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Study on the correspondence between random surface topography and its interface speckle field
Optics Communications 462 (2020) 125308
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Study on the correspondence between random surface topography and its interface speckle field Xiaoyi Chen a ,∗, Yuqin Zhang b , Yujing Han a , Zhenyu Rong a , Li Zhang a , Zhenhua Li c , Chuanfu Cheng b a
School of Physics and Technology, University of Jinan, Jinan, Shandong 250022, China College of Physics and Electronics, Shandong Normal University, Jinan, Shandong 250014, China c College of Physics and Electronic Information, Dezhou University, Dezhou, Shandong 253023, China b
ARTICLE Keywords: Interface speckle Surface topography Correspondence Influence
INFO
ABSTRACT Extracting the information of a random surface based on its speckle field is an important part of non-destructive measurement. To the best of our knowledge, few have emphasised the correspondence between the topography of a random surface and its speckle field. In this paper, Finite Difference Time Domain (FDTD) Solutions software is used to calculate the speckle field at the position of the maximum height of a random surface, called the interface speckle field. An interface speckle field is also extracted experimentally. The results are consistent with the theoretical calculations. The correspondence between the physical quantities of the interface speckle field and the topography of the random surface and the influence of various factors on the correspondence are discussed in detail. Studies indicate that when the lateral correlation length of random surfaces or the wavelength of incident light increases and the roughness or the refractive index of the random surfaces decreases, the correspondence improves. The topography of a random surface is almost identical to the phase distribution of its interface speckle field when its refractive index is small or its roughness is on the order of nanometres.
1. Introduction When an incident light is scattered or reflected by a random surface, random intensity distributions called speckle patterns appear [1]. Applying speckle fields to extract information on surfaces is an important research topic. For example, statistical properties of the speckle intensity of random surfaces made with different materials are used to distinguish the materials [2]; the intensity correlation of the nearfield speckle of single-layer graphene can discriminate its lattice defect density [3]. Numerous studies have shown that speckle fields closer to a surface contain more surface information [4–21]. For example, the intensity correlation of near-field speckles is related to the refractive index of scattering media [4]; the structural features of a random surface affect its near-field coherence length [5,9], and the parameters of random surfaces are successfully extracted based on the intensity correlation of the speckles near surfaces [14–17]. To date, most studies have focussed on the effects of the properties of random surfaces on the statistical properties of their speckle fields. Some references [11,12] also show the topography of random spherical particles and their corresponding speckle intensities and demonstrate that the intensity distributions are affected by the size and the distribution density of the random particles. However, few studies have focussed
on the correspondence between the topography of a random surface and its interface speckle field whose scattering distance equals the maximum height of the random surface. This paper emphasises the correspondence and the influence of random surface roughness, lateral correlation length of the random surface, wavelength of incident light, and medium refractive index on the correspondence. 2. Numerical calculation of interface speckle fields In previous studies [22,23], we calculated speckle fields near random surfaces using Kirchhoff’s approximation theory and determined the correspondence between the distribution of the phase and intensity and the surface topography. The finite difference time domain (FDTD) method [24] is an important method for the numerical calculation of electromagnetic fields. It uses the centre difference quotient instead of the first-order differential quotient of the field to time and space, and the field distribution is obtained through the recursive simulation of the wave propagation process in the time domain. It is usually used to theoretically calculate light fields close to surfaces, especially those in near field. Our research involves near-field regions, so FDTD Solutions software is adopted to numerically calculate interface speckle fields.
∗ Corresponding author. E-mail address: [email protected] (X. Chen).
https://doi.org/10.1016/j.optcom.2020.125308 Received 28 August 2019; Received in revised form 6 January 2020; Accepted 12 January 2020 Available online 15 January 2020 0030-4018/© 2020 Elsevier B.V. All rights reserved.
X. Chen, Y. Zhang, Y. Han et al.
Optics Communications 462 (2020) 125308
Fig. 1. AFM image of a random surface sample.
To study the correspondence between the topography of a random surface and its interface speckle field, it is necessary to build a random surface model. A self-affine fractal surface is a model that can accurately describe many types of rough surfaces [25–27]. The height data of the model used in numerical simulation can be obtained by measuring the height data of a random glass surface with an atomic force microscope (AFM) [28]. We ground an optical flat glass with silicon carbide powder and measured its surface height. A 40 μm × 40 μm AFM image is shown in Fig. 1. According to the height data, surface roughness 𝑤 = 0.6197 ± 0.0021 μm is acquired. The lateral correlation length of a random surface equals the correlation interval at which the normalised auto-correlation function of the surface height drops to 1/e [25]. Based on the height auto-correlation function 𝑅ℎ (𝜌) = ⟨ℎ(𝑟)ℎ(𝑟 + 𝜌)⟩∕𝑤2 , where, 𝜌 is the correlation interval and w is the surface roughness, the lateral correlation length of the sample is 3.457 ± 0.032 μm. In the numerical calculations, the plane where the average height of the sample is located is set as the XY plane; a bundle of monochromatic parallel light polarised in the Y direction and with wavelength 𝜆 = 0.532 μm is perpendicularly incident on the random surface along the Z axis direction. The maximum height of the sample is 1.857 μm, so the distance of the observation plane is 𝑧 = 1.9 μm. We approximate the speckle field at this location as an interface speckle field. The refractive index of the glass medium is n = 1.532. Fig. 2 shows the surface height distribution and the X, Y, and Z components of the intensity of the generated interface speckle. These components refer to the square of the projection of the light vector of any point on the observation plane in the X, Y, and Z directions. Fig. 2 clearly shows that the components have different characteristics. The speckles in Fig. 2(b) are fine and randomly and uniformly distributed. At the edge of Fig. 2(b), there are radially elongated speckles because a longer pulse light wave is used as incident light instead of the ideal plane wave in the simulation. In Fig. 2(c), the long strip structures have a good correspondence with the ‘‘ridges’’ in Fig. 2(a) and the speckles corresponding to the regions with weak height fluctuations are small, uniform, and arranged in a specific direction. There are also strip structures in Fig. 2(d) that are finer and denser than those in Fig. 2(c). Since the Y component speckle intensity more clearly corresponds to the surface topography and its average value is approximately 50 times that of the X component and 10 times that of the Z component, in the following we discuss only the correspondence between the Y component of the speckle field and surface topography.
Fig. 2. (a) Surface height, (b) X, (c) Y, and (d) Z components of the speckle intensity, obtained via numerical calculation.
3. Comparison of surface topography and its interface speckle field Fig. 3 shows the surface height and various physical quantities of the Y component speckle field. In the regions corresponding to the ridges in Fig. 3(a), the fluctuations of all of the physical quantities are the most dramatic, and bright strips appear. The strips in Fig. 3(b) and (c) are narrow and clear and those in Fig. 3(d)–(f) are wide, smooth, and unclear. In the other regions, the light field fluctuates gently, the speckles are relatively uniform and arranged in a certain direction, and the regions are bounded by the strips to form a number of ‘‘patches’’. The patches in Fig. 3(b)–(f) and the patches in Fig. 3(a) are roughly consistent in size and shape but inconsistent in colour due to the periodicity of the physical quantities of the light field. In Fig. 3, the modulus and the intensity have different values but similar characteristics, as do the real and the imaginary parts. Therefore, in the subsequent analysis, we discuss only the intensity, the real part, and the phase of the interface speckle fields. 4. Influence of various factors on the correspondence between surface topography and its interface speckle field 4.1. Influence of the roughness of random surfaces on the correspondence We divided the height data of the sample with roughness 𝑤 = 0.6197 μm by 2 and 10 and obtained different random surfaces with roughness 𝑤 = 0.3100 μm and 𝑤 = 0.0620 μm, respectively. Their interface speckle fields are located at 𝑧 = 0.93 μm and 𝑧 = 0.19 μm, respectively. The other conditions are unchanged. Fig. 4 shows the interface speckle fields of the three samples. From left to right, they are the intensity, phase, and real part, respectively. Generally, as the roughness decreases, the fluctuations of all of the physical quantities become weaker and the shape and size of the ‘‘patches’’ of all of the physical quantities tend to be consistent with those of the ‘‘patches’’ in the height map. In addition, the strips in Fig. 4 become narrower and 2
X. Chen, Y. Zhang, Y. Han et al.
Optics Communications 462 (2020) 125308
Fig. 3. (a) Surface height, (b) modulus, (c) intensity, (d) real part, (e) imaginary part, and (f) phase acquired via numerical calculation.
Fig. 4. The intensity, phase, and real part obtained via numerical calculation when (a1)–(c1) w = 0.6197 μm, (a2)–(c2) w = 0.3100 μm, and (a3)–(c3) w = 0.0620 μm.
3
X. Chen, Y. Zhang, Y. Han et al.
Optics Communications 462 (2020) 125308
Fig. 5. The numerically calculated intensity, phase, and real part with image ranges of (a1)–(c1) 20 μm, (a2)–(c2) 40 μm, and (a3)–(c3) 80 μm.
clearer, and their shape and position tend to coincide with those of the ridges in Fig. 3(a). Interestingly, when the roughness is minimised, phase wrapping disappears in Fig. 4(b3) and Fig. 4(b3) is almost identical to Fig. 3(a). The main reason for this is that when the roughness is very small, the surface scattering ability is so weak that the surface can be regarded as a phase modulation screen. Moreover, the fluctuation in the surface height is less than one wavelength, so there is no phase wrapping and the phase of the light field is similar to the surface topography.
4.3. Influence of the incident light wavelength on the correspondence Fig. 6 shows the interface speckle fields when the wavelength of the incident light is 0.432 μm, 0.532 μm, and 0.632 μm, respectively. In Fig. 6(a1)–(a3), the strips become thicker and the patches become larger and sparser as the wavelength increases, but Fig. 6(a1)–(a3) are roughly similar to each other and have a good correspondence with the surface topography. Conversely, the speckle phase in Fig. 6(b1)– (b3) and the real part in Fig. 6(c1)–(c3) are seriously affected by the wavelength. The phase wrapping becomes severe as the wavelength decreases and Fig. 6(b1) and (c1) further deviate from the surface topography.
4.2. Influence of the correlation length of random surfaces on the correspondence
4.4. Influence of the medium refractive index on the correspondence When the range of the random surfaces and observation planes changes simultaneously and the others are unchanged, the interface speckle fields of random surfaces with different correlation lengths can be obtained. This is experimentally impossible. In the three rows in Fig. 5, from top to bottom, the range is 20 μm × 20 μm, 40 μm × 40 μm, and 80 μm × 80 μm, respectively. In each row, from left to right, they are the intensity, phase, and real part. Fig. 5 demonstrates that the interface speckle fields in Fig. 5(a1)–(c1) are quite different from those in the other figures. The strips disappear in Fig. 5(a1) and the phase in Fig. 5(b1) is approximately even. However, the strips in Fig. 5(a3) are the narrowest and richest and accurately correspond to the ridges in Fig. 3(a). Furthermore, the size of the patches in Fig. 5(b3) is almost equal to that of the patches in Fig. 3(a).
By changing the refractive index of the same random surface, we obtained the interface speckle fields when the refractive indexes are 1.1, 1.3, and 1.532, respectively, as shown in Fig. 7. The aperture diffraction effect is obvious in Fig. 7(a1). Fig. 7(a1)–(a3) are different in fluctuations but approximately similar in features and have a good correspondence with the surface topography. The phase in Fig. 7(b1)– (b3) and the real part in Fig. 7(c1)–(c3) are significantly affected by the refractive index. For example, when the refractive index is 1.1, except for the phase wrapping at two locations, Fig. 7(b1) is almost identical to the surface height map. In addition, the shape and size of each patch in Fig. 7(c1) is almost the same as that of the corresponding patch in the surface topography. 4
X. Chen, Y. Zhang, Y. Han et al.
Optics Communications 462 (2020) 125308
Fig. 6. The intensity, phase, and real part obtained via numerical calculation when the incident light wavelengths are (a1)–(c1) 0.432 μm, (a2)–(c2) 0.532 μm, and (a3)–(c3) 0.632 μm, respectively.
5. Experimental verification
In the experiment, when the direction of polariser P1 was perpendicular to the vertical direction, the intensity on the receiving screen is nearly zero. Conversely, when the direction of polariser P1 was parallel to the vertical direction, the speckle field was almost the same as that recorded without P1. Fig. 9(a) shows the interface speckle intensity recorded without reference light and polariser P1 and with a resolution of 2000 × 2000 pixels. To avoid many calculations, we intercepted the interference pattern of the white rectangular frame of Fig. 9(a) and the enlarged view is Fig. 9(b) with a resolution of 1000 × 1000 pixels. According to the method in reference [23], we extracted the light field from Fig. 9(b). First, the spectrum of Fig. 9(b) was obtained by Fourier transform. The first-order spectrum was selected and its centre was transformed to the origin of the spectrum coordinate. By inverse Fourier transform of the transformed first-order spectrum, the intensity, real part, and imaginary part of the light field were obtained. Based on the relationship between the phase and the real and imaginary parts, the phase of the light field was calculated. It is clear that the speckle intensity in the white rectangular frame of Fig. 9(a) is almost the same as that in Fig. 9(c). Comparing Fig. 9 with Fig. 3 demonstrates that the characteristics of the physical quantities extracted by the experiment are almost the same as those of the physical quantities obtained via numerical calculations.
To experimentally extract an interface speckle field, we designed the Mach–Zehnder interference system as shown in Fig. 8. The vertically polarised green light with a wavelength of 0.532 μm is split into two beams by a splitting prism (SP1). One beam is filtered and expanded by a spatial pinhole filter (SPF) to become a reference light, and the other beam is reflected by a mirror (M1), then scattered by the same random surface as that used in simulation and imaged by a microscope objective lens (MO). The imaging beam and the reference beam interfere with each other after passing through the other splitting prism (SP2), and the interference pattern is recorded by S-CMOS (Zyla5.5, 16-bit, 2560 × 2160 pixels, 6.5 μm × 6.5 μm pixel size). The attenuator A1 adjusts the intensity of the object beam to match that of the reference beam. The polariser P1 is used to obtain the component light field. In the experiment, S-CMOS and the microscope objective is fixed and the random surface is repeatedly adjusted back and forth in the direction of the incident light to find the accurate position of the object plane. When a random surface is just on the location of the object plane, the average size of the recorded speckle is visually the smallest. If the random surface is slightly closer to the imaging system, the average size of the speckle will obviously become larger due to defocusing. If the random surface is slightly further away from the imaging system, the speckle field at a larger distance arrives at the object plane and its average size becomes distinctly larger due to the larger scattering distance. When the average size of the recorded speckle patterns is the smallest, the random surface is just on the object plane and the image of interface speckle field is obtained.
The speckle intensity in the red rectangle in Fig. 9(a) differs from that in the other areas. The main reason may be that the average height of the surface corresponding to the speckle in the red rectangle is far away from that of the surface in the other regions. 5
X. Chen, Y. Zhang, Y. Han et al.
Optics Communications 462 (2020) 125308
Fig. 7. The intensity, phase, and real part obtained via numerical calculation when the refractive indexes of the random surfaces are (a1)–(c1) 1.1, (a2)–(c2) 1.3, and (a3)–(c3) 1.532, respectively.
increases, the roughness decreases, the wavelength of incident light increases or the refractive index of a surface decreases, the scattering ability of the surface becomes weak and this results in a smaller surface area contributing to the light field at arbitrary point of the observation plane. Therefore, the interface speckle field is better able to reflect the topography of the surface. This paper has certain scientific significance for understanding interface speckle fields, for example, the polarisation property, large angle scattered light field located on the surface, formation of a speckle field with a certain evanescent wave component, and the influence of various factors. This paper has application prospects in biomedical imaging and micro-nano topography characterisation, such as roughness, lateral correlation length, fractal index, and other surface parameters.
Fig. 8. Schematic diagram of the experimental device.
6. Discussion and conclusion
CRediT authorship contribution statement
In this paper, interface speckle fields are calculated numerically and extracted experimentally. The experimental results are in complete agreement with the numerical results. The interesting correspondence between the surface topography and its interface speckle field is found and the effect of various factors on the correspondence is discussed. The light field at any point of an observation plane is the interference result of the wavelets from each point of a random surface. However, when the observation plane is extremely close to the surface, the wavelets that contribute to the light field at a point of the observation plane come from a small surface area that is opposite to the point of the observation plane. This is why the interface speckle fields contain surface information. When the lateral correlation length of a surface
Xiaoyi Chen: Conceptualization, Methodology, Software, Writing original draft, Funding acquisition. Yuqin Zhang: Investigation, Software. Yujing Han: Funding acquisition. Zhenyu Rong: Funding acquisition. Li Zhang: Project administration. Zhenhua Li: Resources. Chuanfu Cheng: Resources, Writing - review & editing. Acknowledgements The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NSFC) (Grant nos. 11504096 6
X. Chen, Y. Zhang, Y. Han et al.
Optics Communications 462 (2020) 125308
Fig. 9. (a) Interface speckle intensity. (b) The interface speckle in the white rectangular frame in Fig. 9(a) interferes with the reference light. (c) The speckle intensity, (d) real part, (e) imaginary part, and (f) phase extracted from Fig. 9(b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
and 11704059); the Shandong Provincial Natural Science Foundation, China (Grant no. ZR2017MA047); the Doctoral Foundation of the University of Jinan, China (Grant nos. XBS1407 and XBS1611); and the Scientific Research Foundation of the University of Jinan, China (Grant nos. XKY1407 and XKY1706).
[14] C.X. Liu, Q.R. Dong, H.X. Li, Z.H. Li, X. Li, C.F. Cheng, Measurement of surface parameters from autocorrelation function of speckles in deep fresnel region with microscopic imaging system, Opt. Express 22 (2) (2014) 1302–1312. [15] G.T. Liang, X. Li, M.N. Zhang, Z.H. Li, C.X. Liu, C.F. Cheng, Experimental extraction of rough surface parameters from speckles in the deep Fresnel region with a scanning fibre-optic probe, Eur. Phys. J. D 67 (2013) 030498. [16] C.F. Cheng, C.X. Liu, N.Y. Zhang, T.Q. Jia, R.X. Li, Z.Z. Xu, Absolute measurement of roughness and lateral-correlation length of random surfaces by use of the simplified model of image-speckle contrast, Appl. Opt. 41 (20) (2002) 4148–4156. [17] P. Lehmann, Surface-roughness measurement based on the intensity correlation function of scattered light under speckle pattern illumination, Appl. Opt. 38 (1999) 1144–1152. [18] J. Laverdant, S. Buil, J.P. Hermier, X. Quélin, Near-field intensity correlations on nanoscaled random silver/dielectric films, J. Nanophotonics 4 (1) (2010) 049505. [19] J. Laverdant, S. Buil, B. Bérini, X. Quélin, Polarization dependent near-field speckle of random gold films, Phys. Rev. B 77 (16) (2008) 165406. [20] V. Coello, S.I. Bozhevolnyi, Experimental statistics of near-field intensity distributions at nanostructured surfaces, J. Microsc.-Oxford 202 (Pt 1) (2001) 136–141. [21] J. Kondev, C.L. Henley, D.G. Salinas, Nonlinear measures for characterizing rough surface morphologies, Phys. Rev. E 61 (2000) 104–125. [22] X.Y. Chen, C.F. Cheng, Z.H. Li, H.K. Zhang, M. Liu, M.N. Zhang, Study of the evolutions of speckle intensities and their statistical properties in the extremely deep fresnel diffraction region, Opt. Commun. 329 (2014) 113–118. [23] X.Y. Chen, C.F. Cheng, G.Q. An, Y.J. Han, Z.Y. Rong, L. Zhang, Speckle phase near random surfaces, Opt. Commun. 410 (2018) 310–316. [24] D.M. Sullivan, Electromagnetic Simulation using the FDTD Method, IEEE, New York, 2000. [25] Y.P. Zhao, G.C. Wang, T.M. Lu, Characterization of Amorphous and Crystalline Rough Surface: Principles and Applications, Academic Press, 2001. [26] J.A. Sánchez-Gil, J.V. García-Ramos, E.R. Méndez, Light scattering from selfaffine fractal silver surfaces with nanoscale cutoff: Far-field and near-field calculations, J. Opt. Soc. Amer. A 19 (5) (2002) 902–911. [27] I. Simonsen, D. Vandembroucq, S. Roux, Electromagnetic wave scattering from conducting self-affine surfaces: an analytic and numerical study, J. Opt. Soc. Amer. A 18 (2001) 1101–1111. [28] D.P. Qi, D.L. Liu, S.Y. Teng, N.Y. Zhang, C.F. Cheng, Morphological analysis by atomic force microscope and light scattering study for random scattering screens, Acta Phys. Sin. 49 (7) (2000) 1260–1266.
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7
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Study on the correspondence between random surface topography and its interface speckle field Xiaoyi Chen a ,∗, Yuqin Zhang b , Yujing Han a , Zhenyu Rong a , Li Zhang a , Zhenhua Li c , Chuanfu Cheng b a
School of Physics and Technology, University of Jinan, Jinan, Shandong 250022, China College of Physics and Electronics, Shandong Normal University, Jinan, Shandong 250014, China c College of Physics and Electronic Information, Dezhou University, Dezhou, Shandong 253023, China b
ARTICLE Keywords: Interface speckle Surface topography Correspondence Influence
INFO
ABSTRACT Extracting the information of a random surface based on its speckle field is an important part of non-destructive measurement. To the best of our knowledge, few have emphasised the correspondence between the topography of a random surface and its speckle field. In this paper, Finite Difference Time Domain (FDTD) Solutions software is used to calculate the speckle field at the position of the maximum height of a random surface, called the interface speckle field. An interface speckle field is also extracted experimentally. The results are consistent with the theoretical calculations. The correspondence between the physical quantities of the interface speckle field and the topography of the random surface and the influence of various factors on the correspondence are discussed in detail. Studies indicate that when the lateral correlation length of random surfaces or the wavelength of incident light increases and the roughness or the refractive index of the random surfaces decreases, the correspondence improves. The topography of a random surface is almost identical to the phase distribution of its interface speckle field when its refractive index is small or its roughness is on the order of nanometres.
1. Introduction When an incident light is scattered or reflected by a random surface, random intensity distributions called speckle patterns appear [1]. Applying speckle fields to extract information on surfaces is an important research topic. For example, statistical properties of the speckle intensity of random surfaces made with different materials are used to distinguish the materials [2]; the intensity correlation of the nearfield speckle of single-layer graphene can discriminate its lattice defect density [3]. Numerous studies have shown that speckle fields closer to a surface contain more surface information [4–21]. For example, the intensity correlation of near-field speckles is related to the refractive index of scattering media [4]; the structural features of a random surface affect its near-field coherence length [5,9], and the parameters of random surfaces are successfully extracted based on the intensity correlation of the speckles near surfaces [14–17]. To date, most studies have focussed on the effects of the properties of random surfaces on the statistical properties of their speckle fields. Some references [11,12] also show the topography of random spherical particles and their corresponding speckle intensities and demonstrate that the intensity distributions are affected by the size and the distribution density of the random particles. However, few studies have focussed
on the correspondence between the topography of a random surface and its interface speckle field whose scattering distance equals the maximum height of the random surface. This paper emphasises the correspondence and the influence of random surface roughness, lateral correlation length of the random surface, wavelength of incident light, and medium refractive index on the correspondence. 2. Numerical calculation of interface speckle fields In previous studies [22,23], we calculated speckle fields near random surfaces using Kirchhoff’s approximation theory and determined the correspondence between the distribution of the phase and intensity and the surface topography. The finite difference time domain (FDTD) method [24] is an important method for the numerical calculation of electromagnetic fields. It uses the centre difference quotient instead of the first-order differential quotient of the field to time and space, and the field distribution is obtained through the recursive simulation of the wave propagation process in the time domain. It is usually used to theoretically calculate light fields close to surfaces, especially those in near field. Our research involves near-field regions, so FDTD Solutions software is adopted to numerically calculate interface speckle fields.
