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Focal shift of radially polarized axisymmetric Bessel modulated Gaussian beam with radial variance phase plate
Optik 124 (2013) 5454–5457
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Focal shift of radially polarized axisymmetric Bessel modulated Gaussian beam with radial variance phase plate K. Prabakaran a,∗ , K.B. Rajesh b , T.V.S. Pillai a , R. Chandrasekaran c a b c
University Department, Department of Physics, Anna University of Technology Tirunelveli, Tamil Nadu, India Department of Physics, Chikkanna Government Arts College, Tiruppur, Tamil Nadu, India Department of Physics, Government Arts College, Dharmapuri, Tamil Nadu, India
a r t i c l e
i n f o
Article history: Received 27 October 2012 Accepted 23 March 2013
Keywords: Vector diffraction theory Radially polarized beam Focal shift Axisymmetric Bessel-modulated Gaussian beam
a b s t r a c t Focal shift of the radially polarized axisymmetric QBG beam with radial variance phase plate is investigated theoretically by vector diffraction theory. Axisymmetric Bessel modulated Gaussian beam with quadratic radial dependence (QBG beam) has attracted much attention recently. Calculation results show that focus shifts considerably by changing the phase parameter C that indicates the radial phase variance speed. Under condition of beam parameter of radially polarized axisymmetric QBG beam, there is one focal spot that shifts far away from optical aperture on increasing C. When increases the value of beam parameter, there may occur two focal peaks that also shift remarkably on increasing C. And some novel focal patterns may occur, including focal spot, one ring focus, and dark hollow focus, which is very important in optical tweezers technique. © 2013 Published by Elsevier GmbH.
1. Introduction Recently, a novel class of beam expressed in cylindrical coordinate system, was introduced by Caron and Potvliege [1], namely, the Bessel-modulated Gaussian beams with quadratic radial dependence (QBG beam), it has attracted much attention [2–7]. Various closed-form paraxial solutions of the Helmholtz equation have been described over the years. A particularly interesting class of such solutions, known as Bessel-Gauss beams, has been introduced by Gori et al. in 1987. Bessel-modulated Gaussian beams are another class of simple closed-form solutions of the paraxial wave equation, which are similar to Bessel-Gauss beams in that their amplitude is also a product of a Bessel function and a Gaussian function. In the solutions we are concerned with, however, the argument of the Bessel function is not linear in the transverse coordinate but quadratic .It was shown that such class of beams has familiar collinear geometry of the Gaussian beam and also an interesting non-Gaussian features for certain values of its parameters [1–3]. Particularly, it was demonstrated that the zeroth-order QBG beam, which is referred to as the axisymmetric QBG beam, can be expanded in Laguerre–Gauss modes and has a very flat axial profile when the beam parameterlis of order of unity [1,2]. Belafhal and Dalil-Essakali studied the propagation properties of QBG beams through an unapertured optical paraxial ABCD system [4]. In addition, Mei et al. studied the Bessel-modulated Gaussian light beams
∗ Corresponding author. E-mail address: [email protected] (K. Prabakaran). 0030-4026/$ – see front matter © 2013 Published by Elsevier GmbH. http://dx.doi.org/10.1016/j.ijleo.2013.03.119
passing through a paraxial ABCD optical system with an annular aperture [5]. Hricha, and Belaghal investigated in detail the focusing properties and focal shift of the axisymmetric QBG beam passing through a non-apertured aberration-free lens [2]. The beam propagation factor (M2-factor), far-field distribution, and the kurtosis parameter of such type of beams are also investigated [3,6,7]. In our knowledge, almost all QBG beams in previous papers are in scalar form without considering the polarization distribution of optical field. In fact, the polarization is very important characteristics to alter propagating and focusing properties of beams, and have attracted many researchers recently [8–13]. On the other hand, in high numerical aperture optical systems, polarization characteristics of beam should be considered to provide more accurate results, and vector diffraction theory should be employed to replace scalar diffraction theory. Recently the focal shifts may play a crucial role in achievement of super resolution and in optical trapping [14]. In optical trapping system there are two kinds of forces acting on the particle. One is the optical gradient force, which plays a crucial role in constructing optical trap and its intensity is proportional to the optical intensity gradient; the other kind of force is scattering force, which usually has complex forms because this kind of force is related to the properties of the trapped particles, and whose intensity is proportional to the optical intensity. Opical gradient force is necessary for optical trapping, and its direction is opposite to the optical gradient direction. Tracing the movement of the point of absolute maximum intensity along optical axis has attracted many researchers for several decades [15–17]. It was found that the point of absolute maximum intensity does not coincide with the geometrical focus
K. Prabakaran et al. / Optik 124 (2013) 5454–5457
Fig. 1. Intensity distributions in focal region for NA = 0.9, = 1, and (a) C = 0 (b) C = 1 (c) C = 2 and (d) C = 3, respectively.
