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energy, or EG < hc/l. On this basis it turns out that silicon is useful for wavelengths shorter that 1 m. Germanium is useful at 1.3 m and InSb and InAs at 1.55 m. The total current flowing in a photodiode consists of the actual current generated by the photons, plus what is termed the dark current. This is the current that flows even when no signal is present. Ideally it should be zero, but in practice it is of the order of a few nanoamperes, and it can contribute to the noise. Denoting the dark current by Id the total current is I Iph Id (12.68)
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This current is accompanied by shot noise (a name that is a hangover from vacuum tube days), the mean square spectral density of which is 2qI in A2/Hz with I in amperes. This would be for a diode without any internal amplification such as a PIN diode. An avalanche photo diode (APD) multiplies the signal current by a factor M and at the same time generates excess noise, represented by an excess noise factor, F, so that the mean square spectral density is (2qI)MF. F increases with increase in M (see Jones, 1988, p. 240). Typical values are of the order M 100, F 12. For a PIN diode, M 1 and F 1. The analysis to follow is based on that given in Jones (1988). The equivalent circuit for the input stage of an optical receiver is shown in Fig. 12.14, where R is the parallel combination of diode resistance, input load resistance and preamplifier input resistance, and C is the parallel combination of diode capacitance, preamplifier input capacitance and stray circuit capacitance. Resistance R will generate thermal noise current, the spectral density of which is 4kT/R (k is Boltzmann s constant and T is the temperature, which may be taken as room temperature). The preamplifier transistor will also generate shot noise, which when
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The equivalent input circuit for an optical receiver. {it2} is the mean square spectral density for the current noise source, and 2 {va } the mean square spectral density for the voltage noise source. is is the signal current.
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referred to the input has a spectral density 2q/Iin in A2/Hz, where Iin is the transistor gate (JFET) or base (BJT) current in amperes. Using curly brackets {.} for the spectral density values (see Jones, 1988), the total noise current spectral density at the input is 5i26 5 2qIM 2F 1 t 4kT 1 2qItr A2/Hz R (12.69)
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The pre-amplifier also has a noise voltage component shown as va resulting from the shot noise in the drain or collector current. The mean square noise voltage spectral density is 2qItr/g2 V 2/Hz where Itr is the m transistor drain (JFET) or collector (BJT) current in amperes and gm is the device transconductance in siemens. The total noise voltage spectral density at the input is therefore
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2 5v2 6 5 5i26 ZZL Z 1 n t
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2qItr 2 V /Hz g2 m
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(12.70)
where ZL is the impedance of R and C in parallel. The average signal voltage is MIphR. Increasing R should result in an increase in signal to noise ratio since the signal voltage is proportional to R and the R component of noise current density is inversely proportional to R. However, the bandwidth of the RC input network is inversely proportional to R and therefore sets a limit to how large R can be. An equalizing filter, which has a transfer function, given by Heq(f) 1 j2 f RC can be included in the overall transfer function, which compensates for the input impedance frequency response over the signal bandwidth. The effective spectral density for the mean square 2 2 noise voltage at the input is then 5vn6 ZH( f )Z . The mean square noise 2 voltage Vn at the input is obtained by integrating this expression over the signal bandwidth. Only the result will be given here:
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