Analytical vectorial structure of hollow Gaussian beams in the far field

Guohua Wu; Qihong Lou; Jun Zhou

doi:10.1364/OE.16.006417

Optics Express
Vol. 16,
Issue 9,
pp. 6417-6424
(2008)
•https://doi.org/10.1364/OE.16.006417

Analytical vectorial structure of hollow Gaussian beams in the far field

Guohua Wu, Qihong Lou, and Jun Zhou

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Guohua Wu, Qihong Lou, and Jun Zhou, "Analytical vectorial structure of hollow Gaussian beams in the far field," Opt. Express 16, 6417-6424 (2008)

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- Physical Optics
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About this Article
History
- Original Manuscript: January 7, 2008
- Revised Manuscript: February 26, 2008
- Manuscript Accepted: March 6, 2008
- Published: April 22, 2008

Abstract

The analytical vectorial structure of HGB is investigated in the far field based on the vector plane wave spectrum and the method of stationary phase. The energy flux distributions of HGB in the far-field, which is composed of TE term and TM term, are demonstrated. The physics pictures of HGB is illustrated from the vectorial structure, which is important to understand the theoretical aspects of both scalar and vector HGB propagation.

1. Introduction

Recently, optical beams with zero central intensity, i. e., dark-hollow beams (DHBs), have attracted more and more attention for their unique physical properties in their interaction with atoms, such as small dark spot size, heating-free effect, and may carry spin and orbital angular momentum [1, 2, 3]. DHBs have become a powerful tool in precise manipulation and control of microscopic particles in optical pipes, optical tweezers and spanners [4]. They have also been applied to trap and cool neutral atoms [5]. Various methods have been proposed to generate DHBs, such as transverse-mode selection [6], geometric optical method [7], computer-generated-hologram method [8] and spatial filtering [9]. In theory, several models have been put forward to describe dark hollow beams including TEM _0l beams [10], higher-order bessel beams [11] and hollow Gaussian beam (HGB) [12, 13], etc. HGB as an convenient and powerful tool to describe dark-hollow beams has been investigated with growing interest. The propagation properties of HGB through unapertured [12], apertured optical systems [13] and a turbulent atmosphere [14] has been investigated in details. More recently, Degang Deng et al. has studied the nonparaxial propagation of vectorial HGB in free space [15]. The radiation force produced by highly focused HGB has also been discussed [16]. Spatial filtering has been proposed to obtain HGB from a Fourier transform of a Gaussian beam [9].

The vector angular spectrum method, as an effective method to resolve the Maxwell’s equations, has been investigated with growing interest, recently. The general solution of the Maxwell’s equations can be written as a sum of two terms, i. e., TE term and TM term, in terms of the vector angular spectrum of electromagnetic field [17, 18, 19]. In the far field, the TE and TM terms are orthogonal to each other and can be detached. The isolated TE term may be used to improve the density of optical storage [20]. Based on the vector plane wave spectrum, the analytical vectorial structure of Laguerre-Gaussian beams [19] and radially polarized light beams [21] are investigated. In present paper, the analytical vectorial structure of HGB in the far-field is investigated by means of the vector plane wave spectrum and the method of stationary phase. Based on the analytical vectorial structure of HGB, the energy flux distributions of the HGB are also demonstrated.

2. Analytical vectorial structure

Let us consider a linear homogeneous dielectric, nonconducting medium occupying the halfspace z>0 and all the sources lie in the region z<0. The field is specified at the boundary plane z=0 plane. It is well known that the field throughout this half-space can be expanded into an angular spectrum of plane waves, the propagating electric field toward to half free space z≥0 turns out to be [17, 18, 19]

E_{x} (x, y, z) = \int \int_{- \infty}^{+ \infty} A_{x} (p, q) \exp [i k (px + qy + γ z)] d p d q,

E_{y} (x, y, z) = \int \int_{- \infty}^{+ \infty} A_{y} (p, q) \exp [i k (px + qy + γ z)] d p d q,

E_{z} (x, y, z) = - \int \int_{- \infty}^{+ \infty} [\frac{p}{γ} A_{x} (p, q) + \frac{q}{γ} A_{y} (p, q)] \exp [i k (px + qy + mz)] d p d q,

where k=2π/λ is wavenumber and λ is the wavelength in the medium. The exp(-iωt) has been suppressed in the field expressions, and

γ = {\begin{matrix} {(1 - p^{2} - q^{2})}^{\frac{1}{2}}, if p^{2} + q^{2} \leq 1 \\ {i (p^{2} + q^{2} - 1)}^{\frac{1}{2}}, if p^{2} + q^{2} > 1 \end{matrix} .

The values of p ²+q ²≤1 correspond to the homogeneous part of the field, whereas the values of p ²+q ²>1 correspond to the the evanescent part of the field which propagates along the xy plane but decays exponentially along the positive z direction. The complex factors A_x(p,q) and A_y(p,q) can be determined from the x and y components of the initial electric field,

A_{x} (p, q) = \frac{1}{λ^{2}} \int \int_{- \infty}^{+ \infty} E_{x} (x, y, 0) \exp [- i k (p x + q y)] d x d y,

A_{y} (p, q) = \frac{1}{λ^{2}} \int \int_{- \infty}^{+ \infty} E_{y} (x, y, 0) \exp [- i k (p x + q y)] d x d y .

To find the coefficients A_x and A_y, we consider a HGB with polarization in the x direction, which propagates toward the half-space z≥0 along the z axis. The electric field distribution of HGB at the z=0 plane reads as [12, 13]

E_{x} (x, y, 0) = G_{0} {[\frac{ρ^{2}}{w_{0}^{2}}]}^{n} \exp [- \frac{ρ^{2}}{w_{0}^{2}}],

E_{y} (x, y, 0) = 0,

where n is the beam order of HGB, w ₀ is the beam waist width, G ₀ is a constant and $ρ = \sqrt{x^{2} + y^{2}}$ is the radial coordinate. As indicated by Fig. 1, the beam radius, which is defined as the distance between the position of the maximal radial intensity and the center of the light beam [1], increases with the increasing of the beam order n. Substituting Eqs. (7) and (8) into Eqs. (5) and (6), we find that

A_{x} (p, q) = G_{0} \frac{n!}{π f^{2} 2^{n + 2}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) L_{m} [\frac{p^{2} + q^{2}}{{2 f}^{2}}] \exp [- \frac{p^{2} + q^{2}}{4 f^{2}}],

A_{y} (p, q) = 0,

Fig. 1. Normalized intensity distribution of hollow Gaussian beams as a function of ρ/w ₀ with different beam order n at z=0 plane based on Eq. (7).

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where f=1/kw ₀ is proportional to the ratio of the wavelength to the beam width. In the derivation of the Eq. (9), HGB is expanded into a series of Laguerre-Gaussian modes. As known, the Maxwell’s equation can be separated into transverse and longitudinal field equations and an arbitrary polarized electromagnetic field can be expressed as the sum of two terms [17, 18, 19], E⃗_TE(r⃗) and E⃗_TM(r⃗), namely,

\vec{E} (\vec{r}) = {\vec{E}}_{TE} (\vec{r}) + {\vec{E}}_{TM} (\vec{r}),

where

{\vec{E}}_{TE} (\vec{r}) = {\int \int}_{- \infty}^{+ \infty} \frac{1}{p^{2} + q^{2}} [q A_{x} (p, q) - p A_{y} (p, q)] (q {\hat{e}}_{x} - p {\hat{e}}_{y}) \exp [i k u] d p d q,

