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An interferential method for generating polarization-rotatable cylindrical vector beams
Optics Communications 286 (2013) 6–12
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
An interferential method for generating polarization-rotatable cylindrical vector beams Zhaotai Gu, Cuifang Kuang n, Shuai Li, Yi Xue, Xiang Hao, Zhenrong Zheng, Xu Liu State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 November 2011 Received in revised form 18 July 2012 Accepted 10 August 2012 Available online 18 September 2012
Cylindrical vector beam is one of the polarized beams whose states of polarization are spatially various and cylindrical symmetric. Owing to their peculiar focusing property, cylindrical vector beam has attracted more and more attentions. In this paper, we propose a method to generate cylindrical vector beam with rotatable polarization in free space. Two spatially homogeneous polarized beams are modified properly into spatially varied polarized beam by the vortex phase plate respectively. Then two phase-coded beams are combined together to generate a cylindrical vector beam. In addition, the polarization directions of cylindrical beams can be rotated continuously to a desired direction. The polarization states of the cylindrical vector beam are calculated by Johns matrix. The focusing properties are simulated by MATLAB. & 2012 Elsevier B.V. All rights reserved.
Keywords: Vortex phase plate Cylindrical beam Radial polarization Azimuthal polarization
1. Introduction Different from commonly polarized beams, such as linear, elliptical, and circular polarization beam, whose states of polarization are spatially homogeneous, the polarization state of a special polarized beam depends on the spatial location in the beam cross section. One example is a laser beam whose polarization states are cylindrically symmetry both in amplitude and polarization. Such particular beam is called cylindrical vector beam (CV beam). Two typical CV beams are radially polarized beam and azimuthally polarized beam. Because of the peculiar properties of polarization, CV beam has attracted more and more attentions recently. For example, azimuthally polarized beam converged by high numerical aperture (NA) objective lens can generate a 2D hollow spot in the transverse focal plane [1,2]. Such a doughnut-shaped spot is quite sensitive to the fluorescence molecular orientation while applied as de-excitation spot in stimulated-emission–depletion (STED) microscopy. Thus azimuthally polarization beam can be used as de-excitation beam in molecule orientation microscopy-STED (MOM-STED) [3]. A tighter focus spot can be yielded by focusing a radially polarized beam with a high NA objective lens, due to the creation of strong longitudinal component [1,2,4]. After proper phase encoding, the azimuthally polarized beam also can be used to achieve a sharper focus spot [5]. These focal spots are considered to be suitable tools for optical trapping and manipulation of small particles [6,7]. Other potential applications of CV beam includes lithography [8], electron acceleration [9] and laser machining [10] etc.
Since 1972, a great variety of methods about generating CV beam have been reported. One of the former experiments applied a laser intracavity device to force the laser to oscillate in CV modes [11]. The intracavity devices can be axial birefringent component [12–14] or axial dichroic component [15]. In addition to such mode-discrimination methods, converting commonly polarized beams such as linear polarized beam into CV polarized beam in free space is also an effective method. Such method can be achieved by applying spatially varied polarization rotation devices [16,17] or spatially varied retardation devices [18,19], to yield such cylindrical polarization conversion. CV mode can also be excited by proper misalignment between a single-mode and a multimode fiber [20]. Radially polarized beams were conducted in 1990 by the interferential method [19] based on Mach–Zehnder interferometer. And applying a double l/2 plate polarization rotator can converts the polarization of incident CV beam with a desired angle [1]. The purpose of this paper is going to propose an improved method based on Mach–Zehnder interferometer which has functions of generating CV beams with polarization in a desired direction. In this method, CV beam is generated by modulating two circular polarized beams with proper vortex phase plates and the polarization directions are adjustable by changing the angle difference between two end lines of phase plates. The focusing properties of the generated and pure CV beams are simulated, and the results are proposed in Section 3.
2. System configuration n
Corresponding author. Tel.: þ86 571 8795 3979. E-mail address: [email protected] (C. Kuang).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.08.071
The main idea of the experiment setup for generating a CV beam is illustrated in Fig. 1. A laser beam is coupled into a PMF
Z. Gu et al. / Optics Communications 286 (2013) 6–12
7
Fig. 1. System setup for generating a CV beam. A linear polarized beam 1 passes through a l/2 plate then split into P polarization beam (beam 2) and S polarization beam (beam 3) with a polarization beam splitter (PBS). The orthogonally polarized beams modulated by a matched vortex phase plate (PP) respectively before recombined by another PBS. The cylindrical vector beam is generated after the recombined beam passed through a l/4 plate.
