Polarization-Induced Distortions in Optical Fiber Networks with Polarization-Mode Dispersion and Polarization-Dependent Losses

Transcription

1 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL Polarization-Induced Distortions in Optical Fiber Networks with Polarization-Mode Dispersion and Polarization-Dependent Losses B. Huttner, C. Geiser, and N. Gisin, Member, IEEE Invited Paper Abstract We review the formalism required to investigate the combined effects of polarization-mode dispersion (PMD) and polarization-dependent losses (PDL) in optical fiber networks. The combination of PMD and PDL may lead to anomalous dispersion, which is not correctly described by a direct application of the Jones matrix eigenanalysis (JME) method. This calls for a careful assessment of PMD measurement methods in the presence of PDL. We also present a theoretical analysis of distortions in analog transmissions, and computer simulations of digital transmissions. These show that distributed PDL increases the power penalty of the transmission more than lumped PDL at the end of the channel. Index Terms Optical fiber dispersion, optical fiber losses, optical fiber measurements, optical fiber polarization. I. INTRODUCTION WITH the introduction of long-distance all-optical networks, optical communication systems are becoming more and more complex. Typically, a modern system includes several different components, such as optical amplifiers, wavelength-division multiplexing (WDM) couplers and add drops, optical isolators, and circulators. In contrast with standard single-mode optical fibers, these components may possess significant polarization-dependent loss (PDL). The optical network should thus be modeled as a concatenation of birefringent elements with random polarization-mode couplings, representing essentially the fiber itself, intertwined with several elements with PDL. The study of the interaction between this PDL and the polarization-mode dispersion (PMD) of the fibers is the subject of this paper. The organization is as follows. In Section II, we review the formalism, first developed in [1]. The key concept of principal states of polarization (PSP) is extended to include PDL. The main consequence of PDL is that the two PSP s are not orthogonal anymore, nor do they represent the fastest and slowest propagating polarization states. In Section III, we present some pe- Manuscript received August 16, 1999; revised January 14, This work was supported in part by Swisscom and the Swiss OFES. B. Huttner was with the Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland. He is now with Luciol Instruments, CH1260 Nyon, Switzerland. C. Geiser and N. Gisin are with the Group of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland. Publisher Item Identifier S X(00) culiar results, obtained on carefully chosen concatenations. For example, the polarization-induced dispersion may be larger than the sum of the dispersion from each of the elements building the concatenation [2]. Moreover, we show that we can build a concatenation with zero differential group delay (DGD), but which nevertheless introduces pulse spreading [3]. These results have been tested experimentally. They emphasize that the standard methods of measuring and modeling PMD [4] [7] have to be carefully adapted to the case of PMD PDL [8]. In Section IV, we perform a theoretical analysis of an analog transmission link, with both PMD and PDL. In particular, we show how to calculate the distortions in the radio frequency (RF) signal. In Section V, we turn to computer simulation of realistic digital systems. We calculate the bit error rate (BER) and power penalty. These show that the interaction between PMD and PDL induces more severe distortions than the two effects separately. Moreover, distributed PDL increases the power penalty of the transmission more than the same amount of frequency-independent PDL added at the end of the link. II. FORMALISM This formalism for addressing the effect of PMD and PDL was initially developed in [1]. In this section, we will first recall the main results, referring the reader to [1] for some derivations. Then, we will present a new analysis that enables the separation of the effects of PMD and PDL. A. The Transmission Matrix The standard model for a single-mode telecom fiber is a concatenation of optical fiber trunks, each supporting two polarization modes. We assume that these two modes are uncoupled, all polarization-mode coupling taking place at the junctions between the trunks. Assuming furthermore that the birefringence of the th trunk is independent of wavelength, is equal to DGD between the two modes (this is an excellent approximation for fibers and components used in optical communication, at least over the wavelength range of interest). PDL is introduced through the loss coefficient, defined from the differential loss between the two polarization modes by (1) X/00$ IEEE

2 318 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 where, respectively,, are the intensities of the polarization modes with lowest, respectively, highest, loss. Note that is the th trunk PDL expressed in db. In Jones formalism [9], the polarization state along the fiber is given by the complex-valued two-dimensional (2-D) Jones vector. The evolution of, along a concatenation of trunks is described as follows: where with being the polarization-independent overall loss, and (2) (3) optical frequency in first order. For such states, the Jones vector satisfies the following [7]: where denotes the partial derivative with respect to and is a complex number. In the case of pure PMD, i.e., for all, is real and equal to the differential group delay [7]. When the fiber has PDL, becomes complex. We shall show that its real part is still equal to the differential group delay, and its imaginary part is equal to the frequency derivative of the differential attenuation between the two principal polarization states in the fiber [1]. From (6), we deduce that is an eigenvector of the 2 2 matrix (6) (7) where are the Pauli matrices [9]: (4) where is the inverse matrix of (it always exists, except if one is infinite, which would correspond to a perfect polarizer). To solve this eigenvalue equation, we use (4) to get a recursion relation for Hence, describes the polarization rotations and relative changes in intensity, whereas describes the global attenuation. This global attenuation plays no role in the polarization properties, and will henceforth be omitted. The directions of the vectors and define the two polarization modes of the th trunk. For simplification, we shall assume that and are parallel, though our formalism does not require this (in any case, it is easy to accommodate the more general case by simply combining one element with only PDL