Electromagnetic grazing anomalies. Energy flux extrema
The diffraction of electromagnetic waves at the surface periodic structures accompanied by strong anomalous effects in different diffraction orders is considered in detail for high-contrast interfaces. We restrict discussion by the transverse magnetic polarization of the incident wave (the magneti...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1761232021-02-04T01:29:12Z Electromagnetic grazing anomalies. Energy flux extrema Kats, A.V. Низькоpозмірні та невпорядковані системи The diffraction of electromagnetic waves at the surface periodic structures accompanied by strong anomalous effects in different diffraction orders is considered in detail for high-contrast interfaces. We restrict discussion by the transverse magnetic polarization of the incident wave (the magnetic field is orthogonal to the plane of incidence) and the simplest geometry when the plane of incidence is orthogonal to the grating grooves. The most attention is devoted to the strong maxima and minima of the energy flux density accompanying specific grazing propagation of some diffraction order. Relation to other anomalies, both Rayleigh and the resonance ones is discussed as well. Ретельно проаналізовано дифракцію електромагнітних хвиль на поверхнях з періодичними структурами, що супроводжується великими аномальними ефектами. Розглянуто межі поділу середовищ з високим контрастом властивостей. Дослідження обмежено випадком ТМ поляризації хвилі, що падає на межу (магнітне поле перпендикулярне до площини падіння), та найпростішою геометрією, коли площина падіння ортогональна до штрихів гратки. Найбільшу увагу приділено максимам та мінімумам густини потока енергії, що супроводжують ковзне розповсюдження в деякому дифракційному порядку. Обговорюється зв’язок з іншими аномаліями, як релеївськими, так і резонансними. Детально проанализирована дифракция электромагнитных волн на поверхностях с периодическими структурами, сопровождаемая сильными аномальными эффектами. Рассматриваются границы раздела сред, отличающихся высоким контрастом свойств. Исследование ограничено случаем ТМ поляризации падающей на границу волны (магнитное поле ортогонально плоскости падения) и простейшей геометрией, когда плоскость падения ортогональна штрихам решетки. Наибольшее внимание уделено максимумам и минимумам плотности потока энергии, сопровождающими скользящее распространение в некотором дифракционном порядке. Обсуждается связь с другими аномалиями, как рэлеевскими, так и резонансными. 2019 Article Electromagnetic grazing anomalies. Energy flux extrema / A.V. Kats // Физика низких температур. — 2019. — Т. 45, № 5. — С. 612-619. — Бібліогр.: 23 назв. — англ. 0132-6414 http://dspace.nbuv.gov.ua/handle/123456789/176123 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низькоpозмірні та невпорядковані системи Низькоpозмірні та невпорядковані системи |
| spellingShingle |
Низькоpозмірні та невпорядковані системи Низькоpозмірні та невпорядковані системи Kats, A.V. Electromagnetic grazing anomalies. Energy flux extrema Физика низких температур |
| description |
The diffraction of electromagnetic waves at the surface periodic structures accompanied by strong anomalous
effects in different diffraction orders is considered in detail for high-contrast interfaces. We restrict discussion by
the transverse magnetic polarization of the incident wave (the magnetic field is orthogonal to the plane of incidence) and the simplest geometry when the plane of incidence is orthogonal to the grating grooves. The most attention is devoted to the strong maxima and minima of the energy flux density accompanying specific grazing
propagation of some diffraction order. Relation to other anomalies, both Rayleigh and the resonance ones is discussed as well. |
| format |
Article |
| author |
Kats, A.V. |
| author_facet |
Kats, A.V. |
| author_sort |
Kats, A.V. |
| title |
Electromagnetic grazing anomalies. Energy flux extrema |
| title_short |
Electromagnetic grazing anomalies. Energy flux extrema |
| title_full |
Electromagnetic grazing anomalies. Energy flux extrema |
| title_fullStr |
Electromagnetic grazing anomalies. Energy flux extrema |
| title_full_unstemmed |
Electromagnetic grazing anomalies. Energy flux extrema |
| title_sort |
electromagnetic grazing anomalies. energy flux extrema |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2019 |
| topic_facet |
Низькоpозмірні та невпорядковані системи |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/176123 |
| citation_txt |
Electromagnetic grazing anomalies. Energy flux extrema / A.V. Kats // Физика низких температур. — 2019. — Т. 45, № 5. — С. 612-619. — Бібліогр.: 23 назв. — англ. |
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Физика низких температур |
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AT katsav electromagneticgrazinganomaliesenergyfluxextrema |
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2025-07-15T13:45:04Z |
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2025-07-15T13:45:04Z |
| _version_ |
1837720775882178560 |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 5, pp. 612–619
Electromagnetic grazing anomalies. Energy flux extrema
A.V. Kats
Usikov Institute for Radiophysics and Electronics of National Academy of Sciences of Ukraine
12 Acad. Proskuri, Kharkiv 61085, Ukraine
E-mail: avkats@ire.kharkov.ua
Received January 17, 2019, published online March 26, 2019
The diffraction of electromagnetic waves at the surface periodic structures accompanied by strong anomalous
effects in different diffraction orders is considered in detail for high-contrast interfaces. We restrict discussion by
the transverse magnetic polarization of the incident wave (the magnetic field is orthogonal to the plane of inci-
dence) and the simplest geometry when the plane of incidence is orthogonal to the grating grooves. The most at-
tention is devoted to the strong maxima and minima of the energy flux density accompanying specific grazing
propagation of some diffraction order. Relation to other anomalies, both Rayleigh and the resonance ones is dis-
cussed as well.
