Intuitive model for the scintillations of a partially coherent beam

Anatoly Efimov

doi:10.1364/OE.22.032353

1. Introduction

Understanding the key factors influencing the scintillations of a particular beam propagating in turbulent atmosphere is important for proper design of atmospheric optical systems, such as free-space optical communication. Mathematical expressions for the scintillation index (SI) are often complicated, requiring numerical evaluation and defying intuitive understanding. This particularly applies to a class of beams with reduced spatial coherence—partially coherent beams (PCB) [1]. PCBs are often mentioned to have reduced SI as compared to fully coherent ones, such as Gaussian beams (GB), plane or spherical waves. This advantage, in fact, is a function of respective beam sizes, focusing parameters and the coherence radius r_c of the PCB. Thus, an intuitive and simple model for PCB scintillations would be of great value.

Here we argue that the SI of a PCB $σ_{PCB}^{2}$ (rather its ratio to the Rytov variance $σ_{R}^{2}$ ) as a function of propagation distance allows a clear and intuitive understanding. The parameter, which defines this SI ratio is related to the number of Fresnel zones, which fit on the “working aperture”—the area of the source from which the light can reach the detector. The light from these “Fresnel emitters” propagates through essentially independent atmospheric realizations thus leading to averaging of intensity fluctuations and reduction of the SI.

2. Review of the numerical results

We begin by reviewing the well-known dependencies of the SI for the plane wave, spherical wave, GB and PCB as a function of propagation distance. Relevant expressions are given in Appendix A for the Tatarskii turbulence spectrum with inner scale. Rytov variance $σ_{R}^{2}$ is used as a normalization function for all the other SI functions in this work. Such normalization is meaningful and possible since within Rytov theory of smooth perturbations $σ_{R}^{2}$ enters all the formulas as a multiplicative factor. As a result, the index structure constant $C_{n}^{2}$ cancels out from all the expressions.

Figure 1 shows the dependence of the SI ratio on the propagation distance L for plane and spherical waves, GB and PCB beams for turbulence spectra with inner scale l ₀ set to zero and 5 mm. For the no-inner scale turbulence the Rytov variance is exactly the SI of a plane wave so the blue curve is a constant 1. The spherical wave is known to scintillate 2.5 times less than the plane wave so the green curve is a constant 0.4. The GB curve (red) shows the characteristic transition from the plane wave when the detector is close to the transmitter to the spherical wave when the detector is far away with a minimum falling below the spherical wave curve occurring at Fresnel parameter Ω = ka ² /L ≈ 0.7, where a is the beam intensity radius and k = 2π/λ. Clearly the minimum value of the SI ratio for the GB is independent on the beam aperture, only its location shifts to longer distances with increasing a.

Fig. 1 Scintillation index ratio for plane wave (blue), spherical wave (green), Gaussian beam (red) and PCB (black) for turbulence spectra without (left) and with (right) inner scale l ₀ = 5 mm. Beam intensity radius is 3 mm for curves “1”, 3 cm for “2”, and 10 cm for “3”. Coherence radius r_c = 3 mm, except for the dashed black curve, where r_c = 1 mm.

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The behavior of the PCB curves (black) is most interesting. Not surprisingly, when r_c = a (curves “1”) the behavior is very similar to GB. Further increase of a has two effects on the curve: The position of the minimum (termed here “transition” length L_tr) shifts to longer distances, but not as fast as GB, and becomes deeper as well, see curves “2” and “3”. For the case of smaller r_c = 1 mm (dashed curve) the minimum occurs closer to the transmitter and becomes deeper still. An important observation is that for L < L_tr, the SI ratio for the PCB does not depend on beam radius and for L > L_tr it no longer depends on coherence radius.

The effect of inner scale is shown in fig. 1(right). It is well-known that the effect of the inner scale is important for short ranges where parameter $Q_{m} = 35 L / k l_{0}^{2} \leq 1$ . In this region all the curves bend downwards synchronously, so there are no complicated beam-dependent effects present for non-zero inner scale spectra. We observe also that a range of distances and beam parameters can be found where GB scintillates less than a PCB, although for sufficiently large L PCB always shows smaller SI than any other beam. This fact makes PCB useful for long-range communication. Based on our recent experiments in the open atmosphere [2] we suspect that this advantage remains valid even in strong turbulence in the saturation regime, however this regime is outside the scope of the present work.

