Scintillation index of flat-topped Gaussian laserbeam in strongly turbulent medium

Hamza Gerçekcioğlu1 and Yahya Baykal2,*1Communications and Electronics Department, Prime Ministry Undersecretariat for Maritime Affairs,

Gazi Mustafa Kemal Blv. No.128, 06100 Maltepe Ankara, Turkey2Department of Electronic and Communication Engineering, Çankaya University,

Öğretmenler Cad. No. 14, Yüzüncüyıl, 06530 Balgat, Ankara, Turkey*Corresponding author: y.baykal@cankaya.edu.tr

Received December 20, 2010; revised May 6, 2011; accepted May 22, 2011;posted May 25, 2011 (Doc. ID 139945); published July 6, 2011

In a strongly turbulent medium, the scintillation index of flat-topped Gaussian beams is derived and evaluated. Inthe formulation, unified solution of Rytov method is utilized. Our results correctly reduce to the existing strongturbulence scintillation index of the Gaussian beam, and naturally to spherical and plane wave scintillations.Another checkpoint of our result is the scintillation index of flat-topped Gaussian beams in weak turbulence.Regardless of the order of flatness, scintillations of flat-topped Gaussian beams in strong turbulence are foundto be determined mainly by the small-scale effects. For large-sized beams in moderate and strongly turbulentmedium, flatter beams exhibit smaller scintillations. © 2011 Optical Society of America

OCIS codes: 010.1330, 010.1300, 010.3310, 060.4510.

1. INTRODUCTIONShort link atmospheric optics telecommunication systemshave been used widely in the last decade. Employments ofthese systems are in the access connections of a telecommu-nications infrastructure, which usually span horizontal linkdistances of a few kilometers. Since they determine the per-formance criteria, it is very important to understand thebehavior of the intensity fluctuations in these links, whichare formulated by the scintillation index [1–11]. On the otherhand, flat-topped Gaussian beam incidences [12–15] are in-troduced in weakly turbulent links to understand whetherthey will have advantages in lowering the scintillation noise[16–18]. The current challenge in atmospheric optics telecom-munication is to employ the technology in long links severaltens of kilometers long, e.g., in the backbone networks. Over-coming the turbulence degradation in long links is of consid-erable importance because at such link lengths one has to dealwith strong turbulence. Many researchers studied the effectsof turbulence on the received signal in the strong regime[11,19–23]. These studies cover second-order moments, suchas beam wander, and fourth-order moments, such as the scin-tillation index for spherical, plane wave, and Gaussian beamincidences. Knowing that the type of incidence can vary thebehavior of scintillations [11,21], we reported the annularbeam scintillations in strong turbulence [24] in which weobserved that collimated annular beams do not provide scin-tillation advantage over the spherical, plane wave, and colli-mated Gaussian beam excitations in strong turbulence. In thispaper, we formulate and evaluate the scintillation index offlat-topped Gaussian beams in strongly turbulent atmosphere.Our intention is to seek favorable incident field profilesthat reduce scintillations in long-distance atmospheric opticslinks.

2. FORMULATIONUnified solution of Rytov method for spherical waves, planewaves, and Gaussian beam waves is given in [11,21], wherethe scintillation index is elegantly obtained in weak, moder-ate, and strong turbulence conditions in one solution. Thissolution merges the available asymptotic results valid for theweak and strong scintillations in a unified equation. We haveapplied this unified theory to annular incidence (the details ofwhich formulation are provided in [24]). In this section, theoutline of the formulation of strong turbulence scintillationindex of a flat-topped Gaussian beam is represented.

