1. Introduction

Phase singularities [1, 2] in a scalar optical field are intensity null points satisfying ∮Δϕ.dl ≠0, where ϕ is the phase. Similarly in a vector optical field there are polarization singularities [3, 4]. The singularities associated with linearly polarized fields are called V-point singularities, in which ∮Δγ.dl ≠0, where γ is the azimuth of the state of polarization (SOP). In both the definitions the closed path integrals are evaluated around the respective singularities. The V-point index or Poincare-Hopf index η is defined as η = 1/(2π) ∮Δγ.dl. Radially and azimuthally polarized beams have Poincare-Hopf index +1 and they have found many applications [5–7]. These beams can also be seen as the superposition of two phase-singular beams (charge m=±1) in opposite spin angular momentum states [8, 9]. Also, these beams can be seen to possess point phase defect when viewed in circular polarization state and edge phase dislocation when viewed in linear polarization state [10]. Since phase and polarization of light are closely related, study of polarization singularities demands equal significance as that of phase singularities.

Phase singular beams carry orbital angular momentum (OAM) [2]. Hence their propagation and diffraction through various apertures have been the subject of investigation in recent years [11–17]. Diffraction through triangular apertures [15–17] was also studied. Recently, two-slit interference of vector fields has been investigated to measure the topological charge associated with vector fields [18, 19]. These studies are confined only to field (and intensity) distribution(s) and the polarization distribution in the diffracted field is not studied. Also studies on the diffraction of polarization singularities are scarce in the literature.

In this paper we present experimental studies on diffraction of V-point singularities through equilateral and isosceles right triangular apertures. Diffraction disintegrates +1 index V-point singularity into two monstars/lemons and −1 index V-point singularity into two stars. Positive and negative phase singularities produce different diffraction patterns whereas here the diffraction pattern is insensitive to polarity of the Poincare-Hopf index of the beam. Diffraction of higher order V-point singularities are also studied.

2. Polarization singularities

In polarization singularities one or more of the parameters defining the SOP (azimuth and/or handedness) is/are undefined [20, 21]. V-point singularities are related to vector fields and these are stationary points in a linearly polarized optical field at which both orientation and handedness are undefined. The order of the V-point singularity is defined by the Poincare-Hopf index η. C-point singularity occurs in ellipse fields and is associated with circular polarization in which the azimuth of the SOP is undefined. It is characterized by the singularity index I_C defined as I_C = 1/(2π) ∮Δγ.dl. Lower order C-points are lemon, monster and a star with C-point indices +1, +1 and −1 respectively. In generic light fields, C-points and L lines can be found but no V-points due to lack of symmetry. But in engineered fields, V-points can co-exist with C-points [22, 23]. These singularities can be identified by constructing a complex Stokes field using directly measurable quantities of normalized Stokes parameters S₀; S₁; S₂ and S₃ [20]. The complex Stokes field is constructed as S₁+iS₂=A₁₂ exp(iϕ₁₂). The phase of the Stokes field, ϕ₁₂ with a polarization singularity is similar to the phase structure of an optical vortex and the winding number of the Stokes vortex is defined as σ₁₂=∆ϕ₁₂/(2π). ∆ϕ₁₂ measures the phase acquired around a closed path enclosing the singularity in the anti-clock sense. Both, vector(η) and ellipse(I_C) field singularity indices are related to the Stokes index σ₁₂ as σ₁₂=2η (vector field) and σ₁₂=2I_C (ellipse field). η=1 for radially and azimuthally polarized beams, hence the Stokes index σ₁₂=2 for first order V-point singularities. |σ₁₂| >2 and |η|>1 are higher order V-point singularities. In the Stokes field, phase singularities of even charge (with S₃=0) are V-points and that of all charges (with S₃ ≠0) are C-points [20, 21]. Generalized expressions for V-point singular beams as commonly named in literature [24], can be written as, $[-\sin (m\varphi )\widehat{\mathbf{x}}+\cos (m\varphi )\widehat{\mathbf{y}}]$ for type I, $[\sin (m\varphi )\widehat{\mathbf{x}}+\cos (m\varphi )\widehat{\mathbf{y}}]$ for type II, $[\cos (m\varphi )\widehat{\mathbf{x}}+\sin (m\varphi )\widehat{\mathbf{y}}]$ for type III and $[\cos (m\varphi )\widehat{\mathbf{x}}-\sin (m\varphi )\widehat{\mathbf{y}}]$ for type IV, where m is the azimuthal mode index of the phase singularity.

3. Diffraction of V-points

V-points singular beams can be generated interferometrically [25] or by using spatial light modulators [19, 26]. Consider the experimental setup shown in Fig. 1(a) used to prepare beams with V-point singularities for our experiment. An input light from He-Ne laser is spatially filtered and collimated. A polarizer P, and a quarter wave plate (QP₁) prepare circularly polarized beam which is launched into a Sagnac loop consisting of a polarizing beam splitter (PBS), mirrors M₁, M₂ and M₃. A spiral phase plate (SPP) introduced into a Sagnac loop produces oppositely charged phase vortices in the two counter propagating orthogonally linearly polarized beams. For each orientation of QP₂ (+45° and −45°) and by changing the orientation of the half waveplate (HP) between 0° and −45°, one can realize all four types of V-point singularities for each |η|. Figure 2 shows simulated and experimentally observed Stokes phase variations, polarization distributions of V-point singularities of Poincare-Hopf index |η|=1, 2, and 3. The triangular apertures shown in Fig. 1(b) are inserted one by one in the setup shown in Fig. 1(c) and the diffraction patterns are observed at the back focal plane of the lens. The transmission of the aperture is t(x, y). The field distribution just behind the triangular aperture is given by

 

Fig. 1 Experimental setup: (a) Sagnac Interferometer, (b) Triangular apertures, (c) 2f experimental configuration. L:lens, M:mirror, H/QP_1,2:half/quarter waveplate, PBS:polarizing beam splitter, SPP:spiral phase plate, T:aperture, SC:Stokes camera, PC:computer

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Fig. 2 Simulated(left) and experimentally generated(right) V-point singularities of Poincare-Hopf index |η|=1 to 3. For every |η| the polarization distributions are classified into four types. The polarization distribution is superimposed on the Stokes phase ϕ₁₂.

