Reflection, Transmission, and Absorption Coefficients ... - Springer Link

ISSN 1064-2269, Journal of Communications Technology and Electronics, 2008, Vol. 53, No. 4, pp. 363–376. © Pleiades Publishing, Inc., 2008. Original Russian Text © I.V. Antonets, L.N. Kotov, V.G. Shavrov, V.I. Shcheglov, 2008, published in Radiotekhnika i Elektronika, 2008, Vol. 53, No. 4 pp. 389–402.

ELECTRODYNAMICS AND WAVE PROPAGATION

Reflection, Transmission, and Absorption Coefficients Calculated for the Oblique Incidence of an Electromagnetic Wave on a Plate I. V. Antonets, L. N. Kotov, V. G. Shavrov, and V. I. Shcheglov Received March 2, 2007

Abstract—The averaging and direct methods are applied to calculate the characteristics of propagation of an electromagnetic wave through a parallel-sided plate with an arbitrary conductance in the case when the wave is obliquely incident on the plate’s plane. On the basis of averaged electrodynamic boundary conditions, simplified expressions for the reflection, transmission, and absorption coefficients are derived for the arbitrary conductance of the plate. It is shown that the averaging method facilitates calculations and yields simplified expressions owing to reduction of the order of the system of equations to be solved and to elimination of exponents and trigonometric functions. The error of the averaging method relative to the error of the direct method is several percent for the incidence angle ranging from 0° to 90° and for a plate as thick as 0.35 of the wavelength in the medium. PACS numbers: 78.20.Ci, 41.20.Jb DOI: 10.1134/S1064226908040013

INTRODUCTION The problem of the incidence of an electromagnetic wave on an interface of two or more media has attracted attention of the researchers for a long time. Even the ancient Greeks knew about the equality of the incidence and reflection angles, and a relationship between a refracted ray and the parameters of material media was established upon the discovery of Snell’s law. The problem on the intensities of reflected and transmitted waves was brilliantly solved by Fresnel and completed by Brewster [1–3]. During the subsequent decades, wave propagation in layered media was comprehensively analyzed [4]. The most consistent direct method of calculation involves independent solution of the wave equations in each medium followed by joining the obtained solutions on medium interfaces. In the case of electromagnetic waves, this method yields a system of algebraic equations whose order equals the double number of medium interfaces. For example, in the presence of two, three, four, etc., interfaces, a fourth-order, sixth-order, eighth-order, etc., system is to be solved. When a wave is obliquely incident on an interface, the problem becomes substantially more complicated [5]. The necessary calculations are extremely cumbersome. Therefore, alternative methods that facilitate the analysis have been developed. These methods include a method that takes into account subsequent rereflections and that involves summation of an infinite geometric progression [3]; the characteristicmatrix method, which involves multiplication of the matrices for individual layers [3]; the impedance

method, which involves reduction of the impedances of individual layers to the input impedance of the entire structure [4]; etc. Being universal and suitable for media with arbitrary parameters and for layers of arbitrary thicknesses, these methods, nevertheless, necessitate intricate calculations. At the same time, recently (especially, owing to progress in nanotechnologies [6]), films whose thicknesses do not exceed the incident wavelength have found numerous applications [7–14]. Experimental investigations of such films with the use of electromagnetic microwaves make it possible to reveal the character of conduction, the features of a structure, and the cluster character of growth; to study the percolation process; and to estimate the free path and concentration of electrons [15–20]. Interpretation of the aforementioned experiments can be substantially simplified under the assumption that the distribution of the wave field across the thickness of a thin film is not periodic and exhibits a variation of a sinusoid fragment that is no longer than a quarter-period. The same assumption is the basis of the averaging method, an efficient calculation technique whose classical version [21] involves the substitution of the linear wave distribution across a film’s thickness for the sinusoidal distribution. In this method, the wave fields inside a film are replaced with the arithmetic means of the field values on the surfaces. As a result, there are only two boundary conditions and the problem reduces to a system of two linear equations.

363

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ANTONETS et al. Region 1

Region 3

Region 2

Eref kref

Epref

Href Reflected wave

kpref Epinc θ2

θ1

Einc

Hinc z

kinc

Etr

Hpref

Htr

θ2 θ3

θ2

kpinc

θ1

ktr

Transmitted wave

Hpinc y

Incident wave

b

a

O x

Fig. 1. Geometry of the problem and the orientation of the vectors of propagating waves.

The averaging method was applied for the first time to the problem on a ferrite-filled waveguide [21], and, later, it was used for dielectric, metal, anisotropic, chiral, and other media [22–28]. A detailed generalization of numerous solved problems can be found, for example, in review [29]. Unfortunately, most studies are restricted to derivation of various boundary conditions and less attention is given to calculation of the reflection, transmission, and absorption coefficients. As a rule, only the extreme cases of a dielectric or highly conducting metal are considered. The limits of applicability of the averaging method in the cases where a film’s thickness is commensurable with the wavelength of the incident radiation have not been analyzed consistently. We have tried to compensate partly for this deficiency [30]. In that study, the incidence of an electromagnetic wave on a parallel-sided plate with an arbitrary conductance is considered. Averaged boundary conditions are derived, and the reflection, transmission, and absorption coefficients are calculated. It is shown that the averaging method substantially simplifies computations and final expressions and that the accuracy of the method is several percent for plates as thick as 0.35 of the wavelength in the medium. The oblique incidence of a wave is not considered in [30]. This study logically continues [30] and is devoted to the analysis of the same problems in the case of the oblique incidence of an electromagnetic wave on a plate’s plane at an arbitrary angle to the normal to this plane. The problem is solved by means of the direct and averaging methods. The results obtained with the use of the two methods are compared.

1. BASIC RELATIONSHIPS 1.1. Geometry of the Problem and the Fields of Propagating Waves Consider a plane electromagnetic wave incident on a parallel-sided plate at an arbitrary angle. The geometry of the problem is displayed in Fig. 1, where three spatial regions (1–3) are shown. The regions are filled with a guiding medium and separated by plane parallel interfaces. Region 2 is a parallel-sided plate, and regions 1 and 3 are half-spaces. The wave is incident from region 1 on the interface between regions 1 and 2, transmitted partly into region 2, and reflected partly into region 1. The wave vectors of the incident ( k inc ) and reflected ( k ref ) waves in region 1 make equal angles θ1 with the normal to the interfaces. Wave vectors k pinc and k pref of the incident and reflected waves in region 2 make equal angles θ2 with the normal to the interfaces. Wave vector k tr of the transmitted wave in region 3 makes angle θ3 with the same normal. The time dependence of all waves is assumed to be exp(iωt). We restrict the consideration to the case when the electric filed of the incident wave is oriented in the incidence plane (vertical polarization). Let us assume that the material filling each region is isotropic, i.e., the wave polarization in the entire structure is retained; the electric-field vectors of all waves belong to the incidence plane of the primary wave; and the magnetic fields of all waves are perpendicular to this plane. The Oz coordinate axis is perpendicular to the interfaces, whose coordinates are denoted as a and b. The

