Abstract:
A polarization scrambler, and associated method for polarization scrambling, is provided wherein light is passed through a time-variant retarder. At least after upstream connection or downstream connection of a time-invariant retarder, this time-variant retarder may be represented as a retarder with eigen modes which are uniformly distributed on the Poincaré sphere and with a delay of 5π/6. An exemplary embodiment contains only two electro-optical wave plates, which are actuated by alternating signals containing harmonics. The polarization scrambler which is produced is independent of the input polarization of the light and can be used in devices for detection of polarization mode dispersion.

Description:
BACKGROUND OF THE INVENTION  
         [0001]    Polarization mode dispersion, called PMD, has an adverse effect on high-speed optical data transmission. The Proceedings of the Optical Fiber Communication Conference OFC2001, Mar. 17-22, 2001, Anaheim, Calif., USA, Presentation PD27 disclosed a method for measurement of polarization mode dispersion in which a polarization scrambler is used which can produce a large number of output polarizations, or all possible output polarizations, from a specific input polarization. The output polarizations which can be produced define a three-dimensional, not only flat figure, within the Poincaré sphere. Further polarization scramblers are described in Electronics Letters, Volume 30 (1994)18, pages 1500-1501. These can produce a depolarized output signal from any desired input polarizations. At least for certain input polarizations, the output polarizations which are produced define, however, only a flat figure, and not a three-dimensional figure, within the Poincaré sphere, since depolarization is a necessary precondition, but not in itself a sufficient precondition, for output polarizations to actually be produced which define a three-dimensional figure, and not just a flat figure, within the Poincaré sphere.  
           [0002]    Furthermore, ideally, the output polarizations which are produced should be uncorrelated; that is to say, the correlation matrix of the normalized Stokes vectors of the output polarization should be equal to ⅓ times the 3×3 unit matrix.  
           [0003]    An object of the present invention is, thus, to specify a polarization scrambler, as well as an associated method for polarization scrambling, which emits uncorrelated output polarizations for any given input polarizations.  
         SUMMARY OF THE INVENTION  
         [0004]    The problem is solved by designing a polarization scrambler such that, at least after upstream connection and/or downstream connection of a time-invariant retarder, the polarization scrambler can be regarded as a retarder with eigen modes which are uniformly distributed on the Poincaré sphere, and with a delay of 5π/6.  
           [0005]    In one exemplary embodiment of the present invention, this polarization scrambler is preferably in the form of three cascaded electro-optical wave plates. In a further exemplary embodiment of the present invention, it is preferably in the form of two cascaded electro-optical wave plates.  
           [0006]    Additional features and advantages of the present invention are described in, and will be apparent from, the following Detailed Description of the Invention and the Figures. 
       
    
    
     BRIEF DESCRIPTION OF THE FIGURES  
       [0007]    [0007]FIG. 1 shows a polarization scrambler according to the present invention.  
         [0008]    [0008]FIG. 2 shows an exemplary embodiment of a polarization scrambler according to the present invention.  
         [0009]    [0009]FIG. 3 shows a further exemplary embodiment of a polarization scrambler according to the present invention. 
     
    
     DETAILED DESCRIPTION OF THE INVENTION  
       [0010]    Without restriction to generality, all retarders are regarded as being loss-free in the following text, although the principle of the present invention also applies to lossy retarders. Retarders each have a pair of mutually orthogonally polarized eigen modes. To assist understanding, one of the two eigen modes is in each case referred to as the reference eigen mode in the following text. It is sufficient to quote the reference eigen mode, instead of both eigen modes, since the other eigen mode is simply the mutually orthogonal polarization. Any other eigen mode is subject to a phase delay in the retarder which is greater by the so-called delay of the retarder than that of the reference eigen mode which is otherwise identified or defined as the reference eigen mode by this statement. It is sufficient to consider delays in the main interval −π . . . π, since the effect of two delays which differ by an integer multiple of 2π is identical.  
         [0011]    It is also possible to take yet another step and to consider only positive values in the interval 0 . . . π as delays. This is because negative values can be replaced by positive values when the two eigen modes are interchanged with one another.  
