EEL 4473 Spectral Domain Techniques and Diffraction Theory Spectral Domain Techniques and Diffraction Theory - 2-D Fields 1
References: 1. *R.H. Clark and J. Brown, Diffraction Theory and Antennas, Wiley, 1980 Excellent treatment, easy to read. 2. P.C. Clemmow, The Plane Wave Spectrum Representation of Electromagnetic Fields, Oxford U. Press, 1966 A classic text in the area 3. D.R. Rhodes, Synthesis of Planar Antenna Sources, Clarendon Press, 1974 Most rigorous, hardest to read. 4. C. Scott, Spectral Domain Methods in Electromagnetics, Artech House, 1989 Easiest to read. *These notes are essentially based on reference 1 above. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 2
Elementary example of (two-dimensional) diffraction x P Incident plane wave dx a 2 r r 0 a 2 z Perfectly Conducting Slit Spectral Domain Techniques and Diffraction Theory - 2-D Fields 3
Huygens Principle: Each point on a propagating wavefront can be considered as a secondary source radiating a spherical wave. For the slit, an element of width dx in the aperture radiates (a cylindrical wave in this 2-D case) into the medium to the right of the aperture. For this infinitesimal element, assume that the strength of the secondary source is E o dx. The contribution of element E o dx to the field at point P a distance r away will be of the form: Spectral Domain Techniques and Diffraction Theory - 2-D Fields 4
a 1 dep = CEodx e r k = ω µε C = const. jkr r P For r dx 1 r 1 r, e jk r θ dx jk( r xsinθ ) e 0 θ xsinθ Spectral Domain Techniques and Diffraction Theory - 2-D Fields 5
E P = a 2 a 2 de P = CE o r e jkr a 2 a 2 e jkxsinθ dx = CE o r ae jkr sin πasinθ λ πasinθ λ, λ = 2π k Spectral Domain Techniques and Diffraction Theory - 2-D Fields 6
Huygens Principle gives the field at point P as the superposition of cylindrical waves from all parts of the aperture and produces the interference pattern shown. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 7
Note that the amplitude of the radiated field increases with aperture width, a, while the angular width of the main beam reduces. As the aperture width is increased, the gain of the antenna is increased and its beamwidth is reduced. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 8
Diffraction Theory Using Plane Waves Though there has been much study in diffraction theory, the underlying principle of spherical waves radiated by known fields Huygens Principle has remained. Though spherical waves are a natural physical entity they are mathematically cumbersome. Planes waves on the other hand are a simple mathematical entity but are not physical. We know however that physical fields can be represented by the superposition, be it discrete or continuous, of plane waves traveling in different direction. This superposition is known as the Angular Spectrum Spectral Domain Techniques and Diffraction Theory - 2-D Fields 9
Diffraction Theory Using Plane Waves The angular plane wave spectrum concept is a mathematically simple approach to develop diffraction theory. It is also a fundamentally more precise approach compared with other approaches as approximations are made later in the analysis rather than earlier. In fact, the analysis contains the near field information as well as the far. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 10
Diffraction Theory Using Plane Waves Recall that for temporal signals: Possibly finite in extent (physical) 1 jω t f ( t) = F( ω) e dω 2π Infinite in extent (non -physical) We also know: jωt ( ω) ( ) F = f t e dt Spectral Domain Techniques and Diffraction Theory - 2-D Fields 11
Elementary example of (two-dimensional) diffraction Incident plane wave x Typical plane wave of the angular spectrum θ P( x, z) 0 z Spectral Domain Techniques and Diffraction Theory - 2-D Fields 12
Diffraction Theory Using Plane Waves Suppose, instead of the cylindrical waves of the last example we represent the field diffracted by a slit as a superposition of plane waves one of which is shown. The angle θ describes the directions of the component plane waves. It will be more convenient to use sin θ rather than θ as the angular variable, however. ( θ) = ( sinθ) = ( ) A F F s Spectral Domain Techniques and Diffraction Theory - 2-D Fields 13