and one element with only PMD). We thus introduce the unit vector defining the polarization mode of trunk by with. We make the usual assumptions that the attenuation of the PDL elements and the birefringence of the HiBi fibers are both independent of the wavelength. In this case, phase and group birefringence are equal. This has been well-verified experimentally [10]. Note that since the transmission matrix is made from a product of noncommuting matrices, one cannot separate directly the PMD and the PDL. In all the following, we shall adopt a notation similar to (2): The 2-D, complex components, Jones vectors will be denoted by with an index; the 2 2 transmission matrices will be in boldface; and the three-dimensional (3-D) real vectors will have the usual arrow, or hat for unit vectors. B. Principal States of Polarization Because of the explicit frequency dependence in (2) and (4), even when is independent of the optical frequency,asis the case if the input polarization state is determined by a polarizer or by geometric parameters of the laser, the output polarization state depends on the optical frequency. The principal polarization states [4] are defined as the polarization states such that the outcoming polarization is independent of the (5) The initial condition is:, the unit matrix, for all. Consequently, and since (8) preserves the trace, one has for all. We now use the fact that the Pauli matrices form a basis for the matrices of zero trace, so that there exists a 3-component complex vector such that In the case of pure PMD (i.e., all ), the vector is real and is called the principal state vector. Physically, the direction of the principal state vector is the direction of the fast PSP, while its length is the DGD. When the transmission link includes PDL, the vector becomes complex. The relations between the complex vector, the PSP s on the Poincaré sphere, and the DGD are however more complicated than in the case without PDL [1]. The most important consequence is that the two PSP s are no longer orthogonal. This is easily seen from (9). When there is no PDL, since is real, the matrix is Hermitian. Therefore, it has two orthogonal eigenvectors, the PSP s. When PDL is added, since is a complex vector, the corresponding matrix is no more Hermitian, and its eigenvectors are not necessarily orthogonal. As we shall emphasize below, the nonorthogonality of the two PSP s is responsible for several peculiar features. From (7) and (9), we rewrite the eigenvalue equation for the PSP s as This implies an important relation for (8) (9) (10) (11)

3 HUTTNER et al.: OPTICAL FIBER NETWORKS WITH PMD AND PDL 319 so that the two eigenvalues are opposite:.in the following, we will define as the eigenvalue with positive real part, and the two eigenvectors in Jones formalism by, corresponding to the eigenvalues, respectively. Note that the two vectors, are defined at the output of the system. The DGD, is thus simply the real part of (12) where is the frequency derivative of the differential attenuation of the two PSP s (see Section III-B). Compared to the DGD, we shall call the differential attenuation slope, or DAS. Using (8) and (9), the recursion equation for reads (13) where is defined in (5). This equation generalizes the equation for PMD to the case where some elements along the line have PDL. In the limit of short trunks, we define the local modal birefringence at position along the fiber by (14) where is the length of trunk [ is the difference of the inverse of the group velocities of the two modes in ps/km]. We similarly define the local PDL. Following the same line, we then derive a dynamical equation of evolution for the complex vector (15) This equation generalizes the dynamical equation for PMD [5], [6] to the case where some elements along the line have PDL. The calculation of gives all the information about the PSP s. However, the relation between and the PSP s on the Poincaré sphere is more complicated than for optical links with no PDL. In particular, the decomposition of into real and imaginary parts,, is not the most interesting, as we shall see in the following. It is more relevant to define a new normalized vector by (16) This vector can by decomposed into its real and imaginary parts:. Note that this vector does not have the same direction as. However, it is that will enter into the expression of the PSP s on the Poincaré sphere. The calculation is performed in Appendix A. The two real vectors representing the PSP s are, for the two eigenvalues, respectively, and are given by (17) The overlap between the two Jones vectors, easily obtained from (17) where. is (18) C. Most- and Least-Attenuated States The principal polarization modes, discussed in the previous Section, are characterized by their relative stability in frequency. However, in the presence of PDL, it is also natural to study the polarization modes most and least attenuated. The calculation leading to the dynamical equation for PDL is similar to the one leading to the dynamical evolution of the PSP s [1]. It is briefly outlined in Appendix B. Note that the two polarization modes most and least attenuated are always mutually orthogonal, both at the input and at the output, contrary to the principal polarization modes. In correspondence to the principal state vector, we introduce the PDL vector, parallel to the less-attenuated polarization state on the Poincaré sphere and of norm defined by (19) where is the max/min transmission factor. The PDL vector is directly measurable, on the global link. Moreover, it follows a closed dynamical equation (20) D. Separating PMD and PDL The above formalism emphasizes the evolution of the principal state vector along the fiber. In most cases however, a measurement result only gives a global property of the system. Measurement of distributed PMD, for example, has only recently begun to appear [10] [14], and is still in its infancy. Therefore, it is of practical interest to consider the concatenation globally, and investigate the kind of information available through a standard measurement. For example, using standard polarization measurement, one can measure the Jones matrix of the concatenation. This matrix contains all the relevant information about the birefringence properties of the system. However, the same matrix could also be obtained from different concatenations, corresponding to different physical realizations. In this Section, we introduce the simplest of these realizations, which is a system where PMD and PDL are separated, the PMD element being at the beginning and the PDL element at the end of the fiber section. This separation will prove especially useful in the analysis of analog systems. Let us consider a general concatenation, with random couplings. The transmission matrix of the concatenation is. It turns out that there is a simple way to separate the effects of PMD and PDL, based on the polar decomposition of complex matrices. This theorem states that any complex matrix, say

