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Propagation characteristics of orbital angular momentum and its time evolution carried by a Laguerre-Gaussian beam in supersonic turbulent boundary layer

Lingfei Xu, Yu Xin, Zhichao Zhou, Tianrong Ren, and Bing Han

Open Access Open Access

Abstract

In a supersonic turbulent boundary layer, the characteristics of the refractive index cannot be described well using a statistical approach. Therefore, we theoretically and numerically investigate the evolution of the total orbital angular momentum (OAM) and the spiral spectrum of the OAM mode with respect to space and time when a Laguerre-Gaussian beam propagates through a supersonic turbulent boundary layer. The numerical results of a shock-wave-turbulent boundary layer interaction flow in a supersonic 24° compression ramp with a 2.9 Ma free stream are used. The distribution of the refractive index is obtained by direct numerical simulation of the Navier–Stokes equations. Our results show that the total OAM varies and the spiral spectrum of the OAM mode is asymmetric.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

The orbital angular momentum (OAM) of light propagating in turbulence has become a prevalent research topic owing to its wide range of applications in optical communication or lidar. Recently, numerous high-speed aircraft have been equipped with laser transmitters and receivers, using lasers as a means of communication or detection. Thereby, laser propagation over a very short distance (in the supersonic turbulent boundary layer) has become a new research hotspot [1]. The evolution of the total OAM and its spiral spectrum are two main problems. Sanchez and Oesch [2,3] concluded that turbulence can create well-defined quantum states in photons and that non-zero OAM photons are created in pairs by turbulent flow. They inferred that the total OAM of light is conserved along the direction of propagation. The distribution of refractive index in atmospheric turbulence is often regarded as a random field. Thus, Aksenov and Pogutsa [4] examined the conservation of the total OAM when the vortex beam propagates through atmospheric turbulence using a statistical method.

According to the statistical approach, supposing that the distribution of the refractive index is statistically homogeneous and delta correlated in the direction of propagation (Markov approximation), the power spectrum model of refractive index fluctuation can be described as a 2D Kolmogorov spectrum or Tatarskii spectrum [5–7]. Charnotskii [8] published an excellent study that systematically discussed the ensemble average of the total OAM when light propagates in atmospheric turbulence, and concluded that the ensemble average of the total OAM is conserved in random fields. Numerous studies [9–13] have been carried out on the ensemble average of the total OAM and the spiral spectrum OAM by introducing a spatial power spectrum of refractive index fluctuation. However, the nature of a supersonic turbulent boundary layer is quite different from that of atmospheric turbulence.

Atmospheric turbulence is approximately unbounded (far from the boundary). However, there exists a physical boundary in a supersonic turbulent boundary layer. Moreover, the Reynolds number of a supersonic turbulent boundary layer is finite. Frisch demonstrated that the turbulence can be dealt with using a statistical method only if the flow is both far from boundary and has a very high Reynolds number [14]. Evidently, such conditions cannot be satisfied in the case of a supersonic turbulent boundary layer, which means that the analytical form of the power spectrum model of refractive index fluctuation cannot be deduced.

The Navier-Stokes (NS) equations are the basic equations that describe the motion of the flow. With the help of a numerical method, the physical quantities at any position and any time can be obtained. The numerical results show that structures still exist in the turbulence, but unlike eddies in Kolmogorov cascades theory, the corresponding structures are spatially correlated in 3D and have definite shape [15,16]. Based on the advantage of fluid mechanics, the research area on optical effects caused by a supersonic turbulent boundary layer has developed, which is known as aero-optics [17–19]. In the numerical simulation of the NS equations, the initial and boundary conditions are given by determinist values, so that the results are determinist. Considering the accuracy of the numerical calculation, the results of a direct numerical simulation (DNS) are chosen [20,21].

On account of the reasons described above, the numerical analyses are preferred in this paper. The numerical method proposed by Xu et al. is utilized to calculate the complex amplitude of the scattered beam. They gave a numerical approach to calculate the Rytov perturbation [22] with the help of generalized convolution FFT (GCV-FFT). To be consistent with the numerical method, the deductions of the expression of the total OAM change and the spiral spectrum of OAM are based on the Rytov approximation in Section 2. In Section 3, the detailed process of numerical calculation is presented. The second order Rytov approximation is used. In Section 4, a shock-wave-turbulent boundary layer (SWTBL) interaction flow in a supersonic $24^\circ$ compression ramp with a $2.9$ Ma free stream is used as an example to investigate the spatial and temporal evolution characteristics of the total OAM and its spiral spectrum, respectively. Section 5 presents the conclusion.

2. Derivation of the total OAM’s variation and spiral spectrum of OAM mode based on Rytov approximation

2.1 Total OAM’s variation of Laguerre-Gaussian beam

If $E_0(\mathbf {x})$ denotes the complex amplitude of the optical field without scattering, according to the Rytov approximation, the complex amplitude of the scattered light can be written as

$$E(\mathbf{x})=E_0(\mathbf{x})\exp[\Psi(\mathbf{x})],$$
where $\Psi (\mathbf {x})=\mathop{\sum}\limits _{n}\psi _n(\mathbf {x})$ denotes the complex disturbance term, and $\psi _n(\mathbf {x})$ denotes the Rytov series, which is [23]
$$\psi_n(\mathbf{x})=\frac{1}{E_0(\mathbf{x})}\int_{V'}\ E_0(\mathbf{x'})g_n(\mathbf{x'})G(\mathbf{x-x'}) \ d^3\mathbf{x'},$$
where $G(\mathbf {x-x'})=\frac {1}{4\pi }\frac {\exp (ik|\mathbf {x-x'}|)}{|\mathbf {x-x'}|}$ is Green’s function. $g_n(\mathbf {x})=-\mathop{\sum}\limits _{k=1}^{n-1} \nabla \psi _k\cdot \nabla \psi _{n-k}$ and $g_1(\mathbf {x'})=-k^2[n^2(\mathbf {x'})-1]$.

Therefore, the Rytov approximation can be used in the numerical simulation of the scattering vortex beam, and the complex amplitude of the disturbance is

$$\Psi=\frac{1}{E_0(\mathbf{x})}\int_{V'} E_0(\mathbf{x'})H(\mathbf{x'})G(\mathbf{x-x'}) \ d^3\mathbf{x'},$$
where $H(\mathbf {x'})=\mathop{\sum}\limits _{n} g_n(\mathbf {x'})$. The complex amplitude of a Laguerre-Gaussian(LG) beam is expressed as
$$\begin{aligned} E_0(\rho,\phi,z)=&\left[\frac{2p!}{\pi(|l|+p)!}\right]^\frac{1}{2}\frac{1}{\omega(z)}e^{i(2p+|l|+1)\Psi_G (z)} \left[\frac{\sqrt{2}\rho}{\omega(z)}\right]^{|l|}L_p^{|l|}\left(\frac{2\rho^2}{[\omega(z)]^2}\right) \\ &\times \exp\left(-i\frac{\rho^2Z_N}{[\omega(z)]^2}\right)\exp\left(-\frac{\rho^2}{[\omega(z)]^2}\right)\exp(il\phi)\exp(ikz)\\& =R_{p,l}(\rho,z)\exp(il\phi), \end{aligned}$$
where $Z_N=z/z_R$, $z_R=\frac {1}{2}k\omega _0^2$ is the Fresnel distance of the LG beam. The terms independent of the azimuth angle $\phi$ are defined by $R_{p,l}(\rho ,z)$.

