Leaky rays

So far in our ray description of light propagation we have concentrated mainly on bound rays, which convey power without loss along nonabsorbing optical waveguides, and have assumed that refracting and tunneling rays are associated with a loss of power by radiation. In this chapter we describe the  

We recall from Chapters 1 and 2 that Snell s laws and the ray-path equation tell us that certain rays within the core of a waveguide will undergo refraction at the core-cladding interface. These are the reacting rays. The ray equation further tells us that other rays within the core of a fiber have an associated path in the cladding which extends indefinitely from some finite radial position beyond the core-cladding interface. These are the tunneling rays [1, 2]. In  

In Chapters 1 and 2 we identified refracting rays within the cores of planar waveguides and circular fibers. The important feature of such rays is the bifurcation of the path at each reflection from the core-cladding interface. Here we determine the effect of this on power flow along the path. We consider the simplest example first to emphasize the concepts involved. 

Thus a fraction of T of ray power is lost at each reflection. The ray power attenuation coefficient y is found either by averaging T over a ray half-period Zp, between successive reflections, or by summing the loss at the N reflections in unit length of the fiber. Either way we have [7] 

The transmission coefficient T is found by using the local plane-wave description of a ray. We regard the local plane wave as part of an infinite plane-wave incident on a planar interface between unbounded media, whose refractive indices coincide with the core and cladding indices and of the waveguide, as shown in Fig. l-3(b). For the step interface, Tis identical to the Fresnel transmission coefficient for plane-wave reflection at a planar dielectric interface [6]. In the weak-guidance approximation, when s n, the transmission coefficient is independent of polarization, and is derived in Section 35-6. From Eq. (35-20) we have [7] 

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Snyder, A. W. (1974) Leaky-ray theory of optical waveguides of circular cross-section. Appl. Phys., 4, 273-98. 

We can now calculate the source power carried by bound rays when the fiber is illuminated by the diffuse source. In this part of the chapter we determine the total source power, the total bound-ray power and the radial distribution of bound-ray power over the core cross-section. Later in the chapter we show how to derive the distribution of power among the various bound-ray directions. We assume that the source of Fig. 4-3(a) is placed against the fiber endface in Fig. 4—4, and its surface covers at least the core cross-section. Only the portion of the source within the core cross-section can excite bound rays, so we ignore any effects due to the source outside of this region. The excitation of leaky rays by sources is examined in Chapter 8. 

Only part of the total source power is transmitted to bound rays the rest excites leaky rays, which is discussed in Chapter 8. If we denote the amount of source power transmitted to all bound rays by Pj, then the source (Efficiency S is defined by... 

Outside of the circular region only leaky rays are excited, as discussed in Chapter 8. When 00 = 0 and the beam is on axis, bound rays are excited over the whole core and propagate parallel to the fiber axis. If the beam carries uniform power P per unit cross-sectional area, then the total bound-ray power and source efficiency of Eq. (4-11) are given by... 

Pask, C. and Snyder, A. W., (1974) Illumination of multimode optical fibres-leaky ray analysis. Opto-electronics, 6, 297-304. 

On a uniform fiber, a ray which is bound at the beginning of the fiber remains bound along the fiber. However, on a nonuniform fiber, a ray which is initially bound may become a leaky ray over part of its trajectory and lose power by radiation, if the variation in the parameter /S(z) of Eq. (5-2) is sufficiently large. Otherwise it will remain a bound ray over the length of the fiber even though (z) varies. Here we give a simple upper bound for the radiation loss. 

Fig. 5-4 Step- and graded-profile fibers whose core radius and profile vary along their length, showing (a), (b) a ray which remains bound, and (c), (d) a bound ray which becomes a leaky ray along part of its path.

Consider a step-profile fiber with core and cladding indices and n i, when the core radius p(z) is slowly varying. The angle 9 (z) a ray makes with the axial direction at position z is related to the initial angle 0 (0) by Eq. (5-44) with n (z) = n (0) = n. An initially bound ray may become a leaky ray only if the core radius decreases below its initial value p(0). If the smallest radius anywhere along the fiber is and is the corresponding position, the ray direction at z is given by... 

The total scattering cross-section is by definition independent of 0 and 0 and we have assumed independent of position for simplicity. Second, there is a gain from power scattered into direction 0, 0. Ignoring contributions from leaky rays, the total contribution is from bound rays, whose range of 0 values satisfies Eq. (2-6a). Thus the power gained, dPg, is found by integrating Eq. (5-64) over the cross-section... 

The range of values of invariants for leaky rays is given by Eq. (2-8). However, if we recall the ranges of ray angles for bound, refracting and tunneling rays in... 

For situations where the above assumption cannot be adopted, expressions for the transmission coefficient in the transition region between tunneling and refracting rays are available [4,8]. The values of Tare plotted as curve (i) in Fig. 7-2(b) for a skew leaky ray with I = 0.033 on a clad parabolic fiber. To the left of the vertical dashed line, the curve corresponds to tunneling rays and coincides with the local plane-wave expression of Eq. (7-18) as increases [8]. Similarly, to the r ght of the vertical dashed line, the curve corresponds to refracting rays Ind coincides with the local plane-wave expression of Eq. (7-6) as decreases. A similar transition occurs for skew leaky rays on a step-profile fiber [16]. 

Following the above discussion, the loss of leaky-ray power on noncircular and nonuniform fibers is described by local transmission coefficients for... 

When a leaky ray propagates along a fiber that is absorbing, it is intuitive that the total attenuation is simply the sum of the power attenuation coefficients for... 

Snyder, A. W. and Love, J. D. (1976) Attenuation coefficient for tunnelling leaky rays in graded fibers. Electron. Lett., 12, 324-6. 

Love, J. D. and Winkler, C. (1978) Refracting leaky rays in graded-index fibers. Appl. Opt., 17, 2205-8. 

Adams, M. J., Payne, D. N. and Sladen, F. M. E. (1975) Leaky rays on optical fibers of arbitrary (circularly symmetric) index profiles. Electron. Lett., 11, 238-40. 

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