Abstract

We develop the spatial structure function (SSF) of the phase of an optical beam propagated through an aero-optically active turbulent boundary layer at a flying aircraft, which is characterized by anisotropic spatial fluctuations that arise from pressure fluctuations and are highly non-Kolmogoroff. We use the phase SSF to calculate in closed form the spatial statistics of a monochromatic Gaussian beam transmitted through such a turbulent layer and subsequently propagated in free space, discussing in particular the correlations of its amplitude and intensity fluctuations to arbitrary orders.

© 2017 Optical Society of America

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References

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  1. S. Prasad, “Extended Taylor frozen-flow hypothesis and statistics of optical phase in aero-optics,” J. Opt. Soc. Am. A 34, 931–942 (2017).
    [Crossref]
  2. M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012).
    [Crossref]
  3. A. Smits, B. McKeon, and I. Marusic, “High-Reynolds number wall turbulence,” Annu. Rev. Fluid Mech. 43, 353–375 (2011).
    [Crossref]
  4. M. Kemnetz and S. Gordeyev, “Optical investigations of large-scale boundary-layer structures,” 54th AIAA Aerospace Sciences Meeting, San Diego, California, 4–8 January, 2016, AIAA paper 2016-1460.
  5. V. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. Silverman (McGraw-Hill, 1961).
  6. L. Andrews and R. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 3.
  7. E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The airborne aero-optics laboratory, AAOL,” Proc. SPIE 8395, 839507 (2012).
    [Crossref]
  8. E. Jumper, S. Gordeyev, S. Cavalieri, P. Rollins, M. Whiteley, and M. Krizo, “Airborne aero-optics laboratory—transonic (AAOL-T),” in 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2015), paper 2015-0675.
  9. F. Villars and V. Weisskopf, “The scattering of electromagnetic waves by turbulent atmospheric fluctuations,” Phys. Rev. 94, 232–240 (1954).
    [Crossref]
  10. W. George, P. Beuther, and R. Arndt, “Pressure spectra in turbulent free shear flows,” J. Fluid Mech. 148, 155–191 (1984).
    [Crossref]
  11. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, formula 6.565.4 (Academic, 1965), p. 686.
  12. N. Lebedev, Special Functions and Their Applications (Dover, 1972), Chap. 5.
  13. G. Roussas, A Course in Mathematical Statistics, 2nd ed. (Academic, 1977), Chap. 6.
  14. K. Miller, “On the inverse of the sum of matrices,” Math. Mag. 54(2), 67–72 (1981).
    [Crossref]

2017 (1)

2012 (2)

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The airborne aero-optics laboratory, AAOL,” Proc. SPIE 8395, 839507 (2012).
[Crossref]

2011 (1)

A. Smits, B. McKeon, and I. Marusic, “High-Reynolds number wall turbulence,” Annu. Rev. Fluid Mech. 43, 353–375 (2011).
[Crossref]

1984 (1)

W. George, P. Beuther, and R. Arndt, “Pressure spectra in turbulent free shear flows,” J. Fluid Mech. 148, 155–191 (1984).
[Crossref]

1981 (1)

K. Miller, “On the inverse of the sum of matrices,” Math. Mag. 54(2), 67–72 (1981).
[Crossref]

1954 (1)

F. Villars and V. Weisskopf, “The scattering of electromagnetic waves by turbulent atmospheric fluctuations,” Phys. Rev. 94, 232–240 (1954).
[Crossref]

Andrews, L.

L. Andrews and R. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 3.

Arndt, R.

W. George, P. Beuther, and R. Arndt, “Pressure spectra in turbulent free shear flows,” J. Fluid Mech. 148, 155–191 (1984).
[Crossref]

Beuther, P.

W. George, P. Beuther, and R. Arndt, “Pressure spectra in turbulent free shear flows,” J. Fluid Mech. 148, 155–191 (1984).
[Crossref]

Cavalieri, D.

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The airborne aero-optics laboratory, AAOL,” Proc. SPIE 8395, 839507 (2012).
[Crossref]

Cavalieri, S.

E. Jumper, S. Gordeyev, S. Cavalieri, P. Rollins, M. Whiteley, and M. Krizo, “Airborne aero-optics laboratory—transonic (AAOL-T),” in 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2015), paper 2015-0675.

George, W.

W. George, P. Beuther, and R. Arndt, “Pressure spectra in turbulent free shear flows,” J. Fluid Mech. 148, 155–191 (1984).
[Crossref]

Gordeyev, S.

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The airborne aero-optics laboratory, AAOL,” Proc. SPIE 8395, 839507 (2012).
[Crossref]

E. Jumper, S. Gordeyev, S. Cavalieri, P. Rollins, M. Whiteley, and M. Krizo, “Airborne aero-optics laboratory—transonic (AAOL-T),” in 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2015), paper 2015-0675.

M. Kemnetz and S. Gordeyev, “Optical investigations of large-scale boundary-layer structures,” 54th AIAA Aerospace Sciences Meeting, San Diego, California, 4–8 January, 2016, AIAA paper 2016-1460.

Gradshteyn, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, formula 6.565.4 (Academic, 1965), p. 686.

Jumper, E.

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The airborne aero-optics laboratory, AAOL,” Proc. SPIE 8395, 839507 (2012).
[Crossref]

E. Jumper, S. Gordeyev, S. Cavalieri, P. Rollins, M. Whiteley, and M. Krizo, “Airborne aero-optics laboratory—transonic (AAOL-T),” in 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2015), paper 2015-0675.

Kemnetz, M.

M. Kemnetz and S. Gordeyev, “Optical investigations of large-scale boundary-layer structures,” 54th AIAA Aerospace Sciences Meeting, San Diego, California, 4–8 January, 2016, AIAA paper 2016-1460.

Krizo, M.

E. Jumper, S. Gordeyev, S. Cavalieri, P. Rollins, M. Whiteley, and M. Krizo, “Airborne aero-optics laboratory—transonic (AAOL-T),” in 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2015), paper 2015-0675.

Lebedev, N.

N. Lebedev, Special Functions and Their Applications (Dover, 1972), Chap. 5.

Mani, A.

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

Marusic, I.

A. Smits, B. McKeon, and I. Marusic, “High-Reynolds number wall turbulence,” Annu. Rev. Fluid Mech. 43, 353–375 (2011).
[Crossref]

McKeon, B.

A. Smits, B. McKeon, and I. Marusic, “High-Reynolds number wall turbulence,” Annu. Rev. Fluid Mech. 43, 353–375 (2011).
[Crossref]

Miller, K.

K. Miller, “On the inverse of the sum of matrices,” Math. Mag. 54(2), 67–72 (1981).
[Crossref]

Phillips, R.

L. Andrews and R. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 3.

Prasad, S.

Rollins, P.

E. Jumper, S. Gordeyev, S. Cavalieri, P. Rollins, M. Whiteley, and M. Krizo, “Airborne aero-optics laboratory—transonic (AAOL-T),” in 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2015), paper 2015-0675.

Roussas, G.

G. Roussas, A Course in Mathematical Statistics, 2nd ed. (Academic, 1977), Chap. 6.

Ryzhik, I.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, formula 6.565.4 (Academic, 1965), p. 686.

Smits, A.

A. Smits, B. McKeon, and I. Marusic, “High-Reynolds number wall turbulence,” Annu. Rev. Fluid Mech. 43, 353–375 (2011).
[Crossref]

Tatarski, V.

V. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. Silverman (McGraw-Hill, 1961).

Villars, F.

F. Villars and V. Weisskopf, “The scattering of electromagnetic waves by turbulent atmospheric fluctuations,” Phys. Rev. 94, 232–240 (1954).
[Crossref]

Wang, M.

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

Weisskopf, V.

F. Villars and V. Weisskopf, “The scattering of electromagnetic waves by turbulent atmospheric fluctuations,” Phys. Rev. 94, 232–240 (1954).
[Crossref]

Whiteley, M.

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The airborne aero-optics laboratory, AAOL,” Proc. SPIE 8395, 839507 (2012).
[Crossref]

E. Jumper, S. Gordeyev, S. Cavalieri, P. Rollins, M. Whiteley, and M. Krizo, “Airborne aero-optics laboratory—transonic (AAOL-T),” in 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2015), paper 2015-0675.

Zenk, M.

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The airborne aero-optics laboratory, AAOL,” Proc. SPIE 8395, 839507 (2012).
[Crossref]

Annu. Rev. Fluid Mech. (2)

M. Wang, A. Mani, and S. Gordeyev, “Physics and computation of aero-optics,” Annu. Rev. Fluid Mech. 44, 299–321 (2012).
[Crossref]

A. Smits, B. McKeon, and I. Marusic, “High-Reynolds number wall turbulence,” Annu. Rev. Fluid Mech. 43, 353–375 (2011).
[Crossref]

J. Fluid Mech. (1)

W. George, P. Beuther, and R. Arndt, “Pressure spectra in turbulent free shear flows,” J. Fluid Mech. 148, 155–191 (1984).
[Crossref]

J. Opt. Soc. Am. A (1)

Math. Mag. (1)

K. Miller, “On the inverse of the sum of matrices,” Math. Mag. 54(2), 67–72 (1981).
[Crossref]

Phys. Rev. (1)

F. Villars and V. Weisskopf, “The scattering of electromagnetic waves by turbulent atmospheric fluctuations,” Phys. Rev. 94, 232–240 (1954).
[Crossref]

Proc. SPIE (1)

E. Jumper, M. Zenk, S. Gordeyev, D. Cavalieri, and M. Whiteley, “The airborne aero-optics laboratory, AAOL,” Proc. SPIE 8395, 839507 (2012).
[Crossref]

Other (7)

E. Jumper, S. Gordeyev, S. Cavalieri, P. Rollins, M. Whiteley, and M. Krizo, “Airborne aero-optics laboratory—transonic (AAOL-T),” in 53rd AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2015), paper 2015-0675.

I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products, formula 6.565.4 (Academic, 1965), p. 686.

N. Lebedev, Special Functions and Their Applications (Dover, 1972), Chap. 5.

G. Roussas, A Course in Mathematical Statistics, 2nd ed. (Academic, 1977), Chap. 6.

M. Kemnetz and S. Gordeyev, “Optical investigations of large-scale boundary-layer structures,” 54th AIAA Aerospace Sciences Meeting, San Diego, California, 4–8 January, 2016, AIAA paper 2016-1460.

V. Tatarski, Wave Propagation in a Turbulent Medium, translated by R. Silverman (McGraw-Hill, 1961).

L. Andrews and R. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005), Chap. 3.

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Figures (6)

Fig. 1.
Fig. 1.

Mean beam intensity along the central streamwise direction ( ξ , or x , axis) versus propagation distance, ζ , for a convergent Gaussian beam, with the focusing parameter value, ζ 0 = 2.5 . The strength parameters of the anisotropic aero-optical fluctuations are b K 2 w 0 2 = 0.5 and b K 2 w 0 2 = 0.02 .

Fig. 2.
Fig. 2.

Mean beam intensity along the central spanwise direction ( η , or y , axis) versus propagation distance, ζ , for the same propagation and turbulence strength parameters as in Fig. 1.

Fig. 3.
Fig. 3.

On-axis fractional intensity moment versus propagation distance of a Gaussian beam converging at focal distance, ζ 0 = 2.5 , for different moment orders. The strength parameters of the anisotropic aero-optical fluctuations are b K 2 w 0 2 = 0.5 and b K 2 w 0 2 = 0.02 .

Fig. 4.
Fig. 4.

Same as Fig. 3 except b K 2 w 0 2 = 1 and b K 2 w 0 2 = 0.04 .

Fig. 5.
Fig. 5.

Fractional intensity moment versus order of the moment for a focusing Gaussian beam in its focal plane, ζ = ζ 0 = 2.5 . The turbulence strength parameters are the same as in Fig. 3.

Fig. 6.
Fig. 6.

Same as Fig. 5 except b K 2 w 0 2 = 1 and b K 2 w 0 2 = 0.04 .

Equations (85)

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ϕ ( r ) = k 0 0 σ d ζ n ( r ζ ζ ^ ) d ζ .
D ϕ ( S ) ( r 1 r 2 ) = def E [ ϕ ( r 1 ) ϕ ( r 2 ) ] 2 = k 0 2 0 σ d ζ 0 σ d ζ E [ n ( r 1 ζ ζ ^ ) n ( r 2 ζ ζ ^ ) ] × [ n ( r 1 ζ ζ ^ ) n ( r 2 ζ ζ ^ ) ] = 2 k 0 2 0 σ d ζ 0 σ d ζ [ C n ( S ) ( ( ζ ζ ) ζ ^ ) C n ( S ) ( r 1 r 2 ( ζ ζ ) ζ ^ ) ] ,
C n ( S ) ( r r ) = E [ n ( r ) n ( r ) ] ,
D ϕ ( S ) ( r 1 r 2 ) = 2 k 0 2 σ σ d ζ ( σ | ζ | ) [ C n ( S ) ( ζ ζ ^ ) C n ( S ) ( r 1 r 2 ζ ζ ^ ) ] .
C n ( S ) ( r ) = d 3 k W n ( k ) exp ( i k · r ) .
D ϕ ( S ) ( r 1 r 2 ) = 2 k 0 2 d 3 k W n ( k ) { 1 exp [ i k · ( r 1 r 2 ) ] } × σ σ d ζ ( σ | ζ | ) exp ( i ζ k · ζ ^ ) ,
D ϕ ( S ) ( r 1 r 2 ) = 4 k 0 2 d 3 k W n ( k ) { 1 exp [ i k · ( r 1 r 2 ) ] } × ( 1 cos k ζ σ ) k ζ 2 .
W n ( k ) = B n 2 K x K y K z ( 1 + k x 2 K x 2 + k y 2 K y 2 + k z 2 K z 2 ) γ / 2
ζ ^ = cos θ LOS x ^ cos θ El sin θ Az y ^ + sin θ El z ^
k ζ = k · ζ ^ = k x cos θ LOS k y cos θ El sin θ Az + k z sin θ El .
q i = k i / K i , i = x , y , z ,
k ζ σ = q · P , P = def σ [ K cos θ LOS x ^ + K ( cos θ El sin θ Az y ^ sin θ El z ^ ) ] .
D ϕ ( S ) ( r 1 r 2 ) = 4 k 0 2 σ 2 B n 2 P 2 d 3 q ( 1 cos P q p ) q p 2 ( 1 + q 2 ) γ / 2 [ 1 exp ( i q · R ) ] ,
R = K ( x 1 x 2 ) x ^ + K [ ( y 1 y 2 ) y ^ + ( z 1 z 2 ) z ^ ] .
d ϕ exp ( i q · R ) = 2 π J 0 ( q R ) .
D ϕ ( S ) ( r 1 r 2 ) = 8 π k 0 2 σ 2 B n 2 P 2 d q p ( 1 cos P q p ) q p 2 × 0 d q q ( 1 + q p 2 + q 2 ) γ / 2 [ 1 exp ( i q p R p ) J 0 ( q R ) ] .
D ϕ ( S ) = 8 π k 0 2 σ 2 B n 2 P 2 ( γ 2 ) d q p ( 1 cos P q p ) q p 2 ( 1 + q p 2 ) γ / 2 1 [ 1 exp ( i q p R p ) Γ ( γ / 2 1 ) × ( R ( 1 + q p 2 ) 1 / 2 2 ) γ / 2 1 K γ / 2 1 ( R ( 1 + q p 2 ) 1 / 2 ) ] .
K ν ( x ) = { Γ ( ν ) 2 ( 2 x ) ν + Γ ( ν ) 2 ( 2 x ) ν , | x | 1 , 0 < ν < 1 , Γ ( ν ) 2 ( 2 x ) ν Γ ( ν 1 ) 2 ( 2 x ) ν 2 , | x | 1 , ν > 1 .
D ϕ ( S ) ( r 1 r 2 ) = 8 π k 0 2 σ 2 B n 2 ( γ 2 ) P 2 d q p ( 1 cos P q p ) q p 2 ( 1 + q p 2 ) γ / 2 1 × [ 1 exp ( i q p R p ) + exp ( i q p R p ) ( 1 + q p 2 ) ( γ 4 ) R 2 ] ;
D ϕ ( S ) ( r 1 r 2 ) = 8 π k 0 2 σ 2 B n 2 ( γ 2 ) P 2 d q p ( 1 cos P q p ) q p 2 ( 1 + q p 2 ) γ / 2 1 [ 1 exp ( i q p R p ) exp ( i q p R p ) ( 1 + q p 2 ) γ / 2 1 Γ ( 1 γ / 2 ) Γ ( γ / 2 1 ) R γ 2 ] .
D ϕ ( S ) ( r 1 r 2 ) = { a R p 2 + b R 2 , γ > 4 a R p 2 + b R γ 2 , 3 < γ < 4 ,
a = 4 π k 0 2 σ 2 B n 2 P 2 ( γ 2 ) d q p ( 1 cos P q p ) ( 1 + q p 2 ) γ / 2 1 ; b = 8 π k 0 2 σ 2 B n 2 P 2 ( γ 2 ) ( γ 4 ) d q p ( 1 cos P q p ) q p 2 ( 1 + q p 2 ) γ / 2 2 ; b = 8 π k 0 2 σ 2 B n 2 Γ ( 1 γ / 2 ) P 2 ( γ 2 ) Γ ( γ / 2 1 ) 2 γ 2 d q p ( 1 cos P q p ) q p 2 .
P = σ K z ^ ; R = K ( x 1 x 2 ) x ^ + K ( y 1 y 2 ) y ^ .
D ϕ ( S ) ( r 1 r 2 ) = { b R 2 , R 1 , D = def 8 π k 0 2 σ 2 B n 2 P 2 ( γ 2 ) d q p ( 1 cos P q p ) q p 2 ( 1 + q p 2 ) γ / 2 1 , R 1 ,
D = 8 π k 0 2 B n 2 σ ( γ 2 ) K d u ( 1 cos u ) u 2 = 8 π 2 k 0 2 B n 2 σ K ( γ 2 ) ,
E 0 ( ρ , ζ ) = E 0 1 + i ( ζ ζ 0 ) exp [ ρ 2 w 0 2 ( 1 + i ( ζ ζ 0 ) ]
E ( ρ , 0 ) = E 0 1 i ζ 0 exp [ ρ 2 w 0 2 ( 1 i ζ 0 ) + i ϕ ( ρ ) ] ,
E ( ρ , ζ ) = 1 i π w 0 2 ζ d 2 ρ E ( ρ , 0 ) exp [ i ( ρ ρ ) 2 w 0 2 ζ ] = E 0 i π w 0 2 ζ ( 1 i ζ 0 ) d 2 ρ exp [ ρ 2 w 0 2 ( 1 i ζ 0 ) + i ( ρ ρ ) 2 w 0 2 ζ ] exp [ i ϕ ( ρ ) ] .
χ E ( 2 ) ( ρ 1 , ρ 2 ; ζ ) = E [ E ( ρ 1 , ζ ) E * ( ρ 2 , ζ ) ] .
E exp ( i X ) = exp [ ( 1 / 2 ) E X 2 ] ,
χ E ( 2 ) ( ρ 1 , ρ 2 ; ζ ) = | E 0 | 2 π 2 ζ 2 ( 1 + ζ 0 2 ) d 2 u 1 d 2 u 2 exp [ u 1 2 ( 1 i ζ 0 ) u 2 2 ( 1 + i ζ 0 ) + i ( u 1 u 1 ) 2 ( u 2 u 2 ) 2 ζ ] × exp [ ( 1 / 2 ) D ϕ ( w 0 ( u 1 u 2 ) ) ] ,
χ E ( 2 ) ( ρ 1 , ρ 2 ; ζ ) = | E 0 | 2 π 2 ζ 2 ( 1 + ζ 0 2 ) I x I y ,
I x = d ξ 1 d ξ 2 exp [ ( 1 / 2 ) r _ T M x r _ + λ _ x T r _ ] , I y = d η 1 d η 2 exp [ ( 1 / 2 ) s _ T M y s _ + λ _ y T s _ ] ,
M x = ( 2 i ζ + 2 1 i ζ 0 + b K 2 w 0 2 b K 2 w 0 2 b K 2 w 0 2 2 i ζ + 2 1 + i ζ 0 + b K 2 w 0 2 ) ; M y = ( 2 i ζ + 2 1 i ζ 0 + b K 2 w 0 2 b K 2 w 0 2 b K 2 w 0 2 2 i ζ + 2 1 + i ζ 0 + b K 2 w 0 2 ) .
χ E ( 2 ) ( ρ 1 , ρ 2 ; ζ ) = 4 | E 0 | 2 ζ 2 ( 1 + ζ 0 2 ) det M x det M y × exp { ( 1 / 2 ) [ λ _ x T M x 1 λ _ x + λ _ y T M y 1 λ _ y ] } .
χ E ( 2 ) = | E 0 | 2 [ 1 + ( ζ ζ 0 ) 2 + b K 2 w 0 2 ζ 2 ] [ 1 + ( ζ ζ 0 ) 2 + b K 2 w 0 2 ζ 2 ] × exp [ ( ξ 1 2 + ξ 2 2 ) + b K 2 w 2 ( ζ 0 ) ( ξ 1 ξ 2 ) 2 / 2 1 + ( ζ ζ 0 ) 2 + b K 2 w 0 2 ζ 2 ] × exp [ ( η 1 2 + η 2 2 ) + b K 2 w 2 ( ζ 0 ) ( η 1 η 2 ) 2 / 2 1 + ( ζ ζ 0 ) 2 + b K 2 w 0 2 ζ 2 ] ,
E [ I ( ρ ; ζ ) ] = I 0 [ 1 + ( ζ ζ 0 ) 2 + b K 2 w 0 2 ζ 2 ] [ 1 + ( ζ ζ 0 ) 2 + b K 2 w 0 2 ζ 2 ] × exp [ 2 ξ 2 1 + ( ζ ζ 0 ) 2 + b K 2 w 0 2 ζ 2 ] × exp [ 2 η 2 1 + ( ζ ζ 0 ) 2 + b K 2 w 0 2 ζ 2 ] ,
w , 2 ( ζ ) = w 0 2 [ 1 + ( ζ ζ 0 ) 2 + b K , 2 w 0 2 ζ 2 ] ,
ζ , = ζ 0 1 + b K , 2 w 0 2 ,
min w , 2 = w 0 2 [ 1 + ( 1 + ζ 0 2 ) b K , 2 w 0 2 1 + b K , 2 w 0 2 ] .
b R 2 1 ,
χ E ( 2 n ) ( ρ 1 , , ρ 2 n ; ζ ) = E [ E ( ρ 1 ) E * ( ρ 2 ) E ( ρ 2 n 1 ) E * ( ρ 2 n ) ] ,
χ E ( 2 n ) ( ρ 1 , , ρ 2 n ; ζ ) = | E 0 | 2 n [ π 2 ζ 2 ( 1 + ζ 0 2 ) ] n l = 1 2 n d 2 u l × exp { l = 1 2 n [ u l 2 ( 1 + ( 1 ) l i ζ 0 ) + ( 1 ) l i ( u l u l ) 2 ζ ] } × exp { 1 2 E [ l = 1 2 n ( 1 ) l ϕ ( ρ l ) ] 2 } ,
E [ l = 1 2 n ( 1 ) l ϕ ( ρ l ) ] 2 = b K 2 ( l = 1 2 n ( 1 ) l x l ) 2 + b K 2 ( l = 1 2 n ( 1 ) l y l ) 2 .
χ E ( 2 n ) ( ρ 1 , , ρ 2 n ; ζ ) = | E 0 | 2 n [ π 2 ζ 2 ( 1 + ζ 0 2 ) ] n I x ( 2 n ) I y ( 2 n )
I x ( 2 n ) = ( k = 1 2 n d ξ k ) exp [ ( 1 / 2 ) r _ T M x r _ + λ _ x T r _ ] , I y ( 2 n ) = ( k = 1 2 n d η k ) exp [ ( 1 / 2 ) s _ T M y s _ + λ _ y T s _ ] ,
r _ = ( ξ 1 , , ξ 2 n ) T ; s _ = ( η 1 , , η 2 n ) T ; λ _ x = 2 i ζ ( ξ 1 , ξ 2 , , ξ 2 n 1 , ξ 2 n ) T ; λ _ y = 2 i ζ ( η 1 , η 2 , , η 2 n 1 , η 2 n ) T .
χ I ( 2 ) ( ρ , ρ ; ζ ) = χ E ( 4 ) ( ρ , ρ , ρ , ρ ) = | E 0 | 4 / [ 1 + ( ζ ζ 0 ) 2 ] [ 1 + ( ζ ζ 0 ) 2 + 2 b K 2 w 0 2 ζ 2 ] 1 2 [ 1 + ( ζ ζ 0 ) 2 + 2 b K 2 w 0 2 ζ 2 ] 1 2 × exp { [ ( ξ + ξ ) 2 1 + ( ζ ζ 0 ) 2 + 2 b K 2 w 0 2 ζ 2 + ( η + η ) 2 1 + ( ζ ζ 0 ) 2 + 2 b K 2 w 0 2 ζ 2 ] [ ( ξ ξ ) 2 + ( η η ) 2 1 + ( ζ ζ 0 ) 2 ] } .
χ δ I ( 2 ) ( ρ , ρ ; ζ ) = def E [ | E ( ρ ; ζ ) | 2 | E ( ρ ; ζ ) | 2 ] E [ | E ( ρ ; ζ ) | 2 | ] E [ | E ( ρ ; ζ ) | 2 ] = χ I ( 2 ) ( ρ , ρ ; ζ ) χ E ( 2 ) ( ρ , ρ ; ζ ) χ E ( 2 ) ( ρ , ρ ; ζ ) ,
χ I ( n ) ( ρ 1 , , ρ n ; ζ ) = χ E ( 2 n ) ( ρ 1 , ρ 1 , , ρ n , ρ n ; ζ ) ,
χ I ( n ) ( ρ 1 , , ρ n ; ζ ) = ( 2 | E 0 | ) 2 n [ ζ 2 ( 1 + ζ 0 2 ) ] n det M x ( 2 n ) det M y ( 2 n ) × exp { ( 1 / 2 ) [ λ _ x T ( M x ( 2 n ) ) 1 λ _ x + λ _ y T ( M y ( 2 n ) ) 1 λ _ y ] } ,
e _ 5 = 1 2 ( 0 , 0 , 0 , 0 , 0 , 1 , 0 , 1 ) ; e _ 6 = 1 4 ( 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 ) ; e _ 7 = 1 2 ( 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 ) .
λ _ x T M x ( 8 ) 1 λ _ x = M 11 M 00 M 11 M 01 2 ( λ _ x T e _ 0 ) 2 + 1 d [ ( λ _ x T e _ 2 ) 2 + ( λ _ x T e _ 4 ) 2 + ( λ _ x T e _ 6 ) 2 ] + 1 d * [ ( λ _ x T e _ 3 ) 2 + ( λ _ x T e _ 5 ) 2 + ( λ _ x T e _ 7 ) 2 ] .
λ _ x T M x ( 8 ) 1 λ _ x = ( ξ 1 + ξ 2 + ξ 3 + ξ 4 ) 2 1 + ( ζ ζ 0 ) 2 + 4 b K 2 w 0 2 ζ 2 2 [ ( ξ 1 ξ 2 ) 2 + ( ξ 3 ξ 4 ) 2 ] + ( ξ 1 + ξ 2 ξ 3 ξ 4 ) 2 1 + ( ζ ζ 0 ) 2 .
χ I ( 4 ) = | E 0 | 8 / [ 1 + ( ζ ζ 0 ) 2 ] 3 [ 1 + ( ζ ζ 0 ) 2 + 4 b K 2 w 0 2 ζ 2 ] 1 2 [ 1 + ( ζ ζ 0 ) 2 + 4 b K 2 w 0 2 ζ 2 ] 1 2 × exp { [ ( ξ 1 + ξ 2 + ξ 3 + ξ 4 ) 2 2 [ 1 + ( ζ ζ 0 ) 2 + 4 b K 2 w 0 2 ζ 2 ] + ( η 1 + η 2 + η 3 + η 4 ) 2 2 [ 1 + ( ζ ζ 0 ) 2 + 4 b K 2 w 0 2 ζ 2 ] ] 1 1 + ( ζ ζ 0 ) 2 [ ( ξ 1 ξ 2 ) 2 + ( ξ 3 ξ 4 ) 2 + ( η 1 η 2 ) 2 + ( η 3 η 4 ) 2 + ( 1 / 2 ) ( ξ 1 + ξ 2 ξ 3 ξ 4 ) 2 + ( 1 / 2 ) ( η 1 + η 2 η 3 η 4 ) 2 ] } .
λ _ x = 2 i ( 2 n ) 1 / 2 ξ ζ e _ 0 ; λ _ y = 2 i ( 2 n ) 1 / 2 η ζ e _ 0 .
λ _ x T ( M x ( 2 n ) ) 1 λ _ x = 8 n ξ 2 ζ 2 M 11 M 00 M 11 M 01 2 = 4 n ξ 2 1 + ( ζ ζ 0 ) 2 + n b K 2 w 0 2 ζ 2 ; λ _ y T ( M y ( 2 n ) ) 1 λ _ y = 8 n η 2 ζ 2 M 11 M 00 M 11 M 01 2 = 4 n η 2 1 + ( ζ ζ 0 ) 2 + n b K 2 w 0 2 ζ 2 ;
χ I ( n ) = | E 0 | 2 n / [ 1 + ( ζ ζ 0 ) 2 ] n 1 [ 1 + ( ζ ζ 0 ) 2 + n b K 2 w 0 2 ζ 2 ] 1 2 [ 1 + ( ζ ζ 0 ) 2 + n b K 2 w 0 2 ζ 2 ] 1 2 × exp { 2 n ξ 2 [ 1 + ( ζ ζ 0 ) 2 + n b K 2 w 0 2 ζ 2 ] 2 n η 2 [ 1 + ( ζ ζ 0 ) 2 + n b K 2 w 0 2 ζ 2 ] } .
f I ( n ) = def χ I ( n ) [ χ I ( 1 ) ] n = ( 1 + a ) n / 2 ( 1 + n a ) 1 / 2 ( 1 + a ) n / 2 ( 1 + n a ) 1 / 2 × exp { 2 n ( n 1 ) [ 1 + ( ζ ζ 0 ) 2 ] 2 [ ξ 2 ( 1 + a ) ( 1 + n a ) + η 2 ( 1 + a ) ( 1 + n a ) ] } ,
a = b K 2 w 0 2 ζ 2 1 + ( ζ ζ 0 ) 2 ; a = b K 2 w 0 2 ζ 2 1 + ( ζ ζ 0 ) 2 .
min ξ , η f I ( n ) = ( 1 + a ) n / 2 ( 1 + n a ) 1 / 2 ( 1 + a ) n / 2 ( 1 + n a ) 1 / 2 .
lim ζ ζ 0 min ξ , η f I ( n ) = ( 1 + b K 2 w 0 2 ) n / 2 ( 1 + n b K 2 w 0 2 ) 1 / 2 ( 1 + b K 2 w 0 2 ) n / 2 ( 1 + n b K 2 w 0 2 ) 1 / 2 ,
( 1 / 2 ) r _ T M r _ + λ _ T r _ r _ r _ + u _ ( 1 / 2 ) r _ T M r _ + r _ T ( λ _ M u _ ) ( 1 / 2 ) u _ T M u _ + λ _ T u _ ,
