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100 MHz Amplitude and Polarization Modulated Optical Source for Free-Space Quantum Key Distribution at 850 nm

We report on an integrated photonic transmitter of up to 100 MHz repetition rate, which emits pulses centered at 850 nm with arbitrary amplitude and polarization. The source is suitable for free space quantum key distribution applications. The whole transmitter, with the optical and electronic components integrated, has reduced size and power consumption. In addition, the optoelectronic components forming the transmitter can be space-qualified, making it suitable for satellite and future space missions.


Introduction
In many applications, free space optical (FSO) communications is the technology of choice to transmit information, especially when fiber optical cabling is not easily achievable or its installation is too expensive [1]. Compared to radio frequency (RF) techniques, its main advantages lie in high data rates (up to several Gb/s), minimum free space losses due to the small optical beam divergence and absence of regulatory issues thanks to the low interference level [2][3][4]. Therefore FSO communication is favorable for high data-rate, long-range point-to-point links, where the terminal size, mass, and power consumption are subjected to strong limitations, such is the case of aeronautical or space platforms.
An important issue in today's information society is the security of data transmission against potential intruders, which always put at risk the confidentiality. Current methods to increase security require that two parties wishing to transmit information securely need to exchange or share one or more keys. Once the key has been exchanged, the information can be transferred in a provable secure way using a one-time pad. Therefore, the security of the information transmission is based exclusively on the security of the key exchange. Quantum cryptography, or more precisely Quantum Key Distribution (QKD), guarantees absolutely secure key distribution based on the principles of quantum physics, since it is not possible to measure or reproduce a state (eg. polarization or phase of a photon) without being detected [5]. The key is generated out from the measurement of the information encoded into specific quantum states of a photon. In particular, if a two-dimensional quantum system is used, information is said to be encoded into qubits. For example, a qubit can be created using properties such as the polarization or the phase of a photon.
The first QKD scheme, due to Bennett and Brassard [6,7], employs single photons sent through a quantum channel, plus classical communications over a public channel to generate a secure shared key. This scheme is commonly known as the BB84 protocol. Although single photon sources may be very useful for quantum computing, they are not strictly required for QKD. This, and the relative difficulty of generating true single photons, motivates new approaches based on conventional light sources [8]. Indeed, attenuated laser pulses or faint pulse sources (FPS), which in average emit less than one photon per pulse, are often used as signals in practical QKD devices. The performance limitations of attenuated pulse systems had initially led to believe that single photon sources would be indispensable for building efficient QKD systems. However, the introduction of the decoy-state protocol [9,10] made possible a much tighter bound for the key generation rate, achieving an almost linear dependency of the latter on the channel transmittance. In this way, the technologically much simpler faint pulse systems can offer comparable QKD security with respect to single photon sources. Another key feature of QKD is that the security is linked to the one-time-pad transmission, i.e. the key has to be used once and has to be equal or similar in size to the information being transmitted. It is thus evident the importance of developing faint pulse sources and systems for QKD which can generate high key bit rates. The highest Secure Key Rate reported to date over 20 Km of optical telecom fiber is of 1.02 Mb/s [11] and 14.1 b/s over 200 Km [12], while the achieved speed over 144 Km free space link is of 12.8 Kb/s [13] and 50 Kb/s over 480 m [14].
The goal of QKD is to allow to distant parties to share a common key in the presence of an eavesdropper. Therefore, the most important question of QKD is its security. Therefore, an important aim of this work is to demonstrate a system to generate pulses that differ only in polarization, while being indistinguishable in the other degrees of freedom that characterize the quantum state of photons, such as arrival time, optical frequency, and spatial mode. In other words, to generate pulses which contain no side-channel information correlated to the polarization. We note that previous implementations based on multiple lasers [13][14][15][16] have attempted to achieve time-frequency indistinguishability by laser pre-selection, current and temperature adjustment, and temporal and spectral measurements. Apart from being expensive and cumbersome, this kind of tuning has limited stability due to the inevitable aging of laser diodes. It is worth noting that the temporal and spectral distributions reported to date indicate indistinguishability in the time and frequency bases, but leave open the question of distinguishability based on other pulse characteristics such as chirp.
