1

PLANE WAVE PROPAGATION AND REFLECTION

David R. Jackson Department of Electrical and Computer Engineering

University of Houston Houston, TX 77204-4793

Abstract The basic properties of plane waves propagating in a homogeneous isotropic region are

reviewed, for both lossy and lossless media. These basic properties include the nature of the

electric and magnetic fields, the properties of the wavenumber vector, and the power flow of the

plane wave. Special attention is given to the specific case of a homogenous (uniform) plane

wave, i.e., one having real direction angles, since this case is most often met in practice (for

example, in the far-field of an antenna). Properties such as wavelength, phase and group velocity,

penetration depth, and polarization are discussed for these plane waves.

The theory of reflection of plane waves from layered media is then discussed in a unified

manner, using a transmission line model called the transverse equivalent network. This method

decomposes the incident plane wave into TM and TE parts, and models each separately using a

transmission line equivalent circuit. Basic reflection phenomena such as the law of reflection,

Snells law and the Brewster angle effect come directly from this model. More involved

reflection problems are also easily treated using this approach. As an example, the reflected and

transmitted fields due to a circular-polarized wave incident on the surface of the ocean are

calculated.

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1. INTRODUCTION Planes waves are the simplest solution of Maxwells equations in a homogeneous region of

space, such as free-space (vacuum). In spite of their simplicity, plane waves have played an

important role throughout the development of electromagnetics, starting from the time of the

earliest radio transmissions through the development of modern communications systems. Plane

waves are important for several reasons. First, the far-field radiation from any transmitting

antenna has the characteristics of a plane wave sufficiently far from the antenna. The incoming

wavefield impinging on a receiving antenna can therefore usually be approximated as a plane

wave. Second, the exact field radiated by any source in a region of space can be constructed in

terms of a continuous spectrum of plane waves via the Fourier transform. Understanding the

nature of plane waves is thus important for understanding both the far-field and the exact

radiation from sources.

The theory of plane wave reflection from layered media is also a well-developed area, and

relatively simple expressions suffice for understanding reflection and transmission effects when

layers are present. Problems involving reflections from the earth or sea, for example, are easily

treated using plane-wave theory. Even when the incident wavefront is actually spherical is shape,

as from a transmitting antenna, plane-wave theory may often be approximately used with

accurate results.

Throughout this article, it will be assumed that the regions of interest are homogeneous

(the material properties are constant) and isotropic, which covers most cases of practical interest.

3

2. BASIC PROPERTIES OF A PLANE WAVE Definition of a plane wave The most general definition of a plane wave is an electromagnetic field having the form E E0= ψ x y z, ,b g (1) H H0= ψ x y z, ,b g (2) where E0 and H0 are constant vectors, and the wavefunction ψ is defined as

ψ x y z e

e

j k x k y k z

j

x y z, ,b g d i=

=

− + +

− ⋅k r (3)

where k x y z= + + = −! ! !k k k jx y z ββββ αααα (4) r x y z= + +! ! !x y z , (5) with kx, ky, and kz being complex constants that define a wavenumber vector k. (A time-harmonic

dependence of e j tω is assumed and suppressed.) The vector E0 defines the polarization of the

plane wave. The real and imaginary parts of the wavenumber vector k define the phase vector

ββββ and an attenuation vector αααα . The phase vector has units of radians/meter and gives the

direction of most rapid phase change, while the attenuation vector has units of nepers/meter and

gives the direction of most rapid attenuation. The magnitude of the phase vector gives the phase

change per unit length along the direction of the phase vector, while the magnitude of the

attenuation vector determines the rate of attenuation along the direction of the attenuation vector.

4

Basic properties

In a homogeneous lossless space a plane wave must satisfy Maxwells equations, which in the

time-harmonic form are [1]

∇× =H Ejωε (6)

∇× = −E Hjωµ (7)

∇⋅ =E 0 (8)

∇⋅ =H 0 (9)

where ε and µ are the permittivity and permeability of the space. For free space, ε = ε0 and µ

=µ0 , where µ0 is defined to be 4π × 10-7 Henry/meter and ε0 is determined from the defined

velocity of light (or any plane wave) in vacuum [2], c = 2.99792458 × 108 meters/second, since

c = 1 0 0/ ε µ . This gives the approximate value ε0 = 8.85418781762039 × 10-12 Farads/meter.

A lossy medium can be modeled using a complex effective permittivity, accounting for

conduction loss and/or polarization loss [1]. The complex effective permittivity is expressed as

ε ε σω

= − FHGIKJ! j . (10)

In this equation !ε is the complex permittivity of the material, accounting for polarization loss (if

any), and σ is the conductivity of the medium.