but shifts along optical axis. This phenomenon is referred to as focal shift. Focal shifts have taken an important part in design and optimization of various optical systems and extremely important for image formation in defocused planes, for increasing the depth of focus and in automatic focusing [18]. In this paper the focal shift of the radially polarized axisymmetric QBG beam with radial variance phase plate is investigated theoretically by vector diffraction theory. 2. Principle of focusing system In the system we investigated, on the basis of Richards and Wolf’s vectorial diffraction method widely used for high-NA lens system at arbitrary incident polarization [19]. In the case of the radial incident polarization, adopting the cylindrical coordinates r, z, and the notations of Ref. [12], radial and longitudinal components of the electric field can be written as z) = Er er + Ez ez E(r,
(1)
where er and ez is the unit vector in the radial propagation direction. Er , Ez and are the amplitudes of the two orthogonal components that can be written as
Fig. 2. Intensity distributions in focal region for NA = 0.9, = 2, and (a) C = 0 (b) C = 1, (c) C = 2 and (d) C = 3, respectively.
wavefront of radial vector axisymmetric QBG beam is chosen as exp(iC/˛) [20],the optical field value distribution E0 is in the form,
E0 = J0
sin2 () w2 · NA2
cos1/2 () · E0 · sin 2 · J1 (kr sin )eikz cos d
Er (r, ϕ, z) = A
· exp
sin2 () w2 · NA2
· exp(iC/˛)
(4)
where w = ω0 /r0 is called relative waist width, ω0 is the waist width of the Gaussian beam and r0 is radius of incident optical aperture of focusing optical system. is a beam parameter which is complexvalued in general, and C is phase parameter that indicates the radial phase variance speed. The electric field in the focal region can be written in the same form as Eq. (1); however, the two orthogonal components Er and Ez are different and should be expressed as
˛ Er (r, z) = A
cos
1/2
() · J0
sin2 ()
w2 · NA2
· exp
sin2 ()
w2 · NA2
0
· exp(iC/˛) · P() · sin 2 · J1 (kr sin )eikz cos d
˛ cos1/2 () · J0
Ez (r, z) = 2iA
˛
5455
(2)
sin2 () w2 · NA2
· exp
sin2 ()
w2 · NA2
0
· exp(iC/˛) · P() · sin2 J0 (kr sin )eikz cos d
0
(5)
(6)
˛ cos1/2 () · E0 · sin2 J0 (kr sin )eikz cos d
Ez (r, ϕ, z) = 2iA
(3)
0
where ˛ = arc sin(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 axial coordinates of observation point in focal region, respectively. In this paper, the radial variance phase
The optical intensity in focal region is proportional to the modulus square of Eq. (1). Basing on the above equations, focusing properties of radially polarized axisymmetric QBG beam with radial variance phase plate can be investigated theoretically. 3. Results In the investigation of radially polarized axisymmetric QBG beam, without loss of validity and generality, we perform the
5456
K. Prabakaran et al. / Optik 124 (2013) 5454–5457
broadens along optical axis and the on-axis intensity at the longitudinal ends of focal ring becomes stronger. And the intensity maximum position also shifts simultaneously from on-axis position of the focal ring pattern. From above focal pattern evolution process, we can see that for certain beam parameter = 4, phase parameter C can alter considerably intensity distribution in focal region of radially polarized axisymmetric QBG beam and some novel focal shapes may appear. The intensity distribution in focal region of the radially polarized QBG beam can be altered considerably by beam parameter and phase parameter C, and many novel focal patterns can occur, which can be used to construct tunable optical traps. Therefore, focusing properties of the radially polarized QBG beam shows that this kind of beam may find very wide application in optical tweezers system.
