Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere

doi:10.1364/OPEX.12.004659

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Analysis of reciprocity of cos-Gaussian and cosh- Gaussian laser beams in a turbulent atmosphere

Halil Eyyubo�?lu 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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▸ Article Outline
– Abstract
1. Introduction
2. Propagation of a cos-Gaussian beam in a turbulent atmosphere
3. Limiting cases
4. Results
5. Concluding remarks
– References and links

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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.

1. Introduction

Production and propagation in free space (i.e., in the absence of turbulence) of general Hermite-sinusoidal-Gaussian (HSG) laser beams are studied ^{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]} and applied to complex optical systems.^{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]} HSG beams cover a broad range of special cases of beams such as cosh-Gaussian and cos-Gaussian types. One of the easiest methods for the generation of such special cases is to use an ordinary Gaussian beam as an incident beam on an appropriate transmission or reflection aperture. If a cosh-Gaussian beam is to be generated, a Gaussian beam is made incident on an aperture that has a cosh-Gaussian transmission function. These beams are used in applications for which efficient extraction of energy is required. For example, cosh-Gaussian dependence exhibits a concentration of energy in the outer lobes of a beam that can be used in the space diversity applications in free-space optic (FSO) systems. Also, considerable studies are being made to search for applications of HSG beams by use of ABCD optical systems. Propagation in free space of special cases of HSG beams such as cosh-Gaussian beams,^{3 B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. 164, 165–170 (1999). [CrossRef]} off-axial Hermite-cosh-Gaussian beams,^{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]} elegant Hermite-cosh-Gaussian beams,^{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]} and sinh-Gaussian pulses^{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]} are investigated. Hermite-cos-Gaussian beams that pass through a paraxial optical ABCD system with a hard-edge aperture are also examined.^{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]} An optical resonator with cosh-Gaussian modes is presented.^{8 Y. Song, G. Wangyi, and G. Hong, “Optical resonator with hyperbolic-cosine-Gaussian modes,” Opt. Commun. 221, 241–247 (2003). [CrossRef]} In the presence of atmospheric turbulence, second-order effects exist for Gaussian,^{9–12 Z. I. Feizulin and Y. Kravtsov, “Broadening of a laser beam in a turbulent medium,” Radiophys. Quantum Electron. 10, 33–35 (1967). [CrossRef]} multimode,^{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]} and higher-order beams.^{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 correlation and structure functions of HSG laser beams in a turbulent atmosphere were recently formulated,^{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]} and we found the average intensity and spreading of cosh-Gaussian laser beams in atmospheric turbulence.^{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.}

Atmospheric turbulence plays a significant role in the performance of FSO links that have become competitive in the broadband access networks in recent years. In general, we are interested in investigating the effect of turbulence when different forms of excitation are used. Our main motivation is to understand whether the use of some special form of HSG laser beams in FSO access communication systems will improve system performance. The first step is to determine the received intensity profile that will affect the receiver design. We are also studying the fourth-order effects of HSG beams such as scintillation, but here we report only the second-order results. Specifically we are concerned with determining how the average intensity profile of a cos-Gaussian beam is altered during propagation in practical FSO links. For this purpose, the source plane excitation is described by a cos-Gaussian laser beam. We derived the average intensity profile of this beam at the receiver plane after it has passed through a turbulent atmosphere.

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.

Numerical evaluations are made at various link lengths, turbulence levels, beam displacement parameters, and at the most frequently used FSO wavelengths of 0.85 and 1.55 μm. We observed that a cos-Gaussian beam, following the natural diffraction, is eventually transformed into a cosh-Gaussian beam. We recently formulated^{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.} the average received intensity profile and the spreading of cosh-Gaussian laser beams in a turbulent medium. Combining our earlier results^{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.} with the present findings, we conclude that cos-Gaussian and cosh-Gaussian beams act in a reciprocal manner after propagation in turbulence. Source and propagation parameters that affect this reciprocity have been analyzed.

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

Figure 1 supplies the propagation geometry. The source plane is located at the coordinate ( s , z = 0), location ( p , z = L) refers to the receiver plane, z is the propagation axis, and L is the link length. The sets s = (s_x ,s_y ) and p = (p_x ,p_y ) are the transverse source and receiver coordinates, respectively.

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 V_x = V_xr + iV_xi , Y_x = Y_xr + iY_xi . along the s_x direction and V_y = V_yr + iV_yi , Y_y = Y_yr + iY_yi along the S_y direction, is written as

u_{s} (s_{x}, s_{y}, z = 0) = 0.5 A \exp (- i ϕ) \exp [- 0.5 (\frac{s_{x}^{2}}{α_{s x}^{2}} + \frac{s_{y}^{2}}{α_{s y}^{2}})]

\times {\exp [i (V_{x} s_{x} + V_{y} s_{y})] + \exp [i (Y_{x} s_{x} + Y_{y} s_{y})]}

(1)

where A is the amplitude of the field at the origin of the source plane (i.e., at s_x = s_y = z = 0); ϕ is the constant phase factor, i = (-1)^1/2; α_sx and α_sy are the respective source sizes of the Gaussian beam in the s_x and s_y directions; V_xr , V_xi . denote the real and imaginary components of V_x ; and V_yr , V_yi . denote the real and imaginary components of V_y . Likewise, Y_xr , Y_xi are the real and imaginary components of Y_x , whereas Y_yr , Y_yi refer to the real and imaginary components of Y_y . Here, focal lengths along both s_x and s_y are taken to be infinite.

A cos-Gaussian laser beam is generated by choosing V_x = -Y_x = -V_xr and V_y = -Y_y = -V_yr . Similarly, one can obtain a cosh-Gaussian laser beam by setting V_x = -Y_x = iV_xi and V_y = -Y_y = iV_yi . Thus, use of Eq. (1) with A = 1, the intensity distribution of the cos-Gaussian beam at the exit plane of the laser is

I_{s} (s_{x}, s_{y}, z = 0) = \exp [- (\frac{s_{x}^{2}}{α_{s x}^{2}} + \frac{s_{y}^{2}}{α_{s y}^{2}})] \cos^{2} (V_{x r} s_{x} + V_{y r} s_{y})

(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) = \frac{k \exp (ikL)}{(2 πiL)} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} s u_{s} (s)

\times \exp [\frac{i k {(p - s)}^{2}}{(2 L)} + ψ (s, p) - i 2 π f t]

(3)

where k is the wave number; u_s ( 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) > = \frac{k^{2}}{{(2 π L)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d^{2} s_{1} d^{2} s_{2} u_{s} (s_{1}) u_{s}^{*} (s_{2}) \exp {\frac{ik [{(p - s_{1})}^{2} - {(p - s_{2})}^{2}]}{(2 L)}}

\times < \exp [ψ (s_{1}, p) + ψ^{*} (s_{2}, p)] >

(4)

The ensemble average term within the integrand of Eq. (4) is ^{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 (s_{1}, p) + ψ^{*} (s_{2}, p)] > = \exp [- 0.5 D_{ψ} (s_{1} - s_{2})] = \exp [- ρ_{0}^{- 2} {(s_{1} - s_{2})}^{2}]

(5)

where D_ψ ( s ₁ - s ₂) is the wave structure function, and ρ ₀ = (0.545

C_{n}^{2}

k ² L)^-3/5 is the coherence length of a spherical wave that propagates in the turbulent medium, with

C_{n}^{2}

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 0.307

C_{n}^{2}

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 A. Ishimaru, “Phase fluctuations in a turbulent medium,” Applied Optics 16, 3190–3192 (1977). [CrossRef] [PubMed]}, i.e., when 0.307

C_{n}^{2}

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 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) > = \frac{0.25 k^{2}}{{(2 π L)}^{2}} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d s_{1 x} {d s}_{1 y} {d s}_{2 x} {d s}_{2 y} \exp [\frac{- 0.5 (s_{1 x}^{2} + s_{2 x}^{2})}{α_{s x}^{2}} - \frac{0.5 (s_{1 y}^{2} + s_{2 y}^{2})}{α_{s y}^{2}}]

\times {\exp [i V_{x r} (s_{1 x} + s_{2 x}) + i V_{y r} (s_{1 y} + s_{2 y})] + \exp [- i V_{x r} (s_{1 x} + s_{2 x}) - i V_{y r} (s_{1 y} + s_{2 y})]

+ \exp [i V_{x r} (s_{1 x} - s_{2 x}) + i V_{y r} (s_{1 y} - s_{2 y})] + \exp [- i V_{x r} (s_{1 x} - s_{2 x}) - i V_{y r} (s_{1 y} - s_{2 y})]}

\times \exp [0.5 (\frac{i k}{L}) (s_{1 x}^{2} - 2 p_{x} s_{1 x} - s_{2 x}^{2} + 2 p_{x} s_{2 x} + s_{1 y}^{2} - 2 p_{y} s_{1 y} - s_{2 y}^{2} + 2 p_{y} s_{2 y})]

\times \exp [- ρ_{0}^{- 2} (s_{1 x}^{2} - 2 s_{1 x} s_{2 x} + s_{2 x}^{2} + s_{1 y}^{2} - 2 s_{1 y} s_{2 y} + s_{2 y}^{2})]

(6)

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

< I (p, L) > = 0.5 {(\frac{k}{L})}^{2} ρ_{0}^{4} {(D_{s x} D_{s y})}^{- \frac{1}{2}} \exp {- (\frac{ρ_{0}^{4} k^{2}}{L^{2}}) [\frac{p_{x}^{2}}{(α_{s x}^{2} D_{s x})} + \frac{p_{y}^{2}}{(α_{s y}^{2} D_{s y})}]}

\times (\exp {- 2 ρ_{0}^{2} [\frac{V_{x r}^{2} (ρ_{0}^{2} + 4 α_{s x}^{2})}{(α_{s x}^{2} D_{s x})} + \frac{V_{y r}^{2} (ρ_{0}^{2} + 4 α_{s y}^{2})}{(α_{s y}^{2} D_{s y})}]}

\times \cos [(\frac{2 ρ_{0}^{4} k^{2}}{L^{2}}) (\frac{V_{x r} p_{x}}{D_{s x}} + \frac{V_{s r} p_{y}}{D_{s y}})]

+ \exp {- ρ_{0}^{4} [\frac{V_{x r}^{2}}{{(α}_{s x}^{2} D_{s x})} + \frac{V_{y r}^{2}}{{(α}_{s y}^{2} D_{s y})}]}

\times \cosh {(\frac{2 ρ_{0}^{4} k}{L}) [(\frac{V_{x r} p_{x}}{{(α}_{s x}^{2} D_{s x})} + \frac{V_{y r} p_{y}}{{(α}_{s y}^{2} D_{s y})})]})

(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 α_sx <

V_{xr}^{- 1}

and/or α_sy >

V_{yr}^{- 1}

the cos term becomes quite negligible when k

α_{sx}^{2}

/ L <~ 1, kα _sy ² / L <~ 1 and α _sx >~ ρ ₀ , α_sy >~ 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.

3.1. The average intensity distribution at the receiver plane defined by Eq. (7) is evaluated at zero link length, i.e., for L = 0 . Within this limit, Eq. (7) was found to reduce to the intensity at the source plane as expressed by Eq. (2). In approaching the source plane p_x and p_y should consecutively be replaced by s_x and s_y .

3.2. To determine the limit of a Gaussian beam in free space, we evaluated Eq. (7) with

C_{n}^{2}

= 0 (or alternatively ρ ₀ → ∞) and V_xr = V_yr = 0 . A symmetrical beam was taken by choosing α_sx = α_sy = α_s . Then the average intensity in Eq. (7) simplified to

I (p, L) = [\frac{k^{2} α_{s}^{4}}{(L^{2} + k^{2} α_{s}^{4})}] \exp {- [\frac{k^{2} α_{s}^{2}}{(L^{2} + k^{2} α_{s}^{4})}] (p_{x}^{2} + p_{y}^{2})}

(8)

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

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

α_{s}^{2}

/(L ² + k ²

α_{s}^{4}

) = 2/W ² , 2^0.5 α_s = W₀ , p_x = x, and p_y = 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 V_xr = V_yr = 0 and also constituted symmetry in the x and y directions so that α_sx = α_sy = α_s . With these conditions, Eq. (7) becomes

< I (p, L) > = [\frac{k^{2} α_{s}^{4} ρ_{0}^{2}}{(ρ_{0}^{2} L^{2} + 4 α_{s}^{2} L^{2} + k^{2} α_{s}^{4} ρ_{0}^{2})}] \exp [\frac{- k^{2} α_{s}^{4} ρ_{0}^{2} (p_{x}^{2} + p_{x}^{2})}{(ρ_{0}^{2} L^{2} + 4 α_{s}^{2} L^{2} + k^{2} α_{s}^{4} ρ_{0}^{2})}]

(9)

Equation (9) agrees with Eq. (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

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

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

p_d = 0,

p_{c}^{2}

p_{x}^{2}

p_{y}^{2}

, F = ∞, ζ = 0, A ₀ = 1,and α_o = α_s .

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

C_{n}^{2}

= 0, thus the average intensity in Eq. (7) leads to

I (p, L) = 0.5 k^{2} α_{s x}^{2} α_{s y}^{2} g_{x}^{0.5} g_{y}^{0.5} \exp {- k^{2} [α_{s x}^{2} g_{x} p_{x}^{2} + α_{s y}^{2} g_{y} p_{y}^{2}]} \exp {- L^{2} [α_{s x}^{2} {V_{x r}^{2} g}_{x} + α_{s y}^{2} {V_{y r}^{2} g}_{y}]}

\times (\cos {{2 k}^{2} [α_{s x}^{4} {V_{x r} g}_{x} p_{x} + α_{s y}^{4} {V_{y r} g}_{y} p_{y}]} + \cosh {2 k L [α_{s x}^{2} {V_{x r} g}_{x} p_{x} + α_{s y}^{2} {V_{y r} g}_{y} p_{y}]})

(10)

where g_x = (L ² + k ²

α_{sx}^{4}

)^-1 and g_y = (L ² + k ²

α_{sy}^{4}

)^-1 . After reverting to a single coordinate system in the sense that α_sx = α_sy = w_o / 2^0.5 , p_x = p_y = x / 2^0.5, V_xr = V_yr = Ω_o / 2^0.5, Eq. (10) becomes identical to the intensity equivalent of the field expression of Eq. (14) in Ref. 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 \to \infty) > = 0.5 {(\frac{k}{L})}^{2} ρ_{0}^{4} {(D_{s x} D_{s y})}^{- \frac{1}{2}} \exp {- (\frac{ρ_{0}^{4} k^{2}}{L^{2}}) [\frac{p_{x}^{2}}{(α_{s x}^{2} D_{s x})} + \frac{p_{y}^{2}}{(α_{s y}^{2} D_{s y})}]}

\times \exp {- ρ_{0}^{4} [\frac{V_{x r}^{2}}{(α_{s x}^{2} D_{s x})} + \frac{V_{y r}^{2}}{(α_{s y}^{2} D_{s y})}]}

\times \cosh {(\frac{2 ρ_{0}^{4} k}{L}) [(\frac{V_{x r} p_{x}}{{(α}_{s x}^{2} D_{s x})} + \frac{V_{y r} p_{y}}{{(α}_{s y}^{2} D_{s y})})]}

(11)

By taking ∂ / ∂p_x and ∂ / ∂p_y of Eq. (11) and setting them independently to zero, we found the peak locations to be fixed at p_xp = V_xr L / k and p_yp = V_yr 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

I_{0} = \frac{< I (p_{x} = p_{x p}, p_{y} = p_{y p}, z = L) >}{< I (p_{x} = 0, p_{y} = 0, z = L) >}

(12)

Here the average intensities in both the numerator and the denominator are those of Eq. (11). We can then relate the source and link parameters of a cosh-Gaussian converted cos-Gaussian beam to this ratio in the following manner:

\frac{ρ_{0}^{4} V_{x r}^{2} α_{s x}^{2} L^{2}}{ρ_{0}^{4} L^{2} + 4 ρ_{0}^{2} α_{s x}^{2} L^{2} + ρ_{0}^{4} k^{2} α_{s x}^{4}} = 0.5 ℓ n (2 I_{0}), \frac{ρ_{0}^{4} V_{y r}^{2} α_{s y}^{2} L^{2}}{ρ_{0}^{4} L^{2} + 4 ρ_{0}^{2} α_{s y}^{2} L^{2} + ρ_{0}^{4} k^{2} α_{s y}^{4}} = 0.5 ℓ n (2 I_{0})

(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., α_sx = α_sy and V_xr = V_yr . When dominant, the existence of a cos term in the cos-Gaussian beam dictates that the intensity will peak at s_x = 0.5nπ / V_xr and s_y = 0.5nπ / V_yr , where n is an integer starting from zero. Hence we define normalized intensity at the source plane as follows:

I_{s N} (s_{x}, s_{y}, z = 0) = \frac{I_{s} (s_{x}, s_{y}, z = 0)}{I_{s} (s_{x} = s_{y} = z = 0)}

(14)

A three-dimensional view of I_sN (s_x ,s_y ,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 I_rN (p_x ,p_y ,z = L) is defined as

I_{r N} (p_{x}, p_{y}, z = L) = \frac{< I (p_{x}, p_{y}, z = L) >}{I_{s} (s_{x} = s_{y} = z = 0)}

(15)

The normalized intensity at source plane I_sN (s_x , s_y ,z = 0) and the normalized average intensity at receiver plane I_rN (p_x , p_y , z = L) versus the slanted axis are shown together in Fig. 3(a) for a single cos-Gaussian beam. For cos-Gaussian beam excitation, the beam spreading and the gradual concentration of power within two outer lobes, i.e., the cosh-Gaussian shape, can clearly be identified in Fig. 3(a) for the selected parameters of source size α_sx = α_sy = 5 cm, link length L = 5 km, wavelength λ = 1.55 μm, the real components of the complex displacement parameters associated with the Gaussian part of the beam in the s_x , s_y directions, V_xr = V_yr = 55 m^-1, and the structure constant

C_{n}^{2}

= 1 × 10 ^-15 m^-2/3. Figure 3(b) shows a contour plot of the same I_sN (s_x ,s_y ,z = 0) and I_rN (p_x ,p_y ,z = L) shown in Fig. 3(a).

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 (p_x ,p_y ,z = L) as defined in this way is

I_{r 0} (p_{x}, p_{y}, z = L) = \frac{< I (p_{x}, p_{y}, z = L) >}{Max [< I (p_{x}, p_{y}, z = L) >]}

(16)

We note that, after having transformed into a distinct cosh-Gaussian beam, p_x and p_y in the argument of the intensity function in the denominator of Eq. (16) well approximates p_xp and p_yp derived in Subsection 3.5. Figure 4 displays the variation of I _r0(p_x ,p_y ,z = L) for link lengths L = 0, 2, 10, 20 km. Here the source plane intensity (L = 0) is included and retains the normalization described by Eq. (14). In line with the predictions discussed in Subsection 3.5 and Eq. (11), Fig. 4 confirms, while initially managing to preserve its profile, that the cos-Gaussian beam eventually changes into a cosh-Gaussian beam whose lobe peaks become spaced further apart from the origin in direct proportion to the increasing values of L.

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(p_x , p_y ,z = L) against the different values of the real components of complex displacement parameters V_xr and V_yr . Figure 5 demonstrates that the increases in V_xr and V_yr 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(p_x ,p_y ,z = L) at λ = 1.55 μm and λ = 0.85 μm in the absence of (

C_{n}^{2}

= 0 m^-2/3) and in the presence of (

C_{n}^{2}

= 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 α_sx and α_sy ,
larger beam displacement parameters V_xr and V_yr ,
lower structure constants $C_{n}^{2}$ ,
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.

