Abstract

We present in principle and demonstrate experimentally a new kind of vector fields: elliptic-symmetry vector optical fields. This is a significant development in vector fields, as this breaks the cylindrical symmetry and enriches the family of vector fields. Due to the presence of an additional degrees of freedom, which is the interval between the foci in the elliptic coordinate system, the elliptic-symmetry vector fields are more flexible than the cylindrical vector fields for controlling the spatial structure of polarization and for engineering the focusing fields. The elliptic-symmetry vector fields can find many specific applications from optical trapping to optical machining and so on.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012

2010

2009

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
[CrossRef]

G. M. Lerman, Y. Lilach, and U. Levy, “Demonstration of spatially inhomogeneous vector beams with elliptical symmetry,” Opt. Lett. 34, 1669–1671 (2009).
[CrossRef] [PubMed]

M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A 11, 1–7 (2009).
[CrossRef]

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

2007

2006

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[CrossRef]

2005

2004

2003

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. 71, 233–242 (2003).
[CrossRef]

2000

1959

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

Bandres, M. A.

Bonaccorso, F.

Brown, T. G.

Chávez-Cerda, S.

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. 71, 233–242 (2003).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

Chen, J.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef] [PubMed]

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Chen, W.

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[CrossRef]

Ding, J. P.

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef] [PubMed]

Donato, M. G.

Fan, Y. X.

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Ferrari, A. C.

Gucciardi, P. G.

Guo, C. S.

Gutiérrez-Vega, J. C.

Hnatovsky, C.

Iturbe-Castillo, M. D.

Jones, P. H.

Kozawa, Y.

Krolikowski, W.

Lerman, G. M.

Levy, U.

Li, Y. N.

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Lilach, Y.

Maragò, O. M.

Meneses-Nava, M. A.

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. 71, 233–242 (2003).
[CrossRef]

Ni, W. J.

Qin, J. Q.

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

Rashid, M.

M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A 11, 1–7 (2009).
[CrossRef]

Richards, B.

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

Rode, A. V.

Rodríguez-Dagnino, R. M.

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. 71, 233–242 (2003).
[CrossRef]

Sato, S.

Sayed, R.

Shostka, N.

Shvedov, V. G.

Vasi, S.

Wang, H. T.

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef] [PubMed]

Wang, X. L.

X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010).
[CrossRef] [PubMed]

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
[CrossRef] [PubMed]

Wolf, E.

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

Youngworth, K. S.

Zhan, Q.

Adv. Opt. Photon.

Am. J. Phys.

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, “Mathieu functions, a visual approach,” Am. J. Phys. 71, 233–242 (2003).
[CrossRef]

J. Opt. A

M. Rashid, O. M. Maragò, and P. H. Jones, “Focusing of high order cylindrical vector beams,” J. Opt. A 11, 1–7 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009).
[CrossRef]

W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

X. L. Wang, J. Chen, Y. N. Li, J. P. Ding, C. S. Guo, and H. T. Wang, “Optical orbital angular momentum from the curl of polarization,” Phys. Rev. Lett. 105, 253602 (2010).
[CrossRef]

Proc. Roy. Soc. A

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

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Figures (10)

Fig. 1
Fig. 1

Schematic of experimental setup for generating the vector fields. The main configuration is a 4f system composed of a pair of identical lenses (L1 and L2). A spatial light modulator (SLM) is located at the input plane of the 4f system. Two 1/4 wave plates behind a spatial filter (SF) with two apertures are placed in the Fourier plane of the 4f system. A Ronchi phase grating (G) is placed in the output plane of the 4f system. A linear polarizer may be inserted between G and CCD, to acquire the total or a certain polarized component intensity pattern by CCD.

Fig. 2
Fig. 2

Elliptic coordinate system. Solid and dotted lines show the curves of constant ε and η, respectively.

Fig. 3
Fig. 3

Simulated polarization distributions of the elliptic-symmetry vector fields for different parameters. (a) the polarization map when m = 0.5, n = 0 and δ0 = 0; (b) the polarization map when m = 1, n = 0 and δ0 = 0; (c) the polarization map when m = 1, n = 0 and δ0 = π/2; (d) the polarization map when m = 3, n = 0 and δ0 = 0; (e) the polarization map when m = 0, n = 1 and δ0 = 0; (f) the polarization map when m = 1, n = 1 and δ0 = 0. The arrows represent the local polarization directions. For all the simulations, we select f = 0.3. Any image has a dimension of 6.7 f × 6.7 f.

