2. Conventional indirect off-axis holography review

The word holography comes from the Greek words hólos (whole) and grápho (written or represented) and was first coined by Gabor in 1948 to define a new technique in the optics field for retrieving the amplitude and phase of an unknown field after recording the intensity of a coherent wave disturbance [34] with a reference field, whose amplitude and phase could be properly characterized. The technique was later adapted by Leith and Upatnieks to use an off-axis reference [24].

The term holography has been subsequently employed in the context of antenna metrology and electromagnetic imaging to refer to another technique in which the phase information is directly acquired with the amplitude and a cable reference is employed (direct holography) [35, 36]. Thus, to avoid confusion, the methods described in this chapter will be referred to as indirect off-axis holography, since the phase is indirectly measured.

#### 2.1. Indirect off-axis holography

Indirect off-axis techniques are based on two-step procedures: (1) recording the intensity of the interference pattern formed by the AUT and the reference field and (2) performing the phase retrieval of the unknown field (AUT's field) by means of a filtering process of the recorded pattern or hologram in the spectral domain. Conventional setup is usually implemented as shown in Figure 1 employing a radiated reference field [21, 25] that is obtained from a sample of the source by means of a directional coupler. A variable attenuator (or amplifier) is usually included in the AUT or reference branches in order to balance the power between both branches and to increase the dynamic range of the hologram.

Figure 1. Basic setup for conventional indirect off-axis holography antenna measurement.

Another option is to create a plane reference wave by means of a shaped plane mirror which is employed as the collimator in compact antenna ranges [37]. Nevertheless, correctly shaping the mirror for high frequency indirect holography requires accurate and expensive machining.

The hologram is recorded at each point of the acquisition plane as the squared sum of the fields of the AUT Eaut and the reference antenna Eref as:

$$H(\overrightarrow{r'}) = |E\_{\text{aut}}(\overrightarrow{r'}) + E\_{\text{ref}}(\overrightarrow{r'})|^2. \tag{1}$$

The expression of the hologram can be further developed into

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

where the asterisk is used to denote complex conjugate.

field for retrieving the amplitude and phase of an unknown field after recording the intensity of a coherent wave disturbance [34] with a reference field, whose amplitude and phase could be properly characterized. The technique was later adapted by Leith and Upatnieks to use an

The term holography has been subsequently employed in the context of antenna metrology and electromagnetic imaging to refer to another technique in which the phase information is directly acquired with the amplitude and a cable reference is employed (direct holography) [35, 36]. Thus, to avoid confusion, the methods described in this chapter will be referred to as

Indirect off-axis techniques are based on two-step procedures: (1) recording the intensity of the interference pattern formed by the AUT and the reference field and (2) performing the phase retrieval of the unknown field (AUT's field) by means of a filtering process of the recorded pattern or hologram in the spectral domain. Conventional setup is usually implemented as shown in Figure 1 employing a radiated reference field [21, 25] that is obtained from a sample of the source by means of a directional coupler. A variable attenuator (or amplifier) is usually included in the AUT or reference branches in order to balance the power between both

Another option is to create a plane reference wave by means of a shaped plane mirror which is employed as the collimator in compact antenna ranges [37]. Nevertheless, correctly shaping the mirror for high frequency indirect holography requires accurate and expensive machining. The hologram is recorded at each point of the acquisition plane as the squared sum of the fields

Figure 1. Basic setup for conventional indirect off-axis holography antenna measurement.

indirect off-axis holography, since the phase is indirectly measured.

branches and to increase the dynamic range of the hologram.

of the AUT Eaut and the reference antenna Eref as:

off-axis reference [24].

246 Holographic Materials and Optical Systems

2.1. Indirect off-axis holography

If the expression in Eq. (2) is Fourier-transformed to the spatial frequency domain or k-space, the spectrum of the hologram can be expressed as

$$h(\overrightarrow{k}) = |\varepsilon\_{\rm aut}(\overrightarrow{k})|^2 + |\varepsilon\_{\rm ref}(\overrightarrow{k})|^2 + \varepsilon\_{\rm aut}(\overrightarrow{k}) \otimes \varepsilon\_{\rm ref}^\*( -\overrightarrow{k}) + \varepsilon\_{\rm aut}^\*( -\overrightarrow{k}) \otimes \varepsilon\_{\rm ref}(\overrightarrow{k}),\tag{3}$$

being eaut and eref the Fourier transform (FT) of Eaut and Eref, respectively, and ⊗ is the convolution operator.

