1. Propagation Mechanisms

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1 Contents: 1. Propagation Mechanisms 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 Propagation Mechanisms 1

2 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 Propagation Mechanisms 2

3 The main propagation mechanisms Multipath propagation: Scattering Uplink Line of sight path Downlink Reflections Diffraction Shadowing Propagation Mechanisms 3

4 Spherical coordinate system: Point sources in free space x 3 T( Ω) Definitions and remarks: S 1 O θ φ Ω a h ( Ω) a r ( Ω) a v ( Ω) x 2 S : Sphere of radius 1 centred at O. Ω S: Direction φθ, Azimuthal and coelevation angle, resp. a r ( Ω) : Outward unit normal to S at Ω T( Ω) : Tangent plane to S at Ω Transverse plane to a r ( Ω) x 1 Tx a h ( Ω), a v ( Ω) T( Ω) : Unit vectors pointing in the direction of increment of φ and θ, resp. Propagation Mechanisms 4

5 Point sources in free space Vertically polarized isotropic point source: S Near zone region G T Ω d 0 a h ( Ω) Far zone region λ Polarization plane P T a v ( Ω) a r ( Ω) E v( Ω, d, t) H h( Ω, d, t) Sphere of equal phase d d Propagation Mechanisms 5

6 Vertically polarized isotropic point source (cont d): Field equations: Point sources in free space E v ( Ω, d, t) a v Ω H h ( Ω, d, t) a h Ω ( )E v d ( )H h d ( ) cos( 2πft kd + ϕ v ) Electric field [V/m] ( ) cos( 2πft kd + ϕ v ) Magnetic field [A/m] Propagation Mechanisms 6

7 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 T( Ω). -> Transverse electromagnetic (TEM) wave Z 0 H h ( d) E v ( d) Point sources in free space a h ( Ω)H h ( d) a r ( Ω) a v ( Ω)E v ( d) Propagation Mechanisms 7

8 Electrical characteristics of free-space: ε [F/m] Permittivity μ 0 4π 10 9 [H/m] Permeability Z μ π [ Ω] Intrinsic impedance ε 0 Point sources in free space Wave constants: f Frequency [Hz] k 2πf ( μ 0 ε 0 ) 1 2 2π Propagation constant [ m 1 ] λ c ( μ 0 ε 0 ) [ m/s] Phase velocity [m/s] λ c f Wavelength [m] Propagation Mechanisms 8

9 Wave s Poynting vector: Point sources in free space S r ( Ω, d) a v ( Ω)E v ( d) a h ( Ω)H h ( d) a r ( Ω) E v ( d ) Z xxxxxx 0 a h ( Ω)H h ( d) a r ( Ω)S r ( d) S r ( d) Radiated power: S d : sphere of radius d. a v ( Ω)E v ( d) 1 1 P T -- S 2 r ( Ω, d) ds d -- S 2 r ( d)d 2 dω 2πd 2 S r ( d) constant S Lossless medium ( ) S d Propagation Mechanisms 9

10 Vertically polarized isotropic point source (cont d): It follows from ( ) that: Point sources in free space E v ( d) Z 0 P T P xxxx and 2π d T -- E d v -- H d h ( d) E v Z d E v E v ( Ω, d, t) a v ( Ω) E v cos( 2πft kd + ϕ d v ) H h ( Ω, d, t) a h ( Ω) E v cos( 2πft kd + ϕ Z 0 d v ) Propagation Mechanisms 10

11 Point sources in free space Horizontally polarized isotropic point source: S Near zone region G T Ω d 0 a h ( Ω) Far zone region λ Polarization plane P T a v ( Ω) a r ( Ω) E h( Ω, d, t) H v( Ω, d, t) d d Propagation Mechanisms 11