The OAM per unit power density around the $z$ axis in the observation plane, $L_z(\mathbf {s})$, is defined as

$$L_z\left(\mathbf{s}\right)=Im\left\{\frac{1}{2\omega A}E^\ast(\mathbf{s},z)(\nabla\times\mathbf{x})_{\widehat{z}} E(\mathbf{s},z)\right\},$$
where $A=\iint _{S}\ E^\ast (\mathbf {s},z)E(\mathbf {s},z)\,dxdy$. Then the total OAM is $L_z=\iint _{s}\ L_z\left (\mathbf {s}\right )\,d^2s$. If we switch to the cylindrical coordinates: $(x,y,z)\to (\rho ,\phi ,z)$ then
$$(\nabla\times\mathbf{x})_{\widehat{z}}=y\frac{\partial}{\partial x}-x\frac{\partial}{\partial y}=\frac{\partial}{\partial\phi},$$
and therefore, the total OAM of a light beam on the polar coordinates can be written as [8]
$$L_z=Im\left\{\frac{1}{2\omega A}\iint_{S}\ E^\ast(\mathbf{s},z)\frac{\partial}{\partial\phi}E(\mathbf{s},z)\,d^2s\right\}.$$
Eq. (4) is substituted into Eq. (7). This makes it convenient to obtain the total OAM of a LG beam, denoted by $L_{z0}$, and $L_{z0}=\frac {l}{2\omega }$.

Substituting Eq. (1) into Eq. (7) and using the LG beam as the incident wave function, the total OAM of scattered beam can be expressed as

$$\begin{aligned} L_z&=Im\left\{\frac{1}{2\omega A}\iint_{S}\ E_0^\ast(\rho,\phi,z)\exp(\Psi^\ast)\frac{\partial}{\partial\phi}[E_0(\rho,\phi,z)\exp(\Psi)]\,d^2s\right\} \\ &=Im\left\{{\frac{1}{2\omega A}\iint_{S}\, E_0^\ast(\rho,\phi,z)\exp(\Psi^\ast)\frac{\partial E_0(\rho,\phi,z)}{\partial\phi}\exp(\Psi)\,d^2s}\right. \\ &\left.{+\frac{1}{2\omega A}\iint_{S}\ E_0^\ast(\rho,\phi,z)\exp(\Psi^\ast)E_0(\rho,\phi,z)\frac{\partial\exp(\Psi)}{\partial\phi}\,d^2s}\right\}, \end{aligned}$$
Eq. (8) can be decomposed into two parts, the first is
$$I_1=\frac{1}{2\omega A}\iint_{S}\, E_0^\ast(\rho,\phi,z)\exp(\Psi^\ast)\frac{\partial E_0(\rho,\phi,z)}{\partial\phi}\exp(\Psi)\,d^2s.$$
Paraxial scattering in refractive medium, total power is conserved even for strong scattering [8]. Therefore, the following equation can be easily obtained:
$$Im\left\{I_1\right\}=Im\left\{i\frac{l}{2\omega}\right\}=L_{z0}.$$
The second part of Eq. (8) describes the variant of a beam’s OAM propagating through the turbulence. It can be expressed as
$$\begin{aligned} I_2&=\frac{1}{2\omega A}\iint_{S}\ E_0^\ast(\rho,\phi,z)\exp(\Psi^\ast)E_0(\rho,\phi,z)\frac{\partial \exp(\Psi)}{\partial\phi}\,d^2s \\ &=\frac{1}{2\omega A}\iint_{S}\ E_0^\ast(\rho,\phi,z)\exp(\Psi^\ast)E_0(\rho,\phi,z)\exp(\Psi)\frac{\partial\Psi}{\partial\phi}\,d^2s. \end{aligned}$$
Our main focus is to deduce the expression: $\frac {\partial \Psi }{\partial \phi }$. A simple explicit form can be obtained(See Appendix A)
$$\frac{\partial\Psi}{\partial\phi}=\frac{1}{E_0(\mathbf{x})}\int_{V'}\ G(\mathbf{x-x'})\frac{\partial H(\mathbf{x'})}{\partial\phi'}E_0(\mathbf{x'})\,d^3\mathbf{x'}.$$
Substituting Eq. (12) into Eq. (11), by changing the order of the surface integral for $S$ and the volume integral for $V'$, gives
$$I_2=\frac{1}{2\omega A}\int_{V'}\ \frac{\partial H(\mathbf(x'))}{\partial\phi'}E_0(\mathbf{x'})\,d^3\mathbf{x'}\iint_{S}\ T(\rho,\phi,z)E_0^\ast(\rho,\phi,z)G(\mathbf{x-x'})\,d^2s,$$
where $T(\mathbf {x})=|\exp (\Psi )|^2$ is the spatial distribution of transmittance caused by the turbulence.

According to the expression of Green’s function: $G(\mathbf {x-x'})=G(\mathbf {x'-x})$, the surface integral in Eq. (13) is similar to the Kirchhoff diffraction equation. According to Kirchhoff diffraction [22], in the paraxial approximation (the angular diffraction $\chi$ can be regarded as zero) the diffracted optical field can be written as

$$E(P)=-2ik\iint_{S}\ E(P_0)G(P-P_0)\,d^2s.$$
Therefore, a new wave function $E_{tran}^\ast (\rho ,\phi ,z)$ can be defined
$$E_{tran}^\ast=-2ik\iint_{S}\ T(\mathbf{x})E_0^\ast(\mathbf{x})G(\mathbf{x'-x})\,d^2s,$$
which can be treated as the complex amplitude of a beam experiencing the disturbance of transmittance, $T(\mathbf {x})$, and diffracting backward to point $\mathbf {x'}$. Then, Eq. (13) can be written as
$$\delta L_z=Im\{I_2\}=Im\left\{\frac{i}{4k\omega A}\int_{V'}\ E_{tran}^\ast(\mathbf{x'})\frac{\partial H(\mathbf{x'})}{\partial\phi'}E_0(\mathbf{x'})\,d^3\mathbf{x'}\right\}.$$
$\delta L_z$ is used to denote the variation of the total OAM. Equation (19) shows that the variation of total OAM depends on the scattering potential and the amplitude perturbation of the scattered light. Equation (19) presents the change in the total OAM based on Rytov approximation. Charnotskii also deduced the change in the total OAM based on the paraxial Helmhotz equation, which is expressed as [8]:
$$L(z)-L(0)=C\int_{0}^{z} dz' \iint d^2 \mathbf{s} I(\rho,\phi,z')\frac{\partial n(\rho,\phi,z')}{\partial \phi}$$
where $C$ is a constant. Equation (17) has similar form to Eq. (16).

In general, a beam propagating through inhomogeneous media such as turbulence will maintain its beam-like characteristics, so that the condition of paraxial approximation is satisfied by a scattered beam in the turbulence. Thereby, Eq. (17) is a superior one for theoretical analysis since it does not require perturbation (Rytov) approximation, and is valid for the general beam wave, but not just LG beam. Nevertheless, in this paper, we prefer to deduce the total OAM changing by Rytov approximation. From the aspect of experiment of supersonic turbulent boundary layer, it is difficult to obtain the refractive index distribution in the wind tunnel experiment or other experimental environment. However, the complex amplitude of the disturbance can be easily measured. Therefore, the complex amplitude of disturbance is a more intuitive.

