optics - How to bend light?

As we all know that light travels in rectilinear motion. But can we bend light in parabolic path? If not practically then is it possible in paper? Has anyone succeeded in doing that practically ?

Answer

Light does not, in general circumstances, travel in straight lines (although it does do so in the ones we usually encounter).

For one, light is really a wave and can only approximately be thought of as consisting of independently-propagating rays. This happens when the wavelength of the light is much smaller than the distances it is propagating over, which is usually the case for light (whose wavelength in the visible range is $0.4$ to $0.7\,\mu\textrm{m}$) but is not necessarily the case e.g. for radio waves and when nanoparticles are involved.

In this short-wavelength limit, wave propagation gives way to ray propagation (which is a special, approximate case of the former), and specifically to Fermat's principle for the mathematical description of light. This principle states that light rays starting at $A$ and ending up at $B$ will follow the path that minimizes the travel time $$S=\int_A^B n(s)\textrm{d}s,$$ where $n(s)$ is the (possibly spatially dependant) refraction index along the path.

For a homogeneous medium, this does indeed give straight lines for propagation. For a planar interface between two different media it gives Snell's law for refraction and it also describes reflection. (However, because it does not account for the actual nature of light as an oscillating electric field, this description cannot predict transmission or reflection coefficients.

However, if the medium is not homogeneous, then light will not travel on a straight line, and for complicated inhomogeneities the path can be correspondingly difficult to calculate. For an example, see the formation of mirages or more generally atmospheric refraction. Conversely, if one has a path one wishes a given light ray to take, then it is possible to engineer a refractive index spatial dependence that will make light bend that way. (Of course, whether such a dependence is physically reasonable is another matter; if the path bends too sharply then it may not be possible to find materials with the correspondingly large index and index gradients necessary.)