A related issue which arises in a decoy-state protocol is possible side-channel information indicating the pulse intensity. Intensity level modulation could be achieved rapidly and conveniently by modulating the laser current. This method of modulation, however, induces strong nonlinearities and causes strong phase modulation, which makes it difficult to control the temporal and spectral shape of the output pulses.
In this paper we report the development of a novel integrated pulse source which can reach rates as high as 100 Mb/s at 850nm modulated in amplitude and polarization. For QKD applications, it has been simulated that the source could achieve a Secure Key Rate of the order of 500 Kb/s at 20 Km using decoystate protocol. The source is capable to generate pulses at around 850 nm with at least three different intensity levels (i.e. number of photons per pulse) and four different polarization states. The proposed FPS ensures indistinguishability among the different intensity and polarization pulses and ensures phase incoherence of consecutive generated states. It is based on a single diode emitting a continuous pulse train externally modulated in amplitude and polarization. The wavelength, reduced power consumption, compactness and space qualifiable optoelectronic components constituting the source make it very suitable for space transmission, for free space quantum and classical communication links. One of the foreseen applications is its use to overcome the distance limit of QKD in optical fibers [12,17], by creating a global security network among very distant places on earth through satellite communication [18][19][20].

The integrated faint pulse source
In order to use it for space applications, the proposed integrated FPS source for FSO communication consists of commercially available space-qualified discrete components; single semiconductor laser diode emitting a continuous pulse train at 100 MHz followed by integrated (waveguide) amplitude and polarization lithium niobate (LiNbO 3 ) modulators ( Figure 1).  A distributed feedback (DFB) laser diode (LD) at around 850 nm is driven at 100 MHz train of electrical pulses. The optical pulse of about 400ps is generated via a current pulse of about 1ns duration. In fact, the laser is biased using a DC current of 24mA, far below threshold 36mA, and it is directly modulated using a strong RF current of 50mA (peak value) so that the optical pulse is generated [23]. The generated optical pulses do not have any phase coherence among them due to the fact that the laser is set below and above threshold from one pulse to the subsequent one, thus producing a random phase for each pulse. The output mirror reflectivity (R) of the DFB structure is 30%, the cavity length (L c ) 300µm and the active medium refractive index (n) 3.6 [21,22]. The roundtrip-time (RTT) given by 2L c n/c 0 , where c 0 is the speed of light in vacuum, is ≈ 20ps. In one pulse train period (10ns) the optical pulse power left in the laser cavity, after going below threshold, has bounced back and forth ≈ 500 times. Between pulses, the laser is biased at only 66% of threshold, so that transmission loss through the output mirror is much greater than the round-trip gain. Conservatively assuming a round-trip loss ≥ 1dB, the ≥ 500dB loss from 500 passes will atenuate any coherence to a very low level. At the same time, incoherent spontaneous emission is generated, further obscuring any possible coherence between optical pulses [23]. In terms of partial coherence theory, it is expected a first-order degree of coherence g (1) (τ ) which drops rapidly to zero for τ larger than the pulse duration. This is consistent with the observed spectrum, which implies a coherence length of order 0.75 m during the pulse. Note that the spectrophotometer response is produced by photons coming from the pulses and the coherence length of about 0.75 m which can be calculated from the measured bandwidth, as it was explained above, decreases significantly when the pulse extinguishes, thus making the inter-pulses value much smaller than the distance between consecutive pulses (about 3 m).
In this way, phase incoherence of consecutive generated states, which otherwise would be detrimental for the link security, is achieved. Then, the pulse train is sent through a polarization maintaining fiber (PMF) into an amplitude modulator (AM) (eg. a Mach-Zehnder modulator in LiNbO 3 ) that will randomly generate the three different levels of intensity. Note that if the DFB laser diode were driven in pure continuous wave (CW) mode (no pulse train) and externally modulated to generate the pulses, two potential issues would occur: (i) pulses with different energies (number of photons) would unavoidably have different temporal and spectral shapes due to the nonlinear electro-optic response (optical output as a function of driving voltage) of the amplitude modulator; (ii) there would be phase coherence between the pulses due to the relatively long coherence time (narrow spectrum) of a DFB structure, thus increasing the vulnerability of the QKD transmission [24].