Diffraction Theory Using Plane Waves If the set of plane waves is described by the spectrum function F(sin θ), the contribution of this single plane wave of elemental amplitude F(sinθ) d(sinθ), to the field point P is: Then: de P (x, z) = F ( sinθ )d ( jk( xsinθ +z cosθ ) sinθ )e E P (x, z) = ( ) F sinθ jk( xsinθ +z cosθ ) e d ( sinθ ) For now assume the range of integration is artificially extended beyond its natural limits of ±1 for analytical convenience. We ll later see that waves traveling in directions such that sinθ > 1 is far from artificial. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 14
Define the field over the aperture as Then f( x) = E ( x,0) This is a Fourier integral and can be inverted to give the angular spectrum in terms of the aperture field as P jkxsin ( θ) θ ( θ) EP ( x,0) = f( x) = F sin e d sin 1 jkxsinθ 2π F ( sin θ ) = f ( x) e dx, k = λ λ (see next page for details) Spectral Domain Techniques and Diffraction Theory - 2-D Fields 15
Read details for homework. jkxsinθ ( θ) ( θ) f( x) = F sin e d sin jkxsinθ jkxsinθ jkxsinθ ( ) = sin sin ( θ) ( θ) f x e e F e d jkxsinθ jkxsinθ jkxu f ( x) e dx e F u e du dx jkxsinθ jkxsinθ jkxu f ( x) e dx = F u e e dudx = ( ) ( ) ( ) jkxsinθ jkxsinθ f ( x) e dx = F u e e jkxu dxdu Spectral Domain Techniques and Diffraction Theory - 2-D Fields 16
Read details for homework. 1 ± jω( t t ) Recall: δ ( t t ) o o = e dω 2π f (x)e jkxsinθ dx = F u f (x)e jkxsinθ dx = F u = 2π k ( ) e jkxsinθ e jkxu ( ) 1 k dx du e j ( kx )( u sinθ ) d ( kx) =2πδ ( u sinθ ) F ( u)δ ( u sinθ )du F ( sinθ ) = k 2π Spectral Domain Techniques and Diffraction Theory - 2-D Fields 17 du f (x)e jkxsinθ dx = 2π k F ( sinθ ) = 1 λ f (x)e jkxsinθ dx Which is the desired result.
For the slit example: a Eo x f( x) = 2 0 otherwise thus πasinθ sin a λ λ λ λ πasinθ λ a 2 1 jkxsinθ 1 jkxsinθ F( sin θ ) = f( x) e dx= Eoe dx= Eo a 2 Which has the same angular dependence as the field derived from Huygens principle. Recall: E P = CE r o ae jkr πasinθ sin λ πasinθ λ Spectral Domain Techniques and Diffraction Theory - 2-D Fields 18
This example demonstrates a result that we will later show is generally true; that the angular spectrum gives the far-field of the radiation. Furthermore, the angular spectrum is simply the Fourier transform of the field across the radiating aperture. The result can be extended to three dimensions and the reciprocity theorem to be established allows the technique to be used for receiving as well as transmitting apertures. The results apply to both the near-field and far-field of the aperture, and so this plane wave approach is used extensively in near-field measurements of fields as well as for antennas. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 19
Plane Wave Representation of Two-Dimensional Fields x â θ r Direction of Plane Wave P( x, z) r = xâ x + zâ z â = sinθâ x + cosθâ z let : s = sinθ, c = cosθ â = sâ x + câ z 0 z c = 1 s 2 s and c are the direction cosines of the plane Aperture Plane (x, y) plane Fields are assumed uniform in the y- direction, i.e., the fields are 2-dimensional. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 20
If the electric field at the origin is E o E x, z ( ) = E o e jkâi r then the electric field at P is: = E jk( sx+cz) o e It will be convenient to resolve the field into two orthogonal linearly polarized plane waves, one with the electric field vector parallel to the x-z plane and the other with the electric field vector perpendicular to the plane. The first has its magnetic field vector pointing entirely in the transverse (y) direction and is called the transverse magnetic TM case. The second has its electric field pointing entirely in the transverse (y) direction and is referred to as the transverse electric (TE) case. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 21
Angular Spectrum for TM Fields x E( x,z) = E o e jkâi r = E jk( sx+cz) o e E E z E x H = H y â y P( x, z) 0 θ z E x,z E x E z ( ) = E o e jkâi r = E o e ( x,z) = E o cosθe ( x,z) = E o sinθe E y = 0 H y jk sx+cz ( x,z) = E o η e jk( sx+cz) ( ) jk sx+cz ( ) ( ) jk sx+cz H x = H z = 0 Spectral Domain Techniques and Diffraction Theory - 2-D Fields 22
