4. Broadband indirect off-axis holography

Previous techniques are monochromatic techniques that might not be suitable for characterization of broadband antennas, for whose measurement it is usual to resort to time-domain (TD) techniques [48, 49].

The herein presented technique is an extrapolation of conventional off-axis holography that allows for efficient characterization of broadband antennas by means of amplitude-only acquisitions. Although the data acquisition and phase retrieval are different to the previous methods, as they are carried out in the TD, the physical layout of the elements is identical to the one already presented in Figure 1. This layout is presented in Figure 12(a) again in order to define some relevant distances that will be discussed later.

Figure 12. Broadband indirect off-axis holography: (a) layout of the measurement setup and (b) spectrum of the modified hologram.

During the acquisition process, a frequency sweep is made for each point of the spatial grid and the hologram is acquired over the studied frequency band, Eq. (17); then the spectrum is computed in the TD by means of an inverse FT, Eq. (18):

$$H(\overrightarrow{r},\omega) = |\mathbb{E}\_{\text{aut}}(\overrightarrow{r},\omega)|^2 + |\mathbb{E}\_{\text{ref}}(\overrightarrow{r},\omega)|^2 + \mathbb{E}\_{\text{aut}}(\overrightarrow{r},\omega)\mathbb{E}\_{\text{ref}}^\*(\overrightarrow{r},\omega) + \mathbb{E}\_{\text{aut}}^\*(\overrightarrow{r},\omega)\mathbb{E}\_{\text{ref}}(\overrightarrow{r},\omega) \tag{17}$$

$$h(\overrightarrow{r},t) = \left|e\_{\rm aut}(\overrightarrow{r},t)\right|^2 + \left|e\_{\rm ref}(\overrightarrow{r},t)\right|^2 + e\_{\rm aut}(\overrightarrow{r},t) \otimes e\_{\rm ref}^\*(\overrightarrow{r},t) + e\_{\rm aut}^\*(\overrightarrow{r},t) \otimes e\_{\rm ref}(\overrightarrow{r},t) \tag{18}$$

After filtering the desired image term in the TD by means of a time window Π, defined from t<sup>1</sup> to t2:

$$h\_{\text{filtered}}(\overrightarrow{r},t) = \Pi(t\_1, t\_2) \{ e\_{\text{aut}}(\overrightarrow{r},t) \otimes e\_{\text{ref}}^\*(\overrightarrow{r},t) \},\tag{19}$$

the phase retrieval is performed at each spatial point, simultaneously for all the acquired frequencies as

$$E\_{\text{aut,retiewed}}(\overrightarrow{r},\omega) = \frac{FT\_t\{h\_{\text{filtered}}(\overrightarrow{r},t)\}}{E\_{\text{ref}}^\*(\overrightarrow{r},\omega)}.\tag{20}$$

The subindex t in the FT indicates that the spectrum is computed in the TD.

This allows to retrieve one of the components of the tangential field in the acquisition plane, in order to obtain the FF of the AUT, as in the previous methods, the second tangential component also needs to be retrieved for NF-FF transformation. To do that, the process has to be repeated after a turn of 90<sup>∘</sup> of the AUT to change the acquired polarization.

Main advantages of this method are that position of the image terms can be controlled with the distance between the AUT and the reference antenna, the physical length of the AUT and reference branches, and the separation between the antennas and the acquisition plane, as it will be addressed next. Furthermore, as the phase is retrieved point-by-point in the spatial grid, the technique is compatible with array thinning techniques that allow to drastically reduce the number of acquisition points with the consequent time reduction [22, 31, 50].

On the other hand, the method also presents some disadvantages. As in conventional indirect off-axis holography, the reference antenna has to be previously known in amplitude and phase, also all the components of the setup, mainly the AUT, must be broadband; otherwise their time responses will be spread and may cause overlapping in the spectrum of the recorded hologram [23].

#### 4.1. Main parameter constraints

4. Broadband indirect off-axis holography

define some relevant distances that will be discussed later.

computed in the TD by means of an inverse FT, Eq. (18):

!;ωÞj<sup>2</sup> þ jErefð<sup>r</sup>

!;tÞj<sup>2</sup> þ jerefð<sup>r</sup>

hfilteredðr

Hðr

t<sup>1</sup> to t2:

hologram.

frequencies as

hðr

!;ωÞ¼jEautð<sup>r</sup>

!;tÞ¼jeautð<sup>r</sup>

(TD) techniques [48, 49].

