Abstract

We suggest vortex phase elements to detect the polarization state of the focused incident beam. We analytically and numerically show that only the types of polarization (linear, circular, or cylindrical) can be distinguished in the low numerical aperture (NA) mode. Sharp focusing is necessary to identify the polarization state in more detail (direction or sign). We consider a high NA micro-objective and a diffractive axicon as focusing systems. We show that the diffractive axicon more precisely detects the polarization state than does the micro-objective with the same NA.

© 2015 Optical Society of America

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    [Crossref]

2013 (4)

S. N. Khonina, “Simple phase optical elements for narrowing of a focal spot in high-numerical-aperture conditions,” Opt. Eng. 52(9), 091711 (2013).
[Crossref]

S. N. Khonina, D. A. Savelyev, N. L. Kazanskiy, and V. A. Soifer, “Singular phase elements as detectors for different polarizations,” Proc. SPIE 9066, 90660A (2013).

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

S. N. Khonina and D. A. Savelyev, “High-aperture binary axicons for the formation of the longitudinal electric field component on the optical axis for linear and circular polarizations of the illuminating beam,” J. Exp. Theor. Phys. 117(4), 623–630 (2013).
[Crossref]

2012 (5)

2011 (3)

2010 (5)

2009 (3)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).

L. Rao, J. Pu, Z. Chen, and P. Yei, “Focus shaping of cylindrically polarized vortex beams by a high numerical aperture lens,” Opt. Laser Technol. 41(3), 241–246 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1457 (2009).

2008 (3)

V. V. Kotlyar, A. A. Kovalev, and S. S. Stafeev, “Sharp focus area of radially-polarized Gaussian beam propagation through an axicon,” Prog. Electromagnetics Res. 535–43 (2008).

J. Stadler, C. Stanciu, C. Stupperich, and A. J. Meixner, “Tighter focusing with a parabolic mirror,” Opt. Lett. 33(7), 681–683 (2008).
[Crossref] [PubMed]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

2007 (3)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007).
[Crossref] [PubMed]

V. P. Kalosha and I. Golub, “Toward the subdiffraction focusing limit of optical superresolution,” Opt. Lett. 32(24), 3540–3542 (2007).
[Crossref] [PubMed]

2006 (2)

C. J. Zapata-Rodríguez and A. Sánchez-Losa, “Three-dimensional field distribution in the focal region of low-Fresnel-number axicons,” J. Opt. Soc. Am. A 23(12), 3016–3026 (2006).
[Crossref] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, “Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media,” Phys. Rev. Lett. 96(16), 163905 (2006).
[Crossref] [PubMed]

2005 (2)

A. S. Desyatnikov, L. Torner, and Y. S. Kivshar, “Optical vortices and vortex solitons,” Prog. Opt. 47, 219–319 (2005).

Y. Unno, T. Ebihara, and M. D. Levenson, “Impact of mask errors and lens aberrations on the image formation by a vortex mask,” J. Microlithogr., Microfabr., Microsyst. 4(2), 023006 (2005).

2004 (7)

V. A. Soifer, V. V. Kotlyar, and S. N. Khonina, “Optical microparticle manipulation: Advances and new possibilities created by diffractive optics,” Phys. Part. Nucl. 35(6), 733–766 (2004).

P. Török and P. Munro, “The use of Gauss-Laguerre vector beams in STED microscopy,” Opt. Express 12(15), 3605–3617 (2004).
[Crossref] [PubMed]

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3(2), 293–304 (2004).

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

L. E. Helseth, “Optical vortices in focal regions,” Opt. Commun. 229(1-6), 85–91 (2004).
[Crossref]

N. Davidson and N. Bokor, “High-numerical-aperture focusing of radially polarized doughnut beams with a parabolic mirror and a flat diffractive lens,” Opt. Lett. 29(12), 1318–1320 (2004).
[Crossref] [PubMed]

S. F. Pereira and A. S. van de Nes, “Super-resolution by means of polarisation, phase and amplitude pupil masks,” Opt. Commun. 234(1-6), 119–124 (2004).
[Crossref]

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003).
[Crossref] [PubMed]

A. Ciattoni, G. Cincotti, and C. Palma, “Circularly polarized beams and vortex generation in uniaxial media,” J. Opt. Soc. Am. A 20(1), 163–171 (2003).
[Crossref] [PubMed]

2002 (3)

2001 (3)

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lens,” Opt. Commun. 194(4-6), 225–234 (2001).
[Crossref]

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

M. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219 (2001).

