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Tight focusing of partially coherent radially polarized beam with high NA lens axicon
Optik 124 (2013) 4956–4959
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Tight focusing of partially coherent radially polarized beam with high NA lens axicon C. Mariyal a,c , P. Suresh a,c,∗ , K.B. Rajesh b , T.V.S. Pillai d a
Department of ECE, National College of Engineering, Maruthakulam, Tirunelveli 627151, Tamil Nadu, India Department of Physics, Chikkanna Government Arts College, Tirupur 641602, Tamil Nadu, India University Departments, Anna University, Tirunelveli Region, Tirunelveli 627007, Tamil Nadu, India d University Departments, University College of Engineering, Nagercoil, Tamil Nadu, India b c
a r t i c l e
i n f o
Article history: Received 20 September 2012 Accepted 2 March 2013
Keywords: Partially coherent beam Lens axicon Depth of focus
a b s t r a c t Tight focusing of radially polarized partially coherent vortex beam over a high numerical aperture lens axicon system is introduced and its propagation properties are studied based on vectorial Debye theory. The effect of propagation parameters on the intensity distribution, the polarization property and the coherent property of the beam is illustrated analytically and numerically. It is shown that the correlation length and maximal NA angle has a significant influence on the intensity profile. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction
2. Theory
Partially coherent beams are a topic that has been of considerable theoretical and practical interest. The coherent mode representation of the optical field broached for the first time by Gamo [1], which was later developed by Wolf [2,3], plays a not less important role in contemporary optics. Recently, generation and propagation of radially polarized beams have attracted more and more attentions both from theoretical and experimental viewpoints [4–6]. This focused light has a tighter spot and can be applied in many fields such as high resolution microscopy, lithography, optical data storage, material processing, and optical trap-ping and acceleration [7]. In 2003, Dorn reported experimentally generation of sharp longitudinal field [7]. Vortex beams have unique characteristics the tight focusing of optical vortex beams, such as linearly, circularly, and elliptically polarized vortex beams, have been extensively studied [8–11]. There is also some work about the propagation of partially coherent electromagnetic fields [10,11]. In this paper, we present an extension of the radially polarized beam to the partially coherent one. Its analytical expressions for the beam coherence polarization (BCP) matrix are derived within the paraxial propagation approximation. The effect of the Numerical aperture of the incident field on the shape of intensity distribution and coherence characteristics is analyzed.
The radial or azimuthal polarization beams have the advantage that its unique cylindrical symmetry polarization allows bypassing thermal birefringence induced aberrations which significantly affect linearly, circularly or unpolarized beams. The electric field of a completely coherent radially polarized vortex beam focused by a high NA objective can be expressed as E(r,
, z) = Ex + Ey + Ez
E(r,
, z) = −in+1 E0 [(i(In+1 ei − In−1 e−i ))x+(In+1 ei +In−1 e−i )y + (2In )z]ein
where n is the topological charge, E0 is a constant related to the and z are the cylindrical coordinates intensity of the beam, r, of an observation point in the focal region. The definition of the variables In and In±1 is given by
˛
In (r, z) =
P()
cos sin2 Jn (kr sin ) exp(ikz cos )d.
˛ In±1 (r, z)=
P()
cos sin cos Jn±1 (kr sin ) exp(ikz cos )d
0
(3) where P() is the pupil apodization function and Jn is the first kind Bessel function.
0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.043
(2)
0
Wij (r1 , r2 ) = Ei × (1 , ∗ Corresponding author at: Department of ECE, National College of Engineering, Maruthakulam, Tirunelveli 627151, Tamil Nadu, India. E-mail address: [email protected] (P. Suresh).
(1)
1 , z1 )Ej (2 ,
The intensity distribution I(r, I(r,
, z) = Wxx (r,
, z) + Wyy (r,
2 , z2 )
where (i, j = x, y, z)
, z) is given by , z) + Wzz (r,
, z)
(4)
C. Mariyal et al. / Optik 124 (2013) 4956–4959
4957
Fig. 1. Two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different alpha. (a) ˛ = 50◦ , (b) ˛ = 60◦ and (c) ˛ = 70◦ .
Fig. 2. Contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different alpha. (a) ˛ = 50◦ , (b) ˛ = 60◦ and (c) ˛ = 70◦ in the rz plane.
Fig. 3. Two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different alpha. (a) ˛ = 50◦ , (b) ˛ = 60◦ and (c) ˛ = 70◦ .
