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Scattering of a non-paraxial Bessel light-sheet by a sphere of arbitrary size
Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106869
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Scattering of a non-paraxial Bessel light-sheet by a sphere of arbitrary size Shu Zhang a, Renxian Li a,b,∗, Bojian Wei a, Jiaming Zhang a, Han Sun a, Ningning Song a a b
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China
a r t i c l e
i n f o
Article history: Received 30 October 2019 Revised 28 January 2020 Accepted 28 January 2020 Available online 29 January 2020 Keywords: Scattering Bessel light-sheet Angular spectrum decomposition GLMT
a b s t r a c t The scattering of a non-paraxial Bessel light-sheet by a sphere of arbitrary size is studied in framework of generalized Lorenz-–Mie theory (GLMT). The electrical fields of the Bessel light-sheet are expanded using the vector angular spectrum decomposition method (VASDM), and the beam shape coefficients (BSCs) of the Bessel light-sheet are derived using the method of multipole expansion and vector spherical wave functions (VSWF). The internal and near-surface fields, and absorption and extinction efficiency factors are numerically calculated when a Bessel light-sheet is incident, and the effects of beam’s order, and half-cone angle are mainly discussed. Numerical results show that there is strong point convergence in the forward scattering region under the incident of a Bessel light-sheet, and the internal and near-surface fields of Bessel light-sheet are very sensitive to the beam’s order, half-cone angle, etc. Such results have potential applications in the super-resolution imaging using a light-sheet microscopy. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction Light-sheet has potential applications in imaging microscope [1], particle characterization and sizing [2–4], optical clearing [5,6], non-destructive optical sectioning and imaging of the internal subcellular features [7–9], etc. As one of the biggest advances in microscopy techniques in recent years, light-sheet microscopy, firstly proposed over 100 years ago [10], has many advantages for studying biological samples [7,8,11–14], and provides true optical sectioning by sample illumination with a thin light-sheet. However, standard light-sheet microscopy, which uses Gaussian beam excitations, has disadvantages. Its field of view (FOV) is limited by the depth of the focus of the Gaussian beam, and the excitation beam has a limited penetration depth. To overcome these limitations, Bessel beam [15], which has the properties of self-healing and non-diffraction, is used instead of Gaussian beam in lightsheet microscopy [13,16]. The theoretical prediction of the interaction between light-sheets with particles can help to improve the light-sheet microscopy. Many researches have been devoted to the characteristics of light-sheets and their interaction with particles. In recent years, Gaussian light-sheet [17], Hermite-Gaussian light-sheet, Airy lightsheet [18], and Bessel-–Gaussian light-sheet pincers [19] have been
∗
Corresponding author. E-mail address: [email protected] (R. Li).
https://doi.org/10.1016/j.jqsrt.2020.106869 0022-4073/© 2020 Elsevier Ltd. All rights reserved.
studied and applied in various fields, including optical manipulation [20], particle sizing [21], optical tomography sectioning [22], light-sheet microscopy [16,23–25], etc. Gustavo de Medeiros [26] summarized the state-of-the-art of light-sheet imaging with a focus on mammalian development. A novel type of Bessel and Bessel-Gaussian light-sheets has a tight bending characteristics in the form of pincers [19,27]. The interaction of FBG (Fractional Bessel-Gaussian) light-sheets with a dielectric sub-wavelength spheroid, from the standpoint of optical radiation force and spin torque theories, is investigated in the framework of the dipole approximation [28]. Airy light-sheets, obtained from the higher-order spatial derivatives of the fundamental nonparaxial Airy beam solution in the transverse direction, gave the results of radiated field [29]. The description of adjustable vector Airy light-sheets presented a longitudinal polarization of the vector potential, with particular emphasis on the generation of negative longitudinal optical force and axial spin torque [30]. Generalized Lorenz-–Mie theory (GLMT) [31], which provides a rigorous solution to the scattering of arbitrarily shaped laser beams to facilitate the design of arbitrary beam, is employed for the interaction of a sphere with a elliptic Gaussian sheets [32,33]. A generalization of the Lorenz–Mie theory to the case of light-sheet are given by developing exact analytical closed-form expressions for the optical radiation force and torque components [34]. The approach for the description of scalar and vectorial BGSs (Bessel-Gaussian beam) are presented with particular emphasis on routing the wave fields and their flow of energy at will, along designed curved paths in a
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homogeneous medium [35]. A Bessel sheet is a two-dimensional Bessel beam, and can be generated using a cylindrical lens illuminated by a Bessel beam. During the generation, the non-diffracting and self-healing properties of the Bessel beam remain in the Bessel light-sheet. By using a Bessel beam instead of Gaussian beam in light-sheet microscopy, the FOV and penetration depth [11] can be greatly improved. There is a strategy to select and optimize the linear excitation light-sheet in SPIM (selective plane illumination microscopy) based on spatial resolution, FOV and optical sectioning capability, to create a thin and uniform excitation light-sheet [36]. Bessel plane illumination gave several modes that offer different performance with different imaging metrics [37]. By combining ultra-thin planar illumination produced by scanned Bessel beams with super resolution structured illumination microscopy, it is demonstrated in vivo karyotyping of chromosomes during mitosis and identify different dynamics for the actin cytoskeleton at the dorsal and ventral surfaces of fibroblasts [38]. Florian O.Fahrbach [39] gave a study that combines the benefits of light-sheet microscopy, self-reconstructing Bessel beams, and two-photon fluorescence excitation to improve imaging in large scattering media such as cancer cell clusters. Ming Zhao [40] presented that twophoton Bessel light-sheet SIM (structured illumination microscopy) provides a non-invasive tool for deep organ imaging at cellular level resolution in the live animals by addressing the deep tissue imaging challenges in both excitation and emission paths. Bessel light-sheets have important applications in various fields, including optical manipulation, particle sizing, light-sheet microscopy, etc, and most applications of Bessel light-sheets involve the interaction of the Bessel light-sheet with particles. The purpose of this paper is to investigate the scattering of a Bessel light-sheet by a sphere of arbitrary size, and discuss the effects of beam order l, half-cone angle α 0 , etc. The rest of the paper is organized as below. Firstly, a general theory of the electromagnetic fields of a Bessel light-sheet is given in Section 2. The electric field of a Bessel light-sheet is obtained based on the vector angular spectrum decomposition method (VASDM). The electric fields are also expanded in terms of beam shape coefficients (BSCs) and vector spherical wave functions (VSWFs) using the multipole expansion. The internal and nearsurface fields, and the cross sections and their corresponding efficiencies are given on the basis of GLMT. Section 3 discusses some numerical results of internal and near-surface fields, and the efficiency factors. The effects of beam order l and half-cone angle α 0 are analysed. Section 4 is devoted to the conclusion. 2. Theory
half-cone angle. r is the position vector, and r0 is the position vector of beam center. ex is the unit vector in x-direction. Epw0 is the amplitude of plane wave angular spectrum. δ is the delta function. Substituting Eq. (2) into Eq. (1), we can obtain the analytical expression of the electric field of Bessel light-sheets
E(r, θ , φ ) = ex EB0 eik(z−z0 ) cos α0 il Jl (σ )
(3)
where, Jl (·) is the Bessel function of first kind. EB0 and σ are defined by
EB0 = 2π E pw0 ,
σ = k(y − y0 ) sin α0
Since the term ex
eik·r
(4)
in Eq. (1) can be expanded using VSWF
as:
∞ n
ex eik·r =
(1 ) (1 ) Dmn q mn Mmn + p mn Nmn
(5)
n=1 m=−n
where
Dmn =
(2n + 1)(n − m )! n ( n + 1 ) ( n + m )!
q mn = −in+1 e−imβ p mn = −in+1 e−imβ
(6)
πmn (cos α )eθ (α ) − iτmn (cos α )eφ (α ) · ex (7)
τmn (cos α )eθ (α ) − iπmn (cos α )eφ (α ) · ex (8)
Consider a sphere of radius a and refractive index m1 placed in the field of a non-paraxial Bessel light-sheet as shown in Fig. 1. Suppose that the particle center is located at the origin OO of the coordinate system OO,xyz , and the coordinates of beam center are (y0 , z0 ). The surrounding media is assumed to be vacuum. A Bessel light-sheet propagates along the plane of yz and its electric field is along x direction with the suppressed time harmonic factor e−iωt . The wavelength of the incident light-sheet is λ. Based on the VASDM [41], the electric field of a Bessel light-sheet can be expressed as
E ( r, θ , φ ) =
Fig. 1. The picture displays a graphical representation of the propagation of a nonparaxial Bessel light sheet, propagating along the z-direction (λ = 0.6328 μm, the radius of a sphere a = 1 μm, l = 0 and α0 = 20◦ ).
