We present a systematic study of the superposition of two vector Laguerre-Gaussian (LG) beams. Propagation depended field distribution obtained from the superposition of two vector LG beams has many interesting features of intensity and polarization. Characteristic inhomogeneous polarization distribution of the vector LG beam appears in the form of azimuthally modulated intensity and polarization distributions in the superposition of the beams. We found that the array of polarization singular points, whose number depends upon the azimuthal indices of the two beams, evolves during propagation of the field. The position and number of C-points generated in the field were analyzed using Stokes singularity relations. Novel intensity and polarization patterns obtained from the superposition of two vector LG beams may find applications in the field of molecular imaging, optical manipulation, atom optics, and optical lattices.
©2013 Optical Society of America
The state of polarization of electromagnetic field is one of the most fundamental properties which describe the vectorial character of the field. It is a spatially dependent feature of the field and can have a homogenous or inhomogenous distribution. From the viewpoint of the spatial distribution of polarization, LG beams can be classified into scalar or vector beams [1, 2]. Scalar LG beams are the solutions of the scalar wave equation under the paraxial approximation and have a helical-shaped wavefront, which is associated with the property of orbital angular momentum. Superposition of the beams plays an important role in many optical phenomena. For example, depending on the azimuthal and radial index, superposition of scalar LG beams can produce self-imaging beams, spiral-type beams, and structurally stable beams . In the case of a superposition of two scalar LG beams with unequal topological charges, the resultant beam has many interesting features related to an effective spiral phase, which is different from the phase of either of the beams [4, 5]. Such a superposition of two beams is known to have residual orbital angular momentum . Detailed studies have been performed on the formation of a composite vortex [7, 8] in the superposition of two scalar LG beams. Various applications of the superposition of scalar LG beams have been demonstrated [4, 6, 9–12]. In all of these applications, a commonly observable property is the transformation from azimuthal phase variation to azimuthal intensity variation.
On the other hand, vector LG beam, which has spatially inhomogenous polarization distribution, is known to have rich polarization properties and have the ability to generate new polarization patterns [1, 13–15]. Different from the superposition of scalar LG beams vector LG beam superposition is expected to produce novel intensity and polarization patterns. Because of vectorial character, the superposition of two vector LG beams and associated singularities are not as straightforward as that of scalar LG beams. Recently, two vector beams with different weighting factors have been used under a tight focusing condition, to achieve an arbitrary 3D polarization orientation control . A superposition of vector beams of different mode orders has been used to produce asymmetric beam rotations . Radially and azimuthally polarized beams have also been used in an interferometric setup for improving accuracy of phase change measurements . All such applications require comprehensive understanding of the intensity and polarization properties of superposed vector beams. Motivated by the above works, it is interesting to explore intensity and polarization properties of the superposition of two vector LG beams. The objective of present work is to study the intensity and polarization properties of the vector field generated by the superposition of two vector LG beams, with particular emphasis on polarization singularity.
In this paper, we performed a systematic study on the superposition of two vector LG beams. General expression for total intensity distribution is derived for the superposition of two vector LG beams. Due to the spatially varying polarization distribution, the superposition of two vector LG beams forms a propagation depended elliptically polarized vector field involving polarization singular points. To find the number and the position of C-point in the field, we analyzed the Stokes vortices in the superposed field. We found that vector LG beam has many distinct characters which are absent for the scalar LG beams. Present results give a comparison between superposition properties of scalar and vector LG beams.
2. Vector Laguerre-Gaussian beam
Vector LG beams are paraxial solutions of the vector Helmholtz equation in cylindrical coordinate systems . Vector LG beams have a spatially varying polarization distribution with a cylindrical symmetry. A simple expression for vector LG beam in the paraxial condition can be written as Eq. (1). These four polarization distributions have different angular dependence of the local polarization direction. At each point on the beam cross section, while the local state of polarization is linear there is an azimuthal variation in the polarization direction. The local angle θp(ϕ) of polarization vector for different polarization types of vector LG modes are shown in Table 1. In all of these cases, the direction of polarization vector is undefined at the beam centre, which can also be viewed as an isolated vector point singularity (V-point). In general, a V-point is characterized by its winding number or Poincaŕe-Hopf index, which is quantified by the net rotation angle (divided by 2π) of the polarization vectors surrounding the V-point . In vector LG beams with an azimuthal index of m, the V-point possesses the winding number η = ± m. The sign of the winding number corresponds to the angular dependence of θp for four different types of vector LG beams as shown in Table 1. It is important to note that type I and III have the same sign of η ( = + m). Similarly, type II and IV have the same sign of η ( = -m) opposite that of type I and III.
