Abstract

A generally applicable and computationally efficient description of random irradiance fluctuations induced by single scattering from distributed low-order turbulence (LOT) phase fluctuations is developed for Gaussian beams in the weak scintillation regime. The LOT solution describes irradiance statistics resulting from coarse beam irradiance fluctuations such as beam wander and beam breathing and will generally underestimate the true scintillation owing to the neglect of higher orders. For a subset of beam and turbulence settings that naturally result in non-log-normal irradiance behavior in the weak regime, the LOT solution closely approaches the exact solution and accurately describes the irradiance statistics for any point on the observation plane. For the same settings, beam-wave scintillation theory derived from the Rytov perturbation method yields inaccurate predictions owing to an inherent confinement to log-normal behavior. Examples that naturally exhibit non-log-normal irradiance behavior include focused beams on horizontal paths and collimated beams on ground-to-space paths. The complementary nature of the two scintillation theories (LOT and Rytov) enables a hybrid combination that yields accurate and convenient scintillation predictions for any case exhibiting weak scintillation regardless of irradiance behavior. Comparison of hybrid model predictions with wave optics simulation data reveals excellent agreement.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2005

G. J. Baker and R. S. Benson, "Gaussian beam weak scintillation on ground to space paths; compact descriptions and Rytov method applicability," Opt. Eng. (Bellingham) 44, 106002 (2005).
[CrossRef]

2004

2000

1998

1995

1994

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

P. A. Lightsey, "Scintillation in ground-to-space and retroreflected laser beams," Opt. Eng. (Bellingham) 33, 2535-2543 (1994).
[CrossRef]

W. B. Miller, J. C. Ricklin, and L. C. Andrews, "Effects of the refractive index spectral model on the irradiance variance of a Gaussian beam," J. Opt. Soc. Am. A 11, 2719-2726 (1994).
[CrossRef]

1993

1990

1981

1977

1976

1975

V. L. Mironov, G. Ya. Patrushev, V. V. Pokasov, and L. I. Shchavlev, "Measurements of intensity fluctuations in light beams having angular diversity," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18, 450-452 (1975).

R. L. Fante, "Electromagnetic beam propagation in turbulent media," Proc. IEEE 63, 1669-1692 (1975).
[CrossRef]

1974

1973

1972

J. R. Kerr and R. Eiss, "Transmitter-size and focus effects on scintillation," J. Opt. Soc. Am. 62, 682-684 (1972).
[CrossRef]

V. I. Klyatskin and A. I. Kon, "On the displacement of spatially bounded light beams in a turbulent medium in the Markovian random process approximation," Radiophys. Quantum Electron. 15, 1056-1061 (1972).
[CrossRef]

1971

1970

V. I. Klyatskin and V. I. Tatarski, "The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities," Sov. Phys. JETP 31, 335-339 (1970).

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970).
[CrossRef]

1969

J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt. 8, 1687-1693 (1969).
[CrossRef] [PubMed]

V. I. Tatarski, "Light propagation in a medium with random refractive index inhomogeneities in the Markov approximation," Sov. Phys. JETP 29, 1133-1138 (1969).

1968

J. W. Strohbehn, "Line of sight wave propagation through the turbulent atmosphere," Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

1967

R. A. Schmeltzer, "Means, variances, and covariances for laser beam propagation through a random medium," Q. Appl. Math. 24, 339-354 (1967).

D. L. Fried and J. B. Seidman, "Laser-beam scintillation in the atmosphere," J. Opt. Soc. Am. 57, 181-185 (1967).
[CrossRef]

D. L. Fried, "Scintillation of a ground-to-space laser illuminator," J. Opt. Soc. Am. 57, 980-983 (1967).
[CrossRef]

M. E. Gracheva, "Research into the statistical properties of the strong fluctuations of light when propagated in the lower layer of the atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10, 775-787 (1967).

1965

M. E. Gracheva, and A. S. Gurvich, "On strong fluctuations of the intensity of light when propagating in the lower layer of the atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 8, 717-724 (1965).

Andrews, L. C.

Arnaud, J. A.

Baker, G. J.

G. J. Baker and R. S. Benson, "Gaussian beam weak scintillation on ground to space paths; compact descriptions and Rytov method applicability," Opt. Eng. (Bellingham) 44, 106002 (2005).
[CrossRef]

G. J. Baker and R. S. Benson, "Laser scintillation on ground to space paths: a hybrid model for beam wander effects," in Proceedings of AMOS Technical Conference (AFRL/Det 15, 2004), pp. 205-212.

G. J. Baker and R. S. Benson, "Gaussian beam scintillation on ground-to-space paths: The importance of beam wander," in Free Space Laser Communications IV, J.C.Ricklin and D.G.Voelz, eds., Proc. SPIE 5550, 225-235 (2004).

G. J. Baker, "Gaussian beam weak scintillation: A tour of the D1 region," in Atmospheric Propagation II, C.Y.Young and G.C.Gilbreath eds., Proc. SPIE 5793, 17-27 (2005).

Banakh, V. A.

