Single Matrix Method: Lossless Partial Coherent Case

To include the finite linewidth of the source two possible approaches exist. Either the elements of the Mueller matrix are integrated/summed in the fiequency domain or the same process is done for the elements of the Stokes vector. In both cases the output Stokes vector is averaged in the fiequency domain.

The Mueller matrix for the lossless optical fibre (Eq. (7.1.22)) shows that if the input signal is randomly polarised the output signal is randomly polarised as well and integration in the frequency domain does not change this. So for a randomly polarised partial coherent signal the degree of polarisation along the fibre stays equal to zero.

However, if the input signal is partially polarised (or perfectly polarised as an extreme case) the finite linewidth of the source, the coupling coefficient and the differential propagation constant are causes for the degree of polarisation to degrade

upon propagation along the optical fibre. In this section three figures show the effect of these three entities on the propagation of the degree of polarisation.

As a matter of fact, the finite line width of the source does not determine the final (minimum) degree of polarisation along the optical fibre. But it does determine the length of fibre it takes the signal to reach that minimum degree of polarisation. This effect is shown in Fig. 7.5, where the degree of polarisation is shown as a function of the propagation distance along the optical fibre for three different spectral widths of the source. The input signal was linear polarised (at zero degrees to the optical axis).

1.0 § 0.8 ■5 0.6 cu * 0

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0.4 0.2 0.0 0 200 400 600 800 1000 Propagation Distance (m)

Figure 7.5. Degree of polarisation as a function of the propagation distance along an optical fibre for different spectral line widths of the source, (a) 0^=1 nm, (b) 6X=3 nm and (c) 6X=10 nm.

In the limit, when the spectral line width of the source becomes zero, the degree of polarisation does not degrade upon propagation and stays one. However, for all finite linewidths the degree of polarisation decreases to a constant value upon propagation along the optical fibre (if the line width is very small this might take several km). So for every real physical source the degree of polarisation will degrade upon propagation, as all physical sources have a finite line width.

The degree of polarisation shown in the graphs in Sections 7.2.4 and 7.2.6 (partial coherent cases) are not determined by analytical methods, but by a numerical approach, as it did not seem obvious how to solve the integration of the elements of the Mueller matrix or Stokes vector in the frequency domain. For all the cases of the single matrix method and a partially coherent source the simulated graphs are the result of numerical integration (lossless optical fibre in Section 7.2.4) or numerical

summation (lossy optical fibre in Section 7.2.6) of the elements of the Stokes vector in the frequency domain.

To determine the graphs in Fig. 7.5 it was assumed that the coupling coefficient was frequency independent (C=15) and the differential propagation constant is given by

5p(v) = Px-Py+5xp(v-Vo), (7.2.10)

where - Py =3, 6xp = 10’^^ and Vo=474 THz (i.e. 632.8 nm). It is just stated here that the effect of changes in ôxp are equal to changes in 5v as only the product of these two entities is important in Eq. (7.2.10).

The effect of changes in the coupling coefficient are shown in Fig. 7.6, where the spectral width is taken as 5v=3 nm and the differential propagation constant is frequency dependent and determined by Eq. (7.2.10).

1 . 0 § 0.8 - 0.4 - 0.2 - 0 . 0 2 0 0 400 600 800 0 1000 Propagation Distance (m)

Figure 7.6. Degree of polarisation as a function of the propagation distance along an optical fibre for different coupling coefficients, (a) C = 1 , (b) C=5 and (c) C=15.

From Fig. 7.6 it is clear that changing the coupling coefficient has two distinct effects. Firstly a decrease in the coupling coefficient makes the final degree of polarisation (for long fibre lengths) increase and secondly for decreasing C this constant degree of polarisation is reached more rapidly.

Certain features of the graphs in Fig. 7.6 are probably caused by the numerical approach used to determine the frequency integration for the elements of the Mueller

matrix. One such example is the dip in graph (a) near a propagation distance of a 1000 metres. Using an ordinary summation approach the results showed no such dip, but for consistency only the integral approach is used for the graphs in section 7.2.4.

Finally changes in the differential propagation constant are shown in Fig. 7.7, where again the spectral width is taken as 6v=3 nm and the coupling coefficient is C=5. For the results in Fig. 7.7 only the value of - py in Eq. (7.2.10) is changed, as changes in are already discussed (and equal changes in the spectral width 6v).

1.0 ® 0 . 8 -

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*©

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o 0.4 - 0.2 - 0 . 0 1000 0 2 0 0 400 600 800 Propagation Distance (m)

Figure 7.7. Degree of polarisation as a function of the propagation distance along an optical fibre for various differential propagation constants, (a) - Py=3, (b) p^ - py= 6 and (c) p, - Py=12.

As in Fig. 7.6 for changes in the coupling coefficient, it is clear from Fig. 7.7 that the degree of polarisation is changing with changes in the differential propagation constant. However, an increase in P^ - py causes the final degree of polarisation level to increase as is shown in Fig. 7.7 and the level is reached for the same fibre length.

From Figs. 7.6 and 7.7 it is clear that both the coupling coefficient (C) and the differential propagation constant (P% - Py) determine the constant degree of polarisation level for long optical fibres. The constant level is not determined by analytical methods. However, a phenomenological approach to the simulated results show that the constant degree of polarisation level is given by

DoIL =

Px-Py) + 4 C

Eq. (7.2.11 ) shows that the lower level of the degree of polarisation (DoIL) does not depend on the spectral width of the source (6v), but only on material constants of the optical fibre.