Beam aperture a and coherence radius r_c are the two important parameters defining the SI curve of the PCB. In Fig. 2(left) we plot several curves for the SI ratio of the PCB for a = 1, 3, 10 cm and r_c = 1, 3, 10 mm as a function of the propagation distance. The two arrows point the opposing trends for increasing beam aperture and increasing coherence radius. We again observe that before the minima the SI ratio does not depend on beam radius and after the minima it does not depend on coherence radius.

Fig. 2 Scintillation index ratio for PCB as a function of propagation distance (left) and normalized distance L/L_tr (right). Beam radius a is 1 cm (red curves), 3 cm (green), and 10 cm (blue), while the coherence radius r_c is set to 1 mm (solid lines), 3 mm (dashed), and 10 mm (dot-dashed).

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These observations allow the formulation of an intuitive model as follows.

3. Intuitive model

A PCB source is most conveniently considered as comprising a finite number of mutually incoherent “Elementary Emitters” (EE) with radius r_c. For the case of flat wavefront (focusing parameter F ₀ = ∞) all EEs radiate in the forward direction with half-divergence angle, or numerical aperture NA = λ/4πr_c, which follows from Eq. (7) in Appendix A. Considering the effects of the turbulence on the divergence negligible as compared to regular diffraction we see that a detector placed close to the transmitter will intercept light only from a small number of EEs close to the center of the aperture, Fig. 3. At a distance L ₁ the detector “sees” only a fraction of the whole PCB aperture (termed here “working aperture”), highlighted by the red bracket on the left. Further away, say at a point L ₂, the detector intercepts light from a larger working aperture, which leads to decreased SI. Point L ₃ marks the transition where the detector “sees” the whole aperture and thus all EEs, which number is simply $N_{EE} = 1 + a^{2} / r_{c}^{2} \approx a^{2} / r_{c}^{2}$ . This is the point – the transition length – where the SI ratio assumes its minimum value. Further distancing the detector no longer leads to increased number of EEs for better signal averaging, but rather the angular size of the source begins to reduce. Obviously, at a large propagation distance the source will look like a point source and so the SI curve asymptotically approaches the spherical wave limit, which is what we observe in Fig. 1.

Fig. 3 Conceptual illustration of the intuitive model: Detectors located at L ₁ and L ₂ receive signals only from a fraction of the PCB aperture, determined by the NA of the “elementary emitters”, while the detector located at the transition length L ₃ = L_tr “sees” all the “elementary emitters”.

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The conceptual interpretation depicted in Fig. 3 offers simple understanding of the dynamics of the transition length as a function of the beam radius a and coherence radius r_c observed in Fig. 2. For a fixd r_c the transition length increases with increasing a because the emission from the EEs on the beam edge reach optical axis further away. On the other hand, for fixed a and increasing r_c the beam divergence decreases and again the edge of the beam aperture is “seen” by the on-axis detector located at a longer distance.

The transition length L_tr can thus be defined simply as a distance at which the NA of the EE is equal to the half-angular size of the source λ/4πr_c = a/L giving

L_{tr} = \frac{4 π r_{c} a}{λ} .

Figure 2(right) plots the same curves as in the left panel as a function of normalized propagation distance L/L_tr. Clearly the location of the minima for all the curves with various a and r_c are now coincident. The inset shows the dependence of the location of the SI ratio minima L_min on L_tr for a large number of curves having various a and r_c (red circles) along with the unit-sloped line L_min = L_tr.

Because it’s the “working aperture”, which contributes to the signal on the detector we specify the Fresnel number as

Ω_{w} = {(\frac{γ_{w} L}{\sqrt{L / k}})}^{2} = k L γ_{w}^{2} = \frac{k a_{w}^{2}}{L},

defined as the squared ratio of the “working aperture” radius to the radius of the Fresnel zone

\sqrt{L / k}

, where a_w is the radius and γ_w is the half-angular size of the “working aperture” equal to the lesser of λ/4πr_c or a/L. Indeed, any effective averaging is possible only if the independent emitters are laterally separated by at least the Fresnel zone [3] so that their radiation propagates through substantially independent atmospheric paths. In other words having more than one EEs per Fresnel zone is ineffective for SI reduction. In fact, it is easy to show that in view of the “working” aperture formulation any collimated PCB will have more than one EE per Fresnel zone.