A set of plane waves traveling in different directions is represented by the spectrum function A(θ), such that the electric field amplitude of the plane wave traveling in the direction θ is A(θ) dθ. The angular spectrum can be continuous, discrete, or a mixture of the two. We replace E o by A(θ) dθ and integrate over the range of angles for which the angular spectrum is defined. For example: jk( sx+ cz) (, ) ( ) cos Ex x z = A θ θe dθ It will be convenient to express the angular spectrum in terms of s = sin θ rather than θ. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 23
Let A θ E z E z H y H y s = sinθ, ds = cosθdθ = cdθ, c = 1 s 2 ( ) = F ( s) jk( sx+cz) jk Then : E x ( x,z) = A( θ )cosθe dθ ( sx+cz) 1 = F ( s)cosθe cosθ ds = F ( s)e ( x,z) = E o sinθe ( ) ds jk sx+cz ( ) jk sx+cz jk( sx+cz) ( x,z) = A( θ )sinθe dθ = F s ( x,z) = E o η e jk( sx+cz) ( x,z) = 1 η A ( θ jk( sx+cz) )e dθ = 1 η F s E y = 0, H x = H z = 0 ( ) 1 c Spectral Domain Techniques and Diffraction Theory - 2-D Fields 24 Read details for homework. ( ) s c jk( sx+cz) e ds jk( sx+cz) e ds
E x E z H y ( x,z) = F ( s)e ( x,z) = F s ( ) s c ( x,z) = 1 η F s ( ) ds jk sx+cz ( ) 1 c E y = 0, H x = H z = 0 jk( sx+cz) e ds jk( sx+cz) e ds Note that all the field components for the TM wave have been expressed in terms of one spectrum function F(s) Spectral Domain Techniques and Diffraction Theory - 2-D Fields 25
Evanescent Waves Consider: de x jk( sx+cz) ( x,z) = F ( s)dse Again, this is the contribution of that plane wave to the angular spectrum of amplitude F(s)ds traveling in the direction making the angle θ = sin -1 s to the z-axis. When s < 1, θ lies in the range -π/2 < θ < π/2 and the elemental plane wave is the usual homogeneous type with which we re familiar. The plane wave travels with the speed of light in the medium and transfers power to z > 0. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 26
Evanescent Waves When s > 1, (assuming that s is real) the character of the wave changes since where χ is real and positive. 2 c= 1 s =± jχ Then de x ( x,z) = F ( s)dse jksx e jkcz = F ( s)dse jksx e kχz Where we have chosen the minus sign - jχ = c so that the fields remain finite as z + This is an inhomogeneous plane wave in that the amplitude is no longer constant over planes of constant phase. Inhomogeneous plane waves have the following three properties: Spectral Domain Techniques and Diffraction Theory - 2-D Fields 27
Evanescent Waves a) The direction of propagation of the wave is along the x-axis, parallel to the aperture plane, whether in the positive or negative x-direction depending upon the sign of s. b) The amplitude of the field decreases exponentially in the + z- direction, away from the aperture plane, hence they are evanescent in nature (evanescent means disappearing). For all but the smallest values of χ the evanescent waves become negligibly small at distances more than a few wavelengths from the aperture. The distance from the aperture plane at which the amplitude has decreased to a fraction e -1 of its value at z = 0 is λ z = 2πχ Spectral Domain Techniques and Diffraction Theory - 2-D Fields 28
Evanescent Waves c) The phase velocity is obtained via e jksx e jω t and is v p = 1 s µε is slower than that of a homogeneous plane wave since s > 1. These wave do carry power, but it is not propagated in the region z > 0. It merely travels back and forth across the aperture and can be though of as being stored there. Because they are closely bound to the surface (the aperture plane) these waves are known as electromagnetic surface waves. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 29
Analogy to Total Internal Reflection dielectric air θ > θ i critical incident plane wave aperture plane Spectral Domain Techniques and Diffraction Theory - 2-D Fields 30
