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Effect of spherical aberration on tightly focused cylindrically polarized vortex beams
Optik 125 (2014) 6870–6873
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Effect of spherical aberration on tightly focused cylindrically polarized vortex beams G. Therese Anita a , C. Amala Prathiba Janet b , K. Prabakaran c , T.V.S. Pillai c , K.B. Rajesh d,∗ a
Department of Physics, Arunachala College of Engineering for Women, Nagercoil, Tamil Nadu, India Department of Physics, St. Xavier’s Catholic College of Engineering, Nagercoil, Tamil Nadu, India Department of Physics, Periyar Universit- PG Extension Centre, Dharmapuri, Tamil Nadu, India d Department of Physics, Chikkanna Government Arts College, Tiruppur, Tamil Nadu, India b c
a r t i c l e
i n f o
Article history: Received 5 August 2013 Accepted 5 February 2014 Keywords: Vector diffraction theory Cylindrical vortex beam Polarization Spherical aberration
a b s t r a c t In this paper attention is given to the effects of primary spherical aberration on the cylindrical polarized vortex beam based on the vector diffraction theory. It is observed that by properly choosing the polarization angle and topological charge one can obtain many novel focal patterns suitable for optical tweezers, laser printing and material process. However, it is observed that the focusing objective with spherical aberration generates structural modification and positional shift of the generated focal structure. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction In modern optics study and applications of the optical vortex beam have recently generated great research [1–7]. Recently the focusing properties of a circularly, radially, azimuthally or linearly polarized vortex beam by a high numerical-aperture lens have been discussed [8–13]. The radially and the azimuthally polarized beams are of particular importance in many application fields due to unique cylindrical symmetry of polarization. The size and shape of the focused structure of the vortex beam play an important role in many applications such as in microscopy, lithography, data storage, particle trapping, etc. A deformed focused structure may cause serious problems in optical trapping and microscopy. Deformation in the focused structure can be due to aberrations. Under realistic experimental conditions, it is inevitable to suffer wave front aberrations even for the well corrected objectives [14–16]. When a tightly focused laser propagates through an interface of two different materials, spherical aberration will be induced due to refractive-index mismatch. Marcinkvcius et al. [17] reported that this interface SA increase the size and distorts the shape of photo damaged regions and thus reduces the spatial resolution of micro fabrication, where the focusing depth inside the glass is no more than 250 m. An important investigation was initiated by Braat
∗ Corresponding author. E-mail address: [email protected] (K.B. Rajesh). http://dx.doi.org/10.1016/j.ijleo.2014.08.124 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
et al. [18] who used extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high NA optical beam. Biss and Brown [19] have investigated the effect of primary aberrations on the focused structure of the radially polarized vortex beam. However no detailed studies seem to have been made on the effect of primary aberrations on the tight focusing of cylindrical vortex beam. In view of the importance of the high NA focusing of CVB, in this paper we present the results of our investigation of effect of spherical aberration, on cylindrical vortex beam.
2. Theoretical model A schematic diagram of the suggested method is shown in Fig. 1. The analysis was performed on the basis of Richards and Wolf’s vectorial diffraction method [20] widely used for high-NA focusing systems at arbitrary incident polarization. Instead of a radial polarization or an azimuthally polarization, each point of the optical vortex beam has a polarization rotated by ϕ0 from its radial direction. In this paper, we assume that the cylindrically vortex beam is incident upon a high NA lens. Since this beam can be expressed as a linear combination of the focal fields of radial polarization and azimuthally polarization, we adapted the same analysis method as that in Ref. [21]. The focal field of a cylindrically polarized vortex beam can be written as: E (r, z) = Er er + Ez ez + Eϕ eϕ
(1)
G.T. Anita et al. / Optik 125 (2014) 6870–6873
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3. Results 3.1. Radially polarized vortex beam Fig. 2(a) shows the intensity distribution in the focal plane for
Ф0 = 0◦ , NA = sin(80◦ ) and topological charge n = 1, which corre-
Fig. 1. Schematic diagram of generalized cylindrical vortex beam.
