3. Indirect off-axis holography with mechanical phase shifts

In order to overcome some of the above-mentioned limitations of conventional techniques, two methods allowing either to substitute the phase shifter for mechanical displacements or to control the position of the image terms in setups with radiated reference waves are described in this section.

#### 3.1. Synthesized reference field by means of mechanical shifts

As mentioned before, main advantages of the use of synthesized reference waves are overlapping control of the image terms and that the reference field can be analytically obtained by means of a phase shifter from a sample of the field. Nevertheless, phase shifters can increase the cost of the measurement system or simply not be available for a specific frequency band.

The proposed method aims for the substitution of the phase shifter (see Figure 3) with mechanical displacements of the probe to create the interference-like pattern. The reference branch will provide a constant sample of the source added to the field recorded by the probe by means of a power combiner.

The expression of the hologram in Eq. (2) can be particularized for the case of using synthesized plane waves (Erefðr !Þ ¼ Ae�jk0<sup>r</sup> ) as:

$$H(\overrightarrow{r}) = |E\_{\rm aut}(\overrightarrow{r})|^2 + A^2 + AE\_{\rm aut}(\overrightarrow{r})e^{+\vec{p}\_{0^{r}}} + AE\_{\rm aut}^\*(\overrightarrow{r})e^{-\vec{p}\_{0^{r}}}.\tag{11}$$

On the other hand, the recorded hologram, over a planar conventional acquisition grid, in the setup of Figure 3, when no phase shifter is employed, could be expressed in the following way:

$$H(\overrightarrow{r}) = |E\_{\text{aut}}(\overrightarrow{r}) + \mathbb{C}|^2,\tag{12}$$

where C is the constant reference level added to the power combiner.

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

• In addition, as shown in Figure 3, the reference signal has to be conveyed from the output of the directional coupler and phase shifter to the power combiner, located at the receiver's end. Nevertheless, high frequency equipment (e.g., over 110 GHz) usually requires the use of waveguide sections to convey the signal. In general, these waveguides cannot be arbitrarily bent. Other choice is to convey, by means of flexible cables, a low-frequency signal as reference and, at the end of the cable, resort to a frequency multiplier. However, this approach can suffer from phase inaccuracies due to cable flexing and temperature drift; also

the use of frequency multipliers can increase the cost of the measurement system.

• The use of the modified hologram technique can alleviate the dense sampling demanded at the expenses of an extra measurement for the characterization of the amplitude of the AUT.

• Conventional indirect off-axis holography is a monochromatic technique. Thus, its use for broadband antennas characterization might be unfeasible if each frequency analysis

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 expres-

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.

In order to overcome some of the above-mentioned limitations of conventional techniques, two methods allowing either to substitute the phase shifter for mechanical displacements or to control the position of the image terms in setups with radiated reference waves are described

As mentioned before, main advantages of the use of synthesized reference waves are overlapping control of the image terms and that the reference field can be analytically obtained by means of a phase shifter from a sample of the field. Nevertheless, phase shifters can increase the cost of the measurement system or simply not be available for a specific frequency band.

3. Indirect off-axis holography with mechanical phase shifts

3.1. Synthesized reference field by means of mechanical shifts

!Þ∥<sup>2</sup> to its real and imaginary parts. For both cases the

and the cost of the system can be highly increased.

252 Holographic Materials and Optical Systems

requires an independent spatial acquisition.

sion and the expression that relates ∥Eautðr

in this section.

If small mechanical displacements d ! <sup>¼</sup> d r!=jj<sup>r</sup> !jj<sup>2</sup> are added between each of the points of that conventional acquisition grid, the field of the AUT can be approximated in those new points, disregarding the amplitude variation and taking into account only the phase change by

$$E\_{\rm aut}(\overrightarrow{r} + \overrightarrow{d}) \approx E\_{\rm aut}(\overrightarrow{r})e^{-jk\omega d}.\tag{13}$$

For those new points, the hologram in Eq. (12) can be rewritten as in Eq. (14), yielding an equivalent expression to Eq. (11):

$$H(\overrightarrow{r} + \overrightarrow{d}) = |E\_{\text{unt}}(\overrightarrow{r})e^{-\overline{\eta}\omega d} + \mathbb{C}|^2 = |E\_{\text{unt}}(\overrightarrow{r})|^2 + \mathbb{C}^2 + \mathbb{C}E\_{\text{unt}}(\overrightarrow{r})e^{-\overline{\eta}\omega d} + \mathbb{C}E\_{\text{aut}}^\*(\overrightarrow{r})e^{+\overline{\eta}\omega d}.\tag{14}$$

Therefore, if the mechanical displacements are selected so that the term e�jk0<sup>d</sup> in Eq. (13) introduces appropriate phase shifts, the use of phase shifters can be avoided.