4 320 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 can be decomposed into a unitary matrix positive one, which in our case gives and a Hermitian (21) Since is a unitary matrix, it represents an effective PMD, while represents the effective PDL. This decomposition shows that any concatenation with both PMD and PDL, is equivalent to the concatenation of a purely birefringent element,, followed by an element with only PDL. Note that both elements have an explicit frequency dependence. It is easy to show that the PDL element is given by (22) (Note that the square root of a real positive matrix is defined as the matrix with the same eigenvectors, and the square root of the eigenvalues) and the birefringent one by (23) ( is always invertible, except for a perfect polarizer, which will not be considered here.) The complicated concatenation of several PDL elements intertwined between birefringent elements can thus be replaced by a much simpler system, with only one PMD element followed by one PDL element. Note, however, that is not equal to the concatenation of all the PMD elements, nor is equal to the concatenation of the PDL elements. Moreover, this is not equivalent to having a single element with given PDL axis and frequency-independent PDL value at the end of the trunk. In our case, both the PDL value and the PDL axes of matrix are frequency dependent. The precise frequency-dependence depends on the PMD element at the front. It is easy to obtain the most- and least-attenuated states of the system. Since all the PDL is in matrix, these states correspond also to the most- and least-attenuated states of itself. Therefore, the PDL axes at the output are also given by the eigenaxes of. All the PDL properties of the system are thus given by the properties of alone. For PMD, the problem is more complex. The DGD of a complete system with PDL is given by the imaginary part of the eigenvalues of the matrix [see (9), (10), and (12)]. Here, we define an effective DGD from the eigenvalues of the matrix (since is unitary, these are real). Mathematically, since the calculation of the DGD involves a frequency derivative of the complete transmission matrix, which includes also the frequency dependence of, the real DGD of the system is not equal to the effective one, obtained from the unitary matrix only. However, numerical simulations show that, in general, the real DGD is still approximately equal to the effective one. III. PDL-INDUCED ANAMALOUS DISPERSION Having developed the tools in the previous Section, we now turn to applications. We will show that the interaction PMD PDL generates several peculiar effects. In this section, we present three examples, some of which will be sustained by an experimental verification. In a first example, we show that the DGD of a concatenation may be larger than the sum of the DGD s of its elements. We then calculate explicitly the dispersion generated by the concatenation, and show that it can become even larger than the DGD. Last, we build a concatenation with zero-dgd, but significant pulse broadening. All these examples rely on carefully chosen concatenations. Accordingly, the values of the physical parameters are somewhat unrealistic for real telecom systems. These examples are, therefore, meant to emphasize that the tools developed to understand and measure PMD in optical fiber systems have to be carefully reassessed in the presence of PDL. Results concerning more realistic telecom systems will be given in the next two Sections. A. Anomalous DGD We first present a simple example [2] consisting of one pure PDL element sandwiched between two fibers with PMD. We set the axis of the first and third elements parallel to the axis (on the Poincaré sphere) and that of the central PDL element in the direction (i.e., orthogonal to the PMD fibers in the Poincaré sphere, or at in ordinary space). Solving (13), one obtains (24) where,, and are the PMD delays of the three elements, describes the PDL of the second element, and. In this case, the eigenvalue of (10) is complex, and the DGD corresponds to its real part. The explicit expression is cumbersome, and shall not be given here, but is rather plotted in Fig. 2 (dash-dotted curve) for the experimental system considered later. The effect of the PMD of the central fiber is to modulate the DGD as a function of the wavelength. The maximum value of the DGD is (25) For, i.e., in the presence of PDL, (25) shows that, when is small with respect to and,. This is a surprising result: the presence of PDL in a birefringent optical fiber can increase the DGD above the sum of the DGD of all the elements. In addition to its counterintuitive aspect, this point could have concrete implications for systems that combine birefringent fibers and components with PDL, since the distortion and BER may be larger than expected from simple considerations. In order to verify this result experimentally, we measured the DGD of the above concatenation. The experimental setup is described in Fig. 1. The system we used consisted of a PDL fiber, with PMD ps and PDL of 18 db, corresponding to, sandwiched between two Hibi fibers, with PMD ps. We mounted these fibers on special connectors, which enabled us to rotate the axes of the fibers. Let us emphasize again that a value of 18 db of PDL is certainly not realistic for telecom systems. It was chosen to enable a direct test of the theory. The instrument used to measure the DGD of the concatenation was a HP polarization analyzer (HP8509B) with an external tunable laser (HP8167A). This instrument gives directly the DGD of the device under test as a function of the wavelength [7], and was therefore well suited for our purposes. Moreover, its results are accurate even in the presence of large

