Separation of spin angular momentum in space ...

Appl. Phys. B DOI 10.1007/s00340-013-5523-1

Separation of spin angular momentum in space-variant linearly polarized beam Hao Chen • Zhongliang Yu • Jingjing Hao Zhaozhong Chen • Ji Xu • Jianping Ding • Hui-Tian Wang

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Received: 21 November 2012 / Accepted: 17 May 2013 Ó Springer-Verlag Berlin Heidelberg 2013

Abstract We show that the spin angular momentum (SAM) flux in a space-variant linearly polarized beam can be separated in the focal plane. Such a beam carries only orbital angular momentum (OAM) and develops a net SAM flux upon focusing. The radial splitting of the SAM flux density is mediated by the phase vortex (or OAM) and can be controlled by the topological charge of the phase vortex. Optical trapping experiments verify the separation of the SAM flux density. The proposed approach enriches the manipulation of the angular momentum of light fields and inspires more designs of focus engineering, which would benefit optical micromanipulation of microscopic particles.

1 Introduction The angular momentum (AM) associated with an optical field is exploited in various applications ranging from quantum information and micromanipulation to biosciences [1–4]. The AM can be classified into two categories: spin angular momentum (SAM) and orbital angular momentum (OAM). The former is associated with the polarization in that a circularly polarized light field possesses an SAM value of r per photon where r = ±1, which correspond to left- and righthanded circular polarization, respectively. OAM, in contrast, arises from the azimuthal phase gradient in the optical vortex

Electronic supplementary material The online version of this article (doi:10.1007/s00340-013-5523-1) contains supplementary material, which is available to authorized users. H. Chen  Z. Yu  J. Hao  Z. Chen  J. Xu  J. Ding (&)  H.-T. Wang National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China e-mail: [email protected]

field with a helical phase structure [5–7] in which each photon possesses an OAM value of ‘h, where ‘ is the topological charge indicating the number of 2p cycles in the phase around the circumference. A linearly polarized light beam carries no SAM component, since the circular polarization components are coupled within the beam itself and cancel each other. While the two SAM components in a linearly polarized beam will propagate collinearly in free space, they may split in inhomogeneous media when the beam is refracted, reflected, or scattered via the spin–orbit interaction in which the spatial trajectory of the beam is dependent on the SAM [8–11, 13]. In this paper, we investigate the separation of the SAM flux density of a linearly polarized beam in a focusing system. We propose and demonstrate that a space-variant linearly polarized vector beam, carrying no SAM, can develop both positive and negative SAM fluxes upon focusing. Radial separation of the left- and right-handed circular polarization is used as example for numerical simulations and an optical experiment to show that a net SAM flux density can arise while the total AM is conserved. Our approach demonstrates a desirable tailoring of the AM (i.e., the spatial separation of the AM) and may open new perspectives in shaping of light fields. 2 Principle We can express a linearly polarized beam as a superposition of right-circularly polarized (RCP) and left-circularly ^¼ polarized (LCP) components expressed by R pffiffiffi pffiffiffi ^ ð^x  i^y= 2Þ and L ¼ ð^x þ i^y= 2Þ, respectively. The field of such a beam propagating along ^z can be described by Eðx; yÞ ¼ u0 ðx; yÞei/0 ðx;yÞ ðcos aðx; yÞ^x þ sin aðx; yÞ^yÞ;

ð1Þ

where u0(x,y) and /0(x,y) represent the amplitude and phase profiles of beam, respectively, and a denotes the

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^. This defines a vector polarization angle with respect to x beam as the polarization direction is space variant. Intuitively, any vector beam with a space-variant polarization state described by Eq. (1) would not carry SAM flux in its cross section because the beam is locally linearly polarized. However, the following analysis suggests that this is not true in general. Using the pffiffiffi ^ þ LÞ= ^ ^ transformation relations x ¼ ðR 2 and pffiffiffi ^ ^ ^ y ¼ iðR  LÞ= 2, we can reexpress the field as u0 ðx; yÞei/0 ðx;yÞ iaðx;yÞ ^ ^ pffiffiffi RÞ þ eiaðx;yÞ LÞÞ ðe 2 u0 ðx; yÞ ^ þ eib2 ðx;yÞ LÞÞ; ^ ¼ pffiffiffi ðeib1 ðx;yÞ RÞ 2

