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hv
photonics
Article
Gyrotropic Crystals as a Basis for Creation of Helical
Polychromatic Singular Beams
Yuriy Egorov * and Alexander Rubass
Physics and Technology Institute, V.I. Vernadsky Crimean Federal University, Vernadsky Prospect 4,
295007 Simferopol, Russia; alex.rubass@gmail.com
* Correspondence: yuriyegorov@cfuv.ru
Abstract: In this work, studies are carried out in the field of optical singular beams that have passed
through gyrotropic crystals. We have experimentally shown that singular beams with a helical
intensity distribution are formed when passing through a system of two gyrotropic crystals with
opposite values of the gyration coefficient. It is shown that the system is capable of generating optical
vortices with a double topological charge in one of the components of circular polarization when
light propagates through two gyrotropic crystals.
Keywords: polychromatic singular beams; gyrotropic crystals; topological charge
1. Introduction
In modern singular optics, special attention is paid to beams carrying the so-called
optical vortices and those carrying a topological charge [1]. In the mid-1990s, such beams
in free space were obtained experimentally [2,3]. It was then that an intensive experimental
study of singular beams began. Around the same time, it was discovered that rays with
optical vortices carry angular momentum [4,5]. Particular interest in beams containing
optical vortices has increased significantly due to the fact that such beams can be used to
create optical tweezers and similar objects.
However, the creation of single optical vortices carried by paraxial beams of the
Laguerre–Gauss and Bessel–Gauss types faces serious technical difficulties. The point is
that the traditional method of obtaining optical vortices is based on light diffraction either
Citation: Egorov, Y.; Rubass, A.
on computer-synthesized holograms [2,3] or on spiral phase plates [6,7]. These methods are
Gyrotropic Crystals as a Basis for
based on strict observance of the diffraction conditions near the phase singularity and are
Creation of Helical Polychromatic
critical to the wavelength. In this case, special mention should be made of optical beams
Singular Beams. Photonics 2023, 10,
1044. https://doi.org/10.3390/
that have the property of Fourier invariance [8,9]. However, in [10–12], it was possible
photonics10091044
to avoid such stringent requirements for the formation of an optical vortex due to the
processes of light propagation in a uniaxial anisotropic medium. It has been shown that
Received: 14 August 2023 a circularly polarized beam propagating along the optical axis of an anisotropic medium
Revised: 6 September 2023
is capable of forming optical vortices on the axis with the same localization, regardless of
Accepted: 12 September 2023
the wavelength.
Published: 14 September 2023
Traditional concepts of a linearly polarized paraxial beam passing along the optical
axis of a uniaxial crystal suggest that at the exit from the crystal after passing through the
polarizer, the beam forms an intensity distribution in the form of a Maltese cross [13]. This
Copyright: © 2023 by the authors.
pattern is a distinctive feature of a uniaxial crystal.
Licensee MDPI, Basel, Switzerland. From the point of view of polarization features, the conoscopic pattern arises as a result
This article is an open access article of the superposition of two circularly polarized beams carrying vortices with a double
distributed under the terms and topological charge of the opposite sign. In fact, this picture is a field with perpendicular
conditions of the Creative Commons edge dislocations.
Attribution (CC BY) license (https:// When the beam is tilted relative to the optical axis or when external perturbations are
creativecommons.org/licenses/by/ introduced, the double vortices leave the beam, and the structure of edge dislocations un-
4.0/). dergoes structural transformations. The bulk of theoretical and experimental studies [14,15]
Photonics 2023, 10, 1044. https://doi.org/10.3390/photonics10091044 https://www.mdpi.com/journal/photonicsPage 2
Photonics 2023, 10, 1044 2 of 15
mainly concerns the study of the structure of conoscopic patterns arising in axial beams.
Conoscopic carines are interference patterns in a converging polarized beam obtained in
crossed polarizers, between which the anisotropic medium under study is located. The
conoscopic pattern corresponding to the circular state of polarization on the input face of
an anisotropic crystal is localized in the focal plane of the objective. However, it also has
the shape of concentric rings, and near the beam axis there are four topological dipoles
separated by bright stripes. Any disturbance of the axial extension is considered to be a per-
turbation of the conoscopic pattern. However, this process hides various transformations
of polarization features caused by weak perturbations.
It should be noted that the studies [16–18] are devoted to the study of the structure
of polarization features in low-order Laguerre–Gauss beams that arise after excitation of
the crystal by linearly polarized light. The authors of these papers focused on linearly
polarized beams. Their model is based on two linearly polarized beams carrying optical
vortices (ordinary and extraordinary) propagating at slightly different angles.
As is known, uniaxial and biaxial crystals serve as basic elements for the generation
of optical vortices embedded in various types of singular beams [19]. The most amazing
feature of the crystal is the ability to create stable polychromatic vortices with high energy
efficiency. In contrast to the method of computer holograms [20–22], the crystal forms a
white vortex beam without any additional devices [23].
The purpose of this article is to consider another method that allows one to gen-
erate singular beams carrying screw edge dislocations and optical vortices using two
gyroanisotropic crystals. Before considering the method of generating singular beams
carrying optical vortices using two gyroanisotropic crystals, it is necessary to evaluate the
influence of a gyrotropic crystal on the wave front.
2. Gyroanisotropic Crystals
First, it should be noted that a gyrotropic crystal is a space that does not only have
linear birefringence (that is, space is able to transform circularly polarized light into linearly
polarized, and vice versa), but also circular birefringence (that is, space is able to rotate
linear polarization). Even in simple cases, electrical vector D and magnetic vector H are
expressed in a complex way:
D = ε̂E + g rotE, B = H + g0 rotH . (1)
To estimate the effect of a gyrotropic crystal on a singular beam, we chose the Jones
matrix method, which we have already used for anisotropic crystals in [19]. Consider the
propagation of a beam through an element of a gyrotropic crystal along its optical axis. The
optical axis coincides with the z axis. The input face of the crystal is located on the plane
z = 0. A linearly polarized Gaussian beam is incident normally on the input face of the
crystal. As was shown in [19], a Gaussian beam can be represented as a set of rectilinear
rays distributed in space along a hyperboloid of revolution. Figure 1 shows one of these
beams directed at the crystal at an angle δ to its optical axis. The birefringence axes are in
the same plane as the optical axes and form an angle φ with the y-axis.
The beam passes through the element dz of the crystal. The beam polarization trans-
formation Ein can be written in matrix form:
Eout = ĝEin (2)
where
? ?? γ γ ?
