Millimeter-wave Interferometric Sensing

by Charles R. Dietlein, Amir Zaghloul, and David A. Wikner

ARL-TR-5668 September 2011

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ARL-TR-5668 September 2011

Millimeter-wave Interferometric Sensing

Charles R. Dietlein, Amir Zaghloul, and David A. Wikner

Sensors and Electron Devices Directorate, ARL

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Millimeter-wave Interferometric Sensing

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14. ABSTRACT

This report explores the feasibility of short-range ground-based millimeter-wave sensing systems based on interferometric 

imaging techniques. In comparison with infrared and visible wavelength focal-plane arrays (FPA), millimeter-wave FPAs 

always have fewer detectors and image pixels, poorer spatial resolution, lower signal-to-noise ratio, and an increase of

illumination-based artifacts such as glint in the images. Calibration and scanning schemes induce additional complications. 

These difficulties are ultimately a consequence of the millimeter-wave’s much longer wavelengths compared to visible or 

infrared frequencies. This research effort addresses a specific limitation: low-detector-count FPAs. An alternative approach to 

traditional, high-detector-count, staring FPAs is interferometric (coherence) imaging, wherein an array with a low number of 

coherent detectors is exploited to reconstruct the field distribution in the object plane. This interferometric approach is used 

extensively by radio astronomers at lower frequencies. In this report, we study this technique's validity at 220 GHz for 

imaging terrestrial scenes.

15. SUBJECT TERMS

Millimeter-wave, imaging, coherence, interferometry, standoff

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iii

Contents

List of Figures iv

Summary 1

1. Introduction 3

2. Methods, Assumptions, and Procedures 3

3. Results and Discussion 6

4. Conclusions 11

5. References 12

Distribution List 13
iv

List of Figures

Figure 1. Geometry illustrating the van Cittert-Zernike theorem discussed in the text..................4

Figure 2. Plot of phase difference between detectors for various array sizes (shown in the 

legend in units of meters), as a function of range between the array and object planes. 

Frequency was fixed at 220 GHz. The phase differences φδ << 1 imply a paraxial 

approximation. The black square indicates where the experiments and simulations in 

this report were performed.........................................................................................................6

Figure 3. (Left) Array geometry showing 32 detector positions arranged in a spiral 

configuration, and (right) sampling in the uv plane...................................................................7

Figure 4. (Left) A histogram containing the uv plane coverage shown in Cartesian 

coordinates in figure 3 (right). The histogram contains 496 samples in 50 bins. (Right) A 

histogram showing uv plane coverage for a rectangularly-gridded array, with 

approximately the same number of detectors and physical size as that of figure 3 (left)..........8

Figure 5. (Left) Reconstruction of the simulated point source at a range of 0.5 m from the 

spiral array, and (right) best-fit 2-D Gaussian (opaque surface) with the reconstructed 

object plane (mesh surface), for purposes of determining the FWHM of the point source.......9

Figure 6. (Left) Reconstruction of the two simulated point sources at a range of 0.5 m from 

the spiral array, and (right) best-fit 2-D Gaussians (opaque surface) with the 

reconstructed object plane (mesh surface).................................................................................9

Figure 7. Measured electric field amplitude pattern of two waveguide aperture sources. 

Range to the sources is approximately 0.5 m, and they are separated by approximately 

6 cm. The fringes are tilted slightly, as the sources were not perfectly aligned with the yaxis. The nominal positions of the modeled detectors are depicted by the overlaid dots.......10

Figure 8. (Left) Measured data, reconstructed electric field distribution at object plane and 

(right) simulation results with geometry matching the experimental setup............................11
1

Summary

Millimeter-wave sensing systems designed for imaging almost always produce inferior images

compared to their infrared and visible wavelength counterparts for a variety of fundamental 

reasons. These reasons include low detector/pixel count, poor signal-to-noise ratio, artifacts 

introduced by illumination, and issues arising from calibration or scanning schemes. This 

research effort is intended to address the issue of low-detector-count focal-plane arrays (FPA).

Radio astronomers have dealt with this issue for many decades by using interferometry, or 

coherence imaging. This report details how interferometric imaging techniques can be used in 

the millimeter-wave regime for terrestrial applications.
2

INTENTIONALLY LEFT BLANK.
3

1. Introduction

This research effort explores, through computer simulations and experimental verification, the 

viability of interferometric techniques for millimeter-wave imaging. A fundamental limitation in 

millimeter-wave/terahertz imaging is the lack of high-detector-count FPAs. However, in the 

radio frequency through terahertz regime, coherent (phase-sensitive) detectors are available. 