Eðx; yÞ ¼

ð2Þ

where b1(x, y) = /0(x, y) ? a(x, y) and b2(x, y) = /0(x, y) - a(x, y). Equation (2) indicates that the locally linear polarization can be decomposed into a sum of RCP and LCP beams each accompanied by their own phase profiles. As is well known, the phase profile of the light beam affects the propagation behavior of beam. Since the RCP and LCP components have different phase profiles, they may evolve into different spatial distributions as the beam propagates. As a result, once the RCP and LCP components become spatially separated during the propagation process, a net SAM flux density can develop within the cross section of the beam. Focusing the vector beam can cause the RCP and LCP components to be spatially spilt, which presents an interesting situation. The radial separation of the positive and negative SAM flux densities in the focal region is a feature that could be exploited for optical micromanipulation. Let us consider a vector vortex (VV) beam that has an azimuthal-variant polarization state and a spiral phase profile. We set the parameters in Eq. (1) as /0 ðx; yÞ ¼ nu and aðx; yÞ ¼ mu, with u denoting the azimuth angle in polar coordinates. The VV beam VVnm can be described by VVnm ¼ einu ðcos mux þ sin muyÞ;

ð3Þ

where m and n denote the topological charges of the polarization vortex and helical phase of the VV beam, respectively. We have assumed a uniform beam amplitude profile for simplicity. The OAM of this VV beam is nh per photon (associated with the azimuthal phase factor), but there is no SAM due to the locally linear polarization [14, 15]. The optical field of the VV beam can also be reexpressed in a circular basis as VVnm

1 ^ þ eiðnmÞu LÞ; ^ ¼ pffiffiffi ðeiðnþmÞu R 2

ð4Þ

which is a supposition of the RCP vortex of topological charge n ? m and the LCP vortex of topological charge n - m.

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The focal field of monochromatic light passing through an aplanatic lens can be analyzed in the Debye approximation and is often calculated using the vectorial diffraction integral proposed by Richards and Wolf [16]. Through the established correspondence between the incident beam and optical field formed after focusing, the Cartesian components of the electric field of the focused beam can be readily calculated, and the polarization distribution and AM flux distribution can be obtained analytically. Using the vectorial diffraction integral, the Cartesian components of the electric field of the focused VV beam VVnm after passing through an objective lens with a focal length of f can be expressed by [17–19] 0 1 Ex ðr; u; zÞ B C Eðr; u; zÞ ¼ @ Ey ðr; u; zÞ A Ez ðr; u; zÞ Za pffiffiffiffiffiffiffiffiffiffi ikf ¼ dhl0 ðhÞ sin h cos heikz cos h 2p 0 0a 1 b ðI 2 nþm2 þ Inmþ2 Þ þ 2 ðInþm þ Inm Þ B C  @ 2ia ðInmþ2  Inþm2 Þ þ 2ib ðInþm  Inm Þ A; sin hðInþm1 þ Inmþ1 Þ ð5Þ where a ¼ ðcos h  1Þ=2; b ¼ ðcos h þ 1Þ=2; a ¼ sin1 ðNA=n0 Þ denotes the convergence semiangle determined by the objective lens, and Ip ¼ ip Jp expðipuÞ, where Jp(x) is a Bessel function of the first kind of order p. In calculating Eq. (5), a number aperture of NA = 1.3 and index of refraction of 1.518 between the objective and sample are assumed. To evaluate the local AM density of the beam in the direction of propagation, we consider the AM flux by invoking a general description valid for nonparaxial light in situations such as strong focusing [20, 21]. We eliminate the magnetic field using the Maxwell equation i xB = r 9 E and obtain the z-component of the AM flux density in units of h with cycle-averaged angular momentum and energy components. The total AM flux density jz is given as the sum of the orbital and spin AM flux densities, denoted by lz and sz, respectively, as [22] X Ej ðr  rÞEj E E j¼x;y;z RR RR j z ¼ l z þ sz ¼ þ : ð6Þ i  E Edxdy i  E Edxdy The denominator in the above expression is associated with the energy flux density of the light beam. On substituting the electric field distribution of the light beam into Eq. (6), the OAM and SAM quanta  h of the beam are easily calculated. Such a calculation reveals that the AM flux density of the incident beam VVnm expressed in