Keywords: diffraction, wave, anomaly, resonance, grating, flux.
Introduction
It has been well known since early 1900s that the light
diffraction by metal gratings is accompanied by a number
of strong spectral and angular anomalies which manifest
themselves by the fast dependence of the intensities on the
wavelength and/or angle of incidence. The pioneer' work
on the subject was performed by R. Wood in 1902 [1] with
metal gratings. The first physical interpretation of some of
the observed peculiarities was presented by Lord Rayleigh
in [2]. He associated them with the branch points related to
diffracted waves (i.e., with the transitions from the out-
going wave to the evanescent one and vice versa in differ-
ent diffracted orders). Such explanation is incomplete due
to the fact that some Wood anomalies are to be attributed
to the resonance excitation of surface electromagnetic
wave at the metal\air interface. Such interpretation was
first proposed by U. Fano [3]. The resonantly excited
waves are called the surface plasmon polaritons (SPPs) [4].
The resonance anomaly is still widely discussed due to its
perspective role in nanophotonics. Later, Wood caught site
of one more anomaly relating to anomalously high intensi-
ty of the grazing outgoing wave: “…the spectrum leaving
at grazing emergence, which is the one which governs the
appearance of the anomalous bands, is very bright” [5].
Below the anomalies attributed to the grazing propagating
waves are referred to as GA (Grazing Anomaly).
It is of essence that the Rayleigh anomaly exists for an
arbitrary interface and polarization. However, it is much
more pronounced for the high-dielectric contrast interface
and for TM (transverse magnetic) polarization if the media
we are dealing with are nonmagnetic. In what follows we
restrict the consideration to the nonmagnetic case only.
The results for the magnetic case can be obtained by re-
placing the dielectric permittivity, ε, with the magnetic
permeability, µ, and the TM polarization by the TE one
and vice versa. The resonance anomaly can exist only for
such interfaces that support surface electromagnetic waves
(SEW). GA anomaly is rather universal and is well ex-
pressed for high contrast interfaces for TM polarization. To
our knowledge it was first discussed theoretically in [6].
Consider the main properties of these anomalies. The
branch (Rayleigh) point anomaly is of general type, its
position can be easily obtained from the Bragg diffraction
conditions and it exists for arbitrary polarization and inter-
faces. However, it is more expressed for metals under TM
polarization. At the Rayleigh point the derivative of the
diffracted wave intensity with respect to the wavelength
and angle of incidence turns infinity. The resonance anom-
aly is less general because it is caused by existence of well-
defined eigenmodes of the interface.*
For isotropic and nonmagnetic dissipation-free media
such surface-localized electromagnetic waves do exist un-
der the conditions 0ε < , 0dε > , 0dε + ε < , where ε and
* We restrict consideration to interface of two homogeneous isotropic nonmagnetic media, say, metal and dielectric. If between these
two media exists some third one (even very thin layer), then additional to SPP resonances can occur [7]. For anysotropic media the reso-
nance can be caused by other than SPP surface modes, say, Dyakonov ones, see [8] and citations therein.
© A.V. Kats, 2019
Electromagnetic grazing anomalies. Energy flux extrema
dε denote dielectric permittivity of the metal and the adja-
cent dielectric, respectively. The SPP in-plane wave-
number ( ) /( ) 0d dQ Q
c
ω
= ω = εε ε + ε > , where ω is the
(angular) frequency of the incident wave, exceeds the
wavenumber of the adjacent dielectric volume wave with
the same frequency, ( ) /dk k c= ω = ω ε , Q k> . The
square root symbol stays for the main branch, so that
exp ( /2)Z Z i= φ for exp ( )Z Z i= φ with [0, 2 )φ∈ π .
The SPP is TM polarized, i.e., if it propagates along the
interface 0z = in Ox direction then its magnetic field, H ,
is directed along Oy direction, (0, ,0)H=H , and the elec-
tric field, E, lies in the xOz plane, i.e., plane of incidence,
( ,0, )x zE E=E . The space dependence of the SPP fields in
the dielectric halfspace, 0z ≤ , is given by the ansatz
exp[ ( ) ]iQx ip Q z− , where
( ) 2 2 , /dp k k c= − = ε ωq q . (1)
For dissipation-free media ( )p Q is pure imaginary,
( ) ( )p Q i p Q= , so that the field amplitude decays expo-
nentially with increasing distance from the interface 0z = .
