Spectral shifts and spectral switches of light generated by scattering of arbitrary coherent waves from a quasi-homogeneous media

Jia Li; Liping Chang

doi:10.1364/OE.23.016602

1. Introduction

The measurement of spectra of light is a crucial task in many fields, e.g. astronomy, remote sensing or objective recognization [1–3]. In 1980s, Wolf et al. had a concern about whether the normalized spectrum of light could remain invariant upon the propagation in free space. They showed that the normalized spectrum of light commonly differs from its initial profile unless the spectral degree of the coherence of light across the source satisfies a scaling law [4]. This prediction was further confirmed by experiments where the spectral lines of waves were proven to shift toward longer or shorter wavelengths [5]. Within the accuracy of the Born approximation, it was also shown that the scattering of light from random media can substantially alter the spectral properties of light [6–8]. By extending the first-order Born approximation to the Rytov approximation, the spectrum of light generated by the scattering from random media was found to differ from its initial profile [9]. Further studies were done to compare results between different approximations used in the scattering theory [10]. In addition, it is remarkable to note that the far-zone spectrum generated by the scattering of a polychromatic wave from random media can, in general, be the same in every direction in scattered field, provided that the two-point correlation function of media obeys a scaling law [11].

In effort to modeling scattering effects by a collection of particles, the spectrum of light scattered from multiple particles was analyzed at length. It was found that changes in the spectra of scattered field can be utilized to determine the structural properties of particles [12]. Furthermore, a scaling law was derived to ensure the invariance of spectra of a stochastic electromagnetic beam [13]. Also, a method for determining unknown scatterer was proposed by using the correlation-induced spectral changes [14]. Subsequently, this method was further extended to the case where the scattering of light from a collection of particles was studied [15]. It was indicated in these literatures that the spectral switch of light from the red shift to blue shift can be realized by altering the scattering angle in the far-zone. It was shown in literatures [16–18] that the spectrum of backscattered light from biological tissues shifts toward a higher frequency compared to the initial incident light. This phenomenon is generated from influences of the two-point spatial correlation function of biological tissue. The spectral shifts of light scattered from disordered anisotropic media was first addressed in [19], and then was further extended to the case where the scattering of light from a rotational QH anisotropic media was studied [20]. Apart from the aforementioned investigations, the spectral switches which may be generated from the diffraction of light from a sharp edged aperture were also concerned in [21].

To date, however, the incident light waves considered in aforementioned studies mostly occupy completely coherent properties. The discussion on the scattering of arbitrary coherent beams from QH media, to the best our knowledge, has not been concerned so far. To this effort, we study the case where an arbitrary coherent polychromatic wave interacts with a QH media, while the Born approximation is utilized to treat the weak scattering process. Particularly, a pair of Young’s pinholes is utilized to modulate the coherence of the incident plan wave. Furthermore, substantial concentrations are focused on the spectral properties of scattered field from an arbitrary coherent wave. The dependence of spectral shifts and spectral switches of scattered field on the scattering angle, correlation length of the scatterer and the Young’s configuration parameter is studied at length, respectively. In addition, numerical simulations are performed to better show the derived results.

2. Scattering of an arbitrary coherent wave from a QH media

Let us suppose that a scalar polychromatic plane wave transmits through a pair of Young’s pinholes before interacting with a QH media, as shown in Fig. 1. An opaque screen A with two pinholes located at $Q ({\hat{ρ}}_{1 ⊥})$ and $Q ({\hat{ρ}}_{2 ⊥})$ is utilized to block the incident wave. ${\hat{ρ}}_{1 ⊥}$ and ${\hat{ρ}}_{2 ⊥}$ are the projective position vectors of ${\hat{ρ}}_{1}$ and ${\hat{ρ}}_{2}$ , respectively. $φ_{1}$ and $φ_{2}$ represent the azimuthal angles with regard to ${\hat{ρ}}_{1}$ and ${\hat{ρ}}_{2}$ . The QH scatterer occupies a finite spatial volume D. $P ({\hat{r}}_{1}')$ and $P ({\hat{r}}_{2}')$ are two points where the transmitted wave further interacts with the QH scatterer. ${\hat{r}}_{1}$ and ${\hat{r}}_{2}$ denote the position vectors in scattered field, ${\hat{s}}_{1}$ and ${\hat{s}}_{2}$ are the unit vectors corresponding to ${\hat{r}}_{1}$ and ${\hat{r}}_{2}$ , respectively. $θ_{1}$ and $θ_{2}$ are the scattering angles between ${\hat{s}}_{1}$ , ${\hat{s}}_{2}$ and the z axis, respectively. $〈 U^{(i)} ({\hat{ρ}}_{j}, ϖ) 〉$ (j = 1, 2) is the ensemble average of electric fields of incident plane waves, and $〈 U^{(f)} ({\hat{ρ}}_{j}, ϖ) 〉$ is the ensemble average of electric fields specified at $P ({\hat{r}}_{j}')$ (j = 1, 2). The relation between $〈 U^{(i)} ({\hat{ρ}}_{j}, ϖ) 〉$ and $〈 U^{(f)} ({\hat{ρ}}_{j}, ϖ) 〉$ can be expressed by recalling the elementary wave theory of light waves [20–22]:

