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Demonstrating Arago–Fresnel laws with Bessel beams from vectorial axicons

Keshaan Singh, Wagner Tavares Buono, Sabino Chavez-Cerda, and Andrew Forbes

Open Access Open Access

Abstract

Two-dimensional Bessel beams, both vectorial and scalar, have been extensively studied to date, finding many applications. Here we mimic a vectorial axicon to create one-dimensional scalar Bessel beams embedded in a two-dimensional vectorial field. We use a digital micro-mirror device to interfere orthogonal conical waves from a holographic axicon, and study the boundary of scalar and vectorial states in the context of structured light using the Arago–Fresnel laws. We show that the entire field resembles a vectorial combination of parabolic beams, exhibiting dependence on solutions to the inhomogeneous Bessel equation and asymmetry due to the orbital angular momentum associated rotational diffraction. Our work reveals the rich optical processes involved at the interplay between scalar and vectorial interference, opening intriguing questions on the duality, complementarity, and non-separability of vectorial light fields.

© 2021 Optical Society of America

1. INTRODUCTION

Bessel beams are so-called non-diffracting [1,2] and self-healing [3–6] forms of structured light [7], solutions to the paraxial wave equation in cylindrical symmetry, and have found use in optical trapping [8,9], communication [10–14], imaging microscopy [15], laser machining [16], and quantum studies [17–23] and have even been produced as matter waves [24–26]. It is common to produce Bessel beams by the interference of plane waves traveling on a cone: conical waves [27]. The first creation of such beams used amplitude masks in the form of annular rings, exploiting the far field pattern of the desired Bessel beam, which is created after a Fourier lens. Proposals exist to generalize this approach to tailored sources on a ring [28]. Scalar 2D Bessel beams have since been created using holographic annular rings [29], refractive [30] and holographic [31] axicons, and interferometers [32] and inside laser resonators [33–38]. 1D scalar Bessel beams have also been created by using a four F-Fourier-transformer setup and their propagation characteristics investigated. Even though they are not non-diffracting, they exhibit discrete diffraction similar to that of leaky modes in fiber arrays [39–41]. Vectorial 2D Bessel beams have been created by geometric phase-based liquid crystals [42,43], by spin–orbit interactions [44], and metasurfaces [45], while using space–time control has seen the creation of 1D non-diffracting Bessel beams [46,47]. Here and in the general literature, when one speaks of Bessel beams, the laboratory generation is always some approximation, often in the form of Bessel–Gaussian beams [48].

Young’s double slit experiment, as first conceived and performed, applies for scalar wave fields resulting in an interference pattern of bright and dark fringes. Arago and Fresnel considered the vector characteristics of the field by studying the effect of polarization. They introduced identical polarizers with the same plane of polarization in front of each of the slits and observed the fringes described by Young. Then, in one of the slits, they rotated the polarizer until it reached the perpendicular plane of polarization with respect to the other slit and reported that, at this position, the interference pattern disappeared. From their observations, they stated four laws known as Arago–Fresnel laws [49]: (1) two waves, linearly polarized in the same plane, can interfere; (2) two waves, linearly polarized with perpendicular polarizations, cannot interfere; (3) two waves, linearly polarized with perpendicular polarizations, if derived from perpendicular components of unpolarized light and subsequently brought into the same plane, cannot interfere; and (4) two waves, linearly polarized with perpendicular polarizations, if derived from the same linearly polarized wave and subsequently brought into the same plane, can interfere.

