Coherent vector beams with involved states of polarization (SOP) are widespread in the literature, having applications in laser processing, super-resolution imaging and particle trapping. We report novel vector beams obtained by transforming a Gaussian beam passing through a biaxial crystal, by means of the conical refraction phenomenon. We analyze both experimentally and theoretically the SOP of the different vector beams generated and demonstrate that the SOP of the input beam can be used to control both the shape and the SOP of the transformed beam. We also identify polarization singularities of such beams for the first time and demonstrate their control by the SOP of the input beam.
© 2015 Optical Society of America
The state of polarization (SOP) is one of the fundamental signatures of light fields associated with their vectorial nature. In general, at each point in space the dynamics of the electric field vector of a wave can be described by an ellipse. This ellipse is known as polarization ellipse and it is characterized by the orientation of its major axis through the azimuth angle Φ ∈ [0, π] and by the ellipticity parameter β ∈ [−π/4, π/4] so that tanβ is the ratio of the axes of the polarization ellipse. If β = 0 the light field is linearly polarized, while if β = ±π/4 the SOP will be circular (left handed for ‘−’ and right handed for ‘+’, if we consider an observer looking in the direction from which light is coming). Usually coherent light beams are homogeneously polarized, i.e. the SOP is identical for all points at any transverse plane along the beam propagation. However, there exist light beams possessing non-homogenous polarization, known as vector beams, such as the well known radial or azimuthal polarizations  or even beams with more involved polarization distributions [2–4]. The non-homogeneous polarization distribution of vector beams can lead to singular points where the SOP is exactly circular (C points), lines along which the SOP is linear (L lines) or disclinations where the instantaneous electric field is null [5–10]. Vector beams have been applied to laser material processing, optical imaging, atomic spectroscopy, and optical trapping (see  and the references therein), among many others.
In recent years, there has been a renewed interest in the conical refraction (CR) phenomenon ocurring in biaxial crystals, which can also provide different vector beams [11–23]. In CR, when a focused Gaussian input beam with waist radius w0 propagates along one of the optic axes of a biaxial crystal it appears transformed at the focal plane as a pair of concentric and well resolved bright rings split by a dark (Poggendorff) ring of geometrical radius R0, as shown in Fig. 1(b). This intensity distribution is found as long as the condition ρ0 ≡ R0/w0 ≫ 1 is satisfied [14,24, 25], while other light structures have been found for ρ0 ≤ 1.50 [20,21,23]. R0 is the product of the crystal length, l, and the CR semi-angle α, i.e. R0 = lα . The CR semi-angle α depends on the principal refractive indices of the crystal as , where we have assumed n1 < n2 < n3. The non-uniform polarization distribution of a CR beam at the focal plane under Poggendorff splitting conditions is depicted by blue double arrows in Fig. 1(b). At any point of the CR rings, the electric field is linearly polarized with the azimuth varying so that every pair of diametrically opposite points have orthogonal polarizations. This polarization distribution depends only on the orientation of the plane of optic axes of the biaxial crystal. As a consequence, the intensity distribution along the ring depends on the input SOP, corresponding the one depicted in Fig. 1 to an incident circularly polarized beam. For ρ0 ≫ 1, the SOP of the CR beam at the focal plane is already well known [13,16,18,24].
Here, for the first time to our knowledge, we investigate the CR SOP out of the focal plane, including values of ρ0 ≤ 1.50, when Poggendorff fine splitting vanishes. By means of the Stokes vector formalism, we characterize the resulting novel vector beams of CR and show that the SOP of the input beam can be used to control both the shape and the SOP of the transformed CR beam.
The article is organized as follows. Section 2 is devoted to briefly review the basics of the Stokes vector and CR formalisms needed for the subsequent sections. In Sections 3 and 4, we report for ρ0 ≫ 1 and ρ0 ≤ 1.50, respectively, the generation of novel CR vector beams both theoretically and experimentally. We characterize these vector beams by measuring their Stokes parameters and propose methods to manipulate them as, for instance, making use of the SOP of the input beam. Finally, we summarize the main results of this work and discuss potential applications of these CR vector beams in different fields of optics.
