Revisit: principle of transverse flow measurement by using an optical vortex beam in cylindrical coordinates

Miki Kitazawa; Haruhiko Himura; Takuya Mine; Kyoko Kitamura

doi:10.1364/OSAC.403980

1. Introduction

An optical vortex beam (OVB) [1] possesses a phase distribution along the azimuthal direction in the beam cross section. The beam intensity $I(r)$ at the center of the beam cross-section is null, owing to the nature of its phase singularity. In addition, the beam possesses a helical phase front that carries an orbital angular momentum (OAM). Using these unique characteristics, various types of applications have been proposed, such as optical tweezers [2–6], optical communications [7–12], OAM sorters [13–15] and material processing [16].

Many methods for generating the OVB have been developed, in order to satisfy the number of proposed applications as mentioned above. Most approaches use a hologram [17] or a spatial phase modulator [18]. However, in these cases, some external optical elements other than the laser source are required. In order to develop a compact, single-chip OVB generator, we recently proposed using photonic-crystal surface-emitting lasers (PCSELs). As an initial study, we demonstrated the generation of an OVB using a surface-processed PCSEL in which a quasi-spiral optical phase modulator was fabricated on the emission surface [19].

One of the remarkable phenomena that an OVB is expected to exhibit is an azimuthal Doppler shift $\delta$ [20–22]. The significance of this is that the OVB can be applied to a flow not longitudinally [23,24] but