As it is depicted in Figure 2, the spectrum of the hologram is composed of four different terms: the two zero-frequency harmonics in the center, known as autocorrelation terms, and the cross-correlation or image terms, which contain shifted and distorted (in case of using a nonplanar wave reference field) information about the complex field of the AUT.

Figure 2. Schematic representation of the spectrum of the hologram for an off-axis angle in the x-axis.

Providing no overlap between the autocorrelation terms and the image term corresponding to eautð k ! Þ ⊗ e� refð�k ! Þ exists, the latter can be bandpass-filtered as

$$h\_{\text{filtered}}(\overrightarrow{k}) = \Pi(\overrightarrow{k}\_1, \overrightarrow{k}\_2) \{ e\_{\text{aut}}(\overrightarrow{k}) \otimes e\_{\text{ref}}^\*( - \overrightarrow{k} ) \},\tag{4}$$

where Πð k ! <sup>1</sup>; k ! <sup>2</sup>Þ is a rectangular window defined by its corners at the spectral points k ! <sup>1</sup> and k ! 2 to filter the desired image term.

From the filtered term, the unknown field of the AUT can be easily retrieved back in the spatial domain by removing the effect of the complex conjugate of the reference field as

$$E\_{\text{sat, retrieved}}(\overrightarrow{r}) = \frac{F T^{-1} \{ h\_{\text{filtered}}(\overrightarrow{k}) \}}{E\_{\text{ref}}^{\*}(\overrightarrow{r})} . \tag{5}$$

It is relevant to remark that E� refðr !Þ is a term whose amplitude usually suffers small changes along the spatial domain and, consequently, Eq. (5) can be evaluated without the risk of divisions by zero.

Quality of the phase retrieval will mostly depend on the degree of overlapping between the autocorrelation and cross-correlation terms, which for radiated reference fields is related to the off-axis position of the reference antenna, as it will be addressed next.

At this point it is worth noting two facts: first, the retrieved field corresponds to one of the tangential components of the electric field. In order to obtain the FF pattern of the AUT, both tangential fields are needed [1] and, thus, the process has to be repeated after turning the AUT 90<sup>∘</sup> to acquire the other component [23]. Second, for the sake of simplicity, the offset of the reference antenna has only been introduced in the x-axis (as shown in Figure 1) without loss of generality.

#### 2.1.1. Overlapping control: off-axis reference and sampling requirements

Central position of the image terms is defined by the off-axis angle of the reference antenna as

$$k\_{r,x} = \pm k\_0 \sin\left(\theta\_r\right) \tag{6}$$

being k<sup>0</sup> the propagation vector in vacuum, defined as k<sup>0</sup> ¼ 2π=λ, with λ the wavelength of the fields, and θ<sup>r</sup> the off-axis angle formed by the reference antenna and the normal to the acquisition plane (see Figure 1).

According to Ref. [1], the maximum spatial bandwidth<sup>3</sup> of a radiated field in a planar acquisition is Wk ¼ k0. On the other hand, since the autocorrelation terms are the FT of a squared field, their bandwidth doubles the bandwidth of the original field [21, 28, 38] and, thus, the no overlapping condition is given by

$$k\_{r,x} \ge 3k\_0. \tag{7}$$

Nevertheless, due to the limitations imposed by the topology of the setup, the maximum offaxis angle is limited to 90<sup>∘</sup> , yielding a maximum value of kr;xmax ¼ k0. Therefore, although overlapping can be reduced by employing certain techniques (e.g., filtering after backpropagation of the planar wave spectrum (PWS) of the hologram toward the aperture or employing the so-called modified hologram, described later), it cannot be completely avoided in these setups with radiated reference waves.

<sup>3</sup> Bandwidth is defined for the positive half-space of the spectrum. Since the spectrum of the hologram is symmetric, the total bandwidth is twice the defined bandwidth.

On the other hand, sampling in the spatial domain is related to the extension of the k-space and has to be carefully selected in order to avoid aliasing. According to the Nyquist theorem, the extension of the k-space is related to the sampling step Δx by Ref. [12]:

$$k\_s = \frac{\pi}{\Delta x} \tag{8}$$

As previously mentioned, the image terms are centered in k<sup>0</sup> (kr;<sup>x</sup> ¼ kr;<sup>x</sup>max) and have a bandwidth of Wk ¼ k0, yielding a total extension of ks ¼ 2k0. Therefore, sampling in the spatial domain can be calculated from Eq. (8) as

$$
\Delta \mathbf{x} = \frac{\pi}{k\_s} = \frac{\pi}{2k\_0} = \frac{\lambda}{4} \tag{9}
$$

In practice, the off-axis angle is lower than 90<sup>∘</sup> and the sampling step can be slightly larger. Furthermore, overlapping degree varies depending on the type of reference antenna and the measured AUT. Directive antennas have narrower spectra [28] and the part of the spectrum associated to the squared signals often decays faster as it is computed for the convolution of two signals of bandwidth k<sup>0</sup> [5].

