Abstract

We propose a modified biological spectrum that contains both short length–scale and long length–scale to study light propagation through turbulent biological tissue. Based on the two-scale modified biological spectrum, we derive an analytic expression of the two-frequency mutual coherence function of Laguerre-Gaussian pulsed beam and establish a model of the signal-to-noise ratio (SNR) of Laguerre-Gaussian pulsed beam carrying orbital angular momentum in turbulent biological tissue. The results show that the modified biological spectrum agrees well with experimental results. In addition, the structural length-scale of biological tissue has a significant influence on the bandwidths and SNR of orbital angular momentum states. This work provides theoretical preparation for more accurately medical diagnosis and optical imaging.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]

2018 (2)

Y. Li, Y. Zhang, and Y. Zhu, “Probability distribution of the orbital angular momentum mode of the ultrashort Laguerre-Gaussian pulsed beam propagation in oceanic turbulence,” Results Phys. 11, 698–705 (2018).
[Crossref]

J. Nie, G. Liu, and R. Zhang, “Propagation and spatiotemporal coupling characteristics of ultra-short gaussian vortex pulse,” Opt. Laser Technol. 101, 446C450 (2018).

2017 (3)

2016 (3)

2015 (1)

M. Ornigotti, C. Conti, and A. Szameit, “The effect of orbital angular momentum on nondiffracting ultrashort optical pulses,” Phys. Rev. Lett. 115(10), 100401 (2015).
[Crossref] [PubMed]

2013 (2)

2012 (1)

2010 (1)

2007 (1)

2005 (2)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

M. Xu and R. R. Alfano, “Fractal mechanisms of light scattering in biological tissue and cells,” Opt. Lett. 30(22), 3051–3053 (2005).
[Crossref] [PubMed]

2002 (1)

M. Bloemer, K. Myneni, M. Centini, M. Scalora, and G. D’Aguanno, “Transit time of optical pulses propagating through a finite length medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056615 (2002).
[Crossref] [PubMed]

2001 (2)

1998 (3)

1996 (2)

J. M. Schmitt and G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21(16), 1310–1312 (1996).
[Crossref] [PubMed]

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123(1-3), 5–10 (1996).
[Crossref]

1992 (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

R. W. Ziolkowski and J. B. Judkins, “Propagation characteristics of ultrawide-bandwidth pulsed Gaussian beams,” J. Opt. Soc. Am. A 9(11), 2021–2030 (1992).
[Crossref]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

1989 (1)

1979 (1)

Alfano, R. R.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Andrews, L. C.

Bachmann, D.

Backman, V.

Beck, M.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Bloemer, M.

M. Bloemer, K. Myneni, M. Centini, M. Scalora, and G. D’Aguanno, “Transit time of optical pulses propagating through a finite length medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056615 (2002).
[Crossref] [PubMed]

Centini, M.

M. Bloemer, K. Myneni, M. Centini, M. Scalora, and G. D’Aguanno, “Transit time of optical pulses propagating through a finite length medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056615 (2002).
[Crossref] [PubMed]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Chen, M.

Chen, Y.

Conti, C.

M. Ornigotti, C. Conti, and A. Szameit, “The effect of orbital angular momentum on nondiffracting ultrashort optical pulses,” Phys. Rev. Lett. 115(10), 100401 (2015).
[Crossref] [PubMed]

Cui, M.

D’Aguanno, G.

M. Bloemer, K. Myneni, M. Centini, M. Scalora, and G. D’Aguanno, “Transit time of optical pulses propagating through a finite length medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056615 (2002).
[Crossref] [PubMed]

Darmo, J.

Eick, A. A.

Faist, J.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Freyer, J. P.

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Gao, C.

Gardecki, J. A.

Ghosh, N.

Glaser, A. K.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Gupta, P. K.

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hielscher, A. H.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hyun, C.

Ishimaru, A.

Izquierdo, J. G.

Jacques, S. L.

Johnson, T. M.

Judkins, J. B.

Kong, L.

Kumar, G.

Li, Y.

Y. Li, Y. Zhang, and Y. Zhu, “Probability distribution of the orbital angular momentum mode of the ultrashort Laguerre-Gaussian pulsed beam propagation in oceanic turbulence,” Results Phys. 11, 698–705 (2018).
[Crossref]

Y. Li, L. Yu, and Y. Zhang, “Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean,” Opt. Express 25(11), 12203–12215 (2017).
[Crossref] [PubMed]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Lin, Q.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123(1-3), 5–10 (1996).
[Crossref]

Liu, G.

J. Nie, G. Liu, and R. Zhang, “Propagation and spatiotemporal coupling characteristics of ultra-short gaussian vortex pulse,” Opt. Laser Technol. 101, 446C450 (2018).

