## Abstract

Common randomness arising from turbulence-induced signal fading in reciprocal optical wireless channels is a beneficial resource that can be used to generate secret keys shared by two legitimate parties. The concept of optical wireless channels using common-transverse-spatial-mode coupling (CTSMC) that can maintain perfect fading reciprocity in atmospheric turbulence is first developed in a general manner. Subsequently, by performing Monte Carlo simulations, the Johnson *S _{B}* probability distribution is demonstrated to be appropriate for statistical description of turbulence-induced signal fading in an optical wireless channel constructed by use of two identical CTSMC transceivers, and the nature of correlation between signal fadings detected by two contiguous reception spatial modes is further quantitatively characterized, revealing that rapid spatial decorrelation between signal fadings observed by a legitimate party and an eavesdropper holds for scenarios of practical interest. Finally, the information theoretic capacity for generating secret keys from CTSMC-based optical wireless channels is theoretically formulated and quantitatively examined under different conditions, manifesting that the turbulence strength and average electrical signal-to-noise ratio have a noticeable combined impact on the secret key capacity, especially in the far-field case.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Currently, communication security has become a concern of numerous people over the world. To protect communications against eavesdropping, cryptographic techniques have been widely utilized in various information transmission scenarios. For instance, modern cryptographic systems typically first employ the public key method to distribute shared secret keys between two distant legitimate communicating parties [1], called Alice and Bob later, and subsequently encrypt and decrypt the transmitted messages at the sender and receiver ends, respectively, with use of the shared secret keys. This approach to sharing secret keys between two parties depends on computational complexity of certain mathematical problems, and thus only has computational security. Moreover, public key management infrastructures are commonly costly and may be even unavailable in some circumstances. Over the recent years, quantum key distribution (QKD) technology [2–5], which is, theoretically speaking, unconditionally secure from computational attacks, has been developed to share secret keys between Alice and Bob; however, nowadays, it is still technically demanding and consequently really expensive. On the other hand, it is noted that generation of shared secret keys from common randomness accessible to both Alice and Bob has attracted considerable attention in the scientific community for many years [6–8], because it can provide the potential for making Alice and Bob obtain the same secret keys in the manner of information theoretic security instead of computational security. A representative of this approach is use of stochastic fading in reciprocal wireless channels as a source of common randomness from which shared secret keys are extracted by Alice and Bob [1, 8–17]; it only employs classical hardware devices and thus is much cheaper than QKD. Of course, noise may make Alice and Bob produce key bits that are not completely identical. To correct this discrepancy, key reconciliation and privacy amplification [14, 18] need to be performed after the raw key bits are generated.

Secret key generation based on reciprocal stochastic wireless channels is actually applicable to both radio- and optical-frequency cases. The former can exploit the randomness induced by stochastic multipath fading, and the latter can utilize the randomness caused by atmospheric turbulence. In comparison with radio-frequency wireless systems, optical wireless systems can be rapidly and flexibly deployed in long-distance scenarios without a spectrum license. Unlike QKD systems [2–5], on which atmospheric turbulence produces detrimental effects, interestingly atmospheric turbulence produces beneficial effects for the case where turbulence-induced stochastic signal fading in reciprocal optical wireless channels is treated as a source of common randomness from which shared secret keys are extracted. Up to now, although various aspects of shared secret key generation from reciprocal radio-frequency wireless channels have been dealt with by different researchers, the literature concerning extraction of shared secret keys from reciprocal optical wireless channels in atmospheric turbulence is rather scarce. Recently, Minet *et al*. [19] studied the ‘cooperation’ between separated optical transceivers caused by atmospheric turbulence and thereby perceived the possibility of generating secret keys by measuring the stochastically changing optical signals. Drake *et al*. [16] reported an experimental demonstration of shared secret key extraction from optical phase fluctuations in a wireless channel caused by atmospheric turbulence simulated in the laboratory. In principle, distortion caused by turbulent eddies at different locations and free-space diffraction produce an intricate impact on the instantaneous phase fluctuations of a received wave. The said work lacks rigorous theoretical formulations, which manifest that correlation between the optical phase fluctuations in the two counter-propagating waves can be perfectly maintained even for practical long propagation paths in the atmosphere. In contrast to the work presented in [16], we will treat fluctuations in received light power contained in a special spatial mode, rather than the differential optical phase delays, as a source of common randomness to generate shared secret keys. This will become evident later, and thereupon the phase extraction challenges mentioned in [16] no longer exist in the scheme considered by us. Wang *et al*. [17] have theoretically investigated the rate of shared secret key generation from an air-to-ground single-spatial-mode optical wireless channel by treating the received signal fading as a source of common randomness; the Gamma-Gamma distribution is assumed for the received signal fading in [17] and the channel reciprocity is guaranteed by making use of the single-spatial-mode transmission and reception, which has not been justified until very recently [19, 20]. Although the Gamma-Gamma distribution has been widely employed for describing the statistical behavior of turbulence-induced signal fading observed by power-in-the-bucket (PIB) receivers, it is not necessarily appropriate for characterizing that observed by single-spatial-mode receivers due to the existence of coherent spatial-mode coupling, which will become apparent later.

In this work, attention is primarily paid to theoretical exploration of distinctive aspects of shared secret key generation from common randomness produced by signal fading in reciprocal turbulent optical wireless channels. We begin in Sec. 2 by presenting a general structure of optical wireless channels that can preserve perfect fading reciprocity in atmospheric turbulence, thereby developing a novel concept of common-transverse-spatial-mode coupling (CTSMC). If we distinguish between the reciprocal optical wireless channel and its radio-frequency counterpart from the perspective of sources of common randomness, one fundamental difference between the two cases exists in the aspect of the statistical behavior of signal fading. Consequently, we model the probability distribution of stochastic signal fading in CTSMC-based turbulent optical wireless channels in Sec. 3. Successful secret key generation from reciprocal wireless channels builds on the spatial-decorrelation nature of stochastic signal fading [9–14]; i.e., the signal fadings observed by a legitimate party and an eavesdropper are required to decorrelate rapidly as the separation distance from the former to the latter increases. So far, the spatial-decorrelation nature of signal fading in radio-frequency wireless channels operated in indoor environments has been examined by several researchers [9, 15, 21]. However, the underlying generation mechanism of stochastic signal fading for turbulent optical wireless channels is completely different from that for multipath radio-frequency wireless channels. We treat the problem of spatial decorrelation quantitatively in Sec. 4. To explore the information theoretic limit on the rate of secret key generation from CTSMC-based turbulent optical wireless channels, we develop, in Sec. 5, theoretical expressions for the secret key capacity in terms of our probability distribution model for stochastic signal fading and evaluate the secret key capacity under various channel conditions. Finally, the conclusion is presented in Sec. 6.

## 2. General structure of optical channels preserving perfect fading reciprocity in atmospheric turbulence

Counter-propagation of optical waves through a common channel in the presence of atmospheric turbulence has been addressed by several researchers [19, 20, 22–28]. If instantaneous fluctuations in collected optical signals of two simultaneously counter-propagating waves at their respective receiving planes are always the same, fading reciprocity is thought to hold true and hence two legitimate parties at the two ends of the channel can observe correlated signal fadings. Fading reciprocity is required to treat turbulence-induced signal fading in a bidirectional optical wireless channel as a source of common randomness accessible to both Alice and Bob. The nature of optical wireless channels in atmospheric turbulence has been investigated by numerous authors (see, e.g., [22] and references therein). Even though the point-source point-receiver (PSPR) reciprocity for optical-wave propagation in atmospheric turbulence has been proved for a long time [23–25], it has been found that fading reciprocity of an optical wireless channel in atmospheric turbulence with transceivers of nonzero sizes is perfectly preserved only under special conditions [19, 20]. At this point, we first intend to depict a general structure of bidirectional optical wireless channels in atmospheric turbulence that can theoretically guarantee perfect fading reciprocity in a general enough way, and subsequently present an elaboration of this structure from a novel perspective of spatial mode coupling.