2.2 Spiral spectrum of OAM mode

It is well known that the disturbance of complex amplitude caused by inhomogeneous media will affect the spiral spectrum of the OAM mode. Sanchez and Oesch [2,3] thought that the inhomogeneous media would induce the photons carrying the new OAM mode. Combining the conclusion of Charnotskii’s theoretical work [8] that there is no definite connection between the intrinsic OAM and phase vorticity, we believe that there will not be necessarily a new vortex mode even though a new OAM mode is measured. A simple example can be given: phase tilt occurs on a LG beam. In general, the spiral spectrum of the OAM mode $\{C_m\}$ can be used to describe the OAM mode spread of a scattered LG beam. According to Charnotskii, $C_m\neq 0$ does not mean that a vortex structure $\exp (im\phi )$ exists in the phase of scattered LG beam, but rather that the energy weight of a component whose OAM is $\frac {m}{2\omega }$ in scattered light. Therefore, the relationship between $C_m$ and total OAM can be easily obtained.

$$L_z=\sum_m mC_m.$$
$C_m$ demonstrates more detail information about the effects of inhomogeneous media. The mathematical form of a LG beam’s complex amplitude is a complete orthonormal basis. Equation (4) shows that each OAM mode $m$ is degenerate. Therefore, the complete orthonormal basis should be $\{E_m^p\}$. Then the expansion form of scattered light should be written as
$$E_{scattering}=\sum_{m}\sum_{p}c_m^pE_m^p,$$
where $|c_m^p|^2$ is the energy weight of a certain LG beam mode, $m,p$. Therefore, for a certain OAM mode $m$, the coefficient is the sum of $|c_m^p|^2$ which have the same $m$ and different $p$. That is
$$C_m=\sum_p|c_m^p|^2.$$
$c_m^p$ can be calculated by
$$c_m^p=\iint_{S}\ E(\rho,\phi,z)E_m^{p\ast}(\rho,\phi,z)\,d^2s,$$
hence, the coefficient $C_m$ can be written as:
$$\begin{aligned} C_m=&\sum_p\int\ \rho_1\,d\rho_1\int\ \rho_2R_{p,m}(\rho_1,z)R_{p,m}^{\ast}(\rho_2,z)\,d\rho_2 \\ &\iint\ E^\ast(\rho_1,\phi_1,z)E(\rho_2,\phi_2,z)\exp[im(\phi_2-\phi_1)]\,d\phi_1d\phi_2. \end{aligned}$$
Using the orthogonal relationship(See Appendix B),
$$\sum_pR_{p,m}(\rho_1)R_{p,m}^{\ast}(\rho_2)=\frac{1}{\pi\omega^2}\frac{\delta(\rho_1,\rho_2)}{\rho_1},$$
Eq. (22) can be rewritten as
$$C_m=\frac{1}{\pi\omega^2}\int\ \rho\,d\rho\left|\int_{0}^{2\pi}\ E(\rho,\phi_1,z)\exp(-im\phi_1)\,d\phi_1\right|^2.$$
Eq. (24) is the exact expression for the spiral spectrum of the OAM mode. On the one hand, unlike the multiplexing and demultiplexing of the OAM in FSO communication, $C_m\neq 0$ does not mean that the signal with an OAM of $m$ can be measured by OAM mode sorting method. In FSO communication, the incoherent OAM mode superposition is generally used. Therefore, OAM modes can be measured with the aid of the spiral phase or Fork grating generated by spatial light modulator. However, the scattered light field can be decomposed by spiral harmonic , which means the scattered light field can be regarded as a interference pattern of series light field interferences with different spiral eigen $m$. Thus, the signal with OAM mode $m$ cannot be measured by sorting method of OAM mode demultiplexing. On the other hand, from Rytov approximation, it is clear that in the continuous inhomogeneous medium, the complex amplitude of the disturbance will not tend to zero at a certain position in space, that is, there is no point $\mathbf {x_0}$, where $\exp [\Psi (\mathbf {x_0})]=0$. This shows that spiral spectrum $C_m$ provides no information about the presence or emergence of optical vortices. In fact, $C_m$ is sensitive to the phase tilt or the displacement of the beam. Phase tilt and beam displacement are two typical atmospheric turbulence effects, which also exist in the supersonic turbulent boundary layer. There are a series of coherent structures in the supersonic turbulent boundary layer, which cannot be ignored. Their spatial distribution is not consistent. From the statistical perspective, they are statistically inhomogeneous. Therefore, at every point on the observation surface, the phase tilt of the light field cannot be described by a unified mathematical expression. Therefore, the performance of $C_m$ will be very different from that of atmospheric turbulence.

3. Numerical approach for OAM’s variation and spiral spectrum of OAM mode

The analytical form of the refractive index’s distribution in the supersonic turbulent boundary layer cannot be easily obtained. Therefore, a numerical approach is the most suitable way to study the evolution of light’s OAM in the supersonic turbulent boundary layer. In this section, the numerical approach used to calculate Eqs. (16) and (24) will be presented in detail.

Using DNS method, the distribution of refractive index can be calculated numerically. According to the numerical method proposed by Xu et al. [22], Eq. (2) can be calculated numerically. Noting that the complex amplitude of the vortex beam (LG beam) vanishes at the beam axis, in our numerical approach, we therefore prefer to use second order Rytov approximation to ensure the Rytov series is convergent around the beam axis. $g_2(\mathbf {x})$ can be expressed as:

$$g_2(\mathbf{x})=[\nabla\psi_1(\mathbf{x})]^2.$$
By numerically calculating Eq. (25) and according to Eq. (2), $\psi _2$ can also be obtained numerically. Rytov perturbation can be numerically calculated by
$$\exp(\Psi)=\exp(\psi_1+\psi_2).$$
Thereby, $T(\mathbf {x})$ can also be numerically calculated. According to the numerical result of $T(\mathbf {x})$, the complex amplitude $E_{trans}^{\ast }(\mathbf {x'})$ can be numerically calculated with the aid of Eq. (15). By numerically calculating the volume integral Eq. (16), the numerical result of the variation of the total OAM can be obtained.

Furthermore, based on the numerical result of Rytov perturbation, the complex amplitude of scattered light $E(\mathbf {x})$ can also be get. Therefore, through directly numerically calculating Eq. (24), the spiral spectrum of OAM mode can be easily obtained.

4. Numerical results and discussions

4.1 Numerical results of shock-wave-turbulent boundary layer

A classic example of a SWTBL is used as the scattering medium in this paper. The shape of the solid boundary consists of a straight plate and a $24^{\circ }$ slope. The flow direction is parallel to the plate, and its Mach number is 2.9. According to the forms of the flow motion, the whole flow field can be divided into four regions along the direction of the incoming flow movement: laminar flow, transitional flow, fully developed turbulence, and fully developed turbulence with separated shock. In each region, a subarea is selected as a region of optical transmission. The size of each subarea is $6mm\times 6mm\times 14mm$.

The DNS method proposed by Li et al. [21] is used to solve the NS equations of a SWTBL with determined initial and boundary values. The spatial distribution of density can be calculated numerically by such DNS. In the transitional flow area, a certain velocity perturbation distribution is given artificially. Because of the nonlinearity of the NS equations, such perturbation changes the flow from laminar to turbulent and clutters the spatial distribution of density. The refractive index of the turbulent flow field is obtained using the Gladstone–Dale equation, shown in Fig. 1. Because all the calculation parameters are determined, the numerical results will not change with the number of calculations. In the later numerical discussions, the parameters of the incident LG beam are chosen as: $p=0$, $l=1$, and $\omega _0=1mm$. The wavelength of the LG beam is $633nm$. The initial plane is set as $z=0$.

 figure: Fig. 1.

Fig. 1. SWTBL in a supersonic $24^\circ$ compression ramp with a $2.9$ Ma free stream. Area A is laminar flow, Area B is the transitional flow area, Area C is fully developed turbulence, and Area D is fully developed turbulence with separated shock. $z$ axis is the propagating direction of the light. Each area’s size is $6mm\times 6mm\times 14mm$ with the grids:$121\times 121\times 241$.

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4.2 Numerical discussions for space-evolution of OAM in SWTBL

Area D is selected for numerical discussion. At the corner, a recirculation zone and separate shock layer are generated. Flow field in Area D has complex types of vortex-like structures in this region, such as flow direction vortices and hairpin vortices [21]. Such complex flow motion causes the nonuniform spatial distribution of refractive index, as shown in Fig. 2(a).

 figure: Fig. 2.

