FIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 22.

Transcription

1 FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 22 Optical Receivers Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 1

2 When light from an optical fiber carrying data in the form of optical pulses, is incident onto a photo-detector material, various types of noise are introduced into the already noisy signal and the output electrical signal from the photo-detector has noise which causes pulse distortion in the conversion from the optical to the electrical domain. That is, a rectangular optical pulse does not remain exactly rectangular of the same width as the optical pulse when the optical pulse is converted to an electrical pulse in the photo-detector. The various types of noise introduced by the photo-detector are the shot noise, dark-current noise and thermal noise which have already been discussed. The shot or quantum noise involves Poisson s statistics and is multiplicative in nature. However, the thermal noise is Gaussian in nature and is, hence, additive noise. The presence of these noises in the detected output signal of the photodetector incurs the requirement of defining certain parameters to evaluate the performance efficiency of a photo-detector and also for a qualitative comparison of various photo-detectors, so that the right photo-detector can be used for the right application. One such quantity is the signal-to-noise ratio (SNR) of the receiver which is defined as the ratio of the useful signal power in the received signal to the unwanted noise power in the received signal, at the receiver of the data communication system. However, SNR is more suited for performance evaluation of analog communication system. For a digital communication system, the parameter of interest that is used to evaluate the performance of the system at the receiver is the bit-error-ratio (BER) of the receiver. In a similar manner as the derivation of expression for SNR, we shall derive an expression for the BER for an optical receiver in the subsequent discussions. Data transmission systems may be broadly classified into three types- analog, digital and mixed-signal data transmission systems. Optical communication supports both analog and digital data transmission schemes. Due to this versatility and the wideband nature, optical communication is the most preferred mode of data communication in the modern day. In this section, we shall emphasize on digital data transmission in the optical domain via optical communication link and investigate the performance of the optical receiver with respect to digital data i.e. data in the form of bits (binary digits 0 and 1). In digital amplitude modulation scheme (amplitude shift keying scheme) in the optical domain, the presence of light indicates a binary 1 (or 0) and the absence of light indicates a binary 0 (or 1) in accordance to the positive (or negative) digital logic. Let us assume that the optical source is unbiased i.e. there is equal probability of transmission of 0 and 1. Let us also assume that the overall noise of the system be Gaussian noise which is additive in nature. To make the analysis of BER simpler, we may also assume that the values of the variance of the photo-detector output for the optical intensities in the 0 and 1 level are different, but the individual noise in the two levels be Gaussian in nature. This assumption accounts well for the Poisson shotnoise generated in the photo-detector in the two levels (0 and 1). Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 2

3 Under the above assumptions, the two voltage levels that represent binary 0 and 1 are depicted in the figure 22.1 below: Figure 22.1: 0 and 1 levels of digital data in a photo-detector output. As seen from the above figure, the introduction of noise into the detected signal causes the photo-current to fluctuate about its mean level in the two logic levels which indicate the two binary digits 0 and 1. According to our assumption, the variances of these fluctuations are not equal in the two levels. This fact is obvious from the probability density function shown in the above figure which shows different variance shapes about the mean level for the two logic levels. Obviously, the variance for the logic 0 level is narrower than that for the logic 1 level because the logic 1 level involves shot noise which is appropriately given by a Poisson distribution but the logic 0 level involves only dark-current noise and thermal noise which are Gaussian in nature. The probability density curves shown in the above figure indeed show the extent of the detector current fluctuations about the mean values for the two logic levels 0 and 1. Since we have assumed Gaussian variation of noise for the two levels, we may write probability density function for the current for the two logic levels 0 and 1 as: ( ) (22.1) ( ) (22.2) Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 3