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291 (1999).

1998 (1)

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, “Light field decomposition in angular harmonics by means of diffractive optics,” J. Mod. Opt. 45(7), 1495–1506 (1998).
[Crossref]

1997 (1)

1994 (1)

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The rotor phase filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

1990 (1)

1987 (1)

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
[Crossref]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

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]

1936 (2)

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
[Crossref]

A. H. S. Holbourn, “Angular momentum of circularly polarized light,” Nature 137(3453), 31 (1936).
[Crossref]

Alferov, S. V.

S. N. Khonina, S. V. Karpeev, S. V. Alferov, D. A. Savelyev, J. Laukkanen, and J. Turunen, “Experimental demonstration of the generation of the longitudinal E-field component on the optical axis with high-numerical-aperture binary axicons illuminated by linearly and circularly polarized beams,” J. Opt. 15(8), 085704 (2013).
[Crossref]

S. N. Khonina, S. V. Karpeev, and S. V. Alferov, “Polarization converter for higher-order laser beams using a single binary diffractive optical element as beam splitter,” Opt. Lett. 37(12), 2385–2387 (2012).
[Crossref] [PubMed]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291 (1999).

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, “Mechanical equivalence of spin and orbital angular momentum of light: an optical spanner,” Opt. Lett. 22(1), 52–54 (1997).
[Crossref] [PubMed]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Angelsky, O. V.

O. V. Angelsky, I. I. Mokhun, A. I. Mokhun, and M. S. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(33 Pt 2B), 036602 (2002).
[Crossref] [PubMed]

I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002).
[Crossref] [PubMed]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291 (1999).

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, “Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon,” Phys. Rev. Lett. 92(1), 013601 (2004).
[Crossref] [PubMed]

S. M. Barnett and L. Allen, “Orbital angular momentum and nonparaxial light beams,” Opt. Commun. 110(5-6), 670–678 (1994).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “Meep: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 181687 (2010).
[Crossref]

Berry, M. V.

M. V. Berry and M. R. Dennis, “Polarization singularities in isotropic random vector waves,” Proc. R. Soc. Lond. A 457(2005), 141–155 (2001).
[Crossref]

M. V. Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J. Mod. Opt. 34(11), 1401–1407 (1987).
[Crossref]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A Math. Phys. Sci. 336(1605), 165–190 (1974).
[Crossref]

Beth, R. A.

R. A. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50(2), 115–125 (1936).
[Crossref]

Biener, G.

Bokor, N.

Bomzon, Z.

Brasselet, E.

Chen, Z.

Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012).
[Crossref] [PubMed]

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S. N. Khonina and S. G. Volotovsky, “Controlling the contribution of the electric field components to the focus of a high-aperture lens using binary phase structures,” J. Opt. Soc. Am. A 27(10), 2188–2197 (2010).
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Figures (4)

Fig. 1
Fig. 1 Transmission function of the focusing system. (a) Binary phase of multi-order diffractive optical element. (b) Accordance of diffractive orders to combinations of optical vortices.
Fig. 2
Fig. 2 Detection of orthogonal linear polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.
Fig. 3
Fig. 3 Detection of orthogonal circular polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.
Fig. 4
Fig. 4 Detection of orthogonal cylindrical polarization states in the absence and presence of a vortex phase in an analyzed beam. (a) Without vortex. (b) With vortex.

