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Elimination of fluorescence intensity difference in orientation determination of single molecules by highly focused generalized cylindrical vector beams
Optik 122 (2011) 773–776
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Elimination of fluorescence intensity difference in orientation determination of single molecules by highly focused generalized cylindrical vector beams Xianghui Wang ∗ , Shengjiang Chang, Lie Lin, Linrui Wang, Shujuan Hao Institute of Modern Optics, Nankai University, Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education, Tianjin 300071, China
a r t i c l e
i n f o
Article history: Received 21 September 2009 Accepted 22 May 2010
Keywords: Orientation determination Single molecule Cylindrical vector beam Polarization Confocal microscopy
a b s t r a c t The orientation of the dipole moment, which is one of the important parameters in single-molecule fluorescence spectroscopy and often influenced by subtle changes in the local environment, is determined by highly focused generalized cylindrical vector beams combined with a confocal far-field microscope. As compared to the case of a radially polarized incident beam, the numerical results demonstrate that fluorescence intensity difference among single molecules with different orientations can be effectively overcome for a particular polarization direction of the generalized cylindrical vector beam, which may increases the signal-to-noise ratio of the practical experiments and then ease the difficulty in orientation determination of single molecules. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction In recent years, there is an increasing interest in cylindrical vector beams due to the focus properties of these beams, especially when tightly focused by a high numerical aperture (NA) objective [1–11]. Because of the symmetry of the polarization, the electric field distribution of a focused generalized cylindrical vector beam in the focal plane is usually rotationally symmetrical [1]. In addition, the intensity pattern in the vicinity of the focal spot can be tailored by changing the polarization direction of the cylindrical vector beam [2,3]. Owing to those unique properties, cylindrical vector beams may find wide applications in optical trapping [12], scanning optical microscopy [13], laser cutting of metals [14] and single molecule imaging [5,7]. Among these applications, particular interest has been given to orientation determination of single molecules. The orientation of the dipole moment is one of the important parameters in single-molecule fluorescence spectroscopy. Many photophysical parameters of single molecules, such as fluorescence lifetime and emission intensity, are often influenced by the molecule’s orientation and subtle changes in the local environment can result in reorientation of the dipole moment [15]. Thus, an excitation scheme which allows for an efficient determination of the single-molecule orientation is of important for single molecule experiments [16–19].
∗ Corresponding author. Tel.: +86 02223506422. E-mail address: [email protected] (X. Wang). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.05.022
Usually, cylindrical vector beams can be divided into radial polarization, azimuthal polarization and generalized cylindrical polarization, according to the actual polarization pattern. Theoretical and experimental investigations of the application of a focused radially polarized beam in orientation determination of single molecules have been reported previously [5,7]. The singlemolecule orientation can be determined by scanned imaging of single molecules with a confocal far-field microscope under radial polarization illumination. As compared to the case of a linearly polarized incident beam with annular illumination, the variations of fluorescence patterns of molecules with different orientations under radial polarization illumination are comparatively simpler and the orientation determination is more easily performed. However, molecules with orientation perpendicular to the optical axis show a rather weak fluorescence because the longitudinal component of the excitation field in the focal region is much stronger than the radial component in the case of tightly focusing [1]. Therefore, the fluorescence patterns of those molecules are more easily overlayed by background noise, which undoubtedly increases the difficulty in orientation determination of single molecules [19]. As a matter of fact, a generalized cylindrical vector beam can be decomposed into a linear superposition of radially polarized and azimuthally polarized components. The ratio of the radial and longitudinal components of the focal electric field of the generalized cylindrical vector beam can be adjusted by changing the polarization direction. In this paper, orientation determination of single molecules by focused generalized cylindrical vector beams is studied numerically. Simulations show that for a particular polarization direction, a generalized cylindrical vector beam is more suitable
774
X. Wang et al. / Optik 122 (2011) 773–776
Fig. 1. The polarization pattern of a generalized cylindrical vector beam.
for orientation determination of single molecules as compared to a radially polarized incident beam. 2. Theoretical model Cylindrical vector beams satisfy cylindrical symmetry both in amplitude and polarization [2,3]. The polarization pattern of a generalized cylindrical vector beam is illustrated in Fig. 1. Each point of the beam has a polarization direction rotated by ϕ0 from its radial direction. The Richards and Wolf vectorial diffraction method has been extensively used to analyze the basic property of highly focused polarized beams [20]. Adopting their method, if the optical axis is supposed to be in the z direction (see Fig. 2), the focal electric field of a highly focused generalized cylindrical vector beam can be expressed in the Cartesian coordinate system as [1–3].