Taking the curl of Eq. (7) and then substituting in Eq. (6) gives the vector wave equation

∇× ∇× − =E E 0b g k 2 , (11)

where k is the (possibly complex) wavenumber defined by

k k jk= − =' ' ' ω µε , (12)

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with the square root chosen so that k lies in the fourth quadrant on the complex plane. Using the

definition of the vector Laplacian,

∇ ≡ ∇ ∇⋅ −∇× ∇×2E E Eb g b g , (13)

and the fact that the divergence of the electric field is zero for a time-harmonic field in a

homogeneous region [1], results in the vector Helmholtz equation

∇ + =2 2E E 0k . (14)

In rectangular coordinates, the vector Laplacian is expressed as

∇ = ∇ + ∇ + ∇2 2 2 2E x y z! ! !E E Ex y z . (15)

Hence, all three rectangular components of the electric field satisfy the scalar Helmholtz

equation

∇ + =2 2 0Ψ Ψk (16)

in a homogeneous region.

Substituting Eq. (3) into Eq. (16) gives the result

k k k kx y z2 2 2 2+ + =

or

k k⋅ = k 2 . (17)

This is the separation equation that relates the components of the wavenumber vector k

defined in Eq. (4). Note that the term on the left side of Eq. (17) is not in general equal to 2k ,

since k may be complex.

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Other fundamental relations for a plane wave may be found by substituting Eq. (1) and (2)

into Maxwells equations (6)-(9). Noting that ∇→ − jk for a plane wave, Maxwells equations

reduce to

k H E× = −ωε (18)

k E H× = ωµ (19)

k E⋅ = 0 (20)

k H⋅ = 0 . (21)

Equations (18) and (19) each imply that E H⋅ = 0 , and together they also imply that k E⋅ = 0

and k H⋅ = 0 . That is, all three vectors k , E , and H are mutually orthogonal. Another

interesting property that is true for any plane wave, which may be derived directly from Eq. (18)

or (19), is that

E E H H⋅ = ⋅η2 , (22)

where η is the intrinsic impedance of the space (possibly complex), defined from

η µε

= (23)

(the principle branch of the square root is chosen so that the real part of η is nonnegative). For

vacuum, the intrinsic impedance is often denoted as η0, and has a value of approximately

376.7303 Ω.

Power flow

The complex Poynting vector for a plane wave, giving the complex power flow, is (assuming

peak notation for phasors)

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S E H= ×12

*. (24)

Using Eq. (19), the Poynting vector for a plane wave can be written as

S E k E0 0= × ×1

22

ωµψ*

* *e j . (25)

Using the triple product rule A B C A C B A B C× × = ⋅ − ⋅b g b g b g , this can be rewritten as

S E k E k E0 0 0= − ⋅1

21

22 2 2

ωµψ

ωµψ*

**

* *e j . (26)

The second term in the above equation is not always zero for an arbitrary plane wave, even

though E k0 ⋅ = 0 , since k may be complex in the most general case. If either k or E0 is

proportional to a real vector (i.e., all of the components of the vector have the same phase angle),

then it is easily demonstrated that the second term vanishes. In this case the Poynting vector

becomes

S E k0=1

22 2

ωµψ*

*. (27)

If the medium is also lossless (µ is real), the time-average power flow (coming from the real part

of the complex Poynting vector) is in the direction of the phase vector ββββ . A similar derivation,

casting the Poynting vector in terms of the H0 vector, yields

S H k0=1

22 2

ωεψ , (28)

provided either k or H0 is proportional to a real vector. If the region is lossless (ε is real), the

time-average power flow is then in the direction of the phase vector. Hence, for a lossless region,

the time-average power flow is in the direction of the phase vector ββββ if any of the three vectors

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k, E0 , or H0 is proportional to a real vector. In many practical cases of interest, one of the

three vectors will be proportional to a real vector, and hence the conclusion will be valid.

However, it is always possible to find exceptions, even for free space. One such example is the

plane wave defined by the vectors k = 1 1, ,jb g , E0 = − −2 1 2, , jb g , and

H0 = − + −1 2 4 1 2/ ( ) , ,ωµb gb gj j j , at a frequencyω = c ( k0 1= ). This plane wave satisfies

Maxwells equations (18)-(21). However for this plane wave the vectors ββββ , αααα , and

p = Re Sb g are all in different directions, since the power flow is in the direction of the vector

5 0 4, ,b g One must then be careful to define what is meant by the direction of propagation for

such a plane wave.

Direction angles

One way to characterize a plane wave is through direction angles θ φ,b g in spherical

coordinates. The direction angles (which are in general complex) are defined from the relations

k kx = sin cosθ φ (29)

k ky = sin sinθ φ (30)

k kz = cosθ . (31)

Homogeneous (uniform) plane wave

One important class of plane waves is the class of homogeneous or uniform plane waves. A

homogeneous plane wave is one for which the direction angles are, by definition, real. A

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homogeneous plane wave enjoys certain special properties that are not true in general for all

plane waves. For such a plane wave the wavenumber vector can be written as

k R R= = −k k jk! ! ' ' 'b g , (32)

where !R is a real unit vector defined from

! ! sin cos ! sin sin ! cosR x y z= + +θ φ θ φ θ . (33)

In this case the k vector is proportional to the real vector !R , so the result of Eq. (27) applies.

