k"r
o
-ko
•
The negative imaginairy kz is required and that is the reason, why only the black points in Figure 9 are important. This results in:
For that reason, kxand kyare given by:
kx = k~ +jk~
ky
=
k~+jk;kx
=
-k~ - jk~, ky=
-k~ - jk;.The equation that should be solved was Eq. (7):
In this equation, the integration range may be complex and extends to infinity and therefore, this integral is hard to evaluate.
This gives a need to try to approximate this integral. The two approximations chosen in this report are the paraxial approximation and the far-field approximation.
3.2.1 Narrow angle or paraxial approximation
The inverse Fourier transformation must be approximated, because the integration range of the inverse Fourier integral is complex and extends to infinity and therefore this integral is difficult to evaluate. This approximation should result in a simple transformation.
The inverse Fourier transform is shown in Eq. (7).
kzcan be written in spherical coordinates as is shown:
The Taylor series is given:
kz
=
kocosO=kO(I-~+~-!..-+ J.
2! 4! 6!
De higher order terms can be neglected when 0 « 1 rad (= 56°) as shown in Figure 10.
--cos(9)
Figure 10: Difference of Taylor series of the cosine with the exact cosine.
With increasing
o
the error increases. As a rule of thumb, for 0 =1t/4 (= 45°), the approximation represents the cosine acceptably.For this reasonkzcan be approximated as follows, see [7]:
e+e (
e+eJ
e+ek=~e-e-e=k
z 0 x Y 0 1- xe
Y=k0 1_12 xk2 Y =k- x0 2k Y.0 0 0
Hence, Eq. (7) can be written as
The inverse Fourier transform of the frequency spectrum is:
1 '" '"
f ) f ff(k k) -
jkx- jk Y(x,y
=-()2
x' y e r Y dkxdky •2;r -00--'"
A standard Fourier transfonn pair is used:
With the actual variables substituted this equation becomes:
The convolution of Fourier transfonns is used:
As a result the equation for the paraxial or narrow angle approximation can be obtained:
E ( )
=
jko -jkoz"'f "'fE (, ,) -~k:[(x-x')2+(y-y')2]dx 'd 'x x,y,z e ax x,y e y.
2:rz -00-'"
(10)
Eaxis the amplitude of the aperture andzis chosen to be the propagation direction. The area of integration distances in this equation will be the aperture. In Figure 11, the simulation overview for the paraxial approximation is shown.
"x :
1/
I , , / '
:
~' ",I " /
I " "
---
EDT~-~iJ::~::~:~:C:o.-
,f I ibe· --- ,/ -J.--- --~
z'b: },
,-;a, /.
,-, ,-, ' : .",0
, , ' : /0/
.y :
I I I
Figure 11: Simulation overview for the paraxial approximation.
The aperture is situated in the origin as shown. This aperture radiates a plane wave in the z-direction. In this case the spatial angle is equal to the Cartesian angle. The measurement surface is at a distance of the aperture and the angle,
e,
is defined as the angle between the aperture height and the origin of the measurement surface.The purpose of the paraxial approximation is measuring in a narrow angle around the z-axis. As stated,
e
cannot be larger than 45 0 because of the limited validity of the used Taylor expansion.The equation used for the narrow-angle approximation is obtained with the uniform aperture Eq. (8) substituted in Eq. (10).
'k b a jko [( ,)2 ( ')2]
E(
x x,y,z) =
~e-jkozIIE
oe-2:; x-x +Y-Y dx'dYI2JlZ -b-a
(11)
This equation can be used to derive the propagating E-field by means of a plane-wave spectrum in the positive z-direction. Unless stated otherwise,Eo
=
1 V1m, A=
0.3 m and a large measurement area of x=
4 m and y=
4 m will be used in this report.As an example, for the parameters and uniform aperture sizes ofa
=
1 m and b=
2 m with at a measurement distance of z = 0.044 m are substituted in Eq. (11). This results in a calculated E-field as shown in Figure 12.1.5 ,---,---, 1.5 ,---,---,
At z= 0.044 m, the measurement takes place in the reactive near field of the measurement area.
