Polarization Mode Dispersion (PMD) is a well known problem in optical transmission systems [1
1. N. Gisin, J. P. von der Weid, and J. P. Pellaux, “Polarization mode dispersion in long and short single mode fibers,” J. Lightwave Tech. 9, 821–827 (1991). [CrossRef]
]. It is generated by random fluctuations of the residual birefringence in optical fibers, such that the State of Polarization (SOP) of an optical signal will change randomly over time, in an unpredictable way. If one wishes to keep a correlation between the SOPs at the input and output of a transmission link, some type of active polarization stabilization is needed [2
2. M. Martinelli, P. Martelli, and S. M. Pietralunga, “Polarization stabilization in optical communication systems,” J. Lightwave Technol. 24, 4172–4183 (2006). [CrossRef]
]. A classical example of this situation is found in coherent transmissions, where the polarization state of the received signal must match a fixed SOP of a local oscillator for maximum interference. A more complicated situation is found when different independent polarization states must be mapped into corresponding states at the receiver; this is precisely the case of fiber-optical Quantum Communication systems employing polarization coding, where a quantum bit (qubit) is assigned to the SOP of the transmitted single photon [3
3. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum Cryptography,” Rev. Mod. Phys. 74, 145–195 (2002). [CrossRef]
]. In this case, the sender of the message randomly chooses one out of two independent bases which are mutually non-orthogonal to code its qubits and the receiver must read the data using a similar procedure. Clearly, any random polarization rotations caused to the signal would make transmission of quantum states impossible unless a full polarization control, able to simultaneously control multiple polarization states, can be achieved. In principle, even fully controlling the received SOP, still a timing problem would last, due to the jitter or wander between qubits sent in different polarization states. The origin of this problem is their different arrival times due to the differential group delay (DGD) of the polarization states. However, in most cases, the PMD timing scale is of the order of a few ps, whereas the time resolution of nowadays available photon counting modules is of the order of hundreds of ps. Hence, PMD timing problems can be neglected facing PMD induced polarization fluctuations, the main problem addressed in this paper.
Full control of any polarization state can be achieved if two non-orthogonal polarization states, henceforth called S
1 and S
3, are sent across the fiber as reference states. One can choose, for instance, without loss of generality, the horizontal and +45° linear polarization states, respectively, which are canonically conjugate non-orthogonal states. At the end of the fiber we place a polarization controller which performs a series of two rotations, R
1 and R
3, bringing the output SOP into the desired state. Calling T the Jones matrix representing the transformation over the fiber, our control system requires that the following system of equations holds:
That is, the unitary transformations R
cancel out the effects due to birefringence in the fiber for the input SOPs S
, respectively. Note that R
does not have any effect upon S
, which means that it is a rotation around the axis, in the Poincaré sphere, defined by S
and its orthogonal state. The first two lines of Eq. (1
) can be rewritten in matrix form as follows:
can take any value between 0 and 2π
. Combining this result with the third line of Eq. (1
), we get:
) holds if and only if θ
=0 mod 2π. This means that R
must be equal to the identity, or equivalently, R
, which clearly shows that the effect of the polarization controller is to nullify the polarization rotations generated by T
. Any polarization state passing through the fiber and the two rotators R
will be preserved. Of course, signal and references have all the same optical frequency and the control system must distinguish between them to perform the correct rotations at each controller. This is easily done either by time multiplexing references and signal or by dithering the signal and references at different frequencies and filtering at the receiver.