12 Horizontally polarized isotropic point source (cont d): Isotropic point source: Point sources in free space E h ( Ω, d, t) a h ( Ω) E h cos( 2πft kd + ϕ d h ) H v ( Ω, d, t) a v ( Ω) E h cos( 2πft kd + ϕ Z 0 d h ) Usually, the wave radiated by a source is the superposition of a vertically and of a horizontally polarized spherical wave: E ( Ω, d, t) E h ( Ω, d, t) + E v ( Ω, d, t) Henceforth, we only consider the electric field of the waves. Propagation Mechanisms 12

13 Complex representation of waves Complex representation of spherical waves: xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx E h ( Ω, d, t) a h ( Ω)Re E h exp( jϕ d xxxxxxxxxxxxxxx h ) exp( jkd) exp( j2πft) xxxxxxx E h ( d) [Complex] electric fields E h ( d, t) xxxxxxxxxxxxxxxx xxxxxxx E v ( Ω, d, t) a v ( Ω)Re E v exp( jϕ d v ) exp( jkd) exp( j2πft) xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx E v ( d, t) E v ( d) Time-dependent part Propagation Mechanisms 13

14 Complex representation of waves Concise notation for spherical waves: Ed ( ) E h ( d) E v ( d) E h exp( jϕ d h ) E v exp( jϕ d v ) exp( jkd) exp( jkd) Ed ( ) and E are complex 2-dim. vectors xxxxxxxxxxx exp( jϕ h ) E h E v E exp( jϕ v ) 1 -- exp( jkd) d Ed ( ) E -- 1 exp( jkd) d Propagation Mechanisms 14

15 Polarization Polarization of the electric field vector: E h ( d, t) E ( Ω, d, t) T( Ω) a v ( Ω) a h ( Ω) a r ( Ω) t E v ( d, t) Propagation Mechanisms 15

16 Polarization of the electric field vector: : linearly polarized wave ϕ h ϕ v T( Ω) Polarization E v E h E ( Ω, d, t) a r ( Ω) E h ( d, t) E v ( d, t) Trajectory of E ( d, t) in T( Ω) as a function of t with d fixed. ( π 2) mod π and : circularly polarized wave ϕ h ϕ v T( Ω) E h E v E h a r ( Ω) E ( Ω, d, t) E h ( d, t) CCW, LH ϕh ϕ v π 2 ( mod 2π) CW, RH ϕ h ϕ v 3π 2 ( mod 2π) E v E v ( d, t) Propagation Mechanisms 16

17 Polarization Polarization of the electric field vector: Otherwise the wave is said to be elliptically polarized T( Ω) E ( Ω, d, t) a r ( Ω) E h E h ( d, t) E v ( d, t) E v Propagation Mechanisms 17

18 Electric field pattern Anisotropic sources: Usually, the source does not radiate isotropically: E hv, ϕ hv depend on Ω, i.e. E hv E hv ( Ω), ϕ hv ( Ω) ϕ hv E hv exp( ϕ hv ) E hv E hv ( Ω) [Normalized] electric field pattern of a source: f ( Ω) f h ( Ω) f v ( Ω) with f hv ( Ω) E hv ( Ω) E hv E hv max Ω { E hv ( Ω) } f hv ( Ω) 1 Propagation Mechanisms 18

Electric field pattern Electric field pattern of a vertical f h ( Ω) 0 f v ( Ω) f v ( φθ, ) π cos -- cos( θ) 2

19 Electric field pattern Electric field pattern of a vertical f h ( Ω) 0 f v ( Ω) f v ( φθ, ) π cos -- cos( θ) sin( θ) λ 2-dipole antenna: x 3 θ f v ( φ, θ) φ x 2 x 1 Propagation Mechanisms 19

20 Electric field pattern Spherical wave radiated by an anisotropic source: E( Ω, d) f v Ω xxxxxxx f h ( Ω)E h 1 ( )E d v E( Ω) -- exp( jkd) Propagation Mechanisms 20

