A Brenier

A Majchrowski

E Michalski

acentric biaxial crystal, displayed with Bi2ZnOB2O6

Keywords:

Phase distributions accompanying the conical diffraction through a Pasteur acentric biaxial crystal, displayed with Bi2ZnOB2O6

A Brenier, A Majchrowski,

Introduction

The electric field emerging from a biaxial crystal oriented along an optical axis is a spatially varying vector field (intensity, polarization and phase) caused by the conical diffraction (CD) of the beam instead of the usual double-refraction. So, the beam emerges as a hollow cylinder near the focal plane [START_REF] Landau | Electrodynamique des milieux continus[END_REF][START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[END_REF]. Discovered as early as 1832 by W. Hamilton, this astonishing phenomenon has nowadays numerous applications [START_REF] Turpin | Conical refraction: fundamentals and applications[END_REF]. Let us cite the optical tweezers [START_REF] O'dwyer | Conical diffraction of linearly polarised light controls the angular position of a microscopic object[END_REF], the trapping of Bose-Einstein condensates [START_REF] Turpin | Blue-detuned optical ring trap for Bose-Einstein condensates based on conical refraction[END_REF] and the super resolution microscopy [START_REF] Caron | Conical diffraction illumination opens the way for low phototoxicity super-resolution imaging[END_REF] based on the field intensity, the polarization multiplexing for optical communications [START_REF] Turpin | Free-space optical polarization demultiplexing and multiplexing by means of conical refraction[END_REF], and the polarization metrology [START_REF] Peinado | Optimization, tolerance analysis and implementation of a Stokes polarimeter based on the conical refraction phenomenon[END_REF] using the field polarization. On the other hand, the phase distribution of the field emerging from CD is not exhibited except in Ref. [START_REF] Brenier | Measure by interferential conoscopy of the phase distribution acquired from conical diffraction[END_REF], it is a topic mainly tackled through the orbital angular momentum (OAM) of the beam. Most CD studies are limited to interferograms produced by interference with a linearly polarized reference beam and reveal the OAM [START_REF] O'dwyer | Generation of continuously tunable fractional optical angular momentum using internal conical diffraction[END_REF] in accordance to Berry calculation [START_REF] Berry | Orbital and spin angular momentum in conical diffraction[END_REF]. However, generally speaking, the crucial role of the phase and its singularities were well established in several studies: stability of the vectorial vortices along propagation in fields created with space-variant gratings [START_REF] Niv | Manipulation of the Pancharatnam phase in vectorial vortices[END_REF], phase pattern of fields created by spiral phase plates with fractional step height or spatial light modulator [START_REF] Berry | Optical vortices evolving from helicoidal integer and fractional phase steps[END_REF][START_REF] Leach | Observation of the vortex structure of a non-integer vortex beam[END_REF][START_REF] Götte | Quantum formulation of fractional orbital angular momentum[END_REF]. In the previous cited works on CD, the biaxial crystal is birefringent and centrosymmetric such as the "textbook case" KGd(WO2)4 (KGW) crystal.

The case of acentric crystals is more rarely considered. Let us recall briefly that an acentric crystal (i. e. non-centrosymmetric crystallographic structure) can be isotropic (cubic system) or uniaxial (hexagonal, trigonal and tetragonal) or biaxial (orthorhombic, monoclinic and triclinic) [START_REF] Landau | Electrodynamique des milieux continus[END_REF]; uniaxial and biaxial cases are birefringent media. The acentric crystals exhibit optical activity (they are Pasteur media). The optical activity modifies deeply the CD through a biaxial crystal and was described theoretically adding gyrotropy to the dielectric permittivity, the textbook case being the -iodic acid (-HIO3) crystal [START_REF] Schell | Laser studies of internal conical diffraction. II Intensity patterns in an opically active crystal, -iodic acid[END_REF][START_REF] Belsky | Internal conical refraction of light beams in biaxial gyrotropic crystals[END_REF]. The diffractive theory, including reasonable approximations on refractive indices distribution, was nicely presented by Berry [START_REF] Berry | Chiral conical diffraction[END_REF] making explicit the unitary operator U ̂ acting on the Fourier transform of the input field and allowing the cone diffraction to be described. Other experimental works [START_REF] Brenier | Chirality versus dichroism: competition and role in conical diffraction displayed from the Nd:Bi2ZnOB2O6 acentric biaxial laser crystal[END_REF][START_REF] Brenier | Light propagation properties of the Bi2ZnOB2O6 acentric biaxial crystal: angular orbital momentum from conical diffraction[END_REF] confirm the CD modification and contain the numerical calculation of this operator (in [START_REF] Brenier | Chirality versus dichroism: competition and role in conical diffraction displayed from the Nd:Bi2ZnOB2O6 acentric biaxial laser crystal[END_REF] the operator is not limited to a unitary one).

