Antennas and PropagationAperture Antennas

Alberto Toccafondi

Radiating apertureØ The radiation fields from aperture antennas, such as slots, open-ended

waveguides, horns, reflector and lens antennas, are determined from theknowledge of the fields over the aperture of the antenna

17.1. Field Equivalence Principle 659

Fig. 17.1.1 Radiated fields from an aperture.

The screen can also be a perfectly conducting surface, such as a ground plane, onwhich the aperture opening has been cut. In reflector antennas, the aperture itself isnot an opening, but rather a reflecting surface. Fig. 17.1.2 depicts some examples ofscreens and apertures: (a) an open-ended waveguide over an infinite ground plane, (b)an open-ended waveguide radiating into free space, and (c) a reflector antenna.

Fig. 17.1.2 Examples of aperture planes.

There are two alternative forms of the field equivalence principle, which may be usedwhen only one of the aperture fields Ea or Ha is available. They are:

J s = 0

Jms = −2(n× Ea)(perfect magnetic conductor) (17.1.2)

J s = 2(n×Ha)

Jms = 0(perfect electric conductor) (17.1.3)

They are appropriate when the screen is a perfect electric conductor (PEC) on whichEa = 0, or when it is a perfect magnetic conductor (PMC) on which Ha = 0.

660 17. Radiation from Apertures

Using image theory, the perfect electric (magnetic) conducting screen can be elimi-nated and replaced by an image magnetic (electric) surface current, doubling its valueover the aperture. The image field causes the total tangential electric (magnetic) field tovanish over the screen.

If the tangential fields Ea,Ha were known over the entire aperture plane (screen plusaperture), the three versions of the equivalence principle would generate the same radi-ated fields. But because we consider Ea,Ha only over the aperture, the three versionsgive slightly different results.

In the case of a perfectly conducting screen, the calculated radiation fields (17.4.10)using the equivalent currents (17.1.2) are consistent with the boundary conditions onthe screen.

17.2 Magnetic Currents and Duality

Next, we consider the solution of Maxwell’s equations driven by the ordinary electriccharge and current densities ρ, J, and in addition, by the magnetic charge and currentdensities ρm, Jm.

Although ρm, Jm are fictitious, the solution of this problem will allow us to identifythe equivalent magnetic currents to be used in aperture problems, and thus, establishthe field equivalence principle. The generalized form of Maxwell’s equations is:

∇∇∇×H = J+ jωϵE

∇∇∇ · E = 1ϵρ

∇∇∇× E = −Jm − jωµH

∇∇∇ ·H = 1µρm

(17.2.1)

There is now complete symmetry, or duality, between the electric and the magneticquantities. In fact, it can be verified easily that the following duality transformationleaves the set of four equations invariant :

E −→ HH −→ −Eϵ −→ µµ −→ ϵ

J −→ Jmρ −→ ρm

Jm −→ −Jρm −→ −ρ

A −→ Amϕ −→ ϕm

Am −→ −Aϕm −→ −ϕ

(duality) (17.2.2)

where ϕ,A and ϕm,Am are the corresponding scalar and vector potentials introducedbelow. These transformations can be recognized as a special case (for α = π/2) of thefollowing duality rotations, which also leave Maxwell’s equations invariant:

!E ′ ηJ ′ ηρ′

ηH ′ J ′m ρ′m

"=!

cosα sinα− sinα cosα

"!E ηJ ηρ

ηH Jm ρm

"(17.2.3)

Under the duality transformations (17.2.2), the first two of Eqs. (17.2.1) transforminto the last two, and conversely, the last two transform into the first two.

