Abstract

We show that, in order to attain complete polarization control across a beam, two spatially resolved variable retardations need to be introduced to the light beam. The orientation of the fast axes of the retarders must be linearly independent on the Poincaré sphere if a fixed starting polarization state is used, and one of the retardations requires a range of 2π. We also present an experimental system capable of implementing this concept using two passes on spatial light modulators (SLMs). A third SLM pass can be added to control the absolute phase of the beam. Control of the spatial polarization and phase distribution of a beam has applications in high-NA microscopy, where these properties can be used to shape the focal field in three dimensions. We present some examples of such fields, both theoretically calculated using McCutchen’s method and experimentally observed.

© 2012 OSA

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References

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2012

2011

2010

O. G. Rodríguez-Herrera, D. Lara, K. Bliokh, E. Ostravskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett.104, 253601 (2010).
[CrossRef] [PubMed]

O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Far-field polarization-based sensitivity to sub-resolution displacements of a sub-resolution scatterer in tightly focused fields,” Opt. Express18, 5609–5628 (2010).
[CrossRef] [PubMed]

2009

2007

A. K. Spilman and T. J. Brown, “Stress birefringent, space-variant wave plates for vortex illumination,” Appl. Opt.46, 61–66 (2007).
[CrossRef]

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun.271, 40–47 (2007).
[CrossRef]

2006

2005

2004

2003

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

2002

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun.203, 1–5 (2002).
[CrossRef]

C. W. McCutchen, “Generalized aperture and the three-dimensional diffraction image: erratum,” J. Opt. Soc. Am. A19, 1721–1721 (2002).
[CrossRef]

2001

T. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9, 490–497 (2001).
[CrossRef] [PubMed]

2000

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 1543–1545 (2000).
[CrossRef]

1999

1995

J. L. Pezzaniti and R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng.34, 1569–1573 (1995).
[CrossRef]

1990

1964

1959

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A253, 358–379 (1959).
[CrossRef]

Ballard, S. S.

W. S. Shurcliff and S. S. Ballard, Polarized Light (D. Van Nostrand Company Inc., 1964)

Beresna, M.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett.98, 201101 (2011).
[CrossRef]

Biss, D. P.

Bliokh, K.

O. G. Rodríguez-Herrera, D. Lara, K. Bliokh, E. Ostravskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett.104, 253601 (2010).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980).

Braat, J.

Brown, T. G.

D. P. Biss and T. G. Brown, “Cylindrical vector beam focusing through a dielectric interface,” Opt. Express9, 490–497 (2001).
[CrossRef] [PubMed]

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 1543–1545 (2000).
[CrossRef]

Brown, T. J.

Chen, H.

Chipman, R. A.

J. L. Pezzaniti and R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng.34, 1569–1573 (1995).
[CrossRef]

Choudhury, A.

Compain, E.

Cottrell, D. M.

Courjon, D.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun.203, 1–5 (2002).
[CrossRef]

Dainty, C.

Davis, J. A.

Ding, J.

Dirksen, 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]

Drevillon, B.

Engel, E.

T. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

Ford, D. H.

Gecevicius, M.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett.98, 201101 (2011).
[CrossRef]

Gertus, T.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett.98, 201101 (2011).
[CrossRef]

Goldstein, D. H.

D. H. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2011).

Golub, I.

Grosjean, T.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun.203, 1–5 (2002).
[CrossRef]

Hao, J.

Hell, S. W.

T. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

Hernandez, T. M.

Iglesias, I.

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun.271, 40–47 (2007).
[CrossRef]

Janssen, A. J. E. M.

Kazansky, P. G.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett.98, 201101 (2011).
[CrossRef]

Kenny, F.

Khonina, S. N.

Kimura, W. D.

Klar, T.

T. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

Lara, D.

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]

Lin, J.

McCutchen, C. W.

Moreno, I.

Nesterov, A. V.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D: Appl. Phys.32, 1455–1461 (1999).
[CrossRef]

Niziev, V. G.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D: Appl. Phys.32, 1455–1461 (1999).
[CrossRef]

Ostravskaya, E.

O. G. Rodríguez-Herrera, D. Lara, K. Bliokh, E. Ostravskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett.104, 253601 (2010).
[CrossRef] [PubMed]

Patel, J. S.

Pezzaniti, J. L.