The unit vector !R then gives the direction of time-average power flow, and also points in the

direction of the phase and attenuation vectors. That is, all three vectors point in the same

direction for a homogeneous plane wave. This direction is, unambiguously, the direction of

propagation of the plane wave. The planes of constant phase are also then the planes of constant

amplitude, being the planes perpendicular to the !R vector. That is, the plane wave has a uniform

amplitude across the plane perpendicular to the direction of propagation. If the plane wave is not

homogeneous, corresponding to complex direction angles, then the physical interpretation of the

direction angles is not clear.

From Eq. (18) or (19) it may be easily proven that the fields of a homogeneous plane wave

obey the relation

E H= η . (34)

(Recall that all plane waves obey Eq. (22), which is not in general the same as the above

equation.)

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Any homogeneous plane wave can be decomposed into a sum of two plane waves, with

electric field vectors polarized perpendicular to each other, and with electric field vectors that are

real, to within a multiplicative complex constant. This follows from a simple rotation of

coordinates (the direction of propagation is then z' , with electric field vectors in the x' and y'

directions. the two plane wave then have the fields ( ),x yE H and ( ),y xE H , with

/ /x y y xE H E H η= − = . This decomposition is used later in the discussion of polarization.

Lossless media

Another important special case is when the medium is lossless, so that k ' '= 0 . If Eq. (4) is

substituted into the separation equation (17), the imaginary part of this equation immediately

yields the relation

ββββ αααα⋅ = 0 . (35)

Hence, for a lossless region, the phase and attenuation vectors are always perpendicular. In some

applications (e.g., a Fourier transform solution of radiation from an aperture in a ground plane at

z = 0, or from a planar current source at z = 0 [3]), a plane wave propagating in a lossless region

has the characteristic that two of the wavenumbers, e.g., k x and k y , are real (corresponding to

the transform variables). In this case the third wavenumber kz will be real if k k kx y2 2 2+ < and

will be imaginary if k k kx y2 2 2+ > . That is, all transverse wavenumbers k kx y,d i that lie within a

circle of radius k in the wavenumber plane will be propagating, while all wavenumber outside

the circle will be evanescent. In the first case the power flow is in the direction of the (real) k

vector k k kx y z, ,d i , so that power leaves the aperture from this plane wave. In the second case

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the power flow is in the direction of the transverse wavenumber vector k kx y, ,0d i , so that no

power leaves the aperture for this plane wave component. If the medium is complex, there is no

sharp distinction between propagating and evanescent plane waves. In this case all plane waves

carry power, in the direction of the vector k k kx y z, , Imd i .

Finally, it can be noted that if a homogeneous plane wave is propagating in a lossless region,

then the attenuation vector αααα must be zero, and all wavenumber components k x , k y , and kz

are real.

Summary of Basic Properties

A summary of the basic properties of a plane wave, discussed in the previous sections, is

given in Table 1.

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Table 1. Summary of the basic properties of a plane wave.

INHOMOGENEOUS HOMOGENEOUS PROPERTY Lossy Lossless Lossy Lossless

Direction angles complex complex real real Relation between phase and attenuation vectors

ββββ αααα⋅ = k k' ''

ββββ αααα⋅ = 0

ββββ αααα||

αααα = 0

Impedance relations

E E H H⋅ = ⋅η2

E E H H⋅ = ⋅η2

E E H H⋅ = ⋅η2

E H= η

E E H H⋅ = ⋅η2

E H= η

Direction of power flow vector

not necessarily in ββββ direction

ββββ

if k, E0 or H0 is proportional to a real vector

ββββ

ββββ

Wavenumber vector

k k⋅ = k 2

k k⋅ = k 2

k k⋅ = k 2

k k⋅ =* k 2

k k⋅ = k 2

k k⋅ =* k 2

Orthogonality of vectors

E HE kH k

⋅ =⋅ =⋅ =

000

E HE kH k

⋅ =⋅ =⋅ =

000

E HE kH k

⋅ =⋅ =⋅ =

000

E HE kH k

⋅ =⋅ =⋅ =

000

3. PROPAGATION OF A HOMOGENEOUS PLANE WAVE

The case of a homogeneous plane wave is important enough to warrant further attention. Far

away from a transmitting antenna, the radiation field always has behaves as a homogeneous

plane wave (if the spherical wavefront is approximated as planar). As mentioned, a

homogeneous plane wave is one propagating with real direction angles in space, with a direction

of propagation described by a real unit vector. By a suitable rotation of coordinate axis, the

direction of propagation may be assumed to be along the z axis. Furthermore, the plane way can

always be represented as a sum of two separate plane waves, one with the electric field polarized

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in the x direction, and the other polarized in the y direction. Considering the wave polarized in

the x direction, the fields are

E E exjk z= −

0 (36)

H E eyjk z= −0

η . (37)

The wavenumber k and the intrinsic impedanceη of the space are given by Eqs. (12) and (23),

respectively.

