Cointegración en la metodología de Engle-Granger

To determine the effect of a partial coherent source on the propagation of the degree of polarisation a different source is used. Instead of a coherent laser an incoherent arc lamp is used and the spectral width of the resulting source is determined by the linewidth of an appropriate interference filter. In this case the interference filters had a centre wavelength of 632.8 nm and the linewidth for three different filters was respectively approximately 1, 3 and 10 nm. To determine the propagation of the degree of polarisation along an optical fibre a Stokes vector approach can be used as was discussed in Section 8.1.1. Such a set up is shown in Fig. 8.12 (path A).

Power Supply ccd camera Video Analyser 33x Monitor TLS Arc Lamp 63x Optical Fibre Power Meter Detector

Figure 8.12. Set up to measure the elements of the Stokes vector (Path A) and to determine the mode patterns propagated by the optical fibre in the "few-mode" case (Path B), where P=polariser, IF=interference filter (with a centre wavelength of 632.8 nm and a linewidth o f 1.2,3.0 or 11.3 nm), TLS=telecentric lens system, PH=pinhole and X/4=quarter-wavelength plate.

The other part of the set up shown in Fig. 8.12 (path B) was used to determine the mode patterns that propagated in the optical fibre for this few-mode cases (see Section 8.3.2). In this section this path B of the set up does not play any role.

The Xenon arc lamp (Osram, XBO 150 W/S) with the lamp housing (Speirs Robertson, LH 150) and the power supply (Spectral Energy, LPS 251 SR) was used to provide a broadband source. A variable finite linewidth was accomplished by using

three different interference filters with a centre frequency of roughly 632.8 nm and with respectively linewidths of 1.2 nm (Ealing, 35-8630), 3.0 nm (Melles Griot, 03 FIL 018) and 11.3 nm (Andover Corporation, 633FS10-25). The telecentric lens system and the microscope objective are used to couple the maximum amount of power into the optical fibre. At the output of the optical fibre the elements of the Stokes vector are determined by the method discussed in Section 8.1.1 (a pinhole could be used to select part of the modal pattern as was done for the Michelson approach).

The degree of polarisation for zero propagation length is almost equal to one for all three interference filters because of the polariser (Spindler & Hoyer, 06 3410) in between the source and the optical fibre. The propagation of the degree of polarisation was determined by cutting down the optical fibre from 8.031 m down to 0.299 m. In addition to the finite spectral linewidth measurements the propagation of the degree of polarisation was also measured when the input signal was a white light source (i.e. no interference filter was used). However the degree of polarisation for this white light source at the input of the fibre was not equal to one but close to the value of 0.2, which can be explained by the frequency dependence of the polariser (only perfect near the centre frequency).

The results of the measurement of the propagation of the degree of polarisation for the few-mode partial coherent case with the set up shown in Fig. 8.12 can be found in Figs. 8.13a,b for respectively a linewidth of 3.0 nm and white light.

1.0- 9 •2 0.8 - i

I

Figure 8.13a. Degree of polarisation as a function of the propagation distance along an optical fibre for the few-mode partial coherent case when the linewidth is 3.0 nm.

c 1.0- .2 _{0 .8 -} • i Ô 0 .6 - Q. *o 0 .4 - Of Q _{0 2 4} 0.0 4 0 □ □ □ □ — " ' I '— “t — ^ .r ---1--- 1 2 4 6 8 10 Propagation Distance (m)

Figure 8.13b. Degree of polarisation as a function of the propagation distance along an optical fibre for the few-mode white light case.

The results of only the 3.0 nm interference filter are shown because the measurements showed that it was difficult to distinguish between the degree of polarisation resulting from the three different interference filters, but that a distinct difference occurred between white light and the finite linewidth cases. As the measurements with the three different filters showed similar the propagation of the degree of polarisation is only shown for the interference filter with a 3.0 nm linewidth. However, it is clear from Figs. 8.13 that there is a distinct difference between a signal with a linewidth of 3.0 nm and white light. The average value of the degree of polarisation for these cases are respectively 0.318 and 0.056.

