## Azimuthally polarized surface plasmons as effective terahertz waveguides

Optics Express, Vol. 13, Issue 2, pp. 511-518 (2005)

http://dx.doi.org/10.1364/OPEX.13.000511

Acrobat PDF (170 KB)

### Abstract

Quite recently, it was found that metal wires can effectively guide terahertz radiation. Based on the fact that the absolute values of the relative permittivities of metals in the spectral region of terahertz radiation are huge, we here analyse the properties of this kind of waveguide and explain the related experimental results. In particular, we show that the observed waveguiding is due to the propagation of an azimuthally polarized surface plasmon along the wire. Some related aspects, such as the choice of metal and the slowly decaying modal field, are also discussed. In particular, we show that, if a copper wire with a radius of 0.45 mm is used, the attenuation coefficient is smaller than 2×10^{-3} cm^{-1} in the whole range of 0.1~1 THz.

© 2005 Optical Society of America

## 1. Introduction

## 2. Explanations for THz metal wire waveguides

20. Q. Cao and Ph. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. **88**, 057403 (2002). [CrossRef] [PubMed]

21. S. I. Bozhevolnyi, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. **86**, 3008–3011 (2001). [CrossRef] [PubMed]

22. M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. **93**, 137404 (2004). [CrossRef] [PubMed]

24. U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B **64**, 125420 (2001). [CrossRef]

_{y}, and two electrical field components E

_{x}and E

_{z}. The only transverse magnetic field component H

_{y}indicates the TM polarization. For an SP at a cylindrical metal-dielectric interface, however, there are one magnetic field component H

_{ϕ}and two electric field components E

_{r}and E

_{z}. The only transverse magnetic field component H

_{ϕ}indicates the TM polarization. The latter kind of SP can be reasonably called APSP, because the only magnetic field component H

_{ϕ}is angular.

_{m}and µ

_{m}. And for air, ε=µ=1. We denote by λ

_{0}, c, and R the wavelength in free space, the light speed in free space, and the radius of the metal wire, respectively. We assume a temporal factor exp(-jωt), where ω=2πc/λ

_{0}, is the angular frequency.

**E**/∂ϕ=0 and ∂

**H**/∂ϕ=0 hold in the cylindrical coordinates. By use of these relations, one can separate the electromagnetic field into two families. One is the TE polarization, and the other is the TM polarization. For the former, there are three components E

_{ϕ}, H

_{r}and H

_{z}. For the latter, there are another three components Hϕ, Er and Ez. The two families of TE and TM polarizations are decoupling in this case. We now focus on the TM polarization. By substituting the relations ∂

**E**/∂ϕ=0 and ∂

**H**/∂ϕ=0 into Maxwell’s equations [25], one obtains

_{ϕ}, E

_{r}and E

_{z}are functions of the two variables r and z. For the formulation of the eigenproblem, they can be written as H

_{ϕ}(r,z)=H

_{ϕ}(r)exp(jk

_{0}n

_{eff}z), E

_{r}(r,z)=E

_{r}(r)exp(jk

_{0}n

_{eff}z), and E

_{z}(r,z)=E

_{z}(r)exp(jk

_{0}n

_{eff}z), respectively, where n

_{eff}is the effective index of the eigenmode. Then the operator ∂/∂z can be replaced by jk

_{0}n

_{eff}. In the remainder of this paper, we refer to H

_{ϕ}, E

_{r}and E

_{z}specifically as H

_{ϕ}(r), E

_{r}(r) and E

_{z}(r), respectively. By using the relation ∂/∂z=jk

_{0}n

_{eff}in Eq. (3), one gets

_{0}[(n

_{eff})

^{2}-µε]

^{1/2}r. The suitable solution for E

_{z}has the form I

_{0}(k

_{0}κ

_{m}r) in the metal and K

_{0}(k

_{0}κ

_{a}r) in air, because this solution approximately decays exponentially from the interface, where I

_{0}(.) and K

_{0}(.) are the generalized Bessel functions [26], κ

_{m}=[(n

_{eff})

^{2}-µ

_{m}ε

_{m}]

^{1/2}and κ

_{a}=[(n

_{eff})

^{2}-1]

^{1/2}. Substituting this kind of solution into Eq. (5), and using the relations [26] dI

_{0}(k

_{0}κ

_{m}r)/dr=k

_{0}κ

_{m}I

_{1}(k

_{0}κ

_{m}r) and dK

_{0}(k

_{0}κ

_{a}r)/dr=-k

_{0}κ

_{a}K

_{1}(k

_{0}κ

_{a}r), one can determine the solution form of H

_{ϕ}. By using the continuities of E

_{z}and H

_{ϕ}at the interface r=R, one can obtain the following eigenvalue equation

_{m}. This negative value is due to the electron plasma contribution when the frequency is lower than the plasma frequency. Accordingly, this eigenmode is traditionally called SP. In particular, when R→∞, the relations I

