Abstract:
A polarization-mode-dispersion emulator in accordance with the principles of the present invention includes a plurality of PMD-generating stages aligned in cascade so as to form a clear light-path through the stage concatenation. Each PMD-generating stage includes an optical birefringent crystal or crystals for the purpose of imparting differential group delay and a polarization-rotating plate such as a half-wave waveplate for the purpose of imparting state-of-polarization rotation from one PMD-generating stage to the next. The polarization-rotating plates are mounted to rotation apparatuses and a controller coordinates the relative rotation of each waveplate for the purpose of generating PMD in a controlled manner.

Description:
FIELD OF THE INVENTION 
     This invention relates to optical fiber signal transmission and, in particular, to the generation of polarization-mode dispersion (PMD) to emulate the natural occurrence of PMD in optical fiber; and to use PMD emulation to compensate for PMD generated by optical fiber. 
     BACKGROUND OF THE INVENTION 
     Polarization-Mode Dispersion (PMD) is a fiber-optic telecommunication system impairment which can prevent the transmission of high data rates, such as 10 Gb/s and 40 Gb/s. The effect of PMD originates with the inherent, built-in residual birefringence present in all single-mode optical fiber. Over the course of pulse transmission, PMD interacts with a transmitted optical pulse in such a way as to distort the shape of the pulse. The consequences vary with the degree of pulse distortion, from small penalties in transmission fidelity, to complete system outage. Accordingly, in order to transmit optical pulses at rates above 2.5 Gb/s, either the quality of the optical fiber must be sufficiently high so as to not introduce significant PMD or a PMD-compensating apparatus must be inserted in the transmission system, for example, between the end of the fiber-optic transmission line and the input to the optical receiver. 
     In order to transmit at a rate of 10 Gb/s over legacy fiber (that is, currently installed fiber), a PMD compensator (PMDC) is frequently necessary to recover acceptable system performance. It is generally believed that in order to transmit at a rate of 40 Gb/s and above over the most recently available fiber, a PMDC at the receiver may be essential. Accordingly, a PMDC is a desirable apparatus. A means for laboratory and factory testing of a PMDC is, consequently, a desirable apparatus. Such an apparatus is herein referred to as a PMD Emulator (PMDE). 
     Indeed a PMDC and PMDE are quite similar because both apparatuses must generate PMD; the former apparatus must generate PMD in order to cancel the accrued fiber PMD, the latter apparatus must generate PMD in order to test a PMDC. However, the generation of PMD for a PMDC and PMDE does have practical differences. A PMDC further requires the generation of a control signal which is used to monitor the PMD cancellation, and further requires a feedback system and control algorithm which automatically corrects for changing PMD. A PMDE further requires the precise and repeatable generation of PMD, and may not require the speed of change which may be necessary for a PMDC. A PMDE further requires the synthesis of the PMD effect so as to approximate the PMD of a real fiber as closely as possible. A PMDE further requires a performance which is both known and repeatable so as to test and verify the performance of a PMDC. 
     The term “PMDC” will refer herein to an apparatus which consists of: 1) a PMD generating mechanism; 2) a control-signal generating mechanism; and 3) a feedback control mechanism and algorithm which changes the PMD generating mechanism so as to cancel the PMD of the fiber-optical link. The term “PMDE” will refer solely to a PMD generating mechanism, which includes a means to change the state of PMD. It is recognized that a PMDE can be transformed into a PMDC through the addition of a control-signal generating mechanism and a feedback control mechanism and algorithm. 
     Polarization-mode dispersion is the composite phenomenon of two interleaved effects. One effect is the projection of an input state-of-polarization (SOP) onto a birefringent dielectric system. The other effect is an accrued differential temporal delay between two orthogonal polarization states. FIG. 1 a  illustrates an input optical pulse  100  with an arbitrary input SOP  120 . The pulse  100  is incident upon an optical birefringent medium  110  with orthogonal birefringent axes fast  121  and slow  122 . The terms “fast” and “slow” refer to the speed of the optical pulse as projected on either axis: the pulse on one axis propagates faster than the pulse of the other axis due to the difference in refractive index, the latter which is due to the inherent birefringence of the fiber. The projection of the input SOP  120  onto the birefringent interface  110  results in the formation of two orthogonally polarized pulses  101  and  102 . The balance of energy on the two orthogonal polarization axes is dictated by the relative orientation of the input SOP  120  and the birefringent axes  121 ,  122  at the interface  110 . FIG. 1 a  illustrates the phenomena of polarization projection at a birefringent interface. 
     FIG. 1 b  illustrates an example of “simple” PMD. A short section of optical fiber  130  and the effect of PMD on an optical pulse  100  is herein illustrated. The optical pulse  100  has its SOP  120  projected onto the birefringent axes of the fiber  110 , resulting in pulse  101  on fast axis  121  and pulse  102  on slow axis  122 . The birefringence of fiber  130  causes a relative temporal delay between the two pulses  101  and  102 . This temporal delay is referred to as differential-group delay (DGD). At the end of optical fiber  130  pulses  101  and  102  exhibit a DGD of magnitude Δτ  140 . The magnitude of DGD  140  depends on the magnitude of the birefringence and the length of fiber  130  over which the birefringence does not significantly change. The present instance of a single polarization projection followed by a single differential-group delay stage is denoted as simple, one one-stage, PMD. 