V. A. Banakh, G. M. Krekov, and V. L. Mironov, "Intensity variance and spatial correlation of the intensity of wave beams propagating in a turbulent atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 17, 252-260 (1974).

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused laser beam scintillation in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).
[CrossRef]

V. A. Banakh and I. N. Smalikho, "Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere," Atmos. Oceanic. Opt. 5, 233-237 (1992).

Belmonte, A.

Benson, R. S.

G. J. Baker and R. S. Benson, "Gaussian beam weak scintillation on ground to space paths; compact descriptions and Rytov method applicability," Opt. Eng. (Bellingham) 44, 106002 (2005).
[CrossRef]

G. J. Baker and R. S. Benson, "Laser scintillation on ground to space paths: a hybrid model for beam wander effects," in Proceedings of AMOS Technical Conference (AFRL/Det 15, 2004), pp. 205-212.

G. J. Baker and R. S. Benson, "Gaussian beam scintillation on ground-to-space paths: The importance of beam wander," in Free Space Laser Communications IV, J.C.Ricklin and D.G.Voelz, eds., Proc. SPIE 5550, 225-235 (2004).

Charnotskii, M. I.

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

Churnside, J. H.

Cochetti, F.

Comeron, A.

Consortini, A.

Dios, F.

Dunphy, J. R.

Eiss, R.

Fante, R. L.

R. L. Fante, "Electromagnetic beam propagation in turbulent media," Proc. IEEE 63, 1669-1692 (1975).
[CrossRef]

Frehlich, R.

Fried, D. L.

Gavel, D. T.

E. M. Johansson and D. T. Gavel, "Simulation of stellar speckle imaging," in Amplitude and Intensity Spatial Interferometry II, J.B.Breckenridge, ed., Proc. SPIE2200, 372-383 (1994).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

J. W. Goodman, Statistical Optics (Wiley, 1985).

Gracheva, M. E.

M. E. Gracheva, "Research into the statistical properties of the strong fluctuations of light when propagated in the lower layer of the atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 10, 775-787 (1967).

M. E. Gracheva, and A. S. Gurvich, "On strong fluctuations of the intensity of light when propagating in the lower layer of the atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 8, 717-724 (1965).

Gurvich, A. S.

M. E. Gracheva, and A. S. Gurvich, "On strong fluctuations of the intensity of light when propagating in the lower layer of the atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 8, 717-724 (1965).

Herman, B. J.

B. J. Herman and L. A. Strugala, "Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence," in Propagation of High-Energy Laser Beams through the Earth's Atmosphere, Proc. SPIE 1221, 183-192 (1990).

Hill, R. J.

Hopen, C. Y.

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE, 2001).
[CrossRef]

Ishimaru, A.

A. Ishimaru, "The beam wave case and remote sensing," in Laser Beam Propagation in the Atmosphere, J.W.Strohbehn, ed. (Springer-Verlag, 1978), Chap. 5.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

Johansson, E. M.

E. M. Johansson and D. T. Gavel, "Simulation of stellar speckle imaging," in Amplitude and Intensity Spatial Interferometry II, J.B.Breckenridge, ed., Proc. SPIE2200, 372-383 (1994).

Johnston, R. A.

Jüngling, R.

R. Jüngling, "Simulation gerichteter ausbreitung optischer wellen in turbulenter atmosphäre," Diploma thesis (Institute for Nuclear Theory, Westphalien Wilhelms University, 2001), http://lmb.informatik.uni-freiburg.de/people/juengling/pubs/.

Kerr, J. R.

Khmelevtsov, S. S.

Klyatskin, V. I.

V. I. Klyatskin and A. I. Kon, "On the displacement of spatially bounded light beams in a turbulent medium in the Markovian random process approximation," Radiophys. Quantum Electron. 15, 1056-1061 (1972).
[CrossRef]

V. I. Klyatskin and V. I. Tatarski, "The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities," Sov. Phys. JETP 31, 335-339 (1970).

Kogelnik, H.

Kon, A. I.

V. I. Klyatskin and A. I. Kon, "On the displacement of spatially bounded light beams in a turbulent medium in the Markovian random process approximation," Radiophys. Quantum Electron. 15, 1056-1061 (1972).
[CrossRef]

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarski, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Krekov, G. M.

V. A. Banakh, G. M. Krekov, and V. L. Mironov, "Intensity variance and spatial correlation of the intensity of wave beams propagating in a turbulent atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 17, 252-260 (1974).

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused laser beam scintillation in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).
[CrossRef]

Lane, R. G.

Lataitis, R. J.

Lawrence, R. S.

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970).
[CrossRef]

Lightsey, P. A.

P. A. Lightsey, "Scintillation in ground-to-space and retroreflected laser beams," Opt. Eng. (Bellingham) 33, 2535-2543 (1994).
[CrossRef]

Lutomirski, R.

Miller, W. B.

Mironov, V. L.

V. L. Mironov, G. Ya. Patrushev, V. V. Pokasov, and L. I. Shchavlev, "Measurements of intensity fluctuations in light beams having angular diversity," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18, 450-452 (1975).

V. A. Banakh, G. M. Krekov, and V. L. Mironov, "Intensity variance and spatial correlation of the intensity of wave beams propagating in a turbulent atmosphere," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 17, 252-260 (1974).

V. A. Banakh, G. M. Krekov, V. L. Mironov, S. S. Khmelevtsov, and R. Sh. Tsvik, "Focused laser beam scintillation in the turbulent atmosphere," J. Opt. Soc. Am. 64, 516-518 (1974).
[CrossRef]

Nichelatti, E.

Noll, R. J.

Obukhov, A. M.

A. M. Obukhov, "Effect of weak inhomogeneities in the atmosphere on sound and light propagation," Izv. Akad. Nauk Ser. Geofiz. 2, 155-165 (1953).

Patrushev, G. Ya.

V. L. Mironov, G. Ya. Patrushev, V. V. Pokasov, and L. I. Shchavlev, "Measurements of intensity fluctuations in light beams having angular diversity," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18, 450-452 (1975).

Phillips, R. L.

Pokasov, V. V.

V. L. Mironov, G. Ya. Patrushev, V. V. Pokasov, and L. I. Shchavlev, "Measurements of intensity fluctuations in light beams having angular diversity," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18, 450-452 (1975).

Possi, G.

Ricklin, J. C.

Rodriguez, A.

Roggermann, M. C.

M. C. Roggermann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).

Rubio, J. A.

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarski, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Schmeltzer, R. A.

R. A. Schmeltzer, "Means, variances, and covariances for laser beam propagation through a random medium," Q. Appl. Math. 24, 339-354 (1967).

Seidman, J. B.

Shchavlev, L. I.

V. L. Mironov, G. Ya. Patrushev, V. V. Pokasov, and L. I. Shchavlev, "Measurements of intensity fluctuations in light beams having angular diversity," Izv. Vyssh. Uchebn. Zaved., Radiofiz. 18, 450-452 (1975).

Shelton, J. D.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Smalikho, I. N.

V. A. Banakh and I. N. Smalikho, "Laser beam propagation along extended vertical and slant paths in the turbulent atmosphere," Atmos. Oceanic. Opt. 5, 233-237 (1992).

Strohbehn, J. W.

R. S. Lawrence and J. W. Strohbehn, "A survey of clear-air propagation effects relevant to optical communications," Proc. IEEE 58, 1523-1545 (1970).
[CrossRef]

J. W. Strohbehn, "Line of sight wave propagation through the turbulent atmosphere," Proc. IEEE 56, 1301-1318 (1968).
[CrossRef]

Strugala, L. A.

B. J. Herman and L. A. Strugala, "Method for inclusion of low-frequency contributions in numerical representation of atmospheric turbulence," in Propagation of High-Energy Laser Beams through the Earth's Atmosphere, Proc. SPIE 1221, 183-192 (1990).

Tatarski, V. I.

V. I. Klyatskin and V. I. Tatarski, "The parabolic equation approximation for propagation of waves in a medium with random inhomogeneities," Sov. Phys. JETP 31, 335-339 (1970).

V. I. Tatarski, "Light propagation in a medium with random refractive index inhomogeneities in the Markov approximation," Sov. Phys. JETP 29, 1133-1138 (1969).

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarski, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961).

Titterton, P. J.

Tsvik, R. Sh.

Welsh, B.

M. C. Roggermann and B. Welsh, Imaging through Turbulence (CRC Press, 1996).

Yu, P. T.

Yura, H.

Appl. Opt.

R. Lutomirski and H. Yura, "Propagation of a finite optical beam in an inhomogeneous medium," Appl. Opt. 10, 1652-1658 (1971).
[CrossRef] [PubMed]

L. C. Andrews, R. L. Phillips, and P. T. Yu, "Optical scintillation and fade statistics for a satellite-communications system," Appl. Opt. 34, 7742-7751 (1995) L. C. Andrews, R. L. Phillips, and P. T. Yu,[errata, Appl. Opt. 36, S.6068 (1997)].
[CrossRef] [PubMed]

J. R. Dunphy and J. R. Kerr, "Turbulence effects on target illumination by laser sources: phenomenological analysis and experimental results," Appl. Opt. 16, 1345-1358 (1977).
[CrossRef] [PubMed]

F. Dios, J. A. Rubio, A. Rodriguez, and A. Comeron, "Scintillation and beam-wander analysis in an optical ground station-satellite uplink," Appl. Opt. 43, 3866-3873 (2004).
[CrossRef] [PubMed]

P. J. Titterton, "Power reduction and fluctuations caused by narrow laser beam motion in far field," Appl. Opt. 12, 423-425 (1973).
[CrossRef] [PubMed]

J. H. Churnside and R. J. Lataitis, "Wander of an optical beam in the turbulent atmosphere," Appl. Opt. 29, 926-930 (1990).
[CrossRef] [PubMed]

D. L. Fried, "Statistics of laser beam fade induced by pointing jitter," Appl. Opt. 12, 422-423 (1973).
[CrossRef] [PubMed]

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Figures (9)

Fig. 1
Fig. 1

Regions of distinct on-axis weak scintillation behavior exhibited by Gaussian beams for a fixed path-integrated turbulence strength, as a function of the Fresnel numbers N τ and N L . The labeling for the behavior regions S 1 , P 1 , F 1 , G 1 , and D 1 is adapted from Fig. 4 in Ref. [1]. Certain propagation scenario sets, where the initial beam diameter d 0 is the free parameter, are depicted by the dashed and dotted–dashed lines. The phase screen approximation for a propagated wavefront [Eq. (57)] is valid in the region below and to the right of the dotted line.

