Guiding of light was first dem onstrated by Lord Rayleigh in 1895 [2.1] using w hite light in a je t of w ater. W aveguiding in cylindrical dielectric rods was studied by Snitzer in 1961 [2.2]. W ith the arrival of the laser, interest in optical waveguides increased, b u t losses were
still greater th an 1000 dB/km. In 1966 Kao and Hockham [2.3] suggested th a t dielectric waveguides for communication could be made w ith losses of less th an 20 dB/Km and this was achieved by w orkers a t Corning G lass W orks in 1970. C urrently, losses are around 0.2 dB/Km a t ISOOnm which approaches the Rayleigh scattering limit of the fibre m aterial.
The theory of weakly guiding optical fibres was discussed by Gloge in the early 1970's. Expressions for guided modes, their propagation constant and delay distortion were obtained [2.4,5,6]. Optical power flow is discussed in reference [2.7]. The theory of optical waveguides as given in this thesis follows the notation used by Gloge. O ther useful references for theory are by Marcuse [2.8], Adams [2.9], and Cherin [2.10].
Dispersion lim its the available bandw idth of an optical fibre and so the choice of source and fibre can be im portant if a high bandw idth is required. Time dispersion is discussed by Love [2.11] in term s of interm odal and chromatic dispersion and the effect of a broadband source on fibre bandwidth is discussed by Ohlhaber and Ulyasz [2.12]. Dispersion in singlemode fibres is wavelength dependant and has a
14
m inim um value a t about ISOOnm but, recently, fibres have been produced with the minimum dispersion shifted to 1550nm in order to take advantage of the lower attenuation.
The polarisation properties of optical fibres have become more im portant as a g reater num ber of system s are developed w ith singlemode fibre. These fibres propagate two orthogonal modes which normally do not preserve the polarisation state of the light as it propagates along the fibre. The preservation of polarisation in singlemode optical fibres is discussed by R ashleigh [2.13]. The polarisation optics of light in singlemode fibre depend on fibre im perfections such as bending or tw isting stress which introduce birefringence. The birefringence may be introduced in a controlled m anner in order to control the output polarisation by bending the fibre into a loop. A device described by Lefevre [2.14] consists of two coils of calculated radius which act as 1/4 wave plates to control the ellipticity of the polarisation and a single coil of different radius which acts as a 1/2 wave plate to control the orientation of the output polarisation. Using this device the output polarisation of two beams of a fibre in terfero m eter can be m ade p arallel for m axim um interference to occur.
I
In optical fibres, the light propagates in a finite num ber of modes. If
the losses in a fibre system are mode dependant, for example a t a % splice, th en unw anted am plitude m odulation of the tran sm itted
signal can occur. This in known as modal noise and is discussed by Epworth [2.15].
A review article by W hite [2.16] discusses methods of characterising optical fibres in term s of th eir losses, bandw idth, refractive index profile etc.
In order to be able to use optical fibres for pre-detector signal processing it is necessary to understand the effects th at waveguiding has upon the light. Firstly, as m any optical sources are broadband an im portant param eter for signal processing is the bandw idth over which a fibre propagates one mode. Secondly, it is im portant to know the tim e delay experienced by radiation in a given length of fibre. Finally, a monochromatic source will experience dispersion effects in a multimode fibre and a broadband source will be dispersed even in a singlemode fibre. In the following sections we will derive the basic equations used in fibre optics. The step-index fibre will be considered as the simplest case of an optical fibre. We shall show th at the full set of TE, HE, and EH modes may be approximated by a single se t called lin early polarised m odes. An expression for the approximate num ber of modes in an optical fibre is found. Group and phase velocity dispersion are considered.
2.1.1 The step-index optical fibre - rav model
For m any purposes a ray model is sufficient to derive basic param eters of an optical waveguide. Consider a fibre of spherical sym m etry w ith a core of diam eter 2a. The refractive indices of the core and cladding are n% and ng, respectively, w ith n% > ng, figure 2.1. The angle between the wall of the guide and the ray which ju st undergoes total internal reflection is defined as the critical angle not the angle between the ray and norm al to the surface as would be expected. The angle between the axis of the guide and an incident ray which ju st undergos total internal reflection a t the core-cladding
boundary defines the num erical aperture of the fibre. From Snell's law,we have
Sin (90-8 J = — . (2.1.1.1) Then, the critical angle is given by
n.
Cos 6^ = — . (2.1.1.2) ^2
To m eet this condition it is necessary for the light to enter the fibre w ithin a cone defined by the num erical aperture, sin Om» given by
/~ 2 2
Sin 9^ = ^ / n i - n , . (2.1,1.3)
The angle 0m defines the m aximum acceptance angle of the fibre and the sine of th a t angle defines the num erical aperture. We define a param eter A which is the relative difference betw een core and cladding refractive indices and is expressed
"2
A = l - - ~ . (2.1.1.4)
“i
By making the approximation 2 2
A= . (2.1.1.5)
we can relate the param eter to the critical angle and num erical aperture. By substitution we have
Sin 0^ = nj Sin 0^ = 2A . (2.1.1.6)
For most fibres 0.1 < Sin 0m < 0.2 which gives 5.7° < 0m < 11.5°. The ray model tells us about the acceptance angle of the fibre and the propagation of w avefronts through the fibre b u t does not give a complete picture of the individual modes w ithin a fibre. To learn more about the modes we need a wave theory.