Inner scale effect on scintillation index of flat-topped beam in non-Kolmogorov weak turbulence Zhihong Zeng,1,2 Xiujuan Luo,1,* Aili Xia,1 Yu Zhang,1,2 and Bei Cao1 1
Xi’an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences (CAS), Xi’an 710119, China 2
Graduate University of CAS, Beijing 100049, China *Corresponding author: [email protected]
Received 11 August 2014; revised 30 October 2014; accepted 22 February 2015; posted 23 February 2015 (Doc. ID 220782); published 23 March 2015
A simpler generalized expression of irradiance fluctuation for flat-topped beams is presented based on the Born and Rytov perturbation methods. The theoretical expression of the on-axis scintillation index in non-Kolmogorov weak atmospheric optics links is developed using the generalized von Kármán spectrum model, and using the equivalent structure constant that is different for all power-law exponents. The effect of the inner scale on the on-axis scintillation index is examined comprehensively. It is observed that flat-topped beams happen to possess smaller scintillation indices at larger inner scale. The effects of the power law, flat-topped order, source size of the fundamental Gaussian beam, propagation distance, and wavelength are also analyzed. © 2015 Optical Society of America OCIS codes: (010.1330) Atmospheric turbulence; (010.1300) Atmospheric propagation; (010.3310) Laser beam transmission; (290.5930) Scintillation. http://dx.doi.org/10.1364/AO.54.002630
1. Introduction
Atmospheric turbulence can severely degrade the performance of a remote sensing or laser communication system [1,2]. In order to overcome the effects of atmospheric turbulence, many techniques have been researched, such as the use of partially coherent beams [3–7], incoherent beams [8], and polarized beams [9,10]. Investigation on the shape of the beam has been of increasing interest because of the evidence that other types of incidence may be less influenced by turbulence. A wide variety of beam shapes have been considered, including Gaussian [11], Hermite–Gaussian [7], Hermite–cosh–Gaussian [12], cosh (cos)–Gaussian [13,14], Bessel [15,16], annular [14], dark hollow [17,18], Laguerre–Gaussian [19], and flat-topped beams [20–25]. Specifically, in recent years, some new attempts have been focused on flat-topped beams, which can be applied in inertial
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confinement fusion and material thermal processing for laser beams. Among all the theoretical models for describing flat-topped beams [26–29], the model proposed by Li, where the electric field is a finite sum of fundamental Gaussian beams, has recently been used for the study of atmospheric effects on propagated light intensity and scintillation [21,24,25]. However, for a flat-topped beam propagating through non-Kolmogorov turbulence, to the best of the authors’ knowledge, there is no published work analyzing the impact of the inner scale on scintillation. This has motivated the work herein. On the other hand, various models of the power spectrum for optical turbulence have been developed previously. These models include the Kolmogorov power-law spectrum, the Tatarskii spectrum (with inner scale parameter), the exponential spectrum (with outer scale parameter), the von Kármán spectrum (with both inner scale and outer scale parameters), and the modified atmospheric spectrum (with both inner scale and outer scale parameters). All the corresponding generalized models [30–33], in which
the power law has the range of 3 to 4 or 3 to 5 instead of the value 11/3, have been proposed in recent years. It is shown that the generalized modified atmospheric spectral model for the non-Kolmogorov turbulence characterizes a high wave number “bump” that has been observed in temperature data, but some coefficients in this spectral model have no certain values. In view of this, many researchers adopt the generalized von Kármán spectrum to investigate the scintillation index associated with a beam wave propagating through non-Kolmogorov atmospheric turbulence. Furthermore, in non-Kolmogorov turbulence, Ref. [34] shows that it is inappropriate for the scintillation index of flat-topped beam to use the same structure constant for all the power-law exponents [25], and the problem can be solved by employing the formulated equivalent structure constant [35,36]. In particular, for a flat-topped beam, the generalized expressions of the on-axis scintillation (Eq. (1) in Ref. [25]) are so complicated that the final results are difficult and complex to calculate when the spectra are expanded. In order to develop a simper expression of scintillation, second-order statistical moments are calculated based on the Born and Rytov perturbation methods. We put forward the predigesting formula of the on-axis scintillation index for flat-topped beams and derive the results using the generalized von Kármán spectral model and the equivalent structure constant [36] under weak non-Kolmogorov turbulence. Then, the impact of turbulence inner scales on the scintillation index is scrutinized. Our results could be useful for experimental research of laser communication in horizontal optical links deployed at any altitude. 2. Generalized von Kármán spectrum
The generalized von Kármán spectral model is given by [30] exp −κ 2 ∕κ2l 2 ~ ; Φn κ; α; l0 ; L0 AαCn α 2 κ κ 20 α∕2 0 ≤ κ < ∞;
3 < α < 4;
power-law spectrum in the inertial range κ0 ≪ κ ≪ κ l and reduces to the conventional von Kármán spectrum when α 11∕3. 3. Scintillation Index for Flat-Topped Beam Wave A. Flat-Topped Light Beams
For the flat-topped beam, the electric field at the source (z 0) can be expressed as a finite sum of Gaussian beams with different parameters as follows [21]: U 0 r; 0
1 an exp − bn kr2 ; 2 n1
N X
where r is the radial coordinate, N is the order of the flat-topped beam, k 2π∕λ is the wave number, λ is the wavelength, an
−1n−1 N
N ; n
(3)
r 2 w ; n 0
(4)
and 2 ; bn kw20n
w0n
where Nn denotes a binomial coefficient, w0n is the waist width of the nth Gaussian beam, and w0 is the source size of the fundamental Gaussian beam. As the beam order N increases, the beam profile of the flat-topped beam flattens and the amplitude decreases. When N 1, Eq. (2) reduces to a Gaussian beam. Based on the Huygens–Fresnel integral of the paraxial equation, we can get the complex amplitude, which is also the series of Gaussian beams, at distance z L from the source as [21] U 0 r; L
N X
un r; L;
(5)
n1
(1)
where κ denotes the magnitude of the spatial frequency with units of rad/m; α is the spectral power law; l0 is the turbulence inner scale, which is generally on the order of millimeters; L0 is the turbulence outer scale on the order of meters; and Aα is the generalized amplitude factor, which has the form Aα Γα − 1 · cosαπ∕2∕4π 2 . Γ· is the ~ 2n α is the generalized refractivegamma function. C index structure parameter with units of m3−α. The parameters κ l and κ0 are related to inner scale and outer scale in general by the equations κl cα∕l0 and κ0 C0 ∕L0 , where cα fΓ5 − α∕2 · Aα · 2π∕3g1∕α−5 and the scaling constant C0 is chosen differently according to the application because of the not-well-defined outer scale. In this paper, we set C0 2π. Equation (1) reduces to the non-Kolmogorov
(2)
where un r; L
an 1 bn exp ikL − kr2 ; (6) 2 1 ibn L 1 ibn L
i2 −1. When r0, un 0;L an expikL∕1 ibn L. B. Scintillation Index for Flat-Topped Beam Wave
