Antenna gain of actively compensated free-space optical communication systems under strong turbulence conditions

Juan C. Juarez; David M. Brown; David W. Young

doi:10.1364/OE.22.012551

1. Introduction

Free-space optical communications (FSOC) has been researched as a potential alternative technology to conventional radio frequency (RF) and microwave communications in response to the growing needs for substantially increased data throughput, relief from spectrum planning, and enhanced link security. These basic advantages are a result of the much shorter optical wavelength as compared to RF and microwave systems that allows for significantly increased transmission directionality or antenna gain.

One of the main challenges in fielding FSOC systems are the deleterious effects introduced by atmospheric turbulence [1]. These include: (1) beam broadening beyond natural diffraction effects as the beam propagates from the transmitter to the receiver, (2) spot blurring and broadening in the focal plane at the receiving aperture with increased turbulence, and (3) intensity fades known as scintillation due to interference effects as a result of the aberrated beam wavefront. All three effects introduce substantial link budget penalties for FSOC systems; the first two degrade antenna gain performance as will be presented here while the third imposes an increased dynamic range requirement on the receiver system [2].

For Gigabit/sec class FSOC systems on mobile platforms, active beam control and compensation is required at the transmitting antenna to maintain beam alignment between the transmitter and receiver, and within the receive antenna to place the incoming signal on the data detector or appropriate fiber for remoting to back-end modem hardware. At a minimum, tip/tilt control is required for fine beam pointing and is typically handled with fast steering mirrors [3,4]. Recent demonstrations [5] have shown that adaptive optics (AO) can be beneficial for FSOC by providing additional wavefront compensation allowing for single mode fiber (SMF) coupled systems that support higher data rates.

For characterizing the optical antenna performance as a function of turbulence conditions, the most commonly used metric is the Strehl ratio [6,7], a parameter borrowed from the Astronomy field [8]. This ratio provides a measure of on-axis intensity relative to a diffraction limited beam. It is most commonly used for characterizing the compensation performance of a receive aperture by evaluating signals at the focal plane, where the data detector or receive fiber would normally be placed. The Strehl ratio metric can also be applied to transmit apertures to characterize pre-compensation performance in the far-field, for example, at the plane of the receive aperture.

Strehl ratio models are then critical for developing link budgets for FSOC systems to evaluate operation under different turbulence conditions and to assess the necessary compensation required to meet desired system performance. Due to their development based on weak turbulence theory [1], however, current Strehl ratio models have been found to poorly predict system performance for actively compensated FSOC systems, especially under strong turbulence conditions. As a result, we have developed an approach for simulating the Strehl ratio of systems with both low-order (tip/tilt) and higher-order (adaptive optics) compensation for evaluation under general turbulence conditions. We compare our results to the published models and assess their ranges of validity as a function of turbulence. We then propose a new Strehl ratio model that accurately predicts system performance independent of compensation level for all regimes of turbulence. Finally, this Strehl ratio model is combined with the traditional antenna gain calculation to provide an antenna gain model for FSOC systems that incorporates turbulence induced degradations.

2. Theoretical background

2.1 Turbulence background

Turbulence is present even in the most stable of atmospheric conditions. At a core level, turbulence is based on temperature variations that create localized changes in the refractive index of air. While these perturbations may be small, the cumulative effects can be significant over long distances. The simplest form of turbulence encountered in the atmosphere is dependent on the temperature lapse rate, dT/dh, which is the decrease in air temperature as a function of increasing altitude. For example as a volume of air ascends in altitude, there is less pressure on the air mass and therefore it is able to expand leading to a decreased amount of internal energy and temperature. The opposite is true for an air mass descending into a higher pressure region, thereby causing its internal energy and temperature to increase. More localized motion of the air or wind, also will give rise to increased turbulence due to mixing of these temperature layers. Furthermore, the friction of one air mass on another will also give rise to increased temperature, thereby compounding the effects of turbulence.