Fig. 2. Spatial distribution of refractive index in Area D and corresponding fluctuation of the beam’s irradiance $T(\mathbf {x})$

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According to Section 3, the spatial transmittance distribution $T(\mathbf {x})$ is calculated by a second order Rytov approximation, as shown in Fig. 2(b). Thus, $E_{trans}^{\ast }({\mathbf {x}})$ is also calculated. The variation of the total OAM $\delta L_z$ can be numerically obtained from Eq. (16). Inhomogeneity of the refractive index causes fluctuations in both irradiance and phase. In order to study which is more influential, another numerical calculation is carried out where the fluctuation in irradiance is ignored. Setting $T(\mathbf {x})=1$, there is $E_{trans}^{\ast }({\mathbf {x}})=E_0^{\ast }(\mathbf {x})$. Figure 3 shows the comparison.

 figure: Fig. 3.

Fig. 3. $\delta L_z$of the total OAM along the propagating direction in Area D. The amplitude disturbance $T(\mathbf {s})$ on the planes $z=4 mm$ and $z=8 mm$ are also shown. The amplitude disturbance is relatively flat at a distance of less than $6 mm$, and has obvious fluctuation at a distance of more than $6 mm$. Moreover, such fluctuation has a strong inconsistency in space.

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Because the NS equations are established based on the assumption that the medium is continuous, the spatial distribution of refractive index is continuous. The red line in Fig. 3 shows that the total OAM is invariant if fluctuations in irradiance are ignored, which is in agreement with Charnotskii’s conclusion [8]. In Area D, along the direction of beam propagation, the effect of the separated shock wave becomes increasingly intense, especially at the distances where $z\;>\;6mm$. This effect is demonstrated in the irradiance perturbation of the light. As a result, the variation of the total OAM presents a sharp fluctuation after $z=6mm$, shown as a blue line in Fig. 3.

The numerical results of the spiral spectrum of OAM mode are shown in Fig. 4. The evolution of spiral spectrum along the propagating direction is presented.

 figure: Fig. 4.

Fig. 4. The spiral spectrum of OAM mode in Area D. The propagating distances are $1.4mm$ for Fig. 4(a), $7mm$ for Fig. 4(b), $14mm$ for Fig. 4(c)

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The initial LG beam has a pure spiral spectrum. When the light propagates in the SWTBL, energy appears on the other spiral spectrum. Even if the propagating distance is $1.4mm$, Fig. 4(a) demonstrates the spiral spectrum changes caused by phase tilt or beam displacement. The form of flow motion in Area D are not consistent in space. For example, in the bottom of Area D, the form of flow motion is fully developed turbulence and in the upper of Area D, the form of flow motion is separated shock wave. Such inconsistency leads spatially inconsistent of phase tilt. As a result, the spiral spectrum of the OAM mode appears asymmetric. Furthermore, in Fig. 4(c), the main component in the scattered light is not the initial OAM mode, which was rarely observed in previous studies.

4.3 Time-evolution of OAM in SWTBL

In studies of fluid dynamics, the time evolution of the flow structure is one of the main research objects. Flow structures determine the spatial distribution of the refractive index and their time evolution depends on the form of the flow motion. Therefore, the time evolution of the OAM is concerned in this paper.

In Section 4.1, the SWTBL is divided into four areas according to the form of the flow movement. In this paper, the numerical results of $175$frames are used for each region, and the time interval between frames is $2ms$. The time evolution of the total OAM and the time average of the spiral spectrum in those four regions are numerically calculated. The observed plane is set at $z=14mm$. The time average and variance of the total OAM are numerically calculated and are denoted by $\overline {L_z}$ and $\sigma ^2$, respectively.

Time-evolution of OAM in Area A

Area A is laminar flow, the structures of which is stable with respect to time. The time average of total OAM is $\overline {L_z}=1$, and the time variance of total OAM is $\sigma ^2=0$. Figs. (5a) and (5b) show the statistical characteristics in time of the refractive index in Area A. In Fig. (5b), the time variance of refractive index is zero, so that the total OAM does not change with time in this area as shown in Fig. (5c). For the time average refractive index in this area, along the direction of beam propagation, refractive index does not change, which is $\frac {\partial n}{\partial z}=0$. And the amplitude disturbance in laminar flow is imperceptible. Thus, the total OAM does not change as shown in Fig. (5c). The flow field causes phase tilt. We can see in Fig. (5a) that a refractive index gradient only exists along the $y$ axis and $\frac {\partial n}{\partial y}\approx 10^{-4}$. Such slight phase tilt causes the change in the spiral spectrum. New spiral spectrum information is generated. Also, in this region, the form of flow motion is spatially consistent, $\frac {\partial n}{\partial y}$ is a constant. Thereby, the distribution of spiral spectrum is symmetric as shown in Fig. (5d).

 figure: Fig. 5.

Fig. 5. The time evolution of the OAM in the laminar flow. Figs. (5a) and (5b) show the statistical properties of the refractive index; Fig. (5c) shows the total OAM; Fig. (5d) shows the time average of the spiral spectrum.

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Time-evolution of OAM in Area B

Area B is located in the transition zone of flow field. The time average of total OAM is $\overline {L_z}=1.0124$, and the time variance of total OAM is $\sigma ^2=0.0466$. In this area, the time average of total OAM changes and OAM fluctuates with time. The form of flow motion shifts between laminar flow and turbulence in this region. The instability of the flow motion form, called K-H instability, leads to the destruction of the stratified structure of the flow field. K-H instability also causes an inverse pressure gradient of the flow field. Because the flow is a compressible fluid, the gradient of refractive index becomes irregular. Unlike the case in laminar flow, the gradient of refractive appears in each direction. The inhomogeneity in Area B is more serious, and the amplitude disturbance is more obvious. Thereby, the time average of total OAM changes. Fig. (6b) demonstrates that refractive index is unstable with time, so that the total OAM is also unstable.

 figure: Fig. 6.

Fig. 6. The time evolution of the OAM in the laminar flow. Figs. (6a) and (6b) show the statistical properties of the refractive index; Fig. (6c) shows the total OAM; Fig. (6d) shows the time average of the spiral spectrum.

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Near the boundary wall, non-negligible and deterministic large-scale quasi-ordered structures are generated intermittently, which results the form of flow motion in Area B being spatial unstable. Because the stratified structure is broken, the gradient of refractive index along the $x$ axis appears and varies in the range of $(-2.2\times 10^{-5},3\times 10^{-5})$. Additionally, the gradient of refractive index along the $y$ axis is not a constant. As a result, in this region, the phase tilt of light is spatially inconsistent. The phase tilt leads the spiral spectrum changing and the spatial inconsistency causes the spiral spectrum asymmetry.

Time-Evolution of OAM in Area C

Area C is located in the fully developed turbulence zone. The time average of the total OAM is $\overline {L_z}=1.0146$, and the time variance of total OAM is $\sigma ^2=0.0156$. Considering the time average and variance of the total OAM in Area B, the difference between the time averages is small, however the time variance of the total time is much smaller. In this region, the flow has completely changed into turbulent motion and the status of turbulence is relatively stable, as shown in Figs. (7a) and (7b). The time variance of refractive index in Area C is smaller than in Area B so that $\sigma ^2$ in this area is smaller.

 figure: Fig. 7.

Fig. 7. The time evolution of the OAM in the laminar flow. Figs. (7a) and (7b) show the statistical properties of refractive index; Fig. (7c) shows the total OAM; Fig. (7d) shows the time average of spiral spectrum.

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In the fully developed turbulence, the refractive index gradient still exists in each direction. The spatial distribution of refractive index in the fully developed turbulence becomes completely irregular. Serious amplitude disturbance appears, which leads the total OAM changing.

The form of flow motion in this area can be spatially divided into three types: viscous bottom layer, buffer layer and turbulent layer, so that the gradient of refractive index along the $y$ axis is also not a constant. $\frac {\partial n}{\partial x}$ varies in the range of $(-3\times 10^{-5},3\times 10^{-5})$. In the supersonic turbulence boundary layer, the scale of coherent structures is small [21], so $\frac {\partial n}{\partial x}$ varies with a high spatial frequency. As a result, the spiral spectrum information in this region is more abundant. And also, due to the form of motion being spatially inconsistent, the spiral spectrum is asymmetric.

Time-Evolution of OAM in Area D

In Area D, the form of flow motion is the fully develop turbulence coupling with separated shock. The time average of total OAM is $\overline {L_z}=0.9845$, and the time variance of total OAM is $\sigma ^2=0.0796$. The fluctuation of total OAM becomes larger. $\sigma ^2$ is the largest value in these four regions.

In this region, it can be seen from Figs. (8a) and (8b) that the shock wave oscillation interacts with the turbulent boundary layer, thus the form of motion is very inconsistent in time and space. An irregular Iso-surface structure appears in the time average distribution of refractive index, as shown in Fig. (8a), and the variance of refractive index drastically increased. Thus, the total OAM fluctuates significantly.

 figure: Fig. 8.

Fig. 8. The time evolution of the OAM in the laminar flow. Figs. (8a) and (8b) show the statistical properties of the refractive index; Fig. (8c) shows the total OAM; Fig. (8d) shows the time average of the spiral spectrum.

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$\frac {\partial n}{\partial x}$ varies in the range of $(-1.7\times 10^{-4},1.2\times 10^{-4})$ in this area, which is the maximum in these four areas owing to the shock wave. Additionally, the inconsistency of $\frac {\partial n}{\partial y}$ increases in this area, $\frac {\partial n}{\partial y}$ varies from $-9\times 10^{-5}$ to $1.67\times 10^{-4}$. Strong inconsistent and large phase tilt is caused by shock wave. So that the spiral spectrum information becomes much more abundant and the symmetry of spiral spectrum is broken.

5. Conclusion

In conclusion, the power spectrum model for refractive index fluctuations in a supersonic turbulent boundary layer can hardly be obtained. Therefore, a numerical simulation approach was used to investigate the total OAM variation and the spiral spectrum of the OAM mode when a beam carrying OAM propagates through a supersonic turbulent boundary layer.

The Rytov approximation method is often used to calculate the complex amplitude of a light beam propagating through a turbulence, so the total OAM and spiral spectrum of the OAM mode for a LG beam were calculated using the Rytov approximation in this paper. In our numerical simulation, the second order Rytov approximation was used. A SWTBL was used as a transmission medium. The shape of the solid boundary consisted of a straight plate and a $24^{\circ }$ slope, and the velocity of the incoming flow was 2.9 Ma. The whole flow field was divided into four regions according to the form of the flow motion. We numerically calculated both the space and time evolution of the total OAM and the spiral spectrum of the OAM mode, and the results were discussed in combination with the motion form of the flow field. We found that the disturbance of the amplitude of the light is the main cause of the total OAM changing. Moreover, the spiral spectrum of the OAM mode is asymmetric in the regions where the form of the flow motion is more complex. The numerical results indicated that the inconsistency of the flow motion form in space will lead to the total OAM changing and the spiral spectrum of the OAM mode becoming asymmetric. This conclusion can also be obtained from the time average results.

In summary, the distribution of amplitude disturbance reflects the motion form of the flow field. Owing to the spatial inconsistency of the flow motion form, the distribution of amplitude disturbance is also spatially inconsistent, which results in the OAM of the light field changing. Such spatial inconsistency of the flow motion form is ubiquitous in a supersonic turbulent boundary layer and rare in atmospheric turbulence. These results may enlighten the study of aero-optics to investigate the coupling between flow field structures and light field structures. We are planning to study the OAM transmission characteristics in a flow field with a single structure.

Appendix

A. Proof for Eq. (12)

For $\frac {\partial \Psi }{\partial \phi }$,

$$\begin{aligned} \frac{\partial\Psi}{\partial\phi}&=-\frac{\partial E_0(\rho,\phi,z)/\partial\phi}{\left[E_0(\rho,\phi,z)\right]^2}\int_{V'}\ E_0(\mathbf{x'})H(\mathbf{x'})G(\mathbf{x-x'})\,d^3\mathbf{x'} \\ &+\frac{1}{E_0(\rho,\phi,z)}\int_{V'}\ E_0(\mathbf{x'})H(\mathbf{x'})\frac{\partial G(\mathbf{x-x'})}{\partial\phi}\,d^3\mathbf{x'} \\ &=-il\frac{1}{E_0(\rho,\phi,z)}\int_{V'}\ E_0(\mathbf{x'})H(\mathbf{x'})G(\mathbf{x-x'})\,d^3\mathbf{x'} \\ &+\frac{1}{E_0(\rho,\phi,z)}\int_{V'}\ E_0(\mathbf{x'})H(\mathbf{x'})\frac{\partial G(\mathbf{x-x'})}{\partial\phi}\,d^3\mathbf{x'} \end{aligned}$$
In cylindrical coordinates, Green’s function is
$$G(\mathbf{x-x'})=\frac{1}{4\pi}\frac{\exp\left(ik\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}\right)}{\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}}$$
and the second term on the right hand side of Eq. (27) is
$$\begin{aligned} \int_{V'}\ E_0(\mathbf{x'})H(\mathbf{x'})\frac{\partial G(\mathbf{x-x'})}{\partial\phi}\,d^3\mathbf{x'}=&\int\ \,dz'\int\ \rho\, d\rho\int_{0}^{2\pi}\ E_0(\rho',\phi',z')H(\rho',\phi',z')\times\\ &\frac{\partial}{\partial\phi}\left\{\frac{\exp\left(ik\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}\right)}{4\pi\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}}\right\}\,d\phi' \end{aligned}$$
The complex amplitude of an LG beam, the scattering potential, and Green’s function are periodic functions of the polar angle; thus, the integral for the polar angle can be treated as a convolution, such that
$$\begin{aligned}& \int_{0}^{2\pi}\ E_0(\rho',\phi',z')H(\rho',\phi',z')\frac{\partial}{\partial\phi}\left\{\frac{\exp\left(ik\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}\right)}{4\pi\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}}\right\}\,d\phi'\\& =\{E_0(\rho',\phi,z')H(\rho',\phi,z')\}\ast\frac{\partial}{\partial\phi}\frac{\exp\left(ik\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}\right)}{4\pi\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}}\\& =\frac{\partial}{\partial\phi}\left\{\{E_0(\rho',\phi,z')H(\rho',\phi,z')\}\ast\frac{\exp\left(ik\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}\right)}{4\pi\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}}\right\}\\& =\frac{\partial}{\partial\phi}\{E_0(\rho',\phi,z')H(\rho',\phi,z')\}\ast\frac{\exp\left(ik\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}\right)}{4\pi\sqrt{\rho^2+\rho'^2-2\rho\rho'\cos(\phi-\phi')+(z-z')^2}}\\& =\int_{0}^{2\pi}\ \left[\frac{\partial}{\partial\phi'}E_0(\rho',\phi',z')\right]H(\rho',\phi',z')G(\mathbf{x-x'})\,d\phi'\\& +\int_{0}^{2\pi}\ E_0(\rho',\phi'z')\frac{\partial H(\rho',\phi',z')}{\partial\phi'}G(\mathbf{x-x'})\,d\phi'\\& =il\int_{0}^{2\pi}\ E_0(\rho',\phi',z')H(\rho',\phi',z')G(\mathbf{x-x'})\,d\phi'\\& +\int_{0}^{2\pi}\ E_0(\rho',\phi'z')\frac{\partial H(\rho',\phi',z')}{\partial\phi'}G(\mathbf{x-x'})\,d\phi' \end{aligned}$$
Combined with Eq. (27), then:
$$\frac{\partial\Psi}{\partial\phi}=\frac{1}{E_0(\mathbf{x})}\int_{V'}\ G(\mathbf{x-x'})\frac{\partial H(\mathbf{x'})}{\partial\phi'}E_0(\mathbf{x'})\,d^3\mathbf{x'}$$
Eq. (12) is therefore proven.

B. Proof for Eq. (23)

From Eq. (4), the radius function $R_{p,l}(\rho ,z)$ can be written as

$$\begin{aligned} R_{p,l}(\rho,z)=&\left[\frac{2p!}{\pi(|l|+p)!}\right]^\frac{1}{2}\frac{1}{\omega(z)}e^{i(2p+|l|+1)\Psi_G (z)}\left[\frac{\sqrt{2}\rho}{\omega(z)}\right]^{|l|}L_p^{|l|}\left(\frac{2\rho^2}{[\omega(z)]^2}\right)\times\\ & \exp\left(-i\frac{\rho^2Z_N}{[\omega(z)]^2}\right)\exp\left(-\frac{\rho^2}{[\omega(z)]^2}\right) \end{aligned}$$
Then, the sum of $R_{p,m}$ for $p$ is
$$\begin{aligned} \sum_pR_{p,m}(\rho_1)R_{p,m}^{\ast}(\rho_2)=&\sum_p\frac{2p!}{\pi(|l|+p)!}\frac{1}{\omega^2}\left[\frac{\sqrt{2}\rho_1}{\omega}\right]^{|l|}\left[\frac{\sqrt{2}\rho_2}{\omega}\right]^{|l|}L_{p}^{|l|}\left(\frac{2\rho_1^2}{\omega^2}\right)L_{p}^{|l|}\left(\frac{2\rho_2^2}{\omega^2}\right)\times \\ &\exp\left(-iZ_N\frac{\rho_1^2-\rho_2^2}{\omega^2}\right)\exp\left(-\frac{\rho_1^2+\rho_2^2}{\omega^2}\right) \end{aligned}$$
Let $\frac {2\rho _1^2}{\omega ^2}=x$,and $\frac {2\rho _2^2}{\omega ^2}=y$. Their sum can be written as
$$ \frac{2}{\pi}\exp\left(-iZ_N\frac{x-y}{2}\right)\exp\left(-\frac{x+y}{2}\right)x^{\frac{|l|}{2}}y^{\frac{|l|}{2}}\frac{1}{\omega^2}\sum_{p}\frac{p!}{(|l|+p)!}L_{p}^{|l|}(x)L_{p}^{|l|}(y)$$
For the Laguerre polynomial [24],
$$\sum_{p=0}^{+\infty}\frac{p!}{(|l|+p)!}L_{p}^{|l|}(x)L_{p}^{|l|}(y)z^n=\frac{(xyz)^{-\frac{|l|}{2}}}{1-z}\exp\left(-z\frac{x+y}{1-z}\right)I_{|l|}\left(\frac{2\sqrt{xyz}}{1-z}\right)$$
Then Eq. (34) can be written as
$$\frac{2}{\pi}\exp\left(-iZ_N\frac{x-y}{2}\right)\lim_{z\to1}\frac{1}{(1-z)z^{\frac{|l|}{2}}}\exp\left[-z\frac{1+z}{2(1-z)}(x+y)\right]I_{|l|}\left(\frac{2\sqrt{xyz}}{1-z}\right)$$
where $z\to 1$ infers that $\frac {2\sqrt {xyz}}{1-z}\to +\infty$, and $arg\left (\frac {2\sqrt {xyz}}{1-z}\right )=0$, so that [24]
$$I_{|l|}(t)\sim \frac{e^t}{\sqrt{2\pi t}}\sum_{k=0}^{\infty}\frac{(-1)^k\Gamma(|l|+k+\frac{1}{2})}{(2t)^kk!\Gamma(|l|-k+\frac{1}{2})}+\frac{\exp[-t+(|l|+\frac{1}{2})\pi i]}{\sqrt{2\pi t}}\sum_{k=0}^{\infty}\frac{\Gamma(|l|+k+\frac{1}{2})}{(2t)^kk!\Gamma(|l|-k+\frac{1}{2})}$$
Setting $t=\frac {2\sqrt {xyz}}{1-z}\to \infty$, the second term of Eq. (37) will be zero, and the first term of Eq. (37) is
$$\frac{e^t}{\sqrt{2\pi t}} {_2}F_{0}(|l|+\frac{1}{2},\frac{1}{2}-|l|;-;\frac{1}{2t})$$
where ${_p}F_{q}$ denotes the general hypergeometric function. And for $t\to \infty$, ${_2}F_{0}(|l|+\frac {1}{2},\frac {1}{2}-|l|;-;\frac {1}{2t})\to 1$. Thus,
$$I_{|l|}(t)\sim \frac{e^t}{\sqrt{2\pi t}}$$
Eq. (36) then becomes
$$\begin{aligned} &\frac{2}{\pi}\exp\left(-iZ_N\frac{x-y}{2}\right)\lim_{z\to1}\frac{1}{(1-z)z^{\frac{|l|}{2}}}\exp\left[-z\frac{1+z}{2(1-z)}(x+y)\right]\exp\left(\frac{2\sqrt{xyz}}{1-z}\right)\frac{\sqrt{1-z}}{2\sqrt{\pi}z^{\frac{1}{4}}(xy)^{\frac{1}{4}}}= \\ &\frac{1}{\pi(xy)^{\frac{1}{4}}}\exp\left(-iZ_N\frac{x-y}{2}\right)\lim_{z\to1}\frac{1}{\sqrt{1-z}}\exp\left[-\frac{x+y}{1-z}\right]\exp\left(\frac{2\sqrt{xy}}{1-z}\right)\frac{1}{\sqrt{\pi}} \end{aligned}$$
Re-organizing Eq. (40), results in
$$\frac{1}{\pi(xy)^{\frac{1}{4}}}\exp\left(-iZ_N\frac{x-y}{2}\right)\lim_{z\to1}\frac{1}{\sqrt{\pi}\sqrt{1-z}}\exp\left[-\frac{(\sqrt{x}-\sqrt{y})^2}{1-z}\right]$$
$z\to 1$ infers that $\frac {1}{\sqrt {1-z}}\to +\infty$. According to definition of the Delta function [23]:
$$\delta(x)=\lim_{\mu\to\infty}\frac{\mu}{\sqrt{\pi}}e^{-\mu^2x^2}$$
Replacing $\sqrt {x}$ and $\sqrt {y}$ with $\frac {\sqrt {2}\rho _1}{\omega }$ and $\frac {\sqrt {2}\rho _2}{\omega }$ respectively, then, Eq. (41) can be written as
$$\begin{aligned}&\frac{1}{\pi\sqrt{\rho_1\rho_2}}\exp\left(-iZ_N\frac{\rho_1^2-\rho_2^2}{\omega^2}\right)\lim_{z\to1}\frac{\sqrt{2}}{\sqrt{\pi}\omega\sqrt{1-z}}\exp\left[-\frac{2(\rho_1-\rho_2)^2}{\omega^2(1-z)}\right] \\&=\frac{1}{\pi\rho_1}\delta(\rho_1-\rho_2) \end{aligned}$$
That is,
$$\sum_{p}R_{p,m}(\rho_1)R_{p,m}^{\ast}(\rho_2)=\frac{1}{\pi\omega^2}\frac{\delta(\rho_1,\rho_2)}{\rho_1}$$
Thus, Eq. (23) is proven.

Funding

National Natural Science Foundation of China (61107011, 61675098); China Scholarship Council (2016M601817); Foundation of the Ministry of Education of People's Republic of China (20123219110021).