         [0012]    The 3×3 submatrices of the Müller matrices of retarders are used in the following text to describe retarders, including rows and columns 1 to 3 which are associated with the Stokes vector components 1 to 3, with the numbering from 0 to 3 being used for the rows and columns in the Muller matrices and for the non-normalized Stokes vector components. The 3×3 submatrices accordingly have determinants whose magnitude is 1 or −1 and describe rotations, or mirror-image rotations, in the space of the normalized Stokes vectors. For the sake of simplicity, these are combined by the term rotation matrices in the following text. Normalized Stokes vectors are used to describe the input and output polarizations of retarders.  
         [0013]    In an arrangement according to the present invention for polarization scrambling as shown in FIG. 1, an optical signal OS is supplied at a first input Rsi 1  to a first polarization scrambler Rs 1 , with a first input polarization Si 1  as the polarization P. The polarization P of the optical signal OS is modulated as a function of time t in the first polarization scrambler Rs 1 . To this end, it is driven by at least one preferably electrical control signal ERs. The optical signal OS emerges from the first polarization scrambler Rs 1  at a first output Rso 1 , with a first output polarization So 1  as a polarization P.  
         [0014]    The first polarization scrambler Rs 1  possibly may be represented, at least using mathematical notation, or represented as a chain circuit on an input-side time-invariant retarder Ri with an input-side delay φi as the delay and an input-side reference eigen mode Sri as the reference eigen mode, of a second polarization scrambler Rs 2  and of an output-side time-invariant retarder Ro with an output-side delay φo as the delay and an output-side reference eigen mode Sro as the reference eigen mode, through which the optical signal OS passes in this sequence. Input-side or output-side time-invariant retarders Ri, Ro also may be omitted or may have a zero delay.  
         [0015]    The second polarization scrambler Rs 2  has a second input Rsi 2  as the input, at which the optical signal OS is fed in with a second input polarization Si 2  as the polarization P, and a second output Rso 2  as the output, at which the optical signal OS emerges with a second output polarization So 2  as the polarization P.  
         [0016]    Here and in the following text, designators such as R s1 , R s2 , with the same symbols as the designators for previously introduced polarization scramblers or retarders such as Rs 1 , Rs 2 , but which contain subscript suffices such as  s1, s2 , denote the associated rotation matrices. Likewise, designators such as S i1 , S o1 , S i2 , S o2  which are in the same form as designators for previously introduced polarizations such as Si 1 , So 1 , Si 2 , So 2 , but which contain subscript suffices such as  i1, o1, i2, o2 , denote the associated normalized Stokes vectors.  
         [0017]    The normalized Stokes vector S o1 , S o2  of the first and second output polarizations So 1 , So 2 , respectively, of the first and second polarization scramblers Rs 1 , Rs 2  respectively, according to the present invention has a respective correlation matrix c= 21  S o1 ·S o1   T &gt; or c=&lt;S o2 ·S o2   T&gt;  which is obtained by averaging of the respective product S 01 ·S 01   T  or S 02 ·S 02   T  of the corresponding normalized Stokes vector S o1 , S o2  times its transposed S 01   T , S 02   T  with respect to time t. The correlation matrix C is at least approximately equal to ⅓ times the 3×3 unit matrix  1 , C=(⅓)*1, to be precise independently of the existing first or second input polarization Si 1 , Si 2 .  
         [0018]    The first and/or second polarization scramblers Rs 1 , Rs 2  have an overall reference eigen mode Sr as the reference eigen mode, and an overall delay φ as the delay. According to the present invention, at least the first and possibly also the second polarization scrambler Rs 1 , Rs 2  is in the form of an optical retarder with a Stokes vector Sr, which is at least approximately uniformly distributed on the Poincaré sphere, of the overall reference eigen mode Sr and with an overall delay φ of at least approximately 5π/6. An overall delay φ of −5π/6 is likewise possible and is taken into account in the following text by treating it as an overall delay φ of 5π/6 with the eigen modes at the same time being interchanged; that is to say, with the other eigen mode being chosen as the overall reference eigen mode Sr. Other overall delays φ which are possible according to the present invention, and modulo 2π are equal to 5π/6 or −5π/6, are regarded in the following text as being identical to these, and therefore are not discussed in their own right.  
