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1834 J. Opt. Soc. Am. A/Vol. 27, No. 8 /August 2010 Gerçekcioglu et al.

Annular beam scintillations in strong turbulence

Hamza Gerçekcioglu,1 Yahya Baykal,2,* and Cem Nakiboglu3

1Prime Ministry Undersecretariat for Maritime Affairs, Communications and Elektronics Department, Gazi MustafaKemal Blv. No. 128, 06100 Maltepe, Ankara, Turkey

2Department of Electronic and Communication Engineering, Çankaya University, Ögretmenler Cad. No. 14,Yüzüncüyıl, 06530 Balgat, Ankara, Turkey

3Gazi University, Electrical and Electronics Engineering Department, 06500 Teknikokullar, Ankara, Turkey*Corresponding author: y.baykal@cankaya.edu.tr

Received December 11, 2009; revised June 16, 2010; accepted June 18, 2010;posted June 22, 2010 (Doc. ID 121369); published July 22, 2010

A scintillation index formulation for annular beams in strong turbulence is developed that is also valid in mod-erate and weak turbulence. In our derivation, a modified Rytov solution is employed to obtain the small-scaleand large-scale scintillation indices of annular beams by utilizing the amplitude spatial filtering of the atmo-spheric spectrum. Our solution yields only the on-axis scintillation index for the annular beam and correctlyreduces to the existing strong turbulence results for the Gaussian beam—thus plane and spherical wave scin-tillation indices—and also correctly yields the existing weak turbulence annular beam scintillations. Com-pared to collimated Gaussian beam, plane, and spherical wave scintillations, collimated annular beams seemto be advantageous in the weak regime but lose this advantage in strongly turbulent atmosphere. It is ob-served that the contribution of annular beam scintillations comes mainly from the small-scale effects. At afixed primary beam size, the scintillations of thinner collimated annular beams compared to thicker collimatedannular beams are smaller in moderate turbulence but larger in strong turbulence; however, thinner annularbeams of finite focal length have a smaller scintillation index than the thicker annular beams in strong tur-bulence. Decrease in the focal length decreases the annular beam scintillations in strong turbulence. Examin-ing constant area annular beams, smaller primary sized annular structures have larger scintillations in mod-erate but smaller scintillations in strong turbulence. © 2010 Optical Society of America

OCIS codes: 010.1330, 010.1300, 010.3310, 060.4510.

tetwda

2Ili

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g

. INTRODUCTIONntensity fluctuations of Gaussian beams in weak turbu-ence have been studied for quite some time[1–6]. Theseources basically report the scintillation index of Gauss-an beams, spherical and plane waves. For such beams,eak turbulence results are extended to include the scin-

illation index variations in strong atmospheric turbu-ence [3] as well. In this solution the Rytov method is em-loyed to find the small-scale and large-scale scintillationndices by incorporating an effective atmospheric spec-rum modified by an amplitude spatial filter. This ap-roach yields reasonable scintillation results for Gauss-an, plane, and spherical beams not only in weakurbulence but also in the moderate and strong turbulentegimes. Various aspects of beam degradation in strongurbulence are also studied by many researchers [7–10].mong different types of beams [11], annular beams arelso introduced to weakly turbulent media by findingheir correlation functions [12] and scintillation indices13–15]. Annular beams are interesting in that in weakurbulence, compared to their Gaussian beam counter-arts, finite focal length versions provide advantageouscintillations starting at relatively small propagationengths, and collimated versions become favorable atonger propagation lengths [14]. In the current paper wenvestigated the on-axis scintillations of annular beams introng turbulence by applying the extension of the Rytovethod and finding the small-scale and large-scale scin-

1084-7529/10/081834-6/$15.00 © 2

illation indices for annular beams through the use of theffective atmospheric spectrum that involves the ampli-ude spatial filter. Our motivation is to understandhether annular beams will bring any advantage in re-ucing scintillation noise when they are employed in longtmospheric optical communication links.

. FORMULATIONn the modified Rytov solution in which the strong turbu-ence scintillation index is derived, the total scintillationndex is obtained by the following formula [3]

m2 = exp�mLS2 + mSS

2 � − 1, �1�

here mLS2 and mSS

2 are, respectively, large-scale andmall-scale log-irradiance scintillation indices. Here mLS

2

s formulated by using the large-scale portion of the effec-ive atmospheric spectrum in the Rytov solution of annu-ar scintillation index. Similarly, mSS

2 is formulated by us-ng the small-scale portion of the effective atmosphericpectrum in the Rytov solution of annular scintillation in-ex.The annular beam incident field at the source plane is

iven by [12]

us�s� = us�sx,sy� = �l=1

2

Al exp�− �0.5k�lsx2�

− �0.5k�lsy2��, taking A1 = − A2 = 1,

010 Optical Society of America

wp=lsptttLGm

ba

wtin�

Htlfnt

li���

wLtsc

w−atsanwfaissi

m

Gerçekcioglu et al. Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. A 1835

=exp�− �0.5k�1sx2� − �0.5k�1sy

2��

− exp�− �0.5k�2sx2� − �0.5k�2sy

2��, �2�

here s= �sx ,sy� is the transverse coordinate at the sourcelane, with sx ,sy representing the x and y components; j�−1�0.5, k=2� /� is the wave number, � being the wave-

ength; �1=1/ �k�s12 �+ j /F1, �s1 and F1 are the Gaussian

ource size and focusing parameter of the symmetricalrimary beam; and A1, A2 are the complex amplitudes ofhe fields of the primary and the secondary source beamshat constitute the annular beam. In our evaluations weook A1=−A2=1 as expressed in the second line of Eq. (2).ikewise, �2=1/ �k�s2

2 �+ j /F2, and �s2 and F2 are theaussian source size and focusing parameter of the sym-etrical secondary beam.Using the log-amplitude correlation function of annular

eams in weak turbulence [12], we have formulated thennular beam scintillation index as [14]

m2 = 4� Re��0

L

d��0

�

�d��0

2�

d��G1�L,�,�,��

+ G2�L,�,�,����n���� , �3�

here Re denotes the real part, L is the path length, � ishe distance variable along the propagation axis, � exp�i��s the two-dimensional spatial frequency in polar coordi-ates, � being the magnitude of the spatial frequency,n��� is the unfiltered atmospheric spectrum, and

G1�L,�,�,�� = − D−2�L��l1=1

2

�l2=1

2

Al1Al2

k2

�1 + i�l1L��1 + i�l2

L�

exp−i�L − ��

2k 1 + i�l1�

1 + i�l1L

+1 + i�l2

�

1 + i�l2L��2� , �4�

G2�L,�,�,�� = D�L� −2�l1=1

2

�l2=1

2

Al1Al2

*k2

�1 + i�l1L��1 − i�l2

*L�

exp−i�L − ��

2k 1 + i�l1�

1 + i�l1L

−1 − i�l2

*�

1 − i�l2*L��2� , �5�

D�L� = �l=1

2

Al

1

�1 + i�lL�. �6�

ere * denotes the complex conjugate, and A1, A2 andheir complex conjugates obtained from the corresponding1, l2 indices of the summations in Eqs. (4) and (5) androm the l index in Eq. (6) are as defined in Eq. (2). Weote that Eqs. (3)–(6) are valid only for on-axis points. Inhe evaluation of the annular beam scintillations we fol-

owed the approach in [3] by employing the annular beamncident field given in Eq. (2), and utilized Eq. (3) with

n��� replaced by the effective atmospheric spectrumn,e���, which introduces amplitude spatial filtering.n,e��� is expressed as [3]

�n,e��� = 0.033Cn2�−11/3G��,�0,L0�, �7�

here Cn2 is the structure constant, �0 is the inner scale,

0 is the outer scale, and G�� ,�0 ,L0� is an amplitude spa-ial filter that eliminates the impact of the ineffectivecale sizes on the scintillation under strong fluctuationonditions and is given by [3]