#### 2.2. Modified hologram

Eaut;retrievedðr

refðr

off-axis position of the reference antenna, as it will be addressed next.

2.1.1. Overlapping control: off-axis reference and sampling requirements

It is relevant to remark that E�

248 Holographic Materials and Optical Systems

acquisition plane (see Figure 1).

overlapping condition is given by

in these setups with radiated reference waves.

total bandwidth is twice the defined bandwidth.

axis angle is limited to 90<sup>∘</sup>

3

divisions by zero.

generality.

!Þ ¼ FT�<sup>1</sup>

along the spatial domain and, consequently, Eq. (5) can be evaluated without the risk of

Quality of the phase retrieval will mostly depend on the degree of overlapping between the autocorrelation and cross-correlation terms, which for radiated reference fields is related to the

At this point it is worth noting two facts: first, the retrieved field corresponds to one of the tangential components of the electric field. In order to obtain the FF pattern of the AUT, both tangential fields are needed [1] and, thus, the process has to be repeated after turning the AUT 90<sup>∘</sup> to acquire the other component [23]. Second, for the sake of simplicity, the offset of the reference antenna has only been introduced in the x-axis (as shown in Figure 1) without loss of

Central position of the image terms is defined by the off-axis angle of the reference antenna as

being k<sup>0</sup> the propagation vector in vacuum, defined as k<sup>0</sup> ¼ 2π=λ, with λ the wavelength of the fields, and θ<sup>r</sup> the off-axis angle formed by the reference antenna and the normal to the

According to Ref. [1], the maximum spatial bandwidth<sup>3</sup> of a radiated field in a planar acquisition is Wk ¼ k0. On the other hand, since the autocorrelation terms are the FT of a squared field, their bandwidth doubles the bandwidth of the original field [21, 28, 38] and, thus, the no

Nevertheless, due to the limitations imposed by the topology of the setup, the maximum off-

overlapping can be reduced by employing certain techniques (e.g., filtering after backpropagation of the planar wave spectrum (PWS) of the hologram toward the aperture or employing the so-called modified hologram, described later), it cannot be completely avoided

Bandwidth is defined for the positive half-space of the spectrum. Since the spectrum of the hologram is symmetric, the

fhfilteredð k ! Þg

!Þ is a term whose amplitude usually suffers small changes

kr;<sup>x</sup> ¼ �k<sup>0</sup> sin ðθrÞ (6)

kr;<sup>x</sup> ≥ 3k0: (7)

, yielding a maximum value of kr;xmax ¼ k0. Therefore, although

!Þ : (5)

E� refðr

> The modified hologram technique was first employed for setups with radiated reference fields in Refs. [21, 38] and successively adapted for synthesized reference fields (see Section 2.3) in Refs. [27–29]. The technique consists in the removal of the autocorrelation terms of the hologram prior to the filtering process and can be implemented by means of two different approaches. First of them requires an extra measurement to characterize the amplitude of Eaut (the amplitude of Eref is a priori known) [21, 23, 29, 30, 39]. Second approach, commonly known as opposite-phase holography [28, 40], introduces a hybrid-T component in the setup, which provides simultaneously the complete hologram in the sum port and the autocorrelation terms in the difference port. Another approach, used in imaging applications, is to increase the reference level several times above the level of the AUT's field in order to reduce the autocorrelation terms of the hologram [41, 42].

> Thanks to the removal of the autocorrelation terms, separation between the image terms can be reduced, meaning that physical separation between the AUT and the reference antenna can also be reduced yielding the following advances:


Another advantage of this technique is that since the intensity of Eaut is measured, the final field can be composed with the measured amplitude and the retrieved phase rather than retrieving both, amplitude and phase, from the interferometric pattern as supposed so far. Thus, the quality of the phase retrieval is improved.

Main disadvantage for the modified hologram technique is that an extra measurement for the characterization of the amplitude of the AUT is required.