Although this method can precisely control all polarization states sent throughout the fiber, co-propagation of reference signals along with a quantum channel is very problematic. References extinction ratio must be extremely high to allow photon counting in the quantum channel without too many false counts. An alternative approach is to send the reference signals at different wavelengths, in co-propagating side channels. These channels could also be simultaneously used to synchronize the transmitter and receiver and send disclosed information between them. Here also the filtering needs are high, but the quality of nowadays Wavelength Division Multiplexers (WDM) is good enough to ensure the needed isolation. Synchronization with side channel transmission in quantum key distribution was indeed achieved in recent experiments [5
5. S. Sauge, M. Swillo, S. Albert-Seifried, G. B. Xavier, J. Waldebäck, M. Tengner, D. Ljunggren, and A. Karlsson, “Narrowband polarization-entangled photon pairs distributed over a WDM link for qubit networks,” Opt. Express , 15, 6926–6933 (2007). [CrossRef] [PubMed]
]. Clearly, the fluctuating birefringence, which is the matter of our control system, is frequency dependent so that there are differences between the signal channel SOP, S
, and the two reference channels S
. However, we will show that if the PMD of the fiber is not too high, good quality control can be achieved.
Now we require the polarization S
of a signal at wavelength ω
to be controlled by employing reference signals at S
at different wavelengths, given by ω
. It is also important to stress the dependence of T
on the wavelength, thus we write T
). Hence, we get a new system, similarly to Eq. (1
Now we proceed exactly as done in the monochromatic case. Noticing that T(ω
1)+2Δω(∂T/∂ω) we get:
Where both T
-1 and ∂T/∂ω are calculated at ω
1. The same solution found for the monochromatic case, i.e. θ+ϕ=0 mod 2π, is still valid if the condition below holds:
As the matrix norm in Eq. (6
) is precisely half the DGD value τ
6. B. L. Heffner, “Automated measurement of polarization mode dispersion using Jones Matrix Eigenanalysis”. IEEE Photon. Technol. Lett. 4, 1066–1069 (1992). [CrossRef]
], the condition can be restated as τ
In our experiment we used the optical frequency domain to separate the reference signals from the signal channel. The experimental set-up is presented in Fig. 1
. References and signal were separated by 0.8 nm and launched into the fiber (8.5 km long, 0.54 ps PMD) via a WDM with 3.1 dB insertion loss. A four-plate piezoelectric polarization controller, followed by a second controller, performed the rotations R
, with axes aligned according to the previous description. Control signals were launched from Bob’ side, counter propagating the quantum channel, thus limiting the interference of the side channels to the Rayleigh scattered light. A second filter was used to further reduce the crosstalk to below the dark count level of the single photon counting module in channel 2. At Alice’s side of the link, after passing through all polarization controllers, the control channels were dropped through port 3 of an optical circulator and separated by a second WDM. Channels 1 and 3 were detected (photodiodes D1
) after passing through the corresponding linear polarizers (P1
) with polarization controllers adjusting the axes of R
with respect to the polarizers P1
such that once the polarization is adjusted in channel 1 it is invariant to rotations performed by R
. Efficient control was obtained with received powers as low as -20 dBm in the control channels. At Alice’s side, the quantum channel was launched after a polarization controller and a variable attenuator, so that polarization measurements could be made either with classical light or in single photon regime, by placing a polarimeter at Bob’s end or a polarization controller followed by a linear polarizer and a gated photon counter. The rejection ratio of the polarizers was better than 40 dB. Photon counting measurements were made with 0.2 photons per 2.5 ns gate pulse at 100 kHz rate.
Fig. 1. Experimental set-up: PC: Manual polarization controllers, R: Electrically driven polarization controllers, P: Polarizers, F: Filter, D: Classical photodetectors, C: Single photon counting module, LD: Laser diodes, A: Attenuator, Pol: Polarimeter.
Polarization controllers (40 kHz bandwidth) and fast A/D conversion (500 MHz) allow efficient control under fast variations in the transmission line. Figure 2
presents the detected intensity through a polarizer in channel 2, after a fast polarization rotation was induced in the transmission link. The system recovers very fast to 90% received power (~2.5 ms), and full signal recovery is obtained in times smaller than 10 ms in a worst case scenario. This means that the technique can control polarization states even in fast varying transmission lines.