Recall, if the plane monochromatic electromagnetic
wave with space dependence of the form
( ) ( ) ( ), exp , ,x yi ip z q q∝ ⋅ + = E H q r q q (2)
(where and everywhere else the time dependence is sup-
posed to be of the form exp ( )i t− ω and is omitted) is inci-
dent on the interface from the dielectric medium located at
negative z values, ( )z x−∞ < < ζ , where the surface relief,
( )z x= ζ , presents periodic function with period d ,
( ) ( )x d xζ + = ζ , then the electromagnetic field within the
dielectric medium is the sum of spatial harmonics of the
form
( ) ( ), exp ,
, 2 / , 0, 1, 2, ,
n n n n
n x
i ip z
n d n
∝ ⋅ −
= + = π = ± ±
E H q r q
q q g g e
(3)
xe is the unit vector directed along the Ox axis. In other
words, the diffracted field is given by the Floquet–Fourier
expansion [7,9]. In (3) the sign minus before ( )np q stays
to satisfy the radiation conditions at z = −∞ . Restriction of
the outgoing (and evanescent) waves within the whole
halfspace ( )z x≤ ζ corresponds to the use of the Rayleigh
hypothesis [2] and is not restrictive even for rather deep
gratings, see the recent discussion in [10].
Consequently, if for some specific integer n the condi-
tion n Q=q holds true, then for the appropriate polariza-
tion of this diffracted wave the resonance excitation of SPP
takes place. It is significant that SPP is an evanescent wave
and thus the magnitude of the corresponding diffracted
order can exceed essentially that of the incident wave. Spe-
cifically, in the simplest geometry, when q is orthogonal to
the grating grooves, ( ,0,0)q=q , 0q > , only TM component
of the incident wave can excite the SPP.* Also, it deserves
attention that for the modulated interface the SPP resonance
centre experiences shift as compared with the “naked” con-
dition, n Q=q . However, the SPP resonance in majority of
experimental situations in visible and near infrared spectral
regions seems to be rather evident to attribute.
We would like to underline that the Rayleigh and the
resonance anomalies are related to a specific and rather
sharp dependence of the field amplitudes on the wave-
length and angle of incidence. They can be considered on
the basis of simple qualitative treatment. The treatment of
the third mentioned Wood anomaly cannot be accom-
plished without thorough theoretical approach. The method
for considering this and other diffraction anomalies analyt-
ically was presented in [9], see also a more detailed con-
sideration in [11–13].
Grazing incidence anomaly
In this section we present the brief summary of the re-
sults for the case of the simplest geometry which are essen-
tial for further consideration. For the TM polarization of
interest the magnetic field is orthogonal to the plane of
incidence and thus possesses the y component only, so
that for the incident wave, iH , and for the Fourier–Floquet
expansion of the diffracted field, DH , we have
( )
( ) ( )
exp ,
exp , ,
i
y
D
y n n n
n
H iqx ip q z
H iq x ip q z z x
∞
=−∞
= +
= − ≤ ζ ∑
H e
H e
(4)
where nq q ng= + . Note, the diffracted field in (4) and
below in (5) includes only outgoing (and evanescent)
waves, i.e., here we use the Raylegh hypothesis [2], restrict-
ing the expansion to the terms with z -dependence of the
form exp[ ( ) ]nip q z− only, and omitting those with z -depen-
dence of the alternative form, exp[ ( ) ]nip q z . This guaran-
tees fulfillment of the boundary (radiation) conditions at
z = −∞ . The Raylegh hypothesis is appropriate for shallow
enough gratings (however, the recent investigations [7,9]
have demonstrated that it works well even for rather deep
gratings).
The electric field possesses the x and z components
only, ( ,0, )x zE E=E , and can be easily obtained from (3)
and corresponding Maxwell equation. Specifically, the
diffracted field is
( ) ( )exp ,D
n n n
n
iq x ip q z z x
∞
=−∞
= − ≤ ζ ∑E E . (5)
* Noteworthy, in the simplest geometry the diffraction of TE and TM components of the incident wave become independent processes
and thus can be considered separately.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 5 613
A.V. Kats
At the interface the total fields, i D= +H H H ,
i D= +E E E are to obey the impedance boundary condi-
tions [14],
[ ] ( )for t z x= ξ × = ζE n H , (6)
where the subindex t denotes tangential to the interface
component of the corresponding vector, and n stays for the
unit vector normal to the interface and directed into the
dielectric, i.e., 2[ / ] 1 ( / )z x x x= − − ∂ζ ∂ + ∂ζ ∂n e e .* For
nonmagnetic media the surface impedance ξ in Gauss
units is dimensionless and /dξ = ε ε .