         [0019]    If the stated inventive features do not apply to the second polarization scrambler Rs 2 , then input-side and/or output-side time-invariant retarders Ri, Ro always can be chosen such that the inventive features apply to the first polarization scrambler Rs 1 .  
         [0020]    Input-side and output-side time-invariant retarders Ri, Ro need not actually be present. However, they show that the overall invention follows a single standard idea.  
         [0021]    A general retarder Rg has the general rotation matrix  
               R   g     =                  R   g          (       ϕ   g     ,     S   rg       )                     =                [                   S   rg1   2     +                 (       S   rg2   2     +     S   rg3   2       )        cos                   ϕ   g                             S   rg1            S   rg2          (     1   -     cos                   ϕ   g         )         -                 S   rg3        sin                   ϕ   g                             S   rg1            S   rg3          (     1   -     cos                   ϕ   g         )         +                 S   rg2        sin                   ϕ   g                                 S   rg1            S   rg2          (     1   -     cos                   ϕ   g         )         +                 S   rg3        sin                   ϕ   g                           S   rg2   2     +                 (       S   rg1   2     +     S   rg3   2       )        cos                   ϕ   g                             S   rg2            S   rg3          (     1   -     cos                   ϕ   g         )         -                 S   rg1        sin                   ϕ   g                                 S   rg1            S   rg3          (     1   -     cos                   ϕ   g         )         -                 S   rg2        sin                   ϕ   g                             S   rg2            S   rg3          (     1   -     cos                   ϕ   g         )         +                 S   rg1        sin                   ϕ   g                           S   rg3   2     +                 (       S   rg1   2     +     S   rg2   2       )        cos                   ϕ   g                   ]       ,                               
 
         [0022]    which, in this case, without any restriction to generality, has a positive determinant +1. Its general reference eigen mode Srg is given by the normalized Stokes vector  
         S   rg     =         [           S   rg1           S   rg2           S   rg3           ]     T     =     [           S   rg1               S   rg2               S   rg3           ]                             
 
         [0023]    and its general delay is φ g . For comparison: its Jones matrix is  
       [             cos                     ϕ   g     /   2       +     j                   S   rg1        sin                     ϕ   g     /   2                 j        (       S   rg2     +     j                   S   rg3         )          sin                     ϕ   g     /   2                   j        (       S   rg2     -     j                   S   rg3         )          sin                     ϕ   g     /   2               cos                     ϕ   g     /   2       -     j                   S   rg1        sin                     ϕ   g     /   2               ]                         
 
         [0024]    and the general reference eigen mode Srg is also given by the Jones vector  
           1       2        (     1   +     S   rg1       )                [           1   +     S   rg1                   S   rg2     -     j                   S   rg3               ]       .                         
 
         [0025]    It is advantageous to define the overall reference eigen mode Sr in such a manner that its normalized Stokes vector Sr denotes a point which preferably can be chosen as required on the surface of the Poincaré sphere in a specific polar coordinate system as a function of a longitudinal coordinate α, which lies in the interval 0 . . . 2π, and an azimuth coordinate β, which lies in the interval −π/2 . . . π/2. The uniform distribution according to the present invention of the normalized Stokes vector S r  of the overall reference eigen mode Sr on the surface of the Poincaré sphere then preferably can be achieved, at least approximately, by choosing the longitudinal coordinate α to be at least approximately uniformly distributed in the interval 0 . . . 2π, by choosing the azimuth coordinate β to be at least approximately in the interval −π/2 . . . π/2 with a probability density function which is proportional to the cosine cos(β) of the azimuth component β, and by choosing the longitudinal coordinate ac and the azimuth coordinate β to be at least approximately statistically independent. This results in a first, a second and a third parameter Srp 1 =cos(α)*cos(β), Srp 2 =sin(α)*cos(β), Srp 3 =sin(β), which are each uniformly distributed in the interval −1 . . . 1 and are uncorrelated in pairs. First, second and third parameters Srp 1 , Srp 2 , Srp 3  are the components of the normalized Stokes vector S r  of the overall reference eigen mode Sr in a Cartesian coordinate system with respect to which the longitudinal coordinate a and the azimuth coordinate β form a polar coordinate system.  