Employing the log-amplitude correlation functions for thegeneral beam types [25] and reducing it to flat-toppedGaussian incidence, we have previously reported the on-axisscintillation index of flat-topped Gaussian beams in weakturbulence at the receiver plane as [16]

m2wt ¼ 4πRe

�ZL

0dη

Z∞

0κdκ

Z2π

0dθ½G1ðη; κ; θÞ

þ G2ðη; κ; θÞ�ΦnðκÞ�: ð1Þ

Here, Re represents the real part, η is the variable for the dis-tance along the propagation axis, ðκ; θÞ is the polar coordinaterepresentation of two-dimensional spatial frequency, L is thelink length,ΦnðκÞ is the spectral density of the index of refrac-tion fluctuations,

G1ðL; η; κÞ ¼ −

k2

D2ðLÞXNn1¼1

XNn2¼1

ð−1Þn1þn2

ð1þ iαn1LÞð1þ iαn2

LÞ

×

� N

n1

�� N

n2

�exp

�−

iðL − ηÞð1þ iαn1ηÞκ2

2kð1þ iαn1LÞ

�

× exp�−

iðL − ηÞð1þ iαn2ηÞκ2

2kð1þ iαn2LÞ

�ð2Þ

1540 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 H. Gerçekcioğlu and Y. Baykal

1084-7529/11/081540-05$15.00/0 © 2011 Optical Society of America

G2ðL; η; κÞ ¼ k2

jDðLÞj2XNn1¼1

XNn2¼1

�−1

�n1þn2

×

� N

n1

�� N

n2

�1

ð1þ iαn1LÞð1 − iαn2

LÞ

× exp�−

iðL − ηÞð1þ iαn1ηÞκ2

2kð1þ iαn1LÞ

�

× exp

�iðL − ηÞð1 − iαn2

ηÞκ22kð1 − iαn2

LÞ�; ð3Þ

DðLÞ ¼XNn¼1

ð−1Þn−1�Nn

�1

ð1þ iαnLÞ; ð4Þ

where i ¼ ð−1Þ0:5, αn ¼ 1=ðkα2snÞ ¼ n=ðkα2sÞ, k ¼ 2π=λ is thewave number, λ is the wave length, αsn ¼ αs=

ffiffiffin

prepresents

the source size of the nth beam, and N is the number of Gaus-sian beams forming the flat-topped Gaussian incident field,which is expressed as [13]

usðsx; syÞ ¼XNn¼1

ð−1Þn−1N

�Nn

�exp

�−

�nðs2x þ s2yÞ

2α2s

��; ð5Þ

ðsx; syÞ being the source transverse coordinate.Application of the unified theory [11,21] to the flat-topped

Gaussian incidence, which extends the solution to strong tur-bulence as well, requires that the effective spectral density ofthe index of refraction fluctuations,

Φn;eðκÞ ¼ 0:033C2nκ−11=3

�exp

�−

κ2κ2x

�þ κ11=3ðκ2 þ κ2yÞ11=6

�; ð6Þ

replacesΦnðκÞ in Eq. (1). As known [11,21], the effective spec-tral density introduces amplitude spatial filtering described bythe large-scale κx and the small-scale κy spatial frequency cut-offs. In writing Eq. (6), the inner and outer scales of turbu-lences are taken to be zero and infinity, respectively. WhenEq. (6) is inserted into Eq. (1) and the scintillation index inEq. (1) is solved, the first and the second terms of Eq. (6) yieldthe large-scale m2