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The core of the polarization singular beam is made to go through the centroid of the triangle aperture and the diffraction field is obtained on the Stokes polarization camera (SALSA, Bossa Nova Technologies, USA) with resolution of 1040X1040 pixels. The diffraction through both types of apertures for the beam with Poincare-Hopf index |η|=1, results in the disintegration into two C-point singularities of C-point index |I_C|=1/2 located at different positions. In the case of diffraction of |η|=2 beam, through equilateral triangular aperture, appearance of extra dipoles happens due to lack of symmetry. The total charge in the diffracted field however is conserved. Similarly |η| = 3 beam disintegrates into six C-point singularities symmetrically about the center. The intensity distribution of the diffracted field remains unaltered for positive and negative V-point singularities for a particular value of |η|. But the Stokes phase shows disintegration and the sign conservation. Each V-point singularity with Poincare-Hopf index η disintegrates to C-point singularities with index I_C and redistributed to different positions. Numerically simulated diffracted field distribution of V-point singular beams through equilateral and isosceles right triangle are shown in left and right side of the Fig. 3 respectively. Experimentally observed diffraction patterns of V-point singular beams through equilateral (left side) and isosceles right triangle (right side) apertures are shown in Fig. 4.

 

Fig. 3 Simulated far field pattern: Top row shows the intensity distribution for diffraction through equilateral (left) and isosceles right triangular (right) apertures. Two triangles connecting the diffraction spots in the top three frames are drawn to indicate the superposition of diffraction due to charge +m and −m phase singularities (shown separately in the insets) of V-point singularities. Stokes phase shown in rows 2 to 5 correspond to the diffraction of beam types I–IV. The V-point index for each column is given in the top frame. Note, the incident V-point singularity has disintegrated into C-points of same polarity.

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Fig. 4 Experimentally observed far field patterns: Top row shows the intensity distribution for diffraction through equilateral (left) and isosceles right triangle (right) apertures. Stokes phase shown in rows 2 to 5 correspond to the diffraction of beam types I–IV. The V-point index for each column is given in the top frame. Note the incident V-point singularity has disintegrated into C-points of same polarity.

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4. Discussions

Beam positions (position of the V-point singularity) going through the apertures are as per shown in Fig. 1(b). Equilateral triangle has more symmetry than the isosceles right triangle and hence their diffraction patterns shown in Fig. 4 are different. Diffraction pattern has intensity lobes in case of equilateral triangular aperture and has a fringe like pattern in case of isosceles right triangle. Diffraction disintegrates a V-point singularity of index η into a number of (2η) lowest order C-point singularities. C-point dipoles are also observed in the diffracted field. Referring to Fig. 5, in the equilateral triangular aperture, the Stokes phase variations (modulo 2π) corresponding to |η|=1 and 3 divide the aperture into two and six equal parts respectively. This symmetry is not there for |η|=2 and this may be the reason for the appearance of extra dipoles in the diffraction pattern. These C-point dipoles are absent in case of diffraction through isosceles right triangular aperture.

 

Fig. 5 Stokes phase variation inside the equilateral triangular aperture due to V-point singularities (a) |η|=1, (b) |η|=2 and (c) |η|=3. Note the lack of symmetry in (b)

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Similar thing happens in the diffraction of phase singular beam where higher order phase singularity disintegrates into lowest order phase singularities. Here a higher order V-point disintegrates into lowest order C-points. But the diffraction patterns of phase vortices are different for positive and negative charges and there is a 180 degree flip in the patterns about the centroid of the apertures [14, 17]. It is very interesting to know that for a given |η|, even though the polarization distributions of four beams (2 orthogonal states x 2 polarities = 4) are different, they all produce same far field diffraction pattern. One can see intuitively that the diffraction pattern of a V-point singularity is a simple superposition of the diffraction patterns of a positive and a negative phase singularities. This is because vector beams can be viewed as superposition of right and left circularly polarized phase singular beams of opposite singularity. In Fig. 3, to indicate this trend we have drawn two triangles that connect the diffraction spots for equilateral triangular apertures. In case of isosceles right triangular aperture one can see that the length of the fringes is same whereas it is not so for phase vortex diffraction [17]. Like phase information, the polarization information is also lost in the intensity distribution. This also reveals the close relation between phase and polarization. For eg., what can be achieved through phase, can also be achieved through polarization [7]. This intensity invariance is termed as degeneracy and is observed also in the interference patterns [23].

5. Conclusion

Higher order V-point singularities are generated by Sagnac interferometer and we have experimentally demonstrated the diffraction of V-point singularities through two types of triangular apertures namely, equilateral and isosceles right triangular apertures. It is shown that V-point singularity disintegrates into C-point singularities by diffraction. The diffraction patterns are found to be insensitive to the polarity of the Poincare-Hopf index. Unlike the diffraction of phase singular beams, intensity invariance in the diffraction pattern is observed.