# Optical Receiver Sensitivity Degradation

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The sensitivity analysis in the optical receiver sensitivity tutorial is based on the consideration of receiver noise only. In particular, the analysis assumes that the optical signal incident on the receiver consists of an ideal bit stream such that 1 bits consist of an optical pulse of constant energy while no energy is contained in 0 bits. In practice, the optical signal emitted by a transmitter deviates from this ideal situation. Moreover, it can be degraded during its transmission through the fiber link. An example of such degradation is provided by the noise added at optical amplifiers. The minimum average optical power required by the receiver increases because of such nonideal conditions. This increase in the average received power is referred to as the *power penalty*. In this section we focus on the sources of power penalties that can lead to sensitivity degradation even without signal transmission through the fiber. Several transmission-related power-penalty mechanisms are discussed in another tutorial.

#### 1. Extinction Ratio

A simple source of power penalty is related to the energy carried by 0 bits. Some power is emitted by most transmitters even in the off state. In the case of semiconductor lasers, the off-state power P_{0} depends on the bias current I_{b} and the threshold current I_{th}. If I_{b} < I_{th}, the power emitted during 0 bits is due to spontaneous emission, and generally P_{0} << P_{1} , where P_{1} is the on-state power. By contrast, P_{0} can be a significant fraction of P_{1} if the laser is biased close to but above threshold. The *extinction ratio* is defined as

r_{ex} = P_{0}/P_{1}

The power penalty can be obtained by using

For a p-i-n receiver I_{1} = R_{d}P_{1} and I_{0} = R_{d}P_{0} , where R_{d} is the responsivity (the APD gain can be included by replacing R_{d} with MR_{d}). By using the definition for the receiver sensitivity, the Q factor is given by

In general, σ_{1} and σ_{0} depend on because of the dependence of the shot-noise contribution on the received optical signal. However, both of them can be approximated by the thermal noise σ_{T} when receiver performance is dominated by thermal noise. By using σ_{1} ≈ σ_{0} ≈ σ_{T} in the Q equation above, is given by

This equation shows that increases when r_{ex} ≠ 0. The power penalty is defined as the ratio . It is commonly expressed in decibel (dB) units by using

The figure below shows how the power penalty increases with r_{ex}.

A 1-dB penalty occurs for r_{ex} = 0.12 and increases to 4.8 dB for r_{ex} = 0.5. In practice, for lasers biased below the threshold, r_{ex} is typically below 0.05, and the corresponding power penalty (<0.4 dB) is negligible. Nonetheless, it can become significant if the semiconductor laser is biased above threshold. An expression for can be obtained for APD receivers by including the APD gain and the shot-noise contribution to σ_{1} and σ_{0} in the equation above. The optimum APD gain is lower when r_{ex} ≠ 0. The sensitivity is also reduced because of the lower optimum gain. Normally, the power penalty for an APD receiver is larger by about a factor of 2 for the same value of r_{ex} .

#### 2. Intensity Noise

The noise analysis of the optical receiver in the previous tutorial is based on the assumption that the optical power incident on the receiver does not fluctuate. In practice, light emitted by any transmitter exhibits power fluctuations. Such fluctuations, called intensity noise, were discussed in the context of semiconductor lasers. The optical receiver converts power fluctuations into current fluctuations which add to those resulting from shot noise and thermal noise. As a result, the receiver SNR is degraded and is lower. An exact analysis is complicated, as it involves the calculation of photocurrent statistics. A simple approach consists of adding a third term to the current variance, so that

σ^{2} = σ_{s}^{2} + σ_{T}^{2} + σ_{I}^{2}

where

The parameter r_{I}, defined as , is a measure of the noise level of the incident optical signal. It is related to the *relative intensity noise* (RIN) of the transmitter as

r_{I} is simply the inverse of the SNR of light emitted by the transmitter. Typically, the transmitter SNR is better than 20 dB, and r_{I} < 0.01.

As a result of the dependence of σ_{0} and σ_{1} on the parameter r_{I}, the parameter Q is reduced in the presence of intensity noise. Since Q should be maintained to the same value to maintain the BER, it is necessary to increase the received power. This is the origin of the power penalty induced by intensity noise. To simplify the following analysis, the extinction ratio is assumed to be zero, so that I_{0} = 0 and σ_{0} = σ_{T}. By using and σ^{2} equation above for σ_{1}, Q is given by

where

This Q equation is easily solved to obtain the following expression for the receiver sensitivity:

The power penalty, defined as the increase in when r_{I} ≠ 0, is given by

The following figure shows the power penalty as a function of r_{I} for maintaining Q = 6 and 7 corresponding to a BER of 10^{-9} and 10^{-12}, respectively.