∗ Corresponding author. E-mail address: [email protected] (X. Chen).
https://doi.org/10.1016/j.optcom.2020.125308 Received 28 August 2019; Received in revised form 6 January 2020; Accepted 12 January 2020 Available online 15 January 2020 0030-4018/© 2020 Elsevier B.V. All rights reserved.
X. Chen, Y. Zhang, Y. Han et al.
Optics Communications 462 (2020) 125308
Fig. 1. AFM image of a random surface sample.
To study the correspondence between the topography of a random surface and its interface speckle field, it is necessary to build a random surface model. A self-affine fractal surface is a model that can accurately describe many types of rough surfaces [25–27]. The height data of the model used in numerical simulation can be obtained by measuring the height data of a random glass surface with an atomic force microscope (AFM) [28]. We ground an optical flat glass with silicon carbide powder and measured its surface height. A 40 μm × 40 μm AFM image is shown in Fig. 1. According to the height data, surface roughness 𝑤 = 0.6197 ± 0.0021 μm is acquired. The lateral correlation length of a random surface equals the correlation interval at which the normalised auto-correlation function of the surface height drops to 1/e [25]. Based on the height auto-correlation function 𝑅ℎ (𝜌) = ⟨ℎ(𝑟)ℎ(𝑟 + 𝜌)⟩∕𝑤2 , where, 𝜌 is the correlation interval and w is the surface roughness, the lateral correlation length of the sample is 3.457 ± 0.032 μm. In the numerical calculations, the plane where the average height of the sample is located is set as the XY plane; a bundle of monochromatic parallel light polarised in the Y direction and with wavelength 𝜆 = 0.532 μm is perpendicularly incident on the random surface along the Z axis direction. The maximum height of the sample is 1.857 μm, so the distance of the observation plane is 𝑧 = 1.9 μm. We approximate the speckle field at this location as an interface speckle field. The refractive index of the glass medium is n = 1.532. Fig. 2 shows the surface height distribution and the X, Y, and Z components of the intensity of the generated interface speckle. These components refer to the square of the projection of the light vector of any point on the observation plane in the X, Y, and Z directions. Fig. 2 clearly shows that the components have different characteristics. The speckles in Fig. 2(b) are fine and randomly and uniformly distributed. At the edge of Fig. 2(b), there are radially elongated speckles because a longer pulse light wave is used as incident light instead of the ideal plane wave in the simulation. In Fig. 2(c), the long strip structures have a good correspondence with the ‘‘ridges’’ in Fig. 2(a) and the speckles corresponding to the regions with weak height fluctuations are small, uniform, and arranged in a specific direction. There are also strip structures in Fig. 2(d) that are finer and denser than those in Fig. 2(c). Since the Y component speckle intensity more clearly corresponds to the surface topography and its average value is approximately 50 times that of the X component and 10 times that of the Z component, in the following we discuss only the correspondence between the Y component of the speckle field and surface topography.
Fig. 2. (a) Surface height, (b) X, (c) Y, and (d) Z components of the speckle intensity, obtained via numerical calculation.
3. Comparison of surface topography and its interface speckle field Fig. 3 shows the surface height and various physical quantities of the Y component speckle field. In the regions corresponding to the ridges in Fig. 3(a), the fluctuations of all of the physical quantities are the most dramatic, and bright strips appear. The strips in Fig. 3(b) and (c) are narrow and clear and those in Fig. 3(d)–(f) are wide, smooth, and unclear. In the other regions, the light field fluctuates gently, the speckles are relatively uniform and arranged in a certain direction, and the regions are bounded by the strips to form a number of ‘‘patches’’. The patches in Fig. 3(b)–(f) and the patches in Fig. 3(a) are roughly consistent in size and shape but inconsistent in colour due to the periodicity of the physical quantities of the light field. In Fig. 3, the modulus and the intensity have different values but similar characteristics, as do the real and the imaginary parts. Therefore, in the subsequent analysis, we discuss only the intensity, the real part, and the phase of the interface speckle fields. 4. Influence of various factors on the correspondence between surface topography and its interface speckle field 4.1. Influence of the roughness of random surfaces on the correspondence We divided the height data of the sample with roughness 𝑤 = 0.6197 μm by 2 and 10 and obtained different random surfaces with roughness 𝑤 = 0.3100 μm and 𝑤 = 0.0620 μm, respectively. Their interface speckle fields are located at 𝑧 = 0.93 μm and 𝑧 = 0.19 μm, respectively. The other conditions are unchanged. Fig. 4 shows the interface speckle fields of the three samples. From left to right, they are the intensity, phase, and real part, respectively. Generally, as the roughness decreases, the fluctuations of all of the physical quantities become weaker and the shape and size of the ‘‘patches’’ of all of the physical quantities tend to be consistent with those of the ‘‘patches’’ in the height map. In addition, the strips in Fig. 4 become narrower and 2
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Optics Communications 462 (2020) 125308
Fig. 3. (a) Surface height, (b) modulus, (c) intensity, (d) real part, (e) imaginary part, and (f) phase acquired via numerical calculation.
Fig. 4. The intensity, phase, and real part obtained via numerical calculation when (a1)–(c1) w = 0.6197 μm, (a2)–(c2) w = 0.3100 μm, and (a3)–(c3) w = 0.0620 μm.
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Optics Communications 462 (2020) 125308
Fig. 5. The numerically calculated intensity, phase, and real part with image ranges of (a1)–(c1) 20 μm, (a2)–(c2) 40 μm, and (a3)–(c3) 80 μm.
clearer, and their shape and position tend to coincide with those of the ridges in Fig. 3(a). Interestingly, when the roughness is minimised, phase wrapping disappears in Fig. 4(b3) and Fig. 4(b3) is almost identical to Fig. 3(a). The main reason for this is that when the roughness is very small, the surface scattering ability is so weak that the surface can be regarded as a phase modulation screen. Moreover, the fluctuation in the surface height is less than one wavelength, so there is no phase wrapping and the phase of the light field is similar to the surface topography.
4.3. Influence of the incident light wavelength on the correspondence Fig. 6 shows the interface speckle fields when the wavelength of the incident light is 0.432 μm, 0.532 μm, and 0.632 μm, respectively. In Fig. 6(a1)–(a3), the strips become thicker and the patches become larger and sparser as the wavelength increases, but Fig. 6(a1)–(a3) are roughly similar to each other and have a good correspondence with the surface topography. Conversely, the speckle phase in Fig. 6(b1)– (b3) and the real part in Fig. 6(c1)–(c3) are seriously affected by the wavelength. The phase wrapping becomes severe as the wavelength decreases and Fig. 6(b1) and (c1) further deviate from the surface topography.
4.2. Influence of the correlation length of random surfaces on the correspondence
4.4. Influence of the medium refractive index on the correspondence When the range of the random surfaces and observation planes changes simultaneously and the others are unchanged, the interface speckle fields of random surfaces with different correlation lengths can be obtained. This is experimentally impossible. In the three rows in Fig. 5, from top to bottom, the range is 20 μm × 20 μm, 40 μm × 40 μm, and 80 μm × 80 μm, respectively. In each row, from left to right, they are the intensity, phase, and real part. Fig. 5 demonstrates that the interface speckle fields in Fig. 5(a1)–(c1) are quite different from those in the other figures. The strips disappear in Fig. 5(a1) and the phase in Fig. 5(b1) is approximately even. However, the strips in Fig. 5(a3) are the narrowest and richest and accurately correspond to the ridges in Fig. 3(a). Furthermore, the size of the patches in Fig. 5(b3) is almost equal to that of the patches in Fig. 3(a).
By changing the refractive index of the same random surface, we obtained the interface speckle fields when the refractive indexes are 1.1, 1.3, and 1.532, respectively, as shown in Fig. 7. The aperture diffraction effect is obvious in Fig. 7(a1). Fig. 7(a1)–(a3) are different in fluctuations but approximately similar in features and have a good correspondence with the surface topography. The phase in Fig. 7(b1)– (b3) and the real part in Fig. 7(c1)–(c3) are significantly affected by the refractive index. For example, when the refractive index is 1.1, except for the phase wrapping at two locations, Fig. 7(b1) is almost identical to the surface height map. In addition, the shape and size of each patch in Fig. 7(c1) is almost the same as that of the corresponding patch in the surface topography. 4
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Fig. 6. The intensity, phase, and real part obtained via numerical calculation when the incident light wavelengths are (a1)–(c1) 0.432 μm, (a2)–(c2) 0.532 μm, and (a3)–(c3) 0.632 μm, respectively.
5. Experimental verification
In the experiment, when the direction of polariser P1 was perpendicular to the vertical direction, the intensity on the receiving screen is nearly zero. Conversely, when the direction of polariser P1 was parallel to the vertical direction, the speckle field was almost the same as that recorded without P1. Fig. 9(a) shows the interface speckle intensity recorded without reference light and polariser P1 and with a resolution of 2000 × 2000 pixels. To avoid many calculations, we intercepted the interference pattern of the white rectangular frame of Fig. 9(a) and the enlarged view is Fig. 9(b) with a resolution of 1000 × 1000 pixels. According to the method in reference [23], we extracted the light field from Fig. 9(b). First, the spectrum of Fig. 9(b) was obtained by Fourier transform. The first-order spectrum was selected and its centre was transformed to the origin of the spectrum coordinate. By inverse Fourier transform of the transformed first-order spectrum, the intensity, real part, and imaginary part of the light field were obtained. Based on the relationship between the phase and the real and imaginary parts, the phase of the light field was calculated. It is clear that the speckle intensity in the white rectangular frame of Fig. 9(a) is almost the same as that in Fig. 9(c). Comparing Fig. 9 with Fig. 3 demonstrates that the characteristics of the physical quantities extracted by the experiment are almost the same as those of the physical quantities obtained via numerical calculations.
