OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 24 — Nov. 23, 2009
  • pp: 22235–22239
« Show journal navigation

Magnetic field distribution of a highly focused radially-polarized light beam

Yaoju Zhang and Biaofeng Ding  »View Author Affiliations


Optics Express, Vol. 17, Issue 24, pp. 22235-22239 (2009)
http://dx.doi.org/10.1364/OE.17.022235


View Full Text Article

Acrobat PDF (133 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

A simple expression for the magnetic filed of a highly focused radially polarized light is derived and the incorrect results for the time averaged Poynting vector and the trapping stability for a gold particle presented in the paper “Trapping metallic Rayleigh particles with radial polarization” by Zhan (Opt. Express 12, 3377–3382 (2004)) are corrected.

© 2009 Optical Society of America

It is well-known that when radially polarized beams are tightly focused, a strong longitudinal electric field in the focal region can be generated [1

1. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

] and a shaper circular spot at the focus can be formed [2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

4

4. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [CrossRef]

]. These features are suitable for many applications such as optical data storage [5

5. Y. Zhang and J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009). [CrossRef] [PubMed]

], high-resolution microscopy [3

3. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

], material processing [6

6. V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999). [CrossRef]

], particle acceleration [7

7. S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32(27), 5222–5229 (1993). [CrossRef] [PubMed]

] and optical trapping [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

]. In 2004, Zhan calculated the radiation forces acting on metallic Rayleigh particles using a highly focused radially polarized beam [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

]. To calculate the absorption and scattering forces, the magnetic field in the focal region has to be known. Although Zhan pointed out that the magnetic field is calculated in a similar way to that of the electric field calculation, he didn’t present an obvious expression of the magnetic field distribution, except assuming the magnetic field at the pupil plane is aligned along the azimuthal direction. However, he presented the numerical results of the time averaged Poynting vector and calculations for the absorption and scattering forces based on these results, some of which, we believe, are in error. To analyze the causes of the errors in Ref [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

], we first derive the magnetic field expression of a highly focused radially polarized light beam in this paper and then show the correct distribution of the Pointing vector and the trapping stability for a gold particle in the focal region.

Figure 1 illuminates the geometry of the problem. The incident radially polarized electric field is assumed to have a planar phase front at plane 0 which is the entrance pupil to the optical system. An aplanatic lens produces a converging spherical wave at focal sphere 1 (which may be taken to represent the exit pupil). To obtain diffraction filed distributions, this spherical wave is first expanded into an angular spectrum of plane waves [9

9. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959). [CrossRef]

]. For high-NA systems, the Debye approximation can be used to calculate this angular spectrum. Only those plane waves contribute to the field in the focus the propagation directions of which correspond to the geometric optical rays. In this approximation, diffraction effects due to the edge of the aperture are not considered. The field in the focus can be evaluated by superposing those plane waves, keeping track of the phase and the direction of polarization. Following Quabis et al. [2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

], we can express the electric field near focus as a diffraction integral over the vector field amplitude A(θ)Pe (θ, ϕ), giving
E=ikf2π0αdθ02πdϕsinθcosθA(θ)Pe(θ,ϕ)exp[ikrsinθcos(ϕφ)]exp(ikzcosθ),
(1)
where α is the convergence angle of the lens, f is the focal length of the lens, k = 2πn/λ is the wavenumber in the imaging space with the refractive index n. A(θ) represents the amplitude and phase distribution at the exit pupil. Pe is the electric polarization vector in diffraction field. When the radially polarized beam illuminates the aplanatic lens, the electric polarization vector Pe can be obtained as [2

2. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

]
Pe=[cosθcosϕcosθsinϕsinθ].
(2)

Fig. 1 Ray-tracing model for focusing a radially polarized incoming light

After substituting Eq. (2) into Eq. (1) and finishing the integral over φ, we can obtain the electric field in the focal region. In the cylindrical coordinate system, we find that the azimuthal component of the electric field is zero everywhere in the diffraction field and the non-zero radial and longitudinal components are expressed as
Er(r,z)=η0αcosθA(θ)sin2θJ1(krsinθ)exp(ikzcosθ)dθ,
(3)
Ez(r,z)=2iη0αcosθA(θ)sin2θJ0(krsinθ)exp(ikzcosθ)dθ,
(4)
where η = πfn/λ.

