Simulating thick atmospheric turbulence in the lab with application to orbital angular momentum communication

We describe a procedure by which a long ($\gtrsim 1\,\mathrm{km}$) optical path through atmospheric turbulence can be experimentally simulated in a controlled fashion and scaled down to distances easily accessible in a laboratory setting. This procedure is then used to simulate a 1-km-long free-space communication link in which information is encoded in orbital angular momentum (OAM) spatial modes. We also demonstrate that standard adaptive optics methods can be used to mitigate many of the effects of thick atmospheric turbulence.


Introduction
It is well known that beams of light that contain a phase vortex of the form ψ(r, φ) ∝ exp (iℓφ), where ℓ is an integer, carry orbital angular momentum. A single photon prepared in such a state will have an angular momentum equal to ℓh in addition to any momentum carried by the polarization [1]. In recent years there has been a good deal of excitement based on encoding information onto such OAM beams of light [2][3][4]. Such spatial mode encoding schemes allow for scaling of the bit rate as each pulse may contain more than 2 possible symbols. This scheme also provides an enhancement to the security when used for quantum key distribution [5,6].
Such an encoding scheme generally implies communication over a free-space channel. A known limitation of free-space communication schemes is degradation of the signal due to atmospheric turbulence, and much work has been done to study how this turbulence affects communication systems that utilize spatial mode encoding [7][8][9][10][11][12][13]. However, much of this work focuses on 'weak' turbulence, where the effect is fully described by a random thin phase screen in the receiver aperture. The turbulence obeys 'Kolmogorov statistics' characterized by a single turbulence strength parameter, D/r 0 , where D is the aperture or beam size and r 0 is a correlation length scale known as Fried's parameter [14,15].
In more realistic situations, such as communication along a long horizontal path through which the turbulence is continuously distributed, one will see amplitude fluctuations (or scintillation) that impacts one's ability to communicate in addition to the problems caused by pure phase fluctuations. Some of this degradation can still be compensated for by phase-only adaptive optics (AO) correcting for low spatial order aberrations for a horizontal path of a few kilometers or more [16]. However, for moderate scintillation one will begin to see intensity nulls that are associated with phase vortices [17,18]. These phase vortices or branch points are known to interfere with the performance of AO systems [19], and there is a breakdown in the performance for horizontal paths around 5 km due to this effect [20]. For communication systems that communicate using OAM this presents an additional problem as these phase vortices are precisely the source of the encoding, and randomly generated vortices introduce errors in such a scheme.

Simulating thick turbulence in the laboratory
To more closely examine the effects that a thick horizontal turbulent path might have on OAM-based communication while still allowing information transfer and AO correction, we chose to consider a 1 km path with aperture sizes of the sender and receiver of D = 18.16 cm, at a wavelength of λ = 785 nm corresponding to a Fresnel number of N f = D 2 /(λL) = 42. In addition, the refractive index structure parameter, which quantifies the strength of the refractive index fluctuations due to turbulence at any point, was chosen to be C 2 n = 1.8 × 10 −14 m −2/3 along the path, which represents a typical value for a horizontal path near ground level.
To allow for such a system to be realized in a laboratory setting one must find a way to incorporate the turbulence into the channel, as well as properly scale the system down to more manageable length scales. This was done by finding an equivalent path containing 2 thin phase screens (section 2.1) that can be represented in the lab with spatial light modulators (SLMs). Scaling rules that are invariant under Fresnel propagation are detailed in section 2.2 and the experimental setup is discussed in 2.3.

