## The role of dispersion in the propagation of rotating beams in left-handed materials

Optics Express, Vol. 17, Issue 7, pp. 5645-5655 (2009)

http://dx.doi.org/10.1364/OE.17.005645

Acrobat PDF (290 KB)

### Abstract

We theoretically study the role of dispersion in propagation of rotating beams in left-handed materials (LHMs). By modeling the rotating beam as a superposition of two rotating Laguerre-Gaussian beams with opposite chirality, same magnitude and different frequencies, we demonstrate that the rotation property of the rotating beam in LHM is significantly dependent on the sign and strength of dispersion: In the normal dispersion region, the direction of transverse energy flow is reversed compared to the vacuum, due to the negative refractive index of LHM, while in the anomalous dispersion region it may be parallel or antiparallel to that in the case of vacuum, depending on the strength of dispersion. In addition, we find that the angular momentum density can be parallel or antiparallel to the transverse energy flow in LHM, while the angular momentum flow is always opposite to the transverse energy flow.

© 2009 Optical Society of America

## 1. Introduction

1. J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. Lond. A **336**, 165–190 (1974). [CrossRef]

2. J. F. Nye, “The motion and structure of dislocations in wavefronts,” Proc. R. Soc. Lond. A. **378**, 219–239 (1981). [CrossRef]

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

4. L. Allen, M. J. Padgett, and M. Babiker, “The orbit angular momentum of light,” Prog. Opt. **39**, 291–372 (1999). [CrossRef]

5. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. **42**, 219–276 (2001). [CrossRef]

6. J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. **90**, 133901 (2003). [CrossRef] [PubMed]

7. C. N. Alexeyev and M. A. Yavorsky, “Angular momentum of rotating paraxial light beams,” J. Opt. A: Pure Appl. Opt. **7**, 416–421 (2005). [CrossRef]

8. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Com-mun. **249**, 367–378 (2005). [CrossRef]

9. A. Ya. Bekshaev and M. S. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. **31**, 2199–2201 (2006). [CrossRef] [PubMed]

10. G. Nienhuis, “Polychromatic and rotating beams of light,” J. Phys. B: At. Mol. Opt. Phys. **39**, S529–S544 (2006). [CrossRef]

11. S. J. van Enk and G. Nienhuis, “Photons in polychromatic rotating modes,” Phys. Rev. A **76**, 053825 (2007). [CrossRef]

12. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. **78**, 249–253 (2001). [CrossRef]

13. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein- Dunlop, “Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**, 826–829 (1995). [CrossRef] [PubMed]

14. M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A **54**, 1593–1596 (1996). [CrossRef] [PubMed]

15. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt. Lett. **22**, 52–54 (1997). [CrossRef] [PubMed]

16. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakial, “Controlled rotation of optically trapped microscopic particles,” Science **292**, 912–914 (2001). [CrossRef] [PubMed]

17. M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. **201**, 21–29 (2002). [CrossRef]

18. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakial, “Creation and manipulation of Three-Dimensional optically trapped structures,” Science **296**, 1101–1103 (2002). [CrossRef] [PubMed]

19. Christian H. J. Schmitz, Kai Uhrig, Joachim P. Spatz, and Jennifer E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express **14**, 6604–6612 (2006). [CrossRef] [PubMed]

20. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values ofeand m,” Sov. Phys. Usp. **10**, 509–514 (1968). [CrossRef]

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

22. M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. **121**, 36–40 (1995). [CrossRef]

23. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density,” Opt. Commun. **184**, 67–71 (2000). [CrossRef]

24. H. Luo, Z. Ren, W. Shu, and S. C. Wen, “Reversed propagation dynamics of Laguerre-Gaussian beams in left-handed materials,” Phys. Rev. A **77**, 023812 (2008). [CrossRef]

25. H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, “Rotational Doppler effect in left-handed materials,” Phys. Rev. A **78**, 033805 (2008). [CrossRef]

## 2. Paraxial propagation of a Laguerre-Gaussian beam in LHMs

*k*=

*n*

*ω*/

*c*is wave number,

*n*is the refractive index,

*ω*is the angular frequency,

*c*is the speed of light in vacuum,

*J*is the first kind Bessel function with order

_{l}*l*,

*r*is polar coordinate. The two-dimensional Fourier transformations of Eq. (1) can be easily obtained from an integration table [27]. The function

*u*(

**r**,0) together with Eq. (1) provide the expression of the field in the space

*z*> 0, which yields

*z*is the distance from the beam waist,

*n*are the refractive index of RHM and LHM, respectively,

_{R,L}*k*

_{0}is the wave number in vacuum. The field

*u*(

**r**,

*z*) is the slowly varying envelope amplitude.