4. Conclusion
Fig. 3. Intensity distributions in focal region for NA = 0.9, = 4, and (a) C = 0.0, (b) C = 0.5, (c) C = 1 and (d) C = 1.5, respectively.
integration of Eq. (1) numerically using parameters = 1, A = 1 and NA of the objective is 0.9.For all calculation in the length unit is normalized to and the energy density is normalized to unity. The intensity distributions in focal region for = 1 and different are firstly calculated, and illustrated in Fig. 1. Here it should be noted that the distance unit in all figures in this article is k−1 , where k is wave number. It can be seen from this figure that there is one focal spot when phase parameter C = 0. On increasing C, the focal spot position shifts along optical axis far away from optical aperture, as shown in Fig. 1. From this figure, we can see that the focal shift distance increases on increasing phase parameter C almost linearly. If this focal peak is used to construct one optical trap, phase parameter C may be employed to adjust trap position, can transport micro particles. Now the value of beam parameter is changed to investigate its effect on focal pattern evolution. The intensity distributions for = 2 and different C are illustrated in Fig. 2. It can be seen from this figure that the beam parameter affects focal intensity distribution very considerably. On increasing beam parameter , one focal peak evolves into one dark hollow focus that is defined as those focus whose intensity is lower than that around it, namely, local intensity minimum. There are two optical intensity maximums along optical axis, and one local intensity minimum at geometrical focus position. On increasing phase parameter C, whole focal pattern shifts along optical axis far away from optical aperture, and one intensity peak shrinks so that there is only one optical intensity maximum. Under condition of C = 0, there are two optical intensity maximums. And on increasing C, the focal peak near optical aperture shrinks, while the other focal peak shifts along optical axis. When C changes from 0 to 3, though focal patterns differs very remarkably. The dark hollow focus can act as stable optical trap for those particles whose refractive index is smaller than that of surrounding media, and this condition is very common, especially in life science optical trapping systems [8]. However, from Fig. 3, we observed that there is one focal ring when phase parameter C = 0. Increase C continuously, the focal ring
In conclusion, Focal shift of the radially polarized axisymmetric QBG beam is investigated theoretically by vector diffraction theory in high numerical aperture system. Simulation results show that beam parameter and phase parameter alter the intensity distribution in focal region of radially polarized axisymmetric QBG beam considerably. Focus shifts considerably by changing the phase parameter that indicates the radial phase variance speed. The beam parameter of cylindrical vector QBG beam, there is one focal peak that shifts far away from optical aperture on increasing phase parameter. When beam parameter increases, there occur two focal peaks that also shift remarkably on increasing phase parameter. And some novel focal patterns can appear, such as focal spot, one-ring focus, and dark hollow focus, which can be used to construct stable and alterable optical traps. Radially polarized axisymmetric QBG beam has very promising application due to its particular focusing properties, especially in optical manipulation technique.
