Spectral shifts and spectral switches produced by the scattering system of two anisotropic particles in different distance

Xinyue Du

doi:10.1364/OE.21.022610

1. Introduction

It has been shown theoretically that the spectrum of light wave may experience changes when it propagates in free space [1]. The spectral shift can also be produced by coherence [2], scattering [3], and focusing of light waves [4]. Further investigations indicate that the spectral shifts occur when partially coherent light waves propagate through apertures [5,6], propagate in turbulent atmosphere [7] and negative-phase materials [8]. In some special cases shown by these literatures [5–8], 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 [3,9–12]. We have also studied the spectral shifts produced by the rotation of a quasi-homogeneous anisotropic medium or an anisotropic particle [13,14].

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 $s_{0}$ is incident upon a scattering system formed with two anisotropic particles, which is shown in Fig. 1. The field of the incident light wave is expressed by the form

{U^{(i)} (r^{'}, ω)} = {a (ω)} \exp (i k s_{0} \cdot r^{'}),

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.

Fig. 1 Illustrating the notation relating to the scattering of a plane light wave by a system formed with two anisotropic particles.

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The cross-spectral density function of the incident field at points specified by position vectors ${r^{'}}_{1}$ and ${r^{'}}_{2}$ may be given by the expression ([15], Sec. 4.1)

W^{(i)} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = 〈 U^{(i) *} ({r^{'}}_{1}, ω) U^{(i)} ({r^{'}}_{2}, ω) 〉,

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)} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = S^{(i)} (ω) \exp [i k s_{0} \cdot ({r^{'}}_{2} - {r^{'}}_{1})],

where

S^{(i)} (ω) = 〈 a^{*} (ω) a (ω) 〉

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) = A \exp [- \frac{{(ω - ω_{0})}^{2}}{2 Γ_{0}^{2}}],

where

A

is a positive constant,

ω_{0} = 2 π c / λ_{0}

is the central frequency with

λ_{0}

being the central wavelength,

Γ_{0}

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)} ({r^{'}}_{1}, {r^{'}}_{2}, ω) = \exp [i k s_{0} \cdot ({r^{'}}_{2} - {r^{'}}_{1})],

which indicates that the incident light wave is complete spatial coherence because of

| μ^{(i)} ({r^{'}}_{1}, {r^{'}}_{2}, ω) | = 1.

The cross-spectral density function of the scattered field at points specified by position vectors $r_{1}$ and $r_{2}$ is given by the expression ([15], Sec. 6.2)

W^{(s)} (r_{1}, r_{2}, ω) = 〈 U^{(s) *} (r_{1}, ω) U^{(s)} (r_{2}, ω) 〉,

with

U^{(s)} (r, ω) = U^{(s)} (r s, ω) = \sqrt{S^{(i)} (ω)} \frac{\exp (i k r)}{r} \int_{D} F (r^{'}, ω) \exp [- i k (s - s_{0}) \cdot r^{'}] d^{3} r^{'},

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) = B \exp [- \frac{{x^{'}}^{2}}{2 σ_{x}^{2}} - \frac{{y^{'}}^{2}}{2 σ_{y}^{2}} - \frac{{z^{'}}^{2}}{2 σ_{z}^{2}}],

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

σ_{x} = σ_{y} = σ_{z} .

The anisotropic particle centered at the off-center position

r_{o}

with coordinates

(x_{o}, y_{o}, z_{o})

has the scattering potential

F_{o} (r^{'}, ω) = B \exp [- \frac{{(x^{'} - x_{o})}^{2}}{2 σ_{o x}^{2}} - \frac{{(y^{'} - y_{o})}^{2}}{2 σ_{o y}^{2}} - \frac{{(z^{'} - z_{o})}^{2}}{2 σ_{o z}^{2}}],

where

σ_{o (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) = B \exp (- {r^{'}}^{T} M r^{'}),

where

r^{'} = {(x^{'}, y^{'}, z^{'})}^{T}

and T means the transpose operation.

M

is a

3 \times 3

diagonal matrix

M = \frac{1}{2} [\begin{matrix} σ_{x}^{- 2} & 0 & 0 \\ 0 & σ_{y}^{- 2} & 0 \\ 0 & 0 & σ_{z}^{- 2} \end{matrix}] .

Equation (10) can be rewritten by the tensor form

F_{o} (r^{'}, ω) = B \exp [- {(r^{'} - r_{o})}^{T} M_{o} (r^{'} - r_{o})],

where

r_{o} = {(x_{o}, y_{o}, z_{o})}^{T} .

M_{o}

is a

3 \times 3

diagonal matrix

M_{o} = \frac{1}{2} [\begin{matrix} σ_{o x}^{- 2} & 0 & 0 \\ 0 & σ_{o y}^{- 2} & 0 \\ 0 & 0 & σ_{o z}^{- 2} \end{matrix}] .

Equation (8) can also be rewritten in the tensor form

U^{(s)} (r s, ω) = \sqrt{S^{(i)} (ω)} \frac{\exp (i k r)}{r} \int_{D} F (r^{'}, ω) \exp (- i {r^{'}}^{T} K) d^{3} r^{'},

where

K = k (s - s_{0})

with

s_{0} = {(s_{0 x}, s_{0 y}, s_{0 z})}^{T}

and

s = {(s_{x}, s_{y}, s_{z})}^{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)} (r s, ω) = π^{3 / 2} B \sqrt{S^{(i)} (ω)} \frac{\exp (i k r)}{r} {[\det (M)]}^{- 1 / 2} \exp (- \frac{1}{4} K^{T} M^{- 1} K),

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:

U_{o}^{(s)} (r s, ω) = π^{3 / 2} B \sqrt{S^{(i)} (ω)} \frac{\exp (i k r)}{r} {[\det (M_{o})]}^{- 1 / 2} \exp (- \frac{1}{4} K^{T} M_{o}^{- 1} K) \exp (- i r_{o}^{T} K) .

