1

LECTURE 13: Aperture Antennas – Part II(Rectangular horn antennas. Circular horns.)

1. Rectangular horn antennasHorn antennas are extremely popular in the microwave region (above

1 GHz). Horns provide high gain, low VSWR (with waveguide feeds),relatively wide bandwidth, and they are not difficult to make. There arethree basic types of rectangular horns:

The horns can be flared exponentially, too. This provides bettermatching in a broad frequency band, but is technologically more difficultand expensive.

The rectangular horns are ideally suited for rectangular waveguidefeeders. The horn acts as a gradual transition from a waveguide mode toa free-space mode of the EM wave. When the feeder is a cylindricalwaveguide, the antenna is usually a conical horn.

Why is it necessary to consider the horns separately instead ofapplying the theory of waveguide aperture antennas directly to theaperture of the horn? It is because the so-called phase error occurs due

2

to the difference between the length from the center of the feeder to thecenter of the horn aperture and the length from the center of the feeder tothe horn edge. This complicates the analysis, and makes the results forthe waveguide apertures invalid.

1.1. The H-plane sectoral hornThe following geometry parameters will be used often in the

subsequent analysis.

Cross-section at the H-plane (x-z)of an H-plane sectoral horn

a

Hl

HR

0R

R x

A zHα

22 2

0 2HA

l R = +

(13.1)

0

arctan2H

A

Rα

=

(13.2)

( ) 1

4H

Hl

R A aA

= − −

(13.3)

The two fundamental dimensions for the construction of the horn are Aand HR .

3

The tangential fields arriving at the input of the horn are thetransverse field components of the waveguide dominant mode TE10:

0 cos

/

gj zy

x y g

E E x ea

H E Z

βπ − =

= −(13.4)

where:

2

12

gZ

a

η

λ=

−

is the wave impedance of the TE10 mode;

2

0 12g a

λβ β = −

is the propagation constant of the TE10 mode.

Here, 0 2 /β ω µε π λ= = . The field that is illuminating the aperture ofthe horn is essentially an expanded version of the waveguide field. Notethat the wave impedance of the flared waveguide (i.e. the horn) graduallyapproaches the intrinsic impedance of open space η , as A (the H-planewidth) increases. The complication in analysis arises from the fact thatthe waves arriving at the horn aperture are not in phase due to thedifferent path lengths from the horn apex. The aperture phase variationis given by:

0( )j R Re β− − (13.5)Since the aperture is not flared in the y-direction, the phase is uniform inthis direction.

2 22 20 0 0

0 0

11 1

2

x xR R x R R

R R

= + = + ≈ +

(13.6)

The last approximation holds if 0x R , or 0/ 2A R . Then, one canassume that

2

00

1

2

xR R

R− ≈ (13.7)

Using (13.7), the field at the aperture is approximated as:

4

2

020 cos

y

j xR

aE E x eA

βπ − =

(13.8)

The field at the aperture plane outside the aperture is assumed equal tozero. The field expression (13.8) is substituted in the integral E

yJ (see

Lecture 12):( sin cos sin sin )( , )

y

A

E j x yy a

S

E x y e dx dyβ θ ϕ θ ϕ′ ′+′ ′ ′ ′= ∫∫I (13.9)

2

0

/ 2 / 22 sin cos sin sin

0/ 2 / 2

cosA bj x

RE j x j yy

A b

E x e e dx e dyA

ββ θ ϕ β θ ϕπ+ +′−

′ ′

− −

′ ′ ′= ⋅ ∫ ∫J (13.10)

The second integral has been already encountered but the first integral’ssolution is rather cumbersome. The above integral (13.10) reduces to:

( )00

sin sin sin1 2,2 sin sin

2

Ey

bR

E I bb

β θ ϕπ θ ϕ ββ θ ϕ

=

J (13.11)

where:

[ ]

[ ]