Fig. 2
Fig. 2

Irradiance PDFs derived from wave optics simulation data (circles) consisting of 3000 independent realizations. Error bars are not shown but can be inferred from the vertical scatter in the data. Also shown are the full LOT PDF [Eq. (32)] (heavy solid curve), the LOT beam-wander PDF [Eq. (34)] (thin solid curve), and the log-normal PDF [expression (37)] (dotted–dashed curve) with an irradiance variance and mean equal to that of the wave optics data. The dashed curve in (c) is the LOT PDF with the two astigmatism modes removed. The vertical bars at the bottom margin mark the vacuum irradiance value. (a) and (b) depict horizontal-path focused (HZF) beams (case A in Table 1), and (c) depicts slant-path quasi-focused beams (case B in Table 1); all three cases are in the D 1 region.

Fig. 3
Fig. 3

Irradiance PDF data for two slant-path quasi-focused beams (case B in Table 1) that are close to the D 1 region boundary. All notations and line styles are identical to that in Fig. 2.

Fig. 4
Fig. 4

Scintillation index as a function of initial beam diameter d 0 for horizontal-path (HZ) collimated beams (case C in Table 1). Wave optics simulation data and estimated error bars are shown for the observation plane on-axis point (circles) and the diffractive beam edge, r = w (diamonds). Also shown are predictions from Rytov scintillation theory [Eq. (A8)] (dotted–dashed curves) and the LOT solution [Eq. (31)] (solid curves). The wave optics data consisted of 300 independent realizations.

Fig. 5
Fig. 5

Irradiance PDF data for a horizontal-path focused beam (case A in Table 1) that exhibited strong scintillation. All notations and line styles are identical to that in Fig. 2.

Fig. 6
Fig. 6

(a)–(c) Scintillation index as a function of initial beam diameter d 0 for beams from cases A, B, and D in Table 1, respectively. Wave optics data are shown for the on-axis point (circles) and the diffractive beam edge (diamonds). Also shown are predictions from Rytov scintillation theory [Eq. (A8)] (dotted–dashed curves), the full LOT solution [Eq. (31)] (thin solid curves), and the hybrid model [Eq. (76)] (heavy curves). Predictions from the rigorous curve fit [expression (B1)] (crosses) are shown where applicable. Chart (b) includes the LOT beam-wander prediction [Eq. (40)] (dashed curve) for the on-axis point only. The horizontal bar near the top margin of each chart depicts the range of d 0 values [expressions (65)] where the LOT solution theoretically equals the exact solution. Data symbols highlighted by a square box have irradiance PDF data presented in Figs. 2, 3, 5, 9. G2S, ground to space.

Fig. 7
Fig. 7

Scintillation index as a function of initial beam diameter d 0 for the cases shown in Figs. 6a, 6c, magnified to highlight the D 1 region. Wave optics and hybrid model data are shown using the same styles in Fig. 6. Also shown is the LOT beam-wander prediction [Eq. (40)] (dashed curves).

Fig. 8
Fig. 8

Scintillation index as a function of normalized off-axis distance r w for a slant-path quasi-focused beam (case B in Table 1) in the D 1 region. Shown are wave optics data (circles), the full LOT solution [Eq. (31)] (heavy curve), and Rytov scintillation theory [Eq. (A8)] (dotted–dashed curve).

Fig. 9
Fig. 9

Irradiance PDF data for a slant-path quasi-focused beam (case B in Table 1) outside the D 1 region. All notations and line styles are identical to that in Fig. 2.

Tables (1)

Tables Icon

Table 1 Summary of the Beam Propagation Scenarios Considered in this Paper a

Equations (143)

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N L = ( k w 0 2 2 L ) 1 + κ 0 L ,
N τ = ( k w 0 2 2 z τ ) 1 + κ 0 z τ ,
z τ = 0 L z 1 + κ 0 z 5 3 C n 2 ( z ) d z 0 L 1 + κ 0 z 5 3 C n 2 ( z ) d z ,
N τ 2 1 , N L < 1 ,
U ( x , z ̃ ) = 2 π 1 w ̃ exp ( x 2 w ̃ 2 + j k 1 2 κ ̃ x 2 ) ,
U ( x , z ̃ + Δ z ) = U ( x , z ̃ ) exp ( j k z ̃ z ̃ + Δ z δ n ( x , ζ ) d ζ ) .
k z ̃ z ̃ + Δ z δ n ( x , ζ ) d ζ = k [ θ ̃ x x + θ ̃ y y + Δ κ ̃ 1 2 ( x 2 1 2 w ̃ 2 ) + c ̃ 5 1 2 ( x 2 y 2 ) + c ̃ 6 x y + O ( x w ̃ 3 ) ] .
σ θ ̃ { x , y } 2 = 2.41 w ̃ 1 3 C ̃ n 2 Δ z ,
σ Δ κ ̃ 2 = 1.48 w ̃ 7 3 C ̃ n 2 Δ z ,
σ c ̃ { 5 , 6 } 2 = 0.74 w ̃ 7 3 C ̃ n 2 Δ z .
I ( x , L ) = 2 π 1 w x w y exp ( 2 w x 2 [ ( x δ x ) cos ω ̃ + ( y δ y ) sin ω ̃ ] 2 ) × exp ( 2 w y 2 [ ( x δ x ) sin ω ̃ ( y δ y ) cos ω ̃ ] 2 ) .
δ = ( L z ̃ ) θ ̃ ,
σ δ { x , y } 2 = ( L z ̃ ) 2 σ θ ̃ { x , y } 2 .
w { x , y } = w ̃ { [ 1 + ( κ ̃ + Δ κ ̃ ± c ̃ 5 2 + c ̃ 6 2 ) ( L z ̃ ) ] 2 + [ 2 ( L z ̃ ) k w ̃ 2 ] 2 } 1 2 ,
w { x , y } 2 = w 2 + w ̃ 2 ( L z ̃ ) 2 ( σ Δ κ ̃ 2 + σ c ̃ 5 2 + σ c ̃ 6 2 ) ± μ c ̃ 5 2 + c ̃ 6 2 ,
w = w 0 [ ( 1 + κ 0 L ) 2 + ( 2 L k w 0 2 ) 2 ] 1 2 .
w ̃ = 1 2 arg ( c ̃ 5 + j c ̃ 6 ) .
w { x , y } = [ cos 2 ω ̃ w { x , y } 2 + sin 2 ω ̃ w { y , x } 2 ] 1 2 .
σ δ { x , y } 2 = Δ z 0 ( L z ̃ ) 2 σ θ ̃ { x , y } 2 = 2.41 L 2 w 0 1 3 0 L C n 2 ( z ) ( 1 z L ) 2 [ w ( z ) w 0 ] 1 3 d z ,
σ δ { x , y } 2 = 0.182 L 2 ( d 0 r q ) 5 3 ( λ d 0 ) 2 = L 2 σ θ { x , y } 2 ,
r q 5 3 = 0.423 k 2 0 L C n 2 ( z ) ( 1 z L ) 2 [ ( 1 + κ 0 z ) 2 + ( 2 z k w 0 2 ) 2 ] 1 6 d z .
σ θ { x , y } 2 = 0.182 ( d 0 r q ) 5 3 ( λ d 0 ) 2 .
w { x , y } 2 = w 2 + Δ z 0 w ̃ 2 ( L z ̃ ) 2 ( σ Δ κ ̃ 2 + σ c ̃ 5 2 + σ c ̃ 6 2 ) = w 2 + L 2 w 0 2 ( σ Δ κ 2 + σ c 5 2 + σ c 6 2 ) ,
σ Δ κ 2 = 0.447 d 0 2 ( d 0 r q ) 5 3 ( λ d 0 ) 2 ,
σ c { 5 , 6 } 2 = 0.224 d 0 2 ( d 0 r q ) 5 3 ( λ d 0 ) 2 .
ϕ ( x ) LOT = k [ θ x x + θ y y + Δ κ 1 2 ( x 2 1 2 w 0 2 ) + c 5 1 2 ( x 2 y 2 ) + c 6 x y ] .
I ( x , L ) = 2 π 1 w x w y exp ( 2 w x 2 [ ( x δ x ) cos ω + ( y δ y ) sin ω ] 2 ) exp ( 2 w y 2 [ ( x δ x ) sin ω ( y δ y ) cos ω ] 2 ) .
δ = L θ ,
w { x , y } = w 0 { [ 1 + ( κ 0 + Δ κ ± c 5 2 + c 6 2 ) L ] 2 + ( 2 L k w 0 2 ) 2 } 1 2 .
ω = 1 2 arg ( c 5 + j c 6 ) .
2 σ θ x 0.85 ( λ d 0 ) ,
d 0 σ Δ κ 0.67 ( λ d 0 ) ,
σ I 2 = I 2 I 2 1 ,
p I ( I ̃ ) = d d I ̃ { Prob I I ̃ } .
I ( x , L ) = 2 π 1 w 2 exp [ 2 ( x δ ) 2 w 2 ] .
p I ( I ; r , α ) = 1 2 α I 0 ( I I 0 ) ( 1 2 α ) 1 1 2 π exp ( r 2 α w 2 ) 0 2 π exp ( 2 ln ( I 0 I ) r cos θ α w ) d θ ,
α ( L ) = δ 2 w 2 .
p I ( I ; 0 , α ) = 1 2 α I 0 ( I I 0 ) ( 1 2 α ) 1 ,
p I ( I ; r , α ) 1 I 2 π σ I 2 exp [ 1 2 σ I 2 ( ln I I + 1 2 σ I 2 ) 2 ] , α < 0.04 r 2 w 2 , r w .
p δ ( δ ) = 1 π δ 2 exp ( δ 2 δ 2 ) .
I ( x ) = I v ( x ) p δ ( x ) = I 0 1 + 2 α exp [ 2 r 2 ( 1 + 2 α ) w 2 ] ,
σ I 2 ( r , α ) = I ν 2 ( x ) p δ ( x ) I ( x ) 2 1 = ( 1 + 2 α ) 2 1 + 4 α exp [ 8 α ( 1 + 4 α ) ( 1 + 2 α ) r 2 w 2 ] 1 ,
α = 0.863 ( 1 + N L 2 ) 1 ( d 0 r q ) 5 3 .
σ I 2 ( 0 , L ) = 4 α 2 1 + 4 α = 2.98 ( 1 + N L 2 ) 2 ( d 0 r q ) 10 3 1 + 3.45 ( 1 + N L 2 ) 1 ( d 0 r q ) 5 3 .
( d 0 r q ) σ I 2 ( 0 ) = 1 = 1.22 ,
( d 0 r q ) σ I 2 ( w ) = 1 = 0.39 .
I ( x , L ) = 2 π 1 η w 2 exp ( 2 x 2 η w 2 ) ,
η = [ ( w + Δ w ) w ] 2 = ( 1 + N L 2 ) 1 [ 1 + ( N L + Δ κ k w 0 2 2 ) 2 ] .
p I ( I ; 0 , β ) = 1 + N L 2 2 2 π β [ ( 1 + N L 2 ) ( I 0 I ) 1 ] I 0 I 2 p = 1 2 exp [ 1 2 β ( ( 1 + N L 2 ) ( I 0 I ) 1 + ( 1 ) p N L ) 2 ] ,
β = ( k w 0 2 2 ) 2 σ Δ κ 2 .
σ I 2 ( 0 ) = 4 I 0 2 β N L 2 , N L > 20 .
p I ( I ; 0 , β ) 1 2 π β ( I 0 I 1 ) I 0 I 2 exp ( I 0 I 1 2 β ) , N L 2 1 .
β = 0.276 ( d 0 r q ) 5 3 .
σ I 2 ( 0 ) = I 2 p I ( I ; 0 , β ) d I [ I p I ( I ; 0 , β ) d I ] 2 1 ,
σ I 2 ( 0 ) 0.218 ( d 0 r q ) 5 3 { 1 + 0.0152 ( d 0 r q ) 20 3 [ 1 + 0.053 ( d 0 r q ) 4 3 ] 4 } 1 4 , N L = 2 .
σ I 2 ( 0 ) 0.0502 ( d 0 r q ) 8 3 [ 1 + 0.426 ( d 0 r q ) 3 ] 1 2 , N L 2 1 .
N τ 2 1 ,
N τ N L ,
U ( x , L ) = 1 j λ L exp ( j k κ 0 x 2 2 γ ) U c ( ξ , 0 ) exp [ j k γ 2 L ( x γ ξ ) 2 ] exp { j k 0 L δ n ( 1 + κ 0 z ξ , z ) d z } d 2 ξ .
U ( x , L ) LOT = 1 j λ L exp ( j k κ 0 x 2 2 γ ) U c ( ξ , 0 ) exp [ j k γ 2 L ( x γ ξ ) 2 ] exp { j k ϕ ( ξ ) LOT } d 2 ξ ,
( π w 0 2 ) 1 ( ϕ ϕ ¯ ) 2 d 2 x ( π w 0 2 ) 1 ( ϕ ϕ ¯ ) LOT 2 d 2 x < ( 2 π 10 ) 2 .
( π w 0 2 ) 1 ( ϕ ϕ ¯ ) 2 d 2 x = 1.029 ( d 0 r 0 , g ) 5 3 ,
r 0 , g 5 3 = 0.423 k 2 0 L C n 2 ( z ) 1 + κ 0 z 5 3 d z .
( π w 0 2 ) 1 ( ϕ ϕ ¯ ) LOT 2 d 2 x = ( 0.448 + 0.448 + 0.023 + 0.023 + 0.023 ) ( d 0 r q ) 5 3 ,
r 0 , s 5 3 = 0.423 k 2 0 L C n 2 ( z ) ( 1 z L ) 5 3 d z ,
0.253 ( d 0 r 0 , s ) 5 6 < 2 π 10 d 0 < 3.0 r 0 , s .
N τ 2 1 , N L < 2 , d 0 < 3.0 r 0 , s .
σ I 2 ( 0 , L ) 2.98 ( d 0 r 0 , s ) 10 3 1 + 3.45 ( d 0 r 0 , s ) 5 3 .
w L T 2 = w 2 + 2 α w 2 = w 2 + 2 δ 2 ,
δ 2 = 2 σ δ { x , y } 2 = 0.364 L 2 ( d 0 r q ) 5 3 ( λ d 0 ) 2 .
w L T 2 = ( 1 + g ) w 2 ,
δ 2 RPM = 1 2 w 2 g 0.368 L 2 ( d 0 r 0 , s ) 5 3 ( λ d 0 ) 2 ( 1 + N L 2 ) 1 6 .
σ I 2 ( 0 , L ) = max { σ I 2 ( 0 , L ) RPM , σ I 2 ( 0 , L ) LOT }
σ I 2 ( 0 , L ) LOT > σ I 2 ( 0 , L ) RPM , within the D 1 region ,
σ I 2 ( 0 , L ) LOT < σ I 2 ( 0 , L ) RPM , in the other four regions .
σ I 2 ( r , L ) LOT 6.90 ( 1 + N L 2 ) 1 ( d 0 r 0 , s ) 5 3 ( r w ) 2 , α < 0.04 ( r w ) 2 .
σ I 2 ( r , L ) RPM 6.06 ( 1 + N L 2 ) 5 6 ( d 0 r 0 , s ) 5 3 ( r w ) 2 , a 1 .
α < 0.04 ( r w ) 2 .
σ I 2 ( r , L ) = { σ I 2 ( r , L ) LOT , σ I 2 ( 0 , L ) LOT > σ I 2 ( 0 , L ) RPM , and α > 0.04 ( r w ) 2 σ I 2 ( r , L ) RPM , otherwise } .
s ( a ) = 1 2 + 1 2 tanh ( 2 a a c a decay ) ,
σ I 2 ( a ) = [ 1 s ( a ) ] σ I 2 ( a ) RPM + s ( a ) σ I 2 ( a ) LOT ,
s ( α ) = 1 2 + 1 2 tanh ( 2 α 0.055 ( r w ) 2 0.06 )
E ( x , z ) = exp [ Ψ 0 ( x , z ) + ϵ Ψ 1 ( x , z ) + ϵ 2 Ψ 2 ( x , z ) + ] ,
δ n ( x , z ) = ϵ q ( x , z ) , q 2 1 .
2 Ψ 0 + Ψ 0 ( Ψ 0 + 2 ϵ Ψ 1 + 2 ϵ 2 Ψ 2 ) + k 2 + ϵ 2 Ψ 1 + ϵ 2 Ψ 1 ( Ψ 1 + 2 ϵ Ψ 2 ) + 2 ϵ k 2 q + ϵ 2 2 Ψ 2 + ϵ 2 k 2 q 2 + = 0 .
2 Ψ 0 + ( Ψ 0 ) 2 = k 2 ,
2 Ψ 1 + 2 Ψ 0 Ψ 1 = 2 k 2 q ,
2 Ψ 2 + 2 Ψ 0 Ψ 2 = k 2 q 2 Ψ 1 Ψ 1 .
Ψ 0 n ̂ i 2 ϵ Ψ 1 n ̂ i , 2 ϵ 2 Ψ 2 n ̂ i i { x , y , z } ,
( Ψ 0 ) 2 ϵ 2 ( Ψ 1 ) 2 , 2 ϵ 3 Ψ 1 Ψ 2 ,
2 Ψ 0 ϵ 2 Ψ 1 , ϵ 2 2 Ψ 2 ,
U ( x , z ) = U 0 ( x δ , z ) ,
U = U 0 exp ( ϵ Ψ 1 ) ,
x Ψ 0 2 ϵ x Ψ 1 ,
ϵ Ψ 1 = ln ( U U 0 ) ,
Ψ 0 = ln ( U 0 ) + j mod 2 π ( k z ) .
1 2 x U U x U 0 U 0 x U 0 U 0 .
1 2 x U U 2 ν x 2 ν x ,
δ r 2 ,
α 0.25 ( r w ) 2 .
U ( x , z ) = η 1 2 U 0 ( η 1 2 x , z ) .
1 2 1 η 1 2 .
η 2 = 1 + 2 β + 3 β 2 16 ,
d 0 r 0 , s < 0.87 .
ϵ Ψ 1 ( x , L ) = 2 k 2 0 L d z G ( [ ( x ξ ) 2 + ( L z ) 2 ] 1 2 ) δ n ( ξ , z ) E 0 ( ξ , z ) E 0 ( x , L ) d 2 ξ .
σ 1 g 2 ( L ) = 2.252 k 7 6 L 5 6 0 L C n 2 ( z ) [ ( 1 z L ) 1 + κ 0 z 1 + κ 0 L ] 5 6 d z .
σ 1 2 ( L ) = 2.252 k 7 6 L 5 6 0 L C n 2 ( z ) ( 1 z L ) 5 6 d z ,
σ 1 s 2 ( L ) = 2.252 k 7 6 L 5 6 0 L C n 2 ( z ) [ ( 1 z L ) z L ] 5 6 d z .
σ 1 g 2 0.941 ( k r 0 , s 2 8 L 1 + κ 0 L ) 5 6 ,
I ¯ ( r , L ) = w 0 2 ( 1 + g ) w 2 exp ( 2 r 2 ( 1 + g ) w 2 ) ,