21 Antenna characteristics Far-zone region: d d D2 λ Electric field pattern: As already discussed. : maximal dimension of the antenna in meter. [Normalized] power pattern of a linear [polarized] wave: D λ p( Ω) S( Ω, d) max Ω S( Ω, d) f ( Ω) 2 E max Ω { f ( Ω) 2 E 2 f ( Ω) 2 } E ( Ω) E hv ( Ω) f ( Ω) f hv ( Ω) p( Ω) p hv ( Ω) Propagation Mechanisms 21

22 Gain of a [lossless] linear antenna: Antenna characteristics G max. radiated power/m 2 by the antenna radiated power/m 2 by an isotropic antenna max Ω { S( d, Ω) } S iso ( d) (same input power) S iso ( d) max Ω { S( d, Ω) } S( d, Ω) Propagation Mechanisms 22

23 Antenna characteristics Gain of a [lossless] linear antenna (cont d): S iso ( d) P T G max Ω { S( d, Ω) } S iso ( d) πd 2 S( d, Ω) f ( Ω) 2 E Z 0 d 2 G 2π max Ω{ f ( Ω) 2 E 2 } Z 0 P T 2π E Z 0 P T Propagation Mechanisms 23

24 Antenna characteristics Effective area: A λ G [ m 2 ] 4π A --- G λ π Received power: 1 P R --SA 2 [W] A Ω is the direction of maximum power radiation, i.e. p( Ω) 1 a r ( Ω) G, p( Ω) S S a r ( Ω' ) Ω Ω' P R Propagation Mechanisms 24

25 Isotropic antenna: Gain: Effective area: Antenna characteristics G iso 1 A iso λ 2 ( 4π) Linear antenna: Power pattern: Effective area: Gain: p( Ω) f ( Ω) 2 A ( λ 2 G) ( 4π) G 2π E 2 ( Z 0 P T ) Spherical wave radiated by an linear antenna: E( Ω, d) 60GP T f ( Ω) 1 -- exp( jkd) d Propagation Mechanisms 25

26 Free-space propagation loss Free-space transmission formula: P R P T λ G 4πd T G R Tx Rx Free-space transmission loss [isotropic antennas]: L FS 10log( P T P R ) log( d [km] ) + 20log( f [MHz] ) [db] L FS [ db] Slope: -20 db/decade log( d) log( f ) Propagation Mechanisms 26

27 Uniform plane waves in free-space Approximation of a spherical wave by a plane wave: New reference point r 0 a v ( Ω 0 ) O' a h ( Ω 0 ) r 0 a r ( Ω 0 ) + r r λ H h( Ω 0, d, t) E v( Ω 0, d, t) Vertical polarization d d Propagation Mechanisms 27

28 Approximation of a spherical wave by a plane wave (cont d): Electric field at r: Er ( ) E( Ω) Approximations for» r : r , where d 0 r 0 + r r 0 Uniform plane waves in free-space r 0 d exp( jk r 0 + r ) + r r kr 0 + r kd 0 + k( Ω 0 )r, where Ω 0 is the direction toward O, i.e. a r ( Ω 0 ) r 0. k( Ω 0 ) ka r ( Ω 0 ) is the wave s propagation vector. d 0 3. E( Ω) E( Ω 0 ) Propagation Mechanisms 28

29 Uniform plane waves in free-space Approximation of a spherical wave by a plane wave: 1 ( ) Er ( ) E( Ω 0 ) ---- exp( jkd d 0 ) exp( jkω ( 0 )r ) xxxxxxxxxxxxxxxxx 0 E : Electric field at r 0 Equation of a uniform plane wave propagating in free-space: Er ( ) E exp( j kr ) H( r) E exp( jkr ) Z 0 k k( Ω 0 ) 2π a λ r ( Ω 0 ) ( ) ( ) are the solutions of the Maxwell equations in source-free free-space. Propagation Mechanisms 29

