When the polarisation of the electro-magnetic field is not stationary in the time domain, the effects can be serious for second order interference experiments as the Michelson interferometer.
If the polarisation of the electro-magnetic field is changing in time it is possible that a time delay in the Michelson interferometer causes correlation of two fields with different polarisation. This difference between the polarisation of the fields from both arms in general degrades the fringe visibility of the interference pattern. The field in each arm of the Michelson can be divided in two orthogonal (x and y) components. Upon combination of the fields from the two arms only the parallel components from both arms interfere (the transverse character of the electro-magnetic field forbids interference between orthogonal components). The resulting interference pattern is the sum of the separate x and y component interferograms, i.e.
ly =1, +Iy, (5.2.1)
where the x and y interferograms might be of the form given by Eq. (4.3.11). It is obvious that the significance of the changes depend on the time delay (i) and the spectral width of the field, as the latter is an indication of the stability of the polarisation (see Section 5.1.1). So, as with the coherence of an electro magnetic field the polarisation of the field is assumed stationary (in the time domain) over a finite time delay (x) which should be small compared to l / a ^ , where is the spectral width of the field.
However, if the polarisation is not stationary in the time domain the interference pattern might degrade. This can be explained by dividing the fields in each arm in orthogonal x and y components, which is shown in Fig. 5.1. For the resulting fringe pattern the position of the fringes is not important, so the phase difference between the X components E ' and E'' is set to zero and the relative phase delay between the y
components E' and E" is given by 5 = ô' - Ô".
Figure 5.1. Amplitude and phase relation of the orthogonal x and y components of two distinct electro-magnetic fields.
If the field is quasi-monochromatic the interference patterns in both x and y directions have a perfect fringe visibility, but the fringes in the y direction are shifted with respect to the x direction fringes. If the field is stationary (monochromatic) the phase difference Ô is zero and the resulting fiinge pattern has a visibility equal to one. But introducing a phase difference between the y components degrades the final summed fringe visibility. In Fig. 5.2a it is assumed that 6=tc/4 and the resulting interference pattern has an almost perfect fringe visibility. However, in Fig. 5.2b the value of the phase difference is 5=3ti/4 and the fringe visibility is severely degraded.
Time Delay (arbitrary units) Time Delay (arbitrary units)
Figure 5.2. Effect o f a iion-stationary polarised field on the fringe visibility of a second order coherence experiment, where the two graphs depict the interference pattern when the correlated polarisations are similar (a) and almost orthogonal (b).
the summed x and y interference patterns is equal to zero. In Fig. 5.2 it is assumed that the envelope of the fringe pattern is caused by the finite circular extent of the fields.
So electro-magnetic fields with non-stationary polarisations in the time domain can affect the interference pattern (and the fringe visibility) of second order correlation experiments (e.g. the Michelson interferometer).
5.2.2 Spatial Polarisation Effects and Young's Experim ent
It should come as no surprise that the polarisation effects in interference experiments also affect second order spatial coherence experiments. In this case the polarisation of the electro-magnetic field is non-stationary in the space domain [272-274].
The two electro magnetic fields E' and E" depicted in Fig. 5.1 are in Young's experiment respectively the fields at the two pinholes (see also Fig. 4.4). When the field that illuminates the two pinholes has a stationary polarisation and is monochromatic, the resulting interference pattern (the summation of the x and y components) is not degraded. However, if the polarisation is no longer stationary in the space domain the interference of the combined x and y orientations can be degraded compared to a perfect fringe visibility. Similar interference patterns as in Fig. 5.2 can result if it is assumed that the phase difference at the pinholes is respectively 0=tc/4 and 6=3 7t/ 4 for the graphs (a) and (b). Again, if the phase difference between the y components Ô is equal to 7C the resulting interference pattern has a fringe visibility equal to zero (i.e. orthogonal polarisations do not interfere, see also Sections 4.3.4 and 5.2.1).
In the time domain the polarisation of an electro-magnetic field is said to be stationary if the time delay is small compared to the inverse of the spectral width of the field. In the space domain the polarisation is called stationary when the relative distance between the two pinholes (i.e. (I^ -P2) / c , see Section 4.3.4) is small compared to l / a ^ .
These conditions on the stationarity of the polarisation of an electro-magnetic field are directly related to the quasi-monochromatic conditions (Eq. (4.2.17)) derived in Section 4.2.2. This is not surprising, realising that if the field is quasi- monochromatic, the degree of coherence is stable over a certain region of the space time domain. This guarantees that the field’s polarisation and the degree of polarisation (see Eq. (5.1.3)) are also stationary in this space-time region.
Throughout Section 5.2 it is assumed that the amplitude of the x and y components of the E' and E" fields are equal and stationary. Unequal /non-stationary amplitudes
Part 2
Changes
in
Polarisation
in
Optical Fibres
6
Introduction
For the past two decades there has been a renewed interest in scattering problems, where the term "scattering problems" is used in its broadest sense.
The term scattering is used in both the forward (from here on called propagation) and the backward (from here on called scattering) direction, but usually these are treated separately. Both propagation and scattering depend on the variations in the medium. Propagation effects are governed by slow moving variations which are much larger than the used wavelength. These effects may te random which might be caused by turbulence or changes in temperature. Scattering is caused by discontinuities in the medium which are not slowly moving variations and might be in the range of the wavelength of the source. Most of the time this scattering effect is caused by small particles or perturbations.
Special areas of interest include the changes in the polarisation and the degree of polarisation in these scattering and propagation phenomena. These aspects of the electro-magnetic field are briefly discussed in this introduction for a few different physical situations and links between these are mentioned. In the end, the purpose of this introduction is to show how changes in the polarisation and the degree of polarisation in optical fibres fit in the general picture of propagation and scattering in certain media. So no rigorous discussion about scattering/propagation events in different situations should be expected in this introduction.
