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TitleRecent Advances in Near-Field to Far-Field Transformation Techniques
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Page 1

Recent Advances in Near-Field to
Far-Field Transformation Techniques

Guest Editors: Claudio Gennarelli, Amedeo Capozzoli,
Lars J. Foged, Jeff Fordham, and Daniël Janse van Rensburg

International Journal of Antennas and Propagation

Page 2

Recent Advances in Near-Field to
Far-Field Transformation Techniques

Page 66

753156.fig.0016a.eps


10 International Journal of Antennas and Propagation

ϕ

1

2

AUT

Probe

z
y

x

(a)

1

2

AUT

z

y

ϑ

(b)

Figure 13: Illustrating the need for probe compensation. When the probe rotates around the AUT from position 1 to 2 (a), approximately
the same receiving features of the probe are involved. When the probe moves along the z axis from position 1 to 2 (b), different receiving
features of the probe become relevant.

−60 −40 −20 0 20 40 60
−50
−45
−40
−35
−30
−25
−20
−15
−10
−5

0

ϕ (degrees)

F
ar

fi
el

d
(

d
B

)

Figure 14: Dual horn, horizontal polarization, sum pattern. ϕ cut
of the ϕ component of the far field. Blue solid line: proposed
approach. Red dashed line: approach in [7, 8].

20 40 60 80 100 120 140 160
−50
−45
−40
−35
−30
−25
−20
−15
−10
−5

0

F
ar

fi
el

d
(

d
B

)

ϑ (degrees)

Figure 15: Dual horn, horizontal polarization, sum pattern. ϑ cut of
the ϕ component of the far field. Blue solid line: proposed approach.
Red dashed line: approach in [7, 8].

6

4

2

0

−2

−4

−6

−1 0 1

−5

−10

−15

−20

−30

−25

−35

−40

−50

−45

y


x/λ

(a)

6

4

2

0

−2

−4

−6

y


−1 0 1
x/λ

150

100

50

0

−50

−100

−150

(b)

Figure 16: Dual horn, horizontal polarization, sum pattern. (a)
amplitude of the aperture field distribution. (b) phase of the
aperture field distribution.

in turn, can be linked to the ϑ′ and ϕ′ components hϑ′ and
hϕ′ as

hx′
(
ϑi,ϕi

) = hϑ′
(
ϑi,ϕi

)
cos ϑi cosϕi − hϕ′

(
ϑi,ϕi

)
sinϕi,

hy′
(
ϑi,ϕi

) = hϑ′
(
ϑi,ϕi

)
cos ϑi sinϕi + hϕ′

(
ϑi,ϕi

)
cosϕi,

hz′
(
ϑi,ϕi

) = −hϑ′
(
ϑi,ϕi

)
sin ϑi.

(A.2)

The probe employed in this paper for the measurements
is an a× b sized open-ended rectangular waveguide [24].

Page 67

753156.fig.0020.eps


International Journal of Antennas and Propagation 11

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

−1
−1

−0.8
−0.6
−0.4
−0.2

0

0.2

0.4

0.6

0.8

1

−0.5 0 0.5 1
u/β

v


Figure 17: Dual horn, horizontal polarization, sum pattern.
Amplitude of the ϕ component of the far field.

20 40 60 80 100 120 140 160
−50
−45
−40
−35
−30
−25
−20
−15
−10
−5

0

F
ar

fi
el

d
(

d
B

)

ϑ (degrees)

Figure 18: Dual horn, horizontal polarization, difference pattern.
ϑ cut of the ϕ component of the far field. Blue solid line: proposed
approach. Red dashed line: approach in [7, 8].

The dimensionless components hϑS and hϕS of the probe
effective length in the Oxsyszs reference system, the same
employed in [24], are

hϑS
(
ϑS,ϕS

) =
{

4
π2

[

1+

(
kTE10
β

)



(

1−
(
kTE10
β

))]

+C0

}

× sinϕS
1 +

(
kTE10 /β

)
cos ϑS

1 +
(
kTE10 /β

)
sin
(
β(b/2) sin ϑS

)

β(b/2) sin ϑS
,

hϕS
(
ϑS,ϕS

) = cosϕS cos
(
β
(
a

2

)
sin ϑS

)

×
[

cos ϑS +
(
kTE10 /β

)
+ Γ
(
cos ϑS −

(
kTE10 /β

))

(π/2)2 − (β(a/2) sin ϑS
)2

+C0

]

(A.3)

6

4

2

0

−2

−4

−6

−1 0 1

−5

−10

−15

−20

−30

−25

−35

−40

−50

−45

y


x/λ

(a)

6

4

2

0

−2

−4

−6

y


−1 0 1
x/λ

150

100

50

0

−50

−100

−150

(b)

Figure 19: Dual horn, horizontal polarization, difference pattern.
(a) amplitude of the aperture field distribution. (b) phase of the
aperture field distribution.

−50

−45

−40

−35

−30

−25

−20

−15

−10

−5

−1 −1

−0.8
−0.6
−0.4
−0.2

0

0.2

0.4

0.6

0.8

1

−0.5 0 0.5 1
u/β

v


Figure 20: Dual horn, horizontal polarization, difference pattern.
Amplitude of the ϕ component of the far field.

where ϑS and ϕS are the ϑ and ϕ coordinates of the corre-
sponding reference system, kTE10 is the propagation constant
of the TE10 propagation mode of the a× b sized rectangular
waveguide, and Γ is the reflection coefficient of the TE10
mode from the end of the waveguide, whose value at 10 GHz
has been measured in [24] and is approximately equal to
0.0603 + j0.2837. Moreover, C0 is a real constant whose value
(0.129) has been numerically evaluated according to [24].

During the scanning, the probe is oriented so that the
electric field polarization for the TE10 mode matches the
polarization of the aperture field of interest for the AUT (i.e.,
vertical or horizontal).

Page 131

18 International Journal of Antennas and Propagation

near-field setup has been proposed in this paper. The
method is based on the Gerchberg-Papoulis iterative algo-
rithm used to extrapolate band-limited functions, and it
is a generalization of the approach presented in [17] for
the planar case. Therefore, the proposed method can be
viewed as a continuation of the work developed in [17], not
only extending its applicability, but also introducing new
algorithms to reduce the computational time required to
remove the truncation errors. The convergence of this meth-
od has been mathematically demonstrated. Moreover, a de-
tailed study of the spectral reliable region for each type of
measurement setup and an analysis of critical aspects of the
method has been performed. Finally, the method has been
validated by using both simulated and measured near-field
data, showing that it is possible to reduce the truncation
errors e� ectively. It was noted that the proposed method
works well for planar aperture antennas because the antenna
aperture, in which the fields are assumed to be concentrated,
is well defined.

Acknowledgments

This work was developed with the support of the Spanish
FPU Grant for Ph.D. students and the financing of the
Crocante Project (TEC2008-06736-C03-01/TEC). F. J. Cano-
Fácila would also like to thank DTU Electrical Engineering
for its support during his stay at the Technical University of
Denmark.

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