Similarly, we can express the magnetic field in the focal region
H=iηnπμ0c0αdθ02πdϕcosθsinθA(θ)Pm(θ,ϕ)exp[ikrsinθcos(ϕφ)]exp(ikzcosθ),
(5)
where c and μ0 are the light speed and magnetic permeability in vacuum, respectively, and Pm is the magnetic polarization vector in the diffraction field. According to the relationship between an electric polarization vector and a corresponding magnetic polarization vector in isotropic dielectric, Pm (θ, ϕ) = × Pe (θ, ϕ), where = (– sinθcosϕ, – sinθsinϕ, cosθ) is the unit vector of the wave vector (see Fig. 1), we obtain the magnetic polarization vector in the diffraction field
Pm=[sinϕcosϕ0].
(6)

From Eq. (6) we can find that the magnetic polarization vector in the diffraction field is aligned along the azimuthal direction, which is same as the assuming direction by Zhan.

Substituting Eq. (6) into Eq. (5), the integral over φ can yields the magnetic field. In the cylindrical coordinate system, we obtain this magnetic field
H(r,z)=Hϕ(r,z)e^ϕ=2ηnμ0c0αcosθA(θ))sinθJ1(krsinθ)exp(ikzcosθ)dθe^ϕ.
(7)

It is clear that the magnetic field in the focal region is azimuthally polarized and it has an on-axis null at all distance from the paraxial focus. The radial and longitudinal components of the magnetic field are zero everywhere in the diffraction field. As expected, and in a manner consistent with Maxwell’s equations, the azimuthal magnetic field propagates as a purely transverse polarization through the entire focal region. It is emphasized that the distribution of the magnetic field (Eq. (7)) is different from that of the radial component of the electric field in Eq. (3).

The time averaged Poynting vector can be written as
<S>=Re(E×H*)/2=Re(ErHϕ*)e^z/2Re(EzHϕ*)e^ρ/2.
(8)

Equation (8) shows that the time averaged Poynting vector is not completely along the longitudinal direction. In Fig. 2, we present the axial components <S>z of the Poynting vector obtained by focusing a radially polarized beam with uniform amplitude, respectively. For comparison, the calculation parameters are chosen as same as those in Ref [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

], i.e., n = 1.33, λ = 1.047μm, NA = 0.95n (α = 71.8°), the power of laser is 100 mW, and a following annulus pupil apodization function with NA1 = 0.6 is used.
A(θ)={E0,sin1(NA1)<θ<sin1(NA/n)0,otherwise
(9)
where E0 is a constant. Except some differences in details which are not shown, the formal distribution of Fig. 2 is, on the whole, similar to that in Fig. 3 in Ref [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

]. However, the intensity of <S>z is much larger than that that in Fig. 3 in Ref [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

]. It is well known that the scattering and absorption forces acting on a trapping particle are in direct proportion to the Poynting vector. Therefore the increase of <S>z results in the degradation of the axial trapping stability. To illuminate this, we calculate the radiation force and trapping stability using the Rayleigh scattering theory [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

,10

10. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996). [CrossRef]

] in the following.

Fig. 2 The time averaged Poynting vector based on the exact magnetic field near the focal region of a highly focused radial polarized light beam. (a) 2-dimensional distribution in the xz plane; (b) line scan of (a) at the focal plane.
Fig. 3 Calculated radiation forces on a 19.1 nm (radius) gold particle in the xz plane. (a) Transverse gradient force (Fgrad,x); (b) axial gradient force (Fgrad,z); (c) sum of transverse scattering and absorption forces (Fscat,x + Fabs,x); (d) sum of axial scattering and absorption forces (Fscat,z + Fabs,z). All radiation forces are in unit of pico-Newton (pN).