(polarization maintaining optical fiber), and then collimated into linear polarized beam 1. The l/2 plate is used to adjust the amplitudes of horizontal and vertical components by properly rotating the direction of fast axis. Then a polarization beam splitter (PBS) splits the adjusted beam 1 into S polarization beam (beam 2) and P polarization beam (beam 3) with an intensity ratio of 1:1. Then two orthogonally linear polarized beams are modulated by two matched vortex 0–2p phase plates respectively. In order to recombine beam 4 and beam 5 into the same optical pathway, the position of one mirror should be adjustable with fine adjustment carried out by a PS (piezoelectric stage). Then two beams are recombined by another PBS. After passing through a l/4 plate, two orthogonal polarization beams would be converted into modulated left-handed and modulated right-handed circular polarized beam respectively. Eventually, CV polarization beam (beam 6) is achieved by combining such two modulated circular polarization beams together. Compared with former configuration [19], using PBS to split and recombine the laser beam is an effective way to raise the energy utilization ration. Furthermore, with that configuration, only single l/4 plate is needed. Therefore, our setup not only simplifies the experiment setup, but also reduces optical path difference. In order to simplify the calculation but get the same result, we considered that two orthogonally polarized beams first pass the l/4 plate, and then the phase plates. The polarization states of two orthogonally polarization beams are depicted and calculated by Johns matrix. For P polarization (beam 2), the matrix can be written as , 1 1 EP ¼ pffiffiffi 2 0
ð1Þ
And matrix of S polarization can be written as , ES
1 0 ¼ pffiffiffi 2 1
ð2Þ
The Johns matrix of the l/4 plate shown in the experimental setup (Fig. 1) is 1 1 pffiffiffi 2 i
i 1
ð3Þ
where i is the imaginary unit. Thus, after passing through a l/4 plate, P polarization beam was right-handed circularly polarized. On the contrary, S polarization beam was left-handed circularly polarized. The resulted matrixes of P and S polarization can be
calculated as , 1 i 1 1 1 1 1 pffiffiffi EP ¼ pffiffiffi ¼ 2 i 2 i 1 2 0 , 1 1 ES ¼ pffiffiffi 2 i
i 1
1 0 1 1 pffiffiffi ¼ 2 i 2 1
ð4Þ
ð5Þ
Fig. 1 shows S the structure of the vortex 0–2p phase plate. Its role is to modulate the phase of the beam according to the spatial location in the cross section of the incident beam. Therefore, after modulated by the vortex 0–2p phase plate, the matrixes should be modified with a phase delay parameter. Fig. 2 shows two types of collocation to generate radially and azimuthally polarization beams respectively. The electric vector matrixes of these cases are calculated below. For P polarized beam, electric vector matrix can be written as , 1 1 EP ¼ ei ~ ð6Þ 2 i where ei~ is the phase delay parameter produced by the modulation property of a vortex phase plate. And ~ is the angle deviation from the end line (the line of phase delay 0 or 2p). For the S polarized beam, due to the reversed vortex direction, the electric field vector matrix is , 1 1 ES ¼ ei~ ð7Þ 2 i Then two modulated circular polarized beams are recombined together. The resulting electric field vector according to Fig. 2(a) can be depicted as # " cos ~ , , , 1 1 i~ 1 e E ¼ EP þ ES ¼ ð8Þ þ ei ~ ¼ sin ~ 2 i i As can be seen from Eq. (8), the states of polarization are determined by the angle ~. Thus the states of polarization of the resulting beam are no longer spatially homogeneous. The transient states of polarization in the cross section of the resulted beam were illustrated in Fig. 3(a). Obviously, a radially polarized beam was generated. By rotating a vortex phase plate by 1801 (shown in Fig. 2(b)), the end line of the phase plate is opposite. The phase difference caused by the rotation can be simply expressed by adding the phase delay parameter with a phase difference p. Then, the matrix of S polarization and the resulting matrix can be calculated as , 1 ES ¼ eið~ þ pÞ ð9Þ i
8
Z. Gu et al. / Optics Communications 286 (2013) 6–12
Fig. 2. (a) and (b) are two types of collocation for vortex phase plate and circular polarization beam to generate two typical CV beams. (c) shows the situation that there is an angle difference of 2~0 two end lines of the vortex phase plates.
Fig. 3. (a) Radially and (b) azimuthally polarization. And (c) is the CV beam with ~0 rotation from the purely radially polarization.
,
,
,
E ¼ EP þ ES ¼
# " sin ~ 1 1 i~ 1 e þ eið ~ þ pÞ ¼ cos ~ 2 i i
ð10Þ
In this way, another beam with spatially varied states of polarization is generated. According to Eq. (10), the transient polarized direction in the cross plane of the beam is shown in Fig. 3(b). The figure shows that an azimuthally polarized beam is generated. Fig. 3(a) and (b) shows that the polarization states of beams are cylindrically symmetry both in amplitude and polarization. There is an additional bonus of using phase-modulating methods to generate CV beam. If allocate two vortex phase plates with an angle difference of 2~0 between two end lines of phase plates (shown in Fig. 2(c)), the phase delay parameter of each beam should be modified. Thus the electric vector matrix can be calculated as # " cosð ~ þ ~ 0 Þ , , , 1 1 ið~ ~ 0Þ 1 ið ~ þ ~ 0Þ e E ¼ EP þ ES ¼ þe ¼ sinð ~ þ ~ 0 Þ 2 i i ð11Þ
The matrix (11) expresses a CV beam with polarized direction rotated ~0 from pure radially polarization. The transient polarization states in the transverse section are illustrated in Fig. 3(c). Therefore, the polarization direction of the generated CV beam can be accurately changed by adjusting the angle between two end lines of phase plates. Such method is simpler than that by adding two extra wave plates to produce such a rotation angle. Generally, as depicted by Johns matrixes (6), (7) and (9), the phase plate is used to convert the circular polarized beam into spatially varied polarized beam (beam 4 and beam 5) by spatially varied phase delay. Matrix (8) and (10) show that the phase factors are destructed and the spatial factor are kept after the combination. As a result, beam with cylindrically symmetric polarization states is generated.
