Method for modulating the polarization of optical pulses for generating quantum cryptographic keys, and related polarization modulator

Abstract

A method for modulating the polarization of optical pulses, in particular for generating quantum cryptographic keys, guarantees an optimum stability of the output polarization states, and comprises the steps of : generating a plurality of optical pulses, routing them in a first fibre; separating the horizontal and vertical polarization modes and routing them in respective ends of a second polarization maintaining fibre, forming a ring, so that they travel such ring clockwise and counterclockwise, respectively; inserting a phase modulator inside such ring controlled by a pilot electrical pulse; recomposing the polarization modes in one single beam of optical pulses, and routing the resulting beam through said first polarization maintaining fibre, obtaining at the output a polarization state of the pulses which depends upon the shape of the pilot electrical pulse, wherein a co-propagating optical signal is exposed to a modulation only depending upon the superimposition to the pilot electrical pulse moving in its own direction, and wherein the counter-propagating optical signal is exposed to the whole pilot electrical pulse moving in opposite direction with an integration effect, so as to determine an asymmetry between the effect obtained on the co-propagating optical pulse with respect to the counter-propagating one to code different symbols, and so that said pilot electrical pulse determines different phase shifts to the two components which do not compensate each other.

Description

[0001] Method for modulating the polarization of optical pulses for generating quantum cryptographic keys, and related polarization modulator

[0002] Description

[0003] The present invention relates to a method for modulating the polarization of optical pulses, which are used in the field of quantum key distribution systems for generating quantum cryptographic keys, usable in quantum communication networks and in particular in systems for protecting the communications which take place through such networks. The present invention further relates to a polarization modulator which implements the above-mentioned polarization system.

[0004] The method can be used also for polarization modulators used in the classical communications or in intensity modulators or generators of quantum state generators with other encodings, for example of time-bin type, which use a Sagnac ring interferometer, also simply called Sagnac ring or loop.

[0005] Under optical pulses, or photonic pulses, photons are meant which are generated to produce a quantum state which constitutes the base unit in a quantum communication system. The quantum state is then equivalent to the bit of digital communication protocols, and is also designated as qubit, or quantum bit.

[0006] In particular, the so-called Quantum Key Distribution (QKD) allows two entities, which are separated by a certain distance, to generate, through an exchange of qubit coded by means of photons, a secure cryptographic key known only to them which will enable these two portions to encrypt and decrypt the messages which they will exchange through a classical channel.The security of these keys is intrinsically guaranteed by the principles of quantum mechanics, since a third party, attempting to obtain information about this cryptographic key in order to use it then to decipher illicitly the messages exchanged between said entities, would be detected by them since any measurement process applied to a quantum system, what were to be used to steal the cryptographic key, would inevitably disturb the quantum system itself.

[0007] These keys could be exploited for example to communicate encrypted messages by using secure cryptographic algorithms, such as the Vernam cipher, wherein the encryption key is as long as the text to be encrypted and therefore it cannot be re-used, but it must be regenerated.

[0008] The quantum key distribution is therefore attracting much interest in academic, military and even commercial fields, for the protection and security of confidential communications, but, considering the advantages that this technology promises in improving the security of communications, one must generally face the problem of implementing the quantum key distribution in a communication system, for example in a fibre-optic network or in free space or satellite channels connecting different entities which intend to exchange information protected by a cryptographic key generated by them.

[0009] The practical implementation of the quantum key distribution requires, among other things, a quick modulation of the degrees of freedom of the photons used to generate the cryptographic key, and one of the degrees of freedom which can be used in this regard is polarization.

[0010] In other words, the polarization of one single photon, which generally can be represented as the superimposition of two distinct orthogonal states corresponding to thehorizontal polarization and vertical polarization, can be used to code a qubit.

[0011] However, to obtain a string of polarization states from a plurality of optical pulses indeed it is required to operate a polarization modulation, and the methods known to modulate the photons' polarization generally result to be not optimized, and in some cases even not adequate to guarantee the security of the implemented protocol.

[0012] A way to modulate polarization provides the use of bi-refractive modulators in in-line configuration, as described for example in M. Jofre et al., "100 MHz Amplitude and Polarization Modulated Optical Source for Free-Space Quantum Key Distribution at 850 nm", J. Light. Technol. 28, 2572 (2010); and F. Grünenfelder et al., " Simple and high-speed polarization-based QKD", Appl. Phys. Lett. 112, 051108 (2018 ).

[0013] These modulators operate by exploiting the electro-optical effect of a bi-ref ractive crystal, typically Lithium Niobate (LiNbO3); in the electro-optical crystals the refractive index depends upon a control voltage, applied to a modulator consisting of said crystal, and controlled by an input pilot electrical pulse and, since these crystals are bi-ref ractive, the ordinary and extraordinary refraction indexes vary differently depending upon this voltage. Then, birefringence, and therefore the phase between polarization, can be controlled through the control voltage. However, this type of modulators has the disadvantage of requiring non-standard components, which can make it very expensive and not very effective, and very high voltages to induce the necessary modulation.

[0014] The birefringence of modulator crystals also induces a dispersion of the polarization modes which reduces the degree of polarization for short pulses, by degrading the modulation performances and by reducing the performancesof the quantum key distribution protocol. Moreover, any drift in temperature or in the electrical signal modifies in an uncontrolled way the light polarization, and consequently the efficacy in implementing a quantum key distribution.

[0015] The modulation of phase φ introduced by the modulator, in fact, depends both upon the voltage of the pilot electrical pulse provided to the modulator and upon the temperature of the modulator itself and, to obviate this problem, it is necessary to stabilize then both the temperature and the control voltage of the modulator.

[0016] In order to obviate the above-described problem, it is possible to use a type of modulators wherein a bi-refractive modulator is inserted in a double-passage scheme, by exploiting a Faraday mirror, as proposed for example in I. Lucio-Martinez, et al., " Proof -of -concept of real-world quantum key distribution with quantum frames", New J. Phys. 11, 095001 (2009).

[0017] This scheme suffers from the inconvenience of requiring very high voltages to induce the required modulation, and the dispersion of the polarization modes is not fully compensated, allowing only modest modulation performances.

[0018] An overcoming of the problems described with reference to the state of art highlighted so far is represented by a modulator described in Agnesi, C., Avesani, M., Stance, A., Villoresi, P. & Vallone, G. " All-fiber selfcompensating polarization encoder for quantum key distribution", Opt. Lett. 44, 2398 (2019).

[0019] In this modulator, a polarized light pulse, generally produced by means of a laser, is inj ected in an input port of an optical circulator, to exit then from an output port through a single-mode fibre.

