1. Propagation Mechanisms - Aalborg...

Propagation Mechanisms 1

1. Propagation Mechanisms

Contents:

• The main propagation mechanisms• Point sources in free-space• Complex representation of waves• Polarization• Electric field pattern• Antenna characteristics• Free-space propagation loss• Uniform plane wave in free-space

1. Propagation Mechanisms

Contents (cont’d):

• Uniform plane wave in a lossy medium• Reflection and transmission• Propagation over flat earth• Scattering from rough surfaces• Diffraction over/around obstacles• The uniform geometrical theory of diffraction

Multipath propagation:

Diffraction

Reflections

Line of sight

Shadowing

Scattering

Uplink

Downlink

The main propagation mechanisms

Spherical coordinate system:

Ωar Ω( )

ah Ω( )

av Ω( )

T Ω( ) Definitions and remarks:

• : Sphere of radius 1centred at .

• : Direction

• Azimuthal and coelevationangle, resp.

• : Outward unit normal to at

• : Tangent plane to at

Transverse plane to

• :Unit vectors pointing in thedirection of increment ofand , resp.

Ω S∈φ θ,

ar Ω( ) S Ω

T Ω( ) S Ωar Ω( )

ah Ω( ) av Ω( ), T Ω( )∈

Point sources in free space

Vertically polarized isotropic point source:

av Ω( )ar Ω( )

ah Ω( )

E v Ω d t, ,( )

H h Ω d t, ,( )

Polarization

planeΩ

Far zone

region

Sphere of

equal phase

Near zone

region

Vertically polarized isotropic point source (cont’d):

Field equations:

E v Ω d t, ,( ) av Ω( )E v d( ) 2πft kd ϕv+–( ) Electric field [V/m]cos=

H h Ω d t, ,( ) ah Ω( )H h d( ) 2πft kd ϕv+–( ) Magnetic field [A/m]cos=

Characteristics of the radiated wave:• The surfaces of equal phase are spheres

-> Spherical wave

• The wave amplitude on the equal phase surfaces is constant-> Uniform wave

• The electric and magnetic fields are orthogonal and belong to .-> Transverse electromagnetic (TEM) wave

T Ω( )

Z0H h d( ) E v d( )=

av Ω( )E v d( )

ar Ω( )

ah Ω( )H h d( )

Electrical characteristics of free-space:

• Permittivity

• Permeability

• Intrinsic impedance

Wave constants:

• Frequency [Hz]

• Propagation constant

• Phase velocity [m/s]

• Wavelength [m]

ε0 8.86 1012–

[F/m]⋅≈

μ0 4π 109–

[H/m]⋅≈

-----⎝ ⎠⎛ ⎞ 1 2⁄

≡ 120π Ω[ ]≈

k 2πf μ0ε0( )1 2⁄≡ 2πλ

------= m1–[ ]

c μ0ε0( ) 1 2⁄–≡ 3 108

m/s[ ]⋅≈

λ c f⁄≡

Wave’s Poynting vector:

Radiated power:

: sphere of radius .

av Ω( )E v d( )

ah Ω( )H h d( )xxxxxx

⎧ ⎨ ⎩

Sr d( )

Sr Ω d,( ) av Ω( )E v d( ) ah Ω( )H h d( )× ar Ω( )E v d( )2

------------------= =

ar Ω( )Sr d( )

2--- Sr Ω d,( ) Sdd⟨ | ⟩

2--- Sr d( )d2 Ωd

S∫ 2πd

2Sr d( ) constant= = = =

Lossless medium (✥) Sd d

Vertically polarized isotropic point source (cont’d):

It follows from (✥) that:

andxxxx

⎧ ⎨ ⎩

E v d( )Z0PT

2π-------------

d---⋅ 60PT

d---⋅ E v

d---⋅=≈= H h d( )

-------1

d---⋅=

E v Ω d t, ,( ) av Ω( )E v

d------- 2πft kd ϕv+–( )cos=

H h Ω d t, ,( ) ah Ω( )E v

Z0d--------- 2πft kd ϕv+–( )cos=

Horizontally polarized isotropic point source:

av Ω( )ar Ω( )

ah Ω( )

E h Ω d t, ,( )

H v Ω d t, ,( )

Polarization

Far zone

region

Near zone

region

Horizontally polarized isotropic point source (cont’d):

Isotropic point source:

Usually, the wave radiated by a source is the superposition of a vertically

and of a horizontally polarized spherical wave:

Henceforth, we only consider the electric field of the waves.

