Polarization scrambler and a method for polarization scrambling

A polarization scrambler and associated method for polarization scrambling are provided, wherein light is passed through a retarder whose eigen modes and delay vary with time and which is driven with alternating signals at two harmonic frequencies. One exemplary embodiment with an electrooptical wave plate operates with a horizontal or vertical input polarization and supplies output polarizations which vary with time and have uncorrelated normalized Stokes parameters. The polarization scrambler which is produced can be used in devices for detection of polarization mode dispersion.

BACKGROUND OF THE INVENTION

Polarization mode dispersion, referred to as PMD, adversely affects high-speed optical data transmission. A method for measurement of polarization mode dispersion has been described in the Conference Proceedings of the European Conference on Optical Communication, Amsterdam, NL, Sep. 30–Oct. 4, 2001, Tu.A.3.4, 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 just two-dimensional, 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 given input polarization. At least for certain input polarizations, the output polarizations which are produced define, however, only a two-dimensional, and not a three-dimensional, figure within the Poincaré sphere since depolarization is a necessary but not necessarily sufficient precondition for output polarizations actually being produced which define a three-dimensional figure, and not just a two-dimensional figure, within the Poincaré sphere.

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.

The polarization scrambler should be designed to be as simple as possible and to scramble the polarization within a time which is as small a multiple as possible of the cycle duration of the highest frequency oscillation which is required to drive the polarization scrambler.

An object of the present invention is to specify a polarization scrambler which is as simple as possible but is still ideal, as well as an associated method for polarization scrambling, which emits uncorrelated output polarizations.

SUMMARY OF THE INVENTION

A solution to the problem involves the polarization scrambler being in the form of a retarder with variable eigen modes and a variable delay. Polarization scramblers having eigen modes which can be varied on a great circle of the Poincaré sphere are particularly advantageous.

In one exemplary embodiment of the present invention, this polarization scrambler is formed by an electrooptical wave plate. The control electrodes are preferably provided with sinusoidal signals at a fundamental frequency and at the second harmonic of this fundamental frequency. In this way, the polarization is scrambled within a time which is only twice the cycle duration of the highest frequency oscillation which is required to drive the polarization scrambler. In a further exemplary embodiment of the present invention, the polarization scrambler is provided by a mode converter with mode conversion which can be varied freely in phase and in quadrature. In a further exemplary embodiment of the present invention, an electrooptical wave plate is used with horizontal or vertical input polarization.

DETAILED DESCRIPTION OF THE INVENTION

Without any restriction to generality, the following text regards all retarders as being loss-free, although the principle according to the present invention is also applicable 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 polarization orthogonal to it. 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 normally identified or defined by this statement as the reference eigen mode. 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.

It is even possible to go one step further and to consider only positive values in the interval 0 . . . π as delays. This is because negative values can be replaced by positive values by interchanging the two eigen modes with one another.

Retarders are described in the following text on the basis of 3×3 submatrices of the Müller matrices of retarders, which have rows and columns 1 to 3 which can be associated with the Stokes vector components 1 to 3 using the numbering running from 0 to 3 for rows and columns in the Müller matrices and for 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.

In an arrangement according to the present invention, for polarization scrambling as shown inFIG. 1, a first or second polarization scrambler Rs1, Rs2is supplied with an optical signal OS at a first or second input Rsi1, Rsi2, respectively, with a first or second input polarization Si1, Si2as the polarization P. The polarization P of the optical signal OS is modulated as a function of time t in the polarization scrambler Rs1, Rs2. To do this, it is driven by at least one preferably electrical control signal ERs, ELs, E1s, E2s. The optical signal OS emerges from the polarization scrambler Rs1, Rs2with a first or second output polarization So1, So2, respectively, as the polarization P at a respective first or second output Rso1, Rso2.

Here and in the following text, designations such as Rg, which are equivalent to designations for previously introduced polarization scramblers or retarders such as Rg, but with subscript suffices such asq, may denote the associated rotation matrices. Designations such as Si1, So1, Si2, So2, which are equivalent to designations for previously introduced polarizations such as Si1, So1, Si2, So2, but with subscript suffices such asi1,o1,i2,o2, likewise may denote the associated normalized Stokes vectors.

