Reflected energy flux anomaly under grazing incidence: the Brewster angle analogy

The paper presents thorough theoretical and numerical analysis of the anomalies accompanying light diffraction on periodical structures (gratings). We have developed appropriate theoretical approach allowing to consider strong anomalous effects. Obtained results are presented in the form of analytical expressions for the quntities of interest, both diffracted field amplitudes and the outgoing waves energy fluxes. It is proved existence of the fluxes extrema at the specific grazing angle of incidenceб or wavelength. Namely, the specular reflection can be suppressed even for rather shallow gratings up to approximately total suppression.This effect is accompanied by essential energy redistribution between all outgoing waves depending on the grating profile. It is of essence that the energy maxima exist in all nonspecular diffraction orders at the same point (angle, wavelength) as the minimal specular reflectivity. For small period gratings, such that there do not exist other outgoing waves except the specular one, the reflectance minimum is attended by approximately total absorption of the incident radiation. Thus, we show that the grazing anomaly (GA) can be accompanied by redirection of the incident wave energy into nonspecular diffraction channels and into absorption. The results are applicable in the wide spectral region, from visible and near-infrared to terahertz and high-frequency regions for metals and semiconductors with high permittivity. The anomaly considered is well expressed for high electromagnetic contrast of the adjacent media, say, air and metal or semiconductor. Then the high contrast is due to the high value of the metal/semiconductor dielectric permittivity  , 1  , and the anomaly corresponds to incidence of TM polarized wave. It is shown that the grazing anomaly (GA) is of rather general type and can take place if other than the specular diffraction order experiencies grazing propagation also. This property follows from the results obtained by strict application of the optical reciprocity theorem to the geometry under consideration. The specific case of harmonic relief grating is discussed in detail. It is demomstrated existence of the characteristic inclination, cr a , of the relief inclinatuion for the grating period comparable with the incident radiation wavelength, 1 cr a  , where  stays for the surface impedance, 1    . The condition cr a a , or greater, corresponds to highly expressed GA. The theoretical results are illustrated by numerical applications to gratings on Cu\vacuum (air) interface in THz region. The results obtained can be simply transferred to the TE polarized waves. For this we have to consider the adjacent media with high contrast magnetic properties, i.e., high value of the magnetic permeability  , 1  . This case is of high interest for nowaday applications in nanophotonics and metamaterials development. As compared with other anomalies GA is attributed to the resonance-type behaviour of the energy flux, not wave amplitudes, the latter change monotonically within this anomaly contrary to the well known Rayleigh and resonance anomalies, where the wave amplitude experiences fast