2. Theoretical background
2.1. Stokes vector formalism
The standard tool to analyze the SOP of a light beam is the Stokes vector: S = (S0, S1, S2, S3). For an electric field E = (Ex, Ey) with intensity I the Stokes parameters read :10]:
2.2. Conical refraction theory
The theoretical model predicting the transformation of an input beam once it passes along one of the optic axis of a biaxial crystal is based on the Belsky-Khapalyuk-Berry (BKB) integrals [11,13]. For a uniformly polarized input beam with electric field E and cylindrically symmetric 2D Fourier transform , the electric field amplitude behind the crystal can be written as follows:25]: Eqs. (8) and (9) so it will determine the resulting CR intensity distribution. Accordingly, we will use ρ0 as the control parameter for our investigations. If ρ0 ≫ 1 is fulfilled then the transverse intensity pattern at the focal plane of the system is formed by the two characteristic bright rings split by the dark Poggendorff one, as shown in Fig. 1(b). As the imaging plane is moved away from the focal plane, more involved structures are found including secondary rings and a bright central spot known as the Raman spot, see Fig. 1(a). The intensity maximum of the Raman spot is found at positions given by  Eq. (10) is obtained under the assumption that ρ0 ≫ 1, i.e. in the limit ρ0 → ∞. Out of this approximation, the greater the value of ρ0 is, the better it is predicted by Eq. (10).
3. State of polarization for ρ0 ≫ 1
The first information that can be extracted from Eqs. (7)–(9) with respect to the SOP of the CR beam is that at ρ = 0 there is only contribution of B0, since B1 ∝ J1(ηρ = 0) = 0. Additionally, from Eqs. (8) and (9) note that the SOP of the B0 component is e0. As a consequence, the center of the CR beam will possess always the same SOP as the input beam. This fact, that was already pointed out in [27–29], will be discussed with more detail below. To obtain the Stokes parameters of the CR beam, Eqs. (2)–(4) must be combined with Eqs. (7)–(9). For a CP input beam, the electric field and intensity beyond the crystal become:
For well resolved concentric rings with Poggendorff splitting, i.e. for ρ0 ≫ 1, Eqs. (8), (9), (13) and (16) show that a radially symmetric intensity pattern of CR is obtained only for a CP input beam. Instead, for a LP input beam, a crescent annular intensity pattern appears such that the zero intensity point is obtained for the ring position that possesses orthogonal polarization to the input beam. In both cases, the polarization distribution of the CR pattern is the same. Every point of the rings is linearly polarized and the azimuth rotates continuously along the ring so that every two diametrically opposite points have orthogonal polarizations.
Figure 2 presents the numerically obtained Stokes parameters at transverse sections of the CR beam (ρ0 = 10) at Z = 0 (first and second rows) and Z = 10.92 (third and fourth rows) obtained from a RHCP (first and third rows) and a LP (Φ = 45°) (second and fourth rows) Gaussian input beam. At the focal plane, see first two rows in Fig. 2, the SOP described by the Stokes parameters is the expected: symmetric pattern for the RHCP case and with a node at φ = 270° (Φ = 135°, since we have used φC = 0°). Last column in Fig. 2 demonstrates that at the focal plane the SOP of the CR beams, either RHCP or LP (Φ = 45°) (or any other) is linear, i.e. S3 = 0. In contrast the Raman spot, Z = 10.92, all Stokes parameters are substantially different from zero, as shown in the last two rows in Fig. 2.
Stokes parameters in Fig. 2 clearly identify polarization singularities of CR beams. For RHCP input light, it is a C-point at the center of the CR beam, i.e. it is of circular polarization at any point of the beam center along propagation. The value of the S3 transverse pattern at its center is maximum, while the other two Stokes parameters S1 and S2 have zero values. For LP45° input light, one can identify L-line singularity. In Fig. 2 this line can be identified as a vertical line of zero value of the Stokes parameters S3 and S1 and nonzero value of S2. It belongs to the plane defined by the points of CR ring with linear SOP of LPΦ=45° and LPΦ=135°.