δ
cos 2N δ
+ i sin 2N cos 2φ δ
i sin 2N sin 2φ cos N − sin N
ĝ = γ γ (3)
δ
i sin 2N sin 2φ cos 2N − i sin 2N
δ δ
cos 2φ sin N cos N
2
and where N is the number of layers in a crystal, δ = k∆n L √ 2r is the phase differ-
r + d2
ence between the ordinary E(o) and extraordinary E(e) components of the beam that hasPage 3
Photonics 2023, 10, 1044 3 of 15
passed
Photonics 2023, 10, x FOR through a crystal of length d, ∆n L = n
PEER REVIEW
e − no is linear birefringence, no and ne
are values√for the ordinary and extraordinary refractive indices, k is the wave number,
γ = k∆nC r2 + d2 is the full rotation (gyration) of the electric vector, and ∆nC is the value
for circular birefringence.
Figure 1. Scheme of beam1.propagation
Figure through
Scheme of beam a gyrotropic
propagation uniaxial
through crystal; longitudinal
a gyrotropic (a) A‘B‘
uniaxial crystal; longitudina
and transverse (b) AB.
and transverse (b) AB.
As the thickness The of the
beamcrystal
passeselement
through tends
the to zero, d
element ∆zz of→the0 , and at N
crystal. The→beam∞ , wepolarizatio
have: ?formation in
E can be written in
? matrix
? form:
γ ?
δ
1 + i 2N cos 2φ δ
i 2N sin 2φ 1 −N
ĝ ≈ γ = 1̂ + ∆ˆ N + γ̂ N , (4)
δ
i 2N sin 2φ 1 − i 2Nδ
cos 2φ N 1 out
? ? E = gˆ?E in
?
ˆN = i cos 2φ sin 2φ 0 −1
where matrices are ∆ δ
2N sin 2φ − cos 2φ and γ̂ N = γ
N .
where 1 0
For the total length of the crystal, we have:
δ δ δ ? ? γ γ
cos 2 φ d ?∆ 2 φ.
?Z
Ĝ = lim N →∞ ∏ 1 + ∆ˆ N + γ̂ N ≈ lim N →∞ ∏ exp cos∆ˆ N ++ γ̂i sin? = exp ˆ i)sin sin cos (5) − sin
?
2N N 2N ( z + γ̂
2N( z ) dz N N
gˆN= 0
δ δ δ γ γ
i sin sin 2 φ cos − i sin cos 2 φ sin cos
Define the form Ĝ 2N 2N 2N N N
N N
Ĝ = lim N →∞ ∏ exp ∆ ˆ N + γ̂ N = lim exp ∑ (∆ ˆ N + γ̂ N ) = lim exp ∑ µ N r 2
?
(6)
and where N is the number of layers in a crystal, δ = k Δ nL is the pha
N 1 1
r +d
2 2
E(2φ
o)
E( ) components of the beam
e
!
ence between the ordinary iδN
cos and extraordinary
− γN
+ iδN
sin 2φ
N = ∆N
ˆ + γ̂ N =a crystal2N N 2N (7) no and
µpassed through γN iδNlength d, Δn =
of ne − no is linear birefringence,
iδN
N + 2N sin 2φ −
L
2N cos 2φ
values for the ordinary and extraordinary refractive indices, k is the wave
Rewrite Ĝ in the form:
γ = k ΔnC r 2 + d 2 is the fullc rotation
−s (gyration) of the electric vector, and ΔnC is t
? ?
Ĝ = (8)
for circular birefringence. s∗ c∗
N As theNthickness
? of the crystal
? element tends to zero, Δz → 0 , and at N →
where c = ∑ i 2N
δN
cos 2φ, s = ∑ γNN − iδ
have: N
2N sin 2φ , and * signifies complex conjugation.
1 1
Refine the matrix Ĝ. The matrix Ĝ has its own values:
δ δ γ
? 1+ i 2φ
cos ? i sin 2 φ 1 −
λ −
gˆ ≈ ∗ c 2 N s 2 N N = 1ˆ + Δˆ (9)
+ γˆN ,
= 0,
− s δ λ − c∗ δ γ
N
i sin 2 φ 1− i cos 2 φ 1
2 Ns 2N N
∗ 2
c+c (c + c∗ ) 2.
λ= ± δ cos 2 φ − ssin 2φ γ 0 −(10)1
where matrices are 2Δˆ N = i 2 and γˆN = .
2 N sin 2 φ − cos 2 φ N1 0
For the total length of the crystal, we have:
( ) ( ) (
Gˆ = lim N →∞ ∏ 1 + Δˆ N + γˆN ≈ lim N →∞ ∏ exp Δˆ N + γˆN = exp Δˆ ( z ) + γˆ ( z )
N
d
0Page 4
Photonics 2023, 10, 1044 4 of 15
Write the equation for eigenvectors:
(c − λ)h1 − sh2 = 0, (11)
where h1 and h2 are the eigenvectors of the matrix Ĝ. Then,
q !
1 c−c∗ ( c + c ∗ )2
h1 = q 2 + 2 − s2 , (12)
( c + c ∗ )2 s
2 − s2
q !
1 c∗ −c ( c + c ∗ )2
h2 = q 2 − 2 − s2 . (13)
( c + c ∗ )2 s
2 − s2
Compose from h1 and h2 the matrix H, the columns of which are the vectors h1 and h2 :
r r
c−c∗ + ( c + c ∗ )2 c∗ −c − ( c + c ∗ )2
2 2 − s2 2 2 − s2
? ? r r
h1 ( c + c ∗ )2 ( c + c ∗ )2
H= = 2 − s2 2 + s2 . (14)
h2
s s
r r
( c + c ∗ )2 ( c + c ∗ )2
2 − s2 2 + s2
Matrix H −1 , the inverse of matrix H, has the form:
r
c−c∗ + ( c + c ∗ )2
2 2 − s2
r s r
( c + c ∗ )2 2 ( c + c ∗ )2
−1 2 −s − s2
r2
H = 2
. (15)
c∗ −c − ∗
(c+c )
2
−s
r s 2 2
r
( c + c ∗ )2 ( c + c ∗ )2
2 + s2 2 + s2
Further:
N
G = exp ∑ µ N = H exp ∆
ˆ N H −1 .
?
(16)
1
Using the matrix eigenvalue equation:
ˆ N σi = (exp ηi )σi .
exp ∆
?
(17)
where σi and ηi are the eigenvectors and eigenvalues of the matrix ∆ ˆ N.
Rewrite the matrix Ĝ in the form:
γ+i 2δ sin 2φ
cos Λ + i 2 Λ sin Λ cos 2φ
δ
Λ sin Λ
Ĝ = −γ+i 2δ sin 2φ
. (18)
Λ sin Λ cos Λ − i δ
2Λ sin Λ cos 2φ
where Λ2 = (δ/2)2 + γ2 . Expression (18) is called the generalized Jones matrix.
