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Generation of perfect helical Mathieu vortex beams

Xiaoxiao Li, Zhijun Ren, Fuyang Xu, LvBin Song, Xiang Lv, Yixian Qian, and Ping Yu

Open Access Open Access

Abstract

We introduced a kind of novel perfect optical vortex beam, which we termed herein as perfect helical Mathieu vortex (PHMV) beams. The theoretical mechanism regarding the construction of PHMV beams was divided into two parts: generation of helical Mathieu (HM) beams using the stationary phase method and then Fourier transform of HM beams into the PHMV beams. Accordingly, the experimental system for generating PHMV beams was built as follows. Based on the complex amplitude modulation method, HM beams of different orders and ellipticity were generated using an amplitude–type spatial light modulator (SLM) and a radial–helical phase mask. Subsequently, an achromatic Fourier transform lens was illuminated using the HM beams, and the PHMV beams were presented on the focal plane after the Fourier transform lens. The experimental results were consistent with theoretical predictions. Compared with the classical perfect optical vortex (POV) beams, the PHMV beams still retained the property of ring radius independent of topological charge values. The distribution pattern of the PHMV beams can be controlled by the topological charges and elliptical parameters. Furthermore, two important optical properties of the PHMV beams were theoretically elucidated. First, we proved that the PHMV beams carry a fractional order orbital angular momentum (OAM). Second, we found that the complex amplitudes of any two PHMV beams with the same elliptical parameter but different order numbers are orthogonal to each other.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Although diffraction is an inherent transmission property of light, the nondiffracting beams can propagate in free space without spatial spreading. The concept of nondiffracting Bessel beams was first introduced by Durnin in 1987 [1]. Functions A(ϕ) = exp(imϕ) were later found to lead to Bessel beams, where m is their azimuthal order number (i.e., topological charge value). m = 0 corresponds to zero–order Bessel beams, which have a bright central maximum but cannot be regarded as optical vortices. As a kind of classical nondiffracting optical vortex, high–order Bessel beams of m ≥ 1 are characterized by their concentric rings around a dark central core [2,3]. Given that vortex beams contain a helical phase exp(imϕ) and each photon carries an orbital angular momentum (OAM) [4], they are widely used in optical tweezers [5,6], free space optical communication [7,8], optical fiber communication [9], and other scientific fields [10,11]. However, the bright ring diameter of a classical optical vortex beam increases as the topological charge number increases. Such optical property may cause some technical barriers when using the classical optical vortices in some research fields. For example, when multiple vortex beams with different topological charges are coupled to a fiber with a fixed annular refractive index profile, a vortex with a large topological charge may require a fiber with a large core radius [12,13]. Apparently, for fibers with a given core radius, the coupling efficiency drastically decreases for high–order vortices in an OAM–multiplexed fiber communication system. Thus, these varied ring sizes of vortex beams with different topological charges are not desired [12,13]. Therefore, an optical vortex beam independent of topological charge is necessary in some scientific fields.

Ostrovsky et al. (2013) introduced for the first time the concept of perfect vortex beam [14]. Perfect optical vortex (POV) beams are a kind of beams characterized by a bright ring around a central dark hollow with a radius that does not depend on topological charges and with the highest field gradient on its boundary. Such optical characteristic has important applications in the fields of optical tweezers [15], optical communication [16–18], and optical lattices [19]. Theoretically, POV beams are the Fourier transform of Bessel beams [20]. Hence, a perfect vortex can be generated from the focal plane behind a Fourier transform lens when the vortex beam is incident [14]. Vaity et al. (2015) also used the Fourier transform method to generate POVs on the basis of a new phase mask that was designed by combining an axicon and a helical phase [20]. In this manner, the quality of the perfect vortex beam was further improved. Kovalev et al. (2016) generated a perfect optical vortex with the highest intensity on the ring by using the optimal phase element [21]. Moreover, Kovalev et al. also (2017) introduced an elliptical perfect optical vortex beam with the shape of the beam changing from circle to ellipse, a concept that is an extension of POV beams [22]. Subsequently, Li et al. (2018) generated the elliptic perfect optical vortex beams that the beam shape can be adjusted [23]. In 2021, researchers even generated the generalized perfect optical vortices along arbitrary trajectories [24], and the complex perfect optical vortex array with multiple selective degrees of freedom [25].

In the family of nondiffracting beams, different from Bessel beams, Mathieu beams are another family of nondiffracting beams whose optical field distribution is modulated by topological charge values and elliptical parameters, respectively [26]. As a member of Mathieu beams, a diffraction–free helical Mathieu (HM) beam can be obtained when A(ϕ) = cem(ϕ, q) ± i·sem(ϕ, q) [27,28]. Unlike those of classical Bessel vortex beams, the main optical characteristics of HM vortex beams are their elliptical doughnut shape and more abundant beam profiles. On the basis of optical characteristics, HM vortex can be utilized in optical micro–manipulation that accelerates particles along an ellipsis doughnut rather than along a circle, which is a feature of classical Bessel doughnut beams [29]. Moreover, different from Bessel beams with an integer–order OAM, HM beams carry fractional–order OAM [30,31]. Hence, HM beams can be used in the information transmission, guidance, and transportation of microparticles [32].

In this study, we first generated HM beams using the stationary phase method. Then based on Fourier transform properties of HM beams, we proposed a new kind of perfect vortex beam, which was named herein as perfect helical Mathieu vortex (PHMV) beam. The generating mechanism of PHMV beams was first theoretically derived. The key optical properties of the PHMV beams were also studied and characterized. Finally, the PHMV beams were generated using the corresponding experimental system.

2. Theoretical mechanism of constructing PHMV beams

Mathieu beams have especially abundant beam profiles compared with classical Bessel beams; Bessel beams are a special case of Mathieu beams. The family of Mathieu beams can be described by four kinds of angular and radial Mathieu functions. Angular Mathieu functions are illustrated by Eqs. (1)(a)–1(d). Radial Mathieu functions are given by Eqs. (1)(e)–1(h) [26,33]:

$$c{e_{2m}}(\phi ;q) = \sum\limits_{j = 0}^\infty {{A_{2j}}(q)\cos [2j\phi ]} ,$$
$$c{e_{2m + 1}}(\phi ;q) = \sum\limits_{j = 0}^\infty {{A_{2j + 1}}(q)\cos [(2j + 1)\phi ]} ,$$
$$s{e_{2m + 2}}(\phi ;q) = \sum\limits_{j = 0}^\infty {{B_{2j + 2}}(q)\sin [(2j + 2)\phi ]} ,$$
$$s{e_{2m + 1}}(\phi ;q) = \sum\limits_{j = 0}^\infty {{B_{2j + 1}}(q)\sin [(2j + 1)\phi ]} ,$$
$$J{e_{2m}}(\xi ;q) = \sum\limits_{j = 0}^\infty {{A_{2j}}(q)\cosh [2j\xi ]} ,$$
$$J{e_{2m + 1}}(\xi ;q) = \sum\limits_{j = 0}^\infty {{A_{2j + 1}}(q)\cosh [(2j + 1)\xi ]} ,$$
$$J{o_{2m + 2}}(\xi ;q) = \sum\limits_{j = 0}^\infty {{B_{2j + 2}}(q)\sinh [(2j + 2)\xi ]} ,$$
$$J{o_{2m + 1}}(\xi ;q) = \sum\limits_{j = 0}^\infty {{B_{2j + 1}}(q)\sinh [(2j + 1)\xi ]} ,$$
where ϕ within the variable domain 0 ≤ ϕ ≤ 2π represents a polar angle, and ξ is a radial variable in the domain 0 ≤ ξ ≤∞. The order number m is 0, 1, 2, …, where q is the elliptical parameter that controls the ovality of the HM beams. The normalization constants A and B can be derived recursively from the characteristic equation determined by ϕ and q.

In the cylindrical coordinates (ρ, θ, z), the complex field distribution of optical propagation in free space can be theoretically calculated according to the Fresnel diffraction integral:

$$U(\rho ,\theta ,z) = \frac{k}{{i2\pi z}}\exp (ikz)\int_0^\infty {\int_0^{2\pi } {{U_0}} } (r,\phi )\exp \left[ {i\frac{k}{{2z}}({r^2} + {\rho^2})} \right] \times \exp \left[ {i\frac{k}{z}\rho r\cos (\phi - \theta )} \right]r\textrm{d}r\textrm{d}\phi .$$

The initial incident optical field U0 (r, ϕ) is shown as Eq. (3):

$${U_0}(r,\phi ) = {E_0} \cdot A(\phi ,q) \cdot T(r),$$
where E0 represents the input optical amplitude of collimated laser, A(ϕ, q) is the amplitude modulation function, and T(r) is the phase modulation function.

An axicon with a radial phase distribution was invented in 1954 [34]. One of the most successful techniques to produce Bessel beams is to illuminate an axicon by using a collimated laser beam [35–37]. The radial phase distribution of axicon is the key condition for generating nondiffracting beams based on the stationary phase method. In our scheme, T(r) has the following radial phase distribution [35,36]:

$$T(r) = \left\{ {\begin{array}{cc} {\exp ( - ik\Gamma \cdot r)},&\textrm{ }r \le R,\\ {0},&\textrm{ }r > R. \end{array}} \right.,$$
where r=(x2+y2)1/2 is the radial coordinate, R is the radius of input pupil, Γ is a radial controlling parameter, and k is the wavenumber.

In our proposal, similar to the generation of Mathieu beams based on the stationary phase method [38], HM beams are first generated. In Eq. (3), when

$$A(\phi ,q) = {C_m}(q) \cdot c{e_m}(\phi ,q) \pm i{S_m}(q) \cdot s{e_m}(\phi ,q),$$
where cem(ϕ, q) and sem(ϕ, q) are respectively the even and odd type angular Mathieu functions of mth–order with the elliptical parameter q. The positive and negative signs determine the direction of OAM carried by the HM beams. In this article, the positive sign is taken. Cm (q) and Sm(q) are the weighting constants that depend on q. When Eq. (5) is brought into the integral expression $\int_0^{2\pi } {A(\phi )} \exp [i\frac{k}{z}\rho r\cos (\phi - \theta )]\textrm{d}\phi$, the transverse amplitude distribution of the HM beams can be obtained [27,39]:
$$\begin{aligned} &\int_0^{2\pi } {[{C_m}(q)c{e_m}(\phi ,q)\textrm{ + }is{e_m}(\phi ,q)]} \cdot \exp [{i{k_t}r\cos (\phi - \theta )} ]\textrm{d}\phi \\ &= {C_m}(q)J{e_m}(\xi ,q)c{e_m}(\eta ,q)\textrm{ + }i \cdot {S_m}(q)J{o_m}(\xi ,q)s{e_m}(\eta ,q), \end{aligned}$$
where Jem (·) and Jom (·) are respectively the even and odd type radial Mathieu functions of mth–order; and ξ and η are radial and angular components in elliptical cylindrical coordinates, respectively.

Following the above procedure, Eq. (3) is brought into Eq. (2) to obtain the angular integral:

$$\begin{aligned} &U({\rho ,\theta ,z} )= \frac{{\textrm{ - }ik}}{{2\pi z}}\exp (ikz)\exp (i\frac{{k{\rho ^2}}}{{2z}}) \times {E_0}\\ &\int_0^R {[{C_m}(q)J{e_m}(\xi ,q)c{e_m}(\eta ,q)\textrm{ + }i{S_m}(q)J{o_m}(\xi ,q)s{e_m}(\eta ,q)]} \exp \left[ {ik\left( {\frac{{{r^2}}}{{2z}} - \Gamma r} \right)} \right]r\textrm{d}r. \end{aligned}$$

The stationary phase method can simplify and approximately solve the Fresnel diffraction integral with the form $\int {g(r )\textrm{exp[}ikf(r)\textrm{]}}$ when k → ∞ [40]. We set $f(r )= [{{{r^2}} / {(2z)]}} - \Gamma r$ and $g(r )= {C_m}(q)J{e_m}(\xi ,q)c{e_m}(\eta ,q)\textrm{ + }i{S_m}(q)J{o_m}(\xi ,q)s{e_m}(\eta ,q)$. According to ${ {f^{\prime}(r)} |_{r = {r_0}}} = 0$, the stationary–phase point can be obtained when r0 = Γ·z.