[0020] Subsequently, the pulse meets a polarization controllerwhich converts the light polarization in a balanced superimposition of the horizontal and vertical polarization modes. The light, then, meets a polarization separator (PBS) which separates the two orthogonal polarization modes and inj ects them separately in two different polarization maintaining fibres and, in both cases, the inj ected light is aligned with the so-called slow axis of the polarization maintaining fibre, according to the standard behaviour of the fibre polarization separators.

[0021] Under polarization maintaining optical fibre, or simply PM fibre, a particular single-mode optical fibre is meant which comprises, for its whole length, a birefringence obtained intentionally in the optical fibre, with the purpose of maintaining substantially unaltered the two horizontal and vertical polarization modes of a light polarized during its propagation along the fibre itself. This further determines two different propagation speeds of the two polarization modes, called slow axis and fast axis ( figures 1C and 16A).

[0022] Downstream of the polarization separator a Sagnac effect is determined since the two polarization maintaining fibres re-unite by forming a ring, and the two so-separated horizontal and vertical polarization modes travel in opposed directions, that is clockwise for one mode and counterclockwise for the other mode, causing in this ring, wholly consisting of polarization maintaining elements, a typical interference between the two optical signals, called Sagnac interference. Inside the ring there is a phase modulator which occupies an asymmetrical position in the loop. The phase modulator is controlled by an electrical pulse which activates when one polarization mode or the other one passes, by applying phase φeor φ1, respectively.

[0023] The state of polarization at the output of the device isgiven by the formula:

[0024]

[0025] and then the possible states of qubit depend only upon the difference between the first and the second phase φe, φ1induced by the phase modulator, and this feature makes the drawing of this modulator with Sagnac interferometer immune to any drift due to fluctuations in temperature or electrical voltages.

[0026] This modulator scheme allowed to implement a quantum key distribution scheme according to the BB84 protocol, which provides the use of states not orthogonal therebetween, by using only components of standard type.

[0027] However, this type of modulator has two limitations of some importance.

[0028] In fact, the presence of single-mode fibre and not polarization maintaining fibre before the polarization separator, and then the possible qubit states thereupon they depend, requires that the modulator is calibrated manually through the above-mentioned polarization controller (PC), in order to obtain a balanced superimposition of the horizontal and vertical polarization modes at the input of the polarization separator, and this would forcedly involve the continuous presence of an operator to allow the polarization modulator to operate correctly.

[0029] Moreover, the single-mode fibre guiding the light from the polarization separator induces a unitary transformation of the state of polarization which cannot be known a priori, and this complicates the implementation of the quantum key distribution protocols, since an additional phase for calibrating the so-called polarization bases is introduced.Lastly, it is necessary to consider the fact that in this type of modulator variations in the fibre temperature can change the corresponding unitary transformation.

[0030] The herein described modulator scheme was subj ected to further implementations with some differences, for example by replacing the polarization separator with a displacement crystal (Polarization Beam Displacer, PBD) and a Faraday rotator, as described in Jiang, Y. -F. et al. " Remote Blind State Preparation with Weak Coherent Pulses in the Field", Phys. Rev. Lett. 123, 100503 (2019).

[0031] On the contrary, in Ya-Ping, L. et al. " Experimental realization of a re ference- frame-independent decoy BB84 quantum key distribution based on Sagnac-interferometer", Opt. Lett. 4, 4523 (2019), the polarization separator is maintained, but it does not couple both horizontal and vertical polarization modes along the "slow" axis of the polarization maintaining fibre, but couples one thereof along the fast axis and the other one along the slow axis. This difference requires the introduction of a polarization rotator, like for example a Faraday rotator, to compensate the modal dispersion effects.

[0032] It is to be noted that, in these modulators using a polarization maintaining fibre Sagnac interferometer, the phase modulator is always positioned asymmetrically, so that the pulse travelling clockwise arrives on the modulator at a different time compared to the pulse circulating counterclockwise.

[0033] In the article Stein A., Lopez Grande I. H., L. Castelvero L., Pruneri V. Robust polarization state generation for long-range quantum key distribution (Optics Express Vol.

[0034] 31, Issue 9, pp. 13700-13707 (2023) a polarization modulator was shown which uses a Faraday Mirror (FM) to reverse the polarizations in an unbalanced double-passage interferometer.According to a scheme however similar to those described previously, see S. Wang, et al., Practical gigahertz quantum key distribution robust against channel disturbance, Opt. Lett. 43, 5110 (2018 ), describes a modulator wherein one tries to obtain a stable phase modulator, operating with any input polarization, by using an electro-optical modulator positioned in the centre of the Sagnac interferometer, and then the pulses travelling clockwise and counterclockwise arrive simultaneously on the modulator.

[0035] In literature other schemes have been proposed based on an asymmetrical Sagnac loop to modulate the state of optical pulses.

[0036] In particular, Roberts G. L., Pittaluga M., Minder M., Lucamarini M., Dynes J. F., Yuan Z. L., Shields A. J. Patterning-effect mitigating intensity modulator for secure decoy-state quantum key distribution (Optics Letters Vol. 43, Issue 20, pp. 5110-5113 (2018 ) describes an asymmetrical Sagnac loop which was used as intensity modulator. The authors underline that in order to avoid a limitation on the repetition rate of the optical pulses it is necessary to insert the phase modulator asymmetrically with respect to the Sagnac loop.

[0037] More recently, Scalcon D., Bazzani E., Vallone G., Villoresi P., Avesani M. Low-error encoder for time-bin and decoy states for quantum key distribution (Quantum Physics arXiv: 2311. 02059) proposed a scheme with a Mach-Zehnder interferometer inserted inside a Sagnac loop to generate time-bin states with intensity modulation.

[0038] US patent application No. 2019 / 0260478 Al describes a transmitter scheme for a quantum communication system with a Sagnac interferometer which is used to implement a stable intensity modulator.Chinese patent application No. 109 257 104 A, Chinese patent No. 105 680 955 B and US patent No. 5.311.592 A describe additional known embodiments of methods for modulating the polarization for the optical quantum communications or optical cryptography.

[0039] The article J. Li et al., High-speed robust polarization modulation for quantum key distribution, Optics Lett. Doc ID 374485, to face the above-mentioned problem describes a scheme which uses a polarization maintaining fibre oriented at 45°, a fibre beam separator followed by two polarizers and a Sagnac interferometer. The output is represented by a single-mode fibre.

[0040] Finally, International patent application No. WO 2021 / 078723 Al describes an improved version compared to the scheme proposed by Agnesi et al. mentioned previously, wherein the polarization modulation device consists of a polarization maintaining fibre followed by a fibre polarizing beam splitter (PBS). A first fibre is aligned so that PBS separates the two polarizations corresponding to the fast and slow axes in two subsequent fibres PM2 and PM3: a polarized light pulse (typically generated by a laser) is prepared in a balanced superimposition state of the horizontal ( | H)) and vertical ( I V) ) polarization modes, and is inj ected in the first fibre. Then, the light meets PBS, which represents the beginning of a Sagnac interferometer (Sagnac loop) and the two polarization orthogonal components travel in opposed directions, that is clockwise and counterclockwise.