Analogy to Total Internal Reflection For θi > θcritical, the incident plane wave is totally reflected, but the fields in the air region are not zero, since boundary conditions require that tangential E and H be continuous at the boundary. The fields on the air side of the boundary are evanescent waves which are local to the boundary and carry no energy across it. Recall Snell s law which related the angle of incidence θ i and the angle of transmission θ t : s = sinθ i = ε sinθ t Clearly when θ i > θ critical = sin -1 (ε -1/2 ) then s > 1 but remains real. But these are precisely the conditions we obtained for evanescent waves. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 31
We see that a complete representation of the 2-dimensional fields in z > 0 in terms of an angular spectrum of plane waves F(s), s = sin θ, must include spectral components over the entire range - < s <. If s < 1, the plane wave propagates freely as a homogeneous plane wave into z > 0. This is known as the visible or propagating portion of the angular spectrum. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 32
If s > 1, the plane waves are inhomogeneous and do not propagate into the medium but only store reactive power in the aperture plane, and are evanescent in that they are negligible at more than a few wavelengths from the aperture plane. This is the invisible, non-propagating, or reactive part of the angular spectrum. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 33
Aperture Field: The Fourier Transform of the Angular Spectrum Consider the x-component of the electric field over the aperture plane z = 0: For TM polarization, ( ) = (,0) E x E x jksx ( ) ( ) Eax x = F s e ds The aperture field E a (x) is the Fourier transform of the angular plane wave spectrum F(s). Applying the inversion formula, ax x 1 jkx ax + sin ( ) ( ) θ F s = E x e dx λ Spectral Domain Techniques and Diffraction Theory - 2-D Fields 34
Aperture Field: The Fourier Transform of the Angular Spectrum This means that if we know the tangential component of the electric field over the aperture (at z = 0) we can deduce the angular spectrum in terms of which we know the fields everywhere in the region z > 0. ( ) ( ) E x F s ax This statement is exact, and it applies to TM waves in 2- dimentions. Soon we ll extend this result to TE waves as well as to 3- dimentions. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 35
Return to the slit example Incident plane wave (wholly x-directed) (, ) E x z E e x inc = o jkz The diffracted field will be of the TM type 0 x a 2 a 2 Boundary conditions require that the tangential field be zero on the metal screen, thus: E ax Perfectly Conducting Slit E = 0 x < a o ( x) 2 otherwise z Spectral Domain Techniques and Diffraction Theory - 2-D Fields 36
Return to the slit example F ( s) = 1 λ E ax a as = E o sinc λ λ, F ( s) max = F ( 0) a = E o ( x) e + jkxsinθ dx = E o λ a 2 a 2 e + jkxsinθ sinc ( u ) = sinπu πu λ as a λ, E lim a as sinc o a λ λ λ = E δ s o sin π as a λ dx = E o λ π as λ ( ) This is a plane wave traveling in the θ = 0 direction as it should. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 37
Return to the slit example a For λ finite: a Peak occurs at Eo, Eo = 1here λ Visible Spectrum a λ Amplitude of spectrum small in evanescent region Visible Spectrum a λ Spectrum essentially constant over the propagating range of angles and continues at approximately the same level into the evanescent region. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 38
The example suggests that the angular spectrum and diffraction pattern are equivalent. In fact, at large distances from the aperture, the angular dependence of the fields is approximately that of the angular spectrum, and the approximation improves the larger the distance from the aperture. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 39
Far-Field Approximated by Angular Spectrum Any 2-dimensional TM field propagating into the region z > 0 can be represented by the single spectrum function F(s). As we found for the x-component: x jk( sx+ cz) (, ) ( ) E x z = F s e ds Slide 24 This determines E x everywhere in z > 0 by a single integral though it may be difficult to evaluate. There is one important general result that we can obtain immediately for the field at very large distances from a diffracting aperture of finite size. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 40