References [1] C.F.R. Caron, R.M. Potvliege, Bessel-modulated Gaussian beams with quadratic radial dependence, Opt. Commun. 164 (1999) 83–93. [2] Z. Hricha, A. Belafhal, Focal shift in the axisymmetric Bessel-modulated Gaussian beam, Opt. Commun. 255 (2005) 235–240. [3] X. Wang, B. Lü, The beam propagation factor and far-field distribution of Bessel-modulated Gaussian beams, Opt. Quant. Electron 34 (2002) 1071–1077. [4] A. Belafhal, L. Dalil-Essakali, Collins formula and propagation of Besselmodulated Gaussian light beams through an ABCD optical system, Opt. Commun. 177 (2000) 181–188. [5] Z. Mei, D. Zhao, X. Wei, F. Jing, Q. Zhu, Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture, Optik 116 (2005) 521–526. [6] X. Wang, B. Lü, The beam width of Bessel-modulated Gaussian beams, J. Mod. Opt. 50 (2003) 2107–2115. [7] B. Lü, X. Wang, Kurtosis parameter of Bessel-modulated Gaussian beams propagating through ABCD optical systems, Opt. Commun. 204 (2002) 91–97. [8] R. Oron, S. Blit, N. Davidson, A.A. Friesem, Z. Bomzon, E. Hasman, The formation of laser beams with pure azimuthal or radial polarization, Appl. Phys. Lett. 77 (2000) 3322–3324. [9] X. Gao, S. Hu, H. Gu, J. Wang, Focal shift of three-portion concentric piecewise cylindrical vector beam, Optik 120 (2009) 519–523. [10] Q. Zhan, J.R. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (2002) 324–330. [11] W.T. Tang, E.Y.S. Yew, C.J.R. Sheppard, Polarization conversion in confocal microscopy with radially polarized illumination, Opt. Lett. 34 (2009) 2147–2149. [12] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2000) 77–87. [13] Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photon 1 (2009) 1–57. [14] J. Wang, L. Liu, M. Yun, D. Liu, X. Liu, Superresolution, extended focal depth or focal shift with electrically controllable radial birefringent filter, J. Opt. A: Pure Appl. Opt. 7 (2005) 748.
K. Prabakaran et al. / Optik 124 (2013) 5454–5457 [15] M.P. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, P.E. Bryant, Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap, Opt. Lett. 26 (2001) 863–865. [16] Y. Li, E. Wolf, Focal shifts in diffracted converging spherical waves, Opt. Commun. 39 (1981) 211–215. [17] X. Gao, F. Zhou, W. Xu, F. Gan, Gradient force pattern, focal shift, and focal switch in an apodized optical system, Optik 116 (2005) 99–106.
5457
[18] J. Odeja-Castaneda, L.R. Berriel-Valdos, Arbitrarily high focal depth with finite apertures, Opt. Lett. 13 (1988) 183–185. [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] X. Gao, Q. Zhan, J. Li, S. Hu, J. Wang, S. Zhuang, Cylindrical vector axisymmetric Bessel-modulated Gaussian beam, Opt. Quant. Electron. 41 (2009) 385–396.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Focal shift of radially polarized axisymmetric Bessel modulated Gaussian beam with radial variance phase plate K. Prabakaran a,∗ , K.B. Rajesh b , T.V.S. Pillai a , R. Chandrasekaran c a b c
University Department, Department of Physics, Anna University of Technology Tirunelveli, Tamil Nadu, India Department of Physics, Chikkanna Government Arts College, Tiruppur, Tamil Nadu, India Department of Physics, Government Arts College, Dharmapuri, Tamil Nadu, India
a r t i c l e
i n f o
Article history: Received 27 October 2012 Accepted 23 March 2013
Keywords: Vector diffraction theory Radially polarized beam Focal shift Axisymmetric Bessel-modulated Gaussian beam
a b s t r a c t Focal shift of the radially polarized axisymmetric QBG beam with radial variance phase plate is investigated theoretically by vector diffraction theory. Axisymmetric Bessel modulated Gaussian beam with quadratic radial dependence (QBG beam) has attracted much attention recently. Calculation results show that focus shifts considerably by changing the phase parameter C that indicates the radial phase variance speed. Under condition of beam parameter of radially polarized axisymmetric QBG beam, there is one focal spot that shifts far away from optical aperture on increasing C. When increases the value of beam parameter, there may occur two focal peaks that also shift remarkably on increasing C. And some novel focal patterns may occur, including focal spot, one ring focus, and dark hollow focus, which is very important in optical tweezers technique. © 2013 Published by Elsevier GmbH.