The relief Fourier series expansion is
( ) ( )
0
exp ,
2 0, , 0.
n
n
n n
x ingx
g d
∞
=−∞
∗
−
ζ = ζ
= π > ζ = ζ ζ =
∑ (7)
The condition 0 0ζ = corresponds to the specific choice of
Oz axis origin. The Fourier series coefficients of the inter-
face normal, ( )x=n n , can be expressed in terms of nζ .
Substituting into Eq. (6) the fields representations given
in Eqs. (4), (5), expressing the electric field Fourier ampli-
tudes, nE , in terms of the magnetic ones, nH , and equating
terms with equal space dependence, we arrive at the infi-
nite system of linear algebraic equations for the transfor-
mation coefficients (TCs), /n nh H H= ,
, 0, 1, 2,n m m n
m
D h V n
∞
=−∞
= = ± ±∑ , (8)
where the matrix of the system, ˆ n mD D= , and the right-
hand side column vector, ˆ col{ }nV V= , represent
functionals depending on the problem parameters, specifi-
cally, the relief ( )xζ . The coefficients of the system allow
infinite series expansions with respect to nζ . It is of es-
sence that strong diffraction anomalies take place for rather
shallow gratings such that , 1k d dxζ ζ , see [11–13]
and below, so the expansions are very useful. For shallow
gratings we can restrict series expansions of the coeffi-
cients to the main (linear) terms only, so that
( ) ( )1 ,
, 0, 1, 2, ,
n m n n m n m n mD i
n m
−= β + ξ δ − −α α µ
= ± ±
(9)
( ) ( )0 01 , 0, 1, 2,n n n n nV i n= β − ξ δ + −α α µ = ± ± . (10)
Here n mδ stays for the Kronecker delta-symbol, and
2
, , / ,
1 , Re, Im 0, 0, 1, 2, ,
n n n
n n n
k n g k
n
µ = ζ α = α + κ κ =
β = −α β ≥ = ± ±
(11)
sinα = θ, θ denotes the incidence angle. Noteworthy, the
nondiagonal elements of the matrix ˆ nmD D= possess the
following simple symmetry
fornm mnD D n m∗ = − ≠ .
Below we are dealing with the grazing anomalies. They
correspond to specific dependences (maxima, minima) of
the energetic characteristics (say, intensities) of the dif-
fracted waves in the vicinity of the point where the inci-
dent wave or one of the diffracted waves is propagating at
a small angle with respect to the interface (grazing propa-
gation). The simplest (but of high interest) case here pre-
sents the grazing incidence, 0 1< β (0 1 1< −α ). That
is the specular reflected wave with necessity is the grazing
one. The simplest geometry of such problem is presented
in Fig. 1.
It should be emphasized, that among diffracted waves
only the specular reflected one is close to the correspond-
ing Rayleigh point, 1β , and all the other waves are far
enough from their branch points. That is, the only one di-
agonal element of the matrix ˆ n mD D= , namely,
00D = β+ ξ, is small as compared with unity. Consequent-
ly, it is convenient to decompose the governing system,
Eq. (8), as
ˆˆ ˆDh V= , (12)
00 0 0 0
0
M M
M
D h D h V
≠
+ =∑ , (13)
where and below capital indexes denote all integers except
zero,
* The following considerations can be applied to the case of plane surface of metamaterials with periodically modulated electromag-
netic properties so that the surface impedance is space-periodic, ( )xξ = ξ , ( ) ( )x d xξ + = ξ , cf. [11–13].
Fig. 1. Grazing incidence diffraction. The grating spacing, d , is
supposed to be such that except the specular wave only the minus
first diffraction order presents propagating wave, other diffraction
orders correspond to evanescent ones, i.e., at q k , 1q q g k= + > ,
1q q g k− = − > − and nq k> for all 1, 0n ≠ − .
614 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 5
Electromagnetic grazing anomalies. Energy flux extrema
ˆ , , 1, 2,N MD D N M= = ± ± , (14)
ĥ and V̂ stay for column vectors
{ } { }ˆ ˆcol , col , 1, 2,M Mh h V V M= = = ± ± , (15)
0 0 , 1, 2,M M MV V D h M= − = ± ± . (16)
Let us present Eq. (13) in a more explicit form as well
( ) ( )0 0
0
1 M M M
M
h i h−
≠
β + ξ − −α α µ = β− ξ∑ . (17)
The submatrix D̂ is diagonal dominated due to the fact that
all nondiagonal elements are small as compared with unity
and all diagonal ones are of order unity or greater. Thus, it
can be easily inversed by means of the regular series expan-
sion, see below. Formally, we can express all nonspecular
amplitudes, Mh , in terms of the given parameters of the sys-
tem and unknown at this stage amplitude 0h as follows:
1ˆ ˆ ˆh D V−= , (18)
or, more explicitly,
1
0
ˆ , 1, 2,M L
MLL
h D V M−
≠
= = ± ± ∑ . (19)
Taking into account that in accordance with Eq. (17) and
Eq. (10),
( ) ( )0 0 0 0 01 1L L L L L L LV V D h i i h= − = −α α µ + −α α µ =
( )( )0 01 1 , 1, 2,L Li h L= + −α α µ = ± ± , (20)
rearrange Eq. (19) as
( ) ( )1
0 0
0
ˆ1 1 ,M L L
MLL
h i h D−
≠
= + −α α µ ∑
1, 2,M = ± ± (21)
Substituting this expression into Eq. (17) we arrive at the
closed linear equation for the specular TC,
( ) ( )0 01h hβ+ ξ + + ×
( )( )1
0 0
, 0
ˆ 1 1M L L M
MLM L
D−
−
≠
× −α α −α α µ µ = β− ξ ∑ .