The present work is devoted to the full experimental and theoretical characterization (polarization and phases distribution) of the U ̂ operator describing the CD through the Bi2ZnOB2O6 (BZBO) crystal. Its space group is mm2 (C2v), so it is orthorhombic, biaxial and acentric; it is a Pasteur medium described here as bi-anisotropic. The link with the OAM is provided. The paper is organized as follows. Section 2 is a summary of the theoretical determination of the U ̂ operator in presence of bi-anisotropy without approximations on refractive indices. Section 3 is the description of the experimental set-up which is nothing else than an interferential conoscope. Section 4 is devoted to the experimental results and their agreement with the theoretical modelling and numerical calculations.

Theoretical background describing the CD in a Pasteur bi-anisotropic crystal

The electric field 𝐄 of a monochromatic input wave at the entrance face of the biaxial crystal can be decomposed into plane waves from its 2D-Fourier transform (FT) 𝐄 ̂(0, 𝑘 𝑥 , 𝑘 𝑦 ), 𝑘 𝑥 , 𝑘 𝑦 being the transverse components of the wave-vector in the xyz crystal frame (z is the optical axis direction). The electric field FT inside the crystal has components on the two eigenmodes propagating as exp[𝑖(𝑘 𝑥 𝑥 + 𝑘 𝑦 𝑦 + 𝑘 𝑧± 𝑧)] with:

𝑘 𝑧± = √𝑘 ± 2 -(𝑘 𝑥 2 + 𝑘 𝑦 2 ) (1)
The two eigenmodes (±) and the two eigenvalues 𝑘 ± , assuming that the Pasteur biaxial crystal is bi-anisotropic, can be calculated as functions of (𝑘 𝑥 , 𝑘 𝑦 ) combining the constitutive relations:

𝐄 = 𝛋. 𝐃 + 𝛘. 𝐁 (2)
𝐇 = 𝛄. 𝐃 + 𝛎. 𝐁 (3)
with Maxwell equations [START_REF] Brenier | Light propagation properties of the Bi2ZnOB2O6 acentric biaxial crystal: angular orbital momentum from conical diffraction[END_REF][START_REF] Au | Electromagnetic wave theory[END_REF][START_REF] Berry | The optical singularities of bianisotropic crystals[END_REF], where the tensors 𝛋 and 𝛎 are respectively the inverse dielectric permittivity and the inverse magnetic permeability and the 𝛘 , 𝛄 tensors ensure the coupling between the electric and magnetic fields.

The BZBO crystal is orthorhombic with point group mm2 (C2v). 𝛋 is obtained through the measured three principal refractive indices at 632.8 nm [START_REF] Li | Growth, Structure, and Optical Properties of a Congruent Melting Oxyborate, Bi2ZnOB2O6[END_REF]. The optical activity needs only the antisymmetric part of 𝛘 containing two optical parameters

𝝌 𝑎 = 𝑖 [ 0 0 𝜒 𝑎𝑥𝑧 0 0 0 𝜒 𝑎𝑧𝑥 0 0 ], with 𝜒 𝑎𝑥𝑧
and 𝜒 𝑎𝑧𝑥 real, to ensure that the propagation is lossless (and 𝛄 = -𝝌 𝑎 𝑇 to ensure the reciprocal propagation). In the present work more refined experiments to determine separately these two 𝜒 𝑎𝑥𝑧 and 𝜒 𝑎𝑧𝑥 parameters will not be conducted, but it will be shown as in Ref. [START_REF] Brenier | Light propagation properties of the Bi2ZnOB2O6 acentric biaxial crystal: angular orbital momentum from conical diffraction[END_REF] that using as a try the same value for both of them gives a satisfactory description of all the experimental figures.