LS and the small-scale m2SS scintillation in-

dices, respectively. After solving the integral over κ and sim-plifying, m2

LS and m2SS are found to be

m2LS ¼ 8:7k2C2

nRe

�D−2ðLÞ

XNn1¼1

XNn2¼1

ð−1Þn1þn2

� N

n1

�� N

n2

�1

ð1þ iαn1LÞð1þ iαn2

LÞ

×Z

L

0dη

�iðL − ηÞ

2k

�1þ iαn1

η1þ iαn1

Lþ 1þ iαn2

η1þ iαn2

L

�

þ 0:5�

1

κ2xn1þ 1

κ2xn2

��5=6

− jDðLÞj −2XNn1¼1

XNn2¼1

ð−1Þn1þn2

×

� N

n1

�� N

n2

�1

ð1þ iαn1LÞð1 − iαn2

LÞ

×Z

L

0dη

�iðL − ηÞ

2k

�1þ iαn1

η1þ iαn1

L� 1 − iαn2

η1 − iαn2

L

�

þ 0:5

�1

κ2xn1þ 1κ2xn2

��5=6

�; ð7Þ

m2SS ¼ 1:39k2C2

nRe� ����DðLÞ

����−2 XN

n1¼1

XNn2¼1

ð−1Þn1þn2

ð1þ iαn1LÞð1 − iαn2

LÞ

×

� N

n1

�� N

n2

�X∞n¼0

Γð−nþ 5=6Þðκyn1 κyn2 Þn−5=6

×Z

L

0dη

�−

iðL − ηÞð1þ iαn1ηÞ

2kð1þ iαn1LÞ þ iðL − ηÞð1 − iαn2

ηÞ2kð1 − iαn2

L�

n

− D−2ðLÞXNn1¼1

XNn2¼1

ð−1Þn1þn2

ð1þ iαn1LÞð1þ iαn2

L� N

n1

�

� N

n2

�X∞n¼0

Γð−nþ 5=6Þðκyn1 κyn2 Þn−5=6

×Z

L

0dη

�−

iðL − ηÞð1þ iαn1ηÞ

2kð1þ iαn1LÞ

−

iðL − ηÞð1þ iαn2ηÞ

2kð1þ iαn2LÞ

�n�: ð8Þ

Here, Γð:Þ is the gamma function, κx1 ; κx2……; κxN are re-spectively the large-scale spatial frequency cutoffs of thefirst, second, …, and Nth beams composing the flat-toppedGaussian beam. Similarly, κy1 ; κy2……; κyN are the small-scalespatial frequency cutoffs of the first, second, …, and Nthbeams composing the flat-topped Gaussian beam. We notethat integrating over κ yields an infinite sum in Eq. (8) thatoriginated from the expansion of the exponential terms ofEqs. (2) and (3). In the calculation of the infinite sum, enoughterms are taken such that the contribution of the additionalterm is negligible in constructing the final value of m2

SS inEq. (8). An infinite sum is not observed in Eq. (7) becausewhen Eq. (6) is examined it can be understood that in findingm2

LS, the integration over κ can be taken without the need forthe expansion of the exponential terms of Eqs. (2) and (3).

In finding κxℓ and κyℓ , we assume that each Gaussian beamcomposing the flat-topped Gaussian beam is treated by theturbulent atmosphere individually, thus for each individualGaussian beam, corresponding large-scale and small-scalespatial frequency cutoffs are different. Thus, employing thesame procedure given in [11], large-scale and small-scale spa-tial frequency cutoffs of the ℓth Gaussian beam are found forℓ ¼ 1; 2;…; N to be

κxℓ ¼B−0:58<

:βσRmℓ

12=7 þ 1:12βσ12=5Rh

1þ2:17

B2

α4sℓþB2

i6=7

9=;

1=2 ; ð9Þ

κyℓ ¼ B−0:5

�3

�σRmℓ

�12=5

þ 2:07σ12=5R

�0:5; ð10Þ

where σ2R ¼ 1:23C2nk7=6L11=6 is the scintillation index in weak

turbulence for plane wave, found by the Rytov method, C2n

is the structure constant of the horizontal atmospheric opticslink,

β ¼�13−

0:5B2

α4sℓ þ B2 þ 0:2

�B2

α4sℓ þ B2

�2�6=7

; and B ¼ L=k:

mℓ appearing in Eqs. (9) and (10) is found frommℓ ¼ ðmwtÞ1=2, where m2

wt is the scintillation index given by

H. Gerçekcioğlu and Y. Baykal Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1541

Eqs. (1)–(4) evaluated at n1 ¼ n2 ¼ ℓ, i.e., evaluated for the ℓthGaussian beam. Finally, using the unified Rytov solution[11,21], the scintillation index of the flat-topped Gaussianbeam in strong turbulence is given by

m2 ¼ expðm2LS þm2

SSÞ − 1: ð11Þ

3. RESULTSResults represented in this section are obtained by evaluatingEq. (11). The plots provided are for λ ¼ 1:55 μm,C2

n ¼ 5 × 10−14 m−2=3. The horizontal axes of Figs. 1–5, exceptFigs. 2(b) and 3(b), are the square root of Rytov plane wave

scintillations, which is the measure of turbulence strength. InFig. 1, we check the results obtained by using Eq. (11) with theexisting results [11] for flat-topped Gaussian beam scintilla-tions in strong turbulence. Note that the two curves markedas flat-topped Gaussian in Fig. 1 are actually for Gaussianbeams since N ¼ 1. This is done to emphasize that all thecurves in Fig. 1 are obtained from our flat-topped scintillationindex formula of Eq. (11). Being the limiting cases, the curvesfor the spherical, plane wave, and Gaussian beams correctlymatch the plots in [11], as expected. The on-axis scintillationindex of the collimated Gaussian beam for the power-lawturbulent spectra is known [26] to be fully determined by thetwo-dimensional parameters, σR and N−1