(22)
Let, for brevity,
( )( )1
0 0
, 0
ˆ 1 1M L L M
MLM L
D−
−
≠
Γ = −α α −α α µ µ ∑ , (23)
effξ = ξ + Γ. (24)
Then solution of Eq. (22) for 0h can be presented as
eff
0
eff
h
β− ξ
=
β+ ξ
. (25)
It is of interest that the specular TC form, Eq. (25), co-
incides with the corresponding Fresnel coefficient related
to the unmodulated (plane) interface,
R β− ξ
=
β+ ξ
. (26)
For the nonspecular TCs it follows identically
( )1
0
eff 0
2 ˆ 1 ,M L L
MLL
ih D−
≠
β = −α α µ β + ξ ∑
1, 2,M = ± ± (27)
It is convenient to introduce subsidiary functions MU
so that
( )1
0
0
ˆ 1 , 1, 2,M L L
MLL
U D M−
≠
= −α α µ = ± ± ∑
1, 2,M = ± ± (28)
Then
eff
2 , 1, 2,M M
ih U Mβ
= = ± ±
β+ ξ
(29)
It is of essence that the coefficients MU experience only slow
dependence on the parameters of interest in the vicinity of the
point 0β = , as well as the functions Γ and effξ . Noteworthy,
the quantity Γ can be expressed in terms of MU as
( )01 M M M
M
U −Γ = −α α µ∑ .
In what follows we are dealing with smooth and shal-
low gratings so here we present the main terms of the nec-
essary functions expansions. Let
ˆ ˆ ˆ ˆ( )D B I T= − , (30)
where
ˆ ˆ ˆ, , ,NM NM NMI T T B B= δ = =
, 1, 2,N M = ± ± , (31)
( )1 , ,NM N M N M NM N NM
N
iT B b
b −= −α α µ = δ
, , 1, 2,N Nb N M= β + ξ = ± ± . (32)
Then
1 1 1 1
0
ˆ ˆ ˆ ˆ ˆ ˆ( ) s
s
D I T B T B
∞
− − − −
=
= − =
∑ . (33)
This series expansion converges under the condition
ˆ 1,T < where stays for the matrix norm. It is of essence
that this condition is not restrictive: it allows consideration
of strong anomalies, see [9,11–13] and below. Moreover,
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 5 615
A.V. Kats
strong anomalies hold for ˆ 1T . So, we can make use of
the first terms of the expansion. With accuracy up to the
second-order terms with respect to µ,
( )1 1 3
0
ˆ
ML ML MK KL L
ML K
D T T T b O− −
≠
= δ + + + µ
∑ , (34)
or, more explicitly,
( )1ˆ 1ML M L M L
ML M
iD
b
−
−
δ + −α α µ −
( )( ) 1
0
1 1 1 1M K K L M K K L L
M KK
b
b b
−
− −
≠
− −α α −α α µ µ
∑ .
(35)
After simple rearrangement we obtain alternative expres-
sion,
( )1 1ˆ 1M ML M L M L
ML L
iD b
b
− −
−
δ + −α α µ −
( )( )
0
1 1 1M K K L M K K L
L KK b b − −
≠
− −α α −α α µ µ
∑ . (36)
Consequently, up to the second-order terms it follows from
Eq. (28):
( )1
01M M M MU b− −α α µ +
( )( )1
0
0
1 1 ,L L L M L M L
L
i b− −
≠
+ −α α −α α µ µ
∑
1, 2,M = ± ± . (37)
Noteworthy, here the second-order terms are of essence if
the corresponding Fourier amplitude of the grating, Mµ ,
vanishes or is anomalously small. Under this condition the
anomalous effects in Mth diffraction order are small and
thus of low interest. Therefore, below we can restrict our-
selves with the linear term of MU expansion.