( 1 / 2 ) r _ T M r _ + ( 1 / 2 ) λ _ T M 1 λ _ .
I ( λ _ ) = def d r _ exp [ ( 1 / 2 ) r _ T M r _ + λ _ T r _ ] = d r _ exp [ ( 1 / 2 ) r _ T M r _ ] exp [ ( 1 / 2 ) λ _ T M 1 λ _ ] = I ( 0 ) exp [ ( 1 / 2 ) λ _ T M 1 λ _ ] ,
M = D + v _ v _ T ,
D = diag ( d , d * , d , d * , , d , d * ) , with    d = 2 i ζ + 2 1 i ζ 0 ; v _ = b K 2 w 0 2 ( 1 , 1,1 , 1 , , 1 , 1 ) T , K = K    or    K .
e _ 0 = 1 n ( 1 , 1 , , 1 , 1 ) , v _ = n b K 2 w 0 2 ,
e _ 1 = 1 n ( 1 , 1 , , 1 ) ; e _ 2 = 1 2 ( 1 , 0 , 1 , 0 , 0 , , 0 ) ; e _ 3 = 1 2 ( 0 , 1 , 0 , 1 , 0 , , 0 ) ; e _ 4 = 1 2 ( 0 , 0 , 0 , 0 , 1 , 0 , 1 , 0 , , 0 ) ; e _ 5 = 1 2 ( 0 , 0 , 0 , 0 , 0 , 1 , 0 , 1 , , 0 ) ; 1 4 ( 1 , 0 , 1 , 0 , 1 , 0 , 1 , 0 , , 0 ) ; 1 4 ( 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 , , 0 ) ; 2 n ( n 2 ) ( 1 , 0 , 1 , 0 , , 1 , 0 , ( n 2 ) / 2 , 0 ) ; 2 n ( n 2 ) ( 0 , 1 , 0 , 1 , , 0 , 1 , 0 , ( n 2 ) / 2 )
M 01 = D 10 = e _ 0 T D e _ 1 = i Im d ,
M 00 = e _ 0 T M e _ 0 = Re d + n b K 2 w 0 2 , M 11 = e _ 1 T D e _ 1 = Re d
M 00 M 11 M 01 2 = | d | 2 + n b K 2 w 0 2 Re ( d ) ,
det ( M ) = | d | n 2 ( | d | 2 + n b K 2 w 0 2 Re d ) .
Re d = 2 1 + ζ 0 2 ;    Im d = 2 ζ + 2 ζ 0 1 + ζ 0 2 ;    | d | 2 = 4 [ 1 + ( ζ ζ 0 ) 2 ] ζ 2 ( 1 + ζ 0 2 ) .
M 1 = D 1 w _ w _ T 1 + v _ T w _ , with    w _ = D 1 v _ .
M 1 = [ M 11 e _ 0 e _ 0 T + M 00 e _ 1 e _ 1 T M 01 ( e _ 0 e _ 1 T + e _ 1 e _ 0 T ) ] M 00 M 11 M 01 2 + 1 d j = 1 n / 2 1 e _ 2 j e _ 2 j T + 1 d * j = 1 n / 2 1 e _ 2 j + 1 e _ 2 j + 1 T .
I ( 0 ) = ( 2 π ) n / 2 ( det M ) 1 / 2 ,
I ( λ _ ) = ( 2 π ) n / 2 ( det M ) 1 / 2 exp [ ( 1 / 2 ) λ _ T M 1 λ _ ] ,
E [ l = 1 2 n ( 1 ) l ϕ l ] 2 = 1 2 l = 1 2 n m = 1 2 n ( 1 ) l + m D l m ,
1 2 [ l ( 1 ) l E ϕ l 2 m ( 1 ) m + m ( 1 ) m E ϕ m 2 l ( 1 ) l ] + l m ( 1 ) l + m E ( ϕ l ϕ m ) = 0 + 0 + E [ l ( 1 ) l ϕ l ] 2 ,
D l m = A ( ξ l ξ m ) 2 + B ( η l η m ) 2 ,
1 2 l = 1 2 n m = 1 2 n ( 1 ) l + m D l m = A [ l = 1 2 n ( 1 ) l ξ l ] 2 + B [ l = 1 2 n ( 1 ) l η l ] 2 .
A 2 [ l ( 1 ) l ξ l 2 m ( 1 ) m + m ( 1 ) m ξ m 2 l ( 1 ) l ] + A l m ( 1 ) l + m ξ l ξ m ,
A [ l ( 1 ) l ξ l ] 2 .
E [ l = 1 2 n ( 1 ) l ϕ l ] 2 = A [ l = 1 2 n ( 1 ) l ξ l ] 2 + B [ l = 1 2 n ( 1 ) l η l ] 2 .