At the output face of the crystal with thickness d the 2D-polarization is a vectorial field [𝐄 ̂(𝑑, 𝑘 𝑥 , 𝑘 𝑦 )] 𝑥𝑦 obtained from the following unitary U ̂ operator acting on 𝐄 ̂(0, 𝑘 𝑥 , 𝑘 𝑦 ):

U ̂(𝑘 𝑥 , 𝑘 𝑦 ) = 𝑒𝑥𝑝 {𝑖𝑑𝑃 [ 𝑘 𝑧+ 0 0 𝑘 𝑧- ] 𝑃 -1 } ( 4
)
where P is the transfer matrix from the xy-frame towards the eigen-modes frame.

Let us finally work in a circular basis in which any electric field can be represented by its Jones

vector 𝑒 𝑖Φ [ cos( 𝜃 2 ) sin( 𝜃 2
)𝑒 𝑖𝜑 ] characterized by its polar and azimuthal angle 𝜃and 𝜑 on the Poincaré sphere. Finally the unitary operator can be written as:

U ̂(𝑘 𝑥 , 𝑘 𝑦 ) = [𝑒 𝑖Φ 𝐿 [ cos( 𝜃 2 ) sin( 𝜃 2 )𝑒 𝑖𝜑 ]𝑒 𝑖Φ 𝑅 [ -sin( 𝜃 2 )𝑒 -𝑖𝜑 cos( 𝜃 2 ) ]] (5)
where the four optical parameters in the right hand side are 𝑘 𝑥 , 𝑘 𝑦 dependent, and are obtained by identification with the calculated ones in Eq. ( 4).

Experimental measurement of the Pancharatman phases difference

The BZBO single crystal was grown in Warsaw from stoichiometric melts by means of Kyropoulos technique on [001] oriented seed [START_REF] Brenier | Chirality versus dichroism: competition and role in conical diffraction displayed from the Nd:Bi2ZnOB2O6 acentric biaxial laser crystal[END_REF]. Its thickness was 0.604 cm. The entrance and exit faces were oriented perpendicular to an optical axis making a 41.9° angle with the zaxis. This angle can be calculated from the values of the refractive indices: n1=2.0679, n2=2.105, n3=2.1541 [START_REF] Li | Growth, Structure, and Optical Properties of a Congruent Melting Oxyborate, Bi2ZnOB2O6[END_REF]. The specific rotatory power was measured to be 2.74 rad/cm [START_REF] Brenier | Light propagation properties of the Bi2ZnOB2O6 acentric biaxial crystal: angular orbital momentum from conical diffraction[END_REF].

An interferential conoscope was built (Fig. 1), basically constituted with two arms of the same length, in order to measure the phase of the FT wave emerging from the biaxial crystal and being subjected to CD as it is described by the operator in Eq. 5. It is very close to the one used in Ref. [START_REF] Brenier | Measure by interferential conoscopy of the phase distribution acquired from conical diffraction[END_REF], so only a short summary is given here. The beam of a laser at 632. ) in the plane 0, more precisely:

ℇ 2𝑓 (𝑋, 𝑌) = -𝑖 2𝜋 𝑘 𝑓 exp(2𝑖𝑘𝑓)ℇ 0 ̂(𝑘 𝑥 = 𝑋 𝑘 𝑓 , 𝑘 𝑦 = 𝑌 𝑘 𝑓 ) (6)
ℇ 0 ̂ can be taken as the reference field alone or the CD field alone, or the field resulting from the interference of the two beams.

