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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 6 — Mar. 25, 2013
  • pp: 7343–7353
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Design of cylindrical vector beams based on the rotating Glan polarizing prism

Qi Hu, Zhihua Tan, Xiaoyu Weng, Hanming Guo, Yang Wang, and Songlin Zhuang  »View Author Affiliations


Optics Express, Vol. 21, Issue 6, pp. 7343-7353 (2013)
http://dx.doi.org/10.1364/OE.21.007343


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Abstract

Recently the cylindrically polarized beams have been gained highly attention in the fields of particle manipulation, material processing, nanoscale imaging, etc. So the methods to create the cylindrically polarized beams become more important. Here, based on the principle of the Glan polarizing prism, we design two types of the structures of the cylindrical polarization analyzer that can convert directly a linearly or circularly polarized beam into various cylindrical vector beams. The key optical element in the cylindrical polarization analyzer is the cylindrical polarizing prism with unique structure. We demonstrate the operating principle and the feasibility of the fabrication of the cylindrical polarization analyzer in detail. Analyses show that the cylindrical polarization analyzer designed by us not only have novel structures and excellent characteristics, such as the compact and stabile structures, high extinction ratio, high polarization purity, no requirements on the mode and the wavelength of the incident light (only for the first type), not changing the intensity distribution of the incident light, and easily integrated into the optical systems, but also is easy to be fabricated, especially for the second type.

© 2013 OSA

1. Introduction

In recent years considerable attention has been devoted to cylindrical vector beams including radially and azimuthally polarized beams due to their unique properties and potential applications [1

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

14

14. Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 29(11), 2439–2443 (2012). [CrossRef] [PubMed]

]. For example, the focusing of radially polarized beam is used for particle manipulation [15

15. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

, 16

16. L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37(10), 1694–1696 (2012). [CrossRef] [PubMed]

], Raman spectroscopy [17

17. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004). [CrossRef]

], second-harmonic generation [18

18. K. Yoshiki, R. Kanamaru, M. Hashimoto, N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. 32(12), 1680–1682 (2007). [CrossRef] [PubMed]

], material processing [19

19. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007). [CrossRef]

], and nanoscale imaging [20

20. G. Terakado, K. Watanabe, and H. Kano, “Scanning confocal total internal reflection fluorescence microscopy by using radial polarization in the illumination system,” Appl. Opt. 48(6), 1114–1118 (2009). [CrossRef]

22

22. Y. Xue, C. Kuang, S. Li, Z. Gu, and X. Liu, “Sharper fluorescent super-resolution spot generated by azimuthally polarized beam in STED microscopy,” Opt. Express 20(16), 17653–17666 (2012). [CrossRef] [PubMed]

] owing to the focusing properties with a larger axial field and a higher resolution than that for a linearly polarized illumination under the case of high numerical aperture system [1

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

, 2

2. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

]. Motivated by these applications, various methods for generating cylindrical vector beams have been proposed [23

23. K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]

36

36. A. Shoham, R. Vander, and S. G. Lipson, “Production of radially and azimuthally polarized polychromatic beams,” Opt. Lett. 31(23), 3405–3407 (2006). [CrossRef] [PubMed]

], having different degrees of polarization purity, complexity, efficiency, and cost. Generally, these methods can be classified into two types. One type involves direct generation from the laser cavity [23

23. K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]

25

25. M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32(22), 3272–3274 (2007). [CrossRef] [PubMed]

]. The other one generates the beam outside the laser cavity [26

26. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001). [CrossRef]

36

36. A. Shoham, R. Vander, and S. G. Lipson, “Production of radially and azimuthally polarized polychromatic beams,” Opt. Lett. 31(23), 3405–3407 (2006). [CrossRef] [PubMed]

]. The former usually need one to specially redesign the laser systems allowing for customized tuning and modification of the output polarization or mode. However, it is not always viable or practical to insert the specially designed optical elements inside the resonator of existing lasers, especially in commercial systems, which are often viewed as “black boxes” by the end users. In contrast, the latter permits one to perform the polarization conversion outside the laser, which will bring great convenience and reduce cost in applications.

Among the methods generating the cylindrical vector beam outside the laser cavity, each has its advantages and disadvantages. For example, the method utilizing space-variant subwavelength diffraction gratings [26

26. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001). [CrossRef]

] requires tight fabrication tolerances for the visible spectrum and is sensitive to the illuminating wavelength. The method based on the combination of two orthogonally polarized beams [27

27. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]

29

29. P. Ma, P. Zhou, Y. Ma, X. Wang, R. Su, and Z. Liu, “Generation of azimuthally and radially polarized beams by coherent polarization beam combination,” Opt. Lett. 37(13), 2658–2660 (2012). [CrossRef] [PubMed]

] has stringent requirement on the modes of these two beams (only apply to the TEM01 and TEM10 modes) and the interferometric stability. The method of fiber [30

30. R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express 18(10), 10834–10838 (2010). [CrossRef] [PubMed]

] lacks intrinsic stability and are inefficient unless the input beam is already preshaped. The method based on the liquid crystal elements (e.g., liquid crystal spatial light modulators (SLM)) [31

31. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef] [PubMed]

33

33. M. Bashkansky, D. Park, and F. K. Fatemi, “Azimuthally and radially polarized light with a nematic SLM,” Opt. Express 18(1), 212–217 (2010). [CrossRef] [PubMed]

], theoretically, can generate a cylindrical vector beam with perfectly radial or azimuthal polarization distribution, but, because of the big size of pixel pitch and the fill factor in the liquid crystal elements, it is difficult to generate the cylindrically vector beam with high quality. Moreover, the SLM not only have the highest cost, but also is hardly integrated into the optical systems.