3. Results We perform the integration numerically (Eq. (1)) using parameters = 633 nm, wo = 1 cm, Lc = 1 cm, f = 1 cm, and n = 1. Figs. 1 and 2 show the total intensity distribution of the partially coherent radially polarized vortex beam near the focal region. From Fig. 1, we found that increase in alpha will increase the intensity in the focal region. It can be seen from Fig. 1(a)–(c). Fig. 1(a) shows two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for ˛ = 50◦ . The intensity of total electric field is 0.003, for ˛ = 60◦ the total electric field intensity is increased to 0.006, it can be seen from Fig. 1(b) and for ˛ = 70◦ the total electric field intensity is increased to 0.007, it can be seen from Fig. 1(c). From Fig. 1 it is seen that change in alpha will affect the intensity in the focal region. Fig. 2 illustrates the contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different alpha. Fig. 2(a) shows
total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for ˛ = 50◦ in the rz plane. Fig. 2(b) shows total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for ˛ = 60◦ in the rz plane and Fig. 2(c) for ˛ = 70◦ in the rz plane. From Fig. 2 we can see that the increase in alpha increases the NA. Were NA increases the DOF decreases. For better performance we need a system with high focal depth. Hence inorder to increase the focal depth in the focal region we replaced lens by lens axicon. Were lens axicon is a kind of doublet of aberrated diverging lens and a high NA converging lens. The intensity distribution of the lens axicon is evaluated by replacing the function P() by the function P()T(), where T() is the non-paraxial transmittance function of the thin aberrated diverging lens [12,13].
T () = exp
ik ˇ
sin() sin(˛2 )
4 +
1 2f
sin() sin(˛2 )
2 (5)
4958
C. Mariyal et al. / Optik 124 (2013) 4956–4959
Fig. 4. Contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different alpha. (a) ˛ = 50◦ , (b) ˛ = 60◦ and (c) ˛ = 70◦ in the rz plane.
Fig. 5. Contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different Lc and ˛ = 70◦ . (a) Lc = 3, (b) Lc = 2, (c) Lc = 1 and (d) Lc = 0.5 in the rz plane.
By using the same numerical analysis method, the intensity profile of high NA lens axicon is calculated with the same values and it is shown in Figs. 3 and 4. From Fig. 3 it is observed that the lens axicon generates a uniform focal pattern when compared to the lens
(Fig. 1) of same NA. From Fig. 4 it is observed that the lens axicon generates a smaller focal spot with extended DOF when compared to the lens (Fig. 2) of same NA. The two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different alpha. It can be seen from Fig. 3(a)–(c). Fig. 3(a) shows two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for ˛ = 50◦ . The intensity of total electric field is 0.0002, for ˛ = 60◦ the total electric field intensity is increased to 0.00031, it can be seen from Fig. 3(b) and for ˛ = 70◦ the total electric field intensity is increased to 0.00037, it can be seen from Fig. 3(c). From Fig. 3 it is seen that change in alpha will affect the intensity in the focal region as like lens with uniform focal pattern. Fig. 4 illustrates the contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different alpha. Fig. 4(a) shows total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for ˛ = 50◦ in the rz plane. Fig. 4(b) shows total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for ˛ = 60◦ in the rz plane and Fig. 4(c) for ˛ = 70◦ in the rz plane. On comparing Figs. 2 and 4 the depth of focus is increased when the light pulses illuminated through lens axicon. In order to study the effect of coherent length (Lc), by using the same analysis method as stated above we perform the numerical calculation by fixing alpha constant (˛ = 70◦ ). The coherent length (Lc) is varied from 0.5 cm to 3 cm it is shown in Fig. 5 for high NA lens and Fig. 6 for high NA lens axicon. Fig. 5 illustrates the contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different Lc and ˛ = 70◦ . From Fig. 5 we can see that the focal depth is increased if Lc decreased. Fig. 5(a) shows contour plot for Lc = 3 cm, Fig. 5(b) for Lc = 2 cm, Fig. 5(c) for Lc = 1 cm and Fig. 5(d) for Lc = 0.5 cm. Fig. 6 illustrates the contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different Lc and ˛ = 70◦ . From Fig. 6 we can see that the focal depth is increased if Lc decreased. Fig. 6(a) shows contour plot for Lc = 3 cm, Fig. 6(b) for Lc = 2 cm, Fig. 6(c) for Lc = 1 cm and Fig. 6(d) for Lc = 0.5 cm. From Fig. 6 it is noted that for low Lc, high NA lens axicon system enhances the focused intensity polarization distribution in axial direction. On increase in Lc the DOF decreases. 4. Conclusion The focusing properties of radially polarized partially coherent vortex beam through a high NA lens axicon objective are demonstrated numerically. The coherence length and the maximal NA angle have direct influence on the intensity distribution and the focusing characteristics of the incident beam. By changing coherence length and maximal NA angle, the focusing properties can be altered. It is also shown that when using high NA lens axicon the focusing properties can be altered with large DOF. It is shown that by controlling the coherence of the incident light we can generate the adjustable partially coherent beams with large depth of focus. The generated beams may have applications in atom optics, and optical tweezers. References
Fig. 6. Contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different Lc and ˛ = 70◦ . (a) Lc = 3, (b) Lc = 2, (c) Lc = 1 and (d) Lc = 0.5 in the rz plane.
