I have a Jones Matrix of the form : M= [ A + iB C + iD ; U + iV X + iZ]
The input Jones vector is E1 = [ 0 ; 1] , linearly polarized light along Y-axis
So the output is E2 = M*E1 = [ a + ib ; c + id]
Now I want to normalize E2, what is the best way of doing it ?
I am thinking about calculating the magnitudes of each element of E2 and using that for normalizing
Jones matrices are not necessarily normalized. The intensity of the light after a Jones matrix can be less than went in. Take, for example, a polarizer where all the light in one polarization is lost. You can even have the light get brighter. Many laser gain media have polarization dependent gain and the gain is modelled as a Jones matrix.
So, I am curious how your Jones matrix was constructed, and whether or not you are sure that it needs normalized.
However, let's assume you know your Jones matrix represents a set of lossless optics, and let's assume for mathematical convenience or some other reason you have constructed it in a way that it doesn't preserve intensity but is in every other way correct.
You cannot normalize it by elements, and you cannot normalize it for a particular polarization. If the optics are lossless, then the Jones matrix must be lossless for all polarizations. For an arbitrary Jones vector "v" the light after the optics is "u" = Mv. For lossless optics it must be that u*u = v*v. The correct normalization is (v*v)/(u*u). If everything is correct, the parameters of v will cancel and there will be no arbitrary terms in the normalizer.
Now, without loss of generalization v can be written (e^(i phi), e^(-i phi)). Taking your matrix to be ((A, B),(C,D)) where I am not going to write out the complex components as you did, you will find the normalizer calculated as indicated above is (A*A + B*B + C*C + D*D) + (AB* + CD*)e^(i 2 phi) + (A*B + C*D)e^(-i 2 phi)
This must not depend on phi, so those last two terms had better be zero. This puts a constraint on your Jones matrix. If it preserves energy for all polarizations then it must be antisymmetric. Also, if it preserves energy and is antisymmetric, the normalization you want is (A*A + B*B + C*C + D*D)
@Mike Albert :
Thanks for the explanation. I actually want to normalize the resulting Jones vector, not the matrix.
I want to calculate the power in X and Y polarization directions based on the Jones vector. My input is only having the Y component, but the resultant output has complex numbers on both the rows/components.
Ah, then I'm not sure I understand. What do you mean "normalize"? For a Jones vector |v> the total power is , i.e. multiplied by its own complex conjugate transpose. If you want to know how much power is in each polarization, just look at the modulus, i.e. the complex conjugate square of each component.
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