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Electrically tunable generation of inhomogeneously polarized light beam
Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Electrically tunable generation of inhomogeneously polarized light beam Wenguo Zhu n, Weilong She State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 May 2013 Received in revised form 15 August 2013 Accepted 17 August 2013
We propose a method for generating inhomogeneously polarized light beams using two conjugated electro-optic spiral phase plates and a quarter-wave-plate (QWP). By changing the incident homogeneous polarization state, we can obtain all the states on a higher-order Poincaré sphere (PS). The order m of PS can be an integer or a non-integer, and is electrically tunable. Without the QWP, hybridly polarized vector beams with spatially variant spin angular momentum can be created. & 2013 Published by Elsevier B.V.
Keywords: Electrically tunable Spatially variant spin angular momentum Inhomogeneous polarization
1. Introduction Inhomogeneously polarized beams exhibit spatially variant states of polarization [1]. In recent years, these beams have attracted rapidly growing attention owing to their unique features compared with the homogeneously polarized ones [2,3]. Among all kind of inhomogeneously polarized beams, the radially and azimuthally polarized beams are best known. Under tightly focusing condition, the radially polarized beam has the ability to produce a strong longitudinal electric field component at the focus plane, and the spot associated with this component is sharper than that by a scalar beam [4]. On the other hand, the azimuthally polarized beam can create a hollow dark spot [5]. There is a new kind of inhomogeneously polarized beam called hybridly polarized vector beam [6], which consists of spatially separated linear, circular, and elliptical polarizations, and thus possesses spatially variant spin angular momentum (SAM). Such kind of beam can be focused to generate an interesting polarization distribution with 3D orientation [7]. The peculiar focusing properties of the inhomogeneously polarized beams are useful in many applications such as particle acceleration [8], optical trapping [9], high resolution microscopy [10], and nonlinear optics [11]. A number of methods for producing inhomogeneously polarized beams have been proposed, which can be divided into two kinds: active and passive. For the active method, the inhomogeneously polarized beams are generated directly from the output of novel lasers with
n
Corresponding author. Tel.: þ 86 020 841 12863. E-mail addresses: [email protected] (W. Zhu), [email protected] (W. She).
specially designed or modified laser resonators [12]. However, this kind of method may lead to additional intra-cavity loss and optimization difficulty of laser configuration. The passive method converts traditional laser beams into inhomogeneously polarized ones outside the cavity by using birefringent elements [13,14], segmented wave plates [15], or interferometric techniques [16], among which the interferometric techniques are of great interest owing to their flexibility and versatility [16–18]. In these approaches, two interfering beams with orthogonal polarizations and different intensity profiles construct an inhomogeneously polarized beam [17]. The main limitations of these techniques Q5 are the interferometric instability and the complication of optical path. To get rid of these limitations, we propose a simple method in this letter, where the two interfering beams are actually the two independent polarization components of a single beam. Therefore, a compact optical path and stable outputs can be achieved. In addition, our method is electrically tunable. The key element of our method is the device composed of two conjugated electrooptic spiral phase plates (SPPs), proposed by Chen [19]. The input and output interfaces of the device are coated with two transparent electrodes, so that an even electric field can be applied along the longitudinal direction, i.e., the propagation direction of light beam. Let a horizontally polarized vortex beam with a topologic charge l pass through the device, the polarization state keeps, while the topologic charge becomes l þm with m depending on the applied voltage. If the input vortex beam is vertically polarized, the topologic charge of the output beam will be l m. Therefore, for an incident beam with arbitrary polarization, the output beam can be taken as a linear combination of two orthogonally polarized waves with different topologic charges [18], and various inhomogeneously polarized beams can be obtained under a tunable
0030-4018/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.optcom.2013.08.057
Please cite this article as: W. Zhu, W. She, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.08.057i
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applied voltage. Of particular interest is that arbitrary polarization states represented on a higher-order Poincaré sphere (PS) [20] can be produced when the output beam passes through an additional quarter-wave-plate (QWP). And the order m of PS can be an integer or a non-integer.