[1] H. Gamo, Matrix treatment of partial coherence, in: E. Wolf (Ed.), Progress in Optics III, Amsterdam, North Holland, 1964 (Chapter 3).
C. Mariyal et al. / Optik 124 (2013) 4956–4959 [2] E. Wolf, New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources, JOSA 72 (1982) 343–351. [3] E. Wolf, New theory of partial coherence in the space-frequency domain. Part II: steady-state fields and higher-order correlatrions, JOSA 3 (1986) 76–84. [4] G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams, Opt. Commun. 281 (2008) 732. [5] B. Hao, J. Leger, Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam, Opt. Exp. 15 (2007) 3550. [6] J. Hamazaki, A. Kawamoto, R. Morita, T. Omatsu, Direct production of high-power radially polarized output from a side-pumped Nd:YVO4 bounce amplifier using a photonic crystal mirror, Opt. Exp. 16 (2008) 10762. [7] R. Dorn, S. Quabis, G. Leuchs, Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91 (2003) 233901.
4959
[8] Z. Zhang, J. Pu, X. Wang, Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective, Opt. Lett. 33 (2008) 49. [9] B. Chen, J. Pu, Z. Zhang, X. Wang, Focusing of partially coherent and circularly polarized Bessel-Gaussian beams through a high numerical-aperture objective, Acta Opt. Sin. 29 (2009) 1664 (in Chinese). [10] B. Chen, J. Pu, Tight focusing of elliptically polarized vortex beams, Appl. Opt. 48 (2009) 1288. [11] 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. [12] 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. [13] 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. Exp. 18 (26) (2010) 26799–26805.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Tight focusing of partially coherent radially polarized beam with high NA lens axicon C. Mariyal a,c , P. Suresh a,c,∗ , K.B. Rajesh b , T.V.S. Pillai d a
Department of ECE, National College of Engineering, Maruthakulam, Tirunelveli 627151, Tamil Nadu, India Department of Physics, Chikkanna Government Arts College, Tirupur 641602, Tamil Nadu, India University Departments, Anna University, Tirunelveli Region, Tirunelveli 627007, Tamil Nadu, India d University Departments, University College of Engineering, Nagercoil, Tamil Nadu, India b c
a r t i c l e
i n f o
Article history: Received 20 September 2012 Accepted 2 March 2013
Keywords: Partially coherent beam Lens axicon Depth of focus
a b s t r a c t Tight focusing of radially polarized partially coherent vortex beam over a high numerical aperture lens axicon system is introduced and its propagation properties are studied based on vectorial Debye theory. The effect of propagation parameters on the intensity distribution, the polarization property and the coherent property of the beam is illustrated analytically and numerically. It is shown that the correlation length and maximal NA angle has a significant influence on the intensity profile. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction
2. Theory
Partially coherent beams are a topic that has been of considerable theoretical and practical interest. The coherent mode representation of the optical field broached for the first time by Gamo [1], which was later developed by Wolf [2,3], plays a not less important role in contemporary optics. Recently, generation and propagation of radially polarized beams have attracted more and more attentions both from theoretical and experimental viewpoints [4–6]. This focused light has a tighter spot and can be applied in many fields such as high resolution microscopy, lithography, optical data storage, material processing, and optical trap-ping and acceleration [7]. In 2003, Dorn reported experimentally generation of sharp longitudinal field [7]. Vortex beams have unique characteristics the tight focusing of optical vortex beams, such as linearly, circularly, and elliptically polarized vortex beams, have been extensively studied [8–11]. There is also some work about the propagation of partially coherent electromagnetic fields [10,11]. In this paper, we present an extension of the radially polarized beam to the partially coherent one. Its analytical expressions for the beam coherence polarization (BCP) matrix are derived within the paraxial propagation approximation. The effect of the Numerical aperture of the incident field on the shape of intensity distribution and coherence characteristics is analyzed.