Wavelength, phase velocity, and group velocity

The wavelength λ is defined at the distance required for the plane wave to change phase by 2π

radians, and is given by

λ π= 2k ' . (38)

The phase velocity of the plane wave is defined as the velocity at which a point of constant phase

travels (such as the crest of the wave, where the electric field is the maximum). The phase

velocity is given by

vkp = =ω ω

ω µε' Ree j . (39)

If the permittivity ε is not a function of frequency (true for vacuum, and often approximately true

for nonconducting media, over a certain frequency range), then the phase velocity is constant

with frequency (the plane wave propagates without dispersion). For vacuum the phase velocity is

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c = 2.99792458 × 108 meters/second. The group velocity (velocity of energy flow, and often a

good approximation to the velocity of signal propagation [4], is given by

v ddk d

d

g = =ω

ωω µε

'Re

1

e j. (40)

For a nondispersive lossless media (constant real-valued permittivity) the group velocity is equal

to the phase velocity, and both are equal to

v vp g= = 1µε . (41)

Depth of penetration (skin depth)

For a lossy medium, the plane wave decays as it propagates, since k '' > 0 . The depth of

penetration d p is defined as the distance required to reduce the field level by a factor of

e ≈ 2 718. 28, so that the field is 36.788 % of the starting value (the power is reduced by a factor

of e2, or is at a level of 13.534 % of the starting value). The depth of penetration is given as

dkp = = −1 1

'' Im ω µεe j . (42)

The permittivity is complex for a lossy media, and is represented by Eq. (10). Special cases are

of particular interest. For a low loss medium, ε ε'' '<< , a simple binomial approximation of the

square root in Eq. (42) gives

dk kp = ≈1 1

05'' '. tanδ (43)

where

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k ' '≈ ω µε (44)

and the loss tangent is defined as

tan''

'δ εε

= . (45)

If the loss is due purely to medium conductivity (no polarization loss, so that ! 'ε ε= is a real

number), the penetration depth formula (43) for the low-loss case becomes

d p ≈2σ

εµ

'

. (46)

For a highly lossy medium, ε ε' ''<< , in which case we have

k k' ''''

≈ ≈ ω µε2 . (47)

For a good conductor the conductivity is very large, so that ε σ ω'' /≈ from Eq. (10). In this

case the penetration depth is more commonly referred to as the skin depth δ, since the value

may be very small. (There is no relation to the δ symbol appearing in the loss tangent symbol.)

For a good conductor,

δωµσ

= ≈d p2

. (48)

For pure copper (σ = ×58 107. S / m , µ µ≈ 0 ) at 2.45 GHz, δ µ= 1335. m .

Whether a particular material falls into the low loss or highly lossy categories may

depend on frequency. For example, consider typical seawater, which has roughly !ε r = 78

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(ignoring polarization losses) and σ = 4 0. S / m . Table 2 shows the loss tangent versus

frequency. It is seen that the loss tangent is on the order of unity at microwave frequencies. For

much lower frequencies the seawater is highly lossy, and for much higher frequencies it is a

low loss media. Note that the depth of penetration continues to decrease as the frequency is

raised, even though the medium becomes more of a low loss material at higher frequencies,

according to the definition used (a low loss tangent). At low frequencies the penetration depth is

increasing inversely as the square root of frequency, according to Eq. (48). At high frequencies

the penetration depth approaches a constant, according to Eq. (46).

Table 2. Loss tangent and penetration depth for typical seawater, versus frequency.

frequency (GHz)

loss tangent ( ε ε'' '/ )

penetration depth d p (meters)

0.000001 921800 7.958 0.00001 92180 2.5165 0.0001 9218 0.7958 0.001 921.8 0.2518 0.01 92.18 0.0800

0.1 9.218 0.0266 1.0 0.9218 0.0127 10.0 0.09218 0.01173 100.0 0.009218 0.01172

Summary of Homogeneous Plane Wave Properties

A summary of the properties relating to wavelength, velocity, and depth of penetration for a

homogeneous plane wave is given in Table 3.

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Table 3. Propagation Properties of a homogeneous plane wave.

General Highly Conducting Low Loss Lossless

wavelength 2πk ' 2 2π

ωµσ 2π

ω µε ' 2π

ω µε

phase velocity ωk '

2ωµσ

1

µε ' 1µε

group velocity ddkω

' 2 2ωµσ

1

µε ' 1µε

depth of penetration

1/ ''k 2

ωµσ 2σ

εµ

'

∞

Polarization

The above discussion has assumed a homogeneous plane wave propagating in the z direction,

polarized with the electric field in the x direction. The most general polarization of a wave

propagating in the z direction is one having both x and y components of the field,

E x y= + −! ! ( )E E ex yj k z

0 0d i . (49)

The field components are represented in polar form as

E E ex xj x

0 0= φ , (50)

E E ey yj y

0 0= φ. (51)

The phase difference between the two components is defined as

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φ φ φ= −y x . (52)

