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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 31, Iss. 9 — Sep. 1, 2014
  • pp: 2038–2045
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Partially coherent beam propagation in atmospheric turbulence [Invited]

Greg Gbur  »View Author Affiliations


JOSA A, Vol. 31, Issue 9, pp. 2038-2045 (2014)
http://dx.doi.org/10.1364/JOSAA.31.002038


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Abstract

Partially coherent beams hold much promise in free-space optical communications for their resistance to the deleterious effects of atmospheric turbulence. We describe the basic theoretical and computational tools used to investigate these effects, and review the research to date.

© 2014 Optical Society of America

1. INTRODUCTION

Since the invention of the laser, the possibility of using visible light instead of radio for free-space electromagnetic communications has intrigued researchers. The extreme directionality of lasers allows a great degree of security, while optical frequencies permit a high rate of data transfer. The short wavelength of visible light also gives excellent resolution in applications such as imaging and laser radar. However, atmospheric turbulence, manifesting as random variations of the refractive index of the atmosphere, degrades the quality of an optical beam, resulting in excessive beam spreading of the beam, deviations of the beam direction (beam wander), and intensity fluctuations of the beam at the detector (scintillations). With sufficiently strong turbulence fluctuations, these effects can cause unacceptable data transmission errors over distances of less than a kilometer, severely limiting visible light for practical free-space applications.

However, it has long been known that partially coherent beams—beams that are partially randomized in time and/or space—often prove to be more resistant to the degrading effects of turbulence than their fully coherent counterparts. The earliest work can be dated to soon after birth of the laser and, not coincidentally, to the development of modern optical coherence theory. At the start of the 21st century, renewed interest in free-space optical communications, combined with the discovery of exotic beam classes and new types of partially coherent fields, has spurred much research into the behavior of randomized beams in turbulence.

In this article we review the progress on partially coherent (PC) beam propagation in atmospheric turbulence. In Section 2, we look at the progress in the field from the 1960s to the end of the 20th century, and give a simple description of how PC beams can “resist” turbulence. In Section 3, we discuss the most commonly used analytic model for studying the propagation of light through the atmosphere, and apply it to PC beams. Due to the complexity of turbulence, however, analytic models are often intractable, except in special cases; in Section 4, we describe the multiple phase screen model often used to computationally solve for beam evolution. In Section 5, we review turbulence effects related to the second order (field–field) coherence properties of the field, and in Section 6, we consider effects related to the fourth order (scintillation) properties of the field.

2. HISTORICAL OVERVIEW

The earliest articles discussing the coherence of light in turbulence attempted to build on the classic texts of Tatarski [1

1. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

] and Chernov [2

2. L. A. Chernov, Wave Propagation in a Random Medium (McGraw-Hill, 1960).

] to construct robust and accurate propagation models for coherence functions in random media. Hufnagel and Stanley [3

3. R. E. Hufnagel and N. R. Stanley, “Modulation transfer function associated with image transmission through turbulence media,” J. Opt. Soc. Am. 54, 52–61 (1964). [CrossRef]

] estimated the modulation transfer function for an imaging system seeing through the atmosphere; several propagation models, including a geometric one, were used to estimate the spatial coherence of light at the input aperture. Beran [4

4. M. J. Beran, “Propagation of the mutual coherence function through random media,” J. Opt. Soc. Am. 56, 1475–1480 (1966). [CrossRef]

] introduced an approximate solution for the mutual coherence function in random media by dividing the medium into a series of longitudinal slices. A slightly more general expression for the mutual coherence function was given by Taylor [5

5. L. S. Taylor, “Decay of mutual coherence in turbulent media,” J. Opt. Soc. Am. 57, 304–308 (1967). [CrossRef]

], giving results in accordance with geometrical optics.

These early formulations were all limited to very short propagation distances or, equivalently, a single-scattering approximation. Furthermore, the calculations were done only for exceedingly simple fields, namely, spherical waves, plane waves, or very wide Gaussian beams. In 1971, Lutomirski and Yura [6

6. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971). [CrossRef]

] introduced a new technique, now known as the extended Huygens–Fresnel (eHF) principle (to be discussed in Section 3), that can be used to derive propagation results over a broader range of circumstances and for more general fields. Yura [7

7. H. T. Yura, “Mutual coherence function of a finite cross section optical beam propagating in a turbulent medium,” Appl. Opt. 11, 1399–1406 (1972). [CrossRef]

] applied this method to derive an analytic expression for the mutual coherence function of a finite beam. It was later realized that essentially the same method had been independently derived in 1967 by Feizulin and Kravtsov [8

8. Z. I. Feizulin and Y. A. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967). [CrossRef]

].

