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Spiral polarization rotators, rotating polarization ellipse axes clockwise or counterclockwise, depending on the azimuth angle in the transverse plane, are considered. It is shown that spiral polarization rotators lead to a change in the order of optical vortices with circular polarization. A comparative analysis of spiral rotators of two types (polar and non-polar) is carried out, using a mirror that allows light to pass in the opposite direction through the rotator. The effect of spiral rotators on optical vortices in a resonator is studied. It is shown that spiral rotators can preserve or accumulate changes of the vortex order during the passage of the beam in both directions. The properties of the spiral rotator and the cube-corner reflector with a special phase-correcting coating, as a diffractive polarization-optical element, are compared.

Study of polarization-inhomogeneous beams allows to conclude that, despite the complex structure, such beams have certain types of polarization symmetry [

The beams with axisymmetric polarization structure have two modifications, depending on rotation direction of the polarization ellipse axis and are closely related to optical vortices.

Currently, variety of data on optical vortices [

Much less attention is paid to polarization characteristics and diffraction polarization-optical devices [

The aim of this work is to study spiral polarization rotators, which—like conventional polarization rotators—exist in two forms: polar and non-polar, and their influence on optical vortices, including in optical resonators. This article is a continuation of [

Polarization-symmetric structures with a regular change of polarization in a beam cross section will be denoted by capital letters indicating the direction of rotation of the semi-major axis of the polarization ellipse: P (counterclockwise) and N (clockwise) when viewed towards the ray direction [

Orthogonal polarization structures of P-modification in Cartesian basis are described by Jones vectors of the following type:

Here indices r and a denote orthogonal structures: radially polarized and azimuthally polarized, respectively. For these structures, the plane of vector E oscillations makes s counterclockwise rotations in the transverse beam plane when observed towards the beamand axisymmetric polarization structure order.

Polarization structures of N-modification with the rotation of the plane of vector E oscillations in a clockwise direction are described by Jones vectors of the following type:

An optical vortex with an axisymmetric polarization structure has a spiral phase structure alongside with the axial polarization symmetry with srepetitions of the polarization state in the cross section (in particular, this is the rotation of polarization plane). Optical vortices with an axisymmetric polarization structure of various types vary depending on the number of repetitions of a phase—q (the optical vortex order) and polarization state—s (the order of axisymmetric polarization structure).

The general expression for Jones vector of an optical vortex with axisymmetric polarization structure of P-modification in Cartesian basis is:

and for N-modification:

where

In particular case, if

Here the first index designates the type of circular polarization: right circular (if vector E rotates clockwise when observed towards the beam)—r and left circular (if vector E rotates counter-clockwise)—l; the second index with capital letters designates optical vortex type: right-handed (R) and left-handed (L).

Such polarization optical vortices have optical angular moment 2, −2 and 0, according to [

Optical vortex with arbitrary polarization can be expressed by superposition of two beams with axisymmetric structure of the same order.

Jones matrix of a high-quality mirror for a plane polarization-homogeneous wave is given by:

In case of polarization-inhomogeneous wave transformation of vector Е defined by (6) is preserved for each cross-section point if wave front curvature coincides with mirror curvature or their difference can be neglected.

If coordinates of a light beam point are determined in cylindrical coordinate system by an azimuthal angle

then

and

transforms to

High-quality mirror does not disturb polarization-optical symmetry of beams, because transformations

Transformation of polarization-symmetric structures, in particular, change of the direction of rotation of the plane of E oscillations in the transverse plane, as well as transformation of optical vortices with axisymmetric polarization structure, is possible by means of special polarization-inhomogeneous devices, spiral polarization rotator (SR) being among them.

Jones matrices of spiral rotator of u order are given by:

According to our definition, a positive SR_{P} rotates the polarization ellipse axes counterclockwise by a value depending on azimuth_{N} twists polarization structure clockwise.

Spiral rotators are characterized by the presence of semi-axis of symmetry, from which azimuth should be counted. For points of a transverse plane, superadjacent this semi-axis SR, the state of polarization of an incident light beam is preserved. Thus, unlike standard polarization-uniform rotators, spiral rotators are not invariant to the rotation of the Cartesian polarization basis.

When a linearly polarized wave propagates a SR_{P}, P-modifiedbeams are generated, in particular, if_{N} forms N-modified beams from linear polarization.

SR affects a beam with P-modified axisymmetric polarization structure, changing its order: the new order

Let us consider the effect of a spiral rotator on optical vortices with circular polarization. The order of an optical vortex can either increase or decrease. This is determined by the ratio of the signs of SR and of ellipticity angle

In accordance with our terminology positive SR destroys vertices with Jones vectors

Let us consider circular polarized optical vortex after its spiral phase structure has been destroyed by a spiral rotator. Let the optical vortex be a combination of two axisymmetric beams. In particular, as is well known, the left optical vortex is a superposition of radially polarized and azimuthally polarized beams shifted in phase by 90˚. After the destruction of the phase vortex structure, each of these beams becomes linearly polarized in mutually perpendicular planes with a phase shift of 90˚, which forms circular polarization. Alongside the spatial amplitude distribution is still corresponding to axisymmetric beams with zero intensity on the axis. Due to diffraction, such beams do not retain their amplitude-phase distribution in the far zone: the beam energy “flows” from the periphery to the center.

Like polarization-uniform rotators, spiral rotators are divided into polar, magneto-optical or Faradaytype (further they are designated as SRF) and non-polar, based on natural optical activity, type (SRO).