The normalized Stokes vector So1, So2of the first or second output polarization So1, So2, respectively, of the first or second polarization scrambler Rs1, Rs2, respectively, according to the present invention has a respective correlation matrix

C=⁣〈So1·So1T〉⁢⁢or⁢⁢C=⁣〈So2·So2T〉⁢,
which is produced by averaging the respective product So1·So1Tor So2·So2Tof the corresponding normalized Stokes vector Sol, So2times its transposition So1T, So2Tover time t. The correlation matrix C is at least approximately equal to ⅓ times the 3×3 unit matrix 1, C=(⅓)*1. Its eigen values are accordingly all equal to ⅓. The polarization scrambling thus can be regarded as being of higher quality the larger the smallest eigen value of the correlation matrix C is. Since a loss of, for example, 40% in comparison to the ideal value of ⅓ of this smallest eigen value still may be regarded as being acceptable without any problems, the requirement C=(⅓)*1 may be satisfied only roughly in practice. Corresponding polarization scramblers can be regarded as relatively poor but reliable exemplary embodiments of the present invention.

The polarization scrambler Rs1, Rs2has a reference eigen mode Sr and a delay φ which, according to the present invention, vary with time; that is to say, they are dependent on the time t.

A general retarder Rg has the general rotation matrix

Rg=⁢Rg⁡(φg,Srg)=⁢[Srg12+(Srg22+Srg32)⁢cos⁢⁢φgSrg1⁢Srg2(1-cos⁢⁢φg)-Srg3⁢sin⁢⁢φgSrg1⁢Srg3⁡(1-cos⁢⁢φg)⁢+Srg2⁢sin⁢⁢φgSrg1⁢Srg2⁡(1-cos⁢⁢φg)⁢+Srg3⁢sin⁢⁢φgSrg22+(Srg12+Srg32)⁢cos⁢⁢φgSrg2⁢Srg3⁡(1-cos⁢⁢φg)⁢-Srg1⁢sin⁢⁢φgSrg1⁢Srg3⁡(1-cos⁢⁢φg)⁢-Srg2⁢sin⁢⁢φgSrg2⁢Srg3(1-cos⁢⁢φg)⁢+Srg1⁢sin⁢⁢φgSrg32+(Srg12+Srg22)⁢cos⁢⁢φg],
which here, without any restriction to generality, has a positive determinant +1. Its general reference eigen mode Srg is given by the normalized Stokes vector

Srg=[Srg1Srg2Srg3]T=[Srg1Srg2Srg3]
and its general delay is Φg. For comparison: its Jones matrix is

It is advantageous for the first and second polarization scramblers Rs1, Rs2to be provided by a respective first and second mode converter MC1, MC2with a variable delay φ and with a variable orientation angle ζ. In this case, the orientation ζ is the phase difference between the converted signals and the non-converted signals. The mode converter MC1, MC2can convert a polarization irrespective of its orientation angle ζ to its orthogonal, provided that its delay φ is equal to π. As such, the normalized Stokes vector Srof the reference eigen mode Sr can be chosen on a great circle, which is referred to as the eigen mode great circle GC in the following text, on the Poincaré sphere. This eigen mode great circle GC is assumed to be subdivided by the orientation angle ζ at the angle interval 0 . . . 2π.

