Electromagnetic Waves, Antennas, and RF: Boundary Waves - Reflection, Transmission, and Refraction

Electromagnetic Waves, Antennas, and RF: Boundary Waves - Reflection, Transmission, and Refraction

EM20 showed how a uniform electromagnetic wave propagates inside a homogeneous material. Real propagation paths are rarely uniform forever. A wave eventually meets a boundary: air to glass, air to soil, free space to a radome, dielectric to Conductor, or one layer of a circuit or antenna structure to another.

At that boundary, Maxwell’s Equations do not permit the fields to change arbitrarily. Their allowed discontinuities are fixed by the electromagnetic boundary conditions. Those conditions, together with the plane-wave relation between electric and magnetic fields, determine how much of the incident wave is reflected, how much is transmitted, whether the transmitted wave changes direction, and whether a phase reversal occurs [1, 3, 4, 5].

The central chain developed in this article is

• Maxwell equations;
• boundary conditionse;
• Γ,τ;
• R,T;
• Snell and Fresnel laws;

1 The interface geometry

Let medium 1 occupy z < 0 and medium 2 occupy z > 0. The interface is the plane z = 0, with unit Normal

(1)

pointing from medium 1 into medium 2.

Each medium has constitutive parameters

(2)

and therefore its own propagation constant and intrinsic impedance,

(3)

For the first derivation we use lossless media, so η₁ and η₂ are real. The complex-impedance extension is introduced later.

2 Boundary conditions come directly from Maxwell’s equations

The wave-interface formulas are not separate laws added to Electromagnetism. They follow from the integral forms of Maxwell’s equations by shrinking the integration surface or contour around the interface [1, 2, 3].

2.1 Tangential electric field

Faraday’s law in integral form is

(4)

Take a narrow rectangular contour that straddles the interface. As its height tends to zero, the enclosed area tends to zero, so the magnetic-flux term vanishes for finite fields. The two long sides remain, giving

(5)

Thus the tangential electric field is continuous across an ordinary interface.

2.2 Tangential magnetic field

Ampere-Maxwell law gives

(6)

If a free surface-current density K_s exists at the interface, the limiting contour gives

(7)

For the dielectric interfaces considered first,

(8)

so the tangential magnetic field is also continuous.

2.3 Normal electric flux density

Gauss’s Law applied to a thin pillbox gives

(9)

where ρ_s is free surface-charge density.

2.4 Normal magnetic flux density

Gauss’s law for magnetism similarly gives

(10)

The normal component of B is therefore continuous because magnetic monopole surface charge does not appear in classical electromagnetism.

3 Normal incidence: incident, reflected, and transmitted waves

Consider a linearly polarized plane wave incident normally from medium 1. Choose

(11)

for the incident phasor. The reflected wave propagates in the −z direction,

(12)

and the transmitted wave propagates in the +z direction,

(13)

For a uniform plane wave,

(14)

Therefore

(15)

(16)

and

(17)

The minus sign in H_r is essential: the reflected wave carries energy toward −z.

4 Deriving the normal-incidence reflection coefficient

At z = 0, tangential E is continuous:

(18)

Tangential H is also continuous when K_s = 0:

(19)

Define the electric-field reflection coefficient

(20)

and transmission coefficient

(21)

The first boundary condition gives

(22)

The magnetic-field condition gives

(23)

Substitute τ = 1 + Γ:

(24)

Collect the Γ terms:

(25)

Hence

(26)

Then

(27)

These two compact expressions are the normal-incidence Fresnel amplitude coefficients written in impedance form.

5 What the sign of the reflection coefficient means

If

(28)

then

(29)

The reflected electric field is then reversed in sign relative to the incident electric field at the boundary. For a sinusoid, this is a 180^∘ phase reversal.

If

(30)

then

(31)

and there is no such sign reversal.

If

(32)

then

(33)

The interface is impedance matched and no plane-wave reflection occurs at normal incidence.

6 Power reflection and transmission coefficients

The time-averaged power density of a sinusoidal plane wave in a lossless medium is

(34)

Define the reflected power fraction

(35)

Because the incident and reflected waves are in the same medium,

(36)

For the transmitted power fraction,

(37)

so

(38)

Therefore

(39)

For two lossless media with no absorption at the boundary,

(40)

This is simply energy conservation applied to the interface.

7 Example: air into a dielectric with relative permittivity 4

Let medium 1 be air, approximated as free space,

(41)

and let medium 2 be a nonmagnetic lossless dielectric with

(42)

Its intrinsic impedance is

(43)

Thus

(44)

so

(45)

The reflected electric field has one-third the incident amplitude and is phase reversed.

The transmitted electric-field coefficient is

(46)

The reflected power fraction is

(47)

The transmitted power fraction is

(48)

so

(49)

Notice the important distinction: τ = 2∕3 is a field-amplitude ratio, not a power ratio.

8 Reflection from a lossy medium

The same normal-incidence amplitude formula remains valid when medium 2 has complex intrinsic impedance:

(50)

Now Γ is generally complex. Its magnitude gives the reflected field-amplitude fraction and its argument gives the reflection phase shift:

(51)

The transmitted wave also attenuates after entering medium 2 because

(52)

Thus two physically distinct effects must be separated:

(53)

For a general lossy medium at normal incidence, the time-average power flux should be computed from

(54)

rather than by blindly using the lossless expression |E|²∕(2η) with a complex η.

9 Why refraction occurs at oblique incidence

Now let the incident wave strike the boundary at angle 𝜃_i measured from the normal. The reflected and transmitted waves make angles 𝜃_r and 𝜃_t.

