# Effective refractive index can be equal zero?

Can someone please explain the cases that the Effective refractive index $$n$$ on it can be equal zero? Does such a situation even exist? I am trying to solve an ordinary differential equation to find the dispersion relation for Linearly graded index TM modes but in order to get a solution for the equation $$n$$ must be equal zero.

• Even actual refractive index can be zero (journals.aps.org/prb/abstract/10.1103/PhysRevB.75.155410), sort of, in a narrow frequency range, and with huge losses in real life. So it is possible in principle, but not practical
– Cryo
Jan 11 at 20:01
• that is mean there is a problem on my derivation ,, becouse there is no practical cases N=0 on it ??
– Aya
Jan 11 at 20:05
• I would suspect error before anything else, but zero and negative refractive index is, in principle possible (one has to be very careful about what that means though)
– Cryo
Jan 11 at 20:23

There are a few notable cases where permittivity (and hence $$n$$) can be zero for a specific frequency:
Since all of these cases have losses, $$n$$ does not actually go to exactly zero. But, it's quite close, and when we idealize these systems it is sometimes appropriate to neglect the losses and just assume $$n$$ passes through zero.
In your case, I would advise that when you search for TM modes in whatever structure you have in mind, you should do a calculation including material losses, then allow for attenuation in your mode (either spatial or temporal attenuation, i.e., either complex $$k$$ or complex $$\omega$$). This is slightly more complex but it is more realistic, and as a bonus you get information about how good/bad the mode actually is.