3. Transformation TE and TM Modes in a Gyrotropic Crystal
The above Matrix (18) describes the transformation of the polarization state of a beam
propagating in a gyrotropic crystal. A linearly polarized Gaussian beam passing through a
crystal undergoes structural changes. One example of such a change is shown in Figure 2.Page 5
Photonics 2023,
Photonics 2023, 10,
10, x1044
FOR PEER REVIEW 6 of5 18
of 15
Figure2.2.Field
Figure structureofofa alinearly
Fieldstructure linearlypolarized
polarized Gaussian
Gaussian beam
beam (a)(a) after
after passing
passing through
through a gyrotropic
a gyrotropic
∆n · −-33, ∆n = 3 · 10−5 ,-d5 = 1 cm; (b) long-range order of the field ∆n = 2 · 10−3 ,
crystal with D
crystal with L nL = 2 ⋅ 10
= 2 10 , CD nC = 3 ⋅10 , d = 1cm ; (b) long-range order of Lthe field
∆nnC = -3−5 , d = 1 cm. -5
· 10
=23⋅ 10
D L , D n = 3 ⋅ 10 , d = 1cm .
C
At the same time, it is interesting to consider the transformation of the beam mode
At the same time, it is interesting to consider the transformation of the beam mode
fields depending on their own polarization, which they had in a simple anisotropic crystal.
fields depending on their own polarization, which they had in a simple anisotropic crystal.
As is known [16,17], such beams are TE and TM modes [24] with the following polariza-
As is known [16,17], such beams are TE and TM modes [24] with the following polariza-
tion state:
tion state: ?
sin φ
?
| TEi = G (19)
− cos φ 01
sin φ ?
TE = ? G01 (19)
| TMi = −cos
cos φ
φ G01 , (20)
sin φ
where Gm=0, l =1 is the complex amplitude of the paraxial beam with indices m = 0, l = 1.
The results of the action of the Matrix cos φ on the fields TE and TM are shown in
(18)
TM = G , (20)
Figure 3. The straight lines of the TE and TM sin
Photonics 2023, 10, x FOR PEER REVIEW φ 01that can be observed on the7input
modes of 18 face
of a gyrotropic crystal become twisted in a spiral, and with the opposite twist, as the beam
where G along
propagates thecomplex
is the
m= 0, l =1
crystal. amplitude of the paraxial beam with indices m = 0 , l = 1 .
The results of the action of the Matrix (18) on the fields TE and TM are shown in
Figure 3. The straight lines of the TE and TM modes that can be observed on the input face
of a gyrotropic crystal become twisted in a spiral, and with the opposite twist, as the beam
propagates along the crystal.
−3
Figure3.3.Transformation
Figure TransformationTETE(a)(a) and
and TMTM
(b)(b)
of of mode
mode in aingyrotropic
beams
beams a gyrotropic crystal:
crystal: D nL∆n 10-23 ·, 10 ,
= L2 ⋅=
∆nnC = · 10
=33⋅10 − 5
D C
-5
, d, d== 1 cm.
1cm .
The longer the crystal, the more the direction of the major semiaxis of the polarization
vector twists. Moreover, the linear polarization becomes elliptical. Thus, the TE and TM
modes are not eigenmodes of a gyrotropic crystal. To determine the intrinsic polarizationPage 6
Photonics 2023, 10, 1044 6 of 15
The longer the crystal, the more the direction of the major semiaxis of the polarization
vector twists. Moreover, the linear polarization becomes elliptical. Thus, the TE and TM
modes are not eigenmodes of a gyrotropic crystal. To determine the intrinsic polarization
of the beam, consider the characteristic equation:
−γ+i 2δ sin 2φ
cos Λ + i 2δΛ sin Λ cos 2φ − η Λ sin Λ
γ+i 2δ sin 2φ
= 0. (21)
Λ sin Λ cos Λ − i 2Λ sin Λ cos 2φ − η
δ
It follows that the eigenvalues are:
η = exp(±i Λ). (22)
Therefore, the eigenvectors of Matrix (18) have the form:
γ !
? ?!
δ
2 Λ sin 2φ + i Λ ? − −
1 1 δ
2Λ cos 2φ 1
+
m = ? , m− = , (23)
N − 2δΛ cos 2φ − 1 N δ
sin 2φ − i γ
2Λ Λ
q
where N = 2 − Λδ cos 2φ.
The field structure of eigenmodes is a system of ellipses. Each of these ellipses has
its own ellipticity Q and azimuth angle φ. Write the Stokes parameters S j ( j = 0, 1, 2, 3)
using Expression (23). We receive:
S0 = N 2 ? ?
S1 = − 2Λ
δ
cos 2φ 2 − Λδ cos 2φ
? ? (24)
δ
S2 = 2Λ sin 2φ 2 − Λδ cos 2φ
? ?
γ
S3 = − 2Λ 2 − Λδ cos 2φ .
Therefore, the azimuth angle of the major axis of the ellipse is:
tan 2ψ = SS2 = tan 2φ
1 (25)
or ψ = φ.
The angle of ellipticity χ can be written as:
sin 2χ = S3 . (26)
This is in connection with:
S3 γ
q= = . (27)
S0 Λ
Define the ellipticity of the beam:
b q
Q= = tan χ = p . (28)
a 1 − 1 − q2
Given the very weak gyration of the crystal (which is the case for real crystals δ >> γ)
away from the axes, the ellipticity will be represented as:
γ δ
Q= ≈ . (29)
Λ − δ/2 2γ
Beam eigenmodes in a simple gyrotropic crystal have a nonuniform distribution of
polarization over the beam cross section. Their polarization components do not have phase
singularities (except for the axial case). On the axes, the eigenfields are circularly polarized,
as shown in Figure 4.Page 7
Photonics 2023, 10, x FOR PEER REVIEW 9 of 18
Photonics 2023, 10, 1044 7 of 15
Figure 4. Self-polarized beams m+ (a) and m− (b).
Figure 4. Self-polarized beams m + (a) and m − (b).
As we move away from the center (see Figure 4a,b), the ellipticity of the polarization
stateAs we moveuntil
decreases away thefrom
lightthe center (see
eventually Figure linearly
becomes 4a,b), the ellipticity of the polarization
polarized.
state decreases until the light eventually becomes linearly polarized.
4. Propagation of a Beam through Two Gyroanisotropic Crystals
4. Propagation
Consider of theapropagation
Beam through of aTwo
beamGyroanisotropic Crystals crystals with different
through two gyrotropic
values of circular
Consider the birefringence
propagation of ∆naCbeam
. The rest of thetwo
through crystal parameters
gyrotropic remain
crystals withthe same.
different
valuesFor
of this it is necessary toΔmultiply
case, birefringence
circular two of
nC . The rest matrices [3] with
the crystal the difference
parameters γ1 =
remain the−same.