Acknowledgments

The authors thank Li Xinliang of the Institute of Mechanics, Chinese Academy of Sciences, for providing the turbulence data used in this study. The authors would also like to thank the reviewers for their very helpful comments.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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Figures (8)
Fig. 1.
Fig. 1. SWTBL in a supersonic $24^\circ$ compression ramp with a $2.9$ Ma free stream. Area A is laminar flow, Area B is the transitional flow area, Area C is fully developed turbulence, and Area D is fully developed turbulence with separated shock. $z$ axis is the propagating direction of the light. Each area’s size is $6mm\times 6mm\times 14mm$ with the grids:$121\times 121\times 241$.

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Fig. 2.
Fig. 2. Spatial distribution of refractive index in Area D and corresponding fluctuation of the beam’s irradiance $T(\mathbf {x})$

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Fig. 3.
Fig. 3. $\delta L_z$of the total OAM along the propagating direction in Area D. The amplitude disturbance $T(\mathbf {s})$ on the planes $z=4 mm$ and $z=8 mm$ are also shown. The amplitude disturbance is relatively flat at a distance of less than $6 mm$, and has obvious fluctuation at a distance of more than $6 mm$. Moreover, such fluctuation has a strong inconsistency in space.

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Fig. 4.
Fig. 4. The spiral spectrum of OAM mode in Area D. The propagating distances are $1.4mm$ for Fig. 4(a), $7mm$ for Fig. 4(b), $14mm$ for Fig. 4(c)

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Fig. 5.
Fig. 5. The time evolution of the OAM in the laminar flow. Figs. (5a) and (5b) show the statistical properties of the refractive index; Fig. (5c) shows the total OAM; Fig. (5d) shows the time average of the spiral spectrum.

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Fig. 6.
Fig. 6. The time evolution of the OAM in the laminar flow. Figs. (6a) and (6b) show the statistical properties of the refractive index; Fig. (6c) shows the total OAM; Fig. (6d) shows the time average of the spiral spectrum.

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Fig. 7.
Fig. 7. The time evolution of the OAM in the laminar flow. Figs. (7a) and (7b) show the statistical properties of refractive index; Fig. (7c) shows the total OAM; Fig. (7d) shows the time average of spiral spectrum.

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Fig. 8.
Fig. 8. The time evolution of the OAM in the laminar flow. Figs. (8a) and (8b) show the statistical properties of the refractive index; Fig. (8c) shows the total OAM; Fig. (8d) shows the time average of the spiral spectrum.

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Equations (44)