The first mode converter MC1, formed in this way, is illustrated inFIG. 2as the first polarization scrambler Rs1. This is in the form of a Soleil-Babinet compensator SBC, to be precise based on an electrooptical Soleil-Babinet compensator using LiNbO3with an X cut and a propagation direction which is at least approximately the Z direction. Compensators such as these are known from IEEE J. Lightwave Techn. 6(1988)7, pages 1199–1207. The Soleil-Babinet compensator has a first waveguide WG1, which is produced by titanium diffusion and passes through the entire LiNbO3crystal from the first input Rsi1to the first output Rso1. The optical signal OS is passed through it. A center electrode EM runs above and along the first waveguide WG1. A buffer layer PUF, composed of SiO2, by way of example, can be fitted between the first waveguide WG1and the center electrode EM. A side electrode ER, EL is provided on each of the two sides of the center electrode. The center electrode EM and the side electrode ER, EL also may be subdivided longitudinally along the first waveguide WG1into, in each case, two or more sections EM1, EM2, ER1, ER2, EL1, EL2, which are driven by somewhat different voltages in order to compensate for any circular birefringence which may be present in the first waveguide WG1. The cross section of the X-Y plane is preferably symmetrical with respect to a Z axis running through the center of the first waveguide WG1. An electrical field produced on its own between the two side electrodes ER, EL, with the potential GND on the center electrode EM preferably being the mean value of the potentials ERs, ELs on the two side electrodes ER, EL, produces a first partial delay Φ1. In this case, the potentials ERs, ELs on the two side electrodes ER, EL are respectively a first and a second control signal ERs, ELs. The reference eigen mode Sr is, in this case, a horizontal (or vertical) polarization, corresponding to an orientation angle ζ=0. An electrical field which is present on its own on the center electrode EM, with the potentials of the two side electrodes ERs, ELs preferably being identical and not being the same as the potential GND on the center electrode EM, produces a second partial delay φ2. The reference eigen mode Sr is, in this case, a linear polarization with an angle of elevation of +45° (or −45°), corresponding to an orientation angle ζ=π/2. Combination of these two cases allows any desired delay Φ and any desired linear polarized reference eigen mode Sr to be produced. In this case, the delay Φ is equal to φ=√{square root over (φ12+φ22)} and the orientation angle ζ is equal to ζ=arg (φ1+jφ2), as twice the physical elevation angle of the reference polarization. As shown inFIG. 3, the orientation angle ζ, plotted along the equator of the Poincaré sphere, for example from the positive S1 axis in the direction of the positive S2 axis, thus can be regarded as a point on the S1-S2 great circle on the Poincaré sphere (the great circle which lies on the S1-S2 plane) as the eigen mode great circle GC or the locus of the normalized Stokes vector Srof the reference eigen mode Sr. The normalized Stokes vector Srof the reference eigen mode Sr is, in this case, Sr=[cos ζ sin ζ0]T. The orientation angle ζ and hence the normalized Stokes vector Srof the reference eigen mode Srvary, however, as a function of time t, and are illustrated only in the form of an example inFIG. 3. The Cartesian coordinate system corresponds to the normalized Stokes vectors S1, S2, S3.

According to the present invention, it is advantageous to choose the first and second partial delays φ1, φ2to be equal to linear combinations of Fourier components of a cyclic oscillation at a fundamental frequency f. The inverse 1/f of the fundamental frequency f is then that time period within which the desired polarization scrambling occurs. In order to make it possible to produce the signals for producing the two partial delays φ1, φ2as easily as possible, it is advantageous to use only two Fourier components. The partial delays are then given by the following matrix equation:

R=[cos⁢⁢ζ0-sin⁢⁢ζ0⁢sin⁢⁢ζ0cos⁢⁢ζ0],
where ζ0is the orientation angle offset ζ0. Furthermore, A1, A2, A3, A4are first to fourth coefficients. The first and third coefficients A1, A3have identical mathematical signs. The second and fourth coefficients A2, A4likewise have identical mathematical signs. M1, M2are a first integer and a second integer, respectively. Neither of these must be zero, and they also must have no common divisor. Other but equivalent expressions can be found for the first and second partial delays φ1, φ2if the time t is shifted and/or a specific value is used for the orientation angle offset ζ0, and simple mathematical/trigonometrical conversions are carried out.

