Spatial transient

Radiation from fibers illuminated by a diffuse source 157 

8-5 Spatial transient for clad parabolic-profile fibers 163 

8-8 Example Diffuse illumination of step-profile fibers 167 

8-9 Example Diffuse illumination of clad parabolic-profile fibers 168 

8-11 Example Ray dispersion on clad parabolic-profile fibers 171 

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Limiting Nutrients For most inland waters phosphorus is the limiting nutrient in determining productivity. In some estuaries and in many marine coastal waters, nitrogen appears to be more limiting to algal growth than phosphoms. Deficiency in trace elements occurs usually only as a temporal or spatial transient. 

We begin by determining the power exciting leaky rays on fibers illuminated by a diffuse source, and then show that the spatial transient is accurately and simply described in terms of a single dimensionless parameter which embraces all the physical quantities of the problem. The effect of leaky rays on pulse dispersion is also expressible in terms of this parameter. We then show how... 

The main objective in the first eleven sections is to determine an effective length, or duration, of the spatial transient, as well as pulse shape. Because the conclusions are both simple and important, we summarize them here for ready reference and also to provide perspective for their derivation. 

In Sections 4-21 and 4-22, we showed that the shape of the impulse response on step and clad parabolic profiles is virtually rectangular. This conclusion is valid only in the spatial steady state. In the spatial transient, the power in tunneling rays manifests itself by adding a tail to the pulse. The power in the tail is large close to the source but becomes negligible at the onset of the spatial steady state [5]. 

The duration Zq of the spatial transient depends on wavelength, or, equivalently, on the fiber parameter V, since the attenuation of tunneling rays is a wavelength-dependent phenomenon. Taking zq to be the position where tunneling ray power has decreased by 90 %, we shall show in Section 8-8 that [6,7] ... 

We can, therefore, ignore their presence when determining the spatial transient at distances along the fiber well away from the source. A quantitative analysis, based on a similar approach to that of the present section, fully justifies their omission [8]. 

The extent of the spatial transient depends on the variation of total ray power along the fiber. We ignore refracting rays, for reasons given above, and define P (z) to be the sum of bound and tunneling ray power at distance z. Bound-ray power is conserved along a nonabsorbing fiber, consequently... 

The ratio P (z)/P (0) determines the radiation loss and the duration of the spatial transient. At distance z, the total tunneling ray power is obtained by integrating the product of the attenuation factor and the distribution function of Eq. (8-2a) over the range of invariants in Eq. (8-2b). Thus... 

The spatial transient can extend over considerable distances for larger values of V before approaching this limit. In other words, bound rays alone are insufficient to describe propagation, even on long fibers, as V - oo. 

We can now quantify the duration of the spatial transient, as discussed in Section 8-1. If we assume that the spatial steady state begins when the tunneling-ray power has fallen to 10% of its initial value, then Fig. 8-S(a) shows that this corresponds to G = 0.5. We can then relate G to the duration of the spatial transient through Eq. (8-17), and deduce that ... 

We use the generalized parameter of Eq. (8-21) to repeat the spatial transient calculation of Section 8-5. The development is similar to that of Section 8-8. The values of p and The in the ranges given by Eq. (8-24aX except that ()S) is given by Eq. 

Thus the spatial transient is a factor of2n longer on the clad parabolic prefile than on the step-profile fiber as anticipated in Eq. (8-lb). 

In the present and previous sections we have demonstrated the considerable simplification which results from using generalized parameters to describe the spatial transient when a fiber is illuminated by a diffuse source. A similar simplification occurs when the generalized parameters are used to describe the spatial transient due to collimated-beam excitation [11]. 

The spatial transient on noncircular libers can be determined by summing the power of leaky rays at each position z, but, in general, the noncircular cross-section makes the analysis very complicated. The paths no longer have the fixed periodicity of the axisymmetric fiber, so that, in particular, the ray halfperiod Zp will vary from reflection to reflection. In Chapter 2 we showed that a... 

When the asymmetry is slight, it is sometimes possible to simplify the above analysis by treating the noncircular fiber as a small perturbation of a circular fiber. Thus, for example, the ray invariant I of the circular fiber can be replaced by an approximate invariant l(z) which varies very slowly along the noncircular fiber. The spatial transient on the elliptical, clad parabolic-profile fiber can be determined within this approximation. Details are given elsewhere [13]. 

Snyder, A. W. and Pask, C. (1975) Optical fibre spatial transient and steady state. Opt. Commun., 15, 314-16. 

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