where Er , Ez are the amplitudes of the three orthogonal components and er , ez are their corresponding unit vectors. The two orthogonal components of the electric field is given as: Er =
−iA × cos(0 ) 2
˛
0
2
cos()
0
× sin(2) × A1 × cos( − ϕ) exp(in) × exp[ik(z cos() + r sin() cos( − ϕ)]ddϕ
Ez =
iA × cos(0 )
˛
2
(2)
cos() × sin2 () × A1
0
0
× exp(in) × exp[ik(z cos() + r sin() cos( − ϕ)]ddϕ
Eϕ =
−iA × sin(0 )
˛
0
2
(3)
cos() × sin() × A1
0
× cos( − ϕ) exp(in) × exp[ik(z cos() + r sin() cos( − ϕ)]ddϕ
(4)
where r, ϕ and z are the radial, azimuthal and longitudinal coordinates of the observation point in the focal region respectively. k = 2/ is wave number, and ˛ = arcsin(NA), is the maximal angle determined by the numerical aperture of the lens. A1 denotes the wave front aberration function in the beam which can be expressed as [22]:
A1 = exp I.k.As
sin() sin(˛)
4
(5)
where the spherical aberration coefficient As is in units of the wave length of the beam.
sponds to radially polarized vortex incident beam without spherical aberrations. The generated focal structure agrees well with Fig. 2(d) of [23]. The total intensity is the sum of the radial intensity and longitudinal intensity. In this case, the azimuthal component disappears and only the radial and longitudinal components are present. It is observed that the total intensity is non-zero at the optical axis which is in contrast to the n = 1 case, where the longitudinal component distribution has an on-axial maximum. Fig. 2(b–d) shows the intensity distribution at the focus for different As values. It is observed from the figure, that the intensity distribution in the focal plane undergo significant changes in the presence of spherical aberration. It is noted, that the spherical aberration not only changes the focal structure and intensity distribution of the generated focal segment but also shifted it away from the aperture. It is clearly observed, that as the value of spherical aberration coefficient increases, the shifting of generated focal segment along z-axis is also increased. Fig. 2(b) shows the position of maximum intensity of the generated focal spot at z = 0.5 when As = 0.5. However further increasing of As to 1.5 shifted the focal segment axially to z = 3. It is also observed when As increases, the generated focal segment undergoes structural modification such that the size of the generated focal segment squeezing towards aperture. The modification of the structure is visible when the value of As increases to 1.5. Fig. 2(e–h) shows the corresponding two dimensional intensity distribution calculated at the position of maximum intensity.
3.2. Azimuthally polarized vortex beam Fig. 3(a) plots the intensity distribution in the focal plane for
Ф0 = 90◦ , which correspond to an azimuthally polarized vortex incident beam without spherical aberration. Azimuthally polarized vortex beam has no radial and longitudinal component and the total intensity distribution on the optical axis has a maximum which is in contradiction to n = 0, case where the generated focal segment is a focal hole. We measured the FWHM of the generated focal spot is (0.31) and agrees well with the result of [23]. Hence, such azimuthally polarized vortex beams can obtain a highly confined focal spot. However, it is noted that the presence of spherical aberration generates focal shifting and structural modification. Fig. 3(b)
Fig. 2. Total intensity distribution of radially polarized cylindrical vortex beam (a) As = 0.0, (b) As = 0.5, (c) 1.0, (d) As = 1.5 and two dimensional intensity distribution at the focal plane (e) As = 0.0, (f) As = 0.5, (g) As = 1.0, (h) As = 1.5.
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G.T. Anita et al. / Optik 125 (2014) 6870–6873
Fig. 3. Total intensity distribution of azimuthally polarized cylindrical vortex beam (a) As = 0.0, (b) As = 0.5, (c) 1.0, (d) As = 1.5 and two dimensional intensity distribution at the focal plane (e) As = 0.0, (f) As = 0.5, (g) As = 1.0, (h) As = 1.5.