In the coordinate system, which remains right for the reversewave, the sign of the polar SRF (positive or negative) is reversed, and the sign of the non-polar SRO is retained. However, unlike common Jones matrices in this case azimuth change

Here superscript indicates direct (+) and reverse (–) waves.

Thus, unlike usual Jones matrix in natural-optical rotator, in this case the sign of the rotator in Jones matrix has changed to the opposite.

Consequently, for polar (Faraday) SRF we have:

Mechanism of the sign change remains for negative SR.

Combined affection of spiral rotators and high-quality mirror onto polarization optical vortices is illustrated by

Analyzing the figures one would easily notice the difference in action of Faraday spiral rotator and action of rotator, based on natural optical activity.

If the sign of the optical vortex is opposite to the sign of circular polarization, then after passing twice the positive Faraday spiral rotator in the direct and reverse direction, the order of the optical vortex increases by 2:

However, if the sign of vortex coincides with the sign of circular polarization, vortices will only change the sign:

In case of both positive and negative SRO, based on natural optical activity, only change of optical vortex sign takes place.

Circular polarization sign changes to the opposite in all cases.

The main difference between the effect of spiral rotator, based on natural optical activity, and a magneto-optical one, combined with a plane mirror, is that the first type does not disturb the polarization-wave symmetry of a beam, while the second one disturbs, and orbital moment of a transmitted beam is not preserved.

Let’s perform a thought experiment: a polarization optical vortex passes through a linear two-mirror resonator; a beam splitting cube, which ideally does not change the polarization state, is used to enter the resonator (

As follows from the analysis of

In a four-mirror ring resonator a wave increases optical vortex order after each passage, respectively, its transverse size and orbital angular moment increase. It is obvious that increase of the order corresponds to an increase of diffraction losses on the diaphragms. One can assume that order of optical vortex, generated in the ring resonator, is approximately equal to Fresnel number:

As it is known, the so-called Faraday cells, consisting of two crossed quarter-wave plates and a magneto-optical rotator between them, are used in laser gyroscopes to produce a phase shift between the passing counter-propagating waves. For this effect, it is necessary that the directions of rotation of the vector E of counter-propagating waves be opposite with respect to the magnetic field H of the Faraday rotator. The maximum effect is achieved in the case of circular polarizations. It should be emphasized that if the counter-waves have right and left circular polarization in their own bases, i.e. relative to the vector H the vectors E of the counter-propagating waves rotate in the same direction, then the phase shift is absent.

We show that in the case of spiral Faraday rotators this requirement is optional. This statement is illustrated in

where w is the parameter of the Gaussian mode distribution (radius with respect to the amplitude decrease in e times); ρ—curvature of the wave front; d—the distance between two spiral Faraday rotators, whose thickness is neglected;

Note that this device can be used to identify the sign of an optical vortex, because the change of the waves sign leads to the absence of a phase shift.

Several ways of technical implementation of polarization rotators exist: medium with the natural optical rotation (crystals, solutions), magneto-optical medium under superimposed magnetic field, system, containing two expanded half-wave phase plates [

Application of crystalline spiral phase plates, giving different phase incursions for orthogonal linear polarization states, makes possible to obtain Jones matrix:

Such matrix in circular polarization basis takes form of a spiral polarization rotator:

SR affecting only linear polarization are easiest to be implemented. These plates have radial sectors, each of which has a half-wave phase plate with axes turned at a certain angle. For example, if there are six segments, these are the following angles: 0˚, -120˚, +120˚, 0˚, -120˚, +120˚, counted counterclockwise when facing towards the beam.

Cube-corner reflector (CCR) [

In this case, Jones matrix at CCR output has form for different sectors (

In accordance with our terminology, such CCR is similar to negative SR of the second order.

Ideally, Jones matrix of the second order spiral reflector is:

Worth noting is that technical implementation of SRO in the form of inhomogeneous sugar solution means that sugar concentration on the semi-axis of SRO is zero. In the case of an inhomogeneous magnetic field of the SRF, there is a semiaxis in the transverse plane along which the magnetic field vanishes or does not cause the rotation of the vector E. In contrast to these devices, in the case of the cube-corner reflector, the position of the semiaxis, for which there is no rotation, is not fixed in the transverse plane, but is determined by the orientation of the linear polarization of the incident wave.

The main results are the following:

1) Spiral polarization rotators are divided into two types: polar (magneto-optical) and non-polar (natural-optical), depending on whether their sign changes for the counter-propagating waves.

2) The structure of a circular polarized optical vortex is destroyed in a device, consisting of a polar (magnetic) spiral polarization rotator and a mirror, and optical orbital moment changes. A similar result occurs when spiral polarization rotator is placed into a linear resonator. Therefore, the polar spiral rotators can produces a phase shift (non-reciprocity) between counter-propagating optical vortices with circular polarization, depending on the sign of the optical vortex. For a non-polar spiral rotator, based on natural optical activity, this effect is not observed.

3) Cube-corner reflector with a special phase-correcting coating is a diffraction polarization-optical element with six segments and a kind of like a spiral rotator.

The author declares no conflicts of interest regarding the publication of this paper.

Sokolov, A.L. (2020) Comparative Analysis of the Characteristics of Polar and Non-Polar Spiral Polarization Rotators. Optics and Photonics Journal, 10, 13-27. https://doi.org/10.4236/opj.2020.102002