Polarization mode dispersion measurement

[0002] This invention relates to optical measurements, and particularly to the measurement of the polarization mode dispersion of optical devices, particularly optical fibers used in telecommunication networks. Although the polarization mode dispersion measurement can be made at a single light wavelength, by making the measurements at plural wavelengths, other optical parameters, e.g., chromatic dispersion, can be determined.

[0003] Polarization mode dispersion (PMD) is a distortion mechanism (like chromatic dispersion) that causes optical devices, such as single-mode fibers, optical switches and optical isolators, to distort transmitted light signals. The relative severity of PMD (which is a function of the wavelength of the transmitted light) has increased as techniques for dealing with chromatic dispersion have improved, transmission distances have increased, and bit rates have increased. Negative effects of PMD appear as random signal fading, increased composite second order distortion and increased error rates.

[0004] PMD is due to differential group delay caused by geometrical irregularities and other sources of birefringence in the transmission path of the optical device. For example, a single-mode fiber (SMF) is ideally a homogeneous medium supporting only one mode. In practice, it supports two propagation modes with orthogonal polarizations. When a lightwave source transmits a pulse into a SMF cable, the pulse energy is resolved onto the principal states of polarization of the fiber. The two groups of pulse energy propagate at different velocities and arrive at different times causing pulse broadening and signal distortion.

[0005] Determining the PMD of in-place (installed) optical fibers is useful for determining the capacity of the fibers for transmitting new telecommunication services of ever increasing bandwidth requirements and for the design and control of PMD compensators. Also, because the PMD of any given fiber can change rapidly with time (owing to random asymmetrical stresses along the length of the fiber), repeated PMD measurements of the fiber can be used in a feedback system for compensating for variable PMD distortions.

[0006] The PMD of a fiber is commonly characterized by two specific orthogonal states of polarization called the principal states of polarization (PSPs) and the differential group delay (DGD) between them. This can be described at an optical angular frequency, ω, by the 3-component Stokes vector, {right arrow over (Ω)}=Δτ{right arrow over (q)}, where {right arrow over (q)} is a unit Stokes vector pointing in the direction of the faster PSP, and the magnitude, Δτ, is the DGD. Typical DGD values encountered in transmission systems range between 1 (picosecond) ps and 100 ps.

[0007] Known methods for determining PMD vectors include the Jones Matrix Eigenanalysis (JME) technique and the Müller Matrix Method (MMM). Each of these techniques uses a tunable, continuous-wave laser and a polarimeter to measure the output polarization states for two (or three) different input polarization launches at two optical frequencies. The PMD vector is then calculated for the midpoint frequency. In addition to determining the output PMD vector, the Müller Matrix Method determines the rotation matrix of the fiber at each frequency and thus the input PMD vector can be calculated.

[0008] Shortcomings of these techniques are that they are somewhat difficult to implement, particularly in a feedback system, because they require frequency differentiation of measured data at plural optical frequencies. As described hereinafter, an advantage of the method according to the present invention is that it requires measurements of data at only one optical frequency. Also, while, as hereinafter described, measurements can be made at plural optical frequencies, separate, single frequency measurements are made that do not involve frequency differentiation.

[0009] A further advantage of the invention is that it can be used to accurately measure chromatic dispersion of an optical device in the presence of PMD. Other known methods for measuring chromatic dispersion tend to give inaccurate results if the devices have PMD.

[0010] At the input of an optical device, typically an optical fiber link within an optical fiber telecommunications network, four different light signals, all of the same optical frequency, but having different states of polarization, are transmitted along the fiber and the mean signal delay of each of the light signals is measured. By calculation (described hereinafter), the four mean signal delay measurements yield the desired PMD vector of the optical device. Additionally, by repeating the mean signal delay measurement at multiple optical frequencies (i.e., at a different optical frequency for each set of time delay measurements) determination can be made of the first and higher-order intrinsic (polarization-averaged) chromatic dispersion of the device being measured, as well as the higher-order PMD of the device.

