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BREWSTER-LIKE REFLECTIONLESS TRANSMISSION OF EVANESCENT WAVES THROUGH A PLANE TWO-MEDIA
INTERFACE
Ignatov A.I. * 1, Merzlikin A.M.1
1 Institute for Theoretical and Applied Electromagnetics RAS, Moscow, Russia
Received: 11.04.2023 Accepted: 16.05.2023 Published: 31.05.2023
Abstract
We propose a Brewster-like reflectionless transmission effect of evanescent waves: no evanescent reflected waves are observed at evanescent wave incidence on a plane interface of two media. Similar to the well-known Brewster effect for propagating waves, the proposed Brewster-like effect for evanescent waves occurs when surface impedances of waves in the neighboring media are equal. Contrary to the known Brewster effect for complex incidence angles related to the existence of some surface wave (when surface impedances are equal for evanescent waves to both sides of the two-media interface decaying in the directions away from the interface), the proposed Brewster-like reflectionless transmission corresponds to the surface impedances equality for evanescent waves decaying in the same direction. It was shown that the proposed Brewster-like effect for evanescent waves can occur for both (p- and s-) polarizations and only for magnetic media (^ = 1 for at least one of the two neighboring media). Based on the proposed Brewster-like reflectionless transmission of evanescent waves, we suggest a method of a totally-reflected-wave's phase patterning with a pattern vanishing exactly at Brewster's angle.
Keywords: Brewster effect, evanescent wave, near field, magnetodielectric, surface wave EDN RRYJEX
1. Introduction
The famous Brewster effect has been known for more than two centuries in optics of dielectric media [1]. This effect is one of the basic ones in polarization optics and it has numerous applications, e.g., in polarizers, glare filters, gas lasers, for broadband angular selectivity, for avoiding unwanted light reflection in holography and microscopy [2-7].
Due to the rapid development of electrodynamics of metamaterials (including optics of metamaterials with non-unit magnetic permeability [8-11]) in the last years, the Brewster effect has been intensively studied for magnetodielectric media interfaces [12-19]. For example, the Brewster effect was predicted for s-polarized waves on an interface of two magnetodielectric media with positive refractive indices [12,13,19], for either s- or p-polarized waves on an interface of media with positive and negative refractive indices [18]. In addition, it was shown that under some conditions the Brewster angle at a two-media interface is absent for both polarizations [15] or under some other conditions it exists simultaneously for both polarizations [15].
The extension of many effects formerly known for propagating waves to the case of evanescent optical waves (near field) has been one of the important trends in modern optics (nanooptics) in the last two decades. Thus, a near-field optical microscope [20], near-field optical tweezers [21], near-field molecular spectroscopy [22,23], and near-field holography [24] have been developed. In the current article, we consider transmission of evanescent waves through systems of magnetodielectric layers and, in particular, analyze an analog of the Brewster effect for evanescent waves when no evanescent reflected waves are observed at evanescent wave incidence on a plane interface between two media (below we will refer to this effect as the Brewster-like reflectionless transmission of evanescent waves).
First, we recollect features of the Brewster effect for propagating waves in magnetodielectric layers. Then, we analyze if a Brewster-like reflectionless transmission of evanescent waves exists. Finally, we consider
* ignatovtoha@gmail.com
4
©2023 HTro PAH
a possible application of the Brewster-like reflectionless transmission of evanescent waves.
2. Theory. Description of the Brewster effect for propagating waves in terms of surface impedances
First, let us recollect features of the Brewster effect for propagating waves. The well-known Brewster effect on an interface of two dielectrics proves itself as zero reflection of a p-polarized propagating plane wave incident on this interface at a certain angle (Brewster's angle). Similarly, the variations of the Brewster effect at interfaces of magnetodielectric media (either for p- or for s-polarization) studied in [13-15,18] show themselves in zero reflection of propagating plane waves incident on these interfaces at certain angles. This zero reflection, corresponding to Brewster's angle, meets the equality condition of surface impedances Z of waves in neighboring magnetodielectrics (read [25] on surface impedance). Indeed, let us consider a plane interface of two half-spaces filled with magnetodielectrics with permittivity and permeability ey and and with e2 and (see Fig. 1).
