Visualizing polarization singularities in Bessel-Poincaré beams

We demonstrate that an annulus of light whose polarization is linear at each point, but the plane of polarization gradually rotates by π radians can be used to generate Bessel-Poincaré beams. In any transverse plane this beam exhibits concentric rings of polarization singularities in the form of L-lines, where the polarization is purely linear. Although the Llines are invisible in terms of light intensity variations, we present a simple way to visualize them as dark rings around a sharp peak of intensity in the beam center. To do this we use a segmented polarizer whose transmission axes are oriented differently in each segment. The radius of the first L-line is always smaller than the radius of the central disk of the zero-order Bessel beam that would be produced if the annulus were homogeneously polarized and had no phase circulation along it. ©2015 Optical Society of America OCIS codes: (050.4865) Optical vortices; (140.3300) Laser beam shaping; (260.2130) Ellipsometry and polarimetry; (260.5430) Polarization; (260.6042) Singular optics. References and links 1. P. Jacquinot and B. Roizen-Dossier, “Apodisation,” Prog. Opt. 3, 29–186 (1964). 2. J. Lindberg, “Mathematical concepts of optical superresolution,” J. Opt. 14(8), 083001 (2012). 3. J. Durnin, J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). 4. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization, and superresolution,” Appl. Opt. 43(22), 4322–4327 (2004). 5. T. Grosjean and D. Courjon, “Smallest focal spots,” Opt. Commun. 272(2), 314–319 (2007). 6. H. Dehez, A. April, and M. Piché, “Needles of longitudinally polarized light: guidelines for minimum spot size and tunable axial extent,” Opt. Express 20(14), 14891–14905 (2012). 7. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular Optics: Optical Vortices and Polarization Singularities,” Prog. Opt. 53, 293–363 (2009). 8. Z. Bouchal and M. Olivík, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42(8), 1555–1566 (1995). 9. J. Arlt and K. Dholakia, “Generation of high-order Bessel beams by use of an axicon,” Opt. Commun. 177(1-6), 297–301 (2000). 10. S. Orlov, K. Regelskis, V. Smilgevičius, and A. Stabinis, “Propagation of Bessel beams carrying optical vortices,” Opt. Commun. 209(1-3), 155–165 (2002). 11. D. McGloin and K. Dholakia, “Bessel beams: Diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005). 12. A. Dudley, Y. Li, T. Mhlanga, M. Escuti, and A. Forbes, “Generating and measuring nondiffracting vector Bessel beams,” Opt. Lett. 38(17), 3429–3432 (2013). 13. S. Vyas, Y. Kozawa, and S. Sato, “Generation of radially polarized Bessel-Gaussian beams from c-cut Nd:YVO4 laser,” Opt. Lett. 39(4), 1101–1104 (2014). 14. C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007). 15. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18(10), 10777–10785 (2010). 16. F. Cardano, E. Karimi, L. Marrucci, C. de Lisio, and E. Santamato, “Generation and dynamics of optical beams with polarization singularities,” Opt. Express 21(7), 8815–8820 (2013). #237477 $15.00 USD Received 3 Apr 2015; revised 28 Apr 2015; accepted 28 Apr 2015; published 1 May 2015 © 2015 OSA 4 May 2015 | Vol. 23, No. 9 | DOI:10.1364/OE.23.012444 | OPTICS EXPRESS 12444 17. 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). 18. M. V. Berry and M. R. Jeffrey, “Conical diffraction: Hamilton’s diabolical point at the heart of crystal optics,” Prog. Opt. 50, 13–50 (2007). 19. A. Turpin, Y. Loiko, T. K. Kalkandjiev, H. Tomizawa, and J. Mompart, “On the dual-cone nature of the conical refraction phenomenon,” Opt. Lett. 40(8), 1639–1642 (2015). 20. A. Turpin, Y. V. Loiko, A. Peinado, A. Lizana, T. K. Kalkandjiev, J. Campos, and J. Mompart, “Polarization tailored novel vector beams based on conical refraction,” Opt. Express 23(5), 5704–5715 (2015). 21. A. Turpin, V. Shvedov, C. Hnatovsky, Y. V. Loiko, J. Mompart, and W. Krolikowski, “Optical vault: a reconfigurable bottle beam based on conical refraction of light,” Opt. Express 21(22), 26335–26340 (2013). 22. M. V. Berry, “Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike,” J. Opt. A, Pure Appl. Opt. 6(4), 289–300 (2004). 23. C. F. Phelan, D. P. O’Dwyer, Y. P. Rakovich, J. F. Donegan, and J. G. Lunney, “Conical diffraction and Bessel beam formation with a high optical quality biaxial crystal,” Opt. Express 17(15), 12891–12899 (2009). 24. M. V. Berry, M. R. Jeffrey, and M. Mansuripur, “Orbital and spin angular momentum in conical diffraction,” J. Opt. A, Pure Appl. Opt. 7(11), 685–690 (2005).