Infrared and visible images are acquired with direct (amplitude-only) detectors. Coherent 

measurement of the incident electric field provides additional information regarding the spatial 

amplitude distribution in the object plane. The scope of this research is to explore an 

interferometric imaging array geometry with the goal of demonstrating that a sparsely populated 

coherent array can achieve comparable spatial resolution to that achieved by a traditional directdetection fully populated FPA with similar physical dimensions. The computer program 

MATLAB® is employed to simulate the image amplitude distribution and quantify the spatial 

resolution of the array, and a table-top experiment is configured to verify the simulations.

2. Methods, Assumptions, and Procedures

An imaging interferometer consists of N coherent detectors arranged in an array; we will 

consider planar arrays for simplicity. Each detector measures the amplitude and phase of the 

incident radiation, while each pair of detectors measures one spatial Fourier component of the 

incident radiation. The specific spatial component is defined by the pair’s physical separation, 

called a baseline. For N detectors, there are

2 2

2 N N N

p





















(1)

unique 2-combinations. Each spatial Fourier component is represented by a single point in the 

Fourier plane, known as the uv plane. Image quality is usually determined by the uv plane

sampling. Since each baseline measures a single spatial frequency, the optimum array geometry

contains no redundant spacings. Typical geometries include a logarithmically spaced onedimensional (1-D) array, or a spiral two-dimensional (2-D) array. While image quality is 

primarily dependent on 

uv

plane sampling, it can be improved via standard reconstruction postprocessing algorithms that are used by radio astronomers.

The van Cittert-Zernike theorem explains the fundamentals of interferometric imaging. A more 

detailed version of this discussion can be found in many textbooks (1). Briefly, a source of 

electromagnetic radiation occupies the infinitesimal surface 

dS'

on the 

z  0

plane, at xy
4

coordinates 

x  x'

and 

y  y' . A0 is the amplitude of the source. An array of detectors is in the xy

plane at z = Z0. This geometry is illustrated in figure 1.

x

y

z

y'

x'

dS'

r1

r2

(x1,y1,Z0)

(x2,y2,Z0)

Figure 1. Geometry illustrating the van Cittert-Zernike theorem discussed in the text.

For a pair of detectors at (x1, y1, Z0) and (x2, y2, Z0), the contributions to the total electric field 

provided by the source

dS'

can be expressed by

d  / exp d ' E1  A0r1jt  jkr1 S

(2)

and

d  / exp d ' E2  A0r2jt  jkr2 S , (3)

where 

    

1/ 2

2

0

2 2

rj x'x j  y'y j  Z

and 

k  2 / . The mutual coherence function of the 

electric field at detector positions (x1, y1, Z0) and (x2, y2, Z0) is given by

    

d '

',' exp

1 2

1 2

1,2 S

r r

x y jk r r

C

S

E









, (4)

where σE is the object plane amplitude distribution, and in the paraxial approximation, 

       

0

1 2 1 2

0

2

2

2

2

2

1

2

1

1 2

' '

2 Z

x x x y y y

Z

x y x y

r r

  



  

  . (5)

The phase difference between a pair of detectors in a planar array, due to wave-front curvature, is 

given by 

   

0

2

2

2

2

2

1

2

1

2Z

k x  y  x  y

  . (6)
5

If we define the baselines 

1 2 u  x  x

and 

1 2

v  y  y

, along with the angular coordinates 

0   x'/ Z , 

0   y'/ Z

and the approximation 

2

1 2 0 1/



rr  Z

, the mutual coherence function can be 

written as (2)

C1,2

u,v  expj  E,exp jku  vdd  









. (7)

When the phase shift 

  

exp j

can be ignored, the mutual coherence function above relates the 

coherence in the array plane to the amplitude distribution in the object plane via a Fourier 

transform. When the phase shift is significant, the correction based on the array geometry and 

distance to the object plane can be performed by adding a term to the measured phase at each 

detector, before computing the correlation function per baseline, 

  l l l C  A exp j

. In the 

correlation function, 

Al  Em En

is the product of the measured field amplitudes at detectors 

m

and 

n

, and 

l  m n

is the difference in phase between the two detectors.

For common array sizes in the millimeter-wave/terahertz frequency range, imaging at realistic 

ranges, the far-field condition of 

 1

is not met, and 



must be adjusted in software after 

measurement. The relationship between range, array size, and phase shift is shown in figure 2.

