Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent...
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Correlation and structure functions of Hermite – sinusoidal-Gaussian laser beams in a turbulent atmosphere Yahya Baykal Department of Electronic and Communication Engineering, C ¸ankaya University, O ¨ g ˇ retmenler Cad. No: 14, Yu ¨ zu ¨ ncu ¨ yıl 06530 Balgat, Ankara, Turkey Received October 31, 2003; revised manuscript received January 15, 2004; accepted March 1, 2004 To study the performance of atmospheric optical links by using Hermite– sinusoidal-Gaussian laser beam sources, we derive the log-amplitude and the phase correlation and structure functions of such beams in a turbulent atmosphere. Our formulations correctly reduce to the known higher-order mode correlation and structure functions, which in turn reduce to the fundamental-mode (TEM 00 -mode) results. Several special cases of our formulation are presented, among which the case involving Hermite– cosh-Gaussian dependence is especially noted, since this case is of interest to us owing to the nature of cosh dependence exhibiting the con- centration of the energy in the outer lobes of the beam. © 2004 Optical Society of America OCIS codes: 010.1330, 010.1300, 030.4070, 060.4510. 1. INTRODUCTION Within the past several years, free-space optical (FSO) communication systems are becoming competitive in the broadband access networks. 1–3 Atmospheric turbulence is one of the factors adversely affecting the performance of FSO links. Manifesting itself in the forms of beam wan- der, spreading, and intensity fluctuations, turbulence de- grades the bit error rate in atmospheric optical links. The effect of atmospheric turbulence on the FSO commu- nication systems and its influence on the error perfor- mance have been studied. 4,5 To reduce the effect of turbulence in FSO links, one can employ various techniques. Some of these are the spatial diversity that utilizes multilaser beams separated in space, combined with spatially separated multireceive lenses having large apertures, fine-tuned filtering at the receiver, and automatic gain control. We are interested in finding out how the FSO link per- formance will be affected when the excitation is in the form of a Hermite – sinusoidal-Gaussian beam wave. Hermite – sinusoidal-Gaussian beams are the general form of the beams whose special cases cover a big range of beam types such as Gaussian, Hermite-Gaussian, off-axis Hermite – Gaussian, Hermite – cosh-Gaussian, cosh (sinh)- Gaussian, cosh (sinh) and cosine (sine) beams. The propagation in free space (i.e., in the absence of turbu- lence) of the general Hermite – sinusoidal-Gaussian beams 6,7 and the propagation in free space of some special cases of Hermite – sinusoidal-Gaussian beams 8–12 are ex- amined in detail. It is of interest to understand whether the use of Hermite – sinusoidal-Gaussian beams under certain beam parameters will be beneficial in reducing the turbulence effects. As the first step, formulation of the log- amplitude and phase correlation and structure functions of Hermite – sinusoidal-Gaussian beam excitation in a tur- bulent atmosphere are required, and in this paper we present our formulations and results for these functions. Our formulations correctly reduce to the known higher- order-mode log-amplitude and phase correlation and structure functions, 13 which, in turn, reduce to the funda- mental mode (TEM 00 -mode) results. 14 Our main motiva- tion in this study is to understand whether the use of Hermite – sinusoidal-Gaussian laser beams in FSO access communication systems will improve the bit error rate in a turbulent atmosphere. It is well known that, owing to turbulence, the received signal fluctuates, causing a signal-dependent noise and thus degrading the bit error rate. These fluctuations in general are related to the log- amplitude and phase correlation functions. Specifically, in our current work, we use the log-amplitude correlation function presented in this paper to evaluate the bit error rate of FSO links in weak turbulence when different forms of the Hermite – sinusoidal-Gaussian laser beam are used. It is also of interest to us to evaluate the angle-of- arrival fluctuations (which depend on the phase correla- tion function) in a heterodyne optical communication sys- tem when Hermite – sinusoidal-Gaussian laser beams are used. We note that the structure of turbulence does not change under different types of excitations; i.e., the tur- bulence is still governed by the same spectral density of the index-of-refraction fluctuations. However, when dif- ferent types of laser beam propagate through the turbu- lent atmosphere, the resulting correlations of the received signal vary under different excitations. Our derivation is performed in two stages. In the first stage the log-amplitude and phase correlation and struc- ture functions of the off-axis Hermite – Gaussian beam la- ser excitation in a turbulent atmosphere are derived. Then, in the second stage, off-axis Hermite – Gaussian beam formulation is extended to cover the general Hermite – sinusoidal-Gaussian laser beam excitation in the turbulent atmosphere. 1290 J. Opt. Soc. Am. A/Vol. 21, No. 7/July 2004 Yahya Baykal 1084-7529/2004/071290-10$15.00 © 2004 Optical Society of America
Transcript of Correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent...
1290 J. Opt. Soc. Am. A/Vol. 21, No. 7 /July 2004 Yahya Baykal
Correlation and structure functions ofHermite–sinusoidal-Gaussian
laser beams in a turbulent atmosphere
Yahya Baykal
Department of Electronic and Communication Engineering, Cankaya University, Ogretmenler Cad. No: 14,Yuzuncuyıl 06530 Balgat, Ankara, Turkey
Received October 31, 2003; revised manuscript received January 15, 2004; accepted March 1, 2004
To study the performance of atmospheric optical links by using Hermite–sinusoidal-Gaussian laser beamsources, we derive the log-amplitude and the phase correlation and structure functions of such beams in aturbulent atmosphere. Our formulations correctly reduce to the known higher-order mode correlation andstructure functions, which in turn reduce to the fundamental-mode (TEM00-mode) results. Several specialcases of our formulation are presented, among which the case involving Hermite–cosh-Gaussian dependence isespecially noted, since this case is of interest to us owing to the nature of cosh dependence exhibiting the con-centration of the energy in the outer lobes of the beam. © 2004 Optical Society of America
OCIS codes: 010.1330, 010.1300, 030.4070, 060.4510.
1. INTRODUCTIONWithin the past several years, free-space optical (FSO)communication systems are becoming competitive in thebroadband access networks.1–3 Atmospheric turbulenceis one of the factors adversely affecting the performance ofFSO links. Manifesting itself in the forms of beam wan-der, spreading, and intensity fluctuations, turbulence de-grades the bit error rate in atmospheric optical links.The effect of atmospheric turbulence on the FSO commu-nication systems and its influence on the error perfor-mance have been studied.4,5
To reduce the effect of turbulence in FSO links, one canemploy various techniques. Some of these are the spatialdiversity that utilizes multilaser beams separated inspace, combined with spatially separated multireceivelenses having large apertures, fine-tuned filtering at thereceiver, and automatic gain control.
We are interested in finding out how the FSO link per-formance will be affected when the excitation is in theform of a Hermite–sinusoidal-Gaussian beam wave.Hermite–sinusoidal-Gaussian beams are the generalform of the beams whose special cases cover a big range ofbeam types such as Gaussian, Hermite-Gaussian, off-axisHermite–Gaussian, Hermite–cosh-Gaussian, cosh (sinh)-Gaussian, cosh (sinh) and cosine (sine) beams. Thepropagation in free space (i.e., in the absence of turbu-lence) of the general Hermite–sinusoidal-Gaussianbeams6,7 and the propagation in free space of some specialcases of Hermite–sinusoidal-Gaussian beams8–12 are ex-amined in detail.
It is of interest to understand whether the use ofHermite–sinusoidal-Gaussian beams under certain beamparameters will be beneficial in reducing the turbulenceeffects. As the first step, formulation of the log-amplitude and phase correlation and structure functionsof Hermite–sinusoidal-Gaussian beam excitation in a tur-
1084-7529/2004/071290-10$15.00 ©
bulent atmosphere are required, and in this paper wepresent our formulations and results for these functions.Our formulations correctly reduce to the known higher-order-mode log-amplitude and phase correlation andstructure functions,13 which, in turn, reduce to the funda-mental mode (TEM00-mode) results.14 Our main motiva-tion in this study is to understand whether the use ofHermite–sinusoidal-Gaussian laser beams in FSO accesscommunication systems will improve the bit error rate ina turbulent atmosphere. It is well known that, owing toturbulence, the received signal fluctuates, causing asignal-dependent noise and thus degrading the bit errorrate. These fluctuations in general are related to the log-amplitude and phase correlation functions. Specifically,in our current work, we use the log-amplitude correlationfunction presented in this paper to evaluate the bit errorrate of FSO links in weak turbulence when differentforms of the Hermite–sinusoidal-Gaussian laser beam areused. It is also of interest to us to evaluate the angle-of-arrival fluctuations (which depend on the phase correla-tion function) in a heterodyne optical communication sys-tem when Hermite–sinusoidal-Gaussian laser beams areused. We note that the structure of turbulence does notchange under different types of excitations; i.e., the tur-bulence is still governed by the same spectral density ofthe index-of-refraction fluctuations. However, when dif-ferent types of laser beam propagate through the turbu-lent atmosphere, the resulting correlations of the receivedsignal vary under different excitations.
Our derivation is performed in two stages. In the firststage the log-amplitude and phase correlation and struc-ture functions of the off-axis Hermite–Gaussian beam la-ser excitation in a turbulent atmosphere are derived.Then, in the second stage, off-axis Hermite–Gaussianbeam formulation is extended to cover the generalHermite–sinusoidal-Gaussian laser beam excitation inthe turbulent atmosphere.
2004 Optical Society of America
Yahya Baykal Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1291
Using our results, all the special cases for the log-amplitude and phase correlation and structure functionsof Gaussian, Hermite–Gaussian, off-axis Hermite–Gaussian, Hermite–cosh (sinh)-Gaussian, cosh (sinh)-Gaussian, cosh (sinh) and cosine (sine) beam excitationsin the turbulent atmosphere can be obtained. One ofthese special cases having Hermite–cosh-Gaussian de-pendence is of importance to us at this stage since thecosh dependence concentrates the energy in the outerlobes of the beam. Thus this special case of the log-amplitude and phase correlation and structure functionsof Hermite–cosh-Gaussian excitation are especially de-noted.
2. FORMULATION FOR THE OFF-AXISHERMITE–GAUSSIAN BEAMIn the first stage of our formulation, the excitation istaken as the off-axis Hermite–Gaussian beam. A sche-matic showing the propagation geometry is given in Fig.1. Source and receiver plane geometries are defined bythe coordinates as (s, z 5 0) and (p, z 5 L), where s5 (sx , sy) and p 5 ( px , py) are the transverse sourceand transverse receiver coordinates, respectively, and z isthe propagation axis. Thus z 5 0 (named as the sourceplane) denotes the exit plane of the laser source, and z5 L denotes the receiver plane, where L is the linklength.
By use of the Rytov-method solution, the log-amplitudeand phase fluctuations are evaluated, from which the cor-relation and structure functions for the off-axis Hermite–Gaussian beam are derived. In our derivations, a proce-dure similar to that in Ishimaru’s TEM00-modeformulation14 is followed. The derivation is outlined inAppendix A.
As seen in Eq. (A1), Rytov’s method gives the field inthe turbulent atmosphere as a product of the field in freespace (i.e., in the absence of turbulence) and the field con-tributed owing to atmospheric turbulence. By use of thefield in free space, fluctuations of the complex amplitude(composed of log-amplitude and phase fluctuations) in theturbulent atmosphere are found as in Eq. (A2). The free-space part of the field is found by use of the Huygens–Fresnel principle in which the diffraction integral is ap-plied to the field at the source plane. In the first stage ofour formulation, which is represented in this Section 2,the field at the source plane is taken as the off-axisHermite–Gaussian beam, and the corresponding log-
Fig. 1. Propagation geometry.
amplitude and phase correlation and structure functionsare found. In Section 3 the field at the source plane isrepresented by the superposition of two fields (with differ-ent complex displacement parameters) for the off-axisHermite–Gaussian beams. This superposition yields afield at the source plane for the generalized Hermite–sinusoidal-Gaussian beam. Then the solutions for theoff-axis Hermite–Gaussian beams in the turbulent atmo-sphere obtained in Section 2 are utilized in Section 3 tofind the log-amplitude and phase correlation and struc-ture functions of the generalized Hermite–sinusoidal-Gaussian beam. This two-stage approach allows us toreach a single formulation that covers the generalHermite–sinusoidal-Gaussian beams by employing therelatively simpler off-axis Hermite–Gaussian beam solu-tions. Basic steps of our formulations are presented be-low.
The field at the source plane for the off-axis Hermite–Gaussian beam is written as
un,mOA ~sx , sy , z 5 0 !
5 An,mHn~axsx 1 bx!Hm~aysy 1 by!
3 exp@212 k~axsx
2 1 aysy2!#
3 exp@2i~Vxsx 1 Vysy!#
3 exp~2if !, (1)
where An,m is the amplitude of the field of the mode(n, m) at the origin of the source plane (i.e., at sx 5 sy5 z 5 0); Hn and Hm are Hermite polynomials of ordersn and m determining the field distributions in the sx , sydirections, respectively; ax and ay are complex param-eters characterizing the width of the Hermite polynomialsin the sx , sy directions, respectively; bx , by are complexparameters characterizing the displacement of the Her-mite polynomials in the sx , sy directions, respectively; kis the wave number,
ax 5 1/~kasx2 ! 1 i/Fx , (2)
where asx and Fx are the source size of the Gaussianbeam and the focal length in the sx direction, respectively;and, similarly,
ay 5 1/~kasy2 ! 1 i/Fy , (3)
where asy and Fy are the source size of the Gaussianbeam and the focal length in the sy direction, respectively.Here, i 5 (21)1/2; Vx and Vy are the complex displace-ment parameters (Vx 5 Vxr 1 iVxi , Vy 5 Vyr 1 iVyi ,where Vxr , Vxi denote the real and imaginary compo-nents of Vx and Vyr , Vyi denote the real and imaginarycomponents of Vy) and are associated with the Gaussianpart of the beam in the sx , sy directions, respectively; andf is the constant phase factor.
By use of the Huygens–Fresnel principle, un,mFS (p, z),
which is defined as the field at r 5 (p, z) 5 ( px , py , z)in free space (i.e., in the absence of turbulence), is givenby
1292 J. Opt. Soc. Am. A/Vol. 21, No. 7 /July 2004 Yahya Baykal
un,mFS ~p, z ! 5 k exp~ikz !/~2piz !
3 E2`
`
dsxE2`
`
dsyun,m~sx , sy , z 5 0 !
3 exp$@ik/~2z !#
3 @~sx 2 px!2 1 ~sy 2 py!2#%, (4)
where un,m(sx , sy , z 5 0) is the field at the source plane.Substituting the field at the source plane for the off-axisHermite–Gaussian beam given by Eq. (1) intoun,m(sx , sy , z 5 0) in Eq. (4) and carrying out the inte-gration over the source transverse coordinates by usingEq. (7.374.8) of Ref. 15, one can find the field for the off-axis Hermite–Gaussian beam at r 5 (p, z)5 ( px , py , z) in free space (i.e., in the absence of turbu-lence), un,m
FS,OA(p, z), as
un,mFS,OA~p, z !
5 An,m exp~ikz !exp~2if !~1 2 2iax2zAx /k !n/2
3 ~1 2 2iay2zAy /k !m/2Ax
1/2Ay1/2
3 exp@2iVx2zAx /~2k !#exp@2iVy
2zAy /~2k !#
3 exp~2kaxAxpx2/2!exp~2kayAypy
2/2!
3 exp~2iVxAxpx!exp~2iVyAypy!
3 Hn~b2xpx 1 b1x!Hm~b2ypy 1 b1y!, (5)
where
Ax 5 1/~1 1 iaxz !, (6)
Ay 5 1/~1 1 iayz !, (7)
b2x 5 axAx1/2/$1 2 iz@~2ax
2/k ! 2 ax#%1/2, (8)
b1x 5 b2x@zaxVx 1 kbx~1 1 iaxz !#/~kax!.(9)
b2y and b1y are also given, respectively, by Eqs. (8) and (9)with x replaced by y.
For finding at the receiver plane (z 5 L) the correla-tion function of the log-amplitude fluctuations of the off-axis Hermite–Gaussian beam in a turbulent atmosphere,Eq. (5) is substituted into Eq. (A6), and the integrationsover dpx8 and dpy8 are taken by use of Eq. (7.374.8) of Ref.15. H( px , py , L, kx , ky , z8), thus formulated, is thensubstituted into Eq. (A11), and T1( px , py , L, kx , ky ,z8) for the off-axis Hermite–Gaussian beam is calculated.Consequently, T1( px , py , L, kx , ky , z8) is employed inEq. (A13), and the formulation for the correlation functionof the log-amplitude fluctuations of the off-axis Hermite–Gaussian beam in a turbulent atmosphere is obtained atthe receiver plane (z 5 L) after lengthy algebra.
Our result for the correlation function of the log-amplitude fluctuations of the off-axis Hermite–Gaussianbeam in the turbulent atmosphere is found at the receiverplane (z 5 L) as
Bx~p1 , p2 , L ! 5 p ReH E0
L
dhE0
`
kdkE0
2p
du
3 @G1~p1 , p2 , h, k, u, L !
1 G2~p1 , p2 , h, k, u, L !#Fn~k!J ,
(10)
where Re denotes the real part, k 5 k exp(iu) is the two-dimensional spatial frequency in polar coordinates (kx5 k cos u, ky 5 k sin u),
G1~p1 , p2 , h, k, u, L !
5 $2k2/@E~p1 , 0!E~p2 , 0!#%
3 exp~b3xk2 cos2 u 1 b3yk2 sin2 u!
3 exp@igx~ px12 px2
!k cos u
1 igy~ py12 py2
!k sin u]
3 E~p1 , k!E~p2 , 2k!, (11)
G2~p1 , p2 , h, k, u, L !
5 $k2/@E~p1 , 0!E* ~p2 , 0!#%
3 exp~z3xk2 cos2 u
1 z3yk2 sin2 u)exp
3 @i~gxpx12 gx* px2
!k cos u
1 i~gypy12 gy* py2
!k sin u]
3 exp~z2xk cos u
1 z2yk sin u)E~p1 , k!E* ~p2 , k!, (12)
E~p1 , k! 5 Hn~ g4xpx11 g5xk cos u 1 g7x!
3 Hm~ g4ypy11 g5yk sin u 1 g7y!, (13)
E~p2 , k! 5 Hn~ g4xpx21 g5xk cos u 1 g7x!
3 Hm~ g4ypy21 g5yk sin u 1 g7y!, (14)
gx 5 ~1 1 iaxh!/~1 1 iaxL !,
b3x 5 igx~h 2 L !/k,
g4x 5 ax /(~1 1 iaxL !
3 $1 2 iL@~2ax2/k ! 2 ax#%)1/2,
g5x 5 g4x~h 2 L !/k,
g7x 5 g4x$~LVx /k ! 1 @bx~1 1 iaxL !/ax#%,
z2x 5 @2L~L 2 h!Vx#/
3 (k2asx2 $1 2 ~2L/Fx! 1 ~L2/Fx
2!
1 @L2/~k2asx4 !#%),
z3x 5 @Im~gx!#~L 2 h!/k, (15)
with Im representing the imaginary part. gy , b3y , g4y ,g5y , g7y , z2y , and z3y are given also by the correspondinglines in Eqs. (15) with x replaced by y.
Yahya Baykal Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1293
To find the log-amplitude structure function of the off-axis Hermite–Gaussian beam in the turbulent atmo-sphere, we insert Eq. (10) into Eq. (A22) to obtain
Dx~p1 , p2 , L !
5 p ReH E0
L
dhE0
`
kdkE0
2p
du@G1~p1 , p1 , h, k, u, L !
1 G2~p1 , p1 , h, k, u, L ! 1 G1~p2 , p2 , h, k, u, L !
1 G2~p2, p2, h, k, u, L ! 2 2G1~p1 , p2, h, k, u, L !
2 2G2~p1, p2, h, k, u, L !#Fn~k!J . (16)
The correlation function of the phase fluctuations of theoff-axis Hermite–Gaussian beam is found at the receiverplane (z 5 L) by following a procedure similar to that inthe log-amplitude case. H ( px , py , L, kx , ky , z8) de-rived for the log-amplitude is substituted into Eq. (A20),and T2( px , py , L, kx , ky , z8) for the off-axis Hermite–Gaussian beam is calculated. Consequently, T2( px ,py , L, kx , ky , z8) is employed in Eq. (A21), and the for-mulation for the correlation function of the phase fluctua-tions of the off-axis Hermite–Gaussian beam in the tur-bulent atmosphere is represented at the receiver plane(z 5 L) as
BS~p1 , p2 , L ! 5 2p ReH E0
L
dhE0
`
kdkE0
2p
du
3 @G1~p1 , p2 , h, k, u, L !
2 G2~p1 , p2 , h, k, u, L !#Fn~k!J .
(17)
To find the phase structure function of the off-axisHermite–Gaussian beam in the turbulent atmosphere,we insert Eq. (17) into Eq. (A23) to obtain
DS~p1 , p2 , L !
5 2p ReH E0
L
dhE0
`
kdkE0
2p
du
3 @G1~p1 , p1 , h, k, u, L ! 2 G2~p1 , p1 , h, k, u, L !
1 G1~p2 , p2 , h, k, u, L ! 2 G2~p2 , p2 , h, k, u, L !
2 2G1~p1 , p2 , h, k, u, L !
1 2G2~p1 , p2 , h, k, u, L !#Fn~k!J . (18)
Utilizing Eq. (A24), one can obtain the wave structurefunction of the off-axis Hermite–Gaussian beam in theturbulent atmosphere by the addition of Eqs. (16) and(18).
3. EXTENSION OF THE FORMULATIONFOR THE HERMITE–SINUSOIDAL-GAUSSIAN BEAMIn the second stage of our formulation, the results pre-sented in Section 2 for the off-axis Hermite–Gaussianbeam wave excitation are extended to obtain a solutionfor Hermite–sinusoidal-Gaussian beams. For this pur-pose, the field at the source plane for the Hermite–sinusoidal-Gaussian beam is written as
un,mHSG~sx , sy , z 5 0 !
5 l1An,mHn~axsx 1 bx!Hm~aysy 1 by!
3 exp@212 k~axsx
2 1 aysy2!#exp@2i~Vxsx 1 Vysy!#
3 exp~2if ! 1 l2An,mHn~axsx 1 bx!
3 Hm~aysy 1 by!
3 exp@212 k~axsx
2 1 aysy2!#
3 exp@2i~Yxsx 1 Yysy!#exp~2if !
5 l1un,mOA ~sx , sy , z 5 0 !uVx ,Vy
1 l2un,mOA ~sx , sy , z 5 0 !uYx ,Yy
, (19)
where un,mOA (sx , sy , z 5 0)uVx ,Vy
denotes that the field atthe source plane for the off-axis Hermite–Gaussian beamis evaluated at Vx , Vy and un,m
OA (sx , sy , z 5 0)uYx ,Yyde-
notes that the field at the source plane for the off-axisHermite–Gaussian beam is evaluated at Yx , Yy .
Note that Eq. (19) represents a field at the source planeformed by the superposition of two off-axis Hermite–Gaussian beams. Here l1 and l2 are complex coefficients.Vx , Vy and Yx , Yy are the complex displacement param-eters in the sx , sy directions associated with the Gaussianpart of the first and the second beams, respectively.(Note that Yx 5 Yxr 1 iYxi , Yy 5 Yyr 1 iYyi , whereYxr , Yxi denote the real and imaginary components of Yxand Yyr , Yyi denote the real and imaginary componentsof Yy). By adjusting these complex parameters of l1 , l2 ,Vx , Vy , and Yx , Yy in Eq. (19), we can introduce any in-dependent form of cosh(sinh), cos(sin) dependence intothe Hermite–Gaussian beam excitation; thus we can ob-tain the general form of the Hermite–sinusoidal-Gaussian beam excitation.
The log-amplitude and phase correlation and structurefunctions for each of the two individual off-axis Hermite–Gaussian beams in Eq. (19) are found in Section 2. Inthis section, by using our off-axis Hermite–Gaussianbeam results represented in Section 2, we formulate thelog-amplitude and phase correlation and structure func-tions for the general form of the Hermite–sinusoidal-Gaussian beam excitation given by Eq. (19).
By substituting the field at the source plane for theHermite–sinusoidal-Gaussian beam given by Eq. (19) intoEq. (4) and by using Eq. (5), we find the field for theHermite–sinusoidal-Gaussian beam at r 5 (p, z)5 ( px , py , z) in free space (i.e., in the absence of turbu-lence), un,m
FS,HSG(p, z), as
1294 J. Opt. Soc. Am. A/Vol. 21, No. 7 /July 2004 Yahya Baykal
un,mFS,HSG~p, z ! 5 l1un,m
FS,OA~p, z !uVx ,Vy
1 l2un,mFS,OA~p, z !uYx ,Yy
, (20)
where un,mFS,OA(p, z)uVx ,Vy
is the field for the off-axisHermite–Gaussian beam at r 5 (p, z) 5 ( px , py , z) infree space (i.e., in the absence of turbulence) given by Eq.(5) evaluated at Vx , Vy and
un,mFS,OA~p, z !uYx ,Yy
is the field for the off-axis Hermite–Gaussian beam at r5 (p, z) 5 ( px , py , z) in free space (i.e., in the absenceof turbulence) given by Eq. (5) evaluated at Yx , Yy .
By use of Eqs. (20) and (A3) and expression (A4) in Eq.(A2), the fluctuations for the general Hermite–sinusoidal-Gaussian complex amplitude at z 5 L, i.e., at the re-ceiver plane, are found as Eq. (A5), however, withH( px , py , L, kx , ky , z8) replaced by HN( px , py , L,kx , ky , z8), which is calculated as
HN~ px , py , L, kx , ky , z8!
5 HN1~ px , py , L, kx , ky , , z8!
1 HN2~ px , py , L, kx , ky , z8!, (21)
where, by use of short notation,
HN1~p! 5 l1fV~p!fVY~p!HV~p!, (22)
HN2~p! 5 l2fY~p!fVY~p!HY~p!, (23)
fVY~p! 5 1/@l1fV~p! 1 l2fY~p!#, (24)
fV~p! 5 exp$2iVx2L/@2k~1 1 iaxL !#%
3 exp$2iVy2L/@2k~1 1 iayL !#%
3 exp$2iVxpx /@~1 1 iaxL !#%
3 exp$2iVypy /@~1 1 iayL !#%Hn~ g4xpx
1 g7x!Hm
3 ~ g4ypy 1 g7y!. (25)
fY(p) is also given by Eq. (25) but with Vx and Vy replacedby Yx and Yy , respectively. HV(p) and HY(p) are the Hin Eq. (A6) found for the off-axis Hermite–Gaussian beamevaluated at Vx , Vy and Yx , Yy , respectively. Examin-ing Eq. (21) together with Eqs. (22)–(25), it can be seenthat a complicated solution HN , which is valid for theHermite–sinusoidal-Gaussian beam, is expressed interms of relatively easier solutions of HV(p) and HY(p),which are valid for the off-axis Hermite–Gaussian beams.Thus, through HN( px , py , L, kx , ky , z8) in Eq. (21), theresults for the general form of the Hermite–sinusoidal-Gaussian beam can be obtained by utilizing the resultsfor the off-axis Hermite–Gaussian beams.
Substituting Eq. (21) into Eq. (A11) with HN taking theplace of H, we calculated T1( px , py , L, kx , ky , z8),valid for the general form of the Hermite–sinusoidal-Gaussian beam excitation. This calculated T1 is used in
Eq. (A13), and the correlation function of the log-amplitude fluctuations of the general form of theHermite–sinusoidal-Gaussian beam excitation is found atthe receiver plane (z 5 L) and is presented below:
Bx~p1 , p2 , L ! 5 p ReH E0
L
dhE0
`
kdkE0
2p
du
3 @G1N~p1 , p2 , h, k, u, L !
1 G2N~p1 , p2 , h, k, u, L !#Fn~k!J ,
(26)
where, in short notation,
G1N~p1 , p2!
5 fVY~p1!fVY~p2!@l12fV~p1!fV~p2!G1V
1 l1l2fV~p1!fY~p2!G1VY
1 l2l1fY~p1!fV~p2!G1YV
1 l22fY~p1!fY~p2!G1Y#, (27)
G2N~p1 , p2!
5 fVY~p1!@ fVY~p2!#* $ul1u2fV~p1!@ fV~p2!#* G2V
1 l1~l2!* fV~p1!@ fY~p2!#* G2VY
1 l2~l1!* fY~p1!@ fV~p2!#* G2YV 1 ul2u2fY~p1!
3 @ fY~p2!#* G2Y%, (28)
G1VY 5 $2k2/@E~p1 , 0!EY~p2 , 0!#%exp~b3xk2 cos2 u
1 b3yk2 sin2 u!exp$@igx~ px12 px2
!
1 g1xVY#k cos u 1 @igy~ py12 py2
!
1 g1yVY#k sin u%E~p1 , k!EY~p2 , 2k!, (29)
G1YV 5 $2k2/@EY~p1 , 0!E~p2 , 0!#%exp~b3xk2 cos2 u
1 b3yk2 sin2 u!exp$@igx~ px12 px2
!
2 g1xVY#k cos u 1 @igy~ py12 py2
!
2 g1yVY#k sin u%EY~p1 , k!E~p2 , 2k!, (30)
G2VY 5 (k2/$E~p1 , 0!@EY~p2 , 0!#* %)exp~z3xk2 cos2 u
1 j3yk2 sin2 u!exp$@i~gxpx12 gx* px2
!
1 g2xVY#k cos u 1 @i~gypy12 gy* py2
!
1 g2yVY#k sin u%E~p1 , k!@EY~p2 , k!#* , (31)
Yahya Baykal Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1295
G2YV 5 (k2/$EY~p1 , 0!@E~p2 , 0!#* %)exp~z3xk2 cos2 u
1 z3yk2 sin2 u!exp$@i~gxpx12 gx* px2
!
1 ~ g2xVY!* #k cos u 1 @i~gypy12 gy* py2
!
1 ~ g2yVY!* #k sin u%EY~p1 , k!@E~p2 , k!#* ,
(32)
g1xVY 5 i~L 2 h!~Vx 2 Yx!/@k~1 1 iaxL !#, (33)
g1yVY 5 i~L 2 h!~Vy 2 Yy!/@k~1 1 iayL !#, (34)
g2xVY 5 $i~L 2 h!Vx /@k~1 1 iaxL !#%
1 $i~h 2 L !~Yx!* /@k~1 2 iax* L !#%, (35)
g2yVY 5 $i~L 2 h!Vy /@k~1 1 iayL !#%
1 $i~h 2 L !~Yy!* /@k~1 2 iay* L !#%, (36)
where * denotes the complex conjugate. G1V , G2V aregiven by Eqs. (11) and (12), respectively. G1Y , G2Y arealso given by Eqs. (11) and (12) but with Vx and Vy re-placed by Yx and Yy , respectively. EY denotes either Eq.(13) or (14) but with Vx and Vy replaced by Yx and Yy ,respectively. Note that G1V , G2V and E are parts in thesolution for the correlation function of the off-axisHermite–Gaussian beam in the turbulent atmosphere,and G1Y , G2Y and EY are found by use of G1V , G2V andE, i.e., by use of the off-axis Hermite–Gaussian beam so-lutions. Thus, when Eq. (26), together with Eqs. (27)–(36), is examined, it can be observed that the log-amplitude correlation function of the Hermite–sinusoidal-Gaussian laser beam in the turbulentatmosphere is expressed by utilizing parts of the solutionfor the log-amplitude correlation function of the off-axisHermite–Gaussian beam in the turbulent atmosphere.
To find the log-amplitude structure function for thegeneral form of the Hermite–sinusoidal-Gaussian beamin the turbulent atmosphere, we insert Eq. (26) into Eq.(A22) to obtain
Dx~p1 , p2 , L !
5 p ReH E0
L
dhE0
`
kdkE0
2p
du
3 @G1N~p1 , p1 , h, k, u, L !
1 G2N~p1 , p1 , h, k, u, L !
1 G1N~p2 , p2 , h, k, u, L !
1 G2N~p2 , p2 , h, k, u, L !
2 2G1N~p1 , p2 , h, k, u, L !
2 2G2N~p1 , p2 , h, k, u, L !#Fn~k!J (37)
The correlation function of the phase fluctuations of thegeneral form of the Hermite–sinusoidal-Gaussian beam isfound at the receiver plane (z 5 L) by following a proce-dure similar to that in the log-amplitude case, which isexplained above. HN( px , py , L, kx , ky , z8) derived forthe log-amplitude is substituted into Eq. (A20), andT2( px , py , L, kx , ky , z8) for the general form of the
Hermite–sinusoidal-Gaussian beam is calculated. Con-sequently, T2( px , py , L, kx , ky , z8) is employed in Eq.(A21), and the formulation for the correlation function ofthe phase fluctuations of the general form of theHermite–sinusoidal-Gaussian beam in the turbulent at-mosphere is found at the receiver plane (z 5 L) as
BS~p1 , p2 , L ! 5 2p ReH E0
L
dhE0
`
kdkE0
2p
du
3 @G1N~p1 , p2 , h, k, u, L !
2 G2N~p1 , p2 , h, k, u, L !#Fn~k!J .
(38)
To find the phase structure function for the generalform of the Hermite–sinusoidal-Gaussian beam in theturbulent atmosphere, we insert Eq. (38) into Eq. (A23) toobtain
DS~p1 , p2 , L !
5 2p ReH E0
L
dhE0
`
kdkE0
2p
du
3 FG1N~p1 , p1 , h, k, u, L !
2 G2N~p1 , p1 , h, k, u, L !
1 G1N~p2 , p2 , h, k, u, L !
2 G2N~p2 , p2 , h, k, u, L !
2 2G1N~p1 , p2 , h, k, u, L !
1 2G2N~p1 , p2 , h, k, u, L !]Fn~k!J . (39)
Utilizing Eq. (A24), we obtain the wave structure func-tion for the general form of the Hermite–sinusoidal-Gaussian beam in the turbulent atmosphere by the addi-tion of Eqs. (37) and (39).
4. SPECIAL CASESIn this section we report the limiting and some specialcases of our solutions. In Subsection 4.A, our Hermite–sinusoidal-Gaussian beam results are shown to reduce toour off-axis Hermite–Gaussian beam results in the limit-ing cases. In Subsection 4.B, the equivalence of our so-lutions to the known results are reported. In Subsection4.C, some special cases of interest among the generalform of the Hermite–sinusoidal-Gaussian beams in theturbulent atmosphere are noted.
A. Off-Axis Hermite–Gaussian LimitIn the limit when the field at the source plane for theHermite–sinusoidal-Gaussian beam given by Eq. (19) isreduced to the field at the source plane for the off-axisHermite–Gaussian beam given by Eq. (1), i.e., when l15 1 and l2 5 0, then the log-amplitude and the phasecorrelation and structure functions of the Hermite–sinusoidal-Gaussian beams in the turbulent atmospheregiven by Eqs. (26) and (37)–(39) correctly reduce to thelog-amplitude and the phase correlation and structure
1296 J. Opt. Soc. Am. A/Vol. 21, No. 7 /July 2004 Yahya Baykal
functions of the off-axis Hermite–Gaussian beams in theturbulent atmosphere given by Eqs. (10) and (16)–(18).
In the limit when the second term of the field at thesource plane for the Hermite–sinusoidal-Gaussian beamgiven by Eq. (19) is the exact replica of the first term, i.e.,when l1 5 l2 5 1 and also Yx 5 Vx and Yy 5 Vy , thenthe log-amplitude and the phase correlation and structurefunctions of the Hermite–sinusoidal-Gaussian beams inthe turbulent atmosphere given by Eqs. (26) and (37)–(39)reduce to the log-amplitude and the phase correlation andstructure functions of the off-axis Hermite–Gaussianbeams in the turbulent atmosphere given by Eqs. (10) and(16)–(18), as expected.
B. Fundamental Mode and the Higher-Order Single-Mode LimitsIn the limit when the field at the source plane for the off-axis Hermite–Gaussian beam given by Eq. (1) is reducedto the Hermite–Gaussian excitation, i.e., when l1 5 1and l2 5 0, Vx 5 Vy 5 0, and bx 5 by 5 0 and also whenax 5 ay 5 1/asx 5 1/asy 5 1/as (as is the source size)and Fx 5 Fy 5 `, then the log-amplitude and the phasecorrelation and structure functions of the Hermite–sinusoidal-Gaussian beams in the turbulent atmospheregiven by Eqs. (26) and (37)–(39) correctly reduce tothe log-amplitude and the phase correlation and structurefunctions for the known higher-order single-modeHermite–Gaussian correlation and structure functions.13
(Note that the negative sign is missing in the phase cor-relation and phase structure function expressions inRef. 13.)
In addition to the conditions pointed out in the preced-ing paragraph, if n 5 m 5 0 is also taken, i.e., the beamwave limit is considered, then the log-amplitude and thephase correlation and structure functions of the Hermite–sinusoidal-Gaussian beams in the turbulent atmospheregiven by Eqs. (26) and (37)–(39) reduce to the well-knownGaussian beam wave (TEM00) fundamental moderesults.14 Using the conditions given in this paragraph,and for a plane-wave source (as 5 `), we have evaluatednumerically the correlation function of the log-amplitudefluctuations given by Eq. (26) at the origin of the receiverplane (p1 5 0, p2 5 0, L). Thus-found Bx(0, 0, L)gives the well-known variance of the log-amplitude fluc-tuations for the plane wave, sx
2 5 0.31Cn2k7/6L11/6, where
Cn2 is the structure constant, k 5 2p/l is the wave num-
ber, l is the wavelength, and L is the link length. In Fig.2, Bx(0, 0, L) 5 sx
2 for the plane wave obtained as thelimiting case (when l1 5 1 and l2 5 0, Vx 5 Vy 5 0, n5 m 5 0, bx 5 by 5 0, as 5 `, ax 5 ay 5 0, and Fx5 Fy 5 `) of Eq. (26) is plotted versus the length L of ahorizontal link. The link parameters (the structure con-stant Cn
2, the wavelength l, and the link length L) em-ployed in Fig. 2 are for weak atmospheric turbulence, andtwo links are considered with wavelengths of 850 and1550 nm, which are the most frequently used carrierwavelengths in the current FSO links. As expected, theresults shown in Fig. 2 match the known variance of thelog-amplitude fluctuations for the plane wave, sx
2
5 0.31Cn2k7/6L11/6. Evaluation of the log-amplitude fluc-
tuations (related to the scintillations and the bit errorrate) for the general Hermite–sinusoidal-Gaussian laser
beam parameters will be the subject of our future re-search.
C. Hermite–Sine (Cosine)-Gaussian and Hermite–Sinh(Cosh)-Gaussian LimitsThe special case of the log-amplitude and the phase cor-relation and structure functions of the Hermite–sine-Gaussian laser beam in the turbulent atmosphere are ob-tained by choosing l1 5 2l2 5 1/(2i), Vx 5 2Yx5 2Vxr , and Vy 5 2Yy 5 2Vyr in Eqs. (26) and (37)–(39).
Similarly, one can obtain the log-amplitude and thephase correlation and structure functions of the Hermite–cosine-Gaussian laser beam in the turbulent atmosphereby choosing l1 5 l2 5 1/2, Vx 5 2Yx 5 2Vxr , and Vy5 2Yy 5 2Vyr in Eqs. (26) and (37)–(39).
The special case of the log-amplitude and the phase cor-relation and structure functions of the Hermite–hyperbolic-sine-Gaussian laser beam in the turbulent at-mosphere are obtained by choosing l1 5 2l2 5 1/2, Vx5 2Yx 5 i Vxi , and Vy 5 2Yy 5 i Vyi in Eqs. (26) and(37)–(39).
Especially, the special case of our formulation involvingHermite–hyperbolic-cosine-Gaussian dependence is of in-terest to us owing to the nature of hyperbolic cosine de-pendence exhibiting the concentration of the energy inthe outer lobes of the beam. Thus, in a similar manner,the log-amplitude and the phase correlation and structurefunctions of the Hermite–hyperbolic-cosine-Gaussian la-ser beam in the turbulent atmosphere are obtained bychoosing l1 5 l2 5 1/2, Vx 5 2Yx 5 i Vxi , and Vy5 2Yy 5 i Vyi in Eqs. (26) and (37)–(39).
To find the log-amplitude and the phase correlation andstructure functions of the hyperbolic-cosine-Gaussian la-ser beam in the turbulent atmosphere, one can also takethe condition of n 5 m 5 0 is in addition to the condi-tions listed in obtaining the Hermite–hyperbolic-cosine-Gaussian beam results.
Fig. 2. Variance of the log-amplitude fluctuations Bx(0, 0, L)5 sx
2 for the plane wave obtained as the limiting case (when l15 1 and l2 5 0, Vx 5 Vy 5 0, n 5 m 5 0, bx 5 by 5 0, as5 `, ax 5 ay 5 0, and Fx 5 Fy 5 `) of Eq. (26) versus thelength L of a horizontal link in weak atmospheric turbulence.
Yahya Baykal Vol. 21, No. 7 /July 2004 /J. Opt. Soc. Am. A 1297
5. CONCLUSIONFor investigating the effect of atmospheric turbulence inFSO systems under the general Hermite–sinusoidal-Gaussian beam wave excitation, we derived and pre-sented the formulations for the log-amplitude and phasecorrelation and structure functions of such laser beams ina turbulent atmosphere. To understand the effect ofvarious forms of the Hermite–sinusoidal-Gaussian beamson the system performance in atmospheric turbulence,e.g., the effect on the intensity or power scintillationnoise, or on the angle-of-arrival fluctuations, we need toknow the correlation behavior of the Hermite–sinusoidal-Gaussian beams. Our results correctly reduce to theknown higher-order-mode log-amplitude and phase corre-lation and structure functions,13 which, in turn, reduce tothe fundamental mode (TEM00-mode) results.14
Our formulations presented in this paper cover a widerange of laser beams, such as off-axis Hermite–Gaussian,Hermite–cosh-Gaussian, cosh (sinh)-Gaussian, cosh(sinh) and cosine (sine) beams. Among these, we are es-pecially interested in finding the behavior in atmosphericturbulence for the Hermite–cosh-Gaussian dependence,because through cosh dependence, it is possible to concen-trate the energy of the beam in the outer lobes, yieldingtwo separate beams propagating in the atmosphere.This will be the subject for our future work.
APPENDIX ABy use of the Rytov-method solution, the log-amplitudeand phase fluctuations are evaluated, from which the cor-relation and structure functions of the general laser beamare derived. In the derivations, a procedure similar tothat in Ishimaru’s TEM00-mode formulation14 is followed.Here we outline the derivation.
The field at r 5 (p, z) is given by Rytov’s method as
u~p, z ! 5 un,mFS ~p, z !exp@ c~p, z !#, (A1)
where
c ~p, z ! 5 k2/@2pun,mFS ~p, z !#E
V8d3r8n1~r8!un,m
FS ~r8!
3 exp~ikur 2 r8u!/ur 2 r8u (A2)
is the fluctuations of the complex amplitude in turbu-lence, un,m
FS (p, z) is the field at r 5 (p, z) 5 ( px , py , z)in free space (i.e., in the absence of turbulence), andun,m
FS (r8) is the field at r8 5 (p8, z8) 5 ( px8 , py8 , z8)in free space (i.e., in the absence of turbulence).Here r 5 (p, z) 5 ( px , py , z) and r8 5 (p8, z8)5 ( px8 , py8 , z8) are the three-dimensional vectors. InEq. (A2) the integration is taken over the volume V8, i.e.,d3r8 5 dpx8dpy8dz8.
The refractive index, when it is statistically homoge-neous and isotropic, can be written as
n1~ px8 , py8 , z8! 5 E2`
` E2`
`
exp~ikxpx8
1 ikypy8!dZn~kx , ky , z8!, (A3)
where dZn is the random amplitude of the spectrum ofthe refractive-index fluctuations and kx and ky are thespatial frequencies in the x and y directions, respectively.In writing Eq. (A3), we use notation similar to that in Ref.14 where the integrations are implicit with respect to kxand ky . These integration variables become explicit asdkx1
dky1dkx2
dky2when the ensemble average of the prod-
uct of dZn , i.e., ^dZn (kx1, ky1
, z18)dZn (kx2, ky2
, z28)& isfound by use of the relation in Eq. (A15).
Since, for the major portions of the beam, ( px2px8) and( py2py8) are much smaller than (z2z8), the Green’s func-tion in Eq. (A2) is approximated as
exp~ikur 2 r8u!/ur 2 r8u
' exp(ik$~z 2 z8! 1 @~ px 2 px8!2
1 ~ py 2 py8!2#/@2~z 2 z8!#%)/~z 2 z8!. (A4)
By use of Eq. (A3) and expression (A4) in Eq. (A2), thefluctuations of the complex amplitude at z 5 L, i.e., atthe receiver plane, are found as
C~p, L ! 5 E0
L
dz8E2`
` E2`
`
H~ px , py , L, kx , ky , z8!
3 dZn~kx , ky , z8!, (A5)
where the last two integrations are implicit with respectto kx and ky . Here
H~ px , py , L, kx , ky , z8!
5 k2/@2k~L 2 z8!un,mFS ~p, L !#E
2`
`
dpx8E2`
`
dpy8
3 exp~ikxpx8 1 ikypy8!un,mFS ~p8, z8!exp(ik$~z 2 z8!
1 @~ px 2 px8!2 1 ~ py 2 py8!2#/@2~z 2 z8!#%)/~z 2 z8!,
(A6)
where un,mFS (p8, z8) is the field at r8 5 (p8, z8)
5 ( px8 , py8 , z8) in free space (i.e., in the absence of tur-bulence) and un,m
FS (p, L) is the field at the receiver planein free space (i.e., in the absence of turbulence).
Since
C~p, L ! 5 x~p, L ! 1 iS~p, L !, (A7)
where x and S are the log-amplitude and phase fluctua-tions, respectively,
x~p, L ! 512 @ c~p, L ! 1 c* ~p, L !#, (A8)
S~p, L ! 5 @1/~2i !#@ c~p, L ! 2 c* ~p, L !#.(A9)
By use of Eq. (A5), together with Eq. (A6) in Eq. (A8), thelog-amplitude fluctuations are found as
x~p, L ! 5 E0
L
dz8E2`
` E2`
`
T1~ px , py , L, kx , ky , z8!dZn
3 ~kx , ky , z8!, (A10)
where the last two integrations are implicit with respectto kx and ky . Here
1298 J. Opt. Soc. Am. A/Vol. 21, No. 7 /July 2004 Yahya Baykal
T1~ px , py , L, kx , ky , z8!
512 @H~ px , py , L, kx , ky , z8!
1 H* ~ px , py , L, 2kx , 2ky , z8!#, (A11)
where * denotes the complex conjugate. The correlationfunction of the log-amplitude fluctuations at the receiverplane (z 5 L) is defined by
Bx~p1 , p2 , L ! 5 ^x~p1 , L !x~p2 , L !&, (A12)
where ^•& denote the ensemble average over the mediumstatistics. Substituting Eq. (A10) into Eq. (A12), we find
Bx~p1 , p2 , L !
5 2pE0
L
dhE2`
`
dkxE2`
`
dkyT1
3 ~ px1, py1
, L, kx , ky , h!
3 T1~ px2, py2
, L, 2kx , 2ky , h!Fn~k!, (A13)
where Fn(k) is the spectral density of the index-of-refraction fluctuations and k 5 (kx 1 ky)1/2. In findingEq. (A13), we use the equalities and approximations givenby Eqs. (A14) and (A15), expression (A16), and Eq.(A17):14
dZn* ~kx , ky , z8!
5 dZn~2kx , 2ky , z8!, (A14)
^dZn~kx1, ky1
, z18 !dZn~kx2, ky2
, z28 !&
5 d ~kx11 kx2
!d ~ky11 ky2
!
3 Fn~kx1, ky1
, z18 2 z28 !dkx1dky1
dkx2dky2
,
(A15)
where d is the delta function and Fn is the two-dimensional spectral density of the index-of-refractionfluctuations. Since T1 is a slowly varying function of z18 ,we can replace z18 and z28 in T1 with h 5 (z18 1 z28)/2.Also, Fn is an even function of z18 2 z28 5 zd8 , and it de-creases fast for uzd8 u . correlation distance, so we have
E0
L
dz18E0
L
dz28f~h, zd8 !
5 E0
L
dzd8Ezd8 /2
L2zd8 /2dh@ f~h, zd8 ! 2 f~h, 2zd8 !#
' 2E0
L
dhE0
`
dzd8 f~h, zd8 !. (A16)
Also,
E0
`
dzd8Fn~kx1, ky1
, zd8 ! 5 pFn~k!. (A17)
To find the correlation function of the phase fluctua-tions at the receiver plane (z 5 L), i.e.,
BS~p1 , p2 , L ! 5 ^S~p1 , L !S~p2 , L !&, (A18)
we can follow similar procedure. By use of Eq. (A5), to-gether with Eq. (A6) in Eq. (A9), the phase fluctuationsare found as
S~p, L ! 5 E0
L
dz8E2`
` E2`
`
T2~ px , py , L, kx , ky , z8!
3 dZn~kx , ky , z8!, (A19)
where the last two integrations are implicit with respectto kx and ky . Here
T2~ px , py , L, kx , ky , z8!
5 @1/~2i !#@H~ px , py , L, kx , ky , z8!
2 H* ~ px , py , L, 2kx , 2ky , z8!#. (A20)
Substituting Eq. (A19) into Eq. (A18) and using the sameequalities and approximations14 given by relations (A14)–(A17) for T2 , we find
BS~p1 , p2 , L ! 5 2pE0
L
dhE2`
`
dkxE2`
`
dky
3 T2~ px1, py1
, L, kx , ky , h!
3 T2~ px2, py2
, L, 2kx , 2ky , h!Fn~k!.
(A21)
The structure functions are found from the correspondingcorrelation functions, i.e.,
Dx~p1 , p2 , L ! 5 ^@x~p1 , L ! 2 x~p2 , L !#2&
5 Bx~p1 , p1 , L ! 1 Bx~p2 , p2 , L !
2 2Bx~p1 , p2 , L !, (A22)
DS~p1 , p2 , L ! 5 ^@S~p1 , L ! 2 S~p2 , L !#2&
5 BS~p1 , p1 , L ! 1 BS~p2 , p2 , L !
2 2BS~p1 , p2 , L !, (A23)
where Dx and DS are the log-amplitude and phase struc-ture functions, respectively.
The wave structure function is given by
Dc 5 Dx 1 DS . (A24)
Yahya Baykal can be reached at the address on the titlepage or by phone, 90-312-284-4500, ext. 4023; fax, 90-312-284-8043; or e-mail, [email protected].
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