The main term of the quantity Γ expansion is the square
one,
( )2 21
0
0
1M M M
M
b−
≠
Γ = −α α µ∑ . (38)
Energy flux extremes
The solution obtained allows one to analyze in detail its
dependence on the angle of incidence and all other parame-
ters of the problem. Expressions (25), (29) describe the fast
dependence of the TCs on the angle of incidence through
the quantity cos 1β = θ . Other functions entering the solu-
tion, NU , effξ , etc., are slow ones, and thus for preliminary
analytical considerations can be replaced with constants
relating to their values at 0β = . This fact allows to perform
thorough analytical investigation of the problem. Starting
with the specular reflectivity
( ) ( )
( )
2 2
2 eff eff
0 2 2
eff eff
h
′ ′′β − ξ + ξ
ρ β = =
′ ′′β + ξ + ξ
, (39)
one can see that it possesses specific minimal value at
some point, extrβ = β , such that
extr effβ = ξ . (40)
With high accuracy one can approximate effξ here by it value
at 0β = .
At the extreme point, extrβ = β , we obtain
( ) eff eff
min min extr
eff eff
,
′ξ − ξ
ρ = ρ ρ ≡ ρ β =
′ξ + ξ
. (41)
Here and below the prime (double prime) denotes the real
(imaginary) part of the corresponding quantity. The specu-
lar TC field at the point extrβ = β is as follows
( ) eff eff
0 extr
eff eff
h
ξ − ξ
β =
ξ + ξ
. (42)
Noteworthy, the analogous minimum for TM polarized
wave incidence exists for unmodulated interface, 0Γ = ,
effξ ⇒ ξ (when 0h coincides with the corresponding Fres-
nel reflection coefficient R ) and is discussed in [14]. This
minimum is analogous to the reflectivity minimum from
dielectric medium existing under Brewster angle incidence
(when the reflected and transmitted waves are propagating
at a right angle) [14]. In view of the fact that for 1ε
(which is typical for good metals up to the frequencies of
the visible range), the normal to the interface component of
the wavevector in metal prevails essentially the tangential
one, so the wave in metal can be formally considered as
orthogonal to the interface. Consequently, for grazing inci-
dence the reflected from the metal wave is approximately
orthogonal to the “transmitted” one. Recall, the Brewster
angle of incidence, Brθ , is defined as
Brsin 1θ = ε ε + ,
so that for ε → ∞ one obtains Br /2θ π .
The specular reflectivity minimum, Eq. (41), becomes
deep for relatively high effective losses, i.e., for eff′ξ com-
parable with eff′′ξ . It worth to point out here that eff′ξ in-
cludes both the dissipative and radiative losses relating to
the quantities ′ξ and ′Γ , respectively. The quantity ′Γ is
mainly caused by outgoing (propagating) waves. On the
contrary, minρ approaches unity at vanishing lossess,
eff 0′ξ → . Therefore, the effect of the specular reflection
suppression under consideration is mainly attributed to the
cumulative (both active and radiative) losses maximum, cf.
616 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 5
Electromagnetic grazing anomalies. Energy flux extrema
[15]. However, as it is shown below the point extrβ = β cor-
responds not only to the specular reflection minimum, but
results in well expressed maximal nonspecular efficiencies.
Evidently, if the only propagating diffracted wave is the
specular one, then the grazing minimum is with necessity
accompanied by maximal absorption. Noteworthy, the re-
flectance minimum under grazing incidence can corre-
spond to essential redirection of the energy into
nonspecular diffraction channels corresponding to propa-
gating waves even for shallow gratings. Below this thesis
is illustrated for the simplest case when in addition to the
specular wave only one diffracted order corresponds to
propagating (outgoing) wave. It can be realized if
1 1+α > κ > , the minus first order presents propagating
wave, 1 0−β > , and nβ with 1,0n ≠ − are pure imaginary. In
what follows we consider that 1−β is of order unity, so that the
minus first diffraction order is far from its Rayleigh point.*
It is of interest that normalized intensities of the propa-
gating diffraction orders,
( ) ( )2 2Re
4 ReN
N N N Nh W U
β
ρ = = β
β
, (43)
present strongly nonmonotonic β functions in accordance
with the fast dependence of the subsidiary function intro-
duced, ( )W W= β ,
( )
( ) ( )2 2
eff eff
W β
β =
′ ′′β + ξ + ξ
. (44)
It is easy to see that ( )W β achieves its maximal value,
maxW , strictly at the point extrβ = β , and
( ) ( )max extr
eff eff
1 1
2
W W= β =
′ξ + ξ
. (45)
That is, all intensities simultaneously achieve their maxi-
mal values at the point extrβ = β ,
( ) ( )
2
,max extr
eff eff
2
Re ,N
N N N
U
ρ = ρ β = β
′ξ + ξ
1, 2,N = ± ± . (46)
It seems necessary to check that, first, the total energy
flux outgoing with the propagating waves does not prevail
that of the incident wave, i.e.,
1N
N
ρ ≤∑ ,
where 0ρ stays for ρ. The difference between the sum and
unity is nothing else than the active losses (per unit area).