As we know, the linearly polarized beam is easily generated from the unpolarized beam by utilizing a sheet linear polarizer or a linear polarizing prism. Both the sheet linear polarizer and the linear polarizing prism are very simple optical devices with high conversion efficiency, no requirements on the mode and the wavelength of the incident light, and not changing the intensity distribution of the incident light. Especially for the linear polarizing prism, it has higher extinction ratio than the sheet linear polarizer and can be used for the incident light with high power. Obviously, if like the linear polarizing prism, there is a polarization analyzer that can convert directly the simple polarization, such as linear and circular polarization, into the cylindrical polarization only utilizing several geometrical optical elements (i.g., the prism, the reflector, and the lens), it will be very interesting and convenient in the practical applications. Hereafter, we call this polarization analyzer the cylindrical polarization analyzer.

In this paper, based on the principle of the Glan polarizing prism, we design a cylindrically symmetrical polarizing prism with a coaxial hollow cylinder in order to take the place of the cylindrical sheet of polarizing film. For simplicity, this cylindrically symmetrical polarizing prism is hereafter called the cylindrical polarizing prism. We provide two types of the cylindrical polarization analyzer with novel structures. The structure of the cylindrical polarization analyzer in the first type is similar to that presented by Shoham et. al [36

36. A. Shoham, R. Vander, and S. G. Lipson, “Production of radially and azimuthally polarized polychromatic beams,” Opt. Lett. 31(23), 3405–3407 (2006). [CrossRef] [PubMed]

], but the novel and first reported cylindrical polarizing prism, especially that in the following second type, substitute successfully for the cylindrical sheet of polarizing film and overcome the fabrication difficulty. Based on the first type of the cylindrical polarization analyzer and the transmission matrix method, we derive the Jones matrix of the cylindrical polarization analyzer that is just the required Jones matrices of the azimuthal and the radial analyzers, predicated mathematically by Moh et. al [35

35. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). [CrossRef] [PubMed]

]. In order to further reduce the fabrication difficulty of the cylindrical polarizing prism and make its structures of the first type more compact, the another cylindrical polarizing prism and cylindrical analyzer are designed successfully.

2. Design principle and discussions

Our design principle is illustrated in Fig. 1(a)
Fig. 1 Schematic (a) of the cylindrical polarization analyzer and its equivalent optical path (b). (c) Diagram of the structure of the cylindrical polarizing prism 5. Other optical elements are: the convex 45oconical reflectors 1 and 4, the concave 45oconical reflectors 2 and 3, the cylindrical polarizing prism 5, the spiral phase plate P1, the λ/2 plates P2 and P3..
, where all optical elements are coaxial and cylindrical symmetry and the optical elements inside the dashed rectangle consist the cylindrical polarization analyzer. Figure 1(a) only represents the cross section of the cylindrical polarization analyzer. A collimated vector beam passes through the convex conical reflector 1, the cylindrical polarizing prism 5, the concave conical reflectors 2 and 3 and the convex conical reflector 4 in turn. All reflecting planes of the convex conical reflectors 1 and 4 and the concave conical reflectors 2 and 3 make angles of 45° with the optical axis (same with the caxis) of the cylindrical polarization analyzer. The key element of the cylindrical polarization analyzer is the cylindrical polarizing prism 5 separating the ordinary (o) and extraordinary (e) rays. Shown in Fig. 1(b), the cylindrical polarizing prism 5 is a birefringent crystal and is composed of the Part1 and the Part2. There are a convex conical transmitting surface and an inner cylindrical transmitting surface in the Part1. For the Part2, there are a concave conical transmitting surface and an outer cylindrical transmitting surface. As a whole, the cylindrical polarizing prism 5 has a coaxial hollow cylinder, which is like rotating a Glan polarizing prism with respect to its caxis. Like the Glan polarizing prism, the cylindrical polarizing prism 5 only permits either the o-ray or the e-ray to pass, so the generated cylindrical vector beam will have high polarization purity. The function of the concave conical reflector 2 is to make the divergent light reflected by the convex conical reflector 1 become parallel light again. The aim of the concave conical reflector 3 and the convex conical reflector 4 is to recover the intensity distribution of the incident light by adjusting the relative locations between them. In practical applications, the combination of the concave conical reflector 3 and the convex conical reflector 4 can also be used to construct annular beam.

