Pulse spreading

In Chapters 1 and 2 we introduced the notion of ray transit time. The main contribution to pulse spreading is due to the obvious fact that the ray transit time is different for different ray paths. This effect is known as ray dispersion, and is sometimes referred to as intermodal dispersion, since early investigation used electromagnetic analysis in terms of modes [1], rather than ray theory. In addition to ray dispersion, material dispersion also affects pulse spreading. This effect arises because the materials constituting the fiber have a refractive index which varies with the wavelength of light. 

The main purpose of this chapter is to show that pulse spreading depends on the refractive-index profile, and to demonstrate how it can be minimized by a suitable choice of profile. We first examine planar waveguides, for which it is possible to achieve zero pulse spread, and then circular and nondrcular fibers. 

Initially we assume that the waveguide is composed of materials which are nondispersive. We subsequently determine the modification due to material dispersion. Only bound rays are included in the present chapter, as they characterize the transmission properties of long fibers. The major conclusion of this chapter is that pulse spreading on fibers is minimized if the profile has an approximately parabolic profile. 

Throughout the chapter, we assume that the pulse originates at time t = 0 at the endface z = 0 of a waveguide of arbitrary profile and length z as shown in Fig. 3-1. The pulse is assumed to be composed of bound rays only, all of which are excited simultaneously at z = 0. For convenience, the initial pulse spread is assumed to be zero. In practice, sources excite pulses of finite duration the spread along the fiber is then found by superposition. 

The simplest structures for calculating dispersion are the planar waveguides of Chapter 1. We start with the step profile and progress to graded profiles. 

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The second term corresponds to a simple delay and can be generated by air propagation. The third and higher terms are specific of propagation through material. They induce pulse spreading for optical telecommunications and degradation of the interferences contrast in the frame of interferometry. 

The next thing we note about chromatography is that it is equivalent to tracer injection into a PFTR. Whereas in Chapter 8 we used tracer injection to determine the residence time distribution in a reactor, here we have nearly plug flow (with the pulse spread somewhat by dispersion), but adsorption from the fluid phase onto the solid reduces the flow velocity and increases the residence time to be much longer than x. ... 

Modal Dispersion—Pulse spreading due to multiple light rays traveling different distances and speeds through an optical fiber. 

The diffusion coefficient of isopropanol vapor in helium at 150°C is 0.677 cm2/s. If a narrow pulse of isopropanol can be approximated by a 5-function at time t = 0, to what effective zone width 4 a will this pulse spread by diffusion in (a) one second (b) one minute (c) one day ... 

The pulse spreading has been characterized by an axial diffusion coefficient and an axial Peclet number ... 

Even with completely monochromatic light, pulse spreading can still occur, because the radiation can take various paths, or modes, through the fibre, as sketched in Figure 14.31. It is apparent that a ray that travels along the axis of a fibre will travel less than one that is continually reflected on its journey. [In fact, the dispersion that results cannot be properly understood in terms of the transmission of light rays, and the various modes are better described in terms of the allowed wave patterns that can travel down the fibre.] The resultant pulse broadening, due to the various modes present, is called modal (or intermodal) dispersion. In order to overcome modal dispersion a number of different fibre types have evolved. 

An instantaneous pulse of wavelength 1000 60 pm is introduced into a silica optical fibre. What will be the pulse spread in km after 1 second The refractive indices of silica are as follows n(400 nm) = 1.47000 n(average) = 1.46265 (700nm) = 1.45530. 

Transient thermography, also known as pulse video thermography, uses a heat source such as a xenon flash or a pulse laser to induce a temperature differential within a material. The technique can reveal flaws at depth because the heat pulse spreads out as it penetrates into the specimen, so that late frames are affected by breaks in diffusion paths at great depth [75], With conventional thermography, defects at only a few microns below the surface of the specimen are detectable, and different heat cmissivity. due to the presence of dissimilar materials such as composite and metal, causes difficulties in defect detection and masking of defects due to the high emi.ssivity of metal. A solution to overcome this problem is to cover the material tested with a black coating,... 

Besides the fact that the medium resists the passage of a pressure wave (impedance), an elastic medium possesses another complicating characteristic. A purely sinusoidal pressure wave travels with a characteristic velocity in a medium, i.e., the phase velocity c (= f A). When any simple wave at any frequency travels through a medium at the same phase velocity, this medium is said to be non-dispersive. Actually, a medium is more or less dispersive so that when a multifrequency wave pulse travels in such a medium, the pulse spreads out. 

As the parabohc index profile is not optimum for PMMA-based GI POFs, random mode couplings can affect pulse spreading owing to group delay averaging. 

However, in practical systems, the signal is degraded by various types of noise, interference from adjacent pulses, and the finite extinction ratio ofthe light source. If an optical transmission system has no optical amplifier, thermal and shot noises are the dominant factors in the receiver [6]. Thermal noise is independent ofthe incoming optical power in contrast, shot noise depends on the received optical power. A critical error source is intersymbol interference (ISI), which arises from pulse spreading due to dispersion [7]. The pulse broadening causes the received signal to spread into the adjacent bit period. Thus, errors in the bit decision occur. 

Finally we note that the description of pulse spreading given by Eq. (3-2) is the simplest of a number of possibilities, which ipdqdes the r.m.s. width of the impulse response described in Section 4-20. 

It is possible to reduce pulse spreading from its value for the step profile by grading the core. We recall from Eq. (1-13) that the velocity along a ray is... 

The clad power-law profiles for planar waveguides are defined by Eq. (1-59) and illustrated in Fig. 1-10. Here we determine which of these profiles, i.e. the value of q, gives minimal pulse spreading. Unlike the step-profile waveguide, it is not immediately obvious from the geometry of the sinusoidal-like ray paths... 

Pulse spreading in fibers is investigated in exactly the same manner as for planar waveguides. The added complication is that we must include skew rays... 

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