Fig. 4
Fig. 4

Elliptic-symmetry vector fields for different m = (0.5, 1, 1.5, 2, 2.5, 3) when n = 0 and δ0 = 0. The first row shows the experimentally measured total intensity patterns of the elliptic-symmetry vector fields. The second (fourth) row shows their simulated intensity patterns of the x (y) component. The third (fifth) row shows their measured intensity patterns of the x (y) component. For all the simulations, we select f = 0.3. Any image has a dimension of 6.7 f × 6.7 f.

Fig. 5
Fig. 5

Elliptic-symmetry vector fields for different n = (1, 2, 3, 4, 5) when m ≡ 0 and δ0 = 0. The first row shows the experimentally measured total intensity patterns of the elliptic-symmetry vector fields. The second (fourth) row shows their simulated intensity patterns of the x (y) component. The third (fifth) row shows their measured intensity patterns of the x (y) component. For all the simulations, we select f = 0.3. Any image has a dimension of 6.7 f × 6.7 f.

Fig. 6
Fig. 6

Elliptic-symmetry vector fields for different n = (0.5, 1, 1.5, 2, 2.5, 3) when m ≡ 1 and δ0 = 0. The first row shows the experimentally measured total intensity patterns of the elliptic-symmetry vector fields. The second (fourth) row shows their simulated intensity patterns of the x (y) component. The third (fifth) row shows their measured intensity patterns of the x (y) component. For all the simulations, we select f = 0.3. Any image has a dimension of 6.7 f × 6.7 f.

Fig. 7
Fig. 7

Elliptic-symmetry vector fields for different m = (0.5, 1, 1.5, 2, 2.5, 3) when n ≡ 2 and δ0 = 0. The first row shows the experimentally measured total intensity patterns of the elliptic-symmetry vector fields. The second (fourth) row shows their simulated intensity patterns of the x (y) component. The third (fifth) row shows their measured intensity patterns of the x (y) component. For all the simulations, we select f = 0.3. Any image has a dimension of 6.7 f × 6.7 f.

Fig. 8
Fig. 8

Elliptic-symmetry vector fields for different combinations of m and n with m = n when δ0 = 0. The first row shows the experimentally measured total intensity patterns of the elliptic-symmetry vector fields. The second (fourth) row shows their simulated intensity patterns of the x (y) component. The third (fifth) row shows their measured intensity patterns of the x (y) component. For all the simulations, we select f = 0.3. Any image has a dimension of 6.7 f × 6.7 f.

Fig. 9
Fig. 9

Elliptic-symmetry vector fields for different f in the two cases of (m, n) = (2, 0) and (m, n) = (0, 1.5). Any image has a dimension of 6.7 f × 6.7 f.

Fig. 10
Fig. 10

Tight focusing intensity distributions of cylindrical and elliptic-symmetry vector fields by using an objective with NA = 0.9. The first row corresponds to the vector fields when m = 3 and δ0 = 0, and the second row corresponds to the case when m = 3 and δ0 = π/2. The first column shows the cylindrical vector fields, while other columns show the elliptic-symmetry vector fields for different f. Any image has a dimension of 4λ × 4λ.

Equations (11)

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

t ( x , y ) = [ 1 + γ cos ( 2 π f 0 x + δ ) ] / 2 .
E = A 0 ( cos δ e ^ x + sin δ e ^ y ) .
x = f cosh ε cos η , y = f sinh ε sin η .
x 2 f 2 cosh 2 ε + y 2 f 2 sinh 2 ε = 1 ,
x 2 f 2 cos 2 η y 2 f 2 sin 2 η = 1 .
δ = m φ + 2 n π r / r 0 + δ 0 .
δ = m η + n ε π + δ 0 ,
ε = tanh 1 ( f 2 + x 2 + y 2 ) + ( f 2 + x 2 + y 2 ) 2 4 f 2 x 2 ( f 2 + x 2 + y 2 ) + ( f 2 + x 2 + y 2 ) 2 4 f 2 x 2 ,
η = tan 1 ( f 2 x 2 y 2 ) + ( f 2 + x 2 + y 2 ) 2 4 f 2 x 2 sgn ( y ) ( f 2 + x 2 + y 2 ) ( f 2 + x 2 + y 2 ) 2 4 f 2 x 2 sgn ( x ) + π [ 1 sgn ( x ) sgn ( y ) / 2 sgn ( y ) / 2 ] ,
E = j k F 2 π 0 θ m 0 2 π P ( θ ) M e j k [ z cos θ + r sin θ cos ( φ ϕ ) ] sin θ d ϕ d θ ,
M = [ ( E ρ cos θ cos φ E ϕ sin ϕ ) e ^ x ( E ρ cos θ sin ϕ + E ϕ cos ϕ ) e ^ y E ρ sin θ e ^ z ] ,

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