# Refraction at normal incidence

Gobil
high All,

Just got to thinking, if we have a beam of light normal to a flat surface, and the surface is that of an object which has a variable refractive index across the transverse beam direction, will some of the light be bent away from the normal on the other side of the object?

i.e. if we have a ´flat´lens, with a distribution (lets say Gaussian) of refractive indices in the material transverse to the beam direction, will it act as a normal lens and focus or defocus the beam?

I understand the refractive index causes a phase-shift in the EM waves, but does this also change direction as in the example I describe above?

Many Thanks!

Mentor

Counterintuitive for me :surprised

Gobil
ok, thanks.

But when we talk about the real part of the refractive index changing the phase of the EM wave, do we mean the phase is just changed in the plane of (original) propagation? i.e. it doesn´t really change the phase, but just the propagation direction, and hence if you observe all the phases relative to the plane of incidence they have changed?

..but wait, if the light propagates through a uniform block of glass at normal incidence, there will be a phase change.. confused.

I'm not sure what you are asking- taking the initial phase of the wavefront as 'zero', making a uniform change to the phase does nothing, but making a spatially-dependent phase change does do a lot- you can convert a plane wave to a converging wave, for example.

Or am I not understanding you?

Gobil
well, what I mean is if you propagate 2 beams parallel to each other, one through vacuum, and the other through some uniform medium with a finite refractive index and finite length, will there be a difference in phase between the two waves when they are measured after the block?

yes- that's the principle of a Mach-Zender interferometer.

Gobil
ok, so when we have a uniform block of material and a beam passing through it at normal incidence we have refraction in the form of a phase change of that EM wave. when this medium has a gradient of refractive index transverse to the beam we get a change in wave vector, i.e. the direction of the beam, and also a change in phase, is this correct?

Yes, but they are equivalent- a spatially varying change of phase is equivalent to a change in propagation direction; consider the form exp (ikz), and now let k = k + d(x,y), where d is the amount of phase change. If d(x,y) = d_0, you get an overall constant phase shift (interferometer). If d(x,y) = (x/r)^2 + (y/r)^2, you get a converging spherical wave (if I wrote d correctly....). Thus, a uniform block of glass of varying thickness (say, a lens) is equivalent to a flat slab of glass with a gradient refractive index.

Gobil
great thanks,

so this is the key point, a uniform change in phase (in a uniform medium) will cause phase shifts in the wave, but no directional change. But if the RI varies across the transverse of the beam, the change in phase is different for the different parts of the beam. I have an image in my head of an EM wave having different phase shifts at one 'side' of the amplitude wave the the other, and when this happens, it is essentially bent to the 'faster' area of the medium.

does this sounds right?

Staff Emeritus