Fried’s parameter, r₀, which for a spherical wave is defined as

r_{o} = {[0.423 k^{2} \int_{0}^{L} C_{n}^{2} (z) {(z / L)}^{5 / 3} d z]}^{- 3 / 5}

is often used to describe the cumulative effects of atmospheric turbulence across the entire spectrum [8]. This single metric provides a measure of the atmospheric coherence length at any given time for a particular site, which is why it is often used to compare the strength of turbulence between different times of the day and areas of the world. The value varies from a few meters in very good seeing conditions in the infrared to a few centimeters in difficult seeing at visible wavelengths. Physically, this length can be defined as the diameter of a circular area over which the root mean square wavefront distortion is equal to 1 radian.

By reducing Eq. (1) we can also conclude that the atmospheric coherence length, r₀, is proportional to λ^6/5. This proportionality establishes a mechanism to scale the atmospheric coherence length measured in the visible, for example, to wavelengths out into the infrared.

2.2 Strehl ratio

The Strehl ratio is commonly used as a single, numeric metric for characterizing the performance of optical systems. For an FSOC receiving aperture, the Strehl ratio is defined as the ratio of on-axis intensity in the focal plane with aberrations vs. that without (e.g. diffraction limited). Performance of the system can then be evaluated versus incoming wavefront degradation which is typically characterized by the Fried parameter [8].

For circular apertures with a uniform intensity input, the Fraunhofer integral has an analytic solution that allows for straightforward calculation of the peak, on-axis intensity at the focal plane given by

I_{o} = P \frac{π D^{2}}{4 λ^{2} f^{2}},

where P is the total power (in watts) entering the aperture, D is the aperture diameter, λ is the wavelength, and f is the focal length of the focusing lens [9]. This result gives the maximum on-axis intensity corresponding to the case where zero wavefront aberrations exist, and the Strehl ratio is equal to one. When wavefront aberrations are introduced, the peak intensity decreases and the Strehl ratio becomes less than one. The Strehl ratio is dependent on the wavefront variance, σ², measured in rad² and defined by

σ^{2} = \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} {[W (r, θ) - \bar{W}]}^{2} r d r d θ,

where W(r,θ) represents the aberrated wavefront and

\bar{W}

is the mean over the aperture. For small wavefront aberrations where σ < λ/2π, the Strehl ratio, S, can be estimated using the Maréchal approximation [1], defined as

S \approx \exp (- σ^{2}),

providing a means of estimating system performance based solely on the wavefront variance. Noll [10] later was able to relate the wavefront variance to the receive aperture size and the atmospheric turbulence strength for an uncompensated system as

σ_{}^{2} = 1.03 (D / r_{o})^{5 / 3} .

For weak fluctuation conditions, Andrews and Phillips (A&P) [11] derived improved estimates for the Strehl ratio based on Noll’s relationship between the wavefront variance and the ratio D/r_o defined as

S = {[1 + σ^{2}]}^{- 1} = [1 + (D / r_{o})^{5 / 3}]^{- 1}, D / r_{o} < 1 .

For a general range of wavefront distortions covering both weak and strong fluctuations, A&P’s generalized Strehl ratio result is

S \approx [1 + (D / r_{o})^{5 / 3}]^{- 6 / 5}, 0 \leq D / r_{o} < \infty .

2.3 Zernike polynomials

Wavefront aberrations are phase deviations from the ideal, perfectly flat incoming wavefront that cause the different portions (or rays) of the wavefront to misfocus manifesting as an increase in the long-exposure focal spot size and in turn a decrease in the power coupled into the receive fiber. The pupil plane aberration can be expressed as a wavefront W(x,y) measured in waves. A generalized pupil function, P(x,y), that captures the effects of both apodization and wavefront distortion can then be generated to give the complex function (in radians)

P (x, y) = P_{A} (x, y) e^{i 2 π W (x, y)},

where P_A is the pupil function.