[0011] FIG. 1 is a schematic illustration of a system for implementing the present invention;

[0012] FIG. 2 is similar to FIG. 1 but showing a more specific example of a suitable system for implementing the invention;

[0013] FIG. 3 is a graph showing the wavelength dependence of the signal delay of each of four fixed input polarization signals transmitted through a single mode fiber; and

[0014] FIG. 4 is a graph showing the wavelength dependence of the input PMD vector of the fiber obtained using the data of FIG. 3.

[0015] As above-described, PMD is usually (and herein) described as a three dimensional vector having a magnitude and direction, i.e., having three components. Both an input and an output PMD vector can be determined for any optical device. The PMD vectors are a function of the wavelength of the light transmitted through the device. The present invention involves a new method for making measurements of an optical device and determining the three components of the various PMD vectors. The measurement and calculations made are based upon the known (Mollenauer, L. F. and Gordon, J. P., Optics Lett., Vol. 19, pp. 375-377, 1994) relationship,

[0016] τ_g=τ₀−½{right arrow over (Ω)}·{overscore (s)}  (1)

[0017] where {overscore (s)} and {right arrow over (Ω)} are, respectively, the Stokes vectors of the light and the PMD vector at the fiber input. Equation (1) describes the polarization dependence of the mean signal delay, τ_g, through the fiber as defined by the first moment of the pulse envelope in the time domain with τ₀being a polarization independent delay component. Eq. (1) assumes that τ₀and {right arrow over (≠)} (ω) do not vary significantly over the bandwidth of the signal.

[0018] The definition of mean signal delays involving “moments” of the output signals is known and described, for example, in the afore-cited publication by Mollenauer and Gordon as well as by Elbers, J. P., et al, “Modeling of Polarization Mode Dispersion in Single Mode Fibers,” Electr. Lett., Vol. 33, pp. 1894-1895, October 97; Shieh, W., “Principal States of Polarization for an Optical Pulse,” IEEE Photon. Technol. Lett; Vol. 11, No. 6, p. 677, June 99; and Karlsson, M., “Polarization Mode Dispersion-Induced Pulse Broadening in Optical Fibers,” Opt.

[0019] Lett., Vol. 23, pp. 688-690, '98.

[0020] More precisely, mean signal delay, τ_g, is expressed as the difference of the normalized first moments at fiber output and input, $\begin{array}{cc}{\tau }_g=\frac{W_1\left(z\right)-W_1\left(0\right)}{W},& \left(2\right)\end{array}$

[0021] where z is the distance of propagation in the fiber. Here W=∫dt {right arrow over (E)}†{right arrow over (E)}=∫dω{tilde over (E)}†{tilde over (E)} is the energy of the signal pulse represented by the complex field vector {right arrow over (E)}(z,t) with Fourier transform {tilde over (E)}(z,ω), and W₁(z)=∫dtt{right arrow over (E)}†{right arrow over (E)}=j∫dω{tilde over (E)}†{tilde over (E)}_ω is the first moment.

[0022] Because, in equation (1), there are four unknowns, i.e., τ₀(fiber intrinsic group delay) and the three components of the PMD input vector, {right arrow over (Ω)}, four measurements are necessary to determine {right arrow over (Ω)} and τ₀at a given wavelength. Each set of measurements comprises launching a set of four light signals of respectively different polarization states into the fiber and measuring the mean signal delay of each of the signals by known means.

[0023] A variety of different light polarization states can be used, but for simplicity of the mathematical analysis, a convenient set of polarization states are those coinciding with the Poincaré sphere axes, Ŝ₁,-Ŝ₁, Ŝ₂, and Ŝ₃, i.e., three different linear Polarization states and a circular polarization state. (A “Poincaré sphere” being a graphical representation of all possible Polarization states on a surface of a sphere where each point on the sphere represents a different polarization form; see, for example, W. Shurcliff, Polarized Light: Production and Use, p. 16, Harvard University Press, Cambridge, Mass., 1962. A specific example of selected polarization states is provided hereinafter.)