αmax 2π α =0
β =0
Epw eik·(r−r0 ) | sin α cos α|dα dβ
δ (α − α0 ) ilβ e |sin α0 | cos α0
(1 ) Nmn =
1 d
τmn (cos θ )eθ + iπmn (cos θ )eφ
+er n(n + 1 )Pnm (cos θ )
πmn (cos α ) = m τmn (cos α ) =
(2)
where, α is the angle between the wave vector k and the z-axis. k = (0, k sin α , k cos α ) is the wave vector. Epw is the vector angular spectrum of plane wave, l is the order of beam, and α 0 is the
kr dr
(9)
[r jn (kr )] exp(imφ )
jn (kr ) exp(imφ ) kr
(10)
with
(1)
with
Epw = ex E pw0 (α0 , β )
(1 ) Mmn = iπmn (cos θ )eθ − τmn (cos θ )eφ jn (kr ) exp(imφ )
Pnm (cos α ) sin α
dPnm (cos α ) dα
(11) (12)
where, Pnm (cos α ) represents the associated Legendre polynomials of degree n and order m, jn (·) is the spherical Bessel function of the first kind, and e(r,θ ,φ ) are radial, polar, and azimuthal unit vectors. Substituting Eqs. (5)–(12) into Eq. (1), the electric field of an incident Bessel light-sheet Einc can be expanded in terms of the
S. Zhang, R. Li and B. Wei et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106869
VSWFs
Einc (r, θ , ϕ ) = − with
pmn = qmn =
∞
n
n=1 m=−n
iEmn
(1 ) (1 ) pmn Nmn (kr ) + qmn Mmn ( kr )
(13)
Dmn e−ikz z0 [τmn (cos α0 )I+ + πmn (cos α0 )I− ]
(14)
Dmn e−ikz z0 [πmn (cos α0 )I+ + τmn (cos α0 )I− ]
(15)
I± = π i1+l−m J1+l−m (σ0 ) ± i−1+l−m J−1+l−m (σ0 )
(16) (17)
where, (r, θ , ϕ ) is spherical coordinates corresponding to ( x, y, z) with the origin locating on the particle center, Emn = E0 in Dmn , and kz = k cos α0 . According to GLMT, the internal field is also ex(1 ) (1 ) panded in terms of the VSWFs (Nmn (k, r ) and Mmn (k, r )), and the (3 ) (3 ) scattered field in terms of the VSWFs (Nmn (k, r ) and Mmn (k, r )). ∞ n
(1 ) (1 ) iEmn dmn Nmn (m1 kr ) + cmn Mmn ( m1 kr )
(18)
n=1 m=−n
Es =
∞ n
(3 ) (3 ) iEmn amn Nmn (kr ) + bmn Mmn ( kr )
(19)
n=1 m=−n
with
1 d
(3 ) Mmn = iπmn (cos θ )eθ − τmn (cos θ )eφ hn(1) (kr ) exp(imφ ) (3 ) Nmn =
τmn (cos θ )eθ + iπmn (cos θ )eφ
+er n(n + 1 )Pnm (cos θ )
kr dr
(20)
r hn(1) (kr ) exp(imφ )
hn(1) (kr ) exp(imφ ) kr
(21)
where, hn(1 ) (· ) is the spherical Hankel function of the first kind. By considering the boundary conditions, the partial wave expansion coefficients of the scattering field amn and bmn and internal field cmn and dmn can be obtained
amn = an pmn , bmn = bn qmn
(22)
cmn = cn qmn , dmn = dn pmn
(23)
where, an and bn are the traditional Mie scattering coefficients, and cn and dn are the corresponding internal coefficients [42,43]. Once the BSCs are obtained, we can get the scattering, extinction, and absorption efficiencies
⎧ ⎪ Qsca = ⎪ ⎪ ⎪ ⎨ Qext = ⎪ ⎪ ⎪ ⎪ ⎩
C
4 sca π a2 = k2 a2 C
∞ n
n=1 m=−n
4 ext π a2 = k2 a2 Re
|amn |2 + |bmn |2
∞ n n=1 m=−n
∗
qmn bmn + pmn amn ∗
internal and near-surface fields. Then the absorption and extinction efficiencies are calculated. The effects of the order l and halfcone angle α 0 are particularly emphasized. In the calculation, the refractive index of the particle is m1 = 1.33 [44], the refractive index of the surrounding medium is 1.0, and the wavelength of the incident light-sheet is λ = 0.6328 μm. The radius of the particle is a = 1.0 μm, and the corresponding dimensionless size parameter is ka= 10, with k being the wave number. 3.1. Incident field intensities of Bessel light-sheets
σ0 = −ky0 sin α0
E1 = −
3
(24)
Qabs = Qext − Qsca
where, Csca and Cext are the scattering and extinction cross sections, respectively. The star ∗ denotes the complex conjugate. 3. Numerical results and discussions In this section, numerical calculations to a Bessel light-sheet scattering by an isotropic sphere are performed, and the influences of various parameters on the scattering are investigated. Numerical analysis is started by developing a Python program to compute the internal and near-surface fields, and the characteristics of the incident beam are also analysed for the better understanding of the
Bessel light-sheets, which are characterized by Bessel functions, have different characteristics with different half-cone angle α 0 and order l. For convenience, the electric field intensities |E|2 , which is defined by Eq. (3), of incident Bessel light-sheets are calculated, and the effects of order l and half-cone angle α 0 are analysed. Fig. 2 gives the distributions of the electric field intensities with different α 0 , and panels (a) - (d) correspond to α0 = 0◦ , 20◦ , 45◦ , and 85◦ , respectively. From Eq. (3), a zeroth-order (l = 0) Bessel sheet with α0 = 0◦ is deduced to a plane light-sheet, while a high-order Bessel sheet has zero electric field intensity. This is caused by the characteristics of Bessel functions, namely J0 (0 ) = 1 while Jl (0 ) = 0 (l = 0). This can be seen from Fig. 2(a). For α 0 > 0 (Fig. 2(b)–(d), the curves are formed by a series of peaks. In general, these peaks are symmetrical with respect to the y = 0 axis. A zeroth-order Bessel sheet (l = 0) has maximum central intensity, while a high-order Bessel sheet has null central intensity. In Fig. 2(b) (α0 = 20◦ ), the main peak region of the Bessel sheet with l = 0, presenting distribution pattern of the symmetrical function, is about at y ∈ (−0.6 μm, 0.6 μm) with steep curve change. While the Bessel sheet of l = 1 with slow curve change and l = 2 with slower curve change, are symmetrically distributed with y = 0, the range of y regions corresponding to the main peak is approximately y ∈ (−1 μm, 0 ) or y ∈ (0, 1 μm) and y ∈ (−1.5 μm, 0) or y ∈ (0, 1.5 μm), respectively. In Fig. 2(c) (α0 = 45◦ ) the same distribution mode as Fig. 2(b), the y region corresponding to the main peak of the l = 0 order Bessel sheet is probably at y ∈ (−0.3 μm, 0.3 μm), while l = 1 about at y ∈ (−0.5 μm, 0) ∪ (0, 0.5 μm) and l = 2 about at y ∈ (−0.7 μm, 0) ∪ (0, 0.7 μm). Fig. 2(d) gives the results for α0 = 85◦ . A smaller ranges of y for main peak can be seen comparing with that for smaller α 0 . In general, as α 0 grows, the width of the y region becomes narrower. Fig. 3 gives the distribution of electric field intensity with different l. In Fig. 3(a) (l = 0), the main peaks of different α 0 maintain the same value, except that the intensity for α0 = 0◦ keeps the consistent strength. In Fig. 3(b), the electric field intensity of α0 = 0◦ is 0 along the y-axis. The intensity regions corresponding to the main peak of α0 = 20◦ , 45◦ , and 85◦ are about at y ∈ (−1.0 μm, 0) ∪ (0, 1.0 μm), y ∈ (−0.5 μm, 0) ∪ (0, 0.5 μm), and y ∈ (−0.4 μm, 0) ∪ (0, 0.4 μm), respectively. In Fig. 3(c), the electric field intensity of l = 0 is 0 along the y-axis. The intensity regions corresponding to the main peak of α0 = 20◦ , 45◦ , and 85◦ are about at y ∈ (−1.5 μm, 0) ∪ (0, 1.5 μm), y ∈ (−0.8 μm, 0) ∪ (0, 0.8 μm), and y ∈ (−0.5 μm, 0) ∪ (0, 0.5 μm), respectively. In general, as α 0 is bigger, there is a narrower width of y region and as the l is larger, the width of y region is wider. For the convenience of analysis, the electric fields of Bessel light-sheets in yz plane are given Figs. 4–6 for order l = 0, 1, 2, respectively. According to Eq. (3), a zeroth-order Bessel light-sheet with α0 = 0◦ is deduced to a plane light-sheet with unity amplitude, while a high-order Bessel light-sheet has null electric field for α 0 . Thus, in Figs. 4–6, the results for α0 = 0◦ are not depicted. As shown in Fig. 4, for α0 = 20◦ , the bright and dark stripes are distributed along the y-axis. There is a bright stripe in the region
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Fig. 2. Electric field intensity for different α 0 .
Fig. 3. Electric field intensity for different l.
Fig. 4. Electric field intensities of zeroth-order (l = 0) Bessel beams with α 0 being parameter.
S. Zhang, R. Li and B. Wei et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106869
5
Fig. 5. The same as in Fig. 4, but first-order Bessel beams (l = 1).