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REFLECTION, TRANSMISSION, AND ABSORPTION COEFFICIENTS

Oy axis belongs to the plane of the wave incidence. Then, the Oxy coordinate plane is parallel to the medium interfaces and the Ox axis is perpendicular to the incidence plane. Let Einc0, Eref0, and Etr0 denote the amplitudes of the electric fields of the incident, reflected, and transmitted waves, respectively, and let Epinc0 and Epref0 denote the amplitudes of the electric fields of the waves propagating inside the plate along the negative and positive directions of the Oz axis. The lengths of the wave vectors in media 1–3 are denoted as k1–3, respectively. We assume that there is no magnetic loss in any of the media; i.e., permeability µ is real. The conduction properties of a medium with real conductance σ are taken into account through introduction of a complex permittivity: ε = ε ref + iε σ ,

(1)

(2)

is its imaginary part, which characterizes the conduction properties of a medium. The impedances of the media are determined from the relationship Zn =

µ n µ 0 /ε n ε 0 ,

(3)

where n denotes the number of a medium (1, 2, or 3) and ε0 and µ0 are the electric and magnetic constants in the International System of Units. It is assumed that all the media are homogeneous (i.e., ε1–3, µ1–3, σ1–3 are independent of coordinates) and that there are no charges. The electric and magnetic fields in region 1 are the sums of the respective fields incident on the plate and reflected from it, the electric and magnetic fields in region 2 are the sums of the respective wave fields propagating along the positive and negative directions of the Oz axis, and the electric and magnetic fields in region 3 are the respective fields of the wave transmitted through the plate. All of the waves identically depend on the y coordinate if the condition k 1 sin θ 1 = k 2 sin θ 2 = k 3 sin θ 3 ,

(4)

which coincides with Snell’s law [2, 3], is fulfilled. Next, assume that media 1 and 3, which adjoin the different sides of the plate, are characterized with identical parameters; i.e., k1 = k3,

θ1 = θ3 ,

Z1 = Z3.

The nonzero coordinate components of the fields take the form E 1 y = [ E inc0 exp ( ik 1 z cos θ 1 ) + E ref0 exp ( – ik 1 z cos θ 1 ) ] (6) × cos θ exp ( – ik y sin θ ), 1

1

1

E 1z = [ E inc0 exp ( ik 1 z cos θ 1 ) – E ref0 exp ( – ik 1 z cos θ 1 ) ] (7) × sin θ exp ( – ik y sin θ ), 1

1

1

1 H 1 x = -----[E inc0 exp ( ik 1 z cos θ 1 ) Z1

(5)

(8)

– E ref0 exp ( – ik 1 z cos θ 1 ) exp ( – ik 1 y sin θ 1 ) in region 1 (z > b); E 2 y = [E pinc0 exp ( ik 2 z cos θ 2 ) + E pref0 exp ( – ik 2 z cos θ 2 ) ] cos θ 2 exp ( – ik 1 y sin θ 1 ), E 2z = [E pinc0 exp ( ik 2 z cos θ 2 )

where εr is the real part of the permittivity and ε σ = σ/ε 0 ω

365

– E pref0 exp ( – ik 2 z cos θ 2 )] sin θ 2 exp ( – ik 1 y sin θ 1 ), 1 H 2 x = -----[E pinc0 exp ( ik 2 z cos θ 2 ) Z2

(9)

(10)

(11)

– E pref0 exp ( – ik 2 z cos θ 2 ) ] exp ( – ik 1 y sin θ 1 ) in region 2 (a < z < b); and E 3 y = E tr0 cos θ 1 exp ( ik 1 z cos θ 1 ) exp ( – ik 1 y sin θ 1 ), (12) E 3z = E tr0 sin θ 1 exp ( ik 1 z cos θ 1 ) exp ( – ik 1 y sin θ 1 ),(13) E tr0 - exp ( ik 1 z cos θ 1 ) exp ( – ik 1 y sin θ 1 ) H 3 x = ------Z1

(14)

in region 3 (z < a). Now, after the expressions for the fields in all of the three regions have been obtained, let us determine the field amplitudes and the reflection and transmission coefficients. We apply the direct and averaging methods and, then, compare the results. 1.2. Direct Method In order to solve the problem by means of the direct method, we use the standard boundary conditions, i.e., the continuity of the tangential field components on the medium interfaces. The Oz axis is perpendicular to the interfaces, the electric-field vectors of all waves belong to the incidence (Oyz) plane, and the magnetic-field vectors are aligned with the Ox axis. Therefore, the boundary conditions couple only two field components, Ey and Hx, and yield four equations for two interfaces. Thus, setting a = 0 and b = d, we obtain a system of


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equations coupling wave amplitudes Einc0, Eref0, Epinc0, Epref0, and Etr0: E ref 0 cos θ 1 exp ( – ik 1 d cos θ 1 )

2

– E pinc0 cos θ 2 exp ( ik 2 d cos θ 2 )

(15)

– E pref0 cos – θ 2 exp ( – ik 2 d cos θ 2 )

E ref 0 Z 2 exp ( – ik 1 d cos θ 1 ) + E pinc0 Z 1 exp ( ik 2 d cos θ 2 ) (16) – E pref0Z1 exp ( – ik 2 d cos θ 2 ) = Einc0Z2 exp ( ik 1 d cos θ 1 ), E pinc0 cos θ 2 + E pref0 cos θ 2 – E tr0 cos θ 1 = 0,

(17)

E pinc0 Z 1 – E pref0 Z 1 – E tr0 Z 2 = 0.

(18)

Solution of (15)–(18) yields wave amplitudes Eref0, Etr0, Epinc0, and Epref0 expressed through Einc0: (19)

U E tr0 = ------tr- E inc0 , W

(20)

V ref -E , E pinc0 = ------W inc0

(21)

V ref -E , E pref0 = ------W inc0

(22)

where the impedances are determined from (3) in terms of the constitutive parameters and the following notation is applied: U ref = ( ε 1 µ 2 cos θ2 – ε 2 µ 1 cos θ1 )[ exp ( ik 2 d cos θ 2 ) (23) – exp ( – ik d cos θ ) ] exp ( 2ik d cos θ ), 2

2

1

1

U tr = 4 ε 1 ε 2 µ 1 µ 2 cos θ 1 cos θ 2 exp ( ik 1 d cos θ 1 ), (24) V inc = 2 ε 1 µ 2 ( ε 2 µ 1 cos θ 1 + ε 1 µ 2 cos θ 2 ) × cos θ 1 exp ( ik 1 d cos θ 1 ),

2 E tr0 T s = --------------= U ------tr- , 2 W E inc0

(32)

where Uref , Utr, and W are determined from expressions (23), (24), and (27). In (23)–(27), the explicit form of the exponents is retained to facilitate calculations in the case of complex permittivities. If necessary, these exponents can be replaced with trigonometric functions.