A convenient mathematical representation of wavefront aberrations can be achieved with Zernike polynomials since they have completeness and orthogonality over a circular aperture. We use the convention of Noll [10] where the polynomials are defined in polar coordinates as

Z {}_{i}{(r, θ)} = {\begin{matrix} \sqrt{2 (n + 1)} R_{n}^{m} (r) G^{m} (θ), m \neq 0 \\ R_{n}^{m} (r), m = 0 \end{matrix},

where Zi represents the polynomials when indexed by a single term, m and n are non-negative integers, and m

\leq

n. The index i is a mode ordering number, where the order of the modes is presented by Noll [10]. The radial,

R_{n}^{m} (r)

, and azimuthal,

G^{m} (θ)

, terms are given by [12]

R_{n}^{m} (r) = \sum_{s = 0}^{(n - m) / 2} \frac{{(- 1)}^{s} (n - s)!}{s! (\frac{n + m}{2} - s)! (\frac{n - m}{2} - s)!} r^{n - 2 s},

G^{m} (θ) = {\begin{matrix} \sin (m θ), i odd \\ \cos (m θ), i even \end{matrix} .

An arbitrary wavefront can then be fully defined by a Zernike series given by

W_{} = \sum_{j = 1}^{\infty} a_{j} Z_{j} (r, θ),

where a_j represents the series coefficients. Therefore, due to the Zernike series orthogonality, an arbitrary wavefront can be decomposed into the Zernike series coefficients [12] with the following relationship

a_{i} = \frac{\int_{0}^{2 π} \int_{0}^{1} W (r, θ) Z_{i} (r, θ) r d r d θ}{\int_{0}^{2 π} \int_{0}^{1} Z_{i}^{2} (r, θ) r d r d θ} .

2.4 Wavefront compensation

When the lowest order aberrations in a wavefront are compensated, Noll [10] was able to show the reduction in the wavefront variance as a function of the number of Zernike modes used. For the first J modes of correction, the correction can be written as a Zernike series given by

W_{C} = \sum_{j = 1}^{J} a_{j} Z_{j} .

When the first J modes of correction are applied, the phase variance becomes

σ_{J}^{2} = \frac{1}{π} \int_{0}^{2 π} \int_{0}^{1} {[W (r, θ) - W_{C}]}^{2} r d r d θ .

Substituting Eq. (14) into Eq. (15) yields

σ_{J}^{2} = 〈 W {(r, θ)}^{2} 〉 - \sum_{j = 2}^{J} 〈 {| a_{j} |}^{2} 〉,

where i begins at 2 because i = 1 is the

\bar{W}

term.

As previously mentioned, for an uncompensated system, the residual wavefront variance, σ₁², is 1.03 (D/r₀)^5/3 whereas for a system with tip/tilt correction, σ₃² = 0.134 (D/r₀)^5/3. For compensation orders greater than 10, Eq. (16) can be approximated by

σ_{J}^{2} \approx 0.2944 J^{- \sqrt{3} / 2} (D / r_{o})^{5 / 3} .

Using these results, Roddier [13] applied the wavefront variance model to adaptive optics systems to account for the number of mirror actuators and type of compensation, generalizing it to

σ_{J}^{2} \approx k (D / r_{o})^{5 / 3} N^{- 5 / 6},

where N is the number of mirror actuators channels and k is a coefficient that depends on the type of deformable mirror used. For segmented mirrors (with piston and tip/tilt correction by each segment), k = 0.335 while for actuators with Gaussian influence functions, k = 0.237. Though Eq. (18) uses Zernike modes and Eq. (19) uses actuators, the two are equivalent when beyond seven modes or actuator channels of compensation [8].

Roddier then approximated the Strehl ratio for compensated systems, S_com, using the Maréchal approximation as

S_{c o m} \approx \exp (- k {(D / r_{o})}^{5 / 3} N^{- 5 / 6}),

valid for S_com ≥ 0.3 (−5.2 dB).

Extending the A&P general Strehl ratio model, Eq. (7), by modifying the wavefront variance term as Roddier did results in

S_{c o m} \approx {[1 + k {(D / r_{o})}^{5 / 3} N^{- 5 / 6}]}^{- 6 / 5}, 0 \leq D / r_{o} < \infty .