In this section it is shown that for a lossless optical fibre the degree of polarisation can be decreased depending on the values of the coupling coefficient and the differential propagation constant. The spectral width of the source determines how rapid the final degree of polarisation (DoFL) is reached, but does not influence the actual level. However, a finite spectral width is necessary to make it possible for the degree of polarisation to decrease (if the signal consists of one spectral component the degree of polarisation does not change upon propagation as was discussed in Section 7.2.3). This shows that in the case of a lossless optical fibre it is possible for the degree of polarisation to decrease upon propagation along the optical fibre but not to increase its value.

7.2.5 Single M atrix Method: Lossy Coherent Case

For the lossy single matrix method the degree of polarisation is determined by similar methods as for the lossless optical fibre.

However, this lossy single matrix approach has one more variable than the lossless single matrix, i.e. a% -ay; the differential loss between the two orthogonal polarisations. This differential loss between the orthogonal polarisations should introduce the possibility to increase the degree of polarisation as a signal propagates along an optical fibre. In a range of graphs it is shown how the degree of polarisation changes upon propagation depending on the different variables, which are the input degree of polarisation, the coupling coefficient, the differential propagation constant and the differential loss. For all cases in this section the source is assumed to be coherent (which should give comparable results to the experimental measurements done with a coherent source, i.e. a HeNe laser). The lossy partial coherent case is discussed in Section 7.2.6, where different spectral widths of the input signal are taken into account.

The Mueller matrix describing a lossy optical fibre and a coherent (monochromatic) signal is given by Eq. (7.1.30). Again the degree of polarisation is determined by the elements of the Stokes vector, which are in turn determined by the input Stokes vector and the Mueller matrix of the lossy optical fibre (see discussion about Stokes/Mueller calculus in Section 5.1.3).

Using the theory developed in Section 7.1.4, simulated results for a lossy optical fibre and a coherent source can be found in Fig. 7.8, where the input degree of polarisation is taken as 0, 0.5 and 1. The other variables of the elements of the

Mueller matrix are taken as C=5, - py =3 and - tty =0.05. 1 . 0 ® 0.8

i

1 0.6 0.4 0.2 0.0 0 200 400 600 800 1000 Propagation Distance (m)

Figure 7.8. Degree of polarisation as a function of the propagation distance along a lossy optical fibre for different input degrees of polarisation, (a) DoP=1.0, (b) DoI^O.5 and (c) DoP=0.

In Fig. 7.8 the input signal is linear polarised at zero degrees to the optical axis and the graphs show that independent of the input degree of polarisation the output degree of polarisation for long propagation lengths is equal to one. It should be mentioned that these results are similar to the simulated results from the mode coupling centre model (see Section 7.2.2), so both models result in similar simulated results.

It remains to be seen how the degree of polarisation depends on the other variables, which are, respectively the coupling coefficient (C), the differential propagation constant - py) and the differential loss (a^ -ay ).

Simulated results show that the degree of polarisation does depend on all three variables in a particular way, but a phenomenological approach shows that all three variables are inter-linked and the simulated results depend on a combination of C, P^ - py and a^ - Œy. This particular combination will be called the form factor Y, which is given by

( p .- p ,)(« .-« ,)■

(7.2.12)

The form factor Y is an indication for the general form of the graph, so for equal values of 'F the graphs have similar forms. However, changes in the separate

variables do produce changes in the graphs (even if Y is kept constant), which is discussed at a later stage.

In Fig. 7.9 it is shown how the degree of polarisation changes along a lossy optical fibre depending on Y, where P, - py=3 and - tty =0.05 for all the graphs in Fig. 7.9 and the coupling coefficient is taken as C=l, C=5 and C=15 for graphs (a), (b) and (c) respectively. The input degree of polarisation is zero (i.e. random polarisation). 1.0 - g - 0.8 -

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0.6 - *o

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0 .4 - 0.2 0.0 0 200 400 600 800 1000 Propagation Distance (m)

Figure 7.9. Degree of polarisation as a function of the propagation distance along a lossy optical fibre for different form factors, (a) Y=6.67, (b) 'F=33.33 and (c) Y=100.

From Fig. 7.9 and Eq. (7.2.12) it is clear that if the coupling coefficient is increased the degree of polarisation along the fibre takes a longer propagation distance to reach the final value of one. For both the differential propagation constant and the differential loss it is the other way round, if these are increased perfect polarisation (i.e. a degree of polarisation equal to one) is reached for shorter propagation distances.

However, to show that Y is only an indication for the general form of the simulated graphs additional graphs are shown in Fig. 7.10, where the same values for Y are used. But for the results of Fig. 7.10 the following constants were used for all three graphs: -py=0.03, tty =0.05 and for the separate graphs the following coupling coefficients were used: (a) C=0.01, (b) C=0.05 and (c) C=0.15. For these values the form factor for the graphs in Figs. 7.9 and 7.10 are equal. Again it is assumed that the input signal is randomly polarised (i.e. the degree of

polarisation at the input of the optical fibre is equal to zero). 1.0 - 0.8 •§ 0.6 Jj 0.4

f

® 0.2 0.0 200 0 400 600 800 1000 Propagation Distance (m)

Figure 7.10. Degree of polarisation as a function of the propagation distance along a lossy optical fibre for different form factors, (a) Y=6.67, (b) Y=33.33 and (c) Y=100.

In Fig. 7.10 it can be seen that the graphs do not have to be as smooth as those in Fig. 7.9. But the general form for equal 'F is similar, so the form factor gives useful information about the degree of polarisation of a signal propagating along an optical fibre. As a matter of fact the smooth curves in Fig. 7.9 are the lower limit of the envelope of the curves in Fig. 7.10, so in Fig. 7.10 the sinusoidal oscillation of the graphs "sits" on top of the smooth graphs of Fig. 7.9.

In document Coherence and polarisation phenomena in optical systems and fibres for signal processing (Page 99-106)