When the azimuthal index of a vector LG beam m is equal to zero, then the beam is reduced to a linearly polarized Gaussian beam with the angle of θp(ϕ) = π/2 (for type-I and II) or the angle of θp(ϕ) = 0 (for type-III and IV). For m = 1, types I and III correspond to well-known azimuthally and radially polarized beams respectively, whereas types II and IV correspond to hybrid vector beams [see Fig. 1(a)]. For larger azimuthal index m, vector beams with complicated polarization distribution can be obtained. Figure 1(b) illustrates the intensity and polarization distributions for m = 2.
As we can see, all these field distributions with the same value of m have an identical intensity distribution but polarization patterns are different. This spatially varying polarization distribution together with the cylindrical symmetry provides unique properties to vector LG beams. The azimuthally dependent polarization distribution provides many interesting intensity and polarization features to vector LG beams which we will discuss in the following sections. In all the calculations, we assumed that the wavelength (λ) is 632.8 nm and w0 = 0.3 × 10−3 m.
3. Superposition of two vector Laguerre-Gaussian beams
The total electric field of superposition of two vector LG beams can be written as , where I01 and I02 are the intensities of two vector LG beams. The subscripts 1 and 2 in Ep1,m1 and Ep2,m2 represent the beam 1 and beam 2, respectively. Then, the general expression of the intensity distributions for the superposition of two vector LG beams is given by
To investigate the intensity zeros in the field, we follow the same procedure as given in reference . For the case of p1 = p2 = 0, we can obtain the condition for the formation of dark points around the beam axis. The condition to get dark points (I = 0) requires Q(ϕ)S(z) = −1, which gives the radial position rd of dark points similar to the case for the superposition of scalar LG beams. Because both Q(ϕ) and S(z) are trigonometric functions, the radial position of dark points can be obtained only when either Q(ϕ) = 1 and S(z) = −1, or Q(ϕ) = −1 and S(z) = 1 is satisfied asEq. (4), then |m1−m2| dark points angularly distributes in the beam cross-section. The angular positions ϕd for these dark points are ϕd = 2nπ / |m1 – m2| for Q(ϕ) = 1, or ϕd = (2n + 1)π / |m1−m2| for Q(ϕ) = −1, where n = 0, 1, 2, …, |m1−m2|-1. Thus, the dark points surrounding the beam axis appear only when Q(ϕ) = ± 1 and and the angular position varies according to the sign of Q(ϕ). The position of z-planes which contain dark points (zd) can be obtained from the condition S(z) = ± 1, byEq. (5) is the same as the scalar LG beam case , the propagating field of the superposed vector LG beams shows different behavior compared to that of scalar LG beams as shown in the following sections.
3.1. superposition of two vector Laguerre-Gaussian beams with the same azimuthal indices (m1 = m2)
As we have seen from Eq. (1), four different polarization distributions are possible for single azimuthal index of a vector LG beam. For the superposition of two vector LG beams with the same azimuthal indices, the following two cases arise.
(a) the same sign of η
Figure 2(a) shows the intensity distributions (at z = 0 plane) for the superposition of two vector LG beams with m1 = m2 = 2 and polarization distribution (I1 + I2) for both beams. Here, we assume that the beams have equal intensity distribution, namely, α = 1. As expected in this case, the total intensity distribution is the same as that of the initial beams. Other combinations (II1 + II2, III1 + III2, IV1 + IV2, I1 + III3, and II1 + IV2) which have same sign of η will also exhibit similar behaviors because the local polarizations of both beams are completely identical or orthogonal to each other.
