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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 8, Iss. 8 — Sep. 4, 2013
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Generation of tunable three-dimensional polarization in 4Pi focusing system

Wenguo Zhu and Weilong She  »View Author Affiliations


Optics Express, Vol. 21, Issue 14, pp. 17265-17274 (2013)
http://dx.doi.org/10.1364/OE.21.017265


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Abstract

We show that, by uniformly modulating the amplitude or polarization of one half of the input beam, a tunable three-dimensional (3D) polarization field near the focus of a 4Pi focusing system can be generated. If the input field is radially polarized and modulated by an amplitude-phase modulator, the longitudinal component of the focused field will partially convert to the transversal one according to the modulation factor and a 3D linear polarization state is created. If the input field is circularly polarized in one half and elliptically polarized in another half, the focal field will have elliptical polarization with the normal to the polarization ellipse being 3D controllable, corresponding to a 3D controllable spin angular momentum.

© 2013 OSA

1. Introduction

In recent years, focusing light into a very tight spot is one of the most attractive topics in optics [1

1. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

3

3. S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9(12), 2159–2166 (1992). [CrossRef]

]. The optimization of the shape and size (intensity and phase distribution) of the focal spot has been frequently discussed [4

4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef] [PubMed]

6

6. J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. 14(5), 055004 (2012). [CrossRef]

]. The polarization as an important parameter of the focal field, however, does not receive enough attention. Under tightly focusing condition, the longitude component of the focal field plays an equally important role as the transverse one [1

1. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

]. The field characterized by three-dimensional (3D) polarization is expected to have extensional functionalities and applications [7

7. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000). [CrossRef] [PubMed]

,8

8. X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun 3, 998 (2012). [CrossRef] [PubMed]

]. For example, by employing 3D polarization, it is possible to directly determine the orientation of anisotropic molecule and crystal [7

7. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000). [CrossRef] [PubMed]

]. As was demonstrated recently, 3D polarization enables polarization information encryption with ultra-security, because the ‘key’ polarization orientation is unlimited in 3D space [8

8. X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun 3, 998 (2012). [CrossRef] [PubMed]

]. Moreover, this kind of polarization could also find applications in optical manipulation. A light beam with 3D linear polarization can trap small particles with geometrical or optical anisotropy, and make them aligned in the plane of polarization [9

9. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature 394(6691), 348–350 (1998). [CrossRef]

]. Different from linear polarization, elliptical polarization including circular one carries spin angular momentum (SAM), which can cause a rotation when acting on a small absorbing or birefringent particle [10

10. A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12(8), 083053 (2010). [CrossRef]

12

12. M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A 84(3), 033838 (2011). [CrossRef]

]. And the rotation is confined in the plane perpendicular to the optical axis for a paraxial trapping beam [10

10. A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12(8), 083053 (2010). [CrossRef]

]. In contrast, for a nonparaxial beam with 3D elliptical polarization, the SAM could make the rotation unlimited in 3D space, which significantly enhances the flexibility of optical manipulation [12

12. M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A 84(3), 033838 (2011). [CrossRef]

]. Several methods have been proposed to generate 3D polarization by designing the distribution of input light field [13

13. A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006). [CrossRef] [PubMed]

15

15. W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284(1), 52–56 (2011). [CrossRef]

]. Abouraddy and Toussaint use low order azimuthal spatial harmonics of linear polarization as the input field to create 3D polarization in an optical microscopy [13

13. A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006). [CrossRef] [PubMed]

]. However, the 3D polarization will deviate from the desired one as the observation point moves away from the geometrical focus. Besides, the desired polarization does not occur at the peak intensity of the focal field in most case. For producing a diffraction-limited spherical focal spot with arbitrary 3D polarization, Chen and Zhan have combined the electric dipole radiation and vectorial diffraction method to determine the input field at the pupil plane [14

14. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010). [CrossRef]

,15

15. W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284(1), 52–56 (2011). [CrossRef]

]. But, their input field is complicated in intensity, phase and polarization distributions.

