Abstract

We experimentally investigate the focusing properties of first- and second-order vector beams and vector vortex beams generated by photonic-crystal lasers. When the azimuthal indices of the vector beam (l) and the phase dependence (n) match, strong intensity appears at the center of focus. Our theoretical analyses agree well with the experimental results and predict that the central intensity has circular polarization.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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2011

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, Opt. Express 19, 11963 (2011).
[CrossRef]

2010

2009

2008

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).

S. F. Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

2007

K. Sakai and S. Noda, Electron. Lett. 43, 107 (2007).
[CrossRef]

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).
[CrossRef]

Y. Kozawa and S. Sato, J. Opt. Soc. Am. A 24, 1793 (2007).
[CrossRef]

2006

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

2003

S. W. Hell, Nat. Biotechnol. 21, 1347 (2003).
[CrossRef]

2001

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef]

2000

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

K. S. Youngworth and T. G. Brown, Opt. Express 7, 77 (2000).
[CrossRef]

1996

1995

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef]

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

1990

1972

Y. Mushiake, K. Matsumura, and N. Nakajima, Proc. IEEE 60, 1107 (1972).
[CrossRef]

1959

B. Richards and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

Alexeyev, C.

Allen, L.

S. F. Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Arnold, S. F.

S. F. Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef]

K. S. Youngworth and T. G. Brown, Opt. Express 7, 77 (2000).
[CrossRef]

Chong, C. T.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).

Dorn, R.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Eberler, M.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Fadeyava, T.

Ford, D. H.

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef]

Gahagan, K. T.

Glockl, O.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Gupta, D. N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).
[CrossRef]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef]

Hell, S. W.

S. W. Hell, Nat. Biotechnol. 21, 1347 (2003).
[CrossRef]

Hnatovsky, C.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Iwahashi, S.

Kant, N.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).
[CrossRef]

Kim, D. E.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).
[CrossRef]

Kimura, W. D.

Kitamura, K.

Kozawa, Y.

Krolikowski, W.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Kunishi, W.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Kurosaka, Y.

Leuchs, G.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Lukyanchuk, B.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Matsumura, K.

Y. Mushiake, K. Matsumura, and N. Nakajima, Proc. IEEE 60, 1107 (1972).
[CrossRef]

Miyai, E.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Mushiake, Y.

Y. Mushiake, K. Matsumura, and N. Nakajima, Proc. IEEE 60, 1107 (1972).
[CrossRef]

Nakajima, N.

Y. Mushiake, K. Matsumura, and N. Nakajima, Proc. IEEE 60, 1107 (1972).
[CrossRef]

Noda, S.

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, Opt. Express 19, 11963 (2011).
[CrossRef]

K. Kitamura, K. Sakai, and S. Noda, Opt. Express 18, 4518 (2010).
[CrossRef]

K. Sakai and S. Noda, Electron. Lett. 43, 107 (2007).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Novotny, L.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef]

Ohnishi, D.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Okano, T.

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Padgett, M.

S. F. Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

Quabis, S.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Richards, B.

B. Richards and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

Rode, A.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef]

Sakai, K.

S. Iwahashi, Y. Kurosaka, K. Sakai, K. Kitamura, N. Takayama, and S. Noda, Opt. Express 19, 11963 (2011).
[CrossRef]

K. Kitamura, K. Sakai, and S. Noda, Opt. Express 18, 4518 (2010).
[CrossRef]

K. Sakai and S. Noda, Electron. Lett. 43, 107 (2007).
[CrossRef]

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

Sato, S.

Sheppard, C.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).

Shi, L.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).

Shostka, N.

Shvedov, V.

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

T. Fadeyava, V. Shvedov, N. Shostka, C. Alexeyev, and A. Volyar, Opt. Lett. 35, 3787 (2010).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Suk, H.

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).
[CrossRef]

Swartzlander, G. A.

Takayama, N.

Tidwell, S. C.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Volyar, A.

Wang, H.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Wolf, E.

