Abstract
By constructing a linear combination of several special weight functions, some novel scattering medium was designed to generate far-field distribution with hollow arrays. The construction is divided into two cases, i.e., the linear combination of weight functions with the same types and the linear combination of weight functions with different types. It is shown that the far-field with periodic array distribution that is composed of circular hollow lobes or rectangular hollow lobes may be obtained. In addition, it is shown that the characteristics of the arrays, including the shape of the lobes, the hollow size of the lobes and the distance between the lobes, can be flexibly adjusted by changing the structural parameters of the scattering medium.
© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Due to its potential applications in areas such as medical diagnosis, remote sensing and diffraction tomography [1–3], light wave scattering has become an important method to explore the structural information of scattering media. Therefore, researchers have made a series of theoretical studies, which focused on the influence of structural characteristics of scattering medium on the optical properties of scattered field [4–9]. In 1986, Wolf found that the spectrum of partially spatially coherent light will change in the process of propagating in free space and explained the reason for this phenomenon [10]. After that, he extended this variation phenomenon to the weak scattering process and found that the far-field spectrum will change when poly-chromatic light waves were scattered by a random medium [11]. In the past few decades, great progress has been made in the study of weak scattering. On the one hand, the scatterer has been extended to a wider range of medium models [12–15], including particle scattered model [12], semi-soft boundary medium model [13], anisotropic medium model [14], etc. On the other hand, the incident light wave has been extended from the plane monochromatic wave to some more general light waves [16–19], including polychromatic plane wave [16], partially coherent plane-wave pulse beam [17], electromagnetic light wave [18], etc. Furthermore, the inverse problem of light wave scattering, namely reconstructing the structural characteristics of the scattering medium through the optical characteristics of the scattered field, is also attracted much attention [20–22]. For example, the information of scattering potential of random medium was determined by measurement of the scattered field [23–25].
Due to the fact that the density distribution of the far field is closely related to the structural characteristics of the light field correlation function in the source plane, new light sources can be designed by selecting different realizable correlation functions to obtain various far-field distributions [26–30]. Recently, this method has been generalized to producing array beams, for example, Gaussian schell-model arrays [31] and optical coherence grids [32]. By analogizing with the design of light source, Korotkova firstly designed some media to obtain the desired far-field distribution pattern in the scattered field [33,34]. After that, Li et al. proposed a method to describe a three-dimensional, non-uniformly correlated medium [35]. Ding et al. introduced a scattering medium model to produce the array distribution [36]. Zhu et al. designed a new layered random medium to produce scattered far field with annular coherent array distribution [37]. Pan et al. put forward a method to design new scattering medium by the convolution of two weight functions [38]. In this manuscript, we will propose a kind of random medium model to generate periodic array profile, and discuss the relationship between the properties of far-zone scattered field and the characteristics of scattering medium. Moreover, the possibility of manipulating the far-zone scattered field, including the lobe shape, the hollow size, and the distance between lobes will also be discussed by changing the structural parameters of the scattering medium.
2. Theory
In the weak scattering theory, the properties for a random medium can be described by its correlation function of scattering potentials, which is defined by [39]
In the family of Schell-model media, a kind of basic random media is known as locally homogeneous medium, which was introduced by Silverman in the literature [42]. In addition, a more generalized model, namely, the quasi-homogeneous model medium was discussed by Carter and Wolf by analogizing with the concept of quasi-homogeneous source [43]. In this model, $\tau ({\mathbf r^{\prime}})$ changes more slowly with position ${\mathbf r^{\prime}}$ than the normalized correlation coefficient ${\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )$ changes with the position difference ${{\mathbf r^{\prime}}_2} - {{\mathbf r^{\prime}}_1}$. So that, its scattering potential strength ${I_F}({\mathbf r^{\prime}},\omega )$ almost remains constant, while the model of ${\mu _F}({{{{\mathbf r^{\prime}}}_{}},\omega } )$ varies greatly. In this case, the correlation function of scattering potential takes on the form [44]
Within the accuracy of the first-order Born approximation and the far field approximation, the spectral density of the far-zone scattered field from a QH medium can be expressed as [39]
The far-zone scattered spectral density is solely controlled by the shape of normalized correlation coefficient, while the density of the scattering potential only plays the role of proportionality factor, which is known as reciprocity relations [45]. Therefore, $\tau ({{\mathbf r^{\prime}},\omega } )$ can be chosen at will, and we assume it to be Gaussian [36]
Upon substituting from Eq. (11) into Eq. (7), we can obtain the following expression for scattered spectral density as
If the scattering medium takes the layer structure, the shape of normalized correlation coefficient along the scattering axis (z-axis) does not contribute to the scattered density distribution. In this case, the correlation function can be rewritten as [44]
On substituting from Eq. (14) and Eq. (13) into Eq. (9), we will get the Fourier transform of the normalized correlation coefficients of the two-dimensional components of the transform vector ${\mathbf K}$, i.e.
3. Linear combination of weight functions with the same types
In this section, two special far-field distributions, i.e., circular hollow array and rectangular hollow array, will be obtained through linear combination of weight functions with the same types. Some numerical results will also be presented to illustrate the dependence of far-zone distributions on the characteristics of the scattering medium.
3.1 Scattered field with circular hollow lobe array distribution
To obtain a scattered field with $N \times M$ multi-Gaussian array profile, the spectral degree of coherence should be chosen as [46]
In order to generate a $N \times M$ hollow multi-Gaussian array in the far field, ${p_C}({{\mathbf v}_\rho })$ makes the following transformation
As shown in Eq. (19), Although ${p_{c1}}({{\mathbf v}_\rho }) \ge 0$ and ${p_{c2}}({{\mathbf v}_\rho }) \ge 0$, we have to make sure that the function $P({\mathbf v})$ is always non-negative for any values of the 2D vector ${{\mathbf v}_\rho }$, so we set the parameter ${L_1} > {L_2}$ in the following discussion. On substituting from Eq. (19) together with Eq. (18) into Eq. (14) then into Eq. (15), further substituting the corresponding result into Eq. (16), one can find the far-zone scattered spectral density as follows
In Fig. 1, the influence of parameters M and N on the far-field spectral density distribution is discussed. It follows from Fig. 1(a) and Fig. 1(b) that the number of rows will decrease when the value of parameter M decreases. Moreover, one can find from Fig. 1(a) and Fig. 1(c) that the number of columns will decrease as the value of parameter N decreases. Therefore, the number of rows and columns of the array can be manipulated by changing the values of parameters M and N. In Fig. 2, the influence of parameter ${L_2}$ on the size of circular hollow profile is discussed. It is shown that the hollow area of single lobe will increase with ${L_2}$ increasing gradually when keeps the value of ${L_1}$ unchanged. Therefore, the size of hollow area can be manipulated by changing the value of the parameter ${L_2}$.
In Fig. 3, the influence of parameters ${\delta _x}$ and ${\delta _y}$ on the far-field spectral density distribution is discussed. The numerical result shows that the value of $\delta$ will affect the shape of each lobe. Specifically, when the value of parameter ${\delta _x}$ is equal to the value of parameter ${\delta _y}$, as shown in Fig. 3(a), each lobe in the periodic array is isotropic. However, as shown in Fig. 3(b), when the value of parameter ${\delta _x}$ is smaller than the value of parameter ${\delta _y}$, each lobe in the periodic array is anisotropic and its long axis is in the x direction. As shown in Fig. 3(c), when the value of parameter ${\delta _x}$ is larger than the value of parameter ${\delta _y}$, each lobe in the periodic array is anisotropic and its long axis is in the y direction. Therefore, one can conclude that the shape and direction of each lobe can be manipulated by adjusting the value of $\delta$, and the long axis of each lobe is located in the direction with smaller effective correlation length. In Fig. 4, the dependence of the distance between adjacent lobes on the parameter R is discussed. As shown in Fig. 4(a), when setting ${R_x} = {R_y}$, the distance of adjacent lobes is equal in the x direction and in the y direction. However, as shown in Fig. 4(b), when the value of parameter ${R_x}$ is larger than the value of parameter ${R_y}$, the distance of adjacent lobes is larger in the x direction than in the y direction. When the value of parameter ${R_x}$ is smaller than the value of parameter ${R_y}$, the distance of adjacent lobes is smaller in the x direction than in the y direction, and the corresponding result is displayed in Fig. 4(c).
3. Scattered field with rectangular hollow lobe array distribution
In the above section, the far-field spectral density distribution demonstrates a circular hollow array. Here, let us further consider another case where a hollow array of lobe with a Cartesian symmetrical distribution will be obtained. In this case, we let ${p_R}({{\mathbf v}_\rho })$ take the following form
whereOn substituting from Eq. (23) together with Eq. (22) first into Eq. (14) and into Eq. (15), further substituting the corresponding result into Eq. (16), and one can find the rectangular far-field spectral density distribution as follows
Figure 5 illustrates the influence of parameter ${L_2}$ on the far-field spectral density distribution. Result shows that when the value of ${L_1}$ is fixed, the size of the central hollow of lobes will increase with the value of parameter ${L_2}$ increases. In addition, as the value of ${L_2}$ changes, the shape of the center will gradually transition from circular to rectangular. Figure 6 illustrates the influence of parameter $\delta$ on the far-field spectral density distribution. As shown in Fig. 6(a), we can clearly see that each lobe profile is isotropic when the effective correlation lengths, i.e., ${\delta _x}$ and ${\delta _y}$ take the same value. However, as shown in Fig. 6(b) and Fig. 6(c), when the parameters ${\delta _x}$ and ${\delta _y}$ take different values, each lobe profile in the array will demonstrate anisotropy.
Figure 7 presents the influence of parameter R on the far-field spectral density distribution. We can clearly see that the distance between adjacent lobes in the array can be manipulated by the value of parameter R. As shown in Fig. 7(a), when the parameters ${R_x}$ and ${R_y}$ take same value, the distance between adjacent lobes is equal in any direction. Moreover, it can be clearly seen from Fig. 7(b) and Fig. 7(c) that when the value of parameter ${R_x}$ is larger than the value of parameter ${R_y}$, the distance of adjacent lobes along the x direction is larger than that along the y direction. On the contrary, when the value of parameter ${R_x}$ is smaller than the value of parameter ${R_y}$, the distance of adjacent lobes along the x direction is smaller than that along the y direction.
4. Linear combination of weight functions with different types
In the following, a scattered far field with array lobe distribution may be obtained by introducing the linear superposition of different type of weight function. The superposition of different weight functions makes scattering medium to produce different scattered far fields. In this case, $p({{\mathbf v}_\rho })$ has the following form
where a and b are the weight coefficients. After some calculations, we get the spectral density distribution of the scattered far field asAs shown in Fig. 8, through the linear combination of two different types of weight functions, one can get more complex far-field distribution, i.e., the array scattered field which is composed of rectangular lobes with circular hollow and the array scattered field which is composed of circular lobes with rectangular hollow. Specifically, when the parameters are chosen as $a = 1$, $b ={-} 1$ and ${L_1} > {L_2}$, as shown in Fig. 8(a), the external shape of lobes depends on ${p_c}_1$ and the inner shape of lobes depends on ${p_r}_2$. In this case, a lobe shape of a rectangular hole inside a circle which is like Chinese ancient copper coins. When the parameters are chosen as $a ={-} 1$, $b = 1$ and ${L_2} > {L_1}$, as shown in Fig. 8(b), the external shape of lobes depends on ${p_r}_2$ and the inner shape of lobes depends on ${p_c}_1$. In this case, the lobe shape will demonstrate as a rectangular profile with a circular hollow distribution.
5. Conclusion
In summary, we have discussed a strategy of designing a series of novel random mediums, which can produce scattered field with hollow array distribution, including circular hollow lobes, rectangular hollow lobes, and their combinations. The possibility of manipulating the far field distributions, such as the shape of the lobes, the hollow size of the lobes, and the distance between the lobes, was discussed by adjusting the parameters of scattering medium. It is shown that by properly adjusting the structural parameters of scattering medium, various desired patterns with flexibility manipulating far-zone field can be obtained in the scattered field. This phenomenon may provide a method to produce some special patterns behind the scattering medium, which may have potential applications in areas such as biomedical imaging, medical diagnosis, and light wave manipulation, where the scattering process always exists.
Funding
National Natural Science Foundation of China (11404231, 61775152).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. B. Chu, Laser light scattering (Academic: New York, 1974).
2. Howard and John, “Laser probing of random weakly scattering media,” J. Opt. Soc. Am. A 8(12), 1955–1963 (1991). [CrossRef]
3. P. S. Carney and E. Wolf, “Power-excitation diffraction tomography with partially coherent light,” Opt. Lett. 26(22), 1770–1772 (2001). [CrossRef]
4. J. J. Greffet, M. Cruz-Gutierrez, P. V. Ignatovich, and A. Radunsky, “Influence of spatial coherence on scattering by a particle,” J. Opt. Soc. Am. A 20(12), 2315 (2003). [CrossRef]
5. S. A. Ponomarenko and A. V. Shchegrov, “Spectral changes of light produced by scattering from disordered anisotropic media,” Phys. Rev. E 60(3), 3310–3313 (1999). [CrossRef]
6. O. Korotkova and X. Chen, “Scattering of light from hollow and semi-hollow 3D scatterers with ellipsoidal, cylindrical and cartesian symmetries,” Comp. Opt. 40(5), 635–641 (2016).
7. T. Wang, H. Wu, and Y. Ding, “Changes in the spectral degree of coherence of a light wave on scattering from a particulate medium,” Opt. Commun. 381, 210–213 (2016). [CrossRef]
8. J. Li and L. Chang, “Spectral shifts and spectral switches of light generated by scattering of arbitrary coherent waves from a quasi-homogeneous media,” Opt. Express 23(13), 16602–16616 (2015). [CrossRef]
9. J. Li and O. Korotkova, “Random medium model for cusping of plane waves,” Opt. Lett. 42(17), 3251–3254 (2017). [CrossRef]
10. E. Wolf, “Invariance of the Spectrum of Light on Propagation,” Phys. Rev. Lett. 56(13), 1370–1372 (1986). [CrossRef]
11. 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]
12. T. Wang and D. Zhao, “Spectral Switch of Light Induced by Scattering from a System of Particles,” Prog. Electromagn. Res. Lett. 14, 41–49 (2010). [CrossRef]
13. T. Wang, X. Li, X. Ji, and D. Zhao, “Spectral changes and spectral switches of light waves on scattering from a semisoft boundary medium,” Opt. Commun. 324, 152–156 (2014). [CrossRef]
14. X. Du and D. Zhao, “Scattering of light by Gaussian-correlated quasi-homogeneous anisotropic media,” Opt. Lett. 35(3), 384–386 (2010). [CrossRef]
15. X. Du and D. Zhao, “Rotationally symmetric scattering from anisotropic media,” Phys. Lett. A 375(9), 1269–1273 (2011). [CrossRef]
16. X. Du and D. Zhao, “Reciprocity relations for scattering from quasi-homogeneous anisotropic media,” Opt. Commun. 284(16-17), 3808–3810 (2011). [CrossRef]
17. C. Ding, Y. Cai, Y. Zhang, and L. Pan, “Scattering of a partially coherent plane-wave pulse on a deterministic sphere,” Phys. Lett. A 376(42-43), 2697–2702 (2012). [CrossRef]
18. T. Wang and D. Zhao, “Stokes parameters of an electromagnetic light wave on scattering,” Opt. Commun. 285(6), 893–895 (2012). [CrossRef]
19. Y. Zhang and D. Zhao, “The coherence and polarization properties of electromagnetic rectangular Gaussian Schell-model sources scattered by a deterministic medium,” J. Opt. 16(12), 125709 (2014). [CrossRef]
20. D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(3), 1128–1135 (1994). [CrossRef]
21. G. Gbur and E. Wolf, “Determination of density correlation functions from scattering of polychromatic light,” Opt. Commun. 168(1-4), 39–45 (1999). [CrossRef]
22. C. Ding, Y. Cai, O. Korotkova, Y. Zhang, and L. Pan, “Scattering-induced changes in the temporal coherence length and the pulse duration of a partially coherent plane-wave pulse,” Opt. Lett. 36(4), 517–519 (2011). [CrossRef]
23. M. Lahiri, E. Wolf, D.G. Fischer, and T. Shirai, “Determination of Correlation Functions of Scattering Potentials of Stochastic Media from Scattering Experiments,” Phys. Rev. Lett. 102(12), 123901 (2009). [CrossRef]
24. L. Hong, W. Wang, J. Liu, and W. Liu, “Application of imaging visibility to measurement of correlation coefficient of scattering potential,” Opt. Appl. 41(3), 557–565 (2011).