260 Holographic Materials and Optical Systems

Previous techniques are monochromatic techniques that might not be suitable for characterization of broadband antennas, for whose measurement it is usual to resort to time-domain

The herein presented technique is an extrapolation of conventional off-axis holography that allows for efficient characterization of broadband antennas by means of amplitude-only acquisitions. Although the data acquisition and phase retrieval are different to the previous methods, as they are carried out in the TD, the physical layout of the elements is identical to the one already presented in Figure 1. This layout is presented in Figure 12(a) again in order to

During the acquisition process, a frequency sweep is made for each point of the spatial grid and the hologram is acquired over the studied frequency band, Eq. (17); then the spectrum is

Figure 12. Broadband indirect off-axis holography: (a) layout of the measurement setup and (b) spectrum of the modified

After filtering the desired image term in the TD by means of a time window Π, defined from

the phase retrieval is performed at each spatial point, simultaneously for all the acquired

!;ωÞE� refðr

!;t<sup>Þ</sup> <sup>⊗</sup> <sup>e</sup> � refðr !;tÞ þ <sup>e</sup> � autðr

> !;t<sup>Þ</sup> <sup>⊗</sup> <sup>e</sup> � refðr

!;ωÞ þ <sup>E</sup>�

autðr

!;ωÞErefð<sup>r</sup>

!;t<sup>Þ</sup> <sup>⊗</sup> <sup>e</sup>refð<sup>r</sup>

!;tÞg, (19)

!;ω<sup>Þ</sup> (17)

!;t<sup>Þ</sup> (18)

!;ωÞj<sup>2</sup> <sup>þ</sup> <sup>E</sup>autð<sup>r</sup>

!;tÞj<sup>2</sup> <sup>þ</sup> <sup>e</sup>autð<sup>r</sup>

!;tÞ ¼ <sup>Π</sup>ðt1;t2Þfeautð<sup>r</sup>

As in the previous methods, quality of the retrieved fields depends on how clean the filtering process is. Since the spectrum of the hologram is computed in the TD, position of the image terms is dependent on the starting times of the signals coming from the AUT taut and from the reference antenna tref, and thus, it can be controlled with the distance and the length Leff of the transmission lines employed in the setup.

In order to avoid overlapping two main restrictions have to be fulfilled:

• The length of the elements in the reference branch must be selected so that the image terms of the spectrum are swapped, taut � tref þ Δτ < 0. Thus, the desired term can be easily filtered, as shown in Figure 12(b).

In terms of the distances between elements in the setup, as defined in Figure 12(a), the previous condition yields the following expression considering the worst-case scenario (points in the corners of the acquisition plane closer to the reference antenna for whose tref > taut):

$$
\sqrt{D^2 + W^2 + H^2} - \left(\sqrt{D^2 + (W - L)^2 + H^2} + L\_{\text{eff}}\right) + c\Lambda\tau < 0,\tag{21}
$$

$$L\_{\rm eff} - L > \mathbf{c} \Delta \tau. \tag{22}$$

• The frequency sampling must be selected according to the Nyquist rule:

$$
\Delta f < \frac{1}{2T} = \frac{1}{2(t\_{\text{ref}} - t\_{\text{aut}})}.\tag{23}
$$

#### 4.2. Numerical validation for the characterization of a horn antenna in the Ka-band

For the numerical validation of the method, a 25 dB SGH is characterized in the Ka-band from 26:5 to 40 GHz. The physical layout is shown in Figure 13. The acquisition plane is a square grid of 300 mm side with spatial sampling of 3:7 mm in both directions, that is, λ=2 at 40 GHz, and is located at a distance of D ¼ 260 mm of the aperture of the AUT. A 15 dB horn is employed as reference antenna placed at L ¼ 200 mm from the center of the aperture of the AUT with an off-axis angle of <sup>θ</sup><sup>r</sup> <sup>¼</sup> <sup>37</sup>:5<sup>∘</sup> . A coaxial cable of Leff ≈ 48 cm is employed to connect the directional coupler to the reference antenna.

Figure 13. Setup for the 25 dB SGH antenna characterization in the Ka-band.

Figure 14(a) shows the modified hologram for the three points highlighted in Figure 12(a). The position of the image terms varies depending on the position of the probe in the acquisition plane. Figure 14(b) shows a detail of the retrieved phase in the central part of the frequency band for the worst-case scenario. Apart from some 180<sup>∘</sup> phase shifts, the agreement between the retrieved and directly measured phase is almost complete. Finally, Figure 14(c) depicts the error computed as in Eq. (16). Mean value of the error in the complete frequency band is 2:24%. The large values above 37 GHz are due to the signal level of the reference antenna, which decays in that part of the band.