3. Simulations and discussions The polarization property of beam 6 can be examined via observing the pattern imaged directly by a CCD after passing through an analyzer [21]. Besides, the CV beams can be analyzed
Z. Gu et al. / Optics Communications 286 (2013) 6–12
1.4 N.A Objective Lens
9
y
Q (r0, ϕ0, z0) x
ϕ θ
Incident Beam
z
Focal Plane f Fig. 4. Geometry of the problem. Beam 6 is the outcome beam of system show in Fig. 1. Q(r0,j0,z0) is an observed point in the focal region.
by their focusing properties. The property of focusing a polarized beam by a high numerical aperture (NA) objective lens can be analyzed with the Richards and Wolf vectorial diffraction method [22,23]. Such method has been extended to calculate the focal field of a CV incident beam [2,24,25]. Fig. 4 illustrates the geometry of the problem. By using vectorial diffraction theory, the electric field vector near the spot Q(r0,j0,z0)can be obtained from the generalized Debye integral as ZZ , E ðr 0 , j0 ,z0 Þ ¼ iC sin y A1 ðy, jÞ A2 ðy, jÞ O
2
3 px 6 7 4 py 5 eiDaðy, jÞ eiknðz0 cos y þ r0 sin ycosðjj0 ÞÞ dydj
ð12Þ
pz
,
Where E ðr 0 , j0 ,z0 Þ is the electric field vector at the observed point Q(r0,j0,z0) which expressed in cylindrical coordinates with their origin at the focal point. C is the normalized constant. A1(y,j) is the amplitude function of the input light. A2(y,j) is the 3 3 matrix related to the structure of the imaging lens. [px; py; pz] is a matrix unit vector about the polarization of input light. Da(y,j) is the phase delay factor induced by phase plate. The focusing properties of CV incident beams generated by the methods proposed above are simulated by software of MATLAB according to Eq. (12). In our simulation, amplitude function A1(y,j) is set to be 1. Further, A2(y,j)could be expressed as [25] A2 ðy, jÞ ¼ aðyÞ Vðy, jÞ
ð13Þ
where a(y) is the apodization factor obtained from energy conservation and geometric considerations and V(y,j)is the conversion matrix of the polarization from the object field to the image field. In particular, for the apochromatic lens, pffiffiffiffiffiffiffiffiffiffi aðyÞ ¼ cosy 2 3 1 þðcosy1Þcos2 j ðcosy1Þcosjsinj sinycosj 6 7 2 7 Vðy, jÞ ¼ 6 4 ðcosy1Þcosjsinj 1 þ ðcosy1Þsin j sinysinj 5 sinycosj sinysinj cosy ð14Þ Phase delay parameter Da(y,j) ¼ j for applying a vortex 0–2p phase plate. To simplify the discussion, all beams are focused by an aplanatic oil-immersion lens with NA of 1.4. The refractive index of the oil is n ¼1.518. The simulation outcomes are presented in the form of color map. All aberrations are ignored that the center of each intensity figure is the geometrical focal point. The wavelength of the input light was set to be the unit wavelength while the initial phase was assumed to be the same.
Fig. 5 shows the intensity distribution in the vicinity of focal point of the vortex phase modulated circular polarized beam. The geometry of such focusing properties is illustrated in Fig. 4. We regard the incident beam as the modulated circular polarization , , beam, while e0 and ep are the unit vectors. After being coded by vortex 0–2p phase plate, electric vectors which are axially , , symmetric, will maintain a phase difference p, like e0 and ep shown in Fig. 4. We can anticipate that the x and y components (parallel to the transverse plane) will interfere destructively at the focal point after passing through the objective lens, while z components superpose at the focal point coherently. If the rotation of the circular polarization corresponds to the slope of , , the vortex phase pattern, z component generated by e0 and ep , , will be destructed by the one generated by e p=2 and e 3p=2 (not depicted in the figure). Then, a doughnut-shaped spot shown in Fig. 5(a) is achieved which can be used in STED microscopy [26–28]. In other cases (depicted in Fig. 2), z components will superpose coherently (shown in Fig. 5(b)). Fig. 6(a) is the simulation result of the focused combination beam (matrix (8)). The transient polarized state of such beam is illustrated by Fig. 3(a). The focusing properties of such beam still can be predicted through Fig. 4. In this case, the unit vectors only have the radial component (parallel to section plane shown by arrows). A radially polarized direction can be divided into longitudinal and lateral component after passing through an objective lens. As expected, such beam will generate a strong longitudinal component in the focal point, and the lateral components are still destructively interfered. Identical to the simulation, combination beam generate a much stronger longitudinal component at the focal point. Explained by the simulation result in Fig. 5(b), longitudinal components in the focal point generated by each modulated circular beam are coherently superposed. And the hollow spot shows the lateral components are destructively interfered at focal point. The arrows in the total intensity figure are the transient polarized directions in the focal region. The simulation of focusing a pure radially polarized beam is illustrated in (b). Comparing (a) with (b), the focusing properties of the generated and pure radially polarized beams are identical. Therefore, the method of generating a radial beam presented above is correct and feasible. The combined beam expressed with matrix (10) is obviously an azimuthally polarized beam. For such a beam, no longitudinal component will be generated at the focal point. Thus, the intensity of z component in the vicinity of focus should be zero. Contrary to the radial one, longitudinal components in the focal point generated by each modulated circular beam are interfered destructively due to a phase difference p. Meanwhile, the lateral components still destructively interfered, which makes the
10
Z. Gu et al. / Optics Communications 286 (2013) 6–12
Fig. 5. (a) simulation results of condition that rotation of the circular polarization corresponds to the slope of the vortex phase pattern. (b) shows the focusing simulation results of a single phase-coded circular polarization beam (matrix (6), (7) and (9)).