The paper presents thorough theoretical and numerical analysis of the anomalies accompanying light diffraction on periodical structures (gratings). We have developed appropriate theoretical approach allowing to consider strong anomalous effects. Obtained results are presented in the form of analytical expressions for the quntities of interest, both diffracted field amplitudes and the outgoing waves energy fluxes. It is proved existence of the fluxes extrema at the specific grazing angle of incidenceб or wavelength. Namely, the specular reflection can be suppressed even for rather shallow gratings up to approximately total suppression.This effect is accompanied by essential energy redistribution between all outgoing waves depending on the grating profile. It is of essence that the energy maxima exist in all nonspecular diffraction orders at the same point (angle, wavelength) as the minimal specular reflectivity. For small period gratings, such that there do not exist other outgoing waves except the specular one, the reflectance minimum is attended by approximately total absorption of the incident radiation. Thus, we show that the grazing anomaly (GA) can be accompanied by redirection of the incident wave energy into nonspecular diffraction channels and into absorption. The results are applicable in the wide spectral region, from visible and near-infrared to terahertz and high-frequency regions for metals and semiconductors with high permittivity.
The anomaly considered is well expressed for high electromagnetic contrast of the adjacent media, say, air and metal or semiconductor. Then the high contrast is due to the high value of the metal/semiconductor dielectric permittivity  , 1  , and the anomaly corresponds to incidence of TM polarized wave. It is shown that the grazing anomaly (GA) is of rather general type and can take place if other than the specular diffraction order experiencies grazing propagation also. This property follows from the results obtained by strict application of the optical reciprocity theorem to the geometry under consideration. The specific case of harmonic relief grating is discussed in detail. It is demomstrated existence of the characteristic inclination, cr a , of the relief inclinatuion for the grating period comparable with the incident radiation wavelength, where  stays for the surface impedance, 1   . The condition cr aa , or greater, corresponds to highly expressed GA. The theoretical results are illustrated by numerical applications to gratings on Cu\vacuum (air) interface in THz region.
Introduction. Classification of anomalies. The pioneering work on anomalies in light diffraction on metal gratings was performed by R. Wood in 1902 [1], while the first physical interpretation of some of the observed peculiarities was presented in1907 by Lord Rayleigh [2]. The latter associated them with the transition from the outgoing wave to the evanescent (decaying) one and vice versa in different diffracted orders. However, such explanation was insufficient and other possibility was proposed by U. Fano [3] who attributed some of Wood anomalies to the resonance excitation of the surface electromagnetic waves (surface plasmon polaritons, SPP, [4].) at the metal-air interface. Also, Wood discovered one more anomaly related to the unexpectedly high intensity of the grazing outgoing wave [5]. Below the anomalies attributed to the grazing propagating waves are referred to as GA (Grazing Anomaly), see [6][7][8]. Up to now, Wood anomalies are widely discussed due to their perspective role in nanophysics and, particularly, nanophotonics.
Existing for an arbitrary interface and light polarization, the Rayleigh anomaly is much more pronounced for the high-dielectric contrast interface, for TM (transverse magnetic) polarization and nonmagnetic media. Below, we restrict the consideration to the nonmagnetic case only. The results for the magnetic case can be obtained by replacing the dielectric permittivity,  , with the magnetic permeability,  , and the TM polarization by the TE one and vice versa. It worth mentioning that the resonance anomaly can exist only for such interfaces that support surface electromagnetic waves (SEW) and that GA anomaly is rather universal and is well expressed for high contrast interfaces for TM polarization [6].
Consider briefly 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 interfaces. However, it is more pronounced for metals under TM polarization. At the Rayleigh point the derivative of the diffracted wave intensity with respect to the wavelength or angle of incidence turns infinity. The resonance anomaly is less general because it is caused by existence of well-defined eigenmodes of the interface. For isotropic and nonmagnetic dissipation-free media such surfacelocalized electromagnetic waves do exist under the conditions 0 denote dielectric permittivity of the metal and the adjacent dielectric, respectively. The SPP in-plane wavenumber, frequency of the incident wave, exceeds the wavenumber of the adjacent dielectric volume wave with the same frequency, The square root symbol stays for the main branch, so that where x e is the unit vector directed along the Ox axis. In other words, the diffracted field is given by the Floquet-Fourier expansion, [9,11]. In (3) the sign minus before   n p q stays to satisfy the radiation boundary conditions at z   . Restriction of the outgoing waves (and evanescent decaying ones) within the whole halfspace corresponds to use of the Rayleigh hypothesis, [2], and is not restrictive even for rather deep gratings, see recent discussion in [9,11,12].
If for some specific integer n the condition n Q q holds true, then for the appropriate polarization of this diffracted wave the resonance excitation of SPP takes place. SPP is an evanescent wave so the magnitude of the corresponding diffracted order can exceed that of the incident wave. Specifically, in the simplest geometry, when q is orthogonal to the grating grooves, only TM component of the incident wave can excite the SPP.
We would like to underline that the Rayleigh and the resonance anomalies are related to the specific and rather sharp dependence of the field amplitudes on the wavelength 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 accomplished without a thorough theoretical investigation. This obstacle is caused by the fact that the field amplitude changes monotonically within the anomaly. It can be shown that the corresponding quasiresonance behavior is characteristic for the intensity, not for the field amplitude. The method for considering this and other diffraction anomalies analytically was presented in [8], see also a more detailed consideration in [13][14][15].

Grazing incidence anomaly.
Consider the case of the simplest geometry for TM polarized waves with magnetic field orthogonal to the plane of incidence, so that for the incident wave, i H , and  (4) where n q q ng  . Note, the diffracted field in (4) and below in (5) where the subindex t denotes tangential to the interface component of the corresponding vector,  denotes the surface impedance, and n stays for the unit vector normal to the interface directed into the dielectric. We use Gauss units so that the surface impedance  is dimensionless, and for nonmagnetic media The profile Fourier series expansion is The condition where the matrix of the system, ˆn where sin   ,  denotes the incidence angle, stays for the set of integers.
Consider here the simplest (but of high interest) case of the grazing incidence, 01  That is the specular reflected wave with necessity is the grazing one. The simplest geometry of the problem is such, when only one of the diffracted waves except the specular wave is outgoing from the interface, all other diffraction orders correspond to evanescent waves. This geometry is presented in Fig. 1.
It should be emphasized, that among diffracted waves only the specular reflected one is close to the corresponding Rayleigh point, It is essential that the coefficients In what follows we are dealing with rather smooth and shallow gratings. Under this condition we can restrict the Noteworthy, here the second-order terms are essential if the corresponding Fourier amplitude of the grating, M  , vanishes or is anomalously small. Under this condition, the anomalous effects in M-th diffraction order are small and thus of low interest. Therefore, below we restrict our consideration to the linear term of M U expansion. The main term of the quantity  expansion is the square one, Emphasize here that the results obtained are actually valid for the arbitrary angle of incidence for which all Fig. 1. Grazing  . Only the specular reflected wave can be arbitrary close to the grazing propagation.