         [0026]    There are a large number of different possible ways to vary the longitudinal coordinate α and the azimuth coordinate β as a function of the time t. For example, the longitudinal coordinate α is chosen in accordance with α=Ω 1 t, where Ω 1  is a first angular velocity Ω 1 . The azimuth coordinate β is, for example, chosen in accordance with β=arcsin(2Ω 2 t/π) for −π/2≦Ω 2 t≦π/2 and in accordance with β=arcsin(−2(Ω 2 t−π)/π) for π/2≦Ω 2 t≦3π/2 where Ω 2 t for the range determination is chosen modulo 2π in the interval −π/2 . . . 3π/2, and the function defined in this way is assumed to be an ideal function. In this case, Ω 2  is a second angular velocity Ω 2 , whose magnitude |Ω 2 t| is chosen to be either very large or very small in comparison to the magnitude |Ω 1 | of the first angular velocity Ω 1 . In practice, a simpler function is generally chosen as the azimuth coordinate β, in which the high-frequency harmonics are attenuated or eliminated, at least as far as a specific order. A Fourier series  
         [0027]    β=0.944sin(Ω 2 t),  
         [0028]    β=0.944sin(Ω 2 t)−0.177sin(3Ω 2 t),  
         [0029]    β=0.944sin(Ω 2 t)−0.177sin(3Ω 2 t)+0.081sin(5Ω 2 t),  
         [0030]    β=0.944sin(Ω 2 t)−0.177sin(3Ω 2 t)+0.08sin(5Ω 2 t)−0.049sin(5Ω 2 t)  
         [0031]    terminated after the first, third, fifth or seventh harmonic, for the ideal function are possible examples. In this case, it is advantageous to choose the magnitude |Ω 1 | of the first angular velocity Ω 1  to be equal to an integer multiple of the magnitude |Ω 2 |, |3Ω 2 |, |5Ω 2 |, |7Ω 2 | of the highest multiple of the second angular velocity Ω 2  which occurs in the terminated Fourier breakdown of the azimuth coordinate β.  
         [0032]    In one exemplary embodiment of the present invention, as shown in FIG. 2, the first polarization scrambler Rs 1  is in the form of a chain circuit of a first, a second and a third retarder R 1 , R 2 , R 3 . The first, second and third retarders R 1 , R 2 , R 3  have a first, a second and a third delay φ1, φ2, φ3 and normalized Stokes vectors S r1 , S r2 , S r3  for the first, second and third reference eigen modes Sr 1 , Sr 2 , Sr 3  which can be varied as required on a great circle on the Poincaré sphere with a first, a second and a third angle coordinate ψ1, ψ2, ψ3. In this case, each angle coordinate ψ1, ψ2, ψ3 runs along the great circle in the interval 0 . . . 2π.  
         [0033]    Such first, second and third retarders R 1 , R 2 , R 3  are, for example, the electro-optical wave plates, which are known from Optics Letters 13(1988)6, pages 527-529, using LiNbO 3  with an X cut and Z propagation direction, in which the first, second and third reference eigen modes are linearly polarized and lie on the equator of the Poincaré sphere. Such first, second and third retarders R 1 , R 2 , R 3  have the normalized Stokes vectors  
         S     1   ,   2   ,   3       =     [           cos                   ψ     1   ,   2   ,   3                   sin                   ψ     1   ,   2   ,   3                 0         ]                           
 
         [0034]    for their first, second and third reference eigen modes Sr 1 , Sr 2 , Sr 3 .  
         [0035]    Other options are general retarders, which have been described in the IEEE Journal of Quantum Electronics, Vol. 25, No. 8, August 1989, pages 1898-1906, and which can be produced using LiNbO 3  with an X cut and a Y propagation direction. Those general retarders contain three sets of electrodes for phase shifting, mode conversion in phase and mode conversion in quadrature, so that the first, second and third reference eigen modes can be chosen on any desired great circle on the Poincaré sphere. If one electrode set is omitted, then the remaining two electrode sets each define one of three mutually orthogonally aligned great circles on the Poincaré sphere as the locus of the first, second and third reference eigen modes. If, for example, only mode conversion in phase and in quadrature are allowed, then this results in normalized Stokes vectors  
         S     1   ,   2   ,   3       =     [         0             cos                   ψ     1   ,   2   ,   3                   sin                   ψ     1   ,   2   ,   3               ]                           
 
         [0036]    for the first, second and third reference eigen modes Sr 1 , Sr 2 , Sr 3 . Components such as these are also known from IEEE J. Quantum Electronics, 18 (1982) 4, pages 767-771.  