Fig. 4. Total intensity distribution of flat top profile (a) As = 0.0, (b) As = 0.5, (c) 1.0, (d) As = 1.5 and two dimensional intensity distribution at the focal plane (e) As = 0.0, (f) As = 0.5, (g) As = 1.0, (h) As = 1.5.
shows the generated focal segment for As = 0.5, it is observed from the figure that the generated focal spot shifts away from the aperture and the position of maximum intensity is observed at z = 0.7. It is also observed that further increasing of As value further shifts the focal segment and is shown in Fig. 3(c and d). We also observed that increasing of As value increases the intensity of the side lobe of the generated focal segment which appears to getting squeezed away from the focus when As = 1.5 and is shown in Fig. 3(d). The corresponding two dimensional intensity distribution of azimuthally polarized beam is shown in Fig. 3(e–h). 3.3. Flat-topped focus Fig. 4(a) shows the intensity distribution in the focal plane for
Ф0 = 31.2◦ , which is correspond to a cylindrically polarized vortex incident beam without spherical aberration. It is observed from the figure the generated focal segment is a flat top profile with FWHM of (1.2). Such a flat top profile is high useful to improve uniformity and quality in material processing and microlithography. However we observed that the presence of spherical aberration modifies the intensity distribution and shifted the generated focal structure along z-axis and is shown in Fig. 4(b–d). It is observed
from Fig. 4(b), the generated flat top profile shifted to z = (0.6) with slight structural modification when As = 0.5. However further increase of As value further shifted the focal structure axially away from the focus and completely deformed the flat top profile as shown in Fig. 4(c and d) corresponding to As = 1 and 1.5 respectively. The corresponding two dimensional intensity distributions are shown in Fig. 4(e–h). 4. Conclusion In conclusion, the effects of the spherical aberration on tightly focused cylindrical vortex beam have been investigated using vector diffraction theory. The results show that the presence of spherical aberration generates positional shift, structural modifications and spreading of intensity distribution. Hence we observed that the presence of spherical aberration shifts and modified the focal structure in the case of tightly focused radially and cylindrically polarized vortex beam. However in the case of azimuthally polarized vortex beam, focal shift with slight modification on focal structure is observed. The authors expect such a study is useful for designing objectives for linear confocal or nonlinear microscopy in which weekly aberrated high NA lens system might be used.
G.T. Anita et al. / Optik 125 (2014) 6870–6873
References [1] J.F. Nye, M.V. Berry, Dislocations in wave trains, Proc. R. Soc. Lond. Ser. A 336 (1974) 165–190. [2] M.S. Soskin, M.V. Vasnetsov, Singular Optics, Prog. Opt. (Amst.) 42 (2001) 219–276. [3] V.V. Kotlyar, A.A. Almazov, S.N. Khonina, V.A. Soifer, Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate, J. Opt. Soc. Am. A 22 (2005) 849–861. [4] D.M. Palacios, I.D. Maleev, A.S. Marathay, G.A. Swartzlander, Spatial correlation vortices in partially coherent light theory, Phys. Rev. Lett. 92 (2004) 143905–143908. [5] S.H. Tao, X.C. Yuan, J. Lin, R.E. Burge, Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges, Opt. Express 14 (2006) 535–541. [6] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, Novel optical trap of atoms with a doughnut beam, Phys. Rev. Lett. 78 (1997) 4713–4716. [7] L. Rao, J. Pu, Focusing of partially coherent vortex beams by an aperture lens, Chin. Phys. Lett. 24 (2007) 1252–1255. [8] S. Quabis, R. Dorn, M. Eberler, O. Glockl, G. Leuchs, The focus of light-theoretical calculation and experimental tomographic reconstruction, Appl. Phys. B 72 (2001) 109–113. [9] L.E. Helseth, Optical vortices in focal region, Opt. Commun. 229 (2004) 85–91. [10] T. Grosjean, D. Courjon, Smallest focal spots, Opt. Commun. 272 (2007) 314–319. [11] D. Ganic, X. Gan, M. Gu, Focusing of doughnut laser beams by a high numericalaperture objective in free space, Opt. Express 11 (2003) 2747–2752.