Figure 1 shows a schematic diagram of a bidirectional optical wireless channel with each terminal simultaneously transmitting and receiving wave fields in a common transverse spatial mode; more specifically, Alice’s terminal located at *z* = 0 transmits and receives wave fields in the mode Ψ* _{A}* (

**r**

*), while Bob’s terminal located at*

_{A}*z*=

*L*transmits and receives wave fields in the mode Ψ

*(*

_{B}**r**

*);*

_{B}**r**

*and*

_{A}**r**

*are two-dimensional points in the planes at*

_{B}*z*= 0 and

*z*=

*L*, respectively. Without loss of generality, we assume both ${\iint}_{A}{\mathrm{\Psi}}_{A}\left({\mathbf{r}}_{A}\right){\mathrm{\Psi}}_{A}^{*}\left({\mathbf{r}}_{A}\right){\mathrm{d}}^{2}{\mathbf{r}}_{A}=1$ and ${\iint}_{B}{\mathrm{\Psi}}_{B}\left({\mathbf{r}}_{B}\right){\mathrm{\Psi}}_{B}^{*}\left({\mathbf{r}}_{B}\right){\mathrm{d}}^{2}{\mathbf{r}}_{B}=1$, where the asterisk represents the complex conjugate, the double integrals with subscripts ‘

*A*’ and ‘

*B*’ mean that the integrations are performed over the entire transverse planes at

*z*= 0 and

*z*=

*L*, respectively. Moreover, for the sake of convenience, we assume Alice’s and Bob’s terminals operate with an identical wavelength for the moment. We will discuss the effect of wavelength mismatch on the fading reciprocity at the end of this section. The correlation time of turbulence state variation is much longer than the delay of light propagation in most practical cases. Hence, it can be assumed that the turbulence is frozen during the period of light propagation from a transmitting plane to a receiving plane. By use of the extended Huygens-Fresnel principle [22-24], the incident wave on the plane at

*z*= 0 originating from Bob’s terminal, which radiates an optical field in the mode Ψ

*(*

_{B}**r**

*), can be expressed by*

_{B}*G*(

_{BA}**r**

*,*

_{B}**r**

*) represents the Green’s function for the path from point*

_{A}**r**

*to point*

_{B}**r**

*associated with a single turbulence state, and*

_{A}*P*,

_{t}*is the total light power transmitted by Bob’s terminal. The Green’s function appearing in Eq. (1) takes the mathematical form like that depicted in [23]. The light power of*

_{B}*U*(

_{BA}**r**

*) contained in the spatial mode Ψ*

_{A}*(*

_{A}**r**

*) for a certain instantaneous turbulence state is*

_{A}Notice that, *P _{r}*,

*is indeed the light power collected by Alice’s terminal of the propagated wave field originating from Bob’s terminal. Similarly, the light power collected by Bob’s terminal of the propagated wave field originating from Alice’s terminal for a certain instantaneous turbulence state takes the form*

_{A}*G*(

_{AB}**r**

*,*

_{A}**r**

*) represents the Green’s function for the path from point*

_{B}**r**

*to point*

_{A}**r**

*, and*

_{B}*P*,

_{t}*is the total light power transmitted by Alice’s terminal.*

_{A}The instantaneous signal fading observed by Alice and Bob are characterized by *ζ _{A}* =

*P*,

_{r}*/*

_{A}*P*,

_{t}*= |*

_{B}*c*|

_{BA}^{2}and

*ζ*=

_{B}*P*,

_{r}*/*

_{B}*P*,

_{t}*= |*

_{A}*c*|

_{AB}^{2}, respectively. It has been shown that

*G*(

_{BA}**r**

*,*

_{B}**r**

*) =*

_{A}*G*(

_{AB}**r**

*,*

_{A}**r**

*) even if there exists atmospheric turbulence along a propagation path [23, 24]. With this in mind, by comparing Eqs. (3) and (5), it is straightforward to find that*

_{B}*ζ*≡

_{A}*ζ*when both Ψ

_{B}*(*

_{A}**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) are real valued, regardless of the turbulence; on the other hand, if either one of Ψ*

_{B}*(*

_{A}**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) is complex valued, it is evident that*

_{B}*ζ*≡

_{A}*ζ*in the case of Ψ

_{B}*(*

_{A}**r**) ≡ Ψ

*(*

_{B}**r**) with

**r**denoting a two-dimensional point in a plane perpendicular to the propagation axis, irrespective of the turbulence. We emphasize here that the equality

*ζ*≡

_{A}*ζ*means perfect fading reciprocity.

_{B}To accommodate the preceding reasoning for the fading reciprocity to a practical optical wireless channel with its two transceivers consisting of several optical elements, we regard Ψ* _{A}* (

**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) as two transverse spatial modes defined in the common entrance/exit pupil plane of the terminals at*

_{B}*z*= 0 and

*z*=

*L*, respectively. By assuming that the turbulence inside a terminal is negligible, for Alice’s terminal with given Ψ

*(*

_{A}**r**

*), the corresponding spatial modes at the actual receiving and transmitting planes therein can be formulated by*

_{A}**r**

*to point ${\tilde{\mathbf{r}}}_{A}$ and that from point*

_{A}**r**

*to point ${\widehat{\mathbf{r}}}_{A}$. Here, we implicitly assume that ABCD ray matrices [22, 29] are used to describe propagation paths along which several optical elements exist, and both the actual receiving and transmitting planes are perpendicular to the nominal propagation axis. In a similar way, for Bob’s terminal with given Ψ*

_{A}*(*

_{B}**r**

*), the corresponding spatial modes at the actual receiving and transmitting planes therein take the forms*

_{B}**r**

*to point ${\tilde{\mathbf{r}}}_{B}$ and that from point*

_{B}**r**

*to point ${\widehat{\mathbf{r}}}_{B}$.*

_{B}To make the reciprocal-channel structure shown by Fig. 1 more concrete, below we present reported examples that can satisfy the requirements mentioned above. Minet *et al*. [19] and Shapiro and Puryear [20] have shown that a bidirectional optical wireless channel with two transceivers using a diplexer to form a common entrance/exit pupil and employing single-mode fibers to function as Ψ* _{A}* (

**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) can remain perfect fading reciprocity in the presence of atmospheric turbulence. Moreover, Yura [30] has developed an expression for the signal power of optically coherent detection of a propagated light beam, which essentially has the same form as Eqs. (2) and (4); consequently, besides single-mode fibers, the roles of Ψ*

_{B}*(*

_{A}**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) can also, in principle, be played by the wave functions of the source and local oscillators in a bidirectional optical wireless channel using coherent detection.*

_{B}Notice that, the quantity *c _{BA}* in Eq. (2) can be alternatively interpreted as a coupling coefficient between the two spatial modes Ψ

*(*

_{A}**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) by use of an optical system characterized by*

_{B}*G*(

_{BA}**r**

*,*

_{B}**r**

*) Similar proposition is applicable to*

_{A}*c*in Eq. (4). According to this understanding of

_{AB}*c*and

_{BA}*c*, it is apparent that the optical wireless channel depicted by Fig. 1 should be viewed as a coherent spatial-mode coupling system. Because Ψ

_{AB}*(*

_{A}**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) are common transverse spatial modes employed in wave-field transmission and reception at Alice’s and Bob’s terminals, respectively, for description convenience, here we designate the channel shown by Fig. 1 as CTSMC-based optical wireless channel. Indeed, it is the instantaneous coupling coefficient between Ψ*

_{B}*(*

_{A}**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) rather than the instantaneous light flux entering an entrance pupil that is important for analysis of signal fading of the channel. This is an essential difference between the optical wireless channel using CTSMC and that employing a PIB receiver to act as an incoherent optical reception system. This difference is the fundamental reason for the partial fading reciprocity of the latter case which has been addressed in [27, 28].*

_{B}Strictly speaking, the propagation behavior of optical waves in atmospheric turbulence has wavelength dependence, implying that wavelength mismatch between the optical fields propagating in two opposite directions may degrade the fading reciprocity. However, as illustrated by numerical simulations in [19], the impact of a small wavelength difference between the two optical fields propagating in two opposite directions on the fading reciprocity is substantially negligible. Hence, for CTSMC-based bidirectional optical wireless channels operated at subtly different wavelengths in the two opposite directions, perfect fading reciprocity is approximately remained.