(25)

Now, let us apply the averaging method to solve the same problem. In [30], we obtained boundary conditions describing the fields on the opposite surfaces of the plate. For the geometry considered, these conditions generally have the following form: 2 H yb + H ya⎞ E xb – E xa 1 ∂ = i – ⎛ ωµ 2 µ 0 + -------------- --------2⎞ ⎛ ------------------------------------------⎝ ⎠ ⎝ ⎠ ωε 2 ε d 2 0 ∂x (33) 2 1 ∂ ⎛ H xb + H xa⎞ – -------------- ------------ ------------------------ , ⎠ ωε 2 ε 0 ∂x∂y ⎝ 2 2 H xb + H xa⎞ E yb – E ya 1 ∂ = i ⎛ ωµ 2 µ 0 + -------------- --------2⎞ ⎛ ------------------------------------------⎠ ⎝ ωε 2 ε 0 ∂y ⎠ ⎝ 2 d 2 H yb + H ya⎞ 1 ∂ , – -------------- ------------ ⎛ ----------------------⎠ ωε 2 ε 0 ∂x∂y ⎝ 2 2 E yb + E ya⎞ H xb – H xa 1 ∂ ---------------------- = i ⎛ ωε 2 ε 0 + ---------------- --------2⎞ ⎛ --------------------⎝ ⎠ ⎝ ⎠ ωµ 2 µ 0 ∂x 2 d

E xb + E xa⎞ 1 ∂ - , – ---------------- ------------ ⎛ --------------------⎝ ⎠ ωµ 2 µ 0 ∂x∂y 2 2

V ref = 2 ε 1 µ 2 ( ε 2 µ 1 cos θ 1 – ε 1 µ 2 cos θ 2 ) × cos θ 1 exp ( ik 1 d cos θ 1 ),

(26)

2

W = ( ε 2 µ 1 cos θ 1 + ε 1 µ 2 cos θ 2 ) exp ( ik 2 d cos θ 2 ) (27) 2 – ( ε 2 µ 1 cos θ 1 – ε 1 µ 2 cos θ 2 ) exp ( – ik 2 d cos θ 2 ), k 1 = ω ε1 ε0 µ1 µ0 ,

(28)

k 2 = ω ε2 ε0 µ2 µ0 ,

(29)

ε 2 µ 2 – ε 1 µ 1 sin θ1 ε1 µ1 2 - sin θ1 = ---------------------------------------------. (30) 1 – --------ε2 µ2 ε2 µ2 2

cos θ 2 =

(31)

1.3. Averaging Method

U ref , E ref 0 = --------E W inc0

2

E ref0 U ref 2 ; = -------R s = --------------2 W E inc0 2

= – E inc0 cos θ 1 exp ( ik 1 d cos θ 1 ),

2

From these relationships, we find the energy reflection and transmission coefficients:

(34)

(35)

2 E xb + E xa⎞ H yb – H ya 1 ∂ ---------------------- = i – ⎛ ωε 2 ε 0 + ---------------- --------2⎞ ⎛ --------------------⎝ ⎠ ωµ 2 µ 0 ∂y ⎠ ⎝ 2 d (36) 2 1 ∂ ⎛ E yb + E ya⎞ – ---------------- ------------ ---------------------- . ⎠ ωµ 2 µ 0 ∂x∂y ⎝ 2

In the discussed case of the vertical polarization, the only nonzero components in the Oxy plane are the electric- and magnetic-field components aligned with the Oy and Ox axes, respectively. All the waves are characterized by the identical dependence exp(–ik1ysinθ1) on the y coordinate. Therefore, the quartet of boundary


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conditions (33)–(36) become the following two relationships: k 1 sin θ1⎞ ⎛ H xb + H xa⎞ E yb – E ya = i ⎛ ωµ 2 µ 0 – ------------------- ------------------------ , (37) --------------------⎠ ⎝ 2 d ωε 2 ε 0 ⎠ ⎝ 2

2

E yb + E ya⎞ H xb – H xa - . ---------------------- = iωε 2 ε 0 ⎛ --------------------⎝ ⎠ 2 d

(38)

Taking into account (28), we reduce (37) and (38) after some algebra to the form iωdµ 2 E yb – E ya = ---------------0(ε 2 µ 2 – ε 1 µ 1 sin θ1)(H xb + H xa), 2ε 2 (39) E yb + E ya

2i = – ----------------- ( H xb – H xa ). ωdε 2 ε 0

(40)

Field components Eyb, Eya, Hxb, and Hxa, which enter conditions (39) and (40) and are determined on the plate’s surfaces, can be found from the general relationships valid in media 1 and 3 through setting z = b in expressions (6) and (8) and z = a in expressions (12) and (14):

Relationships (45) and (46) can be modified into the system of equations for Eref0 and Etr0: E ref 0 exp ( – ik 1 b cos θ 1 ) – E tr0 exp ( ik 1 a cos θ 1 ) L–1 = ------------E inc0 exp ( ik 1 b cos θ 1 ), L+1 M–1 = -------------- E inc0 exp ( ik 1 b cos θ 1 ). M+1

LM – 1 E ref 0 = ------------------------------------- exp ( 2ik 1 d cos θ 1 )E inc0 , (51) ( L + 1)( M + 1) –L+M E tr0 = ------------------------------------- exp ( ik 1 d cos θ 1 )E inc0 . ( L + 1)( M + 1)

2

E ya = E tr0 cos θ 1 exp ( ik 1 a cos θ 1 ) exp ( – ik 1 y sin θ 1 ), (42)

(53)

2

(54)

where L and M are determined from (47) and (48). (43)

– E ref 0 exp ( – ik 1 b cos θ 1 ) ] exp ( – ik 1 y sin θ 1 ), E tr0 - exp ( ik 1 a cos θ 1 ) exp ( – ik 1 y sin θ 1 ). H xa = ------Z1

(52)

From these expressions, we find the energy reflection and transmission coefficients:

2 E tr0 –L+M T p = --------------= ------------------------------------ , 2 ( L + 1)( M + 1) E inc0

1 H xb = -----[E inc0 exp ( ik 1 b cos θ 1 ) Z1

(50)

Solving (49), (50) and setting a = 0 and b = d, we obtain

(41)

+ E ref 0 exp ( – ik 1 b cos θ 1 ) cos θ 1 exp ( – ik 1 y sin θ 1 ),

(49)

E ref 0 exp ( – ik 1 b cos θ 1 ) + E tr0 exp ( ik 1 a cos θ 1 )

2 E ref0 LM – 1 = -----------------------------------R p = --------------- , 2 ( L + 1)( M + 1) E inc0

E yb = [E inc0 exp ( ik 1 b cos θ 1 )

367

(44)