In the following, we will derive the Jones matrix of the cylindrical polarization analyzer given by Fig. 1(a) to prove its feasibility. For the sake of analysis conveniently, the equivalent optical path of the cylindrical polarization analyzer is shown in Fig. 1(c) where the cylindrical polarizing prism 5 is not considered temporarily. For a treatment of the refraction and reflection that occurs at the interface, it is convenient to decompose the electric vector of the ray into s- and p-polarized vector components, es and ep, with respect to the meridional plane, respectively, and to rotate the coordinate system so that the new coordinate system will contain components in the (p,s,z) system. This coordinate system is defined in such a way that ez=0. The meridional plane is defined as the plane consisting of the incident ray and the corresponding reflected ray. The s- and p-polarized vectors are perpendicular and parallel to the meridional plane, respectively. As we know, when a ray traverses a interface, the vibration direction of the s-polarized vector of the ray always keeps invariability for both the reflected and the refracted rays, whereas that of the p-polarized vector will rotate with the rotation of the propagation direction of the reflected and the refracted rays. Moreover, the p-polarized vector of the reflected ray will rotate angle of 180o at the meridional plane. Namely, if the (p,s,z) of the incident ray is right handed system, that of the reflected ray will be changed into the left handed system.

Figure 1(c) describes the variations of the s- and p-polarized vectors when a collimated vector beam passes through every optical elements of the cylindrical polarization analyzer. It is easy to derive that the Jones matrices (T1,T2,T3,T4) of the convex conical reflectors 1 and 4 and the concave conical reflectors 2 and 3 are same and can be expressed as

T1=T2=T3=T4=[1001].
(1)

The cylindrical polarizing prism 5 can be considered a linear polarizer for each incident ray. As the polarization directions of the o- and e-rays are separately consistent with the s- and p-polarized vectors of the incident ray, the Jones matrix of the cylindrical polarizing prism 5 is expressed as
Po=[0001],
(2a)
for the case of the e-ray reflected entirely and the o-ray transmitted, and
Pe=[1000],
(2b)
for the case of the o-ray reflected entirely and the e-ray transmitted.

Shown in Fig. 1(c), the standard coordinate system of the cylindrical polarization analyzer is described by xyz before the conical reflectors 1. For each ray, the coordinate system in its meridional plane before the convex conical reflectors 1 is defined as xφyφzφ, where φ is the angle of the positive xφ axis with the positive x axis and the zφ axis is consistent with the z axis. So, the coordinate transformation for rotation around the z axis from xyz to xφyφzφ is

R=[cosφsinφsinφcosφ].
(3)

Then, for the vector beam propagating along the +z axis, the Jones matrices of the cylindrical polarization analyzer in the coordinate system xyz are expressed as
Mo=RTT4T3T2PoT1R,
(4a)
Me=RTT4T3T2PeT1R,
(4b)
where the superscript Tdenotes the transpose. In terms of Eqs. (1)-(4), the Jones matrix of the cylindrical polarization analyzer can be obtained as following
Mo=[sin2φsinφcosφsinφcosφcos2φ],
(5a)
Me=[cos2φsinφcosφsinφcosφsin2φ].
(5b)
The Jones matrices Mo and Me of the cylindrical polarization analyzer are just the required Jones matrices of the azimuthal and the radial analyzers, predicated mathematically by Moh et. al [35

35. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). [CrossRef] [PubMed]

], respectively.

When a linearly or circularly polarized beam Ei incidents on the cylindrical polarization analyzer along the +z axis, the corresponding output fields E=MEi are described as

For the x linearly polarized beam Ei=[10]T,

Eox=sinφ[sinφcosφ]T,
(6a)
Eex=cosφ[cosφsinφ]T.
(6b)

For the y linearly polarized beam Ei=[01]T,

Eoy=cosφ[sinφcosφ]T,
(7a)
Eey=sinφ[cosφsinφ]T.
(7b)

For the circularly polarized beam Ei=[1i]T,

Eoc=iexp(iφ)[sinφcosφ]T,
(8a)
Eec=exp(iφ)[cosφsinφ]T.
(8b)

It is noted that in generally, the radially and azimuthally polarized beams are called the cylindrical vector beams. The radial and the azimuthal polarizations are only dependent on the unit polarization vector [cosφsinφ]T and [sinφcosφ]T, respectively. The so-called cylindrical vector beam is defined as the beam with cylindrical symmetry in polarization [6

6. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

]. Therefore, the cylindrical vector beam is only determined by the cylindrical symmetry in polarization and is irrelevant to the amplitude and the phase modulation in its complex electric field. In terms of this point, although the sine or cosine amplitude modulation or the vortex phase [exp(iφ)] modulation exist naturally in Eqs. (6)-(8), the vector beam generated by the cylindrical polarization analyzer shown in Fig. 1(a) is still called the cylindrical vector beam. Equations (6)-(8) show that, for the case of the e-ray reflected entirely and the o-ray transmitted, the azimuthally polarized beam can be generated by the cylindrical polarization analyzer. For the case of the o-ray reflected entirely and the e-ray transmitted, the radially polarized beam can be generated by the cylindrical polarization analyzer.