_{1}(k

_{0}κ

_{m}R)/I

_{0}(k

_{0}κ

_{m}R)=1 and K

_{1}(k

_{0}κ

_{a}R)/K

_{0}(k

_{0}κ

_{a}R)=1 hold. By use of these properties, one can obtain, for very large R,

_{m}=1, Eq. (8) reduces to n

_{eff}=[ε

_{m}/(1+ε

_{m})]

^{1/2}. Eq. (8) is already enough for some qualitative analyses, though the exact n

_{eff}value of the APSP should be obtained by numerically solving Eq. (7).

_{ϕ}(r). The H

_{ϕ}(r) field is proportional to K

_{1}(k

_{0}κ

_{a}r)/K

_{0}(k

_{0}κ

_{a}R) outside the metal. As we shall point out below, n

_{eff}is approximately equal to 1. As a consequence, κ

_{a}is very small. Because κ

_{a}is very small, then the field distribution K

_{1}(k

_{0}κ

_{a}r)/K

_{0}(k

_{0}κ

_{a}R) decays very slowly with the radial coordinate r. Typically, the modal field has a width of about several tens times of the radius R outside the metal. In the metal, the H

_{ϕ}(r) field is proportional to I

_{1}(k

_{0}κ

_{m}r)/I

_{0}(k

_{0}κ

_{m}R). As we shall see below, |ε

_{m}| is far larger than 1 in the THz spectral region. Using the relations |ε

_{m}|≫1 and n

_{eff}≈1, one can get the relation κ

_{m}≈(-ε

_{m})

^{1/2}. As a consequence, the real part of κ

_{m}is quite large. Typically, it has the order of magnitude of 1.0×10

^{3}or larger. This property leads to the fast decay of the modal field in the metal. Typically, the modal field can exist in the metal for only about 1 µm or less. Therefore, the modal field of an APSP has a very large beam width but with a hollow center.

_{m}of metals are huge in the spectral region of THz radiation. This property can be deduced from the Drude model [25, 27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. **24**, 4493–4499 (1985). [CrossRef] [PubMed]

27. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. **24**, 4493–4499 (1985). [CrossRef] [PubMed]

_{m}) is larger than 10

^{4}and Im(ε

_{m}) is larger than 10

^{5}, where Re(.) and Im(.) indicate the real and the imaginary parts, respectively. By using the properties |ε

_{m}|≫1 and |ε

_{m}|≫µ

_{m}in Eq. (8), one can find that n

_{eff}≈1. This result explains the low loss of metal wire waveguides because Im(n

_{eff}) is very small.

_{eff}≈1 holds for a very wide bandwidth, because |ε

_{m}| is huge in the whole range of THz radiation. This property explains the very low dispersion of metal wire waveguides.

_{z}/E

_{r}| is in the order of magnitude of |κ

_{a}/n

_{eff}| in air. By using the relation n

_{eff}≈1, one can find that |E

_{z}|≪|E

_{r}|. This property implies that the longitudinal field component E

_{z}is negligible in air. As a result, the modal field is an approximate TEM field in air. Note that it is no longer true in the metal. In the metal, |E

_{r}|≪|E

_{z}|. It is worth mentioning that, it is not the radial field component E

_{r}but the longitudinal field component E

_{z}that leads to the attenuation of the eigenmode.

_{z}is negligible, the remaining only electric field component in air is E

_{r}. Then it looks like that the electric field is radially polarized.

_{eff}≈1. These two properties imply that the Fresnel reflectivity at the end of metal wire is negligible. Therefore, the mode can propagate off the end of the waveguide.

23. C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B **10**, 3038–3051 (1974). [CrossRef]

23. C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B **10**, 3038–3051 (1974). [CrossRef]

23. C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B **10**, 3038–3051 (1974). [CrossRef]

## 3. Further discussions

17. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature (London) , **432**, 376–379 (2004). [CrossRef]

_{m}|. In the spectral region of THz radiation, different metals still have different |ε

_{m}| [27

27. M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. **24**, 4493–4499 (1985). [CrossRef] [PubMed]

_{m}| of copper is more than 10 times higher than that of iron [27

**24**, 4493–4499 (1985). [CrossRef] [PubMed]

_{m}|, the better the waveguide. In addition, the relative magnetic permeability µ

_{m}also plays a role.