[0041] The main limitation of this scheme and all schemes mentioned previously is that they use a Sagnac loop with a phase modulator inserted asymmetrically inside the loop thanks to the presence of a delay AL between the pulses circulating clockwise and counterclockwise of the loop. Such delay is required to apply different phases <peand <pi to the optical pulses circulating clockwise andcounterclockwise of the Sagnac loop.

[0042] Such a delay imposes a maximum limit on the frequency at which the optical pulses can be sent: in order to avoid that the counterclockwise component of the N pulse is superimposed to the clockwise component of the N+l pulse, it is necessary that the sending frequency is lower than a value which depends upon the propagation time in the modulator and upon the asymmetry of the Sagnac loop (see the Standard asymmetrical modulation).

[0043] Moreover, the optical pulse uses a certain propagation time in travelling the whole phase modulator (of the order of 500ps for a standard wave guide of different centimetres in length) and, at high repetition rates, this can cause a superimposition between the component travelling clockwise and those travelling counterclockwise.

[0044] Typically, an integrated electro-optical phase modulator is designed in propagating wave technology to increase the modulation speed.

[0045] The propagating wave structure relates to the fact that the electrode is implemented as a transmission line coupled to the input modulating pilot electrical pulse. The modulating pulse travels in the guide with the optical wave in a collinear way. Then, a pilot pulse generates a (microwave) electromagnetic wave propagating along the electrodes in the same direction in which the optical beam propagates.

[0046] Ideally, the phase speeds of both waves are paired, therefore an efficient modulation even for very high frequencies is possible. To summarize, the propagating wave design makes the modulation (microwave) electrical signal to travel in the same direction of the optical pulse inside the LiNbO3material, allowing it to reach a high repetition frequency by accumulating multiple pulsesinside the inner track. This design creates an asymmetry in the modulation between the optical pulses which are copropagating and counter-propagating with respect to the pilot electrical pulse inside the modulator. In fact, it results evident that this modulation method cannot be valid even for a counter-propagating optical pulse since its direction is opposite to that of the electrical pulse inside the modulator.

[0047] The phase modulator works by transforming the electrical pulse into a phase imposed to the optical wave. Assuming, as a first approximation, that the speed of the electrical wave and of the optical wave are equal, the modulator transfer function assumes a simple form, but distinct between the co-propagating case (cop) and counterpropagating case (cont). It can be demonstrated that, given the electrical pulse V*exp (i a t) of angular pulsation a, the transfer function of the phase modulator with respect to a co-propagating or counter-propagating monochrome optical wave is equivalent to

[0048]

[0049] where, herein and hereinafter in the description, imod is the propagation time of the pilot electrical pulse, or of the optical pulse, in the electro-optical phase modulator, and Vnis the voltage of the pilot electrical pulse corresponding to an offset of the optical pulse equal to n.

[0050] It is important to underline that the mentioned transfer functions are a simplification of a generally much more complex model, which expresses the phase imposed by the phase modulator to the optical pulse depending upon the attenuation coefficient a, the length of the modulator L and the effective refractive indices of the pilot electrical pulse and of the resulting optical pulse and ofthe bandwidth saturation.

[0051] The above-mentioned propagation time imo in the electro-optical phase modulator of the pilot electrical pulse, by using as reference the polarization modulators available on the market, typically is in the order of hundreds of ps, for example ~300 ps, whereas Vnis typically around 5 V, even if such values vary from an electro-optical modulator model to the other one.

[0052] It can be noted that, whereas the phase imposed to the copropagating wave is constant, the one imposed to the counter-propagating wave varies as a function of the angular pulsation of the pilot electrical pulse. This imposes a sort of low-pass filter to the phase imposed to the counter-propagating wave.

[0053] The asymmetry between the phase imposed to the propagating pulse and the one imposed to the counter-propagating pulse reflects in the behaviour of the polarization modulator based on Sagnac loop. The phase modulator model described previously can be applied to describe the transfer function of a scheme using the phase modulator inside a Sagnac loop, by combining the two just illustrated transfer functions. For example, considering a device as described in WO 2021 / 078723 Al, if a monochrome optical signal with polarization | H)+ | V) is inj ected, and if a sinusoidal electrical signal with angular pulsation w is used, the outgoing beam has polarization

[0054]

[0055] where < >(&>) is obtained by multiplying the electrical signal by the transfer function GjpOgnac(a>) given by

[0056]

[0057] and id = nfiberAL / c is the time corresponding to the delay line AL which is present in that type of device (nfiber and c are respectively the fibre refractive index and the light speed). It is to be noted that, depending upon the factthat the delay line is placed before or after the phase modulator, the sign of id can be reversed. The function Sine is defined as Sinc

[0058]

[0059] Through the transfer function GjpOgnac(a>) it is possible to calculate the phase as a function of time cp(t) which will be applied to an optical signal when an electrical signal is sent to the phase modulator, which signal has a time trend given by a general function s(t), by obtaining at the output of the device the polarization

[0060] (|tf) + ei cP(t)|7» / 2 (4 )

[0061] In fact, by defining the Fourier transform of the signal s(t) as

[0062]

[0063] the phase cp(t) is obtained from the relationship

[0064]

[0065] Alternatively, by calculating the inverse Fourier transform of the transfer function

[0066]

[0067] the phase cp(t) is obtained then from the convolution

[0068]

[0069] The two preceding expressions are equivalent and can be used to calculate <p(t) starting from the shape of the electrical signal s(t).

[0070] The preceding expressions allow for example to determine the behaviour of the polarization modulator based on Sagnac loop when a pilot electrical pulse in the shape of a squarewave signal of time-length isis sent to the phase modulator.

[0071] It is to be noted that the phase of the co-propagating optical pulse follows exactly the shape of the square wave input signal, whereas for the counter-propagating optical pulse the phase results to be modified by the transfer function.

[0072] Let' s consider an electrical signal formed by a rectangular wave with time-length isand width Vnas illustrated in Figure 2A. In this case, if the time-length of the electrical signal isis greater than double the propagation time in the polarization modulator 2imod, the phase applied to the counter-propagating optical signal follows a trapezoidal shape, with the maximum equal to n. On the contrary, if the time-length of the pilot electrical pulse TSis less than double the propagation time in the polarization modulator 2imod, it is not possible to reach the phase "n", but the phase grows linearly up to the maximum value equal tonis / (2imod) obtained at time t = is. Starting from this instant, the phase remains constant up to t = 2imod to decrease then up to 0 at time t = is+ 2imod.

