In general, digital optical communication systems transmit binary data using two levels of optical power, where the higher power level represents a binary 1 and the lower power level represents a binary 0. These two power levels can be represented as P1 and P0, where P1 > P0 and the units of power are watts.

In optical transmitters, electrical current is converted to optical power (electrical to optical conversion, or E/O), and, in optical receivers, optical power is converted back to electrical current (optical to electrical conversion, or O/E). The electrical currents I0 and I1 are proportional to the corresponding optical power levels. The responsivity, r, of the receiver is the constant of proportionality between the received optical power level and the current with units of amperes per watt. The responsivity can be used to demonstrate the following:

The ratio between the "one" level and the "zero" level, shown in Equation 3, is defined as the "extinction ratio," and is represented by the symbol re. (Note that extinction ratio is defined in some literature as the reciprocal of Equation 3.) In an ideal transmitter, P0 would be zero and thus re would be infinite. In most practical optical transmitters, however, the laser must be biased so that P0 is in the vicinity of the laser threshold, meaning that a finite amount of optical power is emitted at the low level and thus, P0 > 0.

In many cases it is useful to talk in terms of the average optical power, PAVG, and in these instances the following relationships are useful:

Error Probability in Digital Communication Systems

The receiver in a digital communication system must make two decisions: (1) when to sample the received data and (2) whether the sampled value represents a binary 1 or 0. In order to understand the effects of extinction ratio on system performance, the following analysis focuses on the second decision.

Decisions about the Received Signal

If we assume that additive white Gaussian noise (AWGN) is the dominant cause of erroneous decisions, then we can calculate the statistical probability of making an incorrect decision. The probability density function for v(t) with AWGN can be written mathematically as:

where vS is the voltage sent by the transmitter (the mean value of the density function), v(t) is the sampled voltage value in the receiver at time t, and s is the standard deviation of the noise. Equation 7 is illustrated in Figure 1.

Figure 1. AWGN probability density function

In binary signaling, vS can take on one of two voltage levels, which we will call vS0 and vS1, and the probability of making an erroneous decision in the receiver is as follows:

where Pe represents the probability of error and Px | y represents the probability of x given y. If we assume an equal probability of sending vS0 versus vS1 (50% mark density), then PvS0 = PvS1 = 0.5. Also, in order to simplify the example at this point, we will assume that the same noise affects either voltage level (i.e., s0 = s1), which means that Pv(t) > g | vS = vS0 = Pv(t) < g | vS = vS1. Using these assumptions, Equation 8 can be reduced to:

where PROBv(t) is defined in Equation 7. This result is illustrated in Figure 2.

Figure 2. Probability of error for binary signaling

From Figure 2 and Equations 8 and 9, we can see that the probability of error is equal to the area under the tails of the density functions that extend beyond the threshold, g. This area and thus the bit-error ratio (BER) are determined by two factors: (1) the standard deviations of the noise (s0 and s1) and (2) the distance between vS0 and vS1. (Note: This is based on the assumption that g = (vs1,-vs0)/2, which is valid only for s0 = s1.)

The Q Factor

(Note: The optical receiver converts optical power to electrical current. This current is later converted to a voltage--usually by a transimpedance amplifier such as the MAX3864 or MAX3866--before it is applied to the receiver decision circuit.) If we assume that the decision threshold is optimized for minimum BER, then the optical signal-to-noise ratio takes on a useful form, defined as "Q" or the "Q-factor" in the literature2, which is written mathematically as:

Note: this result is derived by using the fact that, at the optimum g, (I1-g)/s1 = (g-I0)/s0 Q, which can be re-arranged to yield goptimum= (I1s0 + I0s1)/(s0 + s1) and then substituted back into (I1-g)/s1 to get equation (10).

The BER can be written mathematically in terms of Q as follows:

where erfc( ) is the complementary error function, defined as3:

The Power Penalty

As noted in Section 2, P0 is ideally equal to zero, making the optimum extinction ratio infinite. When the extinction ratio is not optimum, however, the transmitted power must be increased in order to maintain the same BER at the receiver. This increase in transmitted power due to non-ideal values of extinction ratio is called the "power penalty."

In order to derive a mathematical expression for the power penalty, we can start by noting that, from Equations 10 and 11, if the Q-factor is held constant, then the BER will remain constant. In order to simplify the analysis, we will assume that thermal noise is dominant and equal for both the P0 and P1 power levels, and thus s0 = s1 = sT, and s0 + s1 = 2sT (where sT is the standard deviation of the thermal noise).

Using the following algebra, we can write an expression for the Q-factor in terms of the extinction ratio, re:

Next, we can rearrange Equation 13 in order to express PAVG as a function of re:

From Equation 14, we can make the following important observation: As the extinction ratio is degraded below its ideal value of infinity, the average power must be increased in order to maintain a constant value of Q and hence a constant BER.

The power penalty is defined as the ratio of the average power required for a given value of re to the average power required for the ideal case of re = . This can be defined mathematically as:

where de is the power penalty and PAVG(re) is defined in Equation 14.

The power penalty as a function of extinction ratio is graphed and tabulated below for both linear and logarithmic (dB) ratios.

Figure 3. Power penalty vs. extinction ratio (linear ratios)

Figure 4. Power penalty vs. extinction ratio (dB ratios)

Table 1. Extinction Ratio and Power Penalty

Conclusion

Seemingly small changes in extinction ratio can make a relatively large difference in the power required to maintain a constant BER. This effect is especially acute for extinction ratios less than seven, where a change of one in extinction ratio translates to an approximate 10% change in required average power. This additional required power is aptly termed the "power penalty," as nothing is gained by this increase in power other than the unnecessary privilege of operating at a reduced extinction ratio. Also, the DC offset associated with a low extinction ratio can cause overload problems in the receiver.

*This article is taken from Maxim Application Note HFAN-2.2.0

1 J. M. Senior, Optical Fiber Communications: Principles and Practice, Hertfordshire (UK): Prentice Hall International (UK) Ltd, pp. 625-627, 1992.

N. S. Bergano, F. W. Kerfoot, and C. R. Davidson, "Margin Measurements in Optical Amplifier Systems," in IEEE Photonics Technology Letters, vol. 5, no. 3, pp. 304-306, Mar. 1993.

G. Agrawal, Fiber-Optic Communication Systems, New York, N.Y.: John Wiley & Sons, pp. 172, 1997.