(b) opposite sign of η
Figure 2(b) shows the intensity distributions (at z = 0) for two beams with the same azimuthal indices (m1 = m2 = 2) but opposite sign of η, namely, (I1 + II2). In this case, a petal shaped intensity pattern is obtained which is the same as that of the bright ring lattice obtained for a superposition of two scalar LG beams with equal but opposite topological charges . From Eq. (4), it is clear that the number of petals in the intensity distribution is equal to 2|m|. In other combinations (III1 + IV2, II1 + III2, IV1 + I2), the node positions for the superposed field differ in angle according to the combination of the beams as indicated in Eq. (4).
Although the azimuthal indices of two beams in the above two cases are the same, the superposed field has different intensity structures due to different sign of η. Apart from the value of the azimuthal index m we can infer, from the above results that polarization distribution also plays a major role in deciding the intensity distribution of the superposed field. By choosing different polarization distributions and higher azimuthal indices, a variety of intensity patterns can be obtained.
3.2. Superposition of two vector Laguerre-Gaussian beams having different azimuthal indices (m1≠m2)
Next, we consider the superposition of two vector LG beams with unequal azimuthal indices.
(a) the same sign of η
In this case, due to the difference of azimuthal index m, the intensity distribution of the superposed field is modulated in the azimuthal direction according to the local polarization states of two beams.
Figure 3(a) shows the resultant intensity pattern at z = 0 plane for the superposition of two beams with the same polarization orientations (type-I) and different azimuthal indices m1 = 1 and m2 = 2. In the present example, a single dark point appears at the azimuthal angle π. This is consistent with the condition of dark point formation Q(ϕ)S(z) = −1, where dark point is formed at the z = 0. The radial position is equal to w0, which can be obtained from Eq. (5). For higher values of azimuthal indices, the number of dark points will be larger. The number of dark points in the azimuthal direction depends on the difference of the azimuthal indices |m1-m2|. As shown in Eq. (4), other combinations are also possible which can give the similar situation of different azimuthal indices and the same sign of η.
(b) opposite sign of η
Figure 3(b) shows the intensity pattern for the superposition of two beams with azimuthal indices m1 = 1 (type-I) and m2 = 2 (type-II). The resultant intensity distribution has three dark points around the beam axis. In this combination, according to Eq. (4), the number of dark points in the resultant intensity pattern is |m1 + m2|, which are in contrast to the previous case. Although the time averaged intensity distribution of a scalar LG beams is identical to a vector LG beam with the same mode indices, the phase and polarization properties are different. This difference in the phase and the polarization properties appears in the intensity distribution when the two beams are superposed. In the case of superposition of two scalar LG beams, the spiral phase distribution is responsible for the azimuthal intensity variation. Whereas, in the superposition of vector LG beams, the angular variation of polarization direction is responsible for the azimuthal intensity variation.
4. Propagation of the superposed vector Laguerre-Gaussian beam
In order to further investigate the superposition of two vector LG beams, we study the paraxial propagation of resultant field in free space. The field distribution is calculated in the xy-plane and we assume that beams are propagating in the z direction. Figure 4(a) shows the total intensity distributions for a superposition of two vector LG beams with m1 = m2 = 2 and polarization type-I and II. When the azimuthal indices of two beams are same, the resultant intensity distribution has a petal shape pattern. As previously discussed, this pattern is similar to the superposition of two scalar LG beams with equal and opposite topological charges, where bright and dark petals are obtained and their position remains fixed during propagation . The formation of the petal pattern for vector LG beams can be understood using Eq. (3), which has two terms Q(ϕ) and S(z) with different roles. Q(ϕ) is responsible for the modulation of intensity in the azimuthal direction due to the interference between two beams with different polarization patterns. The tem S(z) provides the Gouy phase shift to the superposed field. This term is unity in the case of the same azimuthal indices and hence there is no contribution to the superposed field. As a result, the intensity patters remain the same during propagation.