Media 1 and Media 2 show, respectively, the spatial evolution of the Stokes parameters along the propagation direction for a RHCP and LP (Φ = 45°) Gaussian input beam. As we move away from the focal plane, the bright rings become wider and the intensity at the Poggendorff dark ring is no longer zero. The intensity of the outer ring decreases while the intensity of the inner ring increases. The outer ring expands and the inner ring becomes smaller in radius. As a result, at Z ≈ 6 a spot in the center of the pattern appears. Finally, the inner ring shrinks into a bright spot at Z = 10.92 for ρ0 = 10, corresponding to the Raman spot. At this point, there can be found an additional type of polarization singularity independently of the polarization state of an input beam. They are clearly distinguishable by expecting Stokes parameter S3 that defines degree of circular polarization of the field. Inspection of S3 far from the focal plane, for instance, at the Raman spot as demonstrated in the last two rows in Fig. 2, reveals alternating annular regions of RHCP and LHCP states. These annular regions of circular polarization are separated by circles of null intensity. At these circles the field is linearly polarized. Therefore, these polarization singularities can be called as L-circles. This behavior has been also observed in the focusing of radially polarized beams and explored in detail in Ref. . For a LP (Φ = 45°) input beam, the central spot is broken by a line of null intensity (L-line) that connects the two points with LP and azimuth Φ = 45° and Φ = 135°, as commented before.
One feature that supplements demonstrate is that polarization singularities, i.e. C-point for CP input beam, L-lines for LP input beam and L-circles, are invariants of CR beam propagation behind the crystal. Another feature demonstrated by the supplement movies is the inversion of the center of the S1 and S2 parameters after the focal plane, which can be understood as a manifestation of the Gouy phase [31,32] of the CR beam. This effect will be more appreciable and discussed in the CR vector beams reported in the following section.
To test the validity of the obtained theoretical results, we have performed the corresponding experimental measurements. Figure 3 shows the experimental set-up. The input beam is obtained from a diode laser at 640nm coupled to a monomode fiber with a collimator (Monocrom C-6305C-4S), yielding a beam waist of w0 = 1.26mm. The M2 factor of the beam has been measured to be 1.0769, which is very close to an ideal Gaussian beam (M2 = 1). Therefore, in what follows we will refer to the beam emerging from the collimator as a Gaussian beam. After the collimator we place a linear polarizer to fix the state of polarization of the Gaussian beam. To generate the different input polarization states (LP with Φ = 45° and RHCP) we use a λ/2 and a λ/4 waveplates. We use lenses with different focal lengths to modify the waist radius w0 of the input beam and to reach different values of ρ0. The beam passes along one of the optic axes of a biaxial crystal, placed always before the expected focal plane of the beam. An additional imaging lens is used to take different planes along the beam propagation direction and transfer them onto the CCD camera. Linear and circular polarizers are used to measure the Stokes parameters of the beam after being transformed by the CR phenomenon. We use a commercially available (CROptics) KGd(WO4)2 biaxial crystals with α = 16.9 mrad and length l = 10.5mm yielding CR ring radius of R0 = 180μm. For more details about the material and the alignment procedure, see e.g. Ref. .
Figure 2(b) shows the obtained experimental Stokes parameters for ρ0 = 10.81. The experimental results agree with the theoretical results presented in Fig. 2. Discrepancy has been observed only for the S3 parameter for the case of a LP input beam (see last image of second rows). This can be explained in terms of the experimental error introduced by the polarization state detector elements (PSA) used, that disturb the beam shape and its position, which is central for the quality of the experimental results. Additionally it must be taken into account that the PSA and PSG used are designed for a working wavelength of 633nm, while our laser system has a wavelength of 640nm.
4. State of polarization for ρ0 ≤ 1.50
For ρ0 ≤ 1.50 CR patterns are significantly different from the double bright concentric rings with clear Poggendorff splitting (occurring for ρ0 ≫ 1 as shown in the previous section). The region ρ0 ≤ 1.50 has been explored recently [20, 21, 23] showing that CR can be used to new CR lasers , to increase the directivity of laser beams , to generate a super-Gaussian beam , to create a three dimensional dark focus  and even to develop a novel scheme for super-resolution microscopy [35, 36]. However, in all these works, no deep insight about the SOP of the generated CR beams has been provided. In what follows by considering the Stokes parameters we uncover the evolution of the SOP and polarization singularities of the CR beams and demonstrate how they depend on the SOP of the input beam. Figure 4 shows the main features and general view of CR beams with ρ0 = [1.50, 0.92, 0.45]. The cross-section of the CR transverse intensity pattern at the focal plane and along the axial direction Z at the beam center are shown in Figs. 4(a) and (b), respectively. Figs. 4(c)–(e) are 2D density plots of the intensity of the CR beams in the Z–ρ plane.