In some cases, this natural modulation for amplitude or phase is beneficial. For example, Eq. (8)a) represents an azimuthally polarized beam with the vortex phase exp(iφ) that can generate sharper focus than other polarized beam including the usually radially polarized beam [8

8. S. Sato and Y. Kozawa, “Hollow vortex beams,” J. Opt. Soc. Am. A 26(1), 142–146 (2009). [CrossRef] [PubMed]

, 10

10. X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010). [CrossRef] [PubMed]

]. When the incident beam is circularly polarized, to obtain a usually cylindrical vector beam [i.e., no vortex phase exp(iφ)], a spiral phase plate with the opposite helicity exp(iφ)can be added at the exit [see the element P1 in Fig. 1(a)] to compensate for the geometric phase [6

6. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

, 35

35. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). [CrossRef] [PubMed]

]. This type of spiral phase plate is available commercially [37].

In addition, it is inconvenient and will increase the cost for the end users and the vendors to fabricate two types of the cylindrical polarization analyzer (i.e., the azimuthal analyzer Mo and the radial analyzer Me). A simple solution to this problem is to only fabricate the radial analyzer. The azimuthal polarization can be obtained through the polarization rotator using two cascaded λ/2 plates [see P2 and P3 in Fig. 1(a)]. If the angle between the fast axes of the two cascaded λ/2 plates is Δφ, the Jones matrix of this combination can be expressed as [6

6. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

]
T=[cos(2Δφ)sin(2Δφ)sin(2Δφ)cos(2Δφ)],
(9)
which is a pure polarization rotation function independent of the incident polarization. The polarization rotation angle can be tuned by adjusting the angle between the fast axes of the two plates.

So far we have derived the Jones matrix of the cylindrical polarization analyzer shown in Fig. 1(a) and demonstrated its feasibility theoretically. As the structure of the cylindrical polarization analyzer shown in Fig. 1(a) is similar to that presented by Shoham et. al [36

36. A. Shoham, R. Vander, and S. G. Lipson, “Production of radially and azimuthally polarized polychromatic beams,” Opt. Lett. 31(23), 3405–3407 (2006). [CrossRef] [PubMed]

], except for the cylindrical polarizing prism instead of the cylindrical sheet of polarizing film and the adding optical elements (the concave conical reflector 3 and the convex conical reflector 4) used to recover the intensity distribution of the incident light, and Shoham et. al carry on preliminary experiment research, we think the design of the cylindrical polarization analyzer shown in Fig. 1(a) is feasible in practice despite not provide any experimental results in this paper. As the fabrication of the cylindrical sheet of polarizing film with high extinction ratio and without a seam is almost impossible, the cylindrical polarization analyzer designed by Shoham et. al hasn't received more attention until now. The cylindrical polarizing prism designed by us is based on the principle of the Glan polarizing prism, so this cylindrical polarizing prism can offer ultra high extinction ratio like the Glan polarizing prism and its fabrication is more feasible than that of the cylindrical sheet of polarizing film.

Of course, we also note the fabrication of the cylindrical polarizing prism 5 shown in Fig. 1(b) is possible, but is not easy. Shown in Fig. 1(b), the cylindrical polarizing prism 5 is composed of the Part1 and the Part2. There are a convex conical transmitting surface and an inner cylindrical transmitting surface in the Part1. For the Part2, there are a concave conical transmitting surface and an outer cylindrical transmitting surface. The convex conical and the inner cylindrical transmitting surfaces of the Part1 must be parallel to the concave conical and the outer cylindrical transmitting surfaces of the Part2, respectively. Moreover, all surfaces need to have optical quality with wavelength precision. These requirements will increase the fabrication difficulty of the cylindrical polarizing prism 5 shown in Fig. 1(b).

In order to further reduce the fabrication difficulty of the cylindrical polarizing prism shown in Fig. 1(b) and make the structures of the cylindrical polarization analyzer more compact, we design another cylindrical polarizing prism and cylindrical polarization analyzer (see Fig. 2
Fig. 2 (a) Schematic of the cylindrical polarization analyzer. (b) Diagram of propagating procedure of the o- and e-rays when the incident light transverses the optical elements 5 and 2. Other optical elements are: the 45o planar reflecting prism 1, the concave α[calculated by Eq. (12)] conical reflector 2, the concave 45o conical reflector 3, the convex 45o conical reflector 4, the cylindrical polarizing prism 5, the transparent optical plate 6, the spiral phase plate P1, the λ/2 plates P2 and P3.. The blue-black dot and the purple short line denote the polarization directions of the o and e-rays, respectively. Note that, except for the 45o planar reflecting prism 1, all other optical elements are coaxial and cylindrical symmetry about the c axis.
). The basis for the design of the new cylindrical polarization analyzer is to reduce the numbers of the optical elements with curved surface, especially for the elements with concave conical surface, because to our knowledge, the convex conical optical element is available commercially, but the concave conical optical element is not. The fabrication of the concave conical optical element is more difficult than that of the convex conical optical element.