G��,�0,L0� = Gx��,�0,L0� + Gy��,�0�

= f���0�g��L0�exp−�2

�x2� +

�11/3

��2 + �y2�11/6

, �8�

here f���0�=exp�−���0 /3.3�2��1+1.802���0 /3.3�0.254���0 /3.3�7/6� and g��L0�=1−exp�−�0.125�L0 /��2�re factors that describe, respectively, the inner-scale andhe outer-scale modifications of the basic Kolmogorovpectrum; and �x and �y are the large-scale (refractive)nd small-scale (diffractive) spatial frequency cutoffs. Weote that in our formulation, �0=0 and L0=� are taken,hich make f���0�=g��L0�=1; thus the filtering comes

rom �x and �y, which are derived below for the primarynd secondary beams of the annular structure. Substitut-ng Eq. (8) into Eq. (7), employing Eqs. (4)–(7) in Eq. (3),olving the integral over �, and simplifying, the large-cale and small-scale log-irradiance scintillation indices,.e., mLS

2 and mSS2 , are found as

mLS2 = 8.7021k2Cn

2 Re�D−2�L��l1=1

2

�l2=1

2 Al1Al2

�1 + i�l1L��1 + i�l2

L�

�0

L

d� i�L − ��

2k 1 + i�l1�

1 + i�l1L

+1 + i�l2

�

1 + i�l2L� +

1

2 1

�xl1

2

+1

�xl2

2 ��5/6

− D�L� −2�l1=1

2

�l2=1

2 Al1Al2

*

�1 + i�l1L��1 − i�l2

*L�

�0

L

d� i�L − ��

2k 1 + i�l1�

1 + i�l1L

−1 − i�l2

* �

1 − i�l2* L�

+1

2 1

�xl1

2 +1

�xl2

2 ��5/6� , �9�

SS2 = 1.385Cn

2k2 Re� D�L� −2�l1=1

2

�l2=1

2 Al1Al2

*

�1 + i�l1L��1 − i�l2

*L�

�t=0

�

�− 1�t�− t + 5/6���yl1�yl2

�t−5/6�0

L

d� i�L − ��

2k

1 + i�l1�

1 + i�l1L

−1 − i�l2

* �

1 − i�l2* L��t

− D−2�L�

wrctttktsrottaeaacstbei[ta

w

att=

=ub

alotbloGiml(dtsatnttstpdRetsfalstbbsbbotvttGtoWatwGnqqbilo

1836 J. Opt. Soc. Am. A/Vol. 27, No. 8 /August 2010 Gerçekcioglu et al.

�l1=1

2

�l2=1

2 Al1Al2

�1 + i�l1L��1 + i�l2

L��t=0

�

�− 1�t�− t + 5/6�

��yl1�yl2

�t−5/6

�0

L

d� i�L − ��

2k 1 + i�l1�

1 + i�l1L

+1 + i�l2

�

1 + i�l2L��t� , �10�

here � . � is the Gamma function, and �x1and �x2

are,espectively, the large-scale (refractive) spatial frequencyutoff for the primary and secondary beams composinghe annular beam. In a similar manner, �y1

and �y2are

he small-scale (diffractive) spatial frequency cutoff forhe primary and secondary beams, respectively. It isnown [3,7] that the cutoff spatial frequencies are foundo eliminate the midrange scale sizes in moderate andtrong atmospheric turbulence, and these cutoffs are di-ectly related to the correlation width and scattering diskf the propagating optical wave. Since, as given by Eq. (2),here are two Gaussian beams in an annular beam struc-ure, the primary and the secondary Gaussian beams aressumed to have different midrange scale sizes to beliminated in moderate and strong turbulence. From thisssumption arises the need to introduce two large-scalend two small-scale spatial frequency cutoffs. The physi-al interpretation of two large-scale and two small-scalepatial frequency cutoffs is that when the primary andhe secondary Gaussian beam composing the annulaream propagate in turbulence, each Gaussian beam willxperience different low-pass and high-pass spatial filter-ng. To find �x1

, �x2, �y1

, and �y2, the same approach as in

3] used for the Gaussian beam is applied individually forhe primary and secondary Gaussian beams forming thennular beam. This leads us to find [3]