30 Planes of equal phase: Uniform plane waves in free-space k k( Ω 0 ) 2π a λ r ( Ω 0 ) a r ( Ω 0 )x λ O a r ( Ω 0 ) α r Planes of equal phase kr constant a r ( Ω 0 )r r cos( α) constant Propagation Mechanisms 30

31 Uniform plane waves in a lossy medium Characteristics of a lossy material: μ Permeability [H/m] ε Permittivity [F/m] κ Conductivity [S/m] Equivalent characterization of a lossy material: μ Permeability [H/m] κ ε eff ε j Effective permittivity [F/m] 2πf (with εκ, 0 ) Comments: We shall retain the symbol ε for the effective permittivity. Henceforth, we only consider non-magnetic material: μ μ 0 Propagation Mechanisms 31

32 Uniform plane waves in a lossy medium Secondary constants of the medium: μ Z -- Intrinsic impedance [ ] ε 1 2 Ω k 2πf ( με) 1 2 k' jk'' Propagation constant [ m 1 ] where ( k', k'' 0) ( ) are still the solutions of the Maxwell equations in an source-free lossy medium: Er ( ) E exp( k'' a r ( Ω 0 )r ) exp( jk' a r ( Ω 0 )r ) H( r) E 1 Z -- exp xxxxxxxxxxxxxxx ( k'' a r ( Ω 0 )r ) exp( jk' a r ( Ω 0 )r ) Z is now complex Attenuation in the direction of propagation Propagation Mechanisms 32

33 Reflection and transmission Perpendicular (transverse magnetic) polarization: ε 1, μ 0 H ih, ki H rh, kr E iv, β θ i θ r E r, v θ t H th, ε 2, μ 0 E t, v kt Propagation Mechanisms 33

34 Reflection and transmission Angle of reflection, angle of transmission: Snell s law: k 1 sin( θ i ) k 1 sin( θ r ) k 2 sin( θ t ) θ i θ r ε θ 1 t asin ---- cos( β) ε 2 Propagation Mechanisms 34

35 Reflection and transmission Perpendicular (transverse magnetic) polarization (cont d): Reflection coefficient: R v E rv, E iv, Transmission coefficient: ε sin( β) ε 1 ε sin( β) + ε 1 ε cos 2 ( β) ε ε cos 2 ( β) ε 1 T v E tv, E iv, ε 2 ε 1 2 ε sin( β) ε sin( β) + ε cos 2 ( β) ε 1 Propagation Mechanisms 35

36 Reflection and transmission Parallel (transverse electric) polarization: ε 1, μ 0 H iv, H rv, kr E ih, ki β θ i θ r E r, h θ t ε 2, μ 0 H tv, E t, h kt Propagation Mechanisms 36

37 Parallel (transverse electric) polarization (cont d): Reflection coefficient: Reflection and transmission R h E rh, E ih, sin( β) sin( β) + ε cos 2 ( β) ε ε cos 2 ( β) ε 1 Transmission coefficient: T h E th, E ih, sin( β) sin( β) + ε cos 2 ( β) ε 1 Propagation Mechanisms 37

38 Reflection and transmission Comments: 1. R hv 1 total reflection; T hv 1 total transmission R hv 1 T hv 1 2. β 0 R v, R h 1 3. β π 2 R h R v ( 1 ε 2 ε 1 ) ( 1 + ε 2 ε 1 ) 4. (ideal conductor) R h 1, R v 1 σ 2 5. σ 1 σ 2 0: β B asin( 1 1+ ε 2 ε 1 ) (Brewster angle) R v 0 Propagation Mechanisms 38

39 Behaviour of as a function of the grazing angle β: R hv R hv Reflection and transmission arg( R hv ) β β Source: Grosskopf Propagation Mechanisms 39

40 Propagation over flat earth Two-path model: h T h T + h R h R h T h T Tx Tx' Direct path Reflected path d d r d d d r ε 0, μ 0 ε 1, μ 0 Rx h R Propagation Mechanisms 40