Compared with Fig. 1(a), the cylindrical polarizing prism 5 in Fig. 2(a) is simplified as the convex 45o conical transmitting prism with negative birefringent crystal. The caxis is its symmetry axis. The convex conical reflector 1 in Fig. 1(a) is changed into the 45o planar reflecting prism 1 in Fig. 2(a). The angle of the reflecting plane of the concave conical reflector 2 with respect to its optical axis (same with the caxis) is 45o in Fig. 1(a), whereas this angle of the concave conical reflector 2 in Fig. 2(a) is α calculated by the Eq. (12). The additional optical element 6 in Fig. 2(a) is a transparent optical plate used for the support of the cylindrical polarizing prism 5, the 45o planar reflecting prism 1, and the convex conical reflector 4 in Fig. 2(a). The other optical elements 3, 4, P1, P2, and P3 are same with those in Fig. 1(a).

The cylindrical vector beam is generated in the following manner. Incident light upon entering the device, is reflected by the planar reflecting prism 1 and will propagate along the c axis of the cylindrical polarizing prism 5 to the interface I. The o and e components of the incident light propagate co-linearly and the refractive index n of the e-ray is equal to the primary refractive index no of the o-ray. Under the proper conditions (will be given later), both the o- and e-rays are reflected entirely at the interface I. The direction of the reflected e- and o-rays will deviate at interface I due to the birefringence of prism 5 [see Fig. 2(a)].

In addition, as the cylindrical polarizing prism 5 is a negative birefringent crystal, i.e., nenno (ne is the primary refractive index of the e-ray), the reflection angle θe of the e-ray at the interface I is bigger than the incident angle θo of the incident light. Therefore, the reflected o-ray at the interface I is still reflected entirely by the interface II, whereas the reflected e-ray at the interface I will be transmitted at the interface II. So, the cylindrical polarizing prism 5 in Fig. 2(a) has the same function with the combination of the convex conical reflector 1 and the cylindrical polarizing prism 5 in Fig. 1(a).

Through selecting the proper angle α of the concave conical reflector 2, we can always make the light reflected by the concave conical reflector 2 propagate along the optical axis (same with the caxis) of the cylindrical polarization analyzer. Obviously, the cylindrical polarization analyzer shown in Fig. 2(a) has the same function with that shown in Fig. 1(a). Compared with the cylindrical polarization analyzer shown in Fig. 1(a), the fabrication of the cylindrical polarizing prism 5 in Fig. 2(a) is easier and more feasible because it is just a convex 45o conical transmitting prism with negative birefringent crystal.

The natural question arises whether the above propagating procedure of the o- and e-rays can happen. Let's take the calcite for example. The calcite is a usual material used for the polarizing prism and its primary refractive indices are no=1.625~1.698 and ne=1.475~1.504 for wavelength λ=350~2000nm. Note that all parameters about no and ne of the calcite are seen in [38]. Shown in Fig. 2(b), as the cylindrical polarizing prism 5 is the convex 45o conical transmitting prism, the incident angle of the incident light is θo=45o that is always bigger than the total reflection angle of the o-ray at the interfaces I and II (about 36o~38o). So, the o-ray is reflected entirely at the interfaces I and II. For the reflected e-ray at the interface I, the following relations are met
nsinθe=nosinθo,
(10a)
n=noneno2sin2θ+ne2cos2θ,
(10b)
θ=πθoθe,
(10c)
where θe is the reflection angle of the e-ray, θ is the angle of the propagating direction of the reflected e-ray at the interface I with respect to the c axis, and n is the corresponding refractive index of the e-ray. At the interface II, there is
nsin(π2θe)=n0sinθi,
(11)
where n0=1 is the refractive index in free space and θi is the refraction angle of the e-ray at the interface II.

In terms of Eqs. (10) and (11), under the case of λ=350~2000nm, it is easy to calculate that n=1.477, θe=51.1o, θi=68o for no=1.625and ne=1.475. For no=1.698and ne=1.504, there are n=1.51, θe=52.8o, and θi=65.6o. The refraction angles of the e-ray at the interface II θi=68o and θi=65.6o mean that the e-ray can be transmitted at the interface II. Therefore, the cylindrical polarization analyzer shown in Fig. 2(a) is feasible.

We note that θi is different for the various wavelengths of the incident light, which means that the output beam of the cylindrical polarization analyzer shown in Fig. 2(a) might be a little divergent or convergent when the angle α of the concave conical reflector 2 is given. The solution to this problem is to limit the range of the wavelength of the incident light for the cylindrical polarization analyzer. For example, for the visible lights with λ=400~780nm, no=1.649~1.683 and ne=1.482~1.497. We can calculate θi=65.9o for no=1.683and ne=1.498 (i.e., λ=400nm) and θi=66.6o for no=1.649and ne=1.482 (i.e., λ=780nm). For the average wavelength λ=588nm, no=1.659, ne=1.486, and θi=66.3o. If the angle α of the concave conical reflector 2 is designed in terms of the average wavelength λ=588nm, the divergent or convergent degree of the output beam is minimum and about ±0.3o. In terms of the geometrical relations of Fig. 2(a), α can be calculated by the following formulae
β+α=π/2,
(11a)
α+π4=(π2θi)+β,
(11b)
namely,
α=3π812θi,
(12)
where β is the incident angle at the surface of the concave conical reflector 2. For λ=588nm (θi=66.3o),α43.3o.