6873
[12] Q. Zhan, J.R. Leeger, Focus shaping using cylindrical vector beams, Opt. Express 10 (2002) 324–331. [13] Q. Zhan, Properties of circularly polarized vortex beams, Opt. Lett. 31 (2006) 867–869. [14] D.P. Biss, T.G. Brown, Primary aberrations in focused radially polarized vortex beams, Opt. Express 12 (3) (2004) 384–393. [15] B.R. Boruah, M.A.A. Neil, Susceptibility to and correction of azimuthal aberrations in singular light beams, Opt. Express 14 (22) (2006) 10377–10385. [16] B.R. Boruah, M.A.A. Neil, Far field computation of an arbitrarily polarized beam using fast Fourier transforms, Opt. Commun. 282 (24) (2009) 4660–4667. [17] A. Marcinkevicius, V. Mizeikis, S. Juodkazis, S. Matsuo, H. Misawa, Effect of refractive index-mismatch on laser microfabrication in silica glass, Appl. Phys. A 76 (2003) 257–260. [18] J.J.M. Braat, P. Dirksen, A. Janssen, A.S. VandeNes, Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high NA aperture optical system, J. Opt. Soc. Am. A 20 (2003) 2281–2292. [19] D.P. Biss, T.G. Brown, Primary aberrations in focused radially polarized vortex beams, Opt. Express 12 (2004) 384–393. [20] B. Richards, E. Wolf, Electromagnetic diffraction in optical systems II. Structure of the Image field in an aplanatic system, Proc. R. Soc. Lond. Ser. A 253 (1959) 358–379. [21] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2000) 77–87. [22] R. Kant, An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion, J. Mod. Opt. 40 (11) (1993) 2293–2310. [23] L. Rao, J. Pu, Z. Chen, P. Yei, Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens, Opt. Laser Technol. 41 (2009) 241–246.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Effect of spherical aberration on tightly focused cylindrically polarized vortex beams G. Therese Anita a , C. Amala Prathiba Janet b , K. Prabakaran c , T.V.S. Pillai c , K.B. Rajesh d,∗ a
Department of Physics, Arunachala College of Engineering for Women, Nagercoil, Tamil Nadu, India Department of Physics, St. Xavier’s Catholic College of Engineering, Nagercoil, Tamil Nadu, India Department of Physics, Periyar Universit- PG Extension Centre, Dharmapuri, Tamil Nadu, India d Department of Physics, Chikkanna Government Arts College, Tiruppur, Tamil Nadu, India b c
a r t i c l e
i n f o
Article history: Received 5 August 2013 Accepted 5 February 2014 Keywords: Vector diffraction theory Cylindrical vortex beam Polarization Spherical aberration
a b s t r a c t In this paper attention is given to the effects of primary spherical aberration on the cylindrical polarized vortex beam based on the vector diffraction theory. It is observed that by properly choosing the polarization angle and topological charge one can obtain many novel focal patterns suitable for optical tweezers, laser printing and material process. However, it is observed that the focusing objective with spherical aberration generates structural modification and positional shift of the generated focal structure. © 2014 Elsevier GmbH. All rights reserved.
1. Introduction In modern optics study and applications of the optical vortex beam have recently generated great research [1–7]. Recently the focusing properties of a circularly, radially, azimuthally or linearly polarized vortex beam by a high numerical-aperture lens have been discussed [8–13]. The radially and the azimuthally polarized beams are of particular importance in many application fields due to unique cylindrical symmetry of polarization. The size and shape of the focused structure of the vortex beam play an important role in many applications such as in microscopy, lithography, data storage, particle trapping, etc. A deformed focused structure may cause serious problems in optical trapping and microscopy. Deformation in the focused structure can be due to aberrations. Under realistic experimental conditions, it is inevitable to suffer wave front aberrations even for the well corrected objectives [14–16]. When a tightly focused laser propagates through an interface of two different materials, spherical aberration will be induced due to refractive-index mismatch. Marcinkvcius et al. [17] reported that this interface SA increase the size and distorts the shape of photo damaged regions and thus reduces the spatial resolution of micro fabrication, where the focusing depth inside the glass is no more than 250 m. An important investigation was initiated by Braat
∗ Corresponding author. E-mail address: [email protected] (K.B. Rajesh). http://dx.doi.org/10.1016/j.ijleo.2014.08.124 0030-4026/© 2014 Elsevier GmbH. All rights reserved.
