Paraxial propagation dynamics of the radially polarized Airy beams in uniaxial crystals orthogonal to the optical axis

Jintao Xie; Jianbin Zhang; Xitao Zheng; Junran Ye; Dongmei Deng

doi:10.1364/OE.26.011309

1. Introduction

Since the finite energy Airy beams were intensively studied and experimentally demonstrated by Siviloglou et al and Christodoulides et al in 2007 [1, 2], the Airy beams have become a research hotspot and been investigated extensively owing to the unique advantages such as self-acceleration [1,2], diffraction-free [3], and self-healing [4]. Up to now, the Airy beams have been used in many applications such as optical cleaning of the micro particles [5], curved plasma channel generation [6,7], light bullet generation [8,9], light-sheet microscopy [10] and so on.

In contrast to the linearly polarized beams, the spot size of the radially polarized beams can be reduced by using an annular aperture so as to achieve the sharper focus effect [11, 12], which can be used in many research field. In addition, cylindrical vector beams with spatial variant states of polarization such as the radially and the azimuthally polarized beams have attracted extensive attention on account of their fascinating properties as compared to scalar light fields, and the radially polarized beams used in different situations have achieved many applications such as optical microfabrication [13], acceleration techniques [14, 15], particles guiding or trapping [16–18], high-resolution microscopy [19] and particularly material processing [20, 21]. And in 2007, the radially polarized elegant light beams [22] and the analytical vectorial structure of radially polarized light beams [23] were investigated deeply by Deng et al, which contributes to the research of the radially polarized light beams a lot. Recently, the radially polarized beams propagating in anisotropic mediums have been deeply investigated by Gu et al [24] and S. N. Khonina et al [25].

On the other hand, it is helpful to study the propagation of laser beams in uniaxial crystal because of its important effects on the design of compensator, the polarizer, amplitude- and phase-modulation devices [26–28]. Over recent years, the Airy beams [29], the circular Airy beam [30, 31], the Airy vortex beams [32], the Airy-Guassian beam [33, 34], the Airy-Guassian vortex beam [35] and the chirped Airy vortex beams [36] propagating in uniaxial crystal have been studied by researchers. But there is no report on the paraxial propagation of the radially polarized Airy beams (RPAiBs) in uniaxial crystals orthogonal to the optical axis so far. Moreover, in 2016, the paper researched by S. N. Khonina et al [37], which clearly exhibits the influences of the relative position between the radiation polarization plane and the crystal optical axis on intensity distributions formed in various vector components of the ordinary and the extraordinary beam, makes the theory of the radially polarized beams propagating in uniaxial crystal more practical and intriguing. Thus, here we intensively investigate the influence of the ratio of the extraordinary refractive index (n_e) to the ordinary refractive index (n_o) on the propagation properties of the RPAiBs.

The paper is organized as follows. In Sec. 2, the analytical expressions for the paraxial propagation of the RPAiBs in uniaxial crystals orthogonal to the optical axis are derived. Then in Sec. 3, the influence of the ratio of n_e to n_o on the propagation properties of the RPAiBs in uniaxial crystals orthogonal to the optical axis are analyzed and described in detail. Finally, the paper is concluded in Sec. 4.

2. Analytic solution of the RPAiBs in uniaxial crystals orthogonal to the optical axis

In the Cartesian coordinate system, the z-axis is set as the propagation axis and the x-axis is the optical axis of the uniaxial crystal. The relative dielectric tensors ε of the uniaxial crystal is

ε = (\begin{matrix} n_{e}^{2} & 0 & 0 \\ 0 & n_{o}^{2} & 0 \\ 0 & 0 & n_{o}^{2} \end{matrix}),

where n_e and n_o are the extraordinary and the ordinary refractive indices of the uniaxial crystal, respectively. The electric fields of the RPAiBs in the input plane can be expressed as