Introduction
In normalized coordinates, the transverse extent of the central disk of the zero-order Bessel beam (J 0 -beam) represents the "practical" far-field limit of spatial light confinement in terms of beating the classical diffraction limit and simultaneously preserving an acceptable ratio of the central peak intensity to the intensity of the secondary maxima [1,2].
A non-diverging J 0 -beam can be produced after a positive lens if a very thin annulus of light is formed in its front focal plane [3].With a low numerical aperture (NA) lens a J 0 -beam is generated provided that two conditions are satisfied: i) the integral of the phase of the electric field along the annulus is zero and ii) the polarization distribution along the annulus is homogeneous.
If a high NA lens is used instead, the resulting transverse intensity distribution is no longer described by a J 0 2 function [4,5] unless i) the NA is equal to one (for experiments performed in vacuum) and ii) the annulus is radially polarized [6].For all other possible annuli of light, the central intensity peak of the corresponding beam will only be broader than that of the canonical J 0 -beam.
Here, we show that while the central disk of the J 0 -beam represents, in the above context, the smallest light spot as far as the transverse intensity distribution is concerned, one can place an arbitrarily small L-type polarization singularity inside it.L-type polarization singularities occur where the handedness of the polarization ellipse is undefined and the polarization on them is purely linear [7].Specifically, we create a sharp annulus of light inside which the plane of the electric field oscillations rotates by π radians as an observation point completes a full circle along the annulus.Using such an annulus we produce a nondiffracting vector beam containing an infinite number of L-type polarization singularities and demonstrate that the radius of the first L-line is always smaller than the radius of the central disk of the J 0 -beam that one would obtain if the annulus were homogeneously polarized and had no phase circulation along it.We reveal the presence of L-lines in the beam cross-section by means of a segmented polarizer with multiple transmission axes.Despite the extensive literature on Bessel beams and different superpositions thereof [3,[8][9][10][11][12][13], this case has not been analyzed until now.

Theory
We first consider the situation when a thin annular aperture is uniformly illuminated with a circularly polarized light beam.Generally, the illuminating beam may have a phase singularity along its axis [7].In this case, integrating the phase of the light field around a path enclosing the axial singularity yields 2πl, where l denotes the orbital number.We assume that 0; 1; 2... l = ± ± .If the annular aperture is placed in the front focal plane of a lens, each point along the aperture produces a plane wave after the lens.The whole aperture then generates a set of interfering planes whose wave vectors k lie on the surface of a cone with apex angle θ .