2. Theory and model The inhomogeneous polarization state generator used consists of two SPPs made of z-cut ZnTe crystal, which has nonvanishing electro-optic coefficients γ41 ¼γ52 ¼ γ63 ¼4.51 pm/V and a refractive index n0 ¼3.06 at λ¼589 nm [21]. A cylindrical device is built up by these two SPPs with their transverse crystalline x–y axes having a relative rotation of 901. In absence of the applied voltage, the device is homogeneously isotropic and will not affect the polarization state of light beam. When the applied voltage is switched on, the device becomes inhomogeneously anisotropic, which will result in a change of the topologic charge of light beam. And the topologic charge becomes lþ m (l–m) for horizontal (vertical) beam component, where l stands for the topologic charge of the incident beam and m ¼ n0 3 γ 63 U=λ with U being the applied voltage. It should be noticed that m can be an integer or a non-integer. We set l¼ 0 in present letter, so that the horizontally and vertically polarized components of the beam emerged from the device have opposite topologic charges. For a scale incident beam with arbitrary polarization Ein ¼ cos ðαÞe^ x þ sin ðαÞexpðiδÞe^ y , where α and δ are two constants, the emerging field can be described by [19] E1 ¼ cos ðαÞexp½imðφπÞe^ x þ sin ðαÞexp½imðφπÞ þ iδe^ y ;
ð1Þ
where φ is the azimuthal angle. Let the light pass through a QWP with fast axis being 451 relative to the horizontal direction, it becomes E2 ¼ cos ðαÞexp½imðφπÞe^ þ sin ðαÞexp½imðφπÞ þ iδi π=2Þe^ þ ; ð2Þ pffiffiffi where e^ 7 ¼ ðe^ x 7 ie^ y Þ= 2. According to Eq. (2), the output beam is the linear combination of right and left circularly polarized (RCP and LCP) components with opposite signs of topologic charge, and can be conveniently described by using a higher-order PS and Stokes parameter representation for an integer m. The higher-order PS is the generalization of the standard one by extending the polarization basis in terms of SAM to the total angular momentum, which includes higher dimensional orbital angular momentum [20]. On the PS, the Cartesian coordinates are represented by higher-order Stokes parameters S1m, S2m, and S3m. Since (S1m)2 þ(S2m)2 þ(S3m)2 ¼ 1 for all polarized beams, any polarization state described by a higherorder PS can be represented by a point on the surface of a unit sphere Σ. When the polarization basis of higher-order PS is chosen to be {Lm, Rm}, where Lm ¼ exp(imφ)ê and Rm ¼exp( imφ)ê þ , the north and south poles on Σ correspond to ( m)-order RCP and m-order LCP optical vortexes, respectively. And the points in equator represent locally linearly polarized beams. Any other points on Σ stand for locally elliptically polarized beams. Fig. 1 shows several typical polarization states on the 1-order PS. When the topologic charge takes non-integer value, the higher-order PS can be extended to represent fractional polarization vortex beams [22]. Therefore, the output beams with integer and non-integer topologic charges can all be described by the higher-order PS. For this we rewrite Eq. (2) as E2 ¼ cos ðαÞexpðimπÞLm þ sin ðαÞexp½iπðm0:5Þ þiδR m :
ð3Þ
Following Ref. [20], the higher-order Stokes parameters can be calculated, which are of the form Sm 1 ¼ sin ð2αÞ sin ðδ þ 2mπÞ;
ð4aÞ
Sm 2 ¼ sin ð2αÞ cos ðδ þ 2mπÞ;
ð4bÞ
Fig. 1. Higher-order PS representation for m¼ 1: the north (south) pole represents RCP (LCP) optical vortex of 1 (þ 1) order. Equatorial points (S11, S21,0) represent locally linearly polarized beams including radially and azimuthally polarized beams. Intermediate points between the poles and equator represent locally elliptically polarized beams.