Two phase screen model
In order to find an equivalent optical path that accurately represents the horizontal turbulent channel, it is necessary that the simulated path faithfully reproduce all the relevant statistical properties of the actual channel. The relevant parameters for OAM communication are the overall strength of the turbulence, the scintillation or amplitude fluctuations, the power fluctuation of the beam due to beam wander and aperture clipping, and the random generation of branch points. We can represent these 4 parameters by using thin Kolmogorov phase screens each with its own value of r 0 . The values of r 0 for each screen, as well as each screen's position along the path, give us four independent parameters that are tuned until the 2 screen path reproduces the same 4 parameters of the full horizontal channel.
The first step in designing a two-phase-screen path to simulate horizontal turbulence is to quantify the 4 statistical parameters of a thick path. The overall strength of turbulence is represented by D/r 0 , just as in the thin-phase-turbulence model. Assuming a constant value of the refractive index structure parameter along the path, C 2 n (z) ≡ C 2 n , then this parameter is given by the expression [14] r 0 = 2.91 6.88 which for our system is found to be r 0 = 24.4 mm. A well known measure for scintillation is the log-amplitude variance [21], given by the expression which for our system is found to be σ 2 χ = 0.197. The normalized power over the aperture Σ of area A Σ , defined by P = 1 A Σ Σ dr exp (2χ(r)), where χ(r) is the log-amplitude of the field, is used to numerically find the normalized power variance in the aperture: where C χ (r, is the log-amplitude covariance function [22]. This value is computed numerically for our system giving σ 2 P = 7.04×10 −3 . Finally we require the average density of branch points created by the turbulent path, ρ BP , to be preserved. This density is found by Monte-Carlo simulation to be ρ BP = 500/m 2 . The second step in designing the two-phase-screen path is to find the values for the position and r 0 for each screen that will give the same values of D/r 0 , σ χ , σ P , and ρ BP of the thick path. One starts with an initial guess for r 01 (i.e. r 0 for screen one), and then solves for the value of r 02 that will give the correct value for D/r 0 . Next, one randomly picks a value for the position of the second screen, z 2 , and finds z 1 such that one gets the correct value for σ χ . z 2 is then varied (along with z 1 to maintain σ χ ) to set the correct value for σ P . Given this solution, ρ BP is computed by Monte-Carlo simulation. If at this point one gets the correct ρ BP a solution has been found, otherwise one starts over with a new choice for r 01 . Using this procedure we found we could simulate our 1 km path with the parameters r 01 = 3.926 cm, r 02 = 3.503 cm, z 1 = 171.7 m and z 2 = 1.538 m (measured from the sender's aperture). As an independent test of this solution, a beam propagation simulation was performed to compare the thick turbulent path represented by 10 random Kolmogorov phase screens with the analogous 2 screen solution. In each simulation a different random realization of turbulence was made and the Strehl ratio, defined as the ratio of the peak intensity to ideal peak intensity of a spot at a focal plane of the receiver, was computed. By repeating this many times, a probability distribution for the Strehl ratio was found and the results are shown in Fig. 1. As can be seen in the plot, the Strehl ratios of the 2 screen and the 'continuous,' 10 screen paths show very good agreement with each other. This demonstrates that the 2 screen model not only reproduces the correct mean values for the statistical parameters of interest (by construction), but can also be expected to give similar distributions of possible measurement outcomes.

Fresnel scaling
The second thing that must be done to effectively simulate a turbulent path in a laboratory setting is to scale the optical paths down to more manageable lengths. In order to ensure that the scaled path still represents the desired physical path, the Fresnel equation must remain invariant under the scaling. Fresnel propagation from one plane to another a distance z away as shown in Fig. 2 is given by: Now if we scale the coordinates using r ′ = α r r, ρ ′ = α ρ ρ, and z ′ = α z z then the above propagation equation becomes: Since we require the Fresnel number to remain constant, α z = α ρ α r . Then we can rewrite equation 5 as: where f ρ = z ′ / 1 − αr αρ and f r = z ′ / 1 − αρ αr . From Eq. 6 we see that the horizontal path between planes A and B can be scaled down (to within a scaling and phase constant) simply by adding a lens with focal length f r at A, propagating a distance z ′ , and then adding another lens with focal length f ρ at B to cancel out the residual quadratic phase.

