## Abstract

We investigate a multiple spatial modes based quantum key distribution (QKD) scheme that employs multiple independent parallel beams through a marine free-space optical channel over open ocean. This approach provides the potential to increase secret key rate (SKR) linearly with the number of channels. To improve the SKR performance, we describe a back-propagation mode (BPM) method to mitigate the atmospheric turbulence effects. Our simulation results indicate that the secret key rate can be improved significantly by employing the proposed BPM-based multi-channel QKD scheme.

© 2016 Optical Society of America

## 1. Introduction

Quantum key distribution (QKD) enables two authentic remote participants, named Alice and Bob, to share a secret key that is unknown to a potential eavesdropper Eve [1]. QKD has been studied over both optical fiber links and free-space optical (FSO) links, and free-space optical channels offer a more flexible physical layer for different applications, especially for establishing secure communication between moving terminals [2,3]. On the other hand, QKD over free-space optical links suffer from several atmospheric effects, e.g. absorption, scattering, and wavefront distortion. Absorption and scattering lead to low transmittance of the signal by photons being absorbed through the channel or scattered out of the direct link. The achievable secret key rate (SKR) of QKD is proportional to the total transmittance through the optical channel in high loss regime. Therefore, to achieve adequate key rates over long distance by a single-mode channel is very challenging. Several high dimensional QKD protocols have been proposed to increase the QKD capacity, such as encoding more than one bit of information in each photon using orbital angular momentum (OAM) of photons instead of polarization states or using pulse position modulation (PPM) to encode more bits in each time frame [4,5]. Unfortunately, the improvement of those spatial encoding methods is not linear but rather logarithmic. The information carried by each photon is log_{2}*M* [bits/photon], where *M* is the number of states used in the protocol. Recently, a multiple spatial modes QKD system with simple parallel channels has been analyzed to obtain the spatial-multiplexing gain, in which the parallel channels were achieved by an overlapping Gaussian beam array [6]. It has been shown that, as an alternative scheme to using mutually orthogonal spatial modes, the parallel channel solution provides almost the same multiplexing gain in vacuum propagation with optimized beam geometry. This scheme provides the possibility to increase SKR linearly with the number of parallel channels. However, the effect of the atmospheric turbulence is unavoidable. The turbulent free space channel will destroy the orthogonality of the spatial modes resulting in crosstalk between parallel channels.

In this paper, we studied a multi-channel QKD scheme which employs an array of QKD transmitters and QKD receivers paired at the focal plane of the transmitter and receiver telescopes, respectively, with each independent channel operating a standard BB84 protocol. This multi-channel scheme also suffers from the crosstalk due to the wavefront distortion caused by the random refractive-index fluctuations along the free space path. In order to mitigate the turbulence effect, we apply two different compensation schemes: a traditional receiver-side adaptive optics (RXAO) configuration and a new back-propagation mode (BPM) method. We demonstrate that both schemes provide substantial decrease of atmospheric turbulence induced crosstalk among the parallel channels, hence they help increasing the SKRs.

The paper is organized as follows. The system diagram, channel modeling, and secret key rate calculation of the multi-channel quantum key distribution system are presented in Section 2. Section 3 presents the turbulence compensation schemes. In Section 3.1, the traditional receiver adaptive optics is applied. The proposed back-propagation mode method is described in Section 3.2, and the temporal effects of turbulence are discussed in Section 3.3. Finally, we conclude the paper in Section 4.

## 2. Multiple spatial modes based quantum key distribution system

Figure 1 shows the system diagram for our multi-channel QKD system. As shown in Fig. 1(a) an array of transmitters is placed at the center of the back focal plane of the expanding (transmitter) telescope. The signal beams propagate through the turbulent free-space channel and are collected by the collecting (receiver) telescope. In our scheme, the system is symmetric, in which the sizes of the expanding telescope and collecting telescope are the same size for potential bi-directional communication. A set of receivers are placed at the conjugate position at the back focal plane of the receiver telescope, so that, without turbulence, we could have multiple independent parallel channels between the sources and their corresponding detectors. For each spatial channel, we consider a standard BB84 protocol employing a weak coherent source (WCS) at 1061 nm with polarization encoding [1]. Each QKD transmitter, which is depicted as a block in the transmitter array in Fig. 1, is composed of four WCS, each emitting photons in one of the four polarization states used in the BB84 protocol, then combined with beamsplitters, and an attenuator is used to control the mean photon number per pulse launching into the channel. Each QKD receiver, which is drawn as a block in the receiver array, has a beamsplitters to passively branch the received photon to one of the two polarization measurement basis, rectilinear basis or diagonal basis, and detected with single photon detectors [7].

