I am currently writing about a military society that enforces its subjects using Acer-class airships. Picture the USS Iowa, except flying and with enough supplies for a year in the air. The only warning sign of their approach would be a disembodied shadow speeding across the ground (because it used active camouflage to blend into the sky). My question is: Is it possible to camouflage even the shadow?
Camouflage seldom truly attempts invisibility, instead aiming to break up patterns so that what you see isn't recognizable as a threat. In that vein, Diffused lighting camouflage has been in use for aircraft since World War II, and is seen in submarines and squid underwater. For aircraft, the common system was Yehudi lights. Because it is difficult to replicate the sheer energy of the sun, you likely would not have a complete lack of a shadow, but the shadow would be less likely to look like an aircraft, and more likely to be disregarded as a fast-moving cloud, or a passing flying animal.
The only true way to eliminate a shadow would be to cover the airship in a "metamaterial cloak". These are carefully engineered materials which force light to refract or reflect in ways the engineer intended, rather than following the simple optical pathway (this works with other waves, metamaterials can be engineered to refract microwaves or even sound). A properly engineered metamaterial cloak could cause the light rays to literally refract all the way around the object, not even leaving an optical "hole" to demonstrate something is there.
Two illustrations of how a metamaterial cloak would work
There are a few difficulties with this approach, however.
Firstly, metamaterials are carefully engineered for a particular wavelength or frequency band. It isn't clear that you could make a metamaterial cloak which would refract 100% of visible wavelengths so you might end up with a ghostly image in (say) the red end of the spectrum, while higher frequencies like blue would be refracted around the object. This also means you would still be visible on radar or using a laser rangefinder once the frequency band is known.
Secondly, for an airship this would add considerable weight, yet be a delicate coating (much like the special coatings on stealth airplanes need frequent and specialized maintenance). The ship might spend a lot of time on the ground being checked out and resurfaced in order for the coating to work at optimum efficiency.
Lastly, a 100% effective metamaterial cloak has the disadvantage of preventing you from looking out. While active sensors like radar could likely penetrate the cloak and receive a return (although the appearance of an unidentified radar emission in an otherwise empty sky would alert a lot of unwelcome attention), passive optical sensors would not be able to "see" out of the cloak for the same reason you can't see into it. The airship will need holes in the cloak, or shutters to allow sensors or even the crew to look outside the ship. Every hole in the cloak is another potential way for the enemy to see you, and as an airship, you would want to have visibility across the entire sphere for protection and even simple navigation.
So metamaterials are a clever work around to the idea of being "invisible", but there must be an understanding of the limitations so you can make reasonable choices in how to use them. In any semi modern seeing, it may be more sensible to use a metamaterial cloak set to microwave frequencies to defeat radar, rather than try to be invisible to the naked eye.
Unless there is something that I am not seeing, the answer ought to be simple. Your airship should have a camera and sensing system on the top side and an array of lights on the other. The camera and sensing system takes in the position of the sun, the location and blockage of clouds or other objects above it, and then sends that data to a central processor. That processor takes the data from above and uses that data to eliminate the shadow through a combination of "sunlight" (sun spectrum) flood lights and projectors.
Depending on the technology level of the world, and the technology level of this ship specifically, you will get different results. If you have a single two-state light sensor on top and a single incandescent bulb below, then the result will not be convincing. If you have an array of different light, uv, and ir sensors along with several cameras and an array of future-tech projectors on your vessel, then your results will be more convincing.
Keep in mind that the shadow simply indicates that there is an object blocking a source of light enough so that you notice. Removing this shadow could be very expensive in terms of energy, but if you only wanted to make it non-obvious, I bet you could make ends meet.
If it keeps the same altitude that our comercial flights, it would need just some smart reflectors and some cloud hunting to avoid detection from most people, even the shadow is completly discarded to be notice.
If the ship is soaring near the top of buildings (Shouldn't do it, thats why cazas and destroyers exist). It becomes complicated, not only has to mimic everything in top and around him without showing a huge thermal footpring but doing it in complete silence too.
I'm not sure what sort of active camouflage you have in mind, but there is one sort of hi-tech camo where this could be possible.
There is a concept designed at MIT where an object is covered in bristles (like a hair brush), and those bristles bend the light around the object, by diffusion I believe. It's supposed to work from multiple angles, so wherever you are in relation to the object, the light and colours from behind are what you see in place of the object. It ought also, to bend the sunlight around the object and make its shadow much less noticeable.
Excerpt from the article linked:
To become invisible, an object must do two things: it has to be able to bend light around itself, so that it casts no shadow, and it must produce no reflection. While naturally occurring materials are unable to do this, a new class of materials called metamaterials is now making it possible.