Fig. 6. The same as in Fig. 4, but second-order Bessel beam (l = 2).
of y ∈ (−0.5 μm, 0.5 μm), the width of which is nearly 1 μm, and the two sides are followed by dark stripes with width of 0.5 μm, and next to both side followed by bright stripes with width of 0.5 μm. Corresponding to α0 = 20◦ of Fig. 2(b), there is a main peak in the region of y ∈ (−0.5 μm, 0.5 μm) and two secondary peak in the region of y ∈ (−1.35 μm, −0.75 μm) ∪ (0.75 μm, 1.35 μm). For α0 = 45◦ , the brightest stripe in Fig. 4(b) along z-axis is the main peak distribution of α0 = 45◦ of Fig. 2(c). The number of secondary peak in Fig. 2(c) is corresponding to the number of bright stripes in Fig. 5. In Fig. 4(c) with α0 = 85◦ , there is a brightest stripe, and the bright and dark stripes on both sides are distributed along both sides of the y-axis, which corresponds to the number of secondary peaks in Fig. 2(d). The width of stripes is narrower than the cases of α0 = 20◦ and α0 = 45◦ . Compared with Fig. 3(b) for l = 0, the larger the α 0 is, the narrower the width of the brightest stripe is, and the more the number of bright and dark stripes are. Different from l = 0 in Fig. 4, there is a dark stripe along y = 0 in Fig. 5. In Fig. 5(a) with α0 = 20◦ , there are two bright stripes in the region of y ∈ (−0.75 μm, −0.25 μm) ∪ (0.25 μm, 0.75 μm) with intensity 0.1, and both sides are dark and bright stripes with the number of 1. According to l = 1 of the Fig. 2(b), there are two main peaks and two secondary peaks. For α0 = 45◦ in Fig. 5(b), there are two bright stripes with a smaller width, compared to the width of α0 = 20◦ . For α0 = 85◦ in Fig. 5(c), there is a bright stripe (actually, there are two stripes.) nearly along z-axis. Correspondingly, for l = 1 in the Fig. 3(b), there is no field along y = 0 and y = 1. But there are some minimal intensity distribution in Fig. 5(c). The electric field intensities for a second-order Bessel lightsheet are given in Fig. 6. Fig. 6(a) displays one dark stripe in the region of y ∈ (−0.5 μm, 0.5 μm), and two bright stripes along zaxis, and two small dark stripes on the both sides. For α0 = 45◦ , there are still one dark stripe with the same width of two bright
stripes, and both sides are the bright and dark stripes in turn. The results for α0 = 85◦ are similar to that for α0 = 45◦ , but having more stripes of bright and dark. According to Fig. 3(c), the width of two main peaks are wide enough compared to the width of main peaks. In general, the bigger the order l is, the less the number of stripes are, and the wider the width of two stripes are. As the increasing of α 0 , there are more stripes along y-axis. And the intensity of incident field maintains the same distribution along z-axis. 3.2. Internal and near-surface fields In this section, the electric field intensities |E|2 of a sphere illuminated by a Bessel light-sheet are discussed. The effect of the order and half-cone angle will be mainly analysed. Figs. 7–9 displays the total field intensity of a sphere illuminated by Bessel lightsheets with different order (l = 0, 1 and 2) and half-cone angle (α0 = 0◦ , 20◦ , 45◦ and 85◦ ). The white circle in the figures denotes the particle. First, we consider the field intensity for a Bessel light-sheet with zeroth-order l = 0, given by Fig. 7. Panels (a) and (d) correspond to the half-cone angle α0 = 0◦ , 20◦ , 45◦ and 85◦ , respectively. A Bessel light-sheet with l = 0 and α0 = 0◦ becomes a plane light-sheet. The incident beam is focused by the sphere, and a photonic nanojet can be seen in the forward direction, as shown in Fig. 7(a). As shown in Fig. 7(b), at the y-axis symmetrical positions, light with a slightly weaker intensity (second peaks) also enters the sphere, and transmits the sphere in the forward direction. Since the intensity of the zeroth-order Bessel light-sheet along the z-axis is large, the extreme region of the intensity on both sides of the z-axis is weaker than the extreme region along the z-axis. Only a strong distribution in the middle can be observed. The area is focused by the sphere. The upper and lower extreme regions are
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Fig. 7. Internal and near-surface electric field intensities of a sphere illuminated by a zeroth-order (l = 0) Bessel light-sheet.
refracted by the particle, and we can see two curves formed by extreme peaks in the vicinity of θ = α0 , but their intensities are weak. Extremely, in the internal field region, while for α0 = 20◦ , there are mainly three pieces of intensity distributions and the value of the strongest reaches 5. While α0 = 45◦ , there are mainly five pieces of intensity distributions and the value of the strongest reaches 1.2. While for α0 = 85◦ , there are mainly seven pieces of intensity distributions, and the maximum intensity reaches 0.5. At the same time, when the order of Bessel light-sheet is 0, there is a symmetrical distribution along the y = 0 axis in the Fig. 3(a). The distribution of intensity in the Fig. 7 has the similar symmetrical phenomenon. Comparing to the Fig. 7(c) with the red line (linestyle with pentagram) in the Figs. 3(b) and 7(d) with the green line (linestyle with dotted line) in the Fig. 3(b), the narrower the main width of the strongest intensity of |E|2 is, the smaller the central intensity distribution is. Meanwhile, the pieces of intensity distribution correspond to the stripes of incident field distribution in Fig. 4. Panels (a)–(d) of Fig. 8 give the distributions of the field intensity for first-order Bessel light-sheets (l = 1) with the half-cone angle being 0◦ , 20◦ , 45◦ and 85◦ , respectively. The strongest regions of the incident sheet completely enters the sphere, and the strongest scattering region is generated on the forward direction of the sphere after the sheet is refracted. In general, there are two tails in each panel in the forward scattering region. There are smaller intensity distributions with the increasing the halfcone angle from 20◦ , 45◦ , to 85◦ . Different from the panels (a)– (d) of Fig. 7, the intensity distribution is weaker and the concentrated intensity distribution extends along z-axis to both sides. There are two strong internal distributions along the z-axis, which
both go through the sphere. For α0 = 20◦ , there are two obvious tails. For α0 = 45◦ , two strong distributions along the z-axis are observed, and the distributions for α0 = 85◦ are closer than that of α0 = 45◦ . Considering the depth of light intensity into the sphere, for α0 = 20◦ , there is the brightest intensity distributions straightly through the sphere. Comparing to the panels 7(a)–(d), there is still a kind of symmetrical characteristic along y = 0 axis, and there is no intensity distribution while y = 0 in Fig. 3(b). Or we can say that the value of intensity in these regions are extremely small. Panels (a)–(d) of Fig. 9 give the field intensities of a secondorder Bessel sheet (l = 2), and the half-cone angle is α0 = 0◦ , 45◦ and 85◦ , respectively. Generally, compared to panels 7(b)–(d) and panels 8(b)–(d), the internal intensity distribution extends further along y = 0 to both sides. When the Bessel light-sheet interacts with the sphere, the two regions with the strongest intensity are partially refracted into the sphere, and some of the light-sheet do not enter the sphere. The light entering the sphere passes through the sphere, and produces a maximum distribution on the forward direction of the sphere. The light out of the sphere directly bypasses the sphere. In internal field region of the panel 9(b), there are two obvious intensity distributions along the internal field contour area for α0 = 20◦ . For the backward scattering field, there is a convergence point with two small tails. For α0 = 45◦ , there are two parts of light intensity distributions away from y = 0 along z-axis. After the interaction of these two parts, there appears two tails with weak intensity along y = 0 in the forward scattering field. And there is also a convergence point in the forward direction of the sphere. However, for α0 = 85◦ , there are apparent standing wave phenomenons in the internal field. The two main light inten-
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7
Fig. 8. Internal and near-surface electric field intensities of a sphere illuminated by a zeroth-order (l = 1) Bessel light-sheet.
sity distributions maintain their characteristic along z-axis, and the light into the sphere and out of sphere have a intensity superposition after scattering. Then, we give the observation of intensity distributions for same half-cone angle α 0 with different order l. Panels 7(a), 8(a) and 9(a) show the field intensity distributions with different order, and there is always a piece of straight intensity distribution along z-axis whatever the half-cone angle α 0 is. Panels 7(b), 8(b), and 9(b) show the field intensity distributions with different order for α0 = 20◦ . We can see that there are internal intensity distribution along the y axis. For l = 0, the main internal intensity is straight along y = 0. For l = 1, the main internal intensity is between y = 0 and y = 0.5 μm or y = 0 and y = −0.5 μm. While for l = 2, the main internal intensity is partly along y = 0.5 μm or y = −0.5 μm. The higher the order is, the farther internal intensity distribution the distance from the y = 0 axis is. Panels 7(c), 8(c), and 9(c) shows the field intensity distribution with different order while α0 = 45◦ . We can see the internal intensity distribution along the y axis is similar with the panels 7(b), 8(b), and 9(d). Panels 7(d), 8(d), and 9(d) show the field intensity distributions with different order while α0 = 85◦ . We can see that the internal intensity distribution is nearly parallel to the y−axis. The reason of this phenomenon could be traced on the standing wave effect. Compared with 7(d), 8(d), and 9(d), it is obvious that larger the half cone angle is, more obvious the standing wave effect is. In general, the half-cone α0 = 20◦ appears better convergence in the backward scattering field. With larger half-cone α 0 , the depth into the sphere is deeper while the α 0 is smaller than 45◦ . But the case of α0 = 85◦ shows obvious standing wave effects.
3.3. Efficiency factor In this section, the absorption and extinction efficiencies are investigated. In the calculation, the wavelength of the incident Bessel light-sheet is λ = 0.6328 μm, and the refractive index is m1 = 1.0 + 1i. The dimensionless size parameter ranges from ka = 0.1 to 25. Fig. 10 shows the absorption efficiency factor Qabs with the order l being parameter. From Fig. 10(a) with l = 0, for α0 = 0◦ , Qabs shows an upward trend for ka < 1 with a maximum value nearly 68, and Qabs shows a downward trend for ka > 1 with a minimum value nearly 35. For α0 = 20◦ , Qabs increases rapidly for ka < 1 with a maximum value 60. Qabs shows a downward trend for ka ∈ (1, 7), and changes gently for ka > 7 with a minimum value about 2. For α0 = 45◦ and 85◦ , Qabs increases rapidly for ka < 1, and shows a downward trend for ka > 1, tends to change gently after exceeding a certain value with a minimum value nearly 0. In Fig. 10(b) for l = 1, different from the case of α0 = 0◦ in Fig. 10(a), all value of Qabs tends to zero with different ka. For α0 = 20◦ , the maximum value of Qabs is nearly 13 at ka = 5, and its minimum value is nearly 0. For α0 = 45◦ , Qabs shows an increasingly upward trend for ka < 3 with a maximum value nearly 16, and shows a downward trend for ka > 3 with a minimum value nearly 0. For α0 = 85◦ , Qabs shows an steeply upward trend for ka < 1 with a maximum value nearly 16, and shows a downward trend for ka > 1 with a minimum value nearly 0. In the Fig. 10(c) for l = 2, the trend of the curve of the different α 0 is similar to the trend of the curve of the different α 0 in Fig. 10(b). Fig. 11 shows the extinction efficiency factor Qext for different l. In the Fig. 11(a) with l = 0, for α0 = 0◦ , Qext shows an upward
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Fig. 9. Internal and near-surface electric field intensities of a sphere illuminated by a second-order (l = 2) Bessel light-sheet.
Fig. 10. Qabs with different l.
Fig. 11. Qext with different l.
S. Zhang, R. Li and B. Wei et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106869
9
Fig. 12. Qabs with different α 0 .
trend for ka < 1 with a maximum value nearly 108, and shows a downward trend for ka > 1 with a minimum value nearly 85. For α0 = 20◦ , Qext increases rapidly for ka < 1 with a maximum value 85, shows a downward trend for ka ∈ (1, 7), and changes gently for ka > 7 with a minimum value about 1. For α0 = 45◦ and 85◦ , Qext increases rapidly for ka < 1, and shows a downward trend for ka > 1, and tends to change gently after exceeding a certain value with a minimum value nearly 0. In Fig. 11(b) for l = 1, different from the α0 = 0◦ in Fig. 11(a), all value of Qext tends to zero with different ka. For α0 = 20◦ , Qext shows an quickly upward trend for ka < 7 with a maximum value nearly 26, and shows a downward trend for ka > 7 with a minimum value nearly 6. For α0 = 45◦ , Qext shows an upward trend for ka < 3 with a maximum value nearly 26, and shows a downward trend for ka > 3 with a minimum value nearly 3. For α0 = 85◦ , Qext shows an upward trend for ka < 1 with a maximum value nearly 23, and shows a downward trend for ka > 1 with a minimum value nearly 2. In Fig. 11(c) for l = 2, the trend of the curve of the different α 0 is similar to the trend of the curve of the different α 0 in Fig. 11(b). In general, for α0 = 0◦ , Qabs and Qext for l = 0 are same to that for a plane sheet. However, Qabs and Qext for l > 0 are 0 caused by the null incident wave. As shown in Figs. 10 and 11, as l increases, the ka, where Qabs and Qext have maximum values, is also increased. For the same l, the value of Qext is larger than that of Qabs , and the larger the value of α 0 , the smaller the position ka where the peak is obtained. Fig. 12 shows the absorption efficiency factor Qabs with different α 0 . From Fig. 12(a) with α0 = 0◦ , for l = 0, Qabs shows an upward trend for ka < 1 with a maximum value nearly 68, and shows a downward trend for ka > 1 with a minimum value nearly 35. For l = 1 and l = 2, the value of Qabs is always 0 with the change of ka. From Fig. 12(a) with α0 = 20◦ , for l = 0, Qabs shows an up-
ward trend for ka < 1 with a maximum value nearly 60, shows a downward trend for ka ∈ (1, 7), and shows a slowly downward trend for ka > 7 with a minimum value nearly 2. For l = 1, Qabs shows an upward trend for ka < 5 with a maximum value nearly 13, and shows an slowly downward trend for ka > 5 with a minimum value nearly 2. For l = 2, Qabs shows an upward trend for ka < 10 with a maximum value nearly 8, and shows an slowly downward trend for ka > 10 with a minimum value nearly 2. From Fig. 12(c) with α0 = 45◦ , for l = 0, Qabs shows an upward trend for ka < 5 with a maximum value nearly 45, shows a downward trend for ka ∈ (1, 5), and shows a slowly downward trend for ka > 5 with a minimum value nearly 2. For l = 1, Qabs shows an upward trend for ka < 2 with a maximum value nearly 16, and shows an slowly downward trend for ka > 2 with a minimum value nearly 2. For l = 2, Qabs shows an upward trend for ka < 4 with a maximum value nearly 8, and shows an slowly downward trend Qabs for ka > 4 with a minimum value nearly 2. For α0 = 85◦ , and different l, the curve in Fig. 12(d) has a similar trend to the curve in Fig. 12(c). Fig. 13 shows the extinction efficiency factor Qext for different α 0 . From Fig. 13(a) with α0 = 0◦ , for l = 0, Qext shows an upward trend for ka < 2 with a maximum value nearly 108, and shows a downward trend Qext for ka > 2 with a minimum value nearly 85. For l = 1 and l = 2, the value of Qext is always 0 with the change of ka. From Fig. 13(b) with α0 = 20◦ , for l = 0, Qext shows an upward trend for ka < 1 with a maximum value nearly 90, shows a downward trend for ka ∈ (2, 8), and shows a slowly downward trend for ka > 8 with a minimum value nearly 4. For l = 1, Qext shows an upward trend for ka < 6 with a maximum value nearly 25, and shows an slowly downward trend for ka > 6 with a minimum value of Qext nearly 4. For l = 2, Qext shows an upward trend for ka < 11 with a maximum value nearly 18, and shows an slowly
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S. Zhang, R. Li and B. Wei et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106869
Fig. 13. Qext with different α 0 .
downward trend for ka > 11 with a minimum value nearly 4. From Fig. 13(c) with α0 = 45◦ , for l = 0, Qext shows an upward trend for ka < 2 with a maximum value nearly 62, shows a downward trend for ka ∈ (1, 5), and shows a slowly downward trend for ka > 5 with a minimum value nearly 2. For l = 1, Qext shows an upward trend for ka < 3 wtih a maximum value nearly 26, and shows an slowly downward trend for ka > 3 with a minimum value nearly 2. For l = 2, Qext shows an upward trend for ka < 4 with a maximum value nearly 15, and shows an slowly downward trend for ka > 4 with a minimum value nearly 2. For α0 = 85◦ , under different l, the curve in Fig. 13(d) has a similar trend to the curve in Fig. 13(c). In general, as shown in Figs. 12 and 13, l affects the value of Qabs and Qext , and α 0 affects the position ka.
CRediT authorship contribution statement
4. Conclusion
References
In summary, we presented a rigorous approach for the interaction of a Bessel light-sheet with a sphere of arbitrary size. The present approach is based on the VASDM and the multipole expansion method to derive the BSCs. The calculation of a sphere illuminated by a Bessel light-sheet is performed for the field distribution under different order l and different half-cone angle α 0 . Meanwhile, we also have a calculation for the absorption and extinction efficiency factors under different order and half-cone angle. Numerical results show that the order influences the internal intensity distribution, and the half-cone angle influences the strength of the standing wave effect. Such results are of use in various fields including imaging of light-sheet microscopy, optimal design and engineering, particle sizing, etc. Declaration of Competing Interest None.
Shu Zhang: Conceptualization, Methodology, Software. Renxian Li: Supervision. Bojian Wei: Writing - original draft, Writing - review & editing. Jiaming Zhang: Software, Validation. Han Sun: Data curation, Writing - original draft. Ningning Song: Visualization. Acknowledgments The authors acknowledge support from the 111 Project (B17035).
