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Optik - International Journal for Light and Electron Optics xxx (xxxx) 169208
Contents lists available at ScienceDirect
Optik - International Journal for Light and
Electron Optics
journal homepage: www.elsevier.com/locate/ijleo
Magneto-optics of helically structured photonic crystals with
wavelength dependence of magneto-optical activity parameter
A.H. Gevorgyan a, S.S. Golik a,b, N.A. Vanyushkin a, I.M. Efimov a
aSchool of Natural Sciences, Far Eastern Federal University, 10 Ajax Bay, Russky Island, 690922 Vladivostok, Russia
bInstitute of Automation and Control Processes, Far East Branch, Russian Academy of Sciences, Vladivostok 690041 Russia
ARTICLE INFO
Keywords:
Cholesteric liquid crystal
Magneto-optical activity
Ellipticity
Rotation
Nonreciprocity
Verdet constant
Isolator
ABSTRACT
In this paper we investigated the magneto-optical properties of helically structured photonic
crystals taking into account the wavelength dependence of magneto-optical activity parameter
and wavelength independence of Verdet constant. We compare the obtained results with the
case of the wavelength independence of magneto-optical activity parameter. We investigated
the peculiarities of spectra of rotation, ellipticity, reflection, photonic density of states, trans-
mission nonreciprocity in these two cases. We showed that in general these results can essen-
tially differ from one to another and wavelength dependence of magneto-optical parameter can-
not be neglected.
1. Introduction
The study of new magnetic materials and magneto-optical effects remains one of the current trends in modern physics. Magneto-
optical effects provide a way for light polarization control and light intensity modulation. However, a magnetic field cannot provide
modulation of light in a wide range both in light intensity and in polarization in homogeneous magneto-active medium layers. There-
fore, mechanisms of enhancement of magneto-optical effects are needed, which are critical for a wide range of magneto-optical de-
vices, such as optical isolators, optical circulators, and modulators, for magnetic field sensors and visualizers, as well as for bio- and
chemo-sensors, etc. The drastic enhancement of Faraday effect takes place in PCs and multilayered structures during multiple reflec-
tion interference due to nonreciprocal character of Faraday effect [1–3]. Enhancement of magneto-optical effects was demonstrated
in magnetic photonic crystals (MPCs) with defect (defects) in their structure [4–6]. MPCs are the PCs formed from magnetic materials
or PCs with defect (defects) from magnetic materials) The mechanisms of magneto-optical effects amplification in optical nanostruc-
tures are of great interest, too [7,8].
Recently, media with a high Verdet constant (on the order of 105rad m−1T−1and higher) are being created on the base of poly-
mer-coated magnetic nanoparticles [9], or on the base of hyperbolic magneto-optical metamaterial based on Au–Ni nanorod arrays
[10].
In recent years, mechanisms of magneto-optical effects enhancement in heterostructures which combine magnetism and natural
chirality [11] or structural chirality [12–14] have been of great interest. However, these works considered the case when the mag-
neto-optical activity parameter gis constant and independent of wavelength.
In this paper we investigated the magneto-optical properties of helically structured PCs (cholesteric liquid crystals (CLCs), chiral
sculptured thin films, chiral ferroelectric liquid crystals, etc.) taking into account exactly the wavelength dependence of magneto-
optical activity parameter. Let us note some new development of photonic crystals (PCs), too (see, for instance [15] and some refer-
ences therein).
https://doi.org/10.1016/j.ijleo.2022.169208
Received 17 March 2022; Received in revised form 16 April 2022; Accepted 26 April 2022
0030-4026/© 20XX
Note: Low-resolution images were used to create this PDF. The original images will be used in the final composition.