The results in Figs. 8.13 show some sort of oscillation for the propagation of the degree of polarisation. These oscillations occurred for all three interference filters and the white light case. The cause of these oscillations might be found to be due to the instability of the used arc lamp. But on the other hand it was shown in Chapter 7 that it was possible to get a certain kind of oscillating behaviour. The measured results shown in Fig. 8.13a are not a conformation of the simulated results given by Figs. 7.11 and 7.12, where a distinct difference occurred in the degree of polarisation for sources with a linewidth of respectively 1, 3 and 10 nm. In addition it was shown in Fig. 7.12 that every value for the degree of polarisation is uniquely determined by the linewidth of the source. As shown in Section 7.2.6 the results of Fig. 7.12 are not unique and changes in the variables showed that different results might be expected

(see Figs. 7.13 and 7.14). Without further knowledge of the exact values of the different variables it has not much use discussing the precise form of the expected propagation of the degree of polarisation. However, the experimental and simulated results (respectively Figs. 8.13 and 7.13) show that it is possible to get oscillatory behaviour for the propagation of the degree of polarisation and that it is not always possible to distinguish between different linewidths by measuring the degree of polarisation.

8.4 DISCUSSION

It has been shown in a range of experiments how the degree of polarisation behaves upon propagation along an optical fibre.

When a multimode optical fibre was used the measurements done showed how the fringe visibility changed upon propagation (see Fig. 8.6), but this would only have been the propagation of the degree of polarisation if the fringe visibility was maximised with respect to orientation. The second problem was caused by the difficulty to get fringes with the resulting speckle pattern from the multi-mode fibre.

The following step was t o use infra-red fibres and a HeNe laser (632.8 nm),

which made it possible to describe the few-mode optical fibre case. Final coherent results (Fig. 8.11) showed that the ultimate value of the degree of polarisation upon propagation did not depend on the degree of polarisation at the input of the optical fibre as was predicted in Section 7.2.5 (see Fig. 7.8).

The introduction of a source with a finite spectral linewidth showed results which seemed not to be a confirmation of the simulated results shown in Fig. 7.11. However, it was shown that for different variables the resulting graphs might be totally different (compare Figs. 7.11 and 7.13). So within limits these finite linewidth results are in agreement with the predicted theoretical results as long as the constant values are appropriately adjusted. Due to a lack of knowledge about the constant values determined by the fibre it is difficult to predict the behaviour of the propagation of the degree of polarisation as a function of the spectral linewidth of the source.

It is obvious that for the results shown in Fig. 7.12 the degree of polarisation is uniquely related to the spectral linewidth of the source, which would make it possible to use optical fibres to pick up a signal in a hostile environment and determine the spectral linewidth at the end of the fibre. However, as other simulated results (Fig. 7.14) and experimental results (Fig. 8.13) showed it might not be as simple as could be concluded from Fig. 7.12. But even for the results found in Fig. 8.13 it is possible to distinguish between a highly coherent source, a source with appreciable

linewidth and a white light source.

With the above discussion in mind it would be interesting to find out if there are optical fibres which behave as predicted by Fig. 7.12. The simulated results seem to suggest that such a fibre might be available if the coupling coefficient, the differential propagation constant and the differential loss have small enough values.

It is also possible that an average taken over an ensemble of optical fibres might give a clearer picture, if it is assumed that the propagation of the degree of polarisation is not a single value but is determined by a certain kind of distribution (as suggested in [100]). This could be one area of future research as very little seems to have been done in this part of the field. The averaging process could take place in either the time or space domain depending on the stability in these domains.

As should be obvious from Chapters 7 and 8, the theoretical and experimental work described in this thesis can be seen as contribution to an area of research with some work done in the past and possibilities for the future.

Part 3