Polarization singularities in superposition of vector beams

Sunil Vyas; Yuichi Kozawa; Shunichi Sato

doi:10.1364/OE.21.008972

1. Introduction

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 [3]. 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 [6]. 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 [16]. A superposition of vector beams of different mode orders has been used to produce asymmetric beam rotations [17]. Radially and azimuthally polarized beams have also been used in an interferometric setup for improving accuracy of phase change measurements [18]. 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 [2]. 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 [19]

E_{p, m} (r, ϕ, z) = {\begin{cases} Y_{p, m} (r, z) [\cos (m - 1) ϕ i_{ϕ} - \sin (m - 1) ϕ i_{r}] (Type - I) \\ Y_{p, m} (r, z) [\cos (m + 1) ϕ i_{ϕ} + \sin (m + 1) ϕ i_{r}] (Type - I I) \\ Y_{p, m} (r, z) [\sin (m - 1) ϕ i_{ϕ} + \cos (m - 1) ϕ i_{r}] (Type - I I I) \\ Y_{p, m} (r, z) [- \sin (m + 1) ϕ i_{ϕ} + \cos (m + 1) ϕ i_{r}] (Type - I V) \end{cases},

where r is the radius, ϕ is the azimuthal angle, z is the distance from the beam waist, p is the radial mode index, m is the azimuthal mode index and, i_ϕ, and i_r are the unit vectors of electric field for the azimuthal and radial directions, respectively. The term Y_p,m (r, z) is given by

Y_{p, m} (r, z) = \sqrt{\frac{2 p!}{π (p + m)!}} \frac{1}{w (z)} {[\frac{\sqrt{2} r}{w (z)}]}^{| m |} L_{p}^{| m |} [\frac{2 r^{2}}{w^{2} (z)}] \exp [- \frac{r^{2}}{w^{2} (z)}] \exp {- \frac{i k r^{2}}{2 R (z)} - i k z + i (2 p + | m | + 1) ψ (z)},

where w₀ is the minimum beam radius at z = 0, R(z) = (z² + z_R²)/z is the radius of curvature of the wavefront, z_R = kw₀²/2 is the Rayleigh length, k is the wave number, (2p + |m| + 1)ψ(z) is the Gouy phase shift with ψ (z) = tan⁻¹(z/z_R), and

w (z) = w_{0} \sqrt{1 + {(z / z_{R})}^{2}}

.

L_{p}^{| m |}

is the generalized Laguerre polynomial. For any azimuthal index m, a vector LG beam can have four different polarization distributions, which are in this study, classified as type-I, II, III and IV as denoted in 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 θ_p(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 [20]. 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 θ_p 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.

Table 1. Angular dependence of the polarization distribution for vector LG modes

View Table

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 θ_p(ϕ) = π/2 (for type-I and II) or the angle of θ_p(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.

Fig. 1 Numerically calculated intensity distributions together with the schematic polarization vector for the vector LG beams with the radial index p = 0, (a) different polarization distributions of the vector LG beam with the azimuthal index m = 1, (b) different polarization distributions of the vector LG beams with the azimuthal index m = 2.

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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 × 10⁻³m.

3. Superposition of two vector Laguerre-Gaussian beams

The total electric field of superposition of two vector LG beams can be written as $E = \sqrt{I_{01}} E_{p_{1}, m_{1}} + \sqrt{I_{02}} E_{p_{2}, m_{2}}$ , where I₀₁ and I₀₂ are the intensities of two vector LG beams. The subscripts 1 and 2 in E_p_1,_m₁ and E_p_2,_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

I = {| E |}^{2} = I_{01} {| E_{p_{1}, m_{1}} |}^{2} + I_{02} {| E_{p_{2}, m_{2}} |}^{2} + 2 \sqrt{I {}_{01}I_{02}} | E_{p_{1}, m_{1}} | | E_{p_{2}, m_{2}} | Q (ϕ) S (z),

where S(z) = cos[(2p₁ - 2p₂ + m₁ - m₂)ψ(z)] is the Gouy phase dependent term. Q(ϕ) is the azimuthally varying term, which arises due to the difference of polarization distribution between two superposed beams. This term provides special features to the intensity and polarization distribution of the resultant field. For a specific azimuthal index m, there are four different types of polarization distributions as mentioned previously and their superposition will give different effects on the resultant field. Superposing two vector LG beams with the azimuthal indices m₁ and m₂ can give sixteen combinations of polarization distributions, which can be summarized in the following six equations by avoiding duplication.