In the presence of turbulence, the wavefront is disturbed after propagation and will not be perfectly focused onto the correct detector. Photons leaked into adjacent channels will increase the quantum bit error rate (QBER) and hence reduce the SKR. To study this effect, the atmospheric turbulence along the propagation path is modeled as a set of transverse random phase screens. Specifically, we studied a marine free-space channel over open ocean environment, a 30 km horizontal path at 50 m height, and it is represented by 16 phase screens in our simulation, which has been proved efficient in our previous work [8]. The marine channel is different from the most typical horizontal path over land and is based on SPAWAR models and measurements [9]. The phase screens are generated based on a modified version of the Kolmogorov spectrum introduced by Andrews, which includes both inner and outer scales, and is given as [10]

*κ*is the spatial frequency of the refractive index distribution,

*κ*= 3.3/

_{l}*l*,

_{0}*κ*= 1/

_{0}*L*,

_{0}*C*

_{n}

^{2}is the refractive index structure parameter,

*l*and

_{0}*L*are the inner and outer scale of the turbulence, respectively. We choose an inner scale of 10 mm and an outer scale of 10 m.

_{0}The SKR of single channel BB84 protocol with WCS has been well studied against general attacks [11]. The differences of a multi-channel scheme originate from the errors due to crosstalk noise introduced by atmospheric turbulence. Apart from all the background noises, which is given by the background noise probability *p _{b}*, photons coming from neighboring channels also contribute to the error rate. The probabilities of having a crosstalk-induced error are characterized by the crosstalk matrix, which can be determined by an auxiliary classical pilot signal with a slightly different wavelength of the single photon signals. The crosstalk is measured by turning on only one of the pilot sources in the array and then measuring the received power that leaks to the other detectors. The crosstalk power is then normalized by the received power measured by the correct detectors, i.e. corresponding detector of the pilot source in use, to give the elements in the crosstalk matrix. The off-axis element in the crosstalk matrix

*p*hence indicates the probability of a photon transfer from channel

_{n→m}*n*to channel

*m*. The total probability of a channel

*m*receives a photon from another channel is

*p*= ∑

_{cross}_{m|m≠n}(

*p*). Therefore, the detection probability of a crosstalk-induced photon in channel

_{n→m}*m*is given as 1-

*exp*(-

*λp*) [12], which includes the source brightness parameter

_{cross}η_{ch}η_{eff}*λ*of WCS, total channel transmittance

*η*, and detection efficiency

_{ch}*η*. Considering the photon number splitting (PNS) attack, the single photon error rate is calculated as

_{eff}*Q*= max [(

^{s}*Q*-

_{det}p_{det}*Q*) / (

^{m}p^{m}*p*-

_{det}*p*), 0], where

^{m}*Q*is the detected error rate,

_{det}p_{det}*Q*is the error rate from multi-photon pulses. Finally, the SKR of a single channel in the multi-mode system is given as,

^{m}p^{m}*h*(·) is binary Shannon entropy function.

*p*,

_{det}*p*, and

^{s}*p*denote the total detection rate, single photon detection rate, and multi-photon detection rate, respectively. The total SKR of the multi-channel QKD is simply the summation of all channels. For simulation purposes, we set the detection efficiency to

^{m}*η*= 0.1 and the background noise probability to

_{eff}*p*= 10

_{b}^{−5}in the calculation of SKR.

For system optimization, there is a tradeoff between crosstalk noise and transmission efficiency. Large transmitter (and receiver) separations can get lower crosstalk. However, due to large propagation angles, the side and/or corner channels will have lower transmittance through the receiver optics. To get the optimal performance, we optimize the separation for a 16 (4 × 4) channels system with both telescopes of 75 cm diameter and detector size of 20 × 20 μm^{2} over the turbulent channel. For each turbulence strength, 1000 channel realizations were generated and results were averaged over all realizations. The optimization results are summarized in Fig. 2. Clearly, a large separation is needed when the transmission channel becomes more turbulent. Once the transmitter separation is determined optimally for the full 16 transmitters array, we can further reduce the crosstalk by turning off some of the transmitters in the array, which increases the separation by two times with four out of 16 transmitters left and 2.8 times with only two being on. In order to have good throughput, we also need to shift the center of the remaining transmitter to the center of the channel as the source position shown in Fig. 3. Figure 3 shows the results of SKR using Eq. (2) for different channel conditions. We optimized source parameter *λ* [13] and transmitter separation of all *C _{n}^{2}* values for each realization, and for all results shown in this paper. We observe that, when the channel becomes more turbulent, the total SKR drops down due to the crosstalk-induced errors.

## 3. Atmospheric turbulence mitigation

Adaptive optics is widely used in FSO communication systems to deal with the wavefront distortion caused by atmospheric turbulence [14–16]. In a free space QKD system, the pilot beam is co-propagated with the signal beams as both the public signal to facilitate the protocol and the reference signal for adaptive optics (AO), shown as the blue lines in Fig. 1.

