FI.,. HEWLETT. Automated Measurement of Polarization Mode Dispersion Using Jones Eigenanalysis

Transcription

1 FI.,. HEWLETT II:~ PCKRD utomated Measurement of Polarization Mode Dispersion Using Jones Eigenanalysis B. L. Heffner Instruments and Photonics Laboratory HPL May, 1992 optical, fiber optics, photonic subsystems, incoherent, photonics, lightwave components Polarization mode dispersion (PMD), which can limit the bandwidth of optical transmission links, has been difficult to measure in a manner independent of human judgment, leading to difficulties in automating the measurement. We demonstrate for the first time that PMD in any linear, time invariant network can be completely characterized by eigenanalysis of Jones matrices measured at a series of discrete wavelengths, even for networks exhibiting polarization-dependent loss. fast, automated system using a tunable laser and an accurate, real-time polarimeter affords temporal accuracy of approximately 2% down to a limit of several femtoseconds, as demonstrated by comparison with other techniques and comparison with known samples. Both the principal states of polarization and the group delay difference are measured as a function of optical frequency. Internal ccession Date Only Copyright Hewlett-Packard Company 1992

2 1 Introduction Thorough characterization of the optical components intended for high-speed transmission links requires accurate, repeatable measurement of polarization mode dispersion (PMD). PMD, which may limit transmission bandwidths in practical systems, is a fundamental characteristic of a network or device under test (DDT) that describes its propensity to split a narrow-band optical input pulse into two temporally separate output pulses according to state of polarization (SOP). The physical mechanism that causes PMD may be localized and stable, as in the birefringent crystals in an optical isolator, or distributed and timevarying, as in the random perturbations in a single-mode fiber. PMD is completely characterized by a wavelength-dependent, three-dimensional polarization dispersion vector, or equivalently by the specification of a pair of principal states of polarization (PSP) and a differential group delay at as a function of wavelength. Several PMD measurement techniques have been reported. Those based on changes in the auto- or cross-correlation of a low-coherence source [1] must employ a wide-spectrum source in order to achieve good temporal resolution, making them unsuitable for measurement of devices whose PMD varies with wavelength. The technique of reference [2], which relates at to the density of extrema in the spectrum of transmission through the DDT in series with a polarizer, yields poor resolution in the variation of at with wavelength and does not identify the PSP. Measurement of the arc described by the output SOP on the Poincare sphere over a series of wavelengths, as in [3] and [4], or measurement of the frequency derivatives of normalized Stokes vectors as in [5], would be difficult to automate because they produce erroneous results when a measurement SOP is near one of the PSP. The technique to be described suffers none of these limitations or disadvantages. 2 Theory R. C. Jones gave an explicit algorithm for experimentally determining the forward transmission Jones matrix T of an unknown linear, time-invariant optical device [6]. The restriction of linearity precludes optical devices that generate new optical frequencies. The restriction of time invariance applies only to the polarization transformation caused by the device, and does not include the absolute optical phase delay. Therefore, this technique can be used to characterize fiber networks even when the phase delay through the fiber is drifting during the measurement. ny Jones vector v can be completely specified by a magnitude, an absolute phase, and a unit vector v which locates the SOP on the Poincare sphere. To measure the Jones matrix of a device, a stimulus optical field of linear polarization parallel to the x axis is first generated, and the resulting response unit vector h is measured through the device. Similarly, stimulus fields of linear polarization parallel to the y axis, and parallel to the bisector of the angle between the positive x and y axes result in response unit vectors v and q, respectively. Three complex ratios independent of the intensities of the three /\. stimulus fields can now be formed from the x and y components of h, v, and q: k 1 = h x / 2

3 h y ' ~ = V x Ivy, and k 3 = qx/~. fourth ratio k 4 = (~- ~)/(kl -~) is then found. To within a complex constant f3, the transmissionjones matrix T is then given [6] by T=f3 [ klk4 k 4 By definition of the PSP, a general device or network has associated with it a pair of input principalstates i(~) which, as the input SOP is held constantwhile the opticalfrequency ~ is changed a small amount, result in a pair of output principal states whose unit vectors are invariant to first order over o, For a general transmission Jones matrix T(~), we can express an output PSP as a magnitude a (o) and absolute phase (o ) times a unit vector y(~) which specifies the SOP of the output PSP: (1) i (~) y(~) = T(~) x(~) = a (o ) e y(~). (2) a(~) and (~) may vary with ~, but y(~) is frequency-invariant to first order by definition of the output PSP. Using primes to denote differentiation with respect to ~, differentiation of (2) (as in [7]) results in,, a., ' J 1., [ Y = T x = -;; + 1 Y + a e y. (3) The first derivative of the absolute phase ' is the group delay 'f g through the network. If the network is not perfectly polarizing, its transmission matrix T is nonsingular and the input can be expressed in terms of the output as we obtain the eigenvalue relation 1 x = T- y. Explicitly setting s: to zero, (4) The imaginary parts of the eigenvalues of the matrix product T' T- 1 are the group delays associated with the PSPs, and the differential delay l::.. T which leads to PMD is given by the difference of the imaginary parts of the two eigenvalues. The output PSPs themselves are the eigenvectors of T'T- 1, which may be nonunitary for networks with polarizationdependent loss, in which case the output PSPs are not necessarily orthogonal. Measurement of T' and T, including measurement of the absolute phase, would allow direct calculation of the two group delays and l::..t, but in practice two restrictions are imposed by the Jones matrix measurement technique previously described. Instead of measuring T' directly, we must approximate it as T' ~ [T(w+l::..w) - T(~ )]Itua for a finite o. If the 3