41 Propagation over flat earth Resulting field at the receiver location: Single polarization E R Ed ( d ) exp( jkd d ) + RE( d r ) exp( jkd r ) Assumptions/approximations: Ed ( d d ) Ed ( r ) Ed ( ) d d d r E --- d h T, «d d r d d 2 h T h R d h R h T + h R Propagation Mechanisms 41

42 Propagation over flat earth Approximated field at the receiver location: E R ( d) E FS ( d) 1 R j4π h T h R + exp λd Field from free-space propagation Tx-Rx Special case: Total reflection ( R 1) E R ( d) π h T h R E FS ( d) sin λd Propagation Mechanisms 42

43 Example [calculated]: 0 Propagation over flat earth 20 20log( d) Breakpoint h T 9m E R ( d ) [db] E R ( 1 ) Free-space propagation h R f c R m 900 MHz 80 40log( d) 100 d BP Two ray model d BP 4h T h R λ Distance d [m] Propagation Mechanisms 43

44 Example [measured]: Propagation over flat earth Measured signal strength Two-path model Free-space propagation Source: COST 231 TD(95)78, J. Wiart Propagation Mechanisms 44

45 Scattering from rough surfaces Rayleigh and Fraunhofer criterions: Conditions for a surface to be flat : β β β β Δh Rayleigh criterion: 4π Δh π sin( β) < -- λ 2 Fraunhofer criterion: 4π Δh π sin( β) < -- λ 8 Δh sin( β) λ Δh sin( β) λ 1 < < Propagation Mechanisms 45

46 Scattering from rough surfaces Rayleigh and Fraunhofer criterions (cont d): Path length difference AB ACB' : 2Δhsin( β) Resulting phase difference: 2π 2Δh sin( β) λ λ ki λ B kr β Δh A 2Δh β B' C 2Δhsin( β) Propagation Mechanisms 46

47 Scattering from rough surfaces 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. Δh A surface will be effectively smooth if either or β is small. λ The Rayleigh and Fraunhofer criteria can be applied to irregular surfaces as well. In this case Δh is replaced in the inequalities by the standard deviation ς of the terrain (see next slide). Propagation Mechanisms 47

48 Scattering from rough surfaces Characterization of random surfaces: Correlation function of the terrain height: E[ hd ( )hd ( + Δd) ] ( E[ hd ( )] 0) (Usual) Gaussian assumption: E[ hd ( )hd ( + Δd) ] ς 2 exp Δd2 l:correlation length (( E[ hd ( )hd ( + l) ]) ς 2 e 1 ) characterizes the horizontal scale roughness ς Var[ hd ( )]: Standard deviation of the terrain height. This number characterizes the vertical scale roughness. l 2 Propagation Mechanisms 48

49 Scattering from rough surfaces Characterization of random surfaces (cont d): Meaning of the terrain parameters ς and l ς 1 hd ( ) hd ( ) ς 2 d ς 1 < ς 2 d l 1 hd ( ) hd ( ) l 2 d l 1 < l 2 d Propagation Mechanisms 49

50 Scattering from rough surfaces Slightly rough surfaces, modified Fresnel reflection coefficients: A surface is slightly rough if ς 1 -- sin( β) < λ 32 (Fraunhofer criterion) In this case the scattering mechanism can be approximated by a reflection with the modified Fresnel reflection coefficients: mod R hv R exp hv 8π 2 ς λ sin( β) At the Fraunhofer limit (the above inequality sign is replaced by equality), mod R hv Rhv 0 926, 0.67dB Propagation Mechanisms 50

51 Scattering from rough surfaces Coherent and diffuse scattering: ki Coherent scattering Diffuse scattering Computation methods: Coherent scattering: Diffuse scattering: Kirchhoff method Small perturbation method Propagation Mechanisms 51