Fn, inverse Fresnelnumbers. For this purpose, Figs. 2(a) and 3(a) are presentedversus σR and Figs. 2(b) and 3(b) are presented versus N−1

Fn.Figure 2 shows that, for relatively small-sized flat-toppedGaussian beams, as the number of beams increases, the scin-tillation index slightly increases in strong turbulence andapproaches a saturation value. In Fig. 3, when the source sizeis increased, flat-topped Gaussian beams formed by a largernumber of beams become advantageous, in terms of scintilla-tions, in moderate and strong turbulence. However, when tur-bulence strength is further increased, this behavior reverses,i.e., flat-topped Gaussian beams formed by larger numbers ofbeams turn out to be disadvantageous. Combining the infor-mations in Figs. 2 and 3, it is observed especially in stronglyturbulent media that the small-scale turbulence effects dom-inate the large-scale turbulence effects in determining thescintillations of flat-topped Gaussian beams. We note that ourstrong turbulence results obtained from Eq. (11) in the weak

Fig. 1. Scintillation index in strong turbulence versus the square rootof the Rytov plane wave scintillation index for plane, spherical, andflat-topped Gaussian beams (N ¼ 1) with αs ¼ 1 cm and αs ¼ 2 cm.

Fig. 2. Scintillation index in strong turbulence of flat-toppedGaussian beams at N ¼ 1; 5; 15 and αs ¼ 1 cm with the large- andsmall-scale scintillation indices versus (a) αR, (b) N−1

Fn.

Fig. 3. Scintillation index in strong turbulence of flat-toppedGaussian beams at N ¼ 1; 5; 10; 15 and αs ¼ 2 cm with the large- andsmall-scale scintillation indices versus (a) αR, (b) N−1

Fn.

1542 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 H. Gerçekcioğlu and Y. Baykal

turbulence limit (σ2R ≪ 1) are found to correctly match theflat-topped Gaussian beam scintillations in weak turbulence[16], which, in turn, match the spherical, plane wave, andGaussian beam scintillations in weak turbulence. Figure 4indicates that, at a fixed turbulence strength in the strongregime, as the flat-topped Gaussian beam, composed of thesame beam number, possesses relatively larger beam size, thescintillations decrease. However, at extremely strong turbu-lence levels, the scintillations for all the beam sizes seemto merge to similar saturation values. In Fig. 4, at certain finitenonzero αs, large values of scintillation index are observed atcertain αR, which can be attributed to the nature of the unifiedRytov solution [11,21] in the moderate turbulence regime.From the curve of αs ¼ 2:8 cm, the peak value of the scintilla-tion index occurs at the Fresnel number of 0.813, which cor-responds to an inverse Fresnel number of N−1

Fn ¼ 1:23. Forvery large αs, i.e., for αs ¼ 25 cm, the peak disappears and thescintillation index returns to the much smaller plane wave re-sults, as expected.

Comparing Fig. 4, which reflects the scintillations offlat-topped Gaussian formed by five beams (N ¼ 5), to Fig. 5,evaluated for 10 beams (N ¼ 10), the conclusion is that thescintillations do not show drastic variations at all the sourcesizes. A similar argument made for Fig. 4 can be repeated forFig. 5. At certain finite nonzero αs, large values of scintillationindex are observed at certain αR. The peak value of the scin-tillation index of αs ¼ 2:4 cm curve occurs at the Fresnel num-ber of 0.705, which corresponds to an inverse Fresnel numberof N−1

Fn ¼ 1:42. For very large αs, i.e., for αs ¼ 25 cm, the peak

disappears and the scintillation index attains the plane wavevalues.