In a first exemplary embodiment, the first input polarization Si1is chosen such that its normalized Stokes vector Si1is at right angles to the eigen mode great circle GC on the Poincaré sphere, which is the locus of the normalized Stokes vector Srof the reference eigen mode Sr. Since this eigen mode great circle GC is, in this case, the S1-S2 great circle, the first input polarization Si1is chosen to be right-hand or left-hand circular. This allows a first output polarization Sol in left-hand or right-hand circular form to be produced; that is to say, the orthogonal to the first input polarization Si1, if the delay φ is equal to π. With a suitable definition, the phase shift of the left-hand or right-hand circular first output polarization So1with respect to the right-hand or left-hand circular first input polarization Si1is the orientation angle ζ.

In order to achieve ideal polarization scrambling with uncorrelated normalized Stokes parameters of the first output polarization So1, the sum magnitude |M1+M2| of the first and second integers M1, M2, that is the magnitude of their sum M1+M2, is chosen to be at least equal to 3 in the first exemplary embodiment. Furthermore, the third and fourth coefficients A3, A4are chosen to be equal respectively to the first and second coefficients A1, A2, that is to say A3=A1, A4=A2. Furthermore, the first coefficient A1is chosen to be at least approximately equal to A1=0.98 radians, and the second coefficient A2is chosen to be at least approximately equal to A2=1.37 radians. Smaller discrepancies between the first and second coefficients A1, A2of these values have a comparatively minor effect on the quality of the polarization scrambling, provided that the sum A1+A2of the first and second coefficients A1, A2remains approximately at the value 2.35 radians. One example of this is the choice of A1=A2=1.12 radians. In this case, the alternating components of the normalized Stokes parameters of the first output polarization So1are likewise uncorrelated, but, on average, the first output polarization Sol has a small but not negligible circular component.

A comparatively low highest available frequency max(|M1f),|M2f|) is obtained when one of the two integers M1, M2is chosen to be equal to 2 and the other is chosen to be equal to 1, or when one of the integers M1, M2is chosen to be equal to −2 and the other is chosen to be equal to −1. The locus curve of the normalized Stokes vector So1of the first output polarization So1on the Poincaré sphere is illustrated inFIG. 3for the situation where M1=2, M2=1, ζ0=0. The first output polarization So1has |M1+M2|= three-part rotational symmetry with respect to the normalized Stokes vector Silof the first input polarization Si1; that is to say, in this case the circular axis corresponding to the Stokes parameter S3. The normalized Stokes vector Si1of the first input polarization Si1was, in this case, chosen to be right-hand circular in accordance with S3=1, S2=S1=0. In many cases, it will be possible to provide horizontal or vertical polarization for the optical signal OS. This can be converted very easily to the desired circular input polarization Si1via an electrooptical quarter-wave plate with eigen modes that are polarized linearly in the 45° and −45° directions. This quarter-wave plate may be integrated on the same LiNbO3substrate as the first polarization scrambler Rs1.

FIG. 4shows the second polarization scrambler Rs2as the basis for further exemplary embodiments of the present invention.

