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Focusing properties of cylindrical vector vortex beams with high numerical aperture objective
G Model IJLEO-53037; No. of Pages 4
ARTICLE IN PRESS Optik xxx (2013) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Focusing properties of cylindrical vector vortex beams with high numerical aperture objective Tingting Wang, Cuifang Kuang ∗ , Xiang Hao, Xu Liu State Key Laboratory of Modern Optical Instrumentations, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 2 September 2012 Accepted 24 January 2013 Keywords: Diffraction theory Polarization Microscopy
a b s t r a c t Focused by a high numerical-aperture objective in free space, the cylindrical vector beam phase-encoded by vortex phase plate with higher topological charge was capable to generate the doughnut-shaped spot in the vicinity of the focal region. The width of the dark focal spot was manipulated by the phase plate with different topological charge. The relationship between the properties of the focal spot and the vortex phase plate was explicitly analyzed for the input beam with different cylindrical vector polarization. Furthermore, the experimental verification was undertaken at the incidence beam = 635 nm with the radial and azimuthal polarization. The experimental results are in excellent agreement with the theoretical calculation. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Polarization, as a vector nature of light, attracts considerable interest of researchers. It has been extensively utilized in various applications, such as data storage, optical lithography, optical measurement and metrology, and optical microscopy. In the past several years, the exploration of the cylindrical vector beam has been a hot research subject on subwavelength focusing [1–3]. With the development of the manufacturing technology and the improvement of the machining precision, both the active [4–7] and passive [8–10] methods were successfully used to generate the radial and azimuthal polarization beams. It was believed that the polarization properties of the incidence beam were crucial to manipulate the intensity distribution of the focal spot [11–13]. Due to the strong longitudinal field component [12,14], the radial polarization beam focuses more sharply than the linear beam under high numerical aperture if the incidence light was centrally obstructed. In contrast, the azimuthally polarized beam was ignored in the subwavelength focusing for the doughnut shaped focal spot. An intriguing phenomenon is that different diffractive optical element (DOE) is introduced to realize focus shaping for the cylindrical vector beams. Neither a three-dimensional beam shaping [15] nor optical needle [16] with long focal depth was attainable by utilizing the binary DOE on the pupil plane. Besides, the spiral phase plate rotating from 0 to 2 also plays an indispensable role, which is capable to generate the doughnut-shaped beam [17], to trap particles [18,19], and to yield Laguerre–Gaussian modes [20]. Although
∗ Corresponding author. E-mail address: [email protected] (C. Kuang).
attention has been paid on the behavior of the incidence beam with vortex 0–2 phase delay on the focal intensity distribution, the higher topological charge phase modulation on the cylindrical vector beam is of neglect. In this paper, the effect of the higher topological charge vortex phase plate on the focusing properties has been studied both for the radial and azimuthal polarization beams. Using the vectorial diffraction theory, the intensity distribution in the vicinity of the focus is calculated rigorously. Furthermore, the experimental verification is carried out based on an optical imaging system. 2. Theory 2.1. Geometry and mathematics The schematic of the focusing setup is shown in Fig. 1. An incidence beam with cylindrical vector polarization, after being phase-encoded by a vortex phase plate, is focused by an aplanatic lens. The aplanatic lens is expected to produce the spherical wavefront without aberration converging to the optical axis to form the focal spot. The Richard–Wolf vector diffraction theory [21] is adopted to calculate the intensity distribution in the vicinity of the focus under the high numerical-aperture condition. When a phase plate with different topological charge is applied in the focusing system, the phase delay to the incidence beam can be represented as: E៝ = E 0 eim ˛(r,ϕ) ,
m = 1, 2, 3, . . .
(1)
where E៝0 and E៝ are the electric field vectors of the incidence beam and the light that passes through the vortex phase plate, respectively. Where m is defined as the topological charge of the vortex
0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.01.070
Please cite this article in press as: T. Wang, et al., Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.070
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Fig. 1. Schematic diagram of the focusing setup of the cylindrical vector beam.
