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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 22610–22616
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Spectral shifts and spectral switches produced by the scattering system of two anisotropic particles in different distance

Xinyue Du  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22610-22616 (2013)
http://dx.doi.org/10.1364/OE.21.022610


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Abstract

We investigate the scattering of polychromatic plane light wave incident upon a system formed with two anisotropic particles in different distance. The analytical expression for the spectrum of the scattered field is derived. Numerical examples show the phenomena of spectral shifts and spectral switches of the scattered field. The influences of the scattering direction and the difference of the particles on the spectral switch are illustrated.

© 2013 OSA

1. Introduction

It has been shown theoretically that the spectrum of light wave may experience changes when it propagates in free space [1

1. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986). [CrossRef] [PubMed]

]. The spectral shift can also be produced by coherence [2

2. E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58(25), 2646–2648 (1987). [CrossRef] [PubMed]

], scattering [3

3. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989). [CrossRef]

], and focusing of light waves [4

4. G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88(1), 013901 (2001). [CrossRef] [PubMed]

]. Further investigations indicate that the spectral shifts occur when partially coherent light waves propagate through apertures [5

5. J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162(1-3), 57–63 (1999). [CrossRef]

,6

6. L. Pan, C. Ding, and X. Yuan, “Spectral shifts and spectral switches of twisted Gaussian Schell-model beams passing through an aperture,” Opt. Commun. 274(1), 100–104 (2007). [CrossRef]

], propagate in turbulent atmosphere [7

7. X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259(1), 1–6 (2006). [CrossRef]

] and negative-phase materials [8

8. Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. A 82(1), 013829 (2010). [CrossRef]

]. In some special cases shown by these literatures [5

5. J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162(1-3), 57–63 (1999). [CrossRef]

8

8. Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. A 82(1), 013829 (2010). [CrossRef]

], the maximum peak and the second peak of the spectral line make a rapid transition, which is considered as the phenomenon of spectral switch.

When light wave scattered by different types of media, the information about the scatterer may be obtained from the measurement of the scattered field. The change in the spectrum of polychromatic light wave is important in inverse problems of scattering. The spectral changes produced by scattering from random or deterministic media have been investigated [3

3. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989). [CrossRef]

,9

9. D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(3), 1128–1135 (1994). [CrossRef]

12

12. T. Wang and D. Zhao, “Effects of source correlation on the spectral shift of light waves on scattering,” Opt. Lett. 38(9), 1545–1547 (2013). [CrossRef] [PubMed]

]. We have also studied the spectral shifts produced by the rotation of a quasi-homogeneous anisotropic medium or an anisotropic particle [13

13. X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett. 36(24), 4749–4751 (2011). [CrossRef] [PubMed]

,14

14. X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012). [CrossRef]

].

In this paper, we study the scattering of polychromatic plane light wave incident upon a system of two anisotropic particles in different distance. The analytical expression for the spectrum of the scattered field is derived. The phenomena of spectral shifts and spectral switches of the scattered field are illustrated by numerical examples.

2. Theoretical analysis

We assume that a polychromatic plane light wave propagating in the direction of a unit vector s0 is incident upon a scattering system formed with two anisotropic particles, which is shown in Fig. 1
Fig. 1 Illustrating the notation relating to the scattering of a plane light wave by a system formed with two anisotropic particles.
. The field of the incident light wave is expressed by the form
{U(i)(r,ω)}={a(ω)}exp(iks0r),
(1)
where a(ω) is a random amplitude with the curly brackets denoting the statistical ensemble. k=ω/c is the wave number associated with the frequency ω and the speed of light in vacuum c. r is a three-dimensional position vector.

The cross-spectral density function of the incident field at points specified by position vectors r1 and r2 may be given by the expression ([15

15. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

], Sec. 4.1)
W(i)(r1,r2,ω)=U(i)(r1,ω)U(i)(r2,ω),
(2)
where the angular brackets denote the ensemble average, the asterisk denotes the complex conjugate. On substituting from Eq. (1) into Eq. (2), we obtain the expression
W(i)(r1,r2,ω)=S(i)(ω)exp[iks0(r2r1)],
(3)
where
S(i)(ω)=a(ω)a(ω)
(4)
represents the spectrum of the incident field. In the following analysis, it is supposed to have a single spectral line of Gaussian profile as follows:
S(i)(ω)=Aexp[(ωω0)22Γ02],
(5)
where A is a positive constant, ω0=2πc/λ0 is the central frequency with λ0 being the central wavelength, Γ0 denotes the linewidth of the spectrum. The spectral degree of coherence of the incident field can be obtained from Eq. (3) by use of the expression
μ(i)(r1,r2,ω)=exp[iks0(r2r1)],
(6)
which indicates that the incident light wave is complete spatial coherence because of |μ(i)(r1,r2,ω)|=1.

The cross-spectral density function of the scattered field at points specified by position vectors r1 and r2 is given by the expression ([15

15. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

], Sec. 6.2)
W(s)(r1,r2,ω)=U(s)(r1,ω)U(s)(r2,ω),
(7)
with
U(s)(r,ω)=U(s)(rs,ω)=S(i)(ω)exp(ikr)rDF(r,ω)exp[ik(ss0)r]d3r,
(8)
where s is the unit vectors of the scattered field, r is the position vector of the scattered field with r=|r|. F(r,ω) is the scattering potential of individual particle.

As shown in Fig. 1, the scattering system is constituted by two anisotropic particles. The anisotropic particle centered at the origin of the coordinate has the scattering potential
F(r,ω)=Bexp[x22σx2y22σy2z22σz2],
(9)
where B is a positive constants and σ(x,y,z) denotes the effective radius in three directions of the scattering potential. It reduces to the case of isotropic particle if σx=σy=σz. The anisotropic particle centered at the off-center position ro with coordinates (xo,yo,zo) has the scattering potential
Fo(r,ω)=Bexp[(xxo)22σox2(yyo)22σoy2(zzo)22σoz2],
(10)
where σo(x,y,z) denotes the effective radius in three directions of the scattering potential of the off-center particle.

Equation (9) can be rewritten by the tensor form
F(r,ω)=Bexp(rTMr),
(11)
where r=(x,y,z)T and T means the transpose operation. M is a 3×3 diagonal matrix
M=12[σx2000σy2000σz2].
(12)
Equation (10) can be rewritten by the tensor form
Fo(r,ω)=Bexp[(rro)TMo(rro)],
(13)
where ro=(xo,yo,zo)T. Mo is a 3×3 diagonal matrix
Mo=12[σox2000σoy2000σoz2].
(14)
Equation (8) can also be rewritten in the tensor form
U(s)(rs,ω)=S(i)(ω)exp(ikr)rDF(r,ω)exp(irTK)d3r,
(15)
where K=k(ss0) with s0=(s0x,s0y,s0z)T and s=(sx,sy,sz)T. On substituting from Eq. (11) into Eq. (15) and after a vector integral operation the analytical expression of the scattered field produced by the anisotropic particle centered at the origin is derived as follows:
U(s)(rs,ω)=π3/2BS(i)(ω)exp(ikr)r[det(M)]1/2exp(14KTM1K),
(16)
where det denotes the determinant. On substituting from Eq. (13) into Eq. (15) the analytical expression of the scattered field produced by the off-center anisotropic particle is derived as follows:

Uo(s)(rs,ω)=π3/2BS(i)(ω)exp(ikr)r[det(Mo)]1/2exp(14KTMo1K)exp(iroTK).
(17)

The cross-spectral density function and the spectrum of the scattered field can then be obtained by the formulae
W(s)(rs1,rs2,ω)=[U(s)(rs1,ω)+Uo(s)(rs1,ω)][U(s)(rs2,ω)+Uo(s)(rs2,ω)],
(18)
and
S(s)(rs,ω)W(s)(rs,rs,ω).
(19)
From Eqs. (16), (17), (19) and using Eq. (5) one readily finds that
S(s)(rs,ω)=π3AB2r2exp[(ωω0)22Γ02]|H(s,ω)|2,
(20)
where
H(s,ω)=[det(M)]1/2exp(14KTM1K)+[det(Mo)]1/2exp(14KTMo1K)exp(iroTK).
(21)
If the two particles have the same effective radius in three dimensions, which means M=Mo, the function H(s,ω) has the form

H(s,ω)=[det(M)]1/2exp(14KTM1K)[1+exp(iroTK)].
(22)

3. Numerical examples

Numerical examples are given relating to the spectrum of the scattered field when a polychromatic plane light wave is incident upon the system of two anisotropic particles in different distance. The direction of the incident field is assumed to be along the z direction, which means s0=(0,0,1)T. The other fixed parameters are chosen as follows: c=3×108m/s, λ0=550nm, and Γ0=102ω0. In the calculation, the off-center anisotropic particle is assumed to move only along the y direction, which indicates ro=(0,d,0)T as shown in Fig. 1.