     FIG. 2 illustrates the concatenation of several simple PMD stages to form a more complex PMD response. FIG. 2 a  illustrates substantially the same PMD as FIG. 1 b  but the orthogonal polarization states are not explicitly indicated. FIG. 2 a  illustrates a pulse  200  input to birefringent fiber segment  130 . The PMD of this fiber segment  130  generates DGD  240  between two output pulses  200  and  201 . FIG. 2 b  illustrates the concatenation effect of two birefringent fiber segments  130  and  131  possessing dissimilar lengths and birefringent orientation. Fiber segment  130  produces two pulses  200  and  201  with DGD  240 . Fiber segment  131  produces two pulses for each pulse input, resulting in four pulses  200 ,  201 ,  202 ,  203 . The time delay between pulse images  200  and  202 , and  201  and  203 , is the DGD  241  of fiber segment  131 . FIG. 2 c  adds a third fiber segment  132  with dissimilar length and birefringent axis orientation. Again, each input pulse  200 ,  201 ,  202 ,  203  to fiber segment  132  is copied and each pair  210 ,  211  is delayed by DGD  242 , forming four pulse pairs  210 ,  211 ,  212 ,  213 . Note that at each interface between fiber segments, the polarization projection alters the balance of energy between that on the incident SOP and that on the projected coordinates; thus, the variation in pulse amplitudes. 
     The fiber within a typical fiber-optical link is composed of tens or hundreds of fiber segments joined in series much as those in FIG. 2 c.  The time-domain representation becomes difficult to extend to such a fiber because of the geometric increase in the number of pulses that is output from a long fiber link. The appropriate alternative representation is in the frequency domain. FIGS. 3 a  and  3   b  illustrate the customary technical representation of the PMD effect in the frequency domain. The production of multiple pulses with various relative temporal delays is Fourier transformed into the spectrum of DGD, FIG. 3 a.  The magnitude of DGD  301  is plotted as a function of frequency  300 . The relative energies of the output pulses and their composite state-of-polarization is represented by the Poincare-sphere representation of Principal States of Polarization (PSP). The PSP is used to represent the overall birefringent axes of a whole fiber link at each frequency. If an input sinusoidal optical wave has an SOP which aligns to the PSP of the fiber which corresponds to the frequency of the optical wave, then the energy of the input optical wave is completely transferred to only one PSP axis. Any other input SOP will cause a splitting of the input pulse energy onto the two orthogonal PSPs of the fiber. FIG. 3 b  illustrates the Poincare sphere  310 , which is a suitable representation of states-of-polarization, and PSP  1  vector of one frequency,  320 , and PSP  2  vector of another frequency,  321 . The direction of the vector is the Principal State of Polarization at one frequency. The length of the vector is the DGD  301  at that frequency  300 . The PSP of the fiber changes for each frequency, mapping a contour of PSPs  330 . 
     FIG. 4 a  illustrates DGD spectrum  301  on frequency axis  300 . The optical signal pulse spectrum  400  is indicated in relation to the DGD spectrum. Four frequencies are considered  401 ,  402 ,  403 ,  404 . FIG. 4 b  illustrates the temporal delay between optical signals at each particular frequency  401 - 404 . Note that this is illustrative, because an optical signal at one particular frequency is a sinusoidal wave and not a pulse; a pulse is used here figuratively. Consider frequency  401  and the result of DGD  301  on a pulse in time. On time axis  410  pulses  420  and  421  experience relative time delay  422  in accordance with the value of DGD at frequency  401  On time axis  410  pulses  423  and  424  experience a relative time delay  425  in accordance with the value of DGD at frequency  402 . On time axis  410  pulses  426  and  427  experience a relative time delay  428  in accordance with the value of DGD at frequency  403 . On time axis  410  pulses  429  and  430  experience a relative time delay  431  in accordance with the value of DGD at frequency  404 . Each impact of distinct relative temporal delays  422 ,  425 ,  428 ,  431 , for each frequency component of optical signal pulse  400  can cause significant pulse distortion. Note that in FIG. 4 b  all pulse heights are all equal. This is for illustrative purposes and does not show the complete effect. 
     FIG. 5 a  illustrates the PSP “spectrum”  330  on the Poincare sphere  310  as it may vary from PSP  1 ,  320 , to PSP  2 ,  321 , as a function of frequency. Four frequencies on the PSP spectrum  330  are indicated,  401 - 404 . Each frequency is coincident with the frequency illustrated in FIG. 4 a.  The pulse input SOP vector is indicated  500 . At each frequency  401 - 404  the input pulse SOP is projected onto the PSP vector. The projection results in a power rebalancing between two orthogonal PSPs. FIG. 5 b  illustrates the combined effect of DGD and SOP-to-PSP projection. Input SOP  500  is projected at frequency  401  in such a manner as to rebalance the pulse energies as indicated by pulses  520 ,  521 . Pulses  520 ,  521  experience DGD  422 . Input SOP  500  is projected at frequency  402  in such a manner as to rebalance the pulse energies as indicated by pulses  522 ,  523 . Pulses  522 ,  523  experience DGD  425 . Input SOP  500  is projected at frequency  403  in such a manner as to rebalance the pulse energies as indicated by pulses  524 ,  525 . Pulses  524 ,  525  experience DGD  428 . Input SOP  500  is projected at frequency  404  in such a manner as to rebalance the pulse energies as indicated by pulses  526 ,  527 . Pulses  526 ,  527  experience DGD  431 . The impact of distinct relative temporal delays at each frequency component of optical signal pulse  400 , coupled with the energy rebalancing due to the SOP-to-PSP projection, can cause significant pulse distortion. FIG. 5 b  illustrates more fully the impact of PMD to an optical pulse. 