Separation of spin angular momentum Fig. 1 Simulated focused fields of the incident beams VV30 -30 and VV-30 -30 (left and right columns). First row intensity distribution in the focal plane; Second row OAM flux density; Third row SAM flux density

Eq. (4) consists of two components: the first has an OAM of ðn þ mÞ h and SAM of þh per photon and the second has an OAM of ðn  mÞ h and SAM of  h per photon. The total AM integrated over the beam cross section yields a net OAM of n h per photon, while the positive and negative SAM fluxes sum to zero. However, the AM distribution in the focused field is somewhat different. Figure 1 shows simulated focused fields of the incident beams VV30 -30 and -30 VV-30 (in the left and right columns, respectively). The intensity distribution, OAM flux density, and SAM flux density are all shown. The intensities distributions show a bright spot in the center accompanied by a relatively faint outer ring. However, in the SAM flux density distributions, the densities in the center and outer ring-shaped region have opposite signs (i.e., the positive and negative SAM are separated). The helical phase (the OAM) of the indent beam mediates this separation of the SAM flux in the focal region (cf. Eq. (4)). It should be noted that in the OAM flux density distributions, the density in the center of the focal plane results from the spin-to-orbital angular momentum conversion [23, 24]. Moreover, our calculations confirm

that the total AM is conserved upon focusing, with a small amount of SAM converted to OAM [25]. In other words, the total OAM increases and the total SAM decreases upon focusing. We can verify the radial splitting of the SAM flux density in the focal plane by observing the motion of particles trapped in the focused field. This involves creating a VV beam VVnm whose polarization state and phase are independently tailored. We previously proposed a method to create such a vector beam [26], and following this scheme, we can produce a VV beam with a polarization vortex and helical phase, as shown in Fig. 2 (m = 2 and n = 1). The cross-sectional intensity is shown in Fig. 2a, and the Stokes parameters, S1, S2, and S3, of the beam are shown in Fig. 2b–d respectively, which indicate that the beam has a locally linear polarization with a vector vortex of topological charge 2. The azimuthal phase variation of the beam can be inferred from the forked features in the interference patterns shown in Fig. 2e, f. This low-order topological charge VV beam is presented as an example that can be easily visualized, although more complicated VV beams can be generated using our setup.

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H. Chen et al. Fig. 2 The created VV beam VV21. a Cross-sectional intensity, and the Stokes parameters a S1, b S2, and c S3 of the beam. The interference patterns between LCP and RCP light with the VV beam, shown in e and f, respectively, imply the desired topological charges of the polarization vortex and helical phase, i.e., m = 2 and n = 1 in this case

(a)

(b)

(e)

(c)

(d)

(f)