Far Field Radiation

Ø When far-field condition holds, the radiated far field is a local plane wave

ds = dlθ ⋅dlφ = rdθ ⋅ rsinθdφ

r

θ

φ

dφ

dθ θ

φ

dlθ

dlφ

r

𝐸 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟𝑓(𝜃,𝜙)

𝐻 𝑟 =1𝜁��×𝐸 𝑟

𝑓(𝜃, 𝜙)= 𝜁𝑁: 𝜃, 𝜙 + 𝐿= 𝜃, 𝜙 𝜃> + 𝜁𝑁= 𝜃,𝜙 − 𝐿: 𝜃,𝜙 𝜙>

𝑓(𝜃, 𝜙)=𝑓: 𝜃, 𝜙 𝜃> + 𝑓= 𝜃, 𝜙 𝜙>

Radiating apertureØ Calculation of the fields to the right (𝑧 > 0) of the aperture plane requires

knowledge of the fields over the entire plane (screen plus aperture.)

n = z

PEC

z𝐽CD(𝑟)

𝐽ED(𝑟)

𝐽ED(𝑟)

𝐽ED(𝑟)

𝐽ED 𝑟 = 𝑛G×𝐻 𝑟 H+∈JKL

𝐽CD 𝑟 = 𝐸 𝑟 ×𝑛GH+∈JKL

z

x

y

r '

𝐸 𝑟 H+MJKL

𝐻 𝑟 H+MJKL

𝐴

𝑆𝐸 𝑟𝐻 𝑟

ε,𝜇

Radiating apertureØ For large apertures (typical dimension much greater than a wave-

length), the approximation of using the fields only over the aperture isfairly adequate

X12010896847260483624120

Electric field [V/m

Radiating apertureØ For large apertures (typical dimension much greater than a wave-

length), the approximation of using the fields only over the aperture 𝐴 isfairly adequate

n = z

PEC

𝐽ED 𝑟 = 𝑛G×𝐻 𝑟 H+∈J

𝐽CD 𝑟 = 𝐸 𝑟 ×𝑛GH+∈J

zz

𝐽ED(𝑟)

Equiv. Principle.

Approx.

𝐽CD(𝑟)

𝐽ED(𝑟)

𝐽ED(𝑟)

𝐽ED 𝑟 = 0

𝐽ED 𝑟 = 0

𝐽CD(𝑟)

𝐽ED(𝑟)

𝐸R 𝑟 H+MJ

𝐻R 𝑟 H+MJ

z

x

y

r '

𝐸 𝑟 H+MJKL

𝐻 𝑟 H+MJKL

𝐴

𝑆

𝐸 𝑟𝐻 𝑟

ε, 𝜇

Radiating aperture

Ø Equivalent magnetic surface currents𝐽CD 𝑟 = 𝐸R 𝑟′ ×��H

+T∈J

ØDenotes with 𝐸R,𝐻R the aperture field, i.e the tangential components of thetotal field on 𝐴

z

xy

r '

φ

φ

θ

𝐸R 𝑟′ H+TMJ

𝐻R 𝑟′ H+TMJ

𝐸R 𝑟′ = 𝐸UR 𝑥T,𝑦T 𝑥G + 𝐸XR(𝑥T, 𝑦′)𝑦G 𝐻R 𝑟′ = 𝐻UR 𝑥T,𝑦T 𝑥G + 𝐻XR(𝑥T, 𝑦′)𝑦G

𝐿 𝜃, 𝜙 =Y𝐽CD 𝑟T 𝑒)*+TZ+𝑑𝑎′J

𝐽CD 𝑟 = 𝐸XR 𝑥T,𝑦T 𝑥G − 𝐸UR(𝑥T, 𝑦′)𝑦G

𝐿 𝜃, 𝜙 = ] ] 𝐸XR 𝑥T,𝑦T 𝑥G − 𝐸UR(𝑥T, 𝑦′)𝑦G 𝑒)*(^_`Ka_b)𝑑𝑥′𝑑𝑦′

c

(c

c

(c

𝑟T Z �� = xTsin 𝜃 cos𝜙 + yTsin 𝜃 sin𝜙

𝑢 = sin𝜃 cos𝜙𝑣 = sin 𝜃 sin𝜙

𝑟T Z �� = xT𝑢 + yT𝑣

Radiating apertureØComponents of the radiation vector 𝐿

z

xy

r '