4 In the above equations, σ 0 and σ 1 are the standard deviations of the current in the two logic levels 0 and 1 respectively from their respective mean current levels I 0 and I 1. We may assume I 0 to be zero, but due to various reasons some amount of photo-current also flows during the logic 0 level and so we assume a general mean current level I 0 for the logic 0 level. Since the data is in binary format, there is a threshold current (or voltage) level which differentiates the two logic levels from each other. That is, if the detected current (or voltage) is greater than the threshold level, the detected bit is assigned as 1 and if the detected current (or voltage) is lower than the threshold level, the detected bit is assigned as 0. Let us assume the photo-current (or voltage) corresponding to the threshold level be denoted by I th (or V th ). However, if a 0 was transmitted by the source but, due to noise and other factors, if the detected signal level exceeds the threshold level the detected bit would be assigned as 1. Similarly, if a 1 was transmitted by the source and the detected signal level drops below threshold level, the detected bit is assigned 0. If any or both of these two cases happen then bit errors are said to have occurred in the data transmission. The quantity that analyses the performance of the communication link with respect to bit errors is the bit-error-ratio (BER) as mentioned earlier. BER is indeed a probability of occurrence of bit errors in the transmission. The following expression can, hence, be written for BER: ( ) ( ) (22.3) We have assumed the source to be unbiased which means there is an equal probability of occurrence of 1 as that of 0 in the stream of data bits. Therefore: The equation 22.3 can be re-written as: (22.4) { ( ) ( )} (22.5) The first probability term in the R.H.S. of equation 22.5 is given by the shaded area under the probability density curve above the threshold level in the figure 22.1 indicated by P 0 (V). That is, ( ) (22.6) Similarly, the second probability term in equation 22.5 is the area of the shaded region indicated by P 1 (V) in figure Therefore: ( ) (22.7) Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 4

5 The integrals in the equations 22.6 and 22.7 cannot be solved in the closed form and are hence represented by certain functions called the error functions because they are used for calculations of bit errors. Since the error functions are used for calculation of errors of two complementary bits, they are more appropriately called the complementary error functions. These complementary error functions are defined as: (22.8) Using the above definition of complementary error function, the probabilities given in equations 22.6 and 22.7 can be written as: ( ) ( ) (22.9) ( ) ( ) (22.10) The value of the above probabilities can be evaluated from the standard complementary error function tables that are available. So, with the knowledge of the threshold current value, the value of the above probabilities can be evaluated from tables and from these values the BER can be calculated using the following relationship: { ( ) ( )} (22.11) The above equation shows that the value of BER depends upon the threshold current value which can also be observed from figure 22.1 as follows- if the threshold current value is shifted lower than that shown, more 0 s would be detected as 1 s and majority 1 s would be detected correctly; if the threshold level is shifted upwards, more number of 1 s would be detected as 0 s but majority 0 s would be detected correctly. Thus the threshold level should be pre-set at an optimum level so that number of bit errors is minimized. For minimum value of bit-error ratio, the shaded areas indicating the error probabilities of 0 and 1 (in figure 22.1) must be equal which requires that the following condition must be satisfied: (22.12) (22.13) The above value of threshold current is the optimum value which ensures a minimum value of BER. If σ 0 =σ 1 (i.e. in presence of thermal noise only), the threshold level lies exactly half-way between the two logic levels. However, for optical communication systems the noises in the two logic levels is assumed to be unequal and so, the two variances cannot be equal which suggests that equation gives the best possible value of threshold current level for minimum BER. Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 5

6 If we substitute the value of the threshold current level given by equation in either the L.H.S. or the R.H.S. of equation we obtain a quantity known as the Q-parameter of the data transmission which is denoted by Q. That is: (22.14) In fact, q-parameter of the data transmission is the quantity that decides the BER of the data transmission. The numerator of the equation is, in fact, the current swing (voltage swing) assigned between the two logic levels 0 and 1. The variances σ 0 and σ 1 are the standard deviations of the photo-current from the preassigned mean current levels of the respective binary bit. The q-parameter of given by equation is also indicative of the noise margin of the data transmission. If we express the optimized BER in terms of the q-parameter of the data transmission using equation 22.11, we obtain: ( ) (22.15) For large values of Q (typically Q>3) the complementary error function may be approximated by an exponential function and the BER may be expressed as: (22.16) The above equation shows that BER is a strong exponential function of Q and the curve plotted between BER and Q looks like the one shown below: Figure 22.2: BER Vs Q-parameter curve of photo-detector It is evident from the above figure that, for a small change in the Q there is a rapid decrease in the BER because (as seen from the figure) the BER-axis is logarithmic which means that the decrease would be in powers of 10. If the tolerable BER of data transmission be at a particular value, the corresponding value of Q can Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 6