[\begin{matrix} E_{x} (x, y, 0) \\ E_{y} (x, y, 0) \end{matrix}] = [\begin{matrix} \frac{x}{w_{1}} Ai (\frac{x}{w_{1}}) Ai (\frac{y}{w_{2}}) \exp (\frac{a x}{w_{1}} + \frac{b y}{w_{2}}) \\ \frac{y}{w_{2}} Ai (\frac{x}{w_{1}}) Ai (\frac{y}{w_{2}}) \exp (\frac{a x}{w_{1}} + \frac{b y}{w_{2}}) \end{matrix}],

where Ai(·) denotes the Airy function, a and b are exponential truncation factors ranging from 0 to 1, w₁ and w₂ represent arbitrary transverse scales in the x-direction and y-direction. Under the paraxial approximation, the propagation of the RPAiBs in uniaxial crystals orthogonal to the optical axis can be written as [28, 29]

\begin{array}{l} E_{x} (x, y, z) = \frac{i k n_{o}}{2 π z} \exp (- i k n_{e} z) \int \int_{- \infty}^{+ \infty} E_{x} (x_{0}, y_{0}, 0) \\ \times \exp {- \frac{i k [n_{o}^{2} {(x - x_{0})}^{2} + n_{o}^{2} {(y - y_{0})}^{2}]}{2 z n_{e}}} d x_{0} d y_{0}, \end{array}

\begin{array}{l} E_{y} (x, y, z) = \frac{i k n_{o}}{2 π z} \exp (- i k n_{o} z) \int \int_{- \infty}^{+ \infty} E_{y} (x_{0}, y_{0}, 0) \\ \times \exp {- \frac{i k n_{o} [{(x - x_{0})}^{2} + {(y - y_{0})}^{2}]}{2 z}} d x_{0} d y_{0}, \end{array}

where z is the propagation distance, and the wave number is k = 2π/λ (λ is the wavelength). Substituting Eq. (2) into Eqs. (3) and (4), the expressions of the RPAiBs propagating in uniaxial crystals orthogonal to the optical axis can be obtained as

E_{x} (x, y, z) = - \frac{k n_{o}}{z} w_{1} w_{2} \sqrt{M_{1} N_{1}} \exp [Q (x, y, z)] (K_{1} + K_{2}),

E_{y} (x, y, z) = - \frac{k n_{o}}{z} w_{1} w_{2} \sqrt{M_{2} N_{2}} \exp [P (x, y, z)] (K_{3} + K_{4}),

where

\begin{array}{l} Q (x, y, z) = - i k n_{e} z + \frac{i}{12} (M_{1}^{3} + N_{1}^{3}) - \frac{1}{2} (a M_{1}^{2} + b N_{1}^{2}) - \frac{i}{2} (a^{2} M_{1} + \frac{x}{w_{1}} M_{1} + b^{2} N_{1} + \frac{y}{w_{2}} N_{1}) + \frac{a x}{w_{1}} + \frac{b y}{w_{2}}, \\ P (x, y, z) = - i k n_{o} z + \frac{i}{12} (M_{2}^{3} + N_{2}^{3}) - \frac{1}{2} (a M_{2}^{2} + b N_{2}^{2}) - \frac{i}{2} (a^{2} M_{2} + \frac{x}{w_{1}} M_{2} + b^{2} N_{2} + \frac{y}{w_{2}} N_{2}) + \frac{a x}{w_{1}} + \frac{b y}{w_{2}}, \\ K_{1} = (- \frac{M_{1}^{2}}{2} - i a M_{1} + \frac{x}{w_{1}}) Ai (- \frac{M_{1}^{2}}{4} - i a M_{1} + \frac{x}{w_{1}}) Ai (- \frac{N_{1}^{2}}{4} - i b N_{1} + \frac{y}{w_{2}}), \\ K_{2} = - i M_{1} A i^{'} (- \frac{M_{1}^{2}}{4} - i a M_{1} + \frac{x}{w_{1}}) Ai (- \frac{N_{1}^{2}}{4} - i b N_{1} + \frac{y}{w_{2}}), \\ K_{3} = (- \frac{N_{2}^{2}}{2} - i b N_{2} + \frac{y}{w_{2}}) Ai (- \frac{N_{2}^{2}}{4} - i b N_{2} + \frac{y}{w_{2}}) Ai (- \frac{M_{2}^{2}}{4} - i a M_{2} + \frac{x}{w_{1}}), \\ K_{4} = - i N_{2} A i^{'} (- \frac{N_{2}^{2}}{4} - i b N_{2} + \frac{y}{w_{2}}) Ai (- \frac{M_{2}^{2}}{4} - i a M_{2} + \frac{x}{w_{1}}), \\ M_{1} = \frac{z n_{e}}{k w_{1}^{2} n_{o}^{2}}, N_{1} = \frac{z}{k w_{2}^{2} n_{e}}, M_{2} = \frac{z}{k w_{1}^{2} n_{o}}, N_{2} = \frac{z}{k w_{2}^{2} n_{o}}, \end{array}