In this paper, we propose a simple method to create tunable 3D polarization field in a 4Pi focusing system. In this system, a radially polarized beam is focused into longitudinal polarized field in the focal plane [2

2. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004). [CrossRef] [PubMed]

,4

4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef] [PubMed]

]. When the amplitude of one half of the input beam is tuned from 1 to −1 by an amplitude-phase modulator (APM), which contains an attenuator and a phase shifter, the polarization vector at focus rotates gradually from longitudinal direction to the transverse one that perpendicular to the amplitude jump line due to the vectorial interference effect. Therefore, the polarization vector can take any 3D direction as long as changing the modulator factor and jump line. When one half of the input field is designed to be circularly polarized and another half elliptically polarized, the resultant focal field near focus has 3D elliptical polarization with 3D SAM. The principle for the generation of this 3D elliptical polarization can be easily understood by the superposition of SAM vectors of plane waves [16

16. C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80(6), 063814 (2009). [CrossRef]

].

2. The generation of 3D linear polarization

Figure 1
Fig. 1 Illustration of the scheme for generating 3D linear polarization. The blue arrows represent the polarization vectors of light field propagating towards the focus, while the light blue arrow represents the resultant polarization vector at geometrical focus. The color rectangle shows the intensity distribution in the xz plane near the focus. And the intensity and polarization distributions within the central focal spot are shown by projections on three orthogonal planes. In calculation, the configure factors (a,ϕ0) = (0.5,-π/2), NA = 0.95.
shows the scheme to produce 3D linear polarization, where two counter-propagating radially polarized beams illuminate two opposing high-NA objectives. The APMs before objectives tunes the amplitudes of one half of the input fields, which causes the power conversion from the longitudinal field component to the transverse one. The polarization distributions of an arbitrary 3D polarization state within the central focal spot is shown by the projection on three orthogonal planes. And the resultant polarization vector at the geometrical focus is shown by the light blue arrow in Fig. 1.

The field after focused by a high-NA objective can be described by Richards-Wolf integral [17

17. B. Richards and E. Wolf, “Electomagnetic diffraction in optical systems: Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Series A 253(1274), 358–379 (1959). [CrossRef]

]. So, for a radially polarized input field, the three components of the focal field can be written as [1

1. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

]

[Ex(r,φ,z)Ey(r,φ,z)Ez(r,φ,z)]=iCπ0α02πX(θ)l0(θ)T(ϕ)sinθexp{ik[zcosθ+rsinθcos(ϕφ)]}[cosθcosϕcosθsinϕsinθ]dϕdθ,
(1)

where C is the normalized factor, α = arcsin(NA/n) with n = 1 in the free space, X(θ) = (cosθ) 1/2 is pupil apodization function for an aplanatic lens. The function l0(θ) describes the amplitude distribution of the Bessel-Gaussian beam, which has the form
l0(θ)=exp[(βsinθsinα)2]J1(2βsinθsinα),
(2)
with β = 1.5 in our configuration. T(ϕ) is the modulation function, with T(ϕ) = TL/R(ϕ) for the modulators at the left and right sides, respectively. TR(ϕ) = TL(ϕ + π), and
TL(ϕ)={1ϕ0ϕ<ϕ0+πaϕ0+πϕ<ϕ0+2π,
(3)
where a is the modulator factor ranging from −1 to 1, ϕ0 is the angle between the amplitude jump line and x-axis,. The electric field near the focus is the superposition of the fields from the left and right objectives. That is [4

4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef] [PubMed]

]
E(r,φ,z)=[ELx(r,φ,z)+ERx(r,φ,z)]ex+[ELy(r,φ,z)+ERy(r,φ,z)]ey+[ELz(r,φ,z)ERz(r,φ,z)]ez.
(4)
It is found that this resultant focal field is linear polarized and can be controlled by the modulation function.