Q (ϕ) = {\begin{cases} \cos (m_{1} - m_{2}) ϕ for I_{1} + I_{2}, I I_{1} + I I_{2}, I I I_{1} + I I I_{2}, I V_{1} + I V_{2} \\ - \sin (m_{1} - m_{2}) ϕ for I_{1} + I I I_{2} \\ \sin (m_{1} - m_{2}) ϕ for I I_{1} + I V_{2} \\ \cos (m_{1} + m_{2}) ϕ for I_{1} + I I_{2}, I I I_{1} + I V_{2} \\ \sin (m_{1} + m_{2}) ϕ for I I_{1} + I I I_{2} \\ - \sin (m_{1} + m_{2}) ϕ for I V_{1} + I_{2} \end{cases}

where the subscripts 1 and 2 represent the beam 1 and 2, respectively. If the sign of η of the two vector LG beams are same, then the term m₁−m₂ always appears in Q(ϕ). On the contrary, if the sign of η are opposite then term m₁ + m₂ appears. The additive and subtractive terms of azimuthal indices will produce a different number of intensity maxima and minima in the intensity distribution.

To investigate the intensity zeros in the field, we follow the same procedure as given in reference [7]. 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 r_d 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 as

r_{d} = \frac{w (z)}{\sqrt{2}} {(\frac{m_{2}!}{m_{1}!} α^{2})}^{\frac{1}{2 (m_{2} - m_{1})}},

where

α = \sqrt{I_{01} / I_{02}}

. The radial position of dark points in the beam cross section depends on w(z) which gives intensity modulation to the beam during propagation. For example, if

Q (ϕ) = \cos (m_{1} - m_{2}) ϕ

is chosen in Eq. (4), then |m₁−m₂| dark points angularly distributes in the beam cross-section. The angular positions ϕ_d for these dark points are ϕ_d = 2nπ / |m₁ – m₂| for Q(ϕ) = 1, or ϕ_d = (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

S (z) = \mp 1

and the angular position varies according to the sign of Q(ϕ). The position of z-planes which contain dark points (z_d) can be obtained from the condition S(z) = ± 1, by

z_{d} = z_{R} \tan (\frac{q π}{| m_{1} - m_{2} |}),

where q = 0, ± 1, ± 2, …, and |q| < |m₁−m₂| / 2. Therefore, the beam waist (q = 0, z_d = 0) always contains dark points in any conditions. Additionally, multiple z-planes on which the dark points appear within the finite z range will be obtained when |m₁−m₂|> 2. Although the radius of dark points for the vector LG beam case presented in Eq. (5) is the same as the scalar LG beam case [7], 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.

Fig. 2 Calculated intensity distributions for the superposition of two vector LG beams (a) the same azimuthal indices (m₁ = 2, m₂ = 2) and type-I polarization distribution, (b) the same azimuthal indices (m₁ = 2, m₂ = 2) with type-I and type-II polarization distributions respectively.

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(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 [4]. 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.

Fig. 3 Calculated intensity distributions for the superposition of two vector LG beams having different azimuthal indices (a) beam 1: (m₁ = 1, polarization type-I), beam 2: (m₂ = 2, polarization type-I), (b) beam 1: (m₁ = 1 polarization type-I), beam 2: (m₂ = 2, polarization type-II).

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(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 [4]. 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.

Fig. 4 Propagated intensity distributions for the superposition of two vector LG beams at different z-planes (a) beam 1: (m₁ = 2, type-I), beam 2: (m₂ = 2, type-II), (b) beam 1: (m₁ = 1 type-I), beam 2 (m₂ = 5, type-I), (c) beam 1 (m₁ = 1, type I), beam 2: (m₂ = 5, type-II).

