Abstract

The propagation and focusing properties of partially coherent vector beams including radially polarized and azimuthally polarized (AP) beams are theoretically and experimentally investigated. The beam profile of a partially coherent radially or AP beam can be shaped by adjusting the initial spatial coherence length. The dark hollow, flat-topped, and Gaussian beam spots can be obtained, which will be useful in trapping particles. The experimental observations are consistent with the theoretical results.

© 2012 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000).
    [CrossRef]
  2. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [CrossRef]
  3. Q. W. Zhan and J. R. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002).
  4. K. Hu, Z. Chen, and J. Pu, “Tight focusing properties of hybridly polarized vector beams,” J. Opt. Soc. Am. A 29, 1099–1104 (2012).
    [CrossRef]
  5. K. Hu, Z. Chen, and J. Pu, “Generation of super-length optical needle by focusing hybridly polarized vector beams through a dielectric interface,” Opt. Lett. 37, 3303–3305 (2012).
    [CrossRef]
  6. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008).
    [CrossRef]
  7. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12, 3377–3382 (2004).
    [CrossRef]
  8. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [CrossRef]
  9. B. Stick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
    [CrossRef]
  10. J. Li, K. I. Ueda, M. Musha, A. Shirakawa, and L. X. Zhong, “Generation of radially polarized mode in Yb fiber laser by using a dual conical prism,” Opt. Lett. 31, 2969–2971 (2006).
    [CrossRef]
  11. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially polarized and azimuthally polarized light using space-invariant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
    [CrossRef]
  12. D. Deng, “Nonparaxial propagation of radially polarized light beams,” J. Opt. Soc. Am. B 23, 1228–1234 (2006).
    [CrossRef]
  13. E. Y. S. Yew and C. J. R. Sheppard, “Tight focusing of radially polarized Gaussian and Bessel-Gauss beams,” Opt. Lett. 32, 3417–3419 (2007).
    [CrossRef]
  14. Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
    [CrossRef]
  15. J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A 20, 1947–1980 (2003).
    [CrossRef]
  16. F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
    [CrossRef]
  17. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
    [CrossRef]
  18. S. R. Seshadri, “Partially coherent Gaussian Schell-model electromagnetic beams,” J. Opt. Soc. Am. A 16, 1373–1380 (1999).
    [CrossRef]
  19. H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101, 361–369 (2010).
    [CrossRef]
  20. Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagetic beams in turbulent atmosphere,” J. Opt. A: Pure Appl. Opt. 9, 1123–1130 (2007).
    [CrossRef]
  21. K. Duan and B. Lu, “Vectorial nonparaxial propagation equation of elliptical Gaussian beams in the presence of a rectangular aperture,” J. Opt. Soc. Am. A 21, 1613–1620 (2004).
    [CrossRef]

2012 (2)

2010 (1)

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101, 361–369 (2010).
[CrossRef]

2008 (1)

2007 (2)

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagetic beams in turbulent atmosphere,” J. Opt. A: Pure Appl. Opt. 9, 1123–1130 (2007).
[CrossRef]

E. Y. S. Yew and C. J. R. Sheppard, “Tight focusing of radially polarized Gaussian and Bessel-Gauss beams,” Opt. Lett. 32, 3417–3419 (2007).
[CrossRef]

2006 (2)

2004 (2)

2003 (3)

Y. Cai, X. Lu, and Q. Lin, “Hollow Gaussian beams and their propagation properties,” Opt. Lett. 28, 1084–1086 (2003).
[CrossRef]

J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A 20, 1947–1980 (2003).
[CrossRef]

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 (2)

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially polarized and azimuthally polarized light using space-invariant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

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

2000 (2)

B. Stick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[CrossRef]

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

1999 (1)

1998 (2)

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Baykal, Y.

Beversluis, M. R.

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

Bomzon, Z.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially polarized and azimuthally polarized light using space-invariant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

Borghi, R.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Brown, T. G.

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

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

Cai, Y.

Chen, Z.

Deng, D.

Dorn, R.

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

Duan, K.

Eyyuboglu, H. T.

Gori, F.

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241–243 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Guattari, G.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Hasman, E.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially polarized and azimuthally polarized light using space-invariant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

Hecht, B.

B. Stick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[CrossRef]

Hu, K.

Kleiner, V.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially polarized and azimuthally polarized light using space-invariant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

Leger, J. R.

Leuchs, G.

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

Li, J.

Lin, Q.

Liu, D.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101, 361–369 (2010).
[CrossRef]

Lu, B.

Lu, X.

Musha, M.

Novotny, L.