�x�

2 =k�x�

L, � = 1,2, �11�

�Y�

2 =k�Y�

L� = 1,2, �12�

here

1

�x�

= 1

3−

1

2� +

1

5�2�6/7 �R

m��12/7

+ 1.12�1

3−

1

2� +

1

5�2

1 + 2.17��

6/7

�R12/5, �13�

�Y�= 3 �R

m��12/5

+ 2.07�R12/5, �14�

� =�1/�k2�s�

4 � + 1/F�2�L2 − L/F�

�1 − L/F��2 + �L/�k�s�2 ��2

, �15�

nd �R2 =1.23Cn

2k7/6L11/6 is the Rytov plane wave scintilla-ion index. In Eqs. (13) and (14), m1= �m�1/2, where m2 ishe scintillation index evaluated by using Eq. (3) for l1l =1, i.e, for the primary Gaussian beam, and m

2 2

�m�1/2, where m2 is the scintillation index evaluated bysing Eq. (3) for l1= l2=2, i.e, for the secondary Gaussianeam.We comment that the fields of the primary and second-

ry Gaussian beams forming the annular beam in turbu-ence are usually statistically correlated; thus the second-rder field moment of the annular beam is not equal tohe sum of the second-order field moment for the primaryeam and that for the secondary beam. Also, the scintil-ation index for the annular beam is not equal to the sumf the scintillation indices of the primary and secondaryaussian beams. Our derivations of Eqs. (9) and (10) are

n line with these comments because in our use of the un-odified classical Rytov method, we start with the annu-

ar incidence as one beam and apply the product solutionas implied by the Rytov solution) to this annular inci-ence. In other words, the scintillation index we find forhe annular beam in weak turbulence is not equal to theum of the scintillation indices of the primary and second-ry Gaussian beams. Moreover, applying the modified Ry-ov method to cover moderate and strong turbulence doesot change this fact, i.e., the scintillation index we find forhe annular beam in strong turbulence is again not equalo the sum of the scintillation indices of the primary andecondary Gaussian beams, because we again start withhe annular incidence as one beam and apply the Rytovroduct solution to this annular incidence, this time un-er the effective atmospheric spectrum. In the modifiedytov method, we need to eliminate the impact of the in-ffective scale sizes on the scintillation under strong fluc-uation conditions, thus we need to define certain large-cale spatial frequency cutoffs and small-scale spatialrequency cutoffs. The method we preferred in the evalu-tion of these spatial frequency cutoffs was to find thearge-scale spatial frequency cutoff and the small-scalepatial frequency cutoff individually for the primary andhe secondary Gaussian beams forming the annulaream. Alternatively, spatial frequency cutoffs could haveeen evaluated by employing the correlation width andcattering disk of the propagating annular beam as oneeam. Evaluations of the spatial frequency cutoffs, eithery the individual treatment of the primary and the sec-ndary Gaussian beams forming the annular beam or byhe treatment of the annular beam in total, are both notiolating the necessity that the annular beam scintilla-ion index in strong turbulence is not equal to the sum ofhe scintillation indices of the primary and secondaryaussian beams. The method applied in the evaluation of

he spatial frequency cutoffs results only in the variationf the accuracy of the obtained scintillation index values.e chose to evaluate large-scale spatial frequency cutoff

nd small-scale spatial frequency cutoff individually forhe primary and the secondary Gaussian beams becausee wanted to utilize the already known cutoffs of theaussian beam [3,7]. With this explanation, it may not beecessary to find the equivalence between the spatial fre-uency cutoffs of the annular beam and the spatial fre-uency cutoffs of the primary and secondary Gaussianeam components of the annular beam. However, exam-ning Eqs. (9) and (10), one can vaguely form an equiva-ence such that for the large-scale spatial frequency cut-ff, the 1/�2 term of the annular beam corresponds to the

x

ato

wat�s+trlEttotofitppGac

tl

3Idat=tafb

msiolbctftawvpavsttwlamlsbblbitactbpFmsbat

Fssp

Fss

Gerçekcioglu et al. Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. A 1837

12 �1/�xl1

2 +1/�xl2

2 � term of the individual beam treatment,nd for the small-scale spatial frequency cutoff, the �y

2

erm of the annular beam corresponds to the �yl1�yl2

termf the individual beam treatment.