4 frequency interval /).w is small enough so that each output PSP suffers nearly the same loss at wand w+ /).w, then a ' /).w / (] ~ 0 and (4) can be rewritten as [ T(w+l\w)T"\w) - (1 + itgiiw)i]y = 0. (5) The second restriction arises from the fact that T(w+/).w)T- 1(w), and therefore its associated eigenvalues Pland P2, can be determined only to within a complex constant, preventing determination of the two group delays individually. When Tg/).W is small we can approximate the eigenvalues as Pk = 1+ irg,k /).w ~ exp( irg,k /).w), and the differential group delay /). T can be expressed as I - Irg( P 1 / P2 ) I.,. _ I Lll - T 1 - '[ 2 - g, g, /).w ' (6) where Pl and P2 are the eigenvalues of T(w+/).w)T- 1(w) and rg denotes the argument function. In fact, the requirement that Tg,k /).w be small can be substantially relaxed: Since all the fundamental measurements are completely insensitive to absolute optical phase, only the quantity!::j. T /).w need be small enough to allow an exponential approximation of the eigenvalues. The exponential approximation is exact when the loss of the DUT is independent of polarization, in which case only the condition /). T /).w < 1f need be satisfied in orderto avoid the ambiguities of the multiple-valued argument function. 3 Experimental results and discussion The experimental apparatus is shown in Fig. 1. tunable laser source was connected to the input of a mechanical polarization synthesizer which was used to sequentially transform a circular SOP to three linear SOPs oriented at 0, 45 and 90 degrees. The output of the polarization synthesizer was directed through the DUT to the polarimeter through short lengths of single-mode fiber which are assumed to introduce negligible polarization dispersion. t any given optical frequency w n ' the polarization synthesizer generated three stimulus SOPs and the polarimeter measured the corresponding three response SOPs, resulting in a Jones matrix T(w J given by (1). The process is repeated for each w n in a sequence from n=1 to n=n. N-1 values of /).T at frequencies (w n+wn _ l)/2 are then calculated from the eigenvalues of successive matrix products T(wJT -1( wn-j using (6). The eigenvectors of T( wjt -1(wn-~ locate the output PSP as a function of frequency. Two birefringent crystal samples of known differential group delay were first measured to confirm the accuracy of this technique. The samples were quartz and x-cut lithium niobate slabs of thickness 1.81 mid and 0.52 mid, leading to expected values of fl T of 50.5± 0.5 fs and 127.3±2.5 fs, respectively, calculated using the manufacturer's values of the birefringences at 1520 nm. Plots of 81 obtained for these crystals using Jones matrix eigenanalysis are shown in Fig. 2. Using a measurement interval of 20 nm,!::j.t varied 4

5 between 52.3 fs and 56.1 fs for the quartz, and between 133.2fs and fs for the lithium niobate, in good agreement with the calculated values. The small discrepancies may be accounted for by PMD in the approximately 3 meters of connecting fiber, which was held in several coils of various orientation. With no crystals present a residual T of 4 to 5 fs was measured. In a second test of accuracy, the differential group delay ofa commercially-available pigtailed optical isolator was measured using three different techniques. In an autocorrelation technique similar to that of [1], the isolator was inserted in the source arm of an HP 8504 precision reflectometer. Working with a 1300-nmsource for best resolution, a single Fresnel reflection from the test arm resulted in a correlation trace with peaks separated by ±.02 mm, corresponding to T =1.22±.07ps. Group delays through the same isolator were next measured using an HP 8703 lightwave component analyzer, which measured the phase response of an intensity-modulated 1550-nmsignal over the modulation range 1 to 20 GHz. By sequentially setting the SOP at the isolator equal to each of its PSP, the differential group delay was measured to be 1.17±.07 ps. Finally, Jones matrix eigenanalysis was used with matrices measured at 3 nm intervals over 1500 to 1560nm. The 20 resulting values of T ranged from to fs, with a mean value of fs, in excellent agreement with the two other techniques. The temporal accuracy which can be achieved using Jones matrix eigenanalysis is related to the accuracy of the SOP measurements upon which the matrices are based, and to the size and uncertainty of the frequency interval used. When frequency measurement error is negligible and transmission through the DUT is not a strong function of polarization, a polarimeterwhich can measure the angle between unit vectors on the Poincare sphere with an accuracy of ± will yield a temporal accuracy of the order of 0 T = ± / w. The polarimeter angular accuracy was =0.007, so choosing a frequency interval to yield T W~ 0.35 results in both a good exponential approximation of the eigenvalues and a fractional accuracy of 0 T / T ~ In measurement topologies where many devices exhibiting PMD are concatenated to form a chain, Jones matrix eigenanalysis can enable one to measure the PMD of a particular DUT without disconnecting that DUT from the chain by exploiting the properties of similar matrices. Suppose we represent the particular DDT of interest (DUTn) by N(W) and the network between the polarization synthesizer and DUTn by M(w), and that we are able to measure only the Jones matrix M to a point immediately before DUT n and the matrix (N M) to a point immediately after DUTn' but that we are not able to measure N directly. Denoting the two optical frequencies bracketing t:..w by the subscripts 1 and 2, we can calculate ~l M 1 (N1M1)1 N2~ ' which is related to N~l N 2 by a similarity transformation. Since similar matrices have the same eigenvalues, using the eigenvalues of M~l M 1 (N1M1)1 N2~ in (6) gives the differential delay of DUT n even when N cannot be directly measured. plot of such measurements is shown in Fig. 3. The differential delay of a pigtailed isolator was first measured (plot ), and then the isolator was spliced to a 2-km single-mode fiber and the combination was measured again (plot B). From these 5