\begin{array}{l} [\begin{matrix} U^{(f)} ({\hat{r}}_{1}', ω) \\ U^{(f)} ({\hat{r}}_{2}', ω) \end{matrix}] = [\begin{matrix} - \frac{i ω \exp (i \frac{ω}{c} R_{11})}{2 π c R_{11}} d S & - \frac{i ω \exp (i \frac{ω}{c} R_{12})}{2 π c R_{12}} d S \\ - \frac{i ω \exp (i \frac{ω}{c} R_{21})}{2 π c R_{21}} d S & - \frac{i ω \exp (i \frac{ω}{c} R_{22})}{2 π c R_{22}} d S \end{matrix}] [\begin{matrix} U^{(i)} ({\hat{ρ}}_{1}, ω) \\ U^{(i)} ({\hat{ρ}}_{2}, ω) \end{matrix}] \\ = - \frac{i ω}{2 π c} [\begin{matrix} - \frac{\exp (i \frac{ω}{c} R_{11})}{R_{11}} U^{(i)} ({\hat{ρ}}_{1}, ω) - \frac{\exp (i \frac{ω}{c} R_{12})}{R_{12}} U^{(i)} ({\hat{ρ}}_{2}, ω) \\ - \frac{\exp (i \frac{ω}{c} R_{21})}{R_{21}} U^{(i)} ({\hat{ρ}}_{1}, ω) - \frac{\exp (i \frac{ω}{c} R_{22})}{R_{22}} U^{(i)} ({\hat{ρ}}_{2}, ω) \end{matrix}], \end{array}

where

R_{α β}

(α = 1, 2; β = 1, 2) is the distance between

P ({\hat{r}}_{α}')

and

Q ({\hat{ρ}}_{β})

, as shown in Fig. 1.

d S

is the pinhole area, c denotes the speed of light propagating in vacuum. In Eq. (1), the incident plane wave is assumed to be polychromatic:

U^{(i)} ({\hat{ρ}}_{j}, ω) = a (ω) \exp (- i \frac{ω}{c} {\hat{s}}_{0} \cdot {\hat{ρ}}_{j}), (j = 1, 2),

where

a (ω)

is the amplitude of the electric field that depends on the frequency.

{\hat{s}}_{0}

is the unit vector that describes the propagating direction of the incident wave. The cross-spectral density function of the transmitted wave specified at

P ({\hat{r}}_{1}')

and

P ({\hat{r}}_{2}')

can be defined as:

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

where the asterisk denotes the complex conjugate and the angled bracket represents the ensemble average over a statistical realization of electric fields. Substituting Eq. (2) into (1) and utilizing Eq. (3), the cross-spectral density function can be further yielded to the following form:

\begin{array}{l} W^{(f)} ({\hat{r}}_{1}', {\hat{r}}_{2}', ω) = {(\frac{ω d S}{2 π c})}^{2} S^{(i)} (ω) {\frac{\exp [- i \frac{ω}{c} (R_{11} - R_{21})]}{R_{21} R_{11}} - \frac{\exp [- i \frac{ω}{c} (R_{12} - R_{21})]}{R_{21} R_{12}} \exp [- i \frac{ω}{c} {\hat{s}}_{0} \cdot ({\hat{ρ}}_{1} - {\hat{ρ}}_{2})] \\ - \frac{\exp [- i \frac{ω}{c} (R_{11} - R_{22})]}{R_{22} R_{11}} \exp [- i \frac{ω}{c} {\hat{s}}_{0} \cdot ({\hat{ρ}}_{2} - {\hat{ρ}}_{1})] + \frac{\exp [- i \frac{ω}{c} (R_{12} - R_{22})]}{R_{12} R_{22}}}, \end{array}

where

S^{(i)} (ω) = 〈 a^{*} (ω) a (ω) 〉

represents the spectrum of the incident field. Furthermore, let us suppose that the divergent angle of the transmitted wave is sufficiently small, and the distance between the opaque screen A and QH scatterer is large enough compared to the wavelength of incident beams. As a result,

R_{α β}

shown in the denominator of Eq. (4) should not substantially differ from a constant R:

R_{α β} \approx R, (α = 1, 2; β = 1, 2) .

In addition, relations between the azimuthal angle and position vector of transmitted waves can be established from Fig. 1, such that:

| {\hat{ρ}}_{1} | \cos φ_{1} = | {\hat{ρ}}_{2} | \cos φ_{2} = R' .

It is also assumed that the transmitted beam is a paraxial wave that propagates along the z axis. As a consequence, the following approximation can be made to

R_{α β}

in the exponential functions of Eq. (4) [22]:

R_{21} - R_{11} \approx \frac{(r'_{2 ⊥} - r'_{1 ⊥}) d}{2 R},

R_{22} - R_{12} \approx \frac{(r'_{2 ⊥} - r'_{1 ⊥}) d}{2 R},

R_{12} - R_{11} \approx \frac{r'_{1 ⊥} d}{R},

R_{22} - R_{21} \approx \frac{r'_{2 ⊥} d}{R},

where d denotes the space between pinholes,

r'_{j ⊥}

(j = 1, 2) represents the modulus of

{\hat{r}}_{j}'

, i.e.

r'_{j ⊥} = | \hat{r}'_{j ⊥} | .

Moreover, the relation between

{\hat{e}}_{0}

and

{\hat{s}}_{0}

can also be established from Fig. 1, such that:

r'_{1 ⊥} = {\hat{r}}_{1}' \cdot {\hat{e}}_{0},

r'_{2 ⊥} = - {\hat{r}}_{2}' \cdot {\hat{e}}_{0},

{\hat{e}}_{0} \cdot {\hat{s}}_{0} = 0.

Substituting Eqs. (5)-(11) into (4), the cross-spectral density function of the transmitted wave can be rewritten as the following representation:

\begin{array}{l} W^{(f)} ({\hat{r}}_{1}', {\hat{r}}_{2}', ω) = {(\frac{ω d S}{2 π c R})}^{2} S^{(i)} (ω) {2 \exp [i ω \frac{({\hat{r}}_{2}' - {\hat{r}}_{1}') \cdot {\hat{e}}_{0}}{2 c R} d] + \exp [- i ω \frac{({\hat{r}}_{2}' + {\hat{r}}_{1}') \cdot {\hat{e}}_{0}}{2 c R} d] \\ + \exp [i ω \frac{({\hat{r}}_{2}' + {\hat{r}}_{1}') \cdot {\hat{e}}_{0}}{2 c R} d]} . \end{array}

On the right-hand side of Eq. (12), the first term within the bracket is contributed by the wave propagated from each pinhole; while the second and third terms are attributed to the interferential effects of incident beams. Particularly, it is noteworthy to emphasize that we utilize a pair of Young’s pinhole to block the incident plane wave for the purpose of altering the spatial coherence of light. Actually, this method was first introduced in [21], which demonstrated that the spectral degree of coherence of light generated from Young’s pinholes can be an arbitrary value. This result can be analytically confirmed by substituting Eq. (12) in our study to Eq. (8) of [21]. Therefore, this method can be utilized here to generate a light wave with arbitrary coherence (either partial or completed coherence) before it scatters upon a QH media.

Fig. 1 Schematic diagram of the scattering theory of light waves from a scatterer, while the Young’s pinholes are used to block the incident plane wave to modulate its spatial coherence.

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As the next step, we account for the weak scattering of the transmitted wave from a spatially QH media. Within the accuracy of the Born approximation, the cross-spectral density function of the far-zone scattered field can be given [23]:

W^{(\infty)} (r {\hat{s}}_{1}, r {\hat{s}}_{2}, ω) = \frac{\exp [i \frac{ω}{c} (r_{2} - r_{1})]}{r_{2} r_{1}} \int_{D} \int_{D} W^{(f)} ({\hat{r}}_{1}', {\hat{r}}_{2}', ω) C_{F} ({\hat{r}}_{1}', {\hat{r}}_{2}', ω) \exp [i \frac{ω}{c} ({\hat{s}}_{2} \cdot {\hat{r}}_{2}' - {\hat{s}}_{1} \cdot {\hat{r}}_{1}')] d^{3} {\hat{r}}_{1}' d^{3} {\hat{r}}_{2}',

where

r_{j} = | {\hat{r}}_{j} |

represents the length of scattered light along

{\hat{r}}_{j},

{\hat{s}}_{1}

and

{\hat{s}}_{1}

denote the unit vectors defined within scattered field. In Eq. (13), the far-zone approximation is utilized to the representation of Green function [1, 23]. The subscript “D” denotes that the integration is taken over the entire scatterer volume.

C_{F}

is the correlation function of the scattering potential, which can be factorized to a product of the strength and normalized correlation coefficient of the scattering potential for a QH media [23–30]:

C_{F} ({\hat{r}}_{1}', {\hat{r}}_{2}', ω) = S_{F} (\frac{{\hat{r}}_{1}' + {\hat{r}}_{2}'}{2}, ω) η_{F} ({\hat{r}}_{2}' - {\hat{r}}_{1}', ω),

where

S_{F} (.)

and

η_{F} (.)

are slow and fast functions with regard to their internal arguments, respectively. Substituting Eqs. (12) and (14) into (13) and changing the integral variables:

{\hat{r}}_{2}' + {\hat{r}}_{1}' = {\hat{R}}_{+},

{\hat{r}}_{2}' - {\hat{r}}_{1}' = {\hat{R}}_{-} .

Accordingly, the cross-spectral density function of scattered field can be rewritten as the following integral form:

\begin{array}{l} W^{(\infty)} (r {\hat{s}}_{1}, r {\hat{s}}_{2}, ω) = {(\frac{ω d S}{2 π c R})}^{2} \frac{\exp [i \frac{ω}{c} (r_{2} - r_{1})]}{r_{2} r_{1}} S^{(i)} (ω) \int_{D} \int_{D} S_{F} (\frac{{\hat{R}}_{+}}{2}) η_{F} ({\hat{R}}_{-}) \\ \times {2 \exp [- \frac{i ω}{2 c} ({\hat{s}}_{2} - {\hat{s}}_{1} + \frac{d}{R} {\hat{e}}_{0}) \cdot {\hat{R}}_{+} - \frac{i ω}{2 c} ({\hat{s}}_{1} + {\hat{s}}_{2}) \cdot {\hat{R}}_{-}] + \exp [- \frac{i ω}{2 c} ({\hat{s}}_{2} - {\hat{s}}_{1}) \cdot {\hat{R}}_{+} - \frac{i ω}{2 c} ({\hat{s}}_{1} + {\hat{s}}_{2} - \frac{d}{R} {\hat{e}}_{0}) \cdot {\hat{R}}_{-}] \\ + \exp [- \frac{i ω}{2 c} ({\hat{s}}_{2} - {\hat{s}}_{1}) \cdot {\hat{R}}_{+} - \frac{i ω}{2 c} ({\hat{s}}_{1} + {\hat{s}}_{2} + \frac{d}{R} {\hat{e}}_{0}) \cdot {\hat{R}}_{-}]} d^{3} {\hat{R}}_{+} d^{3} {\hat{R}}_{-} . \end{array}

By performing the Fourier transforms in Eq. (16), the cross-spectral density function of scattered field is yielded to the following expression:

\begin{array}{l} W^{(\infty)} (r {\hat{s}}_{1}, r {\hat{s}}_{2}, ω) = {(\frac{ω d S}{2 π c R})}^{2} \frac{\exp [i \frac{ω}{c} (r_{2} - r_{1})]}{r_{2} r_{1}} S^{(i)} (ω) {2 {\tilde{S}}_{F} [\frac{ω}{c} ({\hat{s}}_{2} - {\hat{s}}_{1} + \frac{d}{R} {\hat{e}}_{0})] {\tilde{η}}_{F} [\frac{ω}{2 c} ({\hat{s}}_{1} + {\hat{s}}_{2})] \\ + {\tilde{S}}_{F} [\frac{ω}{c} ({\hat{s}}_{2} - {\hat{s}}_{1})] {\tilde{η}}_{F} [\frac{ω}{2 c} ({\hat{s}}_{1} + {\hat{s}}_{2} - \frac{d}{R} {\hat{e}}_{0})] + {\tilde{S}}_{F} [\frac{ω}{c} ({\hat{s}}_{2} - {\hat{s}}_{1})] {\tilde{η}}_{F} [\frac{ω}{2 c} ({\hat{s}}_{1} + {\hat{s}}_{2} + \frac{d}{R} {\hat{e}}_{0})]} . \end{array}

Furthermore, the spectrum of scattered field can be obtained by substituting

{\overset{⇀}{s}}_{1} = {\overset{⇀}{s}}_{2} = \overset{⇀}{s}

into Eq. (17), such that:

\begin{array}{l} S^{(\infty)} (r \hat{s}, ω) = {(\frac{ω d S}{2 π c R r})}^{2} S^{(i)} (ω) {2 {\tilde{S}}_{F} (\frac{d}{R} {\hat{e}}_{0}) {\tilde{η}}_{F} (\frac{ω}{c} \hat{s}) + {\tilde{S}}_{F} (0) {\tilde{η}}_{F} [\frac{ω}{c} (\hat{s} - \frac{d}{2 R} {\hat{e}}_{0})] \\ + {\tilde{S}}_{F} (0) {\tilde{η}}_{F} [\frac{ω}{c} (\hat{s} + \frac{d}{2 R} {\hat{e}}_{0})]}, \end{array}

Equation (18) contains two fundamental terms: the Fourier transforms of the strength and normalized correlation coefficient of the scattering potential of QH media. In particular, the first term within the bracket is contributed by light waves transmitted from each pinhole; while the second and third terms are attributed to the interferential effects caused by Young’s pinholes. Due to the fact that the Fourier transform of the strength of the scattering potential varies rapidly with respect to its internal argument. Therefore, the following relation must be valid:

{\begin{matrix} {\tilde{S}}_{F} (\frac{d}{R} {\hat{e}}_{0}) < < {\tilde{S}}_{F} (0), \\ {\tilde{S}}_{F} (\frac{d}{R} {\hat{e}}_{0}) = {\tilde{S}}_{F} (0), \end{matrix} \begin{matrix} \frac{d}{R} > 0 \\ \frac{d}{R} = 0 \end{matrix} .

Similarly, the Fourier transform of the normalized correlation coefficient of the scattering potential is a slow function with regard to its internal argument [23–32]:

{\tilde{η}}_{F} (\frac{ω}{c} \hat{s}) \approx {\tilde{η}}_{F} [\frac{ω}{c} (\hat{s} - \frac{d}{2 R} {\hat{e}}_{0})] \approx {\tilde{η}}_{F} [\frac{ω}{c} (\hat{s} + \frac{d}{2 R} {\hat{e}}_{0})] .

Alternatively, the spectrum of scattered field can be rewritten to the following optional form:

S^{(\infty)} (r \hat{s}, ω) = {\begin{matrix} {(\frac{ω d S}{π c R r})}^{2} S^{(i)} (ω) {\tilde{S}}_{F} (0) {\tilde{η}}_{F} (\frac{ω}{c} \hat{s}), \\ {(\frac{ω d S}{2 π c R r})}^{2} S^{(i)} (ω) {\tilde{S}}_{F} (0) {{\tilde{η}}_{F} [\frac{ω}{c} (\hat{s} - \frac{d}{2 R} {\hat{e}}_{0})] + {\tilde{η}}_{F} [\frac{ω}{c} (\hat{s} + \frac{d}{2 R} {\hat{e}}_{0})]}, \end{matrix} \begin{matrix} \frac{d}{R} = 0 \\ \frac{d}{R} > 0 \end{matrix} .

Because

{\tilde{S}}_{F} (0)

is a constant, the spectrum of scattered field should be a function that is only proportional to the Fourier transform of the normalized correlation coefficient of the scattering potential. Let us further assume that the strength and normalized correlation coefficient of QH media satisfy the Gaussian distribution, respectively:

S_{F} ({\hat{R}}_{+}) = \frac{A}{{(2 π δ_{s}^{2})}^{3 / 2}} \exp [- \frac{{({\hat{R}}_{+})}^{2}}{2 δ_{s}^{2}}],

η_{F} ({\hat{R}}_{-}) = \frac{B}{{(2 π δ_{η}^{2})}^{3 / 2}} \exp [- \frac{{({\hat{R}}_{-})}^{2}}{2 δ_{η}^{2}}],

where

δ_{s}

denotes the effective size, and

δ_{η}

represents the correlation length [28–30]. Accordingly, the spectrum of scattered field can be given by the following expression by substituting Eq. (22) into (21):

S^{(\infty)} (θ, ω) = {\begin{matrix} {(\frac{ω d S}{2 π c R r})}^{2} S^{(i)} (ω) A B {\exp [- \frac{ω^{2}}{2 c^{2}} δ_{η}^{2} (1 + \frac{d^{2}}{4 R^{2}} - \frac{d}{4 R} \sin θ)] + \exp [- \frac{ω^{2}}{2 c^{2}} δ_{η}^{2} (1 + \frac{d^{2}}{4 R^{2}} + \frac{d}{4 R} \sin θ)]}, \\ {(\frac{ω d S}{π c R r})}^{2} S^{(i)} (ω) A B \exp (- \frac{ω^{2}}{2 c^{2}} δ_{η}^{2}), \end{matrix} \begin{matrix} \frac{d}{R} > 0 \\ \frac{d}{R} = 0 \end{matrix} .

Equation (23) demonstrates that the spectrum of scattered field cannot remain invariant compared to the initial spectrum of incident plane waves. This result is induced by two factors: the interferential effects caused Young’s pinholes and scattering process of light from random media. In what follows we particularly concentrate on the spectral shifts and spectral switches of scattered field. For such purpose, we may recall the expression for the normalized spectrum of a polychromatic field [1,4,11,13]:

S_{N}^{(\infty)} (r \hat{S}, ω) = \frac{S^{(\infty)} (r S, ω)}{\int_{0}^{\infty} S^{(\infty)} (r \hat{S}, ω) d ω} .