This inequality is to be true under rather general condi-
tions, specifically for such β and κ values that we are far
from anomalies relating to all diffraction orders escept the
specular one. If the active losses are absent, then the ine-
quality transforms into the equality. In the specific case of
short-period gratings, such that 2κ > all diffracted orders
except the zeroth one with necessity correspond to evanes-
cent waves. Therefore, the strong specular reflectivity sup-
pression is accompanied by maximal absorption. Under-
line, this conclusion is true under rather specific consitions
and does not describe general case contrary to the state-
ment of [15].
For the specific case shown in Fig. 1 only two diffrac-
tion orders correspond to propagating waves; the specular
and the minus first ones. Consider this specific subcase in
more detail. Suppose additionally that the grating is har-
monic, i.e.,
1 1 0, 0 for 2na n−µ = µ = > µ = ≥ . (47)
Then
( ) ( )2 2
0 1 0 12
1 1
1 1
a
b b
−
−
−α α −α α
Γ +
( ) ( )2 2
0 1 0 12
1 1
1 1
a i −
−
−α α −α α
− +
β β
, (48)
where it is taken into account that 1±β ξ . In the ge-
ometry under discussion
( ) 2
0 11 1±−α α = −α α ± κ = β ακ κ . (49)
Consequently, Γ can be simplified as
2 2
1 1
1 1a i
−
Γ κ −
β β
. (50)
As far as
( )22 2 2 2
1 1 2 2±β = − α ± κ = β ακ − κ −κ κ , (51)
we can express Γ in terms of the dimensionless parameter
of the problem, κ , only, neglecting slow dependence on
the angle of incidence,
( ) ( )
2 2 1 1
2 2
a i
Γ κ −
κ − κ κ + κ
. (52)
In view of the fact that the specular reflectivity possess-
es rather expressed minimal value then for low active loss-
es the incoming energy is to be redirected in other propa-
gating waves. The most interesting case allowing to obtain
rather strong grazing anomalies presents such one that
Γ ξ , (53)
* Alternative case is of interest also, resulting in strong GA as well. The specific case when GA is accompanied by SPP anomaly
relating to some other diffraction order is also of interest. These cases correspond to the double and combined anomalies and will be
considered in forthcoming papers.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 5 617
A.V. Kats
i.e., the case when the effective impedance is mostly caused
by the diffraction rather than the medium properties.* It is of
essence that the supposition presented in Eq. (53) does not
contradict the shallow character of the grating, 2 1,aΓ
in view of the surface impedance smallness, 1ξ . The
characteristic value of the dimensionless grating height, cr ,a
is small, cr 1a = ξ . Under Eq. (53) condition 1,max−ρ
and minρ can be rewritten as
1,max min
2 ,−
′Γ −Γ′Γ
ρ ρ
′ ′Γ + Γ Γ + Γ
, (54)
or, in view of Eq. (50),
2 2
1 1 11
1,max min2 22 2
1 1 1 1 1 1
2
, .−
−
− −
β +β − ββ
ρ ρ
β +β + β β +β + β
(55)
Bearing in mind Eq. (51) we proceed
1,max min
2 2 2 2,
2 2 2 2−
+ κ − + κ
ρ ρ
+ + κ + + κ
. (56)
That is, for rather deep gratings, cra a (but 1a ) the
energy redistribution does not depend on their height,
1,max−ρ and minρ depend on the geometrical parameters
and the wavelength through the dimensionless combination
/dκ = λ only.
It is easy to see that within the accuracy indicated
1,max min 1−ρ +ρ = . That is, all incident energy is redistrib-
uted only between two propagating waves, while the active
losses are negligibly low.
As far as we know the grazing diffraction anomaly un-
der discussion was not considered earlier. However, in
[6,16] one can find the related anomalous effect arising for
such parameters of the diffraction problem that some dif-
fracted order corresponds to the grazing wave propagating
at the specific grazing angle. The anomaly consists in high-
ly enhanced efficiency of this wave accompanied by deep
suppression of the specular reflection. It is worth noting
that this grazing wave enhancement is related to the prob-
lem under consideration by the reciprocity theorem
[17,18]. Namely, reversing the propagation direction of the
minus first order diffracted wave in Fig. 1 we arrive at the
reciprocal diffraction problem. In the latter the correspond-
ing minus first order is related to the grazing wave propa-
gating in the opposite direction to the incident wave in the
primordial problem. In more detail the reciprocity ap-
proach will be discussed in forthcoming papers.
We also present an illustration of other anomalous ef-
fects relating to the interface of metal and isotropic dielec-
tric (vacuum, for simplicity). It is convenient to consider
the effects in terms of the dimensionless normal compo-
nent β of the corresponding diffraction order. This quantity
can be pure real or pure imaginary belonging to positive
half-axis for both cases. The point effβ = −ξ in the β plane,
Fig. 2, corresponds to the relating diffraction order pole
caused by the surface plasmon polariton mode.