x Direction of Plane Wave r ( ) = ( ψ ψ) P: x, z rsin, rcos θ ψ 0 z Spectral Domain Techniques and Diffraction Theory - 2-D Fields 41
Substitute: ( xz, ) = ( rsin ψ, rcosψ) Into: jk( sx+ cz) Ex ( x, z) = F ( s) e ds = = = ( ) F s e ( sinψ ccosψ) jkr s + ds jkr( sinθsinψ+ cosθcosψ) ( sinθ) ( sinθ) F e d jkr cos( θ ψ) ( sinθ) ( sinθ) F e d Spectral Domain Techniques and Diffraction Theory - 2-D Fields 42
Now suppose that P is many wavelengths away from the origin, i.e., kr 1 For most range of ψ evanescent waves will not contribute. As θ varies over the range of real angles the phase in the exponential jkr cos( θ ψ ) e will rotate rapidly through many multiples of 2π except when cos(ψ θ) is stationary. The stationary phase condition occurs when θ =ψ, i.e. for that wave in the angular spectrum traveling in the direction (0, P) Spectral Domain Techniques and Diffraction Theory - 2-D Fields 43
Next suppose that F(s) is a bounded and continuous function of s, i.e., it does not contain any discrete delta function components. This means that the aperture field can be nonzero only over a finite area a very practical condition. Then as θ varies over the integration range, neighboring values of F(s) have approximately the same amplitude but appear in the integrand with opposite phases and almost cancer each other. This happens everywhere except in the direction in which the stationary phase condition is satisfied. The only non-negligible contribution to the integrand will come from the direction θ = ψ and its immediate neighborhood. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 44
f( x) = cos( K g( x)) gx ( ) = cosθ dg π = 0 for θ = dx 3 Spectral Domain Techniques and Diffraction Theory - 2-D Fields 45
We will find that the field at point P is ( ) ( ) E r, ψ = CF sin ψ e jkr, kr x where C is a constant to be determined. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 46
The Principle of Stationary Phase Read details for homework. Used to evaluate integrals of the form: b jkg ( x) I f( x) e dx = a where f(x) and g(x) are real, continuous, and of bounded variation and K is a large, positive, real number. Suppose that g(x) is stationary at only one point x = x o, a < x o < b, i.e., ( ) 0, g x = a< x < b o o Spectral Domain Techniques and Diffraction Theory - 2-D Fields 47
The Method of Stationary Phase Read details for homework. Based on the previous argument, the only significant contribution to the integral I will be in the neighborhood of x o. Thus expand g(x) in a Taylor series about x = x o, g( x) = g(x o ) + g (x o ) =0 ( x x ) o + 1 2 g (x ) ( x x ) 2 + o o and assume that f(x) is slowly varying. Then I = b f (x)e jkg( x) dx f (x o )e jk g( x o )+ a b a = f (x o )e jkg( x o ) e j 1 2 K b a g ( x o ) x x o ( ) 2 dx 1 2 g ( x o ) x x o ( ) 2 dx Spectral Domain Techniques and Diffraction Theory - 2-D Fields 48
The Method of Stationary Phase Read details for homework. Since by assumption there is only one singular point, we may extend the limits of integration to infinity. I f (x o )e jkg( x o ) e j 1 2 K g ( x o ) ( x x o) 2 dx Let ξ 2 = 1 2 K g (x ) ( x x ) 2, 2ξdξ = K g (x o o o )( x x o )dx dx = = 2ξ ( ) dξ = K g (x o ) x x o 2 K g (x o ) assume that g ( x o )>0 dξ 2 1 2 K g (x ) ( x x ) 2 o o K g (x o )( x x ) o dξ Spectral Domain Techniques and Diffraction Theory - 2-D Fields 49
The Method of Stationary Phase Read details for homework. dx = = 2 Kg ( x ) o 1 j Kg ( x ) ( ) o x x jkg xo 2 ( ) 2 o I f( x ) e e dx o dξ 2 jkg ( x ) 2 o j f( xo ) e ξ e d Kg ( xo ) ξ We have the well-known result that (see A. Papoulis, The Fourier Integral, McGraw-Hill, 1962, pp. 141, 220): 2 jξ e dξ = j Spectral Domain Techniques and Diffraction Theory - 2-D Fields 50 π
The Method of Stationary Phase Read details for homework. jkg ( xo) ( ) ξ o o 2 2 jξ jkg x j2π o I f( xo) e e d = f( x ) e Kg ( x ) Kg ( x ) = f( x ) e o π j Kg ( xo ) + 4 2π Kg ( x ) o o Spectral Domain Techniques and Diffraction Theory - 2-D Fields 51
The Method of Stationary Phase Read details for homework. if g (x o ) < 0, g ( x) = g(x o ) + g (x o ) =0 ( x x ) o 1 2 g (x ) ( x x ) 2 + o o Spectral Domain Techniques and Diffraction Theory - 2-D Fields 52