Figure 4(left) plots the SI ratio at the transition length L = L_tr as a function of the Fresnel parameter, Eq. (2), for a large number of a and r_c values of the PCB (red circles) computed using the Banakh formula, Eq. (8), Appendix A, with zero inner scale and infinite focusing parameter. Note that at the transition length the Fresnel parameter 2Ω_w = a/r_c, which is interestingly equal to $\sqrt{N_{EE}}$ . The significance of the ratio a/r_c has been identified previously [4], however those results disagree with the present analysis. The figure also shows the theoretical asymptotic behavior of the SI ratio

\frac{σ_{PCB}^{2}}{σ_{R}^{2}} = 2.52 {(r_{c} / a)}^{7 / 6}, L = L_{tr}, r_{c} / a ≪ 1

in the limit r_c/a ≪ 1 (blue line), derived in Appendix B. A reasonable fit to data suitable for all a and r_c values (blak curve) is also shown.

Fig. 4 Left: Scintillation index ratio of a collimated PCB (red circles) evaluated at L = L_tr for a large number of parameter pairs a and r_c. The asymptotic (blue) and best fit (black) are also shown. Right: SI ratio for all L ⊂ {1, 10⁵} meters and a large number of parameter pairs a and r_c (colored curves) overlayed with a power-law fit (blue line).

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The Fresnel parameter defined in Eq. (2) is universal not only for L = L_tr, but for all L, Fig. 4(right). Here we see that the data points for the SI ratio plotted as a function of Ω_w occupy a tight curve with the same power −7/6 asymptotic for Ω_w ≫ 1. For small Ω_w the curve splits into two branches corresponding to plane (top branch) and spherical (bottom branch) waves.

These results present the best attempt to parameterize the SI ratio of a PCB with a single dimensionless parameter. Unlike the case of a collimated GB where such parameterization is exact, that is the SI ratio of the collimated GB to the Rytov variance is a function of the Fresnel number only, in the case of a PCB such parameterization is only approximate—the numerical points in Fig. 4(right) do not all reside on a single curve, but are somewhat spread around it. Most importantly the presented model offers an intuitive understanding of the role and interplay of all the parameters in the problem.

4. Conclusion

The above analysis identifies a single dimensionless parameter—the properly defined Fresnel number, Eq. (2),—which parameterizes the ratio of the scintillation index of a PCB to the Rytov variance. In the most general sense this parameter is the number of effective Fresnel emitters “seen” by the detector, defined as the ratio of the “working” aperture area to the area of the Fresnel zone. The “working” aperture here is a part of (or whole) transmitter aperture from which the light emission can reach the detector, computed from the simple turbulence-free diffraction based on the coherence radius of the source and its distance to the detector. For a collimated PCB a well-defined minimum in the SI ratio occurs at a transition length where the detector “sees” the whole transmitter aperture. If the detector is located closer to the transmitter it only “sees” a fraction of the aperture—the “working” aperture, while if the detector is further away the angular dimension of the transmitter is reduced. In both cases the SI ratio increases away from the minimum. At the minimum the SI ratio has an analytical asymptotic with respect to the ratio of the coherence radius to the aperture radius to the power 7/6. The analysis is valid for the weak turbulence regime and thus the structure constant $C_{n}^{2}$ does not enter the theory for the scintillation index normalized to Rytov variance, as presented here.

5. Appendix A

In the regime of weak fluctuations the intensity scintillation index $σ_{I}^{2} = \frac{〈 I^{2} 〉}{{〈 I 〉}^{2}} - 1 \approx 4 〈 χ^{2} 〉$ , where χ is log-amplitude of the electric field φ = φ ₀ e ^χ+iS and φ ₀ is unperturbed field. The atmospheric turbulence spectrum with inner scale l ₀ is described by Tatarskii formula $Φ_{n} (κ) = 0.33 C_{n}^{2} κ^{- 11 / 3} exp (- κ^{2} / κ_{m}^{2})$ , where $C_{n}^{2}$ is the index structure constant and κ_m = 5.92/l ₀. The scintillation index of a plane wave in turbulence with zero inner scale l ₀ = 0 is known as Rytov variance $σ_{R}^{2} = 1.23 C_{n}^{2} k^{7 / 6} L^{11 / 6}$ , where k = 2π/λ, and L is the propagation distance. All expressions for the scintillation index of various beams in Rytov theory contain $σ_{R}^{2}$ as a multiplicative factor, so that normalization by it is a standard practice.