The above analyses show that the second type of the cylindrical polarization analyzer [i.e., Fig. 2(a)] is feasible and its fabrication is easier than that of the first type. However, compared with the second type, one major advantage of the first type is that there are not divergent or convergent for the output beams with various wavelengths because the transmitted e-ray always propagates along the direction normal to the surfaces of the cylindrical polarizing prism 5 [see Fig. 1(a)]. The other advantage of the first type is it allows for a safe separation of the back reflected light with a suitable beam stop due to the Glan polarizing prism design. For the second type, as the rejected o-ray will return directly back to the laser and is azimuthal polarized, one had better use the polarization independent optical isolator before the laser to optically isolate the rejected o-ray.

3. Conclusion

Acknowledgments

This work was supported by the National Basic Research Program of China (2011CB707504), the Leading Academic Discipline Project of Shanghai Municipal Government (S30502), the National Natural Science Foundation of China (61178079 and 61137002), the Fok Ying-Tong Education Foundation, China (121010), the Foundation for the Author of National Excellent Doctoral Dissertation of PR China (201033), and the Science and Technology Commission of Shanghai Municipality (STCSM) (11JC1413300).

References and links

1.

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

2.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef] [PubMed]

3.

Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A 24(6), 1793–1798 (2007). [CrossRef] [PubMed]

4.

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

5.

Z. Zhang, J. Pu, and X. Wang, “Tight focusing of radially and azimuthally polarized vortex beams through a uniaxial birefringent crystal,” Appl. Opt. 47(12), 1963–1967 (2008). [CrossRef] [PubMed]

6.

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1(1), 1–57 (2009). [CrossRef]

7.

Y. J. Yoon, W. C. Kim, N. C. Park, K. S. Park, and Y. P. Park, “Feasibility study of the application of radially polarized illumination to solid immersion lens-based near-field optics,” Opt. Lett. 34(13), 1961–1963 (2009). [CrossRef] [PubMed]

8.

S. Sato and Y. Kozawa, “Hollow vortex beams,” J. Opt. Soc. Am. A 26(1), 142–146 (2009). [CrossRef] [PubMed]

9.

K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express 18(5), 4518–4525 (2010). [CrossRef] [PubMed]

10.

X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett. 35(23), 3928–3930 (2010). [CrossRef] [PubMed]

11.

H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20(14), 14891–14905 (2012). [CrossRef] [PubMed]

12.

Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express 19(17), 15947–15954 (2011). [CrossRef] [PubMed]

13.

B. Tian and J. Pu, “Tight focusing of a double-ring-shaped, azimuthally polarized beam,” Opt. Lett. 36(11), 2014–2016 (2011). [CrossRef] [PubMed]

14.

Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A 29(11), 2439–2443 (2012). [CrossRef] [PubMed]

15.

Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express 12(15), 3377–3382 (2004). [CrossRef] [PubMed]

16.

L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett. 37(10), 1694–1696 (2012). [CrossRef] [PubMed]

17.

N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett. 85(25), 6239–6241 (2004). [CrossRef]

18.

K. Yoshiki, R. Kanamaru, M. Hashimoto, N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett. 32(12), 1680–1682 (2007). [CrossRef] [PubMed]

19.

M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process. 86(3), 329–334 (2007). [CrossRef]

20.

G. Terakado, K. Watanabe, and H. Kano, “Scanning confocal total internal reflection fluorescence microscopy by using radial polarization in the illumination system,” Appl. Opt. 48(6), 1114–1118 (2009). [CrossRef]

21.

T. J. Gould, J. R. Myers, and J. Bewersdorf, “Total internal reflection STED microscopy,” Opt. Express 19(14), 13351–13357 (2011). [CrossRef] [PubMed]

22.

Y. Xue, C. Kuang, S. Li, Z. Gu, and X. Liu, “Sharper fluorescent super-resolution spot generated by azimuthally polarized beam in STED microscopy,” Opt. Express 20(16), 17653–17666 (2012). [CrossRef] [PubMed]

23.

K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett. 31(14), 2151–2153 (2006). [CrossRef] [PubMed]

24.

Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett. 30(22), 3063–3065 (2005). [CrossRef] [PubMed]

25.

M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett. 32(22), 3272–3274 (2007). [CrossRef] [PubMed]

26.

Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett. 79(11), 1587–1589 (2001). [CrossRef]

27.

S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt. 29(15), 2234–2239 (1990). [CrossRef] [PubMed]

28.

N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A 22(5), 984–991 (2005). [CrossRef] [PubMed]

29.

