Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar
Additional high-frequency components of the pulsar radiation in Crab Nebula are considered as a result of the resonance with the surface electromagnetic wave at nonlinear reflection from of the neutron star surface. This stimulated scattering consists in generation of the surface periodic relief by...
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Kontorovich, V.M. Spevak, I.S. Gavrikov, V.K. 2019-02-14T14:19:57Z 2019-02-14T14:19:57Z 2018 Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar / V.M. Kontorovich, I.S. Spevak, V.K. Gavrikov // Вопросы атомной науки и техники. — 2018. — № 4. — С. 112-117. — Бібліогр.: 25 назв. — англ. 1562-6016 PACS: 97.60.Jd; 97.60.Gb; 52.38.Bv https://nasplib.isofts.kiev.ua/handle/123456789/147345 Additional high-frequency components of the pulsar radiation in Crab Nebula are considered as a result of the resonance with the surface electromagnetic wave at nonlinear reflection from of the neutron star surface. This stimulated scattering consists in generation of the surface periodic relief by the incident field and diffraction of the radiation of relativistic positrons on the relief which fly from magnetosphere to the star in the accelerating electric field of the polar gap. Додаткові високочастотні компоненти випромінювання пульсара в Крабоподібній туманності розглядаються як результат резонансу з поверхневою електромагнітною хвилею при нелінійному відбитті від поверхні нейтронної зірки. Це − вимушене розсіювання, яке складається в генерації падаючим полем періодичного рельєфу поверхні і дифракції на ньому випромінювання релятивістських позитронів, що летять з магнітосфери до зірки в електричному полі полярного зазору, яке їх прискорює. Дополнительные высокочастотные компоненты излучения пульсара в Крабовидной туманности рассматриваются как результат резонанса с поверхностной электромагнитной волной при нелинейном отражении от поверхности нейтронной звезды. Это − вынужденное рассеяние, которое состоит в генерации падающим полем периодического рельефа поверхности и дифракции на нём излучения релятивистских позитронов, летящих из магнитосферы к звезде в ускоряющем электрическом поле полярного зазора. One of the authors (V.K.) is thankful to Corresponding Member of NAS of Ukraine, prof. D.M. Vavriv for help and support. We also are grateful to prof. A.V. Kats who carefully read the manuscript and made useful comments. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Коллективные процессы в космической плазме Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar Резонансне нелінійне відбиття від нейтронної зірки і додаткові компоненти у випромінюванні пульсара крабоподібної туманності Резонансное нелинейное отражение от нейтронной звезды и дополнительные компоненты в излучении пульсара крабовидной туманности Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar |
| spellingShingle |
Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar Kontorovich, V.M. Spevak, I.S. Gavrikov, V.K. Коллективные процессы в космической плазме |
| title_short |
Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar |
| title_full |
Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar |
| title_fullStr |
Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar |
| title_full_unstemmed |
Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar |
| title_sort |
resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar |
| author |
Kontorovich, V.M. Spevak, I.S. Gavrikov, V.K. |
| author_facet |
Kontorovich, V.M. Spevak, I.S. Gavrikov, V.K. |
| topic |
Коллективные процессы в космической плазме |
| topic_facet |
Коллективные процессы в космической плазме |
| publishDate |
2018 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Резонансне нелінійне відбиття від нейтронної зірки і додаткові компоненти у випромінюванні пульсара крабоподібної туманності Резонансное нелинейное отражение от нейтронной звезды и дополнительные компоненты в излучении пульсара крабовидной туманности |
| description |
Additional high-frequency components of the pulsar radiation in Crab Nebula are considered as a result of the
resonance with the surface electromagnetic wave at nonlinear reflection from of the neutron star surface. This stimulated scattering consists in generation of the surface periodic relief by the incident field and diffraction of the radiation of relativistic positrons on the relief which fly from magnetosphere to the star in the accelerating electric field
of the polar gap.
Додаткові високочастотні компоненти випромінювання пульсара в Крабоподібній туманності розглядаються як результат резонансу з поверхневою електромагнітною хвилею при нелінійному відбитті від поверхні нейтронної зірки. Це − вимушене розсіювання, яке складається в генерації падаючим полем періодичного рельєфу поверхні і дифракції на ньому випромінювання релятивістських позитронів, що летять з магнітосфери до зірки в електричному полі полярного зазору, яке їх прискорює.