Fig. 6. Simulation results of (a) generated radial polarization beam (matrix (8)) and (b) purely radial polarization beam. The arrows in I intensity figure are the transient directions of electric field vector.
intensity distribution of lateral component doughnut-shaped. The focusing simulations of the generated beam and a pure azimuthally polarized beam are shown in Fig. 7. From the simulation results of the two beams, we can find that they totally agree with each other and match the analysis. The results prove that an azimuthally polarized beam has been generated successfully. To verify the resulted matrix (11), we simulate the situation that the angle difference between two end lines of the vortex phase plates is 901 and the results are presented in Fig. 8. As indicated above, inducing an angle difference 901 will lead to optical angular momentum tuned by 451. As can be seen in Fig. 8, according to radially and azimuthally polarized beam, the
transient polarization directions are rotated by 451. In such case, the generated CV beam is consisted of azimuthal and radial components. That means the energy of longitudinal components is decreased and the transverse ones are enhanced. As a result, a flatter focus is generated compared with the one generated by radial polarization. As can be seen from Fig. 9, by tuning the vortex phase plate, the focal spot can be converted continuously from tight focus to hollow spot due to the changes in the amplitude of radial and azimuthal component. In this way, pattern shaping is simply accomplished which may have potentials in applications such as optical tweezers, laser printing and material processing.
Z. Gu et al. / Optics Communications 286 (2013) 6–12
11
Fig. 7. Simulation results of (a) generated and (b) purely azimuthal polarization beam. The arrows in I intensity figure are the transient directions of electric field vector.
Fig. 8. Simulation results in the focal plane with angle difference (2~0) were set to be 901.The arrows in I intensity figure are the transient directions of electric field vector.
vortex phase-coded circular polarization beams. As proved above, the polarization state of the generated beams are cylindrically symmetric both in amplitude and polarization depends on the spatial location and proved to be CV beam. The simulated results demonstrate that the polarization direction of the CV beam can be accurately adjusted by inducing a proper angle difference between two end lines of the vortex phase plates. Thus the amplitudes of radial component and azimuthal component of the generated CV beam are adjustable. Such characteristic can be used to achieve focal pattern shaping which may widely benefit optical tweezers, laser printing and material processing.
Acknowledgments Fig. 9. Normalized intensity profile lines in focal plane generated by adjusted angle difference to 01(radial polarization), 901 and 1801(azimuthal polarization).
4. Conclusions Owing to the peculiar property, CV beams are recommendable for application in the area such as microscopy, optical trapping and manipulating, lithography, electron acceleration and laser machining, etc. Therefore, methods for generating CV beams cause more and more attentions. In this paper, we propose an interferential method which generates CV beam by the use of two
This work was financially supported by grants from National Natural Science Foundation of China (61205160), Doctoral Fund of the Ministry of Education of China (Grant 20110101120061), the Qianjiang Talent Project (Grant 2011R10010), and the Fundamental Research Funds for Central Universities (Grant No. 2012FZA5004). References [1] Q. Zhan, J.R. Leger, Optics Express 10 (7) (2002) 324. [2] K. Youngworth, T. Brown, Optics Express 7 (2) (2000) 77.
the the No. No. the
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M. Reuss, J. Engelhardt, S.W. Hell, Optics Express 18 (2) (2010) 1049. R. Dorn, S. Quabis, G. Leuchs, Physical Review Letters 91 (23) (2003) 233901. X. Hao, C. Kuang, T. Wang, X. Liu, Optics Letters 35 (23) (2010) 3928. Q. Zhan, Optics Express 12 (15) (2004) 3377. S. Sato, Y. Harada, Y. Waseda, Optics Letters 19 (22) (1994) 1807. L.E. Helseth, Roles of Polarization, Phase and Amplitude in Solid Immersion Lens Systems. Arxiv preprint physics/0108064, 2001. B. Hafizi, E. Esarey, P. Sprangle, Physical Review E 55 (3) (1997) 3539. VG Niziev, AV. Nesterov, Journal of Physics D: Applied Physics 32 (1999) 1455. D. Pohl, Applied Physics Letters 20 (7) (1972) 266. K. Yonezawa, Y. Kozawa, S. Sato, Optics Letters 31 (14) (2006) 2151. G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, N. Davidson, Applied Optics 46 (16) (2007) 3304. K. Yonezawa, Y. Kozawa, S. Sato, Japanese Journal of Applied Physics 46 (2007) 5160. J.F. Bisson, J. Li, K. Ueda, Y. Senatsky, Optics Express 14 (8) (2006) 3304. R. Yamaguchi, T. Nose, S. Sato, Japanese Journal of Applied Physics 28 (9) (1989) 1730.