The expressions for the scintillation index of more complex beams, such as spherical wave, Gaussian and partially coherent beams are not analytic, but involve either Hypergeometric functions or single integral expressions. In our numerical modeling we use integral expressions for better transparency and unified structure. Expressions for a plane wave, spherical wave and a Gaussian beam for Tatarskii spectrum with inner scale are written as follows [5]:

σ_{pl}^{2} = 7.07 σ_{R}^{2} \int_{0}^{1} {{[{(1 - ξ)}^{2} + Q_{m}^{- 2}]}^{5 / 12} cos (\frac{5}{6} arctan [Q_{m} (1 - ξ)]) - Q_{m}^{- 5 / 6}} d ξ,

σ_{sp}^{2} = 7.07 σ_{R}^{2} \int_{0}^{1} {{[ξ^{2} {(1 - ξ)}^{2} + Q_{m}^{- 2}]}^{5 / 12} cos (\frac{5}{6} arctan [Q_{m} ξ (1 - ξ)]) - Q_{m}^{- 5 / 6}} d ξ,

σ_{G}^{2} = 7.07 σ_{R}^{2} Re \int_{0}^{1} {{[Q_{m}^{- 1} + i γ (1 - ξ)]}^{5 / 6} - {[Q_{m}^{- 1} - Im (γ) (1 - ξ)]}^{5 / 6}} d ξ,

where

Q_{m} = κ_{m}^{2} L / k = 35 L / k l_{0}^{2}

, γ = (1 − fξ + iΩ⁻¹ ξ)/(1 − f + iΩ⁻¹), f = L/F ₀, and Ω = ka ²/L. Parameters a and F ₀ define the intensity radius and wavefront curvature of the Gaussian beam, as well as the PCB, which is usually modeled by a Gaussian-Schell model (GSM) with the correlation function

Γ (r_{1}, r_{2}) = \bar{φ_{0} (r_{1}) φ_{0} (r_{2})} = φ_{0}^{2} exp {- \frac{r_{1}^{2} + r_{2}^{2}}{2 a^{2}} - \frac{{(r_{1} - r_{2})}^{2}}{4 r_{c}^{2}} + \frac{i k}{2 F_{0}} (r_{1}^{2} - r_{2}^{2})},

where r_c is the coherence radius. For r_c → ∞ Eq. (7) describes the fully coherent Gaussian beam

For the PCB the expression for the scintillation index was derived by Banakh [6] in the case of zero inner scale. Somewhat different formula was obtained by Baykal [7] using extended Huygens-Fresnel approach, rather than Rytov approximation, yielding noticeably different results in the weak fluctuations regime. Therefore we chose to work with Banakh expression modified to include inner scale:

\begin{array}{l} σ_{PCB}^{2} = \frac{7.07}{g_{u}^{5 / 3}} σ_{R}^{2} Re \int_{0}^{1} {{[Ω_{u} {(1 - ξ)}^{2} + i (1 - ξ) [Ω^{2} (1 - f) (1 - ξ f) + k_{u}^{2} ξ] + \frac{g_{u}^{2}}{Q_{m}}]}^{5 / 6} \\ - {[Ω_{u} {(1 - ξ)}^{2} + \frac{g_{u}^{2}}{Q_{m}}]}^{5 / 6}} d ξ, \end{array}

where

k_{u} = \sqrt{1 + a^{2} / r_{c}^{2}}

,

g_{u} = \sqrt{k_{u}^{2} + {(1 - f)}^{2} Ω^{2}}

, and

Ω_{u} = 0.5 Ω (1 + k_{u}^{2})

. Note that Eq. (6) follows from Eq. (8) for r_c → ∞.