1 j K g ( x ) ( ) o x x jkg xo 2 ( ) ( ) o I f ( x ) e e dx o 2 1 2 ξ = K g ( xo) ( x xo), 2 ξdξ = K g ( xo) ( x xo) dx 2 2 1 2 2 j K g ( xo) ( x xo) 2ξ dx = dξ = 2 dξ Kg ( x) x x Kg ( x) x x j 2 = dξ, g ( xo ) < 0 Kg ( x) o Read details for homework. ( ) o o o o Spectral Domain Techniques and Diffraction Theory - 2-D Fields 53
I f (x o )e jkg( x o ) e j 1 2 K = f (x o )e jkg( x o ) j g ( x o ) x x o 2 K g (x o ) ( ) 2 dx Read details for homework. e jξ 2 dω = f (x o )e jkg( x o ) j j2π K g (x o ) = f (x o )e j Kg( x o )+π + π 2 +π 4 2π K g (x o ) = f (x o )e j Kg( x o )+ 4π 4 + 2π 4 +π 4 = 7π 4 = π 4 2π K g (x o ), g (x ) < 0 o Spectral Domain Techniques and Diffraction Theory - 2-D Fields 54
The Method of Stationary Phase Read details for homework. 1 2 π j Kg ( x ) 2 1 j Kg ( xo) sgn [ g ( xo) ] o ξ π + 2 4 o K g ( xo ) I = lim e dx f ( x ) e K sgn ( u) = + 1 u > 0 1 u < 0 This is the stationary phase result for the asymptotic evaluation, as K tends to infinity, of single integrals of the type considered. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 55
Applying the stationary phase to the diffraction integral, identify: 1 2 π j Kg ( x ) 2 1 j Kg ( xo) sgn [ g ( xo) ] o ξ π + 2 4 o K g ( xo ) I = lim e dx f ( x ) e o x K jkr cos( θ ψ), sin sin ( ) = ( θ) E x z F e d x θ x K F( θ) ( θ ψ) ( θ) d ψ, the stationary point, since cos( θ ψ) = 0 at θ = ψ dθ kr f( x) cosθ sin gx ( ) cos Spectral Domain Techniques and Diffraction Theory - 2-D Fields 56
x cos( ), lim jkr θ ψ sin sin kr ( ) = ( θ) E x z F e d cosψf ( sinψ) ( θ) ( ) ( ) ( ) ( ) g( θ) = cos θ ψ, g ( θ) = sin θ ψ, g ( θ) = cos θ ψ g ( ψ) = cos ψ ψ = 1 > 0 π 2π j jkr j2π 4 Ex ( x, z) cosψf( sinψ) e e = cosψf( sinψ) e kr kr = cosψf ( sinψ) 2π 1 e kr g ( x ) j r λ jkr e o π j kr sgn [ g ( xo )] 4 Which specifies the constant C included earlier. jkr Spectral Domain Techniques and Diffraction Theory - 2-D Fields 57
The remaining field components are found in a similar way: jk( sx+ cz) Ex ( x, z) = F ( s) e ds s jk( sx+ cz) Ez ( x, z) = F ( s) e ds c 1 1 jk( sx+ cz) H y ( x, z) = F ( s) e ds η c E = 0, H = H = 0 y x z jλ Ex ( x, z) cosψf( sinψ) e r jλ Ez ( x, z) sinψf( sinψ) e r 1 jλ Hy ( x, z) F( sinψ ) e η r jkr jkr jkr Spectral Domain Techniques and Diffraction Theory - 2-D Fields 58
More rigorous methods, such as the method of steepest descent confirm this result and in fact shows that the result obtained is the first term of an asymptotic series in ascending odd powers of K -1/2, hence our result is indeed a correct asymptotic result. Further details can be found in: P.C. Clemmow, The Plane Wave Spectrum of Electromagnetic Fields, Oxford: Pergamon Press, 1966. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 59
For the sake of economy of symbols, let us rewrite the results replacing ψ by θ. jλ Ex ( x, z) cosθf( sinθ) e r jλ Ez ( x, z) sinθf( sinθ) e r 1 jλ Hy ( x, z) F( sinθ ) e η r jkr jkr jkr Spectral Domain Techniques and Diffraction Theory - 2-D Fields 60
jλ Eθ r F e Hy r r x (, θ) = ( sin θ) jkr = η (, θ) z E θ (, ) E x z r ( x, z) H Ex ( x, z) Prθ (, ) ( x, z) y θ z Spectral Domain Techniques and Diffraction Theory - 2-D Fields 61
We see that the far field for a TM polarized wave radiating in two dimensions is a cylindrical wave (asymptotically as kr ). Note that in this case the axis of the cylinder is the y-axis. Note also that the local character of the field is (asymptotically) a plane wave. Also, Poynting s theorem applied to E θ and H y shows that the power flow in the far field is radial at all points with magnitude: λ Sr r F 2η r (, θ) = ( sinθ) 2 Watts meter-unit width Called the Angular Power Spectrum. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 62
The Antenna Far-Field Theorem a) The far field of an antenna is asymptotically equal to its angular spectrum. b) The angular spectrum for an antenna is the Fourier transform of its aperture field, which is the tangential component of the field in the aperture plane of the antenna. We proved this result for one of the orthogonally polarized components of a 2 dimensional field. The theorem is also true for the 3 dimensional case. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 63