The interference results in the formation of a Bessel beam [3] whose electric field E can be written as: where E 0 is the amplitude of the electric field, where e ρ (e φ ) is the unit vector in the radial (azimuthal) direction, ρ is the radial distance and φ is the azimuthal angle in cylindrical coordinates, ( ) l J k ρ ρ is the Bessel function of the first kind of order l, φ 0 is an arbitrary initial phase, and k ρ = ksin(θ) (k z = kcos(θ)) denotes the radial (axial) component of k.In order to obtain Eq. ( 1), we have redefined the unit circular vector as 1 ( ) 2 where E n is the amplitude of the n-th beam and l n (σ n ) is its orbital (spin) number.It is worth mentioning that the resulting beam described by Eq. ( 2) can be either homogeneously or inhomogeneously polarized, depending on l and σ.Below, we will focus only on two-beam superpositions (m = 2) constructed according to Eq. ( 2).If 1 2 1 σ σ = , the resulting beam is always circularly polarized and therefore belongs to the set of homogeneously polarized beams, which are not the subject of our study.On the other hand, if 1 2 1 σ σ = − , Eq. ( 2) represents an inhomogeneously polarized (i.e., vector) beam whose polarization depends on the transverse coordinates ρ and φ.For instance, by combining two equal-power (i.e, 1 2 0 E E E = = ) Bessel beams with l 1 = −l 2 we produce a subset of vector non-diffracting beams whose polarization is linear locally, but the orientation of the plane in which E oscillates varies in the transverse plane of observation according to: When 1 lσ = − , the set given by Eq. (3) encompasses spirally polarized Bessel beams for all the values of φ 0 , except 0 0, , 2 ϕ π π = ± ± , when the beams are radially polarized [12,13], and 0 / 2; 3 / 2; 3 / 2 , when the Bessel beams become azimuthally polarized.Bessel beams having the polarization distribution similar to that of the so-called "antivortex" Laguerre-Gauss beam [14] are obtained when 1 lσ = and 0 ϕ is arbitrary.We also note that the numbers l and σ in Eq. ( 3) do not imply any longer that the generated vector beam is either circularly polarized or carries orbital angular momentum.
A much richer variety of polarization patterns can be produced when the constituting Bessel beams in Eq. ( 2) have different l.Here, we will analyze two-beam superpositions when i) with φ 0 = 0 in each case.According to Eq. ( 2), the electric fields corresponding to the above situations are: By inserting a half-wave plate into the beam given by Eq. (4a) one transforms it to the beam given by Eq. (4b).The effect of the half-wave plate is to mirror the electric field vector at each point of the beam cross section through the plane formed by the half-wave plate's fast axis and the beam propagation direction z.Hence, the half-wave plate i) inverts the handedness of the polarization ellipse and also ii) rotates the polarization ellipse at each point through twice the angle between its semi-major axis and the half-wave plate's fast axis.
Other superpositions with φ 0 = 0 that result in vector beams can be constructed when iii) with the pertinent electric fields being given by: However, it is sufficient to study only the properties of the beams described by Eqs. ( 4), as the pair (E 11 , E 12 ) can be transformed to the pair (E 21 , E 22 ) using the operation ; l l σ σ → − → − .
The beams given by Eqs.(4a) and (4b) contain polarization singularities in the form of Clines at their axes and an infinite number of coaxial L-surfaces.By definition, C-lines are located where the orientation of the polarization ellipse is undefined, whereas L-surfaces are located where the handedness of the polarization ellipse is undefined [7].The polarization is purely circular along C-lines and purely linear on L-surfaces.C-lines and L-surfaces are intersected by any transverse plane of observation at C-points and along L-lines, respectively.The near-axis electric field of the beams described by Eq. (4a) and Eq.(4b) is shown in Fig. 1(a) and Fig. 1(b), respectively.At any fixed moment of time, the plane of the electric field oscillations rotates by π radians as an observation point completes a full circle along the Llines.In this point the phase of the electric field has a jump discontinuity of π radians.This phase singularity moves along each of the L-lines in time and completes a full circle during a half period of the light wave [dark circles in the bottom panels of Fig. 1].Although the amplitude of the electric field is equal to zero in the point of phase singularity, the intensity averaged over a half period of the light wave has a nonzero value.
Detailed polarization maps of the central regions of the beams described by Eqs.(4a) and (4b) are presented in Fig. 2. From Fig. 2(a) one can see that at ρ = 0 the beam given by Eq. (4a) has a "lemon" C-point [7] in which the polarization is right circular (σ 1 = 1).As ρ increases, the polarization at any fixed value of the angular coordinate φ changes to right elliptical and then to purely linear on the first L-line.As one continues moving past the first L-line along the radial direction, the polarization progressively evolves into left elliptical, left circular, right elliptical, and linear, where the second L-line is located.At any fixed φ, the linear polarizations on odd (i.e., 1st, 3rd, 5th etc) and even (i.e., 2nd, 4th etc) L-lines are orthogonal.Figure 2(b) demonstrates that the beam given by Eq. (4b) contains a "star" C-point [7] in its center.In this case, the polarization in the C-point is left circular (σ 1 = −1).The polarization along the radial direction in the beam given by Eq. (4b) follows exactly the same sequence of transformations as in the beam given by Eq. (4a) if we disregard the fact that now we start with left circular polarization instead of right circular.
