Fibres pulse dispersion

The typical width of the time-of-flight distributions recorded in DOT is of the order of a few ns see Fig. 5.51 and Fig. 5.52, page 109. Therefore a detector IRF width of 150 to 300 ps is normally sufficient. Even longer detector IRFs are sometimes tolerated, especially if the pulse dispersion in long fibre bundles dominates... 

A cracial part of optical tomography instruments are the fibres or fibre bundles used to transmit the light to the sample and back to the detectors. The problem of the fibres is mainly pulse dispersion. The pulse dispersion in multimode fibres increases with the numerical aperture (NA) at which they are used. In particular, the detection fibre bundles, which have to be used at high NA, can introduce an amount of pulse dispersion larger than the transit time spread of the detectors [326, 443]. If the length of the bundles exceeds 1 or 2 meters, a tradeoff between time resolution and NA must often be made. 

In practice, the only feasible solution is often to transfer the light to the poly-chromator slit plane by an optical fibre. The slit is removed, and the numerical aperture at the input of the fibre is reduced to match the numerical aperture of the polyehromator. Because only moderate wavelength resolution is required, a relatively thick fibre (up to 1 mm) can be used. Therefore a reasonably high coupling efficiency with a single fibre can be obtained, even for nondescanned detection systems. The fibre should be not longer than 50 cm to avoid broadening of the IRF by pulse dispersion. 

Due to their high throughput eapability, multimode fibres are frequently used to transmit light in optical systems for TCSPC. Figure 7.22 shows how NA and pulse dispersion can be traded against fibre diameter. 

The light from the source, in this case a laser diode, is transferred to the fibre input cross section by a transfer lens system. The first lens is the laser collimator, with a focal length, fl, which is normally a few mm. If the collimated beam is focused into a fibre by a lens of a longer focal length, 12, all aberrations in the laser beam profile are magnified by a factor M = 12 / fl. This requires a fibre of a eorrespondingly large diameter. However, the NA of the beam coupled into the fibre, and eonsequently the pulse dispersion in the fibre, is reduced by the same ratio. 

If a lens of short foeal length is used, e.g. a seeond laser diode collimator, magnification of the aberrations is avoided. Now the laser ean be coupled into a thin fibre. However, the NA is large, and so is the pulse dispersion. An example is shown in Fig. 7.23. Pulses from a 650 nm, 45 ps diode laser were sent through a 1 mm fibre of 2 m length. The pulse shape shown left is for an NA of 0.3, the right pulse shape is for an NA of < 0.1. 

Fig. 7.23 Pulse dispersion in a multimode fibre of 1 mm diameter and 2 m length. The pulses of a 650 nm, 45 ps diode laser were sent through the fibre and detected by an R3809U MCP PMT. Left NA = 0.3, fwhm = 117 ps. Right NA < 0.1, fwhm = 54 ps...

Another point to be considered is the pulse width of the light source and the pulse dispersion in the optical system. Multimode fibres or fibre bundles used at high NA can easily add a few hundred ps to the IRF widths. It is, of course, not necessary to use a detector that has an IRF width shorter than 30-50% of the pulse dispersion of the optical system. 

A short pulse of light launched into a fibre will tend to spread out, as a result of dispersion. In optical fibres, the dispersion is defined as the delay between the arrival time of the start of a light pulse and its finish time relative to that of the initial pulse. It is measured at half peak amplitude. If the initial pulse has a spread of fj seconds at 50 % amplitude and the final pulse a spread of tf seconds at 50 % amplitode after having travelled d kilometres, the dispersion is given by ... 

Barrel , K. F and Pask, C. (1980) Pulse dispersion in optical fibres of arbitrary refractive-index profile. Appl. Opt., 19, 1298-1305. 

Nonlinear dispersion becomes relevant at sufficient pulse powers. In some fibre stmctures tire interiDlay between tire nonlinear dispersion and tire group velocity dispersion can be used to produce non-dispersive waves called solitons. Solitons, altliough beyond tire scope of tliis treatment, may revolutionize tire communication systems of tire future. A full treatment of soliton tlieory can be found in [4, 261. 

Dispersion will result if the light source is not strictly monochromatic. An initially sharp pulse consisting of a group of wavelengths will spread out as it travels down the fibre, because the refractive index depends on wavelength. Thus, different wavelengths will travel at different speeds. This effect is known as wavelength dispersion. 

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 important characteristic of solitons is their non-dispersive (shape-conserving) motion. Conventional wave packets will lose their shape because the Fourier components of the packet propagate at different velocities. In a non-linear medium the velocity depends not only on the frequency of a wave but also on its amplitude. In favourable circumstances the effect of the amplitude dependence can compensate that of the frequency dependence, resulting in a stable solitary wave. A technical application of this idea is the propagation of soliton-like pulses in fibre optics, which considerably increases the bit rate in data transmission. 

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