Figure 4(b) shows the intensity distributions for the superposition of two vector beams with different azimuthal indices (m1 = 1 and m2 = 5) and the same sign of η (type-I). Figure 4(c) shows the intensity distributions for two beams with m1 = 1 and m2 = 5 but opposite sign of η. (type-I and II). In both cases, it can be seen that the intensity maxima and minima interchange their positions in the beam cross section during propagation. We can consider this variation as a flip in the intensity pattern. These intensity flips appear when |m1-m2| ˃ 2. This behavior is in contrast to the superposition of scalar beams, where dark points always appear in the superposed field.
The change of intensity distribution during propagation is due to the difference between the Gouy phase shifts represented by S(z) in Eq. (3). The peripheral dark point can be found on the z-planes where the condition S(z) = ± 1 is satisfied. This condition for S(z), therefore, interchanges the azimuthal angles of the peripheral dark points, which are determined by Q(ϕ) = in Eq. (3). To further illustrate the intensity flip for larger azimuthal index difference (m1 = 1 with polarization type-I and m2 = 8 with polarization type-IV), Media 1 gives an animated view of the flip in the intensity distribution during propagation between the positions z = -1.13zR to 1.13zR. We can clearly see that the superposed field accompanies an azimuthal intensity modulation as it propagates and the bright points switched to dark points and vice versa. The position of z-plane containing dark points can be deduced from Eq. (6). For the case shown in Media 1 seven planes exist at the distance of z = -4.38zR, −1.25zR, −0.48zR, 0, 0.48zR, 1.25zR, and 4.38zR respectively. From these results, the intrinsic feature of spatially varying polarization distribution in the vector beams is revealed.
5. Polarization singularities and complex Stokes field in superposition of two vector LG beams
So far, we have considered only the intensity distribution and its propagation characteristics for the superposition of two vector LG beams. In this section, we investigate the polarization properties of superposed vector field using Stokes parameters, especially in terms of polarization singularity.
Optical fields can have a variety of polarization singularities each of them have their own characteristic features. Polarization singularities of linearly and elliptically polarized fields and their associated indices can be described by the normalized Stokes parameters, which completely characterize the spatial distribution of the polarization state of a vector field [20–22]. The normalized Stokes parameters can be expressed as
The Stokes parameter analysis of vector field has been used to reveal polarization singularities, which are important elements for characterizing geometry and topology of vector fields [20–29]. Polarization singularities in a monochromatic 2D field are defined by C-points and L-lines. C-points are isolated points of circular polarization where the orientation of the major axis of polarization ellipse is undefined. At these points, S1 = S2 = 0 and S3 = ± 1, where S3 = 1 and S3 = −1 correspond to right-handed and left-handed circular polarizations, respectively. On the other hand, L-lines are lines connecting points where polarization is linear (S3 = 0) and are embedded in an ellipse field where the handedness of elliptical polarization is undefined . In addition to the C-points and L-lines, as mentioned previously, vector singularities (V-points) also exist, where the orientation of a linearly polarized vector field is undefined [28–30].
Polarization distribution of a vector field embedded with singular points can be shown using the complex Stokes fields , which are defined as with , . In the complex Stokes field S12 representation, C-points and V-points appear as phase vortices [20–22]. The L-lines occur at the intersection of counter lines of S2 = 0 and S3 = 0. The stokes index (winding number of Stokes vortices) for vector and elliptic field is given by σ12 = 2η and σ12 = 2Ic, where generally η have integer value and Ic have half integer value [20–22].
We calculated the phase φ12 of complex Stokes field S12 for vector LG beams at z = 0 plane, for two values of azimuthal indices (m = 1, and m = 2) as shown in Fig. 5. For easy identification, the polarization distribution is also shown. The center of the beam contains a Stokes vortex corresponding to a V-point, whose winding number σ12 is equal to ± 2m. From Figs. 5(a) and (b), it is clear that type-I and III have a Stokes vortex with σ12 = + 2m, and types II and IV, have a Stokes vortex with σ12 = −2m.