## 3. Statistical behavior of signal fading in CTSMC-based optical wireless channels

Because we treat instantaneous signal fading in a CTSMC-based bidirectional optical wireless channel as a source of common randomness accessible to both Alice and Bob, the probability distribution model for the signal fading is of paramount significance to characterization of the statistical nature of the source. Until now, many probability density functions for turbulence-induced light-irradiance or light-flux fluctuations with different applicability have been proposed; among these are the log-normal distribution, *K* distribution, Beckman distribution, gamma-gamma distribution, etc [22]. However, they are not necessarily applicable to the signal fading in a CTSMC-based turbulent optical wireless channel. The underlying reason for this is: as opposed to incoherent optical reception systems in which the light irradiance or light flux is of primary concern, instantaneous signal fading in a CTSMC-based optical wireless channel is essentially determined by the squared magnitude of the coupling coefficient between Ψ* _{A}*(

**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) by use of an optical system related to a certain turbulence state. Although various aspects of light-irradiance and light-flux fluctuations caused by atmospheric turbulence have already been studied, turbulence-induced variation behavior of the coupling coefficient between Ψ*

_{B}*(*

_{A}**r**

*) and Ψ*

_{A}*(*

_{B}**r**

*) is still not well explored. Chandrasekaran and Shapiro [31] studied the multiple-spatial-mode multiplexed optical wireless channel impaired by atmospheric turbulence, where the spatial-mode detection conceptually seems like the aforesaid CTSMC and the power-transfer eigenvalues resemble*

_{B}*ζ*and

_{A}*ζ*mentioned in Sec. 2; however, the authors did not deal with the probability distribution of the instantaneous power-transfer eigenvalues. Furthermore, the spatial-mode detection considered by Chandrasekaran and Shapiro [31] is distinguished from that in this work by the following fact: the reception spatial mode in [31] is determined by the wave-field function of the corresponding input spatial mode propagating in vacuum from the transmitting plane to the receiving plane, whereas the reception spatial mode in this work is actually the common transverse spatial mode used by Alice’s and Bob’s terminals. Belmonte [32] suggested the gamma distribution model for the received return-signal fading in a coherent lidar degraded by atmospheric turbulence, where the local-oscillator wave field referred to the pupil plane is postulated to be uniform. It should be pointed out that the probability density functions of all the distribution models mentioned above theoretically have a support of the random fluctuation specified by the range [0, ∞), i.e., the set of all nonnegative real numbers. On the other hand, it is apparent that

_{B}*ζ*and

_{A}*ζ*are nonnegative real numbers inevitably not larger than 1 because they are equal to the squared magnitude of the coupling coefficient between two normalized spatial modes. This fact implies that those probability distributions with a support of all nonnegative real numbers are not very appropriate for statistical description of instantaneous signal fading in a CTSMC-based optical wireless channel; accordingly, we need to find a probability distribution that, on the one hand, has a support with nonnegative and finite lower- and upper bounds and, on the other hand, is proper for describing the statistical behavior of stochastic signal fading.

_{B}We note that much of exiting work dealing with the probability distribution of received optical signals disturbed by atmospheric turbulence follows a procedure that specific distribution formulations are firstly assumed and the corresponding applicability is subsequently examined based on simulation or experimental data. Among various special families of probability distributions, the Johnson *S _{B}* distribution [33] has a support given by the range [0, 1]. To proceed further, we postulate that the Johnson

*S*distribution is suitable for description of the statistical behavior of turbulence-induced stochastic signal fading in a CTSMC-based optical wireless channel. Moreover, for simplicity, thereinafter we only consider the propagation from Bob’s terminal to Alice’ terminal; in fact, by use of the fading reciprocity, it is evident that the probability distribution model for the propagation from Alice’ terminal to Bob’s terminal is the same as the one for the propagation from Bob’s terminal to Alice’s terminal. The Johnson

_{B}*S*probability density function takes the form [33]

_{B}*δ*and

*γ*are two free parameters;

*δ*> 0 and 0 ≤

*ζ*≤ 1. When

_{A}*ζ*obeys the Johnson

_{A}*S*distribution, the mean value and variance of

_{B}*ζ*are and respectively, with

_{A}*n*

^{th}moment of

*ζ*, where the angle brackets represent an ensemble average. It is very hard to develop simple closed-form expressions for ${\overline{\zeta}}_{A}$ and ${\sigma}_{{\zeta}_{A}}^{2}$ in terms of general turbulence conditions. However, it should be expected that both ${\overline{\zeta}}_{A}$ and ${\sigma}_{{\zeta}_{A}}^{2}$ are a function of turbulence conditions. If ${\overline{\zeta}}_{A}$ and ${\sigma}_{{\zeta}_{A}}^{2}$ are known, theoretically speaking, the two free parameters

_{A}*δ*and

*γ*can be determined by numerically solving the system of equations specified by Eqs. (11) and (12); in other words,

*δ*and

*γ*can be associated with turbulence conditions through the formulae for ${\overline{\zeta}}_{A}$ and ${\sigma}_{{\zeta}_{A}}^{2}$. Notice that, although there are obvious differences between the Johnson

*S*probability density function and the gamma-gamma probability density function, the former indeed resembles the latter in the point that they both have two free parameters.

_{B}To validate the applicability of Eq. (10) to description of the statistical behavior of the turbulence-induced signal fading in a CTSMC-based optical wireless channel, we have performed Monte Carlo simulations of beam wave propagation in atmospheric turbulence under various conditions, and subsequently computed the corresponding signal fading samples based on the simulated realizations of the propagated wave fields. The split-step propagation approach based on multiple random phase screens has been employed in the Monte Carlo simulations. Random phase screens, whose statistics are subject to the Kolmogorov turbulence theory [22], are created by employing the fast-Fourier-transform (FFT) technique, and meanwhile the subharmonic method [34] is utilized to improve the large-scale statistics of the random phase screens. The numerical grid comprises 2048×2048 elements with grid spacing properly determined in accordance with the sampling requirements depicted in [34]. We consider horizontal propagation paths near the earth’s ground along which the refractive-index structure constant is unchanged; consequently, random phase screens are located over a path with uniform spacing. Like many published reports dealing with simulations of beam propagation in atmospheric turbulence, we characterize the turbulence strength by Fried’s spherical-wave atmospheric coherence width ${r}_{c}={(0.16{C}_{n}^{2}{k}^{2}L)}^{-3/5}$[34], where ${C}_{n}^{2}$ is the refractive-index structure constant, and *k* = 2*π*/*λ* with *λ* being the wavelength. As in [35], we confirm that Monte Carlo simulations are properly carried out by comparing the average beam irradiance distribution at a receiving plane computed based on the simulation results with that calculated according to the analytical formulae.