Figure 12 shows the plane wave spectrum of the E-field. This spectrum has the general shape of the aperture field, but cannot produce the rapid variation in the field near the comers,
Ixl
= a andlYl
= b,of the aperture. This causes the peaks in the E-field.Because of the paraxial restriction this approximation should not be applied for this aperture at this short measurement distance.
At z= 1.1 m, the E-field calculated with Eq. (11) results in Figure 13. At a larger distance of the aperture antenna, the shape of the illumination becomes less visible in the radiated E-field. On the other hand, the spectral function becomes more visible. Figure 13 perfectly shows the problem of measuring in the near field, i.e. the maximum in this case is aty = 0.5 m. The far field, however, has its maximum aty= 0, this introduces a difference when using the far-field approximation while measuring in the near field. Whether this will result in a large deviation will be determined in the next chapters of this report.
At z= 15 m the E-field calculated with Eq. (11) results in Figure 14. Far away from the aperture plane, the field behaves like an expected sine-function. The amplitude of the field becomes a monotonously decreasing function, by a factor 1/r,of the measurement distance. The field distribution is equal to the calculated frequency spectrum, Eq. (9).
2 ,---,---, 2.---,---.,
-Figure 13: E-field ofa unifonn aperture 1m' 2 matz= 1.1 m forf=1 GHz.
o
O~---'---'In Figure 15, the field distribution in the measurement plane is shown for increasing measurement distancez. Figure 15 clearly shows what happens if the measurement distance is increased. First, the shape of the aperture is clearly visible with peaks all over the surface, then fewer peaks are left but the shape of the aperture is still visible. At the measurement distance of 3 m one dominant peak becomes visible and at a larger distance such as 15 m only this peak remains. Also, the changing shape of the field is shown, i.e., the aperture has larger length than width and in the far field this is shown as an interchanged field pattern. The small side creates a narrow large bundle, which needs space to spread. This explains the large influence of this side at a large distance.
a b c
4 4 1.5 4
1.2 1.5
2 1 2 2
g
0 0.8"§' 0g
0~ 0.6 ';: ~
0.4 0.5
-2 -2 -2
0.2
-4 -4 -4
-4 -2 0 2 4 -4 -2 0 2 4 -4 -2 0 2 4
x[m] x[m] x [m]
d e f
4 1.2 4 4
2 2 0.8 2
g
0 0.80.6g
0 0.6"§'...
0~ 0.4 ~ 0.4 ~
-2 -2 -2
0.2 0.2
-2 0 2 4 -4-4 -2 0 2 4 -2 0 2 4
x[m] x[m] x[m]
Figure15: The field distribution in the measurement plane of 1 m . 2 m at a)z= 0.044 m, b)z
=
0.5 m, c)z= 1.1 m d)z= 3 m, e)z= 5 m andf)z=
15 m.3.2.2 Far-field approximation
The other approximation to simplify the inverse Fourier transformation is the far-field
approximation. Inthis section this approximation is analysed. The far-field region of the uniform aperture dimensionsa= 1 m andb
=
2 m starts atZR=
13 m.In Figure 8 on page 22, the centre of the measurement area is chosen at(O,O,z). In this direction, the uniform-aperture far field has its maximum as demonstrated in Figure 14.
The far-field approximation equation is (see Annex A2):
(12)
Assume uniform aperture sizes ofa= 1 m and b= 2 m with at a measurement distance ofz= 15 m.
The value for the E-field atx= 0 andy= 0 can be derived as follows:
1 0.5 ( )
Ix
=I Ie}
kxx+kyY dxdy=
2sinc(0.5kJsinc(ky)'
-1-0.5
As before
kz
=
kocosB fork; <kg k;=
kgsin2Bkx
=
sinBcoscp ky=
sinB sin cpThe maximum is expected on the z-axis or at
e
= 0:Then the maximum of the E-field will be:
E
=
1 e-/20.9·15f = 2.~.
e-/20.9·15=
0.44V/m.0.3.15 x 9
E
=
0.44Vimis the maximum value positioned atx= 0 andy= 0 and is shown in Figure 14.This far-field approximation is only valid in the far field and the subject of investigation is the near-field to far-field transformation. This is the reason why this far-field approximation will not be used for the analytical analyses and the paraxial approximation will be used in the analyses performed on the uniform aperture in the next chapter. In the next section a physically more realistic behaviour of an antenna will be introduced.