and Ai′(·) denotes the derivative of the Airy function. In Eqs. (5) and (6), it is noted that the propagation of the beams produced by the x-direction electric field (RPAiXBs) in uniaxial crystal whose electric field can be affected by n_e and n_o is similar to the beams propagating in an anisotropic medium while that of the beams produced by the y-direction electric field (RPAiYBs) in uniaxial crystal whose electric field can only be affected by n_o is similar to the beams propagating in an isotropic medium. If n_e = n_o, the anisotropic circumstance for the RPAiXBs turns to the isotropic situation. These descriptions are consistent with the results from Ref. [37]

The intensity of the RPAiXBs and RPAiYBs in uniaxial crystals orthogonal to the optical axis can be shown as

I_{x} = {| E_{x} (x, y, z) |}^{2}, I_{y} = {| E_{y} (x, y, z) |}^{2} .

Then, the total intensity expression of the RPAiBs in uniaxial crystals orthogonal to the optical axis can be written as

I = I_{x} + I_{y} .

Besides, the propagation trajectory of the RPAiXBs in the x-z and y-z planes can be derived from Eq. (5) as

x_{1} = \frac{z^{2} n_{e}^{2}}{4 k^{2} w_{1}^{3} n_{o}^{4}}, y_{1} = \frac{z^{2}}{4 k^{2} w_{2}^{3} n_{e}^{2}},

while that of the RPAiYBs in the x-z and y-z planes can be obtained from Eq. (6) as

x_{2} = \frac{z^{2}}{4 k^{2} w_{1}^{3} n_{o}^{2}}, y_{2} = \frac{z^{2}}{4 k^{2} w_{2}^{3} n_{o}^{2}} .

3. Numerical simulation and discussion

With the analytical expressions for the RPAiBs propagating in uniaxial crystals orthogonal to the optical axis, we further investigate the propagation properties by illustrating numerical examples including the intensity and the radiation forces. In the following simulations, the calculation parameters are chosen as λ = 633nm, w₁ = w₂ = 100µm, a = b = 0.1, n_o = 2.616 (rutile crystal) and $Z r = k w_{1}^{2} / 2$ is the Rayleigh range.

In order to investigate the propagation properties of the RPAiXBs in uniaxial crystals orthogonal to the optical axis, the intensity distributions in different observation planes and the maximum intensity distribution of the RPAiXBs with n_e = 1.5n_o are depicted in Fig. 1. From Figs. 1(a1) and 1(a2), we can find that the propagation trajectory of the RPAiXBs becomes further and further as compared to the x=0 and y=0 plane with the propagation distance increasing. Furthermore, the acceleration of the RPAiXBs in x-direction is much bigger than that in y-direction. In addition, it is interesting that the intensity of the RPAiXBs has a focused performance at about z=22Zr, which can be seen in Figs. 1(a1), 1(a2) and 1(c). For further studying the variation of the intensity during propagating, we need to concentrate on Figs. 1(b1)–1(b4). At the very start, the intensity mainly distributes on the side lobe in the x-direction as shown in Fig. 1(b1). Then, with the propagation distance increasing, the intensity of the side lobe in the x-direction flows into the main lobe gradually but the maximum intensity still decreases because of the diffraction, which can be seen in Fig. 1(b2). When the propagation distance z>9Zr, the effect of the intensity focusing becomes bigger than that of the diffraction so that the maximum intensity increases with propagating. It is intriguing that the intensity becomes biggest at the main lobe during the propagation when z=22Zr, which is manifested in Fig. 1(b3). Afterward, the intensity flows into the side lobe in the x-direction with propagating and at last, the intensity mainly distributes on the side lobe in the x-direction again as depicted in Fig. 1(b4).