To make comparisons in the reverse direction of reciprocity, i.e., cosh-Gaussian beam changing into a cos-Gaussian beam after propagation in turbulence, we used Eq. (17) from Ref. 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.

to provide equivalent graphs of Figs. 4–6 for a cosh-Gaussian source plane excitation. These are Figs. 7–9, where we plotted I _r0 versus the variations in link length, displacement parameters, turbulence levels, and wavelengths of operation. In compliance with the findings of Ref. [16], Figs. 7–9 reflect the concrete process of cosh to cos conversion as well as its dependence on link length, displacement parameters, turbulence levels, and wavelengths of operation. Note that a numerically different source and propagation parameters must be used in Figs. 7–9 from those used in Figs. 4–6 to emphasize the conversion from a cosh-Gausssian beam into a cos-Gaussian beam more clearly. From a collective assessment of Figs. 4–9, we can assert that the conversion from a cosh-Gaussian beam to a cos-Gaussian beam runs at a different pace from that of the conversion from a cos-Gaussian beam to a cosh-Gaussian beam. More specifically, the proportionality of factors that accelerate the transformation from a cosh-Gaussian beam into a cos-Gaussian beam is now somewhat modified and can be listed as follows:

smaller source sizes α_sx and α_sy ,
smaller displacement parameters V_xr and V_yr ,
higher structure constants $C_{n}^{2}$ ,
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

The average intensity profile of a cos-Gaussian beam in a turbulent atmosphere has been formulated and numerically evaluated. We determined that our average intensity formulation correctly reduces to the existing Gaussian beam wave result in turbulence and the cos-Gaussian beam result in free space (i.e., the absence of turbulence). This formulation further stipulates that a cos-Gaussian beam loses its original shape more rapidly, finally turning into a pure cosh-Gaussian beam with an increase in propagation distance, a real part of the complex displacement parameter and wavelength of operation, and with a decrease in turbulence levels and source sizes. This analytic observation is also supported by the related intensity plots.

Combined with the findings of our earlier research,^{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.} we assert that cos-Gaussian and cosh-Gaussian beams act in a reciprocal manner in propagation. This means that given the right set of source and propagation parameters, a cos-Gaussian beam at the source plane will lead to a cosh-Gaussian beam at the receiver plane, whereas a cosh-Gaussian source plane beam excitation will arrive as a cos-Gaussian beam at the receiver plane. We refer to this property as the reciprocity of cos-Gaussian and cosh-Gaussian laser beams in a turbulent atmosphere. The same reciprocity phenomenon is also applicable in the absence of turbulence. Transformation from a cos-Gaussian beam into a cosh-Gaussian beam in turbulence is found to accelerate for smaller source sizes, larger beam displacement parameters, lower structure constants, higher wavelengths, and longer link lengths. However, transformation from a cosh-Gaussian beam into a cos-Gaussian beam in turbulence is accelerated for smaller source sizes, smaller beam displacement parameters, higher structure constants, lower wavelengths, and longer link lengths.

Appendices

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, V_xr and V_yr . 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

I_{1 x} = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} d s_{1 x} d s_{2 x} \exp [- 0.5 \frac{(s_{1 x}^{2} + s_{2 x}^{2})}{α_{s x}^{2}}] \exp [i V_{x r} (s_{1 x} + s_{2 x})]

\times \exp [0.5 (\frac{i k}{L}) (s_{1 x}^{2} - 2 p_{x} s_{1 x} - s_{2 x}^{2} + 2 p_{x} s_{2 x})] \exp [- \frac{(s_{1 x}^{2} - 2 s_{1 x} s_{2 x} + s_{2 x}^{2})}{ρ_{0}^{2}}]

= \int_{- \infty}^{\infty} d s_{1 x} \exp {[- 0.5 α_{s x}^{2} + \frac{j k}{(2 L)} - \frac{1}{ρ_{0}^{2}}] s_{1 x}^{2} + [i V_{x r} - \frac{j k p_{x}}{L} + 2 s_{2 x}] s_{1 x}}

\times \int_{- \infty}^{\infty} d s_{2 x} \exp (\frac{- 0.5 s_{2 x}^{2}}{α_{s x}^{2}}) \exp (i V_{x r} s_{2 x})

\times \exp [0.5 (\frac{i k}{L}) (- s_{2 x}^{2} + 2 p_{x} s_{2 x})] \exp [- \frac{(s_{2 x}^{2} + s_{1 y}^{2} - 2 s_{1 y} s_{2 y} + s_{2 y}^{2})}{ρ_{0}^{2}}]

(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

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

, i.e., in the form of

\int_{- \infty}^{\infty} d x \exp (- p^{2} x^{2} \mp q x) = (\frac{π^{0.5}}{p}) \exp [\frac{q^{2}}{(4 p^{2})}]

(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

I_{1 x} = \frac{π^{0.5}}{{[0.5 α_{s x}^{2} + \frac{1}{ρ_{0}^{2}} - \frac{j k}{(2 L)}]}^{0.5}} \int_{- \infty}^{\infty} d s_{2 x}

\times \exp ({\frac{0.5 α_{s x}^{1} + \frac{1}{ρ_{0}^{2}} + \frac{j k}{(2 L)} - ρ_{0}^{- 4}}{[0.5 α_{s x}^{2} + \frac{1}{ρ_{0}^{2}} - \frac{j k}{(2 L)}]}} s_{2 x}^{2})

\times \exp ({\frac{i V_{x r + \frac{i k p_{x}}{L} + ρ_{0}^{- 2} (i V_{x r} - \frac{i k p_{x}}{L})}}{[0.5 α_{s x}^{2} + \frac{1}{ρ_{0}^{2}} - \frac{j k}{(2 L)}]}} s_{2 x})

(A3)

Equation (A3) is again in the form of Eq. (A2). Thus, by similarly performing the integration over d _s2x and benefiting from the properties explained in the first paragraph before Eq. (A1) of this Appendix A, we finally arrive at the average intensity at the receiver plane given by Eq. (7), where p = (p_x ,p_y ) is the transverse receiver coordinate; k is the wave number; L is the length; ρ ₀ = (0.545

C_{n}^{2}

k ² L)^-3/5 is the coherence length of a spherical wave that propagates in the turbulent medium;

C_{n}^{2}

is the structure constant; α_sx and α_sy are the respective source sizes of the Gaussian beam in the s_x and s_y directions; and V_xr and V_yr denote the real components of V_x and V_y . Here V_x and V_y are the complex displacement parameters along the s_x and s_y directions. D_sx = (

ρ_{0}^{4}

L ² + 4

ρ_{0}^{2}

α_{sx}^{2}

L ² +

ρ_{0}^{4}

k ²

α_{sx}^{4}

)/(

α_{sx}^{4}

L ²), D_sy = (

ρ_{0}^{4}

L ² + 4

ρ_{0}^{2}

α_{sy}^{2}

L ² +

ρ_{0}^{4}

k ²

α_{sy}^{4}

)/(

α_{sy}^{4}

L ²) can be interpreted as parameters related to the beam spread in the p_x , p_y directions, respectively.

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

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]
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]
B. Lü, H. Ma, and B. Zhang, �??Propagation properties of cosh-Gaussian beams,�?? Opt. Commun. 164, 165�??170 (1999). [CrossRef]
D. Zhao, H. Mao, and H. Liu, �??Propagation of off-axial Hermite-cosh-Gaussian laser beams,�?? J. Opt. A 6, 77�??83 (2004). [CrossRef]
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]
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]
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]
Y. Song. G. Wangyi, and G. Hong, �??Optical resonator with hyperbolic-cosine-Gaussian modes,�?? Opt. Commun. 221, 241�??247 (2003). [CrossRef]
Z. I. Feizulin and Y. Kravtsov, �??Broadening of a laser beam in a turbulent medium,�?? Radiophys. Quantum Electron. 10, 33�??35 (1967). [CrossRef]
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]
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]
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]
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]
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]
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]
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.
A. Ishimaru, �??Phase fluctuations in a turbulent medium,�?? Applied Optics 16, 3190-3192 (1977). [CrossRef] [PubMed]
I. S. Gradysteyn, and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, New York, 1980).

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