Figure 14. Phase retrieval process: (a) spectrum of the modified hologram for three different acquisition points, (b) detail of the retrieved phase in the central frequency band, and (c) error for the phase retrieval in the complete frequency band.

The retrieved phase in the acquisition plane at 30 GHz is shown in Figure 15 compared to the phase directly acquired at that frequency. For this frequency, the error of the phase retrieval is 0:25%; thus, the retrieved phase is practically identical to the measured one.

Δf < 1

AUT with an off-axis angle of <sup>θ</sup><sup>r</sup> <sup>¼</sup> <sup>37</sup>:5<sup>∘</sup>

262 Holographic Materials and Optical Systems

decays in that part of the band.

the directional coupler to the reference antenna.

Figure 13. Setup for the 25 dB SGH antenna characterization in the Ka-band.

<sup>2</sup><sup>T</sup> <sup>¼</sup> <sup>1</sup>

For the numerical validation of the method, a 25 dB SGH is characterized in the Ka-band from 26:5 to 40 GHz. The physical layout is shown in Figure 13. The acquisition plane is a square grid of 300 mm side with spatial sampling of 3:7 mm in both directions, that is, λ=2 at 40 GHz, and is located at a distance of D ¼ 260 mm of the aperture of the AUT. A 15 dB horn is employed as reference antenna placed at L ¼ 200 mm from the center of the aperture of the

Figure 14(a) shows the modified hologram for the three points highlighted in Figure 12(a). The position of the image terms varies depending on the position of the probe in the acquisition plane. Figure 14(b) shows a detail of the retrieved phase in the central part of the frequency band for the worst-case scenario. Apart from some 180<sup>∘</sup> phase shifts, the agreement between the retrieved and directly measured phase is almost complete. Finally, Figure 14(c) depicts the error computed as in Eq. (16). Mean value of the error in the complete frequency band is 2:24%. The large values above 37 GHz are due to the signal level of the reference antenna, which

Figure 14. Phase retrieval process: (a) spectrum of the modified hologram for three different acquisition points, (b) detail of the retrieved phase in the central frequency band, and (c) error for the phase retrieval in the complete frequency band.

4.2. Numerical validation for the characterization of a horn antenna in the Ka-band

2ðtref � tautÞ

: (23)

. A coaxial cable of Leff ≈ 48 cm is employed to connect

After the phase is retrieved simultaneously for all the frequencies at each point of the acquisition plane, conventional NF-FF transformation and backpropagation techniques can be applied for the computation of the FF pattern and the fields in the aperture of the AUT [1]. Figure 15(a) shows the copolar pattern of the FF at 30 GHz, while the Ex component of the field in the aperture is shown in Figure 15(b). The black rectangle depicts the position of the aperture whose size is 700 mm · 500 mm.

Figure 15. Retrieved phase of the AUT at 30 GHz compared to the direct measurement: (a) directly acquired phase in the NF, degrees, and (b) retrieved phase, degrees.

Finally, Figure 16 shows the main cuts for <sup>φ</sup><sup>¼</sup> <sup>0</sup><sup>∘</sup> and <sup>φ</sup><sup>¼</sup> <sup>90</sup><sup>∘</sup> of the copolar pattern in Figure 15(a) (blue line labeled as Retrieved NF) compared to the cuts of a direct FF acquisition in an spherical anechoic chamber (labeled as Measured FF) and the cuts obtained for a NF-FF transformation of a field acquired with amplitude and phase (labeled as Measured NF). The valid margin of the NF-FF transformation, in which the data are comparable, is �25<sup>∘</sup> [1]. High level of coincidence

Figure 16. AUT characterization at 30 GHz from the retrieved data: (a) normalized FF copolar pattern in dB and (b) normalized Ex component of the field in the aperture of the AUT in dB.

can be observed between the three measurements. The small differences between the data directly acquired in FF and the transformed data are attributed to the lack of application of probe correction techniques during the NF-FF transformation (Figure 17) [1].

Figure 17. Comparison of the main cuts of the normalized amplitude of the AUT: (a) <sup>φ</sup><sup>¼</sup> <sup>0</sup><sup>∘</sup> and (b) <sup>φ</sup><sup>¼</sup> <sup>90</sup><sup>∘</sup> . The gray shaded areas indicate the valid margin of the NF-FF transformation [23].