     Prior Art exists for a PMD emulator apparatus. FIG. 6 a  illustrates a method which concatenates several segments of highly birefringent fiber  601 ,  602 ,  603 . In order to alter the state of PMD at the output, mechanisms  610 ,  611 ,  612 , are attached along the fiber length which physically rotate the fiber about its longitudinal axis. Two or more fiber segments are used for this apparatus. The relative rotation of the fiber segments  601 ,  602 ,  603  changes the projection of preceding-segment output pulse SOP and following-segment input fiber birefringent axes. The DGD of each fiber segment  601 ,  602 ,  603  is fixed. The limitations of this embodiment include: the birefringence of the fiber segments  601 - 603  is not well controlled during construction and over temperature and aging; the rotation of one fiber segment, e.g.  602 , relative to an adjacent fiber segment, e.g.  603 , is limited to the torsional breaking point of the fiber, and hence rotation is not endless. In sum, the state of PMD is not easily determined in real-time and not easily repeatable. 
     FIG. 6 b  illustrates another apparatus for PMD emulation. Lithium-Niobate (LiNbO) waveguiding polarization controllers  620  transform the SOP from input  630  to output  631 . Highly birefringent fiber segments  604 ,  605 , impart DGD. The LiNbO polarization controllers utilize the electro-optic effect of the LiNbO crystal to alter the SOP of the incoming light  630 . Electrodes  622  are driven by differential voltage  623  to impart an SOP rotation on waveguide  621 . Multiple electrode stages  622  are employed to impart multiple polarization transformations. The limitations of this embodiment include: the actual degree and direction of polarization rotation is not easily monitored and may vary from device to device, the same will change with temperature and aging; the birefringence of the fiber is not well controlled during construction and may change due to environmental effects. In sum, the state of PMD is not easily determined in real-time and not easily repeatable. 
     FIG. 6 c  illustrates another apparatus for PMD emulation. A single LiNbO waveguiding polarization controller  620  is employed. Along the length of waveguide  621 , between electrodes  622 , there is a small degree of DGD which is inherently generated. The design of a device with sufficient number of stages provides for multiple SOP transformation stages and interleaved DGD stages. The limitations of this embodiment include: the SOP transformation from stage to stage is not easily monitored; the actual DGD generation may vary from crystal to crystal; the DGD sections are not easily distinguished from the SOP transformation sections  622 . In sum, the state of PMD is not easily determined in real-time and not easily repeatable. 
     Prior-Art approaches to PMD generation suffer from one or more drawbacks, each of which are sufficient to limit their utility. 
     SUMMARY 
     Therefore, it is the object of the present invention to provide a means and apparatus to generate the effect of PMD in a manner which is known, predictable, repeatable, and sufficient to approximate the behavior of optical fiber. 
     A polarization-mode-dispersion emulation (PMDE) apparatus in accordance with the principles of the present invention includes, briefly and generally, a plurality of PMD-generating stages all positioned to provide a clear light-path through each stage, wherein each PMD-generating stage further includes a first waveplate element, a first birefringent element, and a second waveplate element, all positioned to provide a clear light-path through each element in succession in the order herein listed. To alter the state of PMD which is generated by the apparatus, the waveplate elements are mounted on motorized rotation stages which are operated by a controller that coordinates the waveplate rotation about the axis perpendicular to the birefringent plane. Through rotation of the waveplate elements, the magnitude and modulation of the differential-group delay (DGD) spectrum of the generated PMD can be controlled. 
     The concatenation of PMD-generating stages provides for the interleaving of two complimentary optical effects which generate PMD: the projection of states-of-polarization onto orthogonal birefringent axes and a differential group delay subsequently generated. In accordance with the present invention, the state of PMD which is generated by the apparatus is controlled by the accurate rotation of waveplates, one or more which precedes each birefringent crystal, and by the accurate construction and selection of birefringent crystals. In one preferred embodiment of the present invention, the differential group delay generated by each birefringent crystals located within the apparatus is substantially of the same magnitude. 
     According to another aspect of the present invention, the insertion loss through the PMD-generating apparatus does not substantially vary with the rotation of the waveplates. Typically, the attachment of the waveplates to rotation stages does not provide for zero wobble of the waveplate over 360 degrees of rotation. Any residual wobble imparts displacement on the transiting optical beam which in turn generates rotation-dependent loss. As disclosed in the present invention, use of true zero-order single-plate waveplates minimizes rotation-dependent loss while still providing for the rotation of the state-of-polarization from PMD-generating stage to PMD-generating stage. 