Ex = 2A cos ϕ0 cos ϕr
× J1
v sin
p() l() cos sin exp 0
iu cos
J1
2
sin ˛
v sin sin ˛
× exp
2
sin ˛
J1
P() l() sin exp 0
sin2 ˛
x2
+ y2
iu cos sin ˛
J0
v sin sin ˛
d
(3)
where ϕr and are the azimuthal and polar angles, respectively. ˛ is the maximal angle determined by the NA of the objective. A is proportional to the field amplitude on the exit pupil. Jn is the Bessel
Fig. 2. Focusing of a generalized cylindrical vector beam.
sin ˛
(5)
2
sin sin ˛
2 J1
sin 2 sin ˛
(6)
where is the ratio of the pupil radius and the beam waist. A raster scan image is usually detected by a confocal microscope through recording the fluorescence rate R as a function of the lateral coordinate in order to further obtain orientation determination of single molecules [5,7,18,19]. When there is only a single molecule in the focal region, the fluorescence rate R of this molecule can be given by [5,18]
2
R = c d · E
(2)
(4)
If the objective obeys the sine condition, the pupil apodization function is P() = cos1/2 . For a Bessel–Gauss beam waist at the entrance pupil, l() can be described by [1]
P() l() sin
d
2
0
sin ˛
˛ 2
Ez = 2 iA cos ϕ0
v sin
iu cos
˛
d + 2A sin ϕ0 cos ϕr
iu cos
v=k
l() = exp −
(1)
u = kz sin2 ˛
d
sin ˛
p() l() cos sin exp
sin2 ˛
P() l() sin
0
function of the first kind with order n. The optical co-ordinates u and v are defined by:
0
˛
Ey = 2 A cos ϕ0 sin ϕr
× J1
v sin
iu cos
˛
d − 2A sin ϕ0 sin ϕr
sin ˛
× exp
˛
Fig. 3. Fluorescence patterns for differently oriented single molecules excited by a highly focused radially polarized beam.
(7)
where d is the unit vector along the absorption dipole moment of the molecule. The constant c is determined by the molecule’s absorption cross-section and quantum yield. The single-molecule orientation can be defined by two angles ϕd and ˇ, which are the azimuthal and polar angles of the unit vector d, respectively. Then Eq. (7) can be expressed in the Cartesian coordinate system as R = c(cos ϕd sin ˇEx + sin ϕd sin ˇEy + cos ˇEz )
(8)
3. Results and discussions In the following numerical calculations, it is assumed that an oil immersion objective (NA = 1.4) is used to focus the incident beam and the refractive index of immersion oil is 1.5. In addition, the wavelength of the incident beam is 632 nm and the beam parameter is equal to 1. Fig. 3 shows fluorescence patterns calculated according to Eq. (8) for different values of ϕd and ˇ when ϕ0 = 0◦ , which corresponds to an incoming radially polarized beam. Those fluorescence patterns are all normalized to the maximum fluorescence intensity for molecules oriented at (ϕd , ˇ) = (0◦ , 0◦ ). From the results in Fig. 3, it can be found that the shape of the fluorescence pattern changes with the variation of ˇ, while the fluorescence pattern
X. Wang et al. / Optik 122 (2011) 773–776
775
Fig. 4. Intensity distributions of the radial and longitudinal components of a focused radially polarized beam along the radial direction in the focal plane.