P. Ma, P. Zhou, Y. Ma, X. Wang, R. Su, and Z. Liu, “Generation of azimuthally and radially polarized beams by coherent polarization beam combination,” Opt. Lett. 37(13), 2658–2660 (2012). [CrossRef] [PubMed]

30.

R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express 18(10), 10834–10838 (2010). [CrossRef] [PubMed]

31.

X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007). [CrossRef] [PubMed]

32.

S. C. McEldowney, D. M. Shemo, and R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystal polymers,” Opt. Express 16(10), 7295–7308 (2008). [CrossRef] [PubMed]

33.

M. Bashkansky, D. Park, and F. K. Fatemi, “Azimuthally and radially polarized light with a nematic SLM,” Opt. Express 18(1), 212–217 (2010). [CrossRef] [PubMed]

34.

Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt. 41(22), 4630–4637 (2002). [CrossRef] [PubMed]

35.

K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt. 46(30), 7544–7551 (2007). [CrossRef] [PubMed]

36.

A. Shoham, R. Vander, and S. G. Lipson, “Production of radially and azimuthally polarized polychromatic beams,” Opt. Lett. 31(23), 3405–3407 (2006). [CrossRef] [PubMed]

37.

http://www.rpcphotonics.com/

38.

http://refractiveindex.info/?group=CRYSTALS&material=CaCO3

OCIS Codes
(220.2740) Optical design and fabrication : Geometric optical design
(230.5440) Optical devices : Polarization-selective devices
(260.1180) Physical optics : Crystal optics
(260.5430) Physical optics : Polarization

ToC Category:
Physical Optics

History
Original Manuscript: January 22, 2013
Revised Manuscript: March 7, 2013
Manuscript Accepted: March 7, 2013
Published: March 15, 2013

Citation
Qi Hu, Zhihua Tan, Xiaoyu Weng, Hanming Guo, Yang Wang, and Songlin Zhuang, "Design of cylindrical vector beams based on the rotating Glan polarizing prism," Opt. Express 21, 7343-7353 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-6-7343