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Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
Scattering of a non-paraxial Bessel light-sheet by a sphere of arbitrary size Shu Zhang a, Renxian Li a,b,∗, Bojian Wei a, Jiaming Zhang a, Han Sun a, Ningning Song a a b
School of Physics and Optoelectronic Engineering, Xidian University, Xi’an 710071, China Collaborative Innovation Center of Information Sensing and Understanding, Xidian University, Xi’an 710071, China
a r t i c l e
i n f o
Article history: Received 30 October 2019 Revised 28 January 2020 Accepted 28 January 2020 Available online 29 January 2020 Keywords: Scattering Bessel light-sheet Angular spectrum decomposition GLMT
a b s t r a c t The scattering of a non-paraxial Bessel light-sheet by a sphere of arbitrary size is studied in framework of generalized Lorenz-–Mie theory (GLMT). The electrical fields of the Bessel light-sheet are expanded using the vector angular spectrum decomposition method (VASDM), and the beam shape coefficients (BSCs) of the Bessel light-sheet are derived using the method of multipole expansion and vector spherical wave functions (VSWF). The internal and near-surface fields, and absorption and extinction efficiency factors are numerically calculated when a Bessel light-sheet is incident, and the effects of beam’s order, and half-cone angle are mainly discussed. Numerical results show that there is strong point convergence in the forward scattering region under the incident of a Bessel light-sheet, and the internal and near-surface fields of Bessel light-sheet are very sensitive to the beam’s order, half-cone angle, etc. Such results have potential applications in the super-resolution imaging using a light-sheet microscopy. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction Light-sheet has potential applications in imaging microscope [1], particle characterization and sizing [2–4], optical clearing [5,6], non-destructive optical sectioning and imaging of the internal subcellular features [7–9], etc. As one of the biggest advances in microscopy techniques in recent years, light-sheet microscopy, firstly proposed over 100 years ago [10], has many advantages for studying biological samples [7,8,11–14], and provides true optical sectioning by sample illumination with a thin light-sheet. However, standard light-sheet microscopy, which uses Gaussian beam excitations, has disadvantages. Its field of view (FOV) is limited by the depth of the focus of the Gaussian beam, and the excitation beam has a limited penetration depth. To overcome these limitations, Bessel beam [15], which has the properties of self-healing and non-diffraction, is used instead of Gaussian beam in lightsheet microscopy [13,16]. The theoretical prediction of the interaction between light-sheets with particles can help to improve the light-sheet microscopy. Many researches have been devoted to the characteristics of light-sheets and their interaction with particles. In recent years, Gaussian light-sheet [17], Hermite-Gaussian light-sheet, Airy lightsheet [18], and Bessel-–Gaussian light-sheet pincers [19] have been
∗
Corresponding author. E-mail address: [email protected] (R. Li).
https://doi.org/10.1016/j.jqsrt.2020.106869 0022-4073/© 2020 Elsevier Ltd. All rights reserved.
studied and applied in various fields, including optical manipulation [20], particle sizing [21], optical tomography sectioning [22], light-sheet microscopy [16,23–25], etc. Gustavo de Medeiros [26] summarized the state-of-the-art of light-sheet imaging with a focus on mammalian development. A novel type of Bessel and Bessel-Gaussian light-sheets has a tight bending characteristics in the form of pincers [19,27]. The interaction of FBG (Fractional Bessel-Gaussian) light-sheets with a dielectric sub-wavelength spheroid, from the standpoint of optical radiation force and spin torque theories, is investigated in the framework of the dipole approximation [28]. Airy light-sheets, obtained from the higher-order spatial derivatives of the fundamental nonparaxial Airy beam solution in the transverse direction, gave the results of radiated field [29]. The description of adjustable vector Airy light-sheets presented a longitudinal polarization of the vector potential, with particular emphasis on the generation of negative longitudinal optical force and axial spin torque [30]. Generalized Lorenz-–Mie theory (GLMT) [31], which provides a rigorous solution to the scattering of arbitrarily shaped laser beams to facilitate the design of arbitrary beam, is employed for the interaction of a sphere with a elliptic Gaussian sheets [32,33]. A generalization of the Lorenz–Mie theory to the case of light-sheet are given by developing exact analytical closed-form expressions for the optical radiation force and torque components [34]. The approach for the description of scalar and vectorial BGSs (Bessel-Gaussian beam) are presented with particular emphasis on routing the wave fields and their flow of energy at will, along designed curved paths in a
2
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homogeneous medium [35]. A Bessel sheet is a two-dimensional Bessel beam, and can be generated using a cylindrical lens illuminated by a Bessel beam. During the generation, the non-diffracting and self-healing properties of the Bessel beam remain in the Bessel light-sheet. By using a Bessel beam instead of Gaussian beam in light-sheet microscopy, the FOV and penetration depth [11] can be greatly improved. There is a strategy to select and optimize the linear excitation light-sheet in SPIM (selective plane illumination microscopy) based on spatial resolution, FOV and optical sectioning capability, to create a thin and uniform excitation light-sheet [36]. Bessel plane illumination gave several modes that offer different performance with different imaging metrics [37]. By combining ultra-thin planar illumination produced by scanned Bessel beams with super resolution structured illumination microscopy, it is demonstrated in vivo karyotyping of chromosomes during mitosis and identify different dynamics for the actin cytoskeleton at the dorsal and ventral surfaces of fibroblasts [38]. Florian O.Fahrbach [39] gave a study that combines the benefits of light-sheet microscopy, self-reconstructing Bessel beams, and two-photon fluorescence excitation to improve imaging in large scattering media such as cancer cell clusters. Ming Zhao [40] presented that twophoton Bessel light-sheet SIM (structured illumination microscopy) provides a non-invasive tool for deep organ imaging at cellular level resolution in the live animals by addressing the deep tissue imaging challenges in both excitation and emission paths. Bessel light-sheets have important applications in various fields, including optical manipulation, particle sizing, light-sheet microscopy, etc, and most applications of Bessel light-sheets involve the interaction of the Bessel light-sheet with particles. The purpose of this paper is to investigate the scattering of a Bessel light-sheet by a sphere of arbitrary size, and discuss the effects of beam order l, half-cone angle α 0 , etc. The rest of the paper is organized as below. Firstly, a general theory of the electromagnetic fields of a Bessel light-sheet is given in Section 2. The electric field of a Bessel light-sheet is obtained based on the vector angular spectrum decomposition method (VASDM). The electric fields are also expanded in terms of beam shape coefficients (BSCs) and vector spherical wave functions (VSWFs) using the multipole expansion. The internal and nearsurface fields, and the cross sections and their corresponding efficiencies are given on the basis of GLMT. Section 3 discusses some numerical results of internal and near-surface fields, and the efficiency factors. The effects of beam order l and half-cone angle α 0 are analysed. Section 4 is devoted to the conclusion. 2. Theory
half-cone angle. r is the position vector, and r0 is the position vector of beam center. ex is the unit vector in x-direction. Epw0 is the amplitude of plane wave angular spectrum. δ is the delta function. Substituting Eq. (2) into Eq. (1), we can obtain the analytical expression of the electric field of Bessel light-sheets
E(r, θ , φ ) = ex EB0 eik(z−z0 ) cos α0 il Jl (σ )
(3)
where, Jl (·) is the Bessel function of first kind. EB0 and σ are defined by
EB0 = 2π E pw0 ,
σ = k(y − y0 ) sin α0
Since the term ex
eik·r
(4)
in Eq. (1) can be expanded using VSWF
as:
∞ n
ex eik·r =
(1 ) (1 ) Dmn q mn Mmn + p mn Nmn
(5)
n=1 m=−n
where
Dmn =
(2n + 1)(n − m )! n ( n + 1 ) ( n + m )!
q mn = −in+1 e−imβ p mn = −in+1 e−imβ
(6)
πmn (cos α )eθ (α ) − iτmn (cos α )eφ (α ) · ex (7)
τmn (cos α )eθ (α ) − iπmn (cos α )eφ (α ) · ex (8)
Consider a sphere of radius a and refractive index m1 placed in the field of a non-paraxial Bessel light-sheet as shown in Fig. 1. Suppose that the particle center is located at the origin OO of the coordinate system OO,xyz , and the coordinates of beam center are (y0 , z0 ). The surrounding media is assumed to be vacuum. A Bessel light-sheet propagates along the plane of yz and its electric field is along x direction with the suppressed time harmonic factor e−iωt . The wavelength of the incident light-sheet is λ. Based on the VASDM [41], the electric field of a Bessel light-sheet can be expressed as
E ( r, θ , φ ) =
Fig. 1. The picture displays a graphical representation of the propagation of a nonparaxial Bessel light sheet, propagating along the z-direction (λ = 0.6328 μm, the radius of a sphere a = 1 μm, l = 0 and α0 = 20◦ ).
αmax 2π α =0
β =0
Epw eik·(r−r0 ) | sin α cos α|dα dβ
δ (α − α0 ) ilβ e |sin α0 | cos α0
(1 ) Nmn =
1 d
τmn (cos θ )eθ + iπmn (cos θ )eφ
+er n(n + 1 )Pnm (cos θ )
πmn (cos α ) = m τmn (cos α ) =
(2)
where, α is the angle between the wave vector k and the z-axis. k = (0, k sin α , k cos α ) is the wave vector. Epw is the vector angular spectrum of plane wave, l is the order of beam, and α 0 is the
kr dr
(9)
[r jn (kr )] exp(imφ )
jn (kr ) exp(imφ ) kr
(10)
with
(1)
with
Epw = ex E pw0 (α0 , β )
(1 ) Mmn = iπmn (cos θ )eθ − τmn (cos θ )eφ jn (kr ) exp(imφ )
Pnm (cos α ) sin α
dPnm (cos α ) dα
(11) (12)
where, Pnm (cos α ) represents the associated Legendre polynomials of degree n and order m, jn (·) is the spherical Bessel function of the first kind, and e(r,θ ,φ ) are radial, polar, and azimuthal unit vectors. Substituting Eqs. (5)–(12) into Eq. (1), the electric field of an incident Bessel light-sheet Einc can be expanded in terms of the
S. Zhang, R. Li and B. Wei et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 245 (2020) 106869
VSWFs
Einc (r, θ , ϕ ) = − with
pmn = qmn =
∞
n
n=1 m=−n
iEmn
(1 ) (1 ) pmn Nmn (kr ) + qmn Mmn ( kr )
(13)
Dmn e−ikz z0 [τmn (cos α0 )I+ + πmn (cos α0 )I− ]
(14)
Dmn e−ikz z0 [πmn (cos α0 )I+ + τmn (cos α0 )I− ]
(15)
I± = π i1+l−m J1+l−m (σ0 ) ± i−1+l−m J−1+l−m (σ0 )
(16) (17)
where, (r, θ , ϕ ) is spherical coordinates corresponding to ( x, y, z) with the origin locating on the particle center, Emn = E0 in Dmn , and kz = k cos α0 . According to GLMT, the internal field is also ex(1 ) (1 ) panded in terms of the VSWFs (Nmn (k, r ) and Mmn (k, r )), and the (3 ) (3 ) scattered field in terms of the VSWFs (Nmn (k, r ) and Mmn (k, r )). ∞ n
(1 ) (1 ) iEmn dmn Nmn (m1 kr ) + cmn Mmn ( m1 kr )
(18)
n=1 m=−n
Es =
∞ n
(3 ) (3 ) iEmn amn Nmn (kr ) + bmn Mmn ( kr )
(19)
n=1 m=−n
with
1 d
(3 ) Mmn = iπmn (cos θ )eθ − τmn (cos θ )eφ hn(1) (kr ) exp(imφ ) (3 ) Nmn =
τmn (cos θ )eθ + iπmn (cos θ )eφ
+er n(n + 1 )Pnm (cos θ )
kr dr
(20)
r hn(1) (kr ) exp(imφ )
hn(1) (kr ) exp(imφ ) kr
(21)
where, hn(1 ) (· ) is the spherical Hankel function of the first kind. By considering the boundary conditions, the partial wave expansion coefficients of the scattering field amn and bmn and internal field cmn and dmn can be obtained
amn = an pmn , bmn = bn qmn
(22)
cmn = cn qmn , dmn = dn pmn
(23)
where, an and bn are the traditional Mie scattering coefficients, and cn and dn are the corresponding internal coefficients [42,43]. Once the BSCs are obtained, we can get the scattering, extinction, and absorption efficiencies
⎧ ⎪ Qsca = ⎪ ⎪ ⎪ ⎨ Qext = ⎪ ⎪ ⎪ ⎪ ⎩
C
4 sca π a2 = k2 a2 C
∞ n
n=1 m=−n
4 ext π a2 = k2 a2 Re
|amn |2 + |bmn |2
∞ n n=1 m=−n
∗
qmn bmn + pmn amn ∗
internal and near-surface fields. Then the absorption and extinction efficiencies are calculated. The effects of the order l and halfcone angle α 0 are particularly emphasized. In the calculation, the refractive index of the particle is m1 = 1.33 [44], the refractive index of the surrounding medium is 1.0, and the wavelength of the incident light-sheet is λ = 0.6328 μm. The radius of the particle is a = 1.0 μm, and the corresponding dimensionless size parameter is ka= 10, with k being the wave number. 3.1. Incident field intensities of Bessel light-sheets
σ0 = −ky0 sin α0
E1 = −
3
(24)
Qabs = Qext − Qsca
where, Csca and Cext are the scattering and extinction cross sections, respectively. The star ∗ denotes the complex conjugate. 3. Numerical results and discussions In this section, numerical calculations to a Bessel light-sheet scattering by an isotropic sphere are performed, and the influences of various parameters on the scattering are investigated. Numerical analysis is started by developing a Python program to compute the internal and near-surface fields, and the characteristics of the incident beam are also analysed for the better understanding of the
Bessel light-sheets, which are characterized by Bessel functions, have different characteristics with different half-cone angle α 0 and order l. For convenience, the electric field intensities |E|2 , which is defined by Eq. (3), of incident Bessel light-sheets are calculated, and the effects of order l and half-cone angle α 0 are analysed. Fig. 2 gives the distributions of the electric field intensities with different α 0 , and panels (a) - (d) correspond to α0 = 0◦ , 20◦ , 45◦ , and 85◦ , respectively. From Eq. (3), a zeroth-order (l = 0) Bessel sheet with α0 = 0◦ is deduced to a plane light-sheet, while a high-order Bessel sheet has zero electric field intensity. This is caused by the characteristics of Bessel functions, namely J0 (0 ) = 1 while Jl (0 ) = 0 (l = 0). This can be seen from Fig. 2(a). For α 0 > 0 (Fig. 2(b)–(d), the curves are formed by a series of peaks. In general, these peaks are symmetrical with respect to the y = 0 axis. A zeroth-order Bessel sheet (l = 0) has maximum central intensity, while a high-order Bessel sheet has null central intensity. In Fig. 2(b) (α0 = 20◦ ), the main peak region of the Bessel sheet with l = 0, presenting distribution pattern of the symmetrical function, is about at y ∈ (−0.6 μm, 0.6 μm) with steep curve change. While the Bessel sheet of l = 1 with slow curve change and l = 2 with slower curve change, are symmetrically distributed with y = 0, the range of y regions corresponding to the main peak is approximately y ∈ (−1 μm, 0 ) or y ∈ (0, 1 μm) and y ∈ (−1.5 μm, 0) or y ∈ (0, 1.5 μm), respectively. In Fig. 2(c) (α0 = 45◦ ) the same distribution mode as Fig. 2(b), the y region corresponding to the main peak of the l = 0 order Bessel sheet is probably at y ∈ (−0.3 μm, 0.3 μm), while l = 1 about at y ∈ (−0.5 μm, 0) ∪ (0, 0.5 μm) and l = 2 about at y ∈ (−0.7 μm, 0) ∪ (0, 0.7 μm). Fig. 2(d) gives the results for α0 = 85◦ . A smaller ranges of y for main peak can be seen comparing with that for smaller α 0 . In general, as α 0 grows, the width of the y region becomes narrower. Fig. 3 gives the distribution of electric field intensity with different l. In Fig. 3(a) (l = 0), the main peaks of different α 0 maintain the same value, except that the intensity for α0 = 0◦ keeps the consistent strength. In Fig. 3(b), the electric field intensity of α0 = 0◦ is 0 along the y-axis. The intensity regions corresponding to the main peak of α0 = 20◦ , 45◦ , and 85◦ are about at y ∈ (−1.0 μm, 0) ∪ (0, 1.0 μm), y ∈ (−0.5 μm, 0) ∪ (0, 0.5 μm), and y ∈ (−0.4 μm, 0) ∪ (0, 0.4 μm), respectively. In Fig. 3(c), the electric field intensity of l = 0 is 0 along the y-axis. The intensity regions corresponding to the main peak of α0 = 20◦ , 45◦ , and 85◦ are about at y ∈ (−1.5 μm, 0) ∪ (0, 1.5 μm), y ∈ (−0.8 μm, 0) ∪ (0, 0.8 μm), and y ∈ (−0.5 μm, 0) ∪ (0, 0.5 μm), respectively. In general, as α 0 is bigger, there is a narrower width of y region and as the l is larger, the width of y region is wider. For the convenience of analysis, the electric fields of Bessel light-sheets in yz plane are given Figs. 4–6 for order l = 0, 1, 2, respectively. According to Eq. (3), a zeroth-order Bessel light-sheet with α0 = 0◦ is deduced to a plane light-sheet with unity amplitude, while a high-order Bessel light-sheet has null electric field for α 0 . Thus, in Figs. 4–6, the results for α0 = 0◦ are not depicted. As shown in Fig. 4, for α0 = 20◦ , the bright and dark stripes are distributed along the y-axis. There is a bright stripe in the region
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Fig. 2. Electric field intensity for different α 0 .
Fig. 3. Electric field intensity for different l.