γ2 = γ.
After applying linear algebra, we have
For this case, it is necessary to multiply two matrices [3] with the difference
γ 1 = −γ 2 = γ . After applying linear algebra, ? we have ?
C11 C12
D Ĝ = ∗ ∗ , (30)
−C12 C11
C C12
DGˆ = 11∗
?∗
, (30)
2
?
2 2
? sin2 Λ −
δ 12 C11
C γ ?
C11 = cos Λ + γ − δ /4 + i sin Λ cos Λ cos 2φ − sin Λ sin 2φ , (31)
Λ2 Λ Λ
sin 2 Λ δ γ in Λ cos 2φγ .
+ ( γ=2 i− δ 2sin
/ 4 )Λ cos
δ ? ?
C11 = cos 2 ΛC12 Λ
+ i
sin sin
2φ Λ
+ Λcos Λ cos 2 φ − sin Λ sin 2φ , (32)
(31)
Λ Λ2 Λ Λ
The eigenvalues η of Matrix (6) have the form
r
? sin2 Λ
2 δ 2 δ γ 2
γ
?
2
η = cos Λ +C12γ= − i δsin Λ Λcos
± i2φ +sin sin
/4Λ cos2Λ sin cos2 2Λ
φ +. 2 sin2 Λ. (33)
(32)
Λ Λ Λ Λ Λ
Thus, the eigenvectors can be represented as:
The eigenvalues η of Matrix (6) have the form
? ? ? ?
ps −C12 ps C22 − η
|TEi = , |TMi = . (34)
C11 − ηsin 2 Λ δ −2 C21 γ 2
η = cos Λ + ( γ − δ / 4 )
2 2 2
± i sin Λ cos Λ + 2 sin Λ . 2
(33)
Λ2 Λ Λ
The elements of the column vector (34) are real values. Based on this, the fields are linearly
polarized. The polarization distribution map data are shown in Figure 5.
Thus, the eigenvectors can be represented as:
Their structure near the optical axes looks like TE and TM modes in a simple anisotropic
crystal, but far from the center there are significant differences. We call them pseudo-TE
and pseudo-TM mods [24]. ps −C12 distributions
ps η fields have interesting
C22 −these
TEThe= intensity
, TM = of . (34)
C11 − ηdistribution
features. Figure 6 illustrates the intensity −C21
for differently polarized pseudo-TM
mode components. While the component Ex does not have any singularities, except for
The elements
a simple zeroofon
the column
the vector
axis, the (34) are real
component values.
Ey forms Based
a new on this,
type the fields
of phase are linearly
singularity——a
polarized. The polarization
double helical distribution
edge dislocation. Indeed,map data are
the phase shown incondition
singularity Figure 5. is Ey = 0, that is,Page 8
Photonics 2023, 10, x FOR PEER REVIEW 10 of 18
Photonics 2023, 10, 1044 8 of 15
ps
Figure 5. Self-polarization of the beam in a gyrotropic crystal: (a) pseudo-TE mode TE and (
ps
pseudo-TM mode TM .
Their structure near the optical axes looks like TE and TM modes in a simple aniso
tropic crystal, but far from the center there are significant differences. We call them
pseudo-TE and pseudo-TM mods [24]. The intensity distributions of these fields have in
teresting features. Figure 6 illustrates the intensity distribution for differently polarize
pseudo-TM mode components. While the component Еx does not have any singularitie
except for a simple zero on the axis, the component Е y forms a new type of phase singu
larity——a double helical edge dislocation. Indeed, the phase singularity condition
E y 0 , that is,
Figure 5. Self-polarization of the beam in a gyrotropic 2crystal: (a) pseudo-TE mode |TEi ps and
ps
Figure 5. Self-polarization of the beam in a gyrotropic r
crystal: (a) pseudo-TE mode TE and (b)
(b) pseudo-TM mode |TMi ps . tan 2
tan r 2 ; sin r 2 0 . (3
ps
pseudo-TM mode TM . r 2
2
?
γ r ? ?
2
? ?
2
Their structure tan 2φ = − tan Λ r ; sin Λ = 0.in a simple aniso- (35)
rmodes
Thenear the opticalΛ
first expression axes
(r2 )looksthe
describes like TE and
double TMwhile
helix, the second describes the distributio
tropic crystal, but at the periphery. This complex spiral vortex beam is shownWe
far from the center there are significant differences. call them
in Figure 6.
The first
pseudo-TE and expression
pseudo-TM describes
mods [24].the
Thedouble helix,
intensity while the of
distributions second
these describes
fields havethe
in- distri-
bution at the periphery. This complex spiral vortex beam is shown in Figure
teresting features. Figure 6 illustrates the intensity distribution for differently polarized 6.
pseudo-TM mode components. While the component Еx does not have any singularities,
except for a simple zero on the axis, the component Е y forms a new type of phase singu-
larity——a double helical edge dislocation. Indeed, the phase singularity condition is
E y = 0 , that is,
γ (r2 )
tan 2φ = − tan Λ ( r 2 ) ; sin Λ ( r 2 ) = 0 . (35)
Λ (r2 )
The first expression describes the double helix, while the second describes the distribution
at the periphery. This complex spiral vortex beam is shown in Figure 6.
(a) (b) (c)
Figure 6. Beams with self-polarization of the pseudo-TM mode in a double gyrotropic crystal: Ex
Photonics 2023, 10,(a)
x FOR
andPEER components; (c) spiral vortex phase (∆n L = 2 · 10−3 , ∆nC = 3 · 10−5 , d = 1 cm).
REVIEW
Ey (b) 11
Figure 6. Beams with self-polarization of the pseudo-TM mode in a double gyrotropic crystal: Е
-3
(a) and Е y describes
The first expression the double
(b) components; (c) spiralhelix,
vortexwhile ( D nsecond
phasethe L = 2 ⋅ 10 describes ⋅10-distri-
, D nC = 3the 5
, d = 1cm ).
bution at the periphery. This complex spiral vortex beam is shown in Figure 7.
The first expression describes the double helix, while the second describes the distr
bution at the periphery. This complex spiral vortex beam is shown in Figure 7.
Figure 6. Beams with self-polarization of the pseudo-TM mode in a double gyrotropic crystal: Еx
(a) and Е y (b) components; (c) spiral vortex phase ( D nL = 2 ⋅10-3 , D nC = 3 ⋅ 10-5 , d = 1cm ).
The first expression describes the double helix, while the second describes the distri-
bution at the(a) (b) vortex beam is shown in Figure(c)
periphery. This complex spiral 7.