Though the result is asymptotically true, i.e. for the Rayleigh condition that kr r 2 2a λ is sufficient, where a is the largest dimension of the aperture. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 64
We have formulated our results in terms of the tangential components of the electric field. It is of course perfectly valid to use the tangential magnetic components, though this is not as common. The theorem holds as worded in either case. The Antenna Far Field Theorem is more commonly known in a more abbreviated form: The far field of an antenna is given asymptotically by the Fourier transform of its aperture field. We break it into two statements, a) and b), to emphasize the fact that the Fourier transform relation between the angular spectrum and the aperture field is exact, with the approximation entering only in the evaluation of the far field. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 65
Fraunhofer Diffraction In optics, the far field pattern is referred to as the Fraunhofer diffraction pattern, with the only minor difference being that the fields are specified over a plane rather than over a sphere (circle). r θ x 0 z= ro Aperture Plane Fraunhofer Plane Spectral Domain Techniques and Diffraction Theory - 2-D Fields 66
If the extent of the Fraunhofer plane is small compared with its distance r o from the aperture plane, then x sinθ ro and from our earlier result, jλ Eθ ( r, θ) = ηhy ( r, θ) = F( sinθ) e r using r = z + x 2 2 jkr 2 x jkro 1+ 2 2 jkr λ r o x jλ x j F e F e ro ro ro ro 2 2 2 2 x x = ro + x = ro 1+ r 1 in the exponential 2 o + 2 ro 2ro Spectral Domain Techniques and Diffraction Theory - 2-D Fields 67
Complete 2-Dimensional Fields (TM Polarization) Slide 24: E x E z H y ( x,z) ( x,z) ( x,z) = F TM Note ( ) s 1 s c 1 ηc jk( sx+cz) e ds Which has the asymptotic solution for kr Ex ( x, z) cosθ jλ Ez( x, z) sinθ FTM ( sinθ) e r Hy ( x, z) 1 η jkr Spectral Domain Techniques and Diffraction Theory - 2-D Fields 68
Complete 2-Dimensional Fields (TM Polarization) Or in polar form: λ Eθ r Hy r FTM e r π j kr 4 (, θ) = η (, θ) ( sinθ) Spectral Domain Techniques and Diffraction Theory - 2-D Fields 69
The TE polarized waves (which give the remaining field components) can be constructed in exactly the same way in terms of a second angular spectrum F TE (s), or they can be readily deduced by the principle of duality. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 70
Principle of Duality Suppose the fields E 1 and H 1 satisfy Maxwell s equations: E 1 = jωµ H 1 H 1 = jωε E 1 Make the substitution E 2 = ηh 1 H 2 = 1 E η 1 then ηh 2 = jωµ 1 η 1 E η 2 = jωεη H 2 E 2 Spectral Domain Techniques and Diffraction Theory - 2-D Fields 71
Principle of Duality ηh 2 = jωµ 1 η 1 E η 2 = jωεη H 2 E 2 H 2 = jωµ η 2 E2 E 2 = jωεη 2 H 2 H 2 = jωµ µ ε E 2 = jωε µ ε E 2 = jωε E 2 H 2 = jωµ H 2 Spectral Domain Techniques and Diffraction Theory - 2-D Fields 72
Principle of Duality E 1 = jωµ H 1 H 1 = jωε E 1 E 2 = ηh 1 H 2 = η 1 E 1 H 2 = jωε E 2 E 2 = jωµ H 2 This shows that E 2 and H 2 are also solutions of Maxwell s equations as applied to sinusoidal fields in a lossless, source-free media. This means that if one solution for the fields, E 1, H 1, has been found, then a second solution, E 2, H 2, can be obtained as above. It is important to note that different boundary conditions must apply in order that the two solutions be truly independent. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 73
E x E z H y E 2 =η H 1 ( x,z) ( x,z) ( x,z) H 2 = η 1 E ηh x x,z 1 ηh z η 1 E y 1 = F ( TM s) s jk( sx+cz) e ds c 1 ηc ( ) 1 ( x,z) = F ( TM s) s c ( x,z) 1 ηc jk( sx+cz) e ds Spectral Domain Techniques and Diffraction Theory - 2-D Fields 74
H x H z E y E y H x H z ( x,z) ( x,z) ( x,z) ( x,z) ( x,z) ( x,z) = F ( TM s) 1 = c F ( s ) TM F TE ( s) Note the order of the terms 1 η 1 s η c 1 c 1 1 η s η e jk( sx+cz) e ds Factor out the c so that F TE is the Fourier transform of the aperture field E y consistent with F TM as the Fourier transform of E x. ( ) ds jk sx+cz Spectral Domain Techniques and Diffraction Theory - 2-D Fields 75
