Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment

doi:10.1364/JOSAA.29.001838

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

Editor: Franco Gori
Vol. 29, Iss. 9 — Sep. 1, 2012
pp: 1838–1840

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Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment

Mikhail Charnotskii »View Author Affiliations

JOSA A, Vol. 29, Issue 9, pp. 1838-1840 (2012)
http://dx.doi.org/10.1364/JOSAA.29.001838

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▸ Article Outline
– Abstract
FIXED STRUCTURE CONSTANT MATTERS
PROPER RANGE OF THE SPECTRAL EXPONENT
– References and links

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Abstract

In J. Opt. Soc. Am. A 29, 169 (2012), Gerçekcioğlu and Baykal presented an investigation of the dependence of the scintillation index of flat-topped Gaussian beams on the exponent $α$ of the power-law-type spectra of non-Kolmogorov turbulence. In particular, they found that the scintillation index reaches a maximum at $α \approx 3.2$ . We show that this conclusion is an artifact of their specific calculation, and depends on the choice of the length unit. Gerçekcioğlu and Baykal’s calculations are made for the $3 < α < 5$ range of the spectral exponent. We show that for the homogeneous and isotropic turbulence the correct range is $3 < α < 4$ , when Markov approximation is used.

1. FIXED STRUCTURE CONSTANT MATTERS

One of the main conclusions of Gerçekcioğlu and Baykal’s paper [1

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]

] “Variation of the scintillation index against non-Kolmogorov power law

α

exhibits a peak at the worst power law, which happens to be smaller than the Kolmogorov power law of

11 / 3

” is incorrect.

The authors base this conclusion on calculations shown in Fig. 6, which presents scintillation index

m^{2}

as a function of exponent

α

of the power-law spectrum of turbulence. Figure 6 represents a single propagation scenario for two beam sizes and various “orders of flatness”

N

. For the purpose of this discussion we limit ourselves by the simple collimated Gaussian beam, i.e.,

N = 1

. In order to compare “apples to apples,” when analyzing the impact of the non-Kolmogorov turbulence on the statistics of the propagating optical field, the spectral exponent

α

should be the only variable that is allowed to change, while the rest of the parameters determining the propagation condition remains fixed. At the first glance it looks like the authors did just that by setting wavelength

λ = 1.55 μm

, path length

L = 2 km

, beam width

W_{0}

either 1 cm or 2 cm, and turbulence structure constant

{\tilde{C}}_{n}^{2} = 10^{- 15} m^{3 - α}

. Unfortunately, the physical dimension of the

{\tilde{C}}_{n}^{2}

depends on

α

, and this makes Fig. 6 an “apple-to-orange” comparison, and the aforementioned conclusion incorrect. We emphasize that there is nothing wrong with Eq. (5) that underlies the data presented at Fig. 6. We are concerned with the interpretation of results.

In order to illustrate our point we repeated Gerçekcioğlu and Baykal’s calculations for

N = 1

using centimeters instead of meters for the length measure. While actual path length and wavelength remain unchanged, for structure constant we used

{\tilde{C}}_{n}^{2} = 4.64 \cdot 10^{- 17} {cm}^{3 - α}

. Since

10^{- 15} m^{- 2 / 3} = 4.64 \cdot 10^{- 17} {cm}^{- 2 / 3}

our results should match Gerçekcioğlu and Baykal’s calculations for

α = 11 / 3

. The dashed lines in Fig. 1 show our reproduction of the Gerçekcioğlu and Baykal calculations from their Fig. 6 made for

{\tilde{C}}_{n}^{2} = 10^{- 15} m^{3 - α}

and

N = 1

. The solid lines show our calculations for

{\tilde{C}}_{n}^{2} = 4.64 \cdot 10^{- 17} {cm}^{3 - α}

. As expected, both pairs of curves intersect at

α = 11 / 3

. At the same time values of the scintillation index for other

α

changed dramatically. The maxima of the scintillation index move from

α \approx 3.2

to

α \approx 4.5

, and the ratio of the scintillation index maxima for initial beam width 1 cm and 2 cm changed from being very close to unity to about 0.75. In other words, neither of the conclusions drawn from the calculations for “fixed”

{\tilde{C}}_{n}^{2} = 10^{- 15} m^{3 - α}

remains valid for a different “fixed”

{\tilde{C}}_{n}^{2} = 4.64 \cdot 10^{- 17} {cm}^{3 - α}

, just because the unit of length was changed.

Fig. 1. Dependence of scintillation index on

α

. Dashed curves,

{\tilde{C}}_{n}^{2} = 10^{- 15} m^{3 - α}

. Solid curves,

{\tilde{C}}_{n}^{2} = 4.64 \cdot 10^{- 17} {cm}^{3 - α}

.

The reason for this disastrous effect is very simple. In the spatial domain, spectra considered in [1

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]

] are related to an

α

-dependent family of structure functions of the refractive index fluctuations

D_{n} (ρ⃗) = 〈 {[n (R⃗ + ρ⃗) - n (R⃗)]}^{2} 〉 = C_{n}^{2} ρ^{α - 3} .

(1)

Gerçekcioğlu and Baykal’s calculations for

{\tilde{C}}_{n}^{2} = 10^{- 15} m^{3 - α}

correspond to the different random fields of the refractive index, having one thing in common:

D_{n} (ρ = 1 m) = 10^{- 15}

for all

α

. On the other hand, our calculations for

{\tilde{C}}_{n}^{2} = 4.64 \cdot 10^{- 17} {cm}^{3 - α}

maintain

D_{n} (ρ = 1 cm) = 4.64 \cdot 10^{- 17}

for all

α

, while

D_{n} (ρ = 1 m) = 10^{- 15}

only for

α = 11 / 3

. Simple calculation shows that in our case

{\tilde{C}}_{n}^{2} = 4.64 \cdot 10^{- 23 + 2 α} m^{3 - α}

for other values of

α

. The steep growth of the numeric values of

{\tilde{C}}_{n}^{2}

measured in

m^{3 - α}

with

α

is responsible for the shift of the maxima in Fig. 1. Our choice of

{\tilde{C}}_{n}^{2}

is not better or worse than the choice of Gerçekcioğlu and Baykal, because in both cases the points separation

ρ

is dictated by tradition, and has nothing to do with the propagation physics. The fixed numerical value of

{\tilde{C}}_{n}^{2}

inevitably introduces an additional scale that is not related to the natural physical scales existing in the problem.

Avoiding the dependence of the calculations results on the choice of the units of length is a relatively straightforward task. It is sufficient to use

{\tilde{C}}_{n}^{2}

not as a standalone parameter, but in combination with other dimensional parameters of the problem. This combination should have the physical dimension that is independent on

α

, in particular it can be dimensionless. One example of such combination is the plane wave coherence radius [2

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

]

r_{C} = {(A (α) \frac{π^{2} Γ (2 - \frac{α}{2}) 2^{3 - α}}{Γ (\frac{α}{2}) (α - 2)} {\tilde{C}}_{n}^{2} k^{2} L)}^{\frac{1}{2 - α}},

(2)

having the dimension of length. Here and further on we use the original notations of [1

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]

] for consistency. Other options are the dimensionless scintillation indices for plane or spherical wave [2

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

]:

m_{PW}^{2} = - A (α) π^{2} Γ (1 - \frac{α}{2}) \sin \frac{π α}{4} \frac{2}{α} {\tilde{C}}_{n}^{2} k^{3 - \frac{α}{2}} L^{\frac{α}{2}}, m_{SW}^{2} = - A (α) π^{2} Γ (1 - \frac{α}{2}) \sin \frac{π α}{4} B (\frac{α}{2}, \frac{α}{2}) {\tilde{C}}_{n}^{2} k^{3 - \frac{α}{2}} L^{\frac{α}{2}} .

(3)

It is also possible to use the fixed RMS difference of the refraction index at some distance that naturally emerges in the propagation process, say Fresnel zone

\sqrt{L / k}

. In this case, in order to keep

D_{n} (\sqrt{L / k}) = const

then we have the following equation for

{\tilde{C}}_{n}^{2}

dependence on

α

:

{\tilde{C}}_{n}^{2} {(\frac{L}{k})}^{\frac{α - 3}{2}} = const,

(4)

where the dimensionless right-hand part is determined by the reference values of the path length, wavelength, and structure constant, e.g., for

α = 11 / 3

.

Figure 2 shows dependence of the scintillation index on

α

for

W_{0} = 2 cm

. Four different constraints on the structure function represented by Eqs. (2)–(4) are used. The values of constraints are chosen so that for

λ = 1.55 μm

,

L = 2 km

, and

α = 11 / 3

, the structure constant

{\tilde{C}}_{n}^{2} = 10^{- 15} m^{- 2 / 3}

. This warrants the intersection of all four curves at

α = 11 / 3

. The advantage of these four presentations of the scintillation index variation with

α

in comparison to Fig. 6 of [1

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]

] or Fig. 1 is that the choice of the length unit does not affect the chart. This is trivial for the last three cases, Eqs. (3) and (4), but no matter if

r_{C} = 6.5 cm

or

r_{C} = 0.065 m

, the heavy solid curve in Fig. 2 does not change. That was not the case when the numeric value of

{\tilde{C}}_{n}^{2}

was used as a fixed parameter.

Fig. 2. Dependence of scintillation index on

α

for

W_{0} = 2 cm

. Heavy solid: fixed coherent radius; long-dashed: fixed spherical wave scintillation index; short-dashed: fixed plane wave scintillation index; and thin solid: fixed

D_{n} (\sqrt{L / k})

.

Figure 2 brings up new questions. Clearly, the dependence

m^{2} (α)

is very different when different

{\tilde{C}}_{n}^{2}

related parameters are constant. There is no definite answer to a question that was discussed in many papers lately, including [1

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]

]: how exponent

α

affects the beam scintillation. As Fig. 2 suggests, deviation from Kolmogorov

α = 11 / 3

in any direction can cause growth or lessening of scintillation depending on what quantity is kept fixed. The correct formulation of the question has to include the definition of a parameter that remains unchanged, and this parameter has to be dimensionless, or have a dimension independent on

α

.

2. PROPER RANGE OF THE SPECTRAL EXPONENT $α$

Gerçekcioğlu and Baykal’s claim that their analysis is valid for non-Kolmogorov turbulence with spectral exponent

3 < α < 5

is erroneous, and should be limited to the

3 < α < 4

range.

The relation between the refractive index structure function and the three-dimensional spectrum [3

V. I. Tatarskii, Wave Propagation in Random Media (McGraw-Hill, 1961).

],

D_{n} (r⃗) = {\tilde{C}}_{n}^{2} r^{3 - α} = 2 A (α) {\tilde{C}}_{n}^{2} ∭ d^{3} K K^{- α} [1 - \cos (r⃗ \cdot K⃗)] = 8 π A (α) {\tilde{C}}_{n}^{2} \int_{0}^{\infty} d K K^{2 - α} [1 - \frac{\sin (r K)}{r K}],

(5)

is indeed valid for

3 < α < 5

. The critical values of

α = 5

and

α = 3

correspond to the convergence restrictions at the low and high wavenumbers accordingly in the last integral in (5). The same limits on

α

are also evident in the equation for the coefficient

A (α)

as zero-crossing points.

The propagation model used by Gerçekcioğlu and Baykal (and almost anybody else lately) is based on the Markov approximation [4

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation through Random Media (Springer, 1988).

], when refractive index fluctuations are assumed to be delta-correlated in the propagation direction, e.g.,

z

. In terms of the spectral density, the Markov approximation implies that

Φ_{n} (K⃗) = Φ_{n} (K_{x}, K_{y}, K_{z}) \approx Φ_{n} (K_{x}, K_{y}, 0)

(6)

for calculations of the statistical moment of the propagating field. In particular, the typical spherical wave structure function entering equation for the average irradiance derived with the help of the Markov approximation [4

S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation through Random Media (Springer, 1988).

] is

D (r⃗) = 4 π k^{2} \int_{0}^{L} d z \iint d^{2} κ Φ_{n} (κ⃗, 0) [1 - \cos [κ⃗ \cdot r⃗ (1 - \frac{x}{L})]] = 8 π^{2} \frac{A (α)}{(α - 1)} [\int_{0}^{\infty} d p p^{1 - α} (1 - J_{0} (p))] {\tilde{C}}_{n}^{2} k^{2} L r^{α - 2} = 8 π^{2} \frac{A (α)}{(α - 1)} \frac{2^{2 - α} Γ (\frac{4 - α}{2})}{(α - 2) Γ (\frac{α}{2})} {\tilde{C}}_{n}^{2} k^{2} L r^{α - 2},

(7)

where we use a shortcut notation

κ⃗ \equiv (K_{x}, K_{y})

for two-dimensional wave vectors. The last integral in (7) converges for

2 < α < 4

only, as also evident from the poles at

α = 2

and

α = 4

in the last formula. Similar integrals, but with more complicated variations of the point separation along the propagation path appear in the rigorous scintillation theory [2

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

] based on the path-integral solution for the fourth moment of the optical field. Consequently, restriction

2 < α < 4

should be considered as essential for the use of the Markov approximation.

Equations (5) and (7) have two interesting implications:

• There are legitimate $O (r^{α - 2}) 3 < α < 4$ wave structure functions for $2 < α < 3$ that cannot occur as a result of propagation through isotropic random medium, but can exist in some special classes of anisotropic turbulence or produced by specially manufactured phase screens.
• There are legitimate random media with $4 < α < 5$ where Markov approximation cannot be used as a short-wave propagation model. This can be attributed to the abundance of the large-scale eddies in the turbulent spectrum for large $α$ , which invalidates the assumptions underlining the Markov approximation.

Considering that [1

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]

] is concerned with propagation through turbulence, the discussion should be limited by the

3 < α < 4

range of spectral exponents.

As a final remark, we consider the well-known perturbation theory result for the spherical wave scintillation index [2

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

]:

m^{2} = 4 π k^{2} \int_{0}^{L} d z \iint d^{2} κ Φ_{n} (κ⃗, 0) {1 - \cos [\frac{κ^{2} z}{k} (1 - \frac{z}{L})]} = 8 π^{2} A (α) B (\frac{α}{2}, \frac{α}{2}) [\int_{0}^{\infty} d p p^{1 - α} (1 - \cos p^{2})] {\tilde{C}}_{n}^{2} k^{3 - α / 2} L^{α / 2} .

(8)

Convergence of the last integral in (8) for power-law spectra requires

2 < α < 6

, and the same is true for the plane wave scintillation index.

The perturbation theory result for the scintillation index of collimated Gaussian beam [2

M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]

] in isotropic turbulence has the following form:

m^{2} = 8 π^{2} k^{2} \int_{0}^{L} d z \int_{0}^{\infty} κ d κ Φ_{n} (κ) \exp [- \frac{κ^{2} a^{2}}{1 + Ω^{2}} {(1 - \frac{z}{L})}^{2}] [1 - \cos (\frac{L Ω^{2} + z}{L + L Ω^{2}} \frac{κ^{2} (L - z)}{k})],

(9)

where

Ω = k W_{0}^{2} / L

is the Fresnel number for the initial beam width. Low-pass filter associated with the finite beam width removes the limitation on the convergence for power-law spectra at

κ \to \infty

, but

κ \to 0

limit still requires

α < 6

. This restriction is still valid for the flat-topped beams considered in [1

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]

].

Equations (7)–(9) reveal that constrains on the spectral exponent imposed by the formal convergence for the integral representation of the wave statistics can be more relaxed than ones enacted by the statistics of the turbulence itself and the Markov approximation. Practical importance of this observation for the general field of wave propagation in random media is outside of the scope of these comments.

REFERENCES

1.	H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]
2.	M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]
3.	V. I. Tatarskii, Wave Propagation in Random Media (McGraw-Hill, 1961).
4.	S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation through Random Media (Springer, 1988).

OCIS Codes
(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
(010.7060) Atmospheric and oceanic optics : Turbulence
(030.7060) Coherence and statistical optics : Turbulence
(290.5930) Scattering : Scintillation

ToC Category:
Atmospheric and Oceanic Optics

History
Original Manuscript: February 9, 2012
Revised Manuscript: April 19, 2012
Manuscript Accepted: April 30, 2012
Published: August 9, 2012

Citation
Mikhail Charnotskii, "Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence: comment," J. Opt. Soc. Am. A 29, 1838-1840 (2012)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-9-1838

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References

H. Gerçekcioğlu and Y. Baykal, “Intensity fluctuations of flat-topped beam in non-Kolmogorov weak turbulence,” J. Opt. Soc. Am. A 29, 169–173 (2012). [CrossRef]
M. I. Charnotskii, “Asymptotic analysis of finite-beam scintillation in a turbulent medium,” Waves Random Media 4, 243–273 (1994). [CrossRef]
V. I. Tatarskii, Wave Propagation in Random Media (McGraw-Hill, 1961).
S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics. 4. Wave Propagation through Random Media (Springer, 1988).

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