Дополнительные высокочастотные компоненты излучения пульсара в Крабовидной туманности рассматриваются как результат резонанса с поверхностной электромагнитной волной при нелинейном отражении от
поверхности нейтронной звезды. Это − вынужденное рассеяние, которое состоит в генерации падающим
полем периодического рельефа поверхности и дифракции на нём излучения релятивистских позитронов,
летящих из магнитосферы к звезде в ускоряющем электрическом поле полярного зазора.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/147345 |
| citation_txt |
Resonance nonlinear reflection from neutron star and additional radiation components of crab pulsar / V.M. Kontorovich, I.S. Spevak, V.K. Gavrikov // Вопросы атомной науки и техники. — 2018. — № 4. — С. 112-117. — Бібліогр.: 25 назв. — англ. |
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| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №4(116) 112
RESONANCE NONLINEAR REFLECTION FROM NEUTRON STAR
AND ADDITIONAL RADIATION COMPONENTS OF CRAB PULSAR
V.M. Kontorovich1,2, I.S. Spevak3, V.K. Gavrikov1
1Institute of Radio Astronomy NAS, Kharkov, Ukraine;
2V.N. Karazin Kharkiv National University, Kharkov, Ukraine;
3O.Ya. Usikov Institute for Radiophysics and Electronics NAS, Kharkov, Ukraine
E-mail: vkont@rian.kharkov.ua
Additional high-frequency components of the pulsar radiation in Crab Nebula are considered as a result of the
resonance with the surface electromagnetic wave at nonlinear reflection from of the neutron star surface. This stimu-
lated scattering consists in generation of the surface periodic relief by the incident field and diffraction of the radia-
tion of relativistic positrons on the relief which fly from magnetosphere to the star in the accelerating electric field
of the polar gap.
PACS: 97.60.Jd; 97.60.Gb; 52.38.Bv
INTRODUCTION
Pulsars [1] and neutron stars [2] celebrated their 50-
th anniversary by international conferences in St Peters-
burg with a subtitle «50 years after» and in Cambridge
with a subtitle «The Next Fifty Years»1. The materials
of these conferences include reviews and modern litera-
ture references. The references to the works mentioned
below are contained also in the previous article of one
of coauthors [3]. We use some texts from it in the intro-
ductory part of the present work.
Neutron stars, as it is known, were predicted by
Landau2, connected with supernova ones by Baade and
Zwicky and discovered as pulsars by J. Bell and
A. Hewish and their colleagues after a quarter of a cen-
tury. Appearing as a result of the collapse at supernova
explosion, they possess the strongest magnetic field
1012 Тс, rapid rotation (with a period from seconds to
milliseconds), and are enveloped in the magnetosphere
of the electron-positron pairs. The magnetosphere main-
ly rotates corotationally with the star, but it has a bundle
of open magnetic field lines over the magnetic poles;
the particles accelerate and electromagnetic radiation
goes out along these lines [4]. Particles acceleration
takes place in the gap under the area of open magnetic
field lines where the strong accelerating electric field is
caused by the magnetic field and rotation. The region of
the polar cap is limited by the magnetic field lines that
are tangent to light cylinder where the velocity of the
corotational rotation equals to light velocity.
The neutron star itself [5] is known to correspond to
the nuclear density accompanying the neutronization
reaction p e n n−+ → + . It contains layers possessing
1International Conference Physics of Neutron Stars.
50 years after. St-P, 2017. Pulsar Astrophysics: The
Next Fifty Years; Cambridge, 2017, IAU Symposium
337.
2It is quite natural that word neutron is absent in Lan-
dau's work. The neutron was discovered by Chadwick
the same year and Landau could not yet know about it.
For the results of Landau work the specific role of the
neutron was not so significant. In his work the possibil-
ity of existence of a macroscopic atomic nucleus con-
trolled by gravity was proved. See for details ref [25].
superfluidity, and possibly superconductivity. The prop-
erties of matter at such nuclear densities have not been
studied sufficiently, therefore, there are a number of
differing theoretical models [6].
