Color changes in stochastic light fields propagating innon-Kolmogorov turbulence
Olga Korotkova1,* and Elena Shchepakina2
1Department of Physics, University of Miami, 1320 Campo Sano Drive, Coral Gables, Florida 33146, USA2Department of Technical Cybernetics, Samara State Aerospace University, Molodogvardeiskaya 151, Samara 443001, Russia
The seminal work on the very possibility of optical fieldsgenerated by fluctuating, statistically stationary sourcespropagating in vacuum to exhibit changes in their spec-tral composition was written by Wolf in 1986 [1]. Sincethen, this phenomenon has been widely explored, espe-cially in connection with astronomy [2]. Spectral shifts oflight waves have also been demonstrated to occur onscattering from media, confined in some region in freespace, whether being of a continuous [3,4] or a particu-late [5] nature, and also on propagation within mediawith a variety of refractive properties (see [6]). Some stu-dies discovered the phenomenon that the direction of thespectral shift switched after a light beam passed throughan aperture [7], a convergent lens [8], or an interface be-tween media with positive and negative phase materials[9]. The dependence of spectral shifts in beamlike sto-chastic fields on polarimetric properties of their sourceswas explored in [10,11]. More importantly, spectralchanges in light fields can play an important role for sol-ving inverse problems of determining the properties ofscattering media [12]. We stress here that the sourcecorrelation-induced changes, being an entirely linearphenomenon, must be distinguished from the Doppler-like spectral shifts.While an enormous body of literature exists that dis-
cusses various aspects of optical beam propagation inatmospheric turbulence (see the well-known books[13–15]), only several studies examine specifically thespectral changes [16–21]. These results explored in somedetail modulation in the spectra for beams generated byvarious sources and used the assumption that turbulencehas a classic Kolmogorov structure, being characterizedby the fractal spatial power spectrum with slope −11=3.This Letter deals with spectral changes in light beamspropagating in the non-Kolmogorov turbulence [22], i.e.,with generalized slope α, 3 < α < 5.We begin by reviewing basic equations characterizing
the changes in the spectral composition of optical beamsin random media. Suppose a planar, secondary source[23] located in the plane z ¼ 0 generates a highly direc-tional beam propagating into the half-space z > 0, whichcontains atmosphere governed by non-Kolmogorov sta-tistics. The fluctuations in the beam in the source planemay be characterized by the cross-spectral densityfunction of the form [23] W ð0Þðr01; r02;ωÞ ¼ hU ð0Þ�ðr01;ωÞ×
U ð0Þðr02;ωÞi, where r01 ¼ ðx01; y01; 0Þ, r02 ¼ ðx02; y02; 0Þ, � de-notes complex conjugate, and U ð0Þðr;ωÞ is a memberof a statistical ensemble, denoted by h·i, of monochro-matic realizations at angular frequency ω, or, using therelation ω ¼ cλ=2π, λ being the wavelength of the formWð0Þðr01; r02; λÞ ¼ hU ð0Þ�ðr01; λÞU ð0Þðr02; λÞi. Then the spectraldensity of the field in the source plane isS0ðr0; λÞ ¼ Wðr0; r0; λÞ, and its normalized version hasthe form
S0Nðr; λÞ ¼ S0ðr0; λÞ=
Z∞
0
S0ðr0; λÞdλ:
Upon propagation from the source plane to any planez > 0, the cross-spectral density function takes the form
Wðr1; r2; λÞ ¼Z Z Z Z
W0ðr01; r02; λÞ
×Kðr01; r02; r1; r2; λÞd2r01d2r02; ð1Þ
where Kðr01; r02; r1; r2; λÞ is the propagator, depending onthe Green’s function of the random medium, of the form[14,24]
Kðr01; r02; r1; r2; λÞ ¼�1λz
�2exp
�−πi ðr1 − r01Þ2 − ðr2 − r02Þ2
λz
�
× exp
�−4π4z3λ2
�ðr1 − r2Þ2
þ ðr1 − r2Þðr01 − r02Þ þ ðr01 − r02Þ2�
×Z∞
0
κ3ΦnðκÞdκ�; ð2Þ
where ΦnðκÞ is the one-dimensional power spectrum offluctuations in the refractive index of the turbulent med-ium, having, for the non-Kolmogorov case, the form [22]
where the term ~C2n is a generalized refractive-index struc-
ture parameter with units m3−α,
κ0 ¼2πL0
; κm ¼ cðαÞl0
;
cðαÞ ¼�2π3Γ�5 −
α2
�AðαÞ
� 1α−5;
AðαÞ ¼ 1
4π2 Γðα − 1Þ cos�απ
2
�;
where L0 and l0 are the outer and the inner scales of tur-bulence, respectively, and Γð·Þ is the Gamma function.For the power spectrum (3) the integral in expression(2) becomes
I ¼Z∞
0
κ3ΦnðκÞdκ
¼ AðαÞ2ðα − 2Þ
~C2nκ2−αm β exp
� κ20κ2m
�
× Γ�2 −
α2;κ20κ2m
�− 2κ4−α0 ; ð4Þ
where β ¼ 2κ20 − 2κ2m þ ακ2m and Γð·; ·Þ denotes the incom-plete Gamma function.In what follows we are interested in the evaluation of
the normalized spectral density of the beam at distancez ≥ 0 from the source plane and at any transverse loca-tion ðx; yÞ, given by the expression [23]
SNðr; λÞ ¼ Sðr; λÞ=Z∞
0
Sðr; λÞdλ; ð5Þ
where Sðr; λÞ ¼ Wðr; r; λÞ is the spectral density of thefield at position r ¼ ðx; y; zÞ. Substituting Eq. (1) intoEq. (5) and using the result in Eq. (4), one can tracethe evolution of the spectral density in the medium of in-terest. Further, the shifted central wavelength of thebeam can be found from the expression [21]
λ1ðrÞ ¼Z∞
0
λSðr; λÞdλ=Z∞
0
Sðr; λÞdλ: ð6Þ
The normalized spectral shift at position r may be quan-tified by ϱðrÞ ¼ λ1ðrÞ−λ0
λ0 being blue if its value is positiveand red if its value is negative. Here λ0 is the centralwavelength of the source, which we assume to be posi-tion independent.To illustrate the dependence of the spectral changes
on parameter α numerically, we employ the isotropicGaussian–Schell-model beams [23]. The cross-spectraldensity matrix of such a beam in the source plane z ¼0 has the form
Wð0Þðr01; r02; λÞ ¼ I0ðλÞ exp�−ðr01Þ2 þ ðr02Þ2
4σ2�
× exp
�−ðr01 − r02Þ2
2δ2�;
where the values of the parameters must obey the beamconditions [23]. Without loss of generality we assume
that the initial spectral composition consists of a singleGaussian spectral line, i.e., I0ðλÞ ¼ exp½−ðλ − λ0Þ2=ð2Λ2Þ�, with a peak value of 1, being centered at wave-length λ0, and having rms width Λ. It was found in[25] that, for an arbitrary power spectrum of fluctua-tions in the refractive index, the spectral density of aGaussian–Schell-model beam has the form
Sðr; λÞ ¼ I0ðλÞΔ2ðzÞ exp
�−
r2
2σ2Δ2ðzÞ�;
Δ2ðzÞ ¼ 1þ� λz2πσ
�2�1
4σ2 þ1
δ2�þ 2π2z3I
3σ2 ; ð7Þ
with I being defined in Eq. (4).On substituting Eq. (7) into Eqs. (5) and (6) it is pos-
sible to determine spectral density and spectral shift, re-spectively, in Gaussian–Schell-model beams propagatingin the non-Kolmogorov turbulence. Because of the in-verse dependence of Sðr; λÞ on λ [see Eq. (7)], it is clearthat the spectrum is broadened as the beam propagates.Moreover, since the central wavelength λ1ðrÞ and, hence,the shift ϱðrÞ also generally vary with z and λ, the spec-trum is expected to be shifted, with the shift being diffi-cult to assert qualitatively. Hence, in what follows, weillustrate the spectral shifts by numerical examples.