The substitution of (41)–(44) into (39)–(40) yields the following system of equations coupling wave amplitudes Einc0, Eref0, and Etr0: E inc0 exp ( ik 1 b cos θ 1 ) + E ref0 exp ( – ik 1 b cos θ 1 ) – E tr0 exp ( ik 1 a cos θ 1 ) = L[E inc0 exp ( ik 1 b cos θ 1 ) (45) – E ref 0 exp ( – ik 1 b cos θ 1 ) + E tr0 exp ( ik 1 a cos θ 1 ) ], E inc0 exp ( ik 1 b cos θ 1 ) + E ref0 exp ( – ik 1 b cos θ 1 ) + E tr0 exp ( ik 1 a cos θ 1 ) = M[E inc0 exp ( ik 1 b cos θ 1 ) (46) – E ref 0 exp ( – ik 1 b cos θ 1 ) – E tr0 exp ( ik 1 a cos θ 1 ) ], where iωd ε 1 ε 0 µ 0(ε 2 µ 2 – ε 1 µ 1 sin θ1) -, L = --------------------------------------------------------------------------2ε 2 µ 1 cos θ 1

(47)

2i ε 1 -. M = – ----------------------------------------------ωdε 2 ε 0 µ 1 µ 0 cos θ 1

(48)

1.4. Absorption of Wave Energy in a Conducting Medium Formulas (31), (32), (53), and (54), which have been derived above, describe only wave reflection from a plate and wave transmission through it. At the same time, if the plate’s material is conducting, wave propagation induces currents in it and, owing to these currents, a portion of the energy of the incident radiation is absorbed and conversed into heat. Usually, such loss is described by the absorption coefficient [1, 3, 5]. The terminologies used by different authors in the analysis of these processes are different. Therefore, for consistency, we assume that the absorption coefficient is the quantity A = 1 – R – T. (55) According to the classical definition [1], this expression follows from the law of energy conservation. Obviously, such a definition is universal and does not depend on whether the direct or averaging calculation method is applied.

2

2. COMPARISON OF THE RESULTS OBTAINED VIA USE OF THE DIRECT AND AVERAGING METHODS Let us compare the results obtained with the use of the direct and averaging methods. It is seen from the


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R, T 1.0

R, T 1.0

(a)

4' 4

0.8

3

0.8

3, 3'

4 3

0.6

0.6

4' 1'

0.4

0.4

2'

1

3 4

3'

2 1

1, 1'

2

2 0.2

0.2

2'

1

2 0

15

30

45

60

75

90 θ1, deg

Fig. 2. (1, 2, 1', 2') Reflection coefficient R and (3, 3', 4, 4') transmission coefficient T vs. wave incidence angle at various normalized thicknesses of a dielectric plate: d/λ2 = (1, 1', 3, 3') 0.3 and (2, 2', 4, 4') 0.4.

above calculations that the direct method necessitates solution of system of four equations (15)–(18), whereas, when the averaging method is employed, it suffices to solve simpler system of two equations (49), (50). Formulas (31) and (32) for the reflection and transmission coefficients obtained by means of the direct method and combined with auxiliary formulas (23), (24), (27), and (30) are much more complicated than analogous formulas (53) and (54) obtained by means of the averaging method and combined with auxiliary formulas (47) and (48). The formulas obtained with the help of the averaging method contain neither exponents nor trigonometric functions of the refraction angle. Thus, the averaging method provides for a significant reduction of the computational effort. Next, it is necessary to justify the correctness of the averaging method and to determine its domain of applicability. For this purpose, we apply both methods to analyze the reflection and transmission coefficients as functions of the angle at which the wave is incident on the plate’s surface, functions of the plate’s thickness, and functions of the conductance of the plate’s material. After that, we compare the results obtained. We assume below that the plate is surrounded by an empty space; i.e., the medium parameters in regions 1 and 3 are ε1, 3 = 1, µ1, 3 = 1, and σ1, 3 = 0. In this case, for the frequency ω = 1011 s–1, we have k1 = 333 m–1; i.e., λ1 = λ3 = 1.885 cm. Inside the plate (in region 2), the permeability is assumed to be equal to unity: µ2 = 1. For

0

15

30

45

60

75

90

(b)

1.0

4' 4

0.8

0.6

3

3'

0.4

1

1' 2

0.2

0 75

2' 80

85

90 θ1, deg

Fig. 3. (1, 1', 2, 2') Reflection coefficient R and (3, 3', 4, 4') transmission coefficient T vs. wave incidence angle at various normalized thicknesses of a thick dielectric plate: (a) 0° ≤ θ1 ≤ 90° and (b) 75° ≤ θ1 ≤ 90°.

the case of a dielectric, we assume that ε2 = 5 and σ2 = 0. Then, the wavelength in the plate is λ2 = 0.8430 cm. For the case of a conductor, we assume that ε2 = 1 and σ2 = 16 Ω–1 m–1. Here, the permittivity is set to be equal to unity in order to separate the contribution of conduction. Then, λ2 = 0.6096 cm, εσ = 18.096 (accurate to the sign), and the thickness of the skin layer is δ = 0.0994 cm. Figures 2–6 show the dependences obtained with the direct method according to formulas (31) and (32) (curves with unprimed numbers) and with the averag-


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REFLECTION, TRANSMISSION, AND ABSORPTION COEFFICIENTS R, T 1.0 10 9 8

0.4

9'

0.8

8'

0.6

3'

7'

7

0.4

1 2

1'

2' 4

3

2' 2

3

6'

6

0.2

(a)

10'

0.8

0.6

R 1.0

(a)

369

1'

1

0.2

3'

0

4'

T 1.0

(b)

5'

5 0 1.0

(b)

0.8

4 5

1'

4'

5'

0.6

1

0.8 0.4

6' 0.6

6

3

0.2

1

0.2

1'

0.4

3

A 1.0

2'

7

3'

0

7'

(c)

0.8

4

8'

8 0

2

0.2

0.4 2

2'

3'

0.6

0.8

1.0 d/λ 2

Fig. 4. Reflection coefficient R and transmission coefficient T vs. normalized thickness of a dielectric plate at various wave incidence angles: (a) (1–5, 1'–5') R and (6–10, 6'–10') T for θ1 < θB and (b) (1–4, 1'–4') R and (5–8, 5'–8') T for θ1 > θB.

ing method according to formulas (53) and (54) (curves with primed numbers). 2.1. A Dielectric Plate: Variation of the Incidence Angle Consider the dependences of the reflection and transmission coefficients on the angle of the wave incidence on the plane of a dielectric plate (Figs. 2, 3). Figure 2 shows reflection coefficient R (curves 1, 1', 2, 2') and transmission coefficient T (curves 3, 3', 4, 4') as functions of incidence angle θ1 at two values of the

3

0.6

2 2'

0.4

3'

0

1'

4'

0.2

1 15

30

45

60

75 90 θ1, deg

Fig. 5. (a) Reflection coefficient R, (b) transmission coefficient T, and (c) absorption coefficient A vs. wave incidence angle at various normalized thicknesses of a conducting plate.