Also, it is assumed that the spectrum of the incident waves occupies the Gaussian profile:

S^{(i)} (ω) = \exp [- \frac{{(ω - ω_{0})}^{2}}{2 σ_{0}^{2}}],

where

σ_{0}

represents the spectral width,

ω_{0}

denotes the central frequency. By substituting Eqs. (23) and (25) into (24), the normalized spectrum of scattered field can be derived as:

S_{N}^{(\infty)} (θ, ω) = \frac{ω^{2} \exp [- \frac{1}{2} (\frac{1}{σ_{0}^{2}} + \frac{δ_{η}^{2}}{c^{2}}) ω^{2} + \frac{ω_{0}}{σ_{0}^{2}} ω]}{M^{+} (θ, ω) + M^{-} (θ, ω)}, \frac{d}{R} > 0,

and

S_{N}^{(\infty)} (θ, ω) = \frac{ω^{2} \exp [- \frac{1}{2} (\frac{1}{σ_{0}^{2}} + \frac{δ_{η}^{2}}{c^{2}}) ω^{2} + \frac{ω_{0}}{σ_{0}^{2}} ω]}{\frac{ω_{0}}{{(\frac{1}{2 σ_{0}} + \frac{σ_{0}}{2 c^{2}} δ_{η}^{2})}^{2}} + \frac{π^{\frac{1}{2}} (\frac{ϖ_{0}^{2}}{8 σ_{0}^{4}} + \frac{1}{8 σ_{0}^{2}} + \frac{δ_{η}^{2}}{8 c^{2}})}{{(\frac{1}{2 σ_{0}^{2}} + \frac{δ_{η}^{2}}{2 c^{2}})}^{\frac{5}{2}}} \exp (\frac{ω_{0}^{2}}{2 σ_{0}^{2} + \frac{2 δ_{η}^{2}}{c^{2}} σ_{0}^{4}}) {1 - e r f [\frac{ω_{0}}{{(2 σ_{0}^{2} + \frac{2 δ_{η}^{2}}{c^{2}} σ_{0}^{4})}^{\frac{1}{2}}}]}}, \frac{d}{R} = 0,

with

\begin{array}{l} M^{\pm} (θ, ω) = \frac{ω_{0}}{{[\frac{1}{2 σ_{0}} + \frac{σ_{0}}{2 c^{2}} δ_{η}^{2} (1 + \frac{d^{2}}{4 R^{2}} \pm \frac{d}{4 R} \sin θ)]}^{2}} + \frac{π^{\frac{1}{2}} [\frac{ω_{0}^{2}}{8 σ_{0}^{4}} + \frac{1}{8 σ_{0}^{2}} + \frac{δ_{η}^{2}}{8 c^{2}} (1 + \frac{d^{2}}{4 R^{2}} \pm \frac{d}{4 R} \sin θ)]}{{[\frac{1}{2 σ_{0}^{2}} + \frac{δ_{η}^{2}}{2 c^{2}} (1 + \frac{d^{2}}{4 R^{2}} \pm \frac{d}{4 R} \sin θ)]}^{\frac{5}{2}}} \\ \times \exp [\frac{ω_{0}^{2}}{2 σ_{0}^{2} + \frac{σ_{0}^{4}}{c^{2}} δ_{η}^{2} (1 + \frac{d^{2}}{4 R^{2}} \pm \frac{d}{4 R} \sin θ)}] {1 - e r f [\frac{ω_{0}}{\sqrt{2 σ_{0}^{2} + \frac{σ_{0}^{4}}{c^{2}} δ_{η}^{2} (1 + \frac{d^{2}}{4 R^{2}} \pm \frac{d}{4 R} \sin θ)}}]}, \end{array}

where

e r f (.)

is an error function. It is shown by Eqs. (26)-(28) that the normalized spectrum of light generated by the scattering of an arbitrary coherent wave from a QH media may shift from the initial spectrum of incident plane waves. The spectral shifts are dependent of the correlation length

δ_{η}

, the scattering angle

θ

and the Young’s configuration parameter d/R. It provides us a feasible approach to numerically study the dependence of spectral properties of scattered field on various parameters through numerical simulations.

3. Numerical results and discussions

All parameters used in figures are chosen as follows: $ϖ$ = 3.2 × 1015s-1, σ₀ = 0.6 × 1015s-1, $λ_{0} = 2 π c / ω_{0}$ = 600nm, unless otherwise stated. Figure 2 shows the variation of normalized spectrum of scattered field versus the scattering angle $θ$ , while different $d / R$ are chosen for comparisons. It is shown that the spectral lines of scattered field shifts toward lower frequencies, which are highly dependent of $d / R .$ In particular, when $d / R = 0$ , which corresponds to the single-pinhole situation, the spectral lines of scattered field is solely dependent of the scattering angle $θ .$ However, this result becomes invalid when the Young’s configuration parameter $d / R = 2.$ It is interesting to note that the spectrum may split into two separated counterparts provided that d/R = 2 is satisfied (see Fig. 2(d)).