g ( L ) = 4.35 k 7 6 L 5 6 Λ 5 6 0 L C n 2 ( z ) ( 1 z L ) 5 3 d z = 1.818 ( d 0 r 0 , s ) 5 3 ( 1 + N L 2 ) 5 6 .
Λ = 2 L k w 2 .
σ I 2 ( r , L ) = 8.702 k 2 ( L k ) 5 6 Λ 5 6 0 L C n 2 ( z ) ( 1 z L ) 5 3 { [ 1 + B 2 ( z ) A 2 ( z ) ] 5 12 cos [ 5 6 tan 1 ( B ( z ) A ( z ) ) ] F 1 1 ( 5 6 , 1 , 2 r 2 w 2 ) } d z ,
A ( z ) = Λ ( 1 z L ) , B ( z ) = ϴ + ( 1 ϴ ) z L .
ϴ = 1 + κ 0 L ( 1 + κ 0 L ) 2 + ( 2 L k w 0 2 ) 2 .
N T k w 0 2 2 z T
z T 2 0 L z 2 ( 1 z L ) 1 3 C n 2 ( z ) d z 0 L ( 1 z L ) 5 3 C n 2 ( z ) d z ,
B ( z ) A ( z ) = N L + ( 1 + N L 2 ) N T 1 z z T 1 z L ,
1 B A { N T 1 z z T 1 z L , N L N T 1 , N T 2 1 N L , N L N T 1 , N L 2 1 } .
σ I 2 ( 0 , L ) 0.253 ( d 0 r 0 , s ) 5 3 × { N T 2 , N L N T 1 , N T 2 1 N L 2 , N L N T 1 , N L 2 1 } ,
σ 1 2 < 1 , σ 1 2 Λ 5 6 1 .
σ 1 2 Λ 5 6 ( k r 0 , s 2 8 L ) 5 6 ( 2 L k ) 5 6 ( k d 0 4 L ) 5 3 = ( d 0 r 0 , s ) 5 3 1 .
σ I 2 ( 0 , L )
1 + ( d 0 r 0 , s ) 2 3 [ 2.4 + 17.8 ( d 0 r 0 , s ) 5 3 ] 1 + ( d 0 r 0 , s ) 7 3 [ 53.2 1 + 0.609 ( d 0 r 0 , s ) 5 3 + 7.62 ( d 0 r 0 , s ) 10 3 ] .
2 E ( x , z ) + k 2 [ 1 + δ n ( x , z ) ] 2 E ( x , z ) = 0 , δ n 1 ,
Φ n ( κ ) = 0.033 C n 2 κ ̱ 11 3 ,
x 2 U ( x , z ) + 2 j k U ( x , z ) z + 2 k 2 δ n ( x , z ) U ( x , z ) = 0 ,
U ( x , z ) = 2 π 1 w ( z ) exp ( x 2 w 2 ( z ) + j k 1 2 κ ( z ) x 2 ) ,
w ( z ) = w 0 [ ( 1 + κ 0 z ) 2 + ( 2 z k w 0 2 ) 2 ] 1 2 ,
a { 2 , 3 } 2 = 0.448 ( D r 0 ) 5 3 tilt ,
a 4 2 = 0.023 ( D r 0 ) 5 3 defocus ,
a { 5 , 6 } 2 = 0.023 ( D r 0 ) 5 3 astigmatism .
k 1 ϕ ( x ) = c 1 + c 2 x + c 3 y + c 4 1 2 ( x 2 1 8 D 2 ) + c 5 1 2 ( x 2 y 2 ) + c 6 x y + O ( 2 x D 3 ) .
c { 2 , 3 } 2 = 4 π 2 ( λ D ) 2 a { 2 , 3 } 2 ,
c 4 2 = 192 π 2 D 2 ( λ D ) 2 a 4 2 ,
c { 5 , 6 } 2 = 96 π 2 D 2 ( λ D ) 2 a { 5 , 6 } 2 .
r 0 5 3 = 0.423 k 2 C n 2 Δ z .
U ( x , z ) = 2 π ( w x w y ) 1 2 exp ( x 2 w x 2 y 2 w y 2 ) exp ( j k 1 2 κ x x 2 + j k 1 2 κ y y 2 ) ,
w x ( z + Δ z ) = w x ( z ) { [ 1 + κ x ( z ) Δ z ] 2 + [ 2 Δ z k w x 2 ( z ) ] 2 } 1 2 .
Δ ϕ ( x ) = 1 2 k ( a 00 r 2 cos 2 θ + a 45 r 2 sin 2 θ ) ,
I ( x , z + Δ z ) = 2 π 1 w x w y exp ( 2 ( x cos ω + y sin ω ) 2 w x 2 2 ( x sin ω y cos ω ) 2 w y 2 ) ,
ω { x , y } ( z + Δ z ) = w ( z ) { { 1 + [ κ ( z ) ± a ] Δ z } 2 + [ 2 Δ z k w 2 ( z ) ] } 1 2 .
ω = 1 2 arg ( a 00 + j a 45 ) , a = a 00 2 + a 45 2 .

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