52 Second Green s theorem: Scattering from rough surfaces E s ( r) j e jk s r 4πr k s n ( z ) E( z) S Z ks ( n( z) H ( z) ) e jk s z k s ds Illustration (one-dim. case): E ih, ki r E sh, ks S E iv, ds n( z) z E sv, D z Propagation Mechanisms 52

53 Scattering from rough surfaces Second Green s theorem (cont d): nz ( ): normal unit vector to S at z E( z), H( z) resultant electric and magnetic field at z 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. Propagation Mechanisms 53

54 Kirchhoff scalar approximation for slightly rough surfaces: Definition of a slightly rough (random Gaussian) surface: kl > 6 correlation length > wavelength l 2 Scattering from rough surfaces > 2.76 average radius of curvature > wavelength ςλ 2.76ς ς 2 ς - > RMS terrain slope - 2 < 0.25 l 8 l D x, D y» λ dimension of the surface >> wavelength l 2 Propagation Mechanisms 54

55 Scattering from rough surfaces Coherent scattering matrix: Surface-anchored coordinate system: Impinging wave z ks Scattered wave E ih, E iv, ki θ i n θ s E sh, E sv, y Elementary surface φ s φi D y x D x Propagation Mechanisms 55

56 Scattering from rough surfaces Coherent scattering matrix (cont d): Es S c Ei where is given by S c S c V R h ( θ i ) cos( θ i ) cos( φ s φ i ) R v ( θ i ) sin( φ s φ i ) R h ( θ i ) cos( θ i ) cos( θ s ) cos( φ s φ i ) R v ( θ i ) cos( θ s ) cos( φ s φ i ) with V ζ x j k sin( kd x ζ x ) sin( kd y ζ y ) -----D 2π x D y e kςζ z kd x ζ x kd y ζ y ( sin( θ s ) cos( φ s ) sin( θ i ) cos( φ i )) 2 ( ) 2 2 ζ y ( sin( θ s ) sin( φ s ) sin( θ i ) sin( φ i )) 2 ζ z cos( φ s ) + cos( φ i ) Propagation Mechanisms 56

57 Huygens principle: Diffraction over/around obstacles Surface A da' da Radiating element d' E 0 d Δh d 1 d 2 E R Tx σ Rx Propagation Mechanisms 57

58 Kirchhoff s mathematical formulation of Huygens principle: Kirchhoff s formula for diffraction simplifies in the situation considered above to: E 0 Diffraction over/around obstacles 1 E R E 0 -- exp( jkd) da d A : amplitude of the field generated by the transmitter on the top of the obstacle. Solving the integral above under some further geometrical simplifications yields E R E FS exp( jπ 4) exp( jπ 4) C ( w) 2 Propagation Mechanisms 58

59 Kirchhoff s mathematical formulation of Huygens principle (cont d): : Free-space electric field at Rx if the obstacle were suppressed. E FS w: Fresnel coefficient: C( w) : Cornu spiral: Diffraction over/around obstacles w Δh 2 λ d Approximation for large positive values of w: w d 2 C( w) j π --z2 exp dz 2 E R E FS w πw Propagation Mechanisms 59

60 Diffraction attenuation: Diffraction over/around obstacles Behaviour of the Cornu-spiral Behaviour of as a function of w Im E R E FS exp{ jπ 4} w Re Line of sight No line of sight 5 5 w Source: Grosskopf Propagation Mechanisms 60

61 The uniform geometrical theory of diffraction Geometrical-optics representation of a wave: Investigated special cases: Spherical wave: Plane wave: Er ( ) E exp( jkr) r Er ( ) Eexp( j kr ) Geometrical optics: Er ( ) E 0 ( r) exp( jkφ( r) ) Amplitude term Eikonal Eikonal equation: grad( Φ( r) ) 1 Propagation Mechanisms 61