[0024] By identifying the measured mean signal delays for each polarized state as τ_g1, τ₍₋₁₎, τ_g2, and τ_g3respectively, the calculated components from equation (1) are as follows:

[0025] τ₀=½(τ_g1+τ_g(−1))  (3)

[0026] and the three components of the fiber input PMD Stokes vector {right arrow over (Ω)} are:

[0027] Ω₁=2(τ₀−τ_g1)

[0028] Ω₂=2(τ₀−τ_g2)

[0029] Ω₂=2(τ₀−τ_g3)  (4)

[0030] (Described hereinafter is how the above mathematical relationships, for the “special” Poincaré sphere axial polarizations, can be generalized to any four non-degenerate input polarization states.)

[0031] As above-described, implementation of the present invention involves the launching of a set of different test light signals into a device under test, e.g., an optical fiber, and measuring the mean signal delay for each signal.

[0032] The basic technique according to the invention is illustrated in FIG. 1, which shows a test system including a source 10 of modulated light at a single optical frequency and a polarization controller 12 for adjusting the polarization of the light from the source 10 to four successively different polarization states which are successively introduced into and transmitted through an optical device 14 under test, e.g., a length of optical fiber wound on a coil. The light signals exiting the test device 14 are successively fed into an apparatus 16 for measuring the mean signal delay for each test signal transmitted through the device. The measured data provides, by equations (3) and (4), the polarization independent delay and the input PMD Stokes vector of the device under test at a single optical frequency. The measurements are repeated at successively different optical frequencies for determining the frequency dependence of these factors. Also, the data taken at multiple frequencies allows the determination of the chromatic dispersion of the device under test. This is described hereinafter.

[0033] As noted, the system shown in FIG. 1 measures the PMD vector at the input to the optical device 14. An advantage of the system shown in FIG. 1, in comparison with the prior mentioned known Jones (JME) and Müller (MMM) techniques for measuring input PMD vectors, is that no output polarimeter is required. Conversely, if the output from the optical device is passed into a polarimeter, the rotation matrix of the optical device, R(ω), can be determined. Then the PMD vector at the output of the optical device, {right arrow over (Ω)}_t(ω), can be determined from

[0034] {right arrow over (Ω)}_t(ω)=R(ω){right arrow over (Ω)}(ω),  (5)

[0035] where {right arrow over (Ω)}(ω) is the input PMD vector from eq. (4). For instance, for the previously mentioned launches coinciding Ŝ₁,-Ŝ₁, Ŝ₂, and Ŝ₃, the matrix R(ω) is given by $\begin{array}{cc}R\left(\omega \right)=|\begin{array}{ccc}{T}_{11}& T_{21}& T_{31}\\ T_{12}& T_{22}& T_{32}\\ T_{13}& T_{23}& T_{33}\end{array}|& \left(6\right)\end{array}$

[0036] where T_1j, T_2j, and T_3j(j=1, 2, 3) are the jth components of the output Stokes vectors measured by the polarimeter for launches Ŝ₁, Ŝ₂, and Ŝ₃, respectively.

[0037] Apparatus for performing each of the functions indicated in FIG. 1 are generally known, and different specific arrangements for practicing the invention can be devised by persons of skill. FIG. 2 shows but one example of an arrangement useful for implementing the invention.