et al. [18] who used extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high NA optical beam. Biss and Brown [19] have investigated the effect of primary aberrations on the focused structure of the radially polarized vortex beam. However no detailed studies seem to have been made on the effect of primary aberrations on the tight focusing of cylindrical vortex beam. In view of the importance of the high NA focusing of CVB, in this paper we present the results of our investigation of effect of spherical aberration, on cylindrical vortex beam.
2. Theoretical model A schematic diagram of the suggested method is shown in Fig. 1. The analysis was performed on the basis of Richards and Wolf’s vectorial diffraction method [20] widely used for high-NA focusing systems at arbitrary incident polarization. Instead of a radial polarization or an azimuthally polarization, each point of the optical vortex beam has a polarization rotated by ϕ0 from its radial direction. In this paper, we assume that the cylindrically vortex beam is incident upon a high NA lens. Since this beam can be expressed as a linear combination of the focal fields of radial polarization and azimuthally polarization, we adapted the same analysis method as that in Ref. [21]. The focal field of a cylindrically polarized vortex beam can be written as: E (r, z) = Er er + Ez ez + Eϕ eϕ
(1)
G.T. Anita et al. / Optik 125 (2014) 6870–6873
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3. Results 3.1. Radially polarized vortex beam Fig. 2(a) shows the intensity distribution in the focal plane for
Ф0 = 0◦ , NA = sin(80◦ ) and topological charge n = 1, which corre-
Fig. 1. Schematic diagram of generalized cylindrical vortex beam.
where Er , Ez are the amplitudes of the three orthogonal components and er , ez are their corresponding unit vectors. The two orthogonal components of the electric field is given as: Er =
−iA × cos(0 ) 2
˛
0
2
cos()
0
× sin(2) × A1 × cos( − ϕ) exp(in) × exp[ik(z cos() + r sin() cos( − ϕ)]ddϕ
Ez =
iA × cos(0 )
˛
2
(2)
cos() × sin2 () × A1
0
0
× exp(in) × exp[ik(z cos() + r sin() cos( − ϕ)]ddϕ
Eϕ =
−iA × sin(0 )
˛
0
2
(3)
cos() × sin() × A1
0
× cos( − ϕ) exp(in) × exp[ik(z cos() + r sin() cos( − ϕ)]ddϕ
(4)
where r, ϕ and z are the radial, azimuthal and longitudinal coordinates of the observation point in the focal region respectively. k = 2/ is wave number, and ˛ = arcsin(NA), is the maximal angle determined by the numerical aperture of the lens. A1 denotes the wave front aberration function in the beam which can be expressed as [22]:
A1 = exp I.k.As
sin() sin(˛)
4
(5)
where the spherical aberration coefficient As is in units of the wave length of the beam.
sponds to radially polarized vortex incident beam without spherical aberrations. The generated focal structure agrees well with Fig. 2(d) of [23]. The total intensity is the sum of the radial intensity and longitudinal intensity. In this case, the azimuthal component disappears and only the radial and longitudinal components are present. It is observed that the total intensity is non-zero at the optical axis which is in contrast to the n = 1 case, where the longitudinal component distribution has an on-axial maximum. Fig. 2(b–d) shows the intensity distribution at the focus for different As values. It is observed from the figure, that the intensity distribution in the focal plane undergo significant changes in the presence of spherical aberration. It is noted, that the spherical aberration not only changes the focal structure and intensity distribution of the generated focal segment but also shifted it away from the aperture. It is clearly observed, that as the value of spherical aberration coefficient increases, the shifting of generated focal segment along z-axis is also increased. Fig. 2(b) shows the position of maximum intensity of the generated focal spot at z = 0.5 when As = 0.5. However further increasing of As to 1.5 shifted the focal segment axially to z = 3. It is also observed when As increases, the generated focal segment undergoes structural modification such that the size of the generated focal segment squeezing towards aperture. The modification of the structure is visible when the value of As increases to 1.5. Fig. 2(e–h) shows the corresponding two dimensional intensity distribution calculated at the position of maximum intensity.