[17] M. Stalder, M. Schadt, Optics Letters 21 (23) (1996) 1948. [18] G. Machavariani, Y. Lumer, I. Moshe, A. Meir, S. Jackel, Optics Communications 281 (4) (2008) 732. [19] S.C. Tidwell, D.H. Ford, W.D. Kimura, Applied Optics 29 (15) (1990) 2234. [20] T. Grosjean, D. Courjon, M. Spajer, Optics Communications 203 (1–2) (2002) 1. [21] X.L. Wang, J. Ding, W.J. Ni, C.S. Guo, H.T. Wang, Optics Letters 32 (24) (2007) 3549. [22] E. Wolf, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 253 (1274) (1959) 349. [23] B. Richards, E. Wolf, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 253 (1274) (1959) 358. [24] D. Biss, T. Brown, Optics Express 9 (10) (2001) 490. [25] X. Hao, C. Kuang, T. Wang, X. Liu, Journal of Optics 12 (2010) 115707. [26] K.I. Willig, R.R. Kellner, R. Medda, B. Hein, S. Jakobs, S.W. Hell, Nature Methods 3 (9) (2006) 721. ¨ [27] D. Wildanger, J. Buckers, V. Westphal, S.W. Hell, L. Kastrup, Optics Express 17 (18) (2009) 16100. [28] D. Wildanger, E. Rittweger, L. Kastrup, S.W. Hell, Optics Express 16 (13) (2008) 9614.
Contents lists available at SciVerse ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
An interferential method for generating polarization-rotatable cylindrical vector beams Zhaotai Gu, Cuifang Kuang n, Shuai Li, Yi Xue, Xiang Hao, Zhenrong Zheng, Xu Liu State Key Laboratory of Modern Optical Instrumentation, Zhejiang University, Hangzhou 310027, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 November 2011 Received in revised form 18 July 2012 Accepted 10 August 2012 Available online 18 September 2012
Cylindrical vector beam is one of the polarized beams whose states of polarization are spatially various and cylindrical symmetric. Owing to their peculiar focusing property, cylindrical vector beam has attracted more and more attentions. In this paper, we propose a method to generate cylindrical vector beam with rotatable polarization in free space. Two spatially homogeneous polarized beams are modified properly into spatially varied polarized beam by the vortex phase plate respectively. Then two phase-coded beams are combined together to generate a cylindrical vector beam. In addition, the polarization directions of cylindrical beams can be rotated continuously to a desired direction. The polarization states of the cylindrical vector beam are calculated by Johns matrix. The focusing properties are simulated by MATLAB. & 2012 Elsevier B.V. All rights reserved.
Keywords: Vortex phase plate Cylindrical beam Radial polarization Azimuthal polarization
1. Introduction Different from commonly polarized beams, such as linear, elliptical, and circular polarization beam, whose states of polarization are spatially homogeneous, the polarization state of a special polarized beam depends on the spatial location in the beam cross section. One example is a laser beam whose polarization states are cylindrically symmetry both in amplitude and polarization. Such particular beam is called cylindrical vector beam (CV beam). Two typical CV beams are radially polarized beam and azimuthally polarized beam. Because of the peculiar properties of polarization, CV beam has attracted more and more attentions recently. For example, azimuthally polarized beam converged by high numerical aperture (NA) objective lens can generate a 2D hollow spot in the transverse focal plane [1,2]. Such a doughnut-shaped spot is quite sensitive to the fluorescence molecular orientation while applied as de-excitation spot in stimulated-emission–depletion (STED) microscopy. Thus azimuthally polarization beam can be used as de-excitation beam in molecule orientation microscopy-STED (MOM-STED) [3]. A tighter focus spot can be yielded by focusing a radially polarized beam with a high NA objective lens, due to the creation of strong longitudinal component [1,2,4]. After proper phase encoding, the azimuthally polarized beam also can be used to achieve a sharper focus spot [5]. These focal spots are considered to be suitable tools for optical trapping and manipulation of small particles [6,7]. Other potential applications of CV beam includes lithography [8], electron acceleration [9] and laser machining [10] etc.
Since 1972, a great variety of methods about generating CV beam have been reported. One of the former experiments applied a laser intracavity device to force the laser to oscillate in CV modes [11]. The intracavity devices can be axial birefringent component [12–14] or axial dichroic component [15]. In addition to such mode-discrimination methods, converting commonly polarized beams such as linear polarized beam into CV polarized beam in free space is also an effective method. Such method can be achieved by applying spatially varied polarization rotation devices [16,17] or spatially varied retardation devices [18,19], to yield such cylindrical polarization conversion. CV mode can also be excited by proper misalignment between a single-mode and a multimode fiber [20]. Radially polarized beams were conducted in 1990 by the interferential method [19] based on Mach–Zehnder interferometer. And applying a double l/2 plate polarization rotator can converts the polarization of incident CV beam with a desired angle [1]. The purpose of this paper is going to propose an improved method based on Mach–Zehnder interferometer which has functions of generating CV beams with polarization in a desired direction. In this method, CV beam is generated by modulating two circular polarized beams with proper vortex phase plates and the polarization directions are adjustable by changing the angle difference between two end lines of phase plates. The focusing properties of the generated and pure CV beams are simulated, and the results are proposed in Section 3.