[0038] Light of a desired wavelength is provided by a wavelength tunable laser 20 (Hewlett-Packard 81682A). A directional coupler 22 (Etek SWBC2201PL213) splits the light from the laser into two parts, one of which enters a wavemeter 24 (Hewlett-Packard 86120B) that measures the wavelength of the light, the other of which is coupled to an optical modulator (MOD) 26 (Lucent Technologies X2623C). An electrical signal from an oscillator 28 (1 GHz) (part of Hewlett-Packard 83420A) is also connected to the modulator. The optical modulator imposes a sinusoidal modulation on the optical power of the light. This modulated signal is amplified by an optical amplifier (A) 30 (Lucent Technologies W1724CDDAD), passed through a tunable optical filter 32 (JDS Fitel TB 1500B) tuned to the wavelength of the signal and enters a linear polarizer 34 (Etek FPPD2171LUC02). A polarization controller (PC1) 36 (Fiber Control Industries FPC-1) is used to adjust the polarization of the signal so that it nearly matches the polarization of the linear polarizer 34. This polarizer 34, which is not essential to the measurement, serves to make the results of the measurement insensitive to polarization changes occurring in the components and in the fiber preceding the polarizer. After the polarizer 34, the signal passes though a launch polarizer box 40. This box 40, which functions by switching polarizers into or out of the signal path, selectively modifies the signal polarization from its input polarization to one of four output polarization states.

[0039] Three of those states are provided by putting an appropriate polarizer in the beam path while the fourth state is provided by switching all of the polarizers out of the beam path and using the input signal polarization unmodified. This input polarization can be adjusted to a desired state using polarization controller (PC2) 42 (Fiber Control Industries FPC-1). Alternatively, the fourth polarization could be provided by a fourth switchable polarizer in launch polarizer box 40. After passing through the launch polarizer box 40, the signal is launched into the device under test, shown here as a fiber span 44. After the fiber span, the signal, now modified by the PMD of the fiber span, enters a receiver 46 (part of Hewlett-Packard 83420A) which generates an electrical signal with a voltage proportional to the power in the optical signal. A network analyzer 48 (Hewlett-Packard 8753D) measures the difference between the phase of the signal provided by the oscillator 28 and the phase of the amplitude modulation of the received signal. Prior to entering the receiver 46, the signal can optionally be tapped using a directional coupler 50 (Etek SWBC2210PL213) and the tapped light sent to a polarimeter 52 (Hewlett-Packard 8509B). This polarimeter 52, which is not needed for the inventive measurement techique, is used in the measurement of PMD by a prior technique such as the Müller Matrix Method. This allows comparison of the results obtained by the different techniques. The polarimeter 52 is also convenient when setting the polarization controller 42 to a desired state.

[0040] An example of measurements made according to the invention using the apparatus shown in FIG. 2 follows.

[0041] Four polarization states (linear polarization states at 0°, 60° and 120°, and a circular polarization state) were launched into a 52-km optical fiber span having a known average PMD of 35 ps and a chromatic dispersion of approximately −6 ps/nm at 1555 nm.

[0042] FIG. 3 shows the signal delays (vertical axis) for each input polarization as a function of light wavelength (horizontal axis). (Because the intrinsic signal delays through the 52-km fiber span are about 260 μs, “relative” signal delays are plotted in FIG. 3 where a constant delay has been subtracted from the actual measured delays.) The PMD of the fiber span causes the polarization dependence of the signal delay at each wavelength, and the variations in those relative signal delay curves originate from the change of direction and magnitude of {right arrow over (Ω)} (ω) as well as from underlying chromatic dispersion.

[0043] FIG. 4 shows the input PMD vector {right arrow over (Ω)} (ω) and the relative 263 polarization-independent delays, τ₀, for this fiber, determined from the data of FIG. 3. Here, τ₀is the polarization-independent delay through the fiber span at each wavelength compared to the delay at the first wavelength, 1543.70 nm. The modulation frequency was 1 GHz, providing ω_mΔτ=0.45 at the maximum DGD, where here ω_mis 2π times the modulation frequency. FIG. 4 also shows data for the determination of the chromatic dispersion of the test fiber 44 (FIG. 2). Chromatic dispersion is the change in the intrinsic (polarization-averaged) group delay τ₀through an optical device with wavelength. For instance, if measurements are taken at two wavelengths, λ₁and λ₂, the chromatic dispersion D at the average of these wavelengths, (λ₁+λ₂)/2, can be obtained by dividing the change in τ₀by the change in wavelength: $\begin{array}{cc}D\left[\left({\lambda }_1+{\lambda }_2\right)/2\right]=\frac{{\tau }_0\left({\lambda }_2\right)-{\tau }_0\left({\lambda }_1\right)}{{\lambda }_2-{\lambda }_1}& \left(7\right)\end{array}$