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E ( x ) = E 0 ( x ) exp ⁡ [ Ψ ( x ) ] ,
ψ n ( x ) = 1 E 0 ( x ) ∫ V ′   E 0 ( x ′ ) g n ( x ′ ) G ( x − x ′ )   d 3 x ′ ,
Ψ = 1 E 0 ( x ) ∫ V ′ E 0 ( x ′ ) H ( x ′ ) G ( x − x ′ )   d 3 x ′ ,
E 0 ( ρ , ϕ , z ) = [ 2 p ! π ( | l | + p ) ! ] 1 2 1 ω ( z ) e i ( 2 p + | l | + 1 ) Ψ G ( z ) [ 2 ρ ω ( z ) ] | l | L p | l | ( 2 ρ 2 [ ω ( z ) ] 2 ) × exp ⁡ ( − i ρ 2 Z N [ ω ( z ) ] 2 ) exp ⁡ ( − ρ 2 [ ω ( z ) ] 2 ) exp ⁡ ( i l ϕ ) exp ⁡ ( i k z ) = R p , l ( ρ , z ) exp ⁡ ( i l ϕ ) ,
L z ( s ) = I m { 1 2 ω A E ∗ ( s , z ) ( ∇ × x ) z ^ E ( s , z ) } ,
( ∇ × x ) z ^ = y ∂ ∂ x − x ∂ ∂ y = ∂ ∂ ϕ ,
L z = I m { 1 2 ω A ∬ S   E ∗ ( s , z ) ∂ ∂ ϕ E ( s , z ) d 2 s } .
L z = I m { 1 2 ω A ∬ S   E 0 ∗ ( ρ , ϕ , z ) exp ⁡ ( Ψ ∗ ) ∂ ∂ ϕ [ E 0 ( ρ , ϕ , z ) exp ⁡ ( Ψ ) ] d 2 s } = I m { 1 2 ω A ∬ S E 0 ∗ ( ρ , ϕ , z ) exp ⁡ ( Ψ ∗ ) ∂ E 0 ( ρ , ϕ , z ) ∂ ϕ exp ⁡ ( Ψ ) d 2 s + 1 2 ω A ∬ S   E 0 ∗ ( ρ , ϕ , z ) exp ⁡ ( Ψ ∗ ) E 0 ( ρ , ϕ , z ) ∂ exp ⁡ ( Ψ ) ∂ ϕ d 2 s } ,
I 1 = 1 2 ω A ∬ S E 0 ∗ ( ρ , ϕ , z ) exp ⁡ ( Ψ ∗ ) ∂ E 0 ( ρ , ϕ , z ) ∂ ϕ exp ⁡ ( Ψ ) d 2 s .
I m { I 1 } = I m { i l 2 ω } = L z 0 .
I 2 = 1 2 ω A ∬ S   E 0 ∗ ( ρ , ϕ , z ) exp ⁡ ( Ψ ∗ ) E 0 ( ρ , ϕ , z ) ∂ exp ⁡ ( Ψ ) ∂ ϕ d 2 s = 1 2 ω A ∬ S   E 0 ∗ ( ρ , ϕ , z ) exp ⁡ ( Ψ ∗ ) E 0 ( ρ , ϕ , z ) exp ⁡ ( Ψ ) ∂ Ψ ∂ ϕ d 2 s .
∂ Ψ ∂ ϕ = 1 E 0 ( x ) ∫ V ′   G ( x − x ′ ) ∂ H ( x ′ ) ∂ ϕ ′ E 0 ( x ′ ) d 3 x ′ .
I 2 = 1 2 ω A ∫ V ′   ∂ H ( ( x ′ ) ) ∂ ϕ ′ E 0 ( x ′ ) d 3 x ′ ∬ S   T ( ρ , ϕ , z ) E 0 ∗ ( ρ , ϕ , z ) G ( x − x ′ ) d 2 s ,
E ( P ) = − 2 i k ∬ S   E ( P 0 ) G ( P − P 0 ) d 2 s .
E t r a n ∗ = − 2 i k ∬ S   T ( x ) E 0 ∗ ( x ) G ( x ′ − x ) d 2 s ,
δ L z = I m { I 2 } = I m { i 4 k ω A ∫ V ′   E t r a n ∗ ( x ′ ) ∂ H ( x ′ ) ∂ ϕ ′ E 0 ( x ′ ) d 3 x ′ } .
L ( z ) − L ( 0 ) = C ∫ 0 z d z ′ ∬ d 2 s I ( ρ , ϕ , z ′ ) ∂ n ( ρ , ϕ , z ′ ) ∂ ϕ
L z = ∑ m m C m .
E s c a t t e r i n g = ∑ m ∑ p c m p E m p ,
C m = ∑ p | c m p | 2 .
c m p = ∬ S   E ( ρ , ϕ , z ) E m p ∗ ( ρ , ϕ , z ) d 2 s ,
C m = ∑ p ∫   ρ 1 d ρ 1 ∫   ρ 2 R p , m ( ρ 1 , z ) R p , m ∗ ( ρ 2 , z ) d ρ 2 ∬   E ∗ ( ρ 1 , ϕ 1 , z ) E ( ρ 2 , ϕ 2 , z ) exp ⁡ [ i m ( ϕ 2 − ϕ 1 ) ] d ϕ 1 d ϕ 2 .
∑ p R p , m ( ρ 1 ) R p , m ∗ ( ρ 2 ) = 1 π ω 2 δ ( ρ 1 , ρ 2 ) ρ 1 ,
C m = 1 π ω 2 ∫   ρ d ρ | ∫ 0 2 π   E ( ρ , ϕ 1 , z ) exp ⁡ ( − i m ϕ 1 ) d ϕ 1 | 2 .
g 2 ( x ) = [ ∇ ψ 1 ( x ) ] 2 .
exp ⁡ ( Ψ ) = exp ⁡ ( ψ 1 + ψ 2 ) .
∂ Ψ ∂ ϕ = − ∂ E 0 ( ρ , ϕ , z ) / ∂ ϕ [ E 0 ( ρ , ϕ , z ) ] 2 ∫ V ′   E 0 ( x ′ ) H ( x ′ ) G ( x − x ′ ) d 3 x ′ + 1 E 0 ( ρ , ϕ , z ) ∫ V ′   E 0 ( x ′ ) H ( x ′ ) ∂ G ( x − x ′ ) ∂ ϕ d 3 x ′ = − i l 1 E 0 ( ρ , ϕ , z ) ∫ V ′   E 0 ( x ′ ) H ( x ′ ) G ( x − x ′ ) d 3 x ′ + 1 E 0 ( ρ , ϕ , z ) ∫ V ′   E 0 ( x ′ ) H ( x ′ ) ∂ G ( x − x ′ ) ∂ ϕ d 3 x ′
G ( x − x ′ ) = 1 4 π exp ⁡ ( i k ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 ) ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2
∫ V ′   E 0 ( x ′ ) H ( x ′ ) ∂ G ( x − x ′ ) ∂ ϕ d 3 x ′ = ∫   d z ′ ∫   ρ d ρ ∫ 0 2 π   E 0 ( ρ ′ , ϕ ′ , z ′ ) H ( ρ ′ , ϕ ′ , z ′ ) × ∂ ∂ ϕ { exp ⁡ ( i k ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 ) 4 π ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 } d ϕ ′
∫ 0 2 π   E 0 ( ρ ′ , ϕ ′ , z ′ ) H ( ρ ′ , ϕ ′ , z ′ ) ∂ ∂ ϕ { exp ⁡ ( i k ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 ) 4 π ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 } d ϕ ′ = { E 0 ( ρ ′ , ϕ , z ′ ) H ( ρ ′ , ϕ , z ′ ) } ∗ ∂ ∂ ϕ exp ⁡ ( i k ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 ) 4 π ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 = ∂ ∂ ϕ { { E 0 ( ρ ′ , ϕ , z ′ ) H ( ρ ′ , ϕ , z ′ ) } ∗ exp ⁡ ( i k ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 ) 4 π ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 } = ∂ ∂ ϕ { E 0 ( ρ ′ , ϕ , z ′ ) H ( ρ ′ , ϕ , z ′ ) } ∗ exp ⁡ ( i k ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 ) 4 π ρ 2 + ρ ′ 2 − 2 ρ ρ ′ cos ⁡ ( ϕ − ϕ ′ ) + ( z − z ′ ) 2 = ∫ 0 2 π   [ ∂ ∂ ϕ ′ E 0 ( ρ ′ , ϕ ′ , z ′ ) ] H ( ρ ′ , ϕ ′ , z ′ ) G ( x − x ′ ) d ϕ ′ + ∫ 0 2 π   E 0 ( ρ ′ , ϕ ′ z ′ ) ∂ H ( ρ ′ , ϕ ′ , z ′ ) ∂ ϕ ′ G ( x − x ′ ) d ϕ ′ = i l ∫ 0 2 π   E 0 ( ρ ′ , ϕ ′ , z ′ ) H ( ρ ′ , ϕ ′ , z ′ ) G ( x − x ′ ) d ϕ ′ + ∫ 0 2 π   E 0 ( ρ ′ , ϕ ′ z ′ ) ∂ H ( ρ ′ , ϕ ′ , z ′ ) ∂ ϕ ′ G ( x − x ′ ) d ϕ ′
∂ Ψ ∂ ϕ = 1 E 0 ( x ) ∫ V ′   G ( x − x ′ ) ∂ H ( x ′ ) ∂ ϕ ′ E 0 ( x ′ ) d 3 x ′
R p , l ( ρ , z ) = [ 2 p ! π ( | l | + p ) ! ] 1 2 1 ω ( z ) e i ( 2 p + | l | + 1 ) Ψ G ( z ) [ 2 ρ ω ( z ) ] | l | L p | l | ( 2 ρ 2 [ ω ( z ) ] 2 ) × exp ⁡ ( − i ρ 2 Z N [ ω ( z ) ] 2 ) exp ⁡ ( − ρ 2 [ ω ( z ) ] 2 )
∑ p R p , m ( ρ 1 ) R p , m ∗ ( ρ 2 ) = ∑ p 2 p ! π ( | l | + p ) ! 1 ω 2 [ 2 ρ 1 ω ] | l | [ 2 ρ 2 ω ] | l | L p | l | ( 2 ρ 1 2 ω 2 ) L p | l | ( 2 ρ 2 2 ω 2 ) × exp ⁡ ( − i Z N ρ 1 2 − ρ 2 2 ω 2 ) exp ⁡ ( − ρ 1 2 + ρ 2 2 ω 2 )
2 π exp ⁡ ( − i Z N x − y 2 ) exp ⁡ ( − x + y 2 ) x | l | 2 y | l | 2 1 ω 2 ∑ p p ! ( | l | + p ) ! L p | l | ( x ) L p | l | ( y )
∑ p = 0 + ∞ p ! ( | l | + p ) ! L p | l | ( x ) L p | l | ( y ) z n = ( x y z ) − | l | 2 1 − z exp ⁡ ( − z x + y 1 − z ) I | l | ( 2 x y z 1 − z )
2 π exp ⁡ ( − i Z N x − y 2 ) lim z → 1 1 ( 1 − z ) z | l | 2 exp ⁡ [ − z 1 + z 2 ( 1 − z ) ( x + y ) ] I | l | ( 2 x y z 1 − z )
I | l | ( t ) ∼ e t 2 π t ∑ k = 0 ∞ ( − 1 ) k Γ ( | l | + k + 1 2 ) ( 2 t ) k k ! Γ ( | l | − k + 1 2 ) + exp ⁡ [ − t + ( | l | + 1 2 ) π i ] 2 π t ∑ k = 0 ∞ Γ ( | l | + k + 1 2 ) ( 2 t ) k k ! Γ ( | l | − k + 1 2 )
e t 2 π t 2 F 0 ( | l | + 1 2 , 1 2 − | l | ; − ; 1 2 t )
I | l | ( t ) ∼ e t 2 π t
2 π exp ⁡ ( − i Z N x − y 2 ) lim z → 1 1 ( 1 − z ) z | l | 2 exp ⁡ [ − z 1 + z 2 ( 1 − z ) ( x + y ) ] exp ⁡ ( 2 x y z 1 − z ) 1 − z 2 π z 1 4 ( x y ) 1 4 = 1 π ( x y ) 1 4 exp ⁡ ( − i Z N x − y 2 ) lim z → 1 1 1 − z exp ⁡ [ − x + y 1 − z ] exp ⁡ ( 2 x y 1 − z ) 1 π
1 π ( x y ) 1 4 exp ⁡ ( − i Z N x − y 2 ) lim z → 1 1 π 1 − z exp ⁡ [ − ( x − y ) 2 1 − z ]
δ ( x ) = lim μ → ∞ μ π e − μ 2 x 2
1 π ρ 1 ρ 2 exp ⁡ ( − i Z N ρ 1 2 − ρ 2 2 ω 2 ) lim z → 1 2 π ω 1 − z exp ⁡ [ − 2 ( ρ 1 − ρ 2 ) 2 ω 2 ( 1 − z ) ] = 1 π ρ 1 δ ( ρ 1 − ρ 2 )
∑ p R p , m ( ρ 1 ) R p , m ∗ ( ρ 2 ) = 1 π ω 2 δ ( ρ 1 , ρ 2 ) ρ 1
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