[0073] Generally, in case V = Vn, the maximum phase which is applied to the counter-propagating optical signal can be written as

[0074] n* min [ l, is / (2imod) ] ( 9)

[0075] The standard modulation scheme with asymmetrical Sagnac present in literature is obtained by sending to the phase modulator an electrical pulse with rectangular shape and time-length is, having a sufficiently long time-length to modulate with the same intensity the co-propagating and counter-propagating optical pulses in the asymmetrical device (that is with is> 2imod) and with a sufficiently long delay line so that the condition Td > TS+ imod isimplemented (example in Figure 2A / 2B) to avoid interference between the different symbols.

[0076] The three states:

[0077]

[0078] are respectively obtained by sending the electrical pulse correspondingly to a phase " TT / 2" in synchrony with the copropagating optical pulse ( for the left circular state |L)), by anticipating the electrical symbol of

[0079]

[0080] ( for the right circular state | / ?)) and not sending any electrical pulse ( for the diagonal state \D)), as shown in figure 3A and 3B. In such configuration it is considered that the optical pulse is very narrow (i. e. <100ps) and then with negligible time-length compared to the time-length of the electrical signal. It is important to underline that, in this configuration, with the same electrical voltage value one succeeds in impressing the phase "n / 2" to the propagating pulse and the phase "-n / 2" to the counterpropagating one simply by changing the delay of the electrical signal.

[0081] This scheme type has an intrinsic limitation in the maximum repetition rate, i. e. the number of optical pulses which is possible to modulate in a given time range, it is capable to sustain. To avoid intra- and inter-symbol interference, it is necessary that the separation T between two subsequent optical pulses satisfies the condition T > 2(TS+ Tmoci). The condition obtaining the maximum repetition rate is obtained by putting is= 2imod, id = is + imod = 3imod, therefrom the propagation time of the signal in the modulator is given by T = 6imod. For amodulator with Tmod = 0.4 ns, this constraint imposes a separation between the symbols equal to T = 2.4 ns and a maximum repetition rate = 410 MHz.

[0082] The repetition rate can be increased by reducing the delay id of the interferometer obtaining

[0083] T d < T s + T mod ( 10 )

[0084] but with the risk of causing intra-symbol interference, but by obtaining as minimum propagation period of the signal in the modulator

[0085] Tmin—2 T s Mmod ( 11 )

[0086] The delaytime id can be reduced up to

[0087] Td—5topt + Tmod ( 12 )

[0088] where δtoptis the time-length of the optical signal. However, in this case, since this system cannot be considered ideal and the imperfections are however very difficult to manage, and it results extremely complicated to find the correct delays to avoid the intra-symbol interference.

[0089] Another way to increase the repetition rate is to modulate only the co-propagating optical pulse as described in Yang Li, Yu-Huai Li, Hong-Bo Xie, Zheng-Ping Li, Xiao Jiang, Wen-Qi Cai, Ji-Gang Ren, Juan Yin, Sheng-Kai Liao, and Cheng-Zhi Peng High-speed robust polarization modulation for quantum key distribution (Optics Letters Vol. 44, Issue 21, pp. 5262-5265 (2019). Such technique requires the use of electrical signals with different voltage values and a maximum voltage which must necessarily reach Vn.

[0090] Alternatively, it is possible to modulate the copropagating pulse only, but using a two-stage scheme (by increasing the system complexity) as described in W. Luo et al., Intrinsically Stable 2-GHz Polarization Modulationfor Satellite-Based Quantum Key Distribution ( IEEE Photonics Journal, vol. 14, no. 5, pp. 1-6, Oct. 2022, Art no. 7654706).

[0091] The technical problem underlying the present invention is to provide a method for modulating the polarization of optical pulses allowing to obviate the drawbacks mentioned with reference to the known art.

[0092] By using electrical pulses with specific shape and generally different from rectangular pulses, it is possible to use symmetrical Sagnac loops, by increasing the repetition rate of the optical pulses. Such electrical pulses with specific shape may also be used inside schemes which exploit asymmetrical Sagnac loops.

[0093] Such problem is solved by a method for modulating the polarization as specified above, which comprises the steps of:

[0094] • generating a plurality of optical pulses in a polarizer module by means of each optical pulse comprising a generic state of polarization obtained by the superimposition of the horizontal and vertical polarization, and routing them in a first polarization maintaining fibre;

[0095] • separating, in said first polarization maintaining fibre, said horizontal and vertical polarization modes and routing them in respective ends of a second polarization maintaining fibre forming a ring, so that they travel such ring clockwise and counterclockwise, respectively;

[0096] • inserting a phase modulator controlled by pilot electrical pulses inside the ring formed by the polarization maintaining fibre;

[0097] • sending to the phase modulator pilot electrical pulses travelling inside the phase modulator and resulting to be co-propagating or counter-propagating with respectto the optical modes travelling such ring clockwise and counterclockwise (or vice versa) an which impose different phases to such modes; and

[0098] • recomposing said optical modes in one single beam of optical pulses, and routing the resulting beam through said first polarization maintaining fibre, by obtaining at the output a polarization state of the pulses which depends upon the difference between said phase modulations,

[0099] wherein a co-propagating optical signal is exposed to a modulation only depending upon the superimposition to the pilot electrical pulse moving in its own direction, and wherein the counter-propagating optical signal is exposed to the whole pilot electrical pulse moving in opposite direction with an integration effect, so as to determine an asymmetry between the effect obtained on the copropagating optical pulse with respect to the counterpropagating one to code different symbols, and so that said pilot electrical pulse determines different phase shifts to the two components which do not compensate each other.

[0100] Preferably, in a preferred embodiment example of the method according to the invention, the time-length of the pilot electrical pulses is lower than the time-length of propagation time of the signal in the polarization modulator.

[0101] The invention further relates to a polarization modulator which is operated according to the above-mentioned method.

[0102] The above-defined method and the modulator can be used within the quantum key distribution to generate cryptographic keys of quantum type, and usable according to one of the quantum communication protocols known to secure the reserved data transmission.In this way, it is possible to exploit efficiently the polarization for free space applications, where polarization is the highly preferable degree of freedom.

[0103] Thanks to the stability of the output polarization states, a calibration between transmitter and receiver is not required to obtain a low error rate. The polarization modulator could be used even in a receiving apparatus within a quantum key distribution, to be able to change quickly the measurement base.

[0104] Lastly, it is to be noted that the control voltages of the pilot electrical pulses which are required in the method and modulator according to the present invention can be significantly lower with respect to the technologies of the state of art without Sagnac effect.

[0105] It is further meant that the present invention could be used generally in all applications which require a fast modulation of polarization, intensity or time-bin coding and which currently use asymmetrical Sagnac loop schemes, thus also in the communications of conventional type.