Energy flux extremes. Brewster angle analogy.
Expressions (14), (18) one can see that it possesses specific minimal value at the point, Here and below the prime (double prime) denotes the real (imaginary) part of the corresponding quantity. In The specular TC field at this point is as follows, The  dependence on the angle of incidence in terms of the variable  is illustrated in Fig. 2. As it strictly follows from Eqs. (18), (20), (14), and is easy to see from   Note that reflectivity minimum is of rather general character and exists even for TM polarized wave incidence on unmodulated interfaces, 0  , eff   (when 0 h coincides with the corresponding Fresnel reflection coefficient [16].
These properties present strict analogy to the reflectivity minimum from dielectric media existing under Brewster angle incidence [16]. In view of the fact that for 1

 
(which is typical for good metals up to the frequencies of the visible range), the normal to the interface component of the wavevector in the metal half-space prevails essentially the tangential one, so the wave in the metal region can be formally considered as orthogonal to the interface. Consequently, under grazing incidence the reflected from the metal wave is approximately orthogonal to the "transmitted" one as it holds under Brewster angle incidence. The specular reflectivity minimum, Eq. (21), becomes deep for relatively high effective losses, i.e., for eff   comparable to || eff  (see Fig. 2). On the contrary, it approaches unity for vanishing losses, 0 Therefore, the effect of the specular reflection suppression under consideration is attributed to the cumulative (both active and radiative) losses maximum, cf. [8,18]. However, as it is shown below, the point extr   corresponds not only to the specular reflection minimum but results in well expressed maximal nonspecular efficiencies along with the active losses maximum. Evidently, if the only propagating diffracted wave is the specular one, then the grazing minimum is with necessity accompanied by maximal absorption. It is of interest that normalized intensities of the propagating diffraction orders, present strongly nonmonotonic  functions in accordance with the fast dependence of the subsidiary function, It is easy to see that This property is illustrated in Fig. 4, where the incident angle dependence of the minus first diffraction order intensity, 1   , is shown for the geometry of Fig. 1.  The inequality for the solution presented is to be true under rather general conditions, specifically for such  and  values that are far from anomalies related to all diffraction orders except the specular one. If the active losses are absent, then the inequality transforms into the equality. In the specific case of short-period gratings, such that 2   , all diffracted orders except zeroth one with necessity correspond to evanescent waves. Under such conditions the strong specular reflectivity suppression is accompanied by maximal absorption. The energy redistribution between outgoing waves and the dissipation strongly depends on the parameters of the problem, as one can see from the explicit solution.
Here abovementioned is illustrated for the simplest case when, in addition to the specular wave, only one diffracted order corresponds to the propagating (outgoing) wave. It can be realized if 11     , when the minus first order presents propagating wave, 1 0    , and n  with 1, 0 n  are pure imaginary. Specifically, under such condition, illustraterd in Fig. 1 cf. point L in Fig. 6. Evidently, the absorption vanishes if the medium is dissipation free, 0    . Under rather specific conditions max A can be of order unity, that does not describe general case contrary to the statement in [18].

Harmonic grating
For the case shown in Fig. 1 Since specular reflectivity possesses rather expressed minimum, for relatively low active losses the incoming energy is redirected into other propagating waves. The most interesting case that allows obtaining rather strong grazing anomalies presents such one that,

5.Anomalous diffraction points.
We illustrate position of other points corresponding to diffraction anomalies related to the interface of metal and isotropic lossless dielectric (vacuum, for simplicity). It is convenient to consider them in terms of the dimensionless normal component  of the corresponding diffraction order. The point eff   in the  plane, Fig. 2, shows corresponding diffraction order pole caused by the surface plasmon polariton (SPP) mode. Note, the specific value of eff  for a given grating depends on the "resonance" diffraction order, see Eqs. (14), (15). In Fig. 7, only the vicinity of the corresponding Rayleigh point (that is of main interest in view of the diffraction anomalies),

T. Rokhmanova, A.V. Kats
Вісник ХНУ імені В.Н. Каразіна, серія «Фізика», вип. 30, 2019 39 2. It is shown that the diffraction of TM polarized wave at the high reflecting gratings under grazing incidence can result in deep suppression of the specular reflection accompanied by considerable redirection of the incoming energy to other propagating diffracted waves.
3. It is proved that the suppression of the specular reflection for TM polarization at grazing incidence is analogous to that at the Brewster angle.
4. In the case of arbitrary polarized incident wave, only TM component can experience the anomalous properties discussed. Due to the grating shallownes the TE radiation is not affected by the grating and thereby is nearly totally reflected. Thus, strong polarization transformation of specular reflection can occur.
5. It is worth noticing that essential enhancement of the grazing wave for nongrazing incidence is related to the problem under consideration by the reciprocity theorem, [19,20]. For instance, 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 corresponding minus first order is related to the grazing wave propagating in the opposite direction to the incident wave in the primordial problem.