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Optics Express

  • Editor: Michael Duncan
  • Vol. 12, Iss. 20 — Oct. 4, 2004
  • pp: 4659–4674
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Analysis of reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere

Halil Tanyer Eyyuboğlu and Yahya Baykal  »View Author Affiliations


Optics Express, Vol. 12, Issue 20, pp. 4659-4674 (2004)
http://dx.doi.org/10.1364/OPEX.12.004659


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Abstract

In a turbulent atmosphere, starting with a cos-Gaussian excitation at the source plane, the average intensity profile at the receiver plane is formulated. This average intensity profile is evaluated against the variations of link lengths, turbulence levels, two frequently used free-space optics wavelengths, and beam displacement parameters. We show that a cos-Gaussian beam, following a natural diffraction, is eventually transformed into a cosh-Gaussian beam. Combining our earlier results with the current findings, we conclude that cos-Gaussian and cosh-Gaussian beams act in a reciprocal manner after propagation in turbulence. The rates (paces) of conversion in the two directions are not the same. Although the conversion of cos-Gaussian beams to cosh-Gaussian beams can happen over a wide range of turbulence levels (low to moderate to high), the conversion of cosh-Gaussian beams to cos-Gaussian beams is pronounced under relatively stronger turbulence conditions. Source and propagation parameters that affect this reciprocity have been analyzed.

© 2004 Optical Society of America

1. Introduction

The limiting cases of our formulation are compared with the known Gaussian beam wave solution in the turbulent atmosphere and cos-Gaussian beam solution in free space (i.e., in the absence of turbulence), and we found that there is exact conformity to these limiting cases.

2. Propagation of a cos-Gaussian beam in a turbulent atmosphere

Fig. 1. Propagation geometry.

The sinusoidal-Gaussian beam wave field at the source plane (z = 0), which is essentially a Gaussian beam with complex displacement parameters of Vx = Vxr + iVxi , Yx = Yxr + iYxi . along the sx direction and Vy = Vyr + iVyi , Yy = Yyr + iYyi along the Sy direction, is written as

us(sx,sy,z=0)=0.5Aexp(iϕ)exp[0.5(sx2αsx2+sy2αsy2)]
×{exp[i(Vxsx+Vysy)]+exp[i(Yxsx+Yysy)]}
(1)

where A is the amplitude of the field at the origin of the source plane (i.e., at sx = sy = z = 0); ϕ is the constant phase factor, i = (-1)1/2; αsx and αsy are the respective source sizes of the Gaussian beam in the sx and sy directions; Vxr , Vxi . denote the real and imaginary components of Vx ; and Vyr , Vyi . denote the real and imaginary components of Vy . Likewise, Yxr , Yxi are the real and imaginary components of Yx , whereas Yyr , Yyi refer to the real and imaginary components of Yy . Here, focal lengths along both sx and sy are taken to be infinite.

A cos-Gaussian laser beam is generated by choosing Vx = -Yx = -Vxr and Vy = -Yy = -Vyr . Similarly, one can obtain a cosh-Gaussian laser beam by setting Vx = -Yx = iVxi and Vy = -Yy = iVyi . Thus, use of Eq. (1) with A = 1, the intensity distribution of the cos-Gaussian beam at the exit plane of the laser is

Is(sx,sy,z=0)=exp[(sx2αsx2+sy2αsy2)]cos2(Vxrsx+Vyrsy)
(2)

At the receiver plane (z = L), we represented field u(p,L,t) by applying the Huygens-Fresnel principle as follows:

u(p,L,t)=kexp(ikL)(2πiL)d2sus(s)
×exp[ik(ps)2(2L)+ψ(s,p)i2πft]
(3)

where k is the wave number; us (s) is the field of a cos-Gaussian beam at the source plane(z = 0)as provided by Eq. (1); ψ(s,p)is the solution to the Rytov method that represents the random part of the complex phase of a spherical wave that propagates from the source point (s, z = 0) to the receiver point (p,z = L);f is the frequency; and t denotes time.

The average intensity at the receiver plane is < I(p,L)>=<u(p,L,t)u *(p,L,t)> where the * represents the complex conjugate and the < > indicate the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the atmospheric turbulence. With this definition, Eq. (3) is transformed into

<I(p,L)>=k2(2πL)2d2s1d2s2us(s1)us*(s2)exp{ik[(ps1)2(ps2)2](2L)}
×<exp[ψ(s1,p)+ψ*(s2,p)]>
(4)

The ensemble average term within the integrand of Eq. (4) is 12

12. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979). [CrossRef]

<exp[ψ(s1,p)+ψ*(s2,p)]>=exp[0.5Dψ(s1s2)]=exp[ρ02(s1s2)2]
(5)

where Dψ (s 1 - s 2) is the wave structure function, and ρ 0 = (0.545 Cn2 k 2 L)-3/5 is the coherence length of a spherical wave that propagates in the turbulent medium, with Cn2 being the constant of the structure. Here we state that the Rytov method is known to be valid in weak turbulence, especially when fourth-order moments such as scintillations are considered. Customarily, weak turbulence is associated with Rytov log amplitude variance 0.307Cn2 k 7/6 L 11/6, which is quite smaller than unity. However, here we study the second-order moment by utilizing the wave structure function that is approximated by the phase structure function. Rytov’s phase structure function usually accepted to be valid not only for the case of “weak fluctuations”, but for the case of “strong fluctuations” as well 17

17. A. Ishimaru, “Phase fluctuations in a turbulent medium,” Applied Optics 16, 3190–3192 (1977). [CrossRef] [PubMed]

, i.e., when 0.307Cn2 k 7/6 L 11/6 > 0.5. For completeness and clarity, we note that in order to obtain simpler and viewable analytical results, we have also employed a quadratic approximation 12

12. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979). [CrossRef]

for the Rytov’s phase structure function.

Substituting Eqs. (1) and (5) into Eq. (4) and expanding the transverse source and transverse receiver coordinates into their corresponding x and y components, the average intensity at the receiver plane becomes

<I(p,L)>=0.25k2(2πL)2ds1xds1yds2xds2yexp[0.5(s1x2+s2x2)αsx20.5(s1y2+s2y2)αsy2]
×{exp[iVxr(s1x+s2x)+iVyr(s1y+s2y)]+exp[iVxr(s1x+s2x)iVyr(s1y+s2y)]
+exp[iVxr(s1xs2x)+iVyr(s1ys2y)]+exp[iVxr(s1xs2x)iVyr(s1ys2y)]}
×exp[0.5(ikL)(s1x22pxs1xs2x2+2pxs2x+s1y22pys1ys2y2+2pys2y)]
×exp[ρ02(s1x22s1xs2x+s2x2+s1y22s1ys2y+s2y2)]
(6)

By following the steps outlined in Appendix A, the average intensity at the receiver plane is

<I(p,L)>=0.5(kL)2ρ04(DsxDsy)12exp{(ρ04k2L2)[px2(αsx2Dsx)+py2(αsy2Dsy)]}
×(exp{2ρ02[Vxr2(ρ02+4αsx2)(αsx2Dsx)+Vyr2(ρ02+4αsy2)(αsy2Dsy)]}
×cos[(2ρ04k2L2)(VxrpxDsx+VsrpyDsy)]
+exp{ρ04[Vxr2(αsx2Dsx)+Vyr2(αsy2Dsy)]}
×cosh{(2ρ04kL)[(Vxrpx(αsx2Dsx)+Vyrpy(αsy2Dsy))]})
(7)

where the parameters that appear in Eq. (7) are thoroughly defined in Appendix A. Because of the nature of Eq. (7), the average intensity profile of the receiver plane is initially cos, but source and propagation parameters eventually force it to change to a cosh type. To consider the conditions under which this happens, we compare the cos and cosh terms that include the accompanying exponential factors. Since for all cases of interest αsx < Vxr1 and/or αsy > Vyr1 the cos term becomes quite negligible when k αsx2 / L <~ 1, kαsy 2 / L <~ 1 and αsx >~ ρ 0 , αsy >~ρ 0.

3. Limiting cases

Here Eq. (7) is checked and was determined to reduce to the following limiting cases correctly. Here we note that, although Eq. (7) is considered in the following several limiting cases for which there are already some available results, this kind of testing might be necessary but not sufficient to prove the accuracy of our formulation. We should establish this accuracy by comparing our numerical results with the experimental work. However, to our knowledge no experimental work exists in the literature to compare and validate Eq. (7) for all cases.

3.2. To determine the limit of a Gaussian beam in free space, we evaluated Eq. (7) with Cn2 = 0 (or alternatively ρ 0 → ∞) and Vxr = Vyr = 0 . A symmetrical beam was taken by choosing αsx = αsy = αs . Then the average intensity in Eq. (7) simplified to

I(p,L)=[k2αs4(L2+k2αs4)]exp{[k2αs2(L2+k2αs4)](px2+py2)}
(8)

Equation (8) matches the free-space propagation limit examined in Ref. 14

14. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002). [CrossRef]

. The precise correspondence of Eq. (8) in this paper with Eq. (5) of Ref. 14

14. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002). [CrossRef]

is accomplished by noting that k 2 αs2 /(L 2 + k 2 αs4) = 2/W 2 , 20.5 αs = W0 , px = x, and py = y.

3.3. Next, our formula in Eq. (7) is checked against the existing result of a Gaussian beam in turbulence. To achieve this, we selected Vxr = Vyr = 0 and also constituted symmetry in the x and y directions so that αsx = αsy = αs . With these conditions, Eq. (7) becomes

<I(p,L)>=[k2αs4ρ02(ρ02L2+4αs2L2+k2αs4ρ02)]exp[k2αs4ρ02(px2+px2)(ρ02L2+4αs2L2+k2αs4ρ02)]
(9)

Equation (9) agrees with Eq. (12

12. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979). [CrossRef]

) of Ref. 12

12. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979). [CrossRef]

. To establish this agreement, we adapted the parameters in Eq. (12) of Ref. [12

12. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979). [CrossRef]

] as

pd = 0,pc2 = px2 + py2, F = ∞, ζ = 0, A 0 = 1,and αo = αs .

3.4. Here we compare our results with the cos-Gaussian beam formulation in free space (in the absence of turbulence). For this purpose, we set Cn2 = 0, thus the average intensity in Eq. (7) leads to

I(p,L)=0.5k2αsx2αsy2gx0.5gy0.5exp{k2[αsx2gxpx2+αsy2gypy2]}exp{L2[αsx2Vxr2gx+αsy2Vyr2gy]}
×(cos{2k2[αsx4Vxrgxpx+αsy4Vyrgypy]}+cosh{2kL[αsx2Vxrgxpx+αsy2Vyrgypy]})
(10)

where gx = (L 2 + k 2 αsx4)-1 and gy = (L 2 + k 2 αsy4)-1 . After reverting to a single coordinate system in the sense that αsx = αsy = wo / 20.5 , px = py = x / 20.5, Vxr = Vyr = Ωo / 20.5, Eq. (10) becomes identical to the intensity equivalent of the field expression of Eq. (14) in Ref. 7

7. N. Zhou and G. Zeng, “Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004). [CrossRef]

(when the free-space unapertured option is chosen).

3.5. Here we consider the average intensity distribution at the receiver plane as given in Eq. (7) with the limit of L being too large, i.e., L → ∞. In this case, since the argument of the cosh term is proportional to 1 / L , it becomes the dominating part, hence the entire cos term, together with its accompanying exponential, virtually drop out of the equation. In this limit Eq. (7) scales down to

<I(p,L)>=0.5(kL)2ρ04(DsxDsy)12exp{(ρ04k2L2)[px2(αsx2Dsx)+py2(αsy2Dsy)]}
×exp{ρ04[Vxr2(αsx2Dsx)+Vyr2(αsy2Dsy)]}
×cosh{(2ρ04kL)[(Vxrpx(αsx2Dsx)+Vyrpy(αsy2Dsy))]}
(11)

By taking / ∂px and / ∂py of Eq. (11) and setting them independently to zero, we found the peak locations to be fixed at pxp = Vxr L / k and pyp = Vyr L / k. Equation (11) can be viewed as the expression of a cosh-Gaussian converted cos-Gaussian beam after having traveled sufficiently along the propagation axis. But this conversion will also occur because of other parameters in question. To this end we introduce I 0 as the ratio of average intensities at two locations on the receiver plane

I0=<I(px=pxp,py=pyp,z=L)><I(px=0,py=0,z=L)>
(12)

ρ04Vxr2αsx2L2ρ04L2+4ρ02αsx2L2+ρ04k2αsx4=0.5n(2I0),ρ04Vyr2αsy2L2ρ04L2+4ρ02αsy2L2+ρ04k2αsy4=0.5n(2I0)
(13)

4. Results

Although our formulation is applicable to cos-Gaussian beams that can have asymmetrical attributes along the x and y directions, here we report only the results of symmetrical beams, i.e., αsx = αsy and Vxr = Vyr . When dominant, the existence of a cos term in the cos-Gaussian beam dictates that the intensity will peak at sx = 0.5 / Vxr and sy = 0.5 / Vyr , where n is an integer starting from zero. Hence we define normalized intensity at the source plane as follows:

IsN(sx,sy,z=0)=Is(sx,sy,z=0)Is(sx=sy=z=0)
(14)

A three-dimensional view of IsN (sx ,sy ,z = 0) is presented in Fig. 2(a) for a cos-Gaussian beam with typical source parameters. Figure 2(b), on the other hand, displays a contour plot of the same beam. From Fig. 2(a) we note that a cos-Gaussian beam appears as a Gaussian beam modulated with a cos function, where the summits of the successive lobes are aligned in the direction of the slanted axis. Hence for better assessment, the subsequent plots, where the vertical axis refers to intensity distribution, are drawn as side views cut along the slanted axis.

Fig. 2. (a) Normalized intensity of a cos-Gaussian beam at the source plane and (b) contour plots of the same cos-Gaussian beam.

The normalized average intensity at receiver plane IrN (px ,py ,z = L) is defined as

IrN(px,py,z=L)=<I(px,py,z=L)>Is(sx=sy=z=0)
(15)

For proper investigation of the dependence of average intensity profile on the propagation distance, the real part of complex displacement parameters, turbulence levels, and wavelength of operation, we normalized each profile with respect to its own peak. Normalized average intensity I r0 (px ,py ,z = L) as defined in this way is

Ir0(px,py,z=L)=<I(px,py,z=L)>Max[<I(px,py,z=L)>]
(16)

Fig. 3. (a) Normalized intensity at the source plane and the normalized average intensity at the receiver plane for a typical cos-Gaussian beam and (b) contour plots for the same cos-Gaussian beam.

In Fig. 5 we illustrate the variation of I r0(px , py ,z = L) against the different values of the real components of complex displacement parameters Vxr and Vyr . Figure 5 demonstrates that the increases in Vxr and Vyr serve to accelerate the formation of a cosh-Gaussian beam. This means that for such cases of cos-Gaussian beam excitation, the transformation into a cosh-Gaussian beam occurs at earlier propagation distances.

Figure 6 provides the variation of I r0(px ,py ,z = L) at λ = 1.55 μm and λ = 0.85 μm in the absence of (Cn2 = 0 m-2/3) and in the presence of (Cn2 = 1 × 10-14 m-2/3) turbulence. Here we observe that the presence of turbulence retards the formation of a cosh-Gaussian beam that originates from cos-Gaussian beam excitation. Lowering the wavelength of operation basically has a similar effect.

By considering all the plots in Figs. 4–6 and Eq. (13), we were able to specify the precise source and propagation conditions that govern the course of conversion from a cos-Gaussian beam into a cosh-Gaussian beam. For example, to accelerate the transformation from a cos-Gaussian beam into a cosh-Gaussian beam, we require

  • smaller source sizes αsx and αsy ,
  • larger beam displacement parameters Vxr and Vyr ,
  • lower structure constants Cn2,
  • higher wavelengths λ ,
  • longer link lengths L.
Fig. 4. Dependence of normalized average intensity at the receiver plane on link length.
Fig. 5. Dependence of normalized average intensity at the receiver plane on the real part of a complex displacement parameter.
Fig. 6. Dependence of normalized average intensity at the receiver plane on turbulence level and wavelength of operation.

  • smaller source sizes αsx and αsy ,
  • smaller displacement parameters Vxr and Vyr ,
  • higher structure constants Cn2,
  • lower wavelengths λ ,
  • longer link lengths L.
Fig. 7. Dependence of normalized average intensity at the receiver plane on link length (cosh-Gaussian source excitation case).
Fig. 8. Dependence of normalized average intensity at the receiver plane on the real part of a complex displacement parameter (cosh-Gaussian source excitation case).
Fig. 9. Dependence of normalized average intensity at the receiver plane on turbulence level and wavelength of operation (cosh-Gaussian source excitation case).

5. Concluding remarks

Appendix A

Here we give a more extensive explanation of the algebra leading from Eq. (6) to Eq. (7), along with a detailed physical description of the terms that appear in the expression for the averaged intensity profile. Equation (6) contains four separate exponential terms within the braces. These are basically the same terms and differ only in signs of s 2x, s 2y, Vxr and Vyr . Hence, once the integration has been performed for one exponential, the rest can simply be obtained by analogy. Moreover we note that there is no coupling between the x and the y indices, which means that the integrations with respect to s 1y and s 2y are replicas of the integrations with respect to s 1x .and s 2x. Consider the first exponential term within braces in Eq. (6) and the integration with respect to s 1x, s 2x only. By excluding the terms outside the main integral and the terms associated with the s 1y and s 2y variables, the remainder of the integral in Eq. (6) that we refer to as I 1x appears as

I1x=ds1xds2xexp[0.5(s1x2+s2x2)αsx2]exp[iVxr(s1x+s2x)]
×exp[0.5(ikL)(s1x22pxs1xs2x2+2pxs2x)]exp[(s1x22s1xs2x+s2x2)ρ02]
=ds1xexp{[0.5αsx2+jk(2L)1ρ02]s1x2+[iVxrjkpxL+2s2x]s1x}
×ds2xexp(0.5s2x2αsx2)exp(iVxrs2x)
×exp[0.5(ikL)(s2x2+2pxs2x)]exp[(s2x2+s1y22s1ys2y+s2y2)ρ02]
(A1)

The isolated d s1x integral on the third line of Eq. (A1) is in the form of Eq. 3.323.2 of Ref. 18

18. I. S. Gradysteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1980).

, i.e., in the form of

dxexp(p2x2qx)=(π0.5p)exp[q2(4p2)]
(A2)

After performing the integration over d s1x and the resultant is combined with the other terms inside the d s2x, integral I 1x becomes

I1x=π0.5[0.5αsx2+1ρ02jk(2L)]0.5ds2x
×exp({0.5αsx1+1ρ02+jk(2L)ρ04[0.5αsx2+1ρ02jk(2L)]}s2x2)
×exp({iVxr+ikpxL+ρ02(iVxrikpxL)[0.5αsx2+1ρ02jk(2L)]}s2x)
(A3)

References and links

1.

A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2425–2432 (1998). [CrossRef]

2.

L. W. Casperson and A. A. Tovar, “Hermite sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998). [CrossRef]

3.

B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999). [CrossRef]

4.

D. Zhao, H. Mao, and H. Liu, “Propagation of off-axial Hermite-cosh-Gaussian laser beams,” J. Opt. A 6, 77–83 (2004). [CrossRef]

5.

S. Yu, H. Guo, X. Fu, and W. Hu, “Propagation properties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. 204, 59–66 (2002). [CrossRef]

6.

S. Konar and J. Soumendu, “Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,” Opt. Commun. 236, 7–20 (2004). [CrossRef]

7.

N. Zhou and G. Zeng, “Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004). [CrossRef]

8.

Y. Song, G. Wangyi, and G. Hong, “Optical resonator with hyperbolic-cosine-Gaussian modes,” Opt. Commun. 221, 241–247 (2003). [CrossRef]

9.

Z. I. Feizulin and Y. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967). [CrossRef]

10.

A. I. Kon and V. I. Tatarskii, “On the theory of propagation of partially coherent light beams in a turbulent atmosphere,” Radiophys. Quantum Electron. 15, 1187–1192 (1972). [CrossRef]

11.

M. S. Belen’kii, A. I. Kon, and V. L. Mironov, “Turbulent distortions of the spatial coherence of a laser beam,” Sov. J. Quantum Electron. 7, 287–290 (1977). [CrossRef]

12.

S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. 69, 1297–1304 (1979). [CrossRef]

13.

M. S. Belen’kii and V. L. Mironov, “Phase fluctuations of a multimode laser field in a turbulent atmosphere,” Sov. J. Quantum Electron. 12, 3–6 (1982). [CrossRef]

14.

C. Y. Young, Y. V. Gilchrest, and B. R. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41, 1097–1103 (2002). [CrossRef]

15.

Y. Baykal, “The correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,” J. Opt. Soc. Am. A 21, 1290–1299 (2004). [CrossRef]

16.