E h Ω d t, ,( ) ah Ω( )E h

d------- 2πft kd ϕh+–( )cos=

H v Ω d t, ,( ) av Ω( )E h

Z0d--------- 2πft kd ϕh+–( )cos–=

E Ω d t, ,( ) E h Ω d t, ,( ) E v Ω d t, ,( )+=

Complex representation of spherical waves:

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

E h d t,( )

xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

E v d t,( )

xxxxxxx

⎧ ⎪ ⎨ ⎪ ⎩

xxxxxxx

⎫ ⎪ ⎬ ⎪ ⎭

xxxxxxxxxxxxxxx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

Eh d( )

xxxxxxxxxxxxxxxx⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭

Ev d( )

E h Ω d t, ,( ) ah Ω( )ReE h

d------- jϕh( )exp jkd–( )exp j2πft( )exp⋅

⎩ ⎭⎨ ⎬⎧ ⎫

[Complex] electric fields

E v Ω d t, ,( ) av Ω( )ReE v

d------- jϕv( )exp jkd–( )exp j2πft( )exp⋅

⎩ ⎭⎨ ⎬⎧ ⎫

Time-dependent

Complex representation of waves

Concise notation for spherical waves:

d------- jϕh( )exp jkd–( )exp

d------- jϕv( )exp jkd–( )exp

E h jϕh( )exp⋅

E v jϕv( )exp⋅

d--- jkd–( )exp⋅ ⋅==

E d( )Eh d( )

Ev d( )≡

xxxxxxxxxxx

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

and are complex 2-dim. vectorsE d( ) E

E d( ) E1

d--- jkd–( )exp⋅ ⋅=

Complex representation of waves

Polarization of the electric field vector:

E v d t,( )

E h d t,( ) E Ω d t, ,( )

T Ω( )ar Ω( )

ah Ω( )

av Ω( )

Polarization

Polarization of the electric field vector:• : linearly polarized wave

• and : circularly polarized wave

ϕh ϕv= T Ω( )

Trajectory of in

as a function ofwith fixed.

E d t,( )T Ω( ) t

E Ω d t, ,( ) ar Ω( ) E h d t,( )

E v d t,( )

ϕh ϕv– π 2⁄( ) mod π= E h E v=

CW RH ϕh, ϕv– 3π 2⁄ mod 2π( )=

T Ω( )E Ω d t, ,( )

ar Ω( )E h

CCW LH ϕ, h ϕv– π 2⁄ mod 2π( )=E h d t,( )

E v d t,( )

Polarization

Polarization of the electric field vector:

• Otherwise the wave is said to be elliptically polarized

T Ω( )

E Ω d t, ,( )

ar Ω( )

E v d t,( )

E h d t,( )

Polarization

Anisotropic sources:Usually, the source does not radiate isotropically:

depend on , i.e.

[Normalized] electric field pattern of a source:

E h v ϕh v,⇒ Ω

E h v E h v Ω( )→ ϕh v, ϕh v Ω( )→

E h v ϕh v( )exp⋅ Eh v Eh v Ω( )→=

f Ω( )f h Ω( )

f v Ω( )≡

f h v Ω( )Eh v Ω( )

--------------------≡

E h v maxΩ Eh v Ω( ){ }≡⎩⎪⎪⎨⎪⎪⎧

f h v Ω( )

Electric field pattern

Electric field pattern of a vertical -dipole antenna:λ 2⁄

f v φ θ,( )

π2--- θ( )coscos

θ( )sin-----------------------------------=

f h Ω( ) 0=

f v Ω( ) f v φ θ,( )=

Spherical wave radiated by an anisotropic source:

xxxxxxx

⎧ ⎪ ⎨ ⎪ ⎩

E Ω( )

E Ω d,( )f h Ω( )E h

f v Ω( )E v

d--- jkd–( )exp⋅ ⋅=

Far-zone region:

Electric field pattern:As already discussed.