Tables (5)

Tables Icon

Table 1 Dependence of on-axis intensity components due to superposition in Eq. (4)

Tables Icon

Table 2 Focal intensity distribution for Gaussian beam, NA = 0.25 (red color for x-component, green for y-component, picture size is 8λ × 8λ)

Tables Icon

Table 3 Focal intensity distribution for Gaussian beam, NA = 0.95 (red color for x-component, green for y-component, and blue for z-component; picture size is 2λ × 2λ)

Tables Icon

Table 4 Intensity distributions of Gaussian beam focused by the axicon in Eq. (21) with α0 = 0.95 (red color for x-component, green for y-component, and blue for z-component, picture size 4λ × 4λ)

Tables Icon

Table 5 Intensity distributions in the transverse planes of the Gaussian beam, Laguerre-Gaussian mode (0,1) and Hermite-Gaussian mode (0,1) focused by the axicon with NA = 0.95 (blue color for z-component, yellow for all-components, picture size 6λ × 6λ)

Equations (32)

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E ( ρ , φ , z ) = i f λ 0 α 0 2 π B ( θ , ϕ ) T ( θ ) P ( θ , ϕ ) exp [ i k ( ρ sin θ cos ( ϕ φ ) + z cos θ ) ] sin θ d θ d ϕ ,
P ( θ , ϕ ) = [ 1 + cos 2 ϕ ( cos θ 1 ) sin ϕ cos ϕ ( cos θ 1 ) cos ϕ sin θ sin ϕ cos ϕ ( cos θ 1 ) 1 + sin 2 ϕ ( cos θ 1 ) sin ϕ sin θ sin θ cos ϕ sin θ sin ϕ cos θ ] [ a ( ϕ , θ ) b ( ϕ , θ ) c ( ϕ , θ ) ] ,
B ( θ , ϕ ) = R ( θ ) Ω ( ϕ ) ,
Ω ( ϕ ) = m = M 1 M 2 d m exp ( i m ϕ ) ,
E ( ρ , φ , z ) = i k f m = M 1 M 2 d m i m e i m φ 0 α Q m ( ρ , φ , θ ) R ( θ ) T ( θ ) sin θ exp ( i k z cos θ ) d θ ,
Q m ( ρ , φ , θ ) = [ J m ( t ) + 1 4 [ 2 J m ( t ) e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( cos θ 1 ) i 4 [ e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( cos θ 1 ) i 2 [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] sin θ ] ,
Q m ( ρ , φ , θ ) = [ i 4 [ e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( cos θ 1 ) J m ( t ) + 1 4 [ 2 J m ( t ) + e i 2 φ J m + 2 ( t ) + e i 2 φ J m 2 ( t ) ] ( cos θ 1 ) 1 2 [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] sin θ ] ,
Q m ( ρ , φ , θ ) = 1 2 [ J m ( t ) + 1 2 [ J m ( t ) e ± i 2 φ J m ± 2 ( t ) ] ( cos θ 1 ) ± i { J m ( t ) + 1 2 ( J m ( t ) + e ± i 2 φ J m ± 2 ( t ) ) ( cos θ 1 ) } i e ± i φ J m ± 1 ( t ) sin θ ] ,
Q m ( ρ , φ , θ ) = 1 2 [ i [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] cos θ [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] cos θ 2 J m ( t ) sin θ ] ,
Q m ( ρ , φ , θ ) = 1 2 [ [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] i [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] 0 ] .