The radial coordinates of the L-lines in both the beam given by Eq. (4a) and Eq.(4b) can be found from the equation The radius of the first L-line (j = 1) is always smaller than the radius of the first dark ring ρ 0 of the J 0 -beam in the superpositions described by Eqs.(4).
On the other hand, by simply varying the ratio 1 E / 2 E in these superpositions one can position the first L-line anywhere within the central disk of the constituting J 0 -beam.
Superpositions of Laguerre-Gauss beams that possess similar properties are named Poincaré beams due to the fact that their polarization states cover the entire surface of the Poincaré sphere [15][16][17].Following the established terminology, we will call the beams described by Eqs. ( 4) and ( 5) "Bessel-Poincaré beams."In contrast to Laguerre-Gauss-Poincaré beams, any transverse plane passing through Bessel-Poincaré beams contains an infinite number of concentric rings of L-lines.

Experiment
An elegant way to synthesize the light fields described by Eqs. ( 4) is based on the phenomenon of conical diffraction [18][19][20].A schematic of the beam synthesizer is shown in Fig. 3(a).A Gaussian beam with a vacuum wavelength of λ = 632.8nm is first made circularly polarized using a polarizer (P1) followed by a quarter-wave plate (QWP1).It is then focused with a microscope objective (O1) onto a biaxial crystal (BC) along one of its optic axes.Both the Fig. 3. Generation and analysis of nondiffracting Bessel-Poincaré beams (BPB).(a) Schematic of the beam synthesizer.P1 and QWP1 denote an input polarizer and input quarter-wave plate, respectively; O1 and O2 denote objectives with NA = 0.25 and NA = 0.5, respectively; BC is a ~2.8 mm long KGd(WO 4 ) 2 crystal; IP denotes the image plane where the two sharp annuli of light (see text) are formed with O2; S is a stop whose circular opening has a radius of ~5 mm; L is a plano-convex lens of focal distance f = 70 cm.The inset shows how the BPB1 given by Eq. (4a) can be converted into the BPB2 given by Eq. (4b) using a half-wave plate (HWP).(b) Experimental results on the generation of a nondiffracting BPB1 and revealing the constituting J 0 -and J 1 -beam inside it.QWP2 and P2 denote an analyzing quarter-wave plate and analyzing polarizer, respectively.The intensity distributions of the constituting J 0 -and J 1 -beam are shown at 0 and 5 meters after L.
#237477 -$15.00USD entrance and exit face of the crystal are cut perpendicular to this axis.The waist of the incident beam is positioned behind the crystal.Under our experimental conditions [see captions to Fig. 3], a well defined optical bottle [not shown in Fig. 3] is formed in the beam waist region [21].In the transverse observation plane drawn through the middle of the bottle one would see that the bottle wall is comprised of two closely spaced annuli of light separated by a zero-intensity region (i.e., the Poggendorff ring) [21][22][23].The measured radius of the outer annulus is ~49 μm while its width at the 1/e 2 intensity level is ~2.5 μm.The intensity maxima of the annuli are separated by ~2.5 μm along the radial coordinate ρ.The polarization inside each of the annuli is linear and depends only on the azimuthal coordinate φ [18,22,23].Specifically, the plane of polarization, i.e., the plane of the electric field oscillations, rotates by π radians as an observation point completes a full circle along the annuli [Fig.3(a)], which means that for all diametrically opposite points on the annuli the linear polarizations are orthogonal.The inner and outer annuli have opposite phase [23].
Another microscope objective (O2) is used to form an approximately ten times magnified image of the two annuli in the image plane IP, which coincides with the front focal plane of a positive lens (L).The diffraction patterns from the outer and inner annuli are separated by a dark ring as they propagate from IP to L, as shown in Fig. 3(b).Right before L, the diffraction pattern from the outer annulus is blocked using a stop (S).The radius of the circular opening in the stop S is equal to the radius of the dark ring that separates the diffraction patterns produced by the inner and outer annuli.