Very little is known about the properties of the sur-
face of neutron stars. In the case of Crab Pulsar, it, ap-
parently, has a solid crust undergoing to starquakes. Due
to the colossal force of gravity, the surface is close to
mirror one, but it can contain a regular structure of ele-
vations caused by the influence of a strong magnetic
(and electric) field. In the region of the polar cap the
surface can be substantially perturbed by the incident
radiation. In this region, the upper layer, heated by ac-
celerated particles and radiation, can be in the liquid
state (see references and discussion [4, p. 110; 6]). Ac-
cording to the references, we assume that the boundary
resembles a metal with iron nuclei and collectivized
degenerate electrons and it has high conductivity.
The mechanism of the radio emission of a pulsar in
the Crab Nebula is based on the idea of reflection of
radiation from the surface of a neutron star [7]. The ra-
diation of positrons flying to a star from the magneto-
sphere is reflected. This reflected radiation predomi-
nates in the centimeter frequency range, where a shift of
the interpulse (IP) occurs and high-frequency (hf) com-
ponents HFC1 and HFC2 appear [8, 9]. We consider the
observed shift of the IP as an argument in favor of good
reflecting properties of the surface. The reciprocal mo-
tion of positrons arises in the accelerating electric field
of the gap and was considered earlier in the context of
star surface heating.
Most researchers believe that radio emission occurs
in the interior of the magnetosphere or beyond, near the
"light cylinder" [1, 4, 10]. We are interested in radiation
emanating from the inner gap above the polar cap, for
which there are a number of arguments.
In the model [7] the displacement of the IP is ex-
plained by the mirror reflection in the inclined magnetic
field, and the shift of the IP to 7° means the slope of the
field by 3.5°, which we will use below. The appearance
of hf-components, according to [3], can be due to a non-
linear reflection consisting in the diffraction of the inci-
dent radiation by a periodic structure arising from the
mixing of this radiation with a "material" surface wave
(MSW), which is also excited by it. For definiteness, we
mailto:vkont@rian.kharkov.ua
ISSN 1562-6016. ВАНТ. 2018. №4(116) 113
give expressions for gravitational waves on a liquid
surface.
The wave of "light pressure" arising at the boundary,
bilinear in the amplitudes of the incident 0E ∝
[ ( )]exp i tω−kr and scattered 1 [ ( )E exp i± ∝ −k ± q r
( ) ]i tω− ±Ω combinational3 waves [14, 15]
0 1 0 1 exp( ),свp E E E E i i t∗ ∗
−∝ + ∝ − Ωqr (1)
in turn, swings the surface oscillations, leading to stimu-
lated scattering.
For a liquid medium, solving the linearized equa-
tions of motion of the incompressible liquid
/ t grad pr r∂ ∂ = − +v g ,
taking into account the forces of electromagnetic fields,
with boundary conditions at z z= (see [16]):
2 2 2 2( / / ) ,II I
свp p x y pα z− − ∂ ∂ + ∂ ∂ =
,I II
св nn nnp ≡ Π −Π / ,nt vz∂ ∂ =
where ( ) 2/ 8p p Er ε r p′≡ − ∂ ∂ , nnΠ is the normal
component of the Maxwell stress tensor, p′ , v , and α
are the pressure, velocity and surface tension coeffi-
cient, g is the gravitational acceleration, we find the
Fourier component of the surface oscillation z Ωq , which
is expressed in terms of the Fourier component of the
light pressure
2 2
0( ) / ( ) ( )I II
свq p qz r rΩ Ω = + Ω −Ω q q , (2)
where 0 ( )qΩ is the undisturbed dispersion law of sur-
face waves. The amplitude of the pressure wave at fre-
quency Ω is
2
0 / 8I i
свp iq P Ez ε p= , (3)
where the dimensionless pressure P given in [3, 14, 15]
contains dependences on the wave vectors of electro-
magnetic and surface waves. We find the dispersion
equation for surface waves on the irradiated surface,
taking into account (1) and (3) and including attenuation
due to (small) viscosity / IIn η r= ( II Ir r>> ) in it:
22
02
0
0
( ) ( ) 2
16 ( )
I i
II
iq P E
q q iq
q
ε
n
pr
Ω = ± Ω −
Ω
. (4.7)
At the intensity of the incident field, larger than
threshold, 0 thI I> , that is
2
0 04 ( )
8 Re
i
I
E q
P
η
ε
p
Ω
> , (4)
growth of surface waves and stimulated scattering (SS)
at them take place. The analysis is reduced to investiga-
tion of the quantity Re P proportional to the light pres-
sure.