e x ± i e y = ( e r cos φ e φ sin φ ) ± i ( e r sin φ + e φ cos φ ) = exp ( ± i φ ) [ e r ± i e φ ]
e r = e x cos φ + e y sin φ = 1 2 exp ( i φ ) [ e x i e y ] + 1 2 exp ( i φ ) [ e x + i e y ] ,
e φ = e x sin φ + e y cos φ = 1 2 exp ( i φ ) [ e y + i e x ] + 1 2 exp ( i φ ) [ e y + i e x ] .
E ( ρ , φ , z ) = 1 λ 2 σ 1 σ 2 0 2 π M ( σ , ϕ ) ( F x ( σ , ϕ ) F y ( σ , ϕ ) ) exp [ i k z 1 σ 2 ] exp [ i k σ ρ cos ( φ ϕ ) ] σ d σ d ϕ ,
M ( σ , ϕ ) = [ 1 cos 2 ϕ ( 1 γ ) cos ϕ sin ϕ ( 1 γ ) σ cos ϕ cos ϕ sin ϕ ( 1 γ ) 1 sin 2 ϕ ( 1 γ ) σ sin ϕ ] , γ = 1 σ 2 ,
( F x ( σ , ϕ ) F y ( σ , ϕ ) ) = 0 r 0 0 2 π ( E 0 x ( r , τ ) E 0 y ( r , τ ) ) exp [ i k r σ cos ( τ ϕ ) ] r d r d τ .
E 0 ( r , τ ) = E 0 ( r ) exp ( i m τ ) ,
E ( ρ , φ , z ) = k 2 i 2 m exp ( i m φ ) σ 1 σ 2 S m ( ρ , φ , σ ) ( P x ( σ ) P y ( σ ) ) exp ( i k z 1 σ 2 ) σ d σ ,
S m ( ρ , φ , σ ) = [ 1 4 { 2 J m ( t ) ( 1 + γ ) + [ e i 2 φ J m + 2 ( t ) + e i 2 φ J m 2 ( t ) ] ( 1 γ ) } i 4 [ e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( 1 γ ) i 2 σ [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] i 4 [ e i 2 φ J m + 2 ( t ) e i 2 φ J m 2 ( t ) ] ( 1 γ ) 1 4 { 2 J m ( t ) ( 1 + γ ) [ e i 2 φ J m + 2 ( t ) + e i 2 φ J m 2 ( t ) ] ( 1 γ ) } 1 2 σ [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] ] ,
( P x ( σ ) P y ( σ ) ) = 0 R ( E 0 x ( r ) E 0 y ( r ) ) J m ( k r σ ) r d r .
E 0 ( r ) = exp ( i k α 0 r ) ,
P ( σ ) δ ( σ α 0 ) ,
E ( ρ , φ , z ) i 2 m α 0 exp ( i m φ ) exp ( i k z 1 α 0 2 ) S m ( ρ , φ , α 0 ) ( c x c y ) ,
E ( ρ , φ , z ) α 0 2 ( c x J m ( t ) ( 1 + γ 0 ) + 1 2 { [ e i 2 φ J m + 2 ( t ) ( c x i c y ) + e i 2 φ J m 2 ( t ) ( c x + i c y ) ] } ( 1 γ 0 ) c y J m ( t ) ( 1 + γ 0 ) 1 2 { [ e i 2 φ J m + 2 ( t ) ( c y i c x ) + e i 2 φ J m 2 ( t ) ( c y + i c x ) ] } ( 1 γ 0 ) α 0 [ e i φ J m + 1 ( t ) ( c y + i c x ) + e i φ J m 1 ( t ) ( c y i c x ) ] ) .
| E ( ρ , φ , z ) | 2 ε 2 ( | c x | 2 + | c y | 2 ) J m 2 ( k ε ρ ) ,
| E ( 0 , 0 , z ) | 2 | c x | 2 + | c y | 2 4
| E ( 0 , 0 , z ) | 2 | c y i c x | 2 4
| E ( 0 , 0 , z ) | 2 | c x ± i c y | 2 + | c y ± i c x | 2 16
S m c ( ρ , φ , σ ) = 1 2 [ i γ [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] γ [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] i [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] 2 σ J m ( t ) 0 ] .
S m c ( ρ , φ , α 0 ) = 1 2 [ i γ 0 [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] γ 0 [ e i φ J m + 1 ( t ) + e i φ J m 1 ( t ) ] i [ e i φ J m + 1 ( t ) e i φ J m 1 ( t ) ] 2 α 0 J m ( t ) 0 ] .
| E ( 0 , 0 , z ) | 2 | c r | 2
| E ( 0 , 0 , z ) | 2 | c φ | 2

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