Since two linear polarizations with opposite orientations are actually the same polarization states, We can introduce the polar and azimuthal angles (χll) to describe the orientation of 3D linear polarization, where χl ranges from 0 to π/2, and ψl takes value between 0 and 2π. At the focus, the polarization orientation can be deduced by superposing the fields of plane waves in the angular spectrum from diffractions at two opposing objectives [18

18. S. N. Khonina and I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett. 36(3), 352–354 (2011). [CrossRef] [PubMed]

]. When a = 1, the APMs will not affect the input radially polarized beams, which result in the focal field being longitudinally polarized [4

4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef] [PubMed]

]. If a decreases from 1 to −1, the polarization vector of the focal field will vary gradually form longitudinal direction to the transverse one perpendicular to the amplitude jump line. That means the polar angle of 3D linear polarization χl depends on a. The numerical result is shown in Fig. 2(g)
Fig. 2 The normalized intensity and polarization distributions in the focal plane (a~f) with polarization orientation (a) (χll) = (0,π), (b) (π/6,π), (c) (π/3,π), (d) (π/2,π), (e) (π/2,-π/2), (f) (π/2,-π/4), which correspond to configure factors (a,ϕ0) = (1,π/2), (0.23,π/2), (−0.3,π/2), (−1,π/2), (−1,π), and (−1,5π/4), respectively. (g) The dependence of χl on a. (h) Projection of intensity and polarization distributions on three orthogonal planes within the FWHM region when a = −1, and ϕ0 = 5π/4. The FWHMs along x, y, and z axes are 0.56λ, 0.82λ, and 0.3λ, respectively. In calculation, NA is chosen to be 0.95.
, from which one sees that χl and a almost have a linear relation. The intensity and polarization distributions in the focal plane are plotted in Figs. 2(a)-2(d) for the case of ϕ0 = π/2, a = 1, 0.23, −0.3 and −1, which correspond to the polar angles of 0, π/6, π/3, and π/2, respectively. Due to the rotational symmetry of the input radial polarization, rotating APMs in both sides will cause a rotation of the focal field around z axis. After simple analysis, one can easily get that the azimuthal angle of 3D linear polarization ψl varies with the rotation angle ϕ0 according to the law ψl = ϕ0 + π/2. In Figs. 2(d)-2(f), a is fixed to be −1 and ϕ0 = π/2, π, and 5π/4, which give ψl = –π, –π/2, and –π/4. Therefore, by controlling the configure factors a and ϕ0, arbitrary 3D linear polarization can be created.

The full widths at half-maximum (FWHMs) of the focal spot along x, y, and z axes rely on the modulator factor a, and are equal respectively to 0.56λ and 0.56λ, and 0.3λ at a = 1. When a decreases, The FWHM along x axis will slightly increases, while the FWHMs along y and z axes almost keep the same. It should be noted that the spot size can be significantly reduced by adding an annular aperture [4

4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef] [PubMed]

]. Figure 2(h) shows the projection of intensity and polarization distributions on three orthogonal planes for a = −1 and ϕ0 = 5π/4, where the polarization vectors are nearly uniformly directed at an angle of −45° to x axis and 90° to z axis. Within the FWHM region, the average intensity ratio between the field component along the dominating polarization direction and the total light field is 95.24%. The ratio is always more than 91% for an arbitrary a. It is also found that the peak intensity of the focal field always occurs at geometrical focus. As demonstrated in Fig. 3
Fig. 3 The dependence of intensity at geometrical focus on the modulator factor a. The intensity at the case of a = 1 is set to be 1. In calculation, NA is 0.95.
, the intensity at geometrical focus only depends on the modulator factor a, and takes its maximal value at a = 1. When a decreases, the intensity decreases gradually; and reaches its minimum at a = −0.07 instead of a = 0 because of the vectorial interference between the modulated and unmodulated parts of the input light field. For the same reason, the intensity at geometrical focus for a positive modulator factor |a| is always larger than that for a negative modulator factor -|a|. This kind of tunable 3D polarization may find important applications in optical imaging [19

19. E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82(20), 4014–4017 (1999). [CrossRef]

] and particle manipulation [9

9. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature 394(6691), 348–350 (1998). [CrossRef]

].