{\vec{E}}_{TM} (\vec{r}) = {\int \int}_{- \infty}^{+ \infty} \frac{1}{p^{2} + q^{2}} [q A_{x} (p, q) + p A_{y} (p, q)] (q {\hat{e}}_{x} + p {\hat{e}}_{y} - \frac{b^{2}}{{γ \hat{e}}_{z}}) \exp [i k u] d p d q,

r⃗=xê_x+yê_y+zê_z is the displacement vector and u=px+qy+γz. Equations (11), (12) and (13) show that the divergence condition of the electric field is obeyed and the polarized direction of every plane-wave component is perpendicular to its own wave vector. From the Maxwell equation and Eq. (11), the magnetic field can also be expressed as a sum of two terms, H⃗_TE(r⃗) and H⃗_TM(r⃗), namely,

\vec{H} (\vec{r}) = {\vec{H}}_{TE} (\vec{r}) + {\vec{H}}_{TM} (\vec{r}),

where

{\vec{H}}_{TE} (\vec{r}) = \sqrt{\frac{ε}{μ}} \int \int_{- \infty}^{+ \infty} \frac{1}{p^{2} + q^{2}} [q A_{x} (p, q) - p A_{y} (p, q)]

{\vec{H}}_{T M} (\vec{r}) = - \sqrt{\frac{ε}{μ}} \int \int_{- \infty}^{+ \infty} [p A_{x} (p, q) + q A_{y} (p, q)] \frac{1}{b^{2} γ} (q {\hat{e}}_{x} - p {\hat{e}}_{y}) \exp (i k u) d p d q .

Since z is big enough in the far regime, the condition k(x ²+y ²+z ²)^1/2→∞ is satisfied and the contribution of the evanescent waves to the far field can be omitted. By employing the method of stationary phase [19, 22, 23], the analytical electromagnetic fields of the TE mode for HGB in the far-field may be given by

{\vec{E}}_{T E} (\vec{r}) = - G_{0} \frac{n!}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) \frac{i Z_{R} y z}{r^{2} ρ^{2}} L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}] \exp [- \frac{ρ^{2}}{4 f^{2} r^{2}} + i k r] (y {\hat{e}}_{x} - x {\hat{e}}_{y}),

{\vec{H}}_{T E} (\vec{r}) = - G_{0} \sqrt{\frac{ε}{μ}} \frac{n!}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) \frac{i Z_{R} y z}{r^{3} ρ^{2}} L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}]

where Z_R=kw ² ₀/2 is Rayleigh distance. Similarly, the analytical electromagnetic fields of the TM mode for HGB in the far-field may be given by

{\vec{E}}_{TM} (\vec{r}) = - G_{0} \frac{n!}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) \frac{i Z_{R} x}{r^{2} ρ^{2}} L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}]

{\vec{H}}_{TM} (\vec{r}) = G_{0} \sqrt{\frac{ε}{μ}} \frac{n!}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) \frac{i Z_{R} x}{r ρ^{2}} L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}] \exp [- \frac{ρ^{2}}{4 f^{2} r^{2}} + i k r] (y {\hat{e}}_{x} - x {\hat{e}}_{y}) .

Equations (17) - (20) are the basic results obtained in this paper; they are applicable for both paraxial case and nonparaxial case. As indicated by Eqs. (17) - (20), the TE and TM terms of HGB are orthogonal to each other and can be detached in the far field, which may be used to improve the density of optical storage [20].

3. Energy flux distribution

The energy flux distributions at the z=const plane are given by the z component of their timeaverage Poynting vector. The energy flux distribution at the far-field z=const plane turns out to be

〈 {\vec{S}}_{z} 〉 = \frac{1}{2} Re {[\vec{E} * (\vec{r}) \times \vec{H} (\vec{r})]}_{z}

where Re represents the real part, and the asterisk denotes complex conjugation. From Eqs. (17)-(18), the energy flux distribution of the TE term for HGB at the far-field z=const plane is given by:

{〈 {\vec{S}}_{z} 〉}_{TE} = G_{0}^{2} \sqrt{\frac{ε}{μ}} \frac{{(n!)}^{2}}{2^{2 n + 1}} \frac{Z_{R}^{2} y^{2} z^{3}}{r^{5} ρ^{2}}

From Eqs. (19)-(20), the energy flux distribution of the TM term for HGB in the far-field z=const plane is given by:

{〈 {\vec{S}}_{z} 〉}_{TM} = G_{0}^{2} \sqrt{\frac{ε}{μ}} \frac{{(n!)}^{2}}{2^{2 n + 1}} \frac{Z_{R}^{2} x^{2} z}{r^{3} ρ^{2}}

Therefore, the energy flux distribution of HGB at the far-field z=const plane yields

〈 {\vec{S}}_{z} 〉 = G_{0}^{2} \sqrt{\frac{ε}{μ}} \frac{{(n!)}^{2}}{2^{2 n + 1}} \frac{Z_{R}^{2} z}{r^{3} ρ^{2}} (\frac{y^{2} z^{2}}{r^{2}} + x^{2})

The normalized energy flux distributions of the HGB at the plane z=600λ for different beam order n=4 and n=10 (a) TE term, (b) TM term, (c) whole beam versus x/λ and y/λ are depicted by Fig. 2 and Fig. 3, respectively. Since the medium is homogenous, ε/µ can be set to unity. The constant G ₀ is also set to unity. The used parameter is w ₀=6λ. As shown by Fig. 2 and Fig. 3, in the far field, the TE and TM terms are orthogonal to each other. Also, the dark region in the center disappears and the on-axis energy flux becomes maximal, which differs from its initial ring-shape. This is because HGB is not a pure mode, different modes evolve differently with the propagation distance. The overlap and interfere in propagation between different modes result in the propagation properties of HGB in the far field [12]. With the increasing of the beam order n, the value of the peak on-axis energy flux increases quickly.

Fig. 2. Normalized energy flux distribution of HGB at the plane z=600λ for beam order n=4. (a) TE term, (b) TM term, (c) whole beam.

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Fig. 3. Normalized energy flux distribution of HGB at the plane z=600λ for beam order n=10. (a) TE term, (b) TM term, (c) whole beam.

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4. Conclusion

In conclusion, the analytical vectorial structure of HGB in the far-field is obtained by using of the full vector angular spectrum of electromagnetic and the method of stationary phase. In this presentation, the electric field and magnetic field of HGB is composed as a sum of two terms, i. e., TE term and TM term, which are orthogonal to each other in the far-field. Based on the analytical vectorial structure of HGB, the energy flux distribution of the TE term, the TM term and the whole beam of HGB are studied in the far-field. This work is important to understand the theoretical aspects of both scalar and vector HGB propagation.

Appendix A: Derivation of Eq. (17) by using of the method of stationary phase

Here, consider the asymptotic approximation for large value of k to the double integral [22]

I (k) = \int \int_{S} f (x, y) e^{ikg (x, y)} dxdy,

where f(x,y) and g(x,y) are the real, well-behaved functions. The term f(x,y) will just give rise to ‘amplitude modulation’ of the term e^ikg(x,y). Moreover, when k is large enough, the integrand of the integral on the right side oscillates so rapidly, irrespective of the exact forms of f(x,y) and g(x,y), that various contributions cancel out, except for contributions from the neighborhood of stationary points. The stationary points are points within the domain S of integration where

{\frac{\partial g (x, y)}{\partial x} ∣}_{x_{1}, y_{1}} = {\frac{\partial g (x, y)}{\partial y} ∣}_{x_{1}, y_{1}} = 0,

and points on the boundary of S where

\frac{\partial g}{\partial l} = 0 .