When r = r0 = R, there is z = z0 = R/ Γ; thus, the following expression can be obtained:

$$\begin{aligned} U(\rho ,{z_0}) &\approx \frac{{\sqrt {\lambda z} }}{2}{E_0}[{C_m}(q)J{e_m}(\xi ,q)c{e_m}(\eta ,q)\\ &\textrm{ + }i{S_m}(q)J{o_m}(\xi ,q)s{e_m}(\eta ,q)]{k_r}\exp \left( {\frac{{ik{\rho^2}}}{{2z}}} \right) \times \exp \left\{ { - i\left[ {\frac{{k{\Gamma ^2}{z_0}}}{2}} \right] + \frac{\pi }{4}} \right\}, \end{aligned}$$
where kr = k·Γ is the axial wave number. By ignoring the constant coefficient that does not influence the optical field distribution, the field distribution in the plane of the maximum depth of focus after the axicon can be written as:
$$U(\rho ,{z_0}) \propto {C_m}(q)J{e_m}(\xi ,q)c{e_m}(\eta ,q)\textrm{ + }i{S_m}(q)J{o_m}(\xi ,q)s{e_m}(\eta ,q).$$
where Jem(ξ, q)cem(η, q) and Jom(ξ, q)sem(η, q) are respectively the even and odd modes of Mathieu beams. Equation (9) is the expression of the HM beams [27,31]. Apparently, the mth–order HM beams can be obtained via the superposition of the same order even and odd modes of Mathieu beams. HM beams contain two types of odd–order and even–order. The case of odd–order Mathieu function Jem(ξ, q)cem(η, q) and Jom(ξ, q)sem(η, q) in Eq. (9) admits the Bessel function basis expansions; hence, it can be written as follows [41,42]:
$$J{e_m}(\xi ,q)c{e_m}(\eta ,q) = \sum\limits_{j = 0}^\infty {{A_{2j + 1}}} (q)\cos [(2j + 1)\phi ]{J_{2j + 1}}({k_r}\rho ),$$
$$J{o_m}(\xi ,q)s{e_m}(\eta ,q) = \sum\limits_{j = 0}^\infty {{B_{2j + 1}}} (q)\sin [(2j + 1)\phi ]{J_{2j + 1}}({k_r}\rho ).$$

Similarly, the even–order HM admits the Bessel function basis expansions. However, in this article, to serve as an example, we just study the odd–order PHMV beams. Hence, we do not repeat the case of the even–order PHMV beams.

To generate the PHMV beams, we utilized a Fourier transform lens to act as optical Fourier transform. In cylindrical coordinates, the transform for an arbitrary optical field U(ρ, ϕ, z0) into E(σ, ϕ, zf) can be written as:

$$E(\sigma ,\varphi ,{z_f})\textrm{ = }Fourier\{{U(\rho ,\phi ,{z_0})} \}.$$
where σ and φ are the radial and angular components in the frequency domain, respectively.

After substituting Eq. (10) into Eq. (9), one performs Fourier transform based on Eq. (11). Meanwhile, the following Hankel transform pair, i.e., Fourier transform in cylindrical coordinates is applied as follows [43]:

$$Hankel\{ \cos (m\phi ){J_m}(2\pi {\rho _0}\rho ){\kern 1pt} \} = {( - i)^m}\cos (m\varphi )\frac{{\delta (\sigma - {\rho _0})}}{{2\pi {\rho _0}}},$$
$$Hanke{l^{\textrm{ - }1}}\{ \cos (m\varphi )\frac{{\delta (\sigma - {\rho _0})}}{{2\pi {\rho _0}}}\} = {(i)^m}\cos (m\phi ){J_m}(2\pi {\rho _0}\rho ).$$

Comparing the Bessel function of Eq. (10) and (12), one readily knows ρ0 = kr /2π.

Hence, the following equation can be obtained:

$$E = {( - i)^m}\{ \sum\limits_{j = 0}^\infty {{C_m}(q)A_{2j + 1}^{(m)}} (q)\cos [(2j + 1)\varphi ]{\kern 1pt} \textrm{ + }i\sum\limits_{j = 0}^\infty {{S_m}(q)B_{2j + 1}^{(m)}} (q)\sin [(2j + 1)\varphi ]\} \frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}},{\kern 1pt} {\kern 1pt}$$
where δ(·) is the Dirac delta function that actually represents an optical circle; σ = R0 / (λ f) is the frequency domain parameter; R0 is the circle radius; and f is the focal length of the Fourier transform lens. According to the characteristics of δ–function, the circle radius can be obtained by the following equation:
$${R_0}\textrm{ = }\Gamma \cdot f.$$

Equation (14) shows that both the radial controlling parameter Γ and the focal length f can regulate the radius of the dark core of the PHMV beams.

According to Eq. (1), Eq. (13) can also be written as:

$$E = {( - i)^m}\{ {C_m}(q) \cdot c{e_m}(\varphi ,q)\textrm{ + }i{S_m}(q) \cdot s{e_m}(\varphi ,q)\} \frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}.$$

Equation (15) gives a radial Dirac delta function modulated by the HM function. The radius of the optical ring modulated by the HM function is independent of its topological charge; hence, it is a kind of perfect beams similar to the POV beams.

The analysis above revealed the theoretical mechanism required to obtain the PHMV beams. On the basis of Eq. (15), we also numerically simulated the light field distribution of PHMV with different order numbers and elliptical parameters (see Fig. 1). The PHMV beams are characterized by a discontinuous bright ring around a central dark hollow and with the highest field gradient on its boundary. The radius of the dark center is the same for different PHMV beams with a different order number m and an elliptical parameter q. When q = 0, the ring of the mth–order PHMVs is divided equally into 2 m discontinuous arc (Fig. 1). As q increases, the arc splits further, and the arc length of the PHMV beams also becomes unequal. Such optical characteristics show that the PHMV beams can be modulated using two controlling parameters, i.e., topological charge and elliptical parameter. Undoubtedly, two–parameter PHMV beams have more flexibility and more freedom than classical POV with only one controlling parameter. By changing the topological charge and elliptical parameter, we find that the ring diameter of the PHMV beams does not depend on the topological charge values. This optical characteristic is the same as that of classical perfect vortex [14,20].

 figure: Fig. 1.

Fig. 1. Numerical simulation results of PHMV beams with different parameters.

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3. OAM of PHMV

Carrying OAM is the key property of vortex beams. In Ref. [31], the transfer of OAM into trapped microparticles and the inducement of their rotational motion have been observed in HM beams. Owing to the confinement of gradient force in elliptic cores, the trapped particle moves around the elliptic trajectory of the transverse field of HM beams. The OAM transmission can be modulated by adjusting the elliptical parameter q of the transverse field; hence, the particles can be freely manipulated. In this work, the OAM of the introduced PHMV beams was characterized to identify potential applications of the PHMV beams in various fields. The conjugate complex amplitude of the PHMV beams is given by:

$${E^ \ast } = {i^m}\{ \sum\limits_{j = 0}^\infty {C_m^ \ast (q)A_{2j + 1}^{(m )\ast }} (q)\cos [(2j + 1)\varphi ] - i\sum\limits_{j = 0}^\infty {S_m^ \ast (q)B_{2j + 1}^{(m )\ast }} (q)\sin[(2j + 1)\varphi ]\} \frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}},{\kern 1pt} {\kern 1pt}$$

The derivative expression of its amplitude is:

$$\begin{aligned} \frac{{\partial E}}{{\partial \varphi }} &= {( - i)^m}\sum\limits_{j\textrm{ = }0}^\infty {\{ - (2j + 1){C_m}(q){A_{2j + 1}}(q)\sin [(2j + 1)\varphi ]} \\ &\textrm{ + }i \cdot (2j + 1){S_m}(q){B_{2j + 1}}(q)\cos [(2j + 1)\varphi ]\} \frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}. \end{aligned}$$

The OAM of projection Jz onto the optical axis are given by the following expressions [44–46]:

$$\begin{aligned} &{J_z} = \textrm{Im\{ }\int_0^\infty {\int_0^{2\pi } {{E^ \ast }} } \frac{{\partial E}}{{\partial \varphi }}\sigma \textrm{d}\sigma \textrm{d}\varphi \textrm{\} }\\ &= \mathop {\lim }\limits_{\sigma \to \Gamma /\lambda } \sum\limits_{j = 0}^\infty {(\textrm{2}j\textrm{ + }1)\textrm{\{ }C_m^ \ast (q){S_m}(q)A_{2j + 1}^ \ast (q){B_{2j + 1}}(q)\int_0^{2\pi } {{{\cos }^2}[(2j + 1)\varphi ]} \textrm{d}\varphi } \\ &- {C_m}(q)S_m^ \ast (q){A_{2j + 1}}(q)B_{2j + 1}^ \ast (q)\int_0^{2\pi } {{\textrm{si}}{{\textrm n}^2}[(2j + 1)\varphi ]} \textrm{d}\varphi \textrm{\} }\\ &= \frac{\lambda }{{4{\pi ^2}\Gamma }}[(\textrm{2}j\textrm{ + }1)\sum\limits_{j = 0}^\infty {(C_m^ \ast (q){S_m}(q)A_{2j + 1}^ \ast (q){B_{2j + 1}}(q) + {C_m}(q)S_m^ \ast (q){A_{2j + 1}}(q)B_{2j + 1}^ \ast (q))]} , \end{aligned}$$
where Im represents the imaginary part of the complex number.

The beam intensity of the PHMV beams can be written as:

$$\begin{aligned} &I = \int_0^R {\int_0^{2\pi } {{E^ \ast }} } Er\textrm{d}r\textrm{d}\varphi \\ &= \mathop {\lim }\limits_{\sigma \to \Gamma /\lambda } \sum\limits_{j = 0}^\infty {\{ {C_m}(q)C_m^ \ast (q)A_{2j + 1}^ \ast (q){A_{2j + 1}}(q)\int_0^{2\pi } {si{n^2}[(2j + 1)\varphi ]} } \\ &+ {S_m}(q)S_m^ \ast (q){B_{2j\textrm{ + }1}}(q)B_{2j + 1}^ \ast (q)\int_0^{2\pi } {co{s^2}[(2j + 1)\varphi ]} \} \textrm{d}\varphi \\ &= \frac{\lambda }{{4{\pi ^2}\Gamma }}\sum\limits_{j = 0}^\infty {[{C_m}(q)C_m^ \ast (q)A_{2j + 1}^ \ast (q){A_{2j\textrm{ + }1}}(q) + {S_m}(q)S_m^ \ast (q){B_{2j\textrm{ + }1}}(q)B_{2j + 1}^ \ast (q)]} . \end{aligned}$$

In deriving Eq. (19), the following integral formula is also used.

$$\int_\textrm{0}^\infty {\frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\sigma \textrm{d}\sigma } \textrm{ = }\int_\textrm{0}^\infty {\delta \{ \sigma - \Gamma /\lambda , \sigma - \Gamma /\lambda \} \frac{\sigma }{{k_\textrm{r}^2}}\textrm{d}\sigma } \textrm{ = }\frac{{\Gamma /\lambda }}{{k_\textrm{r}^2}}.$$

Therefore, the OAM’s analytic expression of the PHMV beams can be given by the following form:

$$\frac{{{J_z}}}{I} = \frac{{\sum\limits_{j = 0}^\infty {(2j + 1)[C_m^ \ast (q){S_m}(q)A_{2j + 1}^ \ast (q){B_{2j + 1}}(q) + {C_m}(q)S_m^ \ast (q){A_{2j + 1}}(q)B_{2j + 1}^ \ast (q)]} }}{{\sum\limits_{j = 0}^\infty {[{C_m}(q)C_m^ \ast (q)A_{2j + 1}^ \ast (q){A_{2j + 1}}(q) + {S_m}(q)S_m^ \ast (q){B_{2j + 1}}(q)B_{2j + 1}^ \ast (q)]} }}.$$

Equation (21) shows that the OAM of PHMV beams is a function of two physical parameters m and q. Typically, we calculated the OAM of 1th–order EHMOV beams by using Eq. (21). In Eq. (21), when m=1, there is A1 (q = 0) = B1 (q = 0) = 1 and A2j+1 (q = 0) = B2j+1 (q = 0) = 0 for j > 1. However, A2j+1 (q ≠ 0) and B2j+1 (q ≠ 0) are decimals less than 1 [33]. For example, the calculated OAM of 1th–order EHMOV beams is respectively 1, 1.035, 1.149 and 1.214 when q = 0, 1, 2, 3. The calculated results show that the OAM of the PHMV beams (i.e., q ≠ 0) is in fractional–order similar to that of HM beams [30,31], whereas the OAM of POV beams (i.e., q = 0) is in integral–order [15,16,20,47]. To the PHMV beams, the origin of fractional or non–integer OAM states is easier to understand. The PHMV beams with fractional OAM are a generic superposition of light modes with different integral values of j, where j is infinite number. Besides the order number m of the PHMV beams, the elliptical parameter q is also an important factor to consider in regulating the fractional OAM states of the PHMV beams. Other than one–parameter perfect beams that integral OAM states are fully characterized by the OAM index m, the PHMV beams can be regarded as two–parameter prefect beams, i.e., a new kind of perfect vortex with two physical degrees of freedom. Considering the fact that beams with tunable beam profile or fractional–order OAM play a crucial role in quantum information [48], singular optics [49], and optical communications [31,46,50], the PHMV beams are anticipated to have broader applications than POV beams because their OAM can be flexibly controlled using the two parameters m and q. For example, compared with classical vortexes, the use of PHMV beams in optical communication systems is expected to substantially improve the information–coding ability because PHMV beams carrying fractional OAM can provide a new degree of freedom (i.e., the elliptical parameter q) in photonic information applications.