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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) = $\mp 1$ 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 = -1.13z_R to 1.13z_R. 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 = -4.38z_R, −1.25z_R, −0.48z_R, 0, 0.48z_R, 1.25z_R, and 4.38z_R 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

\begin{matrix} S_{0} = {| E_{x} |}^{2} + {| E_{y} |}^{2} \\ S_{1} = S_{0}^{- 1} ({| E_{x} |}^{2} - {| E_{y} |}^{2}), \\ S_{2} = 2 S_{0}^{- 1} Re (E_{x}^{*} E_{y}) \\ S_{3} = 2 S_{0}^{- 1} Im (E_{x}^{*} E_{y}) \end{matrix}

where E_x and E_y are the x and y components of the local electric field in the Cartesian coordinate basis and S₁² + S₂² + S₃² = 1.

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 [26]. 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 [27], which are defined as $S_{i j} (i, j = 1, 2, 3) = A_{i j} e^{i φ_{i j}}$ with $A_{i j} = \sqrt{S_{i}^{2} + S_{j}^{2}}$ , $φ_{i j} = arc \tan (S_{j} / S_{i})$ . 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.

Fig. 5 Phase φ₁₂ of complex Stokes field S₁₂ (Stokes vortex), (a) for vector LG beams with (m = 1) at the z = 0 plane, (b) for vector LG beams with (m = 2) at the z = 0 plane. Positive sign of the Stokes index (σ₁₂ = + 2m) corresponds to the increase of Stokes phase in the counterclockwise directions and negative sign of Stokes index (σ₁₂ = −2m) corresponds to the clockwise direction.

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By simple coordinate transformation, the electric field of a vector LG beam can be written in the Cartesian coordinates (i_x and i_y are the unit vectors for the x and y directions, respectively) as

E_{p, m} (r, ϕ, z) = {\begin{matrix} Y_{p, m} (r, z) [- \sin (m ϕ) i_{x} + \cos (m ϕ) i_{y}] (Type - I) \\ Y_{p, m} (r, z) [\sin (m ϕ) i_{x} + \cos (m ϕ) i_{y}] (Type - I I) \\ Y_{p, m} (r, z) [\cos (m ϕ) i_{x} + s i n (m ϕ) i_{y}] (Type - I I I) . \\ Y_{p, m} (r, z) [\cos (m ϕ) i_{x} - s i n (m ϕ) i_{y}] (Type - I V) \end{matrix}

By substituting the x and y components of a superposed field of two vector LG beams

E = \sqrt{I_{01}} E_{p_{1}, m_{1}} + \sqrt{I_{02}} E_{p_{2}, m_{2}}

into the definition of S₃ given by Eq. (7), the condition for C-points (S₃ = ± 1) yields

I_{01} {| E_{p_{1}, m_{1}} |}^{2} + I_{02} {| E_{p_{2}, m_{2}} |}^{2} + 2 \sqrt{I {}_{01}I_{02}} | E_{p_{1}, m_{1}} | | E_{p_{2}, m_{2}} | C_{\pm} (ϕ, z) = 0,

where the term C _± (ϕ, z) depends on the combination of the superposition as

C_{\pm} (ϕ, z) = {\begin{cases} \begin{matrix} \cos [(m_{1} - m_{2}) ϕ \pm (m_{1} - m_{2}) ψ (z)] & for & I_{1} + I_{2}, I I I_{1} + I I I_{2} \end{matrix} \\ \begin{matrix} \cos [(m_{1} - m_{2}) ϕ \mp (m_{1} - m_{2}) ψ (z)] & for & I I_{1} + I I_{2}, I V_{1} + I V_{2} \end{matrix} \\ \begin{matrix} - \sin [(m_{1} - m_{2}) ϕ \pm (m_{1} - m_{2}) ψ (z)] & for & I_{1} + I I I_{2} \end{matrix} \\ \begin{matrix} \sin [(m_{1} - m_{2}) ϕ \mp (m_{1} - m_{2}) ψ (z)] & for & I I_{1} + I V_{2} \end{matrix} \\ \begin{matrix} \cos [(m_{1} + m_{2}) ϕ \pm (m_{1} - m_{2}) ψ (z)] & for & I_{1} + I I_{2}, I I I_{1} + I V_{2} \end{matrix} \\ \begin{matrix} \sin [(m_{1} + m_{2}) ϕ \mp (m_{1} - m_{2}) ψ (z)] & for & I I_{1} + I I I_{2} \end{matrix} \\ \begin{matrix} - \sin [(m_{1} + m_{2}) ϕ \mp (m_{1} - m_{2}) ψ (z)] & for & I V_{1} + I_{2} \end{matrix} \end{cases}

In the Eq. (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 r_c in the beam cross-section expressed by

r_{c} = \frac{w (z)}{\sqrt{2}} {(\frac{m_{2}!}{m_{1}!} α^{2})}^{\frac{1}{2 (m_{2} - m_{1})}} .