         [0037]    According to the present invention, the second delay (p 2  is chosen to be equal to φ2=5π/6. The first and third angle coordinates ψ1, ψ3 are chosen, according to the present invention, such that they differ by ±π/2 from the second angle coordinate ψ2, that is to say ψ1=ψ3=ψ2+π/2 or ψ1=ψ3=ψ2−π/2. First and third delays φ1, φ3 are, according to the present invention, equal and opposite to one another, that is to say φ1 =−φ3, and are in each case chosen to lie in the interval −π/2 . . . π/2.  
         [0038]    This results in the desired overall delay φ=5π/6 for the first polarization scrambler Rs 1 . Furthermore, the second angle coordinate ψ2 is chosen as the longitudinal coordinate α and the first delay φ1 is chosen as the azimuth coordinate β, so that this results in an overall reference eigen mode S r , which is distributed uniformly on the surface of the Poincaré sphere.  
         [0039]    The correctness of this statement is evident, for example, for first, second and third retarders R 1 , R 2 , R 3  which are in the form of wave plates, from the equation:  
           R   s1 (5π/6,[cos(α)cos(β)sin(α)cos(β)sin(β)] T ) =R 3 (−β,[cos(α+/π)sin(α+π/2)0] T ) R 2 (5π/6,[cos(α)sin(α)0] T ) R 3 (β,[cos(α+π/2)sin(α+π/2)0] T )  
         [0040]    The first polarization scrambler Rs 1  is once again shown in a further exemplary embodiment of the present invention in FIG. 3. As part of this, the second polarization scrambler Rs 2  is in the form of a chain circuit of a fourth and a fifth retarder R 4 , R 5  whose construction and characteristics are chosen to be analogous to the construction and characteristics of the first, second and third retarders R 1 , R 2 , R 3 .  
         [0041]    According to the present invention, the longitudinal coordinate α plus the angle π/12 is chosen as the fourth angle coordinate ψ4; that is to say, ψ4=α+π/12. According to the present invention, the longitudinal coordinate α minus the angle π/12 is chosen as the fifth angle coordinate ψ5; that is to say, ψ5=α−π/12. According to the present invention, the fourth and fifth delays φ4, φ5 are chosen to be equal to the azimuth coordinate β plus π/2; that is to say, φ4=φ5=β+π/2.  
         [0042]    The second polarization scrambler Rs 2  formed in this way does not have a constant overall delay φ=5π/6, nor does it have a normalized Stokes vector S r  of the overall eigen mode Sr distributed uniformly on the surface of the Poincaré sphere, but is nevertheless based on the principle according to the present invention. In order to verify this, the input-side and output-side time-invariant retarders Ri, Ro are chosen as retarders with the input-side and output-side delays φi=5π/12 and φo=5π/12, respectively, and the input-side and output-side reference eigen modes Si, So, respectively, whose normalized Stokes vectors S i , S o  are equal to the cross-product S i =S o =S r4,ψ4=π/2 ×S r4,ψ4=0  of the normalized Stokes vector S r4,ψ4=π/2  which results for ψ4=π/2 for the fourth reference eigen mode Sr 4  and of the normalized Stokes vector S r4,ψ4=0  which results for ψ4=0 for the fourth reference eigen mode Sr 4 ; with the fourth delay φ4 in each case being subject to the condition 0&lt;φ 4 &lt;π. The fourth and fifth retarders R 4 , R 5  are of the same type, so that FIG. 4 in the last sentence can be replaced by FIG. 5.  