plate’s thickness: d/λ2 = 0.3 (curves 1, 1', 3, 3') and 0.4 (curves 2, 2', 4, 4'). It is seen from Fig. 2 that, as the incidence angle grows, reflection coefficient R first decreases to zero


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(a)

(b) 7'

7

0.8 6' 1'

0.6

2

5'

R

1 3

0.4

6

2' 5

3'

4

4'

4'

5

4

5'

0.2 6

3 2

6'

0 1.0

1

2'

(c)

1' 3'

(d)

0.8 1

6

0.6

6'

T

5'

0.4

0

3 4

4 1'

1 2

2'

4' 5 5' 6'

6

7'

7

(e)

(f) 1

6

0.8 0.7 0.6 A

3'

3

1.0 0.9

5

3

4

0.3

5'

4'

0.2

2

3'

5

2'

1'

1

4' 6

6'

5' 6' 7

1' 2'

0.1 0.2

2

4

3

0.5 0.4

0

1'

4'

5

0.2

3' 2'

2

0.4

7'

3'

0.6

0.8

1.0 0 d/λ 2

0.2

0.4

0.6

0.8

1.0

Fig. 6. (a, b) Reflection coefficient R, (c, d) transmission coefficient T, and (e, f) absorption coefficient A vs. normalized thicknesses of a conducting plate at various wave incidence angles: (a, c, e) θ1 < θC and (b, d, f) θ1 > θC.

and then increases to unity. This behavior is in correspondence with the classical concepts [1–5]. The equality R = 0 is fulfilled when the wave is incident at the Brewster angle θ1 = θB. When the direct calculation method is applied, value θB can be determined from the condition that the first multiplier of expression (23) equals zero. When the averaging calculation method is

applied, value θB can be determined from the condition that the numerator of the fraction in (53) equals zero. Both of the methods yield the same expression: ε2 ( ε1 µ2 – ε2 µ1 ) θ B = arcsin ------------------------------------. 2 2 µ1 ( ε1 – ε2 )


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(56)

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When µ1 = µ2, we obtain from (56) the formula n θ B = arctan -----2 , n1

(57)

where n1, 2 = ε 1, 2 µ 1, 2 are the refractive indexes in media 1 and 2. This formula is traditionally used in optics [1–5]. In the case considered, the value of the aforementioned angle is θ1B = 65.9052° and the refraction angle, which is determined by Snell’s law, is θ2B = 24.0948°. It can easily be seen that θ1B + θ2B = 90°, a result that agrees with the classical theory. The growth 90° likeof reflection coefficient R observed as θ1 wise corresponds to the classical concepts and to the situation known as the case of the grazing incidence [1– 5]. The curve of transmission coefficient í as a function of incidence angle θ1 is specularly symmetric to the curve of dependence R(θ1). This circumstance corresponds to the well-known relationship R + T = 1, which is typical of a lossless dielectric [1–5, 30]. Let us compare the dependences obtained with the direct method (curves 1–4) and the dependences obtained with the averaging method (curves 1'–4'). It is seen that, in the interval 0 ≤ θ1 < θB, the averaging method has an error no worse that 6% near the value θ1 = 0 at the normalized thickness d/λ2 = 0.3 (curves 1 and 1', 3 and 3'). The error is reduced as θ1 grows. At θ1 = θB, both methods yield R = 0 and T = 1: At Brewster’s angle, the wave is not reflected and propagates through the plate with all of the wave energy retained. In the interval θB ≤ θ1 < 90°, the error of the averaging method is no worse than 1%. At the normalized thickness d/λ2 = 0.4 (curves 2 and 2', 4 and 4'), the averaging method has an error of about 50% near the point θ1 = 0 within the interval 0 ≤ θ1 < θB. As value θB is approached, the error gradually lessens. At θ1 = θB, the values of R and T obtained with both methods coincide. In the interval θB ≤ θ1 < 90°, the error of the averaging method is no worse that 1%. Thus, when θ1 < θB, the error of the averaging method is rather low at d/λ2 = 0.3 and substantially worse at d/λ2 = 0.4. This result is in agreement with the result obtained in our study [30] for the case of the normal incidence, where the critical value of the normalized thickness was 0.35. It should be emphasized that, for θ1 > θB, the error of the averaging method is extremely low (no worse than 1% at d/λ2 = 0.3 and no worse than 15% at d/λ2 = 0.4). In order to determine the validity limits of the averaging method in the case under consideration, we have investigated the dependences of the reflection and transmission coefficients on the wave incidence angle for plates so thick that this method yields large errors. The results obtained are illustrated in Fig. 3, which shows dependences (similar to those depicted in Fig. 2) that are calculated at

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d/λ2 = 10 (curves 1 and 1', 3 and 3') and d/λ2 = 100 (curves 2 and 2', 4 and 4'). In Fig. 3a, angle θ1 ranges from 0° to 90°. As θ1 approaches the value 90°, the curves intersect at small angles. Therefore, it is difficult to analyze these curves. As a consequence, Fig. 3b shows the curves from Fig. 3a, but, in Fig. 3b, these curves are extended along the horizontal direction within the interval from 75° to 90°. The general form of curves 1–4 calculated with the use of the direct method corresponds to the classical concepts [1–5]. The oscillating character of these curves results from the interference of waves reflected from the two surfaces of the plate. Incidence angles θ1 corresponding to R = 0 and T = 1 are found from the condition that the reflected waves are in antiphase. Taking into account the π phase shift caused by reflection from the internal surface, we obtain the following expression: 2 ε2 µ2 n - 1 – ------------------- , θ 1 = arcsin --------ε1 µ1 4 ( d/λ 2 )

(58)

where n is an integer ranging within the interval ε1 µ1 2d 2d ------ 1 – --------- ≤ n ≤ ------ . λ2 λ2 ε2 µ2

(59)

In addition, equality (58) can be derived from the condition that the second multiplier in (23) equals zero. It is seen from (59) that, at d/λ2 = 10, there are only three admissible values of n—20, 19, and 18—that provide for the zero R at θ1 = 0°, 44.28°, and 77.08°. Curve 1 for R(θ1) in Fig. 3a has four minima: Three of these correspond to the aforementioned angles, and the fourth minimum corresponds to Brewster’s angle that is specified in (56) and equal to 65.90°. Curve 3 for T(θ1) has maxima of unity at the same angles. In a similar manner, we find that, at d/λ2 = 100, quantity n takes 22 admissible values: from 179 to 200. Hence, with allowance for Brewster’s angle, we obtain 23 zero values of R(θ1) for curve 2 (0°, 12.90°, 18.39°, …, 85.89°) and 23 maxima of unity on curve 3 for T(θ1) at the same angles. At d/λ2 = 10, the averaging method yields (curve 1') values of R close to zero at angles θ1 as large as 87°. However, when the value 90° is approached, the curve exhibits a sharp rise and, as θ1 90°, tends toward unity. Thus, at θ1 = 89.19° and 89.66°, we have R = 0.5 and 0.9, respectively. In this situation, transmission coefficient T (curves 3') tends toward zero. At d/λ2 = 100, curves 2' and 4' approach unity and zero, respectively, when θ1 > 89°. Thus, at θ1 = 89.86°, we have R = 0.5 and T = 0.5, and, at θ1 = 89.89°, we have R = 0.9 and T = 0.1.