Figure 7. Intensity distribution (a) in the E(-) left circularly polarized component and phase distribu-
tion for the left (b) circularly and right (c) circularly polarized components.
Figure 7. Intensity distribution (a) in the E(-) left circularly polarized component and phase d
In contrast to theforcase
bution the left
of a(b) circularly
single and right
crystal, lines(c)
ofcircularly polarized
equal phase havecomponents.
become double
helixes, and the wave front in the vicinity of the singularity is an indirect helicoid. If we
In contrast
change the sign of the to the case of
input polarization a single crystal,
circulation, lines
then the of equaldistribution
intensity phase have inbecome do
the optical vortex will not change, but the direction of the phase helix twist will change.helicoid.
helixes, and the wave front in the vicinity of the singularity is an indirect
change the optical
Although in conventional sign of the input polarization
experiments circulation,
such a helical then the
distribution of intensity distribution i
lines of equal
optical vortex will not change, but the direction of the phase helix twist will change
hough in conventional optical experiments such a helical distribution of lines of e
phase is not detected, it has an unexpected manifestation when a linearly polarized b
propagates through a double gyroanisotropic crystal, which can be represented as a
circularly polarized beams with the opposite direction of rotation of the vector E, soPage 9
Photonics 2023, 10, 1044 9 of 15
phase is not detected, it has an unexpected manifestation when a linearly polarized beam
propagates through a double gyroanisotropic crystal, which can be represented as a set of
circularly polarized beams with the opposite direction of rotation of the vector E, so that at
the output, the field pattern can be represented in the form of two circularly polarized beams
carrying optical vortices with opposite topological charges. Indeed, such a superposition
detects the phase of the vortices so that, as a result, we obtain a quarter-fold intensity
spiral. However, these features are characteristic only of their own rays. They can be
partially “embedded” into the original light beam from a double crystal in the form of a
superposition of eigenbeams.
5. Generate Polychromatic Helical Beams
The question is how to extract a beam with a pure helical edge dislocation from the
combined beam after the crystal. The easiest way to achieve what you want is to launch a
linearly polarized beam into a crystal, that is
? ? ? ?
C11 ? 1
∗ = DĜ . (36)
−C12 0
Comparing Expressions (34) and (35), the y-component in Expression (36) describes
the same helical edge dislocation as in Expression (34). Generally speaking, the images
presented in Figure 8 have been known for quite a long time in crystallography as Airy
spirals, and are used to distinguish right-handed and left-handed crystals. However, our
path shows a way to create spiral singular beams. Thus, from Expression (35), it follows
that the
Photonics 2023, 10, x FOR PEER REVIEW qradius of the first ring dislocation can be found from the following
12 condition:
of 18
Λ0 = δ r0 + γ r0 = π. Provided that the beam waist at the entrance12face
2 2
? ?
Photonics 2023, 10, x FOR PEER REVIEW 2 2 of 18of the
crystal is equal to r0 = ρ, the contribution of the energy flux to the ring dislocation is
negligibly small. The configuration of the spiral beam field is shown in Figure 8.
Figure 8. Intensity distribution in a spiral vortex beam in (a)— Еx , (b) — Е y components (
Intensity distribution
Figure8.8. Intensity in a spiral vortex beam in (a)—E , (b)—E components (∆n =
Figure 2 ⋅ 10 −3 , Δ nC =distribution
Δ nL = − 3 ⋅−10 −5
, d = 1incma ).spiral vortex beam in (a)— Еx x, (b) — Еy y components ( L
2 · 10 3 , ∆n = 3 · 10 5 , d = 1 cm).
Δ nL = 2 ⋅ 10 −3 C, Δ nC = 3 ⋅ 10 −5 , d = 1 cm ).
ItItshould
shouldalso
alsobe
benoted
notedthatthatthis
thisgives
givesus usthe
thetechnical
technicalability
abilityto
togenerate
generatepolychro-
polychromatic
maticIthelical
shouldbeams.
also To
be do
noted this, you
that thisjust need
gives usto focus
the polychromatic
technical
helical beams. To do this, you just need to focus polychromatic light into ability tolight crystal.
intoa acrystal.
generate polychro-
maticThe images
helical shown
beams. To in
do Figure
this, 9
you are the
just result
need to of a
focus computer simulation
polychromatic
The images shown in Figure 9 are the result of a computer simulation of light of
intothea process.
crystal.
the process.
Attention
The was
imagesdrawn
shown to the
in fact
Figure that
9 arethe light
the source
result of a was like
computer a completely
simulation
Attention was drawn to the fact that the light source was like a completely black black
of the body—
process.
body—so
so that all rays
Attention have only
washave
drawn a radial
to the radius
fact that thefor all source
light wavelengths. This
was like means thatblack
a This
completely our rays are
body—
that all rays only a radial radius for all wavelengths. means that our rays are
spatially
so that allcoherent.
rays have only a radial radius for all wavelengths. This means that our rays are
spatially coherent.
spatially coherent.
Figure9.9.
Figure Polychromatic
Polychromatic spiral
spiralvortex
vortexbeam obtained
beam at different
obtained angles
at different relative
angles to theto
relative polarizer axes axes
the polarizer
Δ nL L==
(Figure
(∆n · −10
29.⋅210 , Δ,n∆n
3 −3
Polychromatic=
C C
3 ⋅
= 103−5
·
spiral , −=5 ,1dcm
vortex
10
d = ).
beam
1 obtained
cm). at different angles relative to the polarizer axes
( Δ nL = 2 ⋅ 10 −3 , Δ nC = 3 ⋅ 10 −5 , d = 1 cm ).
6. Optical Vortices
6. Optical Vortices
The beams generated in a double gyroanisotropic crystal have another useful prop-
erty—they can carry
The beams opticalinvortices.
generated a doubleIndeed, let us passcrystal
gyroanisotropic a circularly polarized
have another Gaussian
useful prop-Page 10
Photonics 2023, 10, 1044 10 of 15
6. Optical Vortices
The beams generated in a double gyroanisotropic crystal have another useful property—
they can carry optical vortices. Indeed, let us pass a circularly polarized Gaussian beam
through a double crystal. From a mathematical point of view, this means the following:
? ? ? ?
1 C11 ± iC12
D Ĝ = ∗ =
±i ?C12 ± iC ? 11
cos2 Λ + γ2 + δ4 sinΛ2Λ + i Λδ cos Λ + i Λ
2 2 γ
sin Λ exp(±i2φ)
? (37)
= h ? ? 2 i .
i cos2 Λ + γ2 + δ4 sinΛ2Λ − Λδ cos Λ + i Λ
2
sin Λ exp(±i2φ)
γ ?