The new grid will be a three-dimensional layered grid with as many layers as number of considered phase shifts N<sup>φ</sup> ¼ 2π=Δφ. N<sup>φ</sup> is fixed together with the sampling rate to control the position of the image terms of the hologram, see Eq. (10), and of course, the modified hologram technique (Section 2.2) can also be applied.

Figure 4 shows two different views of the measurement grid generated for the experimental validation of the setup presented next. For those examples, the mechanical displacements are selected to introduce a phase shift of π=2 and thus, N<sup>φ</sup> ¼ 4, as it can be clearly seen in Figure 4(a). Figure 4(b) shows the top view of the grid in which the cyclically repeated pattern can be observed. The orange dots represent the top layer with regular sampling of λ=2, whereas the blue ones are those corresponding to the modified points introduced to generate the phase shifts, yielding a final sampling step of λ=8. The solid line interconnecting the dots indicates the sweep direction. The grid creation process can equivalently be seen as a modification of N<sup>φ</sup> � 1 of every N<sup>φ</sup> points in the sweep axis of a regular grid to introduce the desired phase shifts.

Figure 4. Three-dimensional acquisition grid for the proposed method. Note that a different scale has been used for all the axes: (a) complete grid and (b) detail of the top view.

For the phase retrieval, there are two different options: (i) either Eq. (4) is transformed back to the spatial domain and only the points corresponding to the top layer of the grid are selected, or (ii) the field is retrieved as in Eq. (5) analytically modeling the phase of the reference field. A compensation for the eþjk0<sup>r</sup> term introduced in the modified acquisition points (see Eq. (13)) has to be considered for the latter case.

Despite the approximation in Eq. (13), it is only valid in the FF of the AUT, the maximum phase shift that can be considered is π. This phase shift is associated to a displacement of λ=2, which does not have an influence in the amplitude level; therefore, as it will be proven in the experimental validation, the method provides good results when applied to NF acquisitions.

#### 3.1.1. Experimental validation in the Ku-band for antenna measurement and diagnostics

A small 15 dB standard gain horn (SGH) antenna is characterized at 15 GHz. The measurements are repeated for the case in which a metallic plate blocks part of the antenna aperture as shown in Figure 5. In order to perform antenna diagnostics, in both cases, the retrieved field on the measurement plane is backpropagated to the aperture plane of the AUT. The setup is equivalent to those for imaging applications measured in transmission [41, 45].

The regular acquisition grid is a XY rectangular grid of 700 mm · 700 mm with 10 mm sampling (λ=2 at 15 GHz) in the y-axis and 2:5 mm (λ=8) in the x-axis, placed at z<sup>0</sup> ¼ 620 mm of the aperture of the AUT. As π=2 phase shifts are being considered, three more layers of modified points, as shown in Figure 4, are considered, being the sampling step considering all the points in the y-axis also of λ=8.

If Eq. (10) is applied for the proposed configuration, the central position of the image terms is �2k0. Figure 6(a) shows the recorded NF hologram while its spectrum is shown in Figure 6(b). Since the sweep (and the phase shifts) is made along the y-axis direction, the image terms will appear shifted in the ky axis of the spectrum. The abrupt decay of the autocorrelation terms makes possible to correctly filter the desired image term between 0:4 k<sup>0</sup> and 3:6 k0, and correctly retrieve the field of the AUT.

Figure 5. Ku band SGH with blocked aperture.

For the phase retrieval, there are two different options: (i) either Eq. (4) is transformed back to the spatial domain and only the points corresponding to the top layer of the grid are selected, or (ii) the field is retrieved as in Eq. (5) analytically modeling the phase of the reference field. A compensation for the eþjk0<sup>r</sup> term introduced in the modified acquisition points (see Eq. (13)) has

Figure 4. Three-dimensional acquisition grid for the proposed method. Note that a different scale has been used for all

Despite the approximation in Eq. (13), it is only valid in the FF of the AUT, the maximum phase shift that can be considered is π. This phase shift is associated to a displacement of λ=2, which does not have an influence in the amplitude level; therefore, as it will be proven in the experimental validation, the method provides good results when applied to NF

A small 15 dB standard gain horn (SGH) antenna is characterized at 15 GHz. The measurements are repeated for the case in which a metallic plate blocks part of the antenna aperture as shown in Figure 5. In order to perform antenna diagnostics, in both cases, the retrieved field on the measurement plane is backpropagated to the aperture plane of the AUT. The setup is