#### 2.3. Synthesized reference field off-axis holography

Main differences between optical and microwave holography are stated in Ref. [27]. One of the most important remarks is that in microwave (and mm- and submm-wave bands) the hologram can be (coherently) recorded by scanning the probe across the acquisition plane, meaning that, instead of using radiated reference waves, they can be electronically synthesized and added to the field of the AUT.

Conventional approach to implement synthesized wave setups is schematically shown in Figure 3. A plane wave is synthesized by means of a phase shifter by cyclically modifying the phase of the sample of the field in the output of the directional coupler for each point of the acquisition plane. The synthesized wave is added to the acquired field of the AUT by means of a power combiner in the receiver's end.

Figure 3. Conventional setup for synthesized wave off-axis holography.

In synthesized reference wave setups, position of the image terms is no longer related to the physical position of the reference antenna but to spatial sampling and the phase shifts Δφ, between each point of the acquisition plane, and can be defined as:

$$k\_{\mathbf{r},\mathbf{x}} = \frac{\pm \Delta \phi}{\Delta \mathbf{x}}.\tag{10}$$

The use of electrically synthesized waves removes the limitation imposed by the off-axis angle and makes possible to displace the image terms to the nonvisible part of the spectrum defined by k 2 <sup>x</sup> þ k 2 <sup>y</sup> <sup>&</sup>lt; <sup>k</sup><sup>2</sup> <sup>0</sup> [1].

Considering the nonoverlapping condition imposed in Eq. (7), the sampling step has to be selected depending on the value of the introduced phase shifts, which are typically selected as π=2, 2π=3, or 3π=4, yielding sampling steps of λ=4, λ=6, or λ=8, respectively [5, 28, 29].

This technique has several advantages:

Another advantage of this technique is that since the intensity of Eaut is measured, the final field can be composed with the measured amplitude and the retrieved phase rather than retrieving both, amplitude and phase, from the interferometric pattern as supposed so far.

Main disadvantage for the modified hologram technique is that an extra measurement for the

Main differences between optical and microwave holography are stated in Ref. [27]. One of the most important remarks is that in microwave (and mm- and submm-wave bands) the hologram can be (coherently) recorded by scanning the probe across the acquisition plane, meaning that, instead of using radiated reference waves, they can be electronically synthesized and

Conventional approach to implement synthesized wave setups is schematically shown in Figure 3. A plane wave is synthesized by means of a phase shifter by cyclically modifying the phase of the sample of the field in the output of the directional coupler for each point of the acquisition plane. The synthesized wave is added to the acquired field of the AUT by means of

In synthesized reference wave setups, position of the image terms is no longer related to the physical position of the reference antenna but to spatial sampling and the phase shifts Δφ,

kr;<sup>x</sup> <sup>¼</sup> �Δφ

<sup>Δ</sup><sup>x</sup> : (10)

between each point of the acquisition plane, and can be defined as:

Figure 3. Conventional setup for synthesized wave off-axis holography.

Thus, the quality of the phase retrieval is improved.

characterization of the amplitude of the AUT is required.

2.3. Synthesized reference field off-axis holography

added to the field of the AUT.

250 Holographic Materials and Optical Systems

a power combiner in the receiver's end.


However, synthesized reference field indirect off-axis holography also presents the following limitations:


#### 2.4. Main drawbacks and limitations of indirect off-axis holography

Despite the multiple advantages of conventional indirect off-axis techniques versus complex field measurements, such as robustness and cost reduction [21, 43], and also versus other amplitude-only techniques based on iterative approaches, conventional indirect off-axis techniques exhibit several limitations, which are summarized next:


involves the use of more radiofrequency (RF) components, i.e., phase shifters and power combiners. Implementation of these types of devices is not trivial at high frequency bands and the cost of the system can be highly increased.


Other phase retrieval approaches have been proposed in order to overcome the dense sampling requirements. In Refs. [41] and [46], a new approach, known as phase-shifting, derived from digital inline microscopy, employs three different holograms recorded after introducing phase shifts in the reference field to perform the phase retrieval in the spatial domain; in this case, the phase can be retrieved point-by-point. The method presented in Ref. [47] for a bistatic imaging setup can also be directly employed in antenna measurement setups. In this case, the phase retrieval is performed by solving a set of equations formed by the modified hologram expression and the expression that relates ∥Eautðr !Þ∥<sup>2</sup> to its real and imaginary parts. For both cases the phase is retrieved directly in the spatial domain and, therefore, a sampling rate of λ=2 can be used. An added advantage is that there is no restriction in the position of the reference antenna.