Liu, J. T. C.

Liu, Y. D.

Loriot, V.

Majumder, S. K.

Martínez-Matos, Ó.

Mohanty, S. K.

Mourant, J. R.

Myneni, K.

M. Bloemer, K. Myneni, M. Centini, M. Scalora, and G. D’Aguanno, “Transit time of optical pulses propagating through a finite length medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056615 (2002).
[Crossref] [PubMed]

Nie, J.

J. Nie, G. Liu, and R. Zhang, “Propagation and spatiotemporal coupling characteristics of ultra-short gaussian vortex pulse,” Opt. Laser Technol. 101, 446C450 (2018).

Nishioka, N. S.

Ornigotti, M.

M. Ornigotti, C. Conti, and A. Szameit, “The effect of orbital angular momentum on nondiffracting ultrashort optical pulses,” Phys. Rev. Lett. 115(10), 100401 (2015).
[Crossref] [PubMed]

Parsa, P.

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Radosevich, A. J.

Rogers, J. D.

Rösch, M.

Scalari, G.

Scalora, M.

M. Bloemer, K. Myneni, M. Centini, M. Scalora, and G. D’Aguanno, “Transit time of optical pulses propagating through a finite length medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056615 (2002).
[Crossref] [PubMed]

Schmitt, J. M.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Shen, D.

Sheppard, C. J. R.

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Sreenivasiah, I.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Süess, M. J.

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Szameit, A.

M. Ornigotti, C. Conti, and A. Szameit, “The effect of orbital angular momentum on nondiffracting ultrashort optical pulses,” Phys. Rev. Lett. 115(10), 100401 (2015).
[Crossref] [PubMed]

Tang, J.

Tearney, G. J.

Unterrainer, K.

Vaveliuk, P.

Wang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123(1-3), 5–10 (1996).
[Crossref]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Xu, M.

Xu, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123(1-3), 5–10 (1996).
[Crossref]

Yi, J.

Yin, B.

Young, C. Y.

Yu, L.

Zhang, R.

J. Nie, G. Liu, and R. Zhang, “Propagation and spatiotemporal coupling characteristics of ultra-short gaussian vortex pulse,” Opt. Laser Technol. 101, 446C450 (2018).

Zhang, Y.

Zhang, Z.

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123(1-3), 5–10 (1996).
[Crossref]

Zhu, Y.

Y. Li, Y. Zhang, and Y. Zhu, “Probability distribution of the orbital angular momentum mode of the ultrashort Laguerre-Gaussian pulsed beam propagation in oceanic turbulence,” Results Phys. 11, 698–705 (2018).
[Crossref]

Ziolkowski, R. W.

Appl. Opt. (7)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

Z. Wang, Z. Zhang, Z. Xu, and Q. Lin, “Spectral and temporal properties of ultrashort light pulse in the far zone,” Opt. Commun. 123(1-3), 5–10 (1996).
[Crossref]

Opt. Express (5)

Opt. Laser Technol. (1)

J. Nie, G. Liu, and R. Zhang, “Propagation and spatiotemporal coupling characteristics of ultra-short gaussian vortex pulse,” Opt. Laser Technol. 101, 446C450 (2018).

Opt. Lett. (4)

Optica (3)

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

M. Bloemer, K. Myneni, M. Centini, M. Scalora, and G. D’Aguanno, “Transit time of optical pulses propagating through a finite length medium,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(5 Pt 2), 056615 (2002).
[Crossref] [PubMed]

Phys. Rev. Lett. (2)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94(15), 153901 (2005).
[Crossref] [PubMed]

M. Ornigotti, C. Conti, and A. Szameit, “The effect of orbital angular momentum on nondiffracting ultrashort optical pulses,” Phys. Rev. Lett. 115(10), 100401 (2015).
[Crossref] [PubMed]

Results Phys. (1)

Y. Li, Y. Zhang, and Y. Zhu, “Probability distribution of the orbital angular momentum mode of the ultrashort Laguerre-Gaussian pulsed beam propagation in oceanic turbulence,” Results Phys. 11, 698–705 (2018).
[Crossref]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Other (4)

O. Korotkova, Random Light Beams Theory and Applications (CRC, 2014).

I. S. Gradshteyn and I. M. Ryzhiz, Table of Integrals, Series and Products (Academic Press, 2006).

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

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Figures (8)

