Page 1
Aperture Radiation: Huygen’s Equation
y
zϕx
θ
Rx αx
r(x, y)
Superposition of contributions from radiating patches
ϕy
Aperture in X-Y plane “A”
Radiating patch, assumeuniform plane wave:E⊥H, E/H = 377Ω
ˆxExE =
( ) ( )( ) ( ) ( ) ( )
( )
− π λθ φ ≅ + θ −αλ ∫ j 2 r x,y
eff xx Aphase lag x,y
jE , ,R 1 cos E x,y e dxdyˆ2R
Huygen’s superposition integral
ffS E current surface JHE
→→⎭⎬⎫
radiated by integral of current elements
R1
Page 2
Huygen’s Equation: Geometric approximations
2-D casex r(x) R
x sin ϕx0
ϕx
Aperture in x-y plane
x
( ) yxyxyx yxRsinysinxRyx,r :1, For ϕ−ϕ−≅ϕ−ϕ−≅<<ϕϕ
( ) ( ) ( )( )
∫ϕ+ϕ
λπ
+λπ
•αθ+λ
≅φθ A
yx2jx
K
x
R2j-
dxdyey,xEˆcos1R2
e jR,,E Thusyx
R2
( ) ( )( ) ( ) ( ) ( )
( )∫ πλ−α−θ+
λ≅φθ A
yx, lag phase
y,xr2jxxeff dxdyey,xEˆcos1
R2jR,,E
z
Page 3
Huygen’s Equation: Geometric approximations
( )( ) ( )( )
( )( ) ( ) ( )∫ ϕϕϕϕ≅
∫≅ϕϕ
π
ϕ+ϕλπ
−−
ϕ+ϕλπ
+−
2 yxyx2j
yxx1
x
Ayx2j
x1
yx
dde,EKxvmy,xEx
dxdyey,xEKvm,E
yx
yx
R3
( ) ( ) ( )( )
∫ϕ+ϕ
λπ
+λπ
•αθ+λ
≅φθ A
yx2jx
K
x
R2j-
dxdyey,xEˆcos1R2
e jR,,E Thusyx
Page 4
Huygen’s Equation: Geometric approximations
( )( ) ( ) ( )
( )( ) ( ) ( )∫ ϕϕϕϕ
λ≅
∫λ≅ϕϕ
πϕ+ϕπ−−
λλ
λλϕ+ϕπ+
λλ−
λλ
λλ
2 yxyx2j
yxx2
1
Ayx2j
x21
yx
dde,EK
xvmy,xEx
dydxey,xEKvm,E
yx
yx
R4
This is a Fourier transform pair
( ) ( ) ( ) ( )[ ]∫∫ π+π− == dfefXt x; dtetxfX Recall ft2jft2j
2cos1 , y y, x xLet ≅θ+λ
=λ
∆λλ
( )( ) ( )( )
( )( ) ( ) ( )∫
∫
π
ϕ+ϕλπ
−−
ϕ+ϕλπ
+−
ϕϕϕϕ≅
≅ϕϕ
2 yxyx2j
yxx1
x
A
yx2jx
1yx
dde,EKxvmy,xEx
dxdyey,xEKvm,E
yx
yx
Page 5
Fourier Transform Relations
R5
Thus Aperture (pulse signal)
( )[ ] ( ) [ ]
( ) ( ) [ ]2-
o
2yxx
yx
21-2yxx
21-E
m W 2
,E,S
R at Vm ,E~Vm ,R yxx
η
ϕϕ=ϕϕ∝
ϕϕ↔ττ
↓↓
λλ
( )1Vm)y,x(E − ↔~ ( )[ ] R at Vm ,E 1yxx
−ϕϕ
Page 6
Directivity D(θ, φ) of an Aperture AntennaLet P = radiation intensity andPTR = total power radiated (W) ( ) ( )
]m W[]m W[
R4PRf,,,P ,D 2-
2-
2TR π
φθ∆φθ
( )1, yx <<ϕϕ
R6
( )( )
( )( )
( )∫
∫
πη
λη
θ+≅
ϕ+ϕλπ
A22
xo
2
A
yx2jx
2o
2
R4dy dxy,xE2
1
dy dxey,xE
R22cos1D
yx
( ) ( )( )
( )∫
∫ϕ+ϕ
λπ
λ
θ+π=
A2
x
2
A
yx2jx
2
2
dy dxy,xE
dy dxey,xE cos1D
yx
Page 7
Directive Gain D(θ,φ) of an Aperture Antenna
Bounds on D(ϕx,ϕy), A(ϕx,ϕy)Recall “Schwartz Inequality”
( )( )∫∫∫ ≤ dxg dxfdx g f 222
( ) 2o
A2
x
A o2
x2yx
A4dydx E
Adydx E4,Dλ
π=
••
λ
π≤ϕϕ
∫∫
Therefore: [ ] ( )( )∫∫∫ ≤ A22
x2
Aj
x dydx 1 dydx Edydx eE
Ao(m2) is physical area of aperture
R7
( ) ( )( )
( )∫
∫ϕ+ϕ
λπ
λ
θ+π=
A2
x
2
A
yx2jx
2
2
dy dxy,xE
dy dxey,xE cos1D
yx
Page 8
Directive Gain D(θ, φ) of an Aperture Antenna
( ) 2o
A2
x
A o2
x2yx
A4dydx E
Adydx E4,Dλ
π=
••
λ
π≤ϕϕ
∫∫
( )⇒
η
ϕϕ•
λ
π=
R
yxe2
,A4D But ( ) oyxe Aarea)(effective ,A Rη≤ϕϕ
where radiation efficiency 0.1R ≤η
= η •ηA Re oTherefore A A
Define “aperture efficiency” ηA
( ) 65.0A
maxA o
e
RA ≅
η∆η in practice; = 1 for uniform illumination
R8
Page 9
Uniformly Illuminated Circular Aperture Antennas x
z
y r
φ’
φx
ϕyθ
Aperture coordinates = r, φ′Source coordinates = ϕx, ϕy for θ << 1
T1
( )( )
∫∫
φ
φ•
λ
θ+π=
ϕ+ϕλπ
A2
A
yx2j
2
2
'd dr rE
'd dr re1 cos1D
yx
0–3 dB
–17.6 dB–28.8 dB
0 1.63 4 7 10side lobes
πDλ
sin θ
P = 0
( ) ( )π πθ φ = + θ Λ θ
λ λ⎡ ⎤⎢ ⎥⎣ ⎦
221
D DD(f, , ) 1 cos sin
( )π= = π λ θ =
λ
22
oD
4 A at 0
