708 IEEE TRANSACTIONS ON COMMUNICATIONS. VOL. 40, NO. 4, APRIL 1992 Theory of Polarization Shift Keying Modulation Sergio Benedetto, Senior Member, IEEE, and Pierluigi Poggiolini Abstract-A rigorous analysis of digital coherent optical mod- ulation schemes using the state of polarization as the modulating parameter is presented, which permits to obtain the exact perfor- mance of all the polarization-based modulation schemes proposed in the literature so far, including a differential demodulation scheme, named DPOLSK, which does not require either electro- optic or electronic polarization tracking. Preliminary results involving multilevel transmission schemes based on the state of polarization are introduced. A spectral analysis of POLSK signals is also proposed. I. INTRODUCTION S an alternative to the application of the standard co- A herent modulation techniques like ASK, FSK, PSK and DPSK to coherent optical communications, modulation meth- ods exploiting the vector characteristics of the propagating light radiation have been recently proposed and/or experimen- tally demonstrated in laboratory 131-161. They use the state of polarization (SOP) of a fully polarized lightwave as the information-bearing parameter, exploiting the two orthogonal channels available in free space as well as in a single-mode fiber propagation. In free space, two orthogonal input signals maintain their state of polarization while propagating. In the case of single-mode fibers fed by a monochromatic light source, orthogonal SOP pairs at the input lead to orthogonal output SOP pairs, although the input state of polarization is not maintained in general. Moreover, careful measurements reported in [13], [12], and [14] have shown that depolarization phenomena or polarization dependent losses are of little importance even after relatively long fiber spans. These properties are crucial to digital modulation based on SOP, called POLarization Shift Keying (POLSK). Demodulation and detection is accomplished through the analysis of the SOP. A SOP is fully described by the knowl- edge of the Stokes parameters [ 1, ch. 101. We will call Stokes receiver a coherent heterodyne receiver extracting the Stokes parameters from the IF signals (see Fig. 1). All so far proposed POLSK systems make use of a binary modulation scheme (2-POLSK), i.e., information is sent by switching the polarization of the transmitted lightwave be- tween two linear orthogonal SOP’s. In the three-dimensional space defined by the Stokes parameters two orthogonal SOP’s map onto opposite points with respect to the origin. Thus, de- tection of binary modulation schemes is simply accomplished by looking at the sign of the scalar product of the received SOP Paper approved by the Editor for Modulation Theory and Nonlinear Chan- nels of the IEEE Communications Society. Manuscript received October 26, 1989; revised May 23, 1990. The authors are with the Dipartimento di Elettronica, Politecnico Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy. IEEE Log Number 9107321. TRANSMITER I POLAFUZATION BEAM SPLIlTER % n -D U Fig. 1. Block diagram of the receiver front-end extracting the Stokes parameters from the received signal. vector in the Stokes space with a reference vector representing one of the received SOP’s in the absence of noise. This reference vector depends on the fiber induced changes on the transmitted SOP’s and for this reason all 2-POLSK systems must somehow keep track of it. It is straightforward to see that the system proposed in [3] makes use only of the 52 channel of Fig. 1, while the system outsketched in [4] exploits the SI path. A complete Stokes receiver is used by the system proposed in [5]. A fourth interesting scheme’ (JMPSK) has been presented in [6], which uses a rather different signal processing. All these schemes can be viewed as POLSK systems. The above-mentioned reference recovery is accomplished electro-optically in [3],[4]. This implies that they can make use of only part of the Stokes receiver because it is supposed that active electro-optic controls force the received SOP’s to align with a specific axis of the receiver Stokes space reference. The third and fourth systems, which will be shown to be equivalent, perform an electronic reference recovery by means of long-term averages over the received signals. In terms of performance, approximate results based on signal-to-noise ratios [4] or to additive Gaussian hypothesis applied to the noise perturbing the Stokes parameters 1.51 have been presented. In [SI the exact performance of the system proposed in [3] has been inferred. In 1321 the performances of ‘For this system the authors suggest the use of an unbalanced power splitting between the polarization “channel” bearing only the reference carrier and the orthogonal one carrying the modulated signal. As a result the transmitted SOP’s are no longer orthogonal and the signal tends to become phase modulated instead of polarization modulated. Therefore this version of the scheme gets out of the scope of the present work. Throughout this paper we shall refer strictly to the balanced-power 2-POLSK modulated version of the system. 0090-6778/92$03.00 0 1992 IEEEBENEDETTO AND POGGIOLINI: POLARIZATION SHIFT KEYING MODULATION 701) the system [3] as a function of the depolarization effect taking place in the fiber are evaluated. All the known results place POLSK modulation, in terms of required power, between ASK and DPSK, with a theoretical degradation of 3 dB with respect to DPSK in the absence of laser phase noise. As to the effect of the laser phase noise, a general analytical treatment in coherent optical heterodyne receivers is not avail- able. However, numerical analyses and experimental results have proved the considerable insensitivity to phase noise of receivers based on non-linear memoryless processing of the signal, provided that the IF filter bandwidth is large enough to avoid phase-to-amplitude noise conversion (91-[ll]. POLSK schemes can be thought of belonging to this class of sys- tems, also called PNCHR (phase-noise-canceling heterodyne receivers) [32]. In this