The error in the results presented in the paper [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

] is caused by the use of an incorrect magnetic field formula. The author in Ref. [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

], we guess, most likely consider that the magnetic field distribution is same as the radial component distribution of the electric field. To validate our guess, we calculate the axial trapping stability using the produc of Eq. (3) and the factor of n / μ0c and obtain R = 23.1 and 28.4 for NA1 = 0.01 and 0.6, respectively, which are basically same as 24.9 (NA1 = 0.01) and 26.2 (NA1 = 0.6) of Zhan.

In conclusion, we have obtained a simple formula to calculate the magnetic filed of a highly focused radially polarized light beam and presented the correct results of the time averaged Poynting vector and the axial trapping stability corresponding to those in error in the paper by Zhan [8

8. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

]. The error in his results is most likely caused by the misuse of the radial component of the electric field which is considered as same as the magnetic field distribution. Therefore, his calculations, based on a incorrect magnetic field formula, for the absprption and scattering forces need to be reexamined.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under contract 60777005.

References and links

1.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

2.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “The focus of light-theoretical calculation and experimental tomographic reconstruction,” Appl. Phys. B 72, 109–113 (2001).

3.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

4.

Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [CrossRef]

5.

Y. Zhang and J. Bai, “Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams,” Opt. Express 17(5), 3698–3706 (2009). [CrossRef] [PubMed]

6.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D 32(13), 1455–1461 (1999). [CrossRef]

7.

S. C. Tidwell, G. H. Kim, and W. D. Kimura, “Efficient radially polarized laser beam generation with a double interferometer,” Appl. Opt. 32(27), 5222–5229 (1993). [CrossRef] [PubMed]

8.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

9.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253(1274), 358–379 (1959). [CrossRef]

10.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5–6), 529–541 (1996). [CrossRef]

OCIS Codes
(020.7010) Atomic and molecular physics : Laser trapping
(050.1960) Diffraction and gratings : Diffraction theory
(260.5430) Physical optics : Polarization

ToC Category:
Photonic Crystals

History
Original Manuscript: October 1, 2009
Revised Manuscript: November 6, 2009
Manuscript Accepted: November 9, 2009
Published: November 19, 2009

Citation
Yaoju Zhang and Biaofeng Ding, "Magnetic field distribution of a highly focused radially-polarized light beam," Opt. Express 17, 22235-22239 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-22235


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. K. S. Youngworth and T. G. Brown, "Focusing of high numerical aperture cylindrical-vector beams," Opt. Express 7(2), 77-87 (2000). [CrossRef] [PubMed]
  2. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, "The focus of light-theoretical calculation and experimental tomographic reconstruction," Appl. Phys. B 72, 109-113 (2001).
  3. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized light beam," Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]
  4. Y. Kozawa and S. Sato, "Sharper focal spot formed by higher-order radially polarized laser beams," J. Opt. Soc. Am. A 24(6), 1793-1798 (2007). [CrossRef]
  5. Y. Zhang, and J. Bai, "Improving the recording ability of a near-field optical storage system by higher-order radially polarized beams," Opt. Express 17(5), 3698-3706 (2009). [CrossRef] [PubMed]
  6. V. G. Niziev and A. V. Nesterov, "Influence of beam polarization on laser cutting efficiency," J. Phys. D 32(13), 1455-1461 (1999). [CrossRef]
  7. S. C. Tidwell, G. H. Kim, and W. D. Kimura, "Efficient radially polarized laser beam generation with a double interferometer," Appl. Opt. 32(27), 5222-5229 (1993). [CrossRef] [PubMed]
  8. Q. Zhan, "Trapping metallic Rayleigh particles with radial polarization," Opt. Express 12(15), 3377-3382 (2004). [CrossRef] [PubMed]
  9. B. Richards, and E. Wolf, "Electromagnetic diffraction in optical systes II. Structure of the image field in an aplanatic system," Proc. R. Soc. Lond. A 253(1274), 358-379 (1959). [CrossRef]
  10. Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124(5-6), 529-541 (1996). [CrossRef]

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.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited