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EM theory and its application to microwave remote sensing

Chris Allen (callen@eecs.ku.edu)

Course website URL people.eecs.ku.edu/~callen/823/EECS823.htm

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OutlinePlane wave propagation

Lossless media

Lossy media

Polarization and coherence

Fresnel reflection and transmission

Layered media

EM spectra, bands, and energy sources

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Plane wave propagationPlane wave propagation through lossless and lossy media is fundamental to microwave remote sensing.

Consider the wave equation and plane waves in homogeneous unbounded, lossless media

Plane waves – constant phase and amplitude in the planeHomogeneous – electrical and magnetic parameters do not vary with

throughout the medium

Beginning with Maxwell’s equations and assuming a homogeneous, source-free medium leads to the homogeneous wave equation

whereE is the electric field vector (V/m) [note that bolded symbols denote vectors]

is the medium’s magnetic permeability (H/m) [H: Henrys]

is the medium’s permittivity (F/m) [F: Farads]

2

22

t

E

E

t

t

EH

HE

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Plane wave propagationAssuming sinusoidal time dependence

where is the radian frequency (rad/s) r is the displacement vectorand Re{} is the real operator

E(r,t) satisfies the wave equation if

Using phasor representation (i.e., e.jt is understood) and assuming a rectangular coordinate system, the solution has the general form of

where E0 is a constant vector and

tjeRet, rErE

022 rErE

)]zkykxk(j[exp zyx0 ErE

2z

2y

2x

22 kkkk

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Plane wave propagationA more compact form results from letting

where k is the propagation vector, and

k = |k| is called the wave number (rad/m)

resulting in

Finally reintroducing the time dependence and expressing only the real-time field component yields

This equation represents two waves propagating in opposite directions defined by the propagation vector k

zyx kˆkˆkˆ zyxk

]j[exp0 rkErE

rkErE tcost, 0

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Plane wave propagationRotating the Cartesian coordinate system such that the z axis aligns with k yields

representing two waves propagating in the + z and – z directions.

The argument of the cosine function contains two phase terms: the time phase, t, and the space phase, kz.

If we use the – part of the ± solution we have

representing a wave whose constant phase moves in the positive z direction. As time increases, z must increase to maintain a constant phase argument.

zktcost, 0 ErE

zktcost, 0 ErE

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Plane wave propagationThe time phase component is characterized by where

f is the frequency (Hz) and T is the time period (s).

Similarly the space phase component depends on k where

is the space period (m), or wavelength, in the medium which can also be expressed as

T2f2

2k

f1

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Plane wave propagationConsider now the electric field’s phase for a positive-traveling wave, i.e., t – kz.

A surface on which this phase is constant requires

For any given time t, this surface represents a plane defined by z = constant, on which both the phase and amplitude are constant. As time progresses, this plane of constant phase and amplitude advances along the z axis, hence the name uniform plane wave.

The rate at which this plane advances along the z axisis called the phase velocity, v (m/s)

constantzkt

1

ktd

zdv

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Plane wave propagationGiven an uniform plane E-field solution to the wave equation, the H-field is found using Maxwell’s equations

From the E-field component in the x-axis direction, Ex, is found the H-field component in the y-axis direction, Hy, as

where is the intrinsic impedance () of the medium

Note that Ex and Hy are related through the intrinsic impedance similar to how voltage and current in a circuit are related through Ohm’s law.

Note also the orthogonality of the E, H, and k vectors.

zktcosE

zktcosEk

t,zH 0x0xy

k

kˆ,Hˆ,Eˆ yx zkyHxE

t

H

E

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Plane waves in a lossy mediumA lossy medium is characterized by its permeability, , permittivity, , and conductivity, (S/m) [S: Siemens]. Maxwell’s equations for a source-free medium become

And the corresponding wave equation remains

where the wave number is

Note that for a lossless medium, k is purely real when = 0 and both and are real

t,

t

E

EHH

E

0k22 rErE

jjk

2k

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Plane waves in a lossy mediumFor a lossy medium k is complex

due to 0 or either or are complex

For lossless media the imaginary parts of the permeability and permittivity are zero.

Non-zero imaginary terms ( > 0 and > 0) represent mechanisms for converting a portion of the electromagnetic wave’s energy into heat, resulting in a loss of wave energy.

jjk

jj

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Plane waves in a lossy mediumConsider the complex electric field plane wave propagation along the positive z axis

whereas for the lossless case k was real, in a lossy medium k is complex and is related to the propagation factor or propagation constant, (1/m), by

such that

where and are real quantities and is the attenuation constant (Np/m) and is the phase constant (rad/m)[Np = Neper]

zktj0et,z EE

jjjkj

j

zj0

z0

zkj0 eeez EEEE

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Plane waves in a lossy mediumClearly for a wave travelling along the +z axis

as z increases, the magnitude of the electric field decreases.The real time expression for the x-axis field component is

The attenuation constant is the real part of jk

The phase constant is the imaginary part of of jk

Note: (Neper/m) 8.686 (dB/Neper) = (dB/m)

zjz0 eez EE

ztcoseEt,zE z0xx

m/Np,jjRe

m/rad,jjIm

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Plane waves in a lossy mediumIn a lossy medium, the intrinsic impedance is also complex

giving rise to a non-zero phase relationship between the E and H field components.

While a medium’s loss may be due to its conductivity, or the imaginary components of permittivity or permeability, in most microwave remote sensing applications the magnetization loss () is negligible.Exceptions include the ferrous-rich sands found in Hawaii and soils on the Martian surface.

Therefore the term will be neglected from now on.

A material’s permeability is usually specified relative to that of free space, o, (o = 4 10-7 H/m), as = r o and typically r = 1

j

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Plane waves in a lossy mediumA medium’s loss may be due to its conduction loss ( > 0 but finite) or its polarization loss ( > 0).

Conduction loss is gives rise to a conduction current

where electrical energy is converted to heat energy due to ohmic losses.

2C m/AEJ

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Plane waves in a lossy mediumPolarization loss is due to a displacement current, similar to current through a capacitor. For an ideal dielectric, equal amounts of energy are stored and released during each cycle. For lossy dielectrics, some of the stored energy is converted into heat.

The imaginary part of the displacement current is in-phase from the E field, and hence contributes to real energy loss.

2D m/A

t

E

J

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Plane waves in a lossy medium

For a sinusoidal time variation, there is a transition frequency, t = 2 ft, where these two current components are equal

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Plane waves in a lossy mediumFor dielectric materials, these two loss mechanisms are often combined into a single imaginary component as

The permittivity of dielectric materials is usually specified relative to the permittivity of free space, o, where (o = 8.854 10-12 F/m) as

where

j

or

rroro j

or

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Plane waves in a lossy mediumOften instead of specifying the r, a material’s loss tangent tan where

rrtan

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Plane waves in a lossy mediumClearly since the loss mechanism due to conductivity is frequency dependent, a medium’s loss characteristics may also be frequency dependent.At low frequencies where the conductivity introduces signficant loss, the intrinsic impedance and phase velocity will be frequency dependent.

At frequencies where the conductive loss is dimished, the intrinsic impedance and phase velocity will become frequency independent.

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Plane waves in a lossy mediumLow-loss media (i.e., tan 1)

For the case of low-loss media, the expressions for v, , , and can be simplified to be

where c = 3 108 m/s

where o = 120 377

22

2

8

11

r

o

ro

o 1

2j1

rroo

c11

8

11

1v

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Plane waves in a lossy mediumHigh-loss media (i.e., tan >> 1)

For the case of high-loss media, the expressions for , and can be simplified to

Also, when an electromagnetic wave impinges on a conducting medium, the field amplitude decreases exponentially with depth.

At a depth termed the skin depth, (m), the field amplitude is e-1 of its value at the surface, where

f

f

11

2

j1

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Plane waves in a lossy mediumGeneral lossy mediaFor media that is neither high loss nor low loss, the simplifying approximations and do not apply.For these cases we have

where ko is the free-space wave number and o is the free-space wavelength

It is sometimes useful to refer to a medium’s refractive index, n, where

212

ro 21tan1kImk

f/cwhere2

k oo

o

212

ro 21tan1kRek

rrrr2 norjn

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Plane waves in a lossy medium

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Plane waves in a lossy medium

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Polarization of plane waves and coherenceFor a +z-axis propagating uniform plane wave, the E-field components must lie in the xy plane

For any point on the xy-plane, the E-field varies with time.

The wave’s polarization is associated with the curve the tip of the E-field vector traces out.

A straight line indicates a linear polarization (i.e., y = 0 or 180) with tilt angle

y0y0x ztcosˆEztcosˆEt,z yxE

ztcosˆEˆEt,z 0y0x yxE

xoyo1 EEtan

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Polarization of plane waves and coherenceA circle indicates circular polarization

where Exo = Eyo and y = 90

Left-hand circular polarization (LCP), which results when y = +90, has the E-field rotating once each cycle in the direction the fingers of the left hand point when gripping the z-axis with the thumb in the +z direction.

Right-hand circular polarization (RCP), which results when y = -90, has the E-field rotating in the same direction as the fingers when gripping the z-axis with the right hand so that the thumb is in the +z direction.

y0y0x ztcosˆEztcosˆEt,z yxE

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Polarization of plane waves and coherenceAn elliptical pattern indicates elliptical polarization

where Exo and Eyo > 0 and y 0, 90, or 180 yet these variables remain constant over time.

A wave is unpolarized when the amplitudeand phase relationships of the orthogonalcomponents are time varying.

y0y

0x

ztcosˆE

ztcosˆEt,z

y

xE

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Polarization of plane waves and coherenceSignals produced by single-frequency or multifrequency

transmitters are typically completely polarized

Signals emitted by physical objects, irregular terrain, or inhomogeneous media are usually broadband and are composed of many statistically independent waves with different polarizations

If there is no correlation between the component waves of such a signal it is incoherent or unpolarized

Between these two extremes (completely polarized and unpolarized) are the partially-polarized signals that result when polarized signals are scattered by random targets

While characterization of completely polarized plane waves is fairly straightforward, the characterization of these partially-polarized signals is more challenging

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Polarization of plane waves and coherenceAn analysis technique to evaluate the state of polarization or degree of coherence of a plane wave involves the magnitude of the normalized cross-correlation of the x and y phasor components as

where * denotes the complex conjugate and denotes the average operator over some finite time interval T

For completely polarized signals xy = 1 while for incoherent or unpolarized signals xy = 0.

For partially-polarized or partially-coherent signals

0 < xy < 1

2

y

2

x*yxxy EEEE

T

0Tdt

T

1lim

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Polarization of plane waves and coherenceTo evaluate the degree of polarization (DOP), another measure of a signal’s polarization, involves the Stokes parameters

The DOP is found by

where DOP = 1 for a completely polarized signal, a DOP = 0 for an unpolarized signal, and for partially-polarized signals

0 < DOP < 1

0

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22

21

S

SSSDOP

2

y

2

x0 EES 2

y

2

x1 EES

*yx2 EERe2S *

yx3 EEIm2S

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Polarization of plane waves and coherenceThe Poincaré sphere is a tool for visualizing the continuum of polarization states.

Derived from the Stokes parameters, the sphere maps linear polarizations on the equator (LVP: linear vertical pol.; LHP:

linear horizontal pol.) and the right (RCP) and left (LCP) circular polarizations at the north and south poles.

Points opposite one another on the sphere represent orthogonal polarizations.

Points representing completely-polarized signals lie on the sphere’s surface while points representing partially-polarized waves appear within the sphere.

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Electromagnetic phenomenaSpeed of light

c: speed of light in vacuum (2.99792458 108 m/s 3 108 m/s)n: refractive index of material (n 1)v: speed of light in material (m/s), v = c/n

Wavelength and frequencyf: frequency (Hz): wavelength (m)o: wavelength in free space (m)f: bandwidth in frequency (Hz): bandwidth in wavelength (m)

In vacuum, (n = 1, v = c)

In a medium, (n 1, v c)

2oo

oo

cf,

cf,

f

c,fc

ofn

c

f

v

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Fresnel reflection and transmissionNow consider the electromagnetic interactions as a plane wave impinges on a planar boundary between two different homogeneous media with semi-infinite extent

Properties of interest include reflection, refraction, transmission

Solutions are found by satisfying the requirement for the continuity of tangential E and H fields across the boundary

Simplifying assumptions:• Plane wave propagation• Smooth, planar interface infinite in extent• Linear, isotropic refractive indices• Semi-infinite media

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Fresnel reflection and transmissionSnell’s law

n1: refractive index of medium 1 (incidence side)n2: refractive index of medium 2i: incidence angle (measured from surface normal)r: reflected angle (measured from surface normal)t: transmitted angle (measured from surface normal)

Requirement for continuity of tangential E and H fields across the boundary yields

Reflected

Transmitted

ir

t2i1 sinnsinn

i

t

2

1

sin

sin

n

n

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Fresnel reflection and transmissionFresnel equations• Predicts reflected and transmitted vector field quantities at a plane

interface• Satisfies requirement for continuity of tangential E and H fields

across the boundary• Consequently the interaction is polarization dependent• Arbitrarily polarized incident plane wave

can be decomposed into two orthogonal linear polarizations: perpendicular () and parallel (//)

• Separate solutions for perpendicular and parallel cases

• Perpendicular and parallel refer to E-field orientation with respect to the plane of incidence(plane containing incident, reflected, and transmitted rays)

• These same polarizations have other names as well

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Fresnel reflection and transmissionPerpendicular (horizontal) case

Reflection coefficient (relates to field strength) (sometimes represented by , , or r).

Transmission coefficient (relates to field strength)

Note that 1 + R = T

t1i2

t1i2

0ti

ti

t2i1

t2i1

i

r

coscos

coscos

sin

sin

cosncosn

cosncosn

E

ER

i

t1i2

i2

0ti

it

t2i1

i1

i

t

coscos

cos2

)(sin

cossin2

cosncosn

cosn2

E

ET

i

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Fresnel reflection and transmissionPerpendicular (horizontal) case

Reflectivity (relates to power or intensity)

Transmissivity (relates to power or intensity)(sometimes represented by T)

or

Note that += 1 which satisfies the conservation of energy

2R

2

1i

2t T/cosRe

/cosRe

1

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Fresnel reflection and transmissionParallel (vertical) case

Reflection coefficient (relates to field strength)

Transmission coefficient (relates to field strength)

Note that 1 + R // = T//

t2i1

t2i1

0ti

ti

t1i2

t1i2

i

r//

coscos

coscos

tan

tan

cosncosn

cosncosn

E

ER

i

t2i1

i1

0titi

it

t1i2

i2

i

t//

coscos

cos2

cossin

cossin2

cosncosn

cosn2

E

ET

i

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Fresnel reflection and transmissionParallel (vertical) case

Reflectivity (relates to power or intensity)

Transmissivity (relates to power or intensity)

or

Note that //+//= 1 which satisfies the conservation of energy

2

//// R

2

//i1

t2// T

cosRe

cosRe

//// 1

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Fresnel reflection and transmissionSpecial cases

Normal incidence (i = 0)i = r = t = 0, cos = 1

Brewster angleB: angle where reflection coefficient for parallel (vertical) polarized field goes to zero i.e., at = B, // = 0, // = 1 (note polarization dependence)

21

21// nn

nnRR

21

1// nn

n2TT

1

2B n

ntan

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Fresnel reflection and transmissionSpecial cases

Critical angleC: incidence angle at which total internal reflection occurs (for n1 > n2)

i.e., at C, // = = 1, // = = 0 (note polarization independence)

Evanescent waves exist in medium 2, with imaginary propagation coefficients meaning they decay rapidly with distance z from the boundary.

12C nnsin

zi e)z( EE

22

2i1 nsinn

2

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Fresnel reflection and transmissionNormal incidence reflectioncoefficient for some typicalgeological contacts

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Fresnel reflection and transmissionExample #1Consider the case where a plane wave impinges on a

planar boundary between homogenous

ice ( = 3.14, n = 1.78) and air ( = 1, n = 1).