Following Pancharatman and Berry [START_REF] Pancharatman | Generalized theory of interference, and it, s applications[END_REF][START_REF] Berry | The Adiabatic Phase and Pancharatnam's Phase for Polarized Light[END_REF], the phase difference ∅ 𝑃 (𝑘 𝑥 , 𝑘 𝑦 ) between the reference and the field under study 𝐸 ̂(𝑘 𝑥 , 𝑘 𝑦 ) after CD is the phase retardation introduced by the liquid crystal retarder which maximizes the intensity of their interference for each camera pixel, that is to say:

∅ 𝑃 (𝑘 𝑥 , 𝑘 𝑦 ) = arg < 𝑟𝑒𝑓 ̂(𝑘 𝑥 , 𝑘 𝑦 )|𝐸 ̂(𝑘 𝑥 , 𝑘 𝑦 ) > (7)
The Pancharatman phase ∅ 𝑃 produced by the liquid crystal retarder was increased step by step . All these patterns can be calculated with a good agreement with the model of section 2. When the reference beam is not perfectly aligned with the CD beam and has a slight horizontal or vertical deviation, the resulting modification of the interference pattern can also be calculated in agreement with the experimental data. An example of such a modification is provided in Fig. 2 (a2). The CD beam is the same as in Fig. 2 (a1) but the reference beam is horizontally shifted by 100 µm: it was observed by us as a curiosity that the interference pattern is constituted of "pleasant" concentric circles (also predicted by the theory), while in the general case the patterns are much more complicated.

Experimental and theoretical determination of the CD 𝐔 ̂ operator 4.1 Polarization of the FT electric field after CD

The reference beam is stopped in this subsection and a LCP beam is launched in the arm containing the crystal. The 𝜃(𝑘 𝑥 , 𝑘 𝑦 ) and 𝜑(𝑘 𝑥 , 𝑘 𝑦 ) angular positions of the polarization on the Poincaré sphere of the left column vector of the U ̂ operator in Eq. ( 5) are related to the S1, S2

and S3 Stoke parameters of the exit beam:

𝜃 = acos(𝑆 3 /𝐼) (8) 𝜑 = arg(𝑆 1 + 𝑖𝑆 2 ) (9) (a1) (a2) (b2) (b1)
where 𝐼(𝑘 𝑥 , 𝑘 𝑦 ) is the intensity of the output beam recorded by the camera.

𝑆 1 = 𝐼(0°) -𝐼(90°) 𝑆 2 = 𝐼(45°) -𝐼(135°) (10)
𝑆 3 = 𝐼(𝐿𝐶) -𝐼(𝑅𝐶)
where 𝐼(0°), 𝐼(90°), 𝐼(45°), 𝐼(135°), 𝐼(𝐿𝐶) and 𝐼(𝑅𝐶) are respectively the intensities recorded by the camera through a linear polarizer horizontal, vertical, at 45°, at 135°, and through a LC and a RC polarizers [START_REF] Born | Principles of optics: electromagnetic theory of propagation, interference and diffraction of light[END_REF], all these optical elements being located in the path L3-CCD in Fig. 1. The colour scale is in rad.

(a1) (a2) (b2) (b1)

The 𝜃(𝑘 𝑥 , 𝑘 𝑦 ) and 𝜑(𝑘 𝑥 , 𝑘 𝑦 ) angular distributions obtained experimentally are represented in Fig. 3 (a1) and (a2), respectively. The calculated distributions (Fig. 3 (b1) and (b2) are in agreement with the experimental ones. It can be seen some noise in the experimental data near the transition 0/2 in Fig. 3 (a2): interpenetration of yellow and blue colours due to the difficulty to distinguish 0 and 2 by our apparatus. It could be improved by using zero-order quarter wave plates designed at 632.8 nm in the experimental set-up with low Fresnel reflexion while achromatic wave plates having a non exact /4 retardation with higher unwanted Fresnel reflections were used by us.