merely rotates when the value of ϕd changes. Therefore, the full three-dimensional orientation information can be extracted from a single image by simultaneously evaluating the variation degree of the shape and the rotation of the fluorescence pattern. However, it also can be seen that molecules with larger values of ˇ show a rather weak fluorescence as compared to those with lower values of ˇ. This fluorescence intensity difference among molecules with different orientations can result that the fluorescence patterns of molecules with larger values of ˇ are more likely to be overlayed by background noise, which may decreases the signal-to-noise ratio of the practical experiments and then increase the difficulty in orientation determination of single molecules. Despite that this problem is likely to be overcome by increasing the excitation intensity, fluorescence saturation and photobleaching for molecules with lower values of ˇ may occur when the excitation field is too intense. In fact, the above problem is mainly attributed to the difference between the radial and longitudinal components of the excitation field. According to Eq. (8), molecules with orientation perpendicular to the optic axis are mainly excited by the radial component, whereas molecules with orientation parallel to the z axis only respond to the longitudinal component. Fig. 4 shows the calculated results of the focal intensity distributions in the focal plane. The solid and dashed lines represent the longitudinal and radial components, respectively. Those distributions are all normalized to the maximum intensity of the longitudinal component. It is obvious that the longitudinal component is much stronger than the radial component and the ratio of those two components is about 1:0.33. The relative strength of the longitudinal component can be further enhanced with increasing NA or annular illumination [1,11]. A generalized cylindrical vector beam can be decomposed into a linear superposition of radially polarized and azimuthally polarized components. The ratio of the radial and longitudinal components of the focal electric field can be adjusted by changing the polarization direction and this property has been applied in focus shaping as reported in Refs. [2,3]. For a particular polariza-
Fig. 6. Fluorescence intensity distributions excited by a highly focused generalized cylindrical vector beams for ϕ0 = 45.22◦ .
tion direction, all excitation field components of the focal electric field of a focused generalized cylindrical vector beam may be of comparable magnitude. The intensity distributions of the focal electric field for different values of ϕ0 are illustrated in Fig. 5. From Fig. 5, it is clear that the maximum intensity of the radial component is equal to that of the longitudinal component when ϕ0 = 45.22◦ . Fig. 6 shows fluorescence patterns for single molecules with different orientations when the value of ϕ0 is 45.22◦ . As shown in Fig. 6, the fluorescence intensity for molecules with larger values of ˇ has remarkably increased. At the same time, the shape of the fluorescence pattern also changes with the variation of ˇ and the variation of ϕd merely results in the rotation of the fluorescence pattern, despite that the fluorescence pattern approximately rotates by 45◦ as compared to the case of radial polarization illumination. From the above results, it is obvious that for a particular generalized cylindrical vector beam, not only the three-dimensional orientation can determined by a single scanned image, but also the fluorescence intensity difference among molecules with different orientations can be effectively avoided. Therefore, a generalized cylindrical vector beam at a particular condition is more suitable for orientation determination of single molecules as compared to a radially polarized incident beam. Furthermore, the control of ϕ0 can be conveniently and flexibly performed by inserting a simple polarization rotator consisting of two half-wave plates in the optical path under radial polarization illumination [2,3]. By incorporating this polarization rotator in the existent set-ups, the particular generalized cylindrical vector beam can be easily obtained.
Fig. 5. Intensity distributions of the radial and longitudinal components along the radial direction for (a) ϕ0 = 30◦ , (b) ϕ0 = 45.22◦ , and (c) ϕ0 = 60◦ , respectively.
776
X. Wang et al. / Optik 122 (2011) 773–776
4. Conclusions In summary, we have numerically studied orientation determination of single molecules by a focused generalized cylindrical vector beam. The simulations have shown that when a generalized cylindrical vector beam with appropriate polarization direction is used as the incident beam, it is possible to effectively overcome fluorescence intensity difference among single molecules with different orientations. By incorporating a simple polarization rotator, this particular generalized cylindrical vector beam can be easily converted from a radially polarized beam and then a more efficient excitation scheme for orientation determination of single molecules is likely to be successfully designed. Acknowledgments The work was supported by National 973 Project of China (No. 2007CB310403), the National Natural Science Foundation of China (Nos. 10704043 and 60772105), and Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education, People’s Republic of China. References [1] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2000) 77–87. [2] Q.W. Zhan, J.R. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (2002) 324–331. [3] W.B. Chen, Q.W. Zhan, Three-dimensional focus shaping with cylindrical vector beams, Opt. Commun. 265 (2006) 411–417.