The parameters chosen for Figs. 6 and 7 are respectively formoderate and strong turbulence where the curves are plottedversus the source size of the flat-topped Gaussian beam at linkdistances of 1 and 2 km, respectively. σR is 1 in Fig. 6 and 1.88in Fig. 7. The trend in Figs. 6 and 7 is that the scintillationindex increases in strong turbulence as the size of the flat-topped Gaussian beam becomes substantially large, even-tually reaching a saturation value that is smaller for shorterlink lengths. This trend is the same in strong turbulence forall flat-topped Gaussian incidences composed of any numberof beams. The saturation value in Fig. 6 is not unity since theturbulence is moderate, however, we observe unity saturationin Fig. 7, where the turbulence is in the strong regime. Anotherobservation from Figs. 6 and 7 is that, in strong turbulence, ata fixed source size, as the number of beams increases, i.e., asthe beam becomes flatter, the scintillation index decreases.Figure 6 is closer to the weak scintillation case and the kinksseen at around N−1

Fn ≈ 1 for N ¼ 1 (i.e., Gaussian beam) [11]and flat-topped Gaussian beams [16] are known for pureRytov solution. Saturation to the spherical and plane wave re-sults for large and zero αs, respectively, are observed. Thesecan be repeated for Fig. 7, this time spherical and plane scin-tillations reflect saturation values in strong turbulence. Thecommon trend in Figs. 4–7 is that sharp maxima in the scin-tillation index are seen for the large N when the order of mag-nitude of N−1

Fn is around unity. Such peaks disappear when theflat-topped beam source size is taken very large to approach

Fig. 5. Scintillation index in strong turbulence of flat-topped beamswith N ¼ 10 at selected values of source size.

Fig. 6. Scintillation index in strong turbulence versus the source sizeof flat-topped Gaussian beams with N ¼ 1; 5; 10; 15 at L ¼ 1 km.

Fig. 7. Scintillation index in strong turbulence versus the source sizeof flat-topped Gaussian beams with N ¼ 1; 5; 10; 15 at L ¼ 2 km.

Fig. 4. Scintillation index in strong turbulence of flat-topped beamsat N ¼ 5 at selected values of source size.

H. Gerçekcioğlu and Y. Baykal Vol. 28, No. 8 / August 2011 / J. Opt. Soc. Am. A 1543

the plane wave results. The occurrence of sharp maxima isattributed to the behavior of the unified Rytov solution in theunity inverse Fresnel number range, which is also observedfor flat-topped beams in weak turbulence when pure Rytovsolution is employed. We note that such maxima in thescintillation index for flat-topped Gaussian beams in weakturbulence are also reported in a less emphasized mannerin our previous study, where we have utilized the extendedHuygens–Fresnel principle [27].

4. CONCLUSIONWith the motivation of understanding whether employingflat-topped Gaussian beams may bring some advantages in re-ducing the scintillation noise in long horizontal atmosphericoptics telecommunications links, we evaluated the scintilla-tion index of flat-topped Gaussian incidence in strong turbu-lence. Results indicate that, at large source sizes, thescintillations of flat-topped beams composed of a large num-ber of Gaussian beams are smaller than the scintillations offlat-topped beams composed of a smaller number of Gaussianbeams. When the inverse Fresnel number is around the rangeof unity, the scintillations in strong turbulence exhibit peakvalues that are larger for larger N . This behavior is thoughtto be related to the nature of unified Rytov solution sincethe same trend is also observed in weak turbulence fluctua-tions of flat-topped Gaussian beams of large N when the pureRytov method is applied. Our conclusion is that, under the ap-propriately chosen beam parameters, such as the source sizeand the number of Gaussian beams, flat-topped Gaussianbeams can provide scintillation advantage in strongly turbu-lent atmosphere. Our results can be applied in atmosphericoptics telecommunication links in the backbone networks,which could be a potential alternative to the existing radiolinks, providing incomparably high data rate advantage.

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1544 J. Opt. Soc. Am. A / Vol. 28, No. 8 / August 2011 H. Gerçekcioğlu and Y. Baykal