Phases of the FT electric field after CD

The phase Φ L (k x , k y ) of the left column vector of the U ̂ operator in Eq. ( 5) can be measured launching a LCP beam inside the crystal and extracting the component of the output beam obtained by locating in the path L3-CCD in Fig. 1 a quarter-wave plate with its slow axis at 45° from horizontal, followed by a vertical linear polarizer. The full Jones matrix of both these two elements is

1 √2 [ 1 0 -1 0 ],
and this operator modifies the FT field in Eq. ( 7) as:

|E′ ̂(k x , k y ) > = exp(i k x 2 +k y 2 2k d)e iΦ L [ 1 -1 ] cos ( 𝜃 2 ) /√2 (11)
The reference beam is kept vertically polarized

[ 1 - 1
]. The Pancharatman phase difference resulting of Eq. ( 7) is:

∅ P (k x , k y ) = Φ L (k x , k y ) + k x 2 +k y 2 2k d (12)
Experimentally ∅ P (k x , k y ) is obtained as the phase retardation maximizing the intensity of the interference on each camera pixel. Then the experimental determination of Φ L (k x , k y ) is obtained applying Eq. ( 12). The result for ∅ P (k x , k y ) is represented in Fig. 4 (a1) and it can be compared with its calculated representation in Fig. 4 (b1). The ∅ P calculation is from inserting in Eq. ( 12) the ∅ L calculation based on the model developed in section 2 and represented in Fig.

(c1)

. The reasonable agreement between the two figures Fig. 4 (a1) and (b1) validates the used model. Φ 𝐿 (theoretical). a2, b2 and c2: the same but for Φ 𝑅 . The horizontal and vertical axes are in cm (camera screen). The colour scale is in rad.

(a1) (a2)

(b2) (b1) (c1) (c2)
The phase Φ R (k x , k y ) of the right column vector of the U ̂ operator in Eq. ( 5) can be measured launching a RCP beam inside the crystal and extracting the component of the output beam by locating in the path L3-CCD in Fig. 1 a quarter-wave plate with its slow axis at -45° from horizontal, followed by a vertical linear polarizer (the full Jones matrix is

1 √2 [ 0 -1 0 1 ]). The
Pancharatman phase difference of this exit beam and the VP reference is obtained as previously replacing Φ L by Φ R in Eq. ( 12) and adding a supplementary  phase. The result is represented in Fig. 4 (b2) and it can be compared with its calculated representation in Fig. 4 (c2). The calculated Φ R phase distribution is given in Fig. 4 (c2).

Let us appreciate the complexity of a Pasteur acentric biaxial crystal, displayed in the present work with Bi2ZnOB2O6: the two phases needed for the optical description Φ L and Φ R are different, while in a centrosymmetric birefringent biaxial crystal such as KGd(WO2)4 only one phase is needed [START_REF] Brenier | Measure by interferential conoscopy of the phase distribution acquired from conical diffraction[END_REF]. So, the rotatory power separates the two Φ L and Φ R phases. This is in agreement of the well-known fact that for the peculiar propagation along the optical axis in a Pasteur medium, the two eigen-modes are LCP and RCP and the optical activity separates their refractive indices (leading to the rotation of an incident linear polarization). Quantitatively the specific rotatory power  (rad/cm) is related to the difference k in their wavenumbers: = k/2

(see for example ref. [START_REF] Brenier | Light propagation properties of the Bi2ZnOB2O6 acentric biaxial crystal: angular orbital momentum from conical diffraction[END_REF]), so it is related to the phase difference of the two eigen-modes accumulated during the propagation. The difference between Fig. 4 c1 and c2 is the generalization of this behavior to any propagation directions.

Let us add that neglecting the axial dependence of the refractive indices (such as in Ref. [START_REF] Berry | Chiral conical diffraction[END_REF]) prevents a satisfactory description of these phases.