Fig. 1 - Transmittance of a plane wave through an interface between magnetodielectric with dielectric permittivity e1 and magnetic permeability and magnetodielectric with e2 and
By definition, surface impedance Z of a plane wave incident on a plane surface is the proportionality coefficient between the electric field component Et tangent to this surface and the tangent component Ht of the magnetic field of this wave:
Et = Z [Ht x n], (1)
where n is a normal vector to the surface [25]. Thus, surface impedance is a value determined both by the plane wave and by the surface considered. Surface impedance characterizes a single (with definite wave vector k and definite polarization s- or p-) wave incident on a plane surface. Note that Z of a reflected wave has the opposite sign compared to Z of the incident wave because Ht changes its sign on reflection concerning the xz-plane whereas Et remains unchanged. The Maxwell's boundary conditions are known to require the continuity of electric and magnetic field tangent components at a media interface. Therefore, the equality of the surface impedance of a wave in one medium to the surface impedance of a wave in another - neighboring - medium means that only one wave (either incident or transmitted) in each medium is needed to meet the boundary conditions on the media's common interface. In other words, when the surface impedance of a wave incident on an interface from one medium equals to the surface impedance of the transmitted wave in another medium to the other side of the interface, the reflected wave is absent.
Let us consider the cases of p- and s-polarized waves in more details and derive expressions for surface impedances in magnetodielectric medium. We consider a plane wave with the wave vector k1 incident on a plane interface parallel to the coordinate plane xz (see Fig. 1). The plane of incidence is parallel to the coordinate plane xy and the tangent component of the wave vector equals to kx (this component remains the same for waves in both sides from the interface). First, let us consider p-polarized waves with magnetic fields parallel to the z-axis. The z-component of the magnetic field is Hz = Az exp (ik1yy + ikxx) (here and further we omit the factor exp(-iut)), where k1y = \Jkoe1^1 — is the y-component of the wave vector k1 in the first medium, Az is the magnetic field amplitude and k0 is the wave number in vacuum. From the Maxwell equations, Ex = —(dHz/dy)/ik0e1 = —k1yHz/k0e1. Therefore, the surface impedance of the wave in the first
medium with £i and pi is
z = Ex = _ Ex = kv^ = ^Jkle^i - kl (2)
ip ~ [H x ey1 Hz k0£i k0£i '
The surface impedances equality condition
\Jk0£ipi - kl!ko£i = Zip = Zip = ^kleipi - kl Iko£i
(3)
gives
(kx/ko)l = £l£2 (ElVl - £2MI)/{£1 - 4) ■ (4)
'P
l
In the case of nonmagnetic media, this condition has the form of (kx/k0) = e1el/(e1 + £2) and transforms into the well-known formula tana1 _ (kx/k0) j\J £1 — (kx/k0)2 = yf^Jel, where a1 is the incidence angle of a wave in the medium with dielectric permittivity £\.
By analogy, we consider an s-polarized wave. The electric field is directed along the z-axis and equals to Ez = Az exp (ik1yy + ikxx), where Az is the electric field amplitude. From the Maxwell equations, Hx = (dEz/dy)/ik0p1 = kiyEz/k0p1. Therefore, the surface impedance of the wave in the medium with £1 and is
^ _ Ez = E^ = k0^i = k0^i (5)
1s _ [Hx x ey]z Hx k\y ■s/klEi^i — kl' and the surface impedances equality condition Z1s = Zis gives the formula for Brewster's angle
(kx/k0fs = (£2^1 — £iM2)/(¡A — nl). (6)
Even though the Brewster effect in magnetodielectric media was described and studied more than forty years ago [19], the case when there are only waves evanescent in the y-direction in neighboring layers (i.e. the "complex incidence angle" case when kl > k'^£1p1 and kx > kl£2^2) has been analyzed incompletely. In particular, the case of surface impedances equality for evanescent waves decaying in the same direction (e.g. in the +y-direction) in two neighboring media has not been considered yet. On the contrary, the case of surface impedances equality for evanescent waves to both sides of the two-media interface decaying in the directions away from the interface is well-studied and is commonly related to as the Brewster condition at complex incidence angles [26]. The last case corresponds to a surface wave running along the two-media interface (e.g. surface plasmon-polariton, Zenneck wave etc.) [26,27].