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References

  1. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express7(2), 77–87 (2000). [CrossRef] [PubMed]
  2. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91(23), 233901 (2003). [CrossRef] [PubMed]
  3. Y. Kozawa and S. Sato, “Sharper focal spot formed by higher-order radially polarized laser beams,” J. Opt. Soc. Am. A24(6), 1793–1798 (2007). [CrossRef] [PubMed]
  4. H. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008). [CrossRef]
  5. Z. Zhang, J. Pu, and X. Wang, “Tight focusing of radially and azimuthally polarized vortex beams through a uniaxial birefringent crystal,” Appl. Opt.47(12), 1963–1967 (2008). [CrossRef] [PubMed]
  6. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon.1(1), 1–57 (2009). [CrossRef]
  7. Y. J. Yoon, W. C. Kim, N. C. Park, K. S. Park, and Y. P. Park, “Feasibility study of the application of radially polarized illumination to solid immersion lens-based near-field optics,” Opt. Lett.34(13), 1961–1963 (2009). [CrossRef] [PubMed]
  8. S. Sato and Y. Kozawa, “Hollow vortex beams,” J. Opt. Soc. Am. A26(1), 142–146 (2009). [CrossRef] [PubMed]
  9. K. Kitamura, K. Sakai, and S. Noda, “Sub-wavelength focal spot with long depth of focus generated by radially polarized, narrow-width annular beam,” Opt. Express18(5), 4518–4525 (2010). [CrossRef] [PubMed]
  10. X. Hao, C. Kuang, T. Wang, and X. Liu, “Phase encoding for sharper focus of the azimuthally polarized beam,” Opt. Lett.35(23), 3928–3930 (2010). [CrossRef] [PubMed]
  11. H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express20(14), 14891–14905 (2012). [CrossRef] [PubMed]
  12. Y. Kozawa, T. Hibi, A. Sato, H. Horanai, M. Kurihara, N. Hashimoto, H. Yokoyama, T. Nemoto, and S. Sato, “Lateral resolution enhancement of laser scanning microscopy by a higher-order radially polarized mode beam,” Opt. Express19(17), 15947–15954 (2011). [CrossRef] [PubMed]
  13. B. Tian and J. Pu, “Tight focusing of a double-ring-shaped, azimuthally polarized beam,” Opt. Lett.36(11), 2014–2016 (2011). [CrossRef] [PubMed]
  14. Y. Kozawa and S. Sato, “Focusing of higher-order radially polarized Laguerre-Gaussian beam,” J. Opt. Soc. Am. A29(11), 2439–2443 (2012). [CrossRef] [PubMed]
  15. Q. Zhan, “Trapping metallic Rayleigh particles with radial polarization,” Opt. Express12(15), 3377–3382 (2004). [CrossRef] [PubMed]
  16. L. Huang, H. Guo, J. Li, L. Ling, B. Feng, and Z. Y. Li, “Optical trapping of gold nanoparticles by cylindrical vector beam,” Opt. Lett.37(10), 1694–1696 (2012). [CrossRef] [PubMed]
  17. N. Hayazawa, Y. Saito, and S. Kawata, “Detection and characterization of longitudinal field for tip-enhanced Raman spectroscopy,” Appl. Phys. Lett.85(25), 6239–6241 (2004). [CrossRef]
  18. K. Yoshiki, R. Kanamaru, M. Hashimoto, N. Hashimoto, and T. Araki, “Second-harmonic-generation microscope using eight-segment polarization-mode converter to observe three-dimensional molecular orientation,” Opt. Lett.32(12), 1680–1682 (2007). [CrossRef] [PubMed]
  19. M. Meier, V. Romano, and T. Feurer, “Material processing with pulsed radially and azimuthally polarized laser radiation,” Appl. Phys., A Mater. Sci. Process.86(3), 329–334 (2007). [CrossRef]
  20. G. Terakado, K. Watanabe, and H. Kano, “Scanning confocal total internal reflection fluorescence microscopy by using radial polarization in the illumination system,” Appl. Opt.48(6), 1114–1118 (2009). [CrossRef]
  21. T. J. Gould, J. R. Myers, and J. Bewersdorf, “Total internal reflection STED microscopy,” Opt. Express19(14), 13351–13357 (2011). [CrossRef] [PubMed]
  22. Y. Xue, C. Kuang, S. Li, Z. Gu, and X. Liu, “Sharper fluorescent super-resolution spot generated by azimuthally polarized beam in STED microscopy,” Opt. Express20(16), 17653–17666 (2012). [CrossRef] [PubMed]
  23. K. Yonezawa, Y. Kozawa, and S. Sato, “Generation of a radially polarized laser beam by use of the birefringence of a c-cut Nd:YVO4 crystal,” Opt. Lett.31(14), 2151–2153 (2006). [CrossRef] [PubMed]
  24. Y. Kozawa and S. Sato, “Generation of a radially polarized laser beam by use of a conical Brewster prism,” Opt. Lett.30(22), 3063–3065 (2005). [CrossRef] [PubMed]
  25. M. A. Ahmed, A. Voss, M. M. Vogel, and T. Graf, “Multilayer polarizing grating mirror used for the generation of radial polarization in Yb:YAG thin-disk lasers,” Opt. Lett.32(22), 3272–3274 (2007). [CrossRef] [PubMed]
  26. Z. Bomzon, V. Kleiner, and E. Hasman, “Formation of radially and azimuthally polarized light using space-variant subwavelength metal stripe gratings,” Appl. Phys. Lett.79(11), 1587–1589 (2001). [CrossRef]
  27. S. C. Tidwell, D. H. Ford, and W. D. Kimura, “Generating radially polarized beams interferometrically,” Appl. Opt.29(15), 2234–2239 (1990). [CrossRef] [PubMed]
  28. N. Passilly, R. de Saint Denis, K. Aït-Ameur, F. Treussart, R. Hierle, and J. F. Roch, “Simple interferometric technique for generation of a radially polarized light beam,” J. Opt. Soc. Am. A22(5), 984–991 (2005). [CrossRef] [PubMed]
  29. P. Ma, P. Zhou, Y. Ma, X. Wang, R. Su, and Z. Liu, “Generation of azimuthally and radially polarized beams by coherent polarization beam combination,” Opt. Lett.37(13), 2658–2660 (2012). [CrossRef] [PubMed]
  30. R. Zheng, C. Gu, A. Wang, L. Xu, and H. Ming, “An all-fiber laser generating cylindrical vector beam,” Opt. Express18(10), 10834–10838 (2010). [CrossRef] [PubMed]
  31. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett.32(24), 3549–3551 (2007). [CrossRef] [PubMed]
  32. S. C. McEldowney, D. M. Shemo, and R. A. Chipman, “Vortex retarders produced from photo-aligned liquid crystal polymers,” Opt. Express16(10), 7295–7308 (2008). [CrossRef] [PubMed]
  33. M. Bashkansky, D. Park, and F. K. Fatemi, “Azimuthally and radially polarized light with a nematic SLM,” Opt. Express18(1), 212–217 (2010). [CrossRef] [PubMed]
  34. Q. Zhan and J. R. Leger, “Microellipsometer with radial symmetry,” Appl. Opt.41(22), 4630–4637 (2002). [CrossRef] [PubMed]
  35. K. J. Moh, X. C. Yuan, J. Bu, R. E. Burge, and B. Z. Gao, “Generating radial or azimuthal polarization by axial sampling of circularly polarized vortex beams,” Appl. Opt.46(30), 7544–7551 (2007). [CrossRef] [PubMed]
  36. A. Shoham, R. Vander, and S. G. Lipson, “Production of radially and azimuthally polarized polychromatic beams,” Opt. Lett.31(23), 3405–3407 (2006). [CrossRef] [PubMed]
  37. http://www.rpcphotonics.com/
  38. http://refractiveindex.info/?group=CRYSTALS&material=CaCO3

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