6 two series of matrices, T of the fiber alone was calculated (plot C) using the similarity technique. When the fiber was measured by itself (plot D), the values of '[ agreed very well with the results of the similarity technique, demonstrating its validity and reproducibility. The frequency resolution of this technique allows measurement of the frequency dependence of T, a characteristic of long fibers [8]. 4 Summary Jones matrix eigenanalysis is a new technique which offers measurement of both differential group delay and orientation of the PSP as a function of optical frequency, i.e. the full polarization dispersion frequency response, allowing a complete measurement of polarization mode dispersion. This technique depends on no assumptions about the device or network under test except that it must be linear and its polarization transformation must be constant over a period of several seconds. Eigenanalysis measurement of crystal samples of known PMD and comparison with other PMD measurement techniques yielded agreement within several femtoseconds. When devices are concatenated, the flexible measurement topology of this technique allows PMD measurement of an individual device even when access to the device is limited. utomation allowed the measurement to be performed repeatably in less than 4 seconds. 5 cknowledgment The author thanks Paul Hernday and Harry Chou of Hewlett Packard's Network MeasurementDivision for the loan of measurement equipment and samples. 6 References [1] N. Gisin, J-P Von der Weid and J-P Pellaux, "Polarization mode dispersion of short and long single-mode fibers," IEEE1 Lightwave Technol., LT-9, pp , 1991, and references therein. [2] C. D. Poole, "Measurement of polarization-mode dispersion in single-mode fibers with random mode coupling," Optics Lett., 14, pp , [3] N. S. Bergano, C. D. Poole and R. E. Wagner, "Investigation of polarization dispersion in long lengths of single-mode fiber using multilongitudinal mode lasers," IEEE1. Lightwave Technol., LT-S, pp , [4] D. ndresciani, F. Curti, F. Matera and B. Daino, "Measurement of the group-delay difference between the principal states of polarization on a low-birefringence terrestrial fiber cable," Optics Lett., 12, pp , [5] C. D. Poole, N. S. Bergano, R. E. Wagner and H. J. Schulte, "Polarization dispersion and principal states in a 147-km undersea lightwave cable," IEEE1. Lightwave Techno!., LT 6, pp , [6] R. C. Jones, " new calculus for the treatment of optical systems. VI. Experimental determination of the matrix,"1. Optical Soc. m., 37, pp , [7] C. D. Poole and R. E. Wagner, "Phenomenological approach to polarisation dispersion in long single-mode fibers," Elect. Lett., 22, pp , [8] G J. Foschini and C. D. Poole, "Statistical theory of polarization dispersion in single mode fibers," IEEE1 Lightwave Techno!., LT-9, pp ,

7 SOLENOID-MOUNTED POLRIZERS FOR INSERTION INTO OPEN BEM ~ ~ L80CD L P P P (.C., DEVICE OR ~ L., NETWORK UNDER TEST ~ L 01 TUNBLE LSER SOURCE POLRIMETER Fig.1. pparatus used for measurement of polarization mode dispersion using Jones matrix eigenanalysis. L: lens; P: linear polarizer; C: crystal sample; 01: optical isolator; F: 2-km single-mode fiber. 7

8 150~ , - Ul ~ 100 z o H en c: w 50 n, en H o ~ WVELENGTH (nm) 1560 Fig. 2. Measurements of the differential delay 'f of quartz and lithium niobate slabs. The range of 'f over 5 measurements was less than 4 fs for each crystal. Solid line: lithium niobate; Dashed line: quartz ~ H en rrwn, 500 c: + III III III IS en H 0: t:l mm o 250 mmmfflmmmmmm ls ff1 mffl WVELENGTH (nm) 1560 Fig. 3. Measurements of the differential delay 'f demonstrating the similarity technique. : isolator alone; B: isolator and 2-km fiber; C: 2-km fiber measured through the isolator using the similarity technique; D: 2-km fiber measured alone. 8