[4] C.C. Sun, C.K. Liu, Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation, Opt. Lett. 28 (2003) 99–101. [5] L. Novotny, M.R. Beversluis, K.S. Youngworth, T.G. Brown, Longitudinal field modes probed by single molecules, Phys. Rev. Lett. 86 (2001) 5251–5254. [6] Q.W. Zhan, Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams, J. Opt. A: Pure Appl. Opt. 5 (2003) 229–232. [7] M.R. Beversluis, L. Novotny, S.J. Stranick, Programmable vector point-spread function engineering, Opt. Express 14 (2006) 2650–2656. [8] D.W. Diehl, R.W. Schoonover, T.D. Visser, The structure of focused, radially polarized fields, Opt. Express 14 (2006) 3030–3038. [9] L.E. Helseth, Smallest focal hole, Opt. Commun. 257 (2006) 1–8. [10] S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, The focus of light-theoretical calculation and experimental tomographic reconstruction, Appl. Phys. B 72 (2001) 109–113. [11] S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, Focusing light into a tighter spot, Opt. Commun. 179 (2000) 1–7. [12] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, Novel optical trap of atoms with a doughnut beam, Phys. Rev. Lett. 78 (1997) 4713–4716. [13] K.S. Youngworth, T.G. Brown, Inhomogeneous polarization in scanning optical microscopy, Proc. SPIE 3919 (2000) 75–85. [14] V.G. Niziev, A.V. Nesterov, Influence of beam polarization on laser cutting efficiency, J. Phys. D: Appl. Phys. 32 (1999) 1455–1461. [15] T. Plakhotnik, W.E. Moerner, V. Palm, U.P. Wild, Single molecule spectroscopy: maximum emission rate and saturation intensity, Opt. Commun. 114 (1995) 83–88. [16] J.T. Fourkas, Rapid determination of the three-dimensional orientation of single molecules, Opt. Lett. 26 (2001) 211–213. [17] M. Böhmer, J. Enderlein, Orientation imaging of single molecules by wide-field epifluorescence microscopy, J. Opt. Soc. Am. B 20 (2003) 554–559. [18] B. Sick, B. Hecht, L. Novotny, Orientational imaging of single molecules by annular illumination, Phys. Rev. Lett. 85 (2000) 4482–4485. [19] J. Hohlbein, C.G. Hübner, Simple scheme for rapid three-dimensional orientation determination of the emission dipole of single molecules, Appl. Phys. Lett. 86 (2005) 121104. [20] B. Richards, E. Wolf, Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system, Proc. Roy. Soc. A 253 (1959) 358–379.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Elimination of fluorescence intensity difference in orientation determination of single molecules by highly focused generalized cylindrical vector beams Xianghui Wang ∗ , Shengjiang Chang, Lie Lin, Linrui Wang, Shujuan Hao Institute of Modern Optics, Nankai University, Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education, Tianjin 300071, China
a r t i c l e
i n f o
Article history: Received 21 September 2009 Accepted 22 May 2010
Keywords: Orientation determination Single molecule Cylindrical vector beam Polarization Confocal microscopy
a b s t r a c t The orientation of the dipole moment, which is one of the important parameters in single-molecule fluorescence spectroscopy and often influenced by subtle changes in the local environment, is determined by highly focused generalized cylindrical vector beams combined with a confocal far-field microscope. As compared to the case of a radially polarized incident beam, the numerical results demonstrate that fluorescence intensity difference among single molecules with different orientations can be effectively overcome for a particular polarization direction of the generalized cylindrical vector beam, which may increases the signal-to-noise ratio of the practical experiments and then ease the difficulty in orientation determination of single molecules. © 2010 Elsevier GmbH. All rights reserved.
1. Introduction In recent years, there is an increasing interest in cylindrical vector beams due to the focus properties of these beams, especially when tightly focused by a high numerical aperture (NA) objective [1–11]. Because of the symmetry of the polarization, the electric field distribution of a focused generalized cylindrical vector beam in the focal plane is usually rotationally symmetrical [1]. In addition, the intensity pattern in the vicinity of the focal spot can be tailored by changing the polarization direction of the cylindrical vector beam [2,3]. Owing to those unique properties, cylindrical vector beams may find wide applications in optical trapping [12], scanning optical microscopy [13], laser cutting of metals [14] and single molecule imaging [5,7]. Among these applications, particular interest has been given to orientation determination of single molecules. The orientation of the dipole moment is one of the important parameters in single-molecule fluorescence spectroscopy. Many photophysical parameters of single molecules, such as fluorescence lifetime and emission intensity, are often influenced by the molecule’s orientation and subtle changes in the local environment can result in reorientation of the dipole moment [15]. Thus, an excitation scheme which allows for an efficient determination of the single-molecule orientation is of important for single molecule experiments [16–19].
