Abstract

In this paper, we investigate the field distribution in the focal volume of an aberrated radially polarized beam. Using two different forms of the vectorial diffraction theory, we show that the presence of defocus in the beam displaces both the axially and the radially polarized fields parallel to the optical axis of the focusing lens, while the presence of spherical aberration primarily shifts the longitudinally polarized field only. This facilitates axial separation of the two orthogonally polarized field components, resulting in a significant boost to the ratio of the peak longitudinally polarized field to the peak laterally polarized field in the focal plane. We further show that with an appropriate combination of oppositely signed defocus and spherical aberration, the energy density in the focal volume due to the longitudinally polarized field can be caused to peak at the focal plane. The results obtained are expected to be beneficial to the applications requiring a stronger longitudinally polarized focal field relative to the laterally polarized focal field component.

© 2012 Optical Society of America

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References

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  1. V. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd. 1, 1–36 (1919).
  2. 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, 358 (1959).
    [CrossRef]
  3. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef]
  4. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef]
  5. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24, 1793–1798 (2007).
    [CrossRef]
  6. Q. Tan, K. Cheng, Z. Zhou, and G. Jin, “Diffractive superresolution elements for radially polarized light,” J. Opt. Soc. Am. A 27, 1355–1360 (2010).
    [CrossRef]
  7. S. Yan, B. Yao, W. Zhao, and M. Lei, “Generation of multiple spherical spots with a radially polarized beam in a 4π focusing system,” J. Opt. Soc. Am. A 27, 2033–2037 (2010).
    [CrossRef]
  8. S. Yan, and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
    [CrossRef]
  9. Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
    [CrossRef]
  10. Y. Zhang, T. Suyama, and B. Ding, “Longer axial trap distance and larger radial trap stiffness using a double-ring radially polarized beam,” Opt. Lett. 35, 1281–1283 (2010).
    [CrossRef]
  11. L. Novotny, M. Beversluis, K. Youngworth, and T. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef]
  12. H. Ishitobi, I. Nakamura, N. Hayazawa, Z. Sekkat, and S. Kawata, “Orientational imaging of single molecules by using azimuthal and radial polarizations,” J. Phys. Chem. B 114, 2565–2571 (2010).
    [CrossRef]
  13. Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
    [CrossRef]
  14. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).
  15. B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using fast Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
    [CrossRef]
  16. M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002).
    [CrossRef]
  17. B. R. Boruah and M. A. A. Neil, “Laser scanning confocal microscope with programmable amplitude, phase, and polarization of the illumination beam,” Rev. Sci. Instrum. 80, 013705 (2009).
    [CrossRef]

2010 (5)

2009 (2)

B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using fast Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
[CrossRef]

B. R. Boruah and M. A. A. Neil, “Laser scanning confocal microscope with programmable amplitude, phase, and polarization of the illumination beam,” Rev. Sci. Instrum. 80, 013705 (2009).
[CrossRef]

2008 (1)

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[CrossRef]

2007 (2)

S. Yan, and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

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

2003 (1)

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

2002 (1)

2001 (1)

L. Novotny, M. Beversluis, K. Youngworth, and T. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

2000 (1)

1959 (1)

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, 358 (1959).
[CrossRef]

1919 (1)

V. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd. 1, 1–36 (1919).

Beversluis, M.

L. Novotny, M. Beversluis, K. Youngworth, and T. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Boruah, B. R.

B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using fast Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
[CrossRef]

B. R. Boruah and M. A. A. Neil, “Laser scanning confocal microscope with programmable amplitude, phase, and polarization of the illumination beam,” Rev. Sci. Instrum. 80, 013705 (2009).
[CrossRef]

Brown, T.

L. Novotny, M. Beversluis, K. Youngworth, and T. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

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

Cheng, K.

Ding, B.

Y. Zhang, T. Suyama, and B. Ding, “Longer axial trap distance and larger radial trap stiffness using a double-ring radially polarized beam,” Opt. Lett. 35, 1281–1283 (2010).
[CrossRef]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

Dorn, R.

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

Harman, Z.

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[CrossRef]

Hayazawa, N.

H. Ishitobi, I. Nakamura, N. Hayazawa, Z. Sekkat, and S. Kawata, “Orientational imaging of single molecules by using azimuthal and radial polarizations,” J. Phys. Chem. B 114, 2565–2571 (2010).
[CrossRef]

Ignatowsky, V.

V. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd. 1, 1–36 (1919).

Ishitobi, H.

H. Ishitobi, I. Nakamura, N. Hayazawa, Z. Sekkat, and S. Kawata, “Orientational imaging of single molecules by using azimuthal and radial polarizations,” J. Phys. Chem. B 114, 2565–2571 (2010).
[CrossRef]

Jin, G.

Juškaitis, R.

Kawata, S.

H. Ishitobi, I. Nakamura, N. Hayazawa, Z. Sekkat, and S. Kawata, “Orientational imaging of single molecules by using azimuthal and radial polarizations,” J. Phys. Chem. B 114, 2565–2571 (2010).
[CrossRef]

Keitel, C. H.