A final Strehl ratio model of interest was also presented by A&P [11] based on a ground to space uplink simulation using wave propagation code. This simulation was for the far-field Strehl ratio for a tip/tilt compensated system transmitting a Gaussian beam and defined as a function of W₀/r₀, where W₀ is the Gaussian beam radius. Relating the aperture diameter to the beam diameter as they do, D = 2^3/2 W₀, their Strehl ratio expression becomes

\begin{array}{l} S_{t t} \approx {1 + [0.983 - \frac{0.856}{1 + 0.0071 {(D / r_{o})}^{5 / 3}}] \times {(D / r_{o})}^{5 / 3}}^{- 6 / 5}, \\ 0 \leq D / r_{o} < \infty . \end{array}

2.5 Focal plane transformation

For evaluation of the Strehl ratio for an arbitrary incoming wavefront, it must be transformed to the focal plane. This can be accomplished by modeling the wavefront as it passes through a single focusing lens with a ray-transfer matrix, ABCD [14]. Written in matrix-vector notation the optical system is defined as

(\begin{matrix} y_{2} \\ n_{2} {y^{'}}_{2} \end{matrix}) = (\begin{matrix} A & B \\ C & D \end{matrix}) (\begin{matrix} y_{1} \\ n_{1} {y^{'}}_{1} \end{matrix}),

where y₂, y₂’, and n₂ are the ray height, slope, and refractive index, respectively, after the lens, and y₁, y₁’, and n₁ are the ray height, slope, and refractive index before the lens, respectively. When rays propagate through a spherical lens to its focal plane, the ray-transfer matrix has values: A = 0, B = f, C = -f ⁻¹, and D = 1.

To account for the diffractive effects of the lens on the wavefront, a modified diffraction integral can be employed to simplify computations as compared to the more complex Huygens-Fresnel integral. Lambert [15] shows that the Fresnel diffraction integral can be simplified into a convolution to obtain the transformed optical field

U (A r_{2}) = \frac{1}{i λ B} e^{i π β r_{2}^{2}} [U (r_{1}) \otimes e^{i π α r_{1}^{2}}],

where α is defined as A/(λB) and β as AC/λ. This method allows for rapid numerical computation of the optical field at the focal plane with fast-Fourier transforms.

3. Wavefront simulation

3.1 Turbulence modeling

A common approach for investigating the effects of atmospheric turbulence on propagating laser beams is to use the Split-Step Fourier Method as it is an efficient algorithm for the numerical solution of the non-linear Schrödinger equation. This method principally relies on computing the wave solution in small steps along the propagation direction. Atmospheric turbulence is simulated by employing pseudo-random phase screens with their statistics dictated by the Kolmogorov turbulence spectrum [1]. Beam statistics are then characterized by running hundreds to thousands of independent propagations through the channel.

The approach we used utilizes the Fourier transform for generating the random phase screens as first introduced by McGlamery [16] and described in detail by Schmidt [12]. The turbulence-induced phase aberration presented in a Fourier-integral representation is

ϕ (x, y) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} Ψ (f_{x}, f_{y}) e^{i 2 π (f_{x} x + f_{y} y)} d f_{x} d f {}_{y},

where Ψ(f_x,f_y) is the spatial-frequency-domain representation of the phase. For implementing the simulation into Matlab, the optical phase is written as a Fourier series giving

ϕ (x, y) = \sum_{n = - \infty}^{\infty} \sum_{m = - \infty}^{\infty} c_{n, m} \exp [i 2 π (f_{x_{n}} x + f_{y_{m}} y)],

where c_n,m are the Fourier series coefficients, and

f_{x_{n}}

and

f_{y_{m}}

are the discrete x- and y-directed spatial frequencies. Since atmospherically-induced phase variations are due to numerous independent random inhomegeneities along the beam path, based on the central-limit theorem, the c_n,m coefficients are set to have a Gaussian distribution. Because the Fourier coefficients are complex with zero mean, equal variances, and zero cross-variances [17,18], they obey circular complex Gaussian statistics with zero mean and variance given by

〈 | c_{n, m} |^{2} 〉 = Φ_{ϕ} (f_{x_{n},} f_{y_{m}}) Δ f_{x_{n}} Δ f_{y_{m}} .