5 HUTTNER et al.: OPTICAL FIBER NETWORKS WITH PMD AND PDL 321 The calculated average is ps, and is represented by a dotted line. We then rotated the axis of the PDL fiber by with respect to both Hibi fibers. This is represented by the dashed curve in Fig. 2. The theoretical calculation corresponding to this setup is represented by the dashed-dotted curve, and is also in excellent agreement with the experimental curve. With the experimental values used, the maximum DGD obtained from (25) was ps, which is much higher than the maximum we get with all the fibers aligned. Our theoretical analysis, combined with the above experimental result, already show that special care has to be taken to analyze the effects of combined PMD and PDL. The usual intuition about combining the various elements in the concatenation has to be considered cautiously. Fig. 1. Experimental setup for DGD measurements. This experimental setup implements JME. It measures the Jones matrix of the fiber under test as a function of the wavelength, which enables to obtain the DGD (see [7] for more details). In our case, the fiber under test is a concatenation of three trunks, one trunk with PDL being sandwiched between two Hibi trunks. B. Anomalous Dispersion In Section III-A, we have seen that the DGD of a concatenation of fibers with PMD and PDL may be higher than the sum of the DGD s of the elements. This surprising result may be complemented by the explicit calculation of the dispersion of a pulse. In general, the output pulse can be written as a superposition of the two PSP s with some coefficients and. Such a description is sufficient as long as the spectral spread is small enough for us to validate a first-order development in. This is expected to be the case whenever the coherence time is larger than the DGD. This is, in fact, the interesting domain for optical transmission. In this paper, we only analyze the case of a Gaussian pulse, to show the combined effect of PMD PDL. The influence of chirp could be added to the formalism, but this will not be done here. In the frequency domain, the outgoing field reads (27) Equation (27) explains the physical content of. If we replace it by (12), we see that the time delay for each mode is given by, corresponding to a DGD of, while corresponds to differential attenuation between the two modes, leading to a DAS of. From this, the outgoing intensity can be computed as Fig. 2. Anomalous DGD. The fiber under test is a concatenation of three elements: one HiBi fiber with PMD 5.65 ps, followed by one PDL fiber with differential attenuation 18 db and PMD 1.01 ps, and terminated with another HiBi fiber with PMD 5.65 ps. The full curve represents the DGD when all three axes are aligned, the dotted curve being the theoretical prediction (sum of the three DGD s). The dashed curve represents the anomalous DGD, obtained when the PDL fiber is rotated by 45 with respect to both HiBi fibers. In this case, the maximum DGD can be larger than the sum of all DGD s. The dashed-dotted curve is the corresponding theoretical prediction. PDL. Our results are presented in Fig. 2. We first aligned the axes of the three fibers, to get the sum of the DGD for each component. This corresponds to the full curve of Fig. 2, and is in excellent agreement with the theoretical prediction (26) (28) Because of PDL, the principal polarization states are not orthogonal, in general. Equation (28) thus describes interferences between the fast and slow principal polarization modes. Their overlap has been derived in (18), and has to be included in the calculations. Using (28), one can calculate the time of flight as a function of and. The calculation is rather lengthy, but completely straightforward, involving only integral of Gaussian functions. The result for the slowest and fastest time is (29) Note that this simple result is only correct for a Gaussian pulse, as defined in (27). The addition of chirp would lead to more complicated results. Surprisingly, for given DGD, this can be arbitrarily large. However, the fastest pulse will be more attenuated than a pulse polarized along one of the two principal polarization modes. The physical content of (29) is easily understood in terms of interference between the two PSP components. Indeed, because of PDL, the principal modes are no longer mutually orthogonal, as is the case for pure PMD. Hence, if both

6 322 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 modes are excited by a pulse, the output pulse is distorted by the interference between both principal modes. As an example, assume that the slow mode has one-fourth of the fast-mode intensity, as shown in Fig. 3, and that the interference is destructive. This amounts to cut the back of the fast pulse, resulting in an output pulse whose center is more advanced than that of the fast pulse. The same explanation, with the fast mode now being the weakest, leads to the slowest pulse. For optical transmissions, one can consider that the polarization of a train of pulses is constant for a very large number of pulses. Accordingly, the most relevant parameter is not the time of flight, which will not cause distortions if it fluctuates only very slowly with respect to the pulse repetition rate, but rather the pulse spreading, which may reduce the eye opening. It is again straightforward to compute the largest pulse spreading, which is obtained when the two coefficients, and are equal, and the relative phase between the two terms in (27) is, and reads (30) Fig. 3. Anomalous group velocity. Illustrative example of the displacement of the center of a pulse, due to the interference between the two PSP s. The dashed curve represents the fast PSP, while the dotted one represents the slow PSP. The relative intensities of the PSP s is chosen to be 4. The full curve is the electric field envelope of a pulse, which is the sum of the above two curves. The phase is chosen to give a destructive interference, which removes the back part of the pulse, thus displacing it in the forward direction. A similar setup, with stronger slow pulse, would displace the pulse in the backward direction. This again can be arbitrarily larger than, which is the value obtained for pure PMD. Here again, it is easy to get the physical intuition behind this surprising phenomena in terms of interference between the PSP s. Assume that both principal modes have equal intensity, Gaussian profiles, and have delays of. A destructive interference will destroy the central part of the outcoming pulse, as is illustrated on Fig. 4. Because of this destructive interference, the spread of the output pulse is larger than (in the example of Fig. 4, it is almost twice as large). In Fig. 4, only the absolute value of the electric field is displayed, the intensity being equal to the square of this. The pulse obtained by interference is computed as the absolute value of the difference between the fast and slow pulse electric fields. Note however, that this spread is not of the same kind as the one generated with pure PMD. In the case of PMD, the envelope of the pulse is really wider at the output, while in our case, the spread is only larger due to the destructive interference at the center, as clearly seen in Fig. 4. The envelope of the pulse is not wider than the sum of the envelopes of the two components. However, a most important point here is that this effect does reduce the eye opening in optical transmissions. Since the extra spread is created by interference, it is very sensitive to even minute perturbations of the system. Therefore, one will easily move from