1. Introduction Recently, a novel class of beam expressed in cylindrical coordinate system, was introduced by Caron and Potvliege [1], namely, the Bessel-modulated Gaussian beams with quadratic radial dependence (QBG beam), it has attracted much attention [2–7]. Various closed-form paraxial solutions of the Helmholtz equation have been described over the years. A particularly interesting class of such solutions, known as Bessel-Gauss beams, has been introduced by Gori et al. in 1987. Bessel-modulated Gaussian beams are another class of simple closed-form solutions of the paraxial wave equation, which are similar to Bessel-Gauss beams in that their amplitude is also a product of a Bessel function and a Gaussian function. In the solutions we are concerned with, however, the argument of the Bessel function is not linear in the transverse coordinate but quadratic .It was shown that such class of beams has familiar collinear geometry of the Gaussian beam and also an interesting non-Gaussian features for certain values of its parameters [1–3]. Particularly, it was demonstrated that the zeroth-order QBG beam, which is referred to as the axisymmetric QBG beam, can be expanded in Laguerre–Gauss modes and has a very flat axial profile when the beam parameterlis of order of unity [1,2]. Belafhal and Dalil-Essakali studied the propagation properties of QBG beams through an unapertured optical paraxial ABCD system [4]. In addition, Mei et al. studied the Bessel-modulated Gaussian light beams
∗ Corresponding author. E-mail address: [email protected] (K. Prabakaran). 0030-4026/$ – see front matter © 2013 Published by Elsevier GmbH. http://dx.doi.org/10.1016/j.ijleo.2013.03.119
passing through a paraxial ABCD optical system with an annular aperture [5]. Hricha, and Belaghal investigated in detail the focusing properties and focal shift of the axisymmetric QBG beam passing through a non-apertured aberration-free lens [2]. The beam propagation factor (M2-factor), far-field distribution, and the kurtosis parameter of such type of beams are also investigated [3,6,7]. In our knowledge, almost all QBG beams in previous papers are in scalar form without considering the polarization distribution of optical field. In fact, the polarization is very important characteristics to alter propagating and focusing properties of beams, and have attracted many researchers recently [8–13]. On the other hand, in high numerical aperture optical systems, polarization characteristics of beam should be considered to provide more accurate results, and vector diffraction theory should be employed to replace scalar diffraction theory. Recently the focal shifts may play a crucial role in achievement of super resolution and in optical trapping [14]. In optical trapping system there are two kinds of forces acting on the particle. One is the optical gradient force, which plays a crucial role in constructing optical trap and its intensity is proportional to the optical intensity gradient; the other kind of force is scattering force, which usually has complex forms because this kind of force is related to the properties of the trapped particles, and whose intensity is proportional to the optical intensity. Opical gradient force is necessary for optical trapping, and its direction is opposite to the optical gradient direction. Tracing the movement of the point of absolute maximum intensity along optical axis has attracted many researchers for several decades [15–17]. It was found that the point of absolute maximum intensity does not coincide with the geometrical focus
K. Prabakaran et al. / Optik 124 (2013) 5454–5457
Fig. 1. Intensity distributions in focal region for NA = 0.9, = 1, and (a) C = 0 (b) C = 1 (c) C = 2 and (d) C = 3, respectively.
but shifts along optical axis. This phenomenon is referred to as focal shift. Focal shifts have taken an important part in design and optimization of various optical systems and extremely important for image formation in defocused planes, for increasing the depth of focus and in automatic focusing [18]. In this paper the focal shift of the radially polarized axisymmetric QBG beam with radial variance phase plate is investigated theoretically by vector diffraction theory. 2. Principle of focusing system In the system we investigated, on the basis of Richards and Wolf’s vectorial diffraction method widely used for high-NA lens system at arbitrary incident polarization [19]. In the case of the radial incident polarization, adopting the cylindrical coordinates r, z, and the notations of Ref. [12], radial and longitudinal components of the electric field can be written as z) = Er er + Ez ez E(r,
(1)
where er and ez is the unit vector in the radial propagation direction. Er , Ez and are the amplitudes of the two orthogonal components that can be written as
Fig. 2. Intensity distributions in focal region for NA = 0.9, = 2, and (a) C = 0 (b) C = 1, (c) C = 2 and (d) C = 3, respectively.