phase plate and (r, ϕ) is the polar coordinate of the ray at the cross section of the incidence beam. ␣ is the phase delay by vortex phase plate. Using the Debye integral over the vector field where the spherical wavefront converges, one can express the intensity distribution in the region of focus as [22]:
E៝ (r2 , ϕ2 , z2 ) = iC
⎡
⎢
˝
px
⎤ ⎥
sin() ∗ A1 (, ϕ) ∗ A2 (, ϕ) ∗ ⎣ py ⎦ pz
∗ eim ˛(r,ϕ) ∗ eikn (z2 cos +r2 sin cos (ϕ−ϕ2 )) ddϕ
(2)
where E៝ is the electric field vector at point P (r2 , ϕ2 , z2 ) which is indicated in the cylindrical coordinate system in the imaging field and C is a normalized constant. A1 (, ϕ) is the amplitude of the incidence beam and A2 (, ϕ) is a function related to the aplanatic
−1
is the unit matrix of the polarization lens structure. px , py , pz state of the incidence beam. max is the maximum focusing angle determined by max = sin−1 (NA/n) and n is the refractive index in the imaging region. 2.2. Focusing properties of the cylindrical vector polarization beam The intensity distribution at the focal plane (XY) and through the focus (XZ) is calculated as the modulus squared of Eq. (2) and is shown in Figs. 2 and 3. The focusing aplanatic lens is adopted with numerical aperture NA = 0.8 and refractive index n = 1. It can be seen from Figs. 2 and 3, when the vortex phase plate with topological charge 1 is applied, owning to the vortex phase modulation, both for the radial and azimuthal polarization beams, the peak of the focal spot locates exactly at the center, corresponding to the Gauss-like focal spot shape. What is more, the focus of the azimuthally polarized beam seems sharper than that of the radial polarization light. However, when the topological charge of the phase plate reaches 2 or 3, the doughnut ring shape in the focal region is clearly observed. Under the same topological charge condition, the width of the dark focal spot for the azimuthally polarized beam is narrower than that of the radial polarization beam. In addition, with the increase of the topological charge, neither for the radial nor for the azimuthal polarization beam, the width of the dark focal spot becomes wider. As can be drawn, the width of the dark focal spot would be manipulated by the vortex phase plate with different topological charge. To further give an explicit quantitative analysis of the characteristics of the focal spot, considering the rotation symmetry of the focal spot, the normalized intensity distribution curve along X direction is shown in Fig. 4. When the vortex phase plate with topological charge 1 is utilized, the full width at half maximum (FWHM) of the focal spot for the radial and azimuthal polarization beam is and 0.65 , respectively. When the topological charge of the phase plate increases, both of the two incidence beams have the capacity to generate the doughnut-shaped focal spot, with the intensity in the center of the focal spot down to zero. The peak-peak value is adopted to numerically depict the distinguishing features of the
Fig. 2. Normalized intensity distribution in the horizontal (XY) and longitudinal (XZ) planes of the radial polarization light focused by an objective of NA = 0.8 with different topological charges. The axis units are in the wavelength.
dark focal spot, shown in Table 1. The dependence of the width of the doughnut-shaped focal spot on the topological charge of the vortex phase plate is clearly revealed. One observation is that the doughnut-shaped focal spot of the azimuthally polarized beam is
Fig. 3. Normalized intensity distribution in the horizontal (XY) and longitudinal (XZ) planes of the azimuthally polarized light focused by an objective of NA = 0.8 with different topological charges. The axis units are in the wavelength.
Please cite this article in press as: T. Wang, et al., Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.070
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Fig. 4. Normalized Intensity distribution curves along X direction are shown with the vortex phase plate applied in the system with the topological charge 1 for the radial polarization beam (1st) and for the aziluthally polarized beam (2nd), with topological charge 2 for the radial polarization beam (3rd) and for the azimuthally polarized beam (4th) and with topological charge 3 for the radial polarization beam (5th) and for the azimuthally polarized beam (6th).
