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We present, design and generate a new kind of vector optical fields with linear polarization distributions modeling to electric and magnetic field lines. The geometric configurations of “electric charges” and “magnetic charges” can engineer the spatial structure and symmetry of polarizations of vector optical field, providing additional degrees of freedom assisting in controlling the field symmetry at the focus and allowing engineering of the field distribution at the focus to the specific applications.
© 2013 Optical Society of America
Flexibly manipulating light is always expected due to its important applications in various subjects. Polarization, as an intrinsic nature of light, plays a very important role in the interaction of light with matter. It is undoubtedly of great importance that the state of polarization (SoP) serves as a degree of freedom for manipulating light. The vector optical fields created by engineering the spatial SoP configuration have attracted significant interest due to the unique feature, compared with homogeneously polarized fields [1]. Most previous works focused mainly on generation of vector fields with the cylindrical symmetry of linear polarization [1–7]. More importantly, it has been demonstrated that the vector optical fields with the cylindrical symmetry of linear polarization have many peculiar focusing properties [8–18], such as the sharper focus beyond diffraction limit [10, 15, 18] and the three-dimensional optical cages [13, 14, 16]. In addition, another type of vector optical fields with the elliptical symmetry of linear polarization have also been investigated [19, 20], i.e. the local linear polarization is always along the tangential direction of a series of concentric ellipses with the same eccentricity. In particular, the ellipse’s eccentricity, as a new degree of freedom, was used to control the focal field and to achieve an one-dimensional sharp line with a dimension of ∼λ × 2λ[19, 20].
Vector optical fields constructed by engineering the SoPs have extensive applications in lithography, confocal microscopy, optical trapping, quantum information, and near-field optics. Therefore, arranging the spatial SoP structure of an optical field, purposefully and carefully, is expected to lead to new effects and phenomena that can expand the functionality and enhance the capability of optical systems.
In this paper, we present, design and generate a new kind of local linearly polarized vector fields (L-LP-VFs) with the spatial SoP structures similar to the electric and magnetic field lines, with the aid of theory of electric and magnetic field lines of “electrical charges” and “magnetic charges”. Such a kind of L-LP-VFs break both the cylindrical and elliptical symmetries of linear polarization and enrich the family of vector fields. In particular, the SoP geometric configurations provide additional degrees of freedom assisting in controlling the field symmetry at the focus and allowing controlling of the field distribution at the focus to many potential and specific applications. This work confirmed again the fact that the approach we presented [5, 6, 16] is a concise way to generate the vector optical fields with arbitrary SoP distributions.
As is well known, electric and magnetic field lines may be constructed by tracing a topographic path in the direction of field. The geometry of the field lines of a field can completely specify the direction of field everywhere. An electrostatic field has a nonzero divergence and a zero curl, so electric field lines start at positive charges and end at negative charges (i.e., cannot be closed loops). Here we consider the case of coplanar N charges in the two-dimensional Cartesian coordinate system (x,y). The induced electric field at a point P(x,y) can be written as
where pj is the amount of jth charge located at (xj, yj). êx and êy are the unit vectors along the x and y directions.If we choose the L-LP-VF to possess the form as follows
and where Δ(x,y) is taken as the SoP distribution of L-LP-VF is the same as the electric field lines described by Eq. (1).The question is how to generate the demanded L-LP-VFs with the spatial SoP structure similar to the electric field lines described by Eq. (1). We have presented an approach for generating the vector optical fields in a common path interferometric configuration containing a 4f system, based on the wavefront reconstruction and the Poincáre sphere describing SoP of light [5, 6, 16]. In our approach as shown in Fig. 1, to generate the L-LP-VFs, two λ/4 wave plates placed in the Fourier plane of the 4f system should be used. In principle, the created L-LP-VF could be written as
where δ(x,y) is the additional phase distribution of the transmission function t(x,y) = {1 + γcos[2πf0x + δ(x,y)]}/2 of the holographic grating displayed in a spatial light modulator [5, 6, 16]. Compared Eq. (3) with Eq. (2), we can find that if δ(x,y) is set as δ(x,y) = Δ(x,y), the spatial SoP distribution of the created L-LP-VF is indeed the same as the structure of electric field lines described by Eq. (1).