∗ Corresponding author. Tel.: +86 02223506422. E-mail address: [email protected] (X. Wang). 0030-4026/$ – see front matter © 2010 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2010.05.022
Usually, cylindrical vector beams can be divided into radial polarization, azimuthal polarization and generalized cylindrical polarization, according to the actual polarization pattern. Theoretical and experimental investigations of the application of a focused radially polarized beam in orientation determination of single molecules have been reported previously [5,7]. The singlemolecule orientation can be determined by scanned imaging of single molecules with a confocal far-field microscope under radial polarization illumination. As compared to the case of a linearly polarized incident beam with annular illumination, the variations of fluorescence patterns of molecules with different orientations under radial polarization illumination are comparatively simpler and the orientation determination is more easily performed. However, molecules with orientation perpendicular to the optical axis show a rather weak fluorescence because the longitudinal component of the excitation field in the focal region is much stronger than the radial component in the case of tightly focusing [1]. Therefore, the fluorescence patterns of those molecules are more easily overlayed by background noise, which undoubtedly increases the difficulty in orientation determination of single molecules [19]. As a matter of fact, a generalized cylindrical vector beam can be decomposed into a linear superposition of radially polarized and azimuthally polarized components. The ratio of the radial and longitudinal components of the focal electric field of the generalized cylindrical vector beam can be adjusted by changing the polarization direction. In this paper, orientation determination of single molecules by focused generalized cylindrical vector beams is studied numerically. Simulations show that for a particular polarization direction, a generalized cylindrical vector beam is more suitable
774
X. Wang et al. / Optik 122 (2011) 773–776
Fig. 1. The polarization pattern of a generalized cylindrical vector beam.
for orientation determination of single molecules as compared to a radially polarized incident beam. 2. Theoretical model Cylindrical vector beams satisfy cylindrical symmetry both in amplitude and polarization [2,3]. The polarization pattern of a generalized cylindrical vector beam is illustrated in Fig. 1. Each point of the beam has a polarization direction rotated by ϕ0 from its radial direction. The Richards and Wolf vectorial diffraction method has been extensively used to analyze the basic property of highly focused polarized beams [20]. Adopting their method, if the optical axis is supposed to be in the z direction (see Fig. 2), the focal electric field of a highly focused generalized cylindrical vector beam can be expressed in the Cartesian coordinate system as [1–3].
Ex = 2A cos ϕ0 cos ϕr
× J1
v sin
p() l() cos sin exp 0
iu cos
J1
2
sin ˛
v sin sin ˛
× exp
2
sin ˛
J1
P() l() sin exp 0
sin2 ˛
x2
+ y2
iu cos sin ˛
J0
v sin sin ˛
d
(3)
where ϕr and are the azimuthal and polar angles, respectively. ˛ is the maximal angle determined by the NA of the objective. A is proportional to the field amplitude on the exit pupil. Jn is the Bessel
Fig. 2. Focusing of a generalized cylindrical vector beam.
sin ˛
(5)
2
sin sin ˛
2 J1
sin 2 sin ˛
(6)
where is the ratio of the pupil radius and the beam waist. A raster scan image is usually detected by a confocal microscope through recording the fluorescence rate R as a function of the lateral coordinate in order to further obtain orientation determination of single molecules [5,7,18,19]. When there is only a single molecule in the focal region, the fluorescence rate R of this molecule can be given by [5,18]
2
R = c d · E
(2)
(4)
If the objective obeys the sine condition, the pupil apodization function is P() = cos1/2 . For a Bessel–Gauss beam waist at the entrance pupil, l() can be described by [1]
P() l() sin
d
2
0
sin ˛
˛ 2
Ez = 2 iA cos ϕ0
v sin
iu cos
˛
d + 2A sin ϕ0 cos ϕr
iu cos
v=k
l() = exp −
(1)
u = kz sin2 ˛
d
sin ˛
p() l() cos sin exp
sin2 ˛
P() l() sin
0
function of the first kind with order n. The optical co-ordinates u and v are defined by:
0
˛
Ey = 2 A cos ϕ0 sin ϕr
× J1
v sin
iu cos
˛
d − 2A sin ϕ0 sin ϕr
sin ˛
× exp
˛
Fig. 3. Fluorescence patterns for differently oriented single molecules excited by a highly focused radially polarized beam.