After the AM, the pulses are injected into a polarization modulator (PM) through a PMF. The polarization modulator is in fact a waveguide LiNbO 3 phase modulator where the PMF input axis is oriented at 45 • with respect to the optical axis. In this way, the two orthogonal equal amplitude polarization components of the electromagnetic field that propagates in the crystal experience a refractive index difference, which is proportional to the voltage applied to the modulator. By applying different voltages one can thus change the state of the output polarization, in particular linear +45 • , -45 • , right-handed circular and left-handed circular. The optical pulses present a spectrum within the ac-ceptance bandwidth of the two modulators, so that amplitude and polarization modulation can be achieved with high extinction ratio.
A proper electronic control of the different intensities and polarization states generated, at Alice, for the different states is fundamental in order to perform a QKD transmission. The synchronization and setting of the different optical components of the source is implemented by an automatic control which is split into two working operations. The control system first synchronizes and calibrates the driving signals timings and amplitudes to the AM and PM, and secondly generates the appropriate driving signals for the BB84+Decoy protocol transmission.
For the implementation of a QKD system using decoy-state protocol, besides four different polarization states, the FPS source should generate three intensity levels (optimally 1/2, 1/8 and 0 photons in average per pulse [10]) using a variable optical attenuator (VOA) in order to operate in the single photon regime. The optical pulse duration ≈ 400ps and the pulse peak power 3.5mW which corresponds to 1.4pJ energy per pulse, thus a number of photons per pulse ≈ 6 · 10 6 . In order to get a mean photon number for the signal state which is within an optimum range for the distances of interest [10], the VOA has to introduce an optical attenuation of ≈ 70dB.
The chosen FPS wavelength (850 nm) is optimum for free space operation considering attenuation (due to scattering, absorption and diffraction) and single-photon detector's quantum efficiency [8].

Description of generated states
In a BB84 protocol scheme implementing the decoy-state protocol different pulses should differ in polarization and amplitude while remaining indistinguishable in other characteristics, including temporal shape and frequency spectrum. If the pulses differ in spectrum, for example, an eavesdropper could use spectral measurements to infer the sent polarization without actually measuring it. Removal of this kind of side-channel information is thus critical to the security of the protocol. Since the information is encoded in the polarization state, the statistical similarity between pulses of different polarizations but same intensity level is more relevant than that of different intensity level but same polarization to prevent information leakage from the quantum link. Here we consider the quantum optics of side-channel information, limiting the discussion to pure states and simple measurements. A full treatment including mixed states and generalized measurements will be the subject of a future publication.
We consider a source that produces pulses with amplitudes E l , polarizations p l and pulse shapes Π l (t). Without loss of generality we assume the polarizations and pulses shapes are normalized p * l · p l = dt Π * l (t)Π l (t) = 1. In a classical description, the field envelopes are The corresponding quantum state is a generalized coherent state where |0 is the vacuum state and is a displacement operator, defined in terms of the mode operator A l ≡ dt Π * l (t)a(t), a ≡ (a x , a y ) is a vector of annihilation operators, with [a p (t), a † q (t )] = δ(t − t )δ p,q for p, q ∈ {x, y}. A scaling factor η is included to convert from photon units to field units, chosen such that the positive-frequency part of the quantized electric field isÊ(t) = η −1 a(t). It is easy to check that α l | a(t) |α l = ηE l p l Π l (t), so that the average quantum field α l |Ê(t) |α l = E l p l Π l (t) in agreement with Equation 1.
Quantum mechanics allows measurements on the pulse-shape Π without measurement of the polarization p. For example, the number operator N l ≡ A † l · A l = A † l,x A l,x + A † l,y A l,y counts photons in the mode Π l independent of p l . If the modes {Π l } are different, an eavesdropper could use state-discrimination techniques [25,26] to determine l (and thus the secret key) without disturbing p. This kind of eavesdropping would not be detected by Bob's polarization measurements. For this reason, it is critical to guarantee that this kind of side channel information is not present in the sent optical pulses. The similarity between the various Π l can be quantified by an overlap integral: ≡ S lm δ p,q , so that for example two states with equal amplitudes |E l | = |E m |, α m | N l |α m / α l | N l |α l = |S lm | 2 . Finally, we note that it is possible for pulses to have the same spectra and temporal shape but still be distinguishable, for example if they have different chirp. For this reason, establishing that two (or more) distinct sources produce indistinguishable pulses is not easy.