Link with the CD-induced OAM

The electromagnetic wave emerging after CD carries an OAM [START_REF] Berry | Orbital and spin angular momentum in conical diffraction[END_REF] due to the missing of azimuthal symmetry around the optical axis. Moreover, the optical activity trends to reduce its value. The evaluation of the z-component of the OAM (ℏ/photon), writing the OAM operator acting on the field having two V and H components |𝐸 ̂> = 𝐸 ̂𝑉|𝑉 ̂> +𝐸 ̂𝐻|𝐻 ̂> as 𝐿 ̂𝑧 = -𝑖ℏ 𝜕 𝜕𝜙 (𝜙 is the azimuthal angle in Fourier space), is:

𝑙 𝑧 = ∬(𝐼 𝑉 𝜕 𝜙 ∅ 𝑃𝑉 +𝐼 𝐻 𝜕 𝜙 ∅ 𝑃𝐻 )𝑑𝑘 𝑥 𝑑𝑘 𝑦 ∬(𝐼 𝑉 +𝐼 𝐻 )𝑑𝑘 𝑥 𝑑𝑘 𝑦 (13)
where ∅ 𝑃𝑉 and ∅ 𝑃𝐻 are the Pancharatman phase differences between the field and a vertically and a horizontally polarized references, respectively [START_REF] Brenier | Measure by interferential conoscopy of the phase distribution acquired from conical diffraction[END_REF].

Let us apply Eq. ( 13) in two cases, both launching a LCP beam through the crystal and a Vpolarized reference.

As the first example we chose the linear and uniformly polarized field provided by locating in the path L3-CCD in Fig. 1 a quarter-wave plate with its slow axis at -45° from horizontal followed by a vertical linear polarizer (as at the end of subsection 4.2). This operator modifies the FT field of Eq. ( 7) as:

|𝐸′ ̂(𝑘 𝑥 , 𝑘 𝑦 ) > = -exp(𝑖 𝑘 𝑥 2 +𝑘 𝑦 2 2𝑘 𝑑)𝑒 𝑖(Φ L +φ) 1 √2 sin( 𝜃 2 ) [ 1 -1 ] (14)
The Pancharatman phase difference with the reference is obtained as:

∅ 𝑃 (𝑘 𝑥 , 𝑘 𝑦 ) = Φ L (𝑘 𝑥 , 𝑘 𝑦 ) + 𝜑(𝑘 𝑥 , 𝑘 𝑦 ) + 𝜋 + 𝑘 𝑥 2 +𝑘 𝑦 2 2𝑘 𝑑 (15)
The ∅ 𝑃 (𝑘 𝑥 , 𝑘 𝑦 ) phase is measured by the camera of the interferential conoscope (Fig. 5(a1)). It can be compared with its theoretical calculation (Fig. 5 (b1)).

From Eq. ( 13) in which I H = 0 and the I V (k x , k y ) intensity measured experimentally is presented in Fig 5 (c1) and calculated in Fig. 5 (d1), the OAM of the beam can be determined.

The result is 0.996 and we notice that a phase vortex coincides with the nil field intensity in the centre. (a1) (a2)

(b2) (b1) (c1) (c2) (d1) (d2) (c3) (d3) (b3) (a3)
As the second example we chose the vector field after CD without any polarization selection (the input field is still LCP). This vector field is obtained of course with the U ̂ operator (Eq. ( 5))

and the Pancharatman phase difference with the reference is:

∅ 𝑃 (𝑘 𝑥 , 𝑘 𝑦 ) = Φ 𝐿 (𝑘 𝑥 , 𝑘 𝑦 ) + arg(cos ( 𝜃 2 ) ∓ sin ( 𝜃 2 ) 𝑒 𝑖𝜑 ) + 𝑘 𝑥 2 +𝑘 𝑦 2 2𝑘 𝑑 (16)
where the upper sign is valid for a V reference and the lower for a H reference. The intensities The contribution to the OAM of the V component of the field is 0.219 while the H contribution is 0.224. The full OAM (0.443) is fractional.

Conclusion

Plane waves propagation properties at 632.8 nm of an acentric Pasteur biaxial crystal, BZBO, oriented along one of optical axes for conical diffraction were illustrated. The crystal is described theoretically as bianisotropic and the propagation is fully characterized by a U ̂(𝑘 𝑥 , 𝑘 𝑦 ) unitary operator with four parameters depending on the wave-vector. The parameters representing the polarization on the Poincaré sphere were determined with a polarimetric method. Here two phases are involved instead of one, as in the case of a centrosymmetric birefringent crystal such as KGW. They were determined with an interferential conoscope equipped with a liquid crystal retarder and allowing the measurement of the Pancharatman phase difference between the electric field under study and a vertically or horizontally polarized reference. The link with the OAM is provided in the case of an uniformly polarized field and in the case of a vector field. All the experimental data and their theoretical description are in reasonably good agreement without adjusting any optical parameters.