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Thus, we can draw the conclusion that, when the incidence angle is close to 90° (the grazing-incidence value), the averaging method retains its validity even for an extremely large (exceeding the wavelength by an order of magnitude) thickness of the plate. 2.2. A Dielectric Plate: Variation of the Thickness Now, let us consider the reflection and transmission coefficients as functions of the plate’s thickness at various incidence angles. Since the angular dependences of these coefficients are different on different sides of the point of Brewster’s angle θB = 65.90°, we consider the intervals 0 ≤ θ1 ≤ θB and θB ≤ θ1 ≤ 90° separately. The results obtained are presented in Fig. 4. Figure 4a shows the reflection (R) and transmission (T) coefficients as functions of plate’s normalized thickness d/λ2 for 0 ≤ θ1 ≤ θB: curves 1, 1', 6, and 6' correspond to θ1 = 0°; curves 2, 2', 7, and 7', to θ1 = 15°; curves 3, 3', 8, and 8', to θ1 = 30°; curves 4, 4', 9, and 9', to θ1 = 45°; and curves 5, 5', 10, and 10', to θ1 = 60°. It is seen from the figure that, as incidence angle θ1 grows, reflection coefficient R increases and transmission coefficient T decreases, so that the curves for these coefficients approach the corresponding horizontal axes. As before, the relationship R + T = 1 holds. The general behavior of the curves resembles the behavior of the thickness dependences for R and T that have been analyzed in [30]. The direct method yields periodic dependences, and, in the interval 0 ≤ d/λ2 ≤ 0.35, the characteristics obtained with the averaging method are close to those obtained with the direct method. The discrepancy of the results is no worse that 5%. The upper boundary of the aforementioned interval is independent of incidence angle θ1. When the normalized thickness exceeds the value 0.35, the dependences obtained with the averaging method do not tend to be periodic functions. These dependences smoothly drop and, in the interval d/λ2 ~ 0.7–0.9, approach the dependences obtained with the direct method: The discrepancy does not exceed 15–40%. At d/λ2 = 0.3, the deviation between the curves obtained with the direct and averaging methods (e.g., 1 and 1') does not exceed 5%; while, at d/λ2 = 0.4, the deviation between the same curves reaches 50%. In the angular interval 0 ≤ θ1 ≤ θB, these results are in good agreement with the curves depicted in Fig. 2. Figure 4b shows the reflection (R) and transmission (T) coefficients as functions of plate’s normalized thickness d/λ2 for various incidence angles θ1 exceeding Brewster’s angle θB = 65.90°. Curves 1, 1', 5, and 5' correspond to θ1 = 76°; curves 2, 2', 6, and 6', to θ1 = 80°; curves 3, 3', 7, and 7', to θ1 = 84°; and curves 4, 4', 8, and 8', to θ1 = 88°. It is seen from the figure that, as

incidence angle θ1 grows, reflection coefficient R increases and transmission coefficient T decreases. This behavior corresponds to the dependences of R and T on incidence angle θ1 presented in Fig. 2. As before, the relationship R + T = 1 holds. The periodic behavior of the curves resembles the behavior of the thickness dependences analyzed in [30] and can be attributed to the interference of waves reflected from the two surfaces of the plate. As in the previous case (see Fig. 4a for θ1 < θB), the error of the averaging method in the interval d/λ2 ~ 0.7–0.9 is no worse than 5%. The larger angle θ1, the smaller the discrepancy between the curves calculated with the direct and averaging methods in the interval 0 ≤ d/λ2 ≤ 0.35. Thus, at θ1 = 76°, the deviation is about 40%, while at θ1 = 88°, the discrepancy is reduced to 5%. 2.3. A Conducting Plate: Variation of the Incidence Angle Now, let us consider the dependences of the reflection, transmission, and absorption coefficients on the angle of the wave incidence on the plane of a conducting plate (Fig. 5). 2.3.1. Reflection. Figure 5a shows reflection coefficient R as a function of incidence angle θ1 at three values of the plate’s normalized thickness: d/λ2 = 0.1 (curves 1, 1'), 0.2 (curves 2, 2'), and 0.4 (curves 3, 3'). It is seen from the figure that, as the incidence angle grows, reflection coefficient R first smoothly decreases by a factor of 2 or 3, thereby reaching its minimum near the point θC ~ 75°, and then abruptly increases, thereby 90°. This dependence approaching unity at θ1 resembles that depicted in Fig. 2 (curves 1, 2). However, in the case illustrated in Fig. 5a, R decreases to a value of about 0.1–0.2 rather than zero. The angle 75°, which corresponds to the minimum of dependence R(θ1), is not equal to Brewster’s angle. The latter is undefined here because both of the media are characterized by identical permittivities and permeabilities. In the literature, angle θC is sometimes referred to as the principal incidence angle [5]. The difference between θB and θC can be attributed to the complex permittivity of a conductor. Indeed, when a conductance is introduced into expression (1) for the permittivity with the use of formula (2), we obtain imaginary terms in expression (23) for Ur and in expression (30) for refraction angle θ2. As a result, the absolute value of reflection coefficient (31) contains an imaginary quantity that is nonzero at any value of θ1. The same result is obtained for reflection coefficient R calculated with the averaging method according to formula (53). The comparison of the dependences obtained with the direct (1–3) and averaging (1'–3') methods show that, at the normalized thickness of the plate d/λ2 = 0.1,


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0.2, and 0.4, the discrepancy of the curves is less than 2% (curves 1, 1'), about 10% (curves 2, 2'), and 25% (curves 3, 3'), respectively. Such a discrepancy is retained over the entire range of θ except for the interval of values of θ1 smaller than 90° by 2°–3°. In this interval, all the curves approach unity (the region near the grazing-incidence value) and the discrepancy between them tends toward zero. 2.3.2. Transmission. Now, let us consider similar dependences of transmission coefficient T on θ1 that are depicted in Fig. 5b. The results are obtained at d/λ2 = 0.05 (curves 1, 1'), 0.1 (2, 2'), and 0.2 (3, 3'). It is seen that, as the incidence angle grows, transmission coefficient T first smoothly increases by a factor of 2 or 3, thereby reaching its maximum at θC ~ 75°, and then abruptly decreases, thereby tending toward zero at θ1 90°. This dependence resembles a similar result for a dielectric (Fig. 2, curves 3, 4). However, here, T increases not to unity but to a quantity that decreases as the plate’s normalized thickness grows. This fact can be understood as a natural effect if we take into account that the absorption due to the medium’s conductivity intensifies while the plate’s thickness grows, and, at a sufficiently large thickness, the transmission coefficient tends toward zero. Such absorption is especially noticeable for the skin-layer thickness exceeding a certain value (in our case, δ = 0.0994 cm), i.e., for curves 2 and 3. For the latter curve, the absorption is even more noticeable. Therefore, the curves drop steeply as d/λ2 grows. The comparison of the dependences calculated with the direct (curves 1–3) and averaging (curves 1'–3') methods shows that the discrepancies between the dependences obtained with these two methods is about 2% (1, 1'), 10% (2, 2'), and 50% (3, 3') at the normalized plate’s thicknesses d/λ2 = 0.05, 0.1, and 0.2, respectively. This value of the discrepancy is retained over the entire range of θ1 except for the interval of values of θ1 smaller than 90° by 2°–3°. In this interval, all the curves approach zero (the region near the grazingincidence value).