Fig. 4. Electric field intensities of zeroth-order (l = 0) Bessel beams with α 0 being parameter.
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Fig. 5. The same as in Fig. 4, but first-order Bessel beams (l = 1).
Fig. 6. The same as in Fig. 4, but second-order Bessel beam (l = 2).
of y ∈ (−0.5 μm, 0.5 μm), the width of which is nearly 1 μm, and the two sides are followed by dark stripes with width of 0.5 μm, and next to both side followed by bright stripes with width of 0.5 μm. Corresponding to α0 = 20◦ of Fig. 2(b), there is a main peak in the region of y ∈ (−0.5 μm, 0.5 μm) and two secondary peak in the region of y ∈ (−1.35 μm, −0.75 μm) ∪ (0.75 μm, 1.35 μm). For α0 = 45◦ , the brightest stripe in Fig. 4(b) along z-axis is the main peak distribution of α0 = 45◦ of Fig. 2(c). The number of secondary peak in Fig. 2(c) is corresponding to the number of bright stripes in Fig. 5. In Fig. 4(c) with α0 = 85◦ , there is a brightest stripe, and the bright and dark stripes on both sides are distributed along both sides of the y-axis, which corresponds to the number of secondary peaks in Fig. 2(d). The width of stripes is narrower than the cases of α0 = 20◦ and α0 = 45◦ . Compared with Fig. 3(b) for l = 0, the larger the α 0 is, the narrower the width of the brightest stripe is, and the more the number of bright and dark stripes are. Different from l = 0 in Fig. 4, there is a dark stripe along y = 0 in Fig. 5. In Fig. 5(a) with α0 = 20◦ , there are two bright stripes in the region of y ∈ (−0.75 μm, −0.25 μm) ∪ (0.25 μm, 0.75 μm) with intensity 0.1, and both sides are dark and bright stripes with the number of 1. According to l = 1 of the Fig. 2(b), there are two main peaks and two secondary peaks. For α0 = 45◦ in Fig. 5(b), there are two bright stripes with a smaller width, compared to the width of α0 = 20◦ . For α0 = 85◦ in Fig. 5(c), there is a bright stripe (actually, there are two stripes.) nearly along z-axis. Correspondingly, for l = 1 in the Fig. 3(b), there is no field along y = 0 and y = 1. But there are some minimal intensity distribution in Fig. 5(c). The electric field intensities for a second-order Bessel lightsheet are given in Fig. 6. Fig. 6(a) displays one dark stripe in the region of y ∈ (−0.5 μm, 0.5 μm), and two bright stripes along zaxis, and two small dark stripes on the both sides. For α0 = 45◦ , there are still one dark stripe with the same width of two bright
stripes, and both sides are the bright and dark stripes in turn. The results for α0 = 85◦ are similar to that for α0 = 45◦ , but having more stripes of bright and dark. According to Fig. 3(c), the width of two main peaks are wide enough compared to the width of main peaks. In general, the bigger the order l is, the less the number of stripes are, and the wider the width of two stripes are. As the increasing of α 0 , there are more stripes along y-axis. And the intensity of incident field maintains the same distribution along z-axis. 3.2. Internal and near-surface fields In this section, the electric field intensities |E|2 of a sphere illuminated by a Bessel light-sheet are discussed. The effect of the order and half-cone angle will be mainly analysed. Figs. 7–9 displays the total field intensity of a sphere illuminated by Bessel lightsheets with different order (l = 0, 1 and 2) and half-cone angle (α0 = 0◦ , 20◦ , 45◦ and 85◦ ). The white circle in the figures denotes the particle. First, we consider the field intensity for a Bessel light-sheet with zeroth-order l = 0, given by Fig. 7. Panels (a) and (d) correspond to the half-cone angle α0 = 0◦ , 20◦ , 45◦ and 85◦ , respectively. A Bessel light-sheet with l = 0 and α0 = 0◦ becomes a plane light-sheet. The incident beam is focused by the sphere, and a photonic nanojet can be seen in the forward direction, as shown in Fig. 7(a). As shown in Fig. 7(b), at the y-axis symmetrical positions, light with a slightly weaker intensity (second peaks) also enters the sphere, and transmits the sphere in the forward direction. Since the intensity of the zeroth-order Bessel light-sheet along the z-axis is large, the extreme region of the intensity on both sides of the z-axis is weaker than the extreme region along the z-axis. Only a strong distribution in the middle can be observed. The area is focused by the sphere. The upper and lower extreme regions are
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Fig. 7. Internal and near-surface electric field intensities of a sphere illuminated by a zeroth-order (l = 0) Bessel light-sheet.
refracted by the particle, and we can see two curves formed by extreme peaks in the vicinity of θ = α0 , but their intensities are weak. Extremely, in the internal field region, while for α0 = 20◦ , there are mainly three pieces of intensity distributions and the value of the strongest reaches 5. While α0 = 45◦ , there are mainly five pieces of intensity distributions and the value of the strongest reaches 1.2. While for α0 = 85◦ , there are mainly seven pieces of intensity distributions, and the maximum intensity reaches 0.5. At the same time, when the order of Bessel light-sheet is 0, there is a symmetrical distribution along the y = 0 axis in the Fig. 3(a). The distribution of intensity in the Fig. 7 has the similar symmetrical phenomenon. Comparing to the Fig. 7(c) with the red line (linestyle with pentagram) in the Figs. 3(b) and 7(d) with the green line (linestyle with dotted line) in the Fig. 3(b), the narrower the main width of the strongest intensity of |E|2 is, the smaller the central intensity distribution is. Meanwhile, the pieces of intensity distribution correspond to the stripes of incident field distribution in Fig. 4. Panels (a)–(d) of Fig. 8 give the distributions of the field intensity for first-order Bessel light-sheets (l = 1) with the half-cone angle being 0◦ , 20◦ , 45◦ and 85◦ , respectively. The strongest regions of the incident sheet completely enters the sphere, and the strongest scattering region is generated on the forward direction of the sphere after the sheet is refracted. In general, there are two tails in each panel in the forward scattering region. There are smaller intensity distributions with the increasing the halfcone angle from 20◦ , 45◦ , to 85◦ . Different from the panels (a)– (d) of Fig. 7, the intensity distribution is weaker and the concentrated intensity distribution extends along z-axis to both sides. There are two strong internal distributions along the z-axis, which
both go through the sphere. For α0 = 20◦ , there are two obvious tails. For α0 = 45◦ , two strong distributions along the z-axis are observed, and the distributions for α0 = 85◦ are closer than that of α0 = 45◦ . Considering the depth of light intensity into the sphere, for α0 = 20◦ , there is the brightest intensity distributions straightly through the sphere. Comparing to the panels 7(a)–(d), there is still a kind of symmetrical characteristic along y = 0 axis, and there is no intensity distribution while y = 0 in Fig. 3(b). Or we can say that the value of intensity in these regions are extremely small. Panels (a)–(d) of Fig. 9 give the field intensities of a secondorder Bessel sheet (l = 2), and the half-cone angle is α0 = 0◦ , 45◦ and 85◦ , respectively. Generally, compared to panels 7(b)–(d) and panels 8(b)–(d), the internal intensity distribution extends further along y = 0 to both sides. When the Bessel light-sheet interacts with the sphere, the two regions with the strongest intensity are partially refracted into the sphere, and some of the light-sheet do not enter the sphere. The light entering the sphere passes through the sphere, and produces a maximum distribution on the forward direction of the sphere. The light out of the sphere directly bypasses the sphere. In internal field region of the panel 9(b), there are two obvious intensity distributions along the internal field contour area for α0 = 20◦ . For the backward scattering field, there is a convergence point with two small tails. For α0 = 45◦ , there are two parts of light intensity distributions away from y = 0 along z-axis. After the interaction of these two parts, there appears two tails with weak intensity along y = 0 in the forward scattering field. And there is also a convergence point in the forward direction of the sphere. However, for α0 = 85◦ , there are apparent standing wave phenomenons in the internal field. The two main light inten-
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Fig. 8. Internal and near-surface electric field intensities of a sphere illuminated by a zeroth-order (l = 1) Bessel light-sheet.
sity distributions maintain their characteristic along z-axis, and the light into the sphere and out of sphere have a intensity superposition after scattering. Then, we give the observation of intensity distributions for same half-cone angle α 0 with different order l. Panels 7(a), 8(a) and 9(a) show the field intensity distributions with different order, and there is always a piece of straight intensity distribution along z-axis whatever the half-cone angle α 0 is. Panels 7(b), 8(b), and 9(b) show the field intensity distributions with different order for α0 = 20◦ . We can see that there are internal intensity distribution along the y axis. For l = 0, the main internal intensity is straight along y = 0. For l = 1, the main internal intensity is between y = 0 and y = 0.5 μm or y = 0 and y = −0.5 μm. While for l = 2, the main internal intensity is partly along y = 0.5 μm or y = −0.5 μm. The higher the order is, the farther internal intensity distribution the distance from the y = 0 axis is. Panels 7(c), 8(c), and 9(c) shows the field intensity distribution with different order while α0 = 45◦ . We can see the internal intensity distribution along the y axis is similar with the panels 7(b), 8(b), and 9(d). Panels 7(d), 8(d), and 9(d) show the field intensity distributions with different order while α0 = 85◦ . We can see that the internal intensity distribution is nearly parallel to the y−axis. The reason of this phenomenon could be traced on the standing wave effect. Compared with 7(d), 8(d), and 9(d), it is obvious that larger the half cone angle is, more obvious the standing wave effect is. In general, the half-cone α0 = 20◦ appears better convergence in the backward scattering field. With larger half-cone α 0 , the depth into the sphere is deeper while the α 0 is smaller than 45◦ . But the case of α0 = 85◦ shows obvious standing wave effects.