*ω*propagation in RHMs and LHMs, respectively. Without any loss of generality, we assume that the beam waist locates at the object plane

*z*= 0, which is described by the complex amplitude distribution

*c*is weight coefficient,

_{pl}*w*

_{0}is the beam waist radius,

*R*

_{0}is the beam waist radius of curvature of wavefront,

*L*

_{p}^{|l|}is the generalized Laguerre polynomial. A given mode is usually denoted as LG

*, where*

_{p}^{l}*l*and

*p*are the two integer indices that describe the mode:

*l*refers to the number of 2

*π*phase cycles around the circumference of the mode, so that

*l*is known as the azimuthal index, and (

*p*+ 1) indicates the number of radial nodes in the mode profile. A LG beam is well known to possess orbital angular momentum due to an exp(

*ilφ*) phase term (where

*φ*is the azimuthal phase) in the mode description that gives rise to a well defined orbital angular momentum of

*lh*̄ per photon. This orbital angular momentum

*lh*̄ is distinct from the spin angular momentum due to the polarization state of the light [3

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185–8189 (1992). [CrossRef] [PubMed]

*z*=

_{R}*k*

_{0}

*n*

_{R}*w*

^{2}

_{0}/2 is the corresponding Rayleigh range,

*R*(

*z*) = (

*z*

^{2}+

*z*

_{R}^{2})/

*z*is the radius of curvature of wavefront, and

*ψ*(

*z*) = arctan(

*z*/

*z*) is the Gouy phase. Similarly, the field of LG mode travels in LHMs can be written as

_{R}*z*=

_{L}*k*

_{0}

*n*

_{L}*w*

^{2}

_{0}/2,

*R*(

*z*) = (

*z*

^{2}+

*z*

_{L}^{2})/

*z*and

*ψ*(

*z*) = arctan(

*z*/

*z*) are the beam size, the corresponding Rayleigh range, the radius of curvature of wavefront and the Gouy phase in LHMs, respectively. Because of the negative refractive index, the reversed Gouy-phase shift should be introduced. We have found that, the inverse Gouy-phase shift gives rise to an inverse spiral of Poynting vector [24

_{L}24. H. Luo, Z. Ren, W. Shu, and S. C. Wen, “Reversed propagation dynamics of Laguerre-Gaussian beams in left-handed materials,” Phys. Rev. A **77**, 023812 (2008). [CrossRef]

*n*has a imaginary part. Consider the situation where the LG beam is linearly polarized. The electric and magnetic fields of a LG beam can be written as [28

_{L}28. Rodney Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A **68**, 013806 (2003). [CrossRef]

**e**

*,*

_{x}**e**

*,*

_{y}**e**

*are the unit vectors of corresponding Cartesian axes, axis*

_{z}*x*is supposed to coincide with the direction of the transverse component electric field, and u is shorthand for

*u*(

**r**,

*z*,

*t*). These field expressions neglect terms in each component that are smaller than those retained in accordance with the paraxial approximation. The

*z*components are smaller than the

*x*and

*y*components by a factor of order 1/

*kw*

_{0}. It is readily verified that the fields satisfy Maxwell’s equations. Note that the Cartesian derivatives can be converted to polar

*r*and

*φ*derivatives in the usual way.

## 3. Rotating beams propagation in LHMs

29. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science **292**, 77–79 (2001) [CrossRef] [PubMed]

*ω*

_{e0}is the electronic resonance frequency,

*ω*is the electronic plasma frequency,

_{ep}*γ*is the electronic damping frequency,

_{e}*ω*

_{m0}is the magnetic resonance frequency,

*ω*is the magnetic plasma frequency,and

_{mp}*γ*is the magnetic damping frequency.When

_{m}*ω*

_{e0}=

*ω*

_{m0}=0, the Lorentz model turns to the Drude model. Here, we assume that

*ω*

_{e0}=

*ω*

_{m0},

*ω*=

_{ep}*ω*, and

_{mp}*γ*=

_{e}*γ*.

_{m}*n*is complex denoted by

*n*(

*ω*) =

*η*(

*ω*) +

*iκ*(

*ω*).