tighter than that of the radial polarization light at the same topological charge, for lack of the longitudinal component. In addition, the higher the topological charge, the wider the dark focal spot. Thus potential applications of the cylindrical vector beam with vortex phase plate is in the STED microscopy or in the optical tweezers. 3. Experimental results Actually, similar theoretical derivation has already been proposed by some other researchers, but all previous works [23] lack of conceivable experimental confirmations. To verify the corresponding performance, the experimental demonstration was carried out. The beam of a linearly polarized, single mode pigtailed laser ( = 635 nm) was first expanded and then collimated. The collimated beam then passed through the radial polarization converter (ARCoptix, Switzerland). The converter, which is a nematic crystal cell composed of one uniform and one circularly rubbed alignment layer [23], is capable to convert a linearly polarized incidence beam into light with radial or azimuthal polarization distribution, in addition with phase compensation. After being phase-encoded by the vortex phase plate (PRC photonics, USA, VPP-1) with different topological charge, the cylindrical vector beam was focused by the microscopy objective with NA = 0.8 (Nikon, Japan, Eclipse i80). The measurements of the intensity distribution in the focal region were based on an optical imaging system which is mainly composed of an objective (Nikon, Japan, NA = 0.8) and a charge-coupled device (CCD) camera. The intensity distribution of the focus is collected by the objective and simultaneously displays on the CCD camera. The columns of Fig. 5(a) and (b) show the images recorded by the CCD camera in the focal region which reveal the transverse intensity distribution of the focal spot at different polarization status. By extracting the intensity reading from the image along the center of Table 1 Peak-peak value of the doughnut-shaped focal spot. Topological charge
m=2 m=3 m=4 m=5 m=6 m=7 m=8
Polarization Radial
Azimuthal
1.266 1.9 2.36 2.88 3.36 3.84 4.32
1.6 2.16 2.72 3.24 3.72 4.2
Fig. 5. Optical images recorded by the CCD camera for (a) the radial and (b) the azimuthal polarization beam with topological charge m = 1 and 2, respectively. And the corresponding normalized intensity distributions crossing the center of the spots are plotted, respectively. The black junction presents the peak intensity of the image recorded by the CCD camera. And the red lines crisscross the center of the spots.
the focus, the normalized intensity distribution curve is shown correspondingly. At the topological charge m = 1, the recorded images exhibits the Gaussian focal spot for both the radial and azimuthal light. And the intensity in the center of the focal spot reaches the maximum in the focal plane. The spot size for the radial and azimuthal polarization beam is 0.964 and 0.696 , with the error of 3.6% and 7.08% compared to the simulation results, respectively. However, when the phase plate with topological charge m = 2 is applied in the system, the focal spot changes into the doughnut-ring shape, neither for the radial nor for the azimuthal polarization beam. Ascribing to the adjustment difference between the center of the vortex phase plate and the radial polarization convertor in experimental progress, the dark focal spot reveals little non-uniform intensity distribution. But this will not result in any substantial effects on the experimental results. The peak-peak value of the doughnut-shaped spot for the radial and the azimuthal polarization incidence beam is measured as 1.177 and 1.052 , respectively. The corresponding measurement error is calculated as 7.03% and 5.2%, respectively. The difference of the focal spot size in the calculation and the experimental results can be interpreted as follows: (1) due to the existence of the focal depth of the focusing objective, it is difficult to determine the focal plane exactly, thus the method of direct imaging would induce measurement error of the focal spot size. (2) The resolution of the system is restricted to the accuracy of the CCD camera. (3) The stability of the experimental system plays a crucial part in observation, particularly in the progress of recording. One question is that the intensity in the center of the doughnut-shaped spot is not down to zero as calculated above. In fact, at topological charge m = 2, the intensity in the center of the doughnut-shaped
Please cite this article in press as: T. Wang, et al., Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.070
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spot for the radial and azimuthal polarization beam reaches 24% and 31%, respectively. The errors sources may derive from the pureness of polarization conversion, the electric noise of the CCD camera, the variation the experimental environment, such as the stability of the laser, the vibration of the air turbulence, the stability of the experimental setup and so on. One more important reason is that the deviation of the center of the vortex phase plate and the radial polarization converter. 4. Conclusion Focused by a high-numerical aperture objective with NA = 0.8, we have investigated the influence of the cylindrical vector beam phase-encoded by the vortex phase plate on the focal intensity distribution. Excellent agreement between numerical calculation and experimental results demonstrate the manipulation impact of the phase plate with different topological charge on the focal spot shape and size. When the topological charge of the phase plate exceeds 2, the focal spot shape changes into doughnut-shaped ring. Intriguingly, the dark spot width can be controlled both for the radial and azimuthal beam. The unique specificities of the cylindrical vector beam after being phase-encoded makes it potential to be utilized in the STED microscopy or in the optical tweezers. Acknowledgments This work was financially supported by grants from National Natural Science Foundation of China (No. 61205160), the Qianjiang Talent Project (No. 2011R10010), the Doctoral Fund of Ministry of Education of China (No. 20110101120061 and No. 20120101130006). References [1] G.M. Lerman, U. Levy, Effect of radial polarization and apodization on spot size under tight focusing conditions, Opt. Express 16 (7) (2008) 4567–4581. [2] K. Youngworth, T. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2) (2000) 77–87.