Fig. 1 Schematic of experimental setup for generating L-LP-VFs with their SoP distributions similar to the electric and magnetic field lines. The main configuration is a 4f system composed of a pair of identical lenses (L1 and L2). A spatial light modulator (SLM) is located at the input plane of the 4f system. Two λ/4 waveplates behind a spatial filter (SF) with two apertures are placed in the Fourier plane of the 4f system. A Ronchi phase grating (G) is placed in the output plane of the 4f system.
To confirm experimentally the feasibility of our approach [5, 6, 16] for generating the L-LP-VFs described by Eq. (2), a linearly polarized He-Ne laser at a wavelength of 632.8 nm is expanded and collimated to obtain a nearly uniform intensity distribution and then illuminate the spatial light modulator (SLM) with a transmission function t(x,y) = {1 + γcos[2πf0x + δ(x,y)]}/2, where δ(x,y) = Δ(x,y) is described by Eq. (2b). The input field is diffracted into the ±1th orders carrying the respective wavefronts of exp[±jΔ(x,y)], by the computer-controlled holographic grating displayed at SLM. In the Fourier plane of the 4f system, the focused±1th orders by L1 are spatially filtered by SF and then converted into the right- and left-handed circularly polarized light by two λ/4 waveplates, respectively. The demanded L-LP-VFs are generated in the output plane of the 4f system, by combining the ±1th orders by L2 and G.
We first focus on two typical situations: (i) an electric dipole composed of a pair of unit positive charge (p1 = +1) and unit negative charge (p2 = −1), locate at (x1, y1) = (−d, 0) and (x2, y2) = (d, 0), respectively; (ii) two unit positive charges of (p1 = +1) and (p2 = +1), locate also at (x1, y1) = (−d, 0) and (x2, y2) = (d, 0), respectively. The directions of electric field lines of both cases are shown in the first column of Fig. 2. The simulated SoP distributions of the L-LP-VFs corresponding to the electric field lines of the electric dipole (p1, p2) = (+1, −1) and the dual unit positive charges (p1, p2) = (+1, +1) are indeed in complete agreement with the field lines in the first column of Fig. 2. Here we only show the directions of SoP distributions and the fields lines by the arrows.
Fig. 2 L-LP-VFs with their SoP distributions similar to the field lines of the electric dipole (first row) and the dual equal-positive charges (second row). First column shows the directions of the field lines or the SoP distributions of L-LP-VFs. Second and third columns (fourth and fifth columns) show the simulated and measured intensity patterns of the x-components (y-components), respectively. Any picture has a dimension of 6d × 6d.
We can see from Eq. (3) that the field intensity exhibits the uniform distribution. However, it is still interesting to discuss the dark spots that may occur on the intensity patterns. For the situation of “electric dipole”, the intensity pattern contains two dark spots located at the two “charges”. In contrast, in the situation of dual positive “charges”, in the intensity pattern, besides two dark spots located at the two “charges” there is also an additional dark spot located at (0, 0). For the directions of electric field lines or the SoP distributions shown in the first column, the singularities can be classified into two kinds: the first is located at the “charges”, which is referred to as “intrinsic singularity” labeled by the filled circles and the second is not located at the “charges”, where the zero intensity spot originates from the destructive interference, which can be referred to as “derivative singularity” labeled by the open circles. The “intrinsic singularity” is caused by the polarization uncertainty due to an infinite number of allowed polarizations. In contrast, the “derivative singularity” is caused by the polarization uncertainty due to a finite number of allowed polarizations. After passing through a horizontal polarizer, the intensity patterns of the L-LP-VFs exhibit the inhomogeneous distributions, as simulated patterns in the second column and as measured patterns in the third column. The simulated and measured intensity patterns behind a vertical polarizer are shown in the fourth and fifth column. The extinction locations are in agreement with the SoP distributions shown in the first column. The results confirm the fact that the L-LP-VFs with the spatial SoP distributions similar to the electric field lines can indeed be generated. The generated L-LP-VFs here do not hold a cylindrical symmetry [1–18] and an elliptical symmetry [19, 20] of polarization, whereas exhibit a mirror symmetry of polarization.