_{p,+l}+ LG

*). We assume that all modes have the equal magnitudes and waist parameters, both centered at axis*

_{p,-l}*z*and paraxially propagating in the positive

*z*direction. In the case of equal frequencies, such a combination is well known to be equivalent to an ordinary Hermite-Gaussian beam. For the brevity, we will call this beam a “rotating Hermite-Gaussian” (RHG) beam, which has an unique edge wavefront dislocation and a particular “multi-spot” intensity distribution within the transverse (

*x*,

*y*)-plane [8

8. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Com-mun. **249**, 367–378 (2005). [CrossRef]

*ω*

_{+}=

*ω*

_{0}+ Δ

*ω*/2 and

*ω*

_{-}=

*ω*

_{0}- Δ

*ω*/2 components, respectively.

*ω*

_{0}is the center frequency of RHG beam, Δ

*ω*= (

*ω*

_{+}-

*ω*

_{-}) > 0. In their turn, the field vectors of a paraxial beam can be represented as

*k*

_{±}=

*n*

_{R,L}*ω*

_{±}/

*c*is the wave numbers. Note that, a physically consistent condition Δ

*ω*≪

*ω*

_{±}is satisfied.

**S**

_{⊥}=

**S**

*+*

_{x}**S**

*. Discarding rapidly oscillating and thus unobservable terms, one can write*

_{y}*c*

_{±}= 2

*p*

_{±}+|

*l*

_{±}|+1. We note that in the lossy LHM,

*k*is complex denoted by

*k*=

*τ*+

*iυ*. It is obvious from Eq. (15) that the longitudinal component of Poynting vector attenuates exponentially with distance. In particular, within the plane of analysis

*z*= 0, Eq. (15) can be reduced to the form

*ω*/(

*l*

_{+}-

*l*

_{-}). As usual, positive Ω corresponds to counter-clockwise rotation when seeing against the beam propagation axis (the right-screw rule). One can note that the properties of LHMs cannot reverse the transverse beam pattern rotation. In order to accurately describe this question, we investigate this problem with the help of model examples representing important features of RHG beams. Let us consider two typical patterns of the field, which are formed by a simple two-term superposition of the class LG

_{0,+1}and LG

_{0,-1}, LG

_{1,-1}and LG

_{1,+1}. In the case of LG

_{0,+1}+ LG

_{0,-1}, Ω > 0 and the intensity pattern exhibits anticlockwise spiral. In the another case of LG

_{1,-1}+ LG

_{1,+1}, Ω < 0 and the intensity pattern presents clockwise spiral.

## 4. Transverse energy flow and angular momentum density of rotating beams in LHMs

**S**

*and*

_{x}**S**

*can be divided into the time-invariant and slowly oscillating parts.*

_{y}5. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. **42**, 219–276 (2001). [CrossRef]

**S**

_{⊥}may be written in polar coordinates as

**S**

_{φ}〉 corresponds to counter-clockwise rotation when seeing against the beam propagation axis (the right-screw rule). In particular, within the plane of analysis

*z*= 0, Eq. (21) can be reduced to the form

**S**

*〉 = 0. Furthermore, Eq. (23) can be evaluated approximately by making the Taylor expansion of 1/*

_{r}*η*(

*ω*) in the vicinity of the beam’s central frequency. The average energy flow can be written in the form as the part of known results for non-dispersive media and the correction part from the dispersion

*ω*

^{3}<< Δ

*ω*and its higher order terms become small enough to be negligible. Thus, Eq. (24) can be reduced to

_{0,+1}+ LG

_{0,-1}, we can see from Fig. 2(a) that the direction of transverse energy flow presents anticlockwise spiral in vacuum, where

*l*> 0 and 〈

**S**

_{φ}〉 = [Δ

*ωl*/(2

*r*|

*u*|

^{2})] > 0. However, when such a RHG beam propagates in the normal dispersion region of the LHM,

*η*(

*ω*

_{0}) is negative while

*∂η*/

*∂ω*|

_{ω=ω0}is positive, it is obvious from Eq. (25) that its first term and its second term are both negative. Consequentially, 〈

**S**

_{φ}〉 < 0 and the direction of transverse energy flow is reversed in the normal dispersion region of the LHM due to the negative refractive index (see Fig. 2(a’)). In the other case of LG

_{1, -1}+LG

_{1, +1}, we can see from Fig. 2(b) that the direction of transverse energy flow exhibits clockwise spiral in vacuum, where

*l*< 0 and 〈

**S**

_{φ}〉 = [Δ

*ωl*/(2

*r*|

*u*|

^{2})] < 0. Since in the normal dispersion region of the LHM, 〈

**S**

_{φ}〉 > 0, and the direction of transverse energy flow exhibits anticlockwise spiral (see Fig. 2(b’)).