φ

φ

θ

𝐸R 𝑟′ H+TMJ

𝐻R 𝑟′ H+TMJ

𝐿U 𝜃,𝜙 = ] ] 𝐸XR 𝑥T,𝑦T 𝑒)*(^_`Ka_b)𝑑𝑥′𝑑𝑦′c

(c

c

(c

𝑢 = sin𝜃 cos𝜙𝑣 = sin 𝜃 sin𝜙

𝐿X 𝜃, 𝜙 = − ] ] 𝐸UR 𝑥T,𝑦T 𝑒)*(^_`Ka_b)𝑑𝑥′𝑑𝑦′c

(c

c

(c

Ø Two-dimensional Fourier transforms

𝐹 𝜁, 𝜂 =12𝜋 ] ] 𝑓 𝑥T,𝑦T 𝑒()*(^_pKa_q)𝑑𝑥′𝑑𝑦′

c

(c

c

(c

Ø Two-dimensional Fourier transforms of the aperture fields

𝐿U 𝜃,𝜙 = ] ] 𝐸XR 𝑥T, 𝑦T 𝑒)*(^_`Ka_b)𝑑𝑥T𝑑𝑦T

c

(c

= 2𝜋𝐸rXR(−𝑘𝑢,−𝑘𝑣)c

(c

𝐿X 𝜃, 𝜙 = − ] ] 𝐸UR 𝑥T,𝑦T 𝑒)* ^_`Ka_b 𝑑𝑥T𝑑𝑦Tc

(c

c

(c

= −2𝜋𝐸rUR(−𝑘𝑢,−𝑘𝑣)

Radiating apertureØ Spherical components of the radiation vector 𝐿

z

xy

r '

φ

φ

θ

𝐸R 𝑟′ H+TMJ

𝐻R 𝑟′ H+TMJ

𝐿: = cos𝜃 cos𝜙 𝐿U + cos𝜃 sin𝜙 𝐿X

Ø Radiated field by the aperture magnetic sources

𝐿= = − sin𝜙 𝐿U + cos𝜙 𝐿X

𝐿: = 2𝜋 cos𝜃 cos𝜙 𝐸rXR −𝑘𝑢,−𝑘𝑣 − sin 𝜙 𝐸rUR −𝑘𝑢,−𝑘𝑣

𝐿= = −2𝜋 sin𝜙 𝐸rXR −𝑘𝑢,−𝑘𝑣 + cos𝜙𝐸rUR −𝑘𝑢,−𝑘𝑣

𝐸s 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 𝐿= 𝜃, 𝜙 𝜃> − 𝐿: 𝜃, 𝜙 𝜙>

𝐸s: 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 (−2𝜋) 𝐸rUR −𝑘𝑢,−𝑘𝑣 cos𝜙 + 𝐸rXR −𝑘𝑢,−𝑘𝑣 sin 𝜙

𝐸s= 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 (−2𝜋) cos𝜃 −𝐸rUR −𝑘𝑢,−𝑘𝑣 sin𝜙 + 𝐸rXR −𝑘𝑢,−𝑘𝑣 cos𝜙

Radiating apertureØ Equivalent electric surface currents

𝐽ED 𝑟 = ��×𝐻R 𝑟′ H+T∈J

z

xy

r '

φ

φ

θ

𝐸R 𝑟′ H+TMJ

𝐻R 𝑟′ H+TMJ

𝑁 𝜃, 𝜙 = Y𝐽ED 𝑟T 𝑒)*+TZ+𝑑𝑎′J

𝐽ED 𝑟 = −𝐻XR 𝑥T, 𝑦T 𝑥G + 𝐻UR(𝑥T, 𝑦′)𝑦G

𝑁 𝜃,𝜙 = ] ] −𝐻XR 𝑥T,𝑦T 𝑥G + 𝐻UR(𝑥T, 𝑦′)𝑦G 𝑒)*(^_`Ka_b)𝑑𝑥′𝑑𝑦′

c

(c

c

(c

𝑟T Z �� = xTsin 𝜃 cos𝜙 + yTsin 𝜃 sin𝜙

𝑢 = sin𝜃 cos𝜙𝑣 = sin 𝜃 sin𝜙

𝑟T Z �� = xT𝑢 + yT𝑣

Radiating apertureØComponents of the radiation vector 𝑁

z

xy

r '