3.2. Azimuthally polarized vortex beam Fig. 3(a) plots the intensity distribution in the focal plane for
Ф0 = 90◦ , which correspond to an azimuthally polarized vortex incident beam without spherical aberration. Azimuthally polarized vortex beam has no radial and longitudinal component and the total intensity distribution on the optical axis has a maximum which is in contradiction to n = 0, case where the generated focal segment is a focal hole. We measured the FWHM of the generated focal spot is (0.31) and agrees well with the result of [23]. Hence, such azimuthally polarized vortex beams can obtain a highly confined focal spot. However, it is noted that the presence of spherical aberration generates focal shifting and structural modification. Fig. 3(b)
Fig. 2. Total intensity distribution of radially polarized cylindrical vortex beam (a) As = 0.0, (b) As = 0.5, (c) 1.0, (d) As = 1.5 and two dimensional intensity distribution at the focal plane (e) As = 0.0, (f) As = 0.5, (g) As = 1.0, (h) As = 1.5.
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G.T. Anita et al. / Optik 125 (2014) 6870–6873
Fig. 3. Total intensity distribution of azimuthally polarized cylindrical vortex beam (a) As = 0.0, (b) As = 0.5, (c) 1.0, (d) As = 1.5 and two dimensional intensity distribution at the focal plane (e) As = 0.0, (f) As = 0.5, (g) As = 1.0, (h) As = 1.5.
Fig. 4. Total intensity distribution of flat top profile (a) As = 0.0, (b) As = 0.5, (c) 1.0, (d) As = 1.5 and two dimensional intensity distribution at the focal plane (e) As = 0.0, (f) As = 0.5, (g) As = 1.0, (h) As = 1.5.
shows the generated focal segment for As = 0.5, it is observed from the figure that the generated focal spot shifts away from the aperture and the position of maximum intensity is observed at z = 0.7. It is also observed that further increasing of As value further shifts the focal segment and is shown in Fig. 3(c and d). We also observed that increasing of As value increases the intensity of the side lobe of the generated focal segment which appears to getting squeezed away from the focus when As = 1.5 and is shown in Fig. 3(d). The corresponding two dimensional intensity distribution of azimuthally polarized beam is shown in Fig. 3(e–h). 3.3. Flat-topped focus Fig. 4(a) shows the intensity distribution in the focal plane for
Ф0 = 31.2◦ , which is correspond to a cylindrically polarized vortex incident beam without spherical aberration. It is observed from the figure the generated focal segment is a flat top profile with FWHM of (1.2). Such a flat top profile is high useful to improve uniformity and quality in material processing and microlithography. However we observed that the presence of spherical aberration modifies the intensity distribution and shifted the generated focal structure along z-axis and is shown in Fig. 4(b–d). It is observed
from Fig. 4(b), the generated flat top profile shifted to z = (0.6) with slight structural modification when As = 0.5. However further increase of As value further shifted the focal structure axially away from the focus and completely deformed the flat top profile as shown in Fig. 4(c and d) corresponding to As = 1 and 1.5 respectively. The corresponding two dimensional intensity distributions are shown in Fig. 4(e–h). 4. Conclusion In conclusion, the effects of the spherical aberration on tightly focused cylindrical vortex beam have been investigated using vector diffraction theory. The results show that the presence of spherical aberration generates positional shift, structural modifications and spreading of intensity distribution. Hence we observed that the presence of spherical aberration shifts and modified the focal structure in the case of tightly focused radially and cylindrically polarized vortex beam. However in the case of azimuthally polarized vortex beam, focal shift with slight modification on focal structure is observed. The authors expect such a study is useful for designing objectives for linear confocal or nonlinear microscopy in which weekly aberrated high NA lens system might be used.