At every point along the interface, the boundary conditions must hold for all time. Therefore the incident, reflected, and transmitted waves must have the same phase variation parallel to the interface. If the interface lies along x, phase matching requires

(55)

The first equality gives

(56)

The second gives

(57)

For lossless isotropic media,

(58)

so

(59)

This is Snell’s law, obtained directly from electromagnetic phase continuity along the boundary.

10 TE and TM polarizations at an interface

At oblique incidence, polarization matters because the electric and magnetic fields project differently onto the interface.

10.1 TE polarization

For transverse-electric (TE, or s) polarization, the electric field is perpendicular to the plane of incidence. Applying the tangential E and H boundary conditions gives

(60)

The corresponding electric-field transmission coefficient is

(61)

10.2 TM polarization

For transverse-magnetic (TM, or p) polarization, the magnetic field is perpendicular to the plane of incidence. The electric-field reflection coefficient is

(62)

The electric-field transmission coefficient is

(63)

At normal incidence,

(64)

and both TE and TM reflection coefficients reduce to

(65)

11 Power coefficients at oblique incidence

For lossless media, the normal component of average power flow is proportional to

(66)

Therefore, for either TE or TM polarization,

(67)

and

(68)

For lossless interfaces,

(69)

The cosine factor appears because the relevant conserved quantity is power crossing the interface, not power along the slanted propagation direction.

12 Brewster angle

A special feature appears for TM polarization. At one incidence angle the numerator of Γ_TM can vanish:

(70)

For nonmagnetic lossless media, this occurs at the Brewster angle

(71)

At 𝜃_i = 𝜃_B, an ideal TM-polarized plane wave has zero reflected amplitude.

For air to glass with

(72)

(73)

This effect is polarization selective: the TE reflection coefficient does not vanish at the same angle.

13 Critical angle and total internal reflection

Suppose the wave travels from a higher-index medium into a lower-index medium,

(74)

Snell’s law requires

(75)

At a sufficiently large 𝜃_i, the right side would exceed 1 if interpreted as an ordinary propagating transmitted angle. The limiting incidence angle is the critical angle,

(76)

For

(77)

there is no propagating transmitted plane wave carrying power away from the boundary in the usual refracted direction. Instead, the field in medium 2 is evanescent in the normal direction while the reflection magnitude becomes

(78)

The phase of Γ remains nontrivial even though all time-average normal power is reflected. This phenomenon is total internal reflection.

14 Fresnel reflectance versus angle

For air-to-glass incidence, n₁ = 1 and n₂ = 1.5. The TE and TM reflection coefficients behave differently as the incidence angle grows. The TM reflectance falls to zero at the Brewster angle, while the TE reflectance rises monotonically toward unity near grazing incidence.

15 Standing-wave structure on the incident side

When a reflected wave is present, the total electric field in medium 1 is

(79)

Using

(80)

we obtain

(81)

The reflected and incident fields interfere, producing spatial maxima and minima. For a lossless medium and fixed |Γ| < 1,

(82)

(83)

Their ratio is

(84)

This same mathematical structure later reappears in transmission-line voltage standing-wave ratio (VSWR). The transmission-line result is not a coincidence: it is another manifestation of forward and backward waves meeting an impedance discontinuity.

16 Several important limiting cases

16.1 Matched boundary

If

(85)

then at normal incidence

(86)

No reflection occurs even though other material properties may differ.

16.2 Perfect electric conductor

For an ideal perfect electric conductor, the tangential electric field at the surface must vanish. The reflected field therefore cancels the incident tangential electric field at the boundary:

(87)

Hence

(88)

This is complete reflection with a 180^∘ electric-field phase reversal.

16.3 Very good but finite conductor

A good conductor has a small complex intrinsic impedance compared with free space. Therefore

(89)

but not exactly. A small transmitted field enters the conductor and then decays over the skin depth developed in EM20.

17 The boundary-wave workflow

A reliable interface calculation can be organized as follows:

1. determine 𝜖, μ, and σ in each medium;
2. compute γ and η as needed;
3. identify the incidence angle and polarization;
4. enforce tangential phase matching to obtain reflection and refraction angles;
5. apply the appropriate TE or TM Fresnel coefficient;
6. convert field-amplitude coefficients into power coefficients with the correct impedance and angle factors;
7. if a transmitted medium is lossy, propagate the transmitted field with e^−αz after the interface.

This workflow keeps three distinct physical mechanisms separate:

• reflection at the boundary;
• refraction at the boundary;
• attenuation inside the material

18 Connection to RF systems and antennas

Boundary-wave physics appears throughout RF engineering. Examples include radomes, dielectric antenna substrates, soil and seawater propagation, building penetration, microwave windows, coaxial and waveguide discontinuities, and the transition from an antenna’s near environment into free space.

The reflection coefficient also foreshadows impedance matching in circuit and transmission-line language. At normal incidence,

(90)

has exactly the same algebraic form as the voltage reflection coefficient at a transmission-line load,

(91)

The common structure arises because both problems enforce continuity relations between paired wave variables and their impedance ratios.

19 What EM21 adds to the series

EM20 described propagation within one uniform medium. EM21 adds the physics of crossing between media. The key results are

(92)

for normal incidence,

(93)

(94)

and the TE/TM Fresnel coefficients for oblique incidence.

These results establish the mathematical foundation for multilayer propagation, transmission lines, Standing Waves, impedance matching, waveguides, radomes, and antenna-to-medium coupling. The natural companion is an EM21E1 problem set devoted to boundary-condition derivations, Fresnel coefficients, Brewster angle, total internal reflection, and power bookkeeping.

References

References