About the same time, researchers’ attention began to shift toward the more difficult problem of evaluating the intensity fluctuations of beams with partial spatial coherence in turbulence. Leader [13

13. J. C. Leader, “Intensity fluctuations resulting from partially coherent light propagating through atmospheric turbulence,” J. Opt. Soc. Am. 69, 73–84 (1979). [CrossRef]

] investigated such fluctuations in the context of imaging random rough surfaces, using the “Huygens–Kirchhoff method” (essentially the eHF principle again). Fante [14

14. R. L. Fante, “Intensity fluctuations of an optical wave in a turbulent medium; effect of source coherence,” Opt. Acta 28, 1203–1207 (1981). [CrossRef]

], Banach et al. [15

15. V. A. Banach, 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).

], and Banakh and Buldakov [16

16. V. A. Banakh and V. M. Buldakov, “Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. 54, 423–427 (1983).

] studied the fluctuations of intensity over different receiver response times, noting dramatic differences between a “fast” and a “slow” detector. Fante [17

17. R. L. Fante, “The effect of source temporal coherence on light scintillations in weak turbulence,” J. Opt. Soc. Am. 69, 71–73 (1979). [CrossRef]

] also considered the effect of temporal coherence on scintillation, at least for weak turbulence.

Through the 1990s, the study of atmospheric propagation of PC beams seems to have been largely ignored, with the exception of articles by Wu [18

18. J. Wu, “Propagation of a Gaussian-Schell beam through turbulent media,” J. Mod. Opt. 37, 671–684 (1990). [CrossRef]

] and Wu and Boardman [19

19. J. Wu and A. D. Boardman, “Coherence length of a Gaussian-Schell beam and atmospheric turbulence,” J. Mod. Opt. 38, 1355–1363 (1991). [CrossRef]

]; the former considers the propagation properties of model PC beams, while the latter considers the spatial coherence properties of the beams.

The promise of scintillation reduction by partial coherence has led an increasing number of researchers to study PC beams specifically for optical communications. The first in a new wave of studies were performed by Ricklin and Davidson [25

25. J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19, 1794–1802 (2002). [CrossRef]

,26

26. J. C. Ricklin and F. M. Davidson, “Atmospheric optical communication with a Gaussian-Schell beam,” J. Opt. Soc. Am. A 20, 856–866 (2003). [CrossRef]

], and Korotkova et al. [27

27. O. Korotkova, L. C. Andrews, and R. L. Phillips, “Model for a partially coherent Gaussian beam in atmospheric turbulence with application in Lasercom,” Opt. Eng. 43, 330–341 (2004). [CrossRef]

].

Since this time there has been a vast amount of research on both the propagation and scintillation of PC beams in turbulence. It is to be noted that optimizing such beams for applications is a nontrivial problem: PC beams tend to have less scintillation on propagation, but also have a larger angular spread, resulting in less energy received at the detector. An appropriate balance of these seemingly exclusive traits will depend on the desired system specifications. Optimization is in principle possible; Schulz [28

28. T. J. Schulz, “Iterative transform algorithm for the computation of optimal beams,” J. Opt. Soc. Am. A 21, 1970–1974 (2004). [CrossRef]

,29

29. T. J. Schulz, “Optimal beams for propagation through random media,” Opt. Lett. 30, 1093–1095 (2005). [CrossRef]

] has used variational and iterative methods to find the optimal mode structure of PC beams with either minimum scintillations or minimum spreading in turbulence. However, it is not clear how to produce these optimal beams in practice, and so research continues to find realizable sources with good propagation characteristics. Some of this research will be summarized in the final sections of this review.

3. ANALYTIC PROPAGATION MODELS

The atmosphere is, in a sense, an ideal medium to study using the traditional tools of scattering theory. The small variations of refractive index in atmospheric turbulence induce small phase fluctuations on a beam of light, and these fluctuations only become significant after appreciable propagation distances. Early researchers applied the Born and Rytov approximations to study atmospheric propagation; description of these methods can be found in [30

30. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

], Chapter 13.

In recent years, however, the preferred strategy for studying propagation in turbulence is the eHF method noted previously. It was shown quite early that the eHF method is in good agreement with experimental data in weak and strong turbulence [31

31. V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. S. Tsvik, “Focused-laser-beam scintillations in the turbulent atmosphere,” J. Opt. Soc. Am. 64, 516–518 (1974). [CrossRef]

], even under simplifying phase approximations [32

32. V. A. Banakh and V. L. Mironov, “Phase approximation of the Huygens-Kirchhoff method in problems of laser-beam propagation in the turbulent atmosphere,” Opt. Lett. 1, 172–174 (1977). [CrossRef]

]. The method includes atmospheric turbulence as a phase distortion of the spherical wave in the standard Huygens–Fresnel integral, so that the field U(r,L,ω) after propagation a distance z=L is given by
U(r,L,ω)=ik2πLexp(ikL)U0(ρ,ω)exp[ik|ρr|22L+ψ(r,ρ,L)]d2ρ,
(1)
where U0(ρ,ω) is the field in the source plane at z=0, ρ and r are the transverse coordinates in the source and observation planes, respectively, and ψ(r,ρ,L) is the phase distortion of the Huygens wavelet due to turbulence. The geometry is illustrated in Fig. 1. The field is taken for the moment to be monochromatic at frequency ω and with wavenumber k. In what follows, we will closely follow the notation of Andrews and Phillips [33

33. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

]; we will also suppress the frequency dependence momentarily for brevity.

Fig. 1. Illustration of the geometry of beam propagation through the atmosphere.