We can derive Eqs. (9) and (10) in another way. If werite the effective spatial spectrum by employing thebove mentioned equivalences, i.e., 1

2 �1/�xl1

2 +1/�xl2

2 �erm replacing 1/�x

2 term and �yl1�yl2

term replacing

y2 term in Eqs. (7) and (8), we obtain an effective spatialpectrum of �n,e���=0.033Cn

2�−11/3�exp�−0.5�2��xl1

−2+�xl2

−2���11/3��2+�yl1

�yl2�−11/6�. In Eq. (3), if we replace the unfil-

ered atmospheric spectrum �n��� by this �n,e��� and de-ive the large-scale and small-scale log-irradiance scintil-ation indices, we arrive at the same results as shown byqs. (9) and (10). The interpretation of this could be that

he annular beam experiences spatial filtering with bothwo large-scale and two small-scale spatial frequency cut-ffs. However, spatial filtering of a single beam with bothwo large-scale or two small-scale spatial frequency cut-ffs is not meaningful, because a single beam should beltered by a single effective large-scale and a single effec-ive small-scale spatial frequency cutoff. Thus, as ex-ressed above, viewing the annular beam as being com-osed of two different Gaussian beams with eachaussian beam experiencing filtering by one large-scalend one small-scale spatial frequency cutoff seems to beomforting.

By inserting Eqs. (9) and (10) into Eq. (1), the scintilla-ion index of annular beams in strong atmospheric turbu-ence is found.

. RESULTS AND DISCUSSIONn all the plots in this section, the on-axis scintillation in-ices of annular beams in strong turbulence are evalu-ted versus the square root of Rytov plane wave scintilla-ions. In all the figures, Cn

2 =510−14 m−2/3 and �=2� /k1.55 m are taken. Using Fig. 1, comparisons between

he results of our method and those obtained from thesymptotic theory are made. The curves provided in Fig. 1or the spherical, plane, and Gaussian beams are plottedy using our solution. These curves are in perfect agree-

ig. 1. Scintillation index in strong turbulence for plane,pherical, Gaussian, and collimated annular beams versusquare root of the Rytov plane wave scintillation index.

ent with the spherical, plane, and Gaussian beam re-ults shown in Fig. 9.12 of [3], which were obtained by us-ng the asymptotic theory. Thus, it is shown in Fig. 1 thatur solution correctly matches the existing strong turbu-ence results of the asymptotic theory for the Gaussianeam—thus spherical and plane wave scintillation indi-es [3]—and also reduces correctly to the existing weakurbulence annular beam scintillations, which can beound by using the formulas in [14]. As a consequence ofhe perfect match between our solution and thesymptotic theory [3], as the Rytov variance for a planeave approaches infinity, the limits of mLS

2 and mSS2 pro-

ided by Eqs. (9) and (10) approach the same values asredicted by the asymptotic theory for spherical, plane,nd Gaussian beams. For annular beams, these limitsary with the annular beam parameters, such as theource sizes of the primary and secondary beams, andhese limits need to be obtained numerically for the par-icular annular structure. From Figs. 1 and 2, we see thatithin the chosen parameter range, the collimated annu-

ar beam scintillations are smaller in weak turbulencend higher in strong turbulence when compared to colli-ated Gaussian beam, plane, and spherical wave scintil-

ations. The scintillations, containing both temporal andpatial variations of the received intensity, have complexehavior depending on physical phenomena, such aseam wander and medium statistics governing the turbu-ence strength. Thus, when the scintillation index of aeam of particular shape is compared to the scintillationndex of another beam, it is possible to observe oppositerends in weak and strong turbulence. As seen in Fig. 2,nnular beam scintillations originate basically from theontribution of the small-scale effects. Figure 3 showshat for annular beams with finite focal length, thinnereams possess smaller scintillation indices when com-ared to the thicker annular beams in strong turbulence.igure 4 shows that for annular beams of constant pri-ary beam size, thicker collimated annular beams have

maller scintillation indices than the thinner annulareams in strong turbulence; however, thinner collimatednnular beams possess smaller scintillations than thehicker collimated annular beams in moderate turbu-

ig. 2. Scintillation index in strong turbulence for plane,pherical, and collimated annular beams with the large- andmall-scale scintillation indices versus square root of the Rytovlane wave scintillation index at selected values of source sizes.

llamFiagssadb7nmsstscss

Fbwselected values of secondary source sizes.