Without any real loss of generality, φ x may be chosen as zero, so that φ φ= y . In the time

domain, the field components are then

Ex xE t= 0 cos ωb g (53)

Ey yE t= +0 cos ω φb g . (54)

Using trigonometric identities, Eq. (54) may be expanded into a sum of sin ωtb g and cos ωtb g

functions, and then Eq. (53) may be used to put both cos ωtb g and sin ωtb g in terms of Ex (using

sin cos2 21= − ). After simplification, the result is

2 2x x y yA B C D+ + =E E E E (55)

where

AEE

y

x= 0

0

2

(56)

BEE

y

x= −2 0

0cosφ (57)

C = 1 (58)

D E y= 02 2sin φ . (59)

After simplification, the discriminant of this quadratic curve is

∆ = − = −B ACEE

y

x

2 0

0

224 4 sin φ , (60)

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which is always negative. This curve thus always represents an ellipse. The general form of the

ellipse is shown in Fig. 1. The tilt angle of the ellipse is τ, and the axial ratio AR is defined as the

ratio of the major axis of the ellipse to the minor axis ( AR ≥ 1 ). In the time domain the electric

field vector rotates with the tip of the vector lying on the ellipse. Right-handed elliptical

polarization, or RHEP, corresponds to counterclockwise rotation (the thumb of the right hand

aligns with the direction of propagation, the fingers of the right hand align with the direction of

rotation in time). (This is the IEEE definition, opposite to the usual optics convention.) LHEP

corresponds to rotation in the opposite direction.

x

y

τ

A1

B1A2

B2

A1

B1

A2

B2AR =

Figure 1. Geometry of the polarization ellipse.

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A convenient way to represent the polarization state is with the Poincaré sphere [5]. Using

spherical trigonometric relations, the following results may be derived for the tilt angle of the

ellipse and the axial ratio.

tan tan cos2 2τ γ φ= (61)

sin sin sin2 2ξ γ φ= , (62)

where the parameter ξ is related to the axial ratio as

( )1 0 0cot , 45 45for LHEPfor RHEP.

ARξ ξ−= − ≤ ≤ ++−

∓

(63)

The phase angle φ is defined in Eq. (52). The parameter γ characterizes the ratio of the fields

along the x and y axes, and is defined from

01 o

0

tan , 0 90y

x

EE

γ γ−= ≤ ≤ . (64)

There is no ambiguity in Eq. (62) for ξ, since − ≤ ≤ +90 2 900 0ξ . However, Eq. (61) gives an

ambiguity for τ, since adding multiples of 180o does not change the tangent. To resolve this

ambiguity, Table 4 may be used to determine the appropriate quadrant (1, 2, 3 or 4) that the

angle 2τ is in, based on the Poincaré sphere [5].

Table 4. Quadrant that the angle 2ττττ is in.

cosφ > 0 cosφ < 0

cos2 0γ > 1 4

cos2 0γ < 2 3

21

The following special cases are important.

1) Linear polarization: φ = 0 1800or , or either Ex0 or E y0 is zero.

In this case AR = ∞ (the ellipse degenerates to a straight line).

2) Circular polarization: E Ey x0 0= and φ = ±900 (to within any multiple of 180o ).

In this case the ellipse becomes a circle (either RHCP or LHCP), and AR = 1.

It can also be noted that a wave of arbitrary polarization can be represented as a sum of RHCP

and a LHCP waves, by noting that the arbitrarily polarized wave in Eq. (49) can be written as

E r l= +LNM

OQP+ −LNM

OQP! !1

2120 0 0 0E jE E jEx y x yd i d i (65)

where the unit-amplitude RHCP and LHCP waves are

! ! !r x y= − −12

j e jkzb g (66)

! ! !l x y= + −12

j e jkzb g . (67)

5. PLANE WAVE REFLECTION AND TRANSMISSION

A general plane wave reflection/transmission problem consists of an incident plane wave

impinging on a multilayer structure, as shown in Fig. 2. An important special case is the two-

region problem shown in Fig. 3. The incident plane wave may be either homogeneous or

inhomogeneous, and any of the regions may have an arbitrary amount of loss. The vector

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representing the direction of propagation for the incident wave in Figs. 2 and 3 is a real vector

(as shown) if the incident wave is homogeneous, but the analysis is valid for the general case.

Figure 2. A plane wave reflecting from a multilayer stack of different materials. Each layer has a uniform (constant) set of material parameters. On the right side the equivalent transmission-line model (transverse equivalent network) is shown.

Figure 3. Reflection and transmission from a single interface between two different media. The transmission-line model (transverse equivalent network) for the two-region reflection problem is shown on the right.

inc ref

transθ t

θ i 1

2

n

N

inc ref

trans

Z0(n)

incinc

ref

ref

trans transθ t

θ i 1

2

Z 0(1)

Z 0(2)

23

The key to obtaining a simple solution to such reflection and transmission problems is the use of

a transmission line model of the layered structured, which is based on TE-TM decomposition of

the plane waves, discussed next.