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[CrossRef]

Kozawa, Y.

Lei, M.

Leuchs, G.

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

Massoumian, F.

Nakamura, I.

H. Ishitobi, I. Nakamura, N. Hayazawa, Z. Sekkat, and S. Kawata, “Orientational imaging of single molecules by using azimuthal and radial polarizations,” J. Phys. Chem. B 114, 2565–2571 (2010).
[CrossRef]

Neil, M. A. A.

B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using fast Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
[CrossRef]

B. R. Boruah and M. A. A. Neil, “Laser scanning confocal microscope with programmable amplitude, phase, and polarization of the illumination beam,” Rev. Sci. Instrum. 80, 013705 (2009).
[CrossRef]

M. A. A. Neil, F. Massoumian, R. Juškaitis, and T. Wilson, “Method for the generation of arbitrary complex vector wave fronts,” Opt. Lett. 27, 1929–1931 (2002).
[CrossRef]

Novotny, L.

L. Novotny, M. Beversluis, K. Youngworth, and T. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

Quabis, S.

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

Richards, B.

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, 358 (1959).
[CrossRef]

Salamin, Y. I.

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[CrossRef]

Sato, S.

Sekkat, Z.

H. Ishitobi, I. Nakamura, N. Hayazawa, Z. Sekkat, and S. Kawata, “Orientational imaging of single molecules by using azimuthal and radial polarizations,” J. Phys. Chem. B 114, 2565–2571 (2010).
[CrossRef]

Suyama, T.

Y. Zhang, T. Suyama, and B. Ding, “Longer axial trap distance and larger radial trap stiffness using a double-ring radially polarized beam,” Opt. Lett. 35, 1281–1283 (2010).
[CrossRef]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

Tan, Q.

Wilson, T.

Wolf, E.

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, 358 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Yan, S.

S. Yan, B. Yao, W. Zhao, and M. Lei, “Generation of multiple spherical spots with a radially polarized beam in a 4π focusing system,” J. Opt. Soc. Am. A 27, 2033–2037 (2010).
[CrossRef]

S. Yan, and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Yao, B.

S. Yan, B. Yao, W. Zhao, and M. Lei, “Generation of multiple spherical spots with a radially polarized beam in a 4π focusing system,” J. Opt. Soc. Am. A 27, 2033–2037 (2010).
[CrossRef]

S. Yan, and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Youngworth, K.

L. Novotny, M. Beversluis, K. Youngworth, and T. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

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

Zhang, Y.

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

Y. Zhang, T. Suyama, and B. Ding, “Longer axial trap distance and larger radial trap stiffness using a double-ring radially polarized beam,” Opt. Lett. 35, 1281–1283 (2010).
[CrossRef]

Zhao, W.

Zhou, Z.

J. Opt. Soc. Am. A (3)

J. Phys. Chem. B (1)

H. Ishitobi, I. Nakamura, N. Hayazawa, Z. Sekkat, and S. Kawata, “Orientational imaging of single molecules by using azimuthal and radial polarizations,” J. Phys. Chem. B 114, 2565–2571 (2010).
[CrossRef]

Opt. Commun. (1)

B. R. Boruah and M. A. A. Neil, “Focal field computation of an arbitrarily polarized beam using fast Fourier transforms,” Opt. Commun. 282, 4660–4667 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (2)

S. Yan, and B. Yao, “Radiation forces of a highly focused radially polarized beam on spherical particles,” Phys. Rev. A 76, 053836 (2007).
[CrossRef]

Y. Zhang, B. Ding, and T. Suyama, “Trapping two types of particles using a double-ring-shaped radially polarized beam,” Phys. Rev. A 81, 023831 (2010).
[CrossRef]

Phys. Rev. Lett. (3)

L. Novotny, M. Beversluis, K. Youngworth, and T. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[CrossRef]

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

Y. I. Salamin, Z. Harman, and C. H. Keitel, “Direct high-power laser acceleration of ions for medical applications,” Phys. Rev. Lett. 100, 155004 (2008).
[CrossRef]

Proc. R. Soc. Lond. A (1)

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, 358 (1959).
[CrossRef]

Rev. Sci. Instrum. (1)

B. R. Boruah and M. A. A. Neil, “Laser scanning confocal microscope with programmable amplitude, phase, and polarization of the illumination beam,” Rev. Sci. Instrum. 80, 013705 (2009).
[CrossRef]

Trans. Opt. Inst. Petrograd. (1)

V. Ignatowsky, “Diffraction by a lens having arbitrary opening,” Trans. Opt. Inst. Petrograd. 1, 1–36 (1919).

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

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Figures (11)

Fig. 1.
Fig. 1.

(a) Focusing of a radially polarized beam. (b) Radial polarization profile in the entrance pupil of the lens.

Fig. 2.
Fig. 2.

(a) Plots of ϕd, ϕs, and ϕt=ϕd+ϕs for a=56 and b=36, and (b) a=56 and b=36.

Fig. 3.
Fig. 3.