Following the notion that a flat-topped beam is a finite sum of Gaussian beams, we try to find the relationship between the scintillation index of a flat-topped beam and the scintillation index of a Gaussian beam. Based on the Rytov approximation and the above electric field expression of the flattopped beam, a simpler model of scintillation index can be derived as Eq. (A12). When r 0, the on-axis 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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scintillation index of the flat-topped beam will turn into
Sj1 ;j2 ;j3
0 σ 2I 0;L 8π 2 k2 LRe@
×
1 jU 0 0;Lj2 U 0 0;L
N X N X N X j1 1 j2 1 j3 1
1 uj1 0;Luj2 0;Luj3 0;LSj1 ;j2 ;j3 A; (7)
Γ1 − α∕2 ~ 2n ακ2−α AαC l 2 α 1 3 2 × 2 F 1 1 − ; ; ; −Λj1 ;j3 Lκ l ∕k 2 2 2
2 1 iLκl 1 − 2θ¯0j1 ;j2 ∕3∕kα∕2 − 1 : − iαLκ2l 1 − 2θ¯0j1 ;j2 ∕3∕2k
(12)
Substituting Eq. (12) into Eq. (7), σ 2I 0; L becomes σ 2I 0; L ~ n απ 2 k2 Lκ2−α 4Γ1 − α∕2AαC l PN PN PN j1 1 j2 1 j3 1 uj1 0; Luj2 0; Luj3 0; L × Re jU 0 0; Lj2 U 0 0; L α 1 3 2 × 2 F 1 1 − ; ; ; −Λj1 ;j3 Lκl ∕k 2 2 2
2 1 iLκ l 1 − 2θ¯0j1 ;j2 ∕3∕kα∕2 − 1 − ; (13) iαLκ 2l 1 − 2θ¯0j1 ;j2 ∕3∕2k 2
where Re is the real part, Sj1 ;j2 ;j3
Λj ;j Lκ 2 ε2 κΦn κ; α; l0 ; L0 exp − 1 3 k 0 0
2 iLκ ε1 − θ¯0j1 ;j2 ε dκdε; (8) − exp − k Z 1Z
∞
1 Λj1 ; j3 Λj1 Λj3 − iθ¯j1 − θ¯j3 ; 2 1 θ¯0j1 ;j2 θ¯j1 θ¯j2 iΛj1 Λj2 ; 2
(9)
~ 2n α C (10)
ε 1 − z∕L, Λj Λ0j ∕1 Λ20j , θ¯j Λ20j ∕1 Λ20j , Λ0j jL∕kw20 bj L, and the asterisk () indicates the complex conjugate. One can note that uj 0; L is a complex number and 0 < ReΛj1 ; j3 ; θ¯0j1 ; j2 < 1. Equation (7) can be expanded to the same form as Eq. (1) of Ref. [25]. However, the expression Eq. (7) is much simpler, and it can be used to simplify the computation of the on-axis scintillation index of the flat-topped beam under all kinds of atmospheric spectral models through the similar existent process of the Gaussian beam. When N is taken as unity, Eq. (7) reduces to the on-axis scintillation index of the Gaussian beam with receiver beam parameters Λ Λ1;1 and θ¯ θ¯1;1 . In Eq. (7), we just need to perform the integrations of Sj1 ; j2 ; j3 over κ and ε. Now employing the von Kármán spectrum expression in Eq. (1), Eq. (8) can be expressed as Sj1 ;j2 ;j3
Z 1Z
κ 2 α∕2 0 0 κ κ 0 Λj1 ; j3 Lκ 2 ε2 κ2 − 2 × exp − k κl
iLκ2 κ2 ε1 − θ¯0j1 ; j2 ε − 2 dκdε: − exp − k κl
~ 2n α AαC
∞
2
(11)
Following the same process as in [32] [Eqs. (27)–(29)], Eq. (11) has the form 2632
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where the structure constant in non-Kolmogorov media is expressed as [36]
−0.5Γα2π−11∕6α∕2 λL11∕6−α∕2 C2n ; Γ1 − α∕2Γα∕22 Γα − 1 cosαπ∕2 sinαπ∕4 (14)
where C2n is the structure constant of Kolmogorov turbulence. When N 1, Eq. (13) reduces to the on-axis scintillation index of the Gaussian beam as in [32]. 4. Results and Discussion
In this section, we analyze the influence of some key parameters on the analytical formulas derived in the previous section [Eq. (13)], including inner scale l0 , the flat-topped beam order N, the power law α, Gaussian source size w0 , propagation distance L, and wavelength λ. The effect of the inner scale on the on-axis scintillation index is mainly emphasized; hence most of the figures are considered to cover a subfigure of the scintillation index versus inner scale l0 . In all the figures, the index-of-refraction structure constant in Kolmogorov media is fixed as C2n 10−15 m−2∕3 , because it is linear with the onaxis irradiance fluctuation. Figure 1 plots the on-axis scintillation index versus inner scale l0 and power law α, taking L 1 km, N 5, w0 3 cm, and λ 1.55 μm. Because the inner scale is mere millimeters near the earth surface and several centimeters in the upper regions of the atmosphere, the range of the inner scale is set as 0 to 20 mm. The range of the power law is 3 to 4. Figure 1(a) depicts the scintillation as a function of l0 for several values of α (one of which corresponds to the Kolmogorov’s turbulence
Fig. 1. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for different power law α and (b) as a function of power law α for different inner scale l0 . The calculation parameters are λ 1.55 μm, L 1 km, N 5, and w0 3 cm.
Fig. 2. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for flat-topped order N and (b) as a function of flat-topped order N for different inner scale l0 . The calculation parameters are λ 1.55 μm, w0 3 cm, L 1 km, and α 10∕3.
(α 11∕3)), while Fig. 1(b) represents the scintillation index as a function of α for several values of l0 . It can be seen from Fig. 1(a) that with the increment of inner scale, the scintillation index decreases. This phenomenon is obvious for large spectral index α. As also seen from this figure, the flat-topped beam with large spectral power exponent α has larger on-axis scintillation index at fixed inner scale. Figure 1(b) yields that the scintillation index is a constant for different strengths of inner scale if α 3. The scintillation indices increase linearly with the increment of α. Also, we derive from Fig. 1(b) that, at the same α value, the flat-topped beams have smaller scintillation values propagating through large inner scale turbulence than through small inner scale turbulence, and the detached degree grows as α increases, which means the inner scale can affect the scintillation index especially at large α. The conclusion from the two figures is that, in the weak turbulence regimes, larger inner scale and smaller power law can bring advantage to the propagation of the flat-topped beam. Figure 2 plots the on-axis scintillation index of flattopped beams through non-Kolmogorov weak turbulence as a function of inner scale l0 and flat-topped order N, which correspond to the flat degree of the
beam profile and the magnitude of the amplitude for the flat-topped beam. The fixed parameters except α 10∕3 are the same as in Fig. 1. It can be seen from Fig. 2(a) that, with the increment of inner scale, the scintillations vary like the above curve for α 10∕3 in Fig. 1(a). We note that, for N 1 corresponding to the Gaussian beam, the scintillation index responds with a smaller value compared with the three larger N, at which the scintillation indices at larger N have smaller values for all the inner scale. Figure 2(b) indicates that, for all the flat-topped order N, the scintillation of the flat-topped beam decreases with the increment of inner scale. Furthermore, at fixed l0 , the scintillation indices increase monotonically as the flat-topped order N increases to 6, and then slightly decrease when N increases. Examination of the trend in Fig. 2(b) is also executed for a small Gaussian source size of w0 0.5 cm. This is not plotted separately due to emphasizing the effect of inner scale, though it shows that certain large values of N have small scintillations. The on-axis scintillation index of flat-topped beams through generalized von Kármán spectral turbulence is plotted in Fig. 3 as a function of inner scale l0 and Gaussian source size w0 , setting L 1 km, N 5, α 10∕3, and λ 1.55 μm. As illustrated 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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Fig. 3. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for different Gaussian source size w0 and (b) as a function of Gaussian source size w0 for different inner scale l0 . The calculation parameters are λ 1.55 μm, L 1 km, N 5, and α 10∕3.