3. Generation of 3D elliptical polarization

To create 3D elliptical polarization, the polarization state of the input field should be tunable in ellipticity, which can be achieved by letting the input beam successively pass through a polarizer and a liquid crystal variable retarder (LCVR) [20

20. J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt. 2(3), 216–222 (2000). [CrossRef]

]. The experimental configuration is shown in Fig. 4
Fig. 4 Illustration of the scheme for generating 3D elliptical polarization. The blue arrows represent the SAM vectors of local light fields before and after objectives, while the light blue arrow represents the resultant SAM vector at focus.
. The polarizer transmits the electric field along x axis; the LCVR with its fast axis being 45° to x axis leads to a phase retarder δ. The field from the LCVR is cos(δ/2)ex + isin((δ/2)ey [21

21. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 6th edition, 1983).

]. When δ = ± π/2, it is right (left) handed circular polarizations e ± = 2-1/2(ex ± iey). Now, we consider a LCVR, which is divided into two regions (0<ϕ<π and π<ϕ< 2π) having different phase retarders δ1 and δ2, where δ1 = ± π/2 and -π/2≤δ2π/2, respectively. If the polarizer and LCVR have been rotated around z axis by an angle of ϕ0, the final field becomes
ELCVR(ϕ)=A(ϕ)e++B(ϕ)e={eiϕ02(cosδ12+sinδ12)e++eiϕ02(cosδ12sinδ12)eϕ0ϕ<ϕ0+πeiϕ02(cosδ22+sinδ22)e++eiϕ02(cosδ22sinδ22)eϕ0+πϕ<ϕ0+2π.
(5)
In the laboratory coordinate system, the input fields at the left and right pupil planes are set to be l0(θ)[AL(ϕ)e+ + BL(ϕ)e+] and l0(θ)[AR(ϕ)e+ + BR(ϕ)e+], respectively, where AL(ϕ) = AR(ϕ + π) = A(ϕ), BL(ϕ) = BR(ϕ + π) = B(ϕ). Following [22

22. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef] [PubMed]

], the three components of the fields after left and right objectives can be respectively expressed as
[EL/Rx(r,φ,z)EL/Ry(r,φ,z)EL/Rz(r,φ,z)]=iC2π0α02πX(θ)l0(θ)sinθexp{ik[zcosθ+rsinθcos(ϕφ)]+iϕ}×[(cosθcosϕisinϕ)AL/R(ϕ)+(cosθcosϕ+isinϕ)ei2ϕBL/R(ϕ)(cosθsinϕ+icosϕ)AL/R(ϕ)+(cosθsinϕicosϕ)ei2ϕBL/R(ϕ)sinθ[AL/R(ϕ)+ei2ϕBL/R(ϕ)]]dϕdθ.
(6)
The resultant electric field near the focus can be obtained by summing these two fields. Since the phases of field components are not identical, the resultant focal field will be 3D elliptically polarized.

For a paraxial light field with elliptical polarization, the vector normal to the plane of polarization ellipse is limited almost along z-axis. In contrary, the normal vector can take any 3D direction for a nonparaxial field with 3D elliptical polarization. The 3D orientation of elliptical polarization can be characterized by the local SAM vector, which is in the form of
s=ε0/wIm[(Ey*Ez)ex+(Ez*Ex)ey+(Ex*Ey)ez].
(7)
The local SAM vector is normal to the plane of polarization ellipse; and its length, after normalized to the local intensity, is equal to the “third” Stokes parameter, which characterizes the degree of circular polarization [21

21. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 6th edition, 1983).