This is in the form of a second mode converter MC2as is known from IEEE J. Quantum Electronics, 18 (1982) 4, pages 767–771. The second mode converter MC2is an electrooptical TE-TM mode converter SBA using LiNbO3with an X cut and a Y propagation direction. This has a further waveguide WG2, which is produced by titanium diffusion and through which the optical signal OS is passed from the second input Rsi2to the second output Rso2. Pieces of a first and second electrode E1, E2, respectively, are applied alternately in the form of a comb over the entire length of the further waveguide WG2, whose three-dimensional comb period along the further waveguide WG2is, in each case, equal to the beat length L between the TE and TM modes. The combs which are associated with a respective electrode E1or E2each may be connected to one another on the LiNbO3crystal, or, as shown, outside the LiNbO3crystal. The respective mating piece for the combs of the first and second electrodes E1, E2is a ground electrode E0in the form of a comb. The combs of the first and second electrodes E1, E2are shifted with respect to one another along the further waveguide WG2modulo a TE-TM beat length L by L/4; that is to say, by ¼ of this beat length. Additional separations of L/4 and 3L/4, alternately, therefore are inserted along the further waveguide WG2between combs of the first electrode E1and of the second electrode E2. A voltage which is applied just between the first electrode E1and the ground electrode E0, as the third control signal E1s, produces the first partial delay φ1. In this case, for example, the reference eigen mode Sr is a linear polarization with an elevation angle of +45° (or −45°), corresponding to an orientation angle of ζ=0. A further voltage, which is applied just between the second electrode E2and the ground electrode E0, as the fourth control signal E2s, produces the second partial delay ζ2. In this case, for example, the reference eigen mode Sr is a right-hand (or left-hand) circular polarization, corresponding to an orientation angle of ζ=π/2. Any desired delay φ and any desired reference eigen mode Sr which can be represented on the S2–S2great circle of the Poincaré sphere can be produced by combining these two cases. In this case, φ=√{square root over (φ12+φ22)} and ζ=arg (φ1+jφ2). The orientation angle ζ may be plotted, as shown inFIG. 5, along the S2–S3great circle of the Poincaré sphere, for example from the positive S2axis in the direction of the positive S3axis, and denotes a point thereof as the locus of the reference eigen mode Sr. The normalized Stokes vector Srof the reference eigen mode Sr is, in this case, Sr=[0 cos ζ sin ζ]T.

As in the first exemplary embodiment, the second input polarization Si2in the second exemplary embodiment is chosen such that its normalized Stokes vector Si2is at right angles to that eigen mode great circle GC on the Poincaré sphere which is the locus of the reference eigen mode Sr. Since this eigen mode great circle GC in the second exemplary embodiment is the S2–S3great circle, the second input polarization Si2is chosen as a horizontal or vertical polarization. A vertical or horizontal second output polarization So2can be produced from this; that is to say, the orthogonal of the second input polarization Si2, provided that the delay φ is equal to π. The phase shift between the vertical or horizontal second output polarization So2and the horizontal or vertical second input polarization Si2is the orientation angle ζ, which can once again be plotted on the eigen mode great circle GC; that is to say, in this case on the S2–S3great circle.

The choice of the first and second partial delay φ1, φ2may be made as described further above for the first exemplary embodiment. In contrast to this, first and second numbers M1=−2, M2=−3 were chosen for the illustration of the second exemplary embodiment inFIG. 5. This results, by way of example, in |M1+M2|=5-part part rotational symmetry about the normalized Stokes vector Si2of the second input polarization Si2; that is to say, in this case about the S1axis of the Poincaré sphere. This is shown by the illustration of the locus of the second output polarization So2on the Poincaré sphere as shown inFIG. 5. A negligible orientation angle offset ζ0=0 was once again chosen in this case. The normalized Stokes vector Si2of the second input polarization Si2was, in this case, chosen to be horizontal, in accordance with S1=1, S2=S3=0. The orientation angle ζ and hence the normalized Stokes vector Srof the reference eigen mode Sr once again vary as a function of time t, and are shown only in the form of an example inFIG. 5.

A third exemplary embodiment of the present invention likewise is illustrated byFIG. 4. This differs from the second by having first and second integers M1, M2which are respectively chosen to be equal to M1=−1 and M2=3 or equal to M1=1 and M2−3, and first to fourth coefficients A1, A2, A3, A4which are chosen to be equal to A1=1.09 radians, A2=1.653 radians, A3=0.865 radians, A4=1.114 radians. This situation where M1=−1, M2=3 results in |M1+M2|=2-part rotational symmetry about the normalized Stokes vector Si2of the second input polarization Si2; that is to say, in this case about the S1axis of the Poincaré sphere. This is shown by the illustration of the locus of the second output polarization So2on the Poincaré sphere as shown inFIG. 6. A negligible orientation angle offset ζ0=0 was once again chosen in this case. The normalized Stokes vector Si2of the second input polarization Si2was, in this case, chosen to be horizontal, in accordance with S1=1, S2=S3=0. The orientation angle ζ and hence the normalized Stokes vector Srof the reference eigen mode Sr once again vary as a function of time t, and are shown only in the form of an example inFIG. 6.