Sm 3 ¼ cos ð2αÞ:
ð4cÞ
From Eqs. 4(a), (b), and (c) one finds that, the higher-order Stokes parameters of an output beam depend on the incident polarization state and the topologic charge m. To see more clearly this dependence, we calculate the standard Stokes parameters of incident beam. They are S1i ¼cos (2α), S2i ¼sin (2α)cos δ, and S3i ¼sin (2α)cos δ [21]. It is interesting that the relationship between the higher-order and standard Stokes parameters is quite simple for integer and halfinteger m. For integer m, S1m ¼S3i, S2m ¼S2i, and S3m ¼ S1i, while for half-integer m, S1m ¼ S3i, S2m ¼ S2i, and S3m ¼ S1i. Therefore, we can easily get the incident polarization states needed for the desired output inhomogeneous polarization states. When m¼1, the RCP and LCP incident beam are required to create the radially and azimuthally polarized beams. By changing the incident state of polarization, all the polarization states represented on a PS of arbitrary order can be generated.
3. Application and discussion In the following, we investigate the influence of Pockels effect on the polarization distributions of output beams. To do this, we let a RCP beam of 589 nm pass through a cylindrical device composed of two SPPs and a QWP successively, and set the thickness of the device to be 5 mm. The polarization distributions of the output beams are shown in Fig. 2 for different applied voltages, where U 0 ¼ λ=n0 3 γ 63 ¼ 4:558 kV. One sees from Fig. 2 that, all the local polarizations are linear ones. When U¼mU0 and m is an integer, the local polarization at each point makes an angle mφ with respect to the horizontal axis, as shown in Fig. 2(a). Particularly, when m ¼1, the local polarization at each point is along the direction of radial vector. The angle |mφ| can take a value between 0 and 2π, and every polarization direction appears |m| times in the polarization pattern. When m is a negative integer, the output beam is the so-called π-phase vector beam [23,24], which has a different polarization distribution from that with a positive m (see the first and second column of Fig. 2(a)). It has been shown that focused by an objective with high numerical-aperture, both standard and π-phase vector beams show new features different from the scalar beams [2,23]. By changing the applied voltage, the topologic charge m can take integer or non-integer value. Fig. 2(b) shows the polarization distributions of output beams for different non-integer m, where the local polarization vectors make incomplete rotation around the beam axis. And a line of polarization discontinuity starting from the origin and extending along the radius of φ ¼ 0 emerges in
Please cite this article as: W. Zhu, W. She, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.08.057i
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Fig. 2. The polarization distributions of the output inhomogeneously polarized beams for different applied voltages U. (a) U¼ mU0 with m being positive (first column) and negative (second column) integers. (b) U ¼mU0 with m being positive (first column) and negative (second column) non-integers. The input beam is a RCP one.
Fig. 3. The polarization distributions of hybridly polarized vector beams for different incident polarization states and different m. Upper row: m¼1 (a) diagonally linear polarization, (b) right circular polarization, and (c) left circular polarization. Lower row: diagonally linear polarization (d) m¼ 1, (e) m¼ 0.5 and (f) m¼2.
the polarization patterns. This line has low intensity due to the discontinuity [16,22]. All the output inhomogeneously polarized beams mentioned above have the same local polarization ellipticities e over the cross section of beams. According to the relation SAM p sin ð2arc tan eÞ [21,25], these beams carry a locally identical SAM per photon. The idea to generate light beams with spatially variant SAM is interesting. The hybridly polarized vector beam is one kind
of such beams [6], which can be produced just by removing the QWP. And the output beam is described by Eq. (1), where the horizontal and vertical beam components have an azimuthally dependent phase difference Δδ¼ 2mφ þ2mπ δ, and will form linear, circular, and elliptical polarizations simultaneously in the beam. The polarization distributions of hybridly polarized vector beams for different incident polarization states and different m are plotted in Fig. 3, from which one sees that, the local polarization