Consequently, the radial position of C-points represented by Eq. (11) is identical to radial position of the dark points given by Eq. (5). By contrast, the angular position ϕ_C _± 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),

C_{\pm} (ϕ_{c \pm}, z) = \cos [(m_{1} - m_{2}) ϕ_{c \pm} \pm (m_{1} - m_{2}) ψ (z)] = - 1

is satisfied at C-points. Then, the angular position of each C-point is expressed as

ϕ_{c \pm} = \frac{n π \mp (m_{1} - m_{2}) ψ (z)}{m_{1} - m_{2}} f o r I_{1} + I_{2} a n d I I I_{1} + I I I_{2}

where n = 1, 3, …, 2|m₁-m₂|-1 is an odd integer number. Due to the Gouy phase term (m₁-m₂)ψ(z) in Eq. (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 [7]. 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].

Fig. 6 Calculated field distributions for the superposition of two vector LG beams (m₁ = 1 type-I, and m₂ = 2, type-I). (a) Intensity distributions at different z-planes, (b) polarization distributions at different z-planes. Red and blue color implies right-handed and left handed polarization for elliptical and circular polarization respectively, whereas black color represents linear polarization. (c) phase distribution φ₁₂ obtained from complex Stokes field. C-points are marked with the squares. Positive sign of σ₁₂ of Stokes vortices corresponds to the increase of Stokes phase in the counter-clockwise directions and negative sign of σ₁₂ of Stokes vortices corresponds to the increase of Stokes phase in the clockwise direction.

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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 ϕ_C _± of these C-points are given by $ϕ_{c \pm} = - [π \pm ψ (z)]$ .At the beam waist, both C-points with opposite handedness and the same winding number are located at the same position (r_c = w₀, ϕ_C _± = −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.

Fig. 7 Calculated field distributions for the superposition of two vector LG beams (m₁ = 1 type-I, and m₂ = 2, type-II). (a) Intensity distributions at different z-planes, (b) polarization distributions at different z-planes. Red and blue color implies right-handed and left handed polarization for elliptical and circular polarization respectively, whereas black color represents linear polarization. (c) phase distributions φ₁₂ obtained from complex Stokes field. C-points are marked with the squares. Positive sign of σ₁₂ of Stokes vortices corresponds to the increase of Stokes phase in the counter-clockwise directions and negative sign of σ₁₂ of Stokes vortices corresponds to the increase of Stokes phase in the clockwise direction.

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6. Discussion

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 [32] superposition of waves [33, 34] inhomogeneous media [35–37], spatial polarization distributions [38, 39], imaging [40, 41], laser modes [42], focal field distribution [43], and studies related to angular momentum of light [44].

7. Conclusions

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.

Acknowledgments

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).

References and links

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

2. A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre-Gaussian laser beams,” J. Opt. Soc. Am. A 15(10), 2705–2711 (1998). [CrossRef]

3. J. Courtial, “Self-imaging beams and the Gouy effect,” Opt. Commun. 151(1-3), 1–4 (1998). [CrossRef]

4. S. Franke-Arnold, J. Leach, M. J. Padgett, V. E. Lembessis, D. Ellinas, A. J. Wright, J. M. Girkin, P. Ohberg, and A. S. Arnold, “Optical ferris wheel for ultracold atoms,” Opt. Express 15(14), 8619–8625 (2007). [CrossRef] [PubMed]

5. T. Ando, N. Matsumoto, Y. Ohtake, Yu. Takiguchi, and T. Inoue, “Structure of optical singularities in coaxial superposition of Laguerre-Gaussian mode,” J. Opt. Soc. Am. A 27(12), 2602–2612 (2010). [CrossRef]

6. S. H. Tao, X.-C. Yuan, J. Lin, and R. E. Burge, “Residue orbital angular momentum in interferenced double vortex beams with unequal topological charges,” Opt. Express 14(2), 535–541 (2006). [CrossRef] [PubMed]

