Figure 3 : Error probability versus Q2 (in dB). In order to ensure an error probability less than 10"9, considered as a reference value for telecommunication systems, Q must exceed 6. Note that, due to the signal to noise ratio encountered in digital communication systems, approached expression given in (19) always holds.

Usual receivers performance is limited by thermal noise and the required received power to ensure a given error probability exceeds commonly by 15 or 20 dB the quantum limit. In order to improve this performance, coherent reception appeared as a promising technique. An important research effort was devoted to it up to the beginning of the nineties, before optical fibre amplifiers became available. We will briefly recall the basic principles of coherent receivers. A lot of BER computation methods were developed [Dogliotti, 1979]

4.4 Coherent reception

As the first radio receivers used direct detection through a crystal, heterodyne receiving technique allowed to increase dramatically the sensitivity of the receivers : the idea was to let the incoming modulated wave at angular frequency ws beat in a non linear device called mixer with a sinusoidal signal at angular frequency (0/ produced by the local oscillator. The beat resulted in a signal at the intermediate angular frequency ft),- carrying the modulation of the received signal. Moreover, heterodyning allows all types of modulation to be used, while frequency or phase modulation cannot be detected by a direct receiver which only detects power fluctuations [Salz, 1985, Joindot, 1986],

These receivers very commonly used today in radio can be theoretically implemented in optics. Assuming for example a phase modulated signal which can be written as:

where 0 equals 0 or 7T according to the transmitted data and a local oscillator wave : uL{t) = {2PLcos{coLt+60) (21)

the resulting beat signal (photocurrent) at the photodetector output has an average value / given by :

As far as noise is concerned, due to the fact that power Pj is always large compared to the received signal power, shot noise associated with the local oscillator wave is dominant. It is white with two sided power spectral density (cf. relation (14):

The carrier to noise ratio p, defined as the ratio of the intermediate frequency signal power to the noise power contained in a bandwidth B=l/T determines the error probability. It is given using (22) and (23) by : 71PT_ T)Ps hv hv D' (24)

s s and the bit error rate is given by:

P =lerfcp=lerfM/^~eXp(-p2) f 2 2 V ftv/> pV?r (25)

This relation shows (assuming a perfect detector, 77=1) that the coherent receiver can approach the quantum limit and then provide a significant sensitivity improvement compared to direct detection receivers by eliminating the influence of thermal noise. But practical implementation is not easy. For example, oscillator phase noise is a serious problem, much more difficult to solve than in microwaves, because the linewidth is much larger compared to the bandwidth of the modulating signal. After a lot of efforts had been made to overcome these drawbacks, optical fibre amplifiers associated with direct detection receivers provided the same sensitivity improvement without the aforementioned problems and research about coherent techniques decreased very strongly and was abandoned in most laboratories.

5 Optical amplification

The first optical amplifiers to be studied were semiconductor amplifiers (SOA), which use the same physical phenomena as lasers : pumping in a material through electrical carriers injection (i.e through an electrical current providing the external energy) causes a population inversion. Electrons on the upper overoccupied level fall down onto the fundamental energy level again and emit correlatively photons at a wavelength corresponding to the energy difference between the levels. Most of them add in phase with those of the incident light and contribute to its amplification : this is the stimulated emission process. But other photons are emitted incoherently with the incident light: this is the spontaneous emission. A small part of these photons travel themselves through the amplifier, are amplified and constitute at the amplifier output the amplified spontaneous emission (ASE) noise, i.e the noise generated in the amplification process.

The first theoretical works were published at the beginning of the sixties \Shimoda, 1957] and application to optical communication systems were proposed in the seventies [Personick, 1973], This is only in the beginning of the eighties that progress about semiconductor lasers allowed to consider SOA's as practically implementable devices and a lot of investigations was devoted to them [Yamamoto, 1980].

In optical fiber amplifiers [Desurvire, 1994], the active medium is a piece of rare earth (usually erbium) doped fiber pumped by one (or eventually two) laser diode(s) emitting at a wavelength of 980 or 1480 nm. The amplification bandwidth is around 12 nm, i.e 1500 GHz (and even around 24 nm for fluoride doped amplifiers now under study in the laboratories). The output power of these erbium doped fiber amplifiers (EDFA) can be high (up to 20 dBm), which allows to increase the transmission length, with nevertheless limits due to the counterpart of non linear effects described in section 1. Sometimes, remote pumping amplifiers are used : this is done in some undersea lightwave systems, where the active fiber itself is immerged, while pump, which is the most critical component in terms of reliability, is placed at the end of the link and feeds the amplifier itself through the fiber.

The first publications about EDFA appeared in 1987 and practically usable devices were available less than four years later. Compared to semiconductor amplifiers, fiber amplifiers are easier to implement in practical systems, and present the advantage to be nearly polarization insensitive : they can be used as power emitting amplifiers (boosters), preamplifiers at the receiving end, or in line amplifiers. In this last case they can replace electronic regenerators, as for instance in the last generation of transoceanic undersea lightwave systems. Nevertheless, the counterpart is noise, linear and non linear distortions.

It is important to remark that, although they do not appear as the most promising candidate for amplification in optical communication systems, SOA's exhibit very interesting non linear properties, which make them key devices for optical signal processing (reshaping, sampling...) which will be more and more used in future optical networks.

5.1 Noise characteristics of optical amplifiers

An optical amplifier adds then its own noise to the amplified signal, as any electrical amplifier also does. Optical electromagnetic field associated to ASE noise can be modelized as a white gaussian process and the optical noise power spectral density (p.s.d) per mode at the output of the amplifier is writtten as :

' esa sp s where G is the power gain and nsp the spontaneous emission factor characterizing the amplifier. In the case of a semiconductor amplifier, (26) gives the noise p.s.d while, in the case of a fiber amplifier, the total power in a band B is 2yaaB because each mode is degenerated according to the two possible orthogonal states of polarization. As in the case of an electronic amplifier, the amount of noise due to the amplifier can be characterized by the noise figure. ASE noise can be considered as due to a noise source with a p.s.d JaJG at the input of a noiseless ideal amplifier. Noise at the amplifier output can be viewed as the sum of shot noise and ASE noise and the signal to noise ratio can be written as :

^ c cn v ' c ' n r n where pj„ is the "intrinsic signal to noise ratio" at the amplifier imput (cf. (16)) and Fa the amplifier noise figure expressed as :

If G is large, noise figure equals practically nsp. It is also important to remark that losses in connections to the input fiber must be taken into account in a practical amplifier. If C\ is the transmission factor of the input connector, measured noise figure is : 1

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