[Normalized] power pattern of a linear [polarized] wave:

d d0 2D

λ------≡≥

: maximal dimension ofthe antenna in meter.

D λ≥

p Ω( ) S Ω d,( )maxΩS Ω d,( )---------------------------------≡

f Ω( ) 2E 2

maxΩ f Ω( ) 2E 2{ }------------------------------------------------= f Ω( ) 2

f Ω( ) f h v Ω( )≡

E Ω( ) E h v Ω( )≡

p Ω( ) ph v Ω( )≡

Antenna characteristics

Gain of a [lossless] linear antenna:

(same input power)Gmax. radiated power/m

2by the antenna

radiated power/m2by an isotropic antenna

------------------------------------------------------------------------------------------------------≡

maxΩ S d Ω,( ){ }

d( )----------------------------------------=

S d Ω,( )maxΩ S d Ω,( ){ }

Gain of a [lossless] linear antenna (cont’d):

GmaxΩ S d Ω,( ){ }

d( )----------------------------------------=

d( )PT

------------= S d Ω,( ) f Ω( ) 2E 2

----------------------------=

G 2πmaxΩ f Ω( ) 2E 2{ }

Z0PT------------------------------------------------⋅ 2π E 2

Z0PT-------------⋅= =

Effective area:

Received power:

4π------G= m

2[ ] AG----

4π------=

2---SA=

is the direction of maximumpower radiation, i.e.Ω

p Ω( ) 1= G p Ω( ),

S S ar Ω'( )=

ar Ω( )

Ω Ω'–=

Isotropic antenna:

• Gain:

• Effective area:

Linear antenna:

• Power pattern:

• Effective area:

• Gain:

Spherical wave radiated by an linear antenna:

Aiso λ2

4π( )⁄=

p Ω( ) f Ω( ) 2=

A λ2G( ) 4π( )⁄=

G 2π E 2Z0PT( )⁄⋅=

E Ω d,( ) 60GPT f Ω( ) 1

d--- jkd–( )exp⋅ ⋅=

Free-space transmission formula:

Free-space transmission loss [isotropic antennas]:

PT-------

λ4πd----------⎝ ⎠⎛ ⎞ 2

GT GR=

LFS 10 PT PR⁄( )log≡ 32.4 20 d [km]( )log 20 f [MHz]( ) [dB]log+ +=

d( )log f( )log

Slope: -20 dB/decade

Free-space propagation loss

Approximation of a spherical wave by a plane wave:

E v Ω0 d t, ,( )

H h Ω0 d t, ,( )

av Ω0( )

ar Ω0( )

ah Ω0( )

Vertical polarization

New reference point

Uniform plane waves in free-space

Approximation of a spherical wave by a plane wave (cont’d):

Electric field at :

Approximations for :

1. , where

2. , where is the direction toward , i.e. .

is the wave’s propagation vector.

E r( ) E Ω( ) 1

r0 r+------------------ jk r0 r+–( )exp⋅ ⋅≡

r0 r»

r0 r+------------------

---------≈ 1

-----= d0 r0≡

k r0 r+ kd0 k Ω0( ) r⟨ | ⟩+≈ Ω0 O ar Ω0( ) 1

-----r0=

k Ω0( ) k ar Ω0( )≡

E Ω( ) E Ω0( )≈

Approximation of a spherical wave by a plane wave:

Equation of a uniform plane wave propagating in free-space:

(✮) are the solutions of the Maxwell equations in source-free free-space.