1. RESONANCE WITH SURFACE
ELECTROMAGNETIC WAVE
The stimulated combinational scattering under con-
ditions of resonance with surface electromagnetic wave
3Combinational (Raman) scattering at the surface was
considered in the classical papers by Mandelshtam, An-
dronov and Leontovich [12, 13].
(SEW) was considered in the work by Katz and Maslov
[15]. The pressure analysis was carried out for a sliding
wave with 1 0zk = . Index 1 indicates a combinational
(anti-Stokes) wave of the first order with frequency
ω +Ω . Below we consider the same combinational
waves, but we omit this index in dimensionless quanti-
ties. As follows from the general expressions for scatter-
ing fields, small denominators of the form
11/ ( / )zk k ε− arise in light pressure in the vicinity of
the sliding scattered wave with 1 0zk = for large value
| | 1ε >> .
Large | |ε arises at high conductivity, at that ε is
complex quantity. It is convenient to use a complex
surface impedance ξ , [16], where, 1 /ξ ε∝ ,
| | 1ξ << , instead of ε . We do not discuss the possible
effect of magnetic permeability here. High conductivity
corresponds to the concept of a boundary as a kind of
high-density metal, where the iron nuclei are surrounded
by free electron gas [5, 6]. Then the small denominator
acquires a pole view, 1/ ( )β ξ+ , where 1 /zk kβ = (see
App. A).
a) Forward scattering.
Further on, we measure all the wave numbers in
units of the wave number of the incident wave k . Then
21 ( )xk qβ = − + . Here q is the wave number of the
MSW, sinxk θ= . The hf-component corresponds to the
excitation of the MSW with the value of the (algebraic)
wave number q− and the combination scattering at it
(for details see [3]). There is no fundamental difference
from real values ε here. We explain the large width of
the HFCI component below.
b) Backward scattering.
In this case, we denote the wave number of the
MSW through g ; 21 ( )xk gβ = − + . The presence of
a pole results in maximum Re P at purely imaginary
value ''iβ β≡ corresponding to the resonance with
SEW − the eigen wave of the surface. This purely in-
homogeneous wave does not radiate into space, but we
will see below that it can contribute to scattering at
higher frequencies. The value of pressure at maximum
| |β ξ′′ ′′= can be very large: Re 1/res
maxP ξ ′≈ .
The second maximum Re P arises for real
( )β β ξ ξ′ ′′ ′≡ = − + and corresponds to the propagating
near-surface wave (see App. B): Re 1/W
extrP ξ ′′≈ .
Finally, the special case is the Rayleigh point for
0β = , considered in [3]: Re | | 1RP ε≈ >> . In terms
of impedance 2Re '/RP ξ ξ≈ .
All three characteristic features of Re P are closely
related to the so-called Wood anomalies4 and are basi-
4Wood's anomalies [17] received first physical explana-
tion in the work by Rayleigh [18], who connected them
with a sliding diffraction orders. See also the mono-
graph [19].
ISSN 1562-6016. ВАНТ. 2018. №4(116) 114
cally generated by the diffraction peculiarities near Ray-
leigh angle of incidence for a certain diffraction order.
2. EXPLANATION
OF THE HF-COMPONENTS WIDTH
The width of the hf-components reaches 30°, that is
substantially greater than the width of the IP. This mod-
el presents a simple physical explanation for hf-
components broadening.