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skin layer, absorption coefficient A (curves 1, 1') smoothly decreases to zero as incidence angle θ1 grows to 90°. When the plate is as thick as or thicker than the skin layer, absorption coefficient A (curves 2–4, 2'–4') first smoothly grows by a factor of 1.5 or 2 (the thicker the plate, the more substantial the growth), while the incidence angle increases to a value of about 75°, and then abruptly decreases to zero. In the case of the grazing incidence, the wave practically does not penetrate into the plate; therefore, no energy is absorbed. The comparison of the dependences calculated with the direct (curves 1–4) and averaging (curves 1'–4') methods shows that the discrepancies between the dependences obtained with these two methods are about 2% (1, 1'), 10% (2, 2'), 25% (3, 3'), and 50% (4, 4') at the normalized plate’s thicknesses d/λ2 = 0.05, 0.1, 0.2, and 0.4, respectively. The value of the discrepancy is retained over the entire range of θ except for the interval of values of θ1 smaller than 90° by 2°–3°. In this interval, all the curves approach zero (the region near the grazing-incidence value). The above results show that, relative to the transmission coefficient, the absorption coefficient is as sensitive to application of the averaging method: The discrepancies between the curves calculated with the two methods are the same at the same thickness. 2.4. A Conducting Plate: Variation of the Thickness Now, we consider the reflection, transmission, and absorption coefficients as functions of the thickness of a conducting plate. As has been shown in the previous section, dependences R(θ1), T(θ1), and, for a thick plate, A(θ1), have an extremum near the angle θC = 75°. On different sides of this value, the dependences exhibit different characters. Therefore, we consider the intervals 0 ≤ θ1 ≤ θC and θC ≤ θ1 ≤ 90° separately. The results are presented in Figs. 6–8.

Note that, relative to the reflection coefficient, the transmission coefficient is more sensitive to application of the averaging method: The same deviations are observed at a thickness that is half as large.

2.4.1. Reflection. Figure 6a shows reflection coefficient R as a function of the plate’s normalized thickness d/λ2 for various incidence angles θ1 that are smaller that θC = 75°. The results are obtained at θ1 = 0° (curves 1, 1'), 15° (2, 2'), 30° (3, 3'), 45° (4, 4'), 60° (5, 5'), and 75° (6, 6').

2.3.3. Absorption. Since a conducting medium exhibits absorption, it is necessary to analyze absorption coefficient A that is determined according to formula (55) from the condition of energy conservation and considered as a function of θ1. The calculated dependences are depicted in Fig. 5c. These dependences are obtained at d/λ2 = 0.05 (curves 1, 1'), 0.1 (2, 2'), 0.2 (3, 3'), and 0.4 (4, 4'). It is seen from the figure that, when the plate is substantially thinner than the

The general behavior of the dependences (smooth growth from zero to a constant) resembles the behavior of the thickness dependence for R that has been analyzed in [30] and fits the classical concepts for a highly conductive plate [1–5]. The transition from the rising section to the horizontal section occurs at d/λ2 ~ 0.15, irrespective of the wave incidence angle. This value exceeds the skin-layer thickness (δ = 0.0994 cm) by a factor of ~1.5. It is seen from Fig. 6a that, as incidence


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angle θ1 grows, the decrease of reflection coefficient R is more noticeable at larger θ1. Thus, when the incidence angle grows from 0° to 15° (the transition from curve 1 to 2), R changes by 2%, and, when the incidence angle grows from 60° to 75° (the transition from curve 5 to 6), R changes by almost 50%. Such a difference can be understood if we analyze Fig. 5a. From this figure, it is seen that the steepness of dependence R(θ1) increases as θ1 approaches θC. The comparison of the dependences calculated with the use of the direct (curves 1–6) and averaging (curves 1'–6') methods shows that, when the plate’s normalized thickness d/λ2 is smaller than 0.15 (i.e., smaller than the 1.5 times the thickness of the skin layer), the discrepancy between the dependences obtained with the considered two methods (1, 1') is less than 5%. As d/λ2 continues to grow, curves 1'–6', which are obtained with the averaging method, go above curves 1–6 as far as the maximum point at d/λ2 ~ 0.20– 0.35. After that, curves 1'–6' go down, intersect the horizontal sections of curves 1–6, and tend toward zero. At the intersection points of curves 1'–6' with curves 1–6, the values of R obtained with both methods coincide. The larger wave incidence angle θ1, the smaller normalized thickness d/λ2 at which an intersection occurs. Thus, at θ1 = 0° (curves 1, 1') and 75° (6, 6'), intersections are observed at d/λ2 = 0.8 and 0.5, respectively. Over the entire range 0.15–1.00 of the normalized thickness, the difference of R obtained with the two considered methods is no greater than 50%. Now, let us consider wave reflection at incidence angles θ1 exceeding θC. The corresponding dependences of reflection coefficient R on plate’s normalized thickness d/λ2 are depicted in Fig. 6b. The results are obtained at θ1 = 75° (curves 1, 1'), 80° (2, 2'), 82° (3, 3'), 84° (4, 4'), 86° (5, 5'), 88° (6, 6'), and 89° (7, 7'). The general behavior of the dependences (smooth growth from zero to a constant) resembles the behavior of the thickness dependence observed at θ1 ≤ θC. The transition from the rising section to the horizontal section occurs at d/λ2 ~ 0.10–0.15. The larger incidence angle θ1, the smaller the value of the normalized thickness at which the transition occurs. As angle θ1 grows, the curves for reflection coefficient R rise according to the rise of the right branches of the curves from Fig. 5a. The rate of the rise depends on θ1 only slightly. This result corresponds to the almost linear shape of the aforementioned branches. Curves 1'–7', which are obtained with the averaging method, are compared with curves 1–7, which are obtained with the direct method. Curves 1'–7' resemble similar curves from Fig. 6a. We observe the same rise, passage through the maximum, and subsequent drop with intersection with curves 1–7.