7 be geometrically found out from the above curve, and from the knowledge of the value of Q the threshold level can be appropriately decided. For example, in optical communication, the acceptable BER (without the application of any error correcting schemes) is 10-9 and the corresponding value of Q 6. Let us now revert back to our discussion on the SNR of the data transmission in the optical domain. The operation of the photo-detector may be classified into two domains- high power domain (in the presence of optical signal flux) and low power domain (in the absence of optical signal flux). Owing to the different types of noises present in the two domains, these domains may also be known as- thermal noise dominated operation (low power domain) and shot-noise dominated operation (high power domain). If σ T denotes the total variance of the thermal and dark current noises and σ s denotes the total variance due to shot-noise in the photo-detected signal, then: For thermal noise dominated operation: For shot-noise dominated operation: However, the noise which dominates the operation is also dependent on the relative distance between the transmitter and the receiver in the optical communication link. If the two are very near to each other, some optical power exists even in the low-power domain because this optical power fails to attenuate enough due to the short distance between the transmitter and receiver. In this case the operation would be more of shot-noise dominated than thermal noise. On the other hand, if the transmitter and receiver are very far away, the transmitted optical flux attenuates to very low power levels and even in the presence of optical signal flux (due to its low power) the operation may be of the low-power type and the thermal noise dominates over shot noise. The expression for SNR of an optical communication link (as given in equation 20.9) can be written as: Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 7 (22.17) The terms in above expression represent the same parameters as in equation The above expression is, however, a general expression for the SNR. For a thermal noise dominated operation: ; the SNR can be written as: (22.19) For a p-i-n photo-detector having responsivity R : M=1 and signal current i p =P in.r, where P in is the incident optical signal power. Also, the expression for i T 2 can be substituted from equation Substituting the above values in the expression we obtain the expression for the SNR for thermal noise dominated operation as:

8 (22.20) For a given photo-detector, R and B (bandwidth) are constants and so we may write: (22.21) Thus, for a thermal noise dominated situation, the SNR of the receiver is proportional to the square of the incident optical power. That is, the SNR improves very rapidly with increase in incident optical signal power. Also, the SNR is proportional to the load resistance used in the photo-detector circuit. Therefore a large value of the load resistance is preferred for a better SNR. If SNR=1, then we may say that all the power that is incident onto the photodetector material is converted to equivalent noise by the photo-detector. Therefore, we may define a quantity known as the noise equivalent power (NEP) of the photodetector which determines the amount of incident signal power that would be converted to noise by the receiver. Lesser the value of NEP better is the receiver. NEP is defined (using equation 22.20) as: (22.22) For a shot-noise dominated situation: ; the SNR can hence be written as: (22.23) Substituting the values of the quantities in the equation for a normal p-in photo-detector (using equation 20.5) we have the following expression for SNR: (22.24) Thus, for a given photo-detector with responsivity R operating in high power domain: (22.25) If we concentrate on the two equations and 22.25, we find that at low input optical signal power (when thermal noise dominates the photo-detector output noise) the SNR improves very rapidly with increase in optical power. However, as the input optical power increases and the operation shifts to the high power domain (when shot noise starts to dominate the photo-detector output noise), the improvement of SNR with input optical power is rather slow due to the linear relation between the two. Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of Electrical Engineering, IIT Bombay Page 8