Thinking about this, in principle, if you had a very slight length of the GI material, you could bend the wave to the point that you've bend it in the first diagram, then that material runs out and the ray should continue parallel.  In principle it might work, but how do you intend to design and construct an optical fiber?

Hi Adam. Thanks for your answer. What you said is possible and it is done in GRIN lenses. But I want this bent and horizontal wave be contained in the fiber so that I could transport the wave in the fiber. The gradient can be made and modified using diffusion principals. What I am trying to make is a new way to concentrate sunlight. I have a fantastic idea but it all comes down to controlling the down bending of the light.

Hmm, I don't know.  My impression is that if you create the first say 1 micron length of the fiber to have a lense-like curvature, it can be done, but is this any better than focusing the light directly with a lense into a standard fiber?  Or do you need the rays to stay perfectly horizontal in the tube.

In principle it should work. When the light propagates horizontally it means that it is moving along the interface of two media- below is at a higher index of refraction and above is at lower index. of refraction.  The angle of incidence at the higher index must be the critical angle.  However, there will be constraints. This will work at a particular angle for which the light enters the graded index medium.  At other angles it will either propagate above the interface or bend down and internally reflected. Also, if the design is for a straight fiber, it won't possibly work when the fiber is bent.

Hello Ivan

Thanks for your answer. What I want to see happening is that light travels horizontally inside the GRIN media. The reason is that incoming light is sunlight with a wide range of wavelengths. To have them all travel horizontally I need to have a continuous reduction in refractive index from bottom to top. Having just one interface will only work on one specific wavelength. 

I guess my question is why incoming light at critical angle travel along the interface but in GRIN media -which is essentially made of many layers with interfaces between them- light can not travel horizontally when it gets to the maximum height.

I have written this simple code in excel which simulates 1D travelling of light in a simple GRIN media. Refractive index increases from bottom to top in steps. Using Snell's equation I calculate the angle of light in layers. It shows that at the top, light becomes horizontal inside the GRIN media.

What I don't understand is why light bends down after becoming horizontal. according to snell equation it should not bend in any direction anymore. Or am I wrong.

In fact liquid crystal layer with special alignment may guide the light any way, creating predefined refractive index interface. However the size of the beam and the thickness of the liquid crystal layer are related. And have in mind that liquid crystal devices operates with polarized light.

Thinking  theoretically first you need to use a liquid crystal polymer, then you may need a magnetic field to create an environment that could transfer the light just like in the 2nd graph.

Hi Ruaa

Thanks for your answer. A basic question. How would we create refractive index gradient this way? Also let's say the gradient is there. Would light travel horizontally after getting to the top or would it curve down?

Hi Nasseh,

You welcome. If your work is about controlling the refractive index, the best way is using liquid crystal and magnetic field because its easy to control, you can do this by changing the frequency beside adding some impurities for the liquid crystals to change the density of it. But if you want to improve the refractive index you can use Thin-films, i worked on improving the refractive index  of cars glass by using optical polymers, we get a good results. To help you i need to know more information about the materials used.

It is possible to achieve such effect in a liquid configuration lens utilizing the diffusion among liquids with varying refractive indices. However, if you want to achieve the effect described in sketch 2, you may need to tweak the lens chamber design a little bit.

Hi Yanhui

Thanks for your answer. My main question is to keep light propagating horizontal after being bent. Is it allowed by the laws of physics?

A little background: In GRIN fibers light, after bending to get to the point at top (Horizontal at this point) light does not to horizontal, instead it bend back. 

Hi Nasseh

Sorry for the confusion. I have attached a paper which might be helpful to look your problem from another prospective.

I am not sure if you are familiar with transform optics or metamaterial. The design your proposed is definitely possible with a mapping of refractive indice matrix of the material. However, it might be difficult to realize in experiment depending on the wavelength you are targeting. I also attached some papers that can explain this concept a bit.

Hope them help.

Best

Yanhui

The easiest solution to this problem is to calculate the length of the GRIN lens (or graded index fiber) necesary for the parallel-to-the axis ray propagation condition (as you stated). Let us say that this length is L.  Then, by cutting the fiber at 1L, 3L, 5L, etc., the condition is accomplished. Optically coupling a non-graded (constant) refractive index  material (a step-index fiber in the fiber case) will not deviate the parallel-to-axis beam because of normal incidence to the graded-to-nongraded interface. A good understanding for the calculation of this length could be found on Chapters 4 and 5 of the book:  Introduction to Fiber Optics whose authors are Ghatak and Thyagarajan (Cambridge University Press) . I hope this helps.

Thank you Efrain for your answer. That is a possible solution but not usable for the application I have in mind. The GRIN media has to be continuous. What I need is to be able to guide the beam horizontally inside the GRIN media.

I don't see how you can propagate rays in a straight line through a medium with a transverse index gradient.

If there is a variation in refractive index across the wavefront, then the rate of change of phase in the direction of propagation must also vary across the wavefront.  This will tilt the wavefront and change the direction of propagation.

For example, in a graded index medium with a parabolic profile, the only place where ray curvature falls to zero is on-axis, where the transverse index gradient is locally zero.

See, for example, chapter 2 of A. W. Snyder & J. D. Love, "Optical waveguide theory", Chapman & Hall (1983).

You can reach the same conclusion from Fermat's principle (of least time).  If there is a transverse index gradient, then the time of flight is reduced if the direction of propagation is curved into the low index region.

http://en.wikipedia.org/wiki/Fermat%27s_principle

Alan Robinson you answered my question. Thank you very much.