*η*(

*ω*

_{0}) and

*∂η*/

*∂ω*|

_{ω=ω0}are negative. Obviously, the first term of Eq. (25) is negative but its second term is positive. In this case, Eq. (25) proves that the direction of transverse energy flow could be affected significantly by the strength of the dispersion. As we see from Figs. 3(a) and 3(b), when (

*∂η*/

*∂η*)|

_{ω=ω0}| < 2 |

*η*(

*ω*

_{0})|/

*ω*

_{0}, this point is evident by the fact that Δ

*ω*/

*η*(

*ω*

_{0})| > |

*ω*

_{0}Δ

*ω*(

*∂η*/

*∂ω*)|

_{ω=ω0}/2

*η*

^{2}(

*ω*

_{0})|, which implies that the direction of transverse energy flow is determined by the negative refractive index of the LHM. Figs. 3(a’) and 3(b’) point out that since |(

*∂η*/

*∂ω*)|

_{ω=ω0}| > 2 |

*η*(

*ω*

_{0})|/

*ω*

_{0}, the effect of the anomalous dispersion can reverse the direction of transverse energy flow, which is contrast the case in the normal dispersion region of the LHM. We can conclude that when a small amount of dispersion is introduced in the anomalous dispersion region of LHMs, the effect of dispersion is less than the effect of negative refractive index. With increased strength of the dispersion, one can also clearly observe that the effect of negative refractive index was counteracted by the effect of anomalous dispersion.

**T**, and momentum density

**G**interact with other subsystems via force density

**F**. It is necessary to include the dispersive characteristics of the material in describing the behavior of the electromagnetic wave. Thus the momentum conservation equations for the electromagnetic wave can be written in the form:

*I*is 3 × 3 identity matrix. The electric and magnetic polarization vectors are given by

**P**

_{e}=

*ε*

_{0}(

*ε*- 1)

**E**and

**P**

*= -*

_{m}*μ*

_{0}(

*μ*- 1)

**H**, respectively. Bound electric current

**J**= ∂

**P**

*/*

_{e}*∂t*and bound electric charge

*ρ*= ∇ ·

_{e}**P**

*have been accounted. Similarly, bound magnetic current*

_{e}**M**= ∂

**P**

*/*

_{m}*∂t*and bound magnetic charge

*ρ*= ∇ ·

_{m}**P**

*should be introduced to describe the angular momentum density in LHMs.*

_{m}**E**| =

*c*|

**B**| /

*n*. The momentum density can be reduced to

**e**

*is the unit vector of Poynting vector. It is evident from Eq. (28) that the direction of momentum density is always antiparallel to the energy flow in lossless LHMs, where ∂ (*

_{s}*nω*)/

*∂ω*is positive. Note that Eq. (28) is valid only for lossless medium, and its application to lossy media produces unphysical phenomena such as a negative energy in LHMs. In lossy media, the average momentum flux satisfies relation 〈

**T**〉 =

*η*〈

**S**〉/

*c*[32

32. B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A **75**, 053810 (2007). [CrossRef]

**T**〉 follows exactly the sign of

*η*, the momentum flow of RHG beams is opposite to the power flow in the region where

*η*< 0. Similar to energy velocity, a momentum velocity may also be defined [33

33. R. Loudon, L. Allen, and D. F. Nelson, “Propagation of electromagnetic energy and momentum through an absorbing dielectric,” Phys. Rev. E **55**, 1071–1085 (1997). [CrossRef]

**T**〉 is negative, the momentum density 〈

**G**〉 may be positive or negative in a frequency band with a negative index of refraction. Hence, the momentum density may be parallel or antiparallel to the transverse energy flow in dispersive LHMs. This is in contrast to the results for a lossless LHM, where the angular momentum density is always antiparallel to the energy flow. The result of Eq. (29) prove that: When

*η*

^{2}+

*κ*

^{2}+ 2 (

*ω*

_{0}

*ηκ*/

*γ*) < 0, the angular momentum density is parallel to the transverse energy flow, while when

_{e}*η*

^{2}+

*κ*

^{2}+ 2 (

*ω*

_{0}

*ηκ*/

*γ*) > 0, the angular momentum density is antiparallel to the transverse energy flow. Furthermore, as we see from Fig. 4, the angular momentum flow is always opposite to the transverse energy flow.