Figure 3 shows the generated L-LP-VFs with the SoP distributions similar to the electric field lines of two unequal charges of p2 ≡ −1 and different p1 = +2, +3, +4 and +5, in which p1 and p2 are still located at (x1, y1) = (−d, 0) and (x2, y2) = (d, 0). All the intensity exhibit the uniform distribution excluding some dark spots which correspond to the polarization singularities. The electric field lines and the SoP distributions have two “intrinsic singularities” located at (−d, 0) and (d, 0) for all the four situations. In addition, there is also a “derivative singularity” located at ( , 0), in detail, at (5.83d, 0) for p1 = +2, (3.73d, 0) for p1 = +3, (3d, 0) for p1 = +4 and (2.62d, 0) for p1 = +5. Since any picture has a dimension of 6d × 6d, the “derivative singularity” in the first row disappear for both p1 = +2 and p1 = +3. The intensity patterns behind the horizonal and vertical polarizers exhibit the peculiar patterns with a mirror symmetry about y = 0. Clearly, the simulated results in the second and fourth rows are in well agreement with the experimental ones in the third and fifth rows. The dark spots in the intensity patterns correspond to the polarization singularities.
Fig. 3 L-LP-VFs with their SoP distributions similar to the field lines of the dual unequal charges. First, second, third and fourth columns show the cases of (p1, p2) = (+2, −1), (+3, −1), (+4, −1) and (+5, −1), respectively. The “charges” p1 and p2 are always located at (x1, y1) = (−d, 0) and (x2, y2) = (d, 0), respectively. First row shows the directions of the electric field lines or the SoP distributions. Second and third rows (fourth and fifth rows) show the simulated and measured intensity patterns of the x-components (y-components), respectively. Any picture has a dimension of 6d × 6d.
We now investigate the situations of three and four “charges”. The two situations of three electric “charges” p1, p2 and p3, located at (−d, 0), (d, 0), (0, d), are shown in the first and second columns of Fig. 4. The two situations of four “charges” p1, p2, p3 and p4, located at (−d, 0), (d, 0), (0, d) and (0, −d), are shown in the third and fourth columns of Fig. 4. As shown in the first row, besides three (four) “intrinsic singularities” for the three (four) “charges”, the electric field lines or the corresponding SoP distributions have also two “derivative singularities” located at near (−d/2, d/2) between p1 and p3 and near (d/2, d/2) between p2 and p3 for (p1, p2, p3) = (+1, +1, +1) in the first column, respectively. There is only one “derivative singularity” located at near (0, −0.33d) for (p1, p2, p3) = (+1, +1, −1) in the second column. There are five “derivative singularities” distributed in four corners and center of a square with a side length of ∼0.77d and its center located at (0, 0), for (p1, p2, p3, p4) = (+1, +1, +1, +1) in the third column. In contrast, there is only one “derivative singularity” located at (0, 0), for (p1, p2, p3, p4) = (+1, +1, −1, −1) in the fourth column. Behind the horizonal and vertical polarizers, the extinction patterns exhibit a mirror symmetry about x = 0 only for the three “charges” in the first and second columns, while have a mirror symmetry about x = 0 and y = 0 simultaneously for the four “charges” in the third and fourth columns, respectively. The results reveals that the SoP distributions of the generated L-LP-VFs are in agreement with the theoretical designs.