1 Ex ( x, z) s jk( sx+ cz) Ez( x, z) = FTM ( s) e ds c Hy ( x, z) 1 TM ηc Ey ( x, z) 1 x (, ) = TE ( ) η H ( x, z) 1 ( z) jk sx+ c H x z F s e ds z s η TE E ay E a E ax y Aperture Plane x Spectral Domain Techniques and Diffraction Theory - 2-D Fields 76
jk( sx+ cz) Ey( x, z) = FTE( s) e ds jksx Eay ( x) = Ey ( x,0) = FTE ( s) e ds 1 + jksx FTE ( s) = Eay ( x) e dx λ E ay E a E ax y TE Angular Spectrum x Fourier transform of E ay (x) Spectral Domain Techniques and Diffraction Theory - 2-D Fields 77
The far-field solution is again obtained by applying the duality principle to π λ j kr 4 Eθ ( r, θ) = ηhy( r, θ) FTM ( sinθ) e r or E 2 = ηh 1, H 2 = 1 E η 1 ( ) ηh ( θ r,θ ) = η E r,θ y η E y E y ( r,θ ) = ηh ( θ r,θ ) λ r F sinθ TM ( )e j kr ( r,θ ) = ηh ( θ r,θ ) = λ r cosθ F sinθ TE Spectral Domain Techniques and Diffraction Theory - 2-D Fields F TE s 78 π 4 ( )e j kr π 4 ( ) remember this: 1 c F s TM ( )
Which is the cylindrical wave shown below. x λ Ey r Hθ r FTE e r π j kr 4 (, θ) = η (, θ) = cosθ ( sinθ) r Prθ (, ) (, ) H x z x Hz ( x, z) H θ ( x, z) θ TE z Spectral Domain Techniques and Diffraction Theory - 2-D Fields 79
In summary, the complete electromagnetic field (in two dimensions) radiated into z > 0 is given in terms of two independent angular spectra, with far field asymptotically given for and ( s) ( ) ( x) ( x) e FTM 1 Eax = FTE s λ E ay E r,θ ( ) = â θ E θ r,θ ( ) + â y E y r,θ ( ) = jλ â r θ F TM + jksx dx kr as ( sinθ ) + â y cosθ F ( TE sinθ ) H r,θ ( ) = 1 η âr E r,θ ( ) e jkr Spectral Domain Techniques and Diffraction Theory - 2-D Fields 80
The fields represented by the two spectra are not only complete, they are unique. Uniqueness Theorem of Electromagnetics Consider a source-free volume V completely enclosed by surface A. Suppose there are two solutions, (E 1, H 1 ) and (E 2, H 2 ) which both satisfy Maxwell's equations within V and on A. Then by superposition the difference fields (E 1 E 2, H 1 H 2 ) will also satisfy Maxwell s equations. Now apply the divergence theorem with the surface ˆn the outward normal to Spectral Domain Techniques and Diffraction Theory - 2-D Fields 81
= A V ( E 1 E ) 2 H 1 H 2 t µ 2 ( )iˆnda H 1 H 2 2 + ε 2 E 1 E 2 2 + σ E 1 E 2 2 dv The normal components of the difference fields E 1 E 2 or H 1 H 2 do not enter into the integral on the left-hand-side (see next page). Hence if either tangential E or tangential H is specified uniquely at all points on A, then the integral in the left side is zero. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 82
ˆn B A n B n B t A A t A B ( A B )i ˆn = ( A n ˆn + A ) t ( B n ˆn + B t )i ˆn = A n ˆn B n ˆn =0 + A n ˆn ( ) B t = A n ˆn ˆn = ( A t B t )i ˆn ( ) B t + A t ( B n ˆn ) + A t B t i ˆn =0 + At ( B n ˆn ) i ˆn =0 + ( At B t )i ˆn ˆn i ˆn Spectral Domain Techniques and Diffraction Theory - 2-D Fields 83
But for the right-hand-side to be zero, both ( E 1 E ) 2 0 and ( H 1 H ) 2 0 must be identically zero throughout the entire volume. Hence there is one and only one solution throughout V. That is, the solution is unique within V if either E tangantial or H tangential is specified over the entire surface A. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 84
The uniqueness theorem applied to the source-free region such as the half cylinder z > 0 defined by the aperture plane and the circle of radius r. Aperture A Aperture Plane z = 0 x ax ( ) TM ( ) ( ) ( ) E x F s Eay x FTE s z r Spectral Domain Techniques and Diffraction Theory - 2-D Fields 85
Over the aperture plane the tangential electric field must be completely specified before the angular spectrum F TE and F TM can be determined. For large r the field over the curved surface of the cylinder will have the far-field forms derived previously. These far field components are everywhere tangential to the surface, and as r tends to infinity the amplitudes of all field components tend to zero. Hence the 2-D fields in the half-space z > 0 represented by the angular spectrum are unique. Spectral Domain Techniques and Diffraction Theory - 2-D Fields 86