Abstract

We present and construct a new kind of orthogonal coordinate system, hyperbolic coordinate system. We present and design a new kind of local linearly polarized vector fields, which is defined as the hyperbolic-symmetry vector fields because the points with the same polarization form a series of hyperbolae. We experimentally demonstrate the generation of such a kind of hyperbolic-symmetry vector optical fields. In particular, we also study the modified hyperbolic-symmetry vector optical fields with the twofold and fourfold symmetric states of polarization when introducing the mirror symmetry. The tight focusing behaviors of these vector fields are also investigated. In addition, we also fabricate micro-structures on the K9 glass surfaces by several tightly focused (modified) hyperbolic-symmetry vector fields patterns, which demonstrate that the simulated tightly focused fields are in good agreement with the fabricated micro-structures.

© 2015 Optical Society of America

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References

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    [Crossref]
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2014 (2)

Y. Pan, Y. N. Li, Z. C. Ren, Y. Si, C. H. Tu, and H. T. Wang, “Parabolic-symmetry vector optical fields and their tightly focusing properties,” Phys. Rev. A. 89, 035801 (2014).
[Crossref]

Y. Pan, Y. N. Li, S. M. Li, Z. C. Ren, L. J. Kong, C. H. Tu, and H. T. Wang, “Elliptic-symmetry vector optical fields,” Opt. Express 22, 19302–19313 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2011 (1)

2010 (3)

2009 (1)

2008 (1)

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

2007 (4)

2004 (1)

2003 (2)

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

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

2002 (1)

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. Roy. Soc. A 253, 358–379 (1959).
[Crossref]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Beversluis, M.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Biener, G.

Bomzon, Z.

Bouhelier, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Brown, T. G.

Cao, G. W.

Chen, J.

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]

Chong, C. T.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Ding, J. P.

Dorn, R.

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

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Guo, C. S.

Hao, X.

Hartschuh, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Hasman, E.

Huang, K.

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Kleiner, V.

Kong, L. J.

Kozawa, Y.

Kuang, C. F.

Lerman, G. 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] [PubMed]

Levy, U.

Li, K.

Li, P.

Li, S. M.

Li, Y. N.

Li, Y. P.

Liu, S.

Liu, X.

Lukyanchuk, B.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Ni, W. J.

Novotny, L.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Pan, Y.

Peng, T.

Quabis, S.

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

Ren, Z. C.

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]

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Sato, S.

Sheppard, C.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Shi, L. P.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Shi, P.

Si, Y.

Y. Pan, Y. N. Li, Z. C. Ren, Y. Si, C. H. Tu, and H. T. Wang, “Parabolic-symmetry vector optical fields and their tightly focusing properties,” Phys. Rev. A. 89, 035801 (2014).
[Crossref]

Y. Pan, Y. N. Li, S. M. Li, Z. C. Ren, Y. Si, C. H. Tu, and H. T. Wang, “Vector optical fields with bipolar symmetry of linear polarization,” Opt. Lett. 38, 3700–3703 (2013).
[Crossref] [PubMed]

Tu, C. H.

Wang, H. F.

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

Wang, H. T.

Wang, T. T.

Wang, X. L.

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.

Zhan, Q. W.

Zhang, X. B.

Zhao, J. L.

Adv. Opt. Photon. (1)

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

Nat. Photonics (1)

H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2, 501–505 (2008).
[Crossref]

New J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rev. A. (1)

Y. Pan, Y. N. Li, Z. C. Ren, Y. Si, C. H. Tu, and H. T. Wang, “Parabolic-symmetry vector optical fields and their tightly focusing properties,” Phys. Rev. A. 89, 035801 (2014).
[Crossref]

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] [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]

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Proc. Roy. Soc. A (1)

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 (12)