(7)
where d is the unit vector along the absorption dipole moment of the molecule. The constant c is determined by the molecule’s absorption cross-section and quantum yield. The single-molecule orientation can be defined by two angles ϕd and ˇ, which are the azimuthal and polar angles of the unit vector d, respectively. Then Eq. (7) can be expressed in the Cartesian coordinate system as R = c(cos ϕd sin ˇEx + sin ϕd sin ˇEy + cos ˇEz )
(8)
3. Results and discussions In the following numerical calculations, it is assumed that an oil immersion objective (NA = 1.4) is used to focus the incident beam and the refractive index of immersion oil is 1.5. In addition, the wavelength of the incident beam is 632 nm and the beam parameter is equal to 1. Fig. 3 shows fluorescence patterns calculated according to Eq. (8) for different values of ϕd and ˇ when ϕ0 = 0◦ , which corresponds to an incoming radially polarized beam. Those fluorescence patterns are all normalized to the maximum fluorescence intensity for molecules oriented at (ϕd , ˇ) = (0◦ , 0◦ ). From the results in Fig. 3, it can be found that the shape of the fluorescence pattern changes with the variation of ˇ, while the fluorescence pattern
X. Wang et al. / Optik 122 (2011) 773–776
775
Fig. 4. Intensity distributions of the radial and longitudinal components of a focused radially polarized beam along the radial direction in the focal plane.
merely rotates when the value of ϕd changes. Therefore, the full three-dimensional orientation information can be extracted from a single image by simultaneously evaluating the variation degree of the shape and the rotation of the fluorescence pattern. However, it also can be seen that molecules with larger values of ˇ show a rather weak fluorescence as compared to those with lower values of ˇ. This fluorescence intensity difference among molecules with different orientations can result that the fluorescence patterns of molecules with larger values of ˇ are more likely to be overlayed by background noise, which may decreases the signal-to-noise ratio of the practical experiments and then increase the difficulty in orientation determination of single molecules. Despite that this problem is likely to be overcome by increasing the excitation intensity, fluorescence saturation and photobleaching for molecules with lower values of ˇ may occur when the excitation field is too intense. In fact, the above problem is mainly attributed to the difference between the radial and longitudinal components of the excitation field. According to Eq. (8), molecules with orientation perpendicular to the optic axis are mainly excited by the radial component, whereas molecules with orientation parallel to the z axis only respond to the longitudinal component. Fig. 4 shows the calculated results of the focal intensity distributions in the focal plane. The solid and dashed lines represent the longitudinal and radial components, respectively. Those distributions are all normalized to the maximum intensity of the longitudinal component. It is obvious that the longitudinal component is much stronger than the radial component and the ratio of those two components is about 1:0.33. The relative strength of the longitudinal component can be further enhanced with increasing NA or annular illumination [1,11]. A generalized cylindrical vector beam can be decomposed into a linear superposition of radially polarized and azimuthally polarized components. The ratio of the radial and longitudinal components of the focal electric field can be adjusted by changing the polarization direction and this property has been applied in focus shaping as reported in Refs. [2,3]. For a particular polariza-
Fig. 6. Fluorescence intensity distributions excited by a highly focused generalized cylindrical vector beams for ϕ0 = 45.22◦ .
tion direction, all excitation field components of the focal electric field of a focused generalized cylindrical vector beam may be of comparable magnitude. The intensity distributions of the focal electric field for different values of ϕ0 are illustrated in Fig. 5. From Fig. 5, it is clear that the maximum intensity of the radial component is equal to that of the longitudinal component when ϕ0 = 45.22◦ . Fig. 6 shows fluorescence patterns for single molecules with different orientations when the value of ϕ0 is 45.22◦ . As shown in Fig. 6, the fluorescence intensity for molecules with larger values of ˇ has remarkably increased. At the same time, the shape of the fluorescence pattern also changes with the variation of ˇ and the variation of ϕd merely results in the rotation of the fluorescence pattern, despite that the fluorescence pattern approximately rotates by 45◦ as compared to the case of radial polarization illumination. From the above results, it is obvious that for a particular generalized cylindrical vector beam, not only the three-dimensional orientation can determined by a single scanned image, but also the fluorescence intensity difference among molecules with different orientations can be effectively avoided. Therefore, a generalized cylindrical vector beam at a particular condition is more suitable for orientation determination of single molecules as compared to a radially polarized incident beam. Furthermore, the control of ϕ0 can be conveniently and flexibly performed by inserting a simple polarization rotator consisting of two half-wave plates in the optical path under radial polarization illumination [2,3]. By incorporating this polarization rotator in the existent set-ups, the particular generalized cylindrical vector beam can be easily obtained.