62 The uniform geometrical theory of diffraction Geometrical-optics representation of a wave (cont d): Astigmatic ray tube: Reference point Caustics d 0 d nλ r h Principal radii of curvature da 0 Wave equation: r v da 0 Surfaces of constant phase (wave fronts) da d Axial ray direction of propagation Ed ( ) E( 0) r h r v ( exp( + d) ( r v + d) jkd ) r h Propagation Mechanisms 62

63 The uniform geometrical theory of diffraction Geometrical-optics representation of a wave (cont d): Examples: Spherical wave: Plane wave: r r h r v r Ed ( ) E( 0) ( exp( r + d) jkd ) r h, r v Ed ( ) E( 0) exp( jkd) Propagation Mechanisms 63

64 The uniform geometrical theory of diffraction Key idea of the UTD: Extension of the classical geometrical optics to incorporate rays diffracted by [curved] edges. Keller s law of edge diffraction: ki a w kd a w β i β d Cone of diffracted rays ki β i kd β d a w ki β i π 2 kd a w Propagation Mechanisms 64

65 The uniform geometrical theory of diffraction Shadow boundaries: E t E i + E r + E d ki kr Reflection shadow boundary kd kd E t E i + E d Incident shadow boundary Special case: Plane wave incidence ki θ i θ i kd ki E t E d Incident wave Reflected wave Diffracted wave Propagation Mechanisms 65

66 The uniform geometrical theory of diffraction Edge-anchored coordinate system: E iv, a iv, ki P z 0 z Q β a w Q P d, h β d E d, h E d, v E ih, ϑ d P kd P ih, ϑ i ( 2 m )π Propagation Mechanisms 66

67 The uniform geometrical theory of diffraction Geometrical-optics representation of the incident wave: ( ) Ei( P 0 ) Ei z r ih, : Principal radius of curvature at P 0 of the wavefront in the plane P ih, : Principal radius of curvature at of the wavefront in the plane r iv, P 0 spanned by ki and a iv,. Special case: Plane wave r ih, r iv, Ei z r ih, r iv, r ih, + z, + z exp( ( )( r iv ) jkz ), ( ) Ei( P 0 ) exp( jkz) Propagation Mechanisms 67

68 The uniform geometrical theory of diffraction Geometrical-optics representation of the diffracted wave: where Ed d ( ) DEi( Q) r d r d ( exp( + d)d jkd ) D h ( βϑ, i, ϑ d, m) 0 D : diffraction matrix 0 D v ( βϑ, i, ϑ d, m) r d : depends in particular on the principal radius of curvature of the edge at Q. For a straight edge (curv. radius ): r d r ih, + z Q Special case: Plane wave Ed d ( ) DEi( Q) exp( jkd) d Propagation Mechanisms 68

69 The uniform geometrical theory of diffraction Dyadic diffraction coefficients [diffraction on an edge]: D hv ( βϑ, i, ϑ d, m) exp( jπ 4) m 2πk sin( β) π+ ( ϑ d ϑ i ) cot F[ kdv + ( ϑ 2m d ϑ i )] + π+ ( ϑ d + ϑ i ) cot F[ kdv + ( ϑ 2m d + ϑ i )] + π ( ϑ d ϑ i ) cot F[ kdv - ( ϑ 2m d ϑ i )] π ( ϑ d + ϑ i ) cot F kdv - [ ( ϑ 2m d + ϑ i )] F( u) 2 j uexp( ju) exp( jπ 4) C( u) 2 N + - argmin u integer ( ± π u ( 2πm) ) v + - 2mπN + - u ( u) 2 cos D dz Q sin( β) d + z Q 2 Propagation Mechanisms 69

70 The uniform geometrical theory of diffraction Example: Amplitude rel. to E ih, [db] E t, h E d, h Amplitude rel. to E iv, [db] E t, v E d, v ϑ d [ ] β π 2 m 16 9 ϑ i 55 f d 3 GHz 1 m ϑ d [ ] Propagation Mechanisms 70

Introduction to Condensed Matter Physics

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