Fig. 4 L-LP-VFs with their SoP distributions similar to the field lines of three charges (first and second columns) and four charges (third and four columns). First row shows the directions of the electric field lines or the SoP distributions. Second and third rows (fourth and fifth rows) show the simulated and measured intensity patterns of the x-components (y-components), respectively. Any picture has a dimension of 6d × 6d.
We now explore the L-LP-VFs with their SoP distributions similar to magnetic field lines. Here we focus on a magnetic dipole only, because no magnetic monopole exists. As an example, the magnetic field induced by a magnetic dipole created by a current loop can be written as
where d is the radius of the current loop, which is in the xz plane and centered on the origin. Referencing Eq. (2b) as the electric field lines, if only we set the additional phase as , the SoP distribution of the generated L-LP-VF should be the same as the magnetic field lines. Clearly, the SoP distribution exhibits two opposite polarization-vortex structures surrounding two singularities located at (x1, y1) = (−d, 0) and (x2, y2) = (d, 0). These are another kind of representative L-LP-VFs with dual polarization singularities, which are different from the cylindrical- and elliptical-symmetric L-LP-VFs with a single polarization singularity and the above L-LP-VFs with multiple polarization singularities. It should be pointed out that the L-LP-VFs with the SoP distributions imitating to the magnetic field lines have the two “intrinsic singularities”, whereas have no “derivative singularities”. To enrich this kind of L-LP-VFs, we introduce two new parameters m1 and m2 into Eq. (4), thus Eq. (4) can be rewritten as followsFigure 5 shows six typical situations of (m1, m2) = (1, 1), (m1, m2) = (1, 3), (m1, m2) = (1, 5), (m1, m2) = (3, 1), (m1, m2) = (5, 1), and (m1, m2) = (5, 5). As shown in the first row of Fig. 5, the magnetic field lines or the corresponding SoP distributions have always two singularities located at ( , 0) and ( , 0) given by Eq. (5), which are in well agreement with the zero-intensity spots of the generated L-LP-VFs shown in the simulated and measured components in Fig. 5. The SoP distributions break a cylindrical symmetry and an elliptical symmetry, while exhibit a mirror symmetry about x = 0 and a twofold rotation inversion symmetry. As shown in the second to fifth rows, the extinction patterns behind horizontal and vertical polarizers have a mirror symmetry about both x = 0 and y = 0, which are higher than the symmetry of the SoP distributions. As m1 and/or m2 increase, the x- and y-component fractions of the L-LP-VFs decrease and increase, respectively. In particular, the experimental results in the third and fifth rows are in well agreement with the simulations in the second and fourth row.
Fig. 5 L-LP-VFs with their SoP distributions similar to the field lines of magnetic dipoles for six situations. First row shows the directions of the magnetic field lines or the SoP distributions. Second and third rows (fourth and fifth rows) show the simulated and measured intensity patterns of the x-components (y-components), respectively. Any picture has a dimension of 6d × 6d.
Hereto we focus on the generations of various L-LP-VFs with their SoP distributions similar to the geometric configurations of the electric and magnetic field lines. It is very interesting to explore the tight focusing behaviors of these L-LP-VFs by a high numerical aperture (NA) lens. To treat the tightly focused optical field, as is well known, a basic approach was pioneered by Richards and Wolf [21]. Any vector field can be decomposed to two orthogonally-polarized components. The contribution of each component can be calculated by using the basic approach. Finally the tightly focused field can be given by combining the contributions of the two components.