destructive interference, generating an extra spread, to constructive interference, reducing the spread. The net result is thus an extra source of noise, which will reduce the eye opening. These results should have important consequences for the characterization of optical networks including elements with PDL. In this case, we now see that the calculation of the DGD, as is done with, Jones Matrix Eigenanalysis (JME) method for measuring PMD is not sufficient. The pulse spreading may become significantly more important than predicted by standard Fig. 4. Anomalous pulse spread. When the relative intensities of both PSP s are equal, the result of the destructive interference is to reduce the height of the pulse at the center, thus increasing the spread. The dashed curve represents the fast PSP, while the dotted one represents the slow PSP. The full curve is the electric field envelope of a pulse, which is the sum of the above two curves. Similarly, the spread could be reduced by constructive interference. Since this effect is created by an interference, it is very sensitive to minute fluctuations in the system. This would generate an extra noise in the pulse spread, which may lead to a degradation of the signal. JME. Note, however, that all the information is still in the principal state matrix, measured by JME. One needs to take into account not only the DGD, as in the standard method, but also the imaginary part of the eigenvalues. In fact, as we shall show both theoretically and experimentally in Section IV, we can build a special concatenation of fibers with both PMD and PDL in such a way that the DGD is naught, but which still generates pulse spreading. Analysis of more realistic systems, performed with a computer simulation, is left to Section V.

7 HUTTNER et al.: OPTICAL FIBER NETWORKS WITH PMD AND PDL 323 C. Pulse Spreading with Zero-DGD In this test [3], we analyze a slightly different concatenation, where we simply rotate the third fiber, so that its axis is now on the Poincaré sphere, and replace the central fiber to get an element with PDL, but no birefringence. Theoretical calculation shows that the DGD of such a concatenation is naught. However, a simulation of the interferometric method predicts nonzero PMD [3]. An experiment with the set of parameters of Fig. 5 confirms both predictions, with a very low value of the average DGD (below 0.1 ps), and a PMD value obtained from the interferometric method, which is one order of magnitude larger (for both the simulation and the experiment). When there is no PDL, the two eigenvalues of the PSP matrix [(10)] are real (no attenuation), so that a zero DGD really means no birefringence. However, when the fiber link has PDL, the eigenvalues becomes complex. In this case, the DGD will be zero whenever the eigenvalues are purely imaginary. As we shall see, polarization effects may still create pulse spreading. We consider a fiber section made of three trunks. The first and last trunks are purely birefringent, with birefringence along axes and on the Poincaré sphere, respectively. The second trunk has only PDL, given by along axis. We use the recursion equation, (13), to obtain the principal state vector. The eigenvalue is then given by. After straightforward calculations, we get (31) Since this is a negative value, it corresponds to a purely imaginary. This corresponds to zero DGD over the full wavelength range where our approximations (i.e., wavelength-independent birefringence and PDL) are valid. In Appendix C, we present a more general calculation, with arbitrary birefringence axes,,, and. It shows that the precise directions of the axes are not important. Negative can be obtained for a variety of setups. However, even though the DGD is zero, the device will still cause pulse spreading. Consider a short pulse (short with respect to ), propagating down the fiber. The first HiBi fiber separates the initial pulse into two distinct pulses, separated in time by the DGD of the first trunk, and polarized along and, respectively. When no PDL is present, the second HiBi fiber recombines the two pulses exactly. This device therefore has zero global birefringence. The effect of a PDL element can be understood as follows: a pulse polarized along direction, say, is decomposed along the two axes (axis with low attenuation) and (axis with high attenuation). The effect of the differential attenuation is to reduce the amplitude along, thus creating a rotation of the polarization toward. Since the two components along and experience a different rotation, the second HiBi fiber cannot recombine the pulses fully. For the case under consideration, the output state is therefore composed of three pulses separated by. This time-domain analysis clearly explains the pulse spreading, and the fact that it is predicted by the interferometric method. These theoretical considerations were also tested experimentally. The setup is quite similar to the one in Fig. 1. We first constructed an element with pure PDL and no PMD by concatenating a PDL fiber with a HiBi fiber with exactly opposite birefringence. The PDL of this device was 7 db. Fig. 5. Measurements of a concatenation with zero-dgd. The concatenation is built from one pure PDL element, with 7-dB differential attenuation and axis y, sandwiched between two HiBi fibers with 0.7 ps of PMD each and axes x and 0x, respectively. The predicted DGD of such a concatenation is zero. The full curve is the DGD obtained from a JME measurement, similar to the one in Fig. 1. The dashed curve is the average DGD, at ps, while the dotted curve is the PMD obtained with the interferometric method, at 0.97 ps. The inset gives the interferogram given by the interferometric method, the variance of which gives the PMD. This is a rather high value with respect to telecom components. The point was to choose a value, which would enable to verify easily the theoretical prediction. We then sandwiched this element between two other HiBi fibers, with birefringent axes orthogonal to one another, and at 45 of the PDL axis of the central element. The PMD of these fibers was 0.7 ps each. We measured this device with the interferometric method, using a commercial PMD analyzer. The result is shown in the inset of Fig. 5, and gives a calculated PMD of 0.97 ps. We then performed a standard DGD measurement on the whole device, by means of JME, with a HP polarmetric system. The result is presented in Fig. 5. The average DGD value is an order of magnitude lower than the interferometric PMD. Moreover, even the maximum value of the DGD between 1270 and 1320 nm is much lower than the interferometric PMD. This experimental result is in excellent agreement with the theory. We also tested this idea with a much more complicated concatenation of HiBi fibers and of PDL elements, chosen in such a way that the total DGD of the concatenation is still zero over a whole range of frequency. A simulation of the propagation of pulses through this device shows that, although the DGD is zero, they still experience significant broadening. This is confirmed by the interferometric method, which predicts nonzero PMD, in agreement with the broadening. These results show that the properties of the DGD alone are not enough to fully characterize the birefringence properties of an optical fiber system in the presence of PDL. IV. INFLUENCE ON ANALOG TRANSMISSION The separation between PMD and PDL, which was performed in Section II-D allows for a simple calculation of the distortion induced on analog systems. In such systems, the intensity of the laser is modulated at an RF, to transmit the signal. This intensity modulation generates an optical frequency modulation, the laser chirp, given by (32)