#### 3.1 Receiver adaptive optics scheme

As shown in Fig. 1(b), RXAO is a typical configuration for wavefront correction by measuring the wavefront errors with help of the Shack-Hartmann wavefront sensor (SHWFS) and correcting the distortions with a deformable mirror (DM) in a closed loop. In our simulation, the SHWFS consists of a microlens array of 90 × 90 lenses (6084 over the circular aperture) each with diameter of 250 μm, which are placed just after the entrance aperture with diameter of 22.5 mm; a CCD array with 1875 × 1875 pixels (pixel size 12 × 12 μm^{2}) placed at the back focal plane of the microlens array for analysis of the focused spots; the segmented DM has 3063 actuators over the aperture. The wavefront aberration is expressed by a superposition of Zernike polynomials as

*a*is the coefficient of Zernike polynomial

_{j}*Z*(·) and

_{j}*N*is the number of Zernike polynomials used. The distorted wavefront is reconstructed by calculating the coefficients

*a*using relationship between the Zernike polynomial gradients and the wavefront slopes given by the centroid shift on the CCD [14]. Then the wavefront estimation

_{j}*w*(

*x, y*) is applied to the DM through the control system. We use 54 Zernike polynomials in reconstruction, which includes all Zernike terms with the radial order from 1 to 9 [17]. Figure 4 shows the resulting SKR. Clearly, with the AO corrections, SKRs in the strong turbulence regimes, indicated by Rytov variance

*σ*>1, can be significantly improved. As the turbulence strength

_{R}^{2}*C*(and

_{n}^{2}*σ*) increases, the residual wavefront error increases, which includes the wavefront estimation error

_{R}^{2}*σ*and the DM fitting error

_{WFS}*σ*, hence the SKR drops off due to increasing crosstalk.

_{DM}#### 3.2 Backpropagation mode (BPM) method

Although the RXAO scheme shows improvement in the multi-channel QKD system, the residual wavefront error is limited by the resolution of DM. High resolution DM is needed in very strong turbulent condition. To improve the resolution, a liquid crystal spatial light modulator (LC-SLM) can be used as a promising alternative to the DM [18]. An LC-SLM with a 1000 × 1000 pixel array in a 10 × 10 mm^{2} area is achievable [14]. However, because the LC-SLM is polarization sensitive, we need multiple LC-SLMs or segmented areas on the LC-SLM to modulate single photons in different polarizations for the QKD protocol. Here, we introduce a new turbulence compensation scheme called back-propagation mode (BPM) method. In the BPM scheme, the pilot beam is propagated backward from the receiver to the transmitter to probe the channel. The distorted wavefront is now determined at the transmitter side as shown in Fig. 1(c). For the signal beam, SLMs are used to generate photons in specific spatial mode, which is the conjugation of the determined wavefront. Although there exist amplitude-modulating liquid crystal on silicon (LCOS) devices and several techniques have been also reported that use the phase-only LC-SLM for computer-generated holograms (CGH) to produce complex (both amplitude and phase) modulation of light field [19–21], we chose the phase-only LC-SLMs to modulate the signal beams in our BPM scheme to maintain low implementation complexity. Different from the RXAO, in which the segmented DM applies modal correction with superposition Zernike modes, the LC-SLM is programed to generate the conjugated phase of the piece-wise wavefront measured by the SHWFS directly. It works as a zonal corrector in this way and takes advantage of its high spatial resolution. The wavefront of the signal beam, therefore, is pre-shaped so that after propagation the light will focus at a predetermined position, i.e. the target detector. If we use one probing source for each parallel channel, same number of wavefront sensors are needed which result in high complexity of the transmitter optics. To reduce the complexity, fewer probing sources can be used with linear phase interpolation. If the number of probe sources is less than the number of channels, a linear phase term is added to the conjugation wavefront to force the signal shifting to the position of the target detector at the receiver. Therefore, the pre-shaped wavefront is given as

*x*,

_{det}*y*) is the position of a target detector, (

_{det}*x*,

_{prob}*y*) is the position of the closest probing source to the target detector, and

_{prob}*f*is the focal length of the receiver telescope.

Figure 5 summarizes the SKR results of the proposed BPM scheme. Compared with the RXAO scheme, BPM provides even higher SKRs due to improved wavefront correction. Furthermore, we observe that the 16 probing sources case performs better than using only one probing source. This behavior is based on the fact that the propagation through turbulence is anisoplanatic [14]. There will be different wavefront errors for two beams propagating at slightly different angles, which is known as angular anisoplanatism [22]. If we use only one probe source, the signal beams will propagate through a slightly different path of the probe beam path hence see different wavefront errors. The amount of the angular anisoplanatic aberration is related to the separation angles, which are shown in Fig. 6(a) as the shaded area. Therefore, the corner channels will have more anisoplanatic aberration than the center ones. Also due to the propagation angle, the transmittance of the corner channels is lower. When we use 16 probe beams, with one probe beam for each independent channel, there will not be the angular anisoplanatism effect. Figure 7 shows the intensity pattern at the detector plane for the BPM scheme with *C _{n}*

^{2}of 10

^{−15}(

*σ*= 16). From the first column, the 16 probe beams have better detection quality, brighter and focused spots, while 1 probe beam case shows aberration and low transmittance in the corner and side channels.

