I am not sure, but I expect the explanation may be in re-radiation by the elements in the direction of the grating lobes. Small receive antennas such as half-wave dipoles re-radiate as much power as goes to the load. As the array spacing is increased, the re-radiation from all the elements will be in phase in the direction of each new grating lobe as it appears, resulting in a step reduction in directivity as the array spacing increases and each new grating lobe becomes possible.

Be very careful with definitions here. The directivity of an antenna is the ratio of maximum radiated power density to the average radiated power density. What you are referring to here as N is properly called the Array Factor and refers to the radiated field, not the power density. See Kraus "Antennas" for a full explanation.

Lots of things happen in real array antennas. For a transmit antenna inter-element mutual coupling interacting with inter-element energy coupling in the feed network makes driving the wanted current in each element with the correct amplitude and phase an issue. In a receiving array the incoming wave is received by each element but also scatters to the other elements through mutual coupling. Again the feed network plays a complex role in combining the output of each element to give the array output. Most antenna courses describe antenna operation in transmit mode and use reciprocity to describe the receive mode.

Mutual coupling is a problem unless you are designing a Yagi-Uda array where it is used as the excitation method for all but the driven element.

You will find that the oscillation of directivity with element spacing happens even if the elements are matched and driven at the desired phases and amplitudes. Antennas are reciprocal, so the transmit and receive gains will be the same and show the same variation with spacing.

After 5 days of considering it, I am more certain that in receive mode it is the re-radiation by the individual elements that causes the ripple in directivity. Dipoles re-radiate half the power they intercept, in the same pattern. This may not seem likely, but it is true, see for instance https://physics.stackexchange.com/questions/574662/why-does-an-ideal-antenna-not-absorb-all-incident-power.

Aperture antennas re-radiate, but with the pattern of a Huygens source, which is why you don't see a reflection from the aperture of a waveguide horn, for instance (but there is re-radiation in the forward direction (and other directions) that cancels the continuing incident wave and results in the shadow of the antenna).

I think the coupling effects result in things like scan blindness, where at some directions the impedance of the elements changes so that they are badly matched and the gain is poor.

Reading Nick's question again I wonder if the array model he is using is simply a linear array of omnidirectional sources with assumed uniform amplitude and phase driving each? The radiated field is simply the sum of the fields from each element. His use of the term N to describe the Array Factor may imply this. If so then this model will take no account of mutual coupling and so mutual coupling cannot be the answer.

Is this the case Nick or does your model allow for the interactions between array elements?

It is well known that a conjugate matched receiving antenna will re-radiate (scatter) the same amount of power that it delivers to its load. Put several such antennas in proximity to each other and the drive point impedance of one of them, i.e. the impedance that determines the drive voltage needed to get a given drive current, depends on the relative positions and orientations of the antennas, their load impedances and on the drive voltage of each of the other antennas.

Because it is the directivity rather than the gain that is being discussed, the match of the elements does not affect it, if the match is the same into every element. By definition, directivity does not include the effect of the overall match of the antenna, but gain does.

If the match is different into different elements, this will affect the element weighting, and affect the directivity. In an array, the match is usually the same except for the outer 1 or 2 rows of elements.

We could continue our interesting discussion about the ins and outs of driving array antennas, the subject of my PhD from nearly fifty years ago, but it doesn't help Nick.

The question still remains about how his model works. Is it a simple sum of omni-directional sources or something more sophisticated that could include mutual coupling?

Nick Host asked What is the physical explanation of what is causing directivity fluctuations for a receive antenna array?

It is the re-radiation into the grating lobes by each element that results in the variation of directivity with element spacing.

Nick Host, I believe you are essentially posing the following questions:

Consider a receive array #1 with grating lobes and array #2 without grating lobes, each of which has the same exact mainlobe beamwidth. Why is the directivity at the mainlobe peak of array #1 lower than the directivity at the mainlobe peak of array #2? If the same signal is arriving at both antennas from the mainlobe peak direction, why do they have different power outputs?

A precise answer must be independent of re-radiation, scattering, or other non-idealities of antenna arrays, because presumably, it would also have to apply to unrealizable ideal arrays with isotropic elements separated by more than lambda/2. You are correct that reciprocity implies that if directivity is lower in transmission, it must also be lower in reception. That is one mathematical explanation, albeit not the most intuitive. There are other ways to conceptualize transmit/ receive arrays with grating lobes.