In the Matlab simulation, the randn function is used to generate random numbers with zero mean and unit variance with a Gaussian distribution. The appropriate variance is then obtained by multiplying the random numbers by the square root of the variance given in Eq. (26) to give the random draw for the Fourier series coefficients in Eq. (24).

Finally, to better match up to the theoretical structure function, the power content in the low spatial frequencies is compensated via the subharmonic method described by Lane [19] and also detailed by Schmidt [12].

3.2 Wavefront generation and focal plane evaluation

For the simulation, a linearly spaced 512 by 512 point Cartesian grid with each side equal to the aperture diameter was used. For most of the analysis, an aperture diameter of 10 cm was evaluated, thus giving a spatial resolution of about 0.02 cm. Random wavefronts were generated using the subharmonic phase screen approach described in section 3.1. An example of an aberrated wavefront with r₀ = 2.5 cm is presented in Fig. 1(a). Such wavefronts were then decomposed to different Zernike levels to evaluate the possible level of correction. As an example, Fig. 1(b) presents decomposition of the first 3 modes, commonly referred to as tip/tilt (TT). Figure 1(c) presents decomposition of the first 18 modes, including tip/tilt, and is referred to as AO correction. Note that the higher level of wavefront correction is able to represent the aberrated wavefront with better fidelity as opposed to tip/tilt-only decomposition.

Fig. 1 Pupil plane simulations of (a) random aberrated wavefront with r₀ = 2.5 cm; (b) Zernike decomposition of first 3 modes of the aberrated wavefront; (c) Zernike decomposition of first 18 modes of the aberrated wavefront.

Download Full Size | PDF

The pupil plane wavefronts can then be propagated to the focal plane using the method described in Section 2.5 for evaluation of the Strehl ratio. For our analysis, an optical system with an f/# equal to 4.7 designed for operation at a wavelength of 1.55 μm was chosen to model FSOC terminal systems that are SMF coupled. Such a system would produce focal spots matched to the 10.4 μm mode field diameter of standard SMF-28 fiber. Figure 2 presents the focal plane intensities of the wavefronts presented in Fig. 1. The uncompensated wavefront in Fig. 1(a) yields the aberrated focal spot in Fig. 2(a). Applying the phase conjugate of the tip/tilt Zernike decomposition presented in Fig. 1(b) to the aberrated wavefront yields the focal plane intensity distribution presented in Fig. 2(b). Note that this focal spot is nearly identical to the uncompensated spot (Fig. 2(a)) except that it has been steered to the optical axis (i.e. the center of the frame). Figure 2(c) presents the AO corrected focal spot corresponding to applying the phase conjugate of the Zernike decomposition presented in Fig. 1(c). The higher-order correction removes much of the wavefront aberration resulting in a significantly improved focal spot approaching the airy disk of a diffraction limited beam.

Fig. 2 Focal plane intensities for (a) an uncompensated wavefront with r₀ = 2.5 cm; (b) wavefront with tip/tilt compensation, and; (c) a wavefront with the first 18 Zernike modes of compensation.

Download Full Size | PDF

4. Results and analysis

4.1 Strehl ratio

The wavefront generation approach presented in the previous section was utilized for a simulation study to investigate the Strehl ratio for actively compensated systems under varying turbulence conditions. The level of turbulence was based on the D/r_o ratio and varied between 0.1 and 100 by maintaining the aperture size constant and varying r₀. For each turbulence level, 1000 independent aberrated wavefront realizations were generated, decomposed, and evaluated at the focal plane. As previously stated, the Strehl ratio is the most commonly used metric for evaluating the performance of actively compensated systems, and thus it is the metric we use.

As a first step to validate the simulation approach, the Strehl ratio for uncompensated and tip/tilt corrected beams was evaluated. The results for the simulation for both cases along with several of the relevant models discussed in Sections 2.2 and 2.4 are presented in Fig. 3 as a function of D/r₀. For an uncompensated system, our simulation agrees best with A&P’s general turbulence model, Eq. (7), for all D/r₀ conditions. For weak turbulence conditions (D/r₀ ≤ 1), however, their weak turbulence model, Eq. (6), is a better fit. The simulation and Maréchal approximation not surprisingly only agree for cases where D/r₀ << 1. These results show that the random phase screens are in good agreement with theory, indicating that they are adequately representing the wavefront aberrations at the pupil of our optical system.