Please cite this article as: W. Zhu, W. She, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.08.057i
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ellipse changes gradually with the azimuthal angle φ. By one circle around the beam axis, the local ellipse returns back to its original state when m is an integer. However, it cannot return back to the original state for a non-integer m, and a line of polarization discontinuity will appear in the polarization pattern. As shown by Fig. 3(a)–(c), the local polarization states at φ ¼ 0 correspond to the states of incident polarization. From Fig. 3(a) and (d) one concludes that, the local SAM at each point is opposite for two beams with different parameters (α, δ, m) and (α, δ, m), because two phase differences Δδ and –Δδ are of opposite sign. 4. Conclusion In summary, we have investigated the generation of inhomogeneously polarized light beams using a cylindrical device composed of two conjugated electro-optic SPPs. With an additional QWP after the device, the output polarization states are fully represented on a PS of m order, which can be integer and noninteger, and can be controlled by an applied voltage. All the states represented by an arbitrary order PS can be created by changing the incident homogeneous polarization state. Without the QWP, the output beam is hybridly polarized and carries spatially variant SAM. We believe that this tunable generation of inhomogeneously polarized beams has potential application in optical manipulation. Acknowledgment
References [1] Q. Zhan, Advances in Optics and Photonics 1 (2009) 1. [2] K. Youngworth, T. Brown, Optics Express 7 (2000) 77. [3] H.F. Wang, L.P. Shi, B. Lukyanchuk, C. Sheppard, C.T. Chong, Nature Photonics 2 (2008) 501. [4] R. Dorn, S. Quabis, G. Leuchs, Physical Review Letters 91 (2003) 233901. [5] Q. Zhan, J. Leger, Optics Express 10 (2002) 324. [6] X.L. Wang, Y.N. Li, J. Chen, C.S. Guo, J.P. Ding, H.T. Wang, Optics Express 18 (2010) 10786. [7] G.M. Lerman, L. Stern, U. Levy, Optics Express 18 (2010) 27650. [8] C. Varin, M. Piché, Applied Physics B 74 (2002) S83. [9] Q. Zhan, Optics Express 12 (2004) 3377. [10] A. Abouraddy, K.C. Toussaint, Physical Review Letters 96 (2006) 153901. [11] A. Bouhelier, M. Beversluis, A. Hartschuh, L. Novotny, Physical Review Letters 90 (2003) 013903. [12] H. Kawauchi, Y. Kozawa, and S. Sato, Optics Letters. 33 (2008) 1984. [13] M. Stalder, M. Schadt, Optics Letters 21 (1996) 1948. [14] F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. Lisio, E. Santamato, Applied Optics 51 (2012) C1. [15] G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, Optics Letters 32 (2007) 1468. [16] X.L. Wang, J. Ding, W.J. Ni, C.S. Guo, H.T. Wang, Optics Letters 32 (2007) 3549. [17] S. Liu, P. Li, T. Peng, J. Zhao, Optics Express 20 (2012) 21715. [18] S.C. Tidwell, D.H. Ford, W.D. Kimura, Applied Optics 29 (1990) 2234. [19] L. Chen, W. She, Optics Letters 34 (2009) 178. [20] G. Milione, H.I. Sztul, D.A. Nolan, R.R. Alfano, Physical Review Letters 107 (2011) 053601. [21] A. Yariv, Optical Waves in Crystals, Wiley, New York, 1983. [22] P.H. Jones, M. Rashid, M. Makita, O.M. Maragò, Optics Letters 34 (2009) 2560. [23] B.J. Roxworthy, K.C. Toussaint Jr., New Journal of Physics 12 (2010) 073012. [24] C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, M. Ritsch-Marte, New Journal of Physics 9 (2007) 78. [25] W. Zhu, W. She, Optics Express 2 (2012) 25876.
The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC) (grant 90921009).
Please cite this article as: W. Zhu, W. She, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.08.057i
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Electrically tunable generation of inhomogeneously polarized light beam Wenguo Zhu n, Weilong She State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-sen University, Guangzhou 510275, China
art ic l e i nf o
a b s t r a c t
Article history: Received 21 May 2013 Received in revised form 15 August 2013 Accepted 17 August 2013
We propose a method for generating inhomogeneously polarized light beams using two conjugated electro-optic spiral phase plates and a quarter-wave-plate (QWP). By changing the incident homogeneous polarization state, we can obtain all the states on a higher-order Poincaré sphere (PS). The order m of PS can be an integer or a non-integer, and is electrically tunable. Without the QWP, hybridly polarized vector beams with spatially variant spin angular momentum can be created. & 2013 Published by Elsevier B.V.