After passing through a quarter-wave plate, the field will take the form:
Photonics 2023, 10, x FOR PEER REVIEW 13 of 18
cos Λ + i Λ?sin Λ 2exp
δ γ ? !
? (± i2φ)
? ? ? ?
1 i 1 Λ
D Ĝ = 2i . (38)
cos2 Λ + γ2 + δ4 sinΛ2Λ
2
i 1 ±i
The polarizer cuts out the Еy —component from the spiral vortex field. Thus, the in-
The polarizer cuts out the Ey —component from the spiral vortex field. Thus, the
tensity andand
intensity interference distribution
interference takes
distribution thethe
takes form shown
form in Figure
shown 10.10.
in Figure
Figure 10. Image of a double charged vortex. (a) Intensity profile, (b) phase.
Figure 10. Image of a double charged vortex. (a) Intensity profile, (b) phase
In fact, we have obtained an ordinary vortex with a double charge, similar to those
In fact, we have obtained an ordinary vortex with a double charge, similar to those
that can be obtained on a simple anisotropic crystal [11,19,25].
that can be obtained
Consider in moreon detail
a simpletheanisotropic
mechanismcrystal [11,19,25].
of generation of a double topological charge
Consider in more detail the mechanism of generation
by means of a uniaxial crystal. A typical map of the polarization of a double topological
distribution of charge
a beam
by means of a uniaxial crystal. A typical map of the polarization
that has passed through an anisotropic medium and a polarization filter (a quarter-wave distribution of a beam
that
platehas
andpassed through
a polarizer) is an anisotropic
shown in Figure 11a. Weand
medium polarization
seeathat right-hand filter (a quarter-wave
circular polarization
plate and a polarizer) is shown in Figure 11a. We see that
is located on the beam axis. This exceptional state of polarization surrounds right-hand circular polarization
a family
isoflocated
ellipses.onAthe beam axis. This
characteristic exceptional
property state ofispolarization
of this family surrounds
the typical ordering ofaorientation
family of
ellipses.
directionsA characteristic propertyof
of the major semiaxis ofthe
thisellipse.
familyIfiswe thedraw
typical ordering
integral of orientation
curves—lines di-
tangent
rections of the major semiaxis of the ellipse. If we draw integral curves—lines
at each point to the semi-major axis—we will get a characteristic pattern of spirals (see tangent at
each point
Figure to the
11b). Thesesemi-major axis—we
integral curves are will get a characteristic
analogous to lines of forcepattern
for aof spiralspolarized
linearly (see Figure (at
eachThese
11b). point)integral
electriccurves are analogous
or magnetic field. In to lines
this of they
case, for a linearly
forcerepresent polarized
the lines each
(at of
of force the
point) electric or magnetic
inhomogeneously polarized field.
beam.In this
Suchcase, they represent
pattern of integralthecurves of force
linesnear of the inho-
the singular point
mogeneously polarized beam. Such a pattern of integral curves
is similar to the distribution of lines of equal curvature of the wave front near the near the singular point
waveis
similar to the distribution of lines of equal curvature of the wave front near
caustics [26]. They are called umbilic points. In the future, we will keep this analogy, calling the wave caus-
tics [26].
these They are
patterns called umbilic
surrounding points. In
polarization the future,polarization
singularities we will keep this analogy,
umbilics calling
[11,19,25].
these According
patterns surrounding polarization
to the terminology of J. singularities
Nye [26], an polarization umbilics
umbilic is formed [11,19,25].
in the vicinity of a
singular point called a C-point. In fact, in the general case, we should not be talking about
points, but about lines of space. As in the scalar case, in the vector field, these lines are
formed by the intersection of two surfaces. The intersection of the line by the observation
plane forms a C-point. On the C-line, the field intensity, in the general case, does not vanish.
The authors of [15–19] distinguish, in the form of characteristic features of the vector field,
the so-called L-surfaces. On this surface, only linear polarization of partial waves exists.
L-surfaces always cover C-lines and separate C-lines corresponding to opposite directionsPage 11
ellipses. A characteristic property of this family is the typical ordering of orientation di-
rections of the major semiaxis of the ellipse. If we draw integral curves—lines tangent at
each point to the semi-major axis—we will get a characteristic pattern of spirals (see Figure
11b). These integral curves are analogous to lines of force for a linearly polarized (at each
Photonics 2023, 10, 1044 point) electric or magnetic field. In this case, they represent the lines of force of15the inho-
11 of
mogeneously polarized beam. Such a pattern of integral curves near the singular point is
similar to the distribution of lines of equal curvature of the wave front near the wave caus-
tics [26]. They are called umbilic points. In the future, we will keep this analogy, calling
of polarization, circulation of circular polarization. If at the transition of C-lines the phase
these patterns surrounding polarization singularities polarization umbilics [11,19,25].
of the field changes abruptly by π, then at the transition of the L-surface, the direction of
circulation of the partial fields abruptly changes.
(a) (b)
Figure 11. Distribution map of the polarization state (a) and directions of orientation of the major
semiaxis of the ellipse (b) in the cross section of the fundamental Gaussian beam after passing through
a uniaxialFigure
crystal.11. Distribution map of the polarization state (a) and directions of orientation of the major
semiaxis of the ellipse (b) in the cross section of the fundamental Gaussian beam after passing
through
In our case,aasuniaxial
showncrystal.
in Figure 11a, the pattern of singularities is somewhat different.
In the center of the picture is C—a point that is covered by L-line—as a result of a section
of the observation plane’s C-line and L-surface; however, instead of the L-line, with the
opposite direction of ellipticity, we see a characteristic C-line as a result of section C-plane
surface. On the C-surface, light is circularly polarized. Thus, the overall picture of the
vector field after a uniaxial crystal is presented as a set of C and L-surfaces nested into
each other, which cover the central C-line. Such a difference in the classification introduced
in [15–19] with the vector field we are studying after a uniaxial crystal is due to the fact
that in these works, the structure of random stochastic fields, which arises as a result of
laser beam scattering on random anisotropic inhomogeneities, is studied.
At the same time, we are interested in the process of beam passage through a spatially
homogeneous anisotropic medium, in which field states with an unstable singularity
structure are possible. As soon as a weak polarization perturbation is introduced into the
beam, the picture changes dramatically. A quarter wave plate installed after the crystal can
act as such a polarization perturbation.
It is necessary to point out some characteristic features of the picture obtained as a
result of the action of the perturbation (see Figure 12). First of all, unstable singularities
disappear. The unstable C-point in the center of the picture splits into two single C-points
displaced along the beam ϕ = π/2, 3π/2. In the structure of the Figure 12, there are two
more simple C-points located on the beam ϕ = 0, π. These two singularities arose as a
result of the splitting of the unstable C-line into four symmetrically located C-points: two
shifted along the rays ϕ = 0, π to the beam axis.