According to the principle of stationary phase (reference [22], p. 133; also see reference [24], Chap 9), the asymptotic approximation to Eq. (A. 1) is

I = \frac{2 π i ε}{k \sqrt{∣ α_{1} β_{1} - γ_{1}^{2} ∣}} f (x_{1}, y_{1}) e^{ikg (x_{1}, y_{1})} as k \to \infty,

where

α_{1} = {\frac{\partial^{2} g (x, y)}{\partial x^{2}} ∣}_{x_{1}, y_{1}}, β_{1} = {\frac{\partial^{2} g (x, y)}{\partial y^{2}} ∣}_{x_{1}, y_{1}}, γ_{1} = {\frac{\partial^{2} g (x, y)}{\partial x \partial y}}_{x_{1}, y_{1}} .

ε = {\begin{matrix} 1, & if α_{1} β_{1} > γ_{1}^{2} and α_{1} > 0, \\ - 1, & if α_{1} β_{1} > γ_{1}^{2} and α_{1} < 0, \\ i, & if α_{1} β_{1} < γ_{1}^{2} . \end{matrix}

If it has several stationary points, one can obtain the corresponding asymptotic approximation by summing the contributions. By using of Eq. (A. 4), the far field of the angular spectrum represent of wavefield can be expressed as

E (x, y, z) = \int \int_{\infty}^{+ \infty} A (p, q) e^{ik (px + qy + mz)} dpdq

For a detailed derivations of Eqs. (A. 4) and (A. 7), see reference [22] (p. 128-144, section 3.3). Substituting Eqs. (9) and (10) into Eq. (12) and comparison with Eq. (A. 7), we can obtain Eq. (17). We obtain Eqs. (18)-(20) by using the same procedure.

Acknowledgements

This work is partially supported by the National key Basic Research Project, 863 Project (No. 2005AA846020), National Nature Science Foundation of China (No. 103341101), the National Key Basic Research Project (China), Shanghai Science & Technology Foundation (No. 04D205120, No. 05D222001) and Knowledge Innovation Project of Chinese Academy of Sciences.

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Fig. 1. Normalized intensity distribution of hollow Gaussian beams as a function of ρ/w ₀ with different beam order n at z=0 plane based on Eq. (7).

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Fig. 2. Normalized energy flux distribution of HGB at the plane z=600λ for beam order n=4. (a) TE term, (b) TM term, (c) whole beam.

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Fig. 3. Normalized energy flux distribution of HGB at the plane z=600λ for beam order n=10. (a) TE term, (b) TM term, (c) whole beam.

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Equations (39)

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E_{x} (x, y, z) = \int \int_{- \infty}^{+ \infty} A_{x} (p, q) \exp [i k (px + qy + γ z)] d p d q,

E_{y} (x, y, z) = \int \int_{- \infty}^{+ \infty} A_{y} (p, q) \exp [i k (px + qy + γ z)] d p d q,

E_{z} (x, y, z) = - \int \int_{- \infty}^{+ \infty} [\frac{p}{γ} A_{x} (p, q) + \frac{q}{γ} A_{y} (p, q)] \exp [i k (px + qy + mz)] d p d q,

γ = {\begin{matrix} {(1 - p^{2} - q^{2})}^{\frac{1}{2}}, if p^{2} + q^{2} \leq 1 \\ {i (p^{2} + q^{2} - 1)}^{\frac{1}{2}}, if p^{2} + q^{2} > 1 \end{matrix} .

A_{x} (p, q) = \frac{1}{λ^{2}} \int \int_{- \infty}^{+ \infty} E_{x} (x, y, 0) \exp [- i k (p x + q y)] d x d y,

A_{y} (p, q) = \frac{1}{λ^{2}} \int \int_{- \infty}^{+ \infty} E_{y} (x, y, 0) \exp [- i k (p x + q y)] d x d y .