Keywords: Electrically tunable Spatially variant spin angular momentum Inhomogeneous polarization
1. Introduction Inhomogeneously polarized beams exhibit spatially variant states of polarization [1]. In recent years, these beams have attracted rapidly growing attention owing to their unique features compared with the homogeneously polarized ones [2,3]. Among all kind of inhomogeneously polarized beams, the radially and azimuthally polarized beams are best known. Under tightly focusing condition, the radially polarized beam has the ability to produce a strong longitudinal electric field component at the focus plane, and the spot associated with this component is sharper than that by a scalar beam [4]. On the other hand, the azimuthally polarized beam can create a hollow dark spot [5]. There is a new kind of inhomogeneously polarized beam called hybridly polarized vector beam [6], which consists of spatially separated linear, circular, and elliptical polarizations, and thus possesses spatially variant spin angular momentum (SAM). Such kind of beam can be focused to generate an interesting polarization distribution with 3D orientation [7]. The peculiar focusing properties of the inhomogeneously polarized beams are useful in many applications such as particle acceleration [8], optical trapping [9], high resolution microscopy [10], and nonlinear optics [11]. A number of methods for producing inhomogeneously polarized beams have been proposed, which can be divided into two kinds: active and passive. For the active method, the inhomogeneously polarized beams are generated directly from the output of novel lasers with
n
Corresponding author. Tel.: þ 86 020 841 12863. E-mail addresses: [email protected] (W. Zhu), [email protected] (W. She).
specially designed or modified laser resonators [12]. However, this kind of method may lead to additional intra-cavity loss and optimization difficulty of laser configuration. The passive method converts traditional laser beams into inhomogeneously polarized ones outside the cavity by using birefringent elements [13,14], segmented wave plates [15], or interferometric techniques [16], among which the interferometric techniques are of great interest owing to their flexibility and versatility [16–18]. In these approaches, two interfering beams with orthogonal polarizations and different intensity profiles construct an inhomogeneously polarized beam [17]. The main limitations of these techniques Q5 are the interferometric instability and the complication of optical path. To get rid of these limitations, we propose a simple method in this letter, where the two interfering beams are actually the two independent polarization components of a single beam. Therefore, a compact optical path and stable outputs can be achieved. In addition, our method is electrically tunable. The key element of our method is the device composed of two conjugated electrooptic spiral phase plates (SPPs), proposed by Chen [19]. The input and output interfaces of the device are coated with two transparent electrodes, so that an even electric field can be applied along the longitudinal direction, i.e., the propagation direction of light beam. Let a horizontally polarized vortex beam with a topologic charge l pass through the device, the polarization state keeps, while the topologic charge becomes l þm with m depending on the applied voltage. If the input vortex beam is vertically polarized, the topologic charge of the output beam will be l m. Therefore, for an incident beam with arbitrary polarization, the output beam can be taken as a linear combination of two orthogonally polarized waves with different topologic charges [18], and various inhomogeneously polarized beams can be obtained under a tunable
0030-4018/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.optcom.2013.08.057
Please cite this article as: W. Zhu, W. She, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.08.057i
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applied voltage. Of particular interest is that arbitrary polarization states represented on a higher-order Poincaré sphere (PS) [20] can be produced when the output beam passes through an additional quarter-wave-plate (QWP). And the order m of PS can be an integer or a non-integer.
2. Theory and model The inhomogeneous polarization state generator used consists of two SPPs made of z-cut ZnTe crystal, which has nonvanishing electro-optic coefficients γ41 ¼γ52 ¼ γ63 ¼4.51 pm/V and a refractive index n0 ¼3.06 at λ¼589 nm [21]. A cylindrical device is built up by these two SPPs with their transverse crystalline x–y axes having a relative rotation of 901. In absence of the applied voltage, the device is homogeneously isotropic and will not affect the polarization state of light beam. When the applied voltage is switched on, the device becomes inhomogeneously anisotropic, which will result in a change of the topologic charge of light beam. And the topologic charge becomes lþ m (l–m) for horizontal (vertical) beam component, where l stands for the topologic charge of the incident beam and m ¼ n0 3 γ 63 U=λ with U being the applied voltage. It should be noticed that m can be an integer or a non-integer. We set l¼ 0 in present letter, so that the horizontally and vertically polarized components of the beam emerged from the device have opposite topologic charges. For a scale incident beam with arbitrary polarization Ein ¼ cos ðαÞe^ x þ sin ðαÞexpðiδÞe^ y , where α and δ are two constants, the emerging field can be described by [19] E1 ¼ cos ðαÞexp½imðφπÞe^ x þ sin ðαÞexp½imðφπÞ þ iδe^ y ;
ð1Þ
where φ is the azimuthal angle. Let the light pass through a QWP with fast axis being 451 relative to the horizontal direction, it becomes E2 ¼ cos ðαÞexp½imðφπÞe^ þ sin ðαÞexp½imðφπÞ þ iδi π=2Þe^ þ ; ð2Þ pffiffiffi where e^ 7 ¼ ðe^ x 7 ie^ y Þ= 2. According to Eq. (2), the output beam is the linear combination of right and left circularly polarized (RCP and LCP) components with opposite signs of topologic charge, and can be conveniently described by using a higher-order PS and Stokes parameter representation for an integer m. The higher-order PS is the generalization of the standard one by extending the polarization basis in terms of SAM to the total angular momentum, which includes higher dimensional orbital angular momentum [20]. On the PS, the Cartesian coordinates are represented by higher-order Stokes parameters S1m, S2m, and S3m. Since (S1m)2 þ(S2m)2 þ(S3m)2 ¼ 1 for all polarized beams, any polarization state described by a higherorder PS can be represented by a point on the surface of a unit sphere Σ. When the polarization basis of higher-order PS is chosen to be {Lm, Rm}, where Lm ¼ exp(imφ)ê and Rm ¼exp( imφ)ê þ , the north and south poles on Σ correspond to ( m)-order RCP and m-order LCP optical vortexes, respectively. And the points in equator represent locally linearly polarized beams. Any other points on Σ stand for locally elliptically polarized beams. Fig. 1 shows several typical polarization states on the 1-order PS. When the topologic charge takes non-integer value, the higher-order PS can be extended to represent fractional polarization vortex beams [22]. Therefore, the output beams with integer and non-integer topologic charges can all be described by the higher-order PS. For this we rewrite Eq. (2) as E2 ¼ cos ðαÞexpðimπÞLm þ sin ðαÞexp½iπðm0:5Þ þiδR m :
ð3Þ
Following Ref. [20], the higher-order Stokes parameters can be calculated, which are of the form Sm 1 ¼ sin ð2αÞ sin ðδ þ 2mπÞ;
ð4aÞ
Sm 2 ¼ sin ð2αÞ cos ðδ þ 2mπÞ;
ð4bÞ
Fig. 1. Higher-order PS representation for m¼ 1: the north (south) pole represents RCP (LCP) optical vortex of 1 (þ 1) order. Equatorial points (S11, S21,0) represent locally linearly polarized beams including radially and azimuthally polarized beams. Intermediate points between the poles and equator represent locally elliptically polarized beams.
Sm 3 ¼ cos ð2αÞ:
ð4cÞ
From Eqs. 4(a), (b), and (c) one finds that, the higher-order Stokes parameters of an output beam depend on the incident polarization state and the topologic charge m. To see more clearly this dependence, we calculate the standard Stokes parameters of incident beam. They are S1i ¼cos (2α), S2i ¼sin (2α)cos δ, and S3i ¼sin (2α)cos δ [21]. It is interesting that the relationship between the higher-order and standard Stokes parameters is quite simple for integer and halfinteger m. For integer m, S1m ¼S3i, S2m ¼S2i, and S3m ¼ S1i, while for half-integer m, S1m ¼ S3i, S2m ¼ S2i, and S3m ¼ S1i. Therefore, we can easily get the incident polarization states needed for the desired output inhomogeneous polarization states. When m¼1, the RCP and LCP incident beam are required to create the radially and azimuthally polarized beams. By changing the incident state of polarization, all the polarization states represented on a PS of arbitrary order can be generated.
3. Application and discussion In the following, we investigate the influence of Pockels effect on the polarization distributions of output beams. To do this, we let a RCP beam of 589 nm pass through a cylindrical device composed of two SPPs and a QWP successively, and set the thickness of the device to be 5 mm. The polarization distributions of the output beams are shown in Fig. 2 for different applied voltages, where U 0 ¼ λ=n0 3 γ 63 ¼ 4:558 kV. One sees from Fig. 2 that, all the local polarizations are linear ones. When U¼mU0 and m is an integer, the local polarization at each point makes an angle mφ with respect to the horizontal axis, as shown in Fig. 2(a). Particularly, when m ¼1, the local polarization at each point is along the direction of radial vector. The angle |mφ| can take a value between 0 and 2π, and every polarization direction appears |m| times in the polarization pattern. When m is a negative integer, the output beam is the so-called π-phase vector beam [23,24], which has a different polarization distribution from that with a positive m (see the first and second column of Fig. 2(a)). It has been shown that focused by an objective with high numerical-aperture, both standard and π-phase vector beams show new features different from the scalar beams [2,23]. By changing the applied voltage, the topologic charge m can take integer or non-integer value. Fig. 2(b) shows the polarization distributions of output beams for different non-integer m, where the local polarization vectors make incomplete rotation around the beam axis. And a line of polarization discontinuity starting from the origin and extending along the radius of φ ¼ 0 emerges in
Please cite this article as: W. Zhu, W. She, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.08.057i
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Fig. 2. The polarization distributions of the output inhomogeneously polarized beams for different applied voltages U. (a) U¼ mU0 with m being positive (first column) and negative (second column) integers. (b) U ¼mU0 with m being positive (first column) and negative (second column) non-integers. The input beam is a RCP one.
Fig. 3. The polarization distributions of hybridly polarized vector beams for different incident polarization states and different m. Upper row: m¼1 (a) diagonally linear polarization, (b) right circular polarization, and (c) left circular polarization. Lower row: diagonally linear polarization (d) m¼ 1, (e) m¼ 0.5 and (f) m¼2.
the polarization patterns. This line has low intensity due to the discontinuity [16,22]. All the output inhomogeneously polarized beams mentioned above have the same local polarization ellipticities e over the cross section of beams. According to the relation SAM p sin ð2arc tan eÞ [21,25], these beams carry a locally identical SAM per photon. The idea to generate light beams with spatially variant SAM is interesting. The hybridly polarized vector beam is one kind
of such beams [6], which can be produced just by removing the QWP. And the output beam is described by Eq. (1), where the horizontal and vertical beam components have an azimuthally dependent phase difference Δδ¼ 2mφ þ2mπ δ, and will form linear, circular, and elliptical polarizations simultaneously in the beam. The polarization distributions of hybridly polarized vector beams for different incident polarization states and different m are plotted in Fig. 3, from which one sees that, the local polarization
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ellipse changes gradually with the azimuthal angle φ. By one circle around the beam axis, the local ellipse returns back to its original state when m is an integer. However, it cannot return back to the original state for a non-integer m, and a line of polarization discontinuity will appear in the polarization pattern. As shown by Fig. 3(a)–(c), the local polarization states at φ ¼ 0 correspond to the states of incident polarization. From Fig. 3(a) and (d) one concludes that, the local SAM at each point is opposite for two beams with different parameters (α, δ, m) and (α, δ, m), because two phase differences Δδ and –Δδ are of opposite sign. 4. Conclusion In summary, we have investigated the generation of inhomogeneously polarized light beams using a cylindrical device composed of two conjugated electro-optic SPPs. With an additional QWP after the device, the output polarization states are fully represented on a PS of m order, which can be integer and noninteger, and can be controlled by an applied voltage. All the states represented by an arbitrary order PS can be created by changing the incident homogeneous polarization state. Without the QWP, the output beam is hybridly polarized and carries spatially variant SAM. We believe that this tunable generation of inhomogeneously polarized beams has potential application in optical manipulation. Acknowledgment
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The authors acknowledge the financial support from the National Natural Science Foundation of China (NSFC) (grant 90921009).
Please cite this article as: W. Zhu, W. She, Optics Communications (2013), http://dx.doi.org/10.1016/j.optcom.2013.08.057i