8 324 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 where is the initial laser frequency; is the chirped frequency; represents the chirp amplitude. Since the time delay in the propagation created by PMD is very sensitive to the optical frequency, this frequency modulation induces a modulation in the propagation time, which in turn generates a extra phase modulation of the signal. This phase modulation creates harmonics, which may result in crosstalk between channels (if the harmonics correspond to a different RF). The simplest way to characterize the quality of a links is to define the composite second order (CSO) coefficient [15] where and is the number of frequency products in the RF channel, is calculated from the first harmonic in the RF signal (33) The analysis is quite cumbersome, and was first derived in [16] and [15] for PMD alone. The derivation for the general case, with distributed PDL, is outlined in Appendix D. The analytical calculation gives (34) The first term includes both the chromatic dispersion, being the time delay induced by chromatic dispersion, and the PMD induced distortion, being the principal axes vector of the unitary matrix, defined in Section II-D. The second term corresponds to the interaction PMD PDL, being the PDL vector [17], [18] introduced in (19). Note that in (34), all the terms contain frequency derivatives, except. This is interesting, because it shows that the interaction PMD PDL generates first-order effects in the CSO, whereas only second-order effects were present for the pure PMD case. Apart form this contribution however, we see that the distortion is caused by the frequency dependence of the PDL vector. To our knowledge, (34) is the first general expression taking into account the distortions induced by both PMD and PDL on analog systems. It reduces to the expression of Poole and Darcie [16] for the case of a single trunk of frequency independent PDL at the end of the link. Note, however, that these are two very different cases. In [16], there is one PDL trunk only, while our model deals with any number of trunks, distributed amongst the PMD fibers. As shown in Section II-D, this is equivalent to a frequency-dependent PDL element, placed at the end of the link. V. INFLUENCE ON DIGITAL TRANSMISSION We now turn to digital transmission. The most important parameter characterizing the network is the BER of a transmission. Unfortunately, this parameter depends on a combination of several parameters, including the input laser and the detector, which means that it cannot be obtained from the properties of the transmitting fiber only. Consequently, in order to be able to analyze the BER as a function of PMD and PDL, one has to build a complete transmission system or rely on computer simulations. We use a commercial software (PTDS), with some homemade extensions to address our special needs. This software simulates all the elements of a communication system, including the laser and the detection system. We simulate several concatenations of similar fibers, changing the couplings randomly. We first present the results of simulations of the BER as a function of the DGD of the fiber. These clearly show that, for a given DGD value, the addition of PDL increases the BER. However, this increase is essentially caused by the interaction PMD PDL, and not by the PDL alone. We then turn to the power penalty induced by PDL, and show again an increase caused by PDL. An interesting point is that in some cases, the power penalty can reach an infinite value (which means that the BER is above even for large intensities). Last, we analyze the influence of PDL distributed along the fiber in several trunks versus one big trunk lumped at the end. This analysis shows that distributed PDL is much worse for a system. Note that in the following, we assume that our system has no chromatic dispersion. A. Analysis of the BER For reference, we first analyze a concatenation with no PDL in Fig. 6, which gives the distortions induced by PMD only. We plot the BER induced by a 600-km-long section, as a function of the DGD. This section is built of 600 trunks, each one having a PMD varying from 1 to 2 ps, with random couplings between the trunks. Each point in the figure represents one random choice of the couplings, which leads to a different value of the DGD. The average DGD is 36 ps, in agreement with the theoretical value ( ). We clearly see that the distortions are higher (larger BER) for higher DGD. Of course, even for high DGD, there exist cases with low BER, obtained when the input polarization is along one of the PSP s. The upper curve gives the limit of the BER, obtained when the input polarization is at 45 of the PSP s. We use this curve to show the maximum distortion induced by PMD alone. We now add 200 sections with 0.8 db of PDL each to the same concatenation, randomly placed and oriented with respect to the PMD sections, and analyze the BER in Fig. 7. The PDL is also a random variable, its average following again the same square-root law, to give 11 db ( ). We see that for concatenations with the same DGD as above, the BER can reach much higher values. All the points where the high BER cannot be explained by the DGD alone are represented with a square. To analyze further the cause of the larger BER, we plot the BER as a function of the PDL in Fig. 8, for the same concatenations as in Fig. 7. We see that there is no clear trend: The BER is not correlated to the value of the PDL alone. In addition, the concatenations with anomalous BER, represented by squares in both Figs. 7 and 8, have the same distribution as the other concatenations. In particular, they are not characterized by a higher PDL. This clearly shows that the extra distortion is really created by an interaction between PMD and PDL, but cannot be attributed to any of them separately.