in Fig. 3(a) for fixed Gaussian source sizes, on-axis scintillations become smaller when l0 becomes larger. It can also be inferred from this figure that at fixed l0 , the scintillations increase with the increase of w0 when w0 ≥ 1 cm, but they have larger values at w0 0.5 cm than at w0 1 cm, which is also valid for N > 5. Figure 3(b) plots the on-axis scintillation index as a function of Gaussian source size for several inner scales. With the increase of the value of w0, the scintillation index at fixed l0 first decreases to a minimum value that occurs near w0 0.8 cm, then increases steeply to a maximum value near w0 3 cm, and lastly maintains the same level at w0 > 4 cm after a slight decrease. All the same figures for N 1 ∼ 15 are plotted, and the scintillations vary like the curves of Fig. 9(a) in Ref. [25] when w0 increases. In addition, for all values of N from 1 through 15, the scintillation decreases as inner scale increases, and the degree of the decline grows from around w0 2 cm to its two sides, which means that the inner scale has weak influence on the scintillation of the flat-topped beam at around w0 2 cm. Figure 4 plots the on-axis scintillation index of flat-topped beams as a function of inner scale l0 2634
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Fig. 4. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for different propagation length L and (b) as a function of propagation length L for different inner scale l0 . The calculation parameters are λ 1.55 μm, w0 3 cm, N 5, and α 10∕3.
and propagation distance L, taking w0 3 cm, N 5, α 10∕3, and λ 1.55 μm. We deduce from Fig. 4(a) that with the increase of the value of l0, the scintillation is slightly decreased, the degree of which is almost the same for different L. It can also be deduced that when l0 is fixed, the scintillation increases with the increment of propagation distance. The results can also be obtained in Fig. 4(b). We also note that the flat-topped beams have smaller on-axis scintillation indices at large l0 , although the phenomenon is not obvious. Additionally, examination of the trend in Figs. 4(a) and 4(b) is also executed for other fixed N, w0 , and α, and the whole trend of the variation of scintillation as the inner scale is similar to that in Fig. 4. After combining the information above and taking the relative variance as the criterion of the judgment, it can be revealed that the effect of inner scale is weak for long propagation distances if the other parameters are fixed. In order to further analyze the influence of wavelength on the flat-topped beam on-axis scintillation index based on generalized von Kármán spectral weak turbulence, the scintillation index is plotted in Fig. 5 as a function of inner scale l0 and wavelength λ, taking L 1 km, N 5, α 10∕3, and
Fig. 5. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for different wavelength λ and (b) as a function of wavelength λ for different inner scale l0 . The calculation parameters are L 1 km, w0 3 cm, N 5, and α 10∕3.
w0 3 cm. As illustrated in Fig. 5(a), being valid for all l0 , scintillation indices decrease steadily as the wavelength increases to long-wavelength infrared radiation. In addition, the scintillations become smaller as the inner scale increases, and the velocity of decrease is faster for small λ. The scintillation index for long wavelength of 10.6 μm has a value close to zero, and it is almost invariable with the inner scale increases. More detailed information from Fig. 5(b) is that the scintillation decreases steeply at short wavelength of λ < 2 μm, and then descends slightly as the wavelength increases. As a result, the reduction of scintillation from choosing long wavelength results in the decrease of bit-error-rate and the improvement of free-space optical communication system performance. Moreover, a larger wavelength makes the scintillation index less influenced by the inner scale. The result is also effectual for N 1 ∼ 15. Figures 6 and 7 emphasize the effect of the power law and propagation distance when the inner scale is a nonzero constant. The on-axis scintillation index of the flat-topped beam through generalized von Kármán spectral turbulence is plotted as a function of power law α for different flat-topped order N at
Fig. 6. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of power law α for different flat-topped order N and (b) as a function of power law α for different propagation length L. The calculation parameters are λ 1.55 μm, w0 4 cm, and l0 8 mm.
L 2 km in Fig. 6(a) and for different propagation length L at N 5 in Fig. 6(b), setting l0 8 mm, w0 4 cm, and λ 1.55 μm. As illustrated in Fig. 6(a) for fixed flat-topped order, on-axis scintillations become larger when the power law of nonKolmogorov turbulence becomes larger and the velocity of increase is faster at N 5. It can also be seen from this figure that at all α, the scintillation for Gaussian beams is smaller. Figure 6(b) indicates that with the increment of the spectral power exponent, the scintillation index increases. This phenomenon is obvious for large propagation distance L. As is also seen from this figure, the flat-topped beam with larger propagation distance has a larger on-axis scintillation index at fixed power law. Figures 7(a) and 7(b) are shown at a fixed inner scale in order to examine the variations of the scintillation index against propagation length for flat-topped beams of different N and under different nonKolmogorov realizations, respectively. It is clear from Fig. 7(a) that scintillations become large as the propagation length increases for different N. At L > 3 km, scintillations increase as N increases; at L < 3 km, the scintillations for Gaussian beams have smaller values than for the other curves that 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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Fig. 7. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of propagation length L for different flat-topped order N and (b) as a function of propagation length L for different power law α. The calculation parameters are λ 1.55 μm, w0 4 cm, N 5, and l0 8 mm.
intersect on the different L. Figure 7(b) shows that with the increment of propagation length, the scintillations increase at different α, and being valid for all L, the scintillations become larger for the large power law of non-Kolmogorov turbulence. 5. Conclusions
The concise analytic expressions of the on-axis scintillation index for flat-topped Gaussian beam wave propagation horizontally through weakly nonKolmogorov atmospheric turbulence are derived using the generalized von Kármán spectrum and the equivalent non-Kolmogorov structure constant. The on-axis scintillation index bears no relation to the outer scale parameter under the condition of approximating calculation. The effects of inner scale with values ranging from 0 to 20 mm are comprehensively analyzed, and the effects of power law and propagation length are scrutinized at nonzero inner scale. The scintillation index for the larger inner scale values is always less than that for the smaller inner scale values, and the presence of a nonzero 2636
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inner scale causes scintillation to have a more rapid drop at larger values of the power law, flat-topped order, Gaussian source size, path length, and smaller values of wavelength although the degree of decline is limited. At a fixed inner scale, the scintillation index increases as the power law increases for different flat-topped order and propagation length. Moreover, with increasing flat-topped order, the scintillation index increases sharply up to the maximal point at the value of 6 and then slowly decreases at a certain power law and propagation length. At w0 3 cm, being valid for all the power law and propagation lengths, the scintillation of the Gaussian beam is always smaller than that of the flat-topped beam, which decreases at larger power law and increases at larger propagation length as the flattopped order increases. Examination of the scintillation index versus Gaussian source size yields that, at N 5, as Gaussian source size increases, the scintillation index values tend to be constant after experiencing a lowest value and a peak value. The increment of propagation distance increases the scintillation of the flat-topped beam linearly at fixed inner scale and Gaussian source size. In addition, the long-wavelength infrared radiation beam can substantially reduce the on-axis scintillation. The derived analytical formulas as well as the analyses will be useful in long-distance free-space optical communication through atmospheric channels at high altitudes under weather conditions in which atmosphere is featured as the non-Kolmogorov structure.
Appendix A
This appendix outlines the major steps of derivations in arriving at Eq. (7), for the source beam of Eq. (2). According to Eqs. (11) and (13) of Chap. 8 in [11], the scintillation index, σ 2I , whose radial position is r and axial position is L, can be expressed as σ 2I r; L 2 ReE2 r; r E3 r; r;
(A1)
where E2 r; r and E3 r; r are the two secondorder statistical moments, and Re denotes the real part. The derivations leading up to E2 r; r and E3 r; r, starting from the application of the Rytov approximation for the case of weak fluctuations and the pure Gaussian beam source, are detailed in Chap. 5 of [11]. With due care, it is possible to trace and adopt this derivation for another beam type. Let r1 and r2 denote two points in the transverse plane at z L, the two second-order statistical moments are given by [11] E2 r1 ; r2 hψ 1 r1 ; Lψ 1 r2 ; Li;
(A2)
E3 r1 ; r2 hψ 1 r1 ; Lψ 1 r2 ; Li;
(A3)
where ψ 1 r; L is the first-order Rytov perturbation, and the asterisk in Eq. (A2) refers to the complex conjugate of the quantity. Following the similar process of obtaining Eq. (50) in Chap. 5 of [11], by taking the flat-topped beam as the source beam instead of its pure Gaussian beam and rearranging, we get the first-order spectral representation ψ 1 r; L, ψ 1 r; L
N X 1 u r; Lψ 1n r; L; U 0 r; L n1 n
in Chap. 6 of [11] and find the two second-order statistical moments as E2 r1 ; r2
N X N X 4π 2 k2 L u r ; Lun2 r2 ; L U 0 r1 ; LU 0 r2 ; L n 1 n 1 n1 1 1 2 Z 1Z ∞ κΦn κJ 0 κj1 − θ¯n1 ;n2 εp − 2iΛn1 ;n2 εrj × 0
0
Λn1 ;n2 Lκ2 ε2 dκdε; × exp − k
(A4)
(A8)
where Z ψ 1n r;Lik
L 0
iκ 2 γ n L−z dνκ;z: dz exp iγ n κr− 2k −∞ (A5) ZZ
∞
In Eq. (A5), γ n γ n z is the nth complex path amplitude weighting parameter defined by γ n 1 iαn z∕1 iαn L, and dνκ; z is the random amplitude of the refractive-index fluctuations. The important information is that Eq. (A5) and the first-order spectral representation of the Gaussian beam have the same form by setting γ n γ. Substituting Eq. (A4) into Eqs. (A2) and (A3) and pursuing the derivations and approximations mentioned on pp. 148–150 of [11], E2 r1 ; r2 and E3 r1 ; r2 will transform into E2 r1 ; r2
N X N X 4π 2 k2 u r ; Lun2 r2 ; L U 0 r1 ; LU 0 r2 ; L n 1 n 1 n1 1 1 2 ZL Z∞ dη κΦn κ; ηJ 0 κjγ n1 r1 × 0 0 iκ 2 γ n1 − γ n2 L − η dκ; − γ n2 r2 j exp − (A6) 2k
E3 r1 ; r2
N X N X −4π 2 k2 u r ; Lun2 r2 ; L U 0 r1 ; LU 0 r2 ; L n 1 n 1 n1 1 1 2 ZL Z∞ dη κΦn κ; ηJ 0 κjγ n1 r1 × 0 0 iκ 2 γ γ n2 L − η dκ; − γ n2 r2 j exp − (A7) 2k n1
where J 0 · is a Bessel function of the first kind and order zero, and Φn κ; η is the spectral density of the index of refraction. In order to use the similar existent process of the Gaussian beam to calculate the scintillation index of the flat-topped beam, we realign Eqs. (A6) and (A7) by pursuing the derivations of Eqs. (25) and (26)
E3 r1 ; r2
N X N X −4π 2 k2 L u r ; Lun2 r2 ; L U 0 r1 ; LU 0 r2 ; L n 1 n 1 n1 1 1 2 Z 1Z ∞ κΦn κJ 0 κj1 − θ¯0n1 ;n2 εp − 2iΛ0n1 ;n2 εrj × 0 0 iLκ 2 0 ¯ ε1 − θn1 ;n2 ε dκdε (A9) × exp − k
where Λn1 ;n2 , Λ0n1 ;n2 and θ¯n1 ;n2 ; θ¯0n1 ;n2 are defined as Λn1 ;n2 Λn1 Λn2 − iθ¯n1 − θ¯n2 ∕2; Λ0n ;n Λn − Λn − iθ¯n − θ¯n ∕2;
(A10a)
θ¯n1 ;n2 iΛn1 − Λn2 − iθ¯n1 θ¯n2 ∕2; θ¯0n1 ;n2 iΛn1 Λn2 − iθ¯n1 θ¯n2 ∕2.
(A10b)
1
2
1
2
1
2
In Eq. (A10), θ¯n 1 − θn , Λn Λ0n ∕1 Λ20n , θn 1∕1 Λ20n , Λ0n nL∕kw20 bn L, and p r1 − r2 , r r1 r2 ∕2. θn and Λn are the curvature parameter and Fresnel ratio of the nth Gaussian beam at the receiver plane for vacuum propagation. When n1 n2 n, Λn1 ;n2 , Λ0n1 ;n2 and θ¯n1 ;n2 , θ¯0n1 ;n2 reduce to Λn , 0 and θ¯n , θ¯n iΛn , respectively. Moreover, Eqs. (A8) and (A9) can perfectly reduce to the existing second-order statistical moments for Gaussian beams by taking n 1. By setting r1 r2 r, the scintillation for a flat-topped beam wave based on Eqs. (A8) and (A9) takes the form 0 σ 2I r; L 8π 2 k2 L Re@
N X N X N X 1 jU 0 r; Lj2 U 0 r; L n 1 n 1 n 1 1 2 3 1
× un1 r; Lun2 r; Lun3 r; LSn1 ;n2 ;n3 A (A11) 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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where Sn1 ;n2 ;n3
Z 1Z 0
∞ 0
κΦn κ
Λn1 ;n3 Lκ 2 ε2 × I 0 2Λn1 ;n3 rκεexp − k
iLκ 2 ε1 − θ¯ 0n1 ;n2 ε dκdε: −I 0 2Λ0n1 ;n2 rκεexp − k
(A12) I 0 x J 0 ix is the modified Bessel function. References 1. L. C. Andrews and R. L. Phillips, “Impact of scintillation on laser communication systems: recent advances in modeling,” Proc. SPIE 4489, 23–34 (2002). 2. W. Cheng, J. W. Haus, and Q. Zhan, “Propagation of vector vortex beams through a turbulent atmosphere,” Opt. Express 17, 17829–17836 (2009). 3. G. Gbur and E. Wolf, “Spreading of partially coherent beams in random media,” J. Opt. Soc. Am. A 19, 1592–1598 (2002). 4. T. Shirai, A. Dogariu, and E. Wolf, “Mode analysis of spreading of partially coherent beams propagating through atmospheric turbulence,” J. Opt. Soc. Am. A 20, 1094–1102 (2003). 5. Y. Baykal, H. T. Eyyuboğlu, and Y. Cai, “Scintillations of partially coherent multiple Gaussian beams in turbulence,” Appl. Opt. 48, 1943–1954 (2009). 6. P. Zhou, Y. Ma, X. Wang, H. Ma, X. Xu, and Z. Liu, “Average intensity of a partially coherent rectangular flat-topped laser array propagating in a turbulent atmosphere,” Appl. Opt. 48, 5251–5258 (2009). 7. F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “Partially coherent elegant Hermite–Gaussian beam in turbulent atmosphere,” Appl. Phys. B 103, 461–469 (2011). 8. Y. Baykal and H. T. Eyyuboğlu, “Scintillations of incoherent flat-topped Gaussian source field in turbulence,” Appl. Opt. 46, 5044–5050 (2007). 9. T. Wang and J. Pu, “Propagation of non-uniformly polarized beams in a turbulent atmosphere,” Opt. Commun. 281, 3617–3622 (2008). 10. Y. Gu and G. Gbur, “Reduction of turbulence-induced scintillation by non-uniformly polarized beam arrays,” Opt. Lett. 37, 1553–1555 (2012). 11. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005). 12. H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. 245, 37–47 (2005). 13. X. Chu, C. Qiao, and X. Feng, “The effect of non-Kolmogorov turbulence on the propagation of cosh-Gaussian beam,” Opt. Commun. 283, 3398–3403 (2010). 14. H. T. Eyyuboğlu, “Annular, cosh and cos Gaussian beams in strong turbulence,” Appl. Phys. B 103, 763–769 (2011). 15. H. T. Eyyuboğlu and F. Hardalaç, “Propagation of modified Bessel-Gaussian beams in turbulence,” Opt. Laser Technol. 40, 343–351 (2008).
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16. K. Zhu, S. Li, Y. Tang, Y. Yu, and H. Tang, “Study on the propagation parameters of Bessel–Gaussian beams carrying optical vortices through atmospheric turbulence,” J. Opt. Soc. Am. A 29, 251–257 (2012). 17. Y. Cai and S. He, “Propagation of various dark hollow beams in a turbulent atmosphere,” Opt. Express 14, 1353–1367 (2006). 18. H. T. Eyyuboğlu and Y. Cai, “Non-Kolmogorov spectrum scintillation aspects of dark hollow and flat topped beams,” Opt. Commun. 285, 969–974 (2012). 19. H. T. Eyyuboğlu, Y. Baykal, and A. Falits, “Scintillation behavior of Laguerre Gaussian beams in strong turbulence,” Appl. Phys. B 104, 1001–1006 (2011). 20. A. A. Tovar, “Propagation of flat-topped multi-Gaussian laser beams,” J. Opt. Soc. Am. A 18, 1897–1904 (2001). 21. Y. Baykal and H. T. Eyyuboğlu, “Scintillation index of flat-topped Gaussian beams,” Appl. Opt. 45, 3793–3797 (2006). 22. H. T. Eyyuboğlu, Ç. Arpali, and Y. Baykal, “Flat topped beams and their characteristics in turbulent media,” Opt. Express 14, 4196–4207 (2006). 23. Y. Cai, “Propagation of various flat-topped beams in a turbulent atmosphere,” J. Opt. A 8, 537–545 (2006). 24. H. Gerçekcioğlu and Y. Baykal, “Scintillation index of flattopped Gaussian laser beam in strongly turbulent medium,” J. Opt. Soc. Am. A 28, 1540–1544 (2011). 25. H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flattopped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). 26. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994). 27. Y. Li, “Light beams with flat-topped profiles,” Opt. Lett. 27, 1007–1009 (2002). 28. Y. Li, “New expressions for flat-topped light beams,” Opt. Commun. 206, 225–234 (2002). 29. Y. Cai and Q. Lin, “Light beams with elliptical flat-topped profiles,” J. Opt. A 6, 390–395 (2004). 30. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beam propagation through non-Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). 31. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wave propagating through non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 28, 912–916 (2011). 32. L. Cui, B. Xue, L. Cao, S. Zheng, W. Xue, X. Bai, X. Cao, and F. Zhou, “Irradiance scintillation for Gaussian-beam wave propagating through weak non-Kolmogorov turbulence,” Opt. Express 19, 16872–16884 (2011). 33. X. Yi, Z. Liu, and P. Yue, “Inner- and outer-scale effects on the scintillation index of an optical wave propagating through moderate-to-strong non-Kolmogorov turbulence,” Opt. Express 20, 4232–4247 (2012). 34. M. Charnotskii, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment,” J. Opt. Soc. Am. A 29, 1838–1840 (2012). 35. H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flattopped beam in non-Kolmogorov weak turbulence: reply,” J. Opt. Soc. Am. A 29, 1841–1842 (2012). 36. Y. Baykal and H. Gerçekcioğlu, “Equivalence of structure constants in non-Kolmogorov and Kolmogorov spectra,” Opt. Lett. 36, 4554–4556 (2011).
Xi’an Institute of Optics and Precision Mechanics of Chinese Academy of Sciences (CAS), Xi’an 710119, China 2
Graduate University of CAS, Beijing 100049, China *Corresponding author: [email protected]
Received 11 August 2014; revised 30 October 2014; accepted 22 February 2015; posted 23 February 2015 (Doc. ID 220782); published 23 March 2015
A simpler generalized expression of irradiance fluctuation for flat-topped beams is presented based on the Born and Rytov perturbation methods. The theoretical expression of the on-axis scintillation index in non-Kolmogorov weak atmospheric optics links is developed using the generalized von Kármán spectrum model, and using the equivalent structure constant that is different for all power-law exponents. The effect of the inner scale on the on-axis scintillation index is examined comprehensively. It is observed that flat-topped beams happen to possess smaller scintillation indices at larger inner scale. The effects of the power law, flat-topped order, source size of the fundamental Gaussian beam, propagation distance, and wavelength are also analyzed. © 2015 Optical Society of America OCIS codes: (010.1330) Atmospheric turbulence; (010.1300) Atmospheric propagation; (010.3310) Laser beam transmission; (290.5930) Scintillation. http://dx.doi.org/10.1364/AO.54.002630
1. Introduction
Atmospheric turbulence can severely degrade the performance of a remote sensing or laser communication system [1,2]. In order to overcome the effects of atmospheric turbulence, many techniques have been researched, such as the use of partially coherent beams [3–7], incoherent beams [8], and polarized beams [9,10]. Investigation on the shape of the beam has been of increasing interest because of the evidence that other types of incidence may be less influenced by turbulence. A wide variety of beam shapes have been considered, including Gaussian [11], Hermite–Gaussian [7], Hermite–cosh–Gaussian [12], cosh (cos)–Gaussian [13,14], Bessel [15,16], annular [14], dark hollow [17,18], Laguerre–Gaussian [19], and flat-topped beams [20–25]. Specifically, in recent years, some new attempts have been focused on flat-topped beams, which can be applied in inertial
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confinement fusion and material thermal processing for laser beams. Among all the theoretical models for describing flat-topped beams [26–29], the model proposed by Li, where the electric field is a finite sum of fundamental Gaussian beams, has recently been used for the study of atmospheric effects on propagated light intensity and scintillation [21,24,25]. However, for a flat-topped beam propagating through non-Kolmogorov turbulence, to the best of the authors’ knowledge, there is no published work analyzing the impact of the inner scale on scintillation. This has motivated the work herein. On the other hand, various models of the power spectrum for optical turbulence have been developed previously. These models include the Kolmogorov power-law spectrum, the Tatarskii spectrum (with inner scale parameter), the exponential spectrum (with outer scale parameter), the von Kármán spectrum (with both inner scale and outer scale parameters), and the modified atmospheric spectrum (with both inner scale and outer scale parameters). All the corresponding generalized models [30–33], in which
the power law has the range of 3 to 4 or 3 to 5 instead of the value 11/3, have been proposed in recent years. It is shown that the generalized modified atmospheric spectral model for the non-Kolmogorov turbulence characterizes a high wave number “bump” that has been observed in temperature data, but some coefficients in this spectral model have no certain values. In view of this, many researchers adopt the generalized von Kármán spectrum to investigate the scintillation index associated with a beam wave propagating through non-Kolmogorov atmospheric turbulence. Furthermore, in non-Kolmogorov turbulence, Ref. [34] shows that it is inappropriate for the scintillation index of flat-topped beam to use the same structure constant for all the power-law exponents [25], and the problem can be solved by employing the formulated equivalent structure constant [35,36]. In particular, for a flat-topped beam, the generalized expressions of the on-axis scintillation (Eq. (1) in Ref. [25]) are so complicated that the final results are difficult and complex to calculate when the spectra are expanded. In order to develop a simper expression of scintillation, second-order statistical moments are calculated based on the Born and Rytov perturbation methods. We put forward the predigesting formula of the on-axis scintillation index for flat-topped beams and derive the results using the generalized von Kármán spectral model and the equivalent structure constant [36] under weak non-Kolmogorov turbulence. Then, the impact of turbulence inner scales on the scintillation index is scrutinized. Our results could be useful for experimental research of laser communication in horizontal optical links deployed at any altitude. 2. Generalized von Kármán spectrum
The generalized von Kármán spectral model is given by [30] exp −κ 2 ∕κ2l 2 ~ ; Φn κ; α; l0 ; L0 AαCn α 2 κ κ 20 α∕2 0 ≤ κ < ∞;
3 < α < 4;
power-law spectrum in the inertial range κ0 ≪ κ ≪ κ l and reduces to the conventional von Kármán spectrum when α 11∕3. 3. Scintillation Index for Flat-Topped Beam Wave A. Flat-Topped Light Beams
For the flat-topped beam, the electric field at the source (z 0) can be expressed as a finite sum of Gaussian beams with different parameters as follows [21]: U 0 r; 0
1 an exp − bn kr2 ; 2 n1
N X
where r is the radial coordinate, N is the order of the flat-topped beam, k 2π∕λ is the wave number, λ is the wavelength, an
−1n−1 N
N ; n
(3)
r 2 w ; n 0
(4)
and 2 ; bn kw20n
w0n
where Nn denotes a binomial coefficient, w0n is the waist width of the nth Gaussian beam, and w0 is the source size of the fundamental Gaussian beam. As the beam order N increases, the beam profile of the flat-topped beam flattens and the amplitude decreases. When N 1, Eq. (2) reduces to a Gaussian beam. Based on the Huygens–Fresnel integral of the paraxial equation, we can get the complex amplitude, which is also the series of Gaussian beams, at distance z L from the source as [21] U 0 r; L
N X
un r; L;
(5)
n1
(1)
where κ denotes the magnitude of the spatial frequency with units of rad/m; α is the spectral power law; l0 is the turbulence inner scale, which is generally on the order of millimeters; L0 is the turbulence outer scale on the order of meters; and Aα is the generalized amplitude factor, which has the form Aα Γα − 1 · cosαπ∕2∕4π 2 . Γ· is the ~ 2n α is the generalized refractivegamma function. C index structure parameter with units of m3−α. The parameters κ l and κ0 are related to inner scale and outer scale in general by the equations κl cα∕l0 and κ0 C0 ∕L0 , where cα fΓ5 − α∕2 · Aα · 2π∕3g1∕α−5 and the scaling constant C0 is chosen differently according to the application because of the not-well-defined outer scale. In this paper, we set C0 2π. Equation (1) reduces to the non-Kolmogorov
(2)
where un r; L
an 1 bn exp ikL − kr2 ; (6) 2 1 ibn L 1 ibn L
i2 −1. When r0, un 0;L an expikL∕1 ibn L. B. Scintillation Index for Flat-Topped Beam Wave
Following the notion that a flat-topped beam is a finite sum of Gaussian beams, we try to find the relationship between the scintillation index of a flat-topped beam and the scintillation index of a Gaussian beam. Based on the Rytov approximation and the above electric field expression of the flattopped beam, a simpler model of scintillation index can be derived as Eq. (A12). When r 0, the on-axis 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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scintillation index of the flat-topped beam will turn into
Sj1 ;j2 ;j3
0 σ 2I 0;L 8π 2 k2 LRe@
×
1 jU 0 0;Lj2 U 0 0;L
N X N X N X j1 1 j2 1 j3 1
1 uj1 0;Luj2 0;Luj3 0;LSj1 ;j2 ;j3 A; (7)
Γ1 − α∕2 ~ 2n ακ2−α AαC l 2 α 1 3 2 × 2 F 1 1 − ; ; ; −Λj1 ;j3 Lκ l ∕k 2 2 2
2 1 iLκl 1 − 2θ¯0j1 ;j2 ∕3∕kα∕2 − 1 : − iαLκ2l 1 − 2θ¯0j1 ;j2 ∕3∕2k
(12)
Substituting Eq. (12) into Eq. (7), σ 2I 0; L becomes σ 2I 0; L ~ n απ 2 k2 Lκ2−α 4Γ1 − α∕2AαC l PN PN PN j1 1 j2 1 j3 1 uj1 0; Luj2 0; Luj3 0; L × Re jU 0 0; Lj2 U 0 0; L α 1 3 2 × 2 F 1 1 − ; ; ; −Λj1 ;j3 Lκl ∕k 2 2 2
2 1 iLκ l 1 − 2θ¯0j1 ;j2 ∕3∕kα∕2 − 1 − ; (13) iαLκ 2l 1 − 2θ¯0j1 ;j2 ∕3∕2k 2
where Re is the real part, Sj1 ;j2 ;j3
Λj ;j Lκ 2 ε2 κΦn κ; α; l0 ; L0 exp − 1 3 k 0 0
2 iLκ ε1 − θ¯0j1 ;j2 ε dκdε; (8) − exp − k Z 1Z
∞
1 Λj1 ; j3 Λj1 Λj3 − iθ¯j1 − θ¯j3 ; 2 1 θ¯0j1 ;j2 θ¯j1 θ¯j2 iΛj1 Λj2 ; 2
(9)
~ 2n α C (10)
ε 1 − z∕L, Λj Λ0j ∕1 Λ20j , θ¯j Λ20j ∕1 Λ20j , Λ0j jL∕kw20 bj L, and the asterisk () indicates the complex conjugate. One can note that uj 0; L is a complex number and 0 < ReΛj1 ; j3 ; θ¯0j1 ; j2 < 1. Equation (7) can be expanded to the same form as Eq. (1) of Ref. [25]. However, the expression Eq. (7) is much simpler, and it can be used to simplify the computation of the on-axis scintillation index of the flat-topped beam under all kinds of atmospheric spectral models through the similar existent process of the Gaussian beam. When N is taken as unity, Eq. (7) reduces to the on-axis scintillation index of the Gaussian beam with receiver beam parameters Λ Λ1;1 and θ¯ θ¯1;1 . In Eq. (7), we just need to perform the integrations of Sj1 ; j2 ; j3 over κ and ε. Now employing the von Kármán spectrum expression in Eq. (1), Eq. (8) can be expressed as Sj1 ;j2 ;j3
Z 1Z
κ 2 α∕2 0 0 κ κ 0 Λj1 ; j3 Lκ 2 ε2 κ2 − 2 × exp − k κl
iLκ2 κ2 ε1 − θ¯0j1 ; j2 ε − 2 dκdε: − exp − k κl
~ 2n α AαC
∞
2
(11)
Following the same process as in [32] [Eqs. (27)–(29)], Eq. (11) has the form 2632
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where the structure constant in non-Kolmogorov media is expressed as [36]
−0.5Γα2π−11∕6α∕2 λL11∕6−α∕2 C2n ; Γ1 − α∕2Γα∕22 Γα − 1 cosαπ∕2 sinαπ∕4 (14)
where C2n is the structure constant of Kolmogorov turbulence. When N 1, Eq. (13) reduces to the on-axis scintillation index of the Gaussian beam as in [32]. 4. Results and Discussion
In this section, we analyze the influence of some key parameters on the analytical formulas derived in the previous section [Eq. (13)], including inner scale l0 , the flat-topped beam order N, the power law α, Gaussian source size w0 , propagation distance L, and wavelength λ. The effect of the inner scale on the on-axis scintillation index is mainly emphasized; hence most of the figures are considered to cover a subfigure of the scintillation index versus inner scale l0 . In all the figures, the index-of-refraction structure constant in Kolmogorov media is fixed as C2n 10−15 m−2∕3 , because it is linear with the onaxis irradiance fluctuation. Figure 1 plots the on-axis scintillation index versus inner scale l0 and power law α, taking L 1 km, N 5, w0 3 cm, and λ 1.55 μm. Because the inner scale is mere millimeters near the earth surface and several centimeters in the upper regions of the atmosphere, the range of the inner scale is set as 0 to 20 mm. The range of the power law is 3 to 4. Figure 1(a) depicts the scintillation as a function of l0 for several values of α (one of which corresponds to the Kolmogorov’s turbulence
Fig. 1. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for different power law α and (b) as a function of power law α for different inner scale l0 . The calculation parameters are λ 1.55 μm, L 1 km, N 5, and w0 3 cm.
Fig. 2. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for flat-topped order N and (b) as a function of flat-topped order N for different inner scale l0 . The calculation parameters are λ 1.55 μm, w0 3 cm, L 1 km, and α 10∕3.
(α 11∕3)), while Fig. 1(b) represents the scintillation index as a function of α for several values of l0 . It can be seen from Fig. 1(a) that with the increment of inner scale, the scintillation index decreases. This phenomenon is obvious for large spectral index α. As also seen from this figure, the flat-topped beam with large spectral power exponent α has larger on-axis scintillation index at fixed inner scale. Figure 1(b) yields that the scintillation index is a constant for different strengths of inner scale if α 3. The scintillation indices increase linearly with the increment of α. Also, we derive from Fig. 1(b) that, at the same α value, the flat-topped beams have smaller scintillation values propagating through large inner scale turbulence than through small inner scale turbulence, and the detached degree grows as α increases, which means the inner scale can affect the scintillation index especially at large α. The conclusion from the two figures is that, in the weak turbulence regimes, larger inner scale and smaller power law can bring advantage to the propagation of the flat-topped beam. Figure 2 plots the on-axis scintillation index of flattopped beams through non-Kolmogorov weak turbulence as a function of inner scale l0 and flat-topped order N, which correspond to the flat degree of the
beam profile and the magnitude of the amplitude for the flat-topped beam. The fixed parameters except α 10∕3 are the same as in Fig. 1. It can be seen from Fig. 2(a) that, with the increment of inner scale, the scintillations vary like the above curve for α 10∕3 in Fig. 1(a). We note that, for N 1 corresponding to the Gaussian beam, the scintillation index responds with a smaller value compared with the three larger N, at which the scintillation indices at larger N have smaller values for all the inner scale. Figure 2(b) indicates that, for all the flat-topped order N, the scintillation of the flat-topped beam decreases with the increment of inner scale. Furthermore, at fixed l0 , the scintillation indices increase monotonically as the flat-topped order N increases to 6, and then slightly decrease when N increases. Examination of the trend in Fig. 2(b) is also executed for a small Gaussian source size of w0 0.5 cm. This is not plotted separately due to emphasizing the effect of inner scale, though it shows that certain large values of N have small scintillations. The on-axis scintillation index of flat-topped beams through generalized von Kármán spectral turbulence is plotted in Fig. 3 as a function of inner scale l0 and Gaussian source size w0 , setting L 1 km, N 5, α 10∕3, and λ 1.55 μm. As illustrated 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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Fig. 3. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for different Gaussian source size w0 and (b) as a function of Gaussian source size w0 for different inner scale l0 . The calculation parameters are λ 1.55 μm, L 1 km, N 5, and α 10∕3.
in Fig. 3(a) for fixed Gaussian source sizes, on-axis scintillations become smaller when l0 becomes larger. It can also be inferred from this figure that at fixed l0 , the scintillations increase with the increase of w0 when w0 ≥ 1 cm, but they have larger values at w0 0.5 cm than at w0 1 cm, which is also valid for N > 5. Figure 3(b) plots the on-axis scintillation index as a function of Gaussian source size for several inner scales. With the increase of the value of w0, the scintillation index at fixed l0 first decreases to a minimum value that occurs near w0 0.8 cm, then increases steeply to a maximum value near w0 3 cm, and lastly maintains the same level at w0 > 4 cm after a slight decrease. All the same figures for N 1 ∼ 15 are plotted, and the scintillations vary like the curves of Fig. 9(a) in Ref. [25] when w0 increases. In addition, for all values of N from 1 through 15, the scintillation decreases as inner scale increases, and the degree of the decline grows from around w0 2 cm to its two sides, which means that the inner scale has weak influence on the scintillation of the flat-topped beam at around w0 2 cm. Figure 4 plots the on-axis scintillation index of flat-topped beams as a function of inner scale l0 2634
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Fig. 4. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for different propagation length L and (b) as a function of propagation length L for different inner scale l0 . The calculation parameters are λ 1.55 μm, w0 3 cm, N 5, and α 10∕3.
and propagation distance L, taking w0 3 cm, N 5, α 10∕3, and λ 1.55 μm. We deduce from Fig. 4(a) that with the increase of the value of l0, the scintillation is slightly decreased, the degree of which is almost the same for different L. It can also be deduced that when l0 is fixed, the scintillation increases with the increment of propagation distance. The results can also be obtained in Fig. 4(b). We also note that the flat-topped beams have smaller on-axis scintillation indices at large l0 , although the phenomenon is not obvious. Additionally, examination of the trend in Figs. 4(a) and 4(b) is also executed for other fixed N, w0 , and α, and the whole trend of the variation of scintillation as the inner scale is similar to that in Fig. 4. After combining the information above and taking the relative variance as the criterion of the judgment, it can be revealed that the effect of inner scale is weak for long propagation distances if the other parameters are fixed. In order to further analyze the influence of wavelength on the flat-topped beam on-axis scintillation index based on generalized von Kármán spectral weak turbulence, the scintillation index is plotted in Fig. 5 as a function of inner scale l0 and wavelength λ, taking L 1 km, N 5, α 10∕3, and
Fig. 5. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of inner scale l0 for different wavelength λ and (b) as a function of wavelength λ for different inner scale l0 . The calculation parameters are L 1 km, w0 3 cm, N 5, and α 10∕3.
w0 3 cm. As illustrated in Fig. 5(a), being valid for all l0 , scintillation indices decrease steadily as the wavelength increases to long-wavelength infrared radiation. In addition, the scintillations become smaller as the inner scale increases, and the velocity of decrease is faster for small λ. The scintillation index for long wavelength of 10.6 μm has a value close to zero, and it is almost invariable with the inner scale increases. More detailed information from Fig. 5(b) is that the scintillation decreases steeply at short wavelength of λ < 2 μm, and then descends slightly as the wavelength increases. As a result, the reduction of scintillation from choosing long wavelength results in the decrease of bit-error-rate and the improvement of free-space optical communication system performance. Moreover, a larger wavelength makes the scintillation index less influenced by the inner scale. The result is also effectual for N 1 ∼ 15. Figures 6 and 7 emphasize the effect of the power law and propagation distance when the inner scale is a nonzero constant. The on-axis scintillation index of the flat-topped beam through generalized von Kármán spectral turbulence is plotted as a function of power law α for different flat-topped order N at
Fig. 6. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of power law α for different flat-topped order N and (b) as a function of power law α for different propagation length L. The calculation parameters are λ 1.55 μm, w0 4 cm, and l0 8 mm.
L 2 km in Fig. 6(a) and for different propagation length L at N 5 in Fig. 6(b), setting l0 8 mm, w0 4 cm, and λ 1.55 μm. As illustrated in Fig. 6(a) for fixed flat-topped order, on-axis scintillations become larger when the power law of nonKolmogorov turbulence becomes larger and the velocity of increase is faster at N 5. It can also be seen from this figure that at all α, the scintillation for Gaussian beams is smaller. Figure 6(b) indicates that with the increment of the spectral power exponent, the scintillation index increases. This phenomenon is obvious for large propagation distance L. As is also seen from this figure, the flat-topped beam with larger propagation distance has a larger on-axis scintillation index at fixed power law. Figures 7(a) and 7(b) are shown at a fixed inner scale in order to examine the variations of the scintillation index against propagation length for flat-topped beams of different N and under different nonKolmogorov realizations, respectively. It is clear from Fig. 7(a) that scintillations become large as the propagation length increases for different N. At L > 3 km, scintillations increase as N increases; at L < 3 km, the scintillations for Gaussian beams have smaller values than for the other curves that 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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Fig. 7. On-axis scintillation index of flat-topped beam through non-Kolmogorov weak turbulence (a) as a function of propagation length L for different flat-topped order N and (b) as a function of propagation length L for different power law α. The calculation parameters are λ 1.55 μm, w0 4 cm, N 5, and l0 8 mm.
intersect on the different L. Figure 7(b) shows that with the increment of propagation length, the scintillations increase at different α, and being valid for all L, the scintillations become larger for the large power law of non-Kolmogorov turbulence. 5. Conclusions
The concise analytic expressions of the on-axis scintillation index for flat-topped Gaussian beam wave propagation horizontally through weakly nonKolmogorov atmospheric turbulence are derived using the generalized von Kármán spectrum and the equivalent non-Kolmogorov structure constant. The on-axis scintillation index bears no relation to the outer scale parameter under the condition of approximating calculation. The effects of inner scale with values ranging from 0 to 20 mm are comprehensively analyzed, and the effects of power law and propagation length are scrutinized at nonzero inner scale. The scintillation index for the larger inner scale values is always less than that for the smaller inner scale values, and the presence of a nonzero 2636
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inner scale causes scintillation to have a more rapid drop at larger values of the power law, flat-topped order, Gaussian source size, path length, and smaller values of wavelength although the degree of decline is limited. At a fixed inner scale, the scintillation index increases as the power law increases for different flat-topped order and propagation length. Moreover, with increasing flat-topped order, the scintillation index increases sharply up to the maximal point at the value of 6 and then slowly decreases at a certain power law and propagation length. At w0 3 cm, being valid for all the power law and propagation lengths, the scintillation of the Gaussian beam is always smaller than that of the flat-topped beam, which decreases at larger power law and increases at larger propagation length as the flattopped order increases. Examination of the scintillation index versus Gaussian source size yields that, at N 5, as Gaussian source size increases, the scintillation index values tend to be constant after experiencing a lowest value and a peak value. The increment of propagation distance increases the scintillation of the flat-topped beam linearly at fixed inner scale and Gaussian source size. In addition, the long-wavelength infrared radiation beam can substantially reduce the on-axis scintillation. The derived analytical formulas as well as the analyses will be useful in long-distance free-space optical communication through atmospheric channels at high altitudes under weather conditions in which atmosphere is featured as the non-Kolmogorov structure.
Appendix A
This appendix outlines the major steps of derivations in arriving at Eq. (7), for the source beam of Eq. (2). According to Eqs. (11) and (13) of Chap. 8 in [11], the scintillation index, σ 2I , whose radial position is r and axial position is L, can be expressed as σ 2I r; L 2 ReE2 r; r E3 r; r;
(A1)
where E2 r; r and E3 r; r are the two secondorder statistical moments, and Re denotes the real part. The derivations leading up to E2 r; r and E3 r; r, starting from the application of the Rytov approximation for the case of weak fluctuations and the pure Gaussian beam source, are detailed in Chap. 5 of [11]. With due care, it is possible to trace and adopt this derivation for another beam type. Let r1 and r2 denote two points in the transverse plane at z L, the two second-order statistical moments are given by [11] E2 r1 ; r2 hψ 1 r1 ; Lψ 1 r2 ; Li;
(A2)
E3 r1 ; r2 hψ 1 r1 ; Lψ 1 r2 ; Li;
(A3)
where ψ 1 r; L is the first-order Rytov perturbation, and the asterisk in Eq. (A2) refers to the complex conjugate of the quantity. Following the similar process of obtaining Eq. (50) in Chap. 5 of [11], by taking the flat-topped beam as the source beam instead of its pure Gaussian beam and rearranging, we get the first-order spectral representation ψ 1 r; L, ψ 1 r; L
N X 1 u r; Lψ 1n r; L; U 0 r; L n1 n
in Chap. 6 of [11] and find the two second-order statistical moments as E2 r1 ; r2
N X N X 4π 2 k2 L u r ; Lun2 r2 ; L U 0 r1 ; LU 0 r2 ; L n 1 n 1 n1 1 1 2 Z 1Z ∞ κΦn κJ 0 κj1 − θ¯n1 ;n2 εp − 2iΛn1 ;n2 εrj × 0
0
Λn1 ;n2 Lκ2 ε2 dκdε; × exp − k
(A4)
(A8)
where Z ψ 1n r;Lik
L 0
iκ 2 γ n L−z dνκ;z: dz exp iγ n κr− 2k −∞ (A5) ZZ
∞
In Eq. (A5), γ n γ n z is the nth complex path amplitude weighting parameter defined by γ n 1 iαn z∕1 iαn L, and dνκ; z is the random amplitude of the refractive-index fluctuations. The important information is that Eq. (A5) and the first-order spectral representation of the Gaussian beam have the same form by setting γ n γ. Substituting Eq. (A4) into Eqs. (A2) and (A3) and pursuing the derivations and approximations mentioned on pp. 148–150 of [11], E2 r1 ; r2 and E3 r1 ; r2 will transform into E2 r1 ; r2
N X N X 4π 2 k2 u r ; Lun2 r2 ; L U 0 r1 ; LU 0 r2 ; L n 1 n 1 n1 1 1 2 ZL Z∞ dη κΦn κ; ηJ 0 κjγ n1 r1 × 0 0 iκ 2 γ n1 − γ n2 L − η dκ; − γ n2 r2 j exp − (A6) 2k
E3 r1 ; r2
N X N X −4π 2 k2 u r ; Lun2 r2 ; L U 0 r1 ; LU 0 r2 ; L n 1 n 1 n1 1 1 2 ZL Z∞ dη κΦn κ; ηJ 0 κjγ n1 r1 × 0 0 iκ 2 γ γ n2 L − η dκ; − γ n2 r2 j exp − (A7) 2k n1
where J 0 · is a Bessel function of the first kind and order zero, and Φn κ; η is the spectral density of the index of refraction. In order to use the similar existent process of the Gaussian beam to calculate the scintillation index of the flat-topped beam, we realign Eqs. (A6) and (A7) by pursuing the derivations of Eqs. (25) and (26)
E3 r1 ; r2
N X N X −4π 2 k2 L u r ; Lun2 r2 ; L U 0 r1 ; LU 0 r2 ; L n 1 n 1 n1 1 1 2 Z 1Z ∞ κΦn κJ 0 κj1 − θ¯0n1 ;n2 εp − 2iΛ0n1 ;n2 εrj × 0 0 iLκ 2 0 ¯ ε1 − θn1 ;n2 ε dκdε (A9) × exp − k
where Λn1 ;n2 , Λ0n1 ;n2 and θ¯n1 ;n2 ; θ¯0n1 ;n2 are defined as Λn1 ;n2 Λn1 Λn2 − iθ¯n1 − θ¯n2 ∕2; Λ0n ;n Λn − Λn − iθ¯n − θ¯n ∕2;
(A10a)
θ¯n1 ;n2 iΛn1 − Λn2 − iθ¯n1 θ¯n2 ∕2; θ¯0n1 ;n2 iΛn1 Λn2 − iθ¯n1 θ¯n2 ∕2.
(A10b)
1
2
1
2
1
2
In Eq. (A10), θ¯n 1 − θn , Λn Λ0n ∕1 Λ20n , θn 1∕1 Λ20n , Λ0n nL∕kw20 bn L, and p r1 − r2 , r r1 r2 ∕2. θn and Λn are the curvature parameter and Fresnel ratio of the nth Gaussian beam at the receiver plane for vacuum propagation. When n1 n2 n, Λn1 ;n2 , Λ0n1 ;n2 and θ¯n1 ;n2 , θ¯0n1 ;n2 reduce to Λn , 0 and θ¯n , θ¯n iΛn , respectively. Moreover, Eqs. (A8) and (A9) can perfectly reduce to the existing second-order statistical moments for Gaussian beams by taking n 1. By setting r1 r2 r, the scintillation for a flat-topped beam wave based on Eqs. (A8) and (A9) takes the form 0 σ 2I r; L 8π 2 k2 L Re@
N X N X N X 1 jU 0 r; Lj2 U 0 r; L n 1 n 1 n 1 1 2 3 1
× un1 r; Lun2 r; Lun3 r; LSn1 ;n2 ;n3 A (A11) 1 April 2015 / Vol. 54, No. 10 / APPLIED OPTICS
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where Sn1 ;n2 ;n3
Z 1Z 0
∞ 0
κΦn κ
Λn1 ;n3 Lκ 2 ε2 × I 0 2Λn1 ;n3 rκεexp − k
iLκ 2 ε1 − θ¯ 0n1 ;n2 ε dκdε: −I 0 2Λ0n1 ;n2 rκεexp − k
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