A fourth exemplary embodiment of the present invention is, in turn, illustrated byFIG. 2. In comparison to the first exemplary embodiment, this differs firstly by having first and second integers M1, M2which are respectively chosen to be equal to M1=−1 and M2=3 or equal to M1=1 and M2=−3, and first to fourth coefficients A1, A2, A3, A4which are chosen to be equal to A1=1.804 radians, A2=0.576 radians, A3=2.258 radians, A4=1.524 radians. A further difference from the first exemplary embodiment of the present invention is that the normalized Stokes vector Si1of the first input polarization Si1lies on the eigen mode great circle GC, to be precise at or opposite the point which is identified by the orientation angle offset ζ0, with the orientation angle offset ζ0on the eigen mode great circle GC being plotted from the same point and in the same direction as the orientation angle ζ. In the case of the eigen mode great circle GC on the S1–S2plane here, the normalized Stokes vector Si1of the first input polarization Si1is given by Si1=±[cos ζ0sin ζ00]T. ζ0=0 is once again chosen in the following text. For the situation with horizontal first input polarization Si1corresponding to a normalized Stokes vector Silwhere S1=1, S2=S3=0, the orientation angle ζ and the locus curve of the normalized Stokes vector So1of the first input polarization So1are shown inFIG. 7. This locus curve has |M1+M2|=2-part rotational symmetry about the normalized Stokes vector Si1of the first input polarization Si1; that is to say, in this case about the S1axis of the Poincaré sphere. In this example, the orientation angle ζ runs along the S1–S2great circle, plotted, for example, from the positive S1axis in the direction of the positive S2axis.

The first to fourth coefficients A1, A2, A3, A4of the exemplary embodiments of the present invention described above were optimized on the basis of the respective choice of the first and second integers M1, M2via a simplex method which is called by the function FMINS in the program packet Matlab. The optimization criterion used in this case was that the eigen values of the correlation matrix

C=⁣〈So1·So1T〉⁢⁢or⁢⁢C=⁣〈So2·So2T〉⁢
of the normalized Stokes vector So1, So2of the output polarization So1, So2should as far as possible be of equal magnitude, and/or should as far as possible be equal to ⅓. By virtue of the idea of the present invention of using a delay φ which varies with time and a reference eigen mode Sr which varies with time, this was achieved in all the exemplary embodiments of the present invention described above. Further exemplary embodiments of the present invention easily can be determined in this way. If more than two harmonics of the fundamental frequency f are used in the partial delays φ1, φ2and control signals ELs, ERs, E1s, E2s, this results in additional degrees of freedom.

A fifth exemplary embodiment of the present invention is, in turn, illustrated byFIG. 2. In comparison to the first exemplary embodiment, this has a different first output polarization So1, whose normalized Stokes vector So1is chosen as follows:

In this case, M3, M4and M5are respectively third, fourth and fifth integers. The fifth integer M5is chosen to be equal to 1 or equal to −1. The fourth integer M4is chosen such that it is not equal to zero. The magnitude |M3| of the third integer M3is chosen to be greater than the magnitude |M4| of the fourth integer M4. Possible number pairs are, for example, M3=±2, M4=±1 or M3=±3, M4=±1 or M3=±3, M4=±2. In this case, the mathematical signs of the third to fifth integers M3, M4, M5each may be chosen to be different from one another.

A first output polarization So1such as this can, in principle, be produced from any desired first input polarization Si1via the first polarization scrambler Rs1as shown inFIG. 2. The first and second partial delays φ1, φ2have particularly low magnitudes, however, if the first input polarization Si1is chosen to be right-hand circular Si1=[0 0 1]T. The delay φ is then chosen to be equal to φ=arccos(So13), and the orientation angle ζ=arg(φ1+jφ2) is chosen to be equal to ζ=arg(−So12+jSo11). Finally, the first and second partial delays, respectively, are φ1=φ cos ζ, φ2=φ sin ζ. The required polarization transformations are thus specified completely.