Our strategy to eliminate side-channel information in the pulse shapes is to dissociate pulse generation from the setting of polarization and amplitude levels. As described in the previous section the FPS consists of a single laser diode emitting a continuous train of optical pulses followed by an AM, a PM and a VOA. Considering that the laser operation is the same for each pulse sent, and that both the AM and PM control voltages are held constant over the duration of the pulse, we can assume that the pulse shape does not depend on the sent amplitude and polarization. This assumption is confirmed by measurements shown in Section 4. The complex expression of the pulsed electromagnetic field exiting the FPS can be written as where t is the time, T is the pulse train period and A, φ i , Π are the amplitude, phase, and shape, respectively, of the optical pulse generated by the LD. α i , β i describe the transmission and introduced phase, respectively, of the AM. γ i is the phase difference betweenx andŷ introduced by the polarization modulator in order to generate the different polarization states.
Another security consideration is optical coherence between successive pulses, which could in principle be used for eavesdropping attacks [24]. As the LD is taken below threshold between pulses, each new pulse will start up from vacuum fluctuations, and will have a random overall phase φ i , thus eliminating coherence between successive pulses and thus among states. Similarly, any information contained in the AM phase β i is washed out by the random φ i .

Experimental measurements
Figure 2 (a) shows the train of optical pulses generated by the laser diode when driven by electrical pulses of 1 ns at 100 MHz. The resulting optical pulse duration is about 400 ps. Since the obtained cw train of optical pulses are all generated in the same way, they can be assumed to be indistinguishable thus having no side-channel information. Furthermore, the short optical pulse duration of 400 ps (small duty cycle) has the advantage to increase the signal to noise ratio since the measurement window (detection time) in the receiver can be reduced. The DFB laser diode is driven in direct modulation with a strong RF driving signal with 24mA DC bias current, far below threshold 36mA, thus producing highly similar optical pulses and jitter as low as 100ps, rise time 65ps and fall time 129ps, as shown in Figure 2 (b). From Figure 2 (b) one can see that the ringing of the laser driver current is repeatable from pulse to pulse, thus producing the overlapped temporal profile of several optical pulses, captured in real-time. The traces are indistinguishable by eye, indicating a very small pulse-to-pulse variation of energy, timing, and wave-form. Furthermore, the optical pulse bandwidth is small enough to enter the acceptance bandwidth of the subsequent polarization modulator. Figure 3 (a) shows the three different intensity optical pulses generated after the AM. The attenuations for the medium and low level of intensity pulses are about 4.65dB and 14.76dB, with respect to the high intensity pulse. While Figure 3 (b) shows the four polarization states generated after the PM, as measured with a terminating rotating waveplate polarimeter. The RF modulating signal is driven at 100 MHz, in this way, chirp produced at the pulse edges of the RF driving voltage is avoided and intensity and polarization indistinguishability is obtained. Figure 4 shows pulses with the same polarization but with different intensity levels with the aim of comparing its temporal and spectral indistinguishability. A 8 GHz amplified photodiode and a 4 MHz resolution Fabry-Perot interferometer were used for the temporal and spectral measurements, respectively. In order to compare pulses with different intensity levels, the different pulses are normalized to their own total intensity. Fig. 5 shows a similar comparison, but this time pulses have the same intensity level and different polarization.