Of primary interest here are the second- and fourth-order average field moments, defined as
W2(r1,r2,L)=U(r1,L)U*(r2,L)=U0(ρ1)U0*(ρ2)G0(ρ1,r1)G0*(ρ2,r2)×M2[{ri,ρi},L]d2ρ1d2ρ2
(2)
and
W4(r1,r2,r3,r4,L)=U(r1,L)U*(r2,L)U(r3,L)U*(r4,L)=U0(ρ1)U0*(ρ2)U0(ρ3)U0*(ρ4)×G0(ρ1,r1)G0*(ρ2,r2)G0(ρ3,r3)G0*(ρ4,r4)×M4[{ri,ρi},L]d2ρ1d2ρ2d2ρ3d2ρ4,
(3)
where we have defined
M2[{ri,ρi},L]=exp[ψ(r1,ρ1,L)+ψ*(r2,ρ2,L)],
(4)
M4[{ri,ρi},L]=exp[ψ(r1,ρ1,L)+ψ*(r2,ρ2,L)+ψ(r3,ρ3,L)+ψ*(r4,ρ4,L)],
(5)
where {ri,ρi} represents the set of relevant arguments, and
G0(ρ,r)=ik2πLexp(ikL)exp[ik|ρr|22L]
(6)
is the free-space Green’s function in the Fresnel approximation. The angle brackets represent an ensemble average over a large number of realizations of the turbulence; from these moments we can derive properties such as the average beam width and the average variance of the intensity. It should be noted, however, that specific realizations of these field moments can look very different from the averages.

The average phase terms can be calculated using the method of cumulants (see, for instance, [34

34. H. Cramér, Mathematical Methods of Statistics (Princeton University, 1946).

], Section 15.10). To second order, we may write
exp[Ψ(r,L)]=exp[K1+12K2],
(7)
where
K1=Ψ(r,L),
(8)
K2=Ψ2(r,L)Ψ(r,L)2.
(9)
From these results, it follows that the moments of the phase can be written as
M2[{ri,ρi},L]=exp[2E1(0,0;0,0)+E2(r1,r2;ρ1,ρ2)],
(10)
M4[{ri,ρi},L]=exp[4E1(0,0;0,0)+E2(r1,r2;ρ1,ρ2)+E2(r1,r4;ρ1,ρ4)+E2(r3,r2;ρ3,ρ2)+E2(r2,r4;ρ2,ρ4)+E3(r1,r3;ρ1,ρ3)+E3*(r2,r4;ρ2,ρ4)],
(11)
where we have introduced
E1(0,0;0,0)=2π2k2L0κΦn(κ)dκ,
(12)
E2(r1,r2;ρ1,ρ2)=4π2k2L010κΦn(κ)J0[κ|(1ξ)Δr+ξΔρ|]dκdξ,
(13)
E3(r1,r2;ρ1,ρ2)=4π2k2L010κΦn(κ)J0[κ|(1ξ)Δr+ξΔρ|]exp[iLκ2ξ(1ξ)/k]dκdξ.
(14)
We have now introduced Δrr1r2 and Δρρ1ρ2, as well as the power spectral density of refractive index fluctuations Φn(κ).

To apply these formulas, we must have a model of the atmosphere from which a formula for Φn(κ) can be deduced. The simplest model with some predictive power is illustrated in Fig. 2, and shows the energy cascade theory of turbulence. When wind speeds exceed the critical threshold for turbulence to form, large turbulent eddies of roughly constant refractive index are created with a large characteristic scale L0, called the outer scale. These large eddies undergo a process in which they break up continually into smaller and smaller sizes, eventually reaching a critical minimum size l0, called the inner scale, at which their energy is dissipated completely. The outer scale can range widely from meters to tens of meters, while the inner scale is typically of the order of millimeters.

Fig. 2. Simple illustration of a model for atmospheric turbulence.

Starting from a dimensional argument, Kolmogorov noted that this process could be represented by the simple power spectral density
Φn(κ)=0.033Cn2κ11/3,1/L0κ1/l0.
(15)
Here Cn2 is called the structure parameter and is a measure of the overall turbulence strength. Its values range from 1017m2/3 for weak turbulence to as high as 1013m2/3 for strong turbulence.

The Kolmogorov spectrum is an exceedingly simple expression, but does not treat the inner and outer limits in a satisfactory manner. A more sophisticated model is the Tatarskii spectrum:
Φn(κ)=0.033Cn2κ11/3exp(κ2/κm2),1/L0κ,
(16)
with κm=5.92/l0, which accounts for the inner scale (high spatial frequencies). A yet more detailed spectrum is the von Karman spectrum:
Φn(κ)=0.033Cn2exp(κ2/κm2)(κ2+κ02)11/6,
(17)
with κ0=1/L0, which takes into account both inner and outer scales. Nevertheless, even this analytic model for the spectrum misses important physical details; more advanced ones can be found in [33

33. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 1998).

].

We can now include partial coherence into the equations for the field moments. Considering first the second-order moment, we may average the moment over a set of monochromatic field realizations using the space–frequency representation of the source cross-spectral density W0(r1,r2,ω), i.e.,
W0(ρ1,ρ2,ω)=U0(ρ1,ω)U0*(ρ2,ω)ω,
(18)
where the subscript “ω” represents an average over a specially constructed set of monochromatic field realizations; see [35

35. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

], Chapter 4 for more details. The resulting second-order moment W2(r1,r2,z,ω) is the cross-spectral density of the field after propagating a distance z through the atmosphere.