1. 2.+( ) E r( ) E Ω0( ) 1

----- jkd0–( )exp j k Ω0( ) r⟨ | ⟩–( )exp⋅ ⋅ ⋅≈⇒

: Electric field atE r0

xxxxxxxxxxxxxxxxx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

E r( ) E j k r⟨ | ⟩–( )exp⋅=

H r( ) EZ0

------ j k r⟨ | ⟩–( )exp⋅=(✮)k k Ω0( ) 2π

λ------ ar Ω0( )= =

Planes of equal phase:

ar Ω0( )

ar Ω0( ) x⟨ | ⟩ λ

k r⟨ | ⟩ cons ttan= ar Ω0( ) r⟨ | ⟩ r α( )cos=⇔ cons ttan=

k k Ω0( ) 2πλ

------ ar Ω0( )= =

Planes of equal phase

Characteristics of a lossy material:

• Permeability [H/m]

• Permittivity [F/m]

• Conductivity [S/m]

Equivalent characterization of a lossy material:• Permeability [H/m]

• Effective permittivity [F/m](with )

Comments:•We shall retain the symbol for the effective permittivity.

•Henceforth, we only consider non-magnetic material:

μεκ

εeff ε jκ

2πf---------–=

ε κ, 0≥

εμ μ0≈

Uniform plane waves in a lossy medium

Secondary constants of the medium:

• Intrinsic impedance [ ]

• Propagation constantwhere

(✮) are still the solutions of the Maxwell equations in an source-free lossymedium:

Zμε---⎝ ⎠⎛ ⎞ 1 2⁄

≡ Ω

k 2πf με( )1 2⁄k' jk''–= = m

1–[ ]k' k'', 0≥( )

xxxxxxxxxxxxxxx

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩Attenuation in the direc-

tion of propagation is now

complex

E r( ) E k'' ar Ω0( ) r⟨ | ⟩–( )exp jk' ar Ω0( ) r⟨ | ⟩–( )exp⋅ ⋅=

H r( ) E1

Z--- k'' ar Ω0( ) r⟨ | ⟩–( )exp jk' ar Ω0( ) r⟨ | ⟩–( )exp⋅ ⋅ ⋅=

Uniform plane waves in a lossy medium

Perpendicular (transverse magnetic) polarization:

E t v,

H t h,

E r v,

H r h, kr

E i v,

H i h, ki

ε1 μ0,

ε2 μ0,

Reflection and transmission

Angle of reflection, angle of transmission:

k1 θi( )sin k1 θr( )sin k2 θt( )sin= =

θi θr=

----- β( )cos⎝ ⎠⎜ ⎟⎛ ⎞

asin=⎩⎪⎨⎪⎧

Snell’s law:

Perpendicular (transverse magnetic) polarization (cont’d):•Reflection coefficient:

•Transmission coefficient:

Er v,Ei v,----------≡

----- β( )sinε2

----- cos2 β( )––

----- β( )sinε2

----- cos2 β( )–+

-----------------------------------------------------------------=

Et v,Ei v,---------≡

----- β( )sin

----- β( )sinε2

----- cos2 β( )–+

-----------------------------------------------------------------=

Parallel (transverse electric) polarization:

E i h,

H i v,

E r h,

H r v,kr

E t h,H t v,

ε1 μ0,

ε2 μ0,

Parallel (transverse electric) polarization (cont’d):

•Reflection coefficient:

•Transmission coefficient:

Er h,Ei h,----------≡

β( )sinε2

----- cos2 β( )––

β( )sinε2

----- cos2 β( )–+

-----------------------------------------------------------=

Et h,Ei h,----------≡ 2 β( )sin

β( )sinε2

----- cos2 β( )–+

-----------------------------------------------------------=

Comments:

1. total reflection; total transmission

4. (ideal conductor)

5. : (Brewster angle)

Rh v 1= T h v 1=

β 0→ Rv Rh, 1–→⇒

β π 2⁄= Rh Rv– 1 ε2 ε1⁄–( ) 1 ε2 ε1⁄+( )⁄= =⇒

σ2 ∞→ Rh 1–→ Rv 1→,⇒

σ1 σ2 0= = βB 1 1 ε2 ε1⁄+⁄( )asin≡ Rv 0=⇒

Behaviour of as a function of the grazing angle :Rh v β

Source: Grosskopf

Rh v( )arg

Two-path model:

Direct path

Reflected path

ε0 μ0,

ε1 μ0,

hT hR+

⎩⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎧

hT hR–

⎩⎪⎪⎨⎪⎪⎧

Propagation over flat earth

Resulting field at the receiver location:Single polarization

Assumptions/approximations:

ER E dd( ) jkdd–( )exp RE dr( ) jkdr–( )exp+=

dd-----

dr-----

d---≈ ≈ E dd( ) E dr( ) E d( )≈ ≈⇒ E

hT hR– hT hR+, d« dr dd– 2hT hR

d------------≈⇒

Approximated field at the receiver location:

Special case: Total reflection

ER d( ) EFS d( ) 1 R j4πhT hR

λd------------–⎝ ⎠

⎛ ⎞exp+≈

Field from free-space

propagation Tx-Rx

R 1–=( )

ER d( )EFS d( )------------------ 2 2π

λd------------⎝ ⎠

⎛ ⎞sin≈

Example [calculated]:

−120

−100

Free-space

propagation

Breakpoint

---------------

Distance [m]d

4hT hR

λ----------------≡

f c 900 MHz=

hR 1.6 m=

hT 9 m=

R 1–=

20– d( )log∼

40– d( )log ∼

Two ray model

Example [measured]:

Source: COST 231 TD(95)78, J. Wiart

Measured signal strength

Two-path model

Free-space propagation

Rayleigh and Fraunhofer criterions:

Conditions for a surface to be “flat”:

• Rayleigh criterion:

• Fraunhofer criterion:

Δhβ β

------- β( )sin1

8---<4πΔh

λ------- β( )sin

π2---< ⇔

------- β( )sin1

32------<4πΔh

λ------- β( )sin

π8---< ⇔

Scattering from rough surfaces

Rayleigh and Fraunhofer criterions (cont’d):

Path length difference :

Resulting phase difference:

AB ACB'– 2Δh β( )sin

2π2Δh β( )sin

λ--------------------------

2Δh β( )sin

Rayleigh and Fraunhofer criteria (cont’d):

Comments:

• Investigations have shown that in the microwave range (~ 1GHz), the

Fraunhofer criterion is more appropriate than the one by Rayleigh.

• A surface will be effectively smooth if either or is small.

• The Rayleigh and Fraunhofer criteria can be applied to irregular surfaces

as well. In this case is replaced in the inequalities by the standard

deviation of the terrain (see next slide).

------- β

Characterization of random surfaces:

Correlation function of the terrain height:

(Usual) Gaussian assumption:

• : Correlation lengthcharacterizes the horizontal scale roughness

• : Standard deviation of the terrain height. This number

characterizes the vertical scale roughness.

E h d( )h d Δd+( )[ ] E h d( )[ ] 0=( )

E h d( )h d Δd+( )[ ] ς2 Δd2

l2---------–⎝ ⎠

⎛ ⎞exp=

l E h d( )h d l+( )[ ]( ) ς2⁄ e1–

ς Var h d( )[ ]≡

Characterization of random surfaces (cont’d):

Meaning of the terrain parameters andς l

ς2h d( ) h d( )

ς1 ς2<

l2h d( ) h d( )

l1 l2<

Slightly rough surfaces, modified Fresnel reflection coefficients:

A surface is slightly rough if

(Fraunhofer criterion)

In this case the scattering mechanism can be approximated by a reflection

with the modified Fresnel reflection coefficients:

At the Fraunhofer limit (the above inequality sign is replaced by equality),

ςλ--- β( )sin

32------<

Rh vmod

Rh v 8π2 ςλ---⎝ ⎠⎛ ⎞ 2

β( )sin2

–⎝ ⎠⎛ ⎞exp=

Rh vmod

Rh v⁄ 0 926, 0.67dB–= =

Coherent and diffuse scattering:

