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Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element
Optik 124 (2013) 5039–5041
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element K. Prabakaran a , K.B. Rajesh b,∗ , T.V.S. Pillai a , Z. Jaroszewicz c a b c
Department of Physics, Regional centre, Anna University, Tirunelveli Region,Tirunelveli, Tamil Nadu, India Department of Physics, Chikkanna Government Arts College, Tirupur, Tamilnadu, India Institute of Applied Optics, Department of Physical Optics, Warsaw, Poland and National Institute of Telecommunications, Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 20 March 2012 Accepted 3 March 2013
Keywords: Lens axicon Vector diffraction theory Polarization Diffractive optical element
a b s t r a c t We study the focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam with high numerical aperture lens axicon system is investigated theoretically by vector diffraction theory. The intensity pattern at the focus can be tailored by appropriately adjusting the rotation angle. We show that the high NA lens axicon system can generates a sub wavelength focal spot, focal hole, focal splitting and flat-topped focal shapes with extended depth of focus. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction In recent years, there is an increasing interest in cylindrical vector beams due to the focus properties of these beams, especially when tightly focused by a high numerical aperture (NA) objective [1–3]. Because of the symmetry of the polarization, the electric field distribution of a focused generalized cylindrical vector beam in the focal plane is usually rotationally symmetrical. In addition, the intensity pattern in the vicinity of the focal spot can be tailored by changing the polarization direction of the cylindrical vector beam [2,3]. For example, it has been shown that the longitudinal component of the focus from such a cylindrical beam is much stronger than the transversal component [4]. Youngworth and Brown calculate cylindrical-vector fields, near the focal region of an aplanatic lens [1]. It is showed that, in the particular case of a tightly focused radially polarized beam, the polarization shows large inhomogeneities in the focal region, while the azimuthally polarized beam is purely transverse even at very high numerical apertures. Recently, Intensity distribution in focal region plays an important role in many optical systems, such as in optical tweezers. Focusing an incoming light into a smaller spot or focal hole with long depth of focus (DOF) is always one of the most important topics for optical engineers and scientists. Applications of such beams include microscopy, particle guiding or trapping [5–9], scanning optical microscopy [10], lithography [11], laser cutting of metals [12,13], particle acceleration
∗ Corresponding author. E-mail address: [email protected] (K.B. Rajesh). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.031
[14]. The spot size can be much smaller than the diffraction limit of focused spatially homogeneously polarized beams. Besides an ultrasmall focus, it has been shown that focal distribution with extended depth of focus (DOF) can be achieved with inserting a simple diffractive optical element (DOE) in the optical path under radial polarization illumination [15]. The axicon, energy wise, is the most efficient method for generating beam with large focal depth. Recently we have introduced the possible design of high NA lens axicon to generate sub wavelength beam with large depth of focus [16–18]. The high NA lens axicon is a system of a cemented doublet-lens, where the virtual focal segment created by the aberrated diverging lens can be converted to a real focal segment, of the forward type with a nano scale resolution, by adding a high numerical aperture (NA) converging lens. Here we consider only systems that comprise diverging lens that has third-order spherical aberration and a perfect high NA converging lens. In the paper, we study the focus shaping of tightly focused cylindrically polarized Laguerre Gaussian beams by diffractive optical elements (DOE) with high NA lens axicon. 2. Theory A schematic diagram of the suggested method is shown in Fig. 1. The double ring shaped CV beam with tunable polarization is phase modulated through three belts binary diffractive optical element (BDOE) and then focused through a high NA lens axicon system. The analysis was performed on the basis of Richards and Wolf’s vectorial diffraction method [19] widely used for high-NA lens system at arbitrary incident polarization. In the case of the incident
5040
K. Prabakaran et al. / Optik 124 (2013) 5039–5041
Fig. 1. Focusing of double ring shaped cylindrical vector beam with high NA lens axicon.