Propagation effects in free space are described in a series of publications [74,175- 179,182,185,188,275,276]. One interesting aspect of this propagation is the fact that the measured intensity variance (aj) does not increase upon propagation, as was predicted from a theoretical first order approach [74,175-177,179,182], but becomes constant. This constant level of the intensity variance was explained in the late eighties by a higher order theoretical approach [185,188,275,282]. Without going into toomuch detail about this theory for propagation in free space it is mentioned that a statistical approach for the treatment of changes in the degree of polarisation upon propagation in optical fibres gives the intensity variance as the only variable in a very simple model (see Section 7.2.1). In free space propagation the changes in the degree of polarisation are caused by the number of scattering events within the coherence length of the electro-magnetic field. If only single scattering events (within the
coherence length) occur the degree of polarisation is not changed as long as all the perturbations are identical (it is still possible that the polarisation of the field is changing upon propagation). However, if the perturbations are not identical (different shaped particles) the resulting propagated field can have a degree of polarisation smaller than that of the input field. This depolarisation might also occur if there are more than one scattering events within the coherence length of the electro-magnetic field. The description of depolarisation caused by small particles is not restricted to free space propagation. Similar effects are described in a range of other experiments [139,165,193,195,202,277], where the particles might be suspended in a range of gases or liquids.
Instead of scattering or propagation by perturbations (i.e. particles) in a medium it is also possible to have similar effects at the surface of a medium [141,172,174,278- 282]. In this case the propagation direction can be defined by geometrical optics and scattering effects are mainly found outside this propagation direction. Scattering is caused by the roughness of the surface with respect to the wavelength. As with propagation and scattering in a medium, with surface scattering the degree of polarisation of the output electro-magnetic field is determined by the number of scattering events within the coherence length of the input electro-magnetic field. For single scattering events no depolarisation occurs in the plane of incidence when the surface is plain or the roughness is periodic. If the roughness is not periodic depolarisation might occur for the scattered field. Again, depolarisation is mainly explained when multiple scattering (within the coherence length of the field) is taken into account. This multiple scattering introduces coupling between all polarisation components of the incident and scattered^ropagated electro magnetic field.
Fibre optics is one special area of scattering/propagation research which has been studied extensively during the past decades [75,77,80-86,89-95,98-116,119,121- 124,283-288]. With optical fibres the effects are not restricted to changes in polarisation (or degree of polarisation); also changes in pulse-shape, spatial /temporal coherence and dispersion are important physical aspects. But in this part of the thesis a closer look is taken at specifically the polarisation effects in optical fibres.
With reference to the discussed examples, it should come as no surprise that for optical fibres there is also a regime where no changes in the degree of polarisation occur, but it is still possible that the polarisation of the electro-magnetic field is changing. As with the other scattering/propagation examples the second regime includes changes in the degree of polarisation. In optical fibres the scattering and propagation effects can be caused by both internal and external causes. Examples of internal causes are changes in the refractive index, which might be termed as slow or
as perturbations depending on their relative size with respect to the wavelength of the field. External causes might be changes in temperature or bending of the optical fibre. A short description of various theoretical approaches to predict these polarisation aspects are given below, where only linear aspects are discussed (non-linear effects in optical fibres is a rapidly expanding field of its own.
The first (and probably earliest) method is a phenomenological approach [91] which linked measured results to a series of equations which were not theoretically derived. The biggest disadvantage of this approach is the fact that there is no real physical background for the given equations. But general trends of this approach proved to be correct and were incorporated in more recent theoretical approaches.
A second approach uses a statistical description of the intensity distribution propagating in the different polarisation modes [83,92,102,112,116,285]. Either the variance of the intensity (amplitude) or the dispersion are determined and this information gives an indication about the resulting degree of polarisation for an electro-magnetic field as it propagates along an optical fibre. This approach shows similarities to scattering and propagation effects in free space, where also the variance of the intensity is an important entity. A further discussion of these statistical approaches is given in Chapter 7 (see Section 7.2.1).
More recently a polarisation mode coupling approach is used, where it is assumed that the coupling process has a random distribution [77,82,85,86,89,93,94,100, 104,105,107,108,110,113,115,121,288]. This mode coupling centre (mcc) model uses different Mueller matrices to describe the coupling process in the optical fibre which results in changes in the degree of polarisation. The name of this model refers to the distinct places in the optical fibre where mode coupling takes place. The mode coupling centres are separated by ordinary lengths of fibre in which the field is only propagating. Using this method results in expressions which determine how the degree of polarisation changes upon propagation in an optical fibre. The mcc model and related approaches based on discreet random coupling centres are more rigorous described in Sections 7.1.2 and 7.2.2.
A final approach uses one Mueller matrix to describe the coupling process and the changes in the degree of polarisation in an optical fibre. This single matrix (sm) model incorporates polarisation mode coupling, differential propagation constants, differential loss and spectral linewidth into a single Mueller matrix to describe how the degree of polarisation is changing upon propagation in an optical fibre. This method is explicitly developed in Sections 7.1.3, 7.1.4, 7.2.3, 7.2.4, 7.2.5 and 7.2.6.
A more rigorous theoretical description of changes in the polarisation and degree of polarisation is given in Chapter 7. In this chapter the mode coupling centre method is
reviewed and a full description of both the polarisation variance approach and the single matrix method is given. All theoretical models are accompanied by simulated results, which gives the opportunity to compare these different theoretical approaches.
Experimental methods and results are discussed in Chapter 8. A range of methods is described to measure the degree of polarisation. Some of these experimental methods are used to do experiments ranging from coherent multi-mode to partial coherent few-mode situations.