TE-TM Decomposition

According to a basic electromagnetic theorem [1], the fields in a source-free homogeneous

region can be represented as the sum of two types of fields, a field that is transverse magnetic to

z (TMz) and a field that is transverse electric to z (TEz). The TMz field is defined as one that has

Hz = 0, while the TEz field by definition has Ez = 0. A general plane wave field may therefore be

written as the sum of a TMz plane wave and a TEz plane wave. In free space, the direction z is

rather arbitrary. When a layered media is present, the preferred direction z is perpendicular to the

layers, because this allows for the TMz and TEz plane waves in each region to be modeled as

waves on a transmission line. Standard transmission-line theory may then be conveniently used

to solve plane-wave reflection and transmission problems in a relatively simple manner, without

having to solve the electromagnetic boundary-value problem of matching fields at the interfaces.

The TMz plane wave is commonly referred to as one that is polarized with the electric field in

the place of incidence, while the TEz place wave is polarized with the electric field

perpendicular to the plane of incidence. (The plane of incidence is the y-z plane.) For a

homogeneous plane wave with an incident wave vector in the y-z plane, the polarizations are

shown in Fig. 4.

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y

z

xTM z zTE

E

EH

H

Figure 4. Polarizations of incident TMz and TEz plane waves.

The field components may be found by using Maxwells equations to write the transverse (x

and y) field components in terms of the longitudinal components Ez and Hz. The nonzero

longitudinal component is assumed to be proportional to the wavefunction

( )exp ( )x y zj k x k y k zψ = − + ± . (68)

The plus sign in this equation is chosen for plane waves propagating or decaying in the positive z

direction, while the negative sign is for propagation or decay in the minus z direction. This

representation is convenient for plane-wave reflection problems, since the characteristic

impedance of a transmission line that models a particular region will then have a positive

characteristic impedance for both upward and downward propagating plane waves, in agreement

with the usual transmission line convention.

For the TMz plane wave, the normalized field components may then be written as

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E k k x y z H k x y z

E k k x y z H k x y z

E k k x y z H

x x z x y

y y z y x

z z z

= =

= = −

= − =

∓

∓

Ψ Ψ

Ψ Ψ

Ψ

, , , ,

, , , ,

, , .

b g b gb g b g

c h b g

ωεωε

2 2 0 (69)

For a TEz plane wave the corresponding results are

E k x y z H k k x y z

E k x y z H k k x y z

E H k k x y z

x y x x z

y x y y z

z z z

= − =

= =

= = −

ωµωµ

Ψ Ψ

Ψ Ψ

Ψ

, , , ,

, , , ,

, , .

b g b gb g b g

c h b g

∓

∓

0 2 2 (70)

Note that a plane wave propagating in the z direction (kz = k) has both longitudinal field

components that are zero. Such a plane wave is TEM (transverse electric and magnetic) to the z

direction)

It may be seen from the above equations that the transverse fields obey the relations

E z HtTM TM

tTMZ= ×∓ 0 !c h (71)

E z HtTE TE

tTEZ= ×∓ 0 !c h (72)

where

Z kTM z0 =

ωε (73)

Zk

TE

z0 = ωµ

. (74)

26

Transverse Equivalent Network

From the above relations (71) and (72), the transverse (perpendicular to z) field components

behave as voltages and currents on a transmission line model called the transverse equivalent

network. In particular, the correspondence is given through the relations (i = TM or TE)

E eti

tix y V z= ! ,ψ b g b g (75)

H hti

tix y I z= ! ,ψ b g b g (76)

where the unit vectors are chosen so that

× = ±e h z (77)

for a plane wave propagating in the ±z direction. The transverse wavefunction

ψ tj k x k yx y e x y,b g d i= − +

(78)

is a common transverse phase term that must be the same for all regions (if the transverse

wavenumbers kx or k y were different between two regions, a matching of transverse fields at

the boundary would not be possible). The fact that the transverse wavenumbers are the same in

all regions leads to the law of reflection, which states that the direction angle θ for a reflected

plane wave must equal that for the incident plane wave. It also leads to Snells law, which states

that the direction angles θ inside each of the regions are related to each other, through

n n i Ni isin sin , , ,θ θ= =1 1 1 2… , (79)

where in is the index of refraction (possibly complex) of region i, defined as

0/i i ri rin k k ε µ= = . It is often convenient to express the characteristic impedance for region i

from Eqs. (73) and (74) in terms of the medium intrinsic impedance iη as

27

Z TMi i0 = η θcos (80)

Z TEi i0 = η θsec . (81)

Using Snells law, the impedances then become

Znn

TMi

i0

12

211= −

FHGIKJη θsin (82)

Znn

TE i

i

0

12

211

=

−FHGIKJ

η

θsin. (83)

The above expressions remain valid for lossy media. The square roots are chosen so that the real

part of the characteristic impedances are positive.