One may ask why we use the van Cittert-Zernike theorem, strictly valid only in the paraxial 

approximation, when we are in at least one of the three near-field conditions (3). The reason is 

that the paraxial approximation provides a useful form of the coherence function, because 

 E

can be reconstructed easily through Fourier inversion. Reconstruction of the amplitude 

distribution from the correlation function of each baseline is performed by

            



    

p

l

E l l l l l l C k u v C k u v

1

 , cos   sin   .
6

10-1 100 101 102 10-4

10-2

100

102

104

Range [m]



 [rad]

0.0125

0.025

0.05

0.125

0.25

Figure 2. Plot of phase difference between detectors for various array sizes (shown 

in the legend in units of meters), as a function of range between the array and 

object planes. Frequency was fixed at 220 GHz. The phase differences φδ << 1

imply a paraxial approximation. The black square indicates where the 

experiments and simulations in this report were performed.

3. Results and Discussion

We focused our efforts around f = 220 GHz, primarily because it is one of the few bands with an

acceptable atmospheric transmission window above f = 100 GHz. Additionally, the wavelength 

of approximately λ = 1.4 mm provides adequate spatial resolution for a variety of tasks with 

moderate array sizes. Finally, it is easy to find commercial hardware that operates at this 

frequency. While this approach will work at arbitrary ranges, we constrained our simulations to 

those that could be replicated on a laboratory bench with off-the-shelf equipment. We chose a 

range of approximately 0.5 m and an array size of approximately 20 mm

2

. Figure 3 (left) shows 

one of the simulated array geometries we explored, a planar Archimedean spiral, with 

N  32

detectors. The array’s sampling in the uv plane is shown in figure 3 (right).
7

Figure 3. (Left) Array geometry showing 32 detector positions arranged in a spiral configuration, and (right) 

sampling in the uv plane.

While figure 3 (right) appears to be sampling the Fourier space in a non-uniform way, it is

difficult to visually determine if there is redundant sampling at similar spatial frequencies. A 

simple check can be provided via a histogram of the baseline lengths, which appears in figure 4

(left). There are p = 496 baselines in this array, and the histogram contains 50 bins. It can be seen 

that this geometry is neither a poor choice (e.g., a rectangular gridded array, which samples 

relatively few spatial frequencies) nor an ideal one (in which the histogram would be nominally 

flat from the minimum spatial frequency to the maximum). There are exactly 490 unique 

baseline lengths in the array; a non-redundancy figure of merit (FOM) can be defined as the 

number of unique baseline lengths divided by the total number of baselines in the array. For this 

array, that FOM is 0.988. For comparison, figure 4 (right) contains a histogram of the baseline 

lengths for a rectangular-gridded array with N = 36 detectors (and therefore p = 630 baselines)

and the same approximate physical size as the planar spiral array; inset is the rectangular array 

configuration (cf. figure 3 (left)). Straightforward geometry shows that there are exactly 19

unique baseline lengths in the rectangular-gridded array. The non-redundancy FOM for this array 

is 0.030.

-0.02 -0.01 0 0.01 0.02

-0.02

-0.01

0

0.01

0.02

u [m]

v [m]
8

0 0.005 0.01 0.015 0.02

0

5

10

15

20

25

Baseline length [m]

Bin count

0 0.005 0.01 0.015 0.02 0.025 0.03

0

10

20

30

40

50

60

70

80

Baseline length [m]

Bin count

Figure 4. (Left) A histogram containing the uv plane coverage shown in Cartesian coordinates in figure 3 (right). 

The histogram contains 496 samples in 50 bins. (Right) A histogram showing uv plane coverage for a 

rectangularly-gridded array, with approximately the same number of detectors and physical size as that 

of figure 3 (left).

The spatial resolution of an interferometric array can be approximated by λZ0/b, where b is the 

largest baseline in the array. For the array shown in figure 3 (left), b ≈ 19 mm. At a range of 

0.5 m the spatial resolution should be approximately 3.6 cm. Figure 5 (left) shows a 

reconstructed amplitude distribution of a point source at 0.5 m. To verify the full-width halfmaximum (FWHM) spot size, a fitting code using the Nelder-Mead direct search simplex 

algorithm (4) was written to find the best-fit 2-D Gaussian to the data. The general form of the 

2-D Gaussian is used: f(x, y) = A exp–[a(x – x0)

2

+ 2b(x – x0)(y – y0) + c(y – y0)

2

], where a, b, and 

c are functions of both the rotation angle, and x and y Gaussian widths. Figure 5 (right) shows the 

results of the fitting algorithm. The mesh surface is the simulated data, while the opaque surface 

is the best-fit 2-D Gaussian. The mean of the x and y Gaussian widths was σ ≈ 0.03 m; 

multiplication by

2ln 2

results in a FWHM of 3.55 cm, which is very close to the expected 

3.6 cm.