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A.H. Gevorgyan et al. Optik - International Journal for Light and Electron Optics xxx (xxxx) 169208
2. Models and methodology. Results
Let us consider the light propagation through a planar helically structured PC being in an external magnetic field, directed along
its helix axis (Fig. 1). As is well known, in this case the tensors of dielectric permittivity and magnetic permeability have the forms
[2]:
(1)
where are the principal values of the local dielectric permittivity tensor in the presence of an ex-
ternal magnetic field; gis the parameter of magneto-optical activity, a= 2π/p,pis the pitch of the helix. The fact that the magneto-
optical activity parameter, in general, depends on wavelength follows from the following reasoning. First, Faraday rotation angle is
given by the formula
(2)
where dis the transmission distance and εis the dielectric permittivity. On the other hand, the experimenters use the formula
, where Vis the Verdet constant, and is the external magnetic field induction. Substituting into (2) for gwe
will have
(3)
It follows from (3) that, when considering narrow spectral regions with , the change in g can be neglected and g can be as-
sumed constant, independent of frequency. g will not depend on the wavelength also in the case of . Of course, in the general
case, the dependence of g on wavelength cannot be neglected. Now a little bit about the Verdet constant (for more on the constant
Verdet, see the review article [16] and the references cited therein, as well as [17,18]). It follows from [17,18] that for most magneto-
optical media the Verdet constant undergoes changes mainly near some resonance lines, while in other regions of the spectrum it is
constant. As shown in [18] two of these resonance lines are in the far-ultraviolet and near-infrared regions. So, in the near-ultraviolet,
visible, and far-infrared regions the Verdet constant can be assumed to be constant, independent of wavelength. Below, we will con-
sider exactly the case of and g=g( ) and we will assume a wavelength dependence of g according to formula (3).
The exact analytical solution of Maxwell's equations for helically structured PCs in an external magnetic field in the rotating frame
(the x' and y' axes of which rotate together with the structure; besides, the x' axis is oriented along the local optical axis everywhere,
the y' axis is perpendicular to the x' axis and z' axis directed along medium’s helix axis [2,14]) when light propagates along the helix
axis, is known (see, in particular [2]). In accordance with [2] the dispersion equation (relation between kand ) in the case of light
propagation along media helix axis has the form
(4)
Fig. 1. The geometry of the problem. The large ellipsoids represent the anisotropic molecules, which are rotating continuously forming a helicoidal structure along the
z-axis. pis the helix pitch, dis the helically structured PC layer thickness, ki, krand ktare the wave vectors of the incident, reflected and transmitted waves respectively
and is the external magnetic field induction.
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where are the zcomponents of wave vectors ( ).
Using this exact analytical solution of the Maxwell’s equations, and dispersion Eq. (4) we solve the problem of light reflection,
transmission, rotation of the its plane of polarization, and its ellipticity of polarization in the case of a planar helically structured PC
layer of finite thickness. We will assume that the optical axis of the helically structured PC, which coincides with the zaxis, is perpen-
dicular to the layer boundaries. The helically structured PC layer is located between two isotropic half-spaces z= 0 and z=d, where
dis the thickness of the helically structured PC layer, and the refractive indices of the isotropic half-spaces are the same and equal to
ns. The boundary conditions, consisting in the continuity of the tangential components of the electric and magnetic fields, represent a
system of eight complex linear equations with eight complex unknowns (in more details see [2,19]). By solving this boundary value
problem, one can determine the values for the components of the reflected and transmitted fields
and calculate the energy coefficient of reflection , transmission , rotation and ellipticity
photonic density of states (PDOS) , etc. Here uland vlare the real and imagi-
nary parts of the amplitude of the transmitted wave, the values l= 1, 2 correspond to the diffracting and non-diffracting eigenmodes
of the helically structured PC layer, respectively, , and and are the xand ycomponents of the electric fields of the
transmitted and reflected waves, and are the orts of the x and y axes, correspondingly. The amplitudes of reflected and transmit-
ted fields at incidence on the layer of light with left and right circularly polarized light are defined as: .
In the absence of external magnetic field helically structured PCs exhibit chiral Bragg reflection, almost completely reflecting light
with one circular polarization (with a rotation sign coinciding with the rotation sign of a media helix) and very weakly reflect light
with reverse circular polarization. The polarization-sensitive photonic band gap (PBG) lies between the wavelengths λ1= pnoand λ2
=pne, where and . In the presence of external magnetic field magneto-optical properties of helically structured
PCs in the case when the magneto-optical activity parameter gis constant and independent of wavelength were presented in [12–14].