It is important to recognize that moving a fixed number of array elements farther apart might locally narrow the beam in a subset of the angular range, but it does not increase the total energy reception or transmission capability of the array – it simply partitions its directional (angular) response differently. In other words, increasing element separation eventually splits an unambiguous, wider lobe into multiple narrower lobes, but the peak directivity does not change. The narrower mainlobe and grating lobes present a false illusion of increased directivity. This is easily provable via Parseval’s Theorem. The angular response of an array is the Fourier Transform of the spatial domain function, i.e., the antenna element positions. Moving the elements does not change the input power in the spatial domain, so the radiation patterns (in the angular domain) must have the same total power.

We can verify with this simple MATLAB Phased Array Toolbox script for any separation value.

c = physconst('LightSpeed');

fc = 3e8;

lambda = c/fc;

ang = 0;

separation = lambda/2;

myAnt1 = phased.IsotropicAntennaElement;

myArray1 = phased.ULA(10,separation,'Element',myAnt1);

d = directivity(myArray1,fc,ang,'PropagationSpeed',c)

We can now reason that since our array, with multiple narrow grating lobes, has the equivalent peak directivity of an array with a single wider mainlobe, an array with a single narrow mainlobe replica of a grating lobe in the other array must have higher directivity.

Here are a couple of additional thoughts to consider: if simply repositioning N elements farther apart bought you directivity, we could take a small power transmitter and feed it into our “magic array” with elements spread arbitrarily far apart to produce arbitrarily large EIRP in one direction. Correspondingly, that same arbitrarily wide array could amplify arbitrarily weak signals in any one direction. That is not possible in reality, because, colloquially speaking there is “no free lunch.” Increasing the breadth of an N-element array does not buy you energy/power or directivity, but does it get you anything? The answer is yes. Placing sensing elements over a larger distance or region increases angular resolution (under the condition that you can otherwise resolve any ambiguity, if one exists). This is a consequence of the generalized uncertainty principle: breadth/ narrowness in the time/ spatial domain corresponds to narrowness/ breadth in the frequency/ angular domain.

Finally, it is helpful to think of an array with isotropic elements as being analogous to a spherical, infinitely flexible (non-bursting) balloon filled with an incompressible fluid (like water). Moving the elements around can “squeeze” the pattern in some directions and “swell” the pattern in others like squeezing a balloon contracts the balloon in some places while producing bulges in others. Spreading the elements of our array with elements lambda/2 distance would correspond to transforming our single lobe balloon into a balloon with multiple lobes.

Thanks to all who have provided your insight; Malcolm White A.C. Marvin Mark Mendlovitz . To clarify, the model I'm envisioning is of a uniform linear array with isotropic elements (mutual coupling ignored). As described in [1,2,3] (which use an integration of the total fields radiated by the array), the main lobe directivity (boresight) of such an array increases with element spacing until it reaches a value of N at an element spacing of lambda/2. It continues increasing until around an element spacing of lambda where it dives below N (typically explained as due to the appearance of a grating lobe - energy is being split in two directions). You can continue to increase the element spacings and the main lobe directivity will oscillate around the value N alternating between narrowing lobes and introduction of additional grating lobes.

The oscillatory behavior of the main lobe directivity occurs for arrays on transmit without considering the effects of mutual coupling (element match or reradiation), so it seems like these should not be considered for the array on receive (of course mutual coupling should be considered for real arrays, but I'm just trying to understand the fundamental mechanisms).

So, again, I understand the fluctuations of main lobe directivity when understood on transmit - simply due to the focusing/splitting of energy in space. However, when I try to understand what is physically happening on receive, I'm still not quite understanding (there is no energy splitting here). This fluctuation of directivity is fundamental to arrays and would seem to not primarily depend on the (highly element specific) mutual coupling. Specifically, I'm hoping for a description in terms of the impinging wave and why separating the elements creates the above described oscillatory behavior.

[1] W. L. Stutzman and G. A. Thiele, “Array Antennas,” in Antenna Theory and Design, 3rd ed. Hoboken, NJ: Wiley, 2013, ch. 8, sec. 8.5, pp. 293-294.

[2] R. J. Mailloux, “Pattern Characteristics of Linear and Planar Arrays,” in Phased Array Antenna Handbook, 2nd ed. Norwood, MA: Artech, 2005, ch. 2, sec. 2.2, pp. 80.

[3] R. C. Hansen, “Basic Array Characteristics,” in Phased Array Antennas, 2nd ed. Hoboken, NJ: Wiley, 2009, ch. 2, sec. 2.4, pp. 38.

On receive each element will re-radiate some of the power it receives. For an elemental dipole it is half the power. If the spacing is less than a wavelength (for 0 dreg illumination) then the sum of this has nowhere to go and there is no re-radiation. As each grating lobe becomes possible (its angle is real) then power is re-radiated into any possible grating lobes and gain is reduced. It is the re-radiation into the grating lobes by each element that results in the variation of directivity with element spacing on receive.