The regular acquisition grid is a XY rectangular grid of 700 mm · 700 mm with 10 mm sampling (λ=2 at 15 GHz) in the y-axis and 2:5 mm (λ=8) in the x-axis, placed at z<sup>0</sup> ¼ 620 mm of the aperture of the AUT. As π=2 phase shifts are being considered, three more layers of modified points, as shown in Figure 4, are considered, being the sampling step considering all

If Eq. (10) is applied for the proposed configuration, the central position of the image terms is �2k0. Figure 6(a) shows the recorded NF hologram while its spectrum is shown in Figure 6(b). Since the sweep (and the phase shifts) is made along the y-axis direction, the image terms will appear shifted in the ky axis of the spectrum. The abrupt decay of the autocorrelation terms makes possible to correctly filter the desired image term between 0:4 k<sup>0</sup> and 3:6 k0, and cor-

3.1.1. Experimental validation in the Ku-band for antenna measurement and diagnostics

equivalent to those for imaging applications measured in transmission [41, 45].

to be considered for the latter case.

254 Holographic Materials and Optical Systems

the axes: (a) complete grid and (b) detail of the top view.

the points in the y-axis also of λ=8.

rectly retrieve the field of the AUT.

acquisitions.

Figure 6. (a) Recorded hologram in the modified three-dimensional grid, normalized amplitude in dB. (b) Spectrum of the hologram, normalized amplitude in dB.

The retrieved amplitude and phase of the acquired NF are shown in Figure7(a) and, respectively, for the case in which the horn is not blocked. The backpropagated field in the aperture is shown in Figure 7(c) together with the size of the aperture. The retrieved amplitude and phase for the case of the blocked aperture are shown in Figure 7(d) and. Some discrepancies with respect to the first case can be observed and when the field is backpropagated to the aperture of the AUT, Figure 7(f), the blockage can be clearly detected.

Thus, the proposed method can be successfully applied to antenna measurement and diagnostics with equivalent results to the conventional indirect off-axis method with synthesized reference wave.

Figure 7. Retrieved NF of the AUT without blocking metallic plate (a)–(c) and with blocking metallic plate (d)–(f): (a) and (d) normalized amplitude in dB, (b) and (e) phase in degrees, and (c) and (f) backpropagation of the retrieved field toward the aperture, normalized amplitude in dB.

#### 3.2. Multiplexed holograms with radiated reference field

As explained in Section 2.4, setups with synthesized reference waves for antenna characterization are challenging at high frequency bands, being necessary to resort to setups with radiated reference waves [21, 30]. The main limitation of these setups is that the position of the image terms is determined by the off-axis angle of the reference antenna, see Eq. (6), and will always be below k0, that is, in the visible part of the spectrum. This separation might not be enough to avoid overlapping for certain types of antennas such as nondirective antennas, which have wider spectra [28]. Overlapping can also be observed when the level in the AUT branch is higher than the reference level, due to the differences between the level of the autocorrelation and image terms [30].

The previously presented technique with mechanical phase shifts of the probe cannot be applied when radiated reference waves are employed because the displacements of the probe antenna will introduce the phase shifts in both the reference and the AUT fields, leading to an erroneous approach of the off-axis holography technique.

In order to control the position of the image terms and displace it to the nonvisible part of the spectrum as with synthesized reference waves, this subsection describes a new method for the case of using radiated reference waves. The method consists in multiplexing two subsampled holograms, Eq. (1), obtained from two 180<sup>∘</sup> phase-shifted reference waves. The phase shift can be generated by means of a phase shifter or displacing the reference antenna a distance of λ=2.

The first subsampled hologram, blue grid in Figure 8(a), is acquired in a grid with 2Δx and Δy sampling. The samples are stored in the odd columns of the multiplexed hologram. Then, a displacement of λ=2 is introduced in the reference antenna and the second subsampled hologram, which is stored in the even columns of the multiplexed hologram (orange grid in Figure 8(b)) is acquired over a grid identical to the first one but with an offset of Δx.

Figure 8. (a) Spatial multiplexation of the subsampled grids for the hologram formation. (b) Spectrum of the hologram for the proposed method for an example with Δx ¼ λ=6 sampling.

By combining the two subsampled holograms with 180<sup>∘</sup> phase-shifted references, the amplitude of the final hologram remains almost unchanged with respect to a hologram acquired in the complete grid, while the phase steps of the reference field (Δφ) in the acquisition plane will be increased by a factor of π (see Ref. [30] for a step-by-step proof), leading to the following position of the image terms (see Eqs. (8) and (10)):

3.2. Multiplexed holograms with radiated reference field

the aperture, normalized amplitude in dB.