φ

φ

θ

𝐸R 𝑟′ H+TMJ

𝐻R 𝑟′ H+TMJ

𝑁U 𝜃,𝜙 = − ] ] 𝐻XR 𝑥T,𝑦T 𝑒)*(^_`Ka_b)𝑑𝑥′𝑑𝑦′

c

(c

c

(c

𝑢 = sin𝜃 cos𝜙𝑣 = sin 𝜃 sin𝜙

𝑁X 𝜃, 𝜙 = ] ] 𝐻UR 𝑥T,𝑦T 𝑒)*(^_`Ka_b)𝑑𝑥′𝑑𝑦′

c

(c

c

(c

Ø Two-dimensional Fourier transforms

𝐹 𝜉, 𝜂 =12𝜋 ] ] 𝑓 𝑥T,𝑦T 𝑒()*(^_uKa_q)𝑑𝑥′𝑑𝑦′

c

(c

c

(c

Ø Two-dimensional Fourier transforms of the aperture fields

𝑁U 𝜃,𝜙 = − ] ] 𝐻XR 𝑥T, 𝑦T 𝑒)*(^_`Ka_b)𝑑𝑥T𝑑𝑦T

c

(c

= −2𝜋𝐻vXR(−𝑘𝑢,−𝑘𝑣)c

(c

𝑁X 𝜃, 𝜙 = ] ] 𝐻UR 𝑥T,𝑦T 𝑒)* ^_`Ka_b 𝑑𝑥T𝑑𝑦Tc

(c

c

(c

= 2𝜋𝐻vUR(−𝑘𝑢,−𝑘𝑣)

Radiating apertureØ Spherical components of the radiation vector 𝑁

z

xy

r '

φ

φ

θ

𝐸R 𝑟′ H+TMJ

𝐻R 𝑟′ H+TMJ

𝑁: = cos𝜃 cos𝜙𝑁U + cos𝜃 sin 𝜙𝑁X

Ø Radiated field by the aperture electric sources

𝐿= = − sin 𝜙𝑁U + cos𝜙𝑁X

𝑁: = 2𝜋 cos𝜃 sin𝜙𝐻vUR −𝑘𝑢,−𝑘𝑣 −cos𝜙𝐻vXR −𝑘𝑢,−𝑘𝑣

𝑁= = 2𝜋 sin𝜙𝐻vXR −𝑘𝑢,−𝑘𝑣 + cos𝜙𝐻vUR −𝑘𝑢,−𝑘𝑣

𝐸J 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 𝜁 𝑁: 𝜃,𝜙 𝜃> + 𝑁= 𝜃,𝜙 𝜙>

𝐸J: 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 (2𝜋) cos𝜃 𝜁𝐻vUR −𝑘𝑢,−𝑘𝑣 sin𝜙 − 𝜁𝐻XRx −𝑘𝑢,−𝑘𝑣 cos𝜙

𝐸J= 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 (2𝜋) 𝜁𝐻vUR −𝑘𝑢,−𝑘𝑣 cos𝜙 + 𝜁𝐻vXR −𝑘𝑢,−𝑘𝑣 sin 𝜙

Radiating apertureØRadiated far field by both equivalent sources

𝐸 𝑟 = 𝐸J 𝑟 + 𝐸s 𝑟

𝐸: 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 (−2𝜋) 𝐸rUR cos𝜙 + 𝐸rXR sin𝜙 + cos𝜃 −𝜁𝐻vUR sin 𝜙 + 𝜁𝐻vXR cos𝜙

𝐸= 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 (−2𝜋) cos𝜃 −𝐸rUR sin𝜙 + 𝐸rXR cos𝜙 − 𝜁𝐻vUR cos𝜙 + 𝜁𝐻vXR sin𝜙