20

20

sin cos2

2 2 1 1

sin cos2

2 2 1 1

( , ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Rj

A

Rj

A

I e C s jS s C s jS s

e C t jS t C t jS t

πβ θ ϕβ

πβ θ ϕβ

θ ϕ +

−

′ ′ ′ ′= − − +

′ ′ ′ ′+ − − +

(13.12)

and

01 0

0

1

2

A Rs R u

R A

β πβπβ

′ = − − −

;

02 0

0

1

2

A Rs R u

R A

β πβπβ

′ = + − −

;

01 0

0

1

2

A Rt R u

R A

β πβπβ

′ = − − +

;

5

02 0

0

1

2

A Rt R u

R A

β πβπβ

′ = + − +

;

sin cosu θ ϕ= .( )C x and ( )S x are Fresnel integrals, which are defined as:

2

0

2

0

( ) cos ; ( ) ( )2

( ) sin ; ( ) ( )2

x

x

C x d C x C x

S x d S x S x

π τ τ

π τ τ

= − = −

= − = −

∫

∫(13.13)

More accurate evaluation of EyJ can be obtained if the approximation in

(13.6) is not made, and theyaE is substituted in (13.9) as:

( )2 20 0

2 200

0

0

cos

cos

y

j R x R

a

j R xj R

E E x eA

E e x eA

β

ββ

π

π

− + −

− ++

= =

=

(13.14)

The far fields can be now calculated as (see Lecture 10):

(1 cos )sin4

(1 cos )cos4

j rEy

j rEy

eE j

r

eE j

r

β

θ

β

ϕ

β θ ϕπ

β θ ϕπ

−

−

= + ⋅

= + ⋅

J

J

(13.15)

or

( )

00

sin sin sin1 cos 2

4 2 sin sin2

ˆ ˆ( , ) sin cos

j rb

R eE j E b

br

I

ββ θ ϕ

π θβ ββ π θ ϕ

θ ϕ θ ϕ ϕ ϕ

−

+ =

⋅ +

(13.16)

6

The amplitude pattern of the H-plane sectoral horn is obtained as:

sin sin sin1 cos 2 ( , )

2 sin sin2

b

E Ib

β θ ϕθ θ ϕβ θ ϕ

+ = ⋅

(13.17)

Principal-plane patterns

E-plane ( 90ϕ = ):sin sin sin

1 cos 2( )2 sin sin

2

E

b

Fb

β θ ϕθθ β θ ϕ

+ =

(13.18)

It can be shown that the second factor of (13.18) is exactly the pattern ofa uniform line source of length b along the y-axis.

H-plane ( 0ϕ = ):

1 cos( ) ( )

2

1 cos ( , 0 )

2 ( 0 , 0 )

H HF f

I

I

θθ θ

θ θ ϕθ ϕ

+= ⋅ =

+ == ⋅= =

(13.19)

The H-plane pattern in terms of the ( , )I θ ϕ integral is an approximation,which is a consequence of the phase approximation made in (13.7).Accurate value for ( )Hf θ can be found by integrating numerically thefield as given in (13.14), i.e.

2 20

/ 2sin

/ 2

( ) cosA

j R x j xH

A

xf e e dx

Aβ β θπθ

+′− + ′

−

′ ′∝ ∫ (13.20)

7

8

The directivity of the H-plane sectoral horn is calculated by the generaldirectivity expression for aperture-type antennas (for derivation, seeLecture 12):

2

0 2 2

4

| |A

A

a

S

a

S

E ds

DE ds

πλ

′

=′

∫∫

∫∫(13.21)

The integral in the denominator is proportional to the total radiatedpower:

/ 2 / 222 2

0/ 2 / 2

20

2 | | cos

2

A

b A

rad aS b A

E ds E x dx dyA

AbE

πη+ +

− −

′ ′ ′ ′Π = = =

=

∫∫ ∫ ∫(13.22)

In the solution of the integral in the numerator of (13.21), the field issubstituted with its phase approximated as in (13.8). The final result is:

2

32 4( )H H

H ph t phb A

D Abπε ε ε

λ π λ λ = =

, (13.23)

where:

2

8;tε

π=

[ ] [ ]{ }22 2

1 2 1 2( ) ( ) ( ) ( )64

Hph C p C p S p S p

t

πε = − + − ;

1 21 1

2 1 , 2 18 8

p t p tt t

= + = − + ;

2

0

1 1

8 /

At

Rλ λ =

.