  8 nm in vertical polarization (V) is split in a reference beam, containing a liquid crystal retarder LCC1223 B (Thorlabs), and a second beam (made left or right circularly polarized (LCP or RCP) with a quarter-wave) containing the BZBO crystal under study. The reference can also be made horizontally polarized (HP) adding a half-wave plate. The CD beam can be made LCP or RCP adding a quarter-wave plate and a linear polarizer after the BZBO crystal. A second beamsplitter superimposes the two beams and a convergent lens (L3) makes the interference in the Fourier plane (at abscissa 2 f) where a CCD camera is located.

Fig 1 .

 1 Fig 1. Set-up for interferential conoscopy. BS: beam splitter, QW: quarter-wave plate, LCR: liquid crystal retarder, V: vertical polarization, CP: circular polarization, L1, L2, L3: lenses, M1, M2: Ag mirrors.

(

  22 steps) from 0 to 2. For a few of them in Fig. (2) typical interference patterns obtained from a LCP input field on the BZBO crystal and a VP reference are presented. In Fig. 2 (a1), (a2), (b1) and (b2) the output field is respectively LCP, LCP, RCP and with no polarization selection (let us notice that in the first three cases the field is made vertical polarized before the interference, see subsection 4.2)

Fig. 2

 2 Fig. 2 Typical interference patterns with a VP reference and an output field resulting of a LCP input field through BZBO. a1: the output field is LCP then made VP, a2: the same output field as in a1 but the reference beam is horizontally shifted by 100 µm, b1: the output field is RCP then made VP; b2: the output field has no polarization selection. The horizontal and vertical axes are in cm (camera screen).

Fig. 3 :

 3 Fig. 3: 𝜃(𝑘 𝑥 , 𝑘 𝑦 ) (a1): experimental,(b1): theoretical) and 𝜑(𝑘 𝑥 , 𝑘 𝑦 ) (a2): experimental,(b2): theoretical) angular positions on the Poincaré sphere of the polarization of the left column vector of the U ̂ operator. The horizontal and vertical axes are in cm (camera screen).

Fig. 4 :

 4 Fig. 4: a1 (experimental) and b1 (theoretical) Pancharatman phase difference between the reference beam and a component of the CD beam allowing the Φ 𝐿 determination (see text); c1:

Fig. 5

 5 Fig. 5 a1, b1: experimental and theoretical Pancharatman phase difference between the V reference and the VP charge beam produced by CD; c1, d1: experimental and theoretical intensity of the charge beam produced by CD; a2, b2: experimental and theoretical Pancharatman phase difference between the V reference and the fractional charge vector beam produced by CD; c2, d2: experimental and theoretical intensity of the V component produced by CD. a3, b3: experimental and theoretical Pancharatman phase difference between the H reference and the fractional charge vector beam produced by CD; c3, d3: experimental and theoretical intensity of the H component produced by CD. The horizontal and vertical axes are in cm (camera screen). The colour scale is in rad.

𝐼

  𝑉 and 𝐼 𝐻 are measured (Fig.5(c2) and (c3), respectively) with linear polarizers and are given theoretically (Fig.5 (d (2) and d (3), respectively) by:𝐼 𝑉/𝐻 (𝑘 𝑥 , 𝑘 𝑦 ) = 𝐼 𝐸 (1 ∓ 𝑠𝑖𝑛𝜃𝑐𝑜𝑠𝜑)/2 (17)The Pancharatman phase difference with the V reference is given in Fig.5(a2) (experimental) and 5(b2) (theoretical), while with the H reference it is shown in Fig 5(a3) (experimental) and 5(b3) theoretical).

Acknowledgment: F. Gay is gratefully acknowledged for software writing and optimization.