Here, symmetric CTSMC-based optical wireless channels are considered; in other words, we think that Ψ* _{A}* (

**r**) ≡ Ψ

*(*

_{B}**r**). A symmetric optical wireless channel that is applicable to many system constructions of practical utility can be readily established by use of two transceivers with an identical configuration. Furthermore, we assume that Ψ

*(*

_{A}**r**) and Ψ

*(*

_{B}**r**) are a collimated Gaussian transverse spatial mode defined by

*w*

_{0}represents the mode radius at which the field amplitude decreases to

*e*

^{−1}of that at the center position (i.e., at

**r**=

**0**). We will call

*w*

_{0}the

*e*

^{−1}mode radius later. If single-mode fibers are used to function as Ψ

*(*

_{A}**r**) and Ψ

*(*

_{B}**r**), which are indeed the so-called backpropagated modes of the fibers, the above assumption does not impose severe restrictions on the applicability of our treatments because the backpropagated modes of the fibers can be closely approximated by a Gaussian function for most practical cases [36]. Real optical transceivers usually have a circular pupil of finite extent, which may truncate Ψ

*(*

_{A}**r**) and Ψ

*(*

_{B}**r**); to eliminate this problem, below we postulate that the circular pupil has a large enough radius, thus resulting in a negligible truncation of Ψ

*(*

_{A}**r**) and Ψ

*(*

_{B}**r**). In addition, to render our treatments as general as possible, we choose the parameters of both Gaussian transverse spatial modes and atmospheric turbulence for Monte Carlo simulations in terms of the mode-radius and coherence-width Fresnel parameters defined here by ${q}_{w}={w}_{0}^{2}/{q}_{F}^{2}$ and ${q}_{c}={r}_{c}^{2}/{q}_{F}^{2}$, respectively, with

*q*= (

_{F}*L*/

*k*)

^{1/2}. In fact, similar nondimensional parameters have been employed in analysis of turbulence-induced OAM-mode scrambling [37]. We conjecture that the two free parameters

*δ*and

*γ*may be well determined by use of only

*q*and

_{w}*q*, at least in an approximate sense; this will be confirmed later by simulation results. The mode-radius Fresnel parameter

_{c}*q*can be used to classify beam propagation as three different cases. The first case is

_{w}*q*≪ 1, implying that the receiving plane lies in the far-field region of a transmitted Gaussian beam. The second case is

_{w}*q*≫1, meaning that the receiver is located in the near-field region of a transmitted Gaussian beam. The third case is

_{w}*q*~ 1, i.e., the transition from the first one to the second one.

_{w}In the Monte Carlo simulations, *q _{w}* takes the three typical values of 0.2, 2 and 20. The Rayleigh range for a collimated Gaussian beam is defined by ${Z}_{R}=0.5k{w}_{0}^{2}$[22]. Hence, the case of

*q*= 2 means that the receiving plane is separated from the transmitting plane by a distance of the Rayleigh range. It is apparent that

_{w}*q*= 0.2 and 20 correspond to the situations where the receiving plane is located at a distance of 10 and 0.1 times the Rayleigh range, respectively, from the transmitting plane. Notice that, the Rayleigh range acts as a dividing point between the near-and far-field regions of a collimated Gaussian beam [22]; this underlies our choice of the said three typical values for

_{w}*q*. Moreover,

_{w}*q*takes the values of 10

_{c}^{−0.4}, 1, 10

^{0.4}, 10

^{0.8}, 10

^{1.2}, 10

^{1.6}and 10

^{2}, respectively, in performing the Monte Carlo simulations; these values embrace the cases of weak, moderate and strong turbulence conditions. We let

*λ*be a fixed value of 800 nm and

*L*take the value of 2 km or 10 km. Note that, when the wavelength

*λ*and propagation distance

*L*are fixed, different

*q*and

_{w}*q*correspond to different

_{c}*w*

_{0}and

*r*, respectively. For each simulated random realization of the propagated beam field at the receiving plane, i.e,

_{c}*U*(

_{BA}**r**

*), the light power*

_{A}*P*,

_{r}*collected by Alice is computed by applying a numerical technique to evaluation of the integration in the first step of Eq. (2). For each combination of*

_{A}*q*and

_{w}*q*, we simulated 6000 random realizations of

_{c}*U*(

_{BA}**r**

*) for both*

_{A}*L*= 2 km and

*L*= 10 km, and subsequently computed the corresponding samples of

*P*,

_{r}*and then those of*

_{A}*ζ*, based on which the probability density of

_{A}*ζ*is determined by using a method similar to that stated in [38]. However, unlike the treatment in [38], we do not apply the natural logarithm transformation to

_{A}*ζ*because the Johnson

_{A}*S*probability density function given by Eq. (10) has the constraint that

_{B}*ζ*is bounded between 0 and 1.

_{A}Figure 2 exemplify the probability density functions of the instantaneous signal fading *ζ _{A}* with different

*q*and

_{c}*L*in the far-field, transition, and near-field cases, respectively, where the plus-marks, x-marks, asterisks, triangles, circles and squares denote values determined according to the numerical simulation results, and the dotted and solid curves represent the fit of a Johnson

*SB*probability density function to the values obtained from the numerical simulation results. The Rytov variance [22] is often employed in the literature as a measure of turbulence strength, which is related to

*q*by ${\sigma}_{l}^{2}=\left(1.23/0.16\right){q}_{c}^{-5/6}$; notice that, in Fig. 2,

_{c}*q*= 10

_{c}^{−0.4}, 10

^{0.8}and 10

^{2}correspond to ${\sigma}_{l}^{2}\approx 16.56,1.66\phantom{\rule{0.4em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}0.77$, respectively, which, roughly speaking, represent the strong, moderate and weak turbulence conditions, respectively. It is observed from Fig. 2 that the Johnson

*S*probability distribution basically agrees well with the probability densities estimated according to the numerical simulation results, even though the fitted curves may deviate somewhat from the estimated probability densities within certain sub-ranges of

_{B}*ζ*where the corresponding probability densities are far smaller than the peak value of ${p}_{{\zeta}_{A}}\left({\zeta}_{A}\right)$. Indeed, the same behavior holds for the cases in which

_{A}*q*takes the values of 1, 10

_{c}^{0.4}, 10

^{1.2}and 10

^{1.6}, respectively; nonetheless, the corresponding graphical results are omitted to save space. Hence, it is reasonable, at least in an approximate sense, to use the Johnson

*S*probability distribution to describe the statistical behavior of the instantaneous signal fading

_{B}*ζ*. Furthermore, it is seen from Fig. 2 that, with the same

_{A}*q*and

_{c}*q*, the probability densities yielded from the numerical simulation results for

_{w}*L*= 10 km and

*L*= 2 km show similar variation behavior in terms of

*ζ*, and the same holds for the fitted curves. This finding illustrates that the aforementioned conjecture that

_{A}*δ*and

*γ*may be determined completely by

*q*and

_{w}*q*, at least roughly speaking, is believable.

_{c}It is found from Fig. 2 that the shape of ${p}_{{\zeta}_{A}}\left({\zeta}_{A}\right)$ depends strongly on both *q _{w}* and

*q*. In the far-field case, viz.