Fig. 1
Fig. 1 Power spectrum of refractive index variations in mouse liver tissue fits to the theoretical power spectrum of Eq. (10). The round solid points represent experimental data reported in Ref [25]. The chain-dotted line with circles is theoretical curve in Ref [7]. The two dotted curves are theoretical curves in Ref [9], and the solid line is our theoretical fit. Fitting yields D=3, l c =10.4μm, w s =0.6μm and w L =120μm.
Fig. 2
Fig. 2 Wavelength dependence of the reduced scattering coefficient μ s of rat liver tissue fitted to the power law. The round solid points represent experimental data reported in Ref [26]. The dotted curve is derived from the model in Ref [6], and the chain-dotted line is the fitted curve of our model. Fitting yields D=3.94, l c =2.3μm, w S =0.26μm and w L =12μm.
Fig. 3
Fig. 3 (a) SNR and (b) NSNR of OAM states versus co-moving coordinate ζ=tz/c for different OAM quantum number m 0 .
Fig. 4
Fig. 4 SNR of OAM states versus co-moving coordinate ζ=tz/c for different short length-scale w S .
Fig. 5
Fig. 5 SNR of OAM states versus co-moving coordinate ζ=tz/c for different long length-scale w L .
Fig. 6
Fig. 6 SNR of OAM states versus co-moving coordinate ζ=tz/c for different shape of the distribution D.
Fig. 7
Fig. 7 (a) SNR, (b) NSNR of the OAM states versus co-moving coordinate ζ=tz/c for different characteristic length of heterogeneity l c .
Fig. 8
Fig. 8 SNR of the OAM states versus co-moving coordinate ζ=tz/c for different initial half-pulse width T 0 .

Equations (36)