]. The average spin part of angular momentum per photon is ћσ with
σ=ΩsdV/ΩE*·EdV,
(8)
where the integral region Ω is the whole central spot. Equation (8) is equal to the angular spectrum form given in [16

16. C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80(6), 063814 (2009). [CrossRef]

]. |σ| = 1 associates with circular polarization, while |σ| = 0 means linear polarization. As we will show below, the normalized local SAM vector near the geometrical focus and the average SAM vector of the central spot almost have same orientation but different length.

The orientation of local 3D SAM vector of focal field can be analyzed qualitatively by the superposition of SAM vectors of plane waves from objectives, which is similar to that of 3D linear polarization. The resultant SAM vector at focus can take any 3D direction by controlling the configure factors δ1, δ2, and ϕ0. For example, a transverse SAM vector is created at focus with δ1 = -δ2 = π/2 and ϕ0 = -π/2, as shown in Fig. 4. A light field carrying 3D SAM has 3D elliptical polarization. Figures 5(a)
Fig. 5 (a~c) Projection of intensity and polarization distributions on three orthogonal planes within the central spot with configure factors (a) (δ1,δ2,ϕ0) = (π/2,π/2,/2), (b) (π/2,-π/5,0), and (c) (π/2,-π/2,π/4), respectively. (d) The dependences of χσ (blue) and χs (red) on δ2 for δ1 = ± π/2. (e) The dependences of |σ| (blue) and |s| (red) on δ2 for δ1 = ± π/2. In (d) and (e), the dotted and solid lines correspond to δ1 = ± π/2, respectively. In calculation, NA is chosen to be 0.95.
-5(c) are the intensity and polarizationdistributions within the central spot, which are projected on three orthogonal planes. When the configure factors (δ1,δ2,ϕ0) are equal to (π/2,π/2,-π/2), (π/2,-π/5,0), and (π/2,-π/2,π/4), the fields at geometrical focus are circularly polarized with normalized local SAM vector (χs,ψs) = (0,0), elliptically polarized with (χs,ψs) = (π/4,-π/2), and elliptically polarized with (χs,ψs) = (π/2,-π/4), respectively. One sees from Figs. 5(a)-5(c) that, the focal fields within the central spot have very similar intensity distributions. The FWHMs of the central spot are 0.64λ and 0.3λ along transverse and longitudinal directions, respectively, although with different configure factors. That means the spot size is always as small as that generated by a circular polarized input beam. In the focal plane, the light field is almost uniformly polarized. However, the polarization will deviate slowly from that at the geometrical focus when the observed point moves away from the focal plane. But the average SAM vectors within the whole central spot almost keep the same direction as that of local one at focus. The difference between polar angle of the average SAM vector in central spot and that of local SAM vector at focus (χσ and χs) is small, which is evident in Fig. 5(d), where the dependences of χσ and χs on δ2 are plotted for δ1 = ± π/2. The azimuthal angles of the average and local SAM vectors (ψσ and ψs) are also similar, and ψs=ϕ0π/2 for δ1 = ± π/2. The dependences of |s| and |σ| on δ2 for δ1 = ± π/2 are shown in Fig. 5(e), from which one sees that, |s(δ1,δ2)| = |s(-δ1,-δ2)|, |σ(δ1,δ2)| = |σ(-δ1,-δ2)|, and |σ| is always smaller than |s| because of the polarization deviation. The average SAM |σ| will be much closer to |s| within FWHM region, where the degree of polarization deviation is much smaller. It is well known that, under tightly focusing condition, the longitude component of angular momentum conserves, and SAM will convert to orbital angular momentum [22

22. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef] [PubMed]

]. Therefore, the longitude component of average SAM of focal field is smaller than that of input field. But the conservation law of the transverse component of angular momentum does not hold. For example, for the case of δ1 = -δ2 = ± π/2, an average zero angular momentum of input field can produce a nonzero transverse SAM in the focal field.