_{c}*q*= 0.2, free-space diffraction causes considerable spreading of the transmitted beam in the mode Ψ

_{w}*(*

_{B}**r**

*) even in vacuum; that is, the transverse size of the beam after arriving at*

_{B}*z*= 0 becomes much larger than the transverse size of the reception spatial mode Ψ

*(*

_{A}**r**

*), resulting in a significant coupling loss between the two modes Ψ*

_{A}*(*

_{B}**r**

*) and Ψ*

_{B}*(*

_{A}**r**

*); accordingly, for any turbulence condition, most occurrences of the signal fading*

_{A}*ζ*happen within a range of nonnegative real numbers whose upper bound is much smaller than 1. On the other hand, with

_{A}*q*= 10

_{c}^{0.8}or 10

^{2}, the possibility that occurrences of the signal fading

*ζ*happen at a value near 1 in the case of

_{A}*q*= 2 or 20 is clearly greater than that in the case of

_{w}*q*= 0.2; this phenomenon should be attributed to the fact that the aforesaid coupling loss induced by free-space diffraction in the case of

_{w}*q*= 2 or 20 is considerably lower than the one in the case of

_{w}*q*= 0.2. When turbulence turns very strong (e.g.,

_{w}*q*= 10

_{c}^{−0.4}), turbulence-caused beam field distortions completely dominate the coupling loss between the two modes Ψ

*(*

_{B}**r**

*) and Ψ*

_{B}*(*

_{A}**r**

*), and ${p}_{{\zeta}_{A}}\left({\zeta}_{A}\right)$ with all*

_{A}*q*shows a tendency to achieve its peak at a value of

_{w}*ζ*closely approaching 0 and to decrease drastically with increasing

_{A}*ζ*. Note that,

_{A}*ζ*= 0.0385, 0.8 and 0.998 for

_{A}*q*= 0.2, 2 and 20, respectively, when there is no atmospheric turbulence along the propagation path. By examination of Fig. 2, it is evident that, with given

_{w}*q*and

_{w}*L*,

*ζ*is not upper-bounded by its value in vacuum.

_{A}It is straightforward to find that the shape of the curves corresponding to *q _{c}* = 10

^{2}in Fig. 2(a) seems like a log-normal probability distribution, which is widely used to describe irradiance fluctuations under weak-turbulence conditions, whereas the shape of the curves associated with

*q*= 10

_{c}^{2}in Figs. 2(b) and 2(c) is actually not similar to the commonly used probability distribution for irradiance fluctuations (e.g., log-normal or gamma-gamma distribution). Physical explanations of this phenomenon are as follows. For interpretation purposes, we first define the ratio of the

*e*

^{−1}mode radius

*w*

_{0}to coherence width

*r*as

_{c}*ε*=

*w*

_{0}/

*r*= (

_{c}*q*/

_{w}*q*)

_{c}^{1/2}. Beware that we have assumed Ψ

*(*

_{A}**r**) ≡ Ψ

*(*

_{B}**r**). By recalling Eqs. (2), (3) and (14), one finds that the spatial mode ${\mathrm{\Psi}}_{A}^{*}\left({\mathbf{r}}_{A}\right)$ therein acts as a Gaussian weighting function and the turbulence-induced perturbations of the incident wave field

*U*(

_{BA}**r**

*) with |*

_{A}**r**

*| ≳*

_{A}*w*

_{0}have a negligible impact on the evaluation of

*ζ*. If

_{A}*ε*becomes small enough, there approximately exists only one coherent speckle patch within the circular region |

**r**

*| ≲*

_{A}*w*

_{0}; under this condition, amplitude fluctuations associated with this patch will dominate the variations of

*ζ*and phase fluctuations associated with this patch play a trivial role; hence,

_{A}*ζ*will show the statistical behavior similar to that of irradiance fluctuations. In contrast, when

_{A}*ε*becomes relatively large, e.g., the cases in Figs. 2(b) and 2(c), there may exist two or more independent coherent speckle patches within the region |

**r**

*| ≲*

_{A}*w*

_{0}; under this condition, phase fluctuations related to these patches begin to play an important role in determining the variations of

*ζ*and hence to make the probability distribution of

_{A}*ζ*deviate from that of irradiance fluctuations. In fact, even for

_{A}*q*= 0.2, with decreasing

_{w}*q*,

_{c}*ε*may increase to a value large enough that there are several independent coherent speckle patches within the region |

**r**

*| ≲*

_{A}*w*

_{0}. The above analysis manifests from another point of view that existing probability distribution models for irradiance fluctuations are not completely appropriate for description of the statistical behavior of

*ζ*. The fundamental reason for this fact is that the CTSMC-based optical wireless channel should be regarded as a coherent system which does not directly measure the light irradiance or light flux.

_{A}As a further comment, it should be pointed out that, like the Johnson *S _{B}* distribution, the Beta distribution [39] also has a support given by the range [0, 1], implying that the Beta distribution possesses one of the properties required to appropriately describe the signal fading in CTSMC-based turbulent optical wireless channels. Indeed, we have tried to fit the Beta distribution to our numerical simulation results. However, it has been found that the Johnson

*S*distribution shows a better fit with the numerical simulation results than the Beta distribution.

_{B}## 4. Spatial decorrelation between signal fadings detected by two contiguous reception spatial modes positioned abreast

There exists a requirement for the shared secret key generation under consideration in the aspect of security; specifically, it is required that only Alice and Bob are aware of the shared secret keys extracted from the random signal fading in a wireless channel, which are unknown to an eavesdropper that will be called Eve later. However, for wireless channels, it is common to think that an eavesdropper can observe a version of signal fading from its own spatial position. We note that, in studies of shared secret key generation from radio-frequency wireless channels, it is often assumed that the signal fadings observed by the two legitimate communicating parties are statistically uncorrelated with those observed by Eve, who is separated from Alice and Bob by a distance at least on the order of a half-wavelength [1, 10, 12–14]. Ultrawideband radio signals of frequencies ranging from 2~8 GHz have a half-wavelength within the range of 1.87 ~ 7.5 cm. Experimental measurement results with respect to spatial dependence of ultrawideband radio signals in a home environment revealed that the spatial correlation length lies between 5.08 cm and 15.24 cm [21]. In practical scenarios, generally speaking, Eve cannot be located at a position extremely close to Alice or Bob without exposure of her eavesdropping action. Accordingly, it is acceptable to postulate that the signal fadings observed by Eve are independent of those detected by Alice and Bob for radio-frequency cases. In this section, we investigate the security feature of shared secret key generation from CTSMC-based optical wireless channels through atmospheric turbulence.

For optical wireless channels, an eavesdropper situated in close proximity to a legitimate communicating party may collect a fraction of the light power contained in a propagated optical beam if the beam spot size at the receiving plane is far greater than the entrance pupil size of the legitimate transceiver [40,41]; for an optical wireless channel constructed by use of two identical CTSMC transceivers, this situation may happen when a beam spreads significantly during propagation; in fact, both free-space diffraction and atmospheric turbulence can cause beam spreading. The issue as to secure information transmission through optical wireless channels has been theoretically addressed in [40, 42]. Focusing on an eavesdropping setup in which Eve, located in the spreading region of a propagated beam, collects the optical wave outside a legitimate receiver’s entrance pupil, Endo *et al*. [41] reported an experimental study on secure information transmission through an optical wireless channel in the near-ground atmosphere based on a wiretap channel model whose security is assured by wiretap channel coding. Unlike the wiretap channel coding scheme, to which turbulence-induced signal fading is detrimental, atmospheric turbulence is the origin of beneficial randomness utilized in the shared secret key generation under consideration.