[0044] In FIG. 4, the chromatic dispersion of the fiber corresponds to the slope of the curve plotting τ₀versus wavelength. For example, at a wavelength of 1544.0, the dispersion is 73.3 ps/nm.

[0045] Also shown in FIG. 4 is a measurement of the input {right arrow over (Ω)} (ω) on the same fiber using the aforementioned Müller Matrix Method for determining PMD vectors. The data for the Müller method was taken immediately before the measurement according to the present invention. The two techniques agree to within a maximum discrepancy of 7 ps.

[0046] While it is convenient to launch the specific polarization states, Ŝ₁,-Ŝ₁, Ŝ₂, and Ŝ₃, as previously noted, equations (3) and (4) can be generalized to any four non-degenerate input polarization launches Ŝ_i(i=1,2,3,4) that span Stokes space. The corresponding signal delay measurements, τ_gi, can be grouped into a four-dimensional (4-D) vector {overscore (τ)}_g=(τ_g1,τ_g2,τ_g3,τ_g4), and similarly, the desired delay components can be grouped into a 4-D vector {overscore (τ)}_g=(τ₀,Ω₁,Ω₂,Ω₃) The four launches together with Eq. (1) set up the required four linear equations for the four unknowns represented by {overscore (τ)}, yielding {overscore (τ)}_g=P {overscore (τ)}, where P is a 4×4 matrix he components of the launched polarizations. $\begin{array}{cc}P=\frac{1}{2}|\begin{array}{cccc}2& -S_{11}& -S_{12}& -S_{13}\\ 2& -S_{21}& -S_{22}& -S_{23}\\ 2& -S_{31}& -S_{32}& -S_{33}\\ 2& -S_{41}& -S_{42}& -S_{43}\end{array}|& \left(8\right)\end{array}$

[0047] Here S_ijis the jth component of polarization launch {right arrow over (S)}_i. The equivalent of Eqs. (3) and (4) is then {overscore (τ)}=P⁻¹{overscore (τ)}_g, and so τ₀and the vector components of {right arrow over (Ω)} can be determined by inverting the matrix P at each frequency of interest.

[0048] For this general launch case, Eq. (5) can also be used to determine the output PMD vector when a polarimeter is inserted at the output. In order to determine the rotation matrix of the fiber, R(ω), any two of the four non-degenerate input launches {right arrow over (S)}_ican be used. We use {right arrow over (S)}_aand {right arrow over (S)}_bto designate the two chosen input launches, and {right arrow over (T)}_aand {right arrow over (T)}_bas the corresponding measured output Stokes vectors. We first create a set of three orthogonal vectors at the input, Ŝ₁′, Ŝ₂′, and Ŝ₃′, where Ŝ₁′={right arrow over (S)}_a, Ŝ₃′={right arrow over (S)}_a×{right arrow over (S)}_b/|{right arrow over (S)}_a×{right arrow over (S)}_b|, and Ŝ₂′=Ŝ₃′×Ŝ₁′. Similarly we create a set of three orthogonal vectors at the output, {circumflex over (T)}₁′, {circumflex over (T)}₂′, and {circumflex over (T)}₃′, where {circumflex over (T)}₁′={right arrow over (T)}_a, {circumflex over (T)}₃′={right arrow over (T)}_a×{right arrow over (T)}_b/|{right arrow over (T)}_a×{right arrow over (T)}_b|, and {circumflex over (T)}₂′={circumflex over (T)}₃′×{circumflex over (T)}₁′. The rotation matrix of the fiber is then R=R_TR_S, where $\begin{array}{cc}{R}_S=|\begin{array}{ccc}{S}_{11}^{\prime }& S_{12}^{\prime }& S_{13}^{\prime }\\ S_{21}^{\prime }& S_{22}^{\prime }& S_{23}^{\prime }\\ S_{31}^{\prime }& S_{32}^{\prime }& S_{33}^{\prime }\end{array}|& \left(9\right)\\ \mathrm{and}& \\ R_T=|\begin{array}{ccc}{T}_{11}^{\prime }& T_{21}^{\prime }& T_{31}^{\prime }\\ T_{12}^{\prime }& T_{22}^{\prime }& T_{32}^{\prime }\\ T_{13}^{\prime }& T_{23}^{\prime }& T_{33}^{\prime }\end{array}|.& \left(10\right)\end{array}$