25. D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007). [CrossRef]
26. Z. Mei and O. Korotkova, “Random sources generating ring-shaped beams,” Opt. Lett. 38(2), 91–93 (2013). [CrossRef]
27. Y. Chen, L. Liu, F. Wang, C. Zhao, and Y. Cai, “Elliptical Laguerre-Gaussian correlated Schell-model beam,” Opt. Express 22(11), 13975–13987 (2014). [CrossRef]
28. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef]
29. J. Li, F. Wang, and O. Korotkova, “Random sources for cusped beams,” Opt. Express 24(16), 17779–17791 (2016). [CrossRef]
30. Z. Mei and O. Korotkova, “Cosine-Gaussian Schell-model sources,” Opt. Lett. 38(14), 2578–2580 (2013). [CrossRef]
31. Z. Mei, D. Zhao, and O. Korotkova, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015). [CrossRef]
32. L. Wan and D. Zhao, “Optical coherence grids and their propagation characteristics,” Opt. Express 26(2), 2168–2180 (2018). [CrossRef]
33. O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015). [CrossRef]
34. O. Korotkova, “Can a sphere scatter light producing rectangular density patterns?” Opt. Lett. 40(8), 1709–1712 (2015). [CrossRef]
35. J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016). [CrossRef]
36. Y. Ding and D. Zhao, “Random medium model for producing optical coherence lattice,” Opt. Express 25(21), 25222–25233 (2017). [CrossRef]
37. Z. Zhu, H. Wu, K. Cheng, and T. Wang, “Ring-shaped optical coherence lattice distribution produced by light waves on scattering,” Opt. Commun. 434, 157–162 (2019). [CrossRef]
38. X. Pan, K. Cheng, X. Ji, and T. Wang, “Manipulating far-zone scattered field by convolution of different types of weight function,” Opt. Express 28(11), 16869–16878 (2020). [CrossRef]
39. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).
40. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef]
41. R. Martínez-Herrero, P. M. Mejías, and F. Gori, “Genuine cross-spectral densities and pseudo-modal expansions,” Opt. Lett. 34(9), 1399–1401 (2009). [CrossRef]
42. R. A. Silverman, “Scattering of plane waves by locally homogeneous dielectric noise,” Math. Proc. Camb. Phil. Soc. 54(4), 530–537 (1958). [CrossRef]
43. W. H. Carter and E. Wolf, “Scattering from quasi-homogeneous media,” Opt. Commun. 67(2), 85–90 (1988). [CrossRef]
44. G. Zheng, D. Ye, X. Peng, M. Song, and Q. Zhao, “Tunable scattering density with prescribed weak media,” Opt. Express 24(21), 24169–24178 (2016). [CrossRef]
45. H. Wu, X. Pan, Z. Zhu, X. Ji, and T. Wang, “Reciprocity relations of an electromagnetic light wave on scattering from a quasi-homogeneous anisotropic medium,” Opt. Express 25(10), 11297–11305 (2017). [CrossRef]
46. J. Xu, K. Pan, and D. Zhao, “Random sources generating hollow array beams,” Opt. Express 28(11), 16772–16781 (2020). [CrossRef]