Figure 5 presents density plots of the numerically calculated Stokes parameters for (a) ρ0 = 1.50, (b) ρ0 = 0.92 and (c) ρ0 = 0.45 at Z = 0 (first and second rows) and out of the focal plane (third and fourth rows). For ρ0 = 1.50 and ρ0 = 0.92 these planes give the axial intensity maximum, while for ρ0 = 0.45 we have considered the plane where the cross-section area of the beam is doubled . First and third rows refer to a RHCP Gaussian input beam while second and fourth rows present the case of a LP (Φ = 45°) Gaussian input beam. For ρ0 ≤ 1.50 the position of the Raman spots are not well determined by Eq. (10) and the position of the axial intensity maxima must be determined for each particular case of ρ0. Its value (Z) is indicated at each image.
While the transverse distribution for intensity and polarization of CR beam look different for the ρ0 ≤ 1.50 case with respect to ρ0 ≫ 1, it should be noted that singularities, i.e. topological structure of the CR beam, remains similar. It means that (i) the CR beam center constitutes a C-point for the case of RHCP input beam (see the first and third rows in Fig. 5), (ii) there is L-line polarization singularity for the LPΦ=45° input beam (see the second and fourth rows in Fig. 5) and (iii) there are L-circle singularity for input beam with arbitrary SOP (see the third and fourth rows in Fig. 5).
If we consider the special case of ρ0 = 0.92, the central point at the focal plane is a null-intensity point and therefore all Stokes parameters are also 0 at the beam center. Vanishing intensity at the beam center leads to another feature: at the focal plane the sign of the circular polarization state associated with the Stokes parameter S3 is the opposite with respect to the input beam. In other words, at the focal plane the CR beam is predominantly LHCP for the RHCP input beam that we consider in this article (see first row in Fig. 5 for the case of ρ0 = 0.92). This feature is easily deduced from the mathematical formulation, since for this value of ρ0 the value of the integral B0 at any radial point is null, which leads to dominance of the contribution of integral B1 associated with a SOP orthogonal to the input one. For the case of a LP (Φ = 45°) input beam and for all values of ρ0 investigated, at the focal plane Z = 0 the transverse patterns are crescent-like, with the point of intensity minimum at a point, which is diagonally opposite to the point with maximum intensity. In the case of LPΦ=45° input beam that we consider, the intensity minimum is observed at the bottom, corresponding to the point of LP with Φ = 135°. In contrast to the RCHP case, for all three values of ρ0 the S3 parameter is null, which means that the patterns are completely LP. Out of the focal plane S3 ≠ 0 except at the beam center, where there is an L-line connecting the points with LP Φ = 45° and Φ = 135°.
Additional features of CR beams are revealed by considering the evolution of the transverse pattern of the Stokes parameters along CR beam propagation, as shown in Media 3, Media 4, Media 5, Media 6, Media 7, Media 8 for a RHCP and LP (Φ = 45°) input beam for ρ0 = 1.50, ρ0 = 0.92 and ρ0 = 0.45. For RHCP input light ( Media 3, Media 5, Media 7) the focal plane (Z = 0) is a symmetry plane for S3. In contrast, S1 and S2 are rotated roughly 180° before and after the focal plane, which is associated to the Gouy phase in vector beams [31, 32]. For a LP (Φ = 45°) input beam ( Media 4, Media 6, Media 8) the Stokes parameters S1 and S2 are symmetric with respect to the focal plane and now S3 suffers from a phase shift of 180°. In general, analogous results are obtained with focused Gaussian beams: due to the Gouy phase shift, that induces an additional phase in the electric field of 180° at the focal plane, the S1 and S2 of a Gaussian beam rotate around the Z axis by 180°. However, note that at variance with the results expected for a focused Gaussian beam, the S1 and S2 parameters of CR beams obtained from CP input beams not only rotate along the Z axis but also twist around it at the focal plane. CR beams possess non-integer orbital angular momentum (OAM) that is generated due to the spin-orbit coupling provided by the biaxial crystal . The value and handiness of the OAM depend both on the ellipticity of the input beam and on the ρ0 parameter. For ρ0 ≫ 1 and a CP beam, i.e. with spin angular momentum (SAM) σs = ±1 (+ for LHCP and − for RHCP), the CR beam carries total optical angular momentum ±1/2 and zero SAM. For values of ρ0 out of that condition, the spin-orbit coupling differs. For instance, for ρ0 = 0.924 the SAM of the CR beam has opposite handiness with respect to the input beam, which explains the results discussed above. As reported in Ref. , for ρ0 = 1.50 the OAM of the CR beam is maximum compared with any other value of ρ0. At the focal plane the twist is more appreciable at the central spot. Additionally, the velocity of rotation of the S1 and S2 parameters of that region as the beam approaches to the focal plane (Z = 0) is much faster than the rotation velocity of the outer ring. For the value ρ0 = 0.445 the beam carries almost null OAM, which explains the small twist of the S1 and S2 parameters at the focal plane and its similarities with a homogeneously CP Gaussian beam. As a summary, the combination of both SAM and OAM of the CR vector beams leads to the rotation and twisting of the S1 and S2 Stokes parameters when CP input beams into the biaxial crystal are considered.