Fig. 3 Comparison of Strehl ratio theory and simulation results for uncorrected and tip/tilt corrected wavefronts. The solid lines represent theoretical models while solid markers represent simulation results. Maréchal – Maréchal approximation; Uncom. A&P Weak – Uncompensated weak turbulence (Andrews & Phillips), Uncom A&P Gen. – Uncompensated general turbulence (Andrews & Phillips); TT Roddier – Tip/tilt compensation (Roddier); TT A&P Sim – Tip/tilt compensation (Andrews & Phillips Simulation); TT A&P – Tip/tilt compensation (Andrews & Phillips general turbulence); SR Uncom – Simulation Strehl ratio, uncompensated; SR TT – Simulation Strehl ratio, tip/tilt compensated

Download Full Size | PDF

Next, the simulation results for tip/tilt corrected cases were evaluated. These results are also included in Fig. 3 along with Roddier’s model, Eq. (19), A&P’s tip/tilt simulation-based model, Eq. (21), and A&P’s general turbulence model with Roddier’s modified wavefront variance term, Eq. (20). For tip/tilt compensation, our simulation agrees closely with all three models for turbulence conditions where D/r₀ is less than 3. Across all turbulence conditions, our simulation closely tracks A&P’s tip/tilt simulation result. This is interesting as their result was based on the far-field Strehl ratio for a Gaussian beam, thus highlighting the reciprocity of the optical channel [20]. The disagreement with Roddier’s model is not surprising as he notes it is only valid for Strehl ratios greater than 0.3 (−5.2 dB). The difference with A&P’s general turbulence model highlights the fact that compensation gain degrades with increasing levels of turbulence, eventually approaching an uncompensated wavefront. In considering a system implementation, this would be expected as increasing turbulence levels means increased number of speckles at the pupil and focal planes that tip/tilt compensation alone cannot correct.

Simulation results for higher-order correction were evaluated next with compensation levels of 3, 6, 10, 15, 21, 28, and 36 Zernike modes. Sample results for 6 and 21 modes of correction are presented in Fig. 4 along with the corresponding models from Roddier and A&P. As with the tip/tilt case, both Roddier’s and A&P’s model are valid for weak turbulence conditions and up to D/r₀ less than 3. Beyond this point, the same general trend is observed as in the tip/tilt case. Roddier’s model underestimates the Strehl ratio due to the rapid decay of the exponential function he uses while A&P’s model overestimates it due to the constant difference between their compensated and uncompensated models. Also, in these cases it is more obvious that the simulation result tracks Roddier’s model more closely for a broader range of turbulence conditions until it approaches the A&P’s uncompensated model. At this point a transition region exists between the exponential decay in Roddier’s model and the power law decay in A&P’s uncompensated model. This transition region was found to occur for all simulated cases.

Fig. 4 Comparison of Strehl ratio theory and simulation results for 6 and 21 Zernike modes or actuators channels of correction. The solid lines represent theoretical models while the dotted lines represent simulation results.

Download Full Size | PDF

Based on the results of the simulation study, we propose an update to the Strehl ratio models for compensated systems that covers both weak and strong turbulence conditions. The Strehl ratio is then defined as

\begin{array}{l} S_{c o m} \approx [^{1 + σ_{J}^{2} + 0.5 σ_{J}^{4} + 0.167 σ_{J}^{6}] - 6 / 5} + \\ [^{k_{2}^{- 1} + (k_{2} + 0.35) (^{D / r_{o}) 5 / 3}] - 1.3}, 0 \leq D / r_{o} < \infty . \end{array}

where σ_J² = k₂ (D/r₀)^5/3 and k₂ = 0.24*N^-5/6 similar to Roddier’s wavefront variance definition. The first half of this equation captures exponential behavior under weak turbulence conditions with the first three terms of the Taylor series expansion for the exponential function that Roddier uses. The second half is structured like A&P’s uncompensated turbulence model to capture the power law behavior under strong turbulence conditions.