Experimental setup
A diagram of our experimental setup is presented in Fig. 3 Figure 3. Alice sends a beam prepared in a specific OAM state, ℓ, to Bob. Bob receives the beam after propagation through a channel representing a 1 km turbulence path. The beam is (optionally) corrected using a deformable mirror and sent to a sorter to make a measurement of the of the OAM spectrum of the beam.
using an SLM combined with a 4f system as described in [23,24]. The prepared state is then sent through the simulated 1 km path scaled down as described in section 2.2 to a total length of 1.3 m. The two thin phase screens used to simulate thick turbulence in our setup (section 2.1) were implemented using an SLM in a double pass configuration. In addition, the quadratic phases required for proper scaling of the propagation path were added to the phases on the SLMs. After propagation through the turbulent channel, the beam at Bob's aperture is imaged with a 4f system onto a Thorlabs adaptive optics (AO) kit consisting of a 12 × 12 actuator deformable mirror and a Shack-Hartmann wavefront sensor. After the AO system, the beam is similarly imaged onto the first element (R1) of the OAM sorter. The OAM sorter uses two refractive optical elements (R1 and R2) and a Fourier transforming lens (FT Lens) to spatially separate different OAM modes allowing for the OAM spectrum to be efficiently measured as described in [25].

Experimental results
In order to examine the effects of the turbulent channel on OAM communication, we experimentally measured the OAM spectrum that Bob detects conditioned on what Alice sent. In a perfect channel, if Alice sends OAM mode s, then Bob will measure an OAM spectrum that is simply a Kronecker delta centered at the same mode. However, in an imperfect or turbulent channel, there will be some spreading into neighboring OAM modes to the prepared state. The conditional probability matrix, P (d|s) where d is the detected OAM mode and s is the sent mode, provides a natural expression for this crosstalk induced by the imperfections or turbulence in the channel. P (d|s) is plotted for 3 different scenarios in Fig. 4. Fig. 4a shows P (d|s) when there is no turbulence in the channel, showing only crosstalk due to any misalignment in the system and inherent crosstalk of the sorter [25]. Fig. 4b shows the effects of thick turbulence ensemble averaged over 100 realizations, which act to greatly spread the signal over many neighboring channels. This selective spreading of OAM into neighboring modes rather than randomly into any OAM state is qualitatively similar to what is seen in the thin turbulence regime demonstrated in [9]. For the third case shown in Fig. 4c, adaptive correction was applied to the turbulence with the AO system, allowing much of the signal to be recovered. The phase aberrations induced from each realization of turbulence was sensed and corrected by the AO using the OAM ℓ = 0 mode. Each mode was then sent through the channel and AO system and the OAM spectrum was measured by Bob. This procedure was repeated and averaged over 50 realizations of turbulence. In order to quantify the crosstalk induced by the turbulence as well as the quality of the AO correction, we compute the mutual information between Alice and Bob for all three cases above. The mutual information between Alice and Bob, I (A; B), provides a measure of the channel capacity, as it gives the maximum possible transmission rate that can be extracted by Bob in bits per photon (or classically per detected pulse). The mutual information is given by the expression,

Number of Modes
where N is number of distinct symbols. Fig. 4 qualitatively showed that thick turbulence greatly impacts the quality of the channel. Using Equation 7, we quantify these results by calculating the mutual information as a function of the encoding dimension N. The mutual information for the three cases of no turbulence, thick turbulence, and turbulence with AO correction, is plotted in Fig. 5 as a function of N. One can see that the AO system allows us to cancel roughly half of the loss of channel capacity due to turbulence.  Figure 6. Channel capacity as a function of the spacing between OAM modes used for communication.
Further, since turbulence preferentially scatters power into neighboring OAM modes rather than randomly into all modes, one can increase the channel capacity by choosing to use a less dense set of OAM modes [11]. For instance, rather than encoding every OAM mode, one can encode in every other mode (or third or fourth mode etc.). Changing the encoding is also independent of any AO system one may use, and thus a modified encoding can be used along with AO correction to further enhance the channel capacity. Fig. 6 shows the increase in the mutual information one can obtain for a given number of encoded modes, N. The channel capacity of an ideal 2-bit system is shown for reference. It is worth noting that the use of spatial mode encoding shows an improvement over such a system with very moderate resources (i.e. 3 modes with a channel spacing > 4).

Conclusions
In this work we have demonstrated how one could experimentally simulate a thick horizontal turbulent path as part of a free-space communication channel. This model