Bellow we consider the case when the magneto-optical activity parameter gdepends on wavelength and all calculations were per-
formed for a helically structured PC layer with the following parameters: ε1= 2.29, ε2= 2.143, the helix of the helically structured
PC layer is right-handed, and its pitch is p= 420 nm, .
Let us first investigate and compare the spectra of reflection, transmission, rotation, etc., in two cases, exactly, when g=const and
when g=g(λ). Fig. 2 shows the spectra of rotation (a) at different values of Bext at g=g(λ) and (b) at different values of g at g=const.
Negative values of Bext and g correspond to the positive values when the direction of the external magnetic field is reversed. As can be
seen from Fig. 2, the effect of the dependence of the magneto-optical activity parameter on wavelength has a significant effect on the
Fig. 2. The spectra of rotation (a) at different values of Bext at g=g(λ) and (b) at different values of g at g=const. Helically structured PC layer thickness d= 50p, and
.
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rotation spectra. In particular, an increase in Bext from −0.1–0.1 T results in an upward shift in the rotation spectra (Fig. 2a), while an
increase in g from −0.05–0.05 is accompanied by a downward shift in the rotation spectra (Fig. 2b).
Fig. 3 shows the spectra of ellipticity at different values of Bext at g=g(λ) (solid lines) and at different values of g at g=const (dashed
lines). For curve 1 Bext= 0 T, for solid curve 2 Bext = 5 T and for solid curves 3 Bext =−5 T ( ). Then, for
dashed curve 2 g= 0.721 and for dashed curve 3 g= −0.721. To compare the results at constant g and at wavelength-dependent g,
the parameters are chosen so that in the center of the PBG the parameter g to be the same in both cases. Fig. 3 shows a significant
difference in the ellipticity spectra in the two cases, namely, in the absence of a wavelength dependence of g and in its presence.
Fig. 4 shows the spectra of reflection (a) for incident light with right circular polarization (RCP) and (b) for the one with left circular
polarization (LCP). The parameters and the numbering of curves are the same as in Fig. 3. In the Fig. 4a, the PBG, in the case of the
absence of external magnetic field, is highlighted by vertical dashed lines. Outside the PBG, the reflection coefficient oscillates. For
, the reflection minima are approximately determined from the condition where
At frequencies determined by this condition (they are called the frequencies of the edge modes), strong localization of light takes
place.
Fig. 5 shows the spectra of relative PDOS for the diffracting eigen mode. Here, is the PDOS for an isotropic layer with a re-
fractive index nsand , where cis the speed of light in a vacuum. Again, the parameters and the numbering of curves are the
same as in Fig. 3. Let us note, that the spectra of in the case of wavelength dependence of parameter of magneto-optical activity
g practically coincides at the change of direction of external magnetic field on reverse, while in the case of the absence of wavelength
dependence of parameter g there appears sufficient difference between these spectra at such change of external magnetic field direc-
tion.
Fig. 6. shows the evolution of the reflection spectra with a change in the Bext for incident light with (a) RCP and (b) LCP. As can be
seen from Fig. 6, as in the case of absence of a wavelength dependence of g, in this case we also observe a blue shift of PBG when Bext
increases. But here we observe new peculiarities (see, below). Again, from the presented results, as in the case of absence of a wave-
length dependence of g, it follows that the presence of an external magnetic field leads to the appearance of additional modulation of
Fig. 3. The spectra of ellipticity at different values of Bat g=g(λ) (solid lines) and at different values of g at g=const (dashed lines). For curve 1 Bext = 0 T, for solid
curve 2 Bext = 5 T and for solid curves 3 Bext =−5 T ( ). Then, for dashed curve 2 g= 0.721 and for dashedcurve 3 g= −0.721. d= 50p.