Fig. 1
Fig. 1 Schematic of experimental setup for generating the desired hyperbolic-symmetry vector optical 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 λ/4 wave plates behind a spatial filter (SF) with two apertures are placed in the vicinity of the Fourier plane of the 4f system. A Ronchi phase grating (G) is placed in the output plane of the 4f system. A polarizer may be inserted in the field, then the intensity patterns can be observed by a PC through a CCD.
Fig. 2
Fig. 2 Four kinds of two dimensional orthogonal coordinate systems. (a) The polar coordinate system. The constant r and φ curves are a series of circles and rays, respectively. (b) The parabolic coordinate system. The constant u and v curves are two sets of confocal parabolas. (c) The bipolar coordinate system. The constant u and v curves are two groups of non-concentric circles. (d) The elliptic coordinate system. The constant u and v curves are confocal ellipses and hyperbolas, respectively.
Fig. 3
Fig. 3 The new orthogonal hyperbolic coordinate system. Red and blue hyperbolas show the constant u and v hyperbolas, respectively.
Fig. 4
Fig. 4 Hyperbolic-symmetry vector fields for different m = 1,3,5,7,9 when n = 0 and δ0 = 0. The first row shows the experimentally measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm2.
Fig. 5
Fig. 5 Hyperbolic-symmetry vector fields for different n = 1,3,5,7,9 when m = 0 and δ0 = 0. The first row shows the experimentally measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm2.
Fig. 6
Fig. 6 Hyperbolic-symmetry vector fields for different n = 1,3,5,7,9 when m = 5 and δ0 = 0. The first row shows the measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm2.
Fig. 7
Fig. 7 Hyperbolic-symmetry vector fields for different m = 1,3,5,7,9 when n = 7 and δ0 = 0. The first row shows the measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm2.
Fig. 8
Fig. 8 Hyperbolic-symmetry vector fields for different combinations of m and n with m = n when δ0 = 0. The first row shows the measured total intensity patterns of the hyperbolic-symmetry vector fields. The second (fourth) row shows the simulated intensity patterns of the x (y) component. The third (fifth) row shows the measured intensity patterns of the x (y) component. Any picture has the same dimension of 4 × 4 mm2.
Fig. 9
Fig. 9 Stimulated polarization states of the hyperbolic-symmetry vector fields for different parameters. The first row shows the polarization states of the original vector fields. The second and third rows correspond to the cases of the twofold and fourfold mirror symmetries of polarization states, respectively. The first, second, and third columns correspond to the cases of (m,n,δ0) = (5,0,0),(0,3,0),(5,3,0), respectively.
Fig. 10
Fig. 10 The modified hyperbolic-symmetry vector fields for different n = 1,3,5,7,9 when m = 5 and δ0 = 0. The first and second rows show the original hyperbolic-symmetry vector fields, for comparison. The third and fourth rows show the modified hyperbolic-symmetry vector fields with twofold mirror-symmetric polarization states. The fifth and sixth rows show the modified hyperbolic-symmetry vector fields with fourfold mirror-symmetric polarization states. Any picture has the same dimension of 4 × 4 mm2.
Fig. 11
Fig. 11 Simulated tightly focused field for the modified hyperbolic-symmetry vector fields with different m, n and δ0 by using an objective with NA = 0.9. The first and second rows correspond to the vector fields with the twofold and fourfold mirror-symmetric axes for polarization states, respectively. Any picture has a dimension of 4λ × 4λ.
Fig. 12
Fig. 12 The intensity patterns of the focal fields of three hyperbolic-symmetry vector fields by an objective with NA = 0.75 and the micro-structures on the K9 glass surface fabricated by them. The top row shows the intensity patterns of the tightly focused fields of the three hyperbolic-symmetry vector fields with (m,n,δ0) = (0.5,0,π/2),(0.5,6.5,0),(10,5.5,π/2). The bottom row shows the micro-structures on the K9 glass surface fabricated by the corresponding focal fields in the top row, which are the optical microscopic CCD image under the illumination of white light. Any picture has the same dimension of 10 × 10 μm2.

Equations (11)

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t ( x , y ) = 1 2 + 1 2 cos [ 2 π f 0 x + δ ( x , y ) ] ,
E ( x , y ) = 1 2 A 0 [ e j δ ( x , y ) e ^ + + e j δ ( x , y ) e ^ ] = A 0 [ cos δ ( x , y ) e ^ x + sin δ ( x , y ) e ^ y ] .
u = 1 2 sgn ( x + y ) sgn ( y x ) ( y 2 x 2 ) ,
v = sgn ( x ) sgn ( y ) x y ,
δ = m u + n v + δ 0 .
u = 1 2 r | cos ( 2 φ ) | and v = 1 2 r | sin ( 2 φ ) | .
r = 2 l π m | cos ( 2 φ ) | + n | sin ( 2 φ ) | ,
r φ = 2 l π [ m sgn [ cos ( 2 φ ) ] | sin ( 2 φ ) | 3 / 2 n sgn [ sin ( 2 φ ) ] | cos ( 2 φ ) | 3 / 2 ] [ m | cos ( 2 φ ) | + n | sin ( 2 φ ) | ] 2 | sin ( 4 φ ) | .
| tan ( 2 φ ) | 3 / 2 = n m .
δ = ( m u + n v π / 2 ) sgn ( x ) sgn ( y ) ( π / 2 ) sgn ( y ) + π ,
δ = 2 π ( π / 2 ) sgn ( y ) + ( m u / 2 + π / 4 ) sgn ( y ) sgn ( x y ) + ( m u / 2 3 π / 4 ) sgn ( y ) sgn ( x + y ) + ( m u / 2 π / 2 ) sgn ( x ) sgn ( x y ) ( m u / 2 + π / 2 ) sgn ( x ) sgn ( x + y ) .

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