It is to be noted that the time scales of fluctuations are extremely important in performing averages in the simple manner described. Because the fluctuations of turbulence are typically slow (of the order of milliseconds) compared to the fluctuations of the optical source, the medium may be treated as effectively “frozen” for the time of field averaging, and the averages may be taken independently.

The cross-spectral density may be expanded in the form of a coherent mode representation, as first demonstrated by Wolf [36

36. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part 1: Spectra and cross-spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982). [CrossRef]

]. This representation may be written as
W(r1,r2,ω)=nλn(ω)ϕn(r1,ω)ϕn*(r2,ω),
(19)
where n is a (possibly multiple) sum over a set of so-called coherent modes that are mutually orthogonal when integrated over the domain of interest (in this case, the source plane z=0) that satisfy
W(r1,r2,ω)ϕn(r2,ω)d2r2=λn(ω)ϕn(r1,ω),
(20)
and λn(ω)>0.

The coherent mode representation provides a nice physical picture for the mechanism by which PC beams are more resistant to turbulence; this is illustrated in Fig. 3. A coherent single mode laser will send all of its energy through a single path in the turbulence; this mode will interfere with itself, producing laser speckle and, therefore, intensity fluctuations, if it even hits the detector at all. A PC beam will send energy via multiple modes that travel through different channels toward the detector. Multiple modes will likely hit the detector and, because of their mutual incoherence, their individual speckle patterns will wash out, producing a more regular intensity at the detector and less scintillation.

Fig. 3. Simple illustration of the turbulence resistance of PC beams. A PC beam will simultaneously “sample” multiple paths through the turbulence, and the mutually incoherent beams will not produce interference speckle at the detector. Figure adapted from [37].

The scintillation of an optical beam is typically characterized by the scintillation index, defined as
σI2=I2I21,
(21)
where I is the intensity of the field at a single point, I(r)=|U(r)|2. The denominator of Eq. (21) can be derived from the second-order field moment and the numerator from the fourth-order field moment. For a plane wave in weak turbulence, the scintillation index takes on the specific form
σ12=1.23Cn2k7/6z11/6,
(22)
and is known as the Rytov variance. The Rytov variance is often used as a loose measure of the strength of atmospheric turbulence, with σ121 implying weak turbulence and strong turbulence otherwise.

As was noted in [14

14. R. L. Fante, “Intensity fluctuations of an optical wave in a turbulent medium; effect of source coherence,” Opt. Acta 28, 1203–1207 (1981). [CrossRef]

16

16. V. A. Banakh and V. M. Buldakov, “Effect of the initial degree of spatial coherence of a light beam on intensity fluctuations in a turbulent atmosphere,” Opt. Spectrosc. 54, 423–427 (1983).

], the response time τd of the detector is an important factor in the scintillation measured. If the detector response is much longer than the coherence time τc of the light, i.e., τdτc, the inherent fluctuations in the source intensity will be averaged out, and we may write
U0(ρ1)U0*(ρ2)U0(ρ3)U0*(ρ4)=W0(ρ1,ρ2)W0(ρ3,ρ4),
(23)
i.e., all the intensity fluctuations come from the turbulence, and none from the source.

If τdτc, then the detector will be sensitive to source fluctuations, as well. Most methods of creating PC sources result in fields with Gaussian statistics (see [35

35. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

], Chapter 7), for which we may write
U0(ρ1)U0*(ρ2)U0(ρ3)U0*(ρ4)=W0(ρ1,ρ2)W0(ρ3,ρ4)+W0(ρ1,ρ4)W0(ρ3,ρ2).
(24)
These intensity fluctuations will result in a scintillation that is higher than that of a fully coherent beam.

These observations suggest that any application of PC fields to scintillation reduction must take careful consideration of the time scales involved. To the times already considered we can add the duration of an optical bit in a communications signal, τb, which must be longer than both the coherence time and the detector response. If we also consider the characteristic time of turbulence fluctuations τt, we require the following set of inequalities to be satisfied to have effective scintillation reduction in optical communications:
τcτdτbτt.
(25)
These inequalities also imply that a scintillation-reducing source must have a spectral bandwidth significantly larger than the communications bandwidth.

Though the eHF principle has been shown to agree well with experimental data, there are a number of limitations worth noting. First, the simplest formulas given above, and they are by no means simple, involve light propagating in uniform, homogeneous turbulence throughout the entire source-to-detector path. This excludes situations such as ground-to-air or ground-to-satellite links, which can be accounted for, albeit with additional complexity.

Even for the simplest Kolmogorov model of turbulence, however, the field integrals are generally intractable. In many cases, the integrals are evaluated by making use of a quadratic approximation for the turbulence phase moments, though it has long been known that this approximation leaves out important features of the turbulence [38

38. S. M. Wandzura, “Meaning of quadratic structure functions,” J. Opt. Soc. Am. 70, 745–747 (1980). [CrossRef]

]. Furthermore, the complexity of the models often leads to their misuse, as was discussed recently [39

39. M. Charnotskii, “Common omissions and misconceptions of wave propagation in turbulence: discussion,” J. Opt. Soc. Am. A 29, 711–721 (2012). [CrossRef]

].