q)q(J)q( 11 =Λ
“Lambda function”
where
“Bessel function of first kind”
Page 10
Non-Uniformly Illuminated Circular Apertures
P θB1/2 θNULL #1 First Side - Lobe ηA
0 1.02 λ/D 1.22 λ/D 17.6 dB 1.001 1.27 λ/D 1.63 λ/D 24.6 dB 0.753 1.47 λ/D 2.93 λ/D 30.4 dB 0.56
Moretypical
G(θ)
θ
Ex(r)P = 0P = 1P = 2
0 D/2
( ) ( )= −⎡ ⎤⎢ ⎥⎣ ⎦
P2
x2r
Assume E r 1D
T2
Page 11
Sidelobes and Backlobes of Aperture Antennas
Main lobe
Sidelobes Backlobes
Feed
Spillover
DiffractionBacklobes
Reflector
T3
Page 12
Waveguide Horn Feeds
Pyramidal Horny D
Ey(y)
Ey(x)
⇒
DominantWaveguideMode TE10
⇒
0
Lower Sidelobes(∼25 dB)
G(ϕx)
Null at ϕ = λ/DG(ϕy)
High Sidelobes(∼17.6 dB)
φx
φy
T4
Page 13
“Scalar” Feed
y
ApertureEy(y)
Yields very low sidelobes
λ/4 grooves cut into wall
Side view
∼λ/4 ⇒ open circuitat wall
λ/4 minimizesreturn echo
T5
Page 14
Examples of Parabolic Reflector Antennas
No aperture blockage
“Off-Axis Paraboloid”
Focus
Circularly SymmetricParabolic Reflector
T6
Lateral scan viaphased array line feed
Cylindrical Parabola
Page 15
Spherical Reflector Antennas
Variable-pitchlinear phasedarray, positionedat line focus
Focus at ≥ R/2
Focal plane
R/2Center of curvature
T7
FeedSupport
BeamLine Feed
Illuminated Portione.g. Aricebo (1000’ = D, 600’ is illuminated)
Page 16
Toroidal Parabolic Reflector Antenna
ToroidalReflector
Directionof View
zAxis of Revolution
ω
Feed Surface(No Aperture Blockage)
r(z)
Spinning Feeds
Spin
Antenna Feeds
Toroidal Reflector
Advantage: many rapidly scanning spinning feeds
Toroidal Reflector
z
T8
Page 17
Multifeed Arrays
ParabolaCB
Dx
y z
Focus “A”GA(θ)
GB(θ)
GC(θ)
θ
U1
f length Focal ∆=
sidelobes and G useable withbeams 5-3n ,5.0Df For o≅=
(say ~1 dB gain loss)
Can do much better with good lens systems
( ) direction- xin Df 2∝η
( ) 100055.07xn ,7Df if e.g. 2 ≅•≅=
Page 18
“Scalar” Feed
y
ApertureEy(y)
Yields very low sidelobes
λ/4 grooves cut into wall
Side view
∼λ/4 ⇒ open circuitat wall
λ/4 minimizesreturn echo
U2
Page 19
Multiple-Horn FeedsParabolicReflector
Adjacent Feeds
BA
Cross-over Pointbelow 3 dB, Far-Field
AB
Fourier Transform
Ey(y)Thus
Scalar Feeds
Cross-over ≤ 6 – 10 dBfor Low Sidelobes
Focal Plane
U3
Page 20
Multiple-Horn Feeds
Feeds
1 2
3 4
Poor Coverage
Three-Array Solution
A B C AC A B C
A B C A B
Feed A′ is assembly ofexcited adjacent feeds
21
3 4
U4
Patterns
Page 21
“Near-Field” Antenna Coupling
Near-field of aperture >> near field ofHertzian dipole (r << λ/2π)
Uniform Phase Front
SphericalPhase Front
r z
U5
DrD
≅•λ
λ≅
2Dr
λ>⇒
22D~r field" Far"
Page 22
“Near-Field” Antenna Coupling
r << D2
λ
PT2PT1
Consider near-field link:Say: uniformly illuminated apertures
Ao
Eo(vm-1)
Ao/3
U62T
1r
1T
2r2T1r P
P
3
1
P
P i.e. ty)(reciproci
3
PP Claim ===
wattsA2
2EP o
o
o1T •
η=
3oA
2
2EP
o
o2T •
η=
,3
P
3oA
2
2EP 1T
o
o2r =•
η= ?P1r =
Page 23
“Near-Field” Antenna Coupling, Mode Orthogonality
Only half the power is accepted here!Waves are not a sum of independent “bullets;” they have phase, modal structure (classic wave/particle issue).
U7
y y
Ey(y) Ey(y) receivedIlluminate only half
Eo
[ ]W 2
A
2
E o
o
2o •η
00
Halfgets in
Half reflected,orthogonal to dominant(coupled) mode
=+
( ) oo2
o 2A2E η
2Eo
2Eo