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

B. Stick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[CrossRef]

Pu, J.

Quabis, S.

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

Santarsiero, M.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Seshadri, S. R.

Sheppard, C. J. R.

Shirakawa, A.

Stick, B.

B. Stick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[CrossRef]

Tervo, J.

J. Tervo, “Azimuthal polarization and partial coherence,” J. Opt. Soc. Am. A 20, 1947–1980 (2003).
[CrossRef]

Ueda, K. I.

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Wang, H.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101, 361–369 (2010).
[CrossRef]

Yew, E. Y. S.

Youngworth, K. S.

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

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

Zhan, Q.

Zhan, Q. W.

Zhong, L. X.

Zhou, Z.

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101, 361–369 (2010).
[CrossRef]

Appl. Phys. B (1)

H. Wang, D. Liu, and Z. Zhou, “The propagation of radially polarized partially coherent beam through an optical system in turbulent atmosphere,” Appl. Phys. B 101, 361–369 (2010).
[CrossRef]

Appl. Phys. Lett. (1)

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially polarized and azimuthally polarized light using space-invariant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79, 1587–1589 (2001).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

Z. Chen and J. Pu, “Propagation characteristics of aberrant stochastic electromagetic beams in turbulent atmosphere,” J. Opt. A: Pure Appl. Opt. 9, 1123–1130 (2007).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Opt. Express (4)

Opt. Lett. (5)

Phys. Rev. Lett. (3)

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

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

B. Stick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[CrossRef]

Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, and G. Guattari, “Beam coherence-polarization matrix,” Pure Appl. Opt. 7, 941–951 (1998).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1.

Propagation geometry of a partially coherent radially or AP beam through ABCD optical system in free space.

Fig. 2.
Fig. 2.

Experimental setup for generating a GSM RP beam or an AP beam and measuring its intensity. L1, L2, and L3, thin lenses; RGGP, rotating ground-glass plate; P1 and P2, linear polarizer; BPA, beam profile analyzer; PC-1 and PC-2, personal computer.

Fig. 3.
Fig. 3.

Generated radially partially coherent beam with σ0=1.166mm at z=20cm. (a)–(e) are theoretical calculations; (f)–(j) are corresponding experimental results. The black arrows represent the transmission axis forms an angle ϕ with the x-axis, 0, π/4, π/2, and 3π/4, respectively. The dimensions in the first and second rows are 0.5mm×0.5mm.

Fig. 4.
Fig. 4.

Generated azimuthally partially coherent beam with σ0=1.166mm at z=20cm. (a)–(e) are numerical calculations; (f)–(j) are corresponding experimental results. The black arrows represent the transmission axis of the polarizer forms an angle ϕ with the x-axis, ϕ=0, π/4, π/2, and 3π/4. The dimensions in the first and second rows are 0.5mm×0.5mm.

Fig. 5.
Fig. 5.

Propagation properties of partially coherent RP beam in free space with σ0=1.166mm at several propagation distances with z=20, 40, 60, 80, 100, and 120 cm. (a)–(f) are numerical calculations; (g)–(l) are corresponding experimental results. The dimensions of (a)–(g) are 0.5mm×0.5mm, and (h)–(l) are 0.75mm×0.75mm.

Fig. 6.
Fig. 6.

Propagation properties of partially coherent RP beam in free space with σ0=0.441mm at several propagation distances with z=20, 40, 60, 80, and 100 cm. (a)–(f) are numerical calculations; (g)–(l) are corresponding experimental results. The dimensions of (a)–(g) are 0.5mm×0.5mm, and (h)–(l) are 0.75mm×0.75mm.

Fig. 7.
Fig. 7.

Intensity distributions for partially coherent RP beam at the same propagation distances z=140cm with different coherent length in free space. (a)–(d) are numerical calculations, and the coherent lengths of the first rows are 1.166, 0.903, 0.607, and 0.491 mm, respectively; (e)–(h) are corresponding experimental results. The dimensions in the first and second rows are 0.5mm×0.5mm and 0.75mm×0.75mm, respectively.

Fig. 8.
Fig. 8.

Polarization distributions of the FT beam. The black arrows represent the transmission axis of the polarizer forms an angle ϕ with the x-axis, ϕ=0, π/4, π/2, and 3π/4, respectively. (a)–(e) are numerical calculations, and (f)–(i) are corresponding experimental results, respectively. The dimensions in the first and second rows are 0.5mm×0.5mm and 0.75mm×0.75mm.

Fig. 9.
Fig. 9.