[0049] Here S_1j′, S_2j′, and S_3j′ (j=1, 2, 3) are respectively the jth components of the set of three orthogonal vectors at the input, Ŝ₁′, Ŝ₂′, and Ŝ₃′, while T_1j′, T_2j′, and T_3j′ (j=1, 2, 3) are respectively the jth components of the set of three orthogonal vectors at the output, {circumflex over (T)}₁′, {circumflex over (T)}₂′, and {circumflex over (T)}₃′.

[0050] Instead of determining moments from pulse-shape measurements, the signal delays, τ_gi, can also be obtained (in approximation, i.e., to an acceptable degree of accuracy, e.g., 6% as described hereinafter) by observing phase shifts of the modulation of a sinusoidal amplitude modulated signal and benefiting from the precision of sensitive phase-detection techniques. (Note that Williams [Electron. Lett. Vol. 35, pp. 1578-1579, 1999] has used sinusoidal modulation for the determination of the scalar DGD.) For most purposes, it suffices to only measure changes in τ_giwith polarization and optical frequency and to not resolve the ambiguity presented by the use of a signal with periodic modulation. For a sinusoidal intensity-modulated signal with the assumption of frequency-independent PSP's and DGD, Eq. (1) becomes

[0051] tan ω_m(τ_g−τ₀)=−{right arrow over (q)}·{right arrow over (s)}tan(ω_mΔτ/2),  (11)

[0052] where ω_mis the angular modulation frequency. When the four launch polarizations coincide with the Poincaré sphere axes, Ŝ₁,-Ŝ₁, Ŝ₂, and Ŝ₃, Eq. (3) is still valid for sinusoidal modulation:τ₀=½(τ_g1+τ_g(−1)). Using the relation q₁¹+q₂²+q₃²=1, expressions for the magnitude, Δτ, and components, q_i, of the unit vector, {right arrow over (q)}, pointing in the direction of the PMD vector, {right arrow over (Ω)}, can also be determined and solved exactly.

[0053] In the more general case we can use any four non-degenerate input polarization launches {right arrow over (s)}_i(i=1, a, b, c ) that span Stokes space, where {right arrow over (s)}₁=Ŝ₁, {right arrow over (a)}_a=(a₁, a₂, a₃), {right arrow over (s)}_b=(b₁, b₂, b₃), and {right arrow over (s)}_c=(C₁, C₂, C₃), and measure the corresponding delays τ_g1, τ_ga, τ_gb, and τ_gc. We first express Ŝ₁, Ŝ₂, and Ŝ₃, in terms of {right arrow over (S)}_a, {right arrow over (S)}_b, and {right arrow over (S)}_c,

[0054] Ŝ_i=α_i{right arrow over (s)}_a+β_i{right arrow over (s)}_b+γ_i{right arrow over (s)}_c(i=1,2,3),  (12)