Fig. 2 Normalized spectrum of scattered field $S_{N}^{(\infty)} (θ, ω)$ varies versus the scattering angle $θ$ , while different values of $d / R$ are selected for comparisons. The correlation length $δ_{η} = λ_{0} .$ (a) $d / R = 0,$ (b) $d / R = 0.5,$ (c) $d / R = 1,$ (d) $d / R = 2.$

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Figure 3 displays the variation of normalized spectrum of scattered light versus d/R, while different correlation lengths are chosen for comparisons. It is shown that the peak position of the normalized spectrum is a monotonously reducing function of $d / R$ . Figure 3(a) and 3(b) also show that the red shift emerges when the correlation length of media is comparable to the central wavelength of incident beams, e.g. $δ_{η} = λ_{0}$ (Fig. 3(b)) or $δ_{η} = 2 λ_{0}$ (Fig. 3(a)).

Fig. 3 Normalized spectrum of scattered field $S_{N}^{(\infty)} (θ, ω)$ varies versus $d / R$ , while different correlation length $δ_{η}$ are selected for comparisons. The scattering angle θ = 0. (a) $δ_{η} = 2 λ_{0},$ (b) $δ_{η} = λ_{0},$ (c) $δ_{η} = 0.5 λ_{0},$ (d) $δ_{η} = 0.2 λ_{0} .$

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Apart from the spectral shift, the spectral switch that moves from lower frequencies toward higher frequencies, however, can be produced provided that the correlation length is reduced to a certain value. In particular, when the correlation length is decreased to $δ_{η} = 0.5 λ_{0},$ the blue shift in spectral lines of scattered field occurs (see Fig. 3(c)). Such effect can be substantially enhanced by reducing the correlation length to $δ_{η} = 0.2 λ_{0} .$

The dependence of the normalized spectrum of scattered field on $δ_{η} / λ_{0}$ and the scattering angle is shown in Fig. 4, respectively, while $d / R = 0.5$ is satisfied for all subplots. Similar to Fig. 3, it is worthwhile to note that the spectral distribution narrows when the scattering angle gradually becomes larger. The spectrum is found to have no measurable value in certain scattering angles, e.g. $θ = π / 3$ in Fig. 4(c) or $θ = π / 2$ in Fig. 4(d). In addition, the spectral switch occurs when the correlation length is increased to a certain order of magnitude. This phenomenon is induced by the correlation statistics of spatially random media on spectrum of scattered field [6–9, 14].

Fig. 4 Normalized spectrum of scattered field $S_{N}^{(\infty)} (θ, ω)$ varies versus the correlation length $δ_{η}$ , while different scattering angles are selected for comparisons. The Young’s configuration parameter $d / R = 0.5.$ (a) $θ = 0,$ (b) $θ = π / 6,$ (c) $θ = π / 3,$ (d) $θ = π / 2.$

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Figure 5 plots the normalized spectrum of scattered field versus the scattering angle for $d / R = 2$ (Fig. 5(a)) and $d / R = 0.5$ (Fig. 5(b)), respectively. The initial spectral profile of incident plane waves is also plotted for comparisons. It is shown in Fig. 5(a) that the spectrum of scattered field shifts toward the lower frequency in spectral lines. Also, it is found that the red shift of spectral lines becomes less distinct if the scattering angle increases from $θ = 0$ to $θ = π / 2$ . When $d / R = 0.5$ is satisfied, the spectrum of scattered field shows no dependence on the scattering angle. This result implies that the influence of scattering angles on spectrum of scattered field is further dependent of $d / R .$ Therefore, a conclusion can be drawn from Fig. 5 that the spectrum of scattered field is dependent of both the scattering angle and the Young’s configuration parameter.

Fig. 5 Normalized spectrum of scattered field $S_{N}^{(\infty)} (θ, ω)$ varies versus different scattering angles $θ = 0, π / 4, π / 2$ . The correlation length $δ_{η} = λ_{0} .$ $S^{(i)} (ω)$ denotes the spectrum of the initial incident plane waves. (a) $d / R = 2,$ (b) $d / R = 0.5.$

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Figure 6 shows the normalized spectrum of scattered field versus $d / R$ , while different scattering angles, i.e. θ = 0 and $θ = π / 2$ are chosen for comparisons. The spectrum of scattered field is found to have no dependence on $d / R$ when the scattering angle becomes large enough (see Fig. 6(b)). Also, it is interesting to note that the red shift of spectral lines of scattered field become less obvious if d/R is increased to a certain value.

Fig. 6 Normalized spectrum of scattered field $S_{N}^{(\infty)} (θ, ω)$ varies with different d/R = 0, 1, 2. (a) $θ = 0,$ (b) $θ = π / 2.$

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Figure 7 depicts the spectral switch of scattered field from the red shift (Fig. 7(a)-7(c)) toward the blue shift (Fig. 7(d)), which can be generated by reducing the correlation length from $δ_{η} = λ_{0}$ to $δ_{η} = 0.2 λ_{0},$ while θ = 0 and $d / R = 0.5$ are chosen for the plots. The initial spectral profile of incident plane wave sis also plotted for comparisons. By observing Figs. 5 and 6, it is noteworthy to emphasize that the spectral switch of scattered field can be generated by altering the correlation length of the QH scatterer.