256 Holographic Materials and Optical Systems

erroneous approach of the off-axis holography technique.

terms [30].

distance of λ=2.

As explained in Section 2.4, setups with synthesized reference waves for antenna characterization are challenging at high frequency bands, being necessary to resort to setups with radiated reference waves [21, 30]. The main limitation of these setups is that the position of the image terms is determined by the off-axis angle of the reference antenna, see Eq. (6), and will always be below k0, that is, in the visible part of the spectrum. This separation might not be enough to avoid overlapping for certain types of antennas such as nondirective antennas, which have wider spectra [28]. Overlapping can also be observed when the level in the AUT branch is higher than the reference level, due to the differences between the level of the autocorrelation and image

Figure 7. Retrieved NF of the AUT without blocking metallic plate (a)–(c) and with blocking metallic plate (d)–(f): (a) and (d) normalized amplitude in dB, (b) and (e) phase in degrees, and (c) and (f) backpropagation of the retrieved field toward

The previously presented technique with mechanical phase shifts of the probe cannot be applied when radiated reference waves are employed because the displacements of the probe antenna will introduce the phase shifts in both the reference and the AUT fields, leading to an

In order to control the position of the image terms and displace it to the nonvisible part of the spectrum as with synthesized reference waves, this subsection describes a new method for the case of using radiated reference waves. The method consists in multiplexing two subsampled holograms, Eq. (1), obtained from two 180<sup>∘</sup> phase-shifted reference waves. The phase shift can be generated by means of a phase shifter or displacing the reference antenna a

$$k'\_{\tau} = \pm \frac{\Delta \phi + \pi}{\Delta \mathbf{x}} = \pm \frac{\Delta \phi}{\Delta \mathbf{x}} + \frac{\pi}{\Delta \mathbf{x}} = \pm k\_{\tau} + k\_{s}.\tag{15}$$

Two replicas of the image terms of the hologram appear at a distance of ks <sup>¼</sup> <sup>π</sup> <sup>Δ</sup><sup>x</sup> of the original image terms as shown in Figure 8(b). The replicas are in the nonvisible part of the spectrum, shaded in gray, and have the same information than the original image terms, which are overlapped with the autocorrelation terms. Hence, the field can be retrieved by filtering the desired replica without the need to resort to the modified hologram technique. It has been demonstrated that if the reference field is a plane wave, the original terms are completely canceled allowing a cleaner filtering [30].

Position of the image terms depends on the off-axis angle of the reference antenna, Eq. (6). A common option to convey the reference signal in holography setups is to use mirror reflection (see Figure 9). This option has multiple advances since it is possible to increase the path of the reference field and interfere with a quasi-plane wave. Furthermore, by modifying the position and orientation of the reflector it is possible not only to control the off-axis angle but also to modify the shape of the pattern of the reference field in the acquisition plane, which also influences the shape and the width of the image terms and their replicas in the k-space.

Figure 9. Measurement setup for the lens antenna characterization at 94 GHz. Rear view.

### 3.2.1. Corrections

Two small corrections have to be applied to the retrieved field to compensate the effect that the high frequency replicas introduce in the retrieved field. First, the retrieved phase is contaminated with high frequency noise that can be eliminated by low-pass filtering. Second, since only a fraction of the spectral density of the image term is being considered (the replica), the retrieved amplitude of the AUT is slightly smaller than the one directly acquired. A correction factor can be obtained from the analysis of the reference field, which is known. The spectrum of the reference field is filtered using the same filter that will be used to filter the image term of the complete hologram; then, that filtered part is transformed back to the spatial domain and its amplitude level is compared to the initial amplitude of the reference field. The difference can be used as a correction factor for the retrieved amplitude of the AUT.

#### 3.2.2. Experimental validation: 94 GHz lens antenna NF characterization

The measurement setup shown in Figure 9 has been implemented for the experimental validation of the method. A 64 mm circular lens fed with a horizontally polarized WR10 open-ended waveguide (OEWG) is characterized at 94 GHz. A 20 dB SGH is employed as reference antenna. A plane metallic mirror with a tilt of 22<sup>∘</sup> is placed at 270 mm of the aperture of the reference antenna and used to direct the reference field toward the acquisition plane, at 200 mm of the AUT.