wavefront of radial vector axisymmetric QBG beam is chosen as exp(iC/˛) [20],the optical field value distribution E0 is in the form,
E0 = J0
sin2 () w2 · NA2
cos1/2 () · E0 · sin 2 · J1 (kr sin )eikz cos d
Er (r, ϕ, z) = A
· exp
sin2 () w2 · NA2
· exp(iC/˛)
(4)
where w = ω0 /r0 is called relative waist width, ω0 is the waist width of the Gaussian beam and r0 is radius of incident optical aperture of focusing optical system. is a beam parameter which is complexvalued in general, and C is phase parameter that indicates the radial phase variance speed. The electric field in the focal region can be written in the same form as Eq. (1); however, the two orthogonal components Er and Ez are different and should be expressed as
˛ Er (r, z) = A
cos
1/2
() · J0
sin2 ()
w2 · NA2
· exp
sin2 ()
w2 · NA2
0
· exp(iC/˛) · P() · sin 2 · J1 (kr sin )eikz cos d
˛ cos1/2 () · J0
Ez (r, z) = 2iA
˛
5455
(2)
sin2 () w2 · NA2
· exp
sin2 ()
w2 · NA2
0
· exp(iC/˛) · P() · sin2 J0 (kr sin )eikz cos d
0
(5)
(6)
˛ cos1/2 () · E0 · sin2 J0 (kr sin )eikz cos d
Ez (r, ϕ, z) = 2iA
(3)
0
where ˛ = arc sin(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 axial coordinates of observation point in focal region, respectively. In this paper, the radial variance phase
The optical intensity in focal region is proportional to the modulus square of Eq. (1). Basing on the above equations, focusing properties of radially polarized axisymmetric QBG beam with radial variance phase plate can be investigated theoretically. 3. Results In the investigation of radially polarized axisymmetric QBG beam, without loss of validity and generality, we perform the
5456
K. Prabakaran et al. / Optik 124 (2013) 5454–5457
broadens along optical axis and the on-axis intensity at the longitudinal ends of focal ring becomes stronger. And the intensity maximum position also shifts simultaneously from on-axis position of the focal ring pattern. From above focal pattern evolution process, we can see that for certain beam parameter = 4, phase parameter C can alter considerably intensity distribution in focal region of radially polarized axisymmetric QBG beam and some novel focal shapes may appear. The intensity distribution in focal region of the radially polarized QBG beam can be altered considerably by beam parameter and phase parameter C, and many novel focal patterns can occur, which can be used to construct tunable optical traps. Therefore, focusing properties of the radially polarized QBG beam shows that this kind of beam may find very wide application in optical tweezers system.
4. Conclusion
Fig. 3. Intensity distributions in focal region for NA = 0.9, = 4, and (a) C = 0.0, (b) C = 0.5, (c) C = 1 and (d) C = 1.5, respectively.
integration of Eq. (1) numerically using parameters = 1, A = 1 and NA of the objective is 0.9.For all calculation in the length unit is normalized to and the energy density is normalized to unity. The intensity distributions in focal region for = 1 and different are firstly calculated, and illustrated in Fig. 1. Here it should be noted that the distance unit in all figures in this article is k−1 , where k is wave number. It can be seen from this figure that there is one focal spot when phase parameter C = 0. On increasing C, the focal spot position shifts along optical axis far away from optical aperture, as shown in Fig. 1. From this figure, we can see that the focal shift distance increases on increasing phase parameter C almost linearly. If this focal peak is used to construct one optical trap, phase parameter C may be employed to adjust trap position, can transport micro particles. Now the value of beam parameter is changed to investigate its effect on focal pattern evolution. The intensity distributions for = 2 and different C are illustrated in Fig. 2. It can be seen from this figure that the beam parameter affects focal intensity distribution very considerably. On increasing beam parameter , one focal peak evolves into one dark hollow focus that is defined as those focus whose intensity is lower than that around it, namely, local intensity minimum. There are two optical intensity maximums along optical axis, and one local intensity minimum at geometrical focus position. On increasing phase parameter C, whole focal pattern shifts along optical axis far away from optical aperture, and one intensity peak shrinks so that there is only one optical intensity maximum. Under condition of C = 0, there are two optical intensity maximums. And on increasing C, the focal peak near optical aperture shrinks, while the other focal peak shifts along optical axis. When C changes from 0 to 3, though focal patterns differs very remarkably. The dark hollow focus can act as stable optical trap for those particles whose refractive index is smaller than that of surrounding media, and this condition is very common, especially in life science optical trapping systems [8]. However, from Fig. 3, we observed that there is one focal ring when phase parameter C = 0. Increase C continuously, the focal ring
In conclusion, Focal shift of the radially polarized axisymmetric QBG beam is investigated theoretically by vector diffraction theory in high numerical aperture system. Simulation results show that beam parameter and phase parameter alter the intensity distribution in focal region of radially polarized axisymmetric QBG beam considerably. Focus shifts considerably by changing the phase parameter that indicates the radial phase variance speed. The beam parameter of cylindrical vector QBG beam, there is one focal peak that shifts far away from optical aperture on increasing phase parameter. When beam parameter increases, there occur two focal peaks that also shift remarkably on increasing phase parameter. And some novel focal patterns can appear, such as focal spot, one-ring focus, and dark hollow focus, which can be used to construct stable and alterable optical traps. Radially polarized axisymmetric QBG beam has very promising application due to its particular focusing properties, especially in optical manipulation technique.