Suppose that SS is realized at certain frequency ω1
from the continuous spectrum of positron emission fly-
ing to the star. This means the excitation and buildup of
the MSW with a certain value of the wave vector q1 (for
forward scattering, cf. Fig. 3 in [3]), or g1 (for backward
scattering, see Fig. 1 below). For a higher frequency ω2
of positron emission, a near-surface propagating elec-
tromagnetic wave arises for combinational scattering on
the same MSW g1 if frequency difference 2 1ω ω− ex-
ceeds the (small) surface impedance (Fig. 1).
Thus, there is a contribution to the component HFC2
from the wide region of the positron emission spectrum,
resulting in a large width of the component. Analogous-
ly, such pulse broadening occurs for HFC1 component
due to MSW q1 at forward scattering. If we assume that
SS at frequency ω1 occurs at the surface resonance that
corresponds to non radiated non-uniform electromagnet-
ic wave with a purely imaginary transverse wave num-
ber, then reflected propagating waves appear at higher
frequencies ω2. The set of different frequency combina-
tions explain the large width of the HF-component. The
steeply falling energy spectrum in Crab Pulsar is an
important argument in favor of such scenario.
Fig. 1. A diagram explaining the contribution to the
nonlinear reflection of combinational fields at higher
frequency 2ω from material surface waves ( g ) excited
via resonant stimulate scattering by the incident radia-
tion at lower frequency 1ω . For continuous spectrum of
the radiation incident on the star this explains the large
width of the hf-components. The details is in the text5
CONCLUSIONS
5Let us note that the wave corresponding to ω1 is not
emitted, it has a purely imaginary transverse wave num-
ber. The wave corresponding to 2
2 1 (1 '' / 2)ω ω ξ> + has a
real transverse wave number and contributes to the re-
flection at the frequency ω2.
This work is based on the idea that the radiation of
the return positrons is reflected from the surface of a
neutron star, that was introduced by one of the authors
and S.V. Trofimenko [7, 20 - 22]. The reflection in a
magnetic field inclined to the surface star manifests
itself in the IP shift and the appearance of additional hf-
components discovered and investigated in [8 - 10] by
Moffett, Hankins, Eilek and Jones. The possible effect
of the combinational waves resonance with SEW of the
sliding along the surface is discussed in this paper. The
possibility of forming wide hf-components due to com-
binational scattering of a wide spectrum of radiation of
positrons incident on the star surface is shown. This
allows us (albeit ambiguously) to explain the observed
drift of the HF-component and return, at least partly, to
coupling each of the components to its pole [3]. In par-
ticular, backward scattering in the North Pole on a peri-
odic structure excited at resonance with SEW gives such
an opportunity, since it imitates "drift" towards the
North Pole.
Indeed, let the MSW with a wave number 1g
(Fig. 1) be excited by a wave 1k at resonance with
SEW. Then ( )2
1 1 1 sin '' / 2Ng k θ ξ= + + . In scattering
the wave 2 1k k> on this structure, an anti-Stokes wave
with a tangential component of the wave vector
2 1 2 sinx Nk g k θ+ = − arises. At 2 2xk k+ ≤ these waves will
be propagating, and the equality 2 2xk k+ = corresponds to
/ 2p angle relative to normal. At higher frequencies,
this angle 2ϕ is
( )2 2 2arcsin /xk kϕ +=
or
( )2
2 1 2sin 1 sin '' / 2 /Nk kϕ θ ξ= + + ,
from which it can be seen that the angle decreases with
increasing frequency 2k . This corresponds to the ob-
served "drift" of the component toward the N-pole [9].
At the South Pole (cf. the Fig. 3 in [3]) a similar
"drift" may be occur due to scattering of lower frequen-
cies and the angle 2
Sϕ also will increase in accordance
with observations, but this will require the excitation of
a wide range of material surface waves.
APPENDIX A
Consider the SEW (surface electromagnetic wave)
in terms of the surface impedance. From the Maxwell
equation for the H-wave rot ( )I
x xi cε ω= −H E , and
the Leontovich boundary condition
/ [ , ]II
x xµ ε=E H n ,
n is the normal to the surface, follows:
/y yH z ik Hξ∂ ∂ = − .
where k is the wave number of electromagnetic wave
in the first medium and ξ is a relative impedance.
