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 
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 IIсв 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 / 8
I 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/resmaxP 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/WextrP ξ ′′≈ . 
Finally, the special case is the Rayleigh point for 
0β = , considered in [3]: Re | | 1RP ε≈ >> . In terms 
of impedance 
2
Re '/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 ( )21 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 
( )22 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 ( )Ix xi cε ω= −H E , and 
the Leontovich boundary condition 
/ [ , ]IIx 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 2
Re .
| | | | | |
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.