Two other singularities have shifted along the rays ϕ = π/2, 3π/2 to the periphery. Now,
the singularity distribution pattern is structurally stable to the effects of external perturbations.
To convert vector singularities into optical vortices, it is necessary to install a λ/4 plate
and a polarizer in a series after the crystal. The λ/4 plate converts the circular polarization
to linear polarization, while the polarizer, whose axis is properly oriented, suppresses
linear polarization. As a result, zero electric field strength is formed on the beam axis. The
occurrence of a phase singularity is easy to understand if we turn to the distribution map of
the polarization state (see Figure 11). We will go around C—a point on the beam axis along
a closed contour—and follow the rotation of the major semi-axis of the partial wave ellipse.Page 12
beam, the picture changes dramatically. A quarter wave plate installed after the crystal
can act as such a polarization perturbation.
It is necessary to point out some characteristic features of the picture obtained as a
result of the action of the perturbation (see Figure 12). First of all, unstable singularities
disappear. The unstable C-point in the center of the picture splits into two single C-points
displaced along the beam ϕ = π , 3π . In the structure of the Figure 12, there are two
Photonics 2023, 10, 1044 12 of 15
2 2
more simple C-points located on the beam ϕ = 0, π . These two singularities arose as a
result of the tour
A complete splitting of contour
of the the unstable C-line into
corresponds fourrotation
to the symmetrically located
of the axis of theC-points: two
polarization
shifted along
ellipse by 2π. the rays ϕ = 0, π to the beam axis.
Photonics 2023, 10, x FOR PEER REVIEW 15 of 18
polarization to linear polarization, while the polarizer, whose axis is properly oriented,
suppresses linear polarization. As a result, zero electric field strength is formed on the
beam axis. The occurrence of a phase singularity is easy to understand if we turn to the
distribution map of the polarization state (see Figure 11). We will go around C—a point
on the beam axis along a closed contour—and follow the rotation of the major semi-axis
of the partial wave ellipse. A complete tour of the contour corresponds to the rotation of
the axis ofDistribution
the polarization
12. Distribution
Figure 12.
Figure map
mapof
ellipse
ofthe
by 2π . ofthe
polarizationof
thepolarization beamfield
thebeam fieldthat
thatpassed
passedthrough
throughan ananisotropic
anisotropic
crystal
Therefore, a change in the angle ϕ from
and a quarter-wave plate. The axes of the quarter-wave
crystal and
0 to 2πplate
quarter-wave plate
corresponds
areoriented
are orientedat
to aanchange
atan angleof
angle
in ◦the
of4545to0
tophase
thethe of the
coordinate
coordinate wave
axes.
axes. δ , also from 0 to 2π . This means that in the vicinity of the beam
axis, when passing along a closed contour, a phase difference runs up, which is typical for
fieldsTherefore,
Twowith asingularities
phase
other change in the
singularities. angle
have ϕ from
this
Inshifted case,0 to
along thecorresponds
2π
such arays
singularity 3aπchange
ϕ = π to,belongs totothe the
inthe phase of
left-hand
periphery.
the wavepolarization
circular δ, also from 0component
to 2π. This of means that inAthe
the field. vicinity
typical of the
picture 2ofbeam 2 axis, when
a double passing
helicoid near
Now,
along the
the axis singularity
a closed
shows contour, distribution
the presence a phase patterntopological
of adifference
double is
runsstructurally
up, which
chargestable
isof to vortex.
typical
the the
foreffectsTheof
fields external
with
wave phase
front
perturbations.
singularities.
at the periphery In this case,
is cut such aand
by rings, singularity
each ringbelongs to the to
corresponds left-hand
an unstable ringpolarization
circular dislocation.
component
At convert
To of the
the samevector field.
time, the A typical picture
vortices obtained
singularities of
into optical a double
in anisotropic helicoid
vortices, itcrystals near
is necessary the axis
are surrounded abyλthe
shows
to install nu-
presence of a double topological charge of the vortex. The wave front at the periphery 4is
merous ring dislocations, while in our case, these dislocations are almost never observed.
plate and a polarizer
each in λ
cut
Theby rings,
degree and
of splitting ring series after the
ofa dislocations
corresponds to ancrystal.
depends on The
unstable thering 4dislocation.γ / Λ . Thethe
plate converts
coefficient circular
greater the
At the same time, the vortices obtained in anisotropic crystals
gyration of the crystal, the less noticeable the dislocations, but at the same time, the coef- are surrounded by
numerous ring dislocations, while2 in our case, these dislocations are almost never observed.
Ex dS E y dS , sodepends
2
ficient
The decreases
degree of splitting of /dislocations that in the on final analysis, the
the coefficient γ/Λ.spiral
Thevortex
greaterbeam
the
can disappear
gyration of theunder
crystal, thethe condition Δ nC >>the
less noticeable Δ nLdislocations,
in which the butcircular
at the birefringence
same time, the is
R 2 R 2
much greater
coefficient than the linear
decreases | Ex | dS/ one. Ey dS, so that in the final analysis, the spiral vortex
beam can disappear under the condition ∆nC >> ∆n L in which the circular birefringence
7. much
is Experimental
greater than Obtaining
the linear of Singular
one. Beams in Gyroanisotropic Crystals
At present, our attention is most attracted to polychromatic singular beams. As a re-
7. Experimental Obtaining of Singular Beams in Gyroanisotropic Crystals
sult, we have focused our efforts on the generation of helical edge dislocations embedded
At present, our
in polychromatic attention is most attracted to polychromatic singular beams. As a
beams.
result,The
wescheme
have focused
of the our efforts on the
experimental generation
setup is shownofinhelical
Figuredge
13. dislocations embedded
The key element of the
in polychromatic beams.
installation is a white light source, which is a halogen lamp equipped with a spherical
mirror. scheme
TheThe of the
angular experimental
divergence setup
of the beam is shown in Figure
after passing 13. The
through key
the element
spatial the
lensoffilter
installation is a white light
0 source, which is a halogen lamp equipped with a spherical
becomes less than 4 . The beam then passes through a polarizer to become linearly po- mirror.
The angular divergence of the beam after passing through the spatial lens filter becomes
larized. ◦
less than 4 . The beam then passes through a polarizer to become linearly polarized.