For all the simulations, an aplanatic lens was used with a high NA = 0.9, a radius R0, and a focal length f = R0/NA. All the length units are normalized by the light wavelength λ. As shown in Fig. 6, we first simulate the tight focusing fields of the two typical L-LP-VFs with their SoP distributions similar to the field lines of the electric dipole and the dual-positive charges shown in Fig. 2. One can see that the focal fields exhibit no cylindrical and elliptical symmetry, whereas the mirror symmetry and central inversion symmetry caused by the symmetry of the SoP distribution. The focal fields can be engineered not only by the “charges” but also by the interval between them. For the case of the “electric dipole” in the top row, when d = 0.1 f the focal field has two strong spots separated by a sharp dark line. When d is increased to d = 0.3 f, the two strong spots are close to each other. When d = 0.5 f, the sharp dark line is completely cut to become two dark spots, while the two strong spots are connected. When d = 0.7 f, the focal field exhibits an “H” shape, in which its strong area is a round rectangle. When d = 0.9 f, the focal field becomes an elliptical strong area inside the background of a weak “H”. For the case of the dual positive charges in the bottom row, the focal field exhibits a nearly circular strong spot with a full width at half maximum (FWHM) of 0.74λ × 0.82λ when d = 0.1 f, a date-pit-like strong spot with a size of 0.58λ × 1.22λ (FWHM) when d = 0.3 f, a sharp line with a size of 0.50λ × 1.60λ (FWHM) when d = 0.5 f, and two separated strong spots when d = 0.7 f. When d = 0.9 f, there occur a pair of nearly circular strong spots in the vertical direction and a pair of nearly circular secondary strong spots in the horizontal direction, respectively.
Fig. 6 Intensity distributions of the tightly focused L-LP-VFs with the same SoP distributions as the field lines of dual charges. First and second rows correspond to the cases of electric dipoles and dual-positive charges, for different intervals between two “charges”, respectively. Any picture has a dimension of 4λ × 4λ.
Figure 7 shows the tightly focused fields of the L-LP-VFs with their SoP distributions similar to the field lines of the magnetic dipole with (m1, m2) = (1, 1), for different values of d. Two dark spots are surrounded by a strong racetrack-like shape when d = 0.1 f. The two dark spots have been connected each other when d = 0.2 f. The focal fields exhibit a pattern that a dark area is sandwiched between a pair of “lune” strong areas when d = 0.3 f and d = 0.4 f. A relatively dark elliptical area is surrounded by an ellipse-like strong ring when d = 0.5 f.
Fig. 7 Intensity distributions of the tightly focused L-LP-VFs with the same SoP distributions as the field lines of magnetic dipole with (m1, m2) = (1, 1), for different intervals between two singularities. Any picture has a dimension of 4λ × 4λ.
Figure 8 depicts the tight focusing fields of the L-LP-VFs with their SoP distributions similar to the magnetic field lines for different m1 when d = 0.3 f and m2 ≡ 1. As m1 increases from m2 = 1 to m1 = 5, the pattern of the focal field experiences an evolution as follows: when m1 is changed from m1 = 1 to m1 = 3, the central dark line is cut in the horizontal direction to become two dark spots and a pair of “ear” strong areas. When m1 = 4 and m1 = 5, the focal field patterns are composed of a relatively strong elliptical spot at the center and a pair of strong “ears”, just the central elliptical spot is stronger when m1 = 5.
Fig. 8 Intensity patterns of the tightly focused L-LP-VFs with the same SoP distributions as the magnetic field lines for different m1 when m2 ≡ 1 and d = 0.3 f. Any picture has a dimension of 4λ × 4λ.
In discussion, we should emphasize that any the intensity pattern is normalized by itself maximum. In particular, for the generated L-LP-VFs, only the SoP directions follow the actual directions of the field line patterns, whereas we do not care the similarity of the intensity distributions between the L-LP-VFs and the field lines. In principle, our approach should be able to generate arbitrary locally-elliptically-polarized vector fields, but two λ/4 waveplates in the Fourier plane of the 4f system should probably be replaced by two λ/2 waveplates and the additional phase distribution δ should also be redesigned. The peculiar patterns of the focal fields can be engineered by designing the SoP distributions.
In summary, we present a new kind of vector optical fields with the linear polarization distributions modeling to electric and magnetic field lines. This kind of vector optical fields enriches members of family of vector optical fields. We successfully generate the desired vector optical fields and then explore their novel tight focusing properties. We find the geometric configurations of “electrical charges” and “magnetic charges” of vector optical fields providing additional degrees of freedom assisting in controlling the symmetry property of field at the focus and allowing engineering of the field distribution at the focus to the specific applications, such as lithography, optical trapping and material processing. More importantly, our approach can flexibly create other kinds of vector fields.