As demonstrated by Allion and associates, a beam with transverse angular momentum will result in a lateral shift of the barycenter of intensity [23

23. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009). [CrossRef] [PubMed]

]. However, this shift is not observed in focal field of 4Pi focusing system, even though the field carries transverse SAM. This is because the shifts from left and right objectives are canceled with each other. If only one objective is involved, the barycenter shift is obvious, and can be used to study the geometric spin Hall effect of light. Very recently, a 3D rotational motion of dandelion-shaped microparticle is achieved through a combination of holographic ring trap and conventional optical tweezer, where the horizontal rotation is induced by helicity in the ring trap, while the vertical rotation is caused by the radiation pressure of optical tweezer [24

24. H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett. 102(7), 071103 (2013). [CrossRef]

]. In a different mechanism, the 3D SAM could also cause a 3D rotational motion of both micro- and nano-particles, which has potential applications in microscopic motor and pump [25

25. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001). [CrossRef]

].

4. Conclusion

We have presented an approach for 4Pi microscopy to generate 3D polarization by using APMs or the combination of a polarizer and a LCVR. Both APM and LCVR are divided into two regions with different configure factors. When two counter-propagating radially polarized beams are inputted and modulated by APMs, a tunable 3D linear polarization state is generated in the focal field. When the input fields pass through polarizer and LCVR successively and then focused by two opposing high-NA objectives, the focal field has 3D elliptical polarization with 3D SAM. The normalized local 3D SAM vector has been used to characterize the orientation and helicity of local 3D elliptical polarization. The average SAM vector of focal field has been calculated within the whole central spot, and is shown to have the same orientation as that of local SAM vector at the geometrical focus.

Appendix: the local SAM vector

Following [26

26. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011). [CrossRef]

], the spin part of angular momentum vector has the form
s=ε04wr×[×Im(E*×E)]=ε02wIm[2Ey*Ezy(yEy*Ez)z(zEy*Ez)+yx(Ez*Ex)+zx(Ex*Ey)2Ez*Exx(xEz*Ez)z(zEz*Ex)+zy(Ex*Ey)+xy(Ey*Ez)2Ex*Eyx(xEx*Ey)y(yEx*Ey)+xz(Ey*Ez)+yz(Ez*Ex)].
(9)
Obviously, all the derivative terms have no contribution to the average angular momentum. It has been proved in the paraxial region that, the derivative terms do not give rise to any mechanical effect [27

27. R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. 52, 1045–1052 (2005). [CrossRef]

]. Therefore, we safely preserve only the first term in each components of SAM vector. As will be shown below, this reduced SAM vector is normal to the plane of polarization ellipse; and its length, after normalized to the local intensity, is equal to the “third” Stokes parameter that characterize the degree of circular polarization.

An arbitrary nonparaxial electric field can be written as
E=Exex+Eyey+Ezez=aeiδxex+beiδyey+ceiδzez.
(10)
For convenience, we set δx = 0, Δδ1 = δz-δy, Δδ2 = δz-δx, and Δδ3 = δy-δx, and rewrite Eq. (10) as
E=A[aAex+bcosΔδ3Aey+ccosΔδ2Aez]+iB[bsinΔδ3Bey+csinΔδ2Bez]=Ae1+iBe2,
(11)
where A=a2+b2cos2Δδ3+c2cos2Δδ2, B=b2sin2Δδ3+c2sin2Δδ2, e1 and e2 are two vectors. A new unit vector e3 orthogonal to e2 is found to be
e3=aB2ex+bcsinΔδ1(csinΔδ2ey+bsinΔδ3ez)Ba2B2+b2c2sin2Δδ1,
(12)
which locates at the plane determined by e1 and e2. Now, the electric field can be written in the basis of e2 and e3:
E=12B[(b2sin2Δδ3+c2sin2Δδ2+i2B)e22a2B2+(bcsinΔδ1)2e3]=E2e2+E3e3.
(13)
According to [21

21. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 6th edition, 1983).