Fig. 1 Intensity distributions in different observation planes and the maximum intensity distribution of the RPAiXBs in uniaxial crystals orthogonal to the optical axis with n_e = 1.5n_o.

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To learn about the influence of the ratio of n_e to n_o on the RPAiXBs, the intensity distributions in x-z and y-z planes and the maximum intensity distribution of the RPAiXBs in uniaxial crystals orthogonal to the optical axis with different ratios of n_e to n_o are shown in Fig. 2. The circumstances (anisotropic or isotropic) of the uniaxial crystal for the RPAiXBs are dependent on the ratio of n_e to n_o. From Fig. 2 we can analyze and derive the different effects of the anisotropic and isotropic situations on the propagation of the RPAiXBs. When propagating in isotropic situation (n_e = n_o), the initial maximum intensity is stronger than the after one. Absolutely it has a focused performance in the main lobe at a certain propagation distance as displayed in Fig. 3(b3), but the diffraction of the RPAiXBs with propagating is stronger than the intensity focusing. However, when propagating in anisotropic situation (n_e, n_o), with the ratio of n_e to n_o increasing, the maximum intensity of the main lobe increases so that the maximum intensity during the propagation can be achieved in the main lobe because the intensity focusing stronger than diffraction. It also means that the ratio of n_e to n_o in anisotropic situation can modulate the effect of the intensity focusing, which is performed by the variations of the intensity and position at the main lobe (stronger and closer respectively with the ratio of n_e to n_o increasing). Compared Figs. 1(a1) and 1(a2) with Figs. 2(a1), 2(b1) and 2(a2), 2(b2), it can be clearly seen that the acceleration of the beams in the x-direction becomes bigger while that in the y-direction becomes smaller so that the propagation trajectory turns to be much further and closer as compared to the x=0 and y=0 planes respectively with the increase of the ratio of n_e to n_o. It means that we can control the propagation trajectory of the RPAiXBs in anisotropic situation by varying ratio of the n_e to n_o.

Fig. 2 Intensity distributions in x-z and y-z planes and the maximum intensity distribution of the RPAiXBs in uniaxial crystals orthogonal to the optical axis with different ratios of n_e to n_o.

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Fig. 3 Intensity distributions in different observation planes and the maximum intensity distribution of the RPAiYBs in uniaxial crystals orthogonal to the optical axis with n_o = 2.616.

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Figure 3 represents the intensity distributions in different observation planes and the maximum intensity distribution of the RPAiYBs in uniaxial crystals orthogonal to the optical axis with n_o = 2.616. Similar to the propagation trajectory of the RPAiXBs, in Figs. 3(a1) and 3(a2), we can find that the propagation trajectory of the RPAiYBs becomes further and further as compared to the x=0 and y=0 plane with propagating. However, the acceleration of the RPAiYBs in the x-direction is the same as that in the y-direction, which is different from that of the RPAiXBs. The RPAiYBs also have focused performance depicted in Fig. 3(b3). Combining Figs. 3(b1)–3(b4) with Fig. 3(c), the specific variation of the intensity of the RPAiYBs can be described as below. At the very start, the intensity almost absolutely distributes on the side lobe in the y-direction (Fig. 3(b1)). with the increase of the propagation distance, the intensity of the side lobe in the y-direction flows into the main lobe gradually (Fig. 3(b2)), and after the intensity of the main lobe reaches the biggest extent at about 22Zr (Fig. 3(b3)), the intensity flows into the side lobe in the x-direction (Fig. 3(b4)).