SEW corresponds ( )expyH zκ∝ − with 0κ > (the
axis z is directed into the first medium) whence the
dispersion relation for SEW takes the form
ikκ ξ= ,
ISSN 1562-6016. ВАНТ. 2018. №4(116) 115
which requires a purely imaginary impedance ''iξ ξ= ,
'' 0ξ < , corresponding to the negative 0IIε < (if µ
and Iε are real and positive) [16]. By introducing a
dimensionless transverse component of the wave vector,
β , according to zk kβ≡ or i kκ β≡ − with Im 0β ≥ ,
we rewrite the condition in the form
0β ξ+ = .
We emphasize that the SEW as eigen wave exists for
real and negative value of IIε only (with the remark
made above). In the general case of the complex IIε ,
small denominators of the form ( )β ξ+ arise in the
diffracted fields in the region of Wood anomalies. Re-
spectively, resonance combinational field acquire a
form
1 1 ( )H β ξ± ∝ + .
Then the resonant part of the dimensionless radia-
tion pressure, bilinear in the incident and combinational
field, is
1 ( )P β ξ∝ + ,
and the real part Re P , which determines the instability
increment, is equal to
2 2 2Re .
| | | | | |
P β ξ β ξ
β ξ β ξ β ξ
′ ′ ′ ′+
= = +
+ + +
The above expression has a simple physical interpre-
tation. The first term is proportional to the normal com-
ponent of the energy flux density of the resonance spec-
trum
2 2| | | |outS Hβ β β ξ± ±′ ′∝ ∝ + ,
carried away from the surface. The second term corre-
sponds to the energy flux density directed to the surface,
2Re[ ] | | ,in nS ξ β ξ± ∗ ± ′∝ ∝ +E H
and this flux is completely absorbed by the medium.
Thus, Re P is proportional to the sum of radiative and
dissipative losses. Accordingly, the maxima of Re P
correspond to the maxima of the total losses of the reso-
nance spectrum. The corresponding extremal and singu-
lar points were given at the end of Section 3.
Note for certainty, the radiation (light) pressure used
above is found from the solution of the electrodynamic
problem using Leontovich boundary condition at the
surface, perturbed by a material wave [14 - 15, 3].
APPENDIX B
We illustrate the influence of the sliding waves and
Wood's anomalies by observing the reflection from dif-
fraction grating (according to the data of [23]).
The mechanism of nonlinear reflection of radiation
from the surface of a neutron star, described above, has
much in common with the mechanism of the resonance
diffraction of electromagnetic radiation on the periodic
surface of the conducting medium in the vicinity of
Wood’s anomalies. As in the case considered above, the
interaction between the diffraction components of the
beam leads to an increase in the intensity of Stokes (or
anti-Stokes) components, resulting in a bright near-
surface wave and, correspondingly, decreasing of the
intensity of the specularly reflected radiation (Fig. 2).
We present some results of the laboratory studies of
the resonance diffraction of radiation on a corrugated
metal surface [23]. The radiation source was HCN laser
(wavelength λ =366.7 µm, with beam radius at the laser
output 6.7 mm, and beam divergence 0.9 deg≈ in 1/ e
intensity level.
The experiments were carried out with brass samples
( 60%Cu ), periodic structures (gratings) were prepared
on their surfaces with grooves of different depths h: 16,
24 and 40 µm. The periods d of all gratings are the
same: d = 254 µm.
Fig. 2. Sliding near-surface wave against a background
of incident and reflected waves (numerical simulation).
The bright diffracted wave corresponds to a weakened
reflected one [23]. Courtesy to the authors
The diffraction anomalies were studied in the vicini-
ty of the incidence angle θ corresponding to the Ray-
leigh angle for the −1-th diffraction order:
[ ]( 1) arcsin ( / ) 1 .R dθ θ λ−≈ ≡ − This geometry is prefera-
ble, since in this case there are only two propagating
waves – specularly reflected and minus-first diffraction
component. The remaining diffraction orders are inho-
mogeneous and have not been recorded in the experi-
ment. The power of radiation was measured depending
on the angle of incidence. At angles of incidence lower
than Rayleigh one ( ( 1)
Rθ θ −< ), when Stokes wave 1H−
was inhomogeneous, power of the specularly reflected
radiation was measured, and for angles ( 1)
Rθ θ −> the
dependence of Stokes component power on the sliding
angle ψ (between the grating plane and the recorded
beam) was measured.