4. COMPUTATIONAL PROPAGATION MODELS

The standard method for numerically simulating the propagation of waves in atmospheric turbulence is a multiple phase screen method, in which the extended random medium is described by a collection of thin random phase screens with the appropriate turbulence statistics. We briefly review this method, following the clear discussion of Martin and Flatté [40

40. J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27, 2111–2126 (1988). [CrossRef]

].

A monochromatic scalar wave propagating in an inhomogeneous medium will satisfy the Helmholtz equation:
[2+n2(r)k2]U(r)=0,
(26)
where n(r) is the refractive index of the medium and k=ω/c is the free-space wavenumber. If we restrict ourselves to paraxial fields of the form
U(r)=V(r)eikz,
(27)
where V(r) is a slowly varying function in z, we get the simplified equation
2ikzV+2V+k2(n21)V=0,
(28)
where 2=x2+y2. The refractive index of the atmosphere is quite close to unity; we write n=1+Δn, where Δn1, and then to a good approximation we may write
2ikzV+2V+2k2ΔnV=0.
(29)
This is the parabolic approximation to the wave equation.

The sampling of the screens must be chosen such that their spatial frequencies include the relevant turbulence ranges, notably the inner and outer scale frequencies. The spacing of the screens must also satisfy a pair of additional conditions. They should be spaced such that the real extended medium scintillation is weak over the interscreen distance, i.e., σI2(δz)<0.1. Also, one should require that less than 10% of the total scintillation take place over the interscreen distance, i.e., σI2(δz)<0.1σI2(L).

At this point the argument may seem rather circular: in order to calculate the scintillation of a beam, we need to know the scintillation of the beam! However, the Rytov variance, given by Eq. (22), can usually be used as an estimate of the scintillation for use in the inequalities above.

The simple method discussed here suffers from a significant limitation: the absence of low spatial frequencies. Due to the nature of discrete Fourier transforms, the lowest spatial frequency included in the simulation is roughly equal to the inverse of the width of the simulation domain; any spatial frequencies lower than this are not represented. However, the lowest spatial frequencies are exactly those which result in the wander of the centroid of the beam. These effects become especially important over long propagation distances and strong turbulence. To incorporate them, a variety of so-called subharmonic methods have been introduced, which put the low frequencies back into the simulation “by hand.” Worthwhile discussions can be found in [42

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

44

44. J. Xiang, “Accurate compensation of the low-frequency components for the FFT-based turbulent phase screen,” Opt. Express 20, 681–687 (2012). [CrossRef]

].

To incorporate partial coherence into the simulations, a PC beam can be treated as an incoherent superposition of coherent beams, each of which are propagated through the same realization of turbulence. The coherent mode representation of the beam, formally given in Eq. (19), is one such way to decompose the cross-spectral density. Alternatively, if the beam is Schell-model (see [35

35. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

], Section 5.3), i.e., the degree of coherence μ(r1,r2) is independent of the origin of position,
W(r1,r2)=I(r1)I(r2)μ(r2r1),
(31)
then the degree of coherence can be written in terms of its Fourier representation:
μ(r2r1)=μ˜(K)exp[iK·r2]exp[iK·r1]d2K.
(32)
On substitution into Eq. (31), it is clear that the cross-spectral density may be expressed as an incoherent sum of beams with different inclination phases.

This method produces a source that has no inherent intensity fluctuations, meaning that it is in agreement with the inequalities of Eq. (25). This is the regime of effective scintillation reduction, which makes this formulation appropriate for most practical applications.

5. BEAM PROPAGATION IN TURBULENCE

Despite the fact that the general spreading and propagation characteristics of PC beams have been known for some time, research nevertheless continues. It is spurred by the relatively recent introduction of “special” classes of coherent and PC beams with unusual propagation characteristics, such as Bessel and Airy beams. Furthermore, in seeming contradiction to conventional wisdom, appropriately prepared electromagnetic beams can have unusual polarization changes in turbulence. Finally, the propagation and polarization of light can be used to measure the turbulence itself. In this section, we highlight some of these possibilities.

Extending beyond ordinary Gaussian beams, Cai and He studied the propagation of elliptical Gaussian beams [45

45. Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. 31, 568–570 (2006). [CrossRef]

] and dark hollow beams [46

46. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006). [CrossRef]

] in the atmosphere, finding that in both cases the beams will eventually become circular. More exotic possibilities involve the use of so-called Bessel beams [47

47. Z. Bouchal, “Nondiffracting optical beams: physical properties, experiments, and applications,” Czech. J. Phys. 53, 537–578 (2003). [CrossRef]

], which can be nondiffracting over large propagation distances, and Airy beams [48

48. G. A. Sililoglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007). [CrossRef]

], which can exhibit transverse acceleration. Chen and Pu [49

49. B.-S. Chen and J.-X. Pu, “Propagation of Gauss-Bessel beams in turbulent atmosphere,” Chin. Phys. B 18, 1033–1039 (2009). [CrossRef]

] have investigated Bessel beams in turbulence, while Chu [50

50. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011). [CrossRef]

] has studied Airy beams. Perhaps unsurprisingly, the turbulence always “wins” over sufficiently long distances, erasing the special features of the beam and reducing it to a Gaussian profile. However, Çil et al. [51

51. C. Z. Çil, H. T. Eyyuboğlu, Y. Baykal, O. Korotkova, and Y. Cai, “Beam wander of J0- and I0-Bessel Gaussian beams propagating in turbulent atmosphere,” Appl. Phys. B 98, 195–202 (2010). [CrossRef]

] have noted that the beam wander of Bessel–Gaussian beams can be less than a standard Gaussian.