_{e}## 5. Conclusion

## Acknowledgments

## References and links

1. | J. F. Nye and M. V. Berry, “Dislocation in wave trains,” Proc. R. Soc. Lond. A |

2. | J. F. Nye, “The motion and structure of dislocations in wavefronts,” Proc. R. Soc. Lond. A. |

3. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

4. | L. Allen, M. J. Padgett, and M. Babiker, “The orbit angular momentum of light,” Prog. Opt. |

5. | M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. |

6. | J. E. Curtis and D. G. Grier, “Structure of optical vortices,” Phys. Rev. Lett. |

7. | C. N. Alexeyev and M. A. Yavorsky, “Angular momentum of rotating paraxial light beams,” J. Opt. A: Pure Appl. Opt. |

8. | A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Angular momentum of a rotating light beam,” Opt. Com-mun. |

9. | A. Ya. Bekshaev and M. S. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. |

10. | G. Nienhuis, “Polychromatic and rotating beams of light,” J. Phys. B: At. Mol. Opt. Phys. |

11. | S. J. van Enk and G. Nienhuis, “Photons in polychromatic rotating modes,” Phys. Rev. A |

12. | P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. |

13. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein- Dunlop, “Direct observation of transfer of angular momentum to absorptive particle from a laser beam with a phase singularity,” Phys. Rev. Lett. |

14. | M. E. J. Friese, J. Enger, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Optical angular-momentum transfer to trapped absorbing particles,” Phys. Rev. A |

15. | N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: An optical spanner,” Opt. Lett. |

16. | L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakial, “Controlled rotation of optically trapped microscopic particles,” Science |

17. | M. P. MacDonald, K. Volke-Sepulveda, L. Paterson, J. Arlt, W. Sibbett, and K. Dholakia, “Revolving interference patterns for the rotation of optically trapped particles,” Opt. Commun. |

18. | M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakial, “Creation and manipulation of Three-Dimensional optically trapped structures,” Science |

19. | Christian H. J. Schmitz, Kai Uhrig, Joachim P. Spatz, and Jennifer E. Curtis, “Tuning the orbital angular momentum in optical vortex beams,” Opt. Express |

20. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values ofeand m,” Sov. Phys. Usp. |

21. | H. Luo, W. Shu, F. Li, and Z. Ren, “Focusing and phase compensation of paraxial beams by a left-handed material slab,” Opt. Commun. |

22. | M. J. Padgett and L. Allen, “The Poynting vector in Laguerre-Gaussian laser modes,” Opt. Commun. |

23. | L. Allen and M. J. Padgett, “The Poynting vector in Laguerre-Gaussian beams and the interpretation of angular momentum density,” Opt. Commun. |

24. | H. Luo, Z. Ren, W. Shu, and S. C. Wen, “Reversed propagation dynamics of Laguerre-Gaussian beams in left-handed materials,” Phys. Rev. A |

25. | H. Luo, S. C. Wen, W. Shu, Z. Tang, Y. Zou, and D. Fan, “Rotational Doppler effect in left-handed materials,” Phys. Rev. A |

26. | J. W. Goodman, |

27. | I. S. Gradshteyn and I. M. Ryzhik, |

28. | Rodney Loudon, “Theory of the forces exerted by Laguerre-Gaussian light beams on dielectrics,” Phys. Rev. A |

29. | R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science |

30. | J. D. Jackson, |

31. | J. A. Kong, |

32. | B. A. Kemp, J. A. Kong, and T. M. Grzegorczyk, “Reversal of wave momentum in isotropic left-handed media,” Phys. Rev. A |

33. | R. Loudon, L. Allen, and D. F. Nelson, “Propagation of electromagnetic energy and momentum through an absorbing dielectric,” Phys. Rev. E |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(260.2110) Physical optics : Electromagnetic optics

(260.2160) Physical optics : Energy transfer

(350.5500) Other areas of optics : Propagation

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

**ToC Category:**

Physical Optics

**History**

Original Manuscript: January 9, 2009

Revised Manuscript: March 7, 2009

Manuscript Accepted: March 16, 2009

Published: March 25, 2009

**Citation**

Qiang Lv, Hongyao Liu, Hailu Luo, Shuangchun Wen, Weixing Shu, Yanhong Zou, and Dianyuan Fan, "The role of dispersion in the propagation of rotating beams in left-handed materials," Opt. Express **17**, 5645-5655 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-7-5645

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### References

- J. F. Nye and M. V. Berry, "Dislocation in wave trains," Proc. R. Soc. Lond. A 336, 165-190 (1974). [CrossRef]
- J. F. Nye, "The motion and structure of dislocations in wavefronts," Proc. R. Soc. Lond. A. 378, 219-239 (1981). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992). [CrossRef] [PubMed]
- L. Allen, M. J. Padgett, and M. Babiker, "The orbit angular momentum of light," Prog. Opt. 39, 291-372 (1999). [CrossRef]
- M. S. Soskin and M. V. Vasnetsov, "Singular optics," Prog. Opt. 42, 219-276 (2001). [CrossRef]
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