9 HUTTNER et al.: OPTICAL FIBER NETWORKS WITH PMD AND PDL 325 Fig. 6. PMD-induced distortions. As a reference, we simulate the BER obtained from a fiber link with PMD only. We have 600 trunks with PMD varying from 1 to 2 ps each, leading to an average DGD of 36 ps. We perform 800 simulations with random couplings, and plot the resulting BER as a function of the DGD (crosses). The full curve is the envelope of the highest BER s, obtained when the input polarization is exactly between the two PSP s. Fig. 8. BER as a function of PDL. The points are the same as Fig. 7, but are now plotted with respect to the PDL of the link. We see that there is no clear trend, no increase in the BER occurring when the PDL increases. Moreover, the anomalous points in Fig. 7, represented by the squares, for which the high BER cannot be explained by PMD alone do not have a higher PDL on average. Therefore, the high BER has to be attributed to combined effects of PMD and PDL. Fig. 7. PMD PDL-induced distortions. The setting is the same as in Fig. 6, except that 200 elements with 0.8 db PDL are inserted at random between the PMD elements. The average DGD remains at 36 ps, and the average PDL is 11 db. The squares represent all the points for which PMD alone cannot explain the BER. B. Power Penalty For systems, the most important parameter needed to characterize the network is the power penalty induced by various effects. Therefore, we now compare the power penalty created by a link with only PMD and with both PMD and distributed PDL. The result from our simulation program is presented in Fig. 9. We see that the power penalty is significantly higher for a link with distributed PDL. Another interesting result is that the linear regime in the graph, where an increase in the intensity allows a reduction in the BER, is smaller for links with PDL. If the saturation is reached for a BER below, this may prevent the use of this optical link altogether. Fig. 9. PDL-induced power penalty. In order to obtain the power penalty, we use the standard method of calculating the BER as a function of the intensity, for various input polarizations. For each polarization, plotting log(0 log BER) as a function of I results in a curve, which is close to a straight line in the regions of interest. The difference between the highest (squares) and the lowest (circles) curve, obtained with different polarization states, gives the power penalty induced by polarization effects. Conventionally, it is calculated at a BER of 10. The dotted curves correspond to a system with no PDL, while the full ones correspond to a system with 200 elements with 0.1 db of PDL inserted at random between the PMD elements. Note that this calculation is performed for one particular realization of the random couplings. The dashed-straight line represents a BER value of 10. C. Distributed PDL Versus Lumped PDL As shown above, the main factor for signal distortions is not the PDL itself, but rather the interaction PMD PDL. In order to confirm this point, we now compare two optical links, one with distributed PDL, and one with lumped PDL at the end of the concatenation. The result is that distributed PDL increases the distortions. Therefore, it is not sufficient to specify the global PDL of a given system. The computer simulation uses a similar concatenation as previously, with 200 trunks with 0.1 db of PDL each intertwined with birefringent elements. The average PDL of the link

10 326 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 6, NO. 2, MARCH/APRIL 2000 Fig. 10. Distributed versus lumped PDL. The power penalty induced by PDL is calculated for three cases. Each point on the graph represents one simulation run, with a random choice of the couplings between the trunks. All three cases have the same DGD, 45 ps, and different PDL s. The pluses (+) represent the same fiber as in Fig. 6, with only PMD, but no PDL. The average power penalty for these runs is 1 db (dotted curve). The crosses (2) represent the same concatenation as in Fig. 9 with distributed PDL (200 trunks with 0.1 db each, placed at random between the PMD trunks). In this case, PDL fluctuates between each run, due to the random couplings. The average PDL is 1.4 db, well in agreement with the theoretical value of 0:1 p 200, while the highest value obtained during the various simulation runs is 2.2 db. The circles () represent a concatenation with one single trunk of PDL, with fixed axis and value, placed at the end of the fiber. We chose the highest PDL obtained with the distributed case, i.e., 2.2 db. We clearly see that distributed PDL causes a significantly higher power penalty (1.42 db on average, full curve) than even this worst case (1.06 db, dashed dotted curve). The spread in all these three systems, however, is not significantly different. This result emphasizes that distributed PDL is more detrimental to systems than a similar amount of PDL placed at the end of the link. is 1.4 db, which is quite a realistic value for modern systems. We make many simulations of the link with random couplings, and select the ones with a given value of the DGD, here 45 ps. As a reference, we first remove all PDL elements and calculate the power penalty for several runs. The result is presented in Fig. 10. The power penalty is fluctuating, with an average of 1 db. With the introduction of distributed PDL, the average power penalty increases to above 1.4 db. This is then compared with a lumped PDL of 2.2 db, placed at the end of the link. This value was chosen, because it was the highest PDL value we obtained during the