For quantitatively understanding to what extent an eavesdropper may compromise the secrecy of shared keys extracted from a CTSMC-based turbulent optical wireless channel, it is necessary to examine the variations in the statistical correlation between signal fadings observed by Eve and the legitimate terminals with a changing separation distance from Eve to her legitimate counterparts. Notice that, although the imaginable eavesdropping scenario where Eve uses a beam splitter to intercept an optical wireless channel has been mentioned in the relevant literature [40], this type of suspicious action may be readily detected by using surveillance cameras for a line-of-sight channel; to put it differently, intercepting a beam’s main lobe in concealment is difficult in the case of a line-of-sight optical wireless channel. Accordingly, we do not consider this situation in what follows and only concentrate our attention on the case that an eavesdropper stays away from the legitimate communicating parties and collects the beam wave outside the entrance pupil of the legitimate receivers. Theoretically speaking, Eve can be located at any transverse plane along a propagation path if we do not take physical implementation issues into account. However, intuitively, the signal fadings observed by an eavesdropper lying in a transverse plane far from a legitimate terminal will inevitably be statistically independent of those detected by this legitimate terminal owing to the combined effects of free-space diffraction and turbulence-induced disturbance existing between the eavesdropper and legitimate terminal. It is noted that a CTSMC-based bidirectional optical wireless channel simultaneously operates in the two opposite directions. For this reason, without loss of generality, we assume that an eavesdropper is located in the proximity of Alice. Furthermore, to circumvent too complicated geometrical consideration, here we confine our attention to the worst eavesdropping case in which the entrance pupils of Eve’s and Alice’s receivers are coplanar and Eve employs the same receiver as that of Alice. With the above consideration, the signal fading observed by Eve can be expressed by

*P*,

_{r}*is the light power collected by Eve, Ψ*

_{E}*(*

_{E}**r**

*) = Ψ*

_{A}*(*

_{A}**r**

*−*

_{A}**r**

*) denotes Eve’s reception spatial mode obtained by performing a translation of Alice’s reception spatial mode Ψ*

_{E}*(*

_{A}**r**

*), and*

_{A}**r**

*stands for the center position of Eve’s reception spatial mode. Following the same approach as that depicted in Sec. 3, we can first compute the value of*

_{E}*P*,

_{r}*for each simulated random realization of the propagated beam field, and thereafter obtain a series of random samples of*

_{E}*ζ*. The correlation coefficient between the signal fadings measured by Alice and Eve is

_{E}*ζ*. It is straightforward to conjecture that

_{E}*µ*should depend on the separation distance

*d*between the center positions of Eve’s and Alice’s reception spatial modes, and greater

*d*tends to result in

*µ*with smaller magnitude. Indeed, this has been verified by calculation of

*µ*with different

*d*based on our numerical simulation results, albeit we do not show it graphically for saving space. Notice that,

*d*is actually equal to the magnitude of

**r**

*for the geometry considered above because Alice’s reception spatial mode is centered on the*

_{E}*z*-axis. To quantitatively analyze the dependence of the correlation coefficient

*µ*between

*ζ*and

_{A}*ζ*on the separation distance

_{E}*d*, below we define the spatial correlation distance

*ρ*

_{0}as the separation distance

*d*at which

*µ*falls to

*e*

^{−2}. The signal fadings observed by Alice and Eve can be roughly considered statistically uncorrelated if the separation distance

*d*is larger than

*ρ*

_{0}. As stated before, Eve can collect a nonnegligible fraction of the propagated beam wave outside Alice’s entrance pupil without exposure of her existence only when the beam spot size at the receiving plane becomes much larger than the size of Alice’s entrance pupil; by recalling Eq. (14) and the structure of CTSMC-based optical wireless channels, one can find that, in the near-field cases (e.g.,

*q*= 20) where free-space diffraction causes little beam spreading, the instantaneous beam spot size at the receiving plane may not satisfy the said condition. Consequently, generally speaking, it is the far-field cases that are really relevant to eavesdropping on shared secret keys extracted from CTSMC-based turbulent optical wireless channels. Hence, in what follows, we concentrate our attention on the far-field cases.

_{w}Figure 3 shows the spatial correlation distance scaled by 2*w*_{0} as a function of the base-10 logarithm of *q _{c}* with different

*L*obtained according to the simulations expounded in Sec. 3. One finds from Fig. 3 that the scaled spatial correlation distance enlarges with increasing

*q*, implying that a lower turbulence-strength level may induce a larger spatial correlation distance. Intuitively, weaker turbulence means weaker randomness, hence causing a larger spatial correlation distance. In the area of secret key generation from radio-frequency wireless channels, experimental measurement [15] demonstrates that the statistical correlation between channel responses detected by two spatially separated devices relies on clutter in the surroundings. Indeed, the role of the said clutter in the channel-response decorrelation with increasing spatial distance is conceptually similar to that of turbulence-induced refractive-index disturbance in the spatial decorrelation between signal fadings observed by two contiguous reception spatial modes discussed in this section. Notice that

_{c}*q*= 10

_{c}^{2}corresponds to ${\sigma}_{l}^{2}\approx 0.17$, which is generally attributed to weak turbulence. We can deduce from Fig. 3 that the signal fadings observed by Alice and Eve can be considered roughly uncorrelated even for

*q*= 10

_{c}^{2}if

*d*is approximately larger than 2.6 times the

*e*

^{−1}mode diameter (i.e., 2

*w*

_{0}). Hence, for

*e*

^{−1}mode diameters on the order of several centimeters, the minimum separation distance between the center positions of Eve’s and Alice’s reception spatial modes, at which the signal fadings observed by Eve and Alice almost decorrelate, is also on the order of several centimeters even though the turbulence is relatively weak; in other words, the minimum value of the angle

*α*shown in Fig. 1 that can assure the said decorrelation of signal fadings is on the order of several microradians with

*L*being several kilometers. Note that, in the case of

*q*= 0.2, for a wavelength of 800 nm, 2

_{w}*w*

_{0}is equal to 1.43 cm and 3.19 cm with

*L*= 2 km and 10 km, respectively. Physically speaking,

*µ*can achieve an important value only when the region occupied by the entrance pupils of both Eve and Alice is always fully covered by one coherent speckle patch of the incident optical wave. For numerous cases of practical interest to us, Fried’s atmospheric coherence width is actually on the order of several centimeters [22]. For CTSMC-based optical wireless channels, from a practical implementation perspective, generally speaking, it is difficult for Eve to conceal herself from Alice if Eve is located at a position separated from Alice by a distance of only several centimeters. As a result, analogous to the common treatment in the literature as to secret key generation from radio-frequency wireless channels [1, 10–14], it is reasonable to postulate in scenarios of practical interest that Eve and Alice observe two independent versions of signal fading in the context of secret key generation from CTSMC-based turbulent optical wireless channels.

We emphasize here that the security of shared secret key generation treated in this work is fundamentally determined by the aforementioned spatial correlation distance. By comparing the simulation results for the spatial correlation distance in our case with the measurement results in the case of ultrawideband radio signals reported by [21], we can infer that shared secret key generation from signal fading in a CTSMC-based turbulent optical wireless channel basically has the same security level as that from signal fading in a multipath radio-frequency wireless channel. The latter is currently under intensive research in the area of physical-layer security. Incidentally, it is seen from Fig. 3 that, with the same *q _{w}*, the fitted curve corresponding to

*L*= 10 km basically follows the trend of that corresponding to

*L*= 2 km even though there are certain differences between them. This phenomenon also partly justifies the decision made in Sec. 3 to choose numerical simulation parameters in terms of

*q*and

_{w}*q*.

_{c}## 5. Information theoretic limit on secret key rate

Sharing secret keys between two parties by use of common randomness has been initiated by Maurer [6] and Ahlswede and Csiszár [7]. The highest achievable secret key rate is called the secret key capacity. Two kinds of models for extracting shared secret keys from common randomness, i.e., the source- and channel-type models, have been formulated in [7], wherein some fundamental bounds on the secret key capacity have also been found. In this section, we treat the signal fading in a CTSMC-based turbulent optical wireless channel as a common-randomness source simultaneously observed by Alice and Bob. We consider that a beam wave field in the mode Ψ* _{A}* (

**r**

*) with constant transmitted power is propagated from the pupil plane of Alice’s terminal to that of Bob’s terminal, and meanwhile another beam wave field in the mode Ψ*

_{A}*(*

_{B}**r**

*) with constant transmitted power is propagated from the pupil plane of Bob’s terminal to that of Alice’s terminal; Alice’s and Bob’s terminals measure the signal fading of the channel simultaneously, and hence get correlated observations of the channel signal fading (i.e., the source of common randomness); the two beam wave fields have a subtle wavelength difference whose impact on the fading reciprocity of the channel is negligible as mentioned previously. Further, it is postulated that Alice’s and Bob’s terminals are equipped with identical CTSMC transceivers; an eavesdropper can collect the beam wave outside a legitimate receiver’s entrance pupil, but cannot intercept the beam’s main lobe and meanwhile is unable to approach a legitimate party in such close propinquity that their separation distance is on the order of several centimeters. Below, we build our theoretical development on the said source-type model. When a public channel, by which Alice and Bob can communicate without a rate constraint, is available, with the assumption that the signal fading observed by Eve is statistically independent of that detected by Alice and Bob, in accordance with [6, 7], the secret key capacity*

_{B}*C*is equal to the mutual information between the signal fadings measured by Alice and Bob. As stated in [13], quantization required in practice always diminishes the available common information between Alice and Bob; it, however, gradually approaches the unquantized version of mutual information as the quantization resolution becomes more and more precise. Accordingly, the unquantized version of mutual information indeed acts as a tight upper bound of the achievable key rate if there is a public channel with enough capacity available. Here, our primary intention is to examine how the two nondimensional channel parameters

_{k}*q*and

_{w}*q*impact the secret key capacity. As a result, thereinafter we confine our attention to the unquantized version of mutual information.

