Within the accuracy of the Born approximation, it is shown that the light, which is generated by the scattering of an arbitrary coherent polychromatic wave from a quasi-homogeneous (QH) media can, display both spectral shifts and spectral switches. In our study, a pair of Young’s pinholes is utilized to modulate spatial coherence of the incident plane wave before it interacts with the scatterer. The spectral shifts are found to be highly dependent of the scattering angle, the correlation length of scatterer and the Young’s configuration parameter. Moreover, the spectral shifts can be converted from the red shift to blue one provided that the correlation length of scatterer is small enough. Derived results are confirmed by numerical simulations where influences of various factors on the spectrum are analyzed in detail.
© 2015 Optical Society of America
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 . This prediction was further confirmed by experiments where the spectral lines of waves were proven to shift toward longer or shorter wavelengths . 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 . Further studies were done to compare results between different approximations used in the scattering theory . 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 .
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 . Furthermore, a scaling law was derived to ensure the invariance of spectra of a stochastic electromagnetic beam . Also, a method for determining unknown scatterer was proposed by using the correlation-induced spectral changes . Subsequently, this method was further extended to the case where the scattering of light from a collection of particles was studied . 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 , and then was further extended to the case where the scattering of light from a rotational QH anisotropic media was studied . 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 .
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 and is utilized to block the incident wave. and are the projective position vectors of and , respectively. and represent the azimuthal angles with regard to and . The QH scatterer occupies a finite spatial volume D. and are two points where the transmitted wave further interacts with the QH scatterer. and denote the position vectors in scattered field, and are the unit vectors corresponding to and , respectively. and are the scattering angles between , and the z axis, respectively. (j = 1, 2) is the ensemble average of electric fields of incident plane waves, and is the ensemble average of electric fields specified at (j = 1, 2). The relation between and can be expressed by recalling the elementary wave theory of light waves [20–22]:Fig. 1. 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:Eq. (2) into (1) and utilizing Eq. (3), the cross-spectral density function can be further yielded to the following form:Eq. (4) should not substantially differ from a constant R:Fig. 1, such that:Eq. (4) :Fig. 1, such that:Eqs. (5)-(11) into (4), the cross-spectral density function of the transmitted wave can be rewritten as the following representation: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 , 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 . 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.
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 :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. 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]:Eqs. (12) and (14) into (13) and changing the integral variables:Eq. (16), the cross-spectral density function of scattered field is yielded to the following expression:Eq. (17), such that: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:23–32]:28–30]. Accordingly, the spectrum of scattered field can be given by the following expression by substituting Eq. (22) into (21):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]:Eqs. (23) and (25) into (24), the normalized spectrum of scattered field can be derived as: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 , the scattering angle 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: = 3.2 × 1015s-1, σ0 = 0.6 × 1015s-1, = 600nm, unless otherwise stated. Figure 2 shows the variation of normalized spectrum of scattered field versus the scattering angle , while different are chosen for comparisons. It is shown that the spectral lines of scattered field shifts toward lower frequencies, which are highly dependent of In particular, when , which corresponds to the single-pinhole situation, the spectral lines of scattered field is solely dependent of the scattering angle However, this result becomes invalid when the Young’s configuration parameter 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)).
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 . 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. (Fig. 3(b)) or (Fig. 3(a)).
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 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
The dependence of the normalized spectrum of scattered field on and the scattering angle is shown in Fig. 4, respectively, while 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. in Fig. 4(c) or 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].
Figure 5 plots the normalized spectrum of scattered field versus the scattering angle for (Fig. 5(a)) and (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 to . When 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 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.
Figure 6 shows the normalized spectrum of scattered field versus , while different scattering angles, i.e. θ = 0 and are chosen for comparisons. The spectrum of scattered field is found to have no dependence on 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.
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 to while θ = 0 and 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. 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. , if 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 the relative spectral shift for different scattering angles respectively. Figure 8(b) displays the varying characteristics of relative spectral shift with the correlation length, while different 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 . 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.
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.
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.
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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