Which is the cause of "scan losses" in a linear phased antenna array?

Question

in a linear phased array it is possible to move the peak angle ("beam steering") by applying proper phase shift between the excitation currents:

The second picture represents the array factor for different beams angle. Last picture taken from here. What I see in the second picture is that beam steering increases all the lobes width but does not change the lobe peak. So, the peak value of the array factor is (or at least seems to be) the same for each scan angle.

Always there it is written that if beam steering is used, there will be some "scan losses":

Another characteristic of all active antennas is the loss of aperture gain as the beam is steered away from the boresight direction — defined as Ɵ=0. This characteristic, called scan loss, follows 10*log(cosN(Ɵ)) power, where Ɵ is the scan angle off boresight and N is a numeric value, typically in the 1.3 range, which accounts for the non-ideal isotropic behavior of the embedded element gain. Fig. 6 plots scan loss in dB vs. scan angle, measured in degrees. Note, at the origin, where the boresight angle is zero, there is no scan loss. As the scan angle is increased to 45 degrees, there is 2 dB scan loss. If you increase scan angle to a practical limit of 60 degrees, there is 4 dB scan loss.

The following picture is then shown:

So, which is the scan loss? From my initial picture it doesn't seem to be caused by the array factor. But, if it were caused by the single element pattern, how could we say that it is linked to the scan angle with a fixed equation not dependent on the single element pattern?

Answer 1

Steering any real array off bore-sight induces scan loss, or a loss in directivity. In the image you posted with the multiple beams, that graph is likely normalizing the patterns, hence why they all peak at 0 dB. It does however show the main beam broadening, which is a consequence of steering an array.

For now, forget about the term "array factor" as it might cause confusion. When speaking of an array composed of isotropic elements, the array factor and the antenna pattern are mathematically the same thing. This is not the case with real antennas. But again, terminology not too important to understand the concept of scan loss. We have an antenna pattern and that's that.

Update

Following is an update that corrects and clarifies what others might have read already. It was stated that uniform linear arrays do have scan loss, and it was a result of a mistake in a model. See the comments in Jason's answer for some background.

For both linear and planar uniform arrays, there is no scan loss if the steering occurs along the principal axes, assuming an element spacing of \$\lambda/2\$. This is trivial with a 1-D linear array, and with a planar array, scanning in only azimuth or elevation will preserve the antenna peak gain. It does however, still broaden the beam. Note that this element spacing is a very special case and is not practically achievable.

Let's do a quick example with a 32 x 32 uniform planar array, whose isotropic elements are spaced \$\lambda/2\$ meters apart. Below is a plot showing the nominal unsteered array along with the same array steered 20°, 40°, and 60° off of bore-sight in azimuth only. Each of the arrays are normalized to the un-steered array's peak value. Since we're steering along one of the principal axes, we don't induce a scan loss:

If we now apply the 20°, 40°, and 60° steering angles to both azimuth and elevation axes, it results in a composite off-boresight angle \$\Theta\$. In this case we do see scan loss. For the plot, an azimuth cut is taken at each pattern's peak location:

The scan loss expression given by \$10\ log_{10}(cos^N(\Theta ))\$ is an approximation that's pretty good for initial analysis. After building the antenna you can characterize it more accurately from measurements. Below is a plot of the scan loss for this particular array and you can see that it follows a similar trend as the approximation. Since this is a theoretical array, a smaller value of \$N = 0.9\$ shows a good match:

ULA Scan Loss Special Case: \$d = \lambda/2\$

Here is the reference from Optimum Array Processing, Part IV of Detection, Estimation, and Modulation Theory by Harry L. Van Trees regarding the ULA with \$\lambda/2\$ spacing:

Answer 2

Array factor does not show scan loss. Array factor is merely a multiplier. It's actually a common mistake for engineers to use the array factor as a gain value. Array factor gain does not change with scan angle or element spacing!

To see scan loss, you need to calculate the directivity. However, for a uniform linear array (ULA) of isotropic elements you will not see any scan loss when you calculate directivity. You need to either use a 2-D array or use a non-isotropic element pattern.

Physically speaking, a loss in directivity is caused due to less effective area pointing in the direction of interest. Scan loss occurs when the array has area (e.g. not an isotropic ULA).