It is worth noting that, the OAM vortex microlaser with ultrafast all–optical switching capability is demonstrated in 2020, which is a kind of laser sources capable of emitting vortex beams of various OAM at room temperature with tunable wavelength [51,52]. The generation of high–speed OAM–tunable vortex is a critical step for the ongoing effort of OAM–SAM (spin angular momentum) – WDM (wavelength division multiplexing) for multidimensional high–capacity information transmitting and processing, hence it shows potential for applications in high–speed optical communication. With these advancements in science and technology, our introduced PHMV beams with adjustable OAM open the way for potential applications in optical communication, imaging, micromanipulation, and sensing.

4. Orthogonality of PHMV beams

In OAM–based Mode–Division Multiplexing (MDM) system, the premise of beam demultiplexing is its orthogonality among beams with multiple spatial mode states. The use of orthogonality beams for multiplexing transmission has substantially improved the information capacity of space communication and optical fiber communication systems [16,17,47,53,54]. Bahari et al. (2021) recently proposed a super multiplexed OAM light source on the basis of the quantum Hall effect element [55]. Some researcher even emphasized that the OAM–DM (DM, Division Multiplexing) technique can support an infinite amount of data by using multiple orthogonal OAM vortex beams [55,56]. In this section, the orthogonality of complex amplitudes of the PHMV beams was analyzed. A scalar product of two PHMV beams with different topological charges, namely, the mth–order and the nth–order PHMV beams, was derived to calculate the orthogonality between the complex amplitudes by using the following equation [44,57]:

$$({{E_{mq}},{E_{nq}}} )\textrm{ = }\int\!\!\!\int {{E_{mq}}E_{nq}^ \ast } \sigma \textrm{d}\sigma \textrm{d}\phi ,$$

For PHMV beams, there are:

$${E_{mq}} = \frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\sum\limits_{j = 0}^\infty {{i^{2j + 1}}} \{{{C_m}(q)A_{2j + 1}^{(m)}(q)\cos [(2j + 1)\varphi ]\textrm{ + }i{S_m}{\kern 1pt} (q)B_{2j + 1}^{(m)}(q)\sin [(2j + 1)\varphi ]{\kern 1pt} } \},$$
$$E_{nq}^ \ast{=} \frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\sum\limits_{k = 0}^\infty {{{({ - i} )}^{2k + 1}}} \{{C_n^ \ast (q)A_{2k + 1}^{(n) \ast }(q)\cos [(2k + 1)\varphi ] - i{\kern 1pt} S_n^ \ast (q)B_{2k + 1}^{(n) \ast }(q)\sin [(2k + 1)\varphi ]{\kern 1pt} } \}.$$

Substituting Eq. (23) into Eq. (22) yields the following equations:

$$\begin{aligned} &({E_{mq}},{E_{nq}}) = \int_\textrm{0}^\infty {\frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\sigma \textrm{d}\sigma } \times \sum\limits_{j = 0}^\infty {\sum\limits_{k = 0}^\infty {} } \\ &\times \textrm{\{ }{C_m}(q)C_n^ \ast (q)A_{2j + 1}^{(m)}(q)A_{2k + 1}^{(n) \ast }(q)\int_0^{2\pi } {\cos [(2j + 1)\varphi ]\cos [(2k + 1)\varphi ]} \textrm{d}\varphi \\ & + {S_m}(q)S_n^ \ast (q)B_{2j + 1}^{(m)}(q)B_{2k + 1}^{(n)}(q)\int_0^{2\pi } {\sin [(2j + 1)\varphi ]\sin [(2k + 1)\varphi ]} \textrm{d}\varphi \textrm{\} }\\ &=\int_\textrm{0}^\infty {\frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\sigma \textrm{d}\sigma } \times \sum\limits_{j = 0}^\infty {\sum\limits_{k = 0}^\infty \pi } \\ &\times \textrm{\{ }{C_m}(q)C_n^ \ast (q)A_{2j + 1}^{(m)}(q)A_{2k + 1}^{(n) \ast }(q) + {S_m}{\kern 1pt} (q)S_n^ \ast (q)B_{2j + 1}^{(m)}(q)B_{2k + 1}^{(n)}(q)\textrm{\} }\textrm{.} \end{aligned}$$

The following orthogonality of Mathieu function can be used to simplify Eq. (24):

$$\int_0^{2\pi } {c{e_m}({\varphi ,q} )} c{e_n}({\varphi ,q} )\textrm{d}\varphi = \pi {\delta _{m,n}},$$
$$\int_0^{2\pi } {s{e_m}({\varphi ,q} )} s{e_n}({\varphi ,q} )\textrm{d}\varphi = \pi {\delta _{m,n}},$$
$$\int_0^{2\pi } {c{e_m}({\varphi ,q} )} s{e_n}({\varphi ,q} )\textrm{d}\varphi = 0,$$
$$\int_0^{2\pi } {s{e_m}({\varphi ,q} )} c{e_n}({\varphi ,q} )\textrm{d}\varphi = 0.$$

By substituting Eq. (25) into Eq. (24), the following results can be obtained:

$$\begin{aligned} &({E_{mq}},{E_{nq}}) = \int_\textrm{0}^\infty {\frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\frac{{\delta \{ \sigma - \Gamma /\lambda \} }}{{{k_r}}}\sigma \textrm{d}\sigma } \times \sum\limits_{j = 0}^\infty {\sum\limits_{k = 0}^\infty {} } \\ &\times \left\{ {{C_m}(q)C_n^ \ast (q)A_{2j + 1}^{(m)}(q)A_{2k + 1}^{(n) \ast }(q)\int_0^{2\pi } {\cos [(2j + 1)\varphi ]\cos [(2k + 1)\varphi ]\textrm{d}\varphi } } \right.\\ &\textrm{ + }{S_m}(q)S_n^ \ast (q)B_{2j + 1}^{(m)}(q)B_{2k + 1}^{(n) \ast }(q)\int_0^{2\pi } {\sin [(2j + 1)\varphi ]\sin [(2k + 1)\varphi ]\textrm{d}\varphi } \\ &- i\left[ {{C_m}(q)S_n^ \ast (q)A_{2j + 1}^{(m)}(q)B_{2k + 1}^{(n) \ast }(q)\int_0^{2\pi } {\cos [(2j + 1)\varphi ]{\kern 1pt} \sin [(2k + 1)\varphi ]{\kern 1pt} \textrm{d}\varphi } } \right.\\ &\left. {\left. { - {S_m}(q)C_n^ \ast (q)B_{2j + 1}^{(m)}(q)A_{2k + 1}^{(n) \ast }(q)\int_0^{2\pi } {\sin [(2j + 1)\varphi ]\cos [(2k + 1)\varphi ]\textrm{d}\varphi } } \right]} \right\}\\ &= \left\{ \begin{array}{l} \frac{{\Gamma /\lambda }}{{k_r^2}} \times [|{C_m}(q\textrm{)}{\textrm{|}^2} + |{S_m}(q\textrm{)}{\textrm{|}^2}]\mathop {}\nolimits^{} \mathop {}\nolimits^{} {\kern 4pt}({m = n} ),\\ \textrm{0}\mathop {}\nolimits^{} \mathop {}\nolimits^{} {\kern 4pt}({m \ne n} ). \end{array} \right. \end{aligned}$$

In deriving Eq. (26), Eq. (20) is used again. Equation (26) shows that two PHMV beams with different topological charges are orthogonal to each other when their elliptical parameters are the same. Moreover, Eq. (26) also depicts that two beams with the same topological charge are not orthogonal. Previous studies reported that asymmetric Bessel modes are orthogonal in different scaling factors, whereas the complex amplitudes of different classes of Lommel modes with different scaling factors are orthogonal to each other since Lommel modes are divided into two classes according to even and odd topological charges [43,57]. The PHMV beams maintain different orthogonality and various beam profiles, which indicate potential applications in some scientific fields.

5. Experiments

On the basis of the scheme introduced in the second section, we generated PHMV beams by using a corresponding experimental system. Being similar to the theoretical generating mechanism, the process of generating the PHMV beams includes two stages. First, HM beams are generated via the stationary phase method. Second, the PHMV beams are then generated using Fourier transform of the HM beams.

The actual experimental system is shown in Fig. 2. A He–Ne laser source (632.8 nm) is expanded and incident on the amplitude–modulation and phase–modulation elements. In the beam expanding and collimating system, the focal lengths of L1 and L2 are 10 mm and 200 mm, respectively, and the pinhole size of the pinhole filter is 25 µm. To improve the quality of PHMV beams, two polarizers whose the transmission axes are parallel are respectively installed before and after the SLM.

 figure: Fig. 2.

Fig. 2. Experimental system for generating PHMV beams.

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According to the amplitude distribution |A(ϕ)| given in Eq. (5), the amplitude map for generating the HM beams in Fig. 3 can be loaded to the computer–controlled amplitude–type spatial light modulator (SLM) (GC–SLM–T–XGA, with 1080 × 768 pixel resolution and 22 µm pitch pixel). Apparently, the radius of input pupil R in Eq. (4) is 16.89 mm in the experimental system. The amplitude distribution |A(ϕ)| depends on the order number m and elliptical parameter q. The radial–helical phase masks (see Fig. 4) are tightly installed after the SLM, which is machined by using a laser Write–Direct System. The radial–helical phase function is denoted as Ф = mod[kΓ+Ψ(ϕ), 2π], which is the sum of an axicon function Τ(r) and a helical phase Ψ(ϕ) of HM beams. In our phase mask, Γ=0.0135. The helical phase can be calculated using the equation of Ψ(ϕ) = arctan[ ± Sm(q)se2m+1(ϕ, q)/Cm(q)ce2m+1(ϕ, q)]. After the phase mask, HM beams can be generated. It is no doubt that the premise to generate HM beams based on the stationary phase method is to introduce a radial phase into the helical phase. A perfect vortex beam is then generated by installing an achromatic Fourier transform lens with a focal length f = 30 cm in the position of z0 = R/Γ after the phase mask. The lens is used to perform the Fourier transform of the HM beams. In the back focal plane after the Fourier transform lens, the generated PHMV beams can be recorded using a CCD camera. The perfect vortex beams with different order and elliptical parameters are shown in Fig. 5. The experimental generation of several instances of PHMV beams in turn confirms the theoretical predictions obtained from Eq. (15). Experimental results demonstrated that the stationary phase method combined with the Fourier transform can generate a kind of new perfect optics vortex, i.e., PHMV beams with two physical degrees of freedom. In particular, a POV is essentially the special case of the PHMV beams when m = 0 and q = 0, which is consistent with the fact that the Bessel beam is a special case of the HM beam. Although the HM vortex beams have an elliptical doughnut shape, the PHMV beams have a circle ring that their radii do not depend on topological charges, similar to the classical POV [27,28].

 figure: Fig. 3.

Fig. 3. Amplitude map under different conditions.

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 figure: Fig. 4.

Fig. 4. Radial–helical phase mask.

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 figure: Fig. 5.

Fig. 5. Recorded images of PHMV beams with the same parameters as those in Fig. 1.

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In 2019, Wang et al. proposed a two–dimensional multiplexing scheme with a ring radius and topological charge of POV beam [47]. Using a similar scheme, our system of generating PHMV beams can be improved to generate other kinds of PHMV beams with different ring radii, topological charges, and beam distributions.

6. Summary

To the best of our knowledge, we theoretically introduced a kind of new perfect vortex beams, namely, PHMV beams derived from both the stationary phase method and the Fourier transform. Via the complex amplitude modulation method and the Fourier transform properties of HM beams, we constructed an experimental system to generate a family of PHMV beams for the first time. The experimental results confirmed that the proposed method can produce a kind of new perfect vortex with two physical degrees of freedom. The PHMV beams using the two–parameter has more abundant beam profiles and two controllable degrees of freedom (order number and ellipticity) than the classical POV with one controllable degree of freedom (order number). Moreover, the key optical characteristics of the PHMV beams, particularly their fraction–order OAM and beam orthogonality, were theoretically investigated. The optical vortices of the PHMV beams have the same size of central dark hollow, phase singularities, and fractional–order OAM. In addition, the complex amplitudes of any two PHMV beams with different order numbers are orthogonal to each other. The theoretical predictions and experimental results obtained are expected to pave a way for their potential applications in some scientific fields, including space optical communications and optical capture and manipulation systems.

Funding

National Natural Science Foundation of China (11674288, 11974314).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at the time but may be obtained from the author upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at the time but may be obtained from the author upon reasonable request.

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Figures (5)
Fig. 1.
Fig. 1. Numerical simulation results of PHMV beams with different parameters.

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Fig. 2.
Fig. 2. Experimental system for generating PHMV beams.

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Fig. 3.
Fig. 3. Amplitude map under different conditions.

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Fig. 4.
Fig. 4. Radial–helical phase mask.

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Fig. 5.
Fig. 5. Recorded images of PHMV beams with the same parameters as those in Fig. 1.