Computation methods:

• Coherent scattering: Kirchhoff method• Diffuse scattering: Small perturbation method

Coherent scattering

Diffuse scattering

Second Green’s theorem:

Illustration (one-dim. case):

Es r( ) j–

4πr---------e

j– ksr

⎝ ⎠⎛ ⎞ ks n z( ) E× z( )

ks------ks n z( ) H× z( )( )×– e

j ks z⟨ | ⟩dS

S∫∫×=

n z( )

E s v,

E s h,

E i v,

E i h,

Second Green’s theorem (cont’d):

• : normal unit vector to at

• , resultant electric and magnetic field at

Kirchhoff scalar approximation:Basic assumption:

• The surface boundary is sufficiently smooth

-> In a local region it may be looked upon as an inclined plane.

• Specular reflection occurs at these planes.

-> The resulting field at the surface is the sum of the incident and reflected fields.

• Surface with small slope

-> Reduction from a vector formulation to a scalar formulation.

n z( ) S z

E z( ) H z( ) z

Kirchhoff scalar approximation for slightly rough surfaces:“Definition” of a slightly rough (random Gaussian) surface:

• correlation length > wavelength

• average radius of curvature > wavelength

• RMS terrain slope < 0.25

• dimension of the surface >> wavelength

ςλ------ 2.76> l

2.76ς-------------

8-------> ς

Dx Dy, λ»

Coherent scattering matrix:Surface-anchored coordinate system:

E i v,

E i h,

E s v,E s h,

θiθs

Elementary surface

Scattered waveImpinging wave

Coherent scattering matrix (cont’d):

where is given by

Es Sc Ei=

Sc VRh θi( ) θi( )cos φs φi–( )cos– Rv θi( ) φs φi–( )sin–

Rh θi( ) θi( )cos θs( )cos φs φi–( )cos– Rv θi( ) θs( )cos φs φi–( )cos⋅=

2π------DxDy

k Dxζx( )sin

k Dxζx-----------------------------

k Dyζy( )sin

k Dyζy-----------------------------e

kςζz( )2 2⁄–=

ζx θs( ) φs( )cossin θi( ) φi( )cossin–( ) 2⁄=

ζy θs( ) φs( )sinsin θi( ) φi( )sinsin–( ) 2⁄=

ζz φs( )cos φi( )cos+=

Huygens’ principle:

Tx Rxσ ∞=

Radiating element

Surface A

Diffraction over/around obstacles

Kirchhoff’s mathematical formulation of Huygens’ principle:

Kirchhoff’s formula for diffraction simplifies in the situation considered

above to:

: amplitude of the field generated by the transmitter on the top of theobstacle.

Solving the integral above under some further geometrical simplifications

yields

ER E 01

d--- jkd–( )exp Ad

A∫≈

EFS---------

jπ 4⁄( )exp

2---------------------------

j– π 4⁄( )exp

2------------------------------- C∗ w( )–⋅=

Kirchhoff’s mathematical formulation of Huygens’ principle (cont’d):• : Free-space electric field at Rx if the obstacle were suppressed.

• : Fresnel coefficient:

• : Cornu spiral:

Approximation for large positive values of :

w Δh2

-----1

-----+⎝ ⎠⎛ ⎞≡

C w( )

C w( ) jπ2---z

⎝ ⎠⎛ ⎞exp zd

∫≡

EFS---------

2πw---------------≈ w 0.56≥

Diffraction attenuation:

Behaviour of the Cornu-spiral Behaviour of as a function ofER

EFS--------- w

Source: Grosskopf

5– 5

Line of sight No line of sight

jπ 4⁄{ }exp

2-----------------------------

Geometrical-optics representation of a wave:

Investigated special cases:

• Spherical wave:

• Plane wave:

Geometrical optics:

Eikonal equation:

E r( ) E1

r------- j– k r( )exp=

E r( ) E j– k r⟨ | ⟩( )exp=

E r( ) E0 r( ) j– kΦ r( )( )exp=

Amplitude term Eikonal

grad Φ r( )( ) 1=

The uniform geometrical theory of diffraction

Geometrical-optics representation of a wave (cont’d):Astigmatic ray tube:

Wave equation:

d 0= d nλ=

Surfaces of constant phase(wave fronts)

Axial ray

direction ofpropagation

Principal radii ofcurvature dA0

Caustics

Reference point

E d( ) E 0( )rhrv

rh d+( ) rv d+( )-------------------------------------- jkd–( )exp=

Geometrical-optics representation of a wave (cont’d):Examples:• Spherical wave:

• Plane wave:

rh rv r= = E d( ) E 0( ) rr d+( )

---------------- jkd–( )exp=⇒

rh rv, ∞= E d( ) E 0( ) jkd–( )exp=⇒

Key idea of the UTD:Extension of the classical geometrical optics to incorporate rays diffractedby [curved] edges.

Keller’s law of edge diffraction:

βi π 2⁄=

Cone of diffractedrays

ki aw⟨ | ⟩ kd aw⟨ | ⟩= βi⇔ βd=

Shadow boundaries:Reflection

shadowboundary

Incidentshadow

boundary

Et Ed=

Et Ei Ed+=Et Ei Er Ed+ +=

krkd kd

θiSpecial case:

Plane waveincidence

Incident wave Reflected wave Diffracted wave

Edge-anchored coordinate system:

E i v,

E i h,

E d h,E d v,–

ki dzP0

P i h,

ai v,zQ

P d h,

Geometrical-optics representation of the incident wave:

• : Principal radius of curvature at of the wavefront in the plane

spanned by and .

Special case: Plane wave

Ei z( ) Ei P0( )ri h, ri v,

ri h, z+( ) ri v, z+( )-------------------------------------------- jkz–( )exp=

ri h, P0 P i h,

ri v, P0

ki ai v,

ri h, ri v,, ∞→

Ei z( ) Ei P0( ) jkz–( )exp=

Geometrical-optics representation of the diffracted wave:

Special case: Plane wave

Ed d( ) DEi Q( )rd

rd d+( )d----------------------- jkd–( )exp=

• : diffraction matrix

• : depends in particular on the principal radius of curvature of the edge at .

For a straight edge (curv. radius= ):

DDh β ϑi ϑd m, , ,( )– 0

0 Dv β ϑi ϑd m, , ,( )–≡

rd Q∞ rd ri h, zQ+=

Ed d( ) DEi Q( ) 1

d------- jkd–( )exp=

Dyadic diffraction coefficients [diffraction on an edge]:

Dh v β ϑi ϑd m, , ,( ) jπ 4⁄–( )exp–

2m 2πk β( )sin-------------------------------------- ⋅=

π ϑd ϑi–( )+

2m--------------------------------⎝ ⎠⎛ ⎞cot F kDv

+ ϑd ϑi–( )[ ]π ϑd ϑi–( )–

2m--------------------------------⎝ ⎠⎛ ⎞cot F kDv

- ϑd ϑi–( )[ ]+⋅

π ϑd ϑi+( )+

2m---------------------------------⎝ ⎠⎛ ⎞cot F kDv

+ ϑd ϑi+( )[ ]π ϑd ϑi+( )–

2m--------------------------------⎝ ⎠⎛ ⎞cot F kDv

- ϑd ϑi+( )[ ]+⎩ ⎭⎨ ⎬⎧ ⎫

F u( ) 2 j u ju( )exp1

2------- jπ 4⁄–( )exp C u( )–=

u( ) 22mπN

+|-u–

2-------------------------------⎝ ⎠⎛ ⎞cos

minu integer π± u– 2πm( )⁄( )arg=

d zQ+---------------- β( )sin

Example:

ϑd °[ ]

E t h,

E d h,

E t v,

E d v,

β π 2⁄=

m 16 9⁄=

ϑi 55°=

f 3 GHz=

d 1 m=

ϑd °[ ]A

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