_{m}, one can prove that n

_{eff}≈1-µ

_{m}/(2ε

_{m}). From this expression, one can deduce that, for the same ε

_{m}, a magnetic metal will increase the loss by µ

_{m}times. Therefore, the use of ferromagnetic metal [25, 28], such as iron and nickel, should be avoided. We suggest the use of nonmagnetic metals with very large |ε

_{m}|. Copper, silver and gold are good candidates. Copper, of course, would be preferrable because it is much cheaper than silver and gold. We notice that stainless steel wires are used in the experiments of Ref. [17

17. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature (London) , **432**, 376–379 (2004). [CrossRef]

30. C. Weber and J. Fajans, “Saturation in “nonmagnetic” stainless steel,” Rev. Scientific Instruments **69**, 3695–3696 (1998). [CrossRef]

17. K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature (London) , **432**, 376–379 (2004). [CrossRef]

_{m}value of the used stainless steel. Therefore, it is difficult to concretely calculate the waveguide parameters of the stainless steel wires used there [17

**432**, 376–379 (2004). [CrossRef]

_{m}value should be not smaller than 1 [30

30. C. Weber and J. Fajans, “Saturation in “nonmagnetic” stainless steel,” Rev. Scientific Instruments **69**, 3695–3696 (1998). [CrossRef]

_{m}| value of a stainless steel should be smaller than that of copper, because the |ε

_{m}| value of iron, which is the main chemical composition of stainless steel, is at least 10 times smaller than that of copper [27

**24**, 4493–4499 (1985). [CrossRef] [PubMed]

_{1}(.) and I

_{0}(.) approximately increase exponentially with the increase of the real part of the argument. Unfortunately, the real part of the argument k

_{0}κ

_{m}R is quite large in the investigated case. For example, in the example of Fig. 1, the argument k

_{0}κ

_{m}R is as large as 6.21×10

^{3}-j4.95×10

^{3}. Because the argument k

_{0}κ

_{m}R is quite large, numerical overflow will happen if one directly calculates the functions I

_{1}(k

_{0}κ

_{m}R) and I

_{0}(k

_{0}κ

_{m}R) that appear in Eq. (7). To solve this problem, we use the asymptotic expression [26]

_{1}(k

_{0}κ

_{m}R)/I

_{0}(k

_{0}κ

_{m}R), the diverging factor exp(k

_{0}κ

_{m}R) is removed and the overflow is overcome. Then the eigenvalue equation (7) can be solved by complex-root search methods. Local search methods would be effective. We use the Newton method to solve the eigenvalue problem. Once obtaining the eigenvalue n

_{eff}, one can immediately obtain the complete modal field.

_{eff}of the APSPs at the interfaces of copper wires with different R. The frequency is chosen to be 0.5 THz (i.e., λ

_{0}=0.6 mm). The ε

_{m}value is determined to be ε

_{m}=-6.3×10

^{5}+j2.77×10

^{6}according to a fitted Drude formula for copper [27

**24**, 4493–4499 (1985). [CrossRef] [PubMed]

_{m}=1 is used. The calculated n

_{eff}values, as a function of R, are shown in Fig. 2. One can see that, just as expected, neff approach to the value determined by Eq. (8) for very large R. One can also see that, both the values of Im(n

_{eff}) and Re(n

_{eff})-1 increase with the decrease of the radius R. Thick metal wires are sometimes inconvenient for certain applications, but with lower attenuation. Therefore, one should balance between the size and the attenuation when designing a concrete metal wire waveguide.

_{m}of copper is obtained from the corresponding Drude formula of Ref. [27

**24**, 4493–4499 (1985). [CrossRef] [PubMed]

_{eff}, as a function of frequency is shown in Fig. 3(a). By use of the relation α=k

_{0}Im(n

_{eff}), one can further obtain the attenuate coefficient α. The calculated α value is shown in Fig. 3(b). One can see that the attenuation is smaller than 2×10

^{-3}cm

^{-1}in the range of 0.1~1THz. This result explicitly shows the superiority of a copper wire waveguide.