2. System configuration n
Corresponding author. Tel.: þ86 571 8795 3979. E-mail address: [email protected] (C. Kuang).
0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.08.071
The main idea of the experiment setup for generating a CV beam is illustrated in Fig. 1. A laser beam is coupled into a PMF
Z. Gu et al. / Optics Communications 286 (2013) 6–12
7
Fig. 1. System setup for generating a CV beam. A linear polarized beam 1 passes through a l/2 plate then split into P polarization beam (beam 2) and S polarization beam (beam 3) with a polarization beam splitter (PBS). The orthogonally polarized beams modulated by a matched vortex phase plate (PP) respectively before recombined by another PBS. The cylindrical vector beam is generated after the recombined beam passed through a l/4 plate.
(polarization maintaining optical fiber), and then collimated into linear polarized beam 1. The l/2 plate is used to adjust the amplitudes of horizontal and vertical components by properly rotating the direction of fast axis. Then a polarization beam splitter (PBS) splits the adjusted beam 1 into S polarization beam (beam 2) and P polarization beam (beam 3) with an intensity ratio of 1:1. Then two orthogonally linear polarized beams are modulated by two matched vortex 0–2p phase plates respectively. In order to recombine beam 4 and beam 5 into the same optical pathway, the position of one mirror should be adjustable with fine adjustment carried out by a PS (piezoelectric stage). Then two beams are recombined by another PBS. After passing through a l/4 plate, two orthogonal polarization beams would be converted into modulated left-handed and modulated right-handed circular polarized beam respectively. Eventually, CV polarization beam (beam 6) is achieved by combining such two modulated circular polarization beams together. Compared with former configuration [19], using PBS to split and recombine the laser beam is an effective way to raise the energy utilization ration. Furthermore, with that configuration, only single l/4 plate is needed. Therefore, our setup not only simplifies the experiment setup, but also reduces optical path difference. In order to simplify the calculation but get the same result, we considered that two orthogonally polarized beams first pass the l/4 plate, and then the phase plates. The polarization states of two orthogonally polarization beams are depicted and calculated by Johns matrix. For P polarization (beam 2), the matrix can be written as , 1 1 EP ¼ pffiffiffi 2 0
ð1Þ
And matrix of S polarization can be written as , ES
1 0 ¼ pffiffiffi 2 1
ð2Þ
The Johns matrix of the l/4 plate shown in the experimental setup (Fig. 1) is 1 1 pffiffiffi 2 i
i 1
ð3Þ
where i is the imaginary unit. Thus, after passing through a l/4 plate, P polarization beam was right-handed circularly polarized. On the contrary, S polarization beam was left-handed circularly polarized. The resulted matrixes of P and S polarization can be
calculated as , 1 i 1 1 1 1 1 pffiffiffi EP ¼ pffiffiffi ¼ 2 i 2 i 1 2 0 , 1 1 ES ¼ pffiffiffi 2 i
i 1
1 0 1 1 pffiffiffi ¼ 2 i 2 1
ð4Þ
ð5Þ
Fig. 1 shows S the structure of the vortex 0–2p phase plate. Its role is to modulate the phase of the beam according to the spatial location in the cross section of the incident beam. Therefore, after modulated by the vortex 0–2p phase plate, the matrixes should be modified with a phase delay parameter. Fig. 2 shows two types of collocation to generate radially and azimuthally polarization beams respectively. The electric vector matrixes of these cases are calculated below. For P polarized beam, electric vector matrix can be written as , 1 1 EP ¼ ei ~ ð6Þ 2 i where ei~ is the phase delay parameter produced by the modulation property of a vortex phase plate. And ~ is the angle deviation from the end line (the line of phase delay 0 or 2p). For the S polarized beam, due to the reversed vortex direction, the electric field vector matrix is , 1 1 ES ¼ ei~ ð7Þ 2 i Then two modulated circular polarized beams are recombined together. The resulting electric field vector according to Fig. 2(a) can be depicted as # " cos ~ , , , 1 1 i~ 1 e E ¼ EP þ ES ¼ ð8Þ þ ei ~ ¼ sin ~ 2 i i As can be seen from Eq. (8), the states of polarization are determined by the angle ~. Thus the states of polarization of the resulting beam are no longer spatially homogeneous. The transient states of polarization in the cross section of the resulted beam were illustrated in Fig. 3(a). Obviously, a radially polarized beam was generated. By rotating a vortex phase plate by 1801 (shown in Fig. 2(b)), the end line of the phase plate is opposite. The phase difference caused by the rotation can be simply expressed by adding the phase delay parameter with a phase difference p. Then, the matrix of S polarization and the resulting matrix can be calculated as , 1 ES ¼ eið~ þ pÞ ð9Þ i
8
Z. Gu et al. / Optics Communications 286 (2013) 6–12
Fig. 2. (a) and (b) are two types of collocation for vortex phase plate and circular polarization beam to generate two typical CV beams. (c) shows the situation that there is an angle difference of 2~0 two end lines of the vortex phase plates.
Fig. 3. (a) Radially and (b) azimuthally polarization. And (c) is the CV beam with ~0 rotation from the purely radially polarization.