In Fig. 2
Fig. 2 The solid lines show the spectral shifts and spectral switches produced by the scattering system of two same anisotropic particles having σx=σox=λ0, σy=σoy=2λ0, and σz=σoz=3λ0. We observe in the direction of scattering s=(0,1/2,3/2)T. The off-center anisotropic particle is located with different distance: (a) d=5.01λ0, (b) d=5.04λ0, (c) d=5.046λ0, (d) d=5.05λ0. The dotted lines show the spectrum of the incident field.
, we show the normalized spectrum of the scattered field by solid lines that are obvious different from the normalized spectrum of the incident field shown by dotted lines. This phenomenon is considered as spectral shift. The spectrum of the scattered field can have two peaks named the maximum peak and the second peak. With the variation of the distance between the two particles both the positions and the heights of these two peaks experience changes. When the distance increases from 5.01λ0 to 5.04λ0, the change speed of the heights of peaks is slow. But in the neighborhood of d=5.046λ0 the maximum peak and the second peak make a rapid transition, which indicates the appearance of the spectral switch.

In Fig. 3
Fig. 3 Spectral shifts produced by the scattering system of two same anisotropic particles having σx=σox=λ0, σy=σoy=2λ0, and σz=σoz=3λ0 with distance d. We observe in the direction of scattering: (a) s=(0,1/2,3/2)T, (b) s=(0,3/2,1/2)T.
, we show the spectral shifts produced by the scattering system of two same anisotropic particles with the change of distance d. If the distance is small, the scattering potentials of these particles may superpose, and we consider that the scattering happens independently. Δω=ω0(S)ω0 is the spectral shift between the central frequencies of the scattered field and the incident field. The central frequency ω0(S) of the scattered field is defined as the frequency at which the spectrum of the scattered field takes the maximum. In Fig. 3(a), both the red shift (Δω<0) and the blue shift (Δω>0) occur during the increasing of distance. The positions where the spectral shifts show rapid transitions are the spectral switches. In the calculation of Fig. 3(b), we change the direction of scattering. Comparing Fig. 3(b) with Fig. 3(a), we find the space between the spectral switches is shorter and the extent of spectral switch decreases, which results in the disappearance of blue shift, because of the increasing of the scattering angle. The extent of each spectral switch is the same as shown in Fig. 3(a) also in Fig. 3(b), because the anisotropic particles in the scattering system are the same.

Figure 4
Fig. 4 Spectral shifts produced by the scattering system of two different anisotropic particles with distance d. The anisotropic particle centered at the origin is fixed with the effective radius σx=λ0, σy=2λ0, and σz=3λ0. The solid line is calculated by choosing the effective radius of the off-center anisotropic particle as follows: σox=λ0, σoy=2λ0, and σoz=3.1λ0. The dotted line is calculated by choosing σox=λ0, σoy=2λ0, and σoz=3.5λ0. We observe in the direction of scattering s=(0,1/2,3/2)T.
shows the spectral shifts produced by the scattering system of two different anisotropic particles. We calculate the solid line by setting σoz=3.1λ0. Comparing it with Fig. 3(a), we find the positions where the spectral switches occur are changeless but the extent of each spectral switch is less. It can be explained that the spectral switch is weakened by the difference of the particles. But this effect of weakening is faded by the distance between the two particles because the extent increases with the increasing of d. The dotted line is calculated by setting σoz=3.5λ0. The larger difference of the particles makes the spectral switch disappear.

4. Conclusions

We conclude by saying that we have obtained in this paper the analytical expression for the spectrum of the scattered field when a polychromatic plane light wave is incident upon a system formed with two anisotropic particles in different distance. The derived formula provides us an effective and convenient way to investigate the spectral shifts and spectral switches of the scattered field. Numerical examples show that the space and extent of the spectral switches are influenced by the scattering direction. The spectral switch is weakened or eliminated by the difference of the two anisotropic particles.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (11104055), and the Zhejiang Provincial Natural Science Foundation of China (Y6110271).