Similar results arise for PC beam propagation. Chen et al. [52

52. B. Chen, Z. Chen, and J. Pu, “Propagation of partially coherent Bessel-Gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 40, 820–827 (2008). [CrossRef]

] studied the degradation of a PC Bessel–Gaussian beam in turbulence, finding its reduction to a Gaussian profile. The spreading and directionality of PC Hermite–Gauss beams were investigated by Ji et al. [53

53. X. Ji, X. Chen, and B. Lü, “Spreading and directionality of partially coherent Hermite-Gaussian beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 25, 21–28 (2008). [CrossRef]

]. In recent years, a new analytic class of non-Schell-model PC beams was introduced by Gori and Santarsiero [54

54. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32, 3531–3533 (2007). [CrossRef]

]; the propagation characteristics of such beams were then studied by Tong and Korotkova [55

55. Z. Tong and O. Korotkova, “Nonuniformly correlated light beams in uniformly correlated media,” Opt. Lett. 37, 3240–3242 (2012). [CrossRef]

].

Though it has long been appreciated that the polarization of a uniformly polarized beam is unchanged on propagation, due to the weak scattering of turbulence, the same is not true for beams with a nonuniform state or degree of polarization. Such nonuniformly polarized beams can be considered a coherent or incoherent superposition of different orthogonally polarized modes, each of which propagates through the atmosphere via a different channel. Early research by Salem et al. [56

56. M. Salem, O. Korotkova, A. Dogariu, and E. Wolf, “Polarization changes in partially coherent electromagnetic beams propagating through turbulent atmosphere,” Waves Random Complex Media 14, 513–523 (2004). [CrossRef]

] illustrated that the degree of polarization changes dramatically on propagation; however, unlike in free space, it surprisingly returns to its original value after an appreciable distance. Not long after this work, Korotkova et al. [57

57. O. Korotkova, M. Salem, A. Dogariu, and E. Wolf, “Changes in the polarization ellipse of random electromagnetic beams propagating through the turbulent atmosphere,” Waves Random Complex Media 15, 353–364 (2005). [CrossRef]

] showed that the state of polarization also in general changes on propagation. Similar results apply to the propagation of light through tissue, as was shown by Gao [58

58. W. Gao, “Changes of polarization of light beams on propagation through tissue,” Opt. Commun. 260, 749–754 (2006). [CrossRef]

] and Gao and Korotkova [59

59. W. Gao and O. Korotkova, “Changes in the state of polarization of a random electromagnetic beam propagating through tissue,” Opt. Commun. 270, 474–478 (2007). [CrossRef]

]. Several studies have also looked at the propagation characteristics of coherent radial and azimuthally polarized light; see, for instance, [60

60. Y. Cai, Q. Lin, H. T. Eyyuboğlu, and Y. Baykal, “Average irradiance and polarization properties of a radially or azimuthally polarized beam in a turbulent atmosphere,” Opt. Express 16, 7665–7673 (2008). [CrossRef]

,61

61. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17, 17829–17836 (2009). [CrossRef]

].

A number of new tools have been introduced to model the propagation of PC beams in the atmosphere. Among them is a reformulation of the eHF method in the angular spectrum regime, both in the scalar [62

62. G. Gbur and O. Korotkova, “Angular spectrum representation for the propagation of arbitrary coherent and partially coherent beams through atmospheric turbulence,” J. Opt. Soc. Am. A 24, 745–752 (2007). [CrossRef]

] and electromagnetic [63

63. O. Korotkova and G. Gbur, “Angular spectrum representation for propagation of random electromagnetic beams in turbulent atmosphere,” J. Opt. Soc. Am. A 24, 2728–2736 (2007). [CrossRef]

] cases.

The relationship between partial coherence and a random medium can be exploited to probe the structure of the medium itself. Ponomarenko and Wolf [64

64. S. A. Ponomarenko and E. Wolf, “Solution for the inverse scattering problem for strongly fluctuating media,” Opt. Lett. 27, 1770–1772 (2002). [CrossRef]

] introduced a technique to fully measure the correlation function of turbulence from the correlations of scattered light. McKinney et al. [65

65. J. D. McKinney, M. A. Webster, K. J. Webb, and A. M. Weiner, “Characterization and imaging in optically scattering media by use of laser speckle and a variable-coherence source,” Opt. Lett. 25, 4–6 (2000). [CrossRef]

] tested a strategy for measuring the scattering parameter of an optically diffusive medium using PC light. More recently, a technique called variable coherence tomography was introduced in [66

66. E. Baleine and A. Dogariu, “Variable-coherence tomography for inverse scattering problems,” J. Opt. Soc. Am. A 21, 1917–1923 (2004). [CrossRef]

,67

67. E. Baleine and A. Dogariu, “Variable coherence scattering microscopy,” Phys. Rev. Lett. 95, 193904 (2005). [CrossRef]

] to use variable coherence states to deduce the structure of a random media; this method was adapted to include polarization effects by Tyo and Turner [68

68. J. S. Tyo and T. S. Turner, “Sensing polarization with variable coherence tomography,” J. Opt. Soc. Am. A 25, 2383–2389 (2008). [CrossRef]

]. It has also been shown by Gu and Gbur [69

69. Y. Gu and G. Gbur, “Measurement of atmospheric turbulence strength by vortex beam,” Opt. Commun. 283, 1209–1212 (2010). [CrossRef]

] that the evolution of correlation singularities in turbulence can be used as a crude measure of turbulence strength.