_{R}^{2}#### 3.3 Turbulence temporal effect

A crucial drawback of using LC-SLM is that its frame rate, which is typically limited to only a few hundred hertz, is quite low compared with the DM which can operate at frame rates over one kilohertz. If the LC-SLM is not fast enough to apply the in-time wavefront corrections, the forward signal would not be focused on the correct detector perfectly. We investigate the time lags effect including the transmission latency and LC-SLM frame rate. The transmission latency of a 30 km channel is 0.1 msec. A typical LC-SLM frame rate is 60 Hz, i.e. 16.7 msec to employ a new pattern. Therefore, we have a 16.8 msec time delay between each updates. Following the Taylor’s frozen flow hypothesis [23], we assume that the entire spatial pattern of the random turbulent field is moving transversely across the path and the turbulent eddies do not change significantly through the transverse area. Suppose the transverse wind speed is 5 m/s, the turbulent field will be shifted 83.3 mm during the delay time. Equivalently, this behavior can be modeled as shifting of probe beams, as shown in Fig. 6(b). The propagation path probed by the pilot beam is shifted along with the transverse wind direction, which is indicated by the blue dash line in Fig. 6(b). The propagation path displacement produces the displacement anisoplanatism, which is caused by the time delay between the propagation of the back-propagation probe beams and the forward single photon signal beams, hence also known as the temporal anisoplanatism. As the turbulence medium is moving across the path, the anisoplanatic aberration of all the channels will increase. Therefore, the SKR reduces with increasing time delay. The SKR results are presented in Fig. 8. We also observed that using fewer probing beams is more robust to the temporal effect. The reason for this behavior is that, as we can see from Fig. 6(b), the overlapping channel of the probe beam and signal beam is always bigger in the single probe case than in the multiple probe case, since in the single probe case, the pilot beam probes the whole line-of-sight channel between telescopes. Therefore, there will be less displacement anisoplanatic aberration when we use fewer probe beams. The intensity pattern in Fig. 7 shows the same trends as described above. As the time delay increases, the 16 probe beam case catches more crosstalk due to the anisoplanatic aberration than the single probe beam case. From this aspect, using one probing source is favorable since it also reduces the system complexity. Figure 9 shows the BPM scheme performance with temporal effect compared to the RXAO scheme. In order to have better performance for BPM scheme over the RXAO scheme, the update delay of SLMs needs to be less than 9.6 msec, i.e. a frame rate of 105 Hz, which shown by the dotted line in Fig. 9.

## 4. Conclusion

In conclusion, we have investigated multiple spatial modes based QKD over marine free-space optical channels combined with turbulence compensation schemes. To mitigate the atmospheric turbulence effect, the receiver side adaptive optics with Shack-Hartmann wavefront sensor and deformable mirror and a back-propagation mode method with liquid crystal spatial light modulator are compared in terms of the overall secret key rate for the multi-channel QKD system. Our simulation results indicate that the proposed back-propagation scheme offers better performance over traditional adaptive optics, but requires LC-SLMs with higher frame rate than typical commercial LC-SLM. Given that the SLM has been recently manufactured with rates up to kilohertz [24], the BPM-based multi-channel QKD scheme can be considered as a feasible approach to enable high key rate QKD over strong atmospheric turbulence channels. The transmitter in the BPM scheme can be further improved with reasonable modification. For example, a simple polarization-insensitive configuration can be built with a polarization beam splitter, a half-wave plate, a polarization-sensitive phase-only LC-SLM, and a mirror in a loop structure [25], which prevents time delay of phase pattern update due to polarization selection or reduces the SLM number requirement. Furthermore, the LC-SLM can be replaced by a digital micro-mirror device (DMD). A DMD is an amplitude-only spatial light modulator and its most significant advantage over LC-SLM is frame rate. In fact, DMDs with speeds as high as 32 kHz are commercially available. Generation of a set of 14 spatial modes at a rate of 4 kHz has been reported in [4,26] by dynamically changing the hologram realized by the DMD. Although the limitation of generating arbitrary spatial modes with DMDs needs to be studied further, its high frame rate is well suitable in the BPM scheme for beam shaping.

## Funding

Office of Naval Research (ONR) MURI program (N00014-13-1-0627).

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