The first input polarization Si1, the first output polarization So1, the eigen mode great circle GC corresponding to the S1–S2great circle and the orientation angle ζ are illustrated on the Poincaré sphere for this situation inFIG. 8, to be precise for M3=2, M4=1, M5=1.

A sixth exemplary embodiment of the present invention is, in turn, illustrated byFIG. 2. In comparison to the fifth exemplary embodiment, this has a different first output polarization So1, whose normalized Stokes vector So1is chosen as follows:

In this case, M6, M7and M8are respectively sixth to eighth integers, which are all chosen such that they are not equal to zero. The eighth integer M8is chosen to be equal to 1 or equal to −1. The magnitude |M7| of the seventh integer M7must not be equal to the magnitude |M6| of the sixth integer M6and, furthermore, must not be chosen such that it is equal to twice 2|M6| the magnitude |M6| of the sixth integer M6. Possible numbers are, for example, M6=±3, M7=±1 or M6=+1, M7=±3. In this case, the mathematical signs of the sixth to eighth integers M6, M7, M8may each be chosen to be different to one another.

A first output polarization So1such as this can, in principle, be produced from any desired first input polarization Si1via the first polarization scrambler Rs1as shown inFIG. 2. The first and second partial delays φ1, φ2have particularly low magnitudes, however, if the first input polarization Si1is chosen to be right-hand circular Si1=[0 0 1]T. The delay φ is then chosen to be equal to φ=arccos(So13), and the orientation angle ζ=arg(φ1+jφ2) is chosen to be equal to ζ=arg(−So12+jSo11). Finally, the first and second partial delays, respectively, are φ1=φ cos ζ, φ2=φ sin ζ. The required polarization transformations are thus specified completely.

The first input polarization Si1, the first output polarization So1, the eigen mode great circle GC corresponding to the S1–S2great circle and the orientation angle ζ are illustrated on the Poincaré sphere for this situation inFIG. 9, to be precise for M6=1, M7=2, M8=1.

All the described exemplary embodiments of the present invention have the common feature that ideal polarization scrambling is possible even when the normalized Stokes parameters S1, S2, S3of the Poincaré sphere are rotated with respect to the chosen normalized Stokes vectors Si1, Si2of the input polarizations Si1, Si2, the normalized Stokes vector Srof the reference eigen mode Sr and the resultant normalized Stokes vectors So1, So2of the output polarization So1, So2in the same way.

All the exemplary embodiments of the present invention which make use of a Soleil-Babinet compensator SBC can be implemented using the TE-TM mode converter SBA after cyclically interchanging the normalized Stokes parameters S1, S2, S3to the sequence S3, S1, S2. The cyclic interchanging of the normalized Stokes parameters results in a change in the required input polarization; for example, horizontal/vertical instead of right-hand/left-hand circular. All the exemplary embodiments of the present invention which use the TE-TM mode converter likewise can be implemented using a Soleil-Babinet compensator SBC after cyclically interchanging the normalized Stokes parameters S1, S2, S3to the sequence S2, S3, S1. The cyclic interchanging of the normalized Stokes parameters results in a change in the required input polarization; for example, right-hand/left-hand circular instead of horizontal/vertical.

Further variations relate to rotations of the plane in which the great circle of the Poincaré sphere lies, on which the reference eigen mode Sr can be varied. As such, it is possible to choose an orientation angle offset ζ0which is not equal to zero.

Further variations relate to the replacement of all the polarizations by mutually orthogonal polarizations.

Further variations relate to the replacement of the time t by a shifted time t−t0, where t0is any variable time offset. The time t thus can be regarded as a time zero point, which can be varied as required, in all the above formulae.

The polarization scrambler Rs1, Rs2likewise may be followed or preceded by further retarders. These also may be, for example, short pieces of the further waveguide WG2, possibly with a length equivalent to only a fraction of one beat wavelength L, such as those which unavoidably occur to a greater or lesser extent when sawing off an LiNbO3substrate from which the second mode converter MC2may be produced.

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.