### OF AN OPTICAL FIBRE:

### I. THEORY

### 4.1

### INTRODUCTION

**The excitation of an optical fibre requires no more than the**
**directing of light onto the end face of the fibre core such that at**
**least some of it remains trapped within the core.**

**One simple way to do this is to image an incandescent or high**
**pressure discharge lamp onto the fibre end with a lens system. ** **Usually**
**the light is spectrally filtered, to reduce wavelength dependent effects,**
**and the image space numerical aperture of the lens is chosen to be at**
**least as big as that of the fibre, to maximise the light transmitted by**
**the fibre. ** **Such excitation is commonly labelled "incoherent", regard**
**less of the properties of the primary source, and is assumed to excite**
**all modes of the fibre equally. ** **Even when a laser beam is used in this**
**way the assumption of equal excitation of all modes is usually made, as**
**long as the numerical aperture of the fibre is filled with the incident**
**light, which is now clearly not incoherent (see e.g. [1]).**

**A common alternative is to direct a laser beam onto the end of**
**the fibre, or pass it through an optical system including a spatial**
**filter and so on to the fibre. ** **The axis of the incident beam may be**
**inclined to the fibre axis, and the object is to create a field on the**
**end of the fibre that closely matches a particular mode field (see e.g.**

**[2]). ** **This excitation condition is naturally referred to as "coherent",**
**and results in the launching of a specific mode, or mode type [3-5] .**

**These techniques are commonly used to investigate the**

**in experimental communications systems. ** **It is more common now, however,**
**to use either the light emitting diode (LED) or the semiconductor diode**
**laser for a source in a practical communications system [8]. ** **These**
**devices are discussed in Chapter VI.**

**Generally, LED excitation is classed as incoherent and diode**
**laser excitation as coherent (see e.g. [8]). ** **While the LED derives its**
**radiation from a random process, the recombination of carrier-hole pairs,**
**and does have a more or less isotropic radiation pattern [9], it does**
**not otherwise resemble the classical incoherent sources, the thermal or**
**black-body radiators. ** **The diode lasers "possess few of the attributes**

**normally associated with lasers" as Gooch [10] has said, albeit of the**
**earlier types. ** **However, the problems inherent in these have largely**
**been overcome [8], particularly in the stripe geometry lasers. ** **These**
**points are more fully discussed in Chapter VI.**

**In any case, the practical excitation conditions do not much**
**resemble the idealised totally incoherent or totally coherent cases.**
**These approximations, and the very great difference in the results of**
**the two extreme types of excitation, suggest the need for a general**
**analysis that takes into account the actual coherence properties of the**
**source. Given sufficient information about an actual excitation system,**
**such an analysis should be able to evaluate the modal power distribution,**
**leading to the correct interpretation of experimental results and**

**allowing an assessment of the validity of the totally incoherent or**
**totally coherent approximations.**

**The analyses of partially coherent excitation presented here**
**and in the next chapter deal with the excitation of an optical fibre by**
**some arbitrary partially coherent field in the plane of the entrance**
**face of the fibre [11,12,13]. ** **The details of this field may be derived**
**from a knowledge of the properties of the physical source and**

**excitation system geometry. ** **Hence, without any loss of generality, the**
**source referred to in the analyses is this field, with no reference**
**being made to any specific physical source. ** **In this context, partial**

**coherence refers to the spatial coherence of this field (cf. Chapter II),**
**and it is only the effects due to differing degrees of spatial coherence**
**that are investigated.**

### 4.2

**77**

**In the next section, the mechanism of excitation of an optical**
**fibre is discussed in terms of both electromagnetic mode theory and**
**geometric optics. ** **Previous analyses of totally coherent and totally**
**incoherent excitation are briefly discussed.**

**The excitation by a polychromatic partially coherent field is**
**considered in section 3 using electromagnetic mode theory, with the**
**coherent, incoherent and quasimonochromatic approximations being derived**
**from the general result. The quasimonochromatic formalism is separately**
**developed in section 4, based on the previous analyses of coherent and**

**incoherent excitation [3,14,15] and shown to coincide with the form**
**derived from the general result.**

**In the next chapter, the quasimonochromatic formalism is**
**applied to a step index fibre and used to obtain some numerical results**
**and asymptotic expressions for the bound mode power. ** **A geometric optics**
**approach is also used to obtain results for both step and graded index**
**fibres, for both bound and leaky ray powers.**

**4.2 THE MECHANISM OF EXCITATION **

**(a) ** **Electromagnetic Mode Theory**

**The launching and propagation of electromagnetic energy on an**
**optical fibre may be analysed by considering the normal modes of a**

**circular cross-section, semi-infinite, dielectric rod or waveguide. ** **The**
**subject of dielectric waveguide theory lies outside the scope of this**
**thesis and has been dealt with competently elsewhere (see e.g. [16-20]).**
**Only those aspects of particular relevance are discussed here, and only**
**approximate forms appropriate to the circumstances are considered.**

**The normal modes of a dielectric waveguide form a complete**
**set, composed of the discrete spectrum of bound modes representing**
**trapped or guided power, and a continuum of unbound modes representing**
**the radiation field. ** **The mode fields are solutions of the source-free**
**Maxwell equations, satisfy the boundary conditions of the structure and**
**conform to the radiation condition at infinity. ** **Inside and in the**
**immediate vicinity of the fibre core the radiation field may be**

accurately represented by leaky inodes, which are bound inodes below c ut

off, .i.e. they have complex eigenvalues (see e.g. [ 2 1 J) .

In this study, it is only the guided power that is of interest,

and so only the bound modes are considered. These are mutually

orthogonal, have well defined phase and group velocities, cross- sectional intensity distributions and polarisation, and propagate

unchanged along an ideal loss-less fibre. They are guided by the core,

even though their fields may extend appreciably beyond the core-cladding interface, decaying exponentially in the radial direction.

As the modes form a complete set, any guided field propagating along the fibre may be expanded as a sum over the bound modes, i.e. assuming the fibre axis to lie along the z-axis, and the fibre to extend over 0 ^ z ^ oo (and so only forward propagating modes are included) ,

i (o)t - 3 z)

E (r) = 2 a e (r) e