In this case, the averaging method provides for the same accuracy as in the case of the curves from Fig. 6a. 2.4.2. Transmission. Consider similar dependences of transmission coefficient T on d/λ2 that are depicted in Figs. 6c and 6d. The numbering of curves and the values of angles coincide with those from Figs. 6a and 6b. Figures 6c and 6d correspond to θ1 ≤ θC and θ1 ≥ θC, respectively. All the curves obtained with the direct method and depicted in these figures smoothly drop owing to the attenuation of waves propagating in a conducting medium. The curves in Fig. 6c rise as angle θ1 grows, because reflection is reduced according to Fig. 6a. The curves from Fig. 6d exhibit the opposite behavior: As angle θ1 grows, these drop. This result can be attributed to the fact that, as θ1 approaches 90°, the portion of the wave energy penetrating into the plate is reduced, the reflected portion of the wave energy grows (Fig. 6b), and the portion of the wave energy penetrating through the plate decreases (the grazing incidence). The curves obtained with the averaging method exhibit a similar dependence on incidence angle θ1: As θ1 grows within the intervals θ1 ≤ θC and θ1 ≥ θC, the curves rise and drop, respectively. However, as the plate’s normalized thickness d/λ2 grows to ~0.2, the curves drop; pass through the minimum; and then rise, thereby exhibiting the behavior opposite to that of the curves calculated with the direct method, which approach zero. Thus, the averaging method can be applied for calculation of the transmission coefficient as a function of the plate’s thickness only in the interval 0 ≤ d/λ2 ≤ 0.2. The error of this method amounts to no worse than 2% in the interval 0 ≤ d/λ2 ≤ 0.1, increases to 50% as d/λ2 approaches a value of 0.2, and becomes even worse as d/λ2 continues to increase. The comparison with the results for the reflection coefficient (Figs. 6a, 6b) shows that, in the cases of both a conductor and a dielectric, the transmission coefficient is more sensitive to application of the averaging method than the reflection coefficient and that the difference in the sensitivity is approximately the same in these cases. 2.4.3. Absorption. Consider the dependence of absorption coefficient Ä on the normalized thickness. The results are presented in Figs. 6e and 6f. The numbering of curves and the values of angles coincide with those from Figs. 6a–6d. Figures 6e and 6f correspond to θ1 ≤ θC and θ1 ≥ θC, respectively. As d/λ2 grows, all of the dependences calculated with the use of the direct method (Figs. 6e, 6f), first, increase from zero and, at d/λ2 ~ 0.35, reach a constant value. As the plate’s normalized thickness continues to grow, the absorption becomes independent of this


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thickness: All of the nonreflected wave energy is absorbed, and no wave energy is transmitted through the plate. As angle θ1 increases, the curves in Fig. 6e rise because the wave covers a longer path inside the plate and the absorption of the wave grows. As angle θ1 grows, the curves in Fig. 6f drop because the wave weakly penetrates into the plate in the case of the grazing incidence. The curves obtained with the averaging method exhibit a similar behavior depending on incidence angle θ1: These curves rise and drop as θ1 increases within the intervals θ1 ≤ θC and θ1 ≥ θC, respectively. The difference between the curves obtained with the averaging and direct methods is no greater than 2% in the interval 0 ≤ d/λ2 ≤ 0.1, gradually increases to 50% as d/λ2 approaches 0.2, and continues to grow. The transmission and absorption coefficients are more sensitive to application of the averaging method than the reflection coefficient. CONCLUSIONS The main results of the study are as follows. The propagation of a transverse electromagnetic wave through a thin parallel-sided plate with an arbitrary conductance has been considered in the case of the oblique wave incidence on the plane of the plate. The averaging method and averaged electrodynamic boundary conditions have been applied to derive simplified formulas for the reflection, transmission, and absorption coefficients. In addition, the direct method has been applied to obtain the same coefficients. According to the direct method, the wave equation is solved inside and outside the plate and the solutions are joined on the plate’s surfaces. The comparison of the results obtained with the two methods has shown that the averaging method makes it possible to substantially facilitate calculations, because the order of the system of equations to be solved is reduced, and to simplify the final formulas for the reflection and transmission coefficients owing to the absence of exponents and trigonometric functions. The correctness of the averaging method has been justified and the domain of its applicability has been determined through the analysis of the reflection and transmission coefficients as functions of the angle at which the wave is incident on the plate’s plane, functions of the plate’s thickness, and functions of the conductance of the plate’s material. The analysis has been performed with the use of both methods, and the results have been compared. It has been shown that, when the normalized (by the wavelength in the plate’s material) thickness of a

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dielectric plate is not larger than 0.35 and the incidence angle ranges from zero to Brewster’s angle, the error of the averaging method relative to that of the direct method is no worse than 5%. When the wave is incident at Brewster’s angle, the averaging and direct methods yield the same results. When the incidence angle ranges from Brewster’s angle to 90°, the error of the averaging method is no worse than 1%. The reflection coefficient of a conducting plate can be calculated with the averaging method over the entire range 0°–90° of the incidence angle at an error no worse than 5% when the plate’s normalized thickness is as large as 0.15 and at an error no worse than 50% when the normalized thickness ranges from 0.15 to 1.00. The transmission and absorption coefficients can be calculated with the averaging method over the entire range 0°–90° of the incidence angle at an error no worse than 2% when the plate’s normalized thickness is smaller than 0.1. As the normalized thickness increases to 0.2, the error gradually increases to 50%. As the normalized thickness continues to grow, the error continues to grow. ACKNOWLEDGMENTS This study was supported by the Russian Foundation for Basic Research (project no. 06-02-17302a) and the Council for Grants of the President of the Russian Federation for Support of Leading Scientific Schools (grant no. NSh-8269.2006.2). REFERENCES 1. O. D. Khvol’son, Course of Physics (Gosizdat RSFSR, Berlin, 1923), Vol. 2 [in Russian]. 2. G. S. Landsberg, Optics (Nauka, Moscow, 1976) [in Russian]. 3. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1969; Nauka, Moscow 1970). 4. L. M. Brekhovskikh, Waves in Layered Media (Nauka, Moscow, 1973; Academic, New York, 1980). 5. R. W. Pohl, Einführung in die Optik (Springer, Berlin, 1940; OGIZ, Moscow, 1947). 6. I. P. Suzdalev, Nanotechnology. Physicochemistry of Nanoclusters, Nanostructures, and Nanomaterials (KomKniga, Moscow, 2006) [in Russian]. 7. A. E. Kaplan, Radiotekh. Elektron. (Moscow) 11, 1781 (1964). 8. R. Hasegava, Phys. Rev. Lett. 28, 1376 (1972). 9. A. R. Melnyk and M. J. Harrison, Phys. Rev. B 2, 835 (1970). 10. G. Marchal, P. Mangin, and C. Janot, Thin Solid Films 23 (1), 17 (1974). 11. J. D. Adam, Proc. IEEE 76, 159 (1988). 12. W. C. Ishak, Proc. IEEE 76, 171 (1988). 13. A. K. Sarychev, D. J. Bergman, and Y. Yagil, Phys. Rev. B 51, 5366 (1995).


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