort electromagnetic pulse propagation in nonlinear metamaterials,” Phys. Rev. A |

33. | D. Anderson, M. Bonneal, and M. Lisak, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.3270) Nonlinear optics : Kerr effect

(350.3618) Other areas of optics : Left-handed materials

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 3, 2008

Revised Manuscript: March 14, 2008

Manuscript Accepted: March 18, 2008

Published: March 24, 2008

**Citation**

Yonghua Hu, Shuangchun Wen, Hui Zhuo, Kaiming You, and Dianyuan Fan, "Focusing properties of Gaussian beams by a slab of Kerr-type left-handed metamaterial," Opt. Express **16**, 4774-4784 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-7-4774

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- D. Anderson, M. Bonneal, and M. Lisak, "Variational approach to nonlinear self-focusing of Gaussian laser beams," Phys. Fluids 22, 105-109 (1979). [CrossRef]

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