The radial or azimuthal polarization beams have the advantage that its unique cylindrical symmetry polarization allows bypassing thermal birefringence induced aberrations which significantly affect linearly, circularly or unpolarized beams. The electric field of a completely coherent radially polarized vortex beam focused by a high NA objective can be expressed as E(r,
, z) = Ex + Ey + Ez
E(r,
, z) = −in+1 E0 [(i(In+1 ei − In−1 e−i ))x+(In+1 ei +In−1 e−i )y + (2In )z]ein
where n is the topological charge, E0 is a constant related to the and z are the cylindrical coordinates intensity of the beam, r, of an observation point in the focal region. The definition of the variables In and In±1 is given by
˛
In (r, z) =
P()
cos sin2 Jn (kr sin ) exp(ikz cos )d.
˛ In±1 (r, z)=
P()
cos sin cos Jn±1 (kr sin ) exp(ikz cos )d
0
(3) where P() is the pupil apodization function and Jn is the first kind Bessel function.
0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.043
(2)
0
Wij (r1 , r2 ) = Ei × (1 , ∗ Corresponding author at: Department of ECE, National College of Engineering, Maruthakulam, Tirunelveli 627151, Tamil Nadu, India. E-mail address: [email protected] (P. Suresh).
(1)
1 , z1 )Ej (2 ,
The intensity distribution I(r, I(r,
, z) = Wxx (r,
, z) + Wyy (r,
2 , z2 )
where (i, j = x, y, z)
, z) is given by , z) + Wzz (r,
, z)
(4)
C. Mariyal et al. / Optik 124 (2013) 4956–4959
4957
Fig. 1. Two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different alpha. (a) ˛ = 50◦ , (b) ˛ = 60◦ and (c) ˛ = 70◦ .
Fig. 2. Contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different alpha. (a) ˛ = 50◦ , (b) ˛ = 60◦ and (c) ˛ = 70◦ in the rz plane.
Fig. 3. Two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different alpha. (a) ˛ = 50◦ , (b) ˛ = 60◦ and (c) ˛ = 70◦ .
3. Results We perform the integration numerically (Eq. (1)) using parameters = 633 nm, wo = 1 cm, Lc = 1 cm, f = 1 cm, and n = 1. Figs. 1 and 2 show the total intensity distribution of the partially coherent radially polarized vortex beam near the focal region. From Fig. 1, we found that increase in alpha will increase the intensity in the focal region. It can be seen from Fig. 1(a)–(c). Fig. 1(a) shows two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for ˛ = 50◦ . The intensity of total electric field is 0.003, for ˛ = 60◦ the total electric field intensity is increased to 0.006, it can be seen from Fig. 1(b) and for ˛ = 70◦ the total electric field intensity is increased to 0.007, it can be seen from Fig. 1(c). From Fig. 1 it is seen that change in alpha will affect the intensity in the focal region. Fig. 2 illustrates the contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different alpha. Fig. 2(a) shows
total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for ˛ = 50◦ in the rz plane. Fig. 2(b) shows total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for ˛ = 60◦ in the rz plane and Fig. 2(c) for ˛ = 70◦ in the rz plane. From Fig. 2 we can see that the increase in alpha increases the NA. Were NA increases the DOF decreases. For better performance we need a system with high focal depth. Hence inorder to increase the focal depth in the focal region we replaced lens by lens axicon. Were lens axicon is a kind of doublet of aberrated diverging lens and a high NA converging lens. The intensity distribution of the lens axicon is evaluated by replacing the function P() by the function P()T(), where T() is the non-paraxial transmittance function of the thin aberrated diverging lens [12,13].
T () = exp
ik ˇ
sin() sin(˛2 )
4 +
1 2f
sin() sin(˛2 )
2 (5)
4958
C. Mariyal et al. / Optik 124 (2013) 4956–4959
Fig. 4. Contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different alpha. (a) ˛ = 50◦ , (b) ˛ = 60◦ and (c) ˛ = 70◦ in the rz plane.
Fig. 5. Contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different Lc and ˛ = 70◦ . (a) Lc = 3, (b) Lc = 2, (c) Lc = 1 and (d) Lc = 0.5 in the rz plane.