Fig. 5. Intensity distributions of the radial and longitudinal components along the radial direction for (a) ϕ0 = 30◦ , (b) ϕ0 = 45.22◦ , and (c) ϕ0 = 60◦ , respectively.
776
X. Wang et al. / Optik 122 (2011) 773–776
4. Conclusions In summary, we have numerically studied orientation determination of single molecules by a focused generalized cylindrical vector beam. The simulations have shown that when a generalized cylindrical vector beam with appropriate polarization direction is used as the incident beam, it is possible to effectively overcome fluorescence intensity difference among single molecules with different orientations. By incorporating a simple polarization rotator, this particular generalized cylindrical vector beam can be easily converted from a radially polarized beam and then a more efficient excitation scheme for orientation determination of single molecules is likely to be successfully designed. Acknowledgments The work was supported by National 973 Project of China (No. 2007CB310403), the National Natural Science Foundation of China (Nos. 10704043 and 60772105), and Key Laboratory of Opto-electronic Information Science and Technology, Ministry of Education, People’s Republic of China. References [1] K.S. Youngworth, T.G. Brown, Focusing of high numerical aperture cylindricalvector beams, Opt. Express 7 (2000) 77–87. [2] Q.W. Zhan, J.R. Leger, Focus shaping using cylindrical vector beams, Opt. Express 10 (2002) 324–331. [3] W.B. Chen, Q.W. Zhan, Three-dimensional focus shaping with cylindrical vector beams, Opt. Commun. 265 (2006) 411–417.
[4] C.C. Sun, C.K. Liu, Ultrasmall focusing spot with a long depth of focus based on polarization and phase modulation, Opt. Lett. 28 (2003) 99–101. [5] L. Novotny, M.R. Beversluis, K.S. Youngworth, T.G. Brown, Longitudinal field modes probed by single molecules, Phys. Rev. Lett. 86 (2001) 5251–5254. [6] Q.W. Zhan, Radiation forces on a dielectric sphere produced by highly focused cylindrical vector beams, J. Opt. A: Pure Appl. Opt. 5 (2003) 229–232. [7] M.R. Beversluis, L. Novotny, S.J. Stranick, Programmable vector point-spread function engineering, Opt. Express 14 (2006) 2650–2656. [8] D.W. Diehl, R.W. Schoonover, T.D. Visser, The structure of focused, radially polarized fields, Opt. Express 14 (2006) 3030–3038. [9] L.E. Helseth, Smallest focal hole, Opt. Commun. 257 (2006) 1–8. [10] S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, The focus of light-theoretical calculation and experimental tomographic reconstruction, Appl. Phys. B 72 (2001) 109–113. [11] S. Quabis, R. Dorn, M. Eberler, O. Glöckl, G. Leuchs, Focusing light into a tighter spot, Opt. Commun. 179 (2000) 1–7. [12] T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, H. Sasada, Novel optical trap of atoms with a doughnut beam, Phys. Rev. Lett. 78 (1997) 4713–4716. [13] K.S. Youngworth, T.G. Brown, Inhomogeneous polarization in scanning optical microscopy, Proc. SPIE 3919 (2000) 75–85. [14] V.G. Niziev, A.V. Nesterov, Influence of beam polarization on laser cutting efficiency, J. Phys. D: Appl. Phys. 32 (1999) 1455–1461. [15] T. Plakhotnik, W.E. Moerner, V. Palm, U.P. Wild, Single molecule spectroscopy: maximum emission rate and saturation intensity, Opt. Commun. 114 (1995) 83–88. [16] J.T. Fourkas, Rapid determination of the three-dimensional orientation of single molecules, Opt. Lett. 26 (2001) 211–213. [17] M. Böhmer, J. Enderlein, Orientation imaging of single molecules by wide-field epifluorescence microscopy, J. Opt. Soc. Am. B 20 (2003) 554–559. [18] B. Sick, B. Hecht, L. Novotny, Orientational imaging of single molecules by annular illumination, Phys. Rev. Lett. 85 (2000) 4482–4485. [19] J. Hohlbein, C.G. Hübner, Simple scheme for rapid three-dimensional orientation determination of the emission dipole of single molecules, Appl. Phys. Lett. 86 (2005) 121104. [20] B. Richards, E. Wolf, Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system, Proc. Roy. Soc. A 253 (1959) 358–379.