[3] Q. Zhan, J. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (7) (2002) 324–331. [4] J.-l. Li, et al., Generation of radially polarized mode in Yb fiber laser by using a dual conical prism, Opt. Letters 31 (20) (2006) 2969–2971. [5] Y. Kozawa, S. Sato, Generation of a radially polarized laser beam by use of a conical Brewster prism, Opt. Letters 30 (22) (2005) 3063–3065. [6] R. Oron, et al., The formation of laser beams with pure azimuthal or radial polarization, Appl. Phys. Lett. 77 (21) (2000) 3322–3324. [7] K. Yonezawa, Y. Kozawa, S. Sato, Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal, Opt. Lett. 31 (14) (2006) 2151–2153. [8] S.C. Tidwell, D.H. Ford, W.D. Kimura, Generating radially polarized beams interferometrically, Appl. Opt. 29 (15) (1990) 2234–2239. [9] G. Machavariani, et al., Efficient extracavity generation of radially and azimuthally polarized beams, Opt. Lett. 32 (11) (2007) 1468–1470. [10] K.J. Moh, et al., Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams, Appl. Opt. 46 (30) (2007) 7544–7551. [11] Y. Kozawa, S. Sato, Sharper focal spot formed by higher-order radially polarized laser beams, J. Opt. Soc. Am. A: Opt. Image Sci. 24 (6) (2007) 1793–1798. [12] R. Dorn, S. Quabis, G. Leuchs, Sharper Focus for a Radially Polarized Light Beam, Phys. Rev. Lett. 91 (23) (2003) 233901. [13] S. Quabis, et al., Focusing light to a tighter spot, Opt. Commun. 179 (1/6) (2000) 1–7. [14] B. Hao, J. Leger, Experimental measurement of longitudinal component in the vicinity of focused radially polarized beam, Opt. Express 15 (6) (2007) 3550–3556. [15] W. Chen, Q. Zhan, Three-dimensional focus shaping with cylindrical vector beams, Opt. Commun. 265 (2) (2006) 411–417. [16] H. Wang, et al., Creation of a needle of longitudinally polarized light in vacuum using binary optics, Nat. Photon. 2 (8) (2008) 501–505. [17] T. Watanabe, et al., Generation of a doughnut-shaped beam using a spiral phase plate, Rev. Sci. Instrum. 75 (12) (2004) 5131–5135. [18] K.T. Gahagan, J.G.A. Swartzlander, Optical vortex trapping of particles, Opt. Lett. 21 (11) (1996) 827–829. [19] W.M. Lee, X.C. Yuan, W.C. Cheong, Optical vortex beam shaping by use of highly efficientirregular spiral phase plates for optical micromanipulation, Opt. Lett. 29 (15) (2004) 1796–1798. [20] G.A. Turnbull, et al., The generation of free-space Laguerre–Gaussian modes at millimetre – wave frequencies by use of a spiral phaseplate, Opt. Commun. 127 (4-6) (1996) 183–188. [21] B. Richards, E. Wolf, Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system, Proc. Roy. Soc. London Ser. A 253 (1274) (1959) 358–379. [22] H. Xiang, et al., Effects of polarization on the de-excitation dark focal spot in STED microscopy, J. Opt. 12 (11) (2010) 115707. [23] M. Stalder, M. Schadt, Linearly polarized light with axial symmetry generated by liquid-crystal polarization converters, Opt. Lett. 21 (23) (1996) 1948–1950.