**432**, 376–379 (2004). [CrossRef]

**432**, 376–379 (2004). [CrossRef]

## 4. Conclusions

## References and links

1. | D. M. Mittleman, ed. |

2. | P. R. Smith, D. H. Auston, and M. C. Nuss, “Subpicosecond photoconducting dipole antennas,” IEEE J. Quant. Electron. |

3. | M. Exter and D. Grischkowsky, “Characterization of an optoelectronic terahertz beam system,” IEEE Trans. Microwave Theory Tech. |

4. | P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, “Generation and detection of terahertz pulses from biased semiconductor antennas,” J. Opt. Soc. Am. B |

5. | D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, “T-ray imaging,” IEEE J. Select.Top. Quant. Electron. |

6. | R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, “Chemical recognition of gases and gas mixtures with terahertz waves,” Opt. Lett. |

7. | R. M. Woodward, V. P. Wallace, D. D. Arnone, E. H. Linfield, and M. Pepper, “Terahertz pulsed imaging of skin cancer in the time and frequency domain,” J. Biol. Phys. |

8. | K. Kawase, Y. Ogawa, and Y. Watanabe, “Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,” Opt. Express |

9. | S. Wang and X. -C. Zhang, “Pulsed terahertz tomography,” J. Phys. D |

10. | R. W. McGowan, G. Gallot, and D. Grischkowsky, “Propagation of ultrawideband short pulses of THz radiation through submillimeter-diameter circular waveguides,” Opt. Lett. |

11. | G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, “Terahertz waveguides,” J. Opt. Soc. Am. B |

12. | R. Mendis and D. Grischkowsky, “Plastic ribbon THz waveguides,” J. Appl. Phys. |

13. | S. P. Jamison, R. W. McGown, and D. Grischkowsky, “Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fiber,” Appl. Phys. Lett. |

14. | R. Mendis and D. Grischkowsky, “Undistorted guided-wave propagation of subpicosecond terahertz pulses,” Opt. Lett. |

15. | R. Mendis and D. Grischkowsky, “THz interconnect with low loss and low group velocity dispersion,” IEEE Microwave Wireless Comp. Lett. |

16. | S. Coleman and D. Grischkowsky, “A THz transverse electromagnetic mode two-dimensional interconnect layer incorporating quasi-optics,” Appl. Phys. Lett. |

17. | K. Wang and D. M. Mittleman, “Metal wires for terahertz wave guiding,” Nature (London) , |

18. | H. Raether, |

19. | |

20. | Q. Cao and Ph. Lalanne, “Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. |

21. | S. I. Bozhevolnyi, “Waveguiding in surface plasmon polariton band gap structures,” Phys. Rev. Lett. |

22. | M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. |

23. | C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, “Surface polaritons in a circularly cylindrical interface: surface plasmons,” Phys. Rev. B |

24. | U. Schröter and A. Dereux, “Surface plasmon polaritons on metal cylinders with dielectric core,” Phys. Rev. B |

25. | M. Born and E. Wolf, |

26. | G. N. Watson, |

27. | M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. |

28. | http://hyperphysics.phy-astr.gsu.edu/hbase/tables/magprop.html#c1 |

29. | |

30. | C. Weber and J. Fajans, “Saturation in “nonmagnetic” stainless steel,” Rev. Scientific Instruments |

31. | S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. |

32. | R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. |

33. | Q. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(240.6680) Optics at surfaces : Surface plasmons

(260.3090) Physical optics : Infrared, far

(260.3910) Physical optics : Metal optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: December 15, 2004

Revised Manuscript: January 12, 2005

Published: January 24, 2005

**Citation**

Qing Cao and Jürgen Jahns, "Azimuthally polarized surface plasmons as effective terahertz waveguides," Opt. Express **13**, 511-518 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-2-511