E_{x} (x, y, 0) = G_{0} {[\frac{ρ^{2}}{w_{0}^{2}}]}^{n} \exp [- \frac{ρ^{2}}{w_{0}^{2}}],

E_{y} (x, y, 0) = 0,

A_{x} (p, q) = G_{0} \frac{n!}{π f^{2} 2^{n + 2}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) L_{m} [\frac{p^{2} + q^{2}}{{2 f}^{2}}] \exp [- \frac{p^{2} + q^{2}}{4 f^{2}}],

A_{y} (p, q) = 0,

\vec{E} (\vec{r}) = {\vec{E}}_{TE} (\vec{r}) + {\vec{E}}_{TM} (\vec{r}),

{\vec{E}}_{TE} (\vec{r}) = {\int \int}_{- \infty}^{+ \infty} \frac{1}{p^{2} + q^{2}} [q A_{x} (p, q) - p A_{y} (p, q)] (q {\hat{e}}_{x} - p {\hat{e}}_{y}) \exp [i k u] d p d q,

{\vec{E}}_{TM} (\vec{r}) = {\int \int}_{- \infty}^{+ \infty} \frac{1}{p^{2} + q^{2}} [q A_{x} (p, q) + p A_{y} (p, q)] (q {\hat{e}}_{x} + p {\hat{e}}_{y} - \frac{b^{2}}{{γ \hat{e}}_{z}}) \exp [i k u] d p d q,

\vec{H} (\vec{r}) = {\vec{H}}_{TE} (\vec{r}) + {\vec{H}}_{TM} (\vec{r}),

{\vec{H}}_{TE} (\vec{r}) = \sqrt{\frac{ε}{μ}} \int \int_{- \infty}^{+ \infty} \frac{1}{p^{2} + q^{2}} [q A_{x} (p, q) - p A_{y} (p, q)]

\times [p γ {\hat{e}}_{x} + q γ {\hat{e}}_{y} - (p^{2} + q^{2}) {\hat{e}}_{z}] \exp (i k u) d p d q,

{\vec{H}}_{T M} (\vec{r}) = - \sqrt{\frac{ε}{μ}} \int \int_{- \infty}^{+ \infty} [p A_{x} (p, q) + q A_{y} (p, q)] \frac{1}{b^{2} γ} (q {\hat{e}}_{x} - p {\hat{e}}_{y}) \exp (i k u) d p d q .

{\vec{E}}_{T E} (\vec{r}) = - G_{0} \frac{n!}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) \frac{i Z_{R} y z}{r^{2} ρ^{2}} L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}] \exp [- \frac{ρ^{2}}{4 f^{2} r^{2}} + i k r] (y {\hat{e}}_{x} - x {\hat{e}}_{y}),

{\vec{H}}_{T E} (\vec{r}) = - G_{0} \sqrt{\frac{ε}{μ}} \frac{n!}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) \frac{i Z_{R} y z}{r^{3} ρ^{2}} L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}]

\times \exp [- \frac{ρ^{2}}{4 f^{2} r^{2}} + i k r] (x z {\hat{e}}_{x} + y z {\hat{e}}_{y} - ρ^{2} {\hat{e}}_{z}),

{\vec{E}}_{TM} (\vec{r}) = - G_{0} \frac{n!}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) \frac{i Z_{R} x}{r^{2} ρ^{2}} L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}]

\times \exp [- \frac{ρ^{2}}{4 f^{2} r^{2}} + i k r] (x z {\hat{e}}_{x} + y z {\hat{e}}_{y} - ρ^{2} {\hat{e}}_{z}),

{\vec{H}}_{TM} (\vec{r}) = G_{0} \sqrt{\frac{ε}{μ}} \frac{n!}{2^{n}} \sum_{m = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) \frac{i Z_{R} x}{r ρ^{2}} L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}] \exp [- \frac{ρ^{2}}{4 f^{2} r^{2}} + i k r] (y {\hat{e}}_{x} - x {\hat{e}}_{y}) .