The factor tε explicitly shows the aperture efficiency associated with the

aperture taper. The factor Hphε is the aperture efficiency associated with

the aperture phase distribution.

9

A family of universal directivity curves is given below. From thiscurves, it is obvious that for a given axial length 0R at a givenwavelength, there is an optimal aperture width A corresponding to themaximum directivity.

It can be shown that the optimal directivity is obtained if the relationbetween A and 0R is:

03A Rλ= (13.24)

or

03A R

λ λ= (13.25)

0R

10

1.2. The E-plane sectoral horn

Cross-section at the E-plane (y-z)of an E-plane sectoral horn

b

El

ER

0R

R y

B zEα

The geometry of the E-plane sectoral horn in the E-plane (y-z plane)is analogous to that of the H-plane sectoral horn in the H-plane. Theanalysis is following the same lines as in the previous section. The fieldat the aperture is approximated by (compare with (13.8)):

2

020 cos

y

j yR

aE E x eA

βπ − =

(13.26)

Here, the approximations2 2

2 20 0 0

0 0

11 1

2

y yR R y R R

R R

= + = + ≈ +

(13.27)

and2

00

1

2

yR R

R− ≈ (13.28)

are made, which are analogous to (13.6) and (13.7).

11

The radiation field is obtained as:

( )

[ ]

20 sin sin

2 200

2 2 1 12

4 ˆ ˆsin cos4

cos sin cos(1 cos ) 2 ( ) ( ) ( ) ( )

21 sin cos

2

R Bj r ja R eE j E e

r

a

C r jS r C r jS ra

β ββ θ ϕπβ θ ϕ ϕ ϕπ β π

β θ ϕθβ θ ϕ

− = ⋅ +

+ ⋅ − − + −

(13.29)

The arguments of the Fresnel integrals used in (13.29) are:

1 00

2 00

sin sin ;2 2

sin sin2 2

B Br R

R

B Br R

R

β β θ ϕπ

β β θ ϕπ

= − −

= + −

(13.30)

Principal-plane patternsThe normalized H-plane pattern is found by substituting 0ϕ = in

(13.29).

2

cos sin1 cos 2( )

21 sin

2

a

Ha

β θθθ

β θ

+ = −

(13.31)

The second factor in this expression is the pattern of a uniform phasecosine-amplitude tapered line source.

12

The normalized E-plane pattern is found by substituting 90ϕ = in(13.29).

[ ] [ ]2 22 1 2 1

2 20 0

1 cos( ) ( )

2

( ) ( ) ( ) ( )1 cos

2 4 ( ) ( )

EE f

C r C r S r S r

C r S rθ θ

θθ θ

θ

= =

+= =

− + −+= +

(13.32)

Here, the arguments of the Fresnel integrals are calculated for 90ϕ = :

1 00

2 00

sin ;2 2

sin2 2

B Br R

R

B Br R

R

β β θπ

β β θπ

= − −

= + −

(13.33)

and

0 20

( 0)2

Br r

Rθβθ

π= = = = (13.34)

Similar to the H-plane sectoral horn, the principal E-plane pattern can beaccurately calculated if no approximations for the phase distribution aremade. Then, the function ( )Ef θ has to be calculated by numericalintegration of (compare with (13.20)):

2 20

/ 2sin

/ 2

( )B

j R y j yE

B

f e e dyβ β θθ ′− + ′⋅

−

′∝ ∫ (13.35)

13

14

DirectivityThe directivity of the E-plane sectoral horn is found in a manner

analogous to the H-plane sectoral horn.

2

32 4E EE ph t ph

a BD aB

πε ε ελ π λ λ

= = , (13.36)

where:

2

8tε

π= ;

2 2

20

( ) ( ),

2Eph

C q S q Bq

q Rε

λ+= = .