The functions V(z) and I(z) behave as voltage and current on a transmission line, with

characteristic impedance Z TM0 or Z TE

0 , depending on the case. Hence any plane-wave reflection

and transmission problem reduces to a transmission line problem, giving the exact solution that

satisfies all boundary conditions. One consequence of this is that TMz and TEz plane waves do

not couple at a boundary. If the incident plane wave is TMz, for example, the waves in all regions

will remain TMz plane waves. Hence, the motivation for the TMz-TEz decomposition. The

transverse equivalent network for the multilayer and two-region problems are shown on the right

sides of Figs. 2 and 3, respectively. The network model is the same for either TMz or TEz

polarization, except that the characteristic impedances are different. If an incident plane wave is

a combination of both TMz and TEz waves, the two parts are solved separately and then summed

to get the total reflected or transmitted field.

28

Special Case: Two-Region Problem For the simple two-region problem of Fig. 3, the reflected and transmitted voltages

(modeling the transverse electric fields) are represented as

V z V e jk zzref incb g b g= +Γ 01)(

(84)

V z TV e jk zztrans incb g b g= −02( )

(85)

where the reflection and transmission coefficients are given by the standard transmission line

equations

Γ =−+

Z ZZ Z

02

01

02

01

( ) ( )

( ) ( ) (86)

T = +1 Γ . (87)

Note that a plus sign is used in the exponent of Eq. (84) to account for upward propagation of the

reflected wave.

A. Critical angle

If regions 1 and 2 are lossless, and region 1 is more dense than region 2 ( n n1 2> ), an

incident angle θ θ1 = c will exist for which kz( )2 0= . From Snells law, the angle is

θ cnn

=FHGIKJ

−sin 1 2

1. (88)

When θ θ1 > c , the wavenumber kz( )2 will be purely imaginary, of the form k jz z

( ) ( )2 2= − α . In

this case there is no power flow into the second region, since the phase vector in region 2 has no

z component. In this case 100% of the incident power is reflected back from the interface. There

29

are, however, still fields present in the second region, decaying exponentially with distance z. In

the second region the power flow is in the horizontal direction only.

B. Brewster angle

For lossless layers, it is possible to have 100% of the incident power transmitted into the

second region, with no reflection. This corresponds to a matched transmission line circuit, with

Z Z01

02( ) ( )= . (89).

For nonmagnetic layers, it may be easily shown that this matching equation can only be satisfied

in the TMz case (there is always a nonzero reflection coefficient in the TEz case, unless the trivial

case of identical medium is considered). The angle θ b at which no reflection occurs is called the

Brewster angle. For the TMz case, a simple algebraic manipulation of Eq. (89) yields the result

tanθ εεb = 2

1. (90)

Orthogonality

When treating reflection problems, there is more than one plane wave in at least one of the

regions, a wave traveling in the positive z direction (focusing on the z variation) and a wave

traveling in the negative z direction. Often, it is desired to calculate power flow in such a region.

For a single plane wave, the complex power density in the z direction (the z component of the

complex Poynting vector) is equal to the complex power flowing on the corresponding

transmission line of the transverse equivalent network. When both an incident and a reflected

wave are present, or both TMz and TEz waves are present the following orthogonality results are

30

useful. These theorems related to power flow in the z direction may be proven by direct

calculation using the field components in rectangular coordinates.

1. An orthogonality exists between a homogeneous TMz plane wave and a homogeneous TEz

plane wave propagating in the same direction, in the sense that the complex power density in

the z direction, Sz , is the sum of the two complex power densities S z1 and S z2 . This is true

for a lossy or lossless media.

2. An orthogonality exists between an incident wave and a reflected wave in a lossless media

(both are either TEz or TMz), provided the wavenumber component kz of the two waves is

real. The two waves are orthogonal in the sense that the time-average power density in the z

direction, Re Sz , is the sum of the two time-average power densities Re S z1 and Re S z2 .

To illustrate property 2, consider an incident plane wave traveling in a glass region, impinging

on an air gap that separates the glass region from another identical glass region, as shown in Fig.

5.

glass

glass

air

inc ref

trans

θ

h

31

Figure 5. An incident wave traveling in a semi-infinite region of lossless glass impinges on an air gap separating the glass region from an identical region below. Incident, reflected and transmitted plane waves are shown.

If the incident plane wave is beyond the critical angle, the plane waves in the air region will be

evanescent, with an imaginary vertical wavenumber kz. Each of the two plane waves in the air

region (upward and downward) that constitute a standing-wave field do not, individually, have a

time-average power flow in the z direction, since kz is imaginary. However, there is an overall

power flow in the z direction inside the air region, since there is a transmitted field in the lower

region. The total power flow in the z direction is thus not the sum of the two individual power

flows. The two waves in the air region are not orthogonal, and property 2 does not apply since

the wavenumber kz is not real.

VI. EXAMPLE

A RHCP plane wave at a frequency of 1.0 GHz is incident on the surface of the ocean at an angle

of θ = 30o . Determine the percentage of power that is reflected from the ocean, and characterize

the polarization of the reflected wave (axial ratio, tilt angle, and handedness). Also, determine

the field of the inhomogeneous transmitted plane wave. The parameters of the ocean water are

assumed to be !ε = 78 and σ = 4 S/m.