     According to another aspect of the present invention, the accuracy, predictability, programmability, and repeatability of the generated PMD is a distinguishing feature of the disclosed PMD-generating apparatus. The utilization of birefringent crystals and waveplates, whose optical and mechanical properties are well known and stable, and the utilization of high-accuracy rotation stages provided with the measurement and recording of the rotation orientation, together facilitate the predictability of the generated PMD. Moreover, the replacement of the single birefringent crystals with composite crystals, wherein each composite crystal is designed to substantively eliminate temperature variation of the imparted differential-group delay, further enhances the repeatability of the inventive apparatus. 
     According to another aspect of the present invention, the PMD-generating apparatus of the present invention can be employed to test a PMD compensator (PMDC) apparatus. Through the change of PMD generated by a PMDE, the performance of a PMDC may be determined. 
     According to another aspect of the present invention, the PMD-generating apparatus of the present invention can be employed as one part of a PMD compensator apparatus, wherein the PMD-generating apparatus receives optical signals impaired by the PMD effect, a detector apparatus receives the optical signals distorted by both the original PMD effect and subsequently the PMD-generating apparatus effect, a detector monitor measures the degree of total PMD present on the optical signal, and a controller rotates at least one waveplate located within the PMD-generating apparatus so as to minimize the degree of total PMD present on the optical signal. 
     Additional objects, advantages, and features of the various aspects of the present invention will become apparent from the following description of its preferred embodiments, which description should be taken in conjunction with the accompanying drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The above and further features, aspects, and advantages of the invention will be apparent to those skilled in the art from the following detailed description, taken together with the accompanying drawings in which: 
     FIG. 1 a  illustrates the projection of a polarization state onto two orthogonal axes; 
     and FIG. 1 b  illustrates the subsequent differential temporal delay between the two projection components resulting from a high-PMD fiber segment. 
     FIGS. 2 a,    2   b,  and  2   c  illustrate, respectively, the effect of one, two, and three randomly oriented fiber segments on an optical pulse. 
     FIG. 3 a  illustrates a representative differential-group-delay (DGD) spectrum; and 
     FIG. 3 b  illustrates a representative principal-states-of-polarization (PSP) spectrum as a contour on the Poincare sphere. 
     FIG. 4 a  illustrates a representative optical signal spectrum superimposed on a DGD spectrum; 
     and FIG. 4 b  illustrates the effect of DGD on the optical signal at each of four distinct frequencies. 
     FIG. 5 a  illustrates a representative input state-of-polarization (SOP) in relation to a representative PSP spectrum, as projected on the Poincare sphere; 
     and FIG. 5 b  illustrates, in a representative manner, the combined effect of DGD and PSP-projection on the optical signal at each of four distinct frequencies. 
     FIG. 6 a,    6   b,  and  6   c  illustrate Prior Art embodiments of a polarization-mode-dispersion (PMD) emulator. 
     FIG. 7 a  illustrates a block diagram of a PMD emulator comprised of a plurality of “simple” PMD-generating stages; and 
     FIG. 7 b  illustrates one inventive embodiment of the PMD emulator utilizing a concatenation of birefringent crystals and waveplates. 
     FIG. 8 a  illustrates the details of polarization-projection and DGD effects on an optical pulse; and 
     FIG. 8 b  illustrates the equivalent effect utilizing a preceding and following half-wave waveplate. 
     FIG. 9 a  illustrates a birefringent crystal inclined with respect to a transversing optical beam; 
     FIG. 9 b  illustrates the displacement of a transversing optical beam through an inclined birefringent crystal over a loci of rotation; 
     and FIG. 9 c  illustrates a representative optical power signal as a function of the birefringent crystal rotation. 
     FIG. 10 a  illustrates a birefringent crystal preceded and followed by a waveplate; and 
     FIG. 10 b  illustrates a representative optical power signal as a function of the waveplate rotation. 
     FIGS. 11 a,    11   b  illustrate the replacement of two adjacent waveplates with a single waveplate; and FIG. 11 c  illustrates the correspondence of the crystal concatenation with FIG. 7 b.    
     FIG. 12 a  illustrates a block diagram of a PMD emulator where each stage has the same DGD value; and 
     FIG. 12 b  illustrates just two birefringent crystals with some relative rotation, and the resultant DGD spectrum and DGD locus as a function of the mutual relative rotation. 
     FIG. 13 a  illustrates four birefringent crystals grouped in two pairs, where within each pair the extraordinary axes are aligned; 
     FIG. 13 b  illustrates a representative resultant DGD spectrum; 
     FIG. 13 c  illustrates four birefringent crystals with distinct orientations; and 
     FIG. 13 d  illustrates a representative resultant DGD spectrum that exhibits variation of DGD with frequency. 
     FIGS. 14 a  and  14   b  illustrate representative temperature dependencies of the ordinary and extraordinary indices of refraction for two distinct birefringent crystals; and FIG. 14 c  illustrates the elimination of the birefringent temperature dependence by the appropriate combination of these two crystals. 
     FIGS. 15 a  and  15   b  illustrate block diagrams of a motor drive affixed to a rotation stage, and a rotation encoder and encoder recorder to monitor rotation. 
     FIG. 16 illustrates a PMD emulator configured for double-pass and a preceding optical circulator to discriminate forward-going and backward-going light. 
     FIGS. 17 a,    17   b,  and  17   c  show block diagrams which illustrate methods to test a PMD compensator using a PMD emulator. 
     FIG. 18 shows a block diagram of a PMD compensator built from a PMD generator, a feedback signal generator, and a controller. 