Faisecondary source sizes.

Fcll

Fbwp

Fat

1838 J. Opt. Soc. Am. A/Vol. 27, No. 8 /August 2010 Gerçekcioglu et al.

ence. In Fig. 5, comparing the scintillations of the annu-ar beams of 1 cm primary beam size among themselvesnd the scintillations of the annular beams of 2 cm pri-ary beam size among themselves, the same trend as inig. 4 is observed. Moreover, from Fig. 5, one extracts that

n terms of the scintillation noise, in moderate turbulencennular beams of larger primary beam sizes are advanta-eous over the annular beams of smaller primary beamizes; however, in strong turbulence annular beams ofmaller primary beam sizes are advantageous over thennular beams of larger primary beam sizes. Figure 6 in-icates that when the focal length is decreased, annulaream scintillations in strong turbulence decrease. In Fig., we examine the scintillation index of constant-area an-ular beams and observe that annular beams whose pri-ary beams possess larger source sizes exhibit smaller

cintillation indices in moderate turbulence and largercintillation indices in strong turbulence. Figure 8 is plot-ed to show that at a specifically chosen primary beamize, when comparing Gaussian beam scintillation indi-es, annular beams of any thickness can yield smallercintillation indices in moderate turbulence and largercintillation indices in strong turbulence.

ig. 6. Scintillation index in strong turbulence for annulareams of finite focal length versus square root of the Rytov planeave scintillation index at selected values of source focusingarameters.

ig. 7. Scintillation index in strong turbulence for collimatednnular beams versus square root of the Rytov plane wave scin-illation index at selected values of source sizes of constant area.

ig. 3. Scintillation index in strong turbulence for annulareams of finite focal length versus square root of the Rytov planeave scintillation index at a constant primary source size �s1

and

ig. 4. Scintillation index in strong turbulence for collimatednnular versus square root of the Rytov plane wave scintillationndex at a constant primary source size �s1

and selected values of

ig. 5. Scintillation index in strong turbulence for Gaussian,ollimated annular beams with the large- and small-scale scintil-ation indices versus square root of the Rytov plane wave scintil-ation index at selected dual values of source sizes.

4Usstscsslnssbbbistr

sctl

R

1

1

1

1

1

1

Fcll

Gerçekcioglu et al. Vol. 27, No. 8 /August 2010 /J. Opt. Soc. Am. A 1839

. CONCLUSIONsing the amplitude spatial filtering of the atmospheric

pectrum for large-scale and small-scale log-irradiancecintillation indices, we have formulated and evaluatedhe on-axis scintillation indices of annular beams introng turbulence. Our results show that as compared toollimated Gaussian beam, plane, and spherical wavecintillations, collimated annular beam scintillations aremaller in weak turbulence and higher in strong turbu-ence. Compared to thicker beams, thinner collimated an-ular beams having fixed sized primary beams exhibitmaller scintillations in moderate turbulence but largercintillations in strong turbulence. However, for annulareams of finite focal length, scintillations of thinnereams are smaller in strong turbulence. Larger primaryeam sized annular structures seem to be advantageousn moderate turbulence, however, annular beams withmaller primary beam sizes are advantageous in strongurbulence. Decreasing the focal length of annular beams

ig. 8. Scintillation index in strong turbulence for Gaussian,ollimated annular beams with the large- and small-scale scintil-ation indices versus square root of the Rytov plane wave scintil-ation index at selected values of source sizes.

esults in the decrease of the scintillation indices in

trong turbulence. Among the annular beams possessingonstant area, the ones with larger source size yield scin-illation indices smaller in moderate turbulence, butarger in strong turbulence.

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