[0106] The present invention allows to transform the asymmetrical Sagnac loop schemes into symmetrical schemes, by increasing the performances of the devices. It is to be noted that the present method for modulating the electrical signals can be further used in schemes with strongly or lightly asymmetrical Sagnac loops.

[0107] The present invention will be herein described according to a preferred embodiment example, provided by way of example and not for limitative purpose with reference to the enclosed drawings wherein:

[0108] * figures 1A and IB respectively show a functional scheme illustrating some portions of a polarization modulator which implements the method for modulating the polarization according to the invention by using asymmetrical Sagnac loop;

[0109] * figure 1C shows a cross-sectional view of a position maintaining fibre;

[0110] * figures 2A and 2B, respectively, show the voltage of the pilot electrical pulse (left) and the output phase (right) imposed to an optical signal by a polarization modulator based on an asymmetrical Sagnac loop as a function of time;

[0111] * figures 3A and 3B, respectively, show the shape of the pilot electrical pulses (left) and the corresponding output phase (right) imposed to an optical signal by a polarization modulator based on asymmetrical Sagnac loop as a function of time; by exchanging the instant in which the electrical signal is sent, different polarizations of the optical pulse are obtained;

[0112] * figures 4A and 4B respectively show the voltage of the pilot electrical pulse (left) and the output phase (right) of a polarization modulator based on symmetrical Sagnac loop as a function of time; it is noted that the voltage of the electrical signal must be higher than Vn / 2 to obtain a phase imposed to the optical signal equal to n / 2;

[0113] * figures 5A and 5B, respectively, show the voltages of the pilot electrical pulse (left) and the output phase (right) of a modulator based on symmetrical Sagnac loop as a function of time depending upon the state which one wants to code, |L), |7?> or |D>; the electrical pulses coding |L) and | / ?) are identical in shape, but they have opposite sign and are defined "differential" electrical signals;

[0114] * figure 6 shows the output phase of the polarization modulator based on Sagnac loop when the propagation period of the optical pulses in the polarization modulator is equal to T = is+ imod

[0115] * figure 7 shows the output phase of the polarizationmodulator when the period of the optical pulses is equal to T = (τs+δtopt) / 2+τmodwith an optical pulse having timelength δtopt= 100ps.

[0116] * figure 8 shows the output phase of the polarization modulator when the period T of the optical pulses is slightly higher than τs;

[0117] * figures 9A and 9B respectively show the voltage of the pilot electrical pulse (left) (defined electrical signal of "balanced" type) and the corresponding output phase (right) of the symmetrical polarization modulator as a function of time;

[0118] * figures 10A and 10B respectively show the voltages of the pilot electrical pulses (left) and the corresponding output phase (right) of a symmetrical modulator as a function of time depending upon the state which one wants to code, I, or \D) with electrical signals of "balanced" type;

[0119] * figure 11 shows the output phase of the polarization modulator when the period of the optical pulses is equal to T = τs+ τmod+ δtoptwith an optical pulse with time-length δtopt= 100ps and an electrical signal of "balanced" type;

[0120] * figures 12A and 12B respectively show the voltage of the pilot electrical pulse (left) of "integral" type and the output phase (right) of a symmetrical modulator as a function of time, in a configuration in which a phase "n / 2" is applied to the optical pulse;

[0121] * figures 13A and 13B respectively show the voltage of the pilot electrical pulse (left) of "integral" type and the output phase (right) of a symmetrical modulator as a function of time, in a configuration in which a phase "-n / 2" is applied to the optical pulse;

[0122] * figures 14A and 14B show the voltage of the pilotelectrical pulse (on the left) and the output phase (on the right) of a symmetrical modulator as a function of time depending upon the state which one wants to code, | L), I R) or | D), in a configuration with electrical signal of "integral" type;

[0123] * figures 15A and 15B respectively show the voltage of the pilot electrical pulse (left) and the output phase (right) of an asymmetrical modulator as a function of time in the configuration of electrical signal of "integral" type; and

[0124] * figures 16 to 19 show respective functional schemes illustrating different examples of a polarization modulator which implements the method for modulating the polarization according to the invention by using an asymmetrical Sagnac loop.

[0125] With reference to figures 1A and IB, respective functional schemes of a polarization modulator 1 to implement the method for modulating the polarization according to the invention comprise, respectively, a first single-mode fibre SM ( figure 1A) or a first polarization maintaining fibre PM1 ( figure IB), designated with 110.

[0126] The first fibre 110 receives optical pulses, each one comprising a generic state of polarization obtained by the superimposition of the horizontal and vertical polarization.

[0127] The first polarization maintaining fibre 110 is connected to a fibre polarization separator PBS which is capable of separating said optical pulses, constituting an optical flow which travels the first polarization maintaining fibre PM1, into two distinct flows of optical pulses having, each one, different and respective polarizations, that is Horizontal H and Vertical V polarization.

[0128] Therefore, the first polarization fibre 110 is aligned so that the polarization separator PBS separates the twopolarizations corresponding to the fast and slow axes ( figure 1C), by injecting them respectively in the two ends of a second fibre 102 which, in both examples, is of polarization maintaining type 102 whose separated ends are designated with PM2 and PM3.

[0129] In the present embodiment examples, both polarization modes, in the two outputs of the polarization separator PBS, are aligned with the respective slow axes ( figure 1C) of the fibre, according to the standard behaviour of the fibre polarization separators PBS.

[0130] The ends PM2 and PM3 connect by forming a ring 100. In this example, the ring 100 is symmetrical and therefore a delay segment of polarization maintaining fibre with length ΔL is not present, which conventionally is reduced to zero or is approximated to zero, as it is illustrated by way of not limiting example.

[0131] Fibre connectors 107 are provided which connect the different segments of optical fibre to obtain the ring 100.

[0132] They can be replaced by fibres having different lengths or fusion fibre junctions.

[0133] The second polarization maintaining fibre 102, from its ends PM2 and PM3, implements an example of Sagnac ring interferometer.

[0134] By considering the input polarization, if this is a general polarization

[0135] a\H) + b\V)

[0136] the polarization reaching the polarization separator PBS will be

[0137] a\H) + beiS\V)so, the breakdown of intensity between horizontal H and vertical V polarization does not change. The phase δ is naturally dependent on temperature and fibre mechanical stress.

[0138] However, it is to be noted that if the coherence time of the input pilot pulse is very short, the delay between the horizontal polarization | H) and the vertical polarization I V) could be greater than the coherence time, resulting in a depolarized state at the input of the polarization separator PBS. In each case, the breakdown between the horizontal H and vertical V polarization does not change.

[0139] By using an electrical signal which controls the phase modulator 101 inside the Sagnac loop it is possible to modify the polarization of the optical pulses.