By simple coordinate transformation, the electric field of a vector LG beam can be written in the Cartesian coordinates (ix and iy are the unit vectors for the x and y directions, respectively) asEq. (7), the condition for C-points (S3 = ± 1) yieldsEq. (10), we assumed p1 = p2 = 0. In this case, the condition Eq. (9) is valid only when C ± (ϕ, z) = −1. Note that the plus and minus signs in the subscript of the notation C ± and in the right-hand side of Eq. (10) correspond to C-points with right-handed (S3 = 1) and left-handed (S3 = −1) circular polarizations, respectively. For any of the cases given by Eq. (10), the radial position of C-points rc in the beam cross-section expressed byEq. (11) is identical to radial position of the dark points given by Eq. (5). By contrast, the angular position ϕC ± of C-points depends on the handedness of polarization distribution of the superposed field. For example, for the cases of I1 + I2 and III1 + III2 in Eq. (10), is satisfied at C-points. Then, the angular position of each C-point is expressed asEq. (12), the angular position of each C-point varies depending on the z position, which shows a rotational behavior similar to the peripheral vortices formed by the superposition of two scalar LG beams . However, for the C-points in the superposed vector LG beams, both C-points with opposite handedness (S3 = 1 and S3 = −1) are evenly distributed on a z-plane. In addition, these C-points with opposite handedness will rotate in the opposite direction as the superposed beam propagates. In the following examples, we will see these polarization characteristics in the superposition of two vector LG beams.
(a) the same sign of η
Consider the case of a superposition of two vector LG beams with the same sign of η but different azimuthal indices. Figure 6(a) shows the intensity distribution for the superposition of two vector LG beams (m1 = 1 with type-I, and m2 = 2 with type-I). The spatial polarization distribution corresponding to the superposed field is shown in Fig. 6(b), where the line and ellipse represent linear and elliptical polarizations respectively. At beam waist (z = 0), the polarization is linear at every point. We can see that an elliptically polarized vector field is generated during propagation. A line of linear polarization (L-line), which intersects the beam axis horizontally, divides the elliptic field into two regions with opposite handedness [22, 26].
The position of L-line remains fixed during propagation. It can be seen from Fig. 6(b) that at z = 0 plane where each point on the beam cross-section is linearly polarized, an off-axis V-point (σ12 = + 2) corresponding to a dark point is generated. In general, for the other azimuthal indices |m1-m2| off-axis V-points will be generated. During propagation, linearly polarized field is transformed into the elliptically polarized field. The off-axis V-point transforms into two C-points, lying each in the region of opposite handedness. For easy identification, the C-points are marked with the squares in Fig. 6(b). The number of C-points can be generally expressed as 2|m1-m2| and is two in this case. To determine the winding number of C-points in the superposed field, we plotted the phase of complex Stokes field of the combined beams in Fig. 6(c). As we can see from Fig. 6(c), the Stokes vortices generated in the two regions of opposite handedness have the same winding number (σ12 = + 1). When the superposed field propagates through the beam waist (z = 0), C-points in the two regions move in the opposite directions to each other as shown in Fig. 6(b). According to Eq. (12), the angular positions ϕC ± of these C-points are given by.At the beam waist, both C-points with opposite handedness and the same winding number are located at the same position (rc = w0, ϕC ± = −π), where a pair of C-points is transformed into a dark point (V-point). Thus, at each z-plane containing dark points indicated by Eq. (6), a pair of C-points with opposite handedness and same the winding number is always located at the same position, resulting in dark points. The phase map Fig. 6(c) clearly shows the presence and the motion of a S12 vortex, which corresponds to a C-point with the same winding number in the superposed field. Media 2 shows the Stokes field corresponding to the field shown in Media 1. The change of the radial and angular position of the C-points is evident. C-points with opposite handedness and the same winding number move in the opposite directions and disappear at the point where a dark point is generated.