To get a deep insight on the differences between the RPAiXBs and the RPAiYBs, we plot the intensity distributions in x-z and y-z planes of the RPAiBs with different ratios of n_e to n_o in Fig. 4, and transverse intensity distributions of the RPAiXBs, RPAiYBs and RPAiBs at different propagation distances with n_e = 1.5n_o in Fig. 5. one can see from Fig. 4 that the intensity of the RPAiXBs is always stronger than that of the RPAiYBs. With the ratio of n_e to n_o increasing, the intensity distributing on the RPAiXBs becomes stronger and stronger but that on the RPAiYBs is constant, which induces the RPAiXBs become more and more dominant. Compared Fig. 5(a3) with Fig. 5(b3), it is clear that the RPAiBs also have a focused performance at the main lobe arise from the intensity focusing of the RPAiXBs shown in Fig. 5(b1). Because of the different propagation trajectories of the RPAiXBs and RPAiYBs, with the increase of the propagation distance, the main lobe in the RPAiBs, which maintains one main lobe in a short propagation distance owing to the ultrasmall transverse shift, will evolve into two main lobes which represent the RPAiXBs and RPAiYBs respectively as depicted in Fig. 5(c3). From Figs. 5(c1), 5(c2), 5(d1) and 5(d2) we can see that the intensity of the RPAiXBs is stronger than that of the RPAiYBs in a certain propagation distance (<40Zr when n_e = 1.5n_o) and weaker than that of the RPAiYBs afterward.

Fig. 4 Propagation trajectories of the RPAiBs in the (a1)–(a3) x- and (b1)–(b3) y-directions through uniaxial crystals orthogonal to the optical axis with different ratios of n_e to n_o.

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Fig. 5 Transverse intensity distributions of the (a1)–(d1) RPAiXBs, (a2)–(d2) RPAiYBs and (a3)–(d3) RPAiBs at different propagation distances with n₀ = 1.5n_o.

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Finally, we investigate the gradient force and the scattering force of the RPAiXBs and the RPAiYBs, which are relative to momentum changes of the electromagnetic wave. Assuming that a micro particle with refractive index n₁ is in a stable state, we can describe the gradient force and the scattering force as [38]:

{\vec{F}}_{g r a d} (x, y, z) = \frac{2 π n_{2} r_{0}^{3}}{c} (\frac{m^{2} - 1}{m^{2} + 2}) \nabla I (x, y, z),

{\vec{F}}_{s c a t} (x, y, z) = \frac{8 π n_{2} r_{0}^{6}}{3 c} {(\frac{m^{2} - 1}{m^{2} + 2})}^{2} I (x, y, z) {\vec{e}}_{z},

with

I (x, y, z) = c n_{2} ε_{0} \frac{{| E (x, y, z) |}^{2}}{2},

where m = n₁/n₂ is the relative refractive index of the particle, n₂ is the refractive index of a surrounding medium, r₀ is the radius of the micro particle, c is the light velocity and ε₀ is the permittivity of vacuum.

Here we assume that n₁ = 1.592, r₀ = 50nm. The transverse gradient force and the maximum gradient force distributions of the RPAiXBs (n₂ = n_e) are presented in Fig. 6. From Figs. 6(a1)–6(a5) we can clearly learn about the variation process of the transverse gradient force, which elucidates as following. The transverse gradient force distribution has a hollow portion at the center. When the propagation distance z<13Zr, the gradient force mainly distributes on the left and right portions of the side lobe in the x-direction. Then, when 13Zr<z<22Zr, the gradient force mostly distributes from the under portion of the main lobe to the left portion of the main lobe during propagating. After that, when z>22Zr, the gradient force gradually flows to the side lobe in the x-direction and mainly distributes on the side lobe in the x-direction again but the Over and under portions at last. It is interesting that the gradient force vectors on the side lobe in the x-direction mostly present an up and down distribution which is parallel to the y-axis, and that on the side lobe in the y-direction mostly present a left and right distribution which parallels to the x-axis. Moreover, the gradient force vectors on the main lobe have both distributions. In Fig. 6(b), it is pronounced that the maximum gradient force firstly decreases fast and then increases to a certain value which is smaller than the initial value with propagating. Afterward, it decreases again until equaling to zero. With the ratio of n_e to n_o increasing, the minimum and the maximum become bigger and their corresponding propagation positions become closer.

Fig. 6 (a1)–(a5) The transverse gradient force patterns and flows with n_e = 1.5n_o and (b) the maximum gradient force distribution of the RPAiXBs during the propagation with different ratios of n_e to n_o.