The interference fringes referred to in these laws are in the intensity of the observed light, yet light has many degrees of freedom (DoFs) that can be controlled, for what is now referred to as structured light [7]. Amplitude/intensity is the most well-known DoF of light, with interference fringes observed and understood at least since the time of Thomas Young [50]. Here two waves with a relative phase ($\theta$) difference, $\Delta \theta = {\theta _2} - {\theta _1}$, add to produce a modulation in intensity: $I = |A\exp (i\omega t - ikz)[\exp (- i{\theta _1}) + \exp (- i{\theta _2}{)|^2} = 4{A^2}\mathop {\cos}\nolimits^2 (\Delta \theta /2)$, where $\omega , t, k, z$ and $A$ are, respectively, the temporal frequency, time, spatial frequency, propagation distance, and amplitude. Originally, $\Delta \theta$ was the path length difference described by Thomas Young, but can be a general phase delay in any coordinate for linear, radial, or azimuthal fringes [51]. In the context of interference, one may legitimately ask: fringes in what? Interference of differing frequencies give rise to the familiar beating (fringes in time) observed in electronic and optical signals, angular slits produce fringes in orbital angular momentum (OAM) [52], while fringes in polarization have been known and understood for at least 100 years, following seminal work by Robert Wood [53]. One can understand the polarization fringes by again considering the superposition of two waves, this time of orthogonal (e.g., $\hat{\textbf{x}}$ and $\hat{\textbf{y}}$) polarizations: $\exp (i\omega t - ikz)[{A_{0x}}\exp (- i{\theta _1})\hat{\textbf{x}} + {A_{0y}}\exp (- i{\theta _2})\hat{\textbf{y}}]$. One can show that the components of the final field (${E_x}$ and ${E_y}$) may be written as [54]

$$\frac{{E_x^2}}{{A_{0x}^2}} + \frac{{E_y^2}}{{A_{0y}^2}} - 2\frac{{{E_x}{E_y}}}{{{A_{0x}}{A_{0y}}}}\cos \Delta \theta = \sin \Delta \theta ,$$
which is the well-known equation for the polarization ellipse and represents a general polarization state. Once again, $\Delta \theta$ is the phase difference, which can be spatially varying, so that by changing the relative phases of the waves, one can construct a polarization varying map: fringes in polarization. This topic has a rich history, and the reader is referred to further reading on the topic (see, for example, Refs. [49,55,56] for a more general theoretical treatment and Ref. [51] for a modern experimental treatment).

If instead of slits the experiment is done with two pinholes illuminated with light with the same polarization (scalar case), and a lens is placed at its focal distance, the result is the same: an intensity interference cosine pattern is created just after the lens that transforms the spherical incoming wavefronts into two coherent tilted plane waves, say $\exp [i({k_t}x + {k_z}z)]$ and $\exp [i(- {k_t}x + {k_z}z)]$ that interfere to produce a field with intensity proportional to $|\cos ({k_t}x{)|^2}$. If the pinholes are rotated with respect to their midpoint in such a way that they produce an annular slit, the resulting wave field is proportional to the integral

$$E{(r)_A} \propto {e^{i{k_z}z}}\int_0^\pi \cos ({k_t}r\cos \varphi){\rm d}\varphi ,$$
where the magnitudes of the respective wave vectors of the plane waves are ${k^2} = k_t^2 + k_z^2$ and ${r^2} = {x^2} + {y^2}$. The integral on the right is proportional to the integral representation of the Bessel function of the first kind ${J_0}({k_t}r)$. In other words, the resulting wave field is a Bessel beam. Now, one wonders, what would be the result if the pinholes are illuminated with orthogonal polarizations? The resulting annular slit is composed of two semi-circumferences with orthogonal polarization. Analyzing the field of only one half of the annular slit with one of the polarizations (scalar case), the wave field will now be of the form
$${E_{\frac{1}{2}A}}(x,y) \propto {e^{i{k_z}z}}\int_0^\pi {e^{i{k_t}(x\cos \varphi + y\sin \varphi)}}{\rm d}\varphi ,$$
which is not a Bessel pattern, as can be seen in Fig. 1. Notice that the far field intensity pattern is very similar to that of parabolic non-diffracting beams [57]. The reason for this is that the latter have their spectrum in a semi-circumference for the stationary case and split into two semi-circumferences for the non-stationary case. Strictly speaking, the small difference is due to the phase of the spectrum of the parabolic beams near each extreme of the arc. Also, even if the initial field is not symmetrical, Fig. 1(a), its diffracted far field has symmetry along the transverse $ x $ and $ y $ axes, Fig. 1(b). If a linear azimuthal phase is included in the input field, the symmetry remains only along the $ x $ axis, Fig. 1(c). As will be seen below, along the vertical axis, the 1D profile is still determined by the Bessel function.
 figure: Fig. 1.

Fig. 1. (a) Half annular slit binary amplitude. (b) Diffracted transverse intensity pattern in the far field of the half annular slit. (c) Diffracted transverse intensity pattern in the far field of the half annular slit with an azimuthal phase $\exp (i3\varphi)$.

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Here we use the Arago–Fresnel laws to understand the interference of conical waves created by a segmented vectorial axicon, that is, when one half of the axicon produces a conical wave orthogonal (in polarization) to that produced by the other half of the axicon. In the context of conical interference, we may imagine that the orthogonally polarized contributions will not produce fringes in intensity, but will produce fringes in polarization structure. Further, the situation is conceptually equivalent to a conventional axicon but with a vectorial aperture, allowing only half the field of each polarization component through (e.g., the left portion of the axicon passes the left-handed polarization but blocks the right-handed component, and vice versa), resembling a vectorial version of a scalar parabolic field. We show that the resulting vectorial structure has embedded in it a scalar Bessel beam in one dimension, along the plane of intersection of the two conical waves, while polarization projections reveal the 2D scalar Bessel fields of each polarization component of the field. We find that fields resulting from vectorial axicons endowed with OAM are represented by perturbed Bessel functions along the plane of intersection, while the orthogonal plane exhibits dependence on the Struve functions. Our work contributes to the topical field of structured light, probing the optical processes at play between the boundary of scalar and vectorial light.

2. CONCEPT

It is known that the interference of light waves diffracting from a ring produces, in the far field, an electric field that varies radially according to the Bessel function of the first kind [2]. The oscillatory behavior in the radial direction manifests as a result of the path length induced phase difference of rays traveling along a cone after a Fourier lens. Conversely, this type of field can be produced directly (after some well-defined distance from the near field) through the use of a conical lens known as an axicon. The axicon also allows for azimuthally varying phase profiles associated with OAM to be simultaneously introduced along with radial variation. Let us consider an axicon that acts with spatially inhomogeneous polarization dependence, for instance, acts only on right-circularly polarlized (RCP) light for $y \gt 0$ (top half) and left-circularly polarized (LCP) light for $y \le 0$ (bottom half), which can be characterized by the transmission function