The NF is acquired for a 200 mm cut at y ¼ 0 with λ=2 sampling for the first position of the mirror. Then, the mirror is displaced λ=2 toward the acquisition plane by means of a micropositioner and the second subsampled hologram is acquired. Direct acquisition of the phase and acquisition with conventional off-axis holography with λ=4 sampling have also been made to compare the results to those obtained with the proposed method.

Figure 10(a) shows the hologram for the proposed method and for conventional indirect off-axis holography. An off-axis angle of 22<sup>∘</sup> produces two image terms centered in �0:38k<sup>0</sup> and two replicas at ∓2:38k0, which means that, in the ½�2k0; 2k0� interval, the replicas are swapped and centered at ∓1:64k0, as it can be clearly seen. While the replicas for the proposed method can be filtered, there is some overlapping between the image term and the autocorrelation terms for the conventional case. This is due to the high amplitude level of the AUT, which produces a large autocorrelation term, highly above the level of the image terms of the spectrum.

Figure 10. (a) Spectrum of the hologram and filtering windows, normalized amplitude in dB and (b) percentual error of the phase retrieval.

Figure 10(b) depicts the error of the phase retrieval process calculated as

3.2.1. Corrections

258 Holographic Materials and Optical Systems

Two small corrections have to be applied to the retrieved field to compensate the effect that the high frequency replicas introduce in the retrieved field. First, the retrieved phase is contaminated with high frequency noise that can be eliminated by low-pass filtering. Second, since only a fraction of the spectral density of the image term is being considered (the replica), the retrieved amplitude of the AUT is slightly smaller than the one directly acquired. A correction factor can be obtained from the analysis of the reference field, which is known. The spectrum of the reference field is filtered using the same filter that will be used to filter the image term of the complete hologram; then, that filtered part is transformed back to the spatial domain and its amplitude level is compared to the initial amplitude of the reference field. The difference

The measurement setup shown in Figure 9 has been implemented for the experimental validation of the method. A 64 mm circular lens fed with a horizontally polarized WR10 open-ended waveguide (OEWG) is characterized at 94 GHz. A 20 dB SGH is employed as reference antenna. A plane metallic mirror with a tilt of 22<sup>∘</sup> is placed at 270 mm of the aperture of the reference antenna and used to direct the reference field toward the acquisition plane, at 200 mm of the AUT.

The NF is acquired for a 200 mm cut at y ¼ 0 with λ=2 sampling for the first position of the mirror. Then, the mirror is displaced λ=2 toward the acquisition plane by means of a micropositioner and the second subsampled hologram is acquired. Direct acquisition of the phase and acquisition with conventional off-axis holography with λ=4 sampling have also

Figure 10(a) shows the hologram for the proposed method and for conventional indirect off-axis holography. An off-axis angle of 22<sup>∘</sup> produces two image terms centered in �0:38k<sup>0</sup> and two replicas at ∓2:38k0, which means that, in the ½�2k0; 2k0� interval, the replicas are swapped and centered at ∓1:64k0, as it can be clearly seen. While the replicas for the proposed method can be filtered, there is some overlapping between the image term and the autocorrelation terms for the conventional case. This is due to the high amplitude level of the AUT, which produces a large

been made to compare the results to those obtained with the proposed method.

autocorrelation term, highly above the level of the image terms of the spectrum.

can be used as a correction factor for the retrieved amplitude of the AUT.

Figure 9. Measurement setup for the lens antenna characterization at 94 GHz. Rear view.

3.2.2. Experimental validation: 94 GHz lens antenna NF characterization

$$error[\%] = 100 \frac{\|\mathbf{E\_{measured}} - \mathbf{E\_{retrieved}}\|\_2}{\|\mathbf{E\_{measured}}\|\_2},\tag{16}$$

where Emeasured and Eretrieved are vectors containing the samples of the measured (with amplitude and phase) and the retrieved field (from amplitude-only acquisitions) at the acquisition points, and ∥ � ∥<sup>2</sup> denotes the Euclidean norm. Due to the overlapping with the autocorrelation term, the mean error of the conventional method is 32:8% while the error achieved with the proposed method is only of 5:70%.

Figure 11(a) shows the retrieved amplitude in the acquisition plane with both methods compared to the amplitude directly acquired, whereas in Figure 11(b) the same data are shown for the phase. It can be clearly observed that, while with the proposed method, the retrieved amplitude and phase are in very good agreement with the data from the direct acquisition the retrieved fields with the conventional method exhibit some discrepancies, especially in the areas with larger error (see Figure 10(b)) due to the overlapping of the spectrum.

Figure 11. (a) Amplitude of the AUT, normalized in dB and (b) phase of the AUT in degrees.
