JEOS RP ISSN01 | Page 217

212

J. Eur. Opt. Society-Rapid Publ. 21, 21( 2025)

The phase variations due to expði ~ k R ~ rÞ are the same as for a homogeneous wave with wavevector ~ k R. The constant amplitude term(~ E 0) also appears in( A. 5) as before in( A. 3), but now in( A. 5) is multiplied by a real exponential term given by expð� ~ k I ~ rÞ. Therefore, by introducing the complex wavevector( A. 4) we have defined an inhomogeneous monochromatic wave given by expression( A. 5). In other words, an inhomogeneous plane

wave is defined by the set of the three vectors E~ 0; k ~ R; k ~ I( i. e. E~ 0; Re ~ k and Im ~

k) instead of the two vectors E~ 0 and ~ k which defined a homogeneous plane wave.

When we study plane waves at an interface( within the monochromatic assumption) the calculation of the couples of

values E~ 0 and ~ k or sets of three values E~ 0; Re ~ k and Im ~

k that

define the three coexisting waves( incident, reflected, and transmitted), fully solves the physical problem. Indeed, to directly know the instantaneous values of all the fields, it is only necessary to use( A. 3) together with( A. 1) for each wave.

Wave equation in isotropic absorbing materials

According to Maxwell equations, the wave equation for the spatial part of the electric field in an isotropic medium with relative magnetic permittivity l and complex refractive index n~ ¼ n þ ij( or complex generalized permittivity e, withle =( n + ij) 2), is:

Eð ~ ~ rÞþle x 2

Eð ~ ~ rÞ ¼0: ðA: 6Þ c

The same equation is valid for the magnetic field. It is straightforward to check that the monochromatic inhomogeneous plane waves in the form

E ~ ðÞ¼ ~ r E~ 0 exp i ~ k ~ r; ðA: 7Þ with ~ k ¼ Re ~ k þ iIm ~ k, fulfill this equation provided that

2 ~ k ¼ ~ k ~ k ¼ k x ð

~ k ~ E 0 ¼ 0; ~ k ~ H 0 ¼ 0; ðA: 8Þ

Þ 2 þ ðk y

Þ 2 þ ðk z Þ 2 ¼ le x2

¼ lek 2

0 ¼ P þ iQ ¼ C ðC 2 C; P; Q 2 RÞ; x

c ¼ k 0: ðA: 9Þ Equation( A. 9) is the dispersion equation. Note that this dispersion equation does not contain the quantity ~ k 2 ¼ ~ k ~ k ð2 RÞ but the

2 value ~ k ¼ ~ k ~ k ¼ lek

2

0ð2 CÞ instead. The condition( A. 9) on

~ k ¼ Re ~ k þ iIm ~ k implies:

Re ~

k 2 � Im ~

k 2 � ¼ k 2

0 n2 � j 2 ¼ P ð2 RÞ ðA: 10Þ

c 2

Evanescent waves

One particular case will also be relevant for us. When the medium is transparent( j = 0), the wave equation( A. 6) still allows inhomogeneous waves as a specific solution: according to expression( A. 11) it is only necessary that cos( u) = 0, i. e., Re ~ k? Im ~

k. This kind of inhomogeneous waves are the well-known“ evanescent” waves.

In summary, the wave equation in absorbing materials leads to two conditions( A. 10) and( A. 11) to be fulfilled by the three real-

valued quantities Re ~

k; Im ~ k and the angle u. Itisimportant to grasp the physical meaning of the results we have just obtained: inside an absorbing medium, the complex expressions( A. 7), with the restrictions( A. 8)–( A. 11), represent an electromagnetic field that fulfills the Maxwell equations in the medium. This means that the real physical fields given by( A. 3) and( A. 1) are valid solutions of Maxwell equations, that we name inhomogeneous plane waves since the phase

propagates like a plane wave in one direction( defined by Re ~ k) whiledecaysinanotherdirection( defined by Im ~

k). As a particular case, the results applied to transparent media explain the existence of evanescent waves.

Plane waves reflected and refracted at an interface: computing the wavevectors

Consider the coordinate axes given by Figure 1. The interface is the( X-Y) plane and the plane of incidence is the( X-Z) plane. Dealing only with isotropic materials, the symmetry of the configuration guarantees that the incident, reflected and transmitted wavevectors remain in the X-Z plane. Since the emergent medium is( in our general case) absorbing, we will assume an inhomogeneous wave there. Due to the isotropy of the materials and the symmetry

of the configuration, the direction of the vector Im ~ k which represents the direction of the decay of the transmitted wave( if decay exists), has to point in the direction of our positive Z-axis.

For the two media, incident(“ i”) and transmitted(“ t”), assume that the optical constants are, respectively,

i: n 2 i ¼ e i; l i 1 ðall 2 RÞ; t: ð n t þ ij t

Þ 2 ¼ e t ð2 CÞ; l t 1 ð2 RÞ ðA: 12Þ

For an incidence angle h in the incidence plane( X-Z), we need to calculate the two components( X and Z) of all the wavevectors( all of them have no Y component). This can be done by taking into account that their tangential components( here the X coordinates) have to be the same for the incident, reflected, and transmitted wavevectors.

There are three plane waves in contact at the interface: the incident and reflected ones contained in medium“ i”( with refractive index n i) and with wavevectors ðk x i; 0; kz i Þ and ðkx r; 0; kz r Þ respectively, and the transmitted wave inside medium“ t”( with index n t + ij t), and with wavevector ðk x t; 0; kz t

Þ. Because of the equality of the X components of all the wavevectors, using the same notation as in [ 10 ] wewillwritealltheX components of all the wavevectors in terms of the X component of the incident wavevector k x i:

� ~ ki ¼ k x i; 0; � kz i; ~ kr ¼ k x i; 0; � kz r; ~ kt ¼ k x i; 0; kz t;

k x i; kz i; kz r 2 R; kz t 2 C: ðA: 13Þ

Thus, for the“ i” medium we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k x i

¼ k x r ¼ k 0n i sin ðÞ h; k z i

¼�k z r ¼ � ðk 0 n i Þ 2 � k x 2 i ð2 RÞ: ðA: 14Þ