$$t(\rho ,\varphi) = \left\{{\begin{array}{* {20}{c}}{{e^{- i(\alpha (n - 1)k\rho + l\varphi)}}\langle R|}&\;\;0 \le \varphi \lt \pi ,\\{{e^{- i(\alpha (n - 1)k\rho + l\varphi)}}\langle L|}&\;\;\pi \le \varphi \lt 2\pi ,\end{array}} \right.$$
where $\rho$ and $\varphi$ are, respectively, the radial and azimuthal transverse coordinates, while $\alpha$ is the axicon angle, $n$ is the refractive index, $k = 2\pi /\lambda$ is the wave vector, and $l$ is the topological charge. $\langle R,L|$ represent the transpose of the RCP and LCP Jones matrices. Considering diffraction from each half, the intensity can be shown to be proportional to [31]
$${I_{R(L)}}(x,y,z) \propto z{\left| {\int\limits_{0(\pi)}^{\pi (2\pi)} e^{i{k_t}(x\cos \varphi + y\sin \varphi) + l\varphi}{\rm d}\varphi} \right|^2} ,$$
where ${k_t} = \alpha (n - 1)k$ is the transverse wave vector component introduced by the axicon. Now if we consider only looking along the splitting line (i.e., $y = 0$) and the perpendicular “anti-splitting” line (i.e., $x = 0$), the intensity proportionality (for the $l = 0$ case) reduces to
$${I_{R(L)}}(x,0,z) \propto z{\pi ^2}J_0^2({k_t}x) ,$$
$${I_{R(L)}}(0,y,z) \propto z{\pi ^2}{\left| {{J_0}({k_t}y) \pm i{\textbf{H}_0}({k_t}y))} \right|^2} ,$$
where ${J_n}()$ is the $ n $th order Bessel function of the first kind, and ${\textbf{H}_n}()$ is the $ n $th order Struve function. Here we can see a resulting 1D Bessel intensity profile along the splitting line of the axicon, due to the light diffracted across the split being able to conically interfere with itself. Along the anti-splitting line, intensity is modulated by the Struve function. For the $l = 3$ case, we get
$$\begin{split}{I_{R(L)}}(x,0,z) &\propto | - i\pi {J_3}({k_t}x) \mp 16(({k_t}x{)^{- 2}}\cos ({k_t}x) + {({k_t}x)^{- 3}})\\&\quad - 6({k_t}x{)^{- 1}}\sin (({k_t}x{))|^2} ,\end{split}$$
$$\begin{split}{I_{R(L)}}(0,y,z) &\propto | \pm 8i/3 - \pi {J_3}({k_t}y) - \mp i\pi ({\textbf{H}_{- 1}}({k_t}y)\\&\quad + 4({k_t}y{)^{- 1}}{\textbf{H}_2}({k_t}y{))|^2} .\end{split}$$
Here we note that the function along the splitting line is asymmetric along $y$ due to the rotating diffraction of light associated with OAM, while the anti-parallel case once again shows dependence on Struve functions. Finding a description of the diffractive behavior for general $l$ may prove difficult due to the cylindrically asymmetric nature of the Cartesian symmetry breaking.

3. EXPERIMENTAL SETUP

To generate vectorial fields from polarization dependent axicons, we used the experimental setup shown in Fig. 2. A horizontally polarized Gaussian beam was produced by a 633 nm He–Ne laser, then expanded and collimated using lenses EL (${f} = {2}\;{\rm mm}$) and CL (${f} = {250}\;{\rm mm}$), respectively. A half-wave plate (HWP) was used to rotate the polarization state to diagonal before the expanded beam passed through a polarizing beam splitter (PBS). The horizontally polarized component of the expanded beam was used to illuminate the screen of a digital micro-mirror device (DMD [Texas Instruments DLP6500]). The vertically polarized component, exiting the reflected port of the PBS, was directed by a mirror (M) to overlap the horizontal component at the DMD screen, with an angle of ${\approx} {1.5^ \circ}$ between the two expanded beams. The $125 \times$ magnification, present in the expansion and collimation system, served to project plane waves of approximately constant amplitude (in each polarization component) onto the DMD screen. The DMD was simultaneously addressed (multiplexed) with two holograms ${H_H}$ and ${H_V}$ defined by the following functions [58]:

$$\begin{split}{H_{H/V}}(x,y) &= \frac{1}{2} + \frac{1}{2}\left({\cos (2\pi {G_{H/V}}(x,y) + \pi {\Phi _{H/V}}(x,y))} \right.\\&\quad \left. {- \cos ({A_{H/V}}(x,y))} \right) ,\end{split}$$
$${G_{H/V}}(x,y) = xg_{H/V}^x + yg_{H/V}^y ,$$
$${\Phi _{H/V}}(x,y) = \frac{{\arg ({U_{H/V}}(x,y))}}{\pi} ,$$
$${A_{H/V}}(x,y) = \frac{{{\arcsin} (|{U_{H/V}}(x,y)|)}}{\pi} ,$$
where $U(x,y)$ is a desired complex field encoded into the $+{1}$ diffraction order of the grating defined by spatial carrier frequencies $g_{H/V}^x$ and $g_{H/V}^y$. Since the zeroth diffraction orders of the H and V (horizontally and vertically polarized) components leave the DMD at the same ${\approx} {1.5^ \circ}$ incidence angle, the grating frequencies $g_{H/V}^{x/y}$ can be chosen such that the $+{1}$ diffraction orders of holograms ${H_H}$ and ${H_V}$ from the horizontal and vertical diffraction components spatially overlap [59]. Conventionally, the field before an axicon has a Gaussian profile the width of which will affect the propagation dynamics of the formed Bessel beam, since we are illuminating the DMD with a plane wave we chose to encode the Gaussian profile into the axicon (i.e., $|U(x,y)| = {e^{- ({x^2} + {y^2})/w_0^2}}$, where ${w_0}$ is the beam waist). The axicon was encoded into the phase profile according to $\arg (U(x,y)) = {k_t}\sqrt {{x^2} + {y^2}} + l\,{\arctan} (y/x)$. Directly after the DMD a quarter-wave plate (${\rm QWP}_1$) was used to convert the generating polarizations according to $H \to {\rm LCP}$ and $V \to {\rm RCP}$. The plane at the DMD was imaged through a 4f imaging system including an iris used to filter out the desired combined diffraction order. Additionally a second ${\rm QWP}_2$ and a linear polarizer (LP) were placed in the imaging system to project four Stokes intensities (right, left, horizontally, and diagonally polarized) on to a CCD (FLIR Grasshopper3). The CCD was then translated along the beam axis, using a rail, from the image plane of the DMD ($z = 0 \; {\rm mm}$) to $z = 360 \; {\rm mm}$ to investigate the formation and propagation of the resulting fields.
 figure: Fig. 2.

Fig. 2. Diagram showing how the separation of orthogonal polarization states by a PBS followed by their recombination using a DMD can be used to replicate a vectorial axicon. Inset on the top left shows a representative example of the encoded multiplexed binary hologram, and insets on the bottom right show exemplar CCD images captured at different propagation distances.

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4. RESULTS

The conventional plane of interest when generating non-diffracting beams from an axicon is $z = 0.5{z_{\max }} = 0.5k{w_0}/{k_t}$, where ${z_{\max }}$ is the valid range of the generated field. In our case, we chose $\alpha = 0.02\; {\rm rad}$ (allowing for paraxial treatment), $n = 1.1$, and ${w_0} = 1 \; {\rm mm}$, which results in our plane of interest at $z = 250 \; {\rm mm}$. In Figs. 3(a) and 3(b), we display the appropriately sectioned intensity profiles at the $z = 0.5{z_{\max }}$ plane for the cases of beams generated from vectorial axicons with $l = 0$ and $l = 3$, respectively. It should be noted that, for practical reasons, the DMD is orientated at 45°—the consequence of this is the proportional rotation of the coordinate system in Section 2 in the experimental results. In both cases, we can see good agreement between the theory described in Section 2 and the measured results along the splitting line, while along the anti-splitting line, general qualitative agreement is clear. Note that the total measured intensity corresponds to the sum of the intensities of the diffracted orthogonally polarized field components described by Eqs. (6)–(9).

 figure: Fig. 3.

Fig. 3. Plots showing the theoretical and measured “splitting” and “anti-splitting” intensity cross sections of non-diffracting beams generated by a vectorial axicon for (a) $l = 0$ and (b) $l = 3$ at $z = 250 \; {\rm mm}$ (complete measured transverse intensity profiles are included as insets).

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To verify the physical origin of the observed functional behavior seen in the sectioned profiles, we measured the intensity of the RCP and LCP projections of the generated beam from the axicon plane (i.e., $z = 0 \; {\rm mm}$) to $z = 320 \; {\rm mm}$. Additionally, numerical simulations were generated for comparison using the technique outlined in [60]. The results are shown in Fig. 4.

 figure: Fig. 4.