The penalty is negligible for r_{I} < 0.01 as δ_{I} is below 0.02 dB. Since this is the case for most practical optical transmitters, the effect of transmitter noise is negligible in practice. The power penalty exceeds 2 dB for r_{I} = 0.1 and becomes infinite when r_{I} = 1/Q. An infinite power penalty implies that the receiver cannot operate at the specific BER even if the incident optical power is increased indefinitely. In the case of Bit-error rate versus the Q parameter, an infinite power penalty corresponds to a saturation of the BER curve above the 10^{-9} level for Q = 6, a feature referred to as the BER floor. In this respect, the effect of intensity noise is qualitatively different than the extinction ratio, for which the power penalty remains finite for all values of r_{ex} such that r_{ex} < 1.

The preceding analysis assumes that the intensity noise at the receiver is the same as at the transmitter. This is not typically the case when the optical signal propagates through a fiber link. The intensity noise added by in-line optical amplifiers often becomes a limiting factor for most long-haul lightwave systems. When a multimode semiconductor laser is used, fiber dispersion can lead to degradation of the receiver sensitivity through the mode-partition noise. Another phenomenon that can enhance intensity noise is optical feedback from parasitic reflections occurring all along the fiber link. Such transmission-induced power-penalty mechanisms are considered in another tutorial.

#### 3. Timing Jitter

The calculation of receiver sensitivity is based on the assumption that the signal is sampled at the peak of the voltage pulse. In practice, the decision instant is determined by the clock-recovery circuit. Because of the noisy nature of the input to the clock-recovery circuit, the sampling time fluctuates from bit to bit. Such fluctuations are called *timing jitter*. The SNR is degraded because fluctuations in the sampling time lead to additional fluctuations in the signal. This can be understood by noting that if the bit is not sampled at the bit center, the sampled values is reduced by an amount that depends on the timing jitter Δt. Since Δt is a random variable, the reduction in the sampled value is also random. The SNR is reduced as a result of such additional fluctuations, and the receiver performance is degraded. The SNR can be maintained by increasing the received optical power. This increase is the power penalty induced by timing jitter.

To simplify the following analysis, let us consider a p-i-n receiver dominated by thermal noise σ_{T} and assume a zero extinction ratio. By using I_{0} = 0, the Q factor is given by

where is the average value and σ_{j} is the RMS value of the current fluctuation Δt_{j} induced by timing jitter Δt. If h_{out}(t) governs the shape of the current pulse,

Δt_{j} = I_{1}[h_{out}(0) - h_{out}(Δt)]

where the ideal sampling instant is taken to be t = 0.

Clearly, σ_{j} depends on the shape of the signal pulse at the decision current. A simple choice corresponds to h_{out}(t) = cos^{2}(πBt/2), where B is the bit rate. Here the following is used as many optical receivers are designed to provide that pulse shape.

Since Δt is likely to be much smaller than the bit period T_{B} = 1/B, it can be approximated as

by assuming that BΔt << 1. This approximation provides a reasonable estimate of the power penalty as long as the penalty is not too large. This is expected to be the case in practice. To calculate σ_{j}, the probability density function of the timing jitter Δt is assumed to be Gaussian, so that

where τ_{j} is the RMS value (standard deviation) of Δt. The probability density of Δt_{j} can be obtained by using the two equations above and noting that Δt_{j} is proportional to (Δt)^{2}. The result is

where

The equation above is used to calculate and . The integration over Δi_{j} is easily done to obtain

Noting that , where R_{d} is the responsivity, the receiver sensitivity is given by

The power penalty, defined as the increase in , is given by

The following figure shows how the power penalty varies with the parameter Bτ_{j}, which has the physical significance of the fraction of the bit period over which the decision time fluctuates (one standard deviation).

The power penalty is negligible for Bτ_{j} < 0.1 but increases rapidly beyond Bτ_{j} = 0.1. A 2-dB penalty occurs for Bτ_{j} = 0.16. Similar to the case of intensity noise, the jitter-induced penalty becomes infinite beyond Bτ_{j} = 0.2. The exact value of Bτ_{j} at which the penalty becomes infinite depends on the model used to calculate the jitter-induced power penalty. The δ_{j} equation above is obtained by using a specific pulse shape and a specific jitter distribution. It also assumes Gaussian statistics for the receiver current. Jitter-induced current fluctuations are not Gaussian in nature. A ore accurate calculation shows that the δ_{j} equation above underestimates the power penalty. The qualitative behavior, however, remains the same. In general, the RMS value of the timing jitter should be below 10% of the bit period for a negligible power penalty. A similar conclusion holds ofr APD receivers, for which the penalty is generally larger