Please cite this article in press as: T. Wang, et al., Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.070
ARTICLE IN PRESS Optik xxx (2013) xxx–xxx
Contents lists available at SciVerse ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Focusing properties of cylindrical vector vortex beams with high numerical aperture objective Tingting Wang, Cuifang Kuang ∗ , Xiang Hao, Xu Liu State Key Laboratory of Modern Optical Instrumentations, Zhejiang University, Hangzhou 310027, China
a r t i c l e
i n f o
Article history: Received 2 September 2012 Accepted 24 January 2013 Keywords: Diffraction theory Polarization Microscopy
a b s t r a c t Focused by a high numerical-aperture objective in free space, the cylindrical vector beam phase-encoded by vortex phase plate with higher topological charge was capable to generate the doughnut-shaped spot in the vicinity of the focal region. The width of the dark focal spot was manipulated by the phase plate with different topological charge. The relationship between the properties of the focal spot and the vortex phase plate was explicitly analyzed for the input beam with different cylindrical vector polarization. Furthermore, the experimental verification was undertaken at the incidence beam = 635 nm with the radial and azimuthal polarization. The experimental results are in excellent agreement with the theoretical calculation. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction Polarization, as a vector nature of light, attracts considerable interest of researchers. It has been extensively utilized in various applications, such as data storage, optical lithography, optical measurement and metrology, and optical microscopy. In the past several years, the exploration of the cylindrical vector beam has been a hot research subject on subwavelength focusing [1–3]. With the development of the manufacturing technology and the improvement of the machining precision, both the active [4–7] and passive [8–10] methods were successfully used to generate the radial and azimuthal polarization beams. It was believed that the polarization properties of the incidence beam were crucial to manipulate the intensity distribution of the focal spot [11–13]. Due to the strong longitudinal field component [12,14], the radial polarization beam focuses more sharply than the linear beam under high numerical aperture if the incidence light was centrally obstructed. In contrast, the azimuthally polarized beam was ignored in the subwavelength focusing for the doughnut shaped focal spot. An intriguing phenomenon is that different diffractive optical element (DOE) is introduced to realize focus shaping for the cylindrical vector beams. Neither a three-dimensional beam shaping [15] nor optical needle [16] with long focal depth was attainable by utilizing the binary DOE on the pupil plane. Besides, the spiral phase plate rotating from 0 to 2 also plays an indispensable role, which is capable to generate the doughnut-shaped beam [17], to trap particles [18,19], and to yield Laguerre–Gaussian modes [20]. Although
∗ Corresponding author. E-mail address: [email protected] (C. Kuang).
attention has been paid on the behavior of the incidence beam with vortex 0–2 phase delay on the focal intensity distribution, the higher topological charge phase modulation on the cylindrical vector beam is of neglect. In this paper, the effect of the higher topological charge vortex phase plate on the focusing properties has been studied both for the radial and azimuthal polarization beams. Using the vectorial diffraction theory, the intensity distribution in the vicinity of the focus is calculated rigorously. Furthermore, the experimental verification is carried out based on an optical imaging system. 2. Theory 2.1. Geometry and mathematics The schematic of the focusing setup is shown in Fig. 1. An incidence beam with cylindrical vector polarization, after being phase-encoded by a vortex phase plate, is focused by an aplanatic lens. The aplanatic lens is expected to produce the spherical wavefront without aberration converging to the optical axis to form the focal spot. The Richard–Wolf vector diffraction theory [21] is adopted to calculate the intensity distribution in the vicinity of the focus under the high numerical-aperture condition. When a phase plate with different topological charge is applied in the focusing system, the phase delay to the incidence beam can be represented as: E៝ = E 0 eim ˛(r,ϕ) ,
m = 1, 2, 3, . . .
(1)
where E៝0 and E៝ are the electric field vectors of the incidence beam and the light that passes through the vortex phase plate, respectively. Where m is defined as the topological charge of the vortex
0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.01.070
Please cite this article in press as: T. Wang, et al., Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.070
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ARTICLE IN PRESS
IJLEO-53037; No. of Pages 4
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2
Fig. 1. Schematic diagram of the focusing setup of the cylindrical vector beam.