    
    
     DETAILED DESCRIPTION 
     Reference will now be made in detail to embodiments of the present invention, examples of which are illustrated in the accompanying drawings, wherein the like reference numerals refer to the like elements throughout. 
     FIG. 7 a  is an illustrative embodiment of a PMD-generating apparatus, wherein light which travels along input fiber  710  is collimated by a lens  711  to form a free-space optical beam  700 . PMD-generating apparatus  720  receives optical beam  700 . The PMD-generating apparatus  720  further comprises a plurality of simple PMD-generating stages  721 ,  722 ,  723 ,  724 . There may be any number of simple PMD-generating stages, preferably two or more. The optical beam  701  that transits the PMD-generating apparatus  720  is received by a focusing lens  712 , and is further received by output fiber  713 . The PMD-generating apparatus thereby imparts PMD to light beam  702  from light beam  700 . 
     FIG. 7 b  illustrates a more detailed embodiment of a PMD-generating apparatus. Each simple PMD-generating stage, e.g.  721 , further comprises a first waveplate  730 , a first birefringent crystal  731 , and a second birefringent crystal  732 . First waveplate  730  is preferably, but not limited to, a λ/2, or half-wave, waveplate, where λ is the approximate wavelength of optical beam  700 . 
     The birefringent crystals are preferably uniaxial and are cut so that the plane formed by the extraordinary and ordinary crystal axes is perpendicular to the path of the optical beam  700 . The purpose of two birefringent crystals  731  and  732 , rather that one crystal  731  alone, is to engineer a reduction of the temperature dependence of the PMD-generating stage; this design will be described in the following discussion. The sum length of first and second birefringent crystals  731 ,  732 , is preferably long compared with waveplate  730  so as to exhibit a substantial frequency-dependent polarization transformation. That is, the length of the first and second birefringent crystals  731 ,  732 , is sufficient to produce substantial differential temporal delay between the two orthogonal polarization axes of the crystals. Additionally, the extraordinary axes of birefringent crystals  731  and  732  are preferably aligned with zero or ninety degree difference so as to maintain the production of simple PMD, as well as to maximize the reduction of temperature dependence of the PMD-generating stage. 
     The concatenation of simple PMD-generating stages  721 - 724  may further be terminated with last waveplate  725  for reasons that will be shortly explained. 
     Referring to FIG. 8 a,  a birefringent crystal may impart simple PMD in a like manner to that illustrated in FIG. 1 b  and formerly described. State-of-polarization axes  810  and  811  are aligned, heuristically and with no loss of generality, to the SOP of input light  800 . A birefringent crystal  820  has extraordinary axis  813  and ordinary axis  814  rotated with respect to the input SOP axis  811 . Input light  800  is thereby projected on the face of the birefringent crystal into two orthogonally polarized pulses  801  and  802 . Transit through the crystal imparts differential group delay between pulses  801  and  802 . The light which is output from crystal  820  is then projected back onto the origin polarization axes  810 ,  811 . The result is two pulses  801  ( a, b ) and  802  ( a, b ) which are temporally delayed in relation to one another. Further, pulse  801  ( a, b ) has polarization components  801   a,  aligned with axis  810 , and  801   b,  aligned with axis  811 ; pulse  802  ( a, b ) has polarization components  802   a,  aligned with axis  810 , and  802   b,  aligned with axis  811 . 
     An equivalent polarization transformation and differential time delay is shown in FIG. 8 b.  First waveplate  821  precedes birefringent crystal  820 , and second waveplate  822  follows the same crystal  820 . For the response of the  821 - 820 - 822  configuration to be equivalent to the single rotated  820  birefringent crystal, waveplates  821  and  822  must both be lambda/2, or half-wave, waveplates. The extraordinary axis of the crystal  820 ,  813 , is further rotated to be in alignment with original polarization axis  811 . The extraordinary axis  815  of first waveplate  821  is rotated from axis  811  to one-half the angle subtended between axes  813  and  811  of crystal  820  in FIG. 8 a.  The extraordinary axis  815  of second waveplate  822  is rotated in the opposite direction as first waveplate  821  but with the same magnitude of rotation. The sequence of first waveplate, birefringent crystal, and second waveplate as herein described produces an equivalent effect as the single, rotated crystal of FIG. 8 a.    
     Waveplate  821  transforms the polarization coordinate axes  810 ,  811  into  810 ′ and  811 ′. Pulse  800  is projected onto said axes to form pulses  801 ,  802 . Transit of birefringent crystal  820  imparts differential time delay between pulses  801 ,  802 . Waveplate  822 , with opposite rotation relative to waveplate  821 , restores polarization axes  810 ′,  811 ′, to axes  810 ,  811 . Restoration of said polarization axes further projects pulse  801  into components  801   a,    801   b,  and projects pulse  802  into components  802   a,    802   b,  where the ‘a’ pulses are polarization-aligned to axis  810  and the ‘b’ pulses are polarization-aligned to axis  811 . 