In the present paper, we carefully analyze the case of an evanescent wave incidence on a two magnetodielectric media interface and find out whether a kind of a Brewster-like reflectionless transmission exists in this case of evanescent waves.
3. Results. The Brewster effect for evanescent waves
Let us consider a medium with £1 and (we consider only real-valued dielectric permittivities and magnetic permeabilities in this paper). Let a p-polarized plane wave evanescent in the y-direction with kl > kl£1p1 propagate in this medium. In this case, magnetic field Hz = Az exp(—K1yy + ikxx) and the x-component of the electric field Ex = —(dHz/dy)/ik0£1 = K1yHz/ik0£1 _ —Z1pHz, where K1y = \Jkl — kl£1p1 > 0 and
iko£1. (7)
Zip = -Kiy/iko£i = - kOeipi Ii
By analogy, for an evanescent wave decaying in the y-direction with kl > k'^^p2 in the medium with e2 and Hz = Az exp (—k2yy + ikxx), and Ex _ —Z2pHz = k2yHz/ik0£2, where k2y = \Jkl — k^^ > 0 and Az is the magnetic field amplitude in the second medium. Therefore, in the case of Z1p = Z2p, it can be possible that only one above-mentioned evanescent wave decaying in the y-direction exists in each medium (see Fig. 2) and Maxwell's boundary conditions at the media interface are met.
Let us analyze in details the condition Z1p = Z2p under the assumptions £1,£2,p1,^2 = 0, kl > kl£1p1, kl > kl£2p2, K1y > 0 and «2y > 0. From this condition we obtain £\_£2 > 0 (contrary to the surface plasmon-polariton existence condition when n1y = — \Jkl — k'^£1p1 < 0 and £1£2 < 0). The Brewster condition Z1p = Z2p in this case is
(kx/kofp = £1£2 (£1^2 — £2^1)/i£\ — £D > 0 (8)
Fig. 2 - Dependencies of magnetic field ^-component magnitude (in logarithmic scale) on the ^-coordinate in the case of a p-polarized plane wave with kx = fcoy85/12 incident in the rcy-plane from the left. The black curve corresponds to the case of e2 = 1.0 and y2 = 7.0, the solid red curve - to the case of e2 = 7.0 and y2 = 1.0. In both cases, e = 9.0, y = 1.0, e1 = 5.0 and y1 = 1.0. At y < 0 there is the medium with e and y, at 0 < y < 0.1A the layer with e1 and y1 is placed, and at y > 0.1 A there is the half-space with e2 and y2. The dashed red line is a straight-line segment
connecting points of the solid red curve
and is the same as given above (4) for non-evanescent waves. Then, assuming > ez, without loss of generality, we have e1^2 > £2^1. In turn, from the conditions (kx/k0)2 > £1^1 and (kx/k0)2 > £2^2 we have £2^2 > (both for 0 < £2 < £1 and for £1 <e2 < 0). These conditions
£i£z > 0, £\ > £2, £1^2 > £2^1 and £2^2 > £1^1 (9)
However, they require ^2 = Therefore, the Brewster-like reflectionless transmission for evanescent p-polarized waves cannot be realized for nonmagnetic media.
By analogy, we consider the case of s-polarized waves evanescently decaying in the y-direction with k2x > and k2 > k"^£2^2. In the first medium the electric field is Ez = Az exp(—K1yy + ikxx) and the
x-component of the magnetic field is Hx = Ez/Z1s = —K1yEz/ik0^1, where
Z1S = -iko^1/K1y = -ik0^1 !\Jk2x - .
(10)
It follows from the surface impedances equality condition —iko^1/n1y = Z1s = Z2s = —ikc^2/K2y and with K1y > 0 and n2y > 0 that > 0 and the Brewster condition Z1s = Z2s is
(kx/k0)2s = ^1^2 (£2^1 - £1^2)/(p2 - > 0, (11)
having the same view as for the case of non-evanescent s-polarized waves (6). Therefore, for the last inequality along with conditions k2 > and k2x > k"^£2^2 to be compatible, the following conditions must be
satisfied (without loss of generality):
^1^2 > 0, £1^1 > £2^2 and £1^2 > £2^1 at ^>2 > y^. (12)
Obviously, these inequalities require £2 = e1. As in the case of p-polarized evanescent waves, the Brewster-like reflectionless transmission for s-polarized evanescent waves does not occur in nonmagnetic media.
