Jones Vector superposition of Mobius-light

[lkcl]

Hi,

 

I am looking for a very special and unique mathematical solution which, from another direction of investigation not related to maths, I have reason to believe exists, which is to work out those phases of a "mobius light" configuration that will superimpose properly. For references about the recent experimental work in which "mobius light" was successfully demonstrated, see arxiv:1601.06072, and for the theoretical work dating back to 2009, see Isaac Freund's paper arxiv:0910.1663

 

Please bear in mind: I am a software engineer, I have a CV 15 pages long, I am not a n00b but my mathematical ability is that of an O'Level / A'Level student, whereas my knowledge-derivation, inference, logical reasoning and reverse-engineering skills - and persistence - are extremely high.

 

In looking up the wikipedia page on Jones Calculus https://en.wikipedia.org/wiki/Jones_calculus#The_Jones_vector i noted that there are two parts to a Jones Vector: the elliptical polarisation axis that is rotated (E0x), and an exponent part that performs the adjustment of that axis.

 

Asking last week on maths.stackexchange someone kindly confirmed that if the exponent part of two elliptically-polarised light fields were IDENTICAL, then of course the vectors E0x would simply be superimposed (added) as... well... vectors

 

The more challenging part to answer is: are there any *other* circumstances under which two Jones vectors will superimpose? More specifically: if two fields are arranged in a Mobius-strip configuration, under what circumstances (what phase shifts etc.) would the Jones Vectors of two such *ALWAYS* superimpose, other than in the trivial cases?

 

To do that, first it's necessary to define an MS in terms of Jones Vectors, then it's possible to define the relationship between one MS and another.

 

Also: I read up about euler's theorem, the bit where exp (-i x + -i y) = exp( -i x ) exp( -i y ) and that allowed me to re-express the equation for a Jones Vector that you can clearly see on the wikipedia page, as follows:

 

[math]E_{\hat{x}} = E_{n\hat{x}} e^{-i \left( kz / 2 \right) } e^{-i \left( - \omega t / 2 \right) } e^{-i \left( \theta/2 \right) }[/math]

 

and the relationship between each of the MS's elliptical polarization axes is:

 

[math] E_{n\hat{x}} = E_{0\hat{x}} e^{-i \left( \theta \right) }, \theta = n\tau/12 [/math]

 

And tau is of course 2pi Now, here's the bit where I kinda know the answer, but not enough "math" to fill in the gaps, so to speak. The situation which I suspect the Jones Vectors will superimpose is if:

 

(1) The phases are a multiple of 1/12th tau

 

(2) MS1 is rotated by tau / 4

 

(3) MS2 is rotated by tau / 2

 

(4) The starting phase of MS1 is *HALF* that of MS2.

 

Under these circumstances, if you express MS2 theta in terms of MS1, you can, through Euler's theorem, find that the common factor is the exponential part of MS1 with the rotation of tau/4 doubled to match the rotation tau/2 of MS2, and it's sorted.

 

so there are therefore situations at MS1 n=0 and MS2 n=6, likewise there is one at n=1,2 and another at n=1,8 another at n=2,7 and another at n=3,9 and so on. There's quite a few in fact, all of which fits *precisely* with what I've been working on...

 

... but I'm missing the interim steps, as you can see, but also I could use some help verifying that what I've come up with is in fact correct. I'm not a mathematician, I'm a software engineer.

 

[Phi for All]

!

Moderator Note

Off-topic posts split to Trash.

 

Please stick to the topic.

[lkcl]

ok so i worked out the answer, http://vixra.org/abs/1702.0131 and it's that it's possible to superimpose any two out of only eight unit-vectors around a clock separated by thirty degrees, if you phase-offset one of them by 180 first. this results in the elliptical axis being phase-shifted by 90 degrees, and it results in the EM field being orthogonal (at right angles) at all points. the waves must obviously be of the same frequency.

 

triple superposition is also possible, but amazingly there are only 32 possible permutations (out of a possible 3^12) that will work.

 

i do not know what the actual end-result looks like (what the actual sum is, nor the elliptical axis angle following the superposition) because i am not a mathematician.

 

this is however an extremely important mathematical result that hasn't been published before. castillo in his 2008 paper notes the conditions under which superposition will occur but does not identify the possible candidates, also he assumes that there is no phase delay. the phase delay of 180 degrees is the key to allowing more candidates than the ones that castillo found.

 

it would also be a signlficant advance for mathematics to find out what the superposition result is.