In Fig. 5 we plot the simulation results as a function of D/r₀ for compensation levels of 3, 10, 21, and 36 Zernike modes or actuator channels along with our proposed Strehl ratio model and A&P’s uncompensated general turbulence model for reference. The same parameters for the optical system setup as previously described were used to provide insight for single mode fiber coupled terminals. As can be observed, the proposed model closely matches the simulation results for all cases. The transition from weak to strong turbulence conditions closely tracks for all compensation levels unlike with Roddier’s and A&P’s models as discussed with reference to Figs. 3 and 4. Additionally, the proposed model remains consistent with Roddier’s phase variance term and tracks his model for weak turbulence conditions. Using the Taylor series expansion for the exponential function was found to better fit the simulation data as it provided power law terms that bridge the gap with A&P’s power law, whereas using an exponential function in the first half of Eq. (27) decreased the Strehl ratio too quickly.

Fig. 5 Comparison of Strehl ratio simulation results and proposed model for compensation with 3, 10, 21, and 36 Zernike modes or actuators channels. The solid lines represent theoretical models while the dotted lines represent simulation results.

Download Full Size | PDF

4.2 Antenna gain

The ideal gain of a uniformly illuminated circular aperture relative to an isotropic radiator in dB is given by [21]

G_{a} = 10 \log_{10} {(\frac{π D}{λ})}^{2} (dBi) .

To account for the degradation introduced by atmospheric turbulence, Eq. (29) must be modified to include the Strehl ratio of the aperture as follows

G_{T o t} = 10 \log_{10} [{(\frac{π D}{λ})}^{2} \times S_{c o m}] (dBi) .

where S_com is defined in Eq. (27). Equation (29) then gives an expression for the total antenna gain, G_Tot, of optical antennas as a function of aperture size, compensation level, and atmospheric turbulence level.

As an example of the usefulness of this total gain metric, Fig. 6 presents the total antenna gain at λ = 1.55 μm for different compensation levels as a function of aperture diameter under a turbulence condition of r₀ = 1 cm. These results show that an uncompensated aperture does not provide additional gain with increasing aperture size when its diameter is about twice the length of r₀ (2 cm in this case). With increasing levels of compensation, the total gain initially increases until a maximum at D/r₀ ≈2.4 N ^0.5 [13]. Beyond this point, increases in aperture size lead to reductions in total gain.

Fig. 6 Total antenna gain vs. aperture diameter at λ = 1.55 μm for different compensation levels with turbulence of r₀ = 1 cm.

Download Full Size | PDF

5. Summary

Strehl ratios are critical for developing link budgets for FSOC systems under different turbulence conditions and for evaluating different levels of compensation that may be required for applications of interest. Due to their development based on weak turbulence theory, current Strehl ratio models for actively compensated FSOC terminals have been found to not accurately predict system performance under strong turbulence conditions.

In this paper, we presented an approach for simulating the Strehl ratio of compensated systems with both low-order (tip/tilt) and higher-order (adaptive optics) compensation for evaluation under general turbulence conditions. We compared our results to the published models and assessed their ranges of validity. We then proposed a new Strehl ratio model that better predicts the performance for compensated systems as conditions transition from weak to strong turbulence independent of compensation level. Finally, this Strehl ratio model was incorporated into the general antenna gain equation to provide a total gain model for optical antennas as a function of atmospheric turbulence and compensation level. This total gain metric provides a useful tool for accurately evaluating FSOC systems and link budgets with different compensation architectures and operating under varying turbulence environments.