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Equations (39)

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c e 2 m ( ϕ ; q ) = ∑ j = 0 ∞ A 2 j ( q ) cos ⁡ [ 2 j ϕ ] ,
c e 2 m + 1 ( ϕ ; q ) = ∑ j = 0 ∞ A 2 j + 1 ( q ) cos ⁡ [ ( 2 j + 1 ) ϕ ] ,
s e 2 m + 2 ( ϕ ; q ) = ∑ j = 0 ∞ B 2 j + 2 ( q ) sin ⁡ [ ( 2 j + 2 ) ϕ ] ,
s e 2 m + 1 ( ϕ ; q ) = ∑ j = 0 ∞ B 2 j + 1 ( q ) sin ⁡ [ ( 2 j + 1 ) ϕ ] ,
J e 2 m ( ξ ; q ) = ∑ j = 0 ∞ A 2 j ( q ) cosh ⁡ [ 2 j ξ ] ,
J e 2 m + 1 ( ξ ; q ) = ∑ j = 0 ∞ A 2 j + 1 ( q ) cosh ⁡ [ ( 2 j + 1 ) ξ ] ,
J o 2 m + 2 ( ξ ; q ) = ∑ j = 0 ∞ B 2 j + 2 ( q ) sinh ⁡ [ ( 2 j + 2 ) ξ ] ,
J o 2 m + 1 ( ξ ; q ) = ∑ j = 0 ∞ B 2 j + 1 ( q ) sinh ⁡ [ ( 2 j + 1 ) ξ ] ,
U ( ρ , θ , z ) = k i 2 π z exp ⁡ ( i k z ) ∫ 0 ∞ ∫ 0 2 π U 0 ( r , ϕ ) exp ⁡ [ i k 2 z ( r 2 + ρ 2 ) ] × exp ⁡ [ i k z ρ r cos ⁡ ( ϕ − θ ) ] r d r d ϕ .
U 0 ( r , ϕ ) = E 0 ⋅ A ( ϕ , q ) ⋅ T ( r ) ,
T ( r ) = { exp ⁡ ( − i k Γ ⋅ r ) ,   r ≤ R , 0 ,   r > R . ,
A ( ϕ , q ) = C m ( q ) ⋅ c e m ( ϕ , q ) ± i S m ( q ) ⋅ s e m ( ϕ , q ) ,
∫ 0 2 π [ C m ( q ) c e m ( ϕ , q )  +  i s e m ( ϕ , q ) ] ⋅ exp ⁡ [ i k t r cos ⁡ ( ϕ − θ ) ] d ϕ = C m ( q ) J e m ( ξ , q ) c e m ( η , q )  +  i ⋅ S m ( q ) J o m ( ξ , q ) s e m ( η , q ) ,
U ( ρ , θ , z ) =  -  i k 2 π z exp ⁡ ( i k z ) exp ⁡ ( i k ρ 2 2 z ) × E 0 ∫ 0 R [ C m ( q ) J e m ( ξ , q ) c e m ( η , q )  +  i S m ( q ) J o m ( ξ , q ) s e m ( η , q ) ] exp ⁡ [ i k ( r 2 2 z − Γ r ) ] r d r .
U ( ρ , z 0 ) ≈ λ z 2 E 0 [ C m ( q ) J e m ( ξ , q ) c e m ( η , q )  +  i S m ( q ) J o m ( ξ , q ) s e m ( η , q ) ] k r exp ⁡ ( i k ρ 2 2 z ) × exp ⁡ { − i [ k Γ 2 z 0 2 ] + π 4 } ,
U ( ρ , z 0 ) ∝ C m ( q ) J e m ( ξ , q ) c e m ( η , q )  +  i S m ( q ) J o m ( ξ , q ) s e m ( η , q ) .
J e m ( ξ , q ) c e m ( η , q ) = ∑ j = 0 ∞ A 2 j + 1 ( q ) cos ⁡ [ ( 2 j + 1 ) ϕ ] J 2 j + 1 ( k r ρ ) ,
J o m ( ξ , q ) s e m ( η , q ) = ∑ j = 0 ∞ B 2 j + 1 ( q ) sin ⁡ [ ( 2 j + 1 ) ϕ ] J 2 j + 1 ( k r ρ ) .
E ( σ , φ , z f )  =  F o u r i e r { U ( ρ , ϕ , z 0 ) } .
H a n k e l { cos ⁡ ( m ϕ ) J m ( 2 π ρ 0 ρ ) } = ( − i ) m cos ⁡ ( m φ ) δ ( σ − ρ 0 ) 2 π ρ 0 ,
H a n k e l  -  1 { cos ⁡ ( m φ ) δ ( σ − ρ 0 ) 2 π ρ 0 } = ( i ) m cos ⁡ ( m ϕ ) J m ( 2 π ρ 0 ρ ) .
E = ( − i ) m { ∑ j = 0 ∞ C m ( q ) A 2 j + 1 ( m ) ( q ) cos ⁡ [ ( 2 j + 1 ) φ ]  +  i ∑ j = 0 ∞ S m ( q ) B 2 j + 1 ( m ) ( q ) sin ⁡ [ ( 2 j + 1 ) φ ] } δ { σ − Γ / λ } k r ,
R 0  =  Γ ⋅ f .
E = ( − i ) m { C m ( q ) ⋅ c e m ( φ , q )  +  i S m ( q ) ⋅ s e m ( φ , q ) } δ { σ − Γ / λ } k r .
E ∗ = i m { ∑ j = 0 ∞ C m ∗ ( q ) A 2 j + 1 ( m ) ∗ ( q ) cos ⁡ [ ( 2 j + 1 ) φ ] − i ∑ j = 0 ∞ S m ∗ ( q ) B 2 j + 1 ( m ) ∗ ( q ) sin ⁡ [ ( 2 j + 1 ) φ ] } δ { σ − Γ / λ } k r ,
∂ E ∂ φ = ( − i ) m ∑ j  =  0 ∞ { − ( 2 j + 1 ) C m ( q ) A 2 j + 1 ( q ) sin ⁡ [ ( 2 j + 1 ) φ ]  +  i ⋅ ( 2 j + 1 ) S m ( q ) B 2 j + 1 ( q ) cos ⁡ [ ( 2 j + 1 ) φ ] } δ { σ − Γ / λ } k r .