polarization, adopting the cylindrical coordinates r, z, ϕ and the notations of Ref. [1], the focal field of a generalized cylindrical beam can be written as E (r, ϕ, z) = Er er + Ez ez + Eϕ eϕ
(1)
where Er , Ez and Eϕ are the amplitudes of the three orthogonal components and er , ez and eϕ are their corresponding unit vectors. The three orthogonal components of the electric field is given as
˛
cos1/2 ()A() sin 2J1 (kr sin )eikz cos d
Er (r, ϕ, z) = A cos
(2)
0
˛ 2
Ez (r, ϕ, z) = 2iA cos
cos1/5 ()A() sin J0 (kr sin )eikz cos d
(3)
cos1/2 ()A() sin()J1 (kr sin )eikz cos d
(4)
0
Eϕ (r, ϕ, z) = 2A sin
˛
0
where ˛ = arcsin(NA)/n the maximal angle is determined by the numerical aperture of the objective lens, and n is the index of refraction between the lens and the sample. k = 2/ is the wave number and Jn (x) is the Bessel function of the first kind with order n. r and z are the radial and z coordinates of observation point in focal region, respectively. A() describes the amplitude modulation and for illumination by a double ring shaped cylindrical vector beam with its waist in the pupil, this function is given by [17]. sin
sin A() = ˇ exp − ˇ sin ˛ sin2 ˛ 2
2
Lp1
sin 2 ˇ sin ˛
2
(5)
where ˇ is the parameter that denoted the ratio of pupil diameter to the beam diameter and Lp1 is the generalized Laguerre polynomial. If p = 1, the incident beam is a double-ring shaped CV beam. The effect of phase modulation on the input cylindrical vector beam is evaluated by replacing the function A() by A() P(). For 3 belts Binary diffractive optical element P() is given by
P() =
Fig. 2. Intensity distributions at the focus of the Lens axicon for (a) = 0◦ at z = 0 and (b) are the corresponding contour plot for the total intensity distribution in the r–z plane.
3. Results We perform the integration of Eq. (1) numerically using parameters = 1, ˇ = 1.3 and NA of the objective is 0.9. Here, for simplicity, we assume that the refractive index n = 1 and A = 1. For all calculation in the length unit is normalized to and the energy density is normalized to unity. Fig. 2 illustrates the evolution of threedimensional light intensity distribution of the high NA lens axicon for different polarization angle of the incident phase modulated double ring shaped CV beam. It is observed that by setting = 0◦ , which corresponds to a double ring shaped radially polarized incident beam, one can achieve a sub wavelength focal spot with large focal depth. From Fig. 2a and b we measured the FWHM of the generated sub wavelength focal spot as 0.42 and its focal depth as 14 respectively. The set of three angles of the BDOE optimized to achieve the above mentioned focal segment using traditional global search algorithm are 1 = 17.4◦ , 2 = 58.24◦ , ˛ = 64.19◦ . Such a needle of sub wavelength beam with large focal depth finds its application in near field optical recording and particle acceleration. The intensity distribution for = 90◦ , which corresponds to a azimuthally polarized incident double ring shaped beam for the high NA lens axicon is shown in Fig. 3a and b. In this case, only the azimuthal component is present at the focus and the focal segment resembles a doughnut shape. The FWHM of the generated focal hole is 0.96. The 3D intensity distribution near the focus as shown in Fig. 3b, reveals that the focal depth is 14.2. The set of three angles of the BDOE optimized to achieve the above mentioned focal segment 1 = 17.4◦ , 2 = 58.24◦ , ˛ = 64.19◦ . Such a focal hole segment is highly useful for trapping particles with a dielectric constant lower than the ambient. We also observed that it is possible to split the generated focal spot or focal hole segment axially by properly tuning the phase of the incident double ring shaped CV beam. We also observed that by setting 1 = 40.56◦ , 2 = 62.59◦ , ˛ = 64.19◦ it is possible for us to generated sub wave length double focal spot segments each with FWHM of 0.4 and are axially separated by a longer distance of 22 between them as shown
1, for0 < < 1 , 2 < < ˛, −1for0, 1 < < 2
The intensity distribution of the lens axicon is evaluated by multiply the function P() by the function P()T(), where T() is the non-paraxial transmittance function of the thin aberrated diverging lens [20].
T () = exp
ik
sin() sin(˛)
4 +
1 2f
sin() sin(˛)
2
(7)
Here k = 2/, f is the focal length and is the aberration coefficient. In our calculation we take f = 18.4 mm, = 6.667 × 10−5 mm−3 which results in a equiconcave configure [21].
Fig. 3. Intensity distributions at the focus of the Lens axicon for (a) = 90◦ at z = 0 and (b) are the corresponding contour plot for the total intensity distribution in the r–z plane.
K. Prabakaran et al. / Optik 124 (2013) 5039–5041
5041
on vector diffraction theory. It is observed that the proposed system generates sub wavelength beam with large depth of focus and we expect such a beam with small spot size and long depth of focus can be widely used in application such as optical tweezers, data storage, biomedical imaging, laser drilling, and machining. References
Fig. 4. Intensity distributions in focal region Lens axicon for (a) = 0◦ and (b) = 90◦ .
Fig. 5. Intensity distributions in focal region Lens axicon for (a) = 51◦ at z = 0 and (b) are the corresponding contour plot for the total intensity distribution.
in Fig. 4a. However, for the polarization angle = 90◦ , the same BDOE generates a two sub wave length focal hole segments each with FWHM of 0.3 and are axially separated by a large distance of 22 between them and is shown in Fig. 4b. Such a focal splitting is useful in controlling the position of the optical traps. We also wish to generate a flattop total intensity distribution at the focus by adjusting the weightings of the three field components through controlling of polarization angle . The flattop profile is obtained for the angle = 51◦ . From Fig. 5a, we measured the FWHM of the total intensity of generated flat top focal profile is 0.89 and its corresponding focal depth is 14 which is shown in Fig. 5b. The set of three angles of the BDOE optimized to achieve the above mentioned focal segment are 1 = 30◦ , 2 = 58.24◦ , ˛ = 64.19◦ . This type of beam profile is useful in improved printing filling factor, improved uniformity and quality in materials processing, particle acceleration and microlithography. 4. Conclusion The intensity distributions of the double-ring-shaped cylindrically polarized beam by a high NA lens axicon were calculated based
[1] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2) (2000) 77–87. [2] Q.W. Zhan, J.R. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (7) (2002) 324–331. [3] W.B. Chen, Q.W. Zhan, Three-dimensional focus shaping with cylindrical vector beams, Opt. Commun. 265 (2) (2006) 411–417. [4] 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. [5] 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. [6] K.T. Gahagan, G.A. Swartzlander Jr., Simultaneous trapping of low-index and high- index micro particles observed with an optical-vortex trap, J. Opt. Soc. Am. B 16 (1999) 533–537. [7] Q. Zhan, Trapping metallic Rayleigh particles with radial polarization, Opt. Express 12 (15) (2004) 3377–3382. [8] Q. Zhan, Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams, J. Opt. A: Pure Appl. Opt. 5 (2003) 229–232. [9] Y.Q. Zhao, Q. Zhan, Y.L. Zhang, Y. Li, Creation of a three-dimensional optical chain for controllable particle delivery, Opt. Lett. 30 (2005) 848–850. [10] K.S. Youngworth, T.G. Brown, Inhomogeneous polarization in scanning optical microscopy, Proc. SPIE 3919 (2000). [11] L.E. Helseth, Roles of polarization, phase and amplitude in solid immersion lens systems, Opt. Commun. 191 (2001) 161–172. [12] V.G. Niziev, A.V. Nesterov, Influence of beam polarization on laser cutting efficiency, J. Phys. D 32 (1999) 1455–1461. [13] A.V. Nesterov, V.G. Niziev, Laser beams with axially symmetric polarization, J. Phys. D 33 (1999) 1817–1822. [14] B. Hafizi, E. Esarey, P. Sprangle, Laser-driven acceleration with Bessel beams, Phys. Rev. E 55 (1997) 3539–3545. [15] C. Sun, C. Liu, Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation, Opt. Lett. 28 (2003) 99–101. [16] K.B. Rajesh, Z. Jaroszewicz, P.M. Anbarasan, Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance, Opt. Express 18 (2000) 26799–26805. [17] K.B. Rajesh, N. VeeraBagu Suresh, P.M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, Tight focusing of double ring shaped radially polarized beam with high NA lens axicon, J. Opt. Laser Technol. 43 (2011) 1037–1040. [18] K.B. Rajesh, Z. Jaroszewicz, P.M. Anbarasan, T.V.S. Pillai, N. Veerabagu Suresh, Extending the depth of focus with high NA lens axicon, Optik 122 (2011) 1619–1621. [19] B. Richards, 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) (1959) 358–379. [20] K.B. Rajesh, P.M. Anbarasan, Generation of sub-wavelength and superresolution longitudinally polarized non-diffraction beam using lens axicon, Chin. Opt. Lett. 6 (10) (2008) 785–787. [21] Z. Jaroszewicz, J. Morales, Lens axicon system composed of a diverging aberrated lens and a perfect converging lens, J. Opt. Soc. Am. A 15 (9) (1998) 2383–2390.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam by diffractive optical element K. Prabakaran a , K.B. Rajesh b,∗ , T.V.S. Pillai a , Z. Jaroszewicz c a b c
Department of Physics, Regional centre, Anna University, Tirunelveli Region,Tirunelveli, Tamil Nadu, India Department of Physics, Chikkanna Government Arts College, Tirupur, Tamilnadu, India Institute of Applied Optics, Department of Physical Optics, Warsaw, Poland and National Institute of Telecommunications, Warsaw, Poland
a r t i c l e
i n f o
Article history: Received 20 March 2012 Accepted 3 March 2013
Keywords: Lens axicon Vector diffraction theory Polarization Diffractive optical element
a b s t r a c t We study the focus shaping of tightly focused TEM11 mode cylindrically polarized Laguerre Gaussian beam with high numerical aperture lens axicon system is investigated theoretically by vector diffraction theory. The intensity pattern at the focus can be tailored by appropriately adjusting the rotation angle. We show that the high NA lens axicon system can generates a sub wavelength focal spot, focal hole, focal splitting and flat-topped focal shapes with extended depth of focus. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction In recent years, there is an increasing interest in cylindrical vector beams due to the focus properties of these beams, especially when tightly focused by a high numerical aperture (NA) objective [1–3]. Because of the symmetry of the polarization, the electric field distribution of a focused generalized cylindrical vector beam in the focal plane is usually rotationally symmetrical. In addition, the intensity pattern in the vicinity of the focal spot can be tailored by changing the polarization direction of the cylindrical vector beam [2,3]. For example, it has been shown that the longitudinal component of the focus from such a cylindrical beam is much stronger than the transversal component [4]. Youngworth and Brown calculate cylindrical-vector fields, near the focal region of an aplanatic lens [1]. It is showed that, in the particular case of a tightly focused radially polarized beam, the polarization shows large inhomogeneities in the focal region, while the azimuthally polarized beam is purely transverse even at very high numerical apertures. Recently, Intensity distribution in focal region plays an important role in many optical systems, such as in optical tweezers. Focusing an incoming light into a smaller spot or focal hole with long depth of focus (DOF) is always one of the most important topics for optical engineers and scientists. Applications of such beams include microscopy, particle guiding or trapping [5–9], scanning optical microscopy [10], lithography [11], laser cutting of metals [12,13], particle acceleration
∗ Corresponding author. E-mail address: [email protected] (K.B. Rajesh). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.031
[14]. The spot size can be much smaller than the diffraction limit of focused spatially homogeneously polarized beams. Besides an ultrasmall focus, it has been shown that focal distribution with extended depth of focus (DOF) can be achieved with inserting a simple diffractive optical element (DOE) in the optical path under radial polarization illumination [15]. The axicon, energy wise, is the most efficient method for generating beam with large focal depth. Recently we have introduced the possible design of high NA lens axicon to generate sub wavelength beam with large depth of focus [16–18]. The high NA lens axicon is a system of a cemented doublet-lens, where the virtual focal segment created by the aberrated diverging lens can be converted to a real focal segment, of the forward type with a nano scale resolution, by adding a high numerical aperture (NA) converging lens. Here we consider only systems that comprise diverging lens that has third-order spherical aberration and a perfect high NA converging lens. In the paper, we study the focus shaping of tightly focused cylindrically polarized Laguerre Gaussian beams by diffractive optical elements (DOE) with high NA lens axicon. 2. Theory A schematic diagram of the suggested method is shown in Fig. 1. The double ring shaped CV beam with tunable polarization is phase modulated through three belts binary diffractive optical element (BDOE) and then focused through a high NA lens axicon system. The analysis was performed on the basis of Richards and Wolf’s vectorial diffraction method [19] widely used for high-NA lens system at arbitrary incident polarization. In the case of the incident
5040
K. Prabakaran et al. / Optik 124 (2013) 5039–5041
Fig. 1. Focusing of double ring shaped cylindrical vector beam with high NA lens axicon.
polarization, adopting the cylindrical coordinates r, z, ϕ and the notations of Ref. [1], the focal field of a generalized cylindrical beam can be written as E (r, ϕ, z) = Er er + Ez ez + Eϕ eϕ
(1)
where Er , Ez and Eϕ are the amplitudes of the three orthogonal components and er , ez and eϕ are their corresponding unit vectors. The three orthogonal components of the electric field is given as
˛
cos1/2 ()A() sin 2J1 (kr sin )eikz cos d
Er (r, ϕ, z) = A cos
(2)
0
˛ 2
Ez (r, ϕ, z) = 2iA cos
cos1/5 ()A() sin J0 (kr sin )eikz cos d
(3)
cos1/2 ()A() sin()J1 (kr sin )eikz cos d
(4)
0
Eϕ (r, ϕ, z) = 2A sin
˛
0
where ˛ = arcsin(NA)/n the maximal angle is determined by the numerical aperture of the objective lens, and n is the index of refraction between the lens and the sample. k = 2/ is the wave number and Jn (x) is the Bessel function of the first kind with order n. r and z are the radial and z coordinates of observation point in focal region, respectively. A() describes the amplitude modulation and for illumination by a double ring shaped cylindrical vector beam with its waist in the pupil, this function is given by [17]. sin
sin A() = ˇ exp − ˇ sin ˛ sin2 ˛ 2
2
Lp1
sin 2 ˇ sin ˛
2
(5)
where ˇ is the parameter that denoted the ratio of pupil diameter to the beam diameter and Lp1 is the generalized Laguerre polynomial. If p = 1, the incident beam is a double-ring shaped CV beam. The effect of phase modulation on the input cylindrical vector beam is evaluated by replacing the function A() by A() P(). For 3 belts Binary diffractive optical element P() is given by
P() =
Fig. 2. Intensity distributions at the focus of the Lens axicon for (a) = 0◦ at z = 0 and (b) are the corresponding contour plot for the total intensity distribution in the r–z plane.
3. Results We perform the integration of Eq. (1) numerically using parameters = 1, ˇ = 1.3 and NA of the objective is 0.9. Here, for simplicity, we assume that the refractive index n = 1 and A = 1. For all calculation in the length unit is normalized to and the energy density is normalized to unity. Fig. 2 illustrates the evolution of threedimensional light intensity distribution of the high NA lens axicon for different polarization angle of the incident phase modulated double ring shaped CV beam. It is observed that by setting = 0◦ , which corresponds to a double ring shaped radially polarized incident beam, one can achieve a sub wavelength focal spot with large focal depth. From Fig. 2a and b we measured the FWHM of the generated sub wavelength focal spot as 0.42 and its focal depth as 14 respectively. The set of three angles of the BDOE optimized to achieve the above mentioned focal segment using traditional global search algorithm are 1 = 17.4◦ , 2 = 58.24◦ , ˛ = 64.19◦ . Such a needle of sub wavelength beam with large focal depth finds its application in near field optical recording and particle acceleration. The intensity distribution for = 90◦ , which corresponds to a azimuthally polarized incident double ring shaped beam for the high NA lens axicon is shown in Fig. 3a and b. In this case, only the azimuthal component is present at the focus and the focal segment resembles a doughnut shape. The FWHM of the generated focal hole is 0.96. The 3D intensity distribution near the focus as shown in Fig. 3b, reveals that the focal depth is 14.2. The set of three angles of the BDOE optimized to achieve the above mentioned focal segment 1 = 17.4◦ , 2 = 58.24◦ , ˛ = 64.19◦ . Such a focal hole segment is highly useful for trapping particles with a dielectric constant lower than the ambient. We also observed that it is possible to split the generated focal spot or focal hole segment axially by properly tuning the phase of the incident double ring shaped CV beam. We also observed that by setting 1 = 40.56◦ , 2 = 62.59◦ , ˛ = 64.19◦ it is possible for us to generated sub wave length double focal spot segments each with FWHM of 0.4 and are axially separated by a longer distance of 22 between them as shown
1, for0 < < 1 , 2 < < ˛, −1for0, 1 < < 2
The intensity distribution of the lens axicon is evaluated by multiply the function P() by the function P()T(), where T() is the non-paraxial transmittance function of the thin aberrated diverging lens [20].
T () = exp
ik
sin() sin(˛)
4 +
1 2f
sin() sin(˛)
2
(7)
Here k = 2/, f is the focal length and is the aberration coefficient. In our calculation we take f = 18.4 mm, = 6.667 × 10−5 mm−3 which results in a equiconcave configure [21].
Fig. 3. Intensity distributions at the focus of the Lens axicon for (a) = 90◦ at z = 0 and (b) are the corresponding contour plot for the total intensity distribution in the r–z plane.
K. Prabakaran et al. / Optik 124 (2013) 5039–5041
5041
on vector diffraction theory. It is observed that the proposed system generates sub wavelength beam with large depth of focus and we expect such a beam with small spot size and long depth of focus can be widely used in application such as optical tweezers, data storage, biomedical imaging, laser drilling, and machining. References
Fig. 4. Intensity distributions in focal region Lens axicon for (a) = 0◦ and (b) = 90◦ .
Fig. 5. Intensity distributions in focal region Lens axicon for (a) = 51◦ at z = 0 and (b) are the corresponding contour plot for the total intensity distribution.
in Fig. 4a. However, for the polarization angle = 90◦ , the same BDOE generates a two sub wave length focal hole segments each with FWHM of 0.3 and are axially separated by a large distance of 22 between them and is shown in Fig. 4b. Such a focal splitting is useful in controlling the position of the optical traps. We also wish to generate a flattop total intensity distribution at the focus by adjusting the weightings of the three field components through controlling of polarization angle . The flattop profile is obtained for the angle = 51◦ . From Fig. 5a, we measured the FWHM of the total intensity of generated flat top focal profile is 0.89 and its corresponding focal depth is 14 which is shown in Fig. 5b. The set of three angles of the BDOE optimized to achieve the above mentioned focal segment are 1 = 30◦ , 2 = 58.24◦ , ˛ = 64.19◦ . This type of beam profile is useful in improved printing filling factor, improved uniformity and quality in materials processing, particle acceleration and microlithography. 4. Conclusion The intensity distributions of the double-ring-shaped cylindrically polarized beam by a high NA lens axicon were calculated based
[1] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2) (2000) 77–87. [2] Q.W. Zhan, J.R. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (7) (2002) 324–331. [3] W.B. Chen, Q.W. Zhan, Three-dimensional focus shaping with cylindrical vector beams, Opt. Commun. 265 (2) (2006) 411–417. [4] 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. [5] 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. [6] K.T. Gahagan, G.A. Swartzlander Jr., Simultaneous trapping of low-index and high- index micro particles observed with an optical-vortex trap, J. Opt. Soc. Am. B 16 (1999) 533–537. [7] Q. Zhan, Trapping metallic Rayleigh particles with radial polarization, Opt. Express 12 (15) (2004) 3377–3382. [8] Q. Zhan, Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams, J. Opt. A: Pure Appl. Opt. 5 (2003) 229–232. [9] Y.Q. Zhao, Q. Zhan, Y.L. Zhang, Y. Li, Creation of a three-dimensional optical chain for controllable particle delivery, Opt. Lett. 30 (2005) 848–850. [10] K.S. Youngworth, T.G. Brown, Inhomogeneous polarization in scanning optical microscopy, Proc. SPIE 3919 (2000). [11] L.E. Helseth, Roles of polarization, phase and amplitude in solid immersion lens systems, Opt. Commun. 191 (2001) 161–172. [12] V.G. Niziev, A.V. Nesterov, Influence of beam polarization on laser cutting efficiency, J. Phys. D 32 (1999) 1455–1461. [13] A.V. Nesterov, V.G. Niziev, Laser beams with axially symmetric polarization, J. Phys. D 33 (1999) 1817–1822. [14] B. Hafizi, E. Esarey, P. Sprangle, Laser-driven acceleration with Bessel beams, Phys. Rev. E 55 (1997) 3539–3545. [15] C. Sun, C. Liu, Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation, Opt. Lett. 28 (2003) 99–101. [16] K.B. Rajesh, Z. Jaroszewicz, P.M. Anbarasan, Improvement of lens axicon’s performance for longitudinally polarized beam generation by adding a dedicated phase transmittance, Opt. Express 18 (2000) 26799–26805. [17] K.B. Rajesh, N. VeeraBagu Suresh, P.M. Anbarasan, K. Gokulakrishnan, G. Mahadevan, Tight focusing of double ring shaped radially polarized beam with high NA lens axicon, J. Opt. Laser Technol. 43 (2011) 1037–1040. [18] K.B. Rajesh, Z. Jaroszewicz, P.M. Anbarasan, T.V.S. Pillai, N. Veerabagu Suresh, Extending the depth of focus with high NA lens axicon, Optik 122 (2011) 1619–1621. [19] B. Richards, 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) (1959) 358–379. [20] K.B. Rajesh, P.M. Anbarasan, Generation of sub-wavelength and superresolution longitudinally polarized non-diffraction beam using lens axicon, Chin. Opt. Lett. 6 (10) (2008) 785–787. [21] Z. Jaroszewicz, J. Morales, Lens axicon system composed of a diverging aberrated lens and a perfect converging lens, J. Opt. Soc. Am. A 15 (9) (1998) 2383–2390.