Fundamental optical waveguide query?

Yes, that's true in both the planar waveguide and cylindrical fiber cases.

But what about more complex transverse or radial profiles, in which the index is for example, very high (or, very low) in a central core, drops (or, rises) to an opposite value for a surrounding annular region, then goes to some other intermediate value for another annulus, then maybe more annular layers, then finally settling down to some final cladding index value for the rest of the way to infinity.

My impression is, some of these structures will propagate, some won't. Is there a general theorem for knowing if a varying index profile will be guiding or "anti-guiding", without actually solving for the mode?

Yes, that's true in both the planar waveguide and cylindrical fiber cases.

But what about more complex transverse or radial profiles, in which the index is for example, very high (or, very low) in a central core, drops (or, rises) to an opposite value for a surrounding annular region, then goes to some other intermediate value for another annulus, then maybe more annular layers, then finally settling down to some final cladding index value for the rest of the way to infinity.

My impression is, some of these structures will propagate, some won't. Is there a general theorem for knowing if a varying index profile will be guiding or "anti-guiding", without actually solving for the mode?

For correct modeling solve maxwells equations.

Hmm, I think it'll still always guide at some level, as long as the index is independent of axial position. I don't have a reference for this, though. Do Kawano & Kitoh have a proof for the uniform cylindrical case? (My copy is at home.)

Cheers,

Phil Hobbs

Really living up to the screen name, hmm?

Cheers,

Phil Hobbs

Sorry, but Maxwell's equations are the only reliable way to model "strange" index profiles.

I have absolutely no real knowledge of this subject or intent to get any.

Nevertheless, as you describe the problem, I think thin-film theory. The various cylindrical layers of indexes transform (EE types should think of impedance transformation) the index of the outside as seen by the core. Unfortunately, the transformation is sensitive to the angle of incidence.

Methods have been developed to use analog computers to solve thin-film problems. ODEs can be set up to follow how effective index changes with position in a multilyayer coating. Again, that would be angle sensitive.

How all this gets put together to provide useful results, if it can, is likely to be a lifetime vocation.

Bill

I'm no expert on this, and maybe I'm stating the obvious, but I remember in QM that using the variational principle one can prove that any arbitrarily weak attractive potential has at least one bound state (I vaguely remember this as a homework problem, which means it must be a well-known rule for QM). I sometimes think in QM terms for qualitative descriptions of waveguide modes in fibers; perhaps it applies here. I handwaved using this principle (correctly or not) when someone asked me if light would propagate in fibers whose diameter was significantly smaller than the wavelength... In this line of thinking, any index cross section arbitrarily weakly higher than the values at infinity should guide at least one mode....

Just a thought, Frank

Phil!!!

A waveguide or fiber in which the index in the core is uniform and _lower_ than the index in the cladding(or claddings) absolutely won't guide a (trapped, confined, non-attenuating, whatever) mode.

AES schrieb:

Hmm, what about those waveguide CO2 lasers with an CO2 gas mixture filled small alumina tube? It has a low uniform (approx. 1) index in the "core" and a higher (complex) index in the "cladding" tube. Interestingly, allthoug the diameter with some mm is much larger than the wavelength (10mu) is supports low-loss modes and emits nice single-mode TEM00 like beam patterns.

Or the crystal fibers with air in the holey core and a photonic band-gap structure surrounding? Similar are those planar photonic bandgap waveguides where the guide is also "holey", surrunded by a regular pattern of evenly spaced pillars with higher-index material.

So IMHO any structure with an arbitrary index variation in at least one dimension can guide some kind of modes in this dimension. The question might be if there is some leakage of radiation out of the waveguide. Or similar, are there transmission losses even if the materials are perfect lossless and the waveguide is perfectly homogeneous in the propagation direction?

Anyway I think the effective refractive index or more generally, the group velocity of a certain guided mode can only by calculated by a full field solving for arbitrary index profiles.

Regards Rainer

Okay, that pays me back for my earlier post. ;-) Let me be a bit clearer. My guess would be that if you can put a cylindrical boundary around any region whose average index is higher than that of the distant parts of the cladding, that you'll have at least one guided mode.

Cheers,

Phil Hobbs

On second thought, the cylindrical region would have to be "sufficiently large"--a tiny thread of high index material surrounded by 5 wavelengths of low index, with cladding of an intermediate index, wouldn't guide even though a cylinder of 0.01-wavelength diameter would meet the condition above.

You're right, this is a more interesting problem than I thought.

Cheers,

Phil Hobbs

Uniform planar (aka "slab") waveguide:

• Any losslessly axially propagating pattern inside its core can

• If mode is to propagate losslessly those plane waves must *all* experience TIR at the planar core-cladding interface. (In fact, each individual lowest or higher-order mode of a uniform planar waveguide is made up of exactly one plane wave at a certain angle, and its reflected partner.)

• TIR exists only for interface from greater to lesser index.

Ergo, slab WG with lower index core cannot possibly have a losslessly propagating mode.

Same is true for uniform (i.e., "step function") cylindrical fiber.

Now,about more complex profiles . . . ? That's the question.

In terms of geometric optics ...

n*sin(theta) = constant

where theta = angle of incidence against the planar or cylindrical surfaces of equal index.

As n varies, theta will change to keep this condition true.

Take as boundary conditions the central index n0 and the angle of incidence for rays in the center. It is possible to have n fall below some critical value such that Snell's law requires sin(theta) > 1, in which case the rays will not propagate into this region (TIR).

I am not sure how to describe this in terms of waves, planar or otherwise.

Mark