G.T. Anita et al. / Optik 125 (2014) 6870–6873
References [1] J.F. Nye, M.V. Berry, Dislocations in wave trains, Proc. R. Soc. Lond. Ser. A 336 (1974) 165–190. [2] M.S. Soskin, M.V. Vasnetsov, Singular Optics, Prog. Opt. (Amst.) 42 (2001) 219–276. [3] V.V. Kotlyar, A.A. Almazov, S.N. Khonina, V.A. Soifer, Generation of phase singularity through diffracting a plane or Gaussian beam by a spiral phase plate, J. Opt. Soc. Am. A 22 (2005) 849–861. [4] D.M. Palacios, I.D. Maleev, A.S. Marathay, G.A. Swartzlander, Spatial correlation vortices in partially coherent light theory, Phys. Rev. Lett. 92 (2004) 143905–143908. [5] S.H. Tao, X.C. Yuan, J. Lin, R.E. Burge, Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges, Opt. Express 14 (2006) 535–541. [6] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, Novel optical trap of atoms with a doughnut beam, Phys. Rev. Lett. 78 (1997) 4713–4716. [7] L. Rao, J. Pu, Focusing of partially coherent vortex beams by an aperture lens, Chin. Phys. Lett. 24 (2007) 1252–1255. [8] S. Quabis, R. Dorn, M. Eberler, O. Glockl, G. Leuchs, The focus of light-theoretical calculation and experimental tomographic reconstruction, Appl. Phys. B 72 (2001) 109–113. [9] L.E. Helseth, Optical vortices in focal region, Opt. Commun. 229 (2004) 85–91. [10] T. Grosjean, D. Courjon, Smallest focal spots, Opt. Commun. 272 (2007) 314–319. [11] D. Ganic, X. Gan, M. Gu, Focusing of doughnut laser beams by a high numericalaperture objective in free space, Opt. Express 11 (2003) 2747–2752.
6873
[12] Q. Zhan, J.R. Leeger, Focus shaping using cylindrical vector beams, Opt. Express 10 (2002) 324–331. [13] Q. Zhan, Properties of circularly polarized vortex beams, Opt. Lett. 31 (2006) 867–869. [14] D.P. Biss, T.G. Brown, Primary aberrations in focused radially polarized vortex beams, Opt. Express 12 (3) (2004) 384–393. [15] B.R. Boruah, M.A.A. Neil, Susceptibility to and correction of azimuthal aberrations in singular light beams, Opt. Express 14 (22) (2006) 10377–10385. [16] B.R. Boruah, M.A.A. Neil, Far field computation of an arbitrarily polarized beam using fast Fourier transforms, Opt. Commun. 282 (24) (2009) 4660–4667. [17] A. Marcinkevicius, V. Mizeikis, S. Juodkazis, S. Matsuo, H. Misawa, Effect of refractive index-mismatch on laser microfabrication in silica glass, Appl. Phys. A 76 (2003) 257–260. [18] J.J.M. Braat, P. Dirksen, A. Janssen, A.S. VandeNes, Extended Nijboer–Zernike representation of the vector field in the focal region of an aberrated high NA aperture optical system, J. Opt. Soc. Am. A 20 (2003) 2281–2292. [19] D.P. Biss, T.G. Brown, Primary aberrations in focused radially polarized vortex beams, Opt. Express 12 (2004) 384–393. [20] B. Richards, E. Wolf, Electromagnetic diffraction in optical systems II. Structure of the Image field in an aplanatic system, Proc. R. Soc. Lond. Ser. A 253 (1959) 358–379. [21] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2000) 77–87. [22] R. Kant, An analytical solution of vector diffraction for focusing optical systems with Seidel aberrations I. Spherical aberration, curvature of field, and distortion, J. Mod. Opt. 40 (11) (1993) 2293–2310. [23] L. Rao, J. Pu, Z. Chen, P. Yei, Focus shaping of cylindrically polarized vortex beams by a high numerical-aperture lens, Opt. Laser Technol. 41 (2009) 241–246.