7. S. M. Baumann, D. M. Kalb, L. H. MacMillan, and E. J. Galvez, “Propagation dynamics of optical vortices due to Gouy phase,” Opt. Express 17(12), 9818–9827 (2009). [CrossRef] [PubMed]

8. I. D. Maleev and G. A. Swartzlander Jr., “Composite optical vortices,” J. Opt. Soc. Am. B 20(6), 1169–1176 (2003). [CrossRef]

9. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, “Creation and manipulation of three-dimensional optically trapped structures,” Science 296(5570), 1101–1103 (2002). [CrossRef] [PubMed]

10. A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Size selective trapping with optical “cogwheel” tweezers,” Opt. Express 12(17), 4129–4135 (2004). [CrossRef] [PubMed]

11. D. Yang, J. Zhao, T. Zhao, and L. Kong, “Generation of rotating intensity blades by superposing optical vortex beams,” Opt. Commun. 284(14), 3597–3600 (2011). [CrossRef]

12. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292(5518), 912–914 (2001). [CrossRef] [PubMed]

13. Y. Kozawa and S. Sato, “Focusing property of a double-ring-shaped radially polarized beam,” Opt. Lett. 31(6), 820–822 (2006). [CrossRef] [PubMed]

14. Y. Q. Zhao, Q. Zhan, Y. L. Zhang, and Y. P. Li, “Creation of a three-dimensional optical chain for controllable particle delivery,” Opt. Lett. 30(8), 848–850 (2005). [CrossRef] [PubMed]

15. 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]

16. X. Li, T. H. Lan, C. H. Tien, and M. Gu, “Three-dimensional orientation-unlimited polarization encryption by a single optically configured vectorial beam,” Nat. Commun 3, 998 (2012). [CrossRef] [PubMed]

17. C. Maurer, A. Jesacher, S. Furhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007). [CrossRef]

18. G. M. Lerman and U. Levy, “Radial polarization interferometer,” Opt. Express 17(25), 23234–23246 (2009). [CrossRef] [PubMed]

19. 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]

20. I. Freund, “Polarization singularity indices in Gaussian laser beams,” Opt. Commun. 201(4-6), 251–270 (2002). [CrossRef]

21. I. Freund, “Poincaré vortices,” Opt. Lett. 26(24), 1996–1998 (2001). [CrossRef] [PubMed]

22. I. Freund, M. S. Soskin, and A. I. Mokhun, “Elliptic critical points in paraxial optical fields,” Opt. Commun. 208(4-6), 223–253 (2002). [CrossRef]

23. J. V. Hajnal, “Singularities in the transverse field of electromagnetic waves. I Theory,” Proc. R. Soc. Lond. A Math. Phys. Sci. 414(1847), 433–446 (1987). [CrossRef]

24. J. V. Hajnal, “Singularities in the transverse fields of electromagnetic waves. II. Observation on the electric field,” Proc. R. Soc. Lond. A Math. Phys. Sci. 414(1847), 447–468 (1987). [CrossRef]

25. J. F. Nye and J. V. Hajnal, “The wave structure of monochromatic electromagnetic radiation,” Proc. R. Soc. Lond. A Math. Phys. Sci. 409(1836), 21–36 (1987). [CrossRef]

26. M. R. Dennis, “Polarization singularities in paraxial vector fields: morphology and statics,” Opt. Commun. 213(4-6), 201–221 (2002). [CrossRef]

27. I. Freund, A. I. Mokhun, M. S. Soskin, O. V. Angelsky, and I. I. Mokhun, “Stokes singularity relations,” Opt. Lett. 27(7), 545–547 (2002). [CrossRef] [PubMed]

28. A. I. Mokhun, M. S. Soskin, and I. Freund, “Elliptic critical points: C-points, a-lines, and the sign rule,” Opt. Lett. 27(12), 995–997 (2002). [CrossRef] [PubMed]

29. I. Freund, “Polarization flowers,” Opt. Commun. 199(1-4), 47–63 (2001). [CrossRef]

30. T. H. Lu, Y. F. Chen, and K. F. Huang, “Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields,” Phys. Rev. A 76(6), 063809 (2007). [CrossRef]

31. Y. Luo and B. Lü, “Composite polarization singularities in superimposed Laguerre-Gaussian beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 27(3), 578–584 (2010). [CrossRef] [PubMed]