Acknowledgments
This work is supported by the National Basic Research Program (973 Program) of China under Grant No. 2012CB921900 and the National Natural Science Foundation of China under Grant No. 10934003.
References and links
1. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009) [CrossRef] .
2. J. Arlt and M.J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25, 191–193 (2000) [CrossRef] .
3. Z. Bomzon, G. Biener, V. Kleiner, and E. Hasman, “Radially and azimuthally polarized beams generated by space-variant dielectric subwavelength gratings,” Opt. Lett. 27, 285–287 (2002) [CrossRef] .
4. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9, 78 (2007) [CrossRef] .
5. X. L. Wang, J. P. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector fields with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007) [CrossRef] [PubMed] .
6. X. L. Wang, Y. N. Li, J. Chen, C. S. Guo, J. P. Ding, and H. T. Wang, “A new type of vector fields with hybrid states of polarization,” Opt. Express 18, 10786–10795 (2010) [CrossRef] [PubMed] .
7. S. Liu, P. Li, T. Peng, and J. L. Zhao, “Generation of arbitrary spatially variant polarization beams with a trapezoid Sagnac interferometer,” Opt. Express 20, 21715–21721 (2012) [CrossRef] [PubMed] .
8. K. S. Youngworth and T. G. Brown, “Focusing of high numerical aperture cylindrical vector beams,” Opt. Express 7, 77–87 (2000) [CrossRef] [PubMed] .
9. Q. Zhan and J. Leger, “Focus shaping using cylindrical vector beams,” Opt. Express 10, 324–331 (2002) [CrossRef] [PubMed] .
10. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003) [CrossRef] [PubMed] .
11. N. Bokor and N. Davidson, “Generation of a hollow dark spherical spot by 4π focusing of a radially polarized Laguerre-Gaussian beam,” Opt. Lett. 31, 149–151 (2006) [CrossRef] [PubMed] .
12. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31, 820–822 (2006) [CrossRef] [PubMed] .
13. W. Chen and Q. Zhan, “Three-dimensional focus shaping with cylindrical vector beams,” Opt. Commun. 265, 411–417 (2006) [CrossRef] .
14. N. Bokor and N. Davidson, “A three dimensional dark focal spot uniformly surrounded by light,” Opt. Commun. 279, 229–234 (2007) [CrossRef] .
15. H. F. Wang, L. P. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photon. 2, 501–505 (2008) [CrossRef] .
16. X. L. Wang, J. P. Ding, J. Q. Qin, J. Chen, Y. X. Fan, and H. T. Wang, “Configurable three-dimensional optical cage generated from cylindrical vector beams,” Opt. Commun. 282, 3421–3425 (2009) [CrossRef] .
17. W. Zhang, S. Liu, P. Li, X. Y. Jiao, and J. L. Zhao, “Controlling the polarization singularities of the focused azimuthally polarized beams,” Opt. Express 21, 974–983 (2013) [CrossRef] [PubMed] .
18. L. X. Yang, X. S. Xie, S. C. Wang, and J. Y. Zhou, “Minimized spot of annular radially polarized focusing beam,” Opt. Lett. 38, 1331–1333 (2013) [CrossRef] [PubMed] .
19. G. M. Lerman and U. Levy, “Tight focusing of spatially variant vector optical fields with elliptical symmetry of linear polarization,” Opt. Lett. 32, 2194–2196 (2007) [CrossRef] [PubMed] .
20. G. Lerman, Y. Lilach, and U. Levy, “Demonstration of spatially inhomogeneous vector beams with elliptical symmetry,” Opt. Lett. 34, 1669–1671 (2009) [CrossRef] [PubMed] .
21. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system,” Proc. Roy. Soc. A 253, 358–379 (1959) [CrossRef] .
