Correlation of Intensity Fluctuations for Scattering of a Partially Coherent Plane-Wave Pulse

Abstract

We derived analytical expressions for the correlation of intensity fluctuations of a partially coherent Gaussian Schell-model plane-wave pulse scattered by deterministic and random media. Our results extend the study of correlation of intensity fluctuations at two space points for scattered stationary fields to that at two time points for scattered non-stationary fields.

1. Introduction

Scattering is of great importance in physics, astronomy, chemistry, meteorology, biology, and in other fields. Scattering of electromagnetic fields from a medium which fluctuates both in space and in time has been studied extensively in recent years [1,2,3,4,5,6,7,8,9]. The scattering medium may be deterministic or random, continuous or discrete. The inverse problem, i.e., the problem of finding the properties of the object from the statistical properties of the scattered field, is of considerable interest [10,11,12,13].

Non-stationary light fields, also named stochastic optical pulses or partially coherent pulses, exhibit partial coherence spectrally and temporally, and have attracted much attention due to their important roles in optical telecommunications, optical imaging, fiber optics, etc. The study of partially coherent pulses has been developed from conventional Gaussian correlation function to nonconventional correlation functions. There has been substantial work on the propagation and scattering of partially coherent pulses [14,15,16,17,18].

Correlation of intensity fluctuations, i.e., the Hanbury Brown-Twiss effect, was first introduced to determine the angular diameter of radio stars [19]. Recently, there have been several attempts to study the correlation of intensity fluctuations at two space points of the scattered field, and it was found that information about the scattering potentials of deterministic and random media may be obtained from the measurement of the correlation of intensity fluctuations [20,21,22]. In the present paper, we considered correlation of intensity fluctuations at two time points of the scattered field for a partially coherent Gaussian Schell-model plane-wave pulse. We derive analytical expressions for the correlation of intensity fluctuations of a partially coherent Gaussian Schell-model plane-wave pulse scattered by deterministic and quasi-homogeneous random media.

2. Correlation of Intensity Fluctuations

We begin with a brief review of correlation of intensity fluctuations in the space-time domain at two space-time points, say (r₁, t₁) and (r₂, t₂) [23,24]. The intensity fluctuations are defined by the formula:

⟨ΔI(_rj,_tj)⟩=I(_rj,_tj)−⟨I(_rj,_tj)⟩ (j=1,2),

where:

I(_rj,_tj)=^E*(_rj,_tj)E(_rj,_tj),

is the instantaneous intensity of the field E(r_j, t_j). The asterisk denotes the complex conjugate and the angular bracket denotes the ensemble average. Then the correlation of intensity fluctuations at two space-time points (r₁, t₁) and (r₂, t₂) has the form:

⟨ΔI(_r1,_t1)ΔI(_r2,_t2)⟩=⟨I(_r1,_t1)I(_r2,_t2)⟩−⟨I(_r1,_t1)⟩⟨I(_r2,_t2)⟩.

We note that the first term on the right-hand side of Equation (3) is the fourth-order correlation function. Assuming that the field fluctuations obey Gaussian statistics and the first moment of the field E(r_j, t_j) is zero, the fourth-order correlation function can be expressed in terms of second-order moments by the Gaussian moment theorem:

⟨I(_r1,_t1)I(_r2,_t2)⟩=⟨^E*(_r1,_t1)E(_r1,_t1)⟩⟨^E*(_r2,_t2)E(_r2,_t2)⟩+⟨^E*(_r1,_t1)E(_r2,_t2)⟩⟨^E*(_r2,_t2)E(_r1,_t1)⟩=⟨I(_r1,_t1)⟩⟨I(_r2,_t2)⟩+^{|Γ(_r1,_r2,_t1,_t2)|2}.

On substituting from Equation (4) into Equation (3) we obtain for the correlation of intensity fluctuations the formula:

⟨ΔI(_r1,_t1)ΔI(_r2,_t2)⟩=^{|Γ(_r1,_r2,_t1,_t2)|2},

whereΓ(_r1,_t1,_r2,_t2)is the mutual coherence function of the field. Its normalized version:

Γ(_r1,_r2,_t1,_t2)Γ(_r1,_r1,_t1,_t1)Γ(_r2,_r2,_t2,_t2)=Γ(_r1,_r2,_t1,_t2)⟨I(_r1,_t1)⟩⟨I(_r2,_t2)⟩=γ(_r1,_r2,_t1,_t2),

is the complex degree of coherence of the field. From Equations (5) and (6) it follows that:

⟨ΔI(_r1,_t1)ΔI(_r2,_t2)⟩⟨I(_r1,_t1)⟩⟨I(_r2,_t2)⟩=^{|γ(_r1,_r2,_t1,_t2)|2}.

The term on the left-hand side of Equation (7) may be defined as the normalized correlation of intensity fluctuations of the field. This formula shows that the normalized correlation of intensity fluctuations is equal to the squared modulus of the degree of coherence at two space-time points (r₁,t₁) and (r₂,t₂). 3. Correlation of Intensity Fluctuations of a Partially Coherent Plane-Wave Pulse Scattered by Deterministic and Random Media

We began by considering a temporally partially coherent Gaussian Schell-model plane-wave pulse incident upon a linear scatterer, which occupies a finite domain D. In the space-time domain, the temporal coherence function of the pulses takes the form [14,15]:

Γ(_t1,_t2)=_Γ0exp[−_t12+_t222^T2−^(_t1−_t2)22_Tc2−i_ω0(_t1−_t2)],

where_ω0 is the central frequency of the pulse, T denotes the pulse duration, and T_c describes the temporal coherence length of the pulse. The cross-spectral density function of the pulses in the space-frequency domain can be readily determined by employing the Fourier transform [14,15], which yields:

W(_ω1,_ω2)=_W0exp[−^{(_ω1−_ω0)2}+^{(_ω2−_ω0)2}2^Ω2−^{(_ω1−_ω2)2}2_Ωc2],

where_W0=_Γ0T/2πΩand the spectral widthΩand the spectral coherence width_Ωcof the pulses are connected to the temporal parameters by the equations^Ω2=1/^T2+2/_Tc2and_Ωc=_TcΩ/T.

Suppose that the incident pulses propagate in a direction specified by a real unit vector s₀ (as shown in Figure 1), the cross-spectral density function of the incident light at a pair of points, specified by position vectors r₁ and r₂, is given by the formula:

^W(i)(_r1′,_r2′,_ω1,_ω2)=_W0exp[−^{(_ω1−_ω0)2}+^{(_ω2−_ω0)2}2^Ω2−^{(_ω1−_ω2)2}2_Ωc2]×exp[i(_{k2 s0}⋅_r2′−_{k1 s0}⋅_r1′)],

with_ki=_ωi/_ci(i = 1, 2) and c being the speed of light in vacuum.

We assumed that the medium is a weak scatterer, so that the scattering may be analyzed within the accuracy of the first-order Born approximation. The scattering potential of the medium, at a point specified by a position vector_ri′within the scatterer, is characterized byF(_ri′,_ωi). The resonance frequencies of the medium, i.e., the frequencies of its atomic or molecular transitions, were assumed not to lie within the spectral brand of the incident light. Thus, over the effective frequency range of the incident light, the scattering potential may be approximated byF(_ri′,_ω0).

For a deterministic medium, the scattering potentialF(_ri′,_ω0)is a well-defined function of position. The cross-spectral density function of the scattered light in the far zone, at two points specified by position vectors rs₁ and rs₂, is given by the approximate far-zone formula

^W(s)(r_s1,r_s2,_ω1,_ω2)=∫D∫D^W(i)(_r1′,_r2′,_ω1,_ω2)^F*(_r1′,_ω0)F(_r2′,_ω0)×1^r2exp[ir(_k2−_k1)−i(_{k2 s2}⋅_r2′−_{k1 s1}⋅_r1′)]^d3 _r1′ ^d3 _r2′.

For a random medium, the scattering potential is, of course, a random function of position. The correlation function of the scattering potential at a pair of points, specified by position vectors_r1′and_r2′in the scattering medium, is defined by the formula:

_CF(_r1′,_r2′,_ω0)=⟨^F*(_r1′,_ω0)F(_r2′,_ω0)⟩,

where the angle brackets denote the average value taken over the ensemble of random medium realizations. We then obtain the formula:

^W(s)(r_s1,r_s2,_ω1,_ω2)=∫D∫D^W(i)(_r1′,_r2′,_ω1,_ω2)_CF(_r1′,_r2′,_ω0)×1^r2exp[ir(_k2−_k1)−i(_{k2 s2}⋅_r2′−_{k1 s1}⋅_r1′)]^d3 _r1′ ^d3 _r2′.

The mutual coherence function of the scattered field is obtained by taking the inverse Fourier transform of the cross-spectral density function, which gives:

^Γ(s)(r_s1,r_s2,_ω1,_ω2)=∫_{∫−∞+∞}^W(s)(r_s1,r_s2,_ω1,_ω2)×exp[i(_{ω1 t1}−_{ω2 t2})d_ω1d_ω2.

We now illustrate our analysis by two examples. Suppose first that the pulse is incident on a deterministic spherical scatterer centered at the point r_c = (0,0,d), occupying a finite domain D, with a three-dimensional (soft) Gaussian potential:

F(r,_ω0)=_C0exp[−^x2+^y2+^(z−d)22^σ2],

where C₀ andσare positive constants, which are independent of position but may depend uponω. Given the properties of the scatterer and those of the incident field, we may use Equations (7), (10), (11), (14), and (15) to give the following expressions for the intensity statistics. The average intensity of the scattered field is:

⟨^I(s)(rs,t)⟩=^(2π)3 _C02 ^σ6 _Γ0T^r2(1+4^{σ2 sin2}θ2(1/^T2+2/_Tc2)/^c2)(^T2+4^{σ2 sin2}θ2/^c2)×exp[−^{(t−(2dsin2θ2+r)/c)2T2}+4^{σ2 sin2}θ2/^c2−_ω02[^c2/(4^{σ2 sin2}θ2)+(1/^T2+2/_Tc2)]],

and the normalized correlation of intensity fluctuations (NCIF) has the form:

⟨Δ^I(s)(rs,_t1)Δ^I(s)(rs,_t2)⟩⟨^I(s)(rs,_t1)⟩⟨^I(s)(rs,_t2)⟩=exp[−^(_t1−_t2)2_Tc2^T2[1+4^{σ2 sin2}θ2(1/^T2+2/_Tc2)/^c2][^T2+4^{σ2 sin2}θ2/^c2]],

whereθis the angle between the observation direction and the incident direction, i.e., s·s_{0 =}cosθ. Equations (16) and (17) give the analytical expressions for the average intensity and NCIF of the scattered field. The simple relationships make it easy to deriveσ of the medium if we get the intensity, or the correlation of intensity fluctuations which can be performed by Hanbury Brown-Twiss measurements. We calculated in Figure 2 and Figure 3 the normalized intensity and the NCIF of the scattered partially coherent Gaussian Schell-model plane-wave pulses for different values ofσof the deterministic spherical scatterer, respectively. The calculation parameters areT=15 fs,_Tc=10 fs,λ=800 nm,_ω0=2.36 rad/fsandt=rc . One finds from Figure 2 and Figure 3 that the normalized intensity and the NCIF of the scattered partially coherent Gaussian Schell-model plane-wave pulse are closely related toσ of the deterministic spherical scatterer, which may be useful in studying the inverse problem. Figure 2 shows that the scattering becomes more directional whenσ increases. And Figure 3 shows that the NCIF increases with the increase ofσ.

Next, let us suppose that the pulse is incident on a quasi-homogeneous random medium. The correlation function of the scattering potential of such a medium has the form:

_CF(_r1′,_r2′,_ω0)=_CR(^R′,_ω0)_Cr(^r′,_ω0)=_C0exp[−^(R′)22_σR2−^(r′)22_σr2],

where_C0,_σR, and_σrare positive constants with_σR≫_σr. The function_CR(^R′,_ω0)varies much more slowly with^R′=(_r1′+_r2′)/2than the function_Cr(^r′,_ω0)varies with^r′=_r2′−_r1′. It follows from Equations (7), (10), (13), (14), and (18) that the average intensity and the NCIF of the scattered field are given by the expressions:

⟨^I(s)(rs,t)⟩=^{(2π_{σR σr})3} _{C0 Γ0}T(^T2+8_σR2 ^sin2θ2/^c2)(1+2_σr2 ^sin2θ2(1/^T2+2/_Tc2)/^c2)×exp[−^{(t−r/c)2(T2+8_σR2 sin2θ2/c2)2}+_ω02(1/^T2+2/_Tc2)(1+2(1/^T2+2/_Tc2)_σr2 ^sin2θ2/^c2)],

⟨Δ^I(s)(rs,_t1)Δ^I(s)(rs,_t2)⟩⟨^I(s)(rs,_t1)⟩⟨^I(s)(rs,_t2)⟩=exp[−^(_t1−_t2)2(^T2+8_σR2 ^sin2θ2/^c2)(1/(1/^T2+2/_Tc2)+2_σr2 ^sin2θ2/^c2)(^T4/(_Tc2+2^T2)+(4_σR2−_σr2)^sin2θ2/^c2)].

It follows from the analytical expressions Equations (19) and (20) that for a quasi-homogeneous random medium, one can also derive_σRand_σr from both the average intensity and the NCIF of the scattered field. Figure 4 and Figure 5 illustrate the normalized intensity and the NCIF of the scattered partially coherent Gaussian Schell-model plane-wave for different values of_σRand_σr of the quasi-homogeneous random medium separately. Other calculation parameters are the same as those in Figure 2 and Figure 3. It is shown that_σRand_σrof the quasi-homogeneous random medium determine both the average intensity and the NCIF together; however, the effect of_σris more apparent than that of_σR . Figure 4 shows that the scattering becomes more directional when_σRor_σr increases. Figure 5 shows that the NCIF increases as_σrincreases or_σRdecreases. In applications, the troposphere and confined plasmas sometimes can be modeled as quasi-homogeneous random media.

4. Summary We have derived analytical formulas for the average intensity and the normalized correlation of intensity fluctuations for the scattered field of a partially coherent Gaussian Schell-model plane-wave pulse. We considered two typical scattering objects, a deterministic spherical scatterer and a quasi-homogeneous random medium. Our study extended the research of correlation of intensity fluctuations at two space points of the scattered stationary fields to that at two time points of scattered non-stationary fields.

Figure 1. Scattering problem geometry.s0andsrepresent the incident and scattering directions, respectively.θ is the scattering angle, and D is the domain the scatterer occupies.

Figure 2. The normalized intensity of the scattered partially coherent Gaussian Schell-model plane-wave pulse versusθfor different values ofσof the deterministic spherical scatterer.

Figure 3. The normalized correlation of intensity fluctuations (NCIF) of the scattered partially coherent Gaussian Schell-model plane-wave pulse versustd=t2-t1for different values ofσof the deterministic spherical scatterer withθ=0.5.

Figure 4. The normalized intensity of the scattered partially coherent Gaussian Schell-model plane-wave pulse versusθfor different values ofσRandσrof the quasi-homogeneous random medium.

Figure 5. The NCIF of the scattered partially coherent Gaussian Schell-model plane-wave pulse versustd=t2-t1for different values ofσRandσrof the quasi-homogeneous random medium withθ=0.5.

Author Contributions

Y.Z. (Data curation, Writing-original draft, Methodology); C.D. (Formal analysis, Writing review and editing); L.P. (Project administration, Writing-review and editing), Y.C. (Project administration, Writing-review and editing).

Funding

This research was funded by the National Natural Science Foundation of China, grant number 11474143, 61675094, 61575091 and 91750201; the National Natural Science Fund for Distinguished Young Scholar, grant number 11525418; and the Foundation of Henan Educational Committee, grant number 2016GGJS-116 and 17A140001.

Conflicts of Interest

The authors declare no conflict of interest.

References

1. Wolf, E.; Foley, J.T. Scattering of electromagnetic fields of any state of coherence from space-time fluctuations. Phys. Rev. A 1989, 40, 579-587.

2. Van Dijk, T.; Fischer, D.G.; Visser, T.D.; Wolf, E. Effects of Spatial Coherence on the Angular Distribution of Radiant Intensity Generated by Scattering on a Sphere. Phys. Rev. Lett. 2010, 104, 173902.