The beam after L is a superposition of two equal-power non-diffracting circularly polarized J 0 -and J 1 -beams with the same θ and opposite handednesses, which is nothing else but the Bessel-Poincaré beam given by Eq. ( 4a) when E 1 = E 2 .This beam is abbreviated with BPB1 in the text further down.Correspondingly, the Bessel-Poincaré beam given by Eq. (4b) E 1 = E 2 is called BPB2.The handedness of the J 0 -beam always coincides with the handedness of the input circularly polarized beam [24].To confirm that the BPB1 does consist of a J 0 -and J 1 -beam in orthogonal circular polarizations one may use a quarter-wave plate (QWP2) followed by an analyzing polarizer (P2), as demonstrated in Fig. 3(b).The fast axis of QWP2 must be orthogonal to the fast axis of the input quarter-wave plate QWP1.To reveal the J 0 -beam (J 1 -beam), the transmission axis of P2 must be oriented parallel (perpendicular) to the transmission axis of the input polarizer P1.Under our experimental conditions, the BPB1 propagates without any noticeable diffraction effects over a distance of ~5 m [Fig.3(b)].
Figures 4(a) and 4(b), which represent simulations, demonstrate how L-lines inside both the BPB1 and BPB2 can be traced using a standard polarizer and fully visualized using a segmented polarizer with multiple differently oriented transmission axes.As it was mentioned earlier, the BPB1 is easily converted into the BPB2 by means of a half-wave plate, as shown in the inset of Fig. 3(b).The intensity profiles and propagation properties of these two beams are exactly the same.After a standard polarizer, the distinct intensity minimum near each beam's axis represents a point lying on the first L-line.This implies that in order to see full Llines inside these beams one can use specially designed segmented polarizers whose transmission axes rotate in the same manner as the plane of polarization along the corresponding L-lines.The segmented polarizer in either case, however, must be oriented so that the transmission axis of each segment is orthogonal to the local plane of polarization in the L-lines to be visualized.Taking into account that the linear polarizations along the first and second L-line in both the BPB1 and BPB2 are orthogonal, the segmented polarizers shown in Figs.4(a) and 4(b) allow one to reveal only even or odd L-lines at a time.We also note that while the polarization distributions in the BPB1 and BPB2 are quite different [see Fig. 2(a) and 2(b)], the radial coordinates of L-lines in these beams are exactly the same.
Figure 4(c) presents the pertinent experimental data.The lens L has been removed from the basic setup shown in Fig. 3(a) and after the stop S we now observe a diverging BPB1.A BPB2 is generated by placing a half-wave plate near IP.Approximately 5 m away from the stop S the central regions of the diverging BPB1 and BPB2 are sufficiently broad (i.e., ~4 mm) to perform measurements using a properly matched segmented polarizer without introducing significant inaccuracies caused by the non-perfect alignment of the segments in its center.The top panel of Fig. 4(c) demonstrates several intensity distribution patterns obtained by inserting a standard polarizer into the diverging BPB1 and BPB2 under consideration and changing the orientation of its transmission axis.The bottom panel shows that even a 4-segment polarizer allows one to clearly observe L-lines in these beams.
The graph in Fig. 5(a) quantitatively illustrates the above observations by comparing the intensity profiles of the original BPB1, the BPB1 after the segmented polarizer, and the constituting J 0 -beam.A noteworthy feature of the intensity distribution after the segmented polarizer is that its central disk is ~1.6 times smaller than that of the constituting J 0 -beam, which occurs because this Bessel-Poincaré beam is composed of equal-power J 0 -and J 1beams.It is also noteworthy that the produced central disk has a cusp-shaped intensity profile, in contrast with the smooth, bell-shaped profile of the J 0 -beam.As can be seen from the graph, the experimental data agree with the simulations.