References [1] C.F.R. Caron, R.M. Potvliege, Bessel-modulated Gaussian beams with quadratic radial dependence, Opt. Commun. 164 (1999) 83–93. [2] Z. Hricha, A. Belafhal, Focal shift in the axisymmetric Bessel-modulated Gaussian beam, Opt. Commun. 255 (2005) 235–240. [3] X. Wang, B. Lü, The beam propagation factor and far-field distribution of Bessel-modulated Gaussian beams, Opt. Quant. Electron 34 (2002) 1071–1077. [4] A. Belafhal, L. Dalil-Essakali, Collins formula and propagation of Besselmodulated Gaussian light beams through an ABCD optical system, Opt. Commun. 177 (2000) 181–188. [5] Z. Mei, D. Zhao, X. Wei, F. Jing, Q. Zhu, Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture, Optik 116 (2005) 521–526. [6] X. Wang, B. Lü, The beam width of Bessel-modulated Gaussian beams, J. Mod. Opt. 50 (2003) 2107–2115. [7] B. Lü, X. Wang, Kurtosis parameter of Bessel-modulated Gaussian beams propagating through ABCD optical systems, Opt. Commun. 204 (2002) 91–97. [8] R. Oron, S. Blit, N. Davidson, A.A. Friesem, Z. Bomzon, E. Hasman, The formation of laser beams with pure azimuthal or radial polarization, Appl. Phys. Lett. 77 (2000) 3322–3324. [9] X. Gao, S. Hu, H. Gu, J. Wang, Focal shift of three-portion concentric piecewise cylindrical vector beam, Optik 120 (2009) 519–523. [10] Q. Zhan, J.R. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (2002) 324–330. [11] W.T. Tang, E.Y.S. Yew, C.J.R. Sheppard, Polarization conversion in confocal microscopy with radially polarized illumination, Opt. Lett. 34 (2009) 2147–2149. [12] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2000) 77–87. [13] Q. Zhan, Cylindrical vector beams: from mathematical concepts to applications, Adv. Opt. Photon 1 (2009) 1–57. [14] J. Wang, L. Liu, M. Yun, D. Liu, X. Liu, Superresolution, extended focal depth or focal shift with electrically controllable radial birefringent filter, J. Opt. A: Pure Appl. Opt. 7 (2005) 748.
K. Prabakaran et al. / Optik 124 (2013) 5454–5457 [15] M.P. MacDonald, L. Paterson, W. Sibbett, K. Dholakia, P.E. Bryant, Trapping and manipulation of low-index particles in a two-dimensional interferometric optical trap, Opt. Lett. 26 (2001) 863–865. [16] Y. Li, E. Wolf, Focal shifts in diffracted converging spherical waves, Opt. Commun. 39 (1981) 211–215. [17] X. Gao, F. Zhou, W. Xu, F. Gan, Gradient force pattern, focal shift, and focal switch in an apodized optical system, Optik 116 (2005) 99–106.
5457
[18] J. Odeja-Castaneda, L.R. Berriel-Valdos, Arbitrarily high focal depth with finite apertures, Opt. Lett. 13 (1988) 183–185. [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] X. Gao, Q. Zhan, J. Li, S. Hu, J. Wang, S. Zhuang, Cylindrical vector axisymmetric Bessel-modulated Gaussian beam, Opt. Quant. Electron. 41 (2009) 385–396.