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H. T. Eyyuboglu and Y. Baykal Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2709

Hermite-sine-Gaussian and Hermite-sinh-Gaussianlaser beams in turbulent atmosphere

Halil Tanyer Eyyuboglu and Yahya Baykal

Electronic and Communication Engineering Department, Çankaya University, Ögretmenler Cad. No: 14 Yüzüncüyıl06530 Balgat Ankara, Turkey

Received February 15, 2005; accepted May 4, 2005

Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beam intensities in a turbulent atmosphere are in-vestigated. The received intensity is formulated by applying the extended Huygens–Fresnel principle to gen-eralized Hermite-hyperbolic-Gaussian and Hermite-sinusoidal-Gaussian beam incidences. From this result,the association to different types of Hermite-hyperbolic-Gaussian and Hermite-sinusoidal-Gaussian beams aredefined. The average receiver intensity expressions for Hermite-sine-Gaussian and Hermite-sinh-Gaussian la-ser beams are evaluated and plotted against the variations in source parameters and propagation conditions.It is observed that the propagation of Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in tur-bulence have many similarities to their counterparts, Hermite-cosine-Gaussian and Hermite-cosh-Gaussianlaser beams, that are examined earlier. It is further observed that under certain conditions the main featuresof the previously established reciprocity concept between cosine-Gaussian and cosh-Gaussian beams aremostly applicable to Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams. © 2005 Optical Societyof America

OCIS codes: 010.1330, 010.1300, 010.3310, 010.1290.

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. INTRODUCTIONree-space optical (FSO) telecommunication systemsave become one of the most widely used alternativesmong broadband access applications. An understandingf the degrading effects of atmospheric turbulence in FSOinks and the methods for removal of the degradation dueo turbulence are being investigated in detail.1–5 In ourecent studies, we have introduced various laser beamypes in turbulence among which are the generalermite-sinusoidal-Gaussian (HSG) beams6 and its cer-

ain specific forms such as cosine-Gaussian andosh-Gaussian,7 Hermite-cosh-Gaussian,8 and cosh-aussian laser beams.9 Propagation of such beams in free

pace is reported comprehensively.10–12 Our interest is tonderstand the second- and fourth-order behavior ofhese beams in atmospheric turbulence. In this paper weork on the second order only, investigating specifically

he source and the receiver intensity profiles of Hermite-ine-Gaussian and Hermite-sinh-Gaussian beams whenhe medium is a turbulent atmosphere. However, we keepur formulation more general to cover all types of HSGeams so that we can easily refer to our earlier work7–9

hile making comparisons with the other types HSGeams.In our formulation, an extended Huygens–Fresnel

rinciple is employed. When propagation of a Hermite-ine-Gaussian and a Hermite-sinh-Gaussian laser beamair is examined in turbulence, we observe many similari-ies with the propagation of the previously examined pairf Hermite-cosine-Gaussian and Hermite-cosh-Gaussianaser beams in turbulence. The concept of reciprocity inurbulence that we have recently reported for cosine-aussian and cosh-Gaussian laser beams7 seems to espe-

1084-7529/05/122709-10/$15.00 © 2

ially also apply for Hermite-sine-Gaussian and Hermite-inh-Gaussian laser beams.

Generation of the special cases of the general form ofSG beams involves the excitation by an ordinary Gauss-

an beam of an aperture having the desired appropriateransmission or reflection characteristics.10,13 If a sine-aussian beam is to be generated, a Gaussian beam isade incident on an aperture that has a sine transmis-

ion function. These beams are used in applications forhich efficient extraction of energy is required. For ex-mple, cosh-Gaussian dependence exhibits a concentra-ion of energy in the outer lobes of a beam that can be em-loyed in the space diversity applications in FSOelecommunication links. To our knowledge, introductionf HSG beams to turbulent atmosphere is made only byur group.6–9 We are currently working on the scintilla-ions under HSG beam incidence in turbulence to under-tand whether use of such beams will bring some advan-age in the bit-error-rate performance of FSOelecommunication links.

. FORMULATION. Intensity at the Source Planee envisage a propagation geometry comprising source

nd receiver planes that lie perpendicular to the axis ofropagation z along which a turbulent atmosphere isresent. The x and y coordinates on the source plane areenoted by the vector s decomposed as s= �sx ,sy�. At theeceiver plane, p is the transverse position vector decom-osed as p= �px ,py�. In such a construction, �s� or �sx ,sy�ill mean a location on the source plane z=0, whereas

p ,z=L� will indicate a location on the receiver plane,

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2710 J. Opt. Soc. Am. A/Vol. 22, No. 12 /December 2005 H. T. Eyyuboglu and Y. Baykal

ith L being the link length. On the source plane, theeld distribution of the laser beam is

us�s� = us�sx,sy� = AcHn�axsx + bx�Hm�aysy + by�

�exp�− 0.5�sx2/�sx

2 + sy2/�sy

2 ���l1 exp�i�Vxsx

+ Vysy�� + l2 exp�i�Yxsx + Yysy���. �1�

quation (1) represents a field at the source plane formedy the superposition of two beams where l1, l2 are com-lex coefficients, Ac is the amplitude normalization termnd for convenience will be taken as unity, i= �−1�1/2, andsx and �sy refer to source sizes of the Gaussian beam inhe x and y directions. Hermite polynomials Hn�axsx+bx�nd Hm�aysy+by� define the field distributions of order n,idth ax, and displacement bx for the sx axis and of order, width ay, and displacement by for the sy axis. Vx and Vy

re the complex displacement parameters (Vx=Vxr+ iVxi,y=Vyr+ iVyi where Vxr, Vxi denote the real and imaginary

omponents of Vx and Vyr, Vyi denote the real and imagi-ary components of Vy) associated with the Gaussian partf the first beam in the sx ,sy directions. Similarly Yx andy are the complex displacement parameters (Yx=YxriYxi, Yy=Yyr+ iYyi where Yxr, Yxi denote the real and

maginary components of Yx and Yyr, Yyi denote the realnd imaginary components of Yy) associated with theaussian part of the second beam in the sx ,sy directions.y suitable adjustment of Vx ,Vy ,Yx ,Yy and l1 , l2, the fol-

owing four kinds of laser beam can be obtained:

1. Hermite-cosine-Gaussian beam, when Vx=Vxr, VyVyr, Yx=Yxr=−Vxr, Yy=Yyr=−Vyr, and l1= l2=0.5.2. Hermite-cosh-Gaussian beam, when Vx=−iVxi, Vy

−iVyi, Yx=−Vx= iVxi, Yy=−Vy= iVyi, and l1= l2=0.5.3. Hermite-sine-Gaussian beam, when Vx=Vxr, Vy=Vyr,

x=Yxr=−Vxr, Yy=Yyr=−Vyr, and l1=−0.5i, l2=0.5i.4. Hermite-sinh-Gaussian beam, when Vx=−iVxi, Vy

−iVyi, Yx=−Vx= iVxi, Yy=−Vy= iVyi, and l1=0.5, l2=−0.5.

In this paper we examine primarily the last two types,.e., Hermite-sine-Gaussian and Hermite-sinh-Gaussianaser beams, and references to the two former cases are

ade throughout the text.Using Eq. (1) with the above subitems 1–4 and apply-

ng Is�s�=us�s�us*�s�, where Is�s� is the source intensity

*

istribution and denotes the complex conjugate, the t

� 1 2

�ctsss

ource plane intensities of Hermite-cosine-Gaussian,ermite-cosh-Gaussian, Hermite-sine-Gaussian, andermite-sinh-Gaussian laser beams are obtained in orders

Is cosine�s� = Is�sx,sy� = Hn2�axsx + bx�Hm

2 �aysy + by�

�exp�− �sx2/�sx

2 + sy2/�sy

2 ��cos2��Vxrsx + Vyrsy��,

�2�

Is cosh�s� = Is�sx,sy� = Hn2�axsx + bx�Hm

2 �aysy + by�

�exp�− �sx2/�sx

2 + sy2/�sy

2 ��cosh2��Vxisx + Vyisy��,

�3�

Is sine�s� = Is�sx,sy� = Hn2�axsx + bx�Hm

2 �aysy + by�

�exp�− �sx2/�sx

2 + sy2/�sy

2 ��sin2��Vxrsx + Vyrsy��,

�4�

Is sinh�s� = Is�sx,sy� = Hn2�axsx + bx�Hm

2 �aysy + by�

�exp�− �sx2/�sx

2 + sy2/�sy

2 ��sinh2��Vxisx + Vyisy��.

�5�

. Average Intensity at the Receiver Planehen propagation in turbulence is introduced, the ex-

ended Huygens–Fresnel principle defines the field at theeceiver plane �p ,z=L� as

ur�p,z = L,t� = k exp�ikL�/�2i�L��−�

� �−�

�

d2sus�s�exp�ik�p

− s�2/�2L� + ��s,p� − 2i�ft�, �6�

here k=2� /� is the wavenumber, � being the wave-ength; us�s� is the field at the source plane �p ,z=0� as ex-ressed by Eq. (1); and ��s ,p� is the solution to the Rytovethod representing the random part of the complex

hase of a spherical wave propagating from the sourceoint �s� to the receiver point �p ,z=L�; f is the frequency;nd t denotes time.The average intensity at the receiver plane is obtained

rom �Ir�p ,z=L�= �ur�p ,z=L , t�ur*�p ,z=L , t�, where �

epresents the ensemble average over the medium statis-

ics. On the basis of Eq. (6), �Ir�p ,z=L� is converted into

�Ir�p,z = L� = k2/�2�L�2�−�

� �−�

� �−�

� �−�

�

d2s1d2s2us�s1�us*�s2�exp�ik��p − s1�2 − �p − s2�2�/�2L���exp���s1,p� + �*�s2,p��.

�7�

or the ensemble-average term in Eq. (7), we use14

�exp���s1,p� + �*�s2,p�� = exp�− 0.5D��s1 − s2��

= exp�− �0−2�s1 − s2�2�, �8�

here D �s −s � represents the wave structure function,

0= �0.546 Cn2k2L�−3/5 is the coherence length of a spheri-

al wave propagating in the turbulent medium, and Cn2 is

he structure constant. We note that in this paper wetudy the second-order moment by utilizing the wavetructure function that is approximated by Rytov’s phasetructure function that is accepted to be valid not only for

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H. T. Eyyuboglu and Y. Baykal Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2711

eak fluctuations but also for strong fluctuations asell.15 Substituting Eqs. (1) and (8) into Eq. (7), expand-

ng the transverse source and transverse receiver coordi-ates into their corresponding x and y components, theuadruple integral in Eq. (7) is solved by writing the se-ies expansion of all Hermite polynomials and performingterm-by-term integration by the cyclic use of Eq. 3.462.2

f Ref. 16. Thus the average intensity at the receiverlane develops into

�Ir�p,z = L� = b2�04�DsxDsy�−1/2exp�− �0

4b2�px2/��sx

2 Dsx�

+ py2/��sy

2 Dsy����exp�− 0.5�02�Vx

2��02asx + 1�/Dsx

+ Vy2��0

2asy + 1�/Dsy���Bs1 exp�2i�04b2�Vxpx/Dsx

+ Vypy/Dsy��Sx1Sy1 + Bs2 exp�− 2i�04b2�Vxpx/Dsx

+ Vypy/Dsy��Sx2Sy2� + exp�− 0.25�04�Vx

2/��sx2 Dsx�

SjtlHwottEn

=i(sIvtioThicsif−

+ Vy2/��sy

2 Dsy����Bs3 exp��04b�Vxpx/��sx

2 Dsx�

+ Vypy/��sy2 Dsy���Sx3Sy3 + Bs4

�exp�− �04b�Vxpx/��sx

2 Dsx�

+ Vypy/��sy2 Dsy���Sx4Sy4��, �9�

here

b = k/2L, asx = ��0.5/�sx2 � + �1/�0

2��,

asy = ��0.5/�sy2 � + �1/�0

2��,

Dsx = ��04�asx

2 + b2� − 1�,

Dsy = ��04�asy

2 + b2� − 1�,

Sx1 = lx1=0

�n/2�

lnx1=0

n−2lx1

kx1=0

�lnx1/2�

lnx11=0

lnx1−2kx1

lx2=0

�n/2�

lnx2=0

n−2lx2

kx2=0

��lnx11+lnx2�/2�

�− 1�lx1+lx222n−lx1−lx2−lnx1−lnx2ilnx1+lnx2−2kx1−2kx2Tlx1Tlx2� n

2lx1�� n

2lx2��n − 2lx1

lnx1�

��lnx1 − 2kx1

lnx11��n − 2lx2

lnx2� lnx1!

�lnx1 − 2kx1�!kx1!

�lnx11 + lnx2�!

�lnx11 + lnx2 − 2kx2�!kx2!

� �ax�lnx1+lnx2�bx�2n−2lx1−2lx2−lnx1−lnx2��02�lnx2�Dsx�kx2−lnx11−lnx2�asx − ib�−lnx1+kx1+kx2�Vx − 2bpx�lnx1−2kx1−lnx11��0

2�Vx + 2bpx��asx

− ib� + �Vx − 2bpx��lnx11+lnx2−2kx2. �10�

We acquire Sx2 from Eq. (10) by simply replacing all Vxerms by −Vx terms. Sx3 has the same form as Eq. (10) ex-ept that in the third line of this equation, a minus sign iso be placed in front of the second occurrence of Vx. Fi-ally, Sx4 is the copy of Sx3 where all Vx terms arehanged to −Vx terms. Sy1, Sy2, Sy3, and Sy4 are generatedy swapping all x indices with y indices and all occur-ences of n with m in the respective sum equations of Sx1,x2, Sx3, and Sx4. The remaining terms in Eq. (10) are de-ned as follows:Tlxj=1�3� ¯ · �2lxj−1� for lxj�0, where j=1,2; all ap-

earances in the form of � C1

C2� represent binomial coeffi-

ients; hence � C1

C2�=C1! / ��C1−C2�!C2!�, ! is the factorial no-

ation; [ ] signs appearing in the upper limit of theummation symbols mean that the integral part of the ex-ression within the square brackets is to be taken. Thes1 ,Bs2 ,Bs3 ,Bs4 values are related to the parameters l1nd l2 in a way such that Eqs. (9) and (10) are able to cre-te the average receiver intensities of all beam typesisted as subitems 1–4 in Subsection 2.A. This is describedn Subsection 2.C.

. Association of Equations (9) and (10) to Beam Types. When Bs1= l1 2=0.25, Bs2= l2 2=0.25, Bs3= l1l2

* =0.25,s4= l1

*l2=0.25, and Vx ,Vy are as defined in subitem 1 ofubsection 2.A, that is, Vx=Vxr, Vy=Vyr, Eqs. (9) and (10)ill then generate the average receiver intensity expres-

ion for a Hermite-cosine-Gaussian laser beam. In thisase, examining the summations of Eq. (9) individually,

x1Sy1 and Sx2Sy2 of Eq. (9) will always be a complex con-ugate of each other, and together with the help of its mul-iplying exponentials [appearing on the third and fourthines of Eq. (9)], they will tend toward Hermite-cosine andermite-sine functions, where the former combinationill normally dominate. The Sx3Sy3 and Sx4Sy4 products

f Eq. (9), on the other hand, will always be real and posi-ive quantities, thus including the multiplying exponen-ials [appearing on the sixth, seventh, and eighth lines ofq. (9)]; those terms will merge into Hermite-cosh combi-ations.II. When Bs1= l1 2=0.25, Bs2= l2 2=0.25, Bs3= l1l2

*

0.25, Bs4= l1*l2=0.25, and Vx ,Vy are as defined in sub-

tem 2 of Subsection 2.A, that is, Vx=−iVxi, Vy=−iVyi, Eqs.9) and (10) will then generate the average receiver inten-ity expression of a Hermite-cosh-Gaussian laser beam.n this case, examining the summations of Eq. (9) indi-idually, Sx1Sy1 and Sx2Sy2 of Eq. (9) will always be posi-ive and real, and together with the help of its multiply-ng exponentials [appearing on the third and fourth linesf Eq. (9)] will tend toward a Hermite-cosh combination.he Sx3Sy3 and Sx4Sy4 products of Eq. (9), on the otherand, will be the complex conjugate of each other, depend-

ng on the sum of mode indices being even or odd and in-luding the multiplying exponentials [appearing on theixth, seventh, and eighth lines of Eq. (9)], and will mergento cosine or sine functions. Note that in this case, a sineunction is to come about as a result of the equality 1cos 2x=2 sin2 x.

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2712 J. Opt. Soc. Am. A/Vol. 22, No. 12 /December 2005 H. T. Eyyuboglu and Y. Baykal

III. When Bs1= l1l2* =−0.25, Bs2= l1

*l2=−0.25, Bs3= l1 20.25, Bs4= l2 2=0.25, and Vx ,Vy are as defined in sub-

tem 3 of Subsection 2.A, that is, Vx=Vxr, Vy=Vyr, Eqs. (9)nd (10) will then generate the average receiver intensityxpression of a Hermite-sine-Gaussian laser beam. In thisase, examining the summations of Eq. (9) individually,he characteristics of the Sx1Sy1, Sx2Sy2, Sx3Sy3, andx4Sy4 products will be akin to subitem I above, i.e., thease of a Hermite-cosine-Gaussian laser beam except thatith Sx3Sy3 and Sx4Sy4, a Hermite-sinh-Gaussian combi-ation will be produced from the relation cosh 2x−12 sinh2 x.IV. When Bs1= l1 2=0.25, Bs2= l2 2=0.25, Bs3= l1l2

*

−0.25, Bs4= l1*l2=−0.25, and Vx ,Vy are as defined in sub-

tem 4 of Subsection 2.A, that is, Vx=−iVxi, Vy=−iVyi, Eqs.9) and (10) will then generate the average receiver inten-ity expression of a Hermite-sinh-Gaussian laser beam.n this case, examining the summations of Eq. (9) indi-idually, the characteristics of the Sx1Sy1, Sx2Sy3, Sx3Sy3,nd Sx4Sy4 products will be akin to subitem II above, i.e.,he case of a Hermite-cosh-Gaussian laser beam excepthat with Sx1Sy1 and Sx2Sy2, a Hermite-sinh-Gaussianombination will be produced from the relation cosh 2x1=2 sinh2 x.Here we should add that, in the definitions of

s1 ,Bs2 ,Bs3 ,Bs4 involving the absolute value of l1 and l2r their conjugates, they are meaningful only for the casef Hermite-sine-Gaussian laser beams, since in all othernstances l1 and l2 are real.

V. Once the relevant expressions for the Hermite-osine-Gaussian and Hermite-cosh-Gaussian beams arebtained via these settings, it is also possible to reducehe average received intensity given in Eq. (9) to the cases

ig. 1. Three-dimensional intensity profile of two Hermite-sine-Gndices are n=2, m=0 for the upper plot and n=0, m=0 for the l

f TEM modes in turbulence by equating Vxr, Vyr=0 orxi, Vyi=0. The limiting case of free space, i.e., the ab-ence of turbulence, can be obtained by letting �0→� inq. (9). But we should be cautious that, approaching theEM modes from the side of Hermite-sine-Gaussian orermite-sinh-Gaussian intensity expression is prevented

ince, in these circumstances, the source field expressionf Eq. (1) and subsequently the average receiver plane in-ensity of Eq. (9) will tend toward zero.

. RESULTS AND DISCUSSION. Source Plane Intensity Diagramsor a more convenient presentation of the source plane

ntensity diagrams, the source plane intensity is normal-zed in the following manner:

IsN�sx,sy� = Is�sx,sy�/max��Is�sx,sy���, �11�

here the max operator chooses the peak value in theource intensity expression and Is�sx ,sy� refers to any ofhe Hermite-cosine-Gaussian, Hermite-cosh-Gaussian,ermite-sine-Gaussian, or Hermite-sinh-Gaussian lasereams listed in Eqs. (2)–(5). Although the formulation of-ered in Eqs. (9) and (10) is able to accommodate x and ysymmetries, variable width parameters, and nonzeroisplacements of Hermite polynomials, in the accompany-ng illustrations we have utilized the following settings:sx=�sy, Vxr=Vyr, Vxi=Vyi ax=ay=1/�sx=1/�sy, bx=by=0.Initially, the Hermite-sine-Gaussian beam is investi-

ated. Figure 1 displays the three-dimensional intensityrofile of two Hermite-sine-Gaussian beams, where in thepper plot, mode indices of n=2, m=0, together with the

an beams for �sx=5 cm, Vxr=50 m−1, ax=20 m−1, and bx=0. Modelot.

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H. T. Eyyuboglu and Y. Baykal Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2713

ther parameter settings of �sx=5 cm, Vxr=50 m−1, ax20 m−1, and bx=0 are used, as shown in the inset of Fig.. In the lower part of Fig. 1, these settings are main-ained except that the mode indices are changed to n=0,

=0, which represent the plain sine-Gaussian beam. Inig. 2 the contour plots of two Hermite-sine-Gaussianeams are illustrated, the upper plot with n=2, m=0 andhe lower plot with n=1, m=1, retaining the rest of thearameter settings as in Fig. 1. As detected from Figs. 1nd 2, in the source plane intensities of Hermite-sine-aussian beams, there are sine-modulated profiles ar-

anged in lobe groups whose numbers are determined byhe quantity of the mode index plus one along the related

ig. 2. Contour plots of two Hermite-sine-Gaussian beams, thepper plot with n=2, m=0 and the lower plot with n=1, m=1,etaining the rest of the parameter settings as in Fig. 1.

xis. Figures 1 and 2 also demonstrate that, for equalode indices, there will be perfect symmetry of the inten-

ity pattern with respect to the slanted axis; otherwisehis symmetry will be skewed according to the differencen the mode indices.

Next we explore Hermite-sinh-Gaussian beams. Figureexhibits the contour plots of two Hermite-sinh-Gaussianeams, one with n=2, m=0 and Vxi=Vyi=10 m−1 and thether with n=2, m=1 and Vxi=Vyi=50 m−1 while the re-aining parameters are commonly chosen as �sx=3 cm,

x=33.3 m−1, and bx=0. Employing the same parameter

ig. 3. Contour plots of two Hermite-sinh-Gaussian beams forsx=3 cm, ax=33.3 m−1, and bx=0, one with n=2, m=0 and VxiVyi=10 m−1 and the other with n=2, m=1, and Vxi=Vyi50 m−1.

ig. 4. Two superimposed contour plots of Hermite-cosh-aussian beams that have the same parameter values as inig. 3.

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2714 J. Opt. Soc. Am. A/Vol. 22, No. 12 /December 2005 H. T. Eyyuboglu and Y. Baykal

alues, Fig. 4 shows the two superimposed contour plotsf Hermite-cosh-Gaussian beams. By comparing Figs. 3nd 4, we are able to deduce that, in the limit of VxiVyi→0, the Hermite-cosh-Gaussian beam will reduce topure TEM beam of the given order, whereas the sameill not occur for Hermite-sinh-Gaussian beams. This

onclusion is in line with the predictions of subitem V ofubsection 2.C. Additionally, it is to be witnessed that,imilar to the Hermite-cosh-Gaussian beam, the Hermite-inh-Gaussian beam will continue to preserve the plaininh-Gaussian shape, in spite of higher mode index val-es, where raising the mode indices will merely serve toush the main lobes further away from the origin. Finallye observe that, as long as Gaussian displacement pa-

ameters Vxi and Vyi are greater than the order of 1/�sxnd 1/�sy, that is, if we remain in the region ofyperbolic-Gaussian beams exhibiting a distinct separa-ion of energy in the outer lobes, there will hardly be anyifference in the intensity appearances of Hermite-sinh-aussian and Hermite-cosh-Gaussian beams. This is sim-ly because, for such cases, the arguments of sinh andosh functions will always be larger than the numeral 2,hus sinh�x��cosh�x�.

. Receiver Plane Intensity Diagramsoving away from the source plane, the Hermite-cosine-aussian, Hermite-cosh-Gaussian, Hermite-sine-aussian, and Hermite-sinh-Gaussian laser beams willeep their original shape for some distance; then becausef inevitable spreading, neighboring lobes will start toerge. With a certain choice of source and propagation

arameters, the beam will subsequently develop into itsyperbolic or sinusoidal counterpart, a process we calleciprocity. Basically what happens here is that, the firstroup of terms, that is, the collection comprising the sec-

Fig. 5. Intensity distribution of a Hermite-sinh-G

nd, third, and fourth lines of Eq. (9), will dominate atarlier propagation distances. Then as the propagationange is extended, this dominance will be taken over byhe second group of terms comprising the fifth throughinth lines of Eq. (9). In some situations, the TEM pat-ern will also manifest itself. Eventually, however, as theropagation distance increases, all appearances will di-inish, with the intensity profile turning into a pureaussian shape.For the receiver intensity diagrams, we use the follow-

ng normalization procedure:

�IrN�p,z = L� = Ir�p,z = L�/max��Is�sx,sy���. �12�

igure 5 shows the progress of a Hermite-sinh-Gaussianeam along the axis of propagation at link lengths of L0 (source plane), 2, 5, and 20 km. Here it is seen for thispecific combination of source and medium parametershat the Hermite-sinh-Gaussian beam will eventuallyroduce a pure Gaussian beam without going through thentermediate stage of becoming a Hermite-sine-Gaussianeam. Figure 6 contains the span shots of a Hermite-cosh-aussian beam along the propagation axis for the identi-

al set of source and propagation parameters. By compar-ng Figs. 5 and 6, we conclude that a similar sequence ofntensity patterns are repeated both in the case ofermite-sinh-Gaussian and Hermite-cosh-Gaussianeams. In Figs. 5 and 6 and also in the successive plots,avelength of operation and turbulence strength are, re-

pectively, taken as �=1.55 �m and Cn2 =1�10−15 m−2/3.

To view a Hermite-sinh-Gaussian beam being trans-ormed into a sinosoidal shape during its travel, theource sizes used in Figs. 5 and 6 should be reduced. Fig-res 7 and 8 supply such illustrations. From Figs. 7 and 8,

t is to be understood that, analogous to the previously es-

n beam with n=1, m=0 at L=0, 2, 5, and 20 km.

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H. T. Eyyuboglu and Y. Baykal Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2715

ablished rules of reciprocity,7,8 a Hermite-sinh-Gaussianeam will become a Hermite-cosine-Gaussian or aermite-sine-Gaussian beam depending on the sum ofode indices being odd or even. In the specific case of Fig.