Solution

The geometry of the incident and reflected waves is shown in Fig. 6. The incident plane wave is

represented as

E x uinc = + − − +E j e j k y k zy z0

1! ! b g d i (91)

where

32

k k ky = =0 012

sinθ (92)

k k kz1 0 03

2= =cosθ (93)

! ! cos ! sinu y z= −θ θ . (94)

The reflected wave is represented as

E x vref = + − −E A B e j k y k zy z0

0! ! d i (95)

where

! ! cos ! sinv y z= − −θ θ . (96)

inc ref

θ 1

2

u v

y

z

ocean

Figure 6. Geometry that defines the coordinate system for the example of a plane wave reflecting from the surface of the ocean.

The x component of the incident and reflected waves corresponds to TEz waves, while the u and

v components correspond to TMz waves. (The u and v directions substitute for the y direction in

the previous discussion on polarization.) The transverse equivalent network is shown in Fig. 3.

From Eqs. (82) and (83), the impedances are Z TM1 326 258= . Ω , Z TE

1 435 011= . Ω ,

33

Z jTM2 34 051 13269= +. ( . ) Ω , Z jTE

2 34 089 13346= +. ( . ) Ω . The reflection coefficients are then

ΓTM j= − +0808535 0 066600. ( . ) and ΓTE j= − +0 853161 0 052724. ( . ) .

The coefficient A in Eq. (94) is determined directly from

A jTE= = − +Γ 0853161 0 052724. ( . ) . (97)

To determine the coefficient B, recall that the voltages in the transverse equivalent network

model only the transverse (horizontal) component of the electric field (the y component in the

TM case). Hence,

− = −BE E jTM0 0cos cosθ θref incΓ b g . (98)

Since θ θ θinc ref= = , B is determined directly as

B j jTM TM= = = − −Γ Γb g 0 066600 0 808535. ( . ) . (99)

The percent power reflected is calculated from

P

AZ

BZ

ZjZ

TE TM

TE TM

%

cos

cosref =

+LNMM

OQPP

+−L

NMM

OQPP

F

H

GGGGG

I

K

JJJJJ100

12

12

1

1

2 2

1

1

2

1

θ

θ (100)

which yields

P% .ref = 69 441 . (101)

Hence 30.559 % of the incident power is transmitted into the ocean.

The phase angle between the v and x components of the reflected field is

φ = ∠ −∠ =B A 88 827. o (102)

Hence, the reflected wave is a left-handed elliptically polarized wave.

34

The parameter γ is determined from the ratio of the magnitudes of the v and x components of

the reflected field as

γ =FHGIKJ =

−tan .1 43504BA

o. (103)

From the formulas in the Polarization section, the tilt angle τ of the polarization ellipse from the

x axis (towards the positive v axis) is then

τ = 10 690. o , (104)

and the polarization parameter ξ is

ξ = 43 393. o . (105)

The axial ratio of the reflected wave is then

AR = + =cot .ξ 1058 . (106)

The reflected wave is nearly CP, since the ocean is highly reflecting (if the ocean were replaced

by a perfect conductor, the reflected wave would be LHCP).

The transmitted field in the ocean is represented as

E x y ztrans = + + − −E C D E e j k y k zy z0

2! ! ! d i. (107)

where 2zk is the vertical wavenumber in the ocean, found from Eq. (31) as

( )2

22 2 0 2 0

2

1cos 1 sin 9.582527 3.751642z tk k k n k jn

θ θ

= = − = −

. (108)

From the transverse equivalent network,

C T jTE TE= = + = +1 0146839 0 052724Γ . ( . ) (109)

35

and

D T j j jTM TM= − = + − = −cos cos . ( . )θ θb g e jb g1 0 057678 0165813Γ (110)

The z component of the transmitted field can be determined from the electric Gauss law (20),

which gives

Ek

k Dz

y= − 1

2d i . (111)

Hence,

E j= − +0 0055466 0 00648030. ( . ) . (112)

From the wavenumbers k y and kz2 the direction angles of the transmitted wave may be

determined from Eqs. (29) (31). The results are

φ = 90o (113)

θ t j= +0 045227 0 017679. ( . ) radians . (114)

The transmitted plane wave is inhomogeneous, since the angle θt is complex.

REFERENCES

[1] R. F. Harrington, Time-Harmonic Electromagnetic Fields, McGraw Hill, 1961.

[2] W. H. Hayt, Engineering Electromagnetics, 5th Ed., McGraw-Hill, 1989, Appendix B.

36

[3] P. C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, IEEE Press, 1996, pp. 33-36.

[4] R. E. Collin, Field Theory of Guided Waves, 2nd Ed., IEEE Press, 1991, pp. 231-236.

[5] J. D. Kraus, Antennas, McGraw-Hill, 1988, pp. 75-79.