[0140] After passing through the Sagnac interferometer, the horizontal H and vertical V polarizations reverse at the output of the polarization separator PBS towards the first polarization maintaining fibre 110.

[0141] The segment of first fibre 110 is then entirely crossed again by the return flow from the polarization separator PBS.

[0142] Then, the light will be emitted by the device from the same input end of the first fibre PM1 and with polarization

[0143]

[0144] which does not exhibit a dependence on the phase 6 and where cp(t) is the phase due to the phase modulator when it is controlled by an electrical signal s(t). The relationship between cp(t) and s(t) was described previously:

[0145]

[0146] In the previous expression τmodrepresents the time used by the light to propagate and cross entirely the phase modulator, whereas τdis the delay between the input in the phase modulator of the copropagating optical pulse with respect to the counterpropagating one. In case the Sagnac loop is symmetrical, there is τd= 0.

[0147] The herein described scheme can be implemented by introducing some variants.

[0148] For example, the light in the two outputs of the polarization separator PBS could be aligned with the fast axis of the fibre ends PM2 and PM3, rather than with their slow axis.

[0149] Or, the used polarization separator PBS does not couple both polarization modes along the fibre slow axis, but it could couple one thereof along the fast axis and the other one along the slow axis. This difference requires the possible introduction of a device to rotate polarization, such as for example a Faraday rotator, which rotates polarization and then it is capable of compensating the modal dispersion effects in case said polarization separator PBS aligns the separated polarization modes to the fast and slow axes, or vice versa, of the respective ends PM2, PM3 of the second polarization maintaining fibre 102.

[0150] In this substantially symmetrical configuration of the polarization modulator, it is possible to exploit the asymmetry between the effect obtained on the copropagating optical pulse with respect to the counterpropagating one to code different symbols: the same pilot electrical pulse creates different phase shifts to the two components which do not compensate each other.

[0151] The co-propagating optical signal will be exposed to a modulation which only depends upon the superimposition tothe pilot electrical pulse moving in its own direction. On the contrary, the counter-propagating optical signal will be exposed to the whole pilot electrical pulse moving in opposite direction with an integration effect.

[0152] This principle can be implemented through some variants of the shape of the electrical pulse which allows to modulate polarization, which will be described with reference to figures 2A to 13B.

[0153] In the following diagrams, the state |L> is designated with 10, the state |R> is designated with 20, the state |R> is designated with 30 and the optical pulse is designated with 40.

[0154] According to a first variant (electrical signals of "differential" type), a pilot electrical pulse has a shorter time-length than the propagation time in the polarization modulator, that is τs< τmod. It would be also possible to use τmod< τs< 2τmod, but this would involve an additional disadvantage with respect to the voltage of the pilot electrical pulse to be used. Figures 4A and 4B illustrate the effect of a rectangular pilot electrical pulse with shorter time-length than the propagation time τmodof the electrical (and optical) signal in the phase modulator.

[0155] By using two pilot electrical pulses with the same absolute voltage, in particular a positive pulse and a negative pulse, it is then possible to modulate the different states, as represented in figures 5A and 5B.

[0156] The abnormal shape of the output phase is explained by the fact that, in the area between 0.0 ns and 0.3 ns, the overall phase is the difference between the phase applied to the co-propagating pulse and the phase applied to the counter-propagating pulse which, due to the interferometer symmetry, travel together in the modulator. This effect isthe main innovation of the present method, which exploits the asymmetry between the phase imposed to the copropagating pulse and the one imposed to the counterpropagating pulse in case of pilot electrical pulses of low time-length.

[0157] However, such effect causes a rise in the voltage of the pilot electrical pulse necessary to obtain a modulation of n / 2. The required voltage Vmodis obtained as:

[0158]

[0159] It is to be noted that this voltage increase reduces when the time-length of the pilot electrical pulse τsis small with respect to τmod.

[0160] Moreover, for τs< τmod, there is that Vmod< 2V(π / 2)= Vπ, that is the modulation voltage is always lower than the one requested in case one wants to modulate only the copropagating pulse and however to generate three states.

[0161] Clearly, the time-length of the pilot electrical pulse τscould not be reduced at will since it will be always greater than the time-length δtoptof the produced optical signal; this allows to modulate uniformly the whole optical pulse, therefore

[0162] δtopt≤ τs< 2τmod(16)

[0163] Moreover, the phase which is obtained in the area on the right and left with respect to the previous pilot electrical pulse (that is between -0.2ns and 0ns and between 0.3ns and 0.5ns in figures 3A, 3B) is equal to:

[0164]

[0165] The phase δφ1will be useful in the following description.

[0166] The advantage of this scheme lies in the fact that the period Tminbetween an optical pulse and the other one canbe reduced to

[0167] Tmin= τs+ τmod(18)

[0168] Such condition is obtained by imposing that the phase signal of the adjacent pilot pulses does not temporally superimpose with the area wherein the optical pulse is present (Figure 6).

[0169] The period between an optical pulse and the other one can be further reduced in case one admits that part of the signal referred to the adjacent symbols is present in the modulation area of the pulse, but without touching it, as it can be noted in the example of figure 7. In this case the period between an optical pulse and the other one can be reduced up to

[0170] Tmin= (τs+ δtopt) / 2 + τmod(19)

[0171] where δtoptis the time-length of the optical signal with

[0172] δtopt= 0.1 ns in figure 7.

[0173] Lastly, in case a modest interference between the symbols is acceptable, it is possible to further reduce the period between an optical pulse and the other one to a value equal to

[0174] Tmin= τs(20) HEIGHT="15" WIDTH="29" SRC="imgf000028_0001.tif" / >

[0175] as shown in the example illustrated in figure 8.

[0176] However, in this case, the phase applied to the general k-th pulse results to depend upon the phase applied to the adjacent pulses, and the phase applied to the k-th pulse can be:

[0177] φk= π / 2 or

[0178] φk= π / 2 + δφ1= π / 2(τmod / (τmod- τs / 2)) orφk= π / 2 - δφ1= π / 2((τmod- τs) / (τmod- τs / 2))

[0179] depending on the phase applied to the adjacent pulses. Then, δφ1corresponds to the maximum phase error which is imposed to the signal. Such error is small when it happens that τsis much lower than τmod.

[0180] An alternative to the electrical signals of "differential" type shown previously is an electrical signal of "balanced" type, which allows to use lower voltages of the pilot electrical pulse than the signals of "differential" type. In this case, the pilot electrical pulse can be then represented as a sum of two rectangular signals, a negative one and a positive one, as illustrated in figures 9A and 9B.