, n=2, m=1, making the sum 3 (odd), a Hermite-cosine-aussian beam is generated after propagating a rela-

ively short distance, i.e., L=1 km case plot. Figure 8

Fig. 6. Intensity distribution of a Hermite-cosh-G

Fig. 7. Intensity distribution of a Hermite-sinh-Gau

hows that by changing the mode indices to n=2, m=0,he transformation will be toward a Hermite-sine-aussian beam. It is worth pointing out that the mode in-ex conditions for the creation of a Hermite-cosine-aussian or a Hermite-sine-Gaussian beam will be

nverted if we were to start with a source excitation of aermite-cosh-Gaussian beam as reported in Ref. 8. Fig-

n beam with n=1, m=0 at L=0, 2, 5, and 20 km.

beam with n=2, m=1 at L=0, 0.05, 1, and 250 km.

aussia

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9b=

taaaG9

ssian

2716 J. Opt. Soc. Am. A/Vol. 22, No. 12 /December 2005 H. T. Eyyuboglu and Y. Baykal

res 7 and 8 further reveal that lowering the sum of modendices will retard the speed of the transformation.

To analyze Hermite-sine-Gaussian beams, we plot Fig.that displays the progress of Hermite-sine-Gaussian

eam along the axis of propagation at link lengths of L0 (source plane), 2, 5, and 20 km. Figure 9 shows that at

Fig. 8. Intensity distribution of a Hermite-sinh-Gau

Fig. 9. Intensity distribution of a Hermite-sine-Ga

he intermediate link length �L=2 km�, the beam appearss a sinh-positioned TEM when viewed along the slantedxis. As the beam propagates further, the TEM appear-nce gradually diminishes, turning the beam to a sinh-aussian profile as seen in the L=20 km case plot of Fig.. An illustration of a Hermite-cosine-Gaussian beam is

beam with n=2, m=0 at L=0, 0.05, 1, and 250 km.

beam with n=1, m=0 at L=0, 0.4, 2, and 20 km.

ussian

gucaFaaI�sTs

4WfiGpHochs

bloGbvbHpt

Gdfiba

hB

R

H. T. Eyyuboglu and Y. Baykal Vol. 22, No. 12 /December 2005 /J. Opt. Soc. Am. A 2717

iven in Fig. 10. From Figs. 9 and 10 it is observed thatsing incidences of Hermite-sine-Gaussian or Hermite-osine-Gaussian will not exhibit substantial differenceslong the path of propagation in turbulence. Comparingigs. 9 and 10, it becomes clear that the convergence intopure Gaussian profile will be nearer to completion whenlarger propagation length, i.e., L=50 km, is employed.

t may be interesting to mention that the third pictureL=2 km� present in both Figs. 9 and 10 describes, re-pectively, a sinh-Gaussian and cosh-Gaussian separatedEM intensities whose profiles are not attainable at theource plane.

. CONCLUDING REMARKSe have studied the source and the receiver intensity pro-

les of Hermite-sine-Gaussian and Hermite-sinh-aussian beams in turbulence. In the formulation ofropagation in the random medium, the extendeduygens–Fresnel diffraction integral is used. To compare

ur current results with our previous results for Hermite-osine-Gaussian and Hermite-cosh-Gaussian beams, weave intentionally offered a formulation that covers allpecific types of HSG laser beams in turbulence.

It is to be noted that as HSG beams propagate in tur-ulence, they keep their original shape for a certain linkength, then additional lobes start to appear. Dependingn the source and medium parameters, a Hermite-sine-aussian beam is converted to a Hermite-sinh-Gaussianeam and vice versa. Our earlier work that states the con-ersion between cosine-Gaussian and cosh-Gaussianeams7 and between Hermite-cosine-Gaussian andermite-cosh-Gaussian beams8 that is defined as reci-rocity is observed in a similar manner in the results ofhis paper for Hermite-sine-Gaussian and Hermite-sinh-

Fig. 10. Intensity distribution of a Hermite-cosine-G

aussian beams. Under certain conditions, Hermiteominance will take over to form TEM patterns. For suf-ciently long propagation distances, all these conversionsecome ineffective and the received intensity approachesGaussian profile.

The e-mail address for H. T. Eyyuboglu is.eyyuboglu@cankaya.edu.tr. The e-mail address for Y.aykal is y.baykal@cankaya.edu.tr.

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aussi

1

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1

1

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