On the other hand, Fig. 5(b) presents a visual comparison of the first L-line and the central disk of the J 0 -beam that are contained within the BPB1.The simulations also present the case of an ideal spatially inhomogeneous polarizer whose transmission axis continuously changes its orientation, mimicking the polarization distribution at the L-lines of the beam.Such a variable polarizer is represented by the following Jones matrix: 2 2 cos ( / 2) cos( / 2)sin( / 2) .cos( / 2)sin( / 2) sin ( / 2) One can show that the light intensity distribution right after the variable polarizer is given by ( ) , which clearly depicts a sub-diffractive peak surrounded by a dark ring representing the original L-line.

Conclusion
We have presented two types of vector beams produced by weakly focusing an inhomogeneously polarized annulus of light: the polarization is linear at every point along the annulus, but the plane of the linear polarization rotates by π radians as an observation point moves along it.Each of these beams represents a superposition a circularly polarized J 0 -beam and circularly polarized J 1 -beam with the same propagation constants and opposite handednesses.As a consequence, the produced vector beams are non-diffracting.Nominally, the presented vector beams contain an infinite number of concentric L-lines in any transverse cross-sectional plane, which represent an infinite number of mappings of the Poincaré sphere onto the beams' cross sections.Therefore, these beams are Poincaré beams.The radius of the first L-line in each of these beams is always smaller than the radius of the central disk of the constituting J 0 -beam.Despite the fact that the central disk of the J 0 -beam in normalized optical coordinates represents the smallest light spot as far as the intensity distribution is concerned, one can easily place an L-type polarization singularity of an arbitrarily small radius inside it.We have also shown how Bessel-Poincaré beams can be generated using the phenomenon of conical diffraction.This robust, power efficient and easy to implement method allows one to produce high-quality beams by simply using a biaxial crystal sandwiched between two objective lenses.To visualize the L-lines we use a special segmented polarizer whose transmission axis gradually rotates by π radians.In our experiments, the radius of the first L-line contained within the Bessel-Poincaré beams is ~1.6 times smaller than the radius of the constituting J 0 -beam.
) corresponds to right (left) circular polarization.Using the above notations, the electric field of a complex light field produced by an annular aperture illuminated by a coherent superposition of m coaxial light beams with different l and σ can be presented as a sum:

Fig. 1 .
Fig. 1.The near-axis distribution of the electric field (denoted by red arrows) inside a superposition of two circularly polarized Bessel beams with (a) l 1 = 0, σ 1 = 1; l 2 = 1, σ 2 = −1 and (b) l 1 = 0, σ 1 = −1; l 2 = 1, σ 2 = 1.The instantaneous distribution of the electric field in the constituting Bessel beams is shown in the top panels of (a) and (b).The trajectories of fixed points on the wave fronts of the constituting beams are plotted using green lines.The distribution of linear polarization along the first L-line in the resulting Bessel beam is represented by black double arrows.T is the period and λ is the wavelength of the light wave.

Fig. 2 .
Fig. 2. Polarization maps of (a) the beam described by Eq. (4a) and (4b) the beam described by Eq. (4b).Right-handed states of polarization are in red, left-handed states of polarization are in blue.The first two L-lines shown in both (a) and (b) are in black.

Fig. 4 .
Fig. 4. Revealing the polarization structure of Bessel-Poincaré beams.(a) simulations pertaining to the BPB1.(b) simulations pertaining to the BPB2.(c) experimental data (i.e., CCD images) showing how the first L-line in the BPB1 and BPB2 can be traced using a conventional polarizer with one transmission axis and clearly observed with a segmented polarizer with four differently oriented transmission axes.The different orientations of the transmission axes in both the cases are indicated by double arrows.In (a) and (b), right-handed states of polarization are in red, left-handed states of polarization are in blue.

Fig. 5 .
Fig. 5.The sub-diffraction size of the first L-line inside Bessel-Poincaré beams.(a) experimental (solid lines) and calculated (dotted lines) intensity profiles of the original BPB1 (green), the BPB1 after the segmented polarizer (red), and the constituting J 0 -beam (blue).(b) visual comparison of the revealed first L-line and the central disk of the J 0 -beam, which are contained within the BPB1.The top-right image in the simulations shows the BPB1 after an ideal spatially inhomogeneous polarizer whose transmission axis continuously changes its orientation, mimicking the polarization distribution of the L-lines.The middle and right columns represent a 4-segment polarizer.