References and links

1.

E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986). [CrossRef] [PubMed]

2.

E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett. 58(25), 2646–2648 (1987). [CrossRef] [PubMed]

3.

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989). [CrossRef]

4.

G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. 88(1), 013901 (2001). [CrossRef] [PubMed]

5.

J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun. 162(1-3), 57–63 (1999). [CrossRef]

6.

L. Pan, C. Ding, and X. Yuan, “Spectral shifts and spectral switches of twisted Gaussian Schell-model beams passing through an aperture,” Opt. Commun. 274(1), 100–104 (2007). [CrossRef]

7.

X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun. 259(1), 1–6 (2006). [CrossRef]

8.

Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. A 82(1), 013829 (2010). [CrossRef]

9.

D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(3), 1128–1135 (1994). [CrossRef]

10.

A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998). [CrossRef] [PubMed]

11.

M. Lahiri and E. Wolf, “Spectral changes of stochastic beams scattered on a deterministic medium,” Opt. Lett. 37(13), 2517–2519 (2012). [CrossRef] [PubMed]

12.

T. Wang and D. Zhao, “Effects of source correlation on the spectral shift of light waves on scattering,” Opt. Lett. 38(9), 1545–1547 (2013). [CrossRef] [PubMed]

13.

X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett. 36(24), 4749–4751 (2011). [CrossRef] [PubMed]

14.

X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun. 285(6), 934–936 (2012). [CrossRef]

15.

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

OCIS Codes
(030.1670) Coherence and statistical optics : Coherent optical effects
(290.5850) Scattering : Scattering, particles

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: July 22, 2013
Revised Manuscript: September 10, 2013
Manuscript Accepted: September 10, 2013
Published: September 18, 2013

Citation
Xinyue Du, "Spectral shifts and spectral switches produced by the scattering system of two anisotropic particles in different distance," Opt. Express 21, 22610-22616 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22610


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References

  1. E. Wolf, “Invariance of the spectrum of light on propagation,” Phys. Rev. Lett.56(13), 1370–1372 (1986). [CrossRef] [PubMed]
  2. E. Wolf, “Red shifts and blue shifts of spectral lines emitted by two correlated sources,” Phys. Rev. Lett.58(25), 2646–2648 (1987). [CrossRef] [PubMed]
  3. E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A6(8), 1142–1149 (1989). [CrossRef]
  4. G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett.88(1), 013901 (2001). [CrossRef] [PubMed]
  5. J. Pu, H. Zhang, and S. Nemoto, “Spectral shifts and spectral switches of partially coherent light passing through an aperture,” Opt. Commun.162(1-3), 57–63 (1999). [CrossRef]
  6. L. Pan, C. Ding, and X. Yuan, “Spectral shifts and spectral switches of twisted Gaussian Schell-model beams passing through an aperture,” Opt. Commun.274(1), 100–104 (2007). [CrossRef]
  7. X. Ji, E. Zhang, and B. Lü, “Changes in the spectrum of Gaussian Schell-model beams propagating through turbulent atmosphere,” Opt. Commun.259(1), 1–6 (2006). [CrossRef]
  8. Z. Tong and O. Korotkova, “Spectral shifts and switches in random fields upon interaction with negative-phase materials,” Phys. Rev. A82(1), 013829 (2010). [CrossRef]
  9. D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A11(3), 1128–1135 (1994). [CrossRef]
  10. A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett.23(17), 1340–1342 (1998). [CrossRef] [PubMed]
  11. M. Lahiri and E. Wolf, “Spectral changes of stochastic beams scattered on a deterministic medium,” Opt. Lett.37(13), 2517–2519 (2012). [CrossRef] [PubMed]
  12. T. Wang and D. Zhao, “Effects of source correlation on the spectral shift of light waves on scattering,” Opt. Lett.38(9), 1545–1547 (2013). [CrossRef] [PubMed]
  13. X. Du and D. Zhao, “Spectral shifts produced by scattering from rotational quasi-homogeneous anisotropic media,” Opt. Lett.36(24), 4749–4751 (2011). [CrossRef] [PubMed]
  14. X. Du and D. Zhao, “Frequency shifts of spectral lines induced by scattering from a rotational anisotropic particle,” Opt. Commun.285(6), 934–936 (2012). [CrossRef]
  15. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

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