     There is an important practical advantage to the employment of the scheme of FIG. 8 b,  which is more complicated scheme than that of FIG. 8 a.  This advantage is illustrated in reference to FIGS. 9 a  and  9   b  and the discussion related thereto. FIG. 9 a  illustrates birefringent crystal  820  which is inclined by amount  901  with respect to input light beam  910 . Transit of light beam  910  through the crystal  820 , for a small angle  901 , imparts a displacement  902  between actual output beam  911  and where beam  910  would be  912  in the absence of inclined crystal  820 . The displacement  902  is directly proportional to the length of the crystal  820 . Reduction of the length of crystal  820  would reduce the degree of displacement  902 , but that is contrary to the generation of substantial differential temporal delay. 
     FIG. 9 b  illustrates crystal  820  in a configuration in which the crystal is rotated nominally about the axis normal to the crystal birefringent plane, but where there is persistent inclination of the crystal  820  to the optical axis  910 . Beam  911  output from crystal  820  thereby traces a circle  920  in space. Light which travels through first optical fiber  710 , is coupled by first lens  711  to form collimated beam  910 , and transits crystal  820  produces light beam  911 . Second lens  712  is intended to couple light beam  911  to second optical fiber  713 . However, due to the displacement loci  920  of beam  911 , the optical power which is received by second optical fiber varies with the rotation of crystal  820 . FIG. 9 c  illustrates variation  932  of optical power  931  as a function of crystal rotation  930 . It has been experimentally shown that the modulation depth  933  of the optical power on second optical fiber  713  can be substantial. 
     As a consequence of substantial modulation depth  933 , due to practical difficulties and costs with eliminating the inclination of crystal  820 , the optical system of FIG. 8 b,  redrawn in FIG. 10 a,  may be preferred. Birefringent crystal  820  is preceded by first waveplate  821  and followed by second waveplate  822 . A waveplate in practice can be made as thin as 50 μm, which may be substantially shorter than the birefringent crystal  820 . To the extent the waveplates  821 ,  822  impart displacement of an optical beam due to small inclinations, the magnitude of the displacement may be substantially smaller. To impart the PMD equivalent to a single, rotated birefringent crystal, waveplates  821  and  822  are rotated in concert, and in opposing directions, while birefringent crystal  820  remains fixed. For light which travels along first optical fiber  710 , is collimated by first lens  711  to form beam  910 , and transits simple PMD-generating stage  1000 , output beam  911  is focused by second lens  712  to second optical fiber  713 . The modulation of the optical power  932  received by second optical fiber  713  may exhibit a modulation depth  1030 , FIG. 10 b,  substantially smaller than modulation depth  933 FIG. 9 c.    
     FIG. 7 a  illustrates a PMD-generating apparatus consisting of a plurality of simple PMD-generating stages. Simple PMD-generating stage  1000  of FIG. 10 a  may be used for each simple PMD-generating stage, e.g.  721 . However, FIG. 11 illustrates a possible simplification. Referring to FIG. 11 a,  simple PMD-generating stage  1000  comprises a first waveplate  1110 , a birefringent crystal  1101 , and a second waveplate  1111 . Similarly simple PMD-generating stage  1000 ′ comprises first waveplate  1112 , birefringent crystal  1102 , and second waveplate  1113 . Because stages  1000  and  1000 ′ are adjacent to one another, waveplates  1111  and  1112  are adjacent. FIG. 11 b  illustrates the combination of waveplates  1111  and  1112  into one waveplate  1114 . An equivalence may be established if waveplates  1111 ,  1112 , and  1114  are all half-wave waveplate. In this case, the rotation angle of waveplate  1114  is the sum of inclination angles of waveplates  1111  and  1112 . FIG. 11 c  establishes a correspondence between the system of FIG. 11 b  and FIG. 7 b.  From FIGS. 11 b  to  11   c,  crystal  1101  and waveplate  1110  are grouped as stage  721 ; crystal  1102  and waveplate  1114  are grouped as stage  722 . This grouping is repeated for each birefringent crystal that is present in the PMD-generating apparatus. Lastly, waveplate  1113 , the trailing waveplate, corresponds to waveplate  725 . Waveplate  1113  is required to transform the polarization coordinate system back to the original input system. 
     The relative alignment of waveplates and birefringent crystals is an essential aspect of the predictability of the generated PMD. In accordance with one preferred embodiment of the present invention, alignment may be performed in three stages: 1) alignment of the waveplates to a external standard, 2) alignment of the birefringent crystals to the waveplates, and 3) mutual alignment of the birefringent crystal extraordinary axes. To align the waveplates, first two high-extinction-ratio polarizers are placed in an optical path. The polarizers are mutually rotated to maximally extinguish the optical beam output from the second polarizer. Second, a first waveplate, mounted and fixed on a rotation stage, is inserted between the two polarizers. The waveplate is subsequently rotated so as to again maximize the extinction of the optical beam output from the second polarizer. The rotation angle of the waveplate is recorded, and then the waveplate and rotation stage is removed from the light path. Subsequently, each waveplate, mounted and fixed on individual rotation stages, is inserted into the optical path, between the two polarizers, and the preceding alignment procedure is repeated. 
     For the second stage of the alignment procedure, one aligned waveplate is placed into the optical beam between the two polarizers, and one birefringent crystal is placed on the rotation stage behind the waveplate. The birefringent crystal is mounted onto a portion of the rotation stage which does not move, making the placement of the birefringent crystal stationary. With the birefringent crystal in position, the rotation of the crystal is manually adjusted to again maximize the extinction of the optical beam output from the second polarizer. The birefringent crystal is then fixed into this position. Note that at this point is remains ambiguous whether the ordinary or extraordinary axis of the birefringent crystal has been aligned with the first polarizer axis. Nonetheless, subsequently, each birefringent crystal, mounted to individual rotation stages with pre-aligned waveplates, is inserted into the optical path, between the two polarizers, and the preceding alignment procedure is repeated. 