[0181] The advantage of this type of modulation lies in the fact that the counter-propagating pulse does not undergo modulation if it meets both the positive pulse and the negative pulse: this makes that the output phase is "n / 2" with voltage V(π / 2)in the central area (see figure 9B). It is to be noted that pilot pulses with different timelengths between positive and negative pulse can be used. Moreover, a maximum voltage value equal to V(π / 2)is used.

[0182] By reversing the electrical signal, it is then possible to modulate the different states, as shown in figures 10A and 10B.

[0183] In this case, the period between an optical pulse and the other one can be reduced up to

[0184] Tmin= 2τs+ τmod(21)

[0185] which corresponds to the situation shown in figure 10B.

[0186] If one admits that part of the signal of the subsequent pulse is present inside the area in which the optical signal is modulated, the period between an optical pulseand the other one can be reduced to

[0187] Tmin= τs+ τmod+ δtopt(22)

[0188] where δtoptis the time-length of the optical signal, as shown in figure 11.

[0189] This modulation scheme of balanced type can be used also for any scheme of modulation, for example in line, with double passage or Sagnac, which needs more levels of voltage since it guarantees a constant average value input to the amplifier, and it could remove a patterning or interference phenomenon in the inter-symbol.

[0190] An alternative to the electrical signals of differential type and of balanced type shown previously is a signal of "integral" type, wherein the integral effect on the counter-propagating optical signal is exploited.

[0191] In this configuration, the three symbols are generated by applying the phases "n / 2", "-n / 2" and "0", by using the same voltage but by varying the shape of the electrical pulse between the propagating and counter-propagating one.

[0192] The phase "0" is obtained by not applying any electrical signal. The phase "n / 2" is obtained by applying a rectangular signal having time-length τsas shown in figures 12A and 12B.

[0193] In this regard, it is possible using a pilot electrical pulse having time-length τswith τmod< τs< 2τmod, but even in this case it is more convenient to have a pilot electrical pulse having lower time-length than the propagation time in the modulator, τs< τmod, since the modulation voltage equals to

[0194]

[0195] On the contrary, the phase "-n / 2" is obtained by applying a pilot electrical pulse consisting of a rectangular signalwith the same voltage Vmod, but having longer time-length T ' S, that is T ' S > 2imod and with a central gap having time-length τswhich interrupts the pulse and divides it into two parts, as shown in figures 13A and 13B.

[0196] Such pulse of phase "-n / 2", to make the configuration as symmetrical as possible, has to be anticipated by a period of time equal to (τ's- τs) / 2 with respect to the pulse of phase "n / 2" of figures 12A and 12B.

[0197] This pilot pulse, since it has an interruption at the copropagating optical pulse, acts only on the counterpropagating pulse.

[0198] The time-length T 'Sof this pulse of phase "-n / 2" can be reduced up to τ's= 2τmod+ δtopt, where δtoptis the timelength of the optical pulse, to modulate uniformly the whole pulse, as shown in figures 14A and 14B. In this configuration, the period between an optical pulse and the other one can be reduced to

[0199] Tmin= (τ's+ τs) / 2 + τmod(24)

[0200] If one admits that part of the subsequent pulse is present inside the area wherein the optical pulse is modulated, the period between an optical pulse and the other one can be reduced to

[0201] Tmin= (τ's+ δtopt) / 2 + τmod(25)

[0202] where δtoptis the time-length of the optical signal.

[0203] Since τ's≥ 2τmod+ δtopt, the minimum period between an optical pulse and the other one which can be obtained is equal to

[0204]

[0205] It is to be noted that it is also possible to provide a different voltage between the signal acting on the co-propagating pulse with respect to the one acting on the counter-propagating pulse.

[0206] Alternatively, the three symbols can be generated by applying the phases "n", "-n / 2" and "0", using a higher voltage than the preceding case equal to

[0207]

[0208] The advantage of this solution is that the time-length T 'Sof the pulse of phase "-n / 2" can be reduced to τ's= τmod+ τs / 2.

[0209] As in the other cases described previously, even this type of modulation can be used in asymmetrical configurations (and that is with ΔL ≠ 0) of the Sagnac loop. If the Sagnac loop is asymmetric, the "hole" of the electrical signal used to modulate with phase "-n / 2" will be no more symmetrical with respect toS'. In figures 15A and 15B

[0210] τs= 0.1ns, τ's= 2τmod+ τs= 0.9 ns, τd= τmod+ τs= 0.9 ns

[0211] was considered by obtaining a period

[0212]

[0213] 0.95 ns (28 )

[0214] Lastly, it can be noted that all modulation schemes proposed so far can also be applied, in principle, even to any scheme of polarization modulator exploiting a double passage, in opposite directions of the light in the modulator, as described for example in I. Lucio-Martinez, et al., Proof -of -concept of real-world quantum key distribution with quantum frames, New J. Phys. 11, 095001 (2009) and in all intensity or time-bin modulators exploiting asymmetrical or symmetrical Sagnac loops.

[0215] For sake of completeness with reference to figures 16 to 19, respective polarization modulators 1 are described allowing to implement the method for modulating the polarization according to the present invention by usingan asymmetrical Sagnac loop.

[0216] With reference to figure 16, a first example of polarization modulator 1 using an asymmetrical Sagnac loop comprises a first polarization maintaining fibre PM1 which receives photonic pulses, each one comprising a generic state of polarization obtained from the superimposition of the horizontal and vertical polarization.

[0217] The first polarization maintaining fibre PM1 is connected to a fibre polarization separator PBS which is capable of separating said photonic pulses, which constitute a photonic flow travelling the first polarization maintaining fibre PM1, into two distinct flows of photonic pulses each one having different and respective polarizations, that is Horizontal H and Vertical V polarization ( figure 16A).

[0218] Therefore, the first fibre PM1 is aligned so that the polarization separator PBS separates the two polarizations corresponding to the fast and slow axes of fibre PM1, by injecting them respectively in the two ends of a second polarization maintaining fibre 102 whose separate ends are designated with PM2 and PM3.

[0219] In the present embodiment, both polarization modes, in the two outputs of the polarization separator PBS, are aligned with the respective fibre slow axes ( figure 2A), according to the standard behaviour of the fibre polarization separators PBS.

[0220] The ends PM2 and PM3 are mutually connected by forming a ring 100, thus determining a delay segment PM4 of polarization maintaining fibre, with adequate length L, due to the presence of a a phase modulator 101 in a point of said ring which is separated from said ends by optical paths in said ring having different length, so as to determine, thanks to the difference in length of the twoso-resulting paths, said delay line in polarization maintaining fibre.

[0221] In this way, the second polarization maintaining fibre 102, from its ends PM2 and PM3, implements a ring interferometer of Sagnac type.

[0222] The herein described scheme can be implemented by introducing some variants.

[0223] For example, the light in the two outputs of the polarization separator PBS could be aligned with the fast axis of the fibre ends PM2 and PM3, rather than with their slow axis.