6. SCINTILLATION EFFECTS IN TURBULENCE

A broad research effort has been dedicated to studying the scintillation of a variety of special beam classes over the past decade; from this work has come a number of surprises. In this section, we consider some of the highlights.

A number of coherent beams of special form have been found to have less scintillation than a comparable Gaussian. Strömqvist Vetelino and Andrews [70

70. F. E. Strömqvist Vetelino and L. C. Andrews, “Annular Gaussian beams in turbulent media,” Proc. SPIE 5160, 86–97 (2004). [CrossRef]

] studied an annular Gaussian beam and found it to have superior scintillation characteristics; Baykal [71

71. Y. Baykal, “Log-amplitude and phase fluctuations of higher-order annular laser beams in a turbulent medium,” J. Opt. Soc. Am. A 22, 672–679 (2005). [CrossRef]

] considered higher-order annular beams. The scintillation of elliptical Gaussian beams were investigated by Cai et al. [72

72. Y. Cai, Y. Chen, H. T. Eyyuboğlu, and Y. Baykal, “Scintillation index of elliptical Gaussian beam in turbulent atmosphere,” Opt. Lett. 32, 2405–2407 (2007). [CrossRef]

], and also found to have reduced scintillation under certain circumstances. In weak turbulence, Eyyuboğlu et al. [73

73. H. T. Eyyuboğlu, Y. Baykal, E. Sermutlu, and Y. Cai, “Scintillation advantages of lowest order Bessel-Gaussian beams,” Appl. Phys. B 92, 229–235 (2008). [CrossRef]

] demonstrated that Bessel–Gauss beams can also have somewhat reduced scintillation, as can modified Bessel–Gauss beams [74

74. H. T. Eyyuboğlu, Y. Baykal, E. Sermutlu, O. Korotkova, and Y. Cai, “Scintillation index of modified Bessel-Gaussian beams propagating in turbulent media,” J. Opt. Soc. Am. A 26, 387–394 (2009). [CrossRef]

]. Flat-topped Gaussian beams have also shown some advantages [75

75. H. Gerçekcioğlu and Y. Baykal, “Scintillation index of flat-topped Gaussian laser beam in strongly turbulent medium,” J. Opt. Soc. Am. A 28, 1540–1544 (2011). [CrossRef]

].

Partial coherence can provide further scintillation reduction. Berman et al. [76

76. G. P. Berman, A. R. Bishop, B. M. Chernobrod, D. C. Nguyen, and V. N. Gorshkov, “Suppression of intensity fluctuations in free space high-speed optical communication based on spectral encoding of a partially coherent beam,” Opt. Commun. 280, 264–270 (2007). [CrossRef]

] introduced a design for a communication system based on partial coherence, and suggested suppression by “orders of magnitude.” Baykal et al. [77

77. Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48, 1943–1954 (2009). [CrossRef]

] observed scintillation reduction as a function of decoherence for a beam consisting of multiple incoherent Gaussians. Beams appropriately tailored with nonuniform correlations, of the Gori type mentioned in Section 5, have also been shown to provide scintillation reduction over simpler types of PC beams [78

78. Y. Gu and G. Gbur, “Scintillation of nonuniformly correlated beams in atmospheric turbulence,” Opt. Lett. 38, 1395–1397 (2013). [CrossRef]

]. In the temporal domain, Kiasaleh [79

79. K. Kiasaleh, “On the scintillation index of a multiwavelength Gaussian beam in a turbulent free-space optical communications channel,” J. Opt. Soc. Am. A 23, 557–566 (2006). [CrossRef]

] noted scintillation reduction in a multiwavelength Gaussian beam.

In a study of Bessel-correlated beams, Gu and Gbur [80

80. Y. Gu and G. Gbur, “Scintillation of pseudo-Bessel correlated beams in atmospheric turbulence,” J. Opt. Soc. Am. A 27, 2621–2629 (2010). [CrossRef]

] made the surprising observation that a small array of incoherent beamlets can provide scintillation nearly indistinguishable from a more general PC beam. This suggests that such arrays may be good enough for any scintillation reduction based on partial coherence. An unusual example of such an array was provided by Gu and Gbur [81

81. Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35, 3456–3458 (2010). [CrossRef]

], who used an incoherent collection of Airy beams to achieve reduction.

Experimental tests have confirmed the theoretical improvements. Voelz and Fitzhenry [82

82. D. Voelz and K. Fitzhenry, “Pseudo-partially coherent beam for free-space laser communication,” Proc. SPIE 5550, 218–224 (2004).

