# A phase-noise-compensating transmission method for phase-modulated coherent optical systems

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<ul><li><p>A PHASE-NOISE-COMPENSATING </p><p>PHASE-MODULATED COHERENT TRANSMISSION METHOD FOR </p><p>OPTICAL SYSTEMS S. Betti, F. Curti, G. De Marchis, and E. lannone Fondazione Ugo Bordoni Viale Europa 190-001 44 Rorna Italy </p><p>KEY TERMS Phase noise, coherenr optical system, phase modularion </p><p>ABSTRAa A new method capable of overcoming the limitations in coherenf optical systems that occur when the optical source has a nonnegligible linewidth is proposed. The fransmission of a reference signal derived from a common source allows the use of high-power laser sources which are required in long-haul optical systems. </p><p>INTRODUCTION </p><p>The main problem that limits the feasibility of coherent opti- cal systems is due to the phase noise of the laser sources [l, 21. Technical solutions have been suggested in the direction of improving the source spectral characteristics. For example, a solution that has been adopted in a number of coherent systems [3, 41 is the use of an external cavity [ 5 ] for narrowing </p><p>the spectral linewidth of laser diodes. However, similar solu- tions seem to be very critical and not reliable enough for practical applications. </p><p>In this article we propose an alternative approach and demonstrate how the transmission of a reference signal de- rived from a common optical source and suitably frequency- shifted with respect to the modulated signal allows practical insensitiveness of the system to the phase noise, offering the possibility of using lasers with relatively broad linewidth. Moreover, this technique leads indirectly to an appreciable reduction of the stimulated Brillouin scattering (SBS) [6], allowing the use of high-power laser sources. The proposed system is also compatible with the use of polarization-inde- pendent detection schemes in which a neghgible power penalty has been demonstrated [7]. The system is affected by a 3-dB penalty with respect to the conventional PSK heterodyne receiver due to the power split at the transmitter, but its application to long-haul coherent optical systems, where high-power laser sources are required, seems to be very prom- ising. </p><p>SYSTEM ANALYSIS </p><p>The block diagram of the proposed system is shown in Figure 1. At the transmitter the optical carrier, generated by the laser source LS, is divided into two orthogonally polarized compo- nents by a polarization beam splitter BS: One of these is modulated by the angular modulator PM, the other is shifted </p><p>BS BS </p><p>LS </p><p>0 </p><p>FIBRE </p><p>Figure 1 Block diagram of the proposed PSK transmission system </p><p>MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 1 , No. 5. July 1988 161 </p></li><li><p>by the frequency translator FT from the optical angular frequency w1 to w2. Before launching into the optical fiber, the two signals are recombined by means of another polarization beam splitter as this operation allows no power loss in the coupling. The received signal can therefore be written as </p><p>s ( t ) = ( x ~ , + yAl)exp[ - j ( m l t + + ( t > + 77m(t))l </p><p>+(xA, + yA1,.)exp[ - J ( W 2 f + +(I> + PI1 (1) </p><p>where x and y are the unitary vectors along the x and y axes, 2A' = 2(A: + A f ) is the received optical power, + ( r ) is the transmitter phase noise, m( r ) is the transmitted binary mes- sage, and 8 is the phase mismatch between the two trans- mitted optical signals. At the receiver the polarization beam splitter allows selection of the x and y components of the combined signal and local oscillator (LO) beams, yielding the following expressions for the signals on the electrical branches: </p><p>I , ( t > =Is,( t ) + W>I2 + f l , ( t > </p><p>( 2) </p><p>where s , , s , , I , and 1 ( = 1, ) indicate the polarization compo- nents, respectively, of the signal and the LO along the x and y axes, and n , ( t ) and n , ( t ) are the shot-noise processes on the two branches. </p><p>Neglecting the DC