3 Optical experiments To experimentally observe the radial separation of the SAM flux density during focusing of the VV beam, an anisotropic particle is trapped in the focal region as a probe to detect the OAM and SAM transfer from the electromagnetic field to the particle [12, 24]. Our optical trapping system includes a 100 9 oil-immersed objective (NA = 1.3) and a 532-nm wavelength laser. The VV beam generator and focusing system deliver an effective power of 11 mW to the focal volume. The anisotropic particles are liquid-crystal (LC) particles (4-(trans-4-pentylcyclohexyl) benzonitrile) of various diameters between 2.0 and 3.0 lm. The LC particles are dispersed in a layer of sodium dodecyl sulfate solution sealed between a glass coverslip and a microscope slide prior to trapping. Because most of the LC particles are spherical, the particles with a somewhat nonspherical overall shape must be carefully chosen so that the spinning motion can be discerned. We employ VVnm beams with j m j¼j n j to create a trapping scenario in which a bright ring surrounds a bright spot (cf. Fig. 1). The radius of the bright ring, R0, is proportional to j n þ m þ U=2p j (ascribed to the first term on the right-hand side in Eq. (4)) or to j n  m  U=2p j (ascribed to the second term in Eq. (4)), where U is the geometrical phase of the focused beam ð¼ 2pð1  cos aÞÞ [25]. The radius of the bright ring should be large enough to ensure a sufficient spatial distance between the particles trapped at the center and those in the ring to reduce any possible microrheological interplay. We thus choose j m j¼j n j¼ 30. Figure 3 presents the trapping results for the four VV pffiffiffi ^ þ expðþi60uÞL; ^ VV30 ¼ beams VV30 ¼ ð1= 2Þ½R 30 30 p ffiffi ffi pffiffiffi ^ ^ þ L; ^ VV30 ¼ ð1= 2Þ½expði60uÞR ð1= 2Þ½expði60uÞR 30 pffiffiffi 30 ^ ^ ^ and VV þL, 30 ¼ ð1= 2Þ½R þ expði60uÞL (see Media 1–4 for video viewing). We see two LC particles, one optically trapped at the center and the other in the ring that

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30 V -30

m

0.0 s

6.1 s

12.1 s

16.0 s

-30 V -30

m

0.0 s

4.1 s

10.1 s

19.9 s

30 V 30

m

0.0 s

9.2 s

15.1 s

27.0 s

-30 V 30

m

0.0 s

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13.1 s

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Fig. 3 Intensity distribution of the focused field and successive frames of a video that show the rotation of LC particles trapped by the -30 30 -30 tightly focused beams VV30 -30, VV-30, VV30, and VV30 (see media 1, 2, 3, and 4 for the videos). Two particles, 2.5 and 2.4 lm in diameter, are trapped in the center and outer ring, respectively

rotate in opposite directions, and the particle trapped in the ring also orbits the central particle. This is an explicit demonstration of the radial splitting of the SAM flux density. We also note that the orbital angular speed of the outer particle depends on the rotation of the central particle; the angular speed is higher when the particle moves around the ring in the same direction as the rotation of the central particle. This is mainly a result of the drag in the fluid induced by the rotation of the central particle. The instability in the particle speed is due to experimental factors such as the low light power density attainable in the ring region; the azimuthal optical force is too weak to enable stable and smooth orbital motion. Other possible factors affecting the orbital motion of the outer particle include the intrusion of smaller LC particles and the

Separation of spin angular momentum

eccentric spinning of the central particle. However, the experimental results demonstrate convincingly that the positive and negative SAM fluxes can indeed become separated upon focusing. In addition, the results for VV30 30 and VV30 -30 (Media 3 and 4) indicate that it is not the fluid motion but the OAM that causes the outer particle to revolve around the center.

4 Conclusion In summary, we have proposed that a space-variant, locally linearly polarized vector beam that has only OAM can develop a net flux of positive and negative SAM upon focusing. We provided numerical simulations and optical trapping experimental results to demonstrate the radial splitting of the SAM flux density in the focal region. It is also important to note that other spatial separations of the SAM flux density are possible if proper phase factors are imparted to the RCP and LCP components of the beam. For example, a vector beam defined by b1(x,y) = qx and b2(x,y) = - qx in Eq. (2) has a locally linear polarization that rotates steadily along the x-direction. Such a vector beam can be focused into two spatially separated optical traps with opposite circular polarizations, thereby demonstrating SAM separation in the x-direction. In conclusion, the SAM flux density can be tailored by focusing a vector beam, which is a process that will enrich the optical manipulation of microscopic objects. Acknowledgments This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11074116, 10934003, and 11274158), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20090091110012), and the National Basic Research Program (973 Program) of China (Grant No. 2012CB921900).

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