phase plate and (r, ϕ) is the polar coordinate of the ray at the cross section of the incidence beam. ␣ is the phase delay by vortex phase plate. Using the Debye integral over the vector field where the spherical wavefront converges, one can express the intensity distribution in the region of focus as [22]:
E៝ (r2 , ϕ2 , z2 ) = iC
⎡
⎢
˝
px
⎤ ⎥
sin() ∗ A1 (, ϕ) ∗ A2 (, ϕ) ∗ ⎣ py ⎦ pz
∗ eim ˛(r,ϕ) ∗ eikn (z2 cos +r2 sin cos (ϕ−ϕ2 )) ddϕ
(2)
where E៝ is the electric field vector at point P (r2 , ϕ2 , z2 ) which is indicated in the cylindrical coordinate system in the imaging field and C is a normalized constant. A1 (, ϕ) is the amplitude of the incidence beam and A2 (, ϕ) is a function related to the aplanatic
−1
is the unit matrix of the polarization lens structure. px , py , pz state of the incidence beam. max is the maximum focusing angle determined by max = sin−1 (NA/n) and n is the refractive index in the imaging region. 2.2. Focusing properties of the cylindrical vector polarization beam The intensity distribution at the focal plane (XY) and through the focus (XZ) is calculated as the modulus squared of Eq. (2) and is shown in Figs. 2 and 3. The focusing aplanatic lens is adopted with numerical aperture NA = 0.8 and refractive index n = 1. It can be seen from Figs. 2 and 3, when the vortex phase plate with topological charge 1 is applied, owning to the vortex phase modulation, both for the radial and azimuthal polarization beams, the peak of the focal spot locates exactly at the center, corresponding to the Gauss-like focal spot shape. What is more, the focus of the azimuthally polarized beam seems sharper than that of the radial polarization light. However, when the topological charge of the phase plate reaches 2 or 3, the doughnut ring shape in the focal region is clearly observed. Under the same topological charge condition, the width of the dark focal spot for the azimuthally polarized beam is narrower than that of the radial polarization beam. In addition, with the increase of the topological charge, neither for the radial nor for the azimuthal polarization beam, the width of the dark focal spot becomes wider. As can be drawn, the width of the dark focal spot would be manipulated by the vortex phase plate with different topological charge. To further give an explicit quantitative analysis of the characteristics of the focal spot, considering the rotation symmetry of the focal spot, the normalized intensity distribution curve along X direction is shown in Fig. 4. When the vortex phase plate with topological charge 1 is utilized, the full width at half maximum (FWHM) of the focal spot for the radial and azimuthal polarization beam is and 0.65 , respectively. When the topological charge of the phase plate increases, both of the two incidence beams have the capacity to generate the doughnut-shaped focal spot, with the intensity in the center of the focal spot down to zero. The peak-peak value is adopted to numerically depict the distinguishing features of the
Fig. 2. Normalized intensity distribution in the horizontal (XY) and longitudinal (XZ) planes of the radial polarization light focused by an objective of NA = 0.8 with different topological charges. The axis units are in the wavelength.
dark focal spot, shown in Table 1. The dependence of the width of the doughnut-shaped focal spot on the topological charge of the vortex phase plate is clearly revealed. One observation is that the doughnut-shaped focal spot of the azimuthally polarized beam is
Fig. 3. Normalized intensity distribution in the horizontal (XY) and longitudinal (XZ) planes of the azimuthally polarized light focused by an objective of NA = 0.8 with different topological charges. The axis units are in the wavelength.
Please cite this article in press as: T. Wang, et al., Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.070
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Fig. 4. Normalized Intensity distribution curves along X direction are shown with the vortex phase plate applied in the system with the topological charge 1 for the radial polarization beam (1st) and for the aziluthally polarized beam (2nd), with topological charge 2 for the radial polarization beam (3rd) and for the azimuthally polarized beam (4th) and with topological charge 3 for the radial polarization beam (5th) and for the azimuthally polarized beam (6th).
tighter than that of the radial polarization light at the same topological charge, for lack of the longitudinal component. In addition, the higher the topological charge, the wider the dark focal spot. Thus potential applications of the cylindrical vector beam with vortex phase plate is in the STED microscopy or in the optical tweezers. 3. Experimental results Actually, similar theoretical derivation has already been proposed by some other researchers, but all previous works [23] lack of conceivable experimental confirmations. To verify the corresponding performance, the experimental demonstration was carried out. The beam of a linearly polarized, single mode pigtailed laser ( = 635 nm) was first expanded and then collimated. The collimated beam then passed through the radial polarization converter (ARCoptix, Switzerland). The converter, which is a nematic crystal cell composed of one uniform and one circularly rubbed alignment layer [23], is capable to convert a linearly polarized incidence beam into light with radial or azimuthal polarization distribution, in addition with phase compensation. After being phase-encoded by the vortex phase plate (PRC photonics, USA, VPP-1) with different topological charge, the cylindrical vector beam was focused by the microscopy objective with NA = 0.8 (Nikon, Japan, Eclipse i80). The measurements of the intensity distribution in the focal region were based on an optical imaging system which is mainly composed of an objective (Nikon, Japan, NA = 0.8) and a charge-coupled device (CCD) camera. The intensity distribution of the focus is collected by the objective and simultaneously displays on the CCD camera. The columns of Fig. 5(a) and (b) show the images recorded by the CCD camera in the focal region which reveal the transverse intensity distribution of the focal spot at different polarization status. By extracting the intensity reading from the image along the center of Table 1 Peak-peak value of the doughnut-shaped focal spot. Topological charge
m=2 m=3 m=4 m=5 m=6 m=7 m=8
Polarization Radial
Azimuthal
1.266 1.9 2.36 2.88 3.36 3.84 4.32
1.6 2.16 2.72 3.24 3.72 4.2
Fig. 5. Optical images recorded by the CCD camera for (a) the radial and (b) the azimuthal polarization beam with topological charge m = 1 and 2, respectively. And the corresponding normalized intensity distributions crossing the center of the spots are plotted, respectively. The black junction presents the peak intensity of the image recorded by the CCD camera. And the red lines crisscross the center of the spots.
