Sources of Noise in Optical Receivers

A simplified receiver model:

Overall Signal to Noise Ratio

S		Signal power
	=
N		noise power

		Is2
	=
		in2	+	Na

In a non-multiplying photodiode the noise is due to the random arrivals of photons, it is manifest as shot noise and has the form:

in2

=

2 q B Is

where	q	is the electronic charge, and
	B	is the post-detection bandwidth

We can improve the [S/N] by using the APD.

I_s increases to m.I_s  (m is the mean gain)

Signal power: (m.I_s )²

Problem: avalanche is not 'perfect' (i.e. it is noisy)

This can be interpretted as an increase in the shot noise, so the shot noise expression becomes:

in2

=

2 q B Is m2 F

where: F is the noise factor of the APD.

The noise factor can be expressed as:

F = m^x

and x » 0.3 - 0.5 for silicon APDs and » 1 for germanium.

Thus germanium APDs are very much more noisy than silicon ones.

Overall signal to noise ratio for a receiver:

	S			(m Is )2
			=
	N			2 q B Is m2 F	+	Na

				Is2
			=
				2 q B Is F	+	Na / m2

By taking F to be equal to m^x we can determine the conditions for optimum signal to noise ratio (by minimising the denominator) such that:

N_a = x q B I_s m_o^2+x

where: m_o is the optimum gain

so the optimum signal to noise ratio is:

	S		opt		Is
				=
	N				(2 + x)(mo )x q B

So in a receiver operating at gain = m there will be a noise penalty:

	S/N|	m		S/N of receiver with gain = m
noise penalty			=
	S/N|	mo		optimum S/N

 

	Is2 m2				(2+x)(mo )x q B Is
=				.
	2 q B Is m2+x	+	x q B Is (mo )2+x		Is2

 

	m2(2+x)(mo )x (q B Is )
=
	(2 m2+x + x (mo )2+x )(q B Is )

(substitute m = r.m_o)

	mo2 r2 (2+x) mox
=
	2 r2+x mo2+x + x.mo2+x

 

	r2 (1+x/2)
=
	r2+x + x/2

McIntyre has modelled the avalanche process and produced a closer approximation for F. In the model it is assumed that the ratio of ionisation coefficients of holes and electrons in the avalanche region of the diode is k (a constant: k » 0.02 in silicon and 0.5 in germanium).

the McIntyre expression is:

F = km + (1 - k)(2 - 1/m)

for injected electrons

(and:

F = m/k + ((k - 1)/k)(2 - 1/m)

for injected holes.)

Using this equation we may obtain a better indication of impairment. Repeating the above procedure gives:

	S		opt		Is2
				=
	N				q B Is		2(kmo + (1 - k)(2 - 1/mo )) + ((k(mo2 - 1) + 1)/mo )

The impairment for non-optimum gain is now given by:

S/N|	m		r2		2mo(kmo + (1 - k)(2 - 1/mo )) + (k(mo2 - 1) + 1)
		=
S/N|	mo				2mor2(rkmo + (1 - k)(2 - 1/mo )) + (k(mo2 - 1) + 1)

Notes: the two impairment expressions converge as k 1 and x 1, and are identical for k = 1 = x.

Conclusions

The more exact impairment prediction predicts a lower sensitivity of SNR to avalanche gain for practical silicon APD receivers.

In the case of germanium APD receivers the errors involved in using the less exact expression for avalanche noise are unlikely to result in significant errors for moderate errors in avalanche gain.

See plots and comparison of predictions for germanium receivers and the corresponding predictions for silicon ones.