] introduced a pseudo PC beam for laser communication and laboratory tested its effectiveness. Researchers at the University of Arizona calculated the scintillation reduction characteristics of two- [83

83. A. Peleg and J. V. Moloney, “Scintillation index for two Gaussian laser beams with different wavelengths in weak atmospheric turbulence,” J. Opt. Soc. Am. A 23, 3114–3122 (2006). [CrossRef]

] and multi-Gaussian [84

84. A. Peleg and J. V. Moloney, “Scintillation reduction by use of multiple Gaussian laser beams with different wavelengths,” IEEE Photon. Technol. Lett. 19, 883–885 (2007). [CrossRef]

] beam arrays, and verified their results experimentally [85

85. P. Polynki, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett. 32, 885–887 (2007). [CrossRef]

].

Beams with nontrivial polarization can also achieve surprising reductions. Korotkova [86

86. O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008). [CrossRef]

] noted that even simple unpolarized beams will typically have less scintillation than comparable polarized beams. The polarization properties of light were considered in a pair of studies [87

87. Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16, 15834–15846 (2008). [CrossRef]

,88

88. O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681–690 (2009). [CrossRef]

] on the use of PC beams for laser radar. An electromagnetic “cosine-Gaussian Schell-model beam” has also been investigated [89

89. Z. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21, 27246–27259 (2013). [CrossRef]

].

Curiously, even fully coherent but nonuniformly polarized beams can achieve scintillation reduction. The orthogonal modes of a nonuniformly polarized beam propagate differently through turbulence and do not produce an interference pattern at the detector, acting as an effective two-mode PC source. Scintillation reduction was demonstrated by Gu et al. [90

90. Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34, 2261–2263 (2009). [CrossRef]

], using a beam that would later be recognized as a Poincaré beam [91

91. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef]

]. It was later shown by Gu and Gbur [92

92. Y. Gu and G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Opt. Lett. 37, 1553–1555 (2012). [CrossRef]

] that the benefits of nonuniform polarization even extend to an incoherent array of such beamlets.

As can be seen, many options exist for scintillation reduction. Optimizing the effect, which involves a trade-off in beam spreading and scintillation, will no doubt depend on the desired characteristics of the system it is applied to.

ACKNOWLEDGMENTS

The author’s research is supported by the U.S. Air Force Office of Scientific Research (USAFOSR) under Grant FA9550-13-1-0009. The author wishes to thank Blake Stacey for a careful reading of an early draft of this manuscript.

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74.

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75.

H. Gerçekcioğlu and Y. Baykal, “Scintillation index of flat-topped Gaussian laser beam in strongly turbulent medium,” J. Opt. Soc. Am. A 28, 1540–1544 (2011). [CrossRef]

76.

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77.

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78.

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79.

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80.

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81.

Y. Gu and G. Gbur, “Scintillation of Airy beam arrays in atmospheric turbulence,” Opt. Lett. 35, 3456–3458 (2010). [CrossRef]

82.

D. Voelz and K. Fitzhenry, “Pseudo-partially coherent beam for free-space laser communication,” Proc. SPIE 5550, 218–224 (2004).

83.

A. Peleg and J. V. Moloney, “Scintillation index for two Gaussian laser beams with different wavelengths in weak atmospheric turbulence,” J. Opt. Soc. Am. A 23, 3114–3122 (2006). [CrossRef]

84.

A. Peleg and J. V. Moloney, “Scintillation reduction by use of multiple Gaussian laser beams with different wavelengths,” IEEE Photon. Technol. Lett. 19, 883–885 (2007). [CrossRef]

85.

P. Polynki, A. Peleg, L. Klein, T. Rhoadarmer, and J. Moloney, “Optimized multiemitter beams for free-space optical communications through turbulent atmosphere,” Opt. Lett. 32, 885–887 (2007). [CrossRef]

86.

O. Korotkova, “Scintillation index of a stochastic electromagnetic beam propagating in random media,” Opt. Commun. 281, 2342–2348 (2008). [CrossRef]

87.

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16, 15834–15846 (2008). [CrossRef]

88.

O. Korotkova, Y. Cai, and E. Watson, “Stochastic electromagnetic beams for LIDAR systems operating through turbulent atmosphere,” Appl. Phys. B 94, 681–690 (2009). [CrossRef]

89.

Z. Mei and O. Korotkova, “Electromagnetic cosine-Gaussian Schell-model beams in free space and atmospheric turbulence,” Opt. Express 21, 27246–27259 (2013). [CrossRef]

90.

Y. Gu, O. Korotkova, and G. Gbur, “Scintillation of nonuniformly polarized beams in atmospheric turbulence,” Opt. Lett. 34, 2261–2263 (2009). [CrossRef]

91.

A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincaré beams,” Opt. Express 18, 10777–10785 (2010). [CrossRef]

92.

Y. Gu and G. Gbur, “Reduction of turbulence-induced scintillation by nonuniformly polarized beam arrays,” Opt. Lett. 37, 1553–1555 (2012). [CrossRef]

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(030.7060) Coherence and statistical optics : Turbulence
(060.2605) Fiber optics and optical communications : Free-space optical communication

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: July 16, 2014
Manuscript Accepted: July 18, 2014
Published: August 25, 2014

Citation
Greg Gbur, "Partially coherent beam propagation in atmospheric turbulence [Invited]," J. Opt. Soc. Am. A 31, 2038-2045 (2014)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-31-9-2038


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