By a particular example let us illustrate the fact that a reflected evanescent wave is absent in the case of equal surface impedances. We consider a layer of the thickness d = 0.1A with £1 = 5.0 and y1 = 1.0 between a half-space with £2 = 1.0 and y2 = 7.0 (to the right of the layer) and a half-space with e = 9.0 and y =1.0 (to the left of the layer) (see Fig. 2). The specified values of £1,£2,y1,y2 meet the conditions (9) when the Brewster-like reflectionless transmission for p-polarized evanescent waves can occur. Let a p-polarized wave with the x-component of the wave vector kx be incident from the left half-space. The Brewster-like reflectionless transmission through the interface between the layer with £1 and y1 and the right half-space with £2 and y2 is realized at (kx/k0)2 = £1£2 (£1y2 — £2y1)/(£2 — = 85/12 (Eq. 4). In this case, (kx/k0)2 > £1y1 = 5,
(kx/k0)p > e2y2 = 7 and (kx/k0)2p < ey = 9. As shown in Fig. 2 (black curve), under the condition of (kx/koYp = 85/12, only one evanescent wave (decaying in the y-direction) exists in the layer. On the contrary, if e2 = 7.0 and y2 = 1.0, the Brewster condition is not met at (kx/k0)2 = 85/12, and the field in the layer is a sum of two evanescent waves, one of which decays in the y-direction and the other grows (see the solid red line in Fig. 2 which is not a straight-line segment in the layer).
Thus, the proposed Brewster-like effect manifests itself in the absence of a reflected evanescent wave. On the other hand, contrary to the Brewster effect for propagating waves, the power reflection coefficient is a unity in some range of values of kx around Brewster's angle because of total internal reflection.
4. Discussion. Reflected wave phase tailoring
One of the actively developing areas of modern electromagnetics is electrodynamics of metasurfaces -surfaces structured on a subwavelength scale in a certain way. One of the applications of metasurfaces is manipulating the phase and amplitude of reflected waves along a metasurface plane [28,29]. The described Brewster-like reflectionless transmission of evanescent waves enables manipulating the phase of a wave totally reflected from a layered surface depending on the thicknesses of layers. In this case, the reflected wave phase may depend on the layers' thicknesses only at incidence angles that do not meet the Brewster condition ((4) or (6)) whereas the phase of the reflected wave is independent of the layers' thicknesses if the incident wave satisfies the Brewster condition.
To illustrate this effect, let us consider a plane magnetodielectric layer between two half-spaces of magnetodielectrics (see Fig. 3) again. Let a p-polarized plane wave with x-component kx of the wave vector be incident from the left half-space.
Fig. 3 - The structure demonstrating manipulation of a reflected wave phase. A plane wave with ^-component kx of the wave vector is incident from the left half-space with e and y on a layer with e1 and
If the Brewster condition (4) is met at the interface of the layer and the right half-space, a reflected evanescent wave (growing in the y-direction) is absent in the layer. Therefore, the characteristics of the wave reflected in the left half-space is independent of the layer thickness. Indeed, this can be seen in Fig. 4, where the phase of the reflectance r in the left half-space is shown as a function of kx/k0 for different layer thicknesses d and for e2 = 1.0 and y2 = 7.0 (the r magnitude equals to 1 at the Brewster condition because (kx/k0fp > e2y2). However, if the Brewster condition (4) is not met there is a reflected evanescent wave in the layer growing in the y-direction, which leads to the dependence of r phase on d.
Based on this effect of reflectance phase independence on d at the Brewster condition (4) and reflectance phase dependence on d for other incidence angles, we could tailor the reflected field phase distribution along the reflecting surface (by setting the distribution of d along the surface while the interface between the medium with e and y and the medium with e1 and remains plane) so that this distribution depends on the incidence angle kx/k0. At Brewster's incidence angle, the reflectance from the surface has a uniform phase distribution.