32. I. Freund, “Möbius strip and twisted ribbon in intersecting Gauss-Laguerre beams,” Opt. Commun. 284(16-17), 3816–3845 (2011). [CrossRef]

33. P. Kurzynowski, W. A. Woźniak, M. Zdunek, and M. Borwińska, “Singularities of interference of three waves with different polarization states,” Opt. Express 20(24), 26755–26765 (2012). [CrossRef] [PubMed]

34. P. Senthilkumaran, J. Masajada, and S. Sato, “Interferometry with vortices,” Int. J. Opt. 2012, 517591 (2012).

35. K. Yu. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16(2), 695–709 (2008). [CrossRef] [PubMed]

36. E. Brasselet and S. Juodkazis, “Intangible pointlike tracers for liquid-crystal-based microsensors,” Phys. Rev. A 82(6), 063832 (2010). [CrossRef]

37. F. Flossmann, U. T. Schwarz, M. Maier, and M. R. Dennis, “Stokes parameters in the unfolding of an optical vortex through a birefringent crystal,” Opt. Express 14(23), 11402–11411 (2006). [CrossRef] [PubMed]

38. E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre-Gauss and polarization modes of light,” Appl. Opt. 51(15), 2925–2934 (2012). [CrossRef] [PubMed]

39. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18(10), 10777–10785 (2010). [CrossRef] [PubMed]

40. M. Burresi, R. J. P. Engelen, A. Opheij, D. van Oosten, D. Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102(3), 033902 (2009). [CrossRef] [PubMed]

41. O. V. Angelsky, A. G. Ushenko, Y. G. Ushenko, and Y. Y. Tomka, “Polarization singularities of biological tissues images,” J. Biomed. Opt. 11(5), 054030 (2006). [CrossRef] [PubMed]

42. Y. F. Chen, T. H. Lu, and K. F. Huang, “Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling,” Phys. Rev. Lett. 97(23), 233903 (2006). [CrossRef] [PubMed]

43. R. W. Schoonover and T. D. Visser, “Polarization singularities of focused, radially polarized fields,” Opt. Express 14(12), 5733–5745 (2006). [CrossRef] [PubMed]

44. G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam-Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108(19), 190401 (2012). [CrossRef] [PubMed]

Polarization singularities in superposition of vector beams

Abstract

1. Introduction

2. Vector Laguerre-Gaussian beam

3. Superposition of two vector Laguerre-Gaussian beams

3.1. superposition of two vector Laguerre-Gaussian beams with the same azimuthal indices (m₁ = m₂)

(a) the same sign of η

(b) opposite sign of η

3.2. Superposition of two vector Laguerre-Gaussian beams having different azimuthal indices (m₁≠m₂)

(a) the same sign of η

(b) opposite sign of η

4. Propagation of the superposed vector Laguerre-Gaussian beam

5. Polarization singularities and complex Stokes field in superposition of two vector LG beams

(a) the same sign of η

(b) opposite sign of η

6. Discussion

7. Conclusions

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (7)

Tables (1)

Equations (12)

Optics Express

Type	Angular dependence
I	θ_p = mϕ + π/2
II	θ_p = -mϕ + π/2
III	θ_p = mϕ
IV	θ_p = -mϕ

Abstract

1. Introduction

2. Vector Laguerre-Gaussian beam

3. Superposition of two vector Laguerre-Gaussian beams

3.1. superposition of two vector Laguerre-Gaussian beams with the same azimuthal indices (m1 = m2)

(a) the same sign of η

(b) opposite sign of η

3.2. Superposition of two vector Laguerre-Gaussian beams having different azimuthal indices (m1≠m2)

(a) the same sign of η

(b) opposite sign of η

4. Propagation of the superposed vector Laguerre-Gaussian beam

5. Polarization singularities and complex Stokes field in superposition of two vector LG beams

(a) the same sign of η

(b) opposite sign of η

6. Discussion

7. Conclusions

Acknowledgments

References and links

Supplementary Material (2)

Cited By

Figures (7)

Tables (1)

Equations (12)

Optics Express

3.1. superposition of two vector Laguerre-Gaussian beams with the same azimuthal indices (m₁ = m₂)

3.2. Superposition of two vector Laguerre-Gaussian beams having different azimuthal indices (m₁≠m₂)