Conclusion
It is shown that the diffraction of TM polarized wave at
a high reflecting gratings under grazing incidence can re-
sult in deep suppression of the specular reflection accom-
panied by considerable redirection of the incoming energy
to other propagating diffracted waves. This phenomenon
becomes more pronounced at low temperatures due to
small dissipation. The detailed theoretical analysis of the
problem is presented on the basis of appropriate analytical
approach. The diffraction anomaly considered in the paper
is of general character and can take place for other wave
types under appropriate conditions (high contrast of the
adjacent media properties). In particular, it can hold at the
interface of ordinary dielectric media and for left-handed
media as well. Analogous anomaly does exist and is well
expressed for magnetic high-contrast media interface for
TE polarization.
* Specifically, such condition holds within approximation of ideal metal that is valid in the long wavelength region (low frequency).
Fig. 2. Beta-plane for some diffraction order. Only vicinity of the
corresponding Rayleigh point (that is of main interest in view of
the diffraction anomalies) is shown. With the change of the pa-
rameters of the problem, the β value can be either pure real posi-
tive (propagating wave), or pure imaginary (evanescent wave).
These cases are separated by the Rayleigh (branch) point, R ,
0β = . Other characteristic points, Im( )SPP eiβ = − ξ and
GA eβ = ξ relating to SPP resonance and to the grazing anomaly
(GA) are shown by circles. Here eξ is shortened form of effξ . If
β corresponds to the incident wave, then it is pure real and only
GA point is actually of interest.
618 Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 5
Electromagnetic grazing anomalies. Energy flux extrema
Acknowledgment
The author wishes to acknowledge the constructive
comments by I.S. Spevak.
_______
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(1994).
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14. L.D. Landau and E.M. Lifshits, Electrodynamics of
Continuous Media, Pergamon, Oxford (1977).
15. E.K. Popov, L.B. Mashev, and E.G. Loewen, Appl. Opt. 28,
970 (1989).
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Kuzmenko, and A.V. Kats, Appl. Phys. Lett. 106, 261602
(2015).
17. R.J. Potton, Rep. Prog. Phys. 67, 717 (2004).
18. A.A. Kuzmenko and A.V. Kats, Intern, Young Scientists
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2015, Dnipropetrovsk, Ukraine.
___________________________
Електромагнітні аномалії при ковзному
розповсюдженні. Екстремуми потоків енергії
О.В. Кац
Ретельно проаналізовано дифракцію електромагнітних хвиль
на поверхнях з періодичними структурами, що супроводжується
великими аномальними ефектами. Розглянуто межі поділу
середовищ з високим контрастом властивостей. Дослідження
обмежено випадком ТМ поляризації хвилі, що падає на межу
(магнітне поле перпендикулярне до площини падіння), та
найпростішою геометрією, коли площина падіння ортогона-
льна до штрихів гратки. Найбільшу увагу приділено макси-
мам та мінімумам густини потока енергії, що супроводжують
ковзне розповсюдження в деякому дифракційному порядку.
Обговорюється зв’язок з іншими аномаліями, як релеївськи-
ми, так і резонансними.
Ключові слова: дифракція, хвиля, аномалія, резонанс, гратка,
потік.
Электромагнитные аномалии при скользящем
распространении. Экстремумы потоков энергии
A.В. Кац
Детально проанализирована дифракция электромагнит-
ных волн на поверхностях с периодическими структурами,
сопровождаемая сильными аномальными эффектами. Рас-
сматриваются границы раздела сред, отличающихся высоким
контрастом свойств. Исследование ограничено случаем ТМ
поляризации падающей на границу волны (магнитное поле
ортогонально плоскости падения) и простейшей геометрией,
когда плоскость падения ортогональна штрихам решетки.
Наибольшее внимание уделено максимумам и минимумам
плотности потока энергии, сопровождающими скользящее
распространение в некотором дифракционном порядке. Об-
суждается связь с другими аномалиями, как рэлеевскими, так
и резонансными.
Ключевые слова: дифракция, волна, аномалия, резонанс,
решетка, поток.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2019, v. 45, No. 5 619
https://doi.org/10.1080/14786440209462857
https://doi.org/10.1080/14786440709463661
https://doi.org/10.1098/rspa.1907.0051
https://doi.org/10.1098/rspa.1907.0051
https://doi.org/10.1364/JOSA.31.000213
https://doi.org/10.1364/AO.4.001275
https://doi.org/10.1103/PhysRev.48.928
https://doi.org/10.1080/02726340801921403
http://www.tandfonline.com/toc/uemg20/28/3
https://doi.org/10.1364/OE.17.017102
https://doi.org/10.1007/BF01038021
https://doi.org/10.1007/BF01038021
https://doi.org/10.1103/PhysRevB.65.195406
https://doi.org/10.1364/AO.28.000970
https://doi.org/10.1063/1.4923419
https://doi.org/10.1088/0034-4885/67/5/R03
Introduction
Grazing incidence anomaly
Energy flux extremes
Conclusion
Acknowledgment
|