,
,
,
E ¼ EP þ ES ¼
# " sin ~ 1 1 i~ 1 e þ eið ~ þ pÞ ¼ cos ~ 2 i i
ð10Þ
In this way, another beam with spatially varied states of polarization is generated. According to Eq. (10), the transient polarized direction in the cross plane of the beam is shown in Fig. 3(b). The figure shows that an azimuthally polarized beam is generated. Fig. 3(a) and (b) shows that the polarization states of beams are cylindrically symmetry both in amplitude and polarization. There is an additional bonus of using phase-modulating methods to generate CV beam. If allocate two vortex phase plates with an angle difference of 2~0 between two end lines of phase plates (shown in Fig. 2(c)), the phase delay parameter of each beam should be modified. Thus the electric vector matrix can be calculated as # " cosð ~ þ ~ 0 Þ , , , 1 1 ið~ ~ 0Þ 1 ið ~ þ ~ 0Þ e E ¼ EP þ ES ¼ þe ¼ sinð ~ þ ~ 0 Þ 2 i i ð11Þ
The matrix (11) expresses a CV beam with polarized direction rotated ~0 from pure radially polarization. The transient polarization states in the transverse section are illustrated in Fig. 3(c). Therefore, the polarization direction of the generated CV beam can be accurately changed by adjusting the angle between two end lines of phase plates. Such method is simpler than that by adding two extra wave plates to produce such a rotation angle. Generally, as depicted by Johns matrixes (6), (7) and (9), the phase plate is used to convert the circular polarized beam into spatially varied polarized beam (beam 4 and beam 5) by spatially varied phase delay. Matrix (8) and (10) show that the phase factors are destructed and the spatial factor are kept after the combination. As a result, beam with cylindrically symmetric polarization states is generated.
3. Simulations and discussions The polarization property of beam 6 can be examined via observing the pattern imaged directly by a CCD after passing through an analyzer [21]. Besides, the CV beams can be analyzed
Z. Gu et al. / Optics Communications 286 (2013) 6–12
1.4 N.A Objective Lens
9
y
Q (r0, ϕ0, z0) x
ϕ θ
Incident Beam
z
Focal Plane f Fig. 4. Geometry of the problem. Beam 6 is the outcome beam of system show in Fig. 1. Q(r0,j0,z0) is an observed point in the focal region.
by their focusing properties. The property of focusing a polarized beam by a high numerical aperture (NA) objective lens can be analyzed with the Richards and Wolf vectorial diffraction method [22,23]. Such method has been extended to calculate the focal field of a CV incident beam [2,24,25]. Fig. 4 illustrates the geometry of the problem. By using vectorial diffraction theory, the electric field vector near the spot Q(r0,j0,z0)can be obtained from the generalized Debye integral as ZZ , E ðr 0 , j0 ,z0 Þ ¼ iC sin y A1 ðy, jÞ A2 ðy, jÞ O
2
3 px 6 7 4 py 5 eiDaðy, jÞ eiknðz0 cos y þ r0 sin ycosðjj0 ÞÞ dydj
ð12Þ
pz
,
Where E ðr 0 , j0 ,z0 Þ is the electric field vector at the observed point Q(r0,j0,z0) which expressed in cylindrical coordinates with their origin at the focal point. C is the normalized constant. A1(y,j) is the amplitude function of the input light. A2(y,j) is the 3 3 matrix related to the structure of the imaging lens. [px; py; pz] is a matrix unit vector about the polarization of input light. Da(y,j) is the phase delay factor induced by phase plate. The focusing properties of CV incident beams generated by the methods proposed above are simulated by software of MATLAB according to Eq. (12). In our simulation, amplitude function A1(y,j) is set to be 1. Further, A2(y,j)could be expressed as [25] A2 ðy, jÞ ¼ aðyÞ Vðy, jÞ
ð13Þ
where a(y) is the apodization factor obtained from energy conservation and geometric considerations and V(y,j)is the conversion matrix of the polarization from the object field to the image field. In particular, for the apochromatic lens, pffiffiffiffiffiffiffiffiffiffi aðyÞ ¼ cosy 2 3 1 þðcosy1Þcos2 j ðcosy1Þcosjsinj sinycosj 6 7 2 7 Vðy, jÞ ¼ 6 4 ðcosy1Þcosjsinj 1 þ ðcosy1Þsin j sinysinj 5 sinycosj sinysinj cosy ð14Þ Phase delay parameter Da(y,j) ¼ j for applying a vortex 0–2p phase plate. To simplify the discussion, all beams are focused by an aplanatic oil-immersion lens with NA of 1.4. The refractive index of the oil is n ¼1.518. The simulation outcomes are presented in the form of color map. All aberrations are ignored that the center of each intensity figure is the geometrical focal point. The wavelength of the input light was set to be the unit wavelength while the initial phase was assumed to be the same.