[0055] where the coefficients α_i, β_i, and γ_iare obtained from $\begin{array}{cc}|\begin{array}{ccc}{\alpha }_1& {\beta }_1& {\gamma }_1\\ {\alpha }_2& {\beta }_2& {\gamma }_2\\ {\alpha }_3& {\beta }_3& {\gamma }_3\end{array}|={|\begin{array}{ccc}{a}_1& a_2& a_3\\ b_1& b_2& b_3\\ c_1& c_2& c_3\end{array}|}^{-1}& \left(13\right)\end{array}$

[0056] Substituting Ŝ₁=α₁{right arrow over (s)}_a+β₁{right arrow over (s)}_b+γ₁{right arrow over (s)}_cinto Eq. (11) gives

[0057] tan ω_m(τ_g1−τ₀)=−{right arrow over (q)}·(α₁{right arrow over (s)}_a+β₁{right arrow over (s)}_b+γ₁{right arrow over (s)}_c)tan(ω_mΔτ/2),  (14)

[0058] leading to a transcendental equation for τ₀:

[0059] tanω_m(τ_g1−τ₀)=α₁tanω_m(τ_ga−τ₀+β₁tanω_m(τ_gb−τ₀)+γ₁tanω_m(τ_gc−τ0).  (15)

[0060] A first trial value for τ₀can be obtained by linearizing Eq. (15), $\begin{array}{cc}{\tau }_0=\frac{{\tau }_{\mathrm{g1}}-{\alpha }_1{\tau }_{\mathrm{ga}}-{\beta }_1{\tau }_{\mathrm{gb}}-{\gamma }_1{\tau }_{\mathrm{gc}}}{1-{\alpha }_1-{\beta }_1-{\gamma }_1}& \left(16\right)\end{array}$

[0061] After solving for τ₀and using q₁²+q₂²+q₃²=1, an expression for the magnitude, Δτ, can be found: $\begin{array}{cc}{\tan }^2\left({\omega }_m\Delta \tau /2\right)=\sum _i{\left[{\alpha }_i\tan {\omega }_m\left({\tau }_{\mathrm{ga}}-{\tau }_0\right)+{\beta }_i\tan {\omega }_m\left({\tau }_{\mathrm{gb}}-{\tau }_0\right)+{\gamma }_i\tan {\omega }_m\left({\tau }_{\mathrm{gc}}-{\tau }_0\right)\right]}^2.& \left(17\right)\end{array}$

[0062] The components, q_i, of the PMD vector, {right arrow over (Ω)}, can then be determined from $\begin{array}{cc}-q_i=\frac{{\alpha }_i\tan {\omega }_m\left({\tau }_{\mathrm{ga}}-{\tau }_0\right)+{\beta }_i\tan {\omega }_m\left({\tau }_{\mathrm{gb}}-{\tau }_0\right)+{\gamma }_i\tan {\omega }_m\left({\tau }_{\mathrm{gc}}-{\tau }_0\right)}{\tan \left({\omega }_m\Delta \tau /2\right)}.& \left(18\right)\end{array}$

[0063] Although the above procedure will yield any computational accuracy desired, it is often not necessary. For small modulation frequencies, ω_m, we can approximate tan (x)≅x in Eq. (11), reducing to the earlier expressions (i.e. Eqs. (3) and (8)). These linear expressions are valid for sinusoidal modulation to within 6% as long as ω_mΔτ<π/4. For instance, for the peak DGD we observed here, 70 ps, 6% accuracy will be obtained for modulation frequencies, fm=ωm/2π, less than 1.8 GHz by using the linear expressions.

[0064] As above-explained, the inventive method involves multiple launchings of different polarized light states, determining the mean signal delays of the different polarization states, and calculating the PMD vectors. Basic techniques for measuring the group delay of an optical device are known, see, e.g., Y. Horiuchi, et al., “Chromatic Dispersion Measurements of 4564 km Optical Amplifier Repeater System,” Electronics Letters, Vol. 29, pp. 4-5, Jan. 7, 1993. Additionally, by comparing data taken at adjacent wavelengths with the inventive method, chromatic dispersion of optical devices can be accurately measured even with the presence of PMD.