terms and the beat terms proportional to A t and At at angular frequency wl2 = w2 - w l , Eq (2) yields for I , ( f ) the expression: </p><p>where = wj - wl, wZ3 = w3 - w2 with w 3 , \ c ( t ) , and L, ( = L,. = L ) , respectively, the angular frequency, the phase noise, and the x-component amplitude of the LO. A similar expression can be obtained for Zr( t ) . </p><p>On each branch the two beat terms are selected by means of the ideal bandpass filters 4 and F, . The filters 4, centered at wl3 , are assumed 2 R + kB, wide while F2, centered at w23, are kB,. wide, R being the bit-rate and B , being the sum of the laser linewidths. The positive parameter k has to be chosen to transmit both the modulated and the reference signal undistorted through the filters and to limit the shot- noise. The frequency spacing between the two transmitted signals has been chosen equal to 4( R + kB,.) in order to have negligible spectra overlapping. The beat signal between the component at w23 of I , ( r , and that at of I,(,y), is ex- pressed by </p><p>where oL2 = ol3 - w23 and Nxc,.,(t) is a term due to the shot-noise, which can be considered a Gaussian process. Therefore, Eq. (4) shows that the signal is not affected by the phase noise of the optical sources so that a lot of problems of conventional PSK schemes are avoided. </p><p>As shown in Figure 1 a further mixing with a carrier at the shift frequency wl2, recovered by a PLL, provides two base- band signals. The sum of these signals gives the decision function in which the influence of the received polarization is eliminated. </p><p>10 60 110 </p><p>Ps (ph lb i t ) </p><p>Figure 2 Error probability P , vb. received signal power P, in photons/hit (ph/hit) for various values of the sum of R / E , ~ with the assumption L = 8 </p><p>\ \ </p><p>\ </p><p>\ +&- +-+- </p><p>1 2 5 10 20 50 100 200 500 </p><p>Figure 3 System penalt) vs R / B , ( - proposed system, --- PSK heterodyne system) </p><p>Assuming a Lorentzian laser linewidth, the signal-to-noise ratio S / N and the error probability can be calculated. The system performance in terms of the error probability P, vs. signal power Pr (in photons/bits) has been evaluated for different values of R/B,., where R is the bit rate and is shown in Figure 2, where the absence of a phase-noise-induced floor for P, can be noted. Figure 3 shows the penalty vs. R / B , with respect to the ideal PSK heterodyne system. This penalty is compared with that relative to a conventional PSK heterodyne system which is not feasible for R / B , < 100 [l], while it is confirmed that the proposed technique allows carrying out the transmission even for low values of R / B , with a penalty of a few dB's. </p><p>CONCLUSIONS </p><p>In conclusion, we propose a PSK transmission system in which a reference signal for the phase-noise compensation is transmitted. The system performances have been evaluated and have shown its feasibility even in the presence of a large laser linewidth. Moreover, the adopted technique allows a sensible reduction of stimulated Brillouin scattering [6] and is compatible with a polarization independent receiver. </p><p>162 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 1 , No. 5, July 1988 </p></li><li><p>REFERENCES 1. J. Salz, Coherent Lightwave Communications, A T & T Tech. </p><p>Journul, Vol. 64, No. 10, Dec. 1985, pp. 2153-2209. 2. I. Garrett and G. Jacobsen, Theoretical Analysis of Heterodyne </p><p>Optical Receivers for Transmission Systems Using (Semiconduc- tor) Laser with Non-Negligible Linewidth, IEEE J . Lightwave Tech., LT-4, No. 3, 1986, pp. 323-334. </p><p>3. R. A. Linke, B. L. Kasper, N. A. Olson, and R. C. Alferness, Coherent Lightwave Transmission over 150 Km Fiber Lengths at 400 Mbit/s and 1 Gbit/s Data Rates Using Phase-Modulation, Electr. Lett., Vol. 22, No. 1, 1986, pp. 30-31. </p><p>4. T. Matsumoto, K. Iwashta, and T. Imai, 400 Mb/s Long-Span Optical FSK Transmission Experiment at 1.5 pm, Tech. Digest of IOOC-EOOC 85, Venice, Oct. 1-4, 1985. </p><p>5. R. Wyatt and J. Devlin, 10 kHz Linewidth 1.5 pm InGaAsP External