Fig. 4. The spectra of reflection (a) for incident light with RCP and (b) for one with LCP at different values of Bext at g=g(λ) (solid lines) and at different values of g at
g=const (dashed lines). For curve 1 Bext = 0 T, for solid curve 2 Bext = 5 T and for solid curves 3 Bext =−5 T ( ). Then, for dashed curve
2 g= 0.721 and for dashed curve 3 g= −0.721.
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Fig. 5. The spectra of relative PDOS for diffracting eigen mode at different values of Bext at g=g(λ) (solid lines) and at different values of g at g=const (dashed
lines). For curve 1 B= 0 T, for solid curve 2 Bext = 5 T and for solid curves 3 Bext =−5 T ( ). Then, for dashed curve 2 g= 0.721 and for dashed
curve 3 g= −0.721.
Fig. 6. The evolution of the reflection spectra with a change in the Bext for incident light with (a) RCP and (b) LCP. d= 30p, and .
reflection.Let us note that in the pattern of the evolution of reflection at light / dark inclined lines appear, the contrast of
which increases with the increase of (absolute value). The tilt angle of these lines depends on the polarization of the incident
light. This phenomenon has the following explanation. As it is known, the magnetic field leads to circular birefringence: the light with
orthogonal circular polarizations propagates with different phase and group velocities. And the effective refractive indices at the pres-
ence of external magnetic field become different from the values and . And this difference increases with Bext.
Therefore, the reflection from dielectric boundaries appears, because of the difference between the mean refractive index in this case
and the refractive index of layer surroundings . And thus, the existence of dielectric borders brings to additional modulation
of reflection.
Fig. 7 shows the dependences of the wavelengths and of the first edge modes with and (a) on the Bext at g=g(λ)
and (b) on the parameter g at the absence of wavelength dependence of parameter g. They are approximated with the great accuracy
by the formulas
(5)
where and are some constants, and are the wavelengths of the first edge modes at ( . We note that wave-
lengths of the boundaries of the PBG are determined by similar expressions: , and , respec-
tively, where and , again are some constants. Fig. 7 shows the dependences (a) on the at g=g(λ) and (b) on
the parameter g at the absence of wavelength dependence of the parameter g. They are approximated with the great accuracy by the
formulas
(6)
respectively. And, similarly, a slight broadening of the PBG is also described by a similar type of expressions. As can be seen from
Fig. 8, if in the case of wavelength dependence of parameter of magneto-optical activity gthe PBG bandwidth is maximum at the ab-
sence of external magnetic field, and it decreases with (absolute value), then, in the case of the absence of wavelength depen-
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Fig. 7. The dependences of the wavelengths and of the first edge modes with and (a) on the at g=g(λ) and (b) on the parameter g at the ab-
sence of wavelength dependence of the parameter g.
Fig. 8. The dependences (a) on the at g=g(λ) and (b) on the parameter g at the absence of wavelength dependence of the parameter g.
dence of parameter gwe have the reverse picture, namely, PBG bandwidth is minimum at the absence of external magnetic field, and
it increases with g(absolute value).
We now turn to the study of the features of nonreciprocity. Fig. 9 shows the spectra of nonreciprocity of transmission
for two cases, namely at constant g (dashed lines) and at wavelength-dependent g(solid lines), here
and are transmission coefficients for forward and backward waves. The incident light has linear polarization (curves 1) and
RCP (curves 2). As in Fig. 3, the parameters are chosen so that in the center of the PBG the parameter g to be the same in both cases.
It is easy to see that the following regularity is observed, namely, in the center of the PBG the has the same value in both cases
(with constant gand with the dependence of g on the wavelength). As we move away from the center of the PBG, both toward short
waves and toward long waves, the difference between for the two cases increases. This regularity is also observed in the spectra in
Fig. 9. The spectra of nonreciprocity of transmission for two cases, namely, at constant g (dashed lines) and at wavelength-dependent g (solid
lines). The incident light has linear polarization (curves 1) and RCP (curves 2).
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Figs. 3 and 4, for ellipticity of polarization and for reflection, respectively. And this is natural since we have chosen the problem para-
meters in these cases in such a way that in the center of the PBG the value for the magneto-optical activity parameter would be the
same. With distance from the center of the PBG in the first case gchanges by the Eq. (3) and in the second case it remains constant.