     For the last stage of the alignment procedure, the ambiguity of ordinary or extraordinary axis alignment is resolved. Two rotation-stage assemblies, comprising an aligned waveplate and birefringent crystal, are placed in cascade in the optical beam, between the two polarizers. The optical spectrum of the optical beam output from the second polarizer is analyzed. Amplitude modulation of the optical spectrum indicates that the extraordinary axes of the two birefringent crystals are aligned, whereas no substantial modulation of the optical spectrum indicates that the extraordinary axes of the two birefringent crystals are oriented at 90 degrees from one another. The presence or absence of amplitude modulation of the optical spectrum is recorded. Subsequently, the one of the two rotation-stage assemblies is removed and replaced with another assembly. The determination of amplitude modulation of the optical spectrum is recorded. This procedure is repeated for all remaining rotation-stage assemblies. 
     Following the above-outlined alignment procedure, and any other such procedure that determines the optical axes of all optical components, all waveplate, birefringent crystal, and rotation stage assemblies are mounted in concatenation in an optical beam path. The first and second optical polarizers are removed. 
     Referring now to FIG. 12 a,  a concatenation of simple PMD-generating stages  721 ,  722 ,  723 ,  724 , imparts PMD onto optical beam  702  from optical beam  700 . A maximum value of DGD is attained when all of the PMD-generating stages are aligned, Δτ max  1210 . FIG. 12 a  illustrates a case where the DGD values of all the PMD-generating stages are equal. For N stages, each stage has a DGD value of Δτ max/N,  1211 . The case where all the DGD values of all simple PMD-generating stages are equal is particularly simple to analyze. FIG. 12 b  illustrates two equal-length birefringent crystals  1220 ,  1221 , which are rotated with respect to one another. The maximum DGD value is Δτ max  1210 . For equal length crystals, the DGD values  1233  as a function of frequency  1230  are constant across the free-spectral range  1232 . The absolute value of DGD can range between zero and Δτ max, and is controlled by the relative rotation between the two crystals  1220 ,  1221 . The loci of DGD values  1241  as a function of relative crystal rotation  1240  has the form of: Δτ max|cos (theta 2 −theta 1 )|. The generation of DGD which is independently of frequency is the simplest form of PMD, and one which is essential for the basic testing of PMD compensators. 
     FIG. 13 a  illustrates four simple PMD-generating stages, designed so that the maximum attainable DGD value Δρ max remains equal to that denoted by  1210 . In the case where all DGD values from stages  1301 - 1304  are the same, the imparted PMD is simple to analyze. When all DGD values are the same, then each stage has a DGD of Δτ max/4, 1310. FIGS. 13 a  and  b  indicate an important configuration wherein PMD-generating stages  1301  and  1302  are aligned along the fast axis, and stages  1303  and  1304  are aligned along the fast axis. Rotation is performed wherein the first pair,  1301  and  1302 , is rotated relative to the second pair,  1303  and  1304 . The result is a fixed DGD value  1320  in frequency over the FSR. Note that the FSR  1330  is twice the FSR  1232  in FIG. 12 b  because the per-stage DGD value Δτ max/4,  1310 , of the system in FIG. 13 a  is one-half the per-stage DGD value Δτ max/2,  1211 . 
     FIG. 13 c  illustrates a more complex configuration of the four simple PMD-generating stage configuration. The crystals  1301 ,  1302 ,  1303 ,  1304 , have in general distinct rotations. The resultant DGD spectrum  1321  exhibits modulation over the FSR  1330 . Here is the first example of the preferred embodiment description where complex PMD, one which begins to emulate the true behavior of birefringent fiber, can be generated. A detailed analysis of a PMD-generating apparatus consisting of N like stages shows that the functional form of the DGD Δτ follows                Δ                 τ     =         DC   (       θ   1     ,     θ   2     ,   …                )     +       ∑     n   =   1     N              AC   n     (       θ   1     ,     θ   2     ,   …                )     ·     cos        (       2      π                   n        (     f   -     f   o       )         FSR     )                       (   1   )                                
     where the DC term is function of (θ 1 , θ 2 , . . . ) but not of the frequency, the N AC terms are each functions of (θ 1 , θ 2 , . . . ) but not of frequency, and the frequency dependence follows the cosine form, weighted by the respective AC terms. Note that the N AC terms are in fact correlated and not orthogonal. It can be proved analytically that the AC 13 N and AC 13 N−1 terms are identically zero. Thus when there are only two simple PMD-generating stages, there is no modulation of the DGD spectrum over the FSR. However, with more than two stages, there is modulation of the DGD spectrum. Equation  1  further indicates that there is a maximum rate, 2π(N−2)/FSR, that the DGD spectrum can change in frequency with this emulation apparatus. This has important implications because the design of a PMD emulator may require a substantial amount of DGD modulation across the bandwidth of an optical pulse. 