A family of universal directivity curves ( EDa

λvs. 0R

λ) is given below.

15

The optimal relation between the flared height B and the horn length 0Ris:

02B Rλ= (13.37)

1.3. The pyramidal hornThe pyramidal horn is probably the most popular antenna in the

microwave frequency ranges (from 1≈ GHz up to 18≈ GHz). Thefeeding waveguide is flared in both directions, the E-plane and the H-plane. All results are a combination of the E-plane sectoral horn and theH-plane sectoral horn analysis. The field distribution of the apertureelectric field is:

2 2

2 2

0 02

0 cosE H

y

x yj

R RaE E x e

A

βπ

− + =

(13.38)

The E-plane principal pattern of the pyramidal horn is the same as the E-plane principal pattern of the E-plane sectoral horn. The same holds forthe H-plane patterns of the pyramidal horn and the H-plane sectoral horn.

The directivity of the pyramidal horn can be found rather simply byintroducing the phase efficiency factors of both planes and the taperefficiency factor of the H-plane:

2

4 E HP t ph phD

π ε ε ελ

= , (13.39)

where:

2

8tε

π= ;

[ ] [ ]{ }22 2

1 2 1 2( ) ( ) ( ) ( )64

Hph C p C p S p S p

t

πε = − + − ;

1 21 1

2 1 , 2 1 ,8 8

p t p tt t

= + = − +

2

0

1 1

8 /H

At

Rλ λ =

2 2

20

( ) ( ),

2

Eph E

C q S q Bq

q Rε

λ+= = .

16

The gain of a horn is usually very close to its directivity because theradiation efficiency is very good (low losses). The directivity (and gain)as calculated with (13.39) is very close to measurements. The aboveexpression is a physical optics approximation, and it does not take intoaccount only the multiple diffractions and the diffraction at the edges ofthe horn arising from reflections from the horn interior. Thesephenomena, which are unaccounted for, lead to minor fluctuations of themeasured results about the prediction of (13.39). That is why horns areoften used as gain standards in antenna measurements.

The optimal directivity of an E-plane horn is achieved at 1q = (see

also (13.37)), 0.8Ephε = . The optimal directivity of an H-plane horn is

achieved at 3/8t = (see also (13.24)), 0.79Hphε = . Thus, the optimal

horn has a phase aperture efficiency of0.632P H E

ph ph phε ε ε= = (13.40)

The total aperture efficiency includes the taper factor, too:0.81 0.632 0.51P H E

ph t ph phε ε ε ε⋅= = ⋅ = (13.41)

Therefore, the best achievable directivity for a rectangular waveguidehorn is about half that of a uniform rectangular aperture.

It should be also noted that best accuracy is achieved if Hphε and E

phεare calculated numerically without using the second-order phaseapproximations in (13.7) and (13.28).

17

Optimum horn designUsually, the optimum (from the point of view of maximum gain)

design of a horn is desired because it renders the shortest axial length.The whole design can be actually reduced to the solution of a singlefourth-order equation. For a horn to be realizable,

E H PR R R= = (13.42)must hold.

z

HR

0HR

x

AHα

ab

ER

0ER

y

BEα

It can be shown that

0 / 2

/ 2 / 2

H

H

R A A

R A a A a= =

− −(13.43)

0 / 2

/ 2 / 2

E

E

R B B

R B b B b= =

− −(13.44)

The optimum-gain condition in the E-plane (13.37) is substituted in(13.44) to yield

2 2 0EB bB Rλ− − = (13.45)There is only one physically meaningful solution to (13.45):

( )218

2 EB b b Rλ= + + (13.46)

18

Similarly, the maximum-gain condition for the H-plane of (13.24)together with (13.43) yields

( )2

3 3H

A aA a AR A

A λ λ −−= =

(13.47)

Since E HR R= must be fulfilled, (13.47) is substituted in (13.46), whichgives

( )2 81

2 3

A A aB b b

−= + +

(13.48)