previous simulations (the average was only 1.4 db, but since distributed PDL is a statistical quantity, it does fluctuate). Also note that the PDL element at the end is a single trunk, with fixed PDL axis. We see that the average power penalty is about 1.1 db, higher than the case with no PDL, but significantly smaller than with distributed PDL. Let us emphasize again, that the distributed PDL in all the simulations was below the 2.2 db used in the lumped case. The conclusion is that second-order effects [19], including the frequency dependence of the PDL, create a significant contribution to the power penalty. VI. CONCLUSION The interaction between PMD and PDL is a rather new topic, brought about by recent developments in optical networks. In particular, the new optical components in WDM systems possess some PDL, which must be taken into account. In this paper, we reviewed the formalism to investigate these new effects, and presented several new results. In particular, we presented the dynamical equation for the evolution of the principle states of polarization in the presence of PMD PDL [(15)]. We calculated the maximum pulse spread in the presence of PDL, and showed that it could be much larger than the differential group delay [(30)]. The measurement of the DGD alone is, therefore, not sufficient to estimate the distortions in a network with PMD PDL. However, the interferometric method, which is more directly linked to the pulse spreading gives the correct estimate. We showed that a link with distributed PDL can also be represented as a link with lumped PDL at the end, but that the corresponding PDL element has to be frequency dependent, both in PDL value and axis. This representation allows in particular for a much simpler analysis of analog systems, leading to the calculation of the first harmonic in the RF signal (34). We also performed simulations of digital optical transmissions. We showed that the added distortions are mainly caused by the interaction of PMD and PDL, rather than by PDL alone. Moreover, second-order effects, i.e., effects created by the frequency dependence of the PDL are significant. This was put in evidence by the fact that the distortions induced by distributed PDL, which leads to frequency-dependent effects, are stronger than the distortions induced by one single PDL element of the same value at the end of the link. These analytical and numerical results should now be incorporated

11 HUTTNER et al.: OPTICAL FIBER NETWORKS WITH PMD AND PDL 327 in all simulation tools, and should be taken into account by system engineers. APPENDIX A THE PSP S ON THE POINCARÉ SPHERE The starting point of the calculation is to note that the two components of in (16) are orthogonal: (this is straightforward from the fact that is normalized). Equation (10) is equivalent to (35) Using the relation between the Jones vectors and the 3-D real vectors on the Poincaré sphere, this leads to the two conditions for and From these, we derive the expression (36) (37) (38) where is a constant, which is not yet known. In order to find, we multiply (35) by, and use the relation to get a extra relation for from which we find, which leads to (17). (39) APPENDIX B DYNAMICAL EQUATION FOR PDL The calculations leading to the dynamical equation for PDL are very similar in principle to the ones leading to the calculation of the DGD. We shall, therefore, follow the derivation of Section II-B. The first step is to find an eigenvalue equation for the most and least attenuated states. Using (2), we get (40) where we have already dropped the global attenuation factor of (3). The most and least attenuated in states are, therefore, the eigenstates of [corresponding to the minimum and maximum value of ]. Equivalently, the out states are the eigenstates of, which therefore replaces of (7). We now use (4) to get the recursion equation for, which reads (41) where we now have dropped the explicit dependence in. To solve (41), we use the fact that the Pauli matrices plus the identity form a basis of the matrices, to write (42) In this case, since is a Hermitian matrix, its two eigenvectors are orthogonal. On the Poincaré sphere, the two corresponding 3-D real vectors are oriented along the direction of, with corresponding eigenvalues, where is the norm of. Following precisely the same procedure as in Section II-B, we go to the limit of short trunks [see (14)], and obtain after straightforward calculations the dynamical equation for the scalar and the 3-D real vector A first consequence of these equations is that constant. Since at the input of the fiber (i.e., at, i.e.,,, one obtains (43) (44) remains ) one has (45) for all positions along the fiber. The maximum and minimum attenuations and the PDL axes can then be computed from the solutions of (43) and (44). However, in order to calculate the transmission coefficient for any input state, it is easier to note that Using (42) and (45), we easily obtain (46) (47) Consequently, the transmission coefficient for an output polarization represented by (on the Poincaré sphere) is given by (48) The minimum and maximum transmission coefficients are thus given by (49) These results can be summarized by introducing the PDL vector parallel to the less attenuated polarization state on the Poincaré sphere and of norm defined by:. In this way, from (43) and (44), the PDL vector follows a closed dynamical equation which is (20) in the text. APPENDIX C CALCULATION OF FOR THREE TRUNKS (50) We take a more general concatenation than Section III-C, where the axes of the three trunks are arbitrary, namely,,