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Figure 7 presents the transverse gradient force and the maximum gradient force distributions of the RPAiYBs with n_o = 2.616 (n₂ = n_o). Adopting the analytic methods similar to the Fig. 6, we can easily find that the gradient force mostly distributes on the over and under portions of the side lobe in the y-direction when z<22Zr and gradually distributes both on the main lobe and the side lobe in the x-direction subsequently. At last, it mostly distributes on the over and under portions of the side lobe in the x-direction. Of the gradient force vectors, it can be clearly seen that the results are almost the same as those in Fig. 6. In the case of the maximum gradient force shown in Fig. 6(b), it is apparent that the maximum gradient force firstly decreases fast when z<19Zr and then decreases slowly until equaling to 0.

Fig. 7 (a1)–(a5) The transverse gradient force patterns and flows and (b) the maximum gradient force distribution of the RPAiYBs during the propagation with n_o = 2.616.

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In Fig. 8, we plot the maximum scattering force distributions of the RPAiXBs with different ratios of n_e to n_o (Fig. 8(a)) and the RPAiYBs (Fig. 8(b)) at different propagation distances. Similar to the maximum intensity distribution as depicted in Fig. 3(c), the maximum scattering force decreases at first and then increases to a certain value and decreases again until equaling to zero at last with propagating. Interestingly, the maximum scattering force distribution of the RPAiXBs can be greatly affected by the ratios of n_e to n_o. For instance, the initial maximum scattering force, the minimum and the maximum become bigger, the positions of the minimum and the maximum become closer with the ratio of n_e to n_o increasing. Moreover, when n_e > 1.2n_o, the maximum becomes bigger than the initial value, which can be regarded as the maximum value. In terms of the maximum scattering force distribution of the RPAiYBs shown in Fig. 8(b), its variation properties are almost the same as those of the RPAiXBs with n_e < 1.2n_o.

Fig. 8 (a) Maximum scattering force distributions of the RPAiXBs with different ratios of n_e to n_o and (b) the RPAiYBs with n_o = 2.616 during the propagation.

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4. Conclusion

In this paper, the analytical expressions for the RPAiXBs and the RPAiYBs propagating in uniaxial crystals orthogonal to the optical axis are derived, and their corresponding propagation trajectory, the intensity and the radiation forces have been analyzed and described in detail. It is shown that both the RPAiXBs and RPAiYBs have a focused performance at the main lobe. When n_e, n_o which also means the RPAiXBs propagate in an anisotropic medium, the ratio of n_e to n_o plays an important role on modulating the effect of the intensity focusing and varying the propagation trajectory. With the ratio of n_e to n_o increasing, the intensity focusing position and the focusing intensity of the RPAiXBs at the main lobe becomes closer and stronger respectively. Meanwhile, the acceleration of the RPAiXBs in the x-direction becomes bigger while that in the y-direction becomes smaller so that the propagation trajectory turns to be much further and closer as compared to the x=0 and y=0 planes, respectively. Moreover, the minimum and the maximum of the maximum gradient force and scattering force distribution for the RPAiXBs become bigger and their corresponding propagation positions become closer. In the case of the RPAiYBs whose propagation is similar to the propagation in an isotropic medium, the acceleration in the x-direction always equals to that in the y-direction and the focusing intensity at the main lobe is smaller than the initial maximum intensity. It is worth indicating that the RPAiBs also have a focused performance at the main lobe in a short distance. Afterward, the main lobe of the RPAiBs evolves into two main lobes because of the different propagation trajectories of the RPAiXBs and RPAiYBs. Besides, during the propagation, the intensity of the RPAiXBs transfers from the side lobe in the x-direction to the main lobe and finally returns to the side lobe in the x-direction again, but that of the RPAiYBs transfers from the side lobe in the y-direction to the main lobe and flows to the side lobe in the x-direction at last.

Funding

National Natural Science Foundation of China (NSFC) (11775083 and 11374108); National Training Program of Innovation and Entrepreneurship for Undergraduates.

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Paraxial propagation dynamics of the radially polarized Airy beams in uniaxial crystals orthogonal to the optical axis

Abstract

1. Introduction

2. Analytic solution of the RPAiBs in uniaxial crystals orthogonal to the optical axis

3. Numerical simulation and discussion

4. Conclusion

Funding

References and links

Cited By

Figures (8)

Equations (14)

Optics Express