_{c}To formulate the mutual information in our case specifically, we need to develop the probability density function for the signal fadings measured by Alice and Bob. In practice, optoelectronic detectors are used to measure the instantaneous signal fading of an optical wireless channel. The instantaneous signal fading measured at Alice’s and Bob’s terminals can be expressed by

respectively, where*η*denotes the photodiode responsivity,

*n*and

_{A}*n*are two random variables used to characterize the role of the detector noise at Alice’s and Bob’s terminals, respectively, in measurement of the signal fading. Here, we assume

_{B}*n*and

_{A}*n*are independent zero-mean Gaussian random variables with variances ${\sigma}_{n,A}^{2}$, and ${\sigma}_{n,B}^{2}$, respectively. For reciprocal turbulent optical wireless channels, ${\widehat{\zeta}}_{A}$ and ${\widehat{\zeta}}_{B}$ are equivalent to two noisy observations of a common-randomness source.

_{B}As stated before, by considering the source-type model for shared secret key extraction from common randomness, the secret key capacity *C _{k}* is determined by the mutual information $I({\widehat{\zeta}}_{A};{\widehat{\zeta}}_{B})$ between ${\widehat{\zeta}}_{A}$ and ${\widehat{\zeta}}_{B}$, which is defined by [43]

*v*’ appearing in the above expressions can be specified as ‘

*A*’ or ‘

*B*’; this notation will be used in the following without explicit expositions. According to the definition of differential entropy [43], one finds

*ηζ*is statistically independent of

_{v}*n*, the convolution of the probability density functions for

_{v}*ηζ*and

_{v}*n*leads to the probability density function for ${\widehat{\zeta}}_{v}$ given by

_{v}By introducing Eq. (10) into Eq. (21) and letting ${\zeta}_{v}=t{\overline{\zeta}}_{v}$, one finds

*ζ*; hence,

_{v}*ϑ*can be regarded as the average electrical signal-to-noise ratio (SNR) at the two legitimate terminals. It is really difficult to evaluate the integration of Eq. (22) in a simple closed form. Numerical techniques are needed to calculate this integration. Recalling Eqs. (17) and (18) and keeping in mind that

_{v}*ζ*≡

_{A}*ζ*due to the fading reciprocity, one can formulate the joint probability density function for the joint random variable $({\widehat{\zeta}}_{A},{\widehat{\zeta}}_{B})$ as follows:

_{B}For clarification on Eq. (23), we emphasize here that ${\overline{\zeta}}_{v}$ is equivalent to ${\overline{\zeta}}_{A}$ or ${\overline{\zeta}}_{B}$, and ${\overline{\zeta}}_{A}\equiv {\overline{\zeta}}_{B}$ due to *ζ _{A}* ≡

*ζ*. As could have been expected, it is easy to verify that

_{B}In accordance with the definition of the joint differential entropy [43], it follows that

With Eqs. (19), (20) and (25) in hand, the secret key capacity *C _{k}* can be determined by employing numerical techniques. It can be proved that letting

*η*= 1 in calculation of

*C*with any given

_{k}*ϑ*and

_{A}*ϑ*actually does not cause any loss of generality. Note that we cannot assign an arbitrary value to ${\overline{\zeta}}_{v}$ involved in Eqs. (22) and (23) when evaluating the secret key capacity; i.e., ${\overline{\zeta}}_{v}$ cannot be regarded as an independent degree of freedom. Indeed, ${\overline{\zeta}}_{v}$ is dependent on

_{B}*δ*and

*γ*. Incidentally, Eqs. (17) and (18) actually imply ${\widehat{\zeta}}_{A}={\widehat{\zeta}}_{B}+{z}_{AB}$ with

*z*=

_{AB}*n*−

_{A}*n*, which can lead us to $h({\widehat{\zeta}}_{A}|{\widehat{\zeta}}_{B})=h({z}_{AB}|{\widehat{\zeta}}_{B})$. It is well known that the identity

_{B}*h*(

*X*|

*Y*) =

*h*(

*X*) holds true if the random variables and

*X*and

*Y*are independent. However, this identity cannot be used to simplify $h({z}_{AB}|{\widehat{\zeta}}_{B})$ to a tractable form because

*z*is not completely independent of ${\widehat{\zeta}}_{B}$ even though the term

_{AB}*ηζ*is assumed to have no dependence on both

_{B}*n*and

_{A}*n*. As a consequence, there is a mistake in the third step of Eq. (19) in [17].

_{B}To understand how the channel parameters *q _{w}* and

*q*affect shared secret key generation, below we evaluate the secret key capacity with varying average electrical SNR under different channel conditions. The values for the two parameters

_{c}*δ*and

*γ*corresponding to different conditions are obtained by fitting a Johnson

*S*distribution to the signal fading samples computed according to the relevant simulated beam-wave field realizations (see Sec. 3); here, only the simulation data associated with

_{B}*L*= 10 km is used because, as pointed out in Sec. 3,

*δ*and

*γ*basically only depend on the two nondimensional parameters

*q*and

_{w}*q*. The secret key capacity is plotted as a function of the base-10 logarithm of the average electrical SNR in Fig. 4 for the far-field, transition and near-field cases, respectively, with various turbulence-strength levels. In plotting Fig. 4, we postulate that the variances of the random variables

_{c}*n*and

_{A}*n*in Eqs. (17) and (18) are identical, implying that

_{B}*ϑ*≡

_{A}*ϑ*. One can find from Fig. 4 that the secret key capacity enlarges with an increasing average electrical SNR for all turbulence-strength levels. It is observed from Fig. 4 that the turbulence strength plays an important role in determining the secret key capacity. For instance, with the same average electrical SNR, the secret key capacity associated with

_{B}*q*= 10

_{c}^{2}is smaller than that associated with

*q*= 10

_{c}^{−0.4}. However, an increase in the average electrical SNR may make the secret key capacity associated with a large

*q*become higher than that associated with a small

_{c}*q*. In fact, one can see from Fig. 4 that some curves therein may intersect at a certain value of the average electrical SNR. Hence, the turbulence strength and average electrical SNR produce a combined effect on the secret key capacity. By comparing Figs. 4(a), 4(b) and 4(c), it is found that the curves therein lie closer to each other when

_{c}*q*becomes larger. Specifically, there exists a more noticeable difference in the variation behaviors of the secret key capacity with increasing average electrical SNR for various

_{w}*q*in the far-field case than in the near-field one; the secret key capacities in Fig. 4(c) for different

_{c}*q*achieve values close to each other at log

_{c}_{10}(

*ϑ*) = 2; nevertheless, the secret key capacity in Fig. 4(a) for weak turbulence (e.g.,

*q*= 10

_{c}^{2}) is obviously different from that for strong turbulence (e.g.,

*q*= 10

_{c}^{−0.4}). The findings above imply that the change in the turbulence strength has a more prominent impact on the secret key capacity in the far-field case than in the near-field one.