Cavity Laser with 55 nm Tuning Range, Electron. Lett., Vol. 19, No. 3, 1983, pp. 110-112. </p><p>6. D. Cotter, Suppression of Stimulated Brillouin Scattering During Transmission of High-Power Narrowband Light in Monomode Optical Fibres, Electron. Lett., Vol. 18, No. 15, July 1982, pp. </p><p>7. B. Glance, Polarization Independent Coherent Optical Receiver, 638-640. </p><p>IEEE J . Lightwave Tech., LT-5, No. 2, 1987, pp. 214-276. </p><p>Receiued 4-18-88 </p><p>Microwaoe and Optical Technolou Letters, 1/5, 161 -163 0 1988 John Wiley & Sons, Inc. CCC 0895-2477/88/$4.00 </p><p>SOLUTIONS OF SOME TRANSMISSION LINES Weigan Lin Chengdu Institute of Radio Engineering Chengdu, 61 0054, Sichuan Peoples Republic of China </p><p>Wan Changhua Shanghai Research Institute of Microwave Technology Shanghai Peoples Republic of China </p><p>KEY TERMS Slab line, triangular, square, hexagonal slab line, conformal niupping </p><p>ABSTRACT Solutions of the unscreened slab line, the triungular slab line, the square slab line, and the hexagonal slab line are obtained bb, means of conformul mapping and the continuity of electric potential. Both the method and the results are new. Al l the solutions are believed to he uirtuul!y exact for all ranges of the ratio of the dimensions. </p><p>INTRODUCTION </p><p>The coaxial lines of an outer N-regular polygon concentric with an inner circle are extensively used in microwave filtering [ l , 21, microwave measurement [3], high package transmission, and impedance transformations [4, 51. How to determine their characteristic impedances is permanently the subject of great interest. Over the past years, laborous work has been done for this and remarkable achievements have been gained [6-191. The objective of this paper is to make fuller use of conformal mapping and the continuity of electric potential on the basis of ref. [17] in order to find more accurate and more believable </p><p>solutions for the unscreened slab line, the triangular slab line, the square slab line, and the hexagonal slab line. </p><p>THEORY </p><p>1. General Approximate Formulas for the Characteristic rmped- ance of the N-regular Slab Line. As an auxiliary procedure, we investigate the coaxial system consisting of an outer N-regular polygon and an inner N-regular multifin in Figure l(a), 1, of which is mapped into the upper half-plane of the t-plane by the S c h w a transformation </p><p>According to the corresponding points, we have </p><p>From (2) and (3), we readily derive </p><p>By the Jacobian elliptic function transformation of modulus k , 9 </p><p>t = sn( w , k N ) ( 5 ) </p><p>we map the upper part of the t-plane into the corresponding rectangular region in the w-plane. Thus, we can write the characteristic impedance of the system in Figure l(a) as </p><p>where p and e are the permeability and permittivity of the medium filling the space between the two conductors and K ( k , ) and K ( k , ) are the first kind complete elliptic in- tegrals of modulus k:, and k, , respectively. There exist relations K( k N ) = K ( k h ) and k; = /%. For the arbi- trary equipotential line FGH in the z-plane (i.e., FGH in the w-plane) in Figure 1, it can be proved that JzG/ and lzIIl (the property of IzF( is the same as that of (zHI for symmetry) are extreme values in that both tG and zH satisfy Re(z* dz/dtdt/du) = 0 [7, 17, 181, where Re means the real part of and z* = x -Jy. It is clear that lzGl and Iz/,I are respectively, minimum and maximum for the equipotential lines close to the inner conductor, and (zGI and I z H ) are, respectively, maximum and minimum for those close to the outer conductor. Based on this fact, one can conclude that there exists only one equipotential line on which (zGJ = IzH). If this equipotential line is approximately an arc, we may use it to obtain our results. In fact, the line is virtually circular for smaller k,. Letting the diameter of the quasicircle be d </p><p>MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 1, No. 5, July 1988 163 </p></li></ul>

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