         [0043]    If the fourth retarder R 4  is in the form of a wave plate, then by way of example, S r4,ψ4=π/2 =[0 1 0] T , S r4,ψ4=0 =[1 0 0] T , S i =S o =S r4,ψ=0 =[0 0 −1] T , so that the input-side and output-side time-invariant retarders Ri, Ro are circular retarders.  
         [0044]    In consequence, the first polarization scrambler Rs 1  has a constant overall delay φ=5π/6 and a normalized Stokes vector S r , which is uniformly distributed on the surface of the Poincaré sphere, of the overall eigen mode Sr.  
         [0045]    The correctness of this statement is evident, for example, for fourth and fifth retarders, R 4 , R 5 , which are in the form of wave plates, from the equation:  
           R   s1 (5π/6,[cos(α)cos(β)sin(α)cos(β)sin(β)] T )= R   o (5π/12,[0 0 −1] T ) R   5 (β+π/2,[cos(α−π/12)sin(α−π/12)0] T ) R   4 (β+π/2,[cos(α+π/12)sin(α+π/12)0] T )R 1 (5 π/12, [0 0 −1] T )  
         [0046]    Explained using the example of the general retarder Rg, there are a large number of general retarders Rg whose electrical control signals ERs are proportional to the general delay φg and are proportional to linear combinations of the cosine cos(ψg) and of the sine sin(ψg) of the general angle coordinate ψg. This results in the necessity to find expressions for a first control variable CQ 1 =(β+π/2)·cos(α) and a second control variable CQ 2 =(β+π/2)·sin(α) which have as few harmonics as possible. According to the present invention, suitable expressions for the first and second control variables are:  
         [0047]    CQ 1 =(β+π/2)cos(α)=0.288 sin(ωt) −1.196 sin(3ωt)+0.846 sin(4ωt) and  
         [0048]    CQ 2 =(β+π/2)sin(α)=1.023 cos(ωt)−1.446 cos(3ωt)−0.534 cos(4ωt).  
         [0049]    In this case, ω is a further angular velocity, and t is the time. The first and second control variables CQ 1 , CQ 2  give, at least approximately, a constant overall delay φ=5π/6 and a normalized Stokes vector S r , which is at least approximately uniformly distributed on the surface of the Poincaré sphere, of the overall eigen mode Sr for the polarization scrambler Rs 1 , Rs 2 . In the example of the second polarization scrambler Rs 2 , shown in FIG. 3, control variables which are linear combinations of the first and second control variables CQ 1 , CQ 2  are required to produce the fourth and fifth retarders R 4 , R 5  via lithium niobate modules and/or in accordance with Optics Letters 13(1988)6, pages 527-529 and/or the IEEE Journal of Quantum Electronics, Volume 25, No. 8, August 1989, pages 1898-1906, and/or the IEEE Journal of Quantum Electronics, 18(1982)4, pages 767-771. By way of example, the terms (β+π/2)cos(α±π/12) and (β+π/2)sin(α±π/12) can be calculated very easily via elementary trigonometric conversions, to be precise rotation through ±π/12 in the plane which is defined by the first and second control variables CQ 1 , CQ 2  as Cartesian coordinates. The required control voltages are also proportional to linear combinations of these terms (β+π/2)cos(α±π/12) and (β+π/2)sin(α±π/12).  
         [0050]    Instead of those stated, other first and second control variables CQ 1 , CQ 2  also may be used. Preferably, these are likewise terminated Fourier breakdowns relating to the further angular velocity ω, so that first and second control variables CQ 1 , CQ 2  formed in this way include the harmonic angular velocities kω, where k is an integer.  
         [0051]    In addition to the described exemplary embodiments of the present invention, all other exemplary embodiments are suitable which can be formed by orthogonal transformation of the rotation matrix R s1  of the first polarization scrambler Rs 1 . Instead of this, all unitary transformations would be permissible for a description via Jones matrices.  
         [0052]    Polarization scramblers likewise are suitable which result from the described exemplary embodiments of the present invention by reversal of the beam path of the optical signal (OS), at least provided that all the components that are used are reciprocal.  
         [0053]    Indeed, although the present invention has been described with reference to specific embodiments, those of skill in the art will recognize that changes may be made thereto without departing from the spirit and scope of the present invention as set forth in the hereafter appended claims.