6. Appendix B

To derive the asymptotic behavior of $σ_{PCB}^{2} / σ_{R}^{2}$ ratio for ξ = r_c/a → 0 evaluated at the transition length L = L_tr = 4πar_c/λ we use the expression, which follows from (8):

\frac{σ_{PCB}^{2}}{σ_{R}^{2}} = 2.67 Ω^{5 / 6} {(\frac{1 + k_{u}^{2}}{2 g_{u}^{2}})}^{5 / 6} {\frac{16}{11} Re [{(1 + 2 i \frac{Ω (1 - f)}{1 + k_{u}^{2}})}^{5 / 6} {}_{2}{F_{1}} (- \frac{5}{6}, 1; \frac{17}{6}; z)] - 1}

with the argument of the Gauss Hypergeometric function

z = \frac{2 g_{u}^{2} (f - i k_{u} / Ω) + {(1 - k_{u})}^{2} (k_{u} - i Ω (1 - f))}{2 g_{u}^{2} + {(1 - k_{u})}^{2} (k_{u} - i Ω (1 - f))}

For ξ → 0 we observe that

k_{u}^{2} = ξ^{- 2}

, Ω(L_tr) = (2ξ)⁻¹, Ω_k (L_tr) = ξ/2, and

g_{u}^{2} = \frac{5}{4} ξ^{- 2}

. It will be required later on to keep terms up to quadratic power in ξ in Taylor expansion of the argument of the Hypergeometric function z = 1 − 5iξ − 5ξ ². Further difficulty arises with the expansion of the Hypergeometric function itself since its argument is currently close to the convergence boundary of unity radius. Thus we use the known transformation relation [8] for the Gauss Hypergeometric functions so that the series representation could be used to evaluate its value near zero. For the present case up to the second power of ξ we have

{}_{2}{F_{1}} (a, b; c; z) = \frac{Γ (c) Γ (c - a - b)}{Γ (c - a) Γ (c - b)} {}_{2}{F_{1}} (a, b; a + b - c + 1; 1 - z)

and now the Hypergeometric function can be evaluated up to the third term in a series

{}_{2}{F_{1}} (a, b; c; z) = \sum_{j = 0} \frac{{(a)}_{j} {(b)}_{j}}{{(c)}_{j}} \frac{z^{j}}{j!},

where the Pochhammer symbol (μ)_j is defined as

{(μ)}_{0} = 1; {(μ)}_{j} = μ (μ + 1) \dots (μ + j - 1), j = 1, 2, \dots

Combining all the terms as a power series in ξ we see that under Re the lowest power of ξ is 2 and so the final asymptotic reads

lim_{ξ \to 0} \frac{σ_{PCB}^{2}}{σ_{R}^{2}} = 2.52 ξ^{7 / 6}, at L = L_{tr} = \frac{4 π a r_{c}}{λ} .

Acknowledgments

This work was performed, in part, at the Center for Integrated Nanotechnologies, a U.S. Department of Energy, Office of Basic Energy Sciences user facility. Los Alamos National Laboratory, an affirmative action equal opportunity employer, is operated by Los Alamos National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396.

References and links

1. G. Gbur, “Partially coherent beam propagation in atmospheric turbulence,” J. Opt. Soc. Am. A 31, 2038–2045 (2014). [CrossRef]

2. A. Efimov, K. Velizhanin, and G. Gelikonov, “Simultaneous scintillation measurements of coherent and partially coherent beams in an open atmosphere experiment,” Proc. SPIE 8971, 8971051 (2014).

3. I. I. Kim, H. Hakakha, P. Adhikari, E. Korevaar, and A. K. Majumdar, “Scintillation reduction using multiple transmitters,” Proc, SPIE 2990, 102–113 (1997). [CrossRef]

4. G. P. Berman and A. A. Chumak, “Infuence of phase-diffuser dynamics on scintillations of laser radiation in earths atmosphere: Long-distance propagation,” Phys. Rev. A 79, 0638481 (2009). [CrossRef]

5. A. S. Gurvich, A. I. Kon, V. L. Mironov, and S. S. Khmelevtsov, Laser Radiation in the Turbulent Atmosphere (Nauka, 1976).

6. V. A. Banakh, V. M. Buldakov, and V. L. Mironov, “Intensity fluctuations of a partially coherent light beam in a turbulent atmosphere,” Opt. Spectrosc. 54, 626–629 (1983).

7. Y. Baykal, M. A. Plonus, and S. J. Wang, “The scintillations for weak atmospheric turbulence using a partially coherent source,” Radio Science 18, 551–556 (1983). [CrossRef]

8. J. Pearson, Computation of hypergeometric functions, (Master Thesis, University of Oxford, 2009).

Intuitive model for the scintillations of a partially coherent beam

Abstract

1. Introduction

2. Review of the numerical results

3. Intuitive model

4. Conclusion

5. Appendix A

6. Appendix B

Acknowledgments

References and links

Cited By

Figures (4)

Equations (14)

Optics Express