Plots of axial positions of maximum Ir and maximum IZ in the XZ plane of the focal volume, for increasing values of a, keeping b=0, and then increasing values of b, keeping a=22.

Fig. 4.
Fig. 4.

Plots IZ(u=0,v=0) as functions of a for selected values of b, each normalized by IZ(u=0,v=0) with (a=0,b=0).

Fig. 5.
Fig. 5.

(a) Plots of IZ, Ir, and IT(=IZ+Ir), in the focal plane versus the radial optical coordinate, for (a,b)=(0,0), (b) (a,b)=(22,16), and (c) (a,b)=(56,36). Each plot is normalized by the corresponding IZ(u=0,v=0).

Fig. 6.
Fig. 6.

(a) XZ plane (i) radial, (ii) longitudinal, (iii) total energy densities for (a,b)=(0,0), (b) (a,b)=(22,16), and (c) (a,b)=(56,36). The scale bar along the vertical (X) and along the horizontal (Z) are equal to 24v and 24u, respectively. Each image is self normalized and the center of each image is the paraxial focal point.

Fig. 7.
Fig. 7.

(a) Axial positions of peak longitudinal and peak radial energy densities as a goes from 0 to 22 and then b goes from 0 to 16, (b) as a goes from 0 to 56 and then b goes from 0 to 36.

Fig. 8.
Fig. 8.

(a) XY plane (i) radial, (ii) longitudinal, (iii) total energy densities for (a,b)=(0,0), (b) (a,b)=(22,16), and (c) (a,b)=(56,36). The scale bar along the vertical (Y) and along the horizontal (X) are equal to 7.5v. Each image is self normalized and the center of each image is the paraxial focal point.

Fig. 9.
Fig. 9.

(a) Normalized longitudinal energy density versus u, (b) normalized total energy density versus v, for (a,b)=(0,0), (22,16) and (56,36).

Fig. 10.
Fig. 10.

(a) Poynting vectors in the XZ plane for the optimized pairs (a,b)=(0,0), (b) (a,b)=(22,16), and (c) (a,b)=(56,36).

Fig. 11.
Fig. 11.

(a) XZ plane radial, (b) longitudinal, and (c) total energy densities for (a,b)=(56,36). The scale bars along the vertical (X) and along the horizontal (Z) are equal to 24v and 24u, respectively. Each image is self normalized, and the center of each image is the paraxial focal point.

Tables (1)

Tables Icon

Table 1. For Selected Optimized Pairs of (a,b), the Ratio of Peak Longitudinal Energy Density to the Peak Radial Energy Density in the Focal Plane (LTR=Max(IZ)/Max(Ir)), the Peak Longitudinal Energy Density in the Focal Plane for an Optimized Pair (a,b) Normalized by the Peak Longitudinal Energy Density in the Focal Plane in the Case of a Normal Radially Polarized Beam (ZR=Max(IZ)/Max(IZ(a=0,b=0))), the FWHM of Total Energy Density in the Focal Plane Expressed in the Unit of Radial Optical Coordinate v, and the Axial Separation (ZS) between the Peaks of the Longitudinal Energy Density and the Radial Energy Density in the XZ Plane through Focus, Expressed in the Unit of Axial Optical Coordinate ua

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

EXX=iAπ0α02πRXeik0rPcosϵsinθdθdϕ,EXY=iAπ0α02πRYeik0rPcosϵsinθdθdϕ,EXZ=iAπ0α02πRZeik0rPcosϵsinθdθdϕ,
RX={cosθ+(1cosθ)sin2ϕ}cosθ,RY=(1cosθ)cosϕsinϕcosθ,RZ=sinθcosϕcosθ,rPcosϵ=kxxP+kyyP+kzzP
cosϵ=cosθcosθP+sinθsinθPcos(ϕϕP)
(RX,RY,RZ)(RXeiϕt,RYeiϕt,RZeiϕt),
Er=A0αCRsin(2θ)J1(k0rPsinθ)dθ,
EZ=2iA0αCRsin2θJ0(k0rPsinθ)dθ,
EXX(P)=F[l0(kx,ky)GX(kx,ky)],EXY(P)=F[l0(kx,ky)GY(kx,ky)],EXZ(P)=F[l0(kx,ky)GZ(kx,ky)],
GX(kx,ky)=eikzzPk0kzk0ky2+kzkx2k0kr2,GY(kx,ky)=eikzzPk0kz(kzk0)kxkyk0kr2,GZ(kx,ky)=eikzzPk0kzkxk0.
(EYX,Y,Z)=Rc(F[(QYX,Y,Z)]).
QYX(kx,ky)=l0(kx,ky)GY(kx,ky),QYY(kx,ky)=l0(kx,ky)GX(kx,ky),QYZ(kx,ky)=l0(kx,ky)GZ(kx,ky),
(l0)=Rc(l0)
(EX)=(EXX)+(EYX),(EY)=(EXY)+(EYY),(EZ)=(EXZ)+(EYZ).
(cosϕsinϕ).

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