# The relationship between the aperture field and the target field

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## Feeding line

Let’s considering an antenna in the transmitting mode. The radiation pattern, and the S parameter plot with respect to the frequency, describe its behavior. However the signal generated by the source does not reach directly the antenna. It must pass through a line, which matches the source to the load. It is possible to represent the source and the antenna as two equivalent electric circuits, where the source is represented by the voltage generator S and the series impedance Zs , and the antenna by the transmission line impedance Z0 , and the load impedance Z L (figure 3.2.3.1).

### Losses in the microstrip lines

Often the losses of the transmission line are neglected in the design, as first approximation. However this is not the case for circuits operating at very high frequencies. Actually the losses become very high at terahertz frequencies, and the design of the circuit without taking them into account leads to very large errors. There are two kind of losses that occur in the transmission line circuits: the conductor loss and the dielectric loss. Let’s consider first the conductor loss. The effective width of the feeding line must take into account the thickness of the conductor. It is given by, for W/h> π 2 [18]: We=W+ tπ (1+ ln 2 h t ) 3.2.4.2.1 (a).

#### The feeding networks

The feeding network is extremely important in an array of microstrip antennas. A bad network can compromise the behavior of the entire array. There are essentially two main criteria that feeding networks must satisfy: first the antennas must be feed with the same phase for a broadside radiation; second the impedance of the network connected to the single elements must match that of the source. Let’s analyze the two cases separately.

Phase of the feeding networks

Let’s consider a transmitting array (reciprocity theorem). The feeding signal must reach each element of the array with the same phase. This can be accomplished by imposing that the length of every transmission line, which connects the source to the single antenna element, must be a multiple of the wavelength. If the line which links the port 1 with the source is N λ , and the line from the source to the port 2 is M λ , the phase is the same for both the ports. Nevertheless, this criteria makes the array narrow-band, since it is designed only for the frequency c0 /λ . It is often more convenient, even if more complicated, to design the network such that the distance from the source to every port is the same. The procedure to accomplish such condition will be illustrated. Let’s consider a 2X2 array. An example of feeding network that achieves the same distance from the feeding to the elements is the following (fig 3.2.5.1.1): Fig. 3.2.5.1.1 feeding network of a 2X2 array. The distance from the source to the every element must be the same. By duplicating the array in fig 3.2.5.1.1, it is then possible to design a 2X2 array of 2X2 arrays, resulting in the following 4X4 array (fig 3.2.5.1.2): Fig. 3.2.5.1.2 feeding network of a 4X4 array. It consists of a concatenation of the 2X2 array repeated 4 times. The same procedure can be repeated infinitely to get a generic 2n X2n array.

Impedance of the feeding network

The feeding network is a structure which consists of a series of microstrip lines connected together
to bring the signal from the antennas to the hot electron bolometer, or equivalently, in the transmitting mode, from the HEB to the antennas. An example is given in fig.3.2.5.1.2. The matching condition (eq.3.2.3.2 with Γ=0 ) can be extended to every section of the feeding network: now let’s cut the network in a generic section. In general, the total impedance of the circuit on one side of the cut must be equal to the impedance of the circuit on the other side, to avoid unwanted reflection. At this point, we need to take a special care when the circuit is split in two parts. Let’s do an example by using microstrip lines (fig. 3.2.5.2.1). The HEB is the source S of the signal. It is connected to a feeding network of Z0=50Ω . The width W0 is calculated by inverting eq. 3.2.4.1.2 (or 3.2.4.1.4). We see that the feeding network splits into two parts at the point C. Therefore we have a parallel of the two impedances Z1 , which are connected in series to Z0 : 1 Z0 = 1 Z1 + 1 Z1. . We can then calculate: Z1=100Ω . Finally we can calculate W1 by inverting eq.3.2.4.1.2 (or 3.2.4.1.4). Since Z1>Z 0 , we have W1<W0 . Each split comports a reduction of the width.

Design and simulation of the patch antenna array

The first design parameter we need to define is the operating frequency of the patch antenna. Let’s assume f =600GHz , which is a typical value for astronomical observation with heterodyne receivers. The maximum allowed thickness of the antenna substrate to be operative is h=λr /8 where λr is the wavelength inside the substrate. Thicker substrates present too many losses in term of surface wave, while too thin substrates present too narrow band. Let’ assume as first approximation exactly h=λr /8 . Since λr depends on the dielectric constant of the substrate, we need to define ϵr . High values of ϵr mean small wavelength inside the dielectric and, therefore, thinner substrate. Since it is easier to fabricate thick substrate than thinner substrate (mechanical etching), we opt for a low ϵr value. The best match is given by the “high-density polyethylene” (HDPE) with a dielectric constant of ϵr=2,26 . Thus we have: h=λr /8= c0 8 f √ϵr ≃41μ m 3.2.6.1.

Chapter 1: Introduction
Chapter 2: Background
2.1.1 The diplexer
2.1.2 The focusing optics of the mixer
2.1.3 The mixer
2.1.4 The Local Oscillator
2.1.5 The beam divider
2.2 State of the art of the heterodyne array receivers
2.3 Simulation tools
2.3.1 The simulation softwares
2.3.2 Optimization algorithms
Chapter 3: The array of patch antennas
3.1 The receiver of the mixer
3.2 The array of planar antennas
3.2.1 Rectangular patch
3.2.2 Arrays of antennas
3.2.3 Feeding line
3.2.4 Microstrip line
3.2.4.1 Introduction to microstrip lines
3.2.4.2 Losses in the microstrip lines
3.2.5 The feeding networks
3.2.5.1 Phase of the feeding networks
3.2.5.2 Impedance of the feeding network
3.2.6 Design and simulation of the patch antenna array
3.3 Conclusions
Chapter 4: The transmit-array
4.1 Introduction
4.2 The single transmit-array cell
4.3 The transmit-array concept
4.4 Transmit-array design and simulation
4.5 Experimental measurements
4.6 Summary and conclusions for the transmit-array
Chapter 5: The zone plate
5.1 Introduction
5.2 Physics of the zone plate
5.3 Design and simulation of the zone plate
5.4 Conclusions
Chapter 6: The phase grating
6.1 Introduction
6.2 The Dammann’s grating
6.3 The Fourier grating
6.4 Our phase grating design concept
6.5 The relationship between the aperture field and the target field
6.6 The relationship between the Fourier transform and the matlab FFT
6.7 The otpimization algorithm
6.8 Phase grating design
6.9 Simulations of the phase gratings
6.10 Conclusions
Chapter 7: Conclusions
Appendix: Conferences and presentations
REFERENCES
ABSTRACT
RÉSUMÉ

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