everyday-lifegeometric-opticsopticsrefraction
First, I would like to clarify that I know why a mirage is formed, what I want to know is why is it that when a mirage is formed it appears that a pool of water is present.
Like for a palm tree in a desert, an inverted image is formed accompanied by a virtual pool of water. But where does that pool of water come from? The only object whose image is inverted is the tree isnt it?
The virtual image for Interface 2 here actually means, that the image would have been there if the second interface did not alter the path of light. But due to interface 2, the position of (would have been image, l1) is changed to l. It is "would have been" image, not a virtual object. The virtual object for interface 2 is where the light seems to be coming from, means in exactly opposite direction of l1 along the ray of light. Because, it is a "would have been image", and virtual object is directly opposite, the virtual object and final image are not on the same side.
A gradient index rod lens will form an erect or inverted image, depending on its length. A SELFOC lens is usually designed to transfer an image from the flat surface on one end of a rod, to the flat surface at the other end of the rod. A SELFOC lens is usually designed to produce an inverted image, but when made twice as long (that is, with pitch 1.0) it forms an erect image. The ray paths for a single pixel on one surface go through the rod as shown below. They come to an inverted focus in the middle of the rod, then back to an erect image at the other end of the rod.
This image is one frame from a video on the Stemmer Imaging website, though there are several companies who provide gradient index rod lenses. This sort of device is sometimes called a "Contact Imaging Sensor", and would be one element of a "line scan bar".
Edit #1 Some other approaches might meet the uniform refractive index, single-element condition:
I'm pretty sure two optical surfaces are required, when the refractive index is positive, because we can show that an single optical surface (reflective or positive-refractive) can only form an inverted real image. HOWEVER, negative index materials, in principle, can do it with only a flat, parallel-faced slab. See "Negative refraction makes a perfect lens" by Pendry.
Best Answer
The reflected image of the palm tree is accompanied by the reflected image of the sky above and surrounding it. so in the reflection, you see the palm tree and the sky. There is no pool of water: the light coming towards you from the palm tree and the sky behind it is reflected up away from the ground by a hot layer of air right next to the ground, which acts like a mirror and inverts the image of the palm tree.
Note that at a shallow angle close to sea level, the sea far from you also acts like a mirror, reflecting the blue of the sky, when the surface of the sea is calm and nearly flat. The reflections swirl around a bit because of small waves that "bend" the mirror surface around.
Because there are weak air currents next to the hot ground in the mirage case, the reflections from distant objects get swirled around in a manner that looks like the surface of a pond with small waves in its water.
These effects are easy to demonstrate if you put a table tennis table out in the sun and at one end, you install miniature palm trees. Somewhere near the middle of the table you place a large flat mirror and next to it a very shallow plate with a thin layer of water in it. Then go to the opposite end of the table, get down on your knees, and put your eye right next to the edge of the table and look towards the little palm trees, moving your head until the reflectors are in your line of sight.
With care you can do this experiment with a big flat plate of hot metal in place of the mirror but you might start the table on fire.