     Referring now to FIG. 14, description of FIG. 7 a  item  721  indicated that two birefringent crystals  731 ,  732 , may be preferably employed for the generation of DGD. The purpose of two crystals of distinct material systems is to compensate for the temperature variation that either crystal alone exhibits. FIG. 14 a  illustrates the temperature dependence of the two refractive indices of birefringent crystal  731  with length L 1 . In general the temperature dependence on refraction index for one axis, say  1401 , is different than the orthogonal axis,  1402 . The difference between the two refractive indices, the crystal birefringence, accordingly changes as a function of temperature. To cancel the effect of temperature dependence, a second crystal  732  of length L 2  is used. The second crystal  732  must possess refractive index curves,  1410  and  1411 , that have different temperature dependent slopes than first crystal  731 . Preferably the temperature dependence of second crystal  732  is much stronger than that of the first crystal. Upon combination of the two crystals  731 ,  732 , in the proper manner, and with proper length ratio, temperature dependence along the fast and slow axes remains  1420 ,  1421 , but the slopes of the refractive index change with temperature are the same. Therefore, the difference between the refractive indices, the composite crystal birefringence, remains invariant to temperature. For example, a yttrium ortho-vanadate (YVO4) crystal, which is positive uniaxial, may be combined with a lithium niobate (LiNbO3) crystal, which is negative uniaxial, to produce reduction of temperature dependence. The extraordinary axes of the two crystals are aligned, and the length ratio YVO4 to LiNbO3 must be about 10:1. 
     An advantage of the embodiments of the PMD emulator heretofore disclosed is that the waveplates and/or birefringent crystals may be rotated using a precise rotation apparatus. FIG. 15 a  illustrates a means for the rotation of the simple PMD-generation stage component(s). A optical component is attached to a rotation stage  1500  which has a clear aperture. A motor  1501  is used to drive the rotation of stage  1500 . A motor encoder  1502  is coupled to the motor  1501  so as to encode the revolutions of the motor. An encoder recorder  1503  is attached to the motor encoder  1502  to record the signs generated by said encoder  1502 . Alternatively, FIG. 15 b  illustrates another means for rotation and recording. A optical component of a simple PMD-generation stage is attached to a rotation stage  1500  which has a clear aperture. A motor  1501  is used to drive the rotation stage  1500 . A separate rotation encoder  1510  is further attached to the rotation stage  1500 . The rotation encoder measures the rotation of the rotation stage  1500 . The signals of the rotation encoder  1510  are recorded by encoder recorder  1511 . 
     FIG. 16 illustrates an alternative and preferable embodiment of the invention herein disclosed. PMD emulator  720 , further comprised of a plurality of simple PMD stages,  721 - 724 , is placed between collimating lens  711  and mirror  1601 . Optical circulator  1600  receives forward-going optical beam which travels on first optical fiber  1610 . Second optical fiber  1611  receives light from circulator  1600  which originated from forward-going optical beam  1610 . Coupling lens  711  receives the forward-going beam from second optical fiber  1611  and collimates the light to form forward-going beam  1620 . Forward-going beam  1620  transits PMD emulator  720  and is output to beam  1621 . Mirror  1601  reflects beam  1621 , which is subsequently received by PMD generator  720 . The backward-going beam returns to coupling lens  711  and is received by second optical fiber  1611 . Optical circulator  1600  receives the backward-going beam and redirects the backward-going beam to third optical fiber  1612 . The advantage of this double-pass configuration is the extended production of PMD without the use of additional simple PMD stages. 
     Several uses of a PMD emulator, as disclosed herein, are illustrated in FIGS. 17 a-c.  Referring to FIG. 17 a,  input optical beam  1700  is received by PMD emulator  1701 . The PMD emulator imparts PMD onto the beam, producing impaired optical beam  1700 ′. A PMD compensator  1702  subsequently receives impaired optical beam  1700 ′. The purpose of the PMD compensator  1702  is to substantially restore the impaired optical beam  1700 ′ for subsequent detection. FIG. 17 b  illustrates an optical beam  1700  which is received by PMD emulator  1701 . The resultant impaired optical beam is received by an optical transmission system  1710 . The optical beam  1700 ′, with accumulated PMD and transmission impairments, is subsequently received by a PMD compensator  1702 . FIG. 16 c  illustrates an optical beam  1700  which is received by first PMD emulator  1701 , then first optical transmission system  1710 , second PMD emulator  1701 ′, and second transmission system  1710 ′, producing as a result impaired optical beam  1700 ′. PMD compensator  1702  subsequently receives the impaired optical signal  1700 ′ for substantial restoration. 
     As described in the Background of the Invention, an PMD emulator  1801  in FIG. 18 may be used in combination with a feedback signal generator  1802  and control mechanism and algorithm  1803 . Together blocks  1801 ,  1802 , and  1803  form a closed loop system which can track and correct for the changing state of PMD generated by an optical transmission system. 
     The foregoing description of specific embodiments of the invention has been presented for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise forms disclosed, and many modifications and variations are possible in light of the above teachings. For example, intentional multiple reflections may take place within each birefringent crystal for the purpose of enhancing the accumulated DGD of each stage. The embodiments were chosen and described to best explain the principles of the invention and its practical application, and to thereby enable others skilled in the art to best utilize the invention. It is intended that the scope of the invention be limited only by the claims appended hereto.