Fig. 7 Normalized spectrum of scattered field $S_{N}^{(\infty)} (θ, ω)$ varies with the correlation length $δ_{η}$ = λ₀, 0.5λ₀ and 0.2λ₀. d/R = 0.5, θ = 0, $S^{(i)} (ω)$ denotes the spectrum of incident plane waves.

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To show the spectral switch of scattered field in a quantitative point of view, the relative spectral shift of light can be introduced [6–9,12]:

\frac{δ ω}{ω_{0}} = \frac{ω_{\max} - ω_{0}}{ω_{0}},

where

ω_{\max}

denotes the frequency at which spectrum achieves its maximum value. The variation of relative spectral shifts versus the correlation length is shown in Fig. 8(a), while different scattering angles are selected for comparisons. It is found that the blue shift of spectral lines of scattered field occurs, i.e.

\frac{δ ω}{ω_{0}} \approx 0.03

, if

δ_{η} = 0.2 λ_{0}

is satisfied.. However, the peak position of spectrum gradually moves to the central frequency by increasing the correlation length, until the spectral switch of scattered field is observed. For instance, if the correlation length

δ_{η} = λ_{0},

the relative spectral shift

\frac{δ ω}{ω_{0}} \approx - 0.372, - 0.4, - 0.431

for different scattering angles

θ = 0, π / 4, π / 2,

respectively. Figure 8(b) displays the varying characteristics of relative spectral shift with the correlation length, while different

d / R

are selected for comparisons. It is noteworthy to emphasize that the possibility of generating spectral switches of scattered field from the red shift to the blue shift is highly dependent of the Young’s configuration parameter. It is also remarkable to observe that the spectral switch can be accelerated by selecting relatively large values of

d / R

. This implies that the Young’s pinholes utilized in our study can either modulate the spatial coherence of incident plane waves, or generate the spectral switch in scattered field.

Fig. 8 Relative spectral shift of scattered field $δ ω / ω_{0}$ varies with $δ_{η} / λ_{0}$ , while different scattering angles are selected for comparisons. (a) θ = 0, π/4, π/2, d/R = 1. (b) d/R = 0, 1, 2, θ = 0.

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As a special note, we need to say that the method used to model the partial coherence of light beams entirely differs from those introduced in previous literature. Although [33–36] addressed the scattering theory of partially coherent light upon a random scatterer, it is particularly concerned in their studies that the source beams are assumed to be quasi-homogeneous in the spatial domain. Among their studies, the cross-spectral density function (or matrix) used for describing the partial coherence of beams are all of the Shell-model type. For comparisons, we utilized a Young’s pinhole configuration, which is capable to generate an arbitrary degree of coherence of a planar incident light. We shall emphasize that the proposed model in our study is a generalization of the scattering theory of partially coherent beams, as the Young’s pinholes play an essential role to modulate coherences of transmitted light waves, and no theoretical model (e.g. Schell-model type) is assumed to draw conclusions. Moreover, the presented results in our study are expected to find potential applications in the image processing, free space optics communications and remote sensing etc. For instance, the spectral switch studied in our study can provide spectral information to people to determine the boundary edges or correlation statistics of an unknown scatterer. In addition, the spectral switch showed in Fig. 8 may be further utilized to encoding detailed information of optical images, provided that a series of binary codes are complied to represent either red or blue shift of spectral lines in scattered field.

Conclusion

Within the accuracy of the Born approximation, the spectral shifts and spectral switches of a scattered field are studied by supposing the incident plane wave transmits through Young’s pinholes before interacting with a QH media. In our study, the Young’s configuration is used to modulate the arbitrary coherence of the incident waves. It is shown that the spectrum of scattered field shifts from its initial profile and moves toward the lower frequency. The spectral shift is also found to be dependent of the scattering angle, the correlation length of media and the Young’s configuration parameter. In addition, it is emphasized to note that the generation of spectral switches in scattered field is highly dependent of the correlation length of the scatterer. Our study is anticipated to establish a complete understanding of spectral properties of scattered field, provided that an arbitrary coherent wave is scattered upon a spatially random media.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC No. 61205121, 61304124) and the Natural Science Foundation of Zhejiang Province (No. LY13F010009, LY15F050012).

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Spectral shifts and spectral switches of light generated by scattering of arbitrary coherent waves from a quasi-homogeneous media

Abstract

1. Introduction

2. Scattering of an arbitrary coherent wave from a QH media

3. Numerical results and discussions

Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (33)

Optics Express