〈 {\vec{S}}_{z} 〉 = \frac{1}{2} Re {[\vec{E} * (\vec{r}) \times \vec{H} (\vec{r})]}_{z}

= {〈 {\vec{S}}_{z} 〉}_{TE} + {〈 {\vec{S}}_{z} 〉}_{TM},

{〈 {\vec{S}}_{z} 〉}_{TE} = G_{0}^{2} \sqrt{\frac{ε}{μ}} \frac{{(n!)}^{2}}{2^{2 n + 1}} \frac{Z_{R}^{2} y^{2} z^{3}}{r^{5} ρ^{2}}

\times \sum_{m = 0}^{n} \sum_{b = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) (\begin{matrix} n \\ b \end{matrix}) L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}] L_{b} [\frac{ρ^{2}}{2 f^{2} r^{2}}] \exp [- \frac{ρ^{2}}{2 f^{2} r^{2}}] .

{〈 {\vec{S}}_{z} 〉}_{TM} = G_{0}^{2} \sqrt{\frac{ε}{μ}} \frac{{(n!)}^{2}}{2^{2 n + 1}} \frac{Z_{R}^{2} x^{2} z}{r^{3} ρ^{2}}

\times \sum_{m = 0}^{n} \sum_{b = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) (\begin{matrix} n \\ b \end{matrix}) L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}] L_{b} [\frac{ρ^{2}}{2 f^{2} r^{2}}] \exp [- \frac{ρ^{2}}{2 f^{2} r^{2}}] .

〈 {\vec{S}}_{z} 〉 = G_{0}^{2} \sqrt{\frac{ε}{μ}} \frac{{(n!)}^{2}}{2^{2 n + 1}} \frac{Z_{R}^{2} z}{r^{3} ρ^{2}} (\frac{y^{2} z^{2}}{r^{2}} + x^{2})

\times \sum_{m = 0}^{n} \sum_{b = 0}^{n} (\begin{matrix} n \\ m \end{matrix}) (\begin{matrix} n \\ b \end{matrix}) L_{m} [\frac{ρ^{2}}{2 f^{2} r^{2}}] L_{b} [\frac{ρ^{2}}{2 f^{2} r^{2}}] \exp [- \frac{ρ^{2}}{2 f^{2} r^{2}}] .

I (k) = \int \int_{S} f (x, y) e^{ikg (x, y)} dxdy,

{\frac{\partial g (x, y)}{\partial x} ∣}_{x_{1}, y_{1}} = {\frac{\partial g (x, y)}{\partial y} ∣}_{x_{1}, y_{1}} = 0,

\frac{\partial g}{\partial l} = 0 .

I = \frac{2 π i ε}{k \sqrt{∣ α_{1} β_{1} - γ_{1}^{2} ∣}} f (x_{1}, y_{1}) e^{ikg (x_{1}, y_{1})} as k \to \infty,

α_{1} = {\frac{\partial^{2} g (x, y)}{\partial x^{2}} ∣}_{x_{1}, y_{1}}, β_{1} = {\frac{\partial^{2} g (x, y)}{\partial y^{2}} ∣}_{x_{1}, y_{1}}, γ_{1} = {\frac{\partial^{2} g (x, y)}{\partial x \partial y}}_{x_{1}, y_{1}} .

ε = {\begin{matrix} 1, & if α_{1} β_{1} > γ_{1}^{2} and α_{1} > 0, \\ - 1, & if α_{1} β_{1} > γ_{1}^{2} and α_{1} < 0, \\ i, & if α_{1} β_{1} < γ_{1}^{2} . \end{matrix}

E (x, y, z) = \int \int_{\infty}^{+ \infty} A (p, q) e^{ik (px + qy + mz)} dpdq

= - \frac{2 π i z}{k r^{2}} A (\frac{x}{r}, \frac{y}{r}) e^{ikr} as kr \to \infty .

Abstract

1. Introduction

2. Analytical vectorial structure

3. Energy flux distribution

4. Conclusion

Appendix A: Derivation of Eq. (17) by using of the method of stationary phase

Acknowledgements

References and links

Cited By

Figures (3)

Equations (39)

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