• Acta Photonica Sinica
  • Vol. 50, Issue 9, 0929001 (2021)
Meiling DUAN1, Jiao DU1, Zhiguo ZHAO2, Xiaodong HUANG1, Yanqin GAO1, and Chaoliang DING2
Author Affiliations
  • 1School of Science, North University of China, Taiyuan03005, China
  • 2Henan Key Laboratory of Electromagnetic Transformation and Detection, Luoyang Normal University, Luoyang, Henan471934, China
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    DOI: 10.3788/gzxb20215009.0929001 Cite this Article
    Meiling DUAN, Jiao DU, Zhiguo ZHAO, Xiaodong HUANG, Yanqin GAO, Chaoliang DING. Dynamic Evolution of the Patically Coherent Circular Edge Dislocation Beams Propagating in Biological Tissues[J]. Acta Photonica Sinica, 2021, 50(9): 0929001 Copy Citation Text show less

    Abstract

    The analytical expression for the cross spectral density function of partially coherent circular edge dislocation beams propagating in the deep dermis of mouse tissue is derived based on the generalized Huygens-Fresnel principle, the effects of the initial beam parameters (the beam wavelength λ and the number of circular edge dislocations n) and the propagation distance z on the normalized intensity distribution, phase evolution and propagation trajectory of the beam are investigated. The results show that the central intensity of the partially coherent circular edge dislocation beam with n dislocation number is the largest in the source plane, and 2n secondary peaks symmetrically distribute on both sides. With the increment of propagation distance, the intensity distribution gradually evolves from multi-peak to single-peak, the longer the wavelength is, the smaller the n is, the faster the intensity distribution evolution is. The more the number of dislocations is, the better the beam stability is. Additionally, in the source plane the radius of the innermost ring of n circular edge dislocations decreases as the number of dislocations increases. Owing to the combination of the biological tissue turbulence induction and diffraction effect, the n circular edge dislocation has split into n sets of coherent vortices whose topological charges are "+1" and "-1" from the beginning of the propagation, respectively. As the propagation proceeds, n sets of coherent vortices with topological charges of "+1" and "-1" will be generated. The bigger the wavelength is, the more the n is, the faster the evolution of the beam phase, the more the distribution of the coherent vortices tends to be concentrated, and finally all the coherent vortices are annihilated. The smaller the distance between the coherent vortices, the earlier the annihilation. The longer the wavelength is, the larger the value of n is, the earlier the annihilation of the initial coherent vortices pairs is, and the longer the propagation distance to the total annihilation is. The smaller the wavelength is, the larger the value of n is, the earlier the annihilation of the newly generated coherent vortices pairs is, and the longer the propagation distance to the total annihilation is.
    Es,θ,0=2sw0mLnm2s2w02exp-s2w02exp(imθ)(1)

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    exp(imθ)s2Lnm(s2)=-1n22n+mn!t=0nr=0mirntmrH2t+m-rsxH2n-2t+rsy(2)

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    Es,0=-1n22n+mn!t=0nr=0mirntmrH2t+m-r2sxw0H2n-2t+r2syw0exp-s2w02(3)

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    E(s,0)=(-1)n22nn!ntH2t2sxw0H2n-2t2syw0exp-s2w02(4)

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    W0s1,s2,0=E*E=124n(n!)2t1=0nt2=0nnt1nt2H2t12s1xw0H2n-2t12s1yw0×           H2t22s2xw0H2n-2t22s2yw0exp-s12+s22w02exp-s1-s222σ02(5)

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    Wρ1,ρ2,z=k2πz2W0s1,s2,0exp-ik2zs1-ρ12-s2-ρ22×expψ*ρ1,s1+ψρ2,s2ds1ds2(6)

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    expψ*ρ1,s1+ψρ2,s2=exp-4π2k2z01dt0dκκΦ(κ){1-J0[t(ρ1-ρ2)+(1-t)(s1-s2)κ]}(7)

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    Φ(κ)=4πδn2L02(ς-1)(1+κ2L02)ς(8)

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    exp[ψ*(ρ1,s1)+ψ(ρ2,s2)]exp-1ρ02[(ρ1-ρ2)2+(ρ1-ρ2)(s1-s2)+(s1-s2)2](9)

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    ρ0(z)=0.22(Cn2k2z)-1/2(10)

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    exp[-(x-y)2]Hn(ax)dx=π(1-a2)n2Hnay(1-a2)12(11)

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    xnexp[-(x-β)2]dx=(2i)-nπHniβ(12)

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    Hn(x+y)=12n/2k=0nnkHk(2x)Hn-k(2y)(13)

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    Hn(x)=m=0[n/2](-1)mn!m!(n-2m)!(2x)n-2m(14)

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    Wρ1,ρ2,z=k2πz2124nn!N1N2t1=0nt2=0nnt1nt2AB(15)

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    N1=exp14Dρ12+ρ22ρ04-2z+2ikρ02ρ1xρ2x+ρ1yρ2yρ04z+2ikz-k2ρ02ρ22ρ02z2(16)

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    N2=exp-ρ1-ρ22ρ02exp-ik2zρ12-ρ22expFx2+Fy24G(17)

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    A=c1=0[t1]d1=02t2e1=0[d12]2t2d1-1c1+e12i-(2t1-2c1+d1-2e1)(2t1)!c1!(2t1-2c1)!d1!e1!(d1-2e1)!×πD(1-2w02D)t22-t2[2ρ02+2σ02ρ02σ02w02D2-2D]d1-2e11G2t1-2c1+d1-2e1+1×22w02t1-2c1H2t2-d1ρ1x-ρ2xz-ikρ2xρ02ρ02zw02D2-2DH2t1-2c1+d1-2e1iFx2G(18)

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    B=c2=0[n-t1]d2=02n-2t2e2=0[d22]2n-2t2d2-1c2+e2(2n-2t1)!c2!(2n-2t1-2c2)!d2!e2!(d2-2e2)!×2i-(2n-2t1-2c2+d2-2e2)2-(n-t2)(1-2w02D)n-t22ρ02+2σ02ρ02σ02w02D2-2Dd2-2e2×πD22w02n-2t1-2c2×1G2n-2t1-2c2+d2-2e2+1×H2n-2t2-d2ρ1y-ρ2yz-ikρ2yρ02ρ02zw02D2-2DH2n-2t1-2c2+d2-2e2iFy2G(19)

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    D=1w02+12σ02-ik2z+1ρ02(20)

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    Fx=ikρ1xz-ρ1x-ρ2xρ02+12D1σ02+2ρ02ρ1x-ρ2xρ02-ikρ2xz(21)

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    Fy=ikρ1yz-ρ1y-ρ2yρ02+12D1σ02+2ρ02ρ1y-ρ2yρ02-ikρ2yz(22)

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    G=1w02+12σ02+ik2z+1ρ02-14D1σ02+2ρ022(23)

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    I(ρ,z)=W(ρ,ρ,z)(24)

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    IN=I(ρi,z)I(ρi,z)max(25)

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    μρ1,ρ2,z=Wρ1,ρ2,zIρ1,zIρ2,z1/2(26)

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    Re[μ(ρ1,ρ2,z)]=0

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    Im[μ(ρ1,ρ2,z)]=0

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    Meiling DUAN, Jiao DU, Zhiguo ZHAO, Xiaodong HUANG, Yanqin GAO, Chaoliang DING. Dynamic Evolution of the Patically Coherent Circular Edge Dislocation Beams Propagating in Biological Tissues[J]. Acta Photonica Sinica, 2021, 50(9): 0929001
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