Notice that, *C _{k}* defined by Eq. (19) is in units of bits per channel measurement. If the channel measurement is carried out every

*τ*seconds (i.e., the measurement bandwidth is 1/

_{m}*τ*Hz), the secret key rate is theoretically equal to

_{m}*C*/

_{k}*τ*bits per second under the condition that consecutive channel measurements are statistically uncorrelated. When the frozen-turbulence hypothesis is applicable to characterization of temporal fluctuations in optical waves propagating through the atmosphere, the correlation time of the temporal fluctuations can be estimated according to the wind velocity transverse to the propagation path. The wind velocity may have location dependence in practice [44], which complicates precise calculation of the correlation time. In the area of atmospheric optics, the Greenwood time constant

_{m}*τ*

_{0}[22], which, for the case of constant wind velocity, can be expressed by

*τ*

_{0}= 0.32

*r c*/

*v*

_{⊥}with

*v*

_{⊥}denoting the transverse wind velocity, is often used to quantify the temporal duration over which atmospheric turbulence can be regarded as being hardly changed. Typically,

*τ*

_{0}is on the order of milliseconds for practical atmospheric turbulence [22], implying that the correlation time generally takes a value on the order of milliseconds. With the above statements in mind, we can estimate the secret key rate in units of bits per second for practical scenarios based on the results presented by Fig. 4. For instance, Fig. 4(a) shows that, with log

_{10}(

*ϑ*) = 2,

*C*≈ 1.90, 2.16, 2.28, 2.05, 1.58, 1.07 and 0.68 bits per channel measurement for

_{k}*q*= 10

_{c}^{−0.4}, 10

^{0}, 10

^{0.4}, 10

^{0.8}, 10

^{1.2}, 10

^{1.6}and 10

^{2}, respectively. If we assume the correlation time is approximately 2 ms, meaning that uncorrelated consecutive channel measurements can be achieved by carrying out not more than 500 measurements per second, the maximum secret key rate with log

_{10}(

*ϑ*) = 2 is approximately 950, 1080, 1140, 1025, 790, 535 and 340 bits per second for

*q*= 10

_{c}^{−0.4}, 10

^{0}, 10

^{0.4}, 10

^{0.8}, 10

^{1.2}, 10

^{1.6}and 10

^{2}, respectively. Figure 16 in [16] illustrates that the secret bit generation therein can achieve a rate of roughly 60, 38, 24 and 16 bits per second for ${C}_{n}^{2}L\approx 1.23\times {10}^{-10}$, 5.72×10

^{−11}, 2.65 ×10

^{−11}and 1.23 × 10

^{−11}m

^{1/3}, respectively. In [16], two lasers with wavelengths of 1562.2 nm and 1560.6 nm, respectively, are used to emit two optical waves propagating in opposite directions. We note that, with

*L*= 10 km and

*λ*= (1562.2 + 1560.6)/2 nm = 1561.4 nm, ${C}_{n}^{2}L\approx 1.23\times {10}^{-10}$, 5.72 × 10

^{−11}, 2.65 × 10

^{−11}and 1.23×10

^{−11}

*m*

^{1/3}correspond to

*q*= 10

_{c}^{−0.4}, 10

^{0}, 10

^{0.4}and 10

^{0.8}, respectively. At a first glance, there are prominent differences between our results of secret key rate in units of bits per second and those in [16]. However, we point out that it is indeed difficult to make a rigorous comparison of our results with those in [16], because there is no description of the average electrical SNR related to the secret-key-rate data presented in [16]. Here, we emphasize once again that a lower average electrical SNR makes the mutual information between ${\widehat{\zeta}}_{A}$ and ${\widehat{\zeta}}_{B}$ become smaller and thus the secret key rate has a strong dependence on the average electrical SNR.

As a final comment, in practice, several parallel CTSMC-based optical wireless channels can be arranged to constitute a system resembling the multiple-input-multiple-output (MIMO) channel. It has been recognized that if both different transmitters and different receivers of a MIMO channel in atmospheric turbulence are separated from each other by a sufficient large distance, the spatial correlation between its constituent sub-channels becomes negligible. Hence, the MIMO-based spatial diversity may provide the potential that can be exploited to further increase the secret key capacity. Although a description of how to optimally construct a CTSMC-based MIMO optical wireless channel is beyond the scope of this work, it deserves careful study in the future. In addition, an anonymous reviewer made us perceive a recently published article [45] related to secrecy communication through non-line-of-sight (NLOS) ultra-violet (UV) optical wireless scattering channels; the NLOS scattering nature therein makes it easy for an eavesdropper to collect the optical signal. Unlike the line-of-sight turbulent optical wireless channels, it was stated in [46] that, for single-input-multiple-output (SIMO) NLOS UV optical wireless scattering channels, there may exist relatively strong correlations between the constituent sub-channels even with two receivers separated by a distance of several meters.

## 6. Conclusion

We have theoretically treated the distinctive aspects of extracting shared secret keys from common randomness arising out of turbulence-induced distortions in propagating optical signals. Firstly, we have developed, in a general manner, the concept of CTSMC-based optical wireless channels that can guarantee perfect fading reciprocity in atmospheric turbulence, which is a prerequisite for secret key extraction. Secondly, by performing Monte Carlo numerical simulations, we have demonstrated that the Johnson *S _{B}* probability distribution is appropriate for statistical description of turbulence-induced signal fading in an optical wireless channel formed by use of two identical CTSMC transceivers; additionally, we have quantitatively characterized the nature of spatial decorrelation between signal fadings observed by a legitimate party and an eavesdropper, and found that the legitimate party and eavesdropper only observe independent signal fadings in multitudinous scenarios of practical interest. Finally, we have formulated the information theoretic capacity of secret key generation from a CTSMC-based optical wireless channel in atmospheric turbulence and quantified it under various conditions, discovering that both the turbulence strength and average electrical SNR play an appreciable role in determining the secret key capacity. The obtained results are helpful in understanding the physical nature of inherent common randomness in CTSMC-based optical wireless channels through atmospheric turbulence and can benefit the practical implementation of relevant secret key generation.

We note that shared secret key generation from CTSMC-based optical wireless channels in atmospheric turbulence can be attributed to the broad area of secret-key agreement in the sense of information theoretic security. QKD is an information-theoretic-security approach to secret-key agreement; however, it relies on costly quantum devices and may be impaired by atmospheric turbulence if photonic quantum states thereof are propagated through a wireless channel. We recognize that shared secret key generation from CTSMC-based turbulent optical wireless channels is not unconditionally secure in a rigorous sense, because it actually assumes that an eavesdropper cannot intercept a beam’s main lobe in concealment and cannot be situated at a position very close to legitimate parties without exposure of herself or himself. In contrast, QKD is, in principle, a strictly unconditionally secure approach to secret-key agreement. Nevertheless, imperfections in practical implementation of QKD can cause security loopholes too. If the tradeoff between cost and security level is taken into account, extraction of shared secret keys from CTSMC-based turbulent optical wireless channels can be regarded as a technically viable alternative to QKD in certain situations. Additionally, in contrast to distributing shared secret keys between two parties by the public key method, the scheme dealt with in this work has information theoretic security rather than computational security. One of future efforts deserves to be concentrated on MIMO-based spatial diversity that may provide a potential approach to increasing the rate of secret key generation further.

## Funding

National Natural Science Foundation of China (61475025, 61775022); Development Program of Science and Technology of Jilin Province of China (20180519012JH).

## Acknowledgments

The authors are very grateful to the reviewers for valuable comments.

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