H. T. Eyyuboglu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,”Applied Optics, submitted for publication.

17.

A. Ishimaru, “Phase fluctuations in a turbulent medium,” Applied Optics 16, 3190–3192 (1977). [CrossRef] [PubMed]

18.

I. S. Gradysteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1980).

OCIS Codes
(010.1290) Atmospheric and oceanic optics : Atmospheric optics
(010.1300) Atmospheric and oceanic optics : Atmospheric propagation
(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence
(010.3310) Atmospheric and oceanic optics : Laser beam transmission

ToC Category:
Research Papers

History
Original Manuscript: July 7, 2004
Revised Manuscript: August 15, 2004
Published: October 4, 2004

Citation
Halil Eyyubo�?lu and Yahya Baykal, "Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere," Opt. Express 12, 4659-4674 (2004)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4659


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References

  1. A. A. Tovar and L. W. Casperson, "Production and propagation of Hermite sinusoidal-Gaussian laser beams,�?? J. Opt. Soc. Am. A 15, 2425�??2432 (1998). [CrossRef]
  2. L. W. Casperson and A. A. Tovar, �??Hermite sinusoidal-Gaussian beams in complex optical systems,�?? J. Opt. Soc. Am. A 15, 954�??961 (1998). [CrossRef]
  3. B. Lü, H. Ma, and B. Zhang, �??Propagation properties of cosh-Gaussian beams,�?? Opt. Commun. 164, 165�??170 (1999). [CrossRef]
  4. D. Zhao, H. Mao, and H. Liu, �??Propagation of off-axial Hermite-cosh-Gaussian laser beams,�?? J. Opt. A 6, 77�??83 (2004). [CrossRef]
  5. S. Yu, H. Guo, X. Fu, and W. Hu, �??Propagation properties of elegant Hermite-cosh-Gaussian laser beams,�?? Opt. Commun. 204, 59�??66 (2002). [CrossRef]
  6. S. Konar and J. Soumendu, �??Linear and nonlinear propagation of sinh-Gaussian pulses in dispersive media possessing Kerr nonlinearity,�?? Opt. Commun. 236, 7�??20 (2004). [CrossRef]
  7. N. Zhou and G. Zeng, �??Propagation properties of Hermite-cosine-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,�?? Opt. Commun. 232, 49�??59 (2004). [CrossRef]
  8. Y. Song. G. Wangyi, and G. Hong, �??Optical resonator with hyperbolic-cosine-Gaussian modes,�?? Opt. Commun. 221, 241�??247 (2003). [CrossRef]
  9. Z. I. Feizulin and Y. Kravtsov, �??Broadening of a laser beam in a turbulent medium,�?? Radiophys. Quantum Electron. 10, 33�??35 (1967). [CrossRef]
  10. A. I. Kon and V. I. Tatarskii, �??On the theory of propagation of partially coherent light beams in a turbulent atmosphere,�?? Radiophys. Quantum Electron. 15, 1187�??1192 (1972). [CrossRef]
  11. M. S. Belen�??kii, A. I. Kon, and V. L. Mironov, �??Turbulent distortions of the spatial coherence of a laser beam,�?? Sov. J. Quantum Electron. 7, 287�??290 (1977). [CrossRef]
  12. S. C. H. Wang and M. A. Plonus, �??Optical beam propagation for a partially coherent source in the turbulent atmosphere,�?? J. Opt. Soc. Am. 69, 1297�??1304 (1979). [CrossRef]
  13. M. S. Belen�??kii and V. L. Mironov, �??Phase fluctuations of a multimode laser field in a turbulent atmosphere,�?? Sov. J. Quantum Electron. 12, 3�??6 (1982). [CrossRef]
  14. C. Y. Young, Y. V. Gilchrest, and B. R. Macon, �??Turbulence induced beam spreading of higher order mode optical waves,�?? Opt. Eng. 41, 1097�??1103 (2002). [CrossRef]
  15. Y. Baykal, �??The correlation and structure functions of Hermite-sinusoidal-Gaussian laser beams in a turbulent atmosphere,�?? J. Opt. Soc. Am. A 21, 1290�??1299 (2004). [CrossRef]
  16. H. T. Eyyubo�?lu and Y. Baykal, �??Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,�??Applied Optics, submitted for publication.
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