We have performed experiments in order to prove theoretical findings reported above. Figure 6 shows the experimentally measured Stokes parameters for ρ0 = 1.48 (box (a)), ρ0 = 0.95 (box (b)) and ρ0 = 0.44 (box (c)). For these experiments, the same set-up shown in Fig. 3 was used but in this case taking a 2.3mm long KGd(WO4)2 biaxial crystal yielding CR ring radius of R0 = 39 μm. In general, the theoretical predictions agree with the experimental results. Again, the S3 parameter for a LP input beam (last image of second and fourth rows) is the measurement that mostly differs with respect to the numerical predictions. In addition to the experimental difficulties commented above, i.e. the experimental error introduced by the polarization state detector elements used that disturb the beam shape and its position (being these ones particularly significant for large values of w0), here it must be also taken into account the fact that small changes in the ρ0 can modify quantitatively the CR pattern.
Finally, Fig. 7 presents the calculated transverse patterns and SOP (blue curves) of the vector beams analyzed in this work for a RHCP (first row) and a LP (Φ = 45°) (second row) input Gaussian beam at the focal plane. As it can be observed, for low values of ρ0 the SOP of the generated vector beams is rich and include regions with different ellipticity and azimuth, leading to Poincare beams .
In summary, we have studied in detail the SOP of conical refraction based vector beams for a wide range of ρ0 and for different SOP of the input beam. We have determined the Stokes parameters of the CR beam at different transverse planes along the beam propagation direction and we have shown that both the shape and the SOP of the transformed beams depend on the SOP of the input beam. We have shown that the polarization distribution formed by orthogonal polarizations at any two radially opposite points of the pattern and usually associated to the CR phenomenon remains relevant under the condition ρ0 ≫ 1 and at the focal plane only. For ρ0 ≤ 1.50 and CP input beams we have found that CR beams demonstrate more involved structure of non-homogeneously elliptically polarized states not only with different azimuth but also with different ellipticity. In contrast, for LP input beams, the SOP of the CR has been reported to be completely linear at the focal plane and with variable ellipticity and azimuth out of it. Agreement between the theoretical predictions and the experimental results has been obtained.
Additionally, we have demonstrated for the first time to our knowledge, experimental results on polarization singularities of CR beams and demonstrate how types of polarization singularities can be changed by and controlled with varying SOP of the input light beams. Such polarization singularities as C-points, L-lines and L-circles have been identified for CR beams.
The reported results can be particularly interesting for experiments with tightly focused beams , for the generation of novel polarizations in CR , in optical micromanipulation [26, 33, 34], mode conversion between Heremite–Gauss-like beams and Laguerre–Gauss-like beams  and in super-resolution imaging [35, 36]. It is also promising the generation of polarization-tunable potentials to inject, extract and manipulate ultra-cold atoms [25, 40]. Additionally, by taking into account that the CR beams posses orbital angular momentum  and that the biaxial crystals used are transparent to a wide frequency range, the presented technique could be an alternative for the production of high-frequency vortex beams for molecular-scale super-resolution microscopy . Finally, by using quantum sources, CR vector beams can be a tool for testing new Bell-like inequalities with hybrid polarization-momentum states useful in quantum information technologies .
We acknowledge financial support from the Spanish Ministry of Science and Innovation (MICINN) (contracts FIS2011-23719 and FIS2012-39158-C02-01). A.T. acknowledges financial support through the grant AP2010-2310.
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