-0.01 0 0.01 -0.01

-0.005

0

0.005

0.01

X [m]

Y [m]
9

Figure 5. (Left) Reconstruction of the simulated point source at a range of 0.5 m from the spiral array, and

(right) best-fit 2-D Gaussian (opaque surface) with the reconstructed object plane (mesh surface), for 

purposes of determining the FWHM of the point source. 

Next, we investigated multiple point sources. With the same planar Archimedean spiral array and 

distance to the object plane, we added a second point source 6 cm above the first. The results in 

figure 6 show that the two sources are clearly resolved. A similar best-fit Gaussian code was 

developed to accommodate fitting to multiple Gaussians and was applied to the two-source 

simulation results.

Figure 6. (Left) Reconstruction of the two simulated point sources at a range of 0.5 m from the spiral array, and

(right) best-fit 2-D Gaussians (opaque surface) with the reconstructed object plane (mesh surface).
10

To determine if the simulation results were valid, we constructed a simple table-top experiment 

consisting of a 220-GHz transmitter and receiver. We installed a waveguide directional coupler 

at the transmitter output, and with a pair of attenuators, set the output power at both waveguides 

(coupled and through) to be as close to identical as possible. This measurement was performed 

using a calibrated substitution power meter. The spacing between the waveguide apertures was 

approximately 6 cm. We raster-scanned the receiver over the spatial region where the 32 

detectors were modeled. The field was sampled on a square grid with a pixel dimension of 

0.794 mm per side. The results are shown in figure 7 (amplitude only, phase not shown), with the 

nominal detector positions of the modeled Archimedean spiral overlaid. The interference pattern 

is slightly tilted because the waveguide apertures were not perfectly aligned with the y-axis of

the experiment.

Figure 7. Measured electric field amplitude pattern of two waveguide aperture sources. 

Range to the sources is approximately 0.5 m, and they are separated by approximately 

6 cm. The fringes are tilted slightly, as the sources were not perfectly aligned with 

the y-axis. The nominal positions of the modeled detectors are depicted by the 

overlaid dots.

The measured complex electric field was interpolated to a grid with three times more points in x

and y, and then the electric field amplitude and phase were interpolated to the nearest points of 

the modeled spiral. This produces a 1x32 vector with a measured electric field amplitude and 

phase at each detector’s position in the modeled spiral. This 1x32 complex vector was loaded

into the MATLAB code used in the previous simulations, and the code was started at the 
11

subroutines that process the electric field amplitude and phase from each detector. The next step 

in the processing is to calculate the correlation function for each baseline in the spiral as 

mentioned earlier, 

  l l l C  A exp j

, where 

Al  Em En

is the product of the measured field 

amplitudes at detector locations 

m

and 

n

, and 

l  m n

is the difference in phase between 

the two detector locations. The reconstruction result is shown in figure 8 (left). 

Figure 8. (Left) Measured data, reconstructed electric field distribution at object plane and (right) 

simulation results with geometry matching the experimental setup.

Figure 8 (right) contains the results of the same geometry, but with purely simulated data. The 

agreement between simulation and experiment is very good, and verifies that the concepts

explained in section 3 are valid.

4. Conclusions

We explored and verified the feasibility of interferometric imaging techniques for use at 

millimeter-wave frequencies and short ranges. The paraxial approximation of the van CittertZernike theorem that radio astronomers use is applicable in the situations described by this 

report, as long as the phase front curvature is compensated for. This is equivalent to re-focusing 

the array to the near-field, a technique that has been discussed by some in the radio astronomy 

community (5) for studying astronomical objects in our solar system.
12

5. References

1. Born, M.; Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, 

Interference and Diffraction of Light, 7th ed.; Cambridge University Press, 1999.

2. Thompson, R.; Moran, J.; Swenson, G. Interferometry and Synthesis in Radio Astronomy, 2 

ed.; Wiley-VCH, 2001.

3. Carter, W. H. Three Different Kinds of Fraunhofer Approximations: 1. Propagation of the

Field Amplitude. Radio Science November–December 1988, 23 (6), 1085–1093.

4. Lagarias, J. C.; Reeds, J. A.; Wright, M. H.; Wright, P. E. Convergence Properties of the 

Nelder-mead Simplex Method in Low Dimensions. SIAM J. Optim. 1998, 9 (1), 112–147.

5. Carter, W. H. On Refocusing a Radio Telescope to Image Sources in the Near Field of the 

Antenna Array. IEEE Trans. Antennas Prop. March 1989, 37 (3), 314–319.
13

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