Fig. 2. Signal recovery in channel 2 controlled by channels 1 and 3.
displays the evolution on the Poincaré sphere of the output polarization of channel 2 with and without polarization control, over 2 hours during which the PMD of the fiber was forced to evolve by changing its temperature. Clearly, a polarization based quantum channel would never perform in such an uncontrolled fiber, and that’s the reason why polarization encoding is so difficult to be implemented in optical fibers. When the polarization control is turned on, the stabilization of two non-orthogonal polarization states is clear.
Fig. 3. Left: uncontrolled time evolution of a single output SOP. Right: evolution of two nonorthogonal output SOPs with full control.
Considering the figures used in the experiment, Eq. (6
) gives τ
~0.3, a value not so small if compared to the unity. However, the performance of the experiment is striking, which means that Eq. (6
) seems too restrictive. In fact, it was deduced considering a worst case scenario, as the solution θ
=0 (mod 2π) is not necessarily the unique solution for Eq. (5
). Hence, it is possible that the control algorithm finds better solutions in the general case, somehow relaxing the condition stated in Eq. (6
Fig. 4. (a). Statistics of the deviation angle between the received and target SOP in the Poincaré sphere. (b). Corresponding added power penalties under full polarization control.
A better evaluation of the efficiency of the control scheme can be seen in Fig. 4(a)
, which presents the statistical distribution of the deviation angle (in the Poincaré sphere) between the actual polarization and the target polarization. We observe that all deviation angles are smaller than 10o with a mean value of only 2o. This means that the worst case added channel power loss due to the polarization control in this experiment is given by the squared cosine of half the deviation angle in the Poincaré sphere which was smaller than cos2
(5°)~0.8%. Figure 4(b)
displays the statistical distribution of the power loss that would be added by the polarization control system due to the mismatch between the controlled and target SOPs during the experiment. Of course, this distribution depends on the temporal evolution of the optical fiber transfer function and can be degraded in case of severe fiber vibrations. Moreover, if the PMD of the fiber is much higher than 0.5 ps the accuracy of the control will decrease, and the SOP of the qubits will swing around its target SOP, thus widening the distribution of deviation angles. However, considering the recovery time of the system, the precision of our polarization control system leaves a margin good enough to override the sensitivity of most quantum communications detection systems and is fully compatible with the maximum accepted optical QBER in quantum communications, which is usually around 1%.
Now we must show that the proposed system is able to operate effectively in the single-photon counting regime. One could state, for instance, that detrimental influence of the strong reference beams, such as cross-talk of the Rayleigh-scattered light from the control channels in the fiber or due to insufficient isolation of the WDMs or filters, could create an overwhelmingly high number of noise counts such that the QBER would exceed the maximum acceptable value. In order to evaluate the isolation between the quantum and control channels, we increased the launched power of the side channels to +5 dBm each, 25 dB higher than the minimum needed for an efficient control. In this condition, we still observed that the crosstalk was smaller than the dark counts of our InGaAs single photon counter, which has a dark count probability of 4×10-5
per nanosecond. Now the advantage of using the control signals in a counter- instead of co-propagating scheme is clear. This last setup would require the side channels to work below the minimum acceptable power, thus reducing the efficiency of the control system. In Fig. 5
we show the single photon counts at Bob for two fixed polarization states (linear 0° and 45°) at Alice, corresponding to the same bit value in the two non-orthogonal bases of a BB84 protocol for instance. In each case, Bob uses a polarization controller to change the polarizer angle along the equator of the Poincaré sphere. It can be seen that, when Bob’s polarizer is orthogonal to the state sent by Alice, the total noise counts are entirely comprised of the single photon counter module’s dark counts. This means that all other sources of noise are negligibly small, which demonstrates the feasibility of the proposed setup.
Fig. 5. Single photon counts at Bob for two fixed polarization states (linear 0° and 45°) sent by Alice controlling the polarization through 8.5 km single mode fiber.