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ϕ n ( κ )= A n l c 3 Γ( D/2 ) π 3/2 2 ( 5D )/2 ( 1+ κ 2 l c 2 ) D/2 ,
n Δ S ( r )= n Δ ( r ) G S ( r ),
n Δ L ( r )= n Δ ( r )[ δ( r )+ G L ( r ) ],
D n c ( r )= n Δ S ( r ) n Δ L ( r d r )d r = n Δ S ( r ) n Δ L ( r ).
D n c ( r )= F 1 [ | F[ n Δ ( r ) ] | 2 | F[ G S ( r ) ]( 1+F[ G L ( r ) ] ) | ],
D n c ( r )=4π 0 κ ϕ n c ( κ ) sin( κr ) r dκ.
ϕ n c ( κ )= ϕ n ( κ )| F[ G S ( r ) ]( 1+F[ G L ( r ) ] ) |.
+ exp(a x 2 +ibx) dx= π a exp( b 2 4a ),
ϕ n c ( κ )= A n l c 3 Γ( D/2 ) π 3/2 2 ( 5D )/2 ( 1+ κ 2 l c 2 ) D/2 a S a L π w S w L 2 b S b L exp( w S 2 κ 2 4 b S )[ 1+exp( w L 2 κ 2 4 b L ) ].
a S = 2 b S /( π w S ), a L = 2 b L /( π w L ),
ϕ n c ( κ )= A n l c 3 Γ( D/2 ) π 3/2 2 ( 5D )/2 ( 1+ κ 2 l c 2 ) D/2 exp( w S 2 κ 2 4 b S )[ 1+exp( w L 2 κ 2 4 b L ) ].
u s =4 π 2 k 4 1 1 ϕ n c ( κ )( 1cosθ )( 1+ cos 2 θ ) dcosθ,
v 0 ( r,φ,z;t )= + V j ( ω ) exp( iωt ) 2π U m LG ( r,φ,z;ω+ ω 0 )dω,
U m LG ( r,φ,z;ω )=L G P m 0 ( r,φ,z;ω )exp[ ψ( r,φ,z;ω ) ],
L G P m 0 ( r,φ,z;ω )=exp[ ( 1i z ζ ) r 2 w 0 2 ( 1+ z ζ 2 ) ] r m 0 w ( z ζ ) m 0 +1 L P m 0 [ r 2 w ( z ζ ) 2 ], ×exp[ ikzi( m 0 +1 )arctan( z ζ )i m 0 φ ]
R( r 1 , r 2 ,φ, φ ,z; t 1 , t 2 )= v 0 ( r 1 ,φ,z; t 1 ) v 0 ( r 2 , φ ,z; t 2 ) = 1 ( 2π ) 2 + V j ( ω 1 ) V j ( ω 2 ) ×exp( i ω 1 t+i ω 2 t 2 ) Γ 2 ( r 1 , r 2 ,φ, φ ,z; ω 1 + ω 0 , ω 2 + ω 0 )d ω 1 d ω 2 ,
Γ 2 ( r 1 , r 2 ,φ, φ ,z; k 1 , k 2 )=L G P m 0 ( r 1 ,φ,z; k 1 )L G P m 0 * ( r 2 , φ ,z; k 2 ) × exp[ ψ( r 1 ,φ, k 1 )+ ψ ( r 2 , φ , k 2 ) ] ,
exp[ ψ( r,φ, ω 1 + ω 0 )+ ψ * ( r, φ , ω 2 + ω 0 ) ] =exp{ 2[ 1cos( φ φ ) ] r 2 / ρ b 2 },
ρ b = [ π 2 ( ω 1 ω 2 ) 2 z 0 κ 3 ϕ n c ( κ )dκ/( 3 c 2 ) ] 1/2 .
0 x t μ1 ( 1+βt ) v dt= x μ μ F 2 1 ( v,μ;1+μ;βx ),μ>0,
0 κ 2μ exp( κ 2 / κ l 2 ) ( κ 2 + κ 0 2 ) D/2 dκ= 1 2 κ 0 2μ+1D Γ( μ+ 1 2 )U( μ+ 1 2 ;μ 1 3 ; κ 0 2 κ l 2 ),μ> 1 2 ,
ρ b 2 = ( ω 1 ω 2 ) 2 ,
exp[ ψ( r,φ, ω 1 + ω 0 )+ ψ * ( r, φ , ω 2 + ω 0 ) ] =exp{ Ω ω d 2 },
Γ 2 ( r 1 , r 2 ,φ, φ ,z; k 1 , k 2 )= ( w 0 2zc ) 2 m 0 +2 ( ω 0 + ω c ) 2 m 0 +2 ( 2π ) 2 r 2 m 0 exp[ i m 0 ( φ φ ) ] ×exp[ r 2 w 0 2 2 z 2 c 2 ( ω 0 + ω c ) 2 ]exp( Ω ω d 2 i ω d z c +i r 2 2zc ω d ),
V j ( ω )= π T 0 exp( 1 4 ω 2 T 0 2 ),
R( r,φ, φ ,z;t )= 1 ( 2π ) 2 + Γ 2 ( r,φ, φ ,z; ω c + ω 0 + ω d 2 , ω c + ω 0 ω d 2 ) × V j ( ω 1 ) V j ( ω 2 )exp( i ω d t )d ω c d ω d .
+ exp(a x 2 +ibx) dx= π a exp( b 2 4a ),
+ x n e p x 2 +2qx dx=n! e q 2 /p π p ( q p ) n H=1 n/2 1 ( n2H )!H! ( p 4 q 2 ) H [ p>0 ],
R( r,φ, φ ,z;t )= ( w 0 2zc ) 2 m 0 +2 exp( ω 0 2 T 0 2 [ r w 0 /( zc ) ] 2 2 [ r w 0 /( zc ) ] 2 +2 T 0 2 )exp[ i m 0 ( φ φ ) ] T 0 2 r 2 m 0 ( T 0 2 +8Ω ) × H=1 m 0 +1 ( 2 m 0 +2 )! ( [ r w 0 /( zc ) ] 2 + T 0 2 ) H2 m 0 5/2 ( 2 m 0 +22H )!H! 2 H ( ω 0 T 0 2 ) 2H2 m 0 2 exp{ 2 [ r 2 2zc ( t z c ) ] 2 /( T 0 2 +8Ω ) },
v 0 ( r,φ,z;ω )= T 0 2π P a P, m,t ( z,t ) R P ( r )exp[ i( ω t 0 +mφ ) ],
a P, m,t ( z,t )= T 0 2π R P * ( r ) exp( imφ )exp( -iωt ) v 0 ( r,φ,z;ω )rdrdφdω,
P( m| m 0 ,t )= P | a P, m,t ( z,t ) | 2 ,
P R P ( r ) R P * ( r ) =δ( r, r )/r,
P( m| m 0 ,t )= T 0 ( 2π ) 2 exp[ im( φ φ ) ] v 0 ( r,φ,z;ω ) v 0 * ( r, φ ,z; ω ) ×exp[ -i( ω ω )t ]rdrdφd φ dωd ω .
SNR= P( m 0 | m 0 ,t ) h= P( h| m 0 ,t ) P( m 0 | m 0 ,t ) = P(m= m 0 ,t) h= P(m=h,t) P(m= m 0 ,t) ,
P( m= m 0 ,t )= ( w 0 2zc ) 2 m 0 +2 r 2 m 0 +1 T 0 2 2π ( T 0 2 +8Ω ) H=1 m 0 +1 ( 2 m 0 +2 )! ( [ r w 0 /( zc ) ] 2 + T 0 2 ) H2 m 0 5/2 ( 2 m 0 +22H )!H! 2 H ( ω 0 T 0 2 ) 2H2 m 0 2 ×exp( ω 0 2 T 0 2 [ r w 0 /( zc ) ] 2 2 [ r w 0 /( zc ) ] 2 +2 T 0 2 )exp{ 2 [ r 2 2zc ( t z c ) ] 2 /( T 0 2 +8Ω ) }drdφd φ .

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