As follows from the theoretical consideration of the
problem, the change in the power of the specularly re-
flected radiation has a quasi-resonance character. Near
Rayleigh angle, the specularly reflected radiation is sub-
stantially suppressed, and the suppression increases with
the deepening of the grating grooves. Respectively, the
intensity of Stokes component increases. It is also seen
that minimum of specular reflection at the incidence
angle minθ corresponds to Stokes component maximum
at sliding angle maxψ (in case of diffraction in the −1-st
order, these angles are related by
max mincos sin / dψ θ λ+ = ). It should be noted that the
shape of the curve depends on the depth of the grating
ISSN 1562-6016. ВАНТ. 2018. №4(116) 116
grooves. For shallow gratings ( h d<< ) the angular de-
pendence of reflectivity is almost symmetric and close
to Lorentz curve. At the increase of the groove depth the
symmetry of the wings disappears and the shape of the
curve approaches a form characteristic for Fano reso-
nance [24]. This is stipulated by the presence of two
channels for signal formation: nonresonance (Fresnel)
and resonance ones, caused by interaction of the dif-
fracted order with the grating. The manifestation of this
interaction is the above-mentioned redistribution of en-
ergy between the specularly reflected and Stokes com-
ponents: the incident wave scattering at the grating gen-
erates Stokes component; in turn, scattering of the
Stokes wave itself produces another component which
propagates in the same direction as the specular reflect-
ed wave. Interference of this "additional" component
with the specular reflected wave results in redistribution
of energy between the diffraction components (which is
illustrated by numerical simulation), and a change in the
shape of the Fano resonance curve.
ACKNOWLEDGMENTS
One of the authors (V.K.) is thankful to Correspond-
ing Member of NAS of Ukraine, prof. D.M. Vavriv for
help and support. We also are grateful to prof.
A.V. Kats who carefully read the manuscript and made
useful comments.
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Article received 31.05.2018
ISSN 1562-6016. ВАНТ. 2018. №4(116) 117
РЕЗОНАНСНОЕ НЕЛИНЕЙНОЕ ОТРАЖЕНИЕ ОТ НЕЙТРОННОЙ ЗВЕЗДЫ
И ДОПОЛНИТЕЛЬНЫЕ КОМПОНЕНТЫ В ИЗЛУЧЕНИИ ПУЛЬСАРА КРАБОВИДНОЙ
ТУМАННОСТИ
В.М. Конторович, И.С. Спевак, В.К. Гавриков
Дополнительные высокочастотные компоненты излучения пульсара в Крабовидной туманности рассмат-
риваются как результат резонанса с поверхностной электромагнитной волной при нелинейном отражении от
поверхности нейтронной звезды. Это − вынужденное рассеяние, которое состоит в генерации падающим
полем периодического рельефа поверхности и дифракции на нём излучения релятивистских позитронов,
летящих из магнитосферы к звезде в ускоряющем электрическом поле полярного зазора.
РЕЗОНАНСНЕ НЕЛІНІЙНЕ ВІДБИТТЯ ВІД НЕЙТРОННОЇ ЗІРКИ І ДОДАТКОВІ
КОМПОНЕНТИ У ВИПРОМІНЮВАННІ ПУЛЬСАРА КРАБОПОДІБНОЇ ТУМАННОСТІ
В.М. Конторовіч, І.С. Спєвак, В.К. Гавриков
Додаткові високочастотні компоненти випромінювання пульсара в Крабоподібній туманності розгляда-
ються як результат резонансу з поверхневою електромагнітною хвилею при нелінійному відбитті від повер-
хні нейтронної зірки. Це − вимушене розсіювання, яке складається в генерації падаючим полем періодично-
го рельєфу поверхні і дифракції на ньому випромінювання релятивістських позитронів, що летять з магніто-
сфери до зірки в електричному полі полярного зазору, яке їх прискорює.
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