The cross-spectral density function and the spectrum of the scattered field can then be obtained by the formulae

W^{(s)} (r s_{1}, r s_{2}, ω) = 〈 {[U^{(s)} (r s_{1}, ω) + U_{o}^{(s)} (r s_{1}, ω)]}^{*} [U^{(s)} (r s_{2}, ω) + U_{o}^{(s)} (r s_{2}, ω)] 〉,

and

S^{(s)} (r s, ω) \equiv W^{(s)} (r s, r s, ω) .

From Eqs. (16), (17), (19) and using Eq. (5) one readily finds that

S^{(s)} (r s, ω) = \frac{π^{3} A B^{2}}{r^{2}} \exp [- \frac{{(ω - ω_{0})}^{2}}{2 Γ_{0}^{2}}] {| H (s, ω) |}^{2},

where

\begin{array}{l} H (s, ω) = {[\det (M)]}^{- 1 / 2} \exp (- \frac{1}{4} K^{T} M^{- 1} K) \\ + {[\det (M_{o})]}^{- 1 / 2} \exp (- \frac{1}{4} K^{T} M_{o}^{- 1} K) \exp (- i r_{o}^{T} K) . \end{array}

If the two particles have the same effective radius in three dimensions, which means

M = M_{o},

the function

H (s, ω)

has the form

H (s, ω) = {[\det (M)]}^{- 1 / 2} \exp (- \frac{1}{4} K^{T} M^{- 1} K) [1 + \exp (- i r_{o}^{T} K)] .

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 $s_{0} = {(0, 0, 1)}^{T} .$ The other fixed parameters are chosen as follows: $c = 3 \times 10^{8} m/s,$ $λ_{0} = 550 nm,$ and $Γ_{0} = 10^{- 2} ω_{0} .$ In the calculation, the off-center anisotropic particle is assumed to move only along the y direction, which indicates $r_{o} = {(0, d, 0)}^{T}$ as shown in Fig. 1.

In Fig. 2, 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 $5.01 λ_{0}$ to $5.04 λ_{0},$ the change speed of the heights of peaks is slow. But in the neighborhood of $d = 5.046 λ_{0}$ the maximum peak and the second peak make a rapid transition, which indicates the appearance of the spectral switch.

Fig. 2 The solid lines show the spectral shifts and spectral switches produced by the scattering system of two same anisotropic particles having $σ_{x} = σ_{o x} = λ_{0},$ $σ_{y} = σ_{o y} = 2 λ_{0},$ and $σ_{z} = σ_{o z} = 3 λ_{0} .$ We observe in the direction of scattering $s = {(0, 1 / 2, \sqrt{3} / 2)}^{T} .$ The off-center anisotropic particle is located with different distance: (a) $d = 5.01 λ_{0},$ (b) $d = 5.04 λ_{0},$ (c) $d = 5.046 λ_{0},$ (d) $d = 5.05 λ_{0} .$ The dotted lines show the spectrum of the incident field.

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In Fig. 3, 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. $Δ ω = ω_{0}^{(S)} - ω_{0}$ is the spectral shift between the central frequencies of the scattered field and the incident field. The central frequency $ω_{0}^{(S)}$ 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 ( $Δ ω < 0$ ) and the blue shift ( $Δ ω > 0$ ) 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.

Fig. 3 Spectral shifts produced by the scattering system of two same anisotropic particles having $σ_{x} = σ_{o x} = λ_{0},$ $σ_{y} = σ_{o y} = 2 λ_{0},$ and $σ_{z} = σ_{o z} = 3 λ_{0}$ with distance $d .$ We observe in the direction of scattering: (a) $s = {(0, 1 / 2, \sqrt{3} / 2)}^{T},$ (b) $s = {(0, \sqrt{3} / 2, 1 / 2)}^{T} .$

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Figure 4 shows the spectral shifts produced by the scattering system of two different anisotropic particles. We calculate the solid line by setting $σ_{o z} = 3.1 λ_{0} .$ 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 $σ_{o z} = 3.5 λ_{0} .$ The larger difference of the particles makes the spectral switch disappear.

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 $σ_{x} = λ_{0},$ $σ_{y} = 2 λ_{0},$ and $σ_{z} = 3 λ_{0} .$ The solid line is calculated by choosing the effective radius of the off-center anisotropic particle as follows: $σ_{o x} = λ_{0},$ $σ_{o y} = 2 λ_{0},$ and $σ_{o z} = 3.1 λ_{0} .$ The dotted line is calculated by choosing $σ_{o x} = λ_{0},$ $σ_{o y} = 2 λ_{0},$ and $σ_{o z} = 3.5 λ_{0} .$ We observe in the direction of scattering $s = {(0, 1 / 2, \sqrt{3} / 2)}^{T} .$

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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).

Spectral shifts and spectral switches produced by the scattering system of two anisotropic particles in different distance

Abstract

1. Introduction

2. Theoretical analysis

3. Numerical examples

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Equations (22)

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