], the “third” Stokes parameter is
s3=2Im(E2*E3)|E2|2+|E3|2=2(absinΔδ3)2+(acsinΔδ2)2+(bcsinΔδ1)2a2+b2+c2=2[Im(Ex*Ey)]2+[Im(Ey*Ez)]2+[Im(Ez*Ex)]2|Ex|2+|Ey|2+|Ez|2.
(14)
It can be easily verified that s=(e2×e3)s3[|Ex|2+|Ey|2+|Ez|2]ε0/2w. Therefore, after normalized to the intensity, |s| is equal to s3. And s, e2, and e3 form a right-handed system.

Acknowledgments

The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC) (grant 90921009).

References and links

1.

K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7(2), 77–87 (2000). [CrossRef] [PubMed]

2.

N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett. 29(17), 1968–1970 (2004). [CrossRef] [PubMed]

3.

S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A 9(12), 2159–2166 (1992). [CrossRef]

4.

Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37(8), 1286–1288 (2012). [CrossRef] [PubMed]

5.

H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics 2(8), 501–505 (2008). [CrossRef]

6.

J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt. 14(5), 055004 (2012). [CrossRef]

7.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85(21), 4482–4485 (2000). [CrossRef] [PubMed]

8.

X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun 3, 998 (2012). [CrossRef] [PubMed]

9.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature 394(6691), 348–350 (1998). [CrossRef]

10.

A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys. 12(8), 083053 (2010). [CrossRef]

11.

W. Zhu and W. She, “Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal,” Opt. Express 20(23), 25876–25883 (2012). [CrossRef] [PubMed]

12.

M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A 84(3), 033838 (2011). [CrossRef]

13.

A. F. Abouraddy and K. C. Toussaint Jr., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett. 96(15), 153901 (2006). [CrossRef] [PubMed]

14.

W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt. 12(4), 045707 (2010). [CrossRef]

15.

W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun. 284(1), 52–56 (2011). [CrossRef]

16.

C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A 80(6), 063814 (2009). [CrossRef]

17.

B. Richards and E. Wolf, “Electomagnetic diffraction in optical systems: Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Series A 253(1274), 358–379 (1959). [CrossRef]

18.

S. N. Khonina and I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett. 36(3), 352–354 (2011). [CrossRef] [PubMed]

19.

E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett. 82(20), 4014–4017 (1999). [CrossRef]

20.

J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt. 2(3), 216–222 (2000). [CrossRef]

21.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 6th edition, 1983).

22.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett. 99(7), 073901 (2007). [CrossRef] [PubMed]

23.

A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett. 103(10), 100401 (2009). [CrossRef] [PubMed]

24.

H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett. 102(7), 071103 (2013). [CrossRef]

25.

P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett. 78(2), 249–251 (2001). [CrossRef]

26.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011). [CrossRef]

27.

R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt. 52, 1045–1052 (2005). [CrossRef]

OCIS Codes
(050.1970) Diffraction and gratings : Diffractive optics
(180.6900) Microscopy : Three-dimensional microscopy
(260.5430) Physical optics : Polarization

ToC Category:
Physical Optics

History
Original Manuscript: June 3, 2013
Revised Manuscript: July 1, 2013
Manuscript Accepted: July 2, 2013
Published: July 11, 2013

Virtual Issues
Vol. 8, Iss. 8 Virtual Journal for Biomedical Optics

Citation
Wenguo Zhu and Weilong She, "Generation of tunable three-dimensional polarization in 4Pi focusing system," Opt. Express 21, 17265-17274 (2013)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-14-17265