Substituting in the expression for the horn’s gain

2

4apG AB

π ελ

= (13.49)

gives the relation between A, the gain G and the aperture efficiency 2apε :

22

4 1 8 ( )

2 3apA a a

G A b bπ ε

λ −= + +

(13.50)

2 2 44 3

2 2

3 30

8 32ap ap

bG GA aA A

λ λπε π ε

⇒ − + − = (13.51)

Equation (13.51) is the optimum pyramidal horn design equation. Theoptimum-gain value of 0.51apε = is usually used, which makes the

equation a fourth-order polynomial equation. Its roots can be foundanalytically (which is not particularly easy), and numerically. In anumerical solution, the first guess is usually set at (0) 0.45A Gλ= .

Horn antennas operate well over bandwidth of about 50%. However,performance is optimal only at a given frequency. To understand betterthe frequency dependence of the directivity and the aperture efficiency,the plot of these curves for an X-band (8.2 GHz to 12.4 GHz) horn fedby WR90 waveguide is given below.

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Directivity and aperture efficiency of standard gain rectangular horn forWR90 ( 0.9a = in. = 2.286 cm and 0.4b = in. = 1.016 cm):

The gain increases with frequency, which is typical for apertureantennas. However, the curve shows saturation at higher frequencies.This is due to the decrease of the aperture efficiency, which is a result ofan increased phase error.

20

The pattern of a “large” pyramidal horn ( 10.525f = GHz, feeder iswaveguide WR90):

21

Comparison of the E-plane patterns of a waveguide open end, “small”pyramidal horn and “large” pyramidal horn:

Note the multiple side lobes and the significant back lobe. They are dueto diffraction at the horn edges, which are perpendicular to the E field.To reduce edge diffraction, enhancements are proposed for hornantennas such as

• Corrugated horns• Aperture-matched horns

Corrugated horns taper the E field in the x-direction, thus, reducingside-lobes and diffraction from edges. The overall main beam becomessmooth and nearly rotationally symmetrical (esp. for A B≈ ). This isimportant when the horn is used as a feed to a reflector antenna.

22

23

Comparison of the H-plane patterns of a waveguide open end, “small”pyramidal horn and “large” pyramidal horn:

24

2. Circular apertures2.1. A uniform circular aperture

The uniform circular aperture is approximated by a circular openingin a ground plane illuminated by a uniform plane wave normally incidentfrom behind.

x y

z

Ea

The field distribution is described as:

0ˆ ,aE xE aρ′= ≤ (13.52)The radiation integral is:

ˆ0

a

E j r rx

S

E e dsβ ′⋅ ′= ∫∫J (13.53)

The integration point is at:ˆ ˆcos sinr x yρ ϕ ρ ϕ′ ′ ′ ′ ′= + (13.54)

In (13.54), cylindrical notations are used.( )( )

ˆ sin cos cos sin sin

sin cos

r r ρ θ ϕ ϕ ϕ ϕρ θ ϕ ϕ

′ ′ ′ ′⇒ ⋅ = + =′ ′= −

(13.55)

Hence, (13.53) becomes:2

sin cos( )0

0 0

0 00

2 ( sin )

aE jx

a

E e d d

E J d

πβρ θ ϕ ϕ ϕ ρ ρ

π ρ βρ θ ρ

′ ′− ′ ′ ′= =

′ ′ ′=

∫ ∫

∫

J

(13.56)

Here 0J is the Bessel function of the first kind of order zero. Thefollowing is true:

25

0 1( ) ( )xJ x dx xJ x=∫ (13.57)

Applying (13.57) to (13.56) leads to:

0 12 ( sin )sin

Ex

aE J aπ β θ

β θ=J (13.58)

In this case, the equivalent magnetic current formulation of theequivalence principle is used (see Lecture 10). The far field is obtainedas:

( )( ) 2 1

0

ˆ ˆcos cos sin2

2 ( sin )ˆ ˆcos cos sin2 sin

j rEx

j r

eE j

r

e J aj E a

r a

β

β

θ ϕ ϕ θ ϕ βπ

β θθ ϕ ϕ θ ϕ β ππ β θ

= − =

= −

J(13.59)

Principal-plane patterns

E-plane ( 0ϕ = ): 12 ( sin )( )

sin

J aE

aθβ θθ

β θ= (13.60)

H-plane ( 90ϕ = ): 12 ( sin )( ) cos

sin

J aE

aϕβ θθ θ

β θ= ⋅ (13.61)

The 3-D amplitude pattern:2 2 1

( )

2 ( sin )( , ) 1 sin sin

sinf

J aE

aθ

β θθ ϕ θ ϕβ θ

= − ⋅ (13.62)

The larger the aperture, the less significant is the cosθ factor in (13.61)because the main beam in the 0θ = direction is very narrow and in thissmall solid angle cos 1θ ≈ .

26

Example plot of the principal-plane patterns for 3a λ= :

-15 -10 -5 0 5 10 15

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

6*pi*sin(theta)

E-planeH-plane

The half-power angle for the ( )f θ factor is obtained atsin 1.6aβ θ = . So, the HPBW for large apertures (a λ ) is given by

1/ 21.6 1.6

2 2arcsin 2 58.42

HPBWa a a

λθβ β

= ≈ =

, deg (13.63)

For example, if the diameter of the aperture is 10a λ= , then5.84HPBW = .

The side-lobe level of any uniform circular aperture is 0.1332 (-17.5dB).

Any uniform aperture has unity taper aperture efficiency, and itsdirectivity can be found only in terms of its physical area:

22 2

4 4u pD A a

π π πλ λ

= = (13.64)

27

2.2. Tapered circular aperturesMany practical circular aperture antennas can be approximated as

radially symmetric apertures with field amplitude distribution, which istapered from the center toward the aperture edge. Then, the radiationintegral (13.56) has a more general form:

0 00

2 ( ) ( sin )a

E E J dπ ρ ρ βρ θ ρ′ ′ ′ ′= ∫J (13.65)

In (13.65), it is still assumed that the field has axial symmetry, i.e. it doesnot depend on ϕ′ . Often used approximation is the parabolic taper oforder n:

2

( ) 1

n

aEa

ρρ ′ ′ = −

(13.66)

This is substituted in (13.65) to calculate the respective component of theradiation integral:

( )2

0 00

2 1 ( sin )

naE E J d

a

ρθ π ρ βρ θ ρ ′ ′ ′ ′= −

∫J (13.67)

The following relation is used to solve (13.67):1

20 11

0

2 !(1 ) ( ) ( )

nn

nn

nx xJ bx dx J b

b++− =∫ (13.68)

In our case, /x aρ′= and sinb aβ θ= . Then ( )E θJ reduces to2

0( ) ( , )1

E aE f n

n

πθ θ=+

J , (13.69)

where:

( )1

11

2 ( 1)! ( sin )( , )

sin

nn

n

n J af n

a

β θθβ θ

++

++= (13.70)

is actually the normalized pattern function (neglecting the angular factorssuch as cosϕ and cos sinθ ϕ .

28

The aperture taper efficiency is calculated to be:2

22

11

2 (1 ) (1 )1 2 1

t

CC

n

C C CC

n n

ε

− + + =− −+ +

+ +

(13.71)

Here, C denotes the pedestal height. The pedestal height is the edgefield illumination relative to the illumination at the center.

The properties of several common tapers are given in the tablesbelow. The parabolic taper ( 1n = ) provides lower side lobes incomparison with the uniform distribution ( 0n = ) but it has a broadermain beam. There is always a trade-off between low side-lobe levelsand high directivity (small HPBW). More or less optimal solution isprovided by the parabolic-on-pedestal aperture distribution. Moreover,this distribution approximates very closely the real case of circularreflector antennas, where the feed antenna pattern is intercepted by thereflector only out to the reflector rim.

29