ØRadiated far field is found in terms of the aperture field spectra

Radiated far field – Huygens apertures (I)

Ø Huygens sources: at all points on the aperture electric and magnetic field arerelated by the uniform plane-wave relationship

[𝑛G×𝐸 𝑟 ]H+∈J

= 𝑛G×[𝜁𝐻 𝑟 ×𝑛G]H+∈J

𝐸: 𝑟𝐸= 𝑟

= −𝑗𝑘𝑒({*+

4𝜋𝑟 −4𝜋 𝐶(𝜃) 𝑐𝑜𝑠𝜙 𝑠𝑖𝑛𝜙−𝑠𝑖𝑛𝜙 𝑐𝑜𝑠𝜙

𝐸rUR

𝐸rXR

𝐸XR 𝑥T,𝑦T = −𝜁𝐻UR(𝑥T, 𝑦T)𝐸UR 𝑥T, 𝑦T = 𝜁𝐻XR(𝑥T, 𝑦T)

𝐸rXR −𝑘𝑢,−𝑘𝑣 = −𝜁𝐻vXR(−𝑘𝑢,−𝑘𝑣)𝐸rUR −𝑘𝑢,−𝑘𝑣 = 𝜁𝐻vUR(−𝑘𝑢,−𝑘𝑣)

𝑛G𝐽CD(𝑟)

𝐽ED(𝑟)𝜁

𝐸: 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 (−4𝜋)1 + cos𝜃

2 𝐸rUR cos𝜙 + 𝐸rXR sin 𝜙

𝐸= 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 (−4𝜋)1 + cos𝜃

2 𝐸rXR cos𝜙 − 𝐸rUR sin 𝜙

Ø Generalized Huygens sources: at all points on the aperture electric andmagnetic field are related by

[𝑛G×𝐸 𝑟 ]H+∈J

= 𝑛G×[𝑍�𝐻 𝑟 ×𝑛G]H+∈J

Radiated far field – Huygens apertures (II)

Ø Huygens sources: at all points on the aperture electric and magnetic fieldare related by the uniform plane-wave relationship

[𝑛G×𝐸 𝑟 ]H+∈J

= 𝑛G×[𝜁𝐻 𝑟 ×𝑛G]H+∈J

𝐸 𝑟 = −𝑗𝑘𝑒()*+

4𝜋𝑟 𝑓(𝜃, 𝜙)

𝑓 𝜃, 𝜙 = −4𝜋 𝐶(𝜃) 𝐸rUR𝑐𝑜𝑠𝜙 + 𝐸rXR𝑠𝑖𝑛𝜙 𝜃>+ −𝐸rUR𝑠𝑖𝑛𝜙+ 𝐸rXR𝑐𝑜𝑠𝜙 𝜙>

Ø Radiation intensity

𝑈 𝜃, 𝜙 =12𝜁

𝑘�

4𝜋 � 𝑓 𝜃, 𝜙�

𝑓 𝜃, 𝜙�= 4𝜋 � 𝐶 𝜃 � 𝐸rUR

�+ 𝐸rXR

�

𝑛G𝐽CD(𝑟)

𝐽ED(𝑟)𝜁

Radiated far field – Huygens apertures (III)

Ø Radiation intensity 𝑈 𝜃, 𝜙 =12𝜁

𝑘�

4𝜋 � 𝑓 𝜃, 𝜙�

𝑓 𝜃, 𝜙�= 4𝜋 � 𝐶 𝜃 � 𝐸rUR

�+ 𝐸rXR

�

𝐸rUR�+ 𝐸rXR

�= 𝐸rUR𝑥G + 𝐸rXR𝑦G

�= 𝐸r�R

�

𝐸r�R(−𝑘𝑢; −𝑘𝑣)�=

12𝜋 Y 𝐸�R 𝑥T,𝑦T 𝑒)* `U_KbX_ 𝑑𝑥T𝑑𝑦T

Kc

(c

�

𝑛G𝐽CD(𝑟)

𝐽ED(𝑟)𝜁