Fig. 5 shows the intensity distribution in the vicinity of focal point of the vortex phase modulated circular polarized beam. The geometry of such focusing properties is illustrated in Fig. 4. We regard the incident beam as the modulated circular polarization , , beam, while e0 and ep are the unit vectors. After being coded by vortex 0–2p phase plate, electric vectors which are axially , , symmetric, will maintain a phase difference p, like e0 and ep shown in Fig. 4. We can anticipate that the x and y components (parallel to the transverse plane) will interfere destructively at the focal point after passing through the objective lens, while z components superpose at the focal point coherently. If the rotation of the circular polarization corresponds to the slope of , , the vortex phase pattern, z component generated by e0 and ep , , will be destructed by the one generated by e p=2 and e 3p=2 (not depicted in the figure). Then, a doughnut-shaped spot shown in Fig. 5(a) is achieved which can be used in STED microscopy [26–28]. In other cases (depicted in Fig. 2), z components will superpose coherently (shown in Fig. 5(b)). Fig. 6(a) is the simulation result of the focused combination beam (matrix (8)). The transient polarized state of such beam is illustrated by Fig. 3(a). The focusing properties of such beam still can be predicted through Fig. 4. In this case, the unit vectors only have the radial component (parallel to section plane shown by arrows). A radially polarized direction can be divided into longitudinal and lateral component after passing through an objective lens. As expected, such beam will generate a strong longitudinal component in the focal point, and the lateral components are still destructively interfered. Identical to the simulation, combination beam generate a much stronger longitudinal component at the focal point. Explained by the simulation result in Fig. 5(b), longitudinal components in the focal point generated by each modulated circular beam are coherently superposed. And the hollow spot shows the lateral components are destructively interfered at focal point. The arrows in the total intensity figure are the transient polarized directions in the focal region. The simulation of focusing a pure radially polarized beam is illustrated in (b). Comparing (a) with (b), the focusing properties of the generated and pure radially polarized beams are identical. Therefore, the method of generating a radial beam presented above is correct and feasible. The combined beam expressed with matrix (10) is obviously an azimuthally polarized beam. For such a beam, no longitudinal component will be generated at the focal point. Thus, the intensity of z component in the vicinity of focus should be zero. Contrary to the radial one, longitudinal components in the focal point generated by each modulated circular beam are interfered destructively due to a phase difference p. Meanwhile, the lateral components still destructively interfered, which makes the
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Z. Gu et al. / Optics Communications 286 (2013) 6–12
Fig. 5. (a) simulation results of condition that rotation of the circular polarization corresponds to the slope of the vortex phase pattern. (b) shows the focusing simulation results of a single phase-coded circular polarization beam (matrix (6), (7) and (9)).
Fig. 6. Simulation results of (a) generated radial polarization beam (matrix (8)) and (b) purely radial polarization beam. The arrows in I intensity figure are the transient directions of electric field vector.
intensity distribution of lateral component doughnut-shaped. The focusing simulations of the generated beam and a pure azimuthally polarized beam are shown in Fig. 7. From the simulation results of the two beams, we can find that they totally agree with each other and match the analysis. The results prove that an azimuthally polarized beam has been generated successfully. To verify the resulted matrix (11), we simulate the situation that the angle difference between two end lines of the vortex phase plates is 901 and the results are presented in Fig. 8. As indicated above, inducing an angle difference 901 will lead to optical angular momentum tuned by 451. As can be seen in Fig. 8, according to radially and azimuthally polarized beam, the
transient polarization directions are rotated by 451. In such case, the generated CV beam is consisted of azimuthal and radial components. That means the energy of longitudinal components is decreased and the transverse ones are enhanced. As a result, a flatter focus is generated compared with the one generated by radial polarization. As can be seen from Fig. 9, by tuning the vortex phase plate, the focal spot can be converted continuously from tight focus to hollow spot due to the changes in the amplitude of radial and azimuthal component. In this way, pattern shaping is simply accomplished which may have potentials in applications such as optical tweezers, laser printing and material processing.
Z. Gu et al. / Optics Communications 286 (2013) 6–12
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Fig. 7. Simulation results of (a) generated and (b) purely azimuthal polarization beam. The arrows in I intensity figure are the transient directions of electric field vector.
Fig. 8. Simulation results in the focal plane with angle difference (2~0) were set to be 901.The arrows in I intensity figure are the transient directions of electric field vector.
vortex phase-coded circular polarization beams. As proved above, the polarization state of the generated beams are cylindrically symmetric both in amplitude and polarization depends on the spatial location and proved to be CV beam. The simulated results demonstrate that the polarization direction of the CV beam can be accurately adjusted by inducing a proper angle difference between two end lines of the vortex phase plates. Thus the amplitudes of radial component and azimuthal component of the generated CV beam are adjustable. Such characteristic can be used to achieve focal pattern shaping which may widely benefit optical tweezers, laser printing and material processing.
Acknowledgments Fig. 9. Normalized intensity profile lines in focal plane generated by adjusted angle difference to 01(radial polarization), 901 and 1801(azimuthal polarization).
4. Conclusions Owing to the peculiar property, CV beams are recommendable for application in the area such as microscopy, optical trapping and manipulating, lithography, electron acceleration and laser machining, etc. Therefore, methods for generating CV beams cause more and more attentions. In this paper, we propose an interferential method which generates CV beam by the use of two
This work was financially supported by grants from National Natural Science Foundation of China (61205160), Doctoral Fund of the Ministry of Education of China (Grant 20110101120061), the Qianjiang Talent Project (Grant 2011R10010), and the Fundamental Research Funds for Central Universities (Grant No. 2012FZA5004). References [1] Q. Zhan, J.R. Leger, Optics Express 10 (7) (2002) 324. [2] K. Youngworth, T. Brown, Optics Express 7 (2) (2000) 77.
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