J z = Im{  ∫ 0 ∞ ∫ 0 2 π E ∗ ∂ E ∂ φ σ d σ d φ }  = lim σ → Γ / λ ⁡ ∑ j = 0 ∞ ( 2 j  +  1 ) {  C m ∗ ( q ) S m ( q ) A 2 j + 1 ∗ ( q ) B 2 j + 1 ( q ) ∫ 0 2 π cos 2 [ ( 2 j + 1 ) φ ] d φ − C m ( q ) S m ∗ ( q ) A 2 j + 1 ( q ) B 2 j + 1 ∗ ( q ) ∫ 0 2 π si n 2 [ ( 2 j + 1 ) φ ] d φ }  = λ 4 π 2 Γ [ ( 2 j  +  1 ) ∑ j = 0 ∞ ( C m ∗ ( q ) S m ( q ) A 2 j + 1 ∗ ( q ) B 2 j + 1 ( q ) + C m ( q ) S m ∗ ( q ) A 2 j + 1 ( q ) B 2 j + 1 ∗ ( q ) ) ] ,
I = ∫ 0 R ∫ 0 2 π E ∗ E r d r d φ = lim σ → Γ / λ ⁡ ∑ j = 0 ∞ { C m ( q ) C m ∗ ( q ) A 2 j + 1 ∗ ( q ) A 2 j + 1 ( q ) ∫ 0 2 π s i n 2 [ ( 2 j + 1 ) φ ] + S m ( q ) S m ∗ ( q ) B 2 j  +  1 ( q ) B 2 j + 1 ∗ ( q ) ∫ 0 2 π c o s 2 [ ( 2 j + 1 ) φ ] } d φ = λ 4 π 2 Γ ∑ j = 0 ∞ [ C m ( q ) C m ∗ ( q ) A 2 j + 1 ∗ ( q ) A 2 j  +  1 ( q ) + S m ( q ) S m ∗ ( q ) B 2 j  +  1 ( q ) B 2 j + 1 ∗ ( q ) ] .
∫ 0 ∞ δ { σ − Γ / λ } k r δ { σ − Γ / λ } k r σ d σ  =  ∫ 0 ∞ δ { σ − Γ / λ , σ − Γ / λ } σ k r 2 d σ  =  Γ / λ k r 2 .
J z I = ∑ j = 0 ∞ ( 2 j + 1 ) [ C m ∗ ( q ) S m ( q ) A 2 j + 1 ∗ ( q ) B 2 j + 1 ( q ) + C m ( q ) S m ∗ ( q ) A 2 j + 1 ( q ) B 2 j + 1 ∗ ( q ) ] ∑ j = 0 ∞ [ C m ( q ) C m ∗ ( q ) A 2 j + 1 ∗ ( q ) A 2 j + 1 ( q ) + S m ( q ) S m ∗ ( q ) B 2 j + 1 ( q ) B 2 j + 1 ∗ ( q ) ] .
( E m q , E n q )  =  ∫ ∫ E m q E n q ∗ σ d σ d ϕ ,
E m q = δ { σ − Γ / λ } k r ∑ j = 0 ∞ i 2 j + 1 { C m ( q ) A 2 j + 1 ( m ) ( q ) cos ⁡ [ ( 2 j + 1 ) φ ]  +  i S m ( q ) B 2 j + 1 ( m ) ( q ) sin ⁡ [ ( 2 j + 1 ) φ ] } ,
E n q ∗ = δ { σ − Γ / λ } k r ∑ k = 0 ∞ ( − i ) 2 k + 1 { C n ∗ ( q ) A 2 k + 1 ( n ) ∗ ( q ) cos ⁡ [ ( 2 k + 1 ) φ ] − i S n ∗ ( q ) B 2 k + 1 ( n ) ∗ ( q ) sin ⁡ [ ( 2 k + 1 ) φ ] } .
( E m q , E n q ) = ∫ 0 ∞ δ { σ − Γ / λ } k r δ { σ − Γ / λ } k r σ d σ × ∑ j = 0 ∞ ∑ k = 0 ∞ × {  C m ( q ) C n ∗ ( q ) A 2 j + 1 ( m ) ( q ) A 2 k + 1 ( n ) ∗ ( q ) ∫ 0 2 π cos ⁡ [ ( 2 j + 1 ) φ ] cos ⁡ [ ( 2 k + 1 ) φ ] d φ + S m ( q ) S n ∗ ( q ) B 2 j + 1 ( m ) ( q ) B 2 k + 1 ( n ) ( q ) ∫ 0 2 π sin ⁡ [ ( 2 j + 1 ) φ ] sin ⁡ [ ( 2 k + 1 ) φ ] d φ }  = ∫ 0 ∞ δ { σ − Γ / λ } k r δ { σ − Γ / λ } k r σ d σ × ∑ j = 0 ∞ ∑ k = 0 ∞ π × {  C m ( q ) C n ∗ ( q ) A 2 j + 1 ( m ) ( q ) A 2 k + 1 ( n ) ∗ ( q ) + S m ( q ) S n ∗ ( q ) B 2 j + 1 ( m ) ( q ) B 2 k + 1 ( n ) ( q ) }  .
∫ 0 2 π c e m ( φ , q ) c e n ( φ , q ) d φ = π δ m , n ,
∫ 0 2 π s e m ( φ , q ) s e n ( φ , q ) d φ = π δ m , n ,
∫ 0 2 π c e m ( φ , q ) s e n ( φ , q ) d φ = 0 ,
∫ 0 2 π s e m ( φ , q ) c e n ( φ , q ) d φ = 0.
( E m q , E n q ) = ∫ 0 ∞ δ { σ − Γ / λ } k r δ { σ − Γ / λ } k r σ d σ × ∑ j = 0 ∞ ∑ k = 0 ∞ × { C m ( q ) C n ∗ ( q ) A 2 j + 1 ( m ) ( q ) A 2 k + 1 ( n ) ∗ ( q ) ∫ 0 2 π cos ⁡ [ ( 2 j + 1 ) φ ] cos ⁡ [ ( 2 k + 1 ) φ ] d φ  +  S m ( q ) S n ∗ ( q ) B 2 j + 1 ( m ) ( q ) B 2 k + 1 ( n ) ∗ ( q ) ∫ 0 2 π sin ⁡ [ ( 2 j + 1 ) φ ] sin ⁡ [ ( 2 k + 1 ) φ ] d φ − i [ C m ( q ) S n ∗ ( q ) A 2 j + 1 ( m ) ( q ) B 2 k + 1 ( n ) ∗ ( q ) ∫ 0 2 π cos ⁡ [ ( 2 j + 1 ) φ ] sin ⁡ [ ( 2 k + 1 ) φ ] d φ − S m ( q ) C n ∗ ( q ) B 2 j + 1 ( m ) ( q ) A 2 k + 1 ( n ) ∗ ( q ) ∫ 0 2 π sin ⁡ [ ( 2 j + 1 ) φ ] cos ⁡ [ ( 2 k + 1 ) φ ] d φ ] } = { Γ / λ k r 2 × [ | C m ( q ) | 2 + | S m ( q ) | 2 ] ⁡ ⁡ ( m = n ) , 0 ⁡ ⁡ ( m ≠ n ) .
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