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 and applied to complex optical systems.² 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 off-axial Hermite-cosh-Gaussian beams,⁴ elegant Hermite-cosh-Gaussian beams,⁵ and sinh-Gaussian pulses⁶ are investigated. Hermite-cos-Gaussian beams that pass through a paraxial optical ABCD system with a hard-edge aperture are also examined.⁷ An optical resonator with cosh-Gaussian modes is presented.⁸ In the presence of atmospheric turbulence, second-order effects exist for Gaussian,^9–12 multimode,¹³ and higher-order beams.¹⁴ The correlation and structure functions of HSG laser beams in a turbulent atmosphere were recently formulated,¹⁵ and we found the average intensity and spreading of cosh-Gaussian laser beams in atmospheric turbulence.¹⁶

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 formulated¹⁶ the average received intensity profile and the spreading of cosh-Gaussian laser beams in a turbulent medium. Combining our earlier results¹⁶ 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.

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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

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

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

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

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

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, 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 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

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

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 < $V_{\mathit{\text{xr}}}^{-1}$ and/or α_sy > $V_{\mathit{\text{yr}}}^{-1}$ the cos term becomes quite negligible when k ${\alpha }_{\mathit{\text{sx}}}^2$ / 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

Equation (8) matches the free-space propagation limit examined in Ref. 14. The precise correspondence of Eq. (8) in this paper with Eq. (5) of Ref. 14 is accomplished by noting that k ² ${\alpha }_s^2$ /(L ² + k ² ${\alpha }_s^4$) = 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

Equation (9) agrees with Eq. (12) of Ref. 12. To establish this agreement, we adapted the parameters in Eq. (12) of Ref. [12] 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

where g_x = (L ² + k ² ${\alpha }_{\mathit{\text{sx}}}^4$)^-1 and g_y = (L ² + k ² ${\alpha }_{\mathit{\text{sy}}}^4$)^-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 (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

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

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:

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:

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.

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The normalized average intensity at receiver plane I_rN (p_x ,p_y ,z = L) is defined as

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

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.

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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

 

Fig. 4. Dependence of normalized average intensity at the receiver plane on link length.

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Fig. 5. Dependence of normalized average intensity at the receiver plane on the real part of a complex displacement parameter.

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Fig. 6. Dependence of normalized average intensity at the receiver plane on turbulence level and wavelength of operation.

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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 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:

 

Fig. 7. Dependence of normalized average intensity at the receiver plane on link length (cosh-Gaussian source excitation case).

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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).

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Fig. 9. Dependence of normalized average intensity at the receiver plane on turbulence level and wavelength of operation (cosh-Gaussian source excitation case).

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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 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.