J. L. Pezzaniti and R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng.34, 1569–1573 (1995).
[CrossRef]

Poirier, S.

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]

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. London Ser. A253, 358–379 (1959).
[CrossRef]

Rodríguez-Herrera, O. G.

Sand, D.

Sheppard, C. J. R.

Shurcliff, W. S.

W. S. Shurcliff and S. S. Ballard, Polarized Light (D. Van Nostrand Company Inc., 1964)

Spajer, M.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun.203, 1–5 (2002).
[CrossRef]

Spilman, A. K.

Suh, S.-W.

Tidwell, S. C.

Vohnsen, B.

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun.271, 40–47 (2007).
[CrossRef]

Wang, H.-t.

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. London Ser. A253, 358–379 (1959).
[CrossRef]

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980).

Xu, J.

Youngworth, K. S.

K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7, 1543–1545 (2000).
[CrossRef]

Zhan, Q.

Zhang, B.-f.

Zhuang, Z.

Adv. Opt. Photon.

Appl. Opt.

Appl. Phys. Lett.

M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Appl. Phys. Lett.98, 201101 (2011).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. D: Appl. Phys.

V. G. Niziev and A. V. Nesterov, “Influence of beam polarization on laser cutting efficiency,” J. Phys. D: Appl. Phys.32, 1455–1461 (1999).
[CrossRef]

Opt. Commun.

T. Grosjean, D. Courjon, and M. Spajer, “An all-fiber device for generating radially and other polarized light beams,” Opt. Commun.203, 1–5 (2002).
[CrossRef]

I. Iglesias and B. Vohnsen, “Polarization structuring for focal volume shaping in high-resolution microscopy,” Opt. Commun.271, 40–47 (2007).
[CrossRef]

Opt. Eng.

J. L. Pezzaniti and R. A. Chipman, “Linear polarization uniformity measurements taken with an imaging polarimeter,” Opt. Eng.34, 1569–1573 (1995).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

T. Klar, E. Engel, and S. W. Hell, “Breaking Abbe’s diffraction limit in fluorescence microscopy with stimulated emission depletion beams of various shapes,” Phys. Rev. E64, 066613 (2001).
[CrossRef]

Phys. Rev. Lett.

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

O. G. Rodríguez-Herrera, D. Lara, K. Bliokh, E. Ostravskaya, and C. Dainty, “Optical nanoprobing via spin-orbit interaction of light,” Phys. Rev. Lett.104, 253601 (2010).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A253, 358–379 (1959).
[CrossRef]

Other

W. S. Shurcliff and S. S. Ballard, Polarized Light (D. Van Nostrand Company Inc., 1964)

D. H. Goldstein, Polarized Light, 3rd ed. (CRC Press, 2011).

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1980).

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Figures (13)

Fig. 1
Fig. 1

Any cylindrically polarized vector beam includes a phase vortex. (a) Includes a vortex in its absolute phase, while (b) has a flat phase distribution. (b) Was calculated by resolving (a) into its x- and y-components, and removing the phase discontinuity present in each of the components. Differences in the polarization, as well as the absolute phase, arise out of this flattening, for example in the upper left quadrant of the pupil.

Fig. 2
Fig. 2

Resolving a radially polarized beam into its x- and y-components reveals the phase discontinuity at the centre more clearly; (a) and (b) show the x- and y-components of a radial distribution, which was previously shown in Fig. 1(a).

Fig. 3
Fig. 3

Illustration of two rotations around two perpendicular axes of the Poincaré sphere. Perpendicular axes on the sphere represent polarization states which have their angle of ellipticity at 45° with respect to each other.

Fig. 4
Fig. 4

Schematic diagram of the setup built which can create any polarization state at any point across a laser beam, limited by the pixel size of the spatial light modulator used.

Fig. 5
Fig. 5

Stokes vector distribution of the polarization state in the pupil of the high-NA lens which results in two perpendicularly polarized spots in the focal field. A uniform retardance of π/2 is used on the first SLM, while five wrapped waves of tilt in retardance are applied on the second. The incident beam is uniformly vertically polarized.

Fig. 6
Fig. 6

Theoretical (a) and experimental (b) intensities for the split focal field. The right image was acquired using the auxiliary camera (D2 in Fig. 4), after re-imaging the field at low-NA.

Fig. 7
Fig. 7

Magnitude and phase of the three components of the electric field of the intensity distribution in Fig. 6(a).