From the formulas presented earlierNormal reflectivity = -11.0 dB

Normal transmissivity = -0.4 dB

Critical angle, C = 34.4°

Brewster angle, B = 29.4°

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Fresnel reflection and transmissionFresnel reflection and transmission coefficients vs. incidence angle at an ice-air boundary (ice = 3.14, air = 1)

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Fresnel reflection and transmissionReflectivity and transmissivity expressed in linear units vs. incidence angle at an ice-air boundary (ice = 3.14, air = 1)

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Fresnel reflection and transmissionReflectivity and transmissivity expressed in decibels vs. incidence angle at an ice-air boundary (ice = 3.14, air = 1)

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Fresnel reflection and transmissionExample #2Consider the case where a plane wave impinges on a

planar boundary between homogenous ice ( = 3.14, n = 1.78) and rock ( = 5, n = 2.24).

From the formulas presented earlierNormal reflectivity = -19.1 dB

Normal transmissivity = -0.1 dB

Critical angle, C = NA

Brewster angle, B = 51.6°

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Fresnel reflection and transmissionFresnel reflection and transmission coefficients vs. incidence angle at an ice-rock boundary (ice = 3.14, rock = 5)

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Fresnel reflection and transmissionReflectivity and transmissivity expressed in linear units vs. incidence angle at an ice-rock boundary (ice = 3.14, rock = 5)

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Fresnel reflection and transmissionReflectivity and transmissivity expressed in decibels vs. incidence angle at an ice-rock boundary (ice = 3.14, rock = 5)

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Layered media

Now consider the case of multiple, planar layers and the associated composite reflection and transmission characteristics.

Consider first the simplest case, a single layer of thickness d1 sandwiched between two semi-infinite layers.

Layer 0, 0 1, 0

Layer 2, 2 1, 2

Layer 1, 1 1, 1

z = 0

z = -d1

R

T

z

xy

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Layered media (1 of 5)

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Layered media (2 of 5)

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Layered media (3 of 5)

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Layered media (4 of 5)

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Layered media (5 of 5)

A similar approach can be developed for the V-polarized case.

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Layered media – example (1 of 4)

Multilayer example: 1-layer case

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Layered media – example (2 of 4)

Wavenumbers in media 1 and 2 are

Matching tangential E-field components at each interface requires

Matching tangential H-field components at each interface requires

022

012

1z sinkk 022

022

2z sinkk

12z12z11z11z dkj2

dkj2

dkj1

dkj1 eCeAeCeA

111z000z CAkCAk

12z12z11z11z dkj2

dkj22z

dkj1

dkj11z eCeAkeCeAk

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Layered media – example (3 of 4)

These boundary matching conditions can be written in matix form as

which leads to

11z

11z

dkj1

dkj1

010

0

eC

eAB

C

A

22z

22z

11z

11z

dkj2

dkj2

12dkj1

dkj1

eC

eAB

eC

eA

12z

12z

11z11z

11z11z

22z

dkj12

dkj

1z

2zdkjdkj

01

dkj01

dkj

0z

1z

dkj

1201

TeR

eT

k

k1

2

1

eeR

eRe

k

k1

2

1

0

eTBB

R

1

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Layered media – example (4 of 4)

And the R variables in the matrices are defined as

To find the reflection and transmission coefficient for this 1-layer structure, we solve for R and T

1z0z

1z0z01 kk

kkR

2z1z

2z1z12 kk

kkR

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Electromagnetic spectrum

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Radio spectrum

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Atmospheric transmission at radio frequencies

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Reserved frequencies and band designations

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Generic energy band diagram

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Quantum energy levelsPhoton contains energy related to the electromagnetic frequency, (Hz)

E = hwhere h is Planck’s constant

h = 6.625 × 10-34 J sh = 4.136 × 10-15 eV s

[Note: 1 eV = 1.6 × 10-19 J ]At 1 GHz ( = 109 Hz),

E = 4.136 × 10-6 eV or 6.625 × 10-25 J

At 10 GHz ( = 1010 Hz),E = 4.136 × 10-5 eV or 6.625 × 10-24 J

Therefore at 1 GHz, a 10-mW signal contains more than 1.5 × 1022 photons

For comparison, one photon of visible light ( 500 nm, = 600 THz)E = 2.5 eV or 4 × 10-19 J

Therefore, a 10-mW signal would contain more than 2.5 × 1016 photons

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Relating EM band frequencies and wavelengths to mechanisms

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EM source mechanisms