3. Tong, Z.; Korotkova, O. Theory of weak scattering of stochastic electromagnetic fields from deterministic and random media. Phys. Rev. A 2010, 82, 033836.

4. Gbur, G. Designing directional cloaks from localized fields. Opt. Lett. 2015, 40, 986-989.

5. Wang, Y.; Yan, S.; Kuebel, D.; Visser, T.D. Dynamic control of light scattering using spatial coherence. Phys. Rev. A 2015, 92, 013806.

6. Li, J.; Chang, L. Spectral shifts and spectral switches of light generated by scattering of arbitrary coherent waves from a quasi-homogeneous media. Opt. Express 2015, 23, 16602-16616.

7. Wang, Y.; Kuebel, D.; Visser, T.D.; Wolf, E. Creating von Laue patterns in crystal scattering with partially coherent sources. Phys. Rev. A 2016, 94, 033812.

8. Wang, T.; Wu, H.; Ding, Y.; Ji, X.; Zhao, D. Changes in the spectral degree of coherence of a light wave on scattering from a particulate medium. Opt. Commun. 2016, 381, 210-213.

9. Hurwitz, E.; Gbur, G. Optically switchable directional invisibility. Opt. Lett. 2017, 42, 1301-1304.

10. Zhao, D.; Korotkova, O.; Wolf, E. Application of correlation-induced spectral changes to inverse scattering. Opt. Lett. 2007, 32, 3483-3485.

11. Lahiri, M.; Wolf, E.; Fischer, D.G.; Shirai, T. Determination of Correlation Functions of Scattering Potentials of Stochastic Mediafrom Scattering Experiments. Phys. Rev. Lett. 2009, 102, 123901.

12. Wu, H.; Pan, X.; Zhu, Z.; Ji, X.; Wang, T. Reciprocity relations of an electromagnetic light wave on scattering from a quasi-homogeneous anisotropic medium. Opt. Express 2017, 25, 11297-11305.

13. Zhao, D.; Wang, T. Direct and inverse problems in the theory of light scattering. Prog. Opt. 2012, 57, 261-308.

14. Pääkkönen, P.; Turunen, J.; Vahimaa, P.; Friberg, A.T.; Wyrowski, F. Partially coherent Gaussian pulses. Opt. Commun. 2002, 204, 53-58.

15. Lajunen, H.; Vahimaa, P.; Tervo, J. Theory of spatially and spectrally partially coherent pulses. J. Opt. Soc. Am. A 2005, 22, 1536-1545.

16. Ding, C.; Cai, Y.; Korotkova, O.; Zhang, Y.; Pan, L. Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse. Opt. Lett. 2011, 36, 517-519.

17. Ding, C.; Cai, Y.; Zhang, Y.; Pan, L. Scattering-induced changes in the degree of polarization of a stochastic electromagnetic plane-wave pulse. J. Opt. Soc. Am. A 2012, 29, 1078-1090.

18. Tang, M.; Zhao, D.; Zhu, Y.; Ang, L.K. Electromagnetic sinc Schell-model pulses in dispersive media. Phys. Lett. A 2016, 380, 794-797.

19. Wu, G.; Visser, T.D. Correlation of intensity fluctuations in beams generated by quasi-homogeneous sources. J. Opt. Soc. Am. A 2014, 31, 2152-2159.

20. Xin, Y.; He, Y.; Chen, Y.; Li, J. Correlation between intensity fluctuations of light scattered from a quasi-homogeneous random media. Opt. Lett. 2010, 35, 4000-4002.

21. Jacks, H.C.; Korotkova, O. Intensity-intensity fluctuations of stochastic fields produced upon weak scattering. J. Opt. Soc. Am. A 2011, 28, 1139-1144.

22. Kuebel, D.; Visser, T.D.; Wolf, E. Application of the Hanbury Brown-Twiss effect to scattering from quasi-homogeneous media. Opt. Commun. 2013, 294, 43-48.

23. Wolf, E. Introduction to the Theory of Coherence and Polarization of Light; Cambridge University Press: New York, NY, USA, 2007.

24. Mandel, L.; Wolf, E. Optical Coherence and Quantum Optics; Cambridge University Press: New York, NY, USA, 1995.