[0224] Otherwise, the used polarization separator PBS does not couple both polarization modes along the fibre slow axis, but it could couple one along the fast axis and the other one along the slow axis. This difference requires the possible introduction of a device to rotate polarization, as for example a Faraday rotor, which rotates the polarization and thus is capable of compensating the modal dispersion effects in case said polarization separator PBS aligns the separated polarization modes to the fast and slow axes, or vice versa, of the respective ends PM2, PM3 of the second polarization maintaining fibre 102.

[0225] To obtain an optimum calibration and be certain that the polarization input to the polarization separator PBS is correct, it is possible to insert beam separators 112, 113 inside the ring, connected to respective photodiodes 114, 115 to measure the intensity of the polarization modes, as it is shown in figure 17. With reference to figure 18, it is also possible to use one single beam separator 112, connected to two photodiodes 114, 115.

[0226] With reference to figure 19, an additional variant of the modulator 1, and in particular of its ring 100, is constituted by the presence of an additional delay line120, interlaced to a segment of ring 100 by means of two switches 121, of variable type, which allows to transform the modulator 1 from a polarization modulator to a phase modulator independent from polarization.

[0227] Even this solution can be applied also to the ring described with reference to the state of art.

[0228] To the above-described method for modulating the polarization and the related polarization modulator a person skilled in the art, with the purpose of satisfying additional and contingent needs, could introduce several additional modifications and variants, in particular in the shape of electrical pulses to be sent to the phase modulator, all however comprised within the protective scope of present invention, as defined by the enclosed claims.

Claims

CLAIMS1. Method for modulating the polarization or phase of optical pulses, in particular for generating quantum cryptographic keys, which comprises the steps of:• generating a plurality of optical pulses, each optical pulse comprising a generic state of polarization obtained by superimposing horizontal and vertical polarization modes, and routing them in a first polarization maintaining fibre;• separating said horizontal and vertical polarization modes and routing them in respective ends of a second polarization maintaining fibre forming a ring, so that they travel such ring clockwise and counterclockwise, respectively;• inserting a phase modulator controlled by pilot electrical pulses inside the ring formed by the polarization maintaining fibre;• sending to the phase modulator pilot electrical pulses travelling inside the phase modulator and resulting to be co-propagating or counter-propagating with respect to the optical modes travelling such ring clockwise and counterclockwise (or vice versa); and• recomposing said polarization modes in one single beam of optical pulses, and routing the resulting beam through said first polarization maintaining fibre, thereby obtaining at the output a polarization state of the pulses which depends upon the pilot electrical pulses,wherein a co-propagating optical signal is exposed to a modulation only depending upon the superimposition to the pilot electrical pulse moving in its own direction, and wherein the counter-propagating optical signal is exposed to the whole pilot electrical pulse moving in opposite direction with an integration effect, so as to determine an asymmetry between the effect obtained on the co-propagating optical pulse with respect to the counterpropagating one so as to codify different symbols, and so that said pilot electrical pulse determines different phase shifts to the two components which do not compensate each other.

2. The method for modulating the polarization of optical pulses according to claim 1, wherein the pilot electrical pulses have a shorter time-length of the propagation time of the optical pulses in the polarization modulator.

3. The method for modulating the polarization of optical pulses according to claim 1 or 2, wherein the pilot electrical pulses comprise a sequence of two pilot electrical pulses with the same absolute voltage, one positive and one negative.

4. The method for modulating the polarization of optical pulses according to any one of the preceding claims, wherein the period between an optical pulse and the other one can be reduced up to the sum of the time-length of the pilot electrical pulse and the propagation time of the optical pulse in the polarization modulator, so that a phase signal of the adjacent pilot pulses does not temporally superimpose with the area wherein the optical pulse is present.

5. The method for modulating the polarization of optical pulses according to any one of claims 1 to 3, wherein the period between an optical pulse and the other one can be reduced up to the sum of the time-length of the pilot electrical pulse, and wherein the phase applied to a generic k-th pilot electrical pulse depends upon the phase applied to the adjacent pulses and corresponds to the maximum phase error which is imposed to the signal.

6. The method for modulating the polarization of optical pulses according to claim 1 or 2, wherein the pilotelectrical pulses are the sum of two rectangular signals, one negative and one positive.

7. The method for modulating the polarization of optical pulses according to claim 6, wherein said rectangular signals have different time-lengths between positive and negative pulse.

8. The method for modulating the polarization of optical pulses according to claim 6 or 7, wherein the period between an optical pulse and the other one can be reduced up to the sum of the double of the time-length of the pilot electrical pulse and the propagation time of the optical pulse in the polarization modulator, so that a phase signal of the adjacent pilot pulses does not temporally superimpose with the area wherein the optical pulse is present.

9. The method for modulating the polarization of optical pulses according to claim 1 or 2, wherein the pilot electrical pulses comprise a sequence of electrical pulses with equal voltage, but by varying the shape of the electrical pulse between propagating and counterpropagating one.

10. The method for modulating the polarization of optical pulses according to claim 9, wherein the phase "0" is obtained by not applying any electrical signal, wherein the phase "n / 2" is obtained by applying a rectangular signal, and wherein the phase "-n / 2" is obtained by applying an electrical pulse consisting of a rectangular signal with the same voltage as that of phase "n / 2", but lasting more than double the propagation time of the optical pulse in the polarization modulator, but divided into two portions by a central gap wherein the value of the electrical pulse is zero.

11. The method for modulating the polarization of opticalpulses according to claim 10, wherein the pulse of phase "-n / 2", is advanced by a period equal to half the difference between the time-length of the electrical pulse of phase "n / 2" and the time-length of the electrical pulse of phase "-n / 2".

12. The method for modulating the polarization of optical pulses according to claim 10 or 11, wherein the time-length of the electrical pulse of phase "-n / 2" can be reduced up to the sum of the double of propagation time of the optical pulse in the polarization modulator and of the time-length of the optical pulse produced in the phase "n / 2".

13. The method for modulating the polarization of optical pulses according to claim 1 or 2, wherein the pilot electrical pulses comprise a sequence of electrical pulses having different shape and voltage to act differently between propagating and counter-propagating optical pulse.

14. The method for modulating the polarization of optical pulses according to any one of the preceding claims, wherein the optical pulses at the two outlets of the polarization separator are aligned with the fast axis of the fibre ends, rather than with their slow axis.

15. The method for modulating the polarization of optical pulses according to any one of the preceding claims, wherein the polarization separator couples the polarization modes one along the fast axis and the other one along the slow axis.

16. A polarization modulator of optical pulses, in particular for generating quantum cryptographic keys, implemented according to a method for modulating the polarization of optical pulses according to any one of the preceding claims.

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