5. Conclusions
In the paper we proposed the Brewster-like reflectionless transmission of evanescent waves through a two-media interface. If an evanescent wave is incident on a plane interface between two media (in the case
Fig. 4 - Phase of the reflectance r in the left half-space (see Fig. 2 or Fig. 3) as functions of kx/k0. The black curve corresponds to the layer thickness d = 0, the red curve - d = 0.05A, the blue curve - d = 0.1A
of incident wave decaying along the normal to the interface in the direction to the interface), the reflected evanescent wave, decaying in the direction opposite to that of the incident wave, may be absent. Similar to the known Brewster effect for propagating waves, the proposed Brewster-like effect for evanescent waves occurs when surface impedances of waves in the neighboring media are equal. Contrary to the known Brewster effect for complex incidence angles related to existence of some surface wave (when surface impedances are equal for evanescent waves to both sides of the two-media interface decaying in the directions away from the interface), the proposed Brewster-like reflectionless transmission corresponds to the surface impedances equality for evanescent waves decaying in the same direction.
We considered both the cases of p- and s-polarized waves. It was shown that the Brewster-like reflectionless transmission for evanescent waves could occur for any of these polarizations. In the case of each of these polarizations, it is necessary that at least one of the two neighboring media has magnetic properties (i.e. has magnetic permeability different from unity) for the reflectionless transmission to take place.
Based on the Brewster-like reflectionless transmission effect for evanescent waves, we proposed a method of tailoring of totally-reflected-wave phase distribution in the plane of a reflecting surface. If we take such system of plane layers as a reflecting system, that the Brewster condition for evanescent waves is met at the interfaces between these layers, the reflected wave phase will be independent of the number and the thicknesses of these layers at the Brewster condition. However, at other angles the reflected wave's phase will depend on the characteristics of this layered system. This fact could be used for the development of a structured layered system with layers' thicknesses varying along the layers plane, thus, providing a desired totally-reflected-wave phase distribution. This distribution would be observed only for incidence angles different from Brewster's angle, whereas the phase distribution would be uniform at Brewster's angle.
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БЕЗОТРАЖАТЕЛЬНОЕ ПРОХОЖДЕНИЕ (ЭФФЕКТ БРЮСТЕРА)
ЭВАНЕСЦЕНТНЫХ ВОЛН ЧЕРЕЗ ПЛОСКУЮ ГРАНИЦУ ДВУХ СРЕД
Игнатов А.И.1*, Мерзликин А.М.1
1 Федеральное государственное бюджетное учреждение науки Институт теоретической и прикладной электродинамики Российской академии наук, Москва, Россия
* ignatovtoha@gmail.com Аннотация
Мы описываем и теоретически анализируем эффект безотражательного прохождения эванесцентных волн через границу сред, подобный эффекту Брюстера для распространяющихся волн. Данный эффект выражается в том, что если на плоскую границу раздела двух сред падает эванесцентная волна (когда падающая волна затухает в направлении по нормали к границе раздела двух сред), то отраженная эванесцентная волна, затухающая в направлении, противоположном направлению затухания падающей волны, отсутствует. Назовем данный эффект безотражательным прохождением Брюстера для эванесцентных волн. Подобно известному эффекту Брюстера для распространяющихся волн, безотражательное прохождение Брюстера для эванесцентных волн возникает, когда поверхностные импедансы волн в граничащих средах равны. Известный эффект Брюстера для комплексного угла падения связан с существованием поверхностной волны, когда поверхностные импедансы равны для эванесцентных волн по обе стороны границы, затухающих в направлениях от границы двух сред. При этом описываемое нами безотражательное прохождение Брюстера соответствует равенству поверхностных импедансов для эванесцентных волн, затухающих в одном и том же направлении. Для плоских границ двух сред было теоретически показано, что эффект Брюстера для эванесцентных волн может иметь место как для р-, так и для s- поляризации, причем только если хотя бы одна из двух граничащих сред обладает магнитными свойствами (^ = 1). На основе рассмотренного эффекта безотражательного прохождения Брюстера предложен метод настройки фазового распределения волны, испытавшей полное отражение, в плоскости отражающей поверхности, притом что данное фазовое распределение превращается в однородное при условии безотражательного прохождения Брюстера.
Ключевые слова: эффект Брюстера, эванесцентная волна, ближнее поле, магнитоди-электрик, поверхностная волна