Fig. 4. CCD images showing the formation of non-diffracting polarization components produced by a vectorial axicon for (a) $l = 0$ and (b) $l = 3$ (simulated comparisons are included as insets).

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For the $l = 0$ case, we can see how the diffraction across the splitting line occurs immediately with propagation from the axicon plane, thus allowing the interference of conical (now triangular in one dimension) rays along the splitting line. This interference leads to the formation of identical Bessel structures along the splitting line for each of the orthogonal components, resulting in a scalar 1D Bessel beam. When OAM ($l = 3$) is introduced, the diffraction across the splitting line is accompanied by the rotational energy/intensity flow imparted by the azimuthal phase factor. The resulting asymmetric structures are non-diffracting over the valid range and occupy opposing ends of the splitting line, resulting in a 1D vectorial structure resembling a perturbed Bessel function along the splitting line. It was also observed that the rotation direction is changed when $l = - 3$. Videos showing the formation of these beams in total intensity and through orthogonal polarization projections can be found in Visualization 1. To investigate the structure of the complete 2D vectorial field, we acquired sets of reduced Stokes intensity measurements [61] at the $z = 0.5{z_{\max }}$ plane and reconstructed the state of polarization (SOP).

From the results presented in Fig. 5, we can see that the transverse profiles of the optical fields generated by diffraction through a vectorial axicon are indeed structured in polarization. Naturally, the diagonal polarization projections represent conventional scalar Bessel beams for both cases due to this projection “seeing” a complete axicon, while the horizontal projection contains a discontinuity for $l = 0$, as it sees an axicon with a constant $0.5\pi$ phase difference between the halves; in the $l = 3$ case, this distorts the projection.

 figure: Fig. 5.

Fig. 5. SOP and Stokes intensities measured at $z = 0.5{z_{\max }}$ for (a) $l = 0$ and (b) $l = 3$ (red [blue] ellipses represent RCP [LCP]).

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5. DISCUSSION AND CONCLUSION

This work can be framed in the context of classical entanglement [62–69], where the non-separability of vectorial structured light resembles that of its quantum entangled analog. Here the initial beam is fully vectorial but spatially separated into two distinct parts, simulating a prior reported situation that revealed counterintuitive non-separability dynamics [70], where the local non-separability changed dramatically even though the global non-separability remained the same. In our case, the situation can be thought of as analogous to Young’s double slit experiment, but with a vectorial twist courtesy of the Arago–Fresnel laws. Each half of the axicon is a “slit,” passing through a polarization state orthogonal to the other. Orthogonal polarization states do interfere, but the fringes are seen in the polarization structure itself and not the intensity. At the interface of the two waves, we observe the Bessel beam; orthogonal to this plane, we do not. We imagine that this simple arrangement may be useful to unravel the vectorial dynamics at play in interference when viewed from a visibility and distinguishability perspective [71].

The beams we present simultaneously demonstrate the first, second, and fourth Arago–Fresnel laws. The first (second) law is verified by the presence (lack) of intensity fringes along (across) the interface due to the identical (orthogonal) planes of polarization of overlapping waves. The fourth law manifests in the polarization fringes across the interface, which occur due to initially orthogonal planes of polarization being projected into an identical plane during Stokes intensity measurements.

In conclusion, we have observed how the diffraction dynamics introduced by a vectorial axicon lead to 1D scalar and vector Bessel structures embedded into non-diffracting vector fields, due to self-interference from polarization similar conical waves. We have also observed how OAM embedded into the vectorial axicons interact with the Cartesian symmetry breaking, resulting in spatially periodic perturbations to the generated 1D structures. We have framed the work in the context of the well-known Arago–Fresnel laws, testing the interplay between scalar and vectorial processing in the creation of structured light fields.

Funding

Department of Science and Innovation, South Africa; Council for Scientific and Industrial Research, South Africa.

Disclosures

The authors declare no conflicts of interest.