References and links

1. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

2. J. C. Juarez, D. W. Young, J. E. Sluz, and L. B. Stotts, “High-sensitivity DPSK receiver for high-bandwidth free-space optical communication links,” Opt. Express 19(11), 10789–10796 (2011). [CrossRef] [PubMed]

3. D. S. Grinch, J. A. Cunningham, and D. S. Fisher, “Laser system for cooperative pointing and tracking of moving terminals over long distance,” Proc. SPIE 6238, 623803 (2006). [CrossRef]

4. H. R. Burris, M. S. Ferraro, W. Freeman, P. G. Goetz, R. Mahon, C. I. Moore, J. L. Murphy, J. Overfield, W. S. Rabinovich, W. R. Smith, M. R. Suite, L. M. Thomas, and B. B. Xu, “Tactical network demonstration with free space lasercomm,” Proc. SPIE 7923, 792305 (2011). [CrossRef]

5. L. B. Stotts, L. C. Andrews, P. C. Cherry, J. J. Foshee, P. J. Kolodzy, W. K. McIntire, M. Northcott, R. L. Phillips, H. A. Pike, B. Stadler, and D. W. Young, “Hybrid Optical RF Airborne Communications,” Proc. IEEE 97(6), 1109–1127 (2009). [CrossRef]

6. T. Weyrauch, M. A. Vorontsov, J. Gowens, and T. G. Bifano, “Fiber coupling with adaptive optics for free-space optical communication,” Proc. SPIE 4489, 177–184 (2002). [CrossRef]

7. J. C. Juarez, D. W. Young, R. A. Venkat, D. M. Brown, A. M. Brown, R. L. Oberc, J. E. Sluz, H. A. Pike, and L. B. Stotts, “Analysis of link performance for the FOENEX laser communications system,” Proc. SPIE 8380, 838007 (2012). [CrossRef]

8. R. K. Tyson, Introduction to Adaptive Optics (SPIE, 2000).

9. M. Born and E. Wolf, Principles of Optics (Pergamon, 1975).

10. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66(3), 207–211 (1976). [CrossRef]

11. L. C. Andrews, R. L. Phillips, R. J. Sasiela, and R. R. Parenti, “Strehl ratio and scintillation theory for uplink Gaussian-beam waves: beam wander effects,” Opt. Eng. 45(7), 076001 (2006). [CrossRef]

12. J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB (SPIE, 2010).

13. F. Roddier, “Maximum gain and efficiency of adaptive optics systems,” Publ. Astron. Soc. Pac. 110(749), 837–840 (1998). [CrossRef]

14. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 2007).

15. A. J. Lambert and D. Fraser, “Linear systems approach to simulation of optical diffraction,” Appl. Opt. 37(34), 7933–7939 (1998). [CrossRef] [PubMed]

16. B. L. McGlamery, “Restoration of turbulence-degraded images,” J. Opt. Soc. Am. 57(3), 293–297 (1967). [CrossRef]

17. W. A. Coles, J. P. Filice, R. G. Frehlich, and M. Yadlowsky, “Simulation of wave propagation in three-dimensional random media,” Appl. Opt. 34(12), 2089–2101 (1995). [CrossRef] [PubMed]

18. B. M. Welsh, “A Fourier series based atmospheric phase screen generator for simulating anisoplanatic geometries and temporal evolution,” Proc. SPIE 3125, 327–338 (1997). [CrossRef]

19. R. G. Lane, A. Glindemann, and J. C. Dainty, “Simulation of a Kolmogorov phase screen,” Waves Random Media 2(3), 209–224 (1992). [CrossRef]

20. D. L. Fried and H. T. Yura, “Telescope-performance reciprocity for propagation in a turbulent medium,” J. Opt. Soc. Am. 62(4), 600–602 (1972). [CrossRef]

21. B. J. Klein and J. J. Degnan, “Optical antenna gain. 1: Transmitting antennas,” Appl. Opt. 13(9), 2134–2141 (1974). [CrossRef] [PubMed]

Antenna gain of actively compensated free-space optical communication systems under strong turbulence conditions

Abstract

1. Introduction

2. Theoretical background

2.1 Turbulence background

2.2 Strehl ratio

2.3 Zernike polynomials

2.4 Wavefront compensation

2.5 Focal plane transformation

3. Wavefront simulation

3.1 Turbulence modeling

3.2 Wavefront generation and focal plane evaluation

4. Results and analysis

4.1 Strehl ratio

4.2 Antenna gain

5. Summary

References and links

Cited By

Figures (6)

Equations (29)

Optics Express