How large is the smallest object that can be detected at a given wavelength? What is the cross section of the smallest object that can be detected with say visible light ($\lambda$ ~380 - 750 nm) or X - band radar ($\lambda$ ~20 - 50 mm).
Does the object need to have one side larger then $\frac{\lambda}{2}$ or something like that to get a reflection?
 A: Seems like in principle, there should be no minimum size. For example, a small hand-held radio (cell phone for those of you in the 21st century) can detect, generate, and scatter EM waves with wavelength much longer than the object itself.
For optical and X-ray scales, one would probably be limited by quantum or thermal effects, which may in principle be mitigated at very low temperatures. But even so, a single atom is MUCH MUCH smaller than 500nm, and can absorb and emit light at that wavelength.
A: The Rayleigh criterion determines the maximum minimum size of objects - in terms of the maximum minimum angular resolution $\theta$ - which can be resolved at a give wavelength $\lambda$ and with a lens of a given diameter $D$:
$$ sin(\theta) = 1.22 \frac{\lambda}{D} $$
The spatial resolution is given by:
$$ \Delta l = 1.22 \frac{f\lambda}{D} $$
where $f$ is the focal length. For visible light this gives a limit of $\sim $ 200 nm for the minimum resolvable distance.
Of course, this is all based on classical optics. With newer metamaterial based lenses one can do much better than the Rayleigh limit.
A: Detecting an object requires you to get enough of a signal from it that you can discriminate that signal from background noise. This is not necessarily dependent on the size of the object, although it can be.
For example, if you are trying to detect an object which has very low reflectivity at your given wavelength, you might get a stronger signal if the object is larger. Of course this also depends on the geometry of the object, and how much of the reflected light is reflected towards your detector. Ultimately the deciding factor in whether you can detect an object is the ratio of the signal strength to the noise in your detection system. Engineers typically call this the "Signal to noise ratio" or SNR.
The Rayleigh limit will describe the minimum distance between two point sources such that an imaging system can distinguish one point from the other, rather than seeing them as a single point. However, this has nothing to do with the fundamental requirement of detecting the presence of an object in the first place. It is quite possible to, for example, detect medium-wavelength infrared light (with a wavelength of 3 to 8 microns) emanating from a pinhole or the tip of an optical fiber which is only a couple microns across. In this case an imaging system would indeed see a spot of light with a radius given by:
$$
\begin{array}{rl}
 \Delta l = 1.22 \frac{f\lambda}{D} &\mbox{for a circular aperture}\\
 \Delta l = \frac{f\lambda}{D} &\mbox{for a square aperture}
\end{array}
$$
Simply by detecting this spot, you would have detected the presence of the object, although you would not be able to say anything about its size or shape, because it is below the resolution of your device.
A good way to think about this is to get away from the concept of an imaging system. Imagine you've got a lens focusing light onto a photodiode, instead of an imaging detector like the CCD in a camera. With a single photodiode, you can't determine the size or shape of anything. All you can do is detect a signal, or the lack of a signal. This is mostly independent of resolving power or the size of the object being detected.