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References

  1. K. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000). [CrossRef] [PubMed]
  2. N. Bokor and N. Davidson, “Toward a spherical spot distribution with 4π focusing of radially polarized light,” Opt. Lett.29(17), 1968–1970 (2004). [CrossRef] [PubMed]
  3. S. Hell and E. H. K. Stelzer, “Properties of a 4Pi confocal fluorescence microscope,” J. Opt. Soc. Am. A9(12), 2159–2166 (1992). [CrossRef]
  4. Z. Chen and D. Zhao, “4Pi focusing of spatially modulated radially polarized vortex beams,” Opt. Lett.37(8), 1286–1288 (2012). [CrossRef] [PubMed]
  5. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008). [CrossRef]
  6. J. Wang, W. Chen, and Q. Zhan, “Creation of uniform three-dimensional optical chain through tight focusing of space-variant polarized beams,” J. Opt.14(5), 055004 (2012). [CrossRef]
  7. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett.85(21), 4482–4485 (2000). [CrossRef] [PubMed]
  8. X. Li, T.-H. Lan, C.-H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat Commun3, 998 (2012). [CrossRef] [PubMed]
  9. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser trapped microscopic particles,” Nature394(6691), 348–350 (1998). [CrossRef]
  10. A. Picón, A. Benseny, J. Mompart, J. R. Vázquez de Aldana, L. Plaja, G. F. Calvo, and L. Roso, “Transferring orbital and spin angular momenta of light to atoms,” New J. Phys.12(8), 083053 (2010). [CrossRef]
  11. W. Zhu and W. She, “Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal,” Opt. Express20(23), 25876–25883 (2012). [CrossRef] [PubMed]
  12. M. Mansuripur, “Spin and orbital angular momenta of electromagnetic waves in free space,” Phys. Rev. A84(3), 033838 (2011). [CrossRef]
  13. A. F. Abouraddy and K. C. Toussaint., “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96(15), 153901 (2006). [CrossRef] [PubMed]
  14. W. Chen and Q. Zhan, “Diffraction limited focusing with controllable arbitrary three-dimensional polarization,” J. Opt.12(4), 045707 (2010). [CrossRef]
  15. W. Chen and Q. Zhan, “Three dimensional polarization control in 4Pi microscopy,” Opt. Commun.284(1), 52–56 (2011). [CrossRef]
  16. C.-F. Li, “Spin and orbital angular momentum of a class of nonparaxial light beams having a globally defined polarization,” Phys. Rev. A80(6), 063814 (2009). [CrossRef]
  17. B. Richards and E. Wolf, “Electomagnetic diffraction in optical systems: Structure of the image field in an aplanatic system,” Proc. Roy. Soc. London Series A253(1274), 358–379 (1959). [CrossRef]
  18. S. N. Khonina and I. Golub, “Optimization of focusing of linearly polarized light,” Opt. Lett.36(3), 352–354 (2011). [CrossRef] [PubMed]
  19. E. J. Sánchez, L. Novotny, and X. S. Xie, “Near-field fluorescence microscopy based on two-photon excitation with metal tips,” Phys. Rev. Lett.82(20), 4014–4017 (1999). [CrossRef]
  20. J. M. Bueno, “Polarimetry using liquid-crystal variable retarders: theory and calibration,” J. Opt. A, Pure Appl. Opt.2(3), 216–222 (2000). [CrossRef]
  21. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 6th edition, 1983).
  22. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, “Spin-to-orbital angular momentum conversion in a strongly focused optical beam,” Phys. Rev. Lett.99(7), 073901 (2007). [CrossRef] [PubMed]
  23. A. Aiello, N. Lindlein, C. Marquardt, and G. Leuchs, “Transverse angular momentum and geometric spin Hall effect of light,” Phys. Rev. Lett.103(10), 100401 (2009). [CrossRef] [PubMed]
  24. H. Shpaisman, D. B. Ruffner, and D. G. Grier, “Light-driven three-dimensional rotational motion of dandelion-shaped microparticles,” Appl. Phys. Lett.102(7), 071103 (2013). [CrossRef]
  25. P. Galajda and P. Ormos, “Complex micromachines produced and driven by light,” Appl. Phys. Lett.78(2), 249–251 (2001). [CrossRef]
  26. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt.13(5), 053001 (2011). [CrossRef]
  27. R. Zambrini and S. M. Barnett, “Local transfer of angular momentum to matter,” J. Mod. Opt.52, 1045–1052 (2005). [CrossRef]

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