Abstract

The propagation and focusing properties of light beams continue to remain a research interest owning to their promising applications in physics, chemistry and biological sciences. One of the main challenges to these applications is the control of polarization distribution within the focal volume. In this work, we propose and experimentally demonstrate a method for generating a focused beam with arbitrary homogeneous polarization at any transverse plane. The required input field at the pupil plane of a high numerical aperture objective lens can be found analytically by solving an inverse problem with the Richard-Wolf vectorial diffraction method, and can be experimentally created with a vectorial optical field generator. Focused fields with various polarizations are successfully generated and verified using a Stokes parameter measurement to demonstrate the capability and versatility of proposed technique.

© 2016 Optical Society of America

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References

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2016 (1)

2014 (1)

2013 (3)

2012 (3)

2007 (1)

S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

2002 (2)

1995 (1)

T. Y. F. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52(5), 4116–4125 (1995).
[Crossref] [PubMed]

1994 (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 Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Bokor, N.

K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields,” Opt. Commun. 288, 13–16 (2013).
[Crossref]

Carnicer, A.

Chen, Z.

Cheng, J.

Cheng, W.

Cottrell, D. M.

Dainty, C.

Davis, J. A.

Ding, J.

Han, W.

Hell, S. W.

Hernandez, T. M.

Jahn, K.

K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields,” Opt. Commun. 288, 13–16 (2013).
[Crossref]

Jureller, J. E.

Juvells, I.

Kenny, F.

Lara, D.

Maluenda, D.

Martínez-Herrero, R.

Moreno, I.

Park, S.

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 Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Rodríguez-Herrera, O. G.

Sand, D.

Scherer, N. F.

Toussaint, K. C.

Tripathi, S.

Tsang, T. Y. F.

T. Y. F. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52(5), 4116–4125 (1995).
[Crossref] [PubMed]

Volkmer, A.

Wichmann, J.

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 Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Xie, X. S.

Yang, Y.

Zeng, T.

Zhan, Q.

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

K. Jahn and N. Bokor, “Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields,” Opt. Commun. 288, 13–16 (2013).
[Crossref]

Opt. Express (5)

Opt. Lett. (5)

Phys. Rev. A (1)

T. Y. F. Tsang, “Optical third-harmonic generation at interfaces,” Phys. Rev. A 52(5), 4116–4125 (1995).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A Math. Phys. Sci. (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 Math. Phys. Sci. 253(1274), 358–379 (1959).
[Crossref]

Science (1)

S. W. Hell, “Far-Field Optical Nanoscopy,” Science 316(5828), 1153–1158 (2007).
[Crossref] [PubMed]

Other (1)

R. W. Boyd, Nonlinear Optics (Academic, 1992).

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Figures (11)

Fig. 1
Fig. 1 Focusing of a vectorial optical beam.
Fig. 2
Fig. 2 Synthesis of elliptically polarized focused beam with m = 1, α = π/4 and ε = π/8. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the ideal input beam. (d)-(f) Intensity distributions with polarization map of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.
Fig. 3
Fig. 3 Synthesis of linearly polarized focused beam with m = 1, α = π/4 and ε = 0. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the ideal input beam. (d)-(f) Intensity distributions with polarization map of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.
Fig. 4
Fig. 4 Synthesis of circularly polarized focused beam with m = 1, α = 0 and ε = π/4. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the ideal input beam. (d)-(f) Intensity distributions with polarization map of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.
Fig. 5
Fig. 5 Schematic diagram of the experiment setup. HWP, half wave plate; P, polarizer; L, lens; M, mirror; QWP, quarter wave plate; SF, spatial filter; BS, beam splitter; MO, microscope objective; OP, observation plane.
Fig. 6
Fig. 6 Experimental results of the synthesized elliptically polarized focused beam with m = 1, α = π/4 and ε = π/8. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the experimentally generated input beam. (d)-(f) Intensity distributions of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.
Fig. 7
Fig. 7 (a) Stokes images of the elliptically polarized focused field at z = 2.5λ. (b) Theoretical and experimental Pi values for the different transverse planes.
Fig. 8
Fig. 8 Experimental results of the synthesized linearly polarized focused beam with m = 1, α = π/4 and ε = 0. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the experimentally generated input beam. (d)-(f) Intensity distributions of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.
Fig. 9
Fig. 9 Experimental results of the synthesized circularly polarized focused beam with m = 1, α = 0 and ε = π/4. (a) Intensity distribution with polarization map, histograms of (b) ellipticity and (c) elevation angle of the experimentally generated input beam. (d)-(f) Intensity distributions of the focused beam at different transverse planes in the focal region for z = 1.5λ, z = 2.5λ, and z = 5λ.
Fig. 10
Fig. 10 (a) Stokes images of the linearly polarized focused field at z = 2.5λ. (b) Theoretical and experimental Pi values for the different transverse planes.
Fig. 11
Fig. 11 (a) Stokes images of the circularly polarized focused field at z = 2.5λ. (b) Theoretical and experimental Pi values for the different transverse planes.

Equations (10)

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E i = e i m φ ( f r e r + f φ e φ ) ,
E ( r , ϕ , z ) = i A 0 θ max 0 2 π P ( θ ) [ f r ( θ , φ ) e r + f φ ( θ , φ ) e φ ] × e i k r sin θ cos ( ϕ φ ) e i k z cos θ e i m φ sin θ d θ d φ ,
A = π f l 0 λ ,
{ e a = ( cos α cos ε i sin α sin ε ) e x + ( sin α cos ε + i cos α sin ε ) e y e b = ( sin α cos ε + i cos α sin ε ) e x + ( cos α cos ε + i sin α sin ε ) e y .
e r ( θ , φ ) = sin φ e x + cos φ e y = { [ cos ε sin ( φ α ) i sin ε cos ( φ α ) ] e a [ cos ε cos ( φ α ) + i sin ε sin ( φ α ) ] e b e φ ( θ , φ ) = cos θ cos φ e x + cos θ sin φ e y + sin θ e z = { cos θ [ cos ε cos ( φ α ) i sin ε sin ( φ α ) ] e a cos θ [ cos ε sin ( φ α ) i sin ε cos ( φ α ) ] e b sin θ e z
E t = e i m φ × { f r cos θ [ cos ε cos ( φ α ) i sin ε sin ( φ α ) ] + f φ [ cos ε sin ( φ α ) i sin ε cos ( φ α ) ] e a f r cos θ [ cos ε sin ( φ α ) i sin ε cos ( φ α ) ] + f φ [ cos ε cos ( φ α ) + i sin ε sin ( φ α ) ] e b f r sin θ e z
{ f r = h ( θ , φ ) f φ = cos ε sin ( φ α ) i sin ε cos ( φ α ) cos ε cos ( φ α ) + i sin ε sin ( φ α ) cos θ h ( θ , φ )
E = i A π 0 θ max 0 2 π e i k [ z cos θ + r sin θ cos ( φ ϕ ) ] e i m φ h ( θ , φ ) × { cos θ / [ cos ε cos ( φ α ) + i sin ε sin ( φ α ) ] e a 0 e b sin θ e z
{ S 0 = I ( 0 , 0 ) + I ( 90 , 90 ) S 1 = I ( 0 , 0 ) I ( 90 , 90 ) S 2 = I ( 45 , 45 ) I ( 135 , 135 ) S 3 = I ( 45 , 0 ) I ( 135 , 0 )
P i = S i 2 ( x 0 , y 0 ) S 0 2 ( x 0 , y 0 ) i = 1 , 2 , 3

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