Now a little about mechanism of nonreciprocity in helically structured PCs being in external magnetic field. As is well known, the
magnetic field leads to circular birefringence: the light with orthogonal circular polarizations propagates with different phase and
group velocities. This circular birefringence arises due to the spin-orbital coupling of magneto-optical materials, and the resulting
split of electronic energy levels with different orbital angular momentum. Due to the magnetic-field-induced Larmor precession of
electron orbits instead of one eigenfrequency of the electron ( ),two ( and ) arise, corresponding to the right-hand and the left-
hand circular oscillations when there is a magnetic field in the medium. As a result, magneto active medium exhibits circular birefrin-
gence with refractive indices , and the two orthogonal circularly polarized lights propagate in the medium with different
phase and group velocities. This birefringence is nonreciprocal. On the other hand, it is well known that helically structured PCs ex-
hibit reciprocal circular birefringence, and the two orthogonal circularly polarized lights propagate in the medium with different
phase and group velocities. So that, in helically structured PCs subjected in magnetic field directed along helix axis, when the waves
propagate, say, in the direction of the magnetic field, the structural birefringence and magnetic birefringence effects (for example, ro-
tations) are added (or subtracted), however, when the waves propagate in the backward direction, the structural birefringence and
magnetic birefringence effects are now subtracted (added), causing a new type of nonreciprocal transmission [2,12,20].
3. Conclusions
In conclusion, we investigated the magneto-optical properties of helically structured PCs taking into account the wavelength de-
pendence of magneto-optical activity parameter and wavelength independence of Verdet constant. We investigated and compared
the spectra of reflection, transmission, rotation, etc., in two cases, exactly, when g=const and when g=g(λ). It was shown that if an
increase in from −0.1–0.1 T results in an upward shift in the rotation spectra in the case of wavelength dependence of magneto-
optical activity parameter, then an increase in g from −0.05–0.05 is accompanied by a downward shift in the rotation spectra in the
case of wavelength independence of this parameter. Then we showed that if in the case of wavelength dependence of parameter of
magneto-optical activity g the PBG bandwidth is maximum at the absence of external magnetic field, and it decreases with (ab-
solute value), then, in the case of the absence of wavelength dependence of parameter gwe have the reverse picture, namely, PBG
bandwidth is minimum at the absence of external magnetic field, and it increases with g(absolute value). The obtained results show
that, in general, the results in these two cases can essentially differ from one to another and wavelength dependence of magneto-
optical parameter cannot be neglected.
CRediT authorship contribution statement
A.H. Gevorgyan proposed the idea and contributed to the data analysis and wrote the manuscript. N.A. Vanyushkin and I.M. Efi-
mov developed procedures for investigating, contributed to the Software, to the Data curation and to the data analysis. S.S. Golik co-
ordinated the project. All authors contributed to the discussion of this work, to the Validation, Writing - Review & Editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to
influence the work reported in this paper.
Acknowledgment
The work was supported by the Foundation for the Advancement of Theoretical Physics and Mathematics “BASIS”(investigation
of the spectra of rotation and ellipticity at different values of B and g) (Grant №21-1-1-6-1) and by the project of the Ministry of Sci-
ence and Higher Education of the Russian Federation (investigation of the spectra of reflection, wavelength dependence of the pho-
tonic band gap bandwidth and the study of the features of nonreciprocity) FZNS-2020-0003 No. 0657-2020-0003.
Author agreement
We declare that this manuscript is original, has not been published before and is not currently being considered for publication
elsewhere. We confirm that the manuscript has been read and approved by all named authors and that there are no other persons who
satisfied the criteria for authorship but are not listed. We further confirm that the order of authors listed in the manuscript has been
approved by all of us. We understand that the Corresponding Author is the sole contact for the Editorial process. He is responsible for
communicating with the other authors about progress, submissions of revisions and final approval of proofs.
Conflict of interest
No potential conflict of interest was reported by the authors.
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