B. Richards and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef]

K. S. Youngworth and T. G. Brown, Opt. Express 7, 77 (2000).
[CrossRef]

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Appl. Opt.

Electron. Lett.

K. Sakai and S. Noda, Electron. Lett. 43, 107 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Laser Photon. Rev.

S. F. Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

Nat. Biotechnol.

S. W. Hell, Nat. Biotechnol. 21, 1347 (2003).
[CrossRef]

Nat. Photon.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).

Nature

E. Miyai, K. Sakai, T. Okano, W. Kunishi, D. Ohnishi, and S. Noda, Nature 441, 946 (2006).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef]

Opt. Commun.

S. Quabis, R. Dorn, M. Eberler, O. Glockl, and G. Leuchs, Opt. Commun. 179, 1 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Lett. A

D. N. Gupta, N. Kant, D. E. Kim, and H. Suk, Phys. Lett. A 368, 402 (2007).
[CrossRef]

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Phys. Rev. Lett.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, Phys. Rev. Lett. 86, 5251 (2001).
[CrossRef]

C. Hnatovsky, V. Shvedov, W. Krolikowski, and A. Rode, Phys. Rev. Lett. 106, 123901 (2011).
[CrossRef]

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Phys. Rev. Lett. 75, 826 (1995).
[CrossRef]

Proc. IEEE

Y. Mushiake, K. Matsumura, and N. Nakajima, Proc. IEEE 60, 1107 (1972).
[CrossRef]

Proc. R. Soc. A

B. Richards and E. Wolf, Proc. R. Soc. A 253, 358 (1959).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Experimental setup. The lower-left images are far-field patterns of two types of beams generated by photonic-crystal lasers taken from [10,20].

Fig. 2.
Fig. 2.

Measured intensity profiles at the beam foci.

Fig. 3.
Fig. 3.

(a) Incident vector beams (p=0). The black arrows show the electric polarization. Total electric-field-intensity distributions in the focal plane are shown for vector vortex beams with azimuthal mode indices for spiral phase shifts of (b) n=0; (c) n=1; (d) n=2; and (e) n=3. The center of each image is the defined focal spot (x=0, y=0), and the side of each square has dimensions of four wavelengths. The images surrounded by dashed lines indicate the beams that were investigated experimentally.

Fig. 4.
Fig. 4.

Calculated intensity distributions for beams with different electric polarization at the focal plane for the case of p=0, l=2, n=0, NA=0.9, and total intensity distribution for NA=0.7. Each profile is normalized by the maximum intensity of the total distribution. In the case of the z component, the maximum intensity is twice that of the total distribution.

Fig. 5.
Fig. 5.

Time dependence of the total electric-field-intensity profile and electric-field vectors for the case of p=0, l=2, n=2, and NA=0.9. Black arrows show the electric polarization.

Equations (10)

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

Ebeam cross.=upl(r,ϕ)exp(inϕ)(er(0)er+eϕ(0)eϕ).
er(0)=cos{(l1)ϕ+σ},eϕ(0)=sin{(l1)ϕ+σ},
e(s)=(ex(s)ey(s)ez(s))=iAπ0α02πsinθcos12θl0(θ)eik(zscosθ+ρssinθcos(ϕϕs))Edϕdθ.
E=exp(inϕ){er(0)(cosθcosϕcosθsinϕsinθ)+eϕ(0)(sinϕcosϕ0)}.
l0(θ)=(2β0sinθsinα)|l|Lp|l|{2(β0sinθsinα)2}exp{(β0sinθsinα)2}.
Lp|l|(x)=m=0p(1)m(p+l)!(pm)!(l+m)!m!xm,
Ebeam cross.=u(r)[exp(ilϕ+iσ)e++exp(ilϕiσ)e].
eϕ=sinϕexcosϕey,er=cosϕex+sinϕey
e+=(exiey)/2,e=(ex+iey)/2.
Ebeam cross.=u(r)[exp{i(n+l)ϕ+iσ}e++exp{i(nl)ϕiσ}e].

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