3.3. Efficiency factor In this section, the absorption and extinction efficiencies are investigated. In the calculation, the wavelength of the incident Bessel light-sheet is λ = 0.6328 μm, and the refractive index is m1 = 1.0 + 1i. The dimensionless size parameter ranges from ka = 0.1 to 25. Fig. 10 shows the absorption efficiency factor Qabs with the order l being parameter. From Fig. 10(a) with l = 0, for α0 = 0◦ , Qabs shows an upward trend for ka < 1 with a maximum value nearly 68, and Qabs shows a downward trend for ka > 1 with a minimum value nearly 35. For α0 = 20◦ , Qabs increases rapidly for ka < 1 with a maximum value 60. Qabs shows a downward trend for ka ∈ (1, 7), and changes gently for ka > 7 with a minimum value about 2. For α0 = 45◦ and 85◦ , Qabs increases rapidly for ka < 1, and shows a downward trend for ka > 1, tends to change gently after exceeding a certain value with a minimum value nearly 0. In Fig. 10(b) for l = 1, different from the case of α0 = 0◦ in Fig. 10(a), all value of Qabs tends to zero with different ka. For α0 = 20◦ , the maximum value of Qabs is nearly 13 at ka = 5, and its minimum value is nearly 0. For α0 = 45◦ , Qabs shows an increasingly upward trend for ka < 3 with a maximum value nearly 16, and shows a downward trend for ka > 3 with a minimum value nearly 0. For α0 = 85◦ , Qabs shows an steeply upward trend for ka < 1 with a maximum value nearly 16, and shows a downward trend for ka > 1 with a minimum value nearly 0. In the Fig. 10(c) for l = 2, the trend of the curve of the different α 0 is similar to the trend of the curve of the different α 0 in Fig. 10(b). Fig. 11 shows the extinction efficiency factor Qext for different l. In the Fig. 11(a) with l = 0, for α0 = 0◦ , Qext shows an upward
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Fig. 9. Internal and near-surface electric field intensities of a sphere illuminated by a second-order (l = 2) Bessel light-sheet.
Fig. 10. Qabs with different l.
Fig. 11. Qext with different l.
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9
Fig. 12. Qabs with different α 0 .
trend for ka < 1 with a maximum value nearly 108, and shows a downward trend for ka > 1 with a minimum value nearly 85. For α0 = 20◦ , Qext increases rapidly for ka < 1 with a maximum value 85, shows a downward trend for ka ∈ (1, 7), and changes gently for ka > 7 with a minimum value about 1. For α0 = 45◦ and 85◦ , Qext increases rapidly for ka < 1, and shows a downward trend for ka > 1, and tends to change gently after exceeding a certain value with a minimum value nearly 0. In Fig. 11(b) for l = 1, different from the α0 = 0◦ in Fig. 11(a), all value of Qext tends to zero with different ka. For α0 = 20◦ , Qext shows an quickly upward trend for ka < 7 with a maximum value nearly 26, and shows a downward trend for ka > 7 with a minimum value nearly 6. For α0 = 45◦ , Qext shows an upward trend for ka < 3 with a maximum value nearly 26, and shows a downward trend for ka > 3 with a minimum value nearly 3. For α0 = 85◦ , Qext shows an upward trend for ka < 1 with a maximum value nearly 23, and shows a downward trend for ka > 1 with a minimum value nearly 2. In Fig. 11(c) for l = 2, the trend of the curve of the different α 0 is similar to the trend of the curve of the different α 0 in Fig. 11(b). In general, for α0 = 0◦ , Qabs and Qext for l = 0 are same to that for a plane sheet. However, Qabs and Qext for l > 0 are 0 caused by the null incident wave. As shown in Figs. 10 and 11, as l increases, the ka, where Qabs and Qext have maximum values, is also increased. For the same l, the value of Qext is larger than that of Qabs , and the larger the value of α 0 , the smaller the position ka where the peak is obtained. Fig. 12 shows the absorption efficiency factor Qabs with different α 0 . From Fig. 12(a) with α0 = 0◦ , for l = 0, Qabs shows an upward trend for ka < 1 with a maximum value nearly 68, and shows a downward trend for ka > 1 with a minimum value nearly 35. For l = 1 and l = 2, the value of Qabs is always 0 with the change of ka. From Fig. 12(a) with α0 = 20◦ , for l = 0, Qabs shows an up-
ward trend for ka < 1 with a maximum value nearly 60, shows a downward trend for ka ∈ (1, 7), and shows a slowly downward trend for ka > 7 with a minimum value nearly 2. For l = 1, Qabs shows an upward trend for ka < 5 with a maximum value nearly 13, and shows an slowly downward trend for ka > 5 with a minimum value nearly 2. For l = 2, Qabs shows an upward trend for ka < 10 with a maximum value nearly 8, and shows an slowly downward trend for ka > 10 with a minimum value nearly 2. From Fig. 12(c) with α0 = 45◦ , for l = 0, Qabs shows an upward trend for ka < 5 with a maximum value nearly 45, shows a downward trend for ka ∈ (1, 5), and shows a slowly downward trend for ka > 5 with a minimum value nearly 2. For l = 1, Qabs shows an upward trend for ka < 2 with a maximum value nearly 16, and shows an slowly downward trend for ka > 2 with a minimum value nearly 2. For l = 2, Qabs shows an upward trend for ka < 4 with a maximum value nearly 8, and shows an slowly downward trend Qabs for ka > 4 with a minimum value nearly 2. For α0 = 85◦ , and different l, the curve in Fig. 12(d) has a similar trend to the curve in Fig. 12(c). Fig. 13 shows the extinction efficiency factor Qext for different α 0 . From Fig. 13(a) with α0 = 0◦ , for l = 0, Qext shows an upward trend for ka < 2 with a maximum value nearly 108, and shows a downward trend Qext for ka > 2 with a minimum value nearly 85. For l = 1 and l = 2, the value of Qext is always 0 with the change of ka. From Fig. 13(b) with α0 = 20◦ , for l = 0, Qext shows an upward trend for ka < 1 with a maximum value nearly 90, shows a downward trend for ka ∈ (2, 8), and shows a slowly downward trend for ka > 8 with a minimum value nearly 4. For l = 1, Qext shows an upward trend for ka < 6 with a maximum value nearly 25, and shows an slowly downward trend for ka > 6 with a minimum value of Qext nearly 4. For l = 2, Qext shows an upward trend for ka < 11 with a maximum value nearly 18, and shows an slowly
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Fig. 13. Qext with different α 0 .
downward trend for ka > 11 with a minimum value nearly 4. From Fig. 13(c) with α0 = 45◦ , for l = 0, Qext shows an upward trend for ka < 2 with a maximum value nearly 62, shows a downward trend for ka ∈ (1, 5), and shows a slowly downward trend for ka > 5 with a minimum value nearly 2. For l = 1, Qext shows an upward trend for ka < 3 wtih a maximum value nearly 26, and shows an slowly downward trend for ka > 3 with a minimum value nearly 2. For l = 2, Qext shows an upward trend for ka < 4 with a maximum value nearly 15, and shows an slowly downward trend for ka > 4 with a minimum value nearly 2. For α0 = 85◦ , under different l, the curve in Fig. 13(d) has a similar trend to the curve in Fig. 13(c). In general, as shown in Figs. 12 and 13, l affects the value of Qabs and Qext , and α 0 affects the position ka.
CRediT authorship contribution statement
4. Conclusion
References
In summary, we presented a rigorous approach for the interaction of a Bessel light-sheet with a sphere of arbitrary size. The present approach is based on the VASDM and the multipole expansion method to derive the BSCs. The calculation of a sphere illuminated by a Bessel light-sheet is performed for the field distribution under different order l and different half-cone angle α 0 . Meanwhile, we also have a calculation for the absorption and extinction efficiency factors under different order and half-cone angle. Numerical results show that the order influences the internal intensity distribution, and the half-cone angle influences the strength of the standing wave effect. Such results are of use in various fields including imaging of light-sheet microscopy, optimal design and engineering, particle sizing, etc. Declaration of Competing Interest None.
Shu Zhang: Conceptualization, Methodology, Software. Renxian Li: Supervision. Bojian Wei: Writing - original draft, Writing - review & editing. Jiaming Zhang: Software, Validation. Han Sun: Data curation, Writing - original draft. Ningning Song: Visualization. Acknowledgments The authors acknowledge support from the 111 Project (B17035).
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