the focus, the normalized intensity distribution curve is shown correspondingly. At the topological charge m = 1, the recorded images exhibits the Gaussian focal spot for both the radial and azimuthal light. And the intensity in the center of the focal spot reaches the maximum in the focal plane. The spot size for the radial and azimuthal polarization beam is 0.964 and 0.696 , with the error of 3.6% and 7.08% compared to the simulation results, respectively. However, when the phase plate with topological charge m = 2 is applied in the system, the focal spot changes into the doughnut-ring shape, neither for the radial nor for the azimuthal polarization beam. Ascribing to the adjustment difference between the center of the vortex phase plate and the radial polarization convertor in experimental progress, the dark focal spot reveals little non-uniform intensity distribution. But this will not result in any substantial effects on the experimental results. The peak-peak value of the doughnut-shaped spot for the radial and the azimuthal polarization incidence beam is measured as 1.177 and 1.052 , respectively. The corresponding measurement error is calculated as 7.03% and 5.2%, respectively. The difference of the focal spot size in the calculation and the experimental results can be interpreted as follows: (1) due to the existence of the focal depth of the focusing objective, it is difficult to determine the focal plane exactly, thus the method of direct imaging would induce measurement error of the focal spot size. (2) The resolution of the system is restricted to the accuracy of the CCD camera. (3) The stability of the experimental system plays a crucial part in observation, particularly in the progress of recording. One question is that the intensity in the center of the doughnut-shaped spot is not down to zero as calculated above. In fact, at topological charge m = 2, the intensity in the center of the doughnut-shaped
Please cite this article in press as: T. Wang, et al., Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.070
G Model IJLEO-53037; No. of Pages 4
ARTICLE IN PRESS T. Wang et al. / Optik xxx (2013) xxx–xxx
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spot for the radial and azimuthal polarization beam reaches 24% and 31%, respectively. The errors sources may derive from the pureness of polarization conversion, the electric noise of the CCD camera, the variation the experimental environment, such as the stability of the laser, the vibration of the air turbulence, the stability of the experimental setup and so on. One more important reason is that the deviation of the center of the vortex phase plate and the radial polarization converter. 4. Conclusion Focused by a high-numerical aperture objective with NA = 0.8, we have investigated the influence of the cylindrical vector beam phase-encoded by the vortex phase plate on the focal intensity distribution. Excellent agreement between numerical calculation and experimental results demonstrate the manipulation impact of the phase plate with different topological charge on the focal spot shape and size. When the topological charge of the phase plate exceeds 2, the focal spot shape changes into doughnut-shaped ring. Intriguingly, the dark spot width can be controlled both for the radial and azimuthal beam. The unique specificities of the cylindrical vector beam after being phase-encoded makes it potential to be utilized in the STED microscopy or in the optical tweezers. Acknowledgments This work was financially supported by grants from National Natural Science Foundation of China (No. 61205160), the Qianjiang Talent Project (No. 2011R10010), the Doctoral Fund of Ministry of Education of China (No. 20110101120061 and No. 20120101130006). References [1] G.M. Lerman, U. Levy, Effect of radial polarization and apodization on spot size under tight focusing conditions, Opt. Express 16 (7) (2008) 4567–4581. [2] K. Youngworth, T. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2) (2000) 77–87.
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Please cite this article in press as: T. Wang, et al., Focusing properties of cylindrical vector vortex beams with high numerical aperture objective, Optik - Int. J. Light Electron Opt. (2013), http://dx.doi.org/10.1016/j.ijleo.2013.01.070