(b) opposite sign of η
Next we considered the superposition of two vector LG beams, where the signs of η are opposite and the azimuthal indices are different. Figure 7(a) shows the intensity distribution for two beams with (m1 = 1, and m2 = 2) and type-I and type-II polarization distributions. Figure 7(b) shows the spatial distribution of polarization at different z-planes. In contrast to the previous case, the polarization distribution is entirely different and the number of C-points generated is equal to 2|m1 + m2|. In this case, three L-lines are present which divides the total field into the regions of opposite handedness and each region has its own C-point. Figure 7(c) shows the phase map for complex Stokes field S12. In contrast to the previous case at z = 0 plane, three off-axis V-points (σ12 = −2) are generated. In general |m1 + m2| off-axis V-points will be generated. During propagation these three off-axis V-points are transformed into six C-points, which can be generally expressed as 2|m1 + m2|. The numbers of C-points in the right and left handed polarization region are equal. Figure 7(c) shows that the Stokes vortices generated in the regions of opposite handedness have the same winding number (σ12 = −1) which is opposite in sign to the previous case as shown in Fig. 6(c). The comparison between Figs. 6 and 7 reveals that the azimuthal indices of two beams are same but the polarization distribution of the resultant field and the associated singular points are different. The number of polarization singular points depends not only on the azimuthal indices but also on the sign of η. By superposing two vector LG beams with different polarization distributions, different number of polarization singular points can be generated.
Most of previous studies related to polarization singularities have been performed on either random vector fields or the superposition of orthogonally polarized scalar LG beams [26, 27, 31]. Present results have show that arrays of polarization singularity can be generated in a controlled manner by properly choosing azimuthal index and the polarization distribution of two superposed vector LG beams. Propagation depended vector field generated by a superposition of vector LG beams can be used to study the dynamics of polarization singularity. Present results also give a simple comparison between the scalar and vector LG beams. Both scalar and vector LG beams have identical intensity distributions for the same mode indices. The intensity distribution of superposition of two vector LG beams with the same azimuthal indices and opposite sign of η have a similar pattern as that of superposition of the two scalar LG beams with equal and opposite topological charges. In the case of the superposition of two scalar LG beams having different topological charges, the optical vortices are evolved due to an effective spiral phase and they are symmetrically distributed in the azimuthal direction. In contrast, in the case of the superposition of two vector LG beams, V-points and L-lines are evolved their number depends not only the azimuthal indices but also on the sign of η. During propagation of the field V-points transforms into the two C-points of same winding number. Both superpositions of scalar and vector LG beams produce an array of dark points, which are symmetrically distributed in the azimuthal direction. The dark points are associated with the optical vortices for scalar beams whereas for the vector beams they are associated with the axially symmetric polarization distribution having V-points. We believe that present study may help in understanding the mechanism of the appearance of polarization singularity in Möbius strip  superposition of waves [33, 34] inhomogeneous media [35–37], spatial polarization distributions [38, 39], imaging [40, 41], laser modes , focal field distribution , and studies related to angular momentum of light .
Polarization singularities have been explored in the intricate polarization distribution obtained from the superposition of two vector LG beams. Spatially varying polarization distribution is responsible for the intensity and polarization modulation in the propagated field. Expressions for the total intensity distribution of superposition of two vector LG beams have been derived and intensity distributions and its propagation characteristics for different cases are studied. Superposition of two vector LG beams can produce a propagation depended vector field with an array of polarization singular points. In the case of the superposition of the two vector LG beams with the same sign of η, |m1-m2| peripheral V-points are evolved. During propagation these peripheral V-points are transformed into 2|m1-m2| C-points. Whereas in the case of the superposition of two the vector LG beams with opposite sign of η, |m1 + m2| peripheral V-points are evolved. During propagation these peripheral V-points are transformed into 2|m1 + m2| C-points. Radial and angular positions of C-points in the superposed field have been obtained from Stokes singularity relations. Novel structures of intensity and polarization can be generated from various combinations of azimuthal indices. Present results may have an important implication in understanding the geometry and topological properties of vector beams during propagation.
This work was supported in part by Japan Society for the Promotion of Science, and Japan Science and Technology Agency (JST), Core Research for Evolutional Science and Technology (CREST).
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