Fig. 8
Fig. 8

Polarization state in the pupil of the high-NA lens which results in four perpendicularly polarized spots in the focal field. Ten wrapped waves of tilt in retardance are applied by each SLM. These tilts are perpendicular to each other.

Fig. 9
Fig. 9

Irradiance of the focused field, theoretically calculated using McCutchen’s method (a), and experimentally measurement after re-imaging the field at high–magnification (b).

Fig. 10
Fig. 10

Magnitude and phase of the components of the field in the focus using the polarization state shown in Fig. 8.

Fig. 11
Fig. 11

Modified setup which includes both absolute phase and complete polarization control spatially over the pupil distribution.

Fig. 12
Fig. 12

Retardance of the pupil polarization distribution (a), defined using a Z 3 3 Zernike polynomial with amplitude π/2 (in the Born and Wolf normalization [31]). A schematic diagram of the local coordinate systems and polarization states is shown in (b).

Fig. 13
Fig. 13

The magnitude and phase of the focal field components shown in a polar coordinate system. Note the triangular symmetry of the E⃗p and E⃗s components.

Equations (22)

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M ret ( δ , 0 ° ) = ( 1 0 0 0 0 1 0 0 0 0 cos δ sin δ 0 0 sin δ cos δ ) .
M ret ( δ , 45 ° ) = M rot ( 45 ° ) M ret ( δ , 0 ° ) M rot ( 45 ° ) ,
M rot ( θ ) = ( 1 0 0 0 0 cos 2 θ sin 2 θ 0 0 sin 2 θ cos 2 θ 0 0 0 0 1 ) .
s in = ( 1 1 0 0 ) .
s out = M ret ( δ , 45 ° ) s in .
s out = ( 1 cos δ 0 sin δ ) .
j in = [ 0 1 ] .
J R ( δ , θ ) = [ cos 2 θ + exp ( i δ ) sin 2 θ ( 1 exp ( i δ ) ) cos θ sin θ ( 1 exp ( i δ ) ) cos θ sin θ exp ( i δ ) cos 2 θ + sin 2 θ ] .
J R ( δ , θ ) = J rot ( θ ) J Ret ( δ ) J rot ( θ ) ,
J rot ( θ ) = [ cos θ sin θ sin θ cos θ ] ,
J Ret ( δ ) = [ 1 0 0 exp ( i δ ) ] .
J R ( δ 1 , 45 ° ) = 1 2 [ 1 + exp ( i δ 1 ) 1 exp ( i δ 1 ) 1 exp ( i δ 1 ) 1 + exp ( i δ 1 ) ] .
J R ( δ 2 , 0 ° ) = [ 1 0 0 exp ( i δ 2 ) ] .
j out = J R ( δ 2 , 0 ° ) J R ( δ 1 , 45 ° ) j in .
j out = 1 2 [ 1 0 0 exp ( i δ 2 ) ] [ 1 + exp ( i δ 1 ) 1 exp ( i δ 1 ) 1 exp ( i δ 1 ) 1 + exp ( i δ 1 ) ] [ 0 1 ] , j out = 1 2 [ 1 exp ( i δ 1 ) ( 1 + exp ( i δ 1 ) ) exp ( i δ 2 ) ] .
j out ( x , y ) = exp ( i ϕ ( x , y ) ) 2 [ 1 exp ( i δ 1 ( x , y ) ) ( 1 + exp ( i δ 1 ( x , y ) ) ) exp ( i δ 2 ( x , y ) ) ] .
j S L M 1 = J H W P J S L M J Mirror J S L M + J H W P + j in .
j S L M 1 = J rot ( 22.5 ° ) J Ret ( π ) J rot ( 22.5 ° ) J Ret ( δ 1 / 2 ) . J Mirror J Ret ( δ 1 / 2 ) J rot ( 22.5 ° ) J Ret ( π ) J rot ( 22.5 ° ) j in .
J Ret ( π ) J rot ( θ ) = J rot ( θ ) J Ret ( π ) ,
J Ret ( α ) J Mirror = J Mirror J Ret ( α ) ,
J Ret ( α ) J Ret ( β ) = J Ret ( β ) J Ret ( α ) = J Ret ( α + β ) .
j S L M 1 = J Mirror J rot ( 45 ° ) J Ret ( δ 1 ) J rot ( 45 ° ) j in ,

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