Propagation properties of radially and AP partially coherent beam with the same coherent length at several propagation distances. (a)–(d) are numerical calculations, and (e)–(h) and (i)–(l) are corresponding experimental results of the radially and AP partially coherent beam. The dimensions in the first, second, and third rows are 0.5mm×0.5mm, 0.5mm×0.5mm, and 0.75mm×0.75mm, respectively.

Fig. 10.
Fig. 10.

Focusing properties of partially coherent RP beam with different coherent lengths. (a)–(c) are numerical calculation, and (d)–(f) are corresponding experimental results. The dimensions in the first, second, and third rows are 0.25mm×0.25mm, 0.25mm×0.25mm, and 0.375mm×0.375mm, respectively.

Equations (36)

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

Er(x,y,0)=E0[xw0exp(r2w2)ex+yw0exp(r2w2)ey],
Eθ(x,y,0)=E0[yw0exp(r2w2)ex+xw0exp(r2w2)ey].
Wαβ(r1,r2,z)=Eα(r1,z)Eβ*(r2,z)(α,β=x,y),
I(r,z)=Wxx(r,r,z)+Wyy(r,r,z).
Wrαβ(r10,r20,0)=E02w2exp(r102+r202w2)gαβ(r10,r20)α1β2,
Wθαβ(r10,r20,0)=E02w2exp(r102+r202w2)gαβ(r10,r20)α1β2.
gαβ(r10,r20)=exp[(x10x20)2σαβ2(y10y20)2σαβ2],
Wαβ(r1,r2,z)=1λ2B2+Wαβ(r10,r20,0)exp{ik2B[A(x102x202+y102y202)+D(x12x22+y12y22)2(x1x10x2x20+y1y10y2y20)]}dx10dy10dx20dy20,
Wrxx(r1,r2,z)=E021λ2B2w2π2axbxGxxexp[r124ax(ikB)2]exp(Sx12r124bx+Sx22r224bx+Sx1Sx22bxr1r2),
Sx1=1ax1σxx2ikB,
Sx2=ikB,
Gxx=12ax1bxσxx2[1+12bx(Sx1x1+Sx2x2)2]+12ax12bx(ikx1B)(Sx1x1+Sx2x2),
ax=1w2+1σxx2+ikA2B,
bx=1w2+1σxx2ikA2B1ax1σxx4,
Wryy(r1,r2,z)=E021λ2B2w2π2aybyGyyexp[r124ay(ikB)2]exp(Sy12r124by+Sy22r224by+Sy1Sy22byr1r2),
Sy1=1ay1σyy2ikB,
Sy2=ikB,
Gyy=12ay1byσyy2[1+12by(Sy1y1+Sy2y2)2]+12ay12by(iky1B)(Sy1y1+Sy2y2),
ay=1w2+1σyy2+ikA2B,
by=1w2+1σyy2ikA2B1ay1σyy4,
Wrxy(r1,r2,z)=E021λ2B2w2π2axybxyGxyexp[r124axy(ikB)2]exp(Sxy12rxy124bxy+Sxy22r224bxy+Sxy1Sxy22bxyr1r2),
Sxy1=1axy1σxy2ikB,
Sxy2=ikB,
Gxy=[12axy1bxyσxy2(Sxy1x1+Sxy2x2)+12axy(ikx1B)]12bxy(Sy1y1+Sy2y2),
axy=1w2+1σxy2+ikA2B,
bxy=1w2+1σxy2ikA2B1axy1σxy4,
Wryx(r1,r2,z)=E021λ2B2w2π2ayxbyxGyxexp[r124ayx(ikB)2]exp(Syx12ryx124byx+Syx22r224byx+Syx1Syx22byxr1r2),
Syx1=1ayx1σyx2ikB,
Syx2=ikB,
Gyx=[12ayx1byxσyx2(Sxy1y1+Sxy2y2)+12ayx(iky1B)]12byx(Syx1y1+Syx2y2),
ayx=1w2+1σyx2+ikA2B,
byx=1w2+1σyx2ikA2B1ayx1σyx4.
Wθxx(r,z)=Wryy(r,z),Wθyy(r,z)=Wrxx(r,z),Wθxy(r,z)=Wθyx(r,z)=Wrxy(r,z).
Ir(r,z)=Wrxx(r,z)cos2ϕ+Wryy(r,z)sin2ϕ+Wrxy(r,z)sin2ϕ,
Iθ(r,z)=Wθxx(r,z)cos2ϕ+Wθyy(r,z)sin2ϕ+Wθxy(r,z)sin2ϕ.
(ABCD)=(1f01)(101/f1)(1f01)=(1f1/f1).

Metrics