Data Availability

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

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Supplementary Material (1)

NameDescription
Visualization 1       Videos depicting the total intensity and right-/left-circular projections of beams produced by a segmented axicon over a set propagation distance.

Data Availability

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

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Figures (5)
Fig. 1.
Fig. 1. (a) Half annular slit binary amplitude. (b) Diffracted transverse intensity pattern in the far field of the half annular slit. (c) Diffracted transverse intensity pattern in the far field of the half annular slit with an azimuthal phase $\exp (i3\varphi)$.

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Fig. 2.
Fig. 2. Diagram showing how the separation of orthogonal polarization states by a PBS followed by their recombination using a DMD can be used to replicate a vectorial axicon. Inset on the top left shows a representative example of the encoded multiplexed binary hologram, and insets on the bottom right show exemplar CCD images captured at different propagation distances.

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Fig. 3.
Fig. 3. Plots showing the theoretical and measured “splitting” and “anti-splitting” intensity cross sections of non-diffracting beams generated by a vectorial axicon for (a) $l = 0$ and (b) $l = 3$ at $z = 250 \; {\rm mm}$ (complete measured transverse intensity profiles are included as insets).

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Fig. 4.
Fig. 4. CCD images showing the formation of non-diffracting polarization components produced by a vectorial axicon for (a) $l = 0$ and (b) $l = 3$ (simulated comparisons are included as insets).

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Fig. 5.
Fig. 5. SOP and Stokes intensities measured at $z = 0.5{z_{\max }}$ for (a) $l = 0$ and (b) $l = 3$ (red [blue] ellipses represent RCP [LCP]).

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Equations (13)

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E x 2 A 0 x 2 + E y 2 A 0 y 2 − 2 E x E y A 0 x A 0 y cos ⁡ Δ θ = sin ⁡ Δ θ ,
E ( r ) A ∝ e i k z z ∫ 0 π cos ⁡ ( k t r cos ⁡ φ ) d φ ,
E 1 2 A ( x , y ) ∝ e i k z z ∫ 0 π e i k t ( x cos ⁡ φ + y sin ⁡ φ ) d φ ,
t ( ρ , φ ) = { e − i ( α ( n − 1 ) k ρ + l φ ) ⟨ R | 0 ≤ φ < π , e − i ( α ( n − 1 ) k ρ + l φ ) ⟨ L | π ≤ φ < 2 π ,
I R ( L ) ( x , y , z ) ∝ z | ∫ 0 ( π ) π ( 2 π ) e i k t ( x cos ⁡ φ + y sin ⁡ φ ) + l φ d φ | 2 ,
I R ( L ) ( x , 0 , z ) ∝ z π 2 J 0 2 ( k t x ) ,
I R ( L ) ( 0 , y , z ) ∝ z π 2 | J 0 ( k t y ) ± i H 0 ( k t y ) ) | 2 ,
I R ( L ) ( x , 0 , z ) ∝ | − i π J 3 ( k t x ) ∓ 16 ( ( k t x ) − 2 cos ⁡ ( k t x ) + ( k t x ) − 3 ) − 6 ( k t x ) − 1 sin ⁡ ( ( k t x ) ) | 2 ,
I R ( L ) ( 0 , y , z ) ∝ | ± 8 i / 3 − π J 3 ( k t y ) − ∓ i π ( H − 1 ( k t y ) + 4 ( k t y ) − 1 H 2 ( k t y ) ) | 2 .
H H / V ( x , y ) = 1 2 + 1 2 ( cos ⁡ ( 2 π G H / V ( x , y ) + π Φ H / V ( x , y ) ) − cos ⁡ ( A H / V ( x , y ) ) ) ,
G H / V ( x , y ) = x g H / V x + y g H / V y ,
Φ H / V ( x , y ) = arg ⁡ ( U H / V ( x , y ) ) π ,
A H / V ( x , y ) = arcsin ( | U H / V ( x , y ) | ) π ,
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