We use the following parameters for the source and theatmosphere, unless other parameters are specified in thefigure captions: λ0 ¼ 0:5435 × 10−6 m, Λ ¼ λ0=3, σ ¼10−2 m, δ ¼ 10−3 m, ~C2
n ¼ 10−13 m3−α, L0 ¼ 1 m, andl0 ¼ 10−3 m.
Figure 1 shows the changes in the on-axis behavior ofthe normalized spectral density SN in the source planeand in the field, calculated at three different distancesfrom the source. We note that, for very steep power spec-tra (α ¼ 3:01 and α ¼ 3:1), the spectral density graduallyrecovers with propagation distance from the source. Thiseffect is delayed for a Kolmogorov power spectrum(α ¼ 3:67) and completely disappears for flatter powerspectra (α ¼ 4:99).
In Fig. 2 we show the density plot of the actual centralwavelength λ1 and the contour plot of its normalized shiftϱ as a function of z and r ¼ jrj for several values of slopeα of the atmospheric power spectrum. In the region closeto the z axis, the blueshift of the spectrum is well pro-nounced for α ¼ 4:9, but is somewhat compensated by
Fig. 1. Normalized spectral density SN (unitless) as a functionof λ for r ¼ jrj ¼ 0 and different α: α ¼ 3:01 (dashed), α ¼ 3:1(dotted), α ¼ 3:67 (dotted–dashed), and α ¼ 4:9 (solid thin) atz ¼ 0:5 km, z ¼ 1 km, and z ¼ 5 km. Solid thick curve showsthe normalized spectral density in the source plane.
the atmosphere. The best compensation is provided inthe case of α ¼ 3:1. For the Kolmogorov’s turbulence(α ¼ 3:67), the compensation is partial, in accordancewith [21]. Although the propagating beam is shown formost of its transverse cross section, the approximationused in derivation of Eq. (2) prevents rigorous anal-ysis of regions far from the beam axis (see [26–28]). How-ever, we show the full cross section so the reader canbest see the change in the color in the neighborhoodof the axis.In summary, we have analyzed the dependence of
correlation-induced spectral changes in light beamspropagating in non-Kolmogorov atmospheric turbulenceon fractal dimension α (3 < α < 5). The provided numer-ical example involved a stochastic beam with initial cen-tral wavelength in the green zone, which visuallydemonstrated spectral shifts and switches. We have dis-covered that, in comparison with classic Kolmogorov tur-bulence (αK ¼ 11=3) in which the spectral changes areknown to be partially suppressed, spectral changes aremitigated much stronger for 3 < α < αK and weakerfor αK < α < 5. In particular, we found that the atmo-sphere reconstructs the original spectral compositionof the beam most strongly for α ¼ 3:1, the value of thepower spectrum at altitudes 2–8 km, and least stronglyfor α ¼ 4:99, which corresponds to higher layers of theatmosphere (8–20 km) [29]. While along the optical axisthe original blueshift can be suppressed only on propa-gation, resulting in a blue–red spectral switch, for off-axis
positions initial redshift is a possibility, which turns toblueshift after propagation, resulting in a red–blueswitch. Thus, both types of spectral switches have beenfound.
The ability of non-Kolmogorov turbulence to stronglymitigate spectral changes originally caused by sourcecorrelations may have crucial consequences for spectro-scopic measurements relating to astrophysics and atmo-spheric remote sensing.
O. Korotkova’s research was funded by the U.S. AirForce Office of Scientific Research (USAFOSR) throughgrant FA 95500810102 and the U.S. Office of NavalResearch (ONR) through grant N0018909P1903.E. Shchepakina’s research was funded by the RussianFoundation of Fundamental Investigations through grant10-08-00154a.
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Fig. 2. Density plots of actual spectral shift λ1 overlapped withcontour plots of normalized spectral shift ϱ ¼ λ1−λ0
λ0 as a functionof z (horizontal axis, in meters) and r (vertical axis, in meters)for (a) α ¼ 3:01, (b) α ¼ 3:10, (c) α ¼ 3:67, and (d) α ¼ 4:99.