Sort: Journal | Reset

### References

- D. M. Mittleman, ed. Sensing with Terahertz Radiation (Springer, Heidelberg, 2002).
- P. R. Smith, D. H. Auston, and M. C. Nuss, �??Subpicosecond photoconducting dipole antennas,�?? IEEE J. Quant. Electron. 24, 255-260 (1998). [CrossRef]
- M. Exter and D. Grischkowsky, �??Characterization of an optoelectronic terahertz beam system,�?? IEEE Trans. Microwave Theory Tech. 38, 1684-1691 (1990). [CrossRef]
- P. U. Jepsen, R. H. Jacobsen, and S. R. Keiding, �??Generation and detection of terahertz pulses from biased semiconductor antennas,�?? J. Opt. Soc. Am. B 13, 2424-2436 (1996). [CrossRef]
- D. M. Mittleman, R. H. Jacobsen, and M. C. Nuss, �??T-ray imaging,�?? IEEE J. Select.Top. Quant. Electron. 2, 679-692 (1996). [CrossRef]
- R. H. Jacobsen, D. M. Mittleman, and M. C. Nuss, �??Chemical recognition of gases and gas mixtures with terahertz waves,�?? Opt. Lett. 21, 2011-2013 (1996). [CrossRef] [PubMed]
- R. M. Woodward, V. P. Wallace, D. D. Arnone, E. H. Linfield, and M. Pepper, �??Terahertz pulsed imaging of skin cancer in the time and frequency domain,�?? J. Biol. Phys. 29, 257-261 (2003). [CrossRef]
- K. Kawase, Y. Ogawa, and Y. Watanabe, �??Non-destructive terahertz imaging of illicit drugs using spectral fingerprints,�?? Opt. Express 11, 2549-2554 (2003). [CrossRef] [PubMed]
- S. Wang and X. �??C. Zhang, �??Pulsed terahertz tomography,�?? J. Phys. D 37, R1-R36 (2004). [CrossRef]
- R. W. McGowan, G. Gallot, and D. Grischkowsky, �??Propagation of ultrawideband short pulses of THz radiation through submillimeter-diameter circular waveguides,�?? Opt. Lett. 24, 1431-1433 (1999). [CrossRef]
- G. Gallot, S. P. Jamison, R. W. McGowan, and D. Grischkowsky, �??Terahertz waveguides,�?? J. Opt. Soc. Am. B 17, 851-863 (2000). [CrossRef]
- R. Mendis and D. Grischkowsky, �??Plastic ribbon THz waveguides,�?? J. Appl. Phys. 88, 4449-4451 (2000). [CrossRef]
- S. P. Jamison, R. W. McGown, and D. Grischkowsky, �??Single-mode waveguide propagation and reshaping of sub-ps terahertz pulses in sapphire fiber,�?? Appl. Phys. Lett. 76, 1987-1989 (2000). [CrossRef]
- R. Mendis and D. Grischkowsky, �??Undistorted guided-wave propagation of subpicosecond terahertz pulses,�?? Opt. Lett. 26, 846-848 (2001). [CrossRef]
- R. Mendis and D. Grischkowsky, �??THz interconnect with low loss and low group velocity dispersion,�?? IEEE Microwave Wireless Comp. Lett. 11, 444-446 (2001). [CrossRef]
- S. Coleman and D. Grischkowsky, �??A THz transverse electromagnetic mode two-dimensional interconnect layer incorporating quasi-optics,�?? Appl. Phys. Lett. 83, 3656-3658 (2003). [CrossRef]
- K. Wang and D. M. Mittleman, �??Metal wires for terahertz wave guiding,�?? Nature (London), 432, 376-379 (2004). [CrossRef]
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings, (Springer, Berlin, 1988).
- <a href="http://www.surfaceplasmonoptics.org">http://www.surfaceplasmonoptics.org</a>
- Q. Cao and Ph. Lalanne, �??Negative role of surface plasmons in the transmission of metallic gratings with very narrow slits,�?? Phys. Rev. Lett. 88, 057403 (2002). [CrossRef] [PubMed]
- S. I. Bozhevolnyi, �??Waveguiding in surface plasmon polariton band gap structures,�?? Phys. Rev. Lett. 86, 3008-3011 (2001). [CrossRef] [PubMed]
- M. I. Stockman, �??Nanofocusing of optical energy in tapered plasmonic waveguides,�?? Phys. Rev. Lett. 93, 137404 (2004). [CrossRef] [PubMed]
- C. A. Pfeiffer, E. N. Economou, and K. L. Ngai, �??Surface polaritons in a circularly cylindrical interface: surface plasmons,�?? Phys. Rev. B 10, 3038-3051 (1974). [CrossRef]
- U. Schröter and A. Dereux, �??Surface plasmon polaritons on metal cylinders with dielectric core,�?? Phys. Rev. B 64, 125420 (2001). [CrossRef]
- M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon Press, Oxford, 1975).
- G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, Cambridge, UK 1966).
- M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, �??Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,�?? Appl. Opt. 24, 4493-4499 (1985). [CrossRef] [PubMed]
- <a href="http://hyperphysics.phy-astr.gsu.edu/hbase/tables/magprop.html#c1">http://hyperphysics.phy-astr.gsu.edu/hbase/tables/magprop.html#c1</a>
- <a href="http://www.stainless-rebar.org/grades.htm">http://www.stainless-rebar.org/grades.htm</a>
- C. Weber and J. Fajans, �??Saturation in �??nonmagnetic�?? stainless steel,�?? Rev. Scientific Instruments 69, 3695- 3696 (1998). [CrossRef]
- S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, �??Focusing light to a tighter spot,�?? Opt. Commun. 179, 1-7 (2000). [CrossRef]
- R. Dorn, S. Quabis, and G. Leuchs, �??Sharper focus for a radially polarized light beam,�?? Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
- Q. Zhan and J. R. Leger, �??Focus shaping using cylindrical vector beams,�?? Opt. Express 10, 324-331 (2002). [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.