Figure Schemeof
13. Scheme
Figure 13. ofthe
theexperimental
experimentalsetup:
setup:1—halogen
1—halogenlamp;
lamp;2—spatial
2—spatiallens filter;
lens 3,9—polarizers;
filter; 3,9—polariz-
4,6,8—lenses; 5—LiNbO
ers; 4,6,8—lenses; 5—LiNbO
3 crystal; 7—SiO
3 crystal; 7—SiO
2 crystal; 10—CCD camera, Ĉ—unit vector
2 crystal; 10—CCD camera, Ĉ —unit vector of optical axes.
of optical
axes.
After that, the beam is focused by a lens with a focal length of 3 cm into a LiNbO3
crystal. Next, we focus the beam again, but now into the SiO2 crystal. The optical axes of
the crystals are directed along the beam axes. The beam image is projected onto the screenPage 13
Photonics 2023, 10, 1044 13 of 15
After that, the beam is focused by a lens with a focal length of 3 cm into a LiNbO3
crystal. Next, we focus the beam again, but now into the SiO2 crystal. The optical axes of
Photonics 2023, 10, x FOR PEER REVIEW
the crystals are directed along the beam axes. The beam image is projected onto the 16 of 18
screen
of a CCD camera and processed by a computer.
We considered the intensity distribution of the singular beam as a function of the
direction of the
Figure polarizer
14 shows axesimages
typical α. of a singular beam. When the angle is α = 90◦0 , a pure
Figure
spiral 14beam
vortex showsdislocation
typical images of a singular
is embedded beam.
in the When
beam. It isthe
important α=
angle is to 90 that
note , a pure
the
spiral vortex are
dislocations beam
notdislocation
washed out is in
embedded in the light,
polychromatic beam.but important
It isare to note
seen as clear the
thatlines
spiral
dislocations are not washed
with four branches. out inofpolychromatic
The method generation oflight, but are seen
polychromatic as clearspiral
singular spiralvortex
lines
with four branches. The method of generation of polychromatic singular spiral
beams described in this paper can be applied by other researchers to analyze the proper- vortex
beams
ties of described in this angular
spin and orbital paper can be applied[5,9,22]
momentum by other researchers to analyze the properties
of spin and orbital angular momentum [5,9,22].
Figure 14. Scheme
Figure 14. Scheme of of the
the distribution
distribution of
of the
the light
light flux over the
flux over angular spectrum
the angular spectrum in
in the
the Bessel–Gauss
Bessel–Gauss
beam (∆n L = 1.8 · 10−3−3 , ∆nC = 2.8 · 10
−5
−5 , d = 0.5 cm).
beam ( Δ nL = 1.8 ⋅ 10 , Δ nC = 2.8 ⋅ 10 , d = 0.5 cm ).
8. Conclusions
8. Conclusions
A type of monochromatic and polychromatic singular beams carrying helical edge
A type and
dislocations of monochromatic
optical vorticesand has polychromatic
been theoretically singular beams carrying
and experimentally helical Such
studied. edge
dislocations
beams can beand optical
created vortices
using natural has been theoretically and experimentally
objects—gyroanisotropic crystals. A linearly studied. Such
polarized
beams can be created
monochromatic using natural objects—gyroanisotropic
or polychromatic beam of light passing through crystals.twoA linearly polarized
gyroanisotropic
monochromatic
crystals or polychromatic
with opposite signs of the beam of light
gyration passing through
and polarization two gyroanisotropic
coefficients creates spiral
crystalsedge
vortex withdislocations—also
opposite signs of called
the gyration and polarization
Airy rings. The expressionscoefficients
that form creates spiral
the basis ofvor-
the
described phenomenon are written
tex edge dislocations—also down.
called Airy These
rings. Theexpressions
expressions enable
that us
formto analyze
the basis various
of the
cases of thephenomenon
described propagation are of singular beams.These
written down. As experimental
expressions objects,
enable us wetoused a system
analyze variousof
casescrystals:
two LiNbO3 andofSiO
of the propagation 2 , whose
singular optical
beams. Asaxes are directed
experimental alongwe
objects, beama axes.
theused system The of
beam of light produced
two crystals: LiNbO3 and by SiO
a halogen
2, whose lamp is transformed
optical by this system
axes are irected along the beama axes.
in such way that
The
abeam
polychromatic phase singularity
of light produced by a halogen withlamp
a clearly defined central
is transformed by thisspiral line isinbuilt
system suchinto
a wayit.
The method described in this article can be applied by other researchers
that a polychromatic phase singularity with a clearly efined central spiral line is built to analyze
the
intoproperties
it. of spin and orbital moments in free space [27,28], to analyze the shapes and
properties of beams that carryinathis
The method described topological charge,
article can to studybyanisotropic
be applied media [29,30],
other researchers and
to analyze
thestudy
to the properties
properties of spin andof orbital
topological charges
moments [31–36]
in free spaceboth in anisotropic
[27,28], to analyze the media
shapesandand in
weakly
propertiesturbulent
of beams atmospheric
that carrymedia.
a topological charge, to study anisotropic media [29,30],
Thestudy
and to resultstheobtained in this
properties publicationcharges
of topological can be used in modern
[31–36] both in photonics,
anisotropicfor example,
media and
to develop improved configurations
in weakly turbulent atmospheric media. of the shape and types of optical beams, to find states
(C-lines
Theand L-surfaces)
results obtainedof polarization of anisotropic
in this publication mediainthat
can be used cannot
modern be createdfor
photonics, byexam-
other
means, and to overcome any technical limitations associated with
ple, to develop improved configurations of the shape and types of optical beams, to find the improvement of
design instruments and apparatus, including those for medical research.
states (C-lines and L-surfaces) of polarization of anisotropic media that cannot be created
by other means, and to overcome any technical limitations associated with the improve-
ment of design instruments and apparatus, including those for medical research.
Author Contributions: Conceptualization, Y.E. and A.R.; methodology, Y.E.; validation, Y.E. and
A.R.; formal analysis, Y.E. and A.R.; investigation, Y.E. and A.R.; resources, Y.E. and A.R.; writing—
original draft preparation, Y.E.; writing—review and editing, Y.E. and A.R.; supervision, Y.E.; pro-
ject administration, Y.E. and A.R. All authors have read and agreed to the published version of thePage 14
Photonics 2023, 10, 1044 14 of 15
Author Contributions: Conceptualization, Y.E. and A.R.; methodology, Y.E.; validation, Y.E. and A.R.;
formal analysis, Y.E. and A.R.; investigation, Y.E. and A.R.; resources, Y.E. and A.R.; writing—original
draft preparation, Y.E.; writing—review and editing, Y.E. and A.R.; supervision, Y.E.; project adminis-
tration, Y.E. and A.R. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board: Not applicable.
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study.
Data Availability Statement: The data presented in this study are available upon request from the
respective author.
Conflicts of Interest: The authors declare no conflict of interest.
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