To experimentally extract an interface speckle field, we designed the Mach–Zehnder interference system as shown in Fig. 8. The vertically polarised green light with a wavelength of 0.532 μm is split into two beams by a splitting prism (SP1). One beam is filtered and expanded by a spatial pinhole filter (SPF) to become a reference light, and the other beam is reflected by a mirror (M1), then scattered by the same random surface as that used in simulation and imaged by a microscope objective lens (MO). The imaging beam and the reference beam interfere with each other after passing through the other splitting prism (SP2), and the interference pattern is recorded by S-CMOS (Zyla5.5, 16-bit, 2560 × 2160 pixels, 6.5 μm × 6.5 μm pixel size). The attenuator A1 adjusts the intensity of the object beam to match that of the reference beam. The polariser P1 is used to obtain the component light field. In the experiment, S-CMOS and the microscope objective is fixed and the random surface is repeatedly adjusted back and forth in the direction of the incident light to find the accurate position of the object plane. When a random surface is just on the location of the object plane, the average size of the recorded speckle is visually the smallest. If the random surface is slightly closer to the imaging system, the average size of the speckle will obviously become larger due to defocusing. If the random surface is slightly further away from the imaging system, the speckle field at a larger distance arrives at the object plane and its average size becomes distinctly larger due to the larger scattering distance. When the average size of the recorded speckle patterns is the smallest, the random surface is just on the object plane and the image of interface speckle field is obtained.
The speckle intensity in the red rectangle in Fig. 9(a) differs from that in the other areas. The main reason may be that the average height of the surface corresponding to the speckle in the red rectangle is far away from that of the surface in the other regions. 5
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Optics Communications 462 (2020) 125308
Fig. 7. The intensity, phase, and real part obtained via numerical calculation when the refractive indexes of the random surfaces are (a1)–(c1) 1.1, (a2)–(c2) 1.3, and (a3)–(c3) 1.532, respectively.
increases, the roughness decreases, the wavelength of incident light increases or the refractive index of a surface decreases, the scattering ability of the surface becomes weak and this results in a smaller surface area contributing to the light field at arbitrary point of the observation plane. Therefore, the interface speckle field is better able to reflect the topography of the surface. This paper has certain scientific significance for understanding interface speckle fields, for example, the polarisation property, large angle scattered light field located on the surface, formation of a speckle field with a certain evanescent wave component, and the influence of various factors. This paper has application prospects in biomedical imaging and micro-nano topography characterisation, such as roughness, lateral correlation length, fractal index, and other surface parameters.
Fig. 8. Schematic diagram of the experimental device.
6. Discussion and conclusion
CRediT authorship contribution statement
In this paper, interface speckle fields are calculated numerically and extracted experimentally. The experimental results are in complete agreement with the numerical results. The interesting correspondence between the surface topography and its interface speckle field is found and the effect of various factors on the correspondence is discussed. The light field at any point of an observation plane is the interference result of the wavelets from each point of a random surface. However, when the observation plane is extremely close to the surface, the wavelets that contribute to the light field at a point of the observation plane come from a small surface area that is opposite to the point of the observation plane. This is why the interface speckle fields contain surface information. When the lateral correlation length of a surface
Xiaoyi Chen: Conceptualization, Methodology, Software, Writing original draft, Funding acquisition. Yuqin Zhang: Investigation, Software. Yujing Han: Funding acquisition. Zhenyu Rong: Funding acquisition. Li Zhang: Project administration. Zhenhua Li: Resources. Chuanfu Cheng: Resources, Writing - review & editing. Acknowledgements The authors gratefully acknowledge the support of the National Natural Science Foundation of China (NSFC) (Grant nos. 11504096 6
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Optics Communications 462 (2020) 125308
Fig. 9. (a) Interface speckle intensity. (b) The interface speckle in the white rectangular frame in Fig. 9(a) interferes with the reference light. (c) The speckle intensity, (d) real part, (e) imaginary part, and (f) phase extracted from Fig. 9(b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
and 11704059); the Shandong Provincial Natural Science Foundation, China (Grant no. ZR2017MA047); the Doctoral Foundation of the University of Jinan, China (Grant nos. XBS1407 and XBS1611); and the Scientific Research Foundation of the University of Jinan, China (Grant nos. XKY1407 and XKY1706).
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