QKD performance analysis
Given the experimental data on the classical optical performance of the proposed source, a low Quantum Bit Error Rate (QBER) as well as a high Secure Key Rate in the order of Mb/s are expected. A simulation-based analysis of the expected rates and performances of a QKD BB84, implementing decoy-state protocol, is derived below as a demonstration of the potentials for applications of the proposed FPS. In a BB84 only single photon pulses contribute to the secure key while in a 3state decoy protocol one can obtain a lower bound for the secure key generation rate as where q depends on the implementation (1/2 for the BB84 protocol), N µ is the total number of detected signal pulses, t is the time duration of the QKD transmission, µ represents the intensity of the signal states, Q µ is the gain of the signal states, E µ is the total QBER, Q 1 is the gain of single photon states, e 1 is the error rate of single photon states, f (x) is the bi-direction error correction efficiency (taken as 1.16 [10], for an error rate of 1%) as a function of error rate, and H 2 (x) is the binary Shannon information function, given by     6 Results and discussion Table 1 summarizes the characteristics of the driving RF and corresponding optical pulses for the three levels of intensity, suitable for a decoy-state protocol. We believe that, should they be needed, larger intensity attenuation could be achieved by improved DC voltage bias of the AM. The AM driving RF signal and the corresponding AM output quality largely demonstrate the 100 MHz and even beyond capability of the source. The modulator "ON" window has a duration of at least 5 ns, much larger than that of the optical pulse. Therefore, only the amplitude of the optical pulse changes, while the temporal and spectral shape remain unaltered. In addition low driving voltages are needed, making the design suitable for electronic integration with low electrical power consumption drivers. Table 2 summarizes the RF voltages driving the PM generating the four orthogonal states. In the same table, cross polarization extinction ratio (PER) values for the four different polarization states are given. The PER values obtained (>25dB) are significantly higher than those required for a low QBER (20dB). As for the AM case, low driving voltages are needed, suitable for integration with low power consumption and inexpensive electronics.
As expected, Figure 4 and 5 show the high degree of similarity of the pulses, independently of their polarization or intensity state, indicating minimal pulse distortion due to the AM and the PM. It has to be noticed that the small dif-    ferences for the different intensity pulses are due to measurement errors. Nevertheless, as commented in section 3 polarization statistical similarity is more important than intensity statistical similarity. Furthermore, information on the absolute or relative phase between pulses is not contained in these four figures. However, by design, the phase of each pulse varies at random between pulses due to the fact that, as already mentioned, pulses are generated by taking continuously the laser diode above and below threshold, as explained in section 3.
In the simulation the detectors efficiency was set to 50%, free space loss 0.1dB/Km, 5dB were accounted for the loss due to the transmitting and receiving optical systems, background yield 1 × 10 −5 and detector misalignment error of 1%. Note that the background yield Y 0 includes the detector dark count and other background contributions from stray light, including scattered light from timing pulses [10,13], being for larger distances the major cause of secure key rate drop. The parameters, derived from the values presented above, and results for the simulated BB84 transmission, implementing the decoy-state protocol as well as for free space distance of 20 Km, are shown in Table 3. The simulation has been completed using values taken from [10,11], achieving a theoretical Secure Key Rate of 559.80 Kb/s, which is consistent with the free space achieved value of 50 Kb/s (over 480 m) reported for a 10 MHz source in [14] taking into account that the presented source emits pulses with a repetition rate one order of magnitude larger.
The laser diode is DC biased at 24mA presenting a DC resistance of 3Ω accounting for 1.7mW. In addition, an impedance matching circuit has been designed to 50Ω for RF modulation where the electrical pulses of 50mA, 1ns wide, at 100MHz account for 12.5mW. The modulators do not have a termination resistor, basically, they present an open-ended transmission line with an equivalent loss resistor and parallel capacitor of 5Ω and 10pF, respectively. Considering the worst case situation where the source is at maximum modulation speed and using the maximum driving voltages, the power consumption for the AM is 2.7mW and for the PM is 7.7mW. Thus, the overall power consumption of the integrated module is potentially very low.

Conclusion
We have shown that, starting from commercially available and space-qualifiable components, it is possible to build an integrated transmitter capable of generating the several intensity and polarization states required for decoy-state QKD.
The experimental demonstration has been carried out at 850 nm with 100 MHz modulation rates. However, taking into consideration that the modulators bandwidth can go well beyond 10 GHz and operate also at other wavelengths (e.g. 1550 nm), the source can be easily scalable to higher bit rates, the upper limit being probably given by the laser diode itself, and other transmission systems (e.g. optical fibers). Although we believe that the proposed source is of general use in polarization modulation optical systems, especially free-space links, we have focused our demonstration in preparation for a QKD experiment using decoy-state protocol, where the indistinguishability of the pulses, both in the frequency and time domain, is the key for the security of the link. Given the relatively low driving voltages of the modulators, the proposed transmitter is potentially low power consumption and also highly integrable.