By using the same numerical analysis method, the intensity profile of high NA lens axicon is calculated with the same values and it is shown in Figs. 3 and 4. From Fig. 3 it is observed that the lens axicon generates a uniform focal pattern when compared to the lens
(Fig. 1) of same NA. From Fig. 4 it is observed that the lens axicon generates a smaller focal spot with extended DOF when compared to the lens (Fig. 2) of same NA. The two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different alpha. It can be seen from Fig. 3(a)–(c). Fig. 3(a) shows two dimensional intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for ˛ = 50◦ . The intensity of total electric field is 0.0002, for ˛ = 60◦ the total electric field intensity is increased to 0.00031, it can be seen from Fig. 3(b) and for ˛ = 70◦ the total electric field intensity is increased to 0.00037, it can be seen from Fig. 3(c). From Fig. 3 it is seen that change in alpha will affect the intensity in the focal region as like lens with uniform focal pattern. Fig. 4 illustrates the contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different alpha. Fig. 4(a) shows total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for ˛ = 50◦ in the rz plane. Fig. 4(b) shows total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for ˛ = 60◦ in the rz plane and Fig. 4(c) for ˛ = 70◦ in the rz plane. On comparing Figs. 2 and 4 the depth of focus is increased when the light pulses illuminated through lens axicon. In order to study the effect of coherent length (Lc), by using the same analysis method as stated above we perform the numerical calculation by fixing alpha constant (˛ = 70◦ ). The coherent length (Lc) is varied from 0.5 cm to 3 cm it is shown in Fig. 5 for high NA lens and Fig. 6 for high NA lens axicon. Fig. 5 illustrates the contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens for different Lc and ˛ = 70◦ . From Fig. 5 we can see that the focal depth is increased if Lc decreased. Fig. 5(a) shows contour plot for Lc = 3 cm, Fig. 5(b) for Lc = 2 cm, Fig. 5(c) for Lc = 1 cm and Fig. 5(d) for Lc = 0.5 cm. Fig. 6 illustrates the contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different Lc and ˛ = 70◦ . From Fig. 6 we can see that the focal depth is increased if Lc decreased. Fig. 6(a) shows contour plot for Lc = 3 cm, Fig. 6(b) for Lc = 2 cm, Fig. 6(c) for Lc = 1 cm and Fig. 6(d) for Lc = 0.5 cm. From Fig. 6 it is noted that for low Lc, high NA lens axicon system enhances the focused intensity polarization distribution in axial direction. On increase in Lc the DOF decreases. 4. Conclusion The focusing properties of radially polarized partially coherent vortex beam through a high NA lens axicon objective are demonstrated numerically. The coherence length and the maximal NA angle have direct influence on the intensity distribution and the focusing characteristics of the incident beam. By changing coherence length and maximal NA angle, the focusing properties can be altered. It is also shown that when using high NA lens axicon the focusing properties can be altered with large DOF. It is shown that by controlling the coherence of the incident light we can generate the adjustable partially coherent beams with large depth of focus. The generated beams may have applications in atom optics, and optical tweezers. References
Fig. 6. Contour plot of total intensity distribution of incident radially polarized partially coherent vortex beam focused through high NA lens axicon for different Lc and ˛ = 70◦ . (a) Lc = 3, (b) Lc = 2, (c) Lc = 1 and (d) Lc = 0.5 in the rz plane.
[1] H. Gamo, Matrix treatment of partial coherence, in: E. Wolf (Ed.), Progress in Optics III, Amsterdam, North Holland, 1964 (Chapter 3).
C. Mariyal et al. / Optik 124 (2013) 4956–4959 [2] E. Wolf, New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources, JOSA 72 (1982) 343–351. [3] E. Wolf, New theory of partial coherence in the space-frequency domain. Part II: steady-state fields and higher-order correlatrions, JOSA 3 (1986) 76–84. [4] G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, Spatially-variable retardation plate for efficient generation of radially- and azimuthally-polarized beams, Opt. Commun. 281 (2008) 732. [5] B. Hao, J. Leger, Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam, Opt. Exp. 15 (2007) 3550. [6] J. Hamazaki, A. Kawamoto, R. Morita, T. Omatsu, Direct production of high-power radially polarized output from a side-pumped Nd:YVO4 bounce amplifier using a photonic crystal mirror, Opt. Exp. 16 (2008) 10762. [7] R. Dorn, S. Quabis, G. Leuchs, Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91 (2003) 233901.
4959
[8] Z. Zhang, J. Pu, X. Wang, Focusing of partially coherent Bessel-Gaussian beams through a high-numerical-aperture objective, Opt. Lett. 33 (2008) 49. [9] B. Chen, J. Pu, Z. Zhang, X. Wang, Focusing of partially coherent and circularly polarized Bessel-Gaussian beams through a high numerical-aperture objective, Acta Opt. Sin. 29 (2009) 1664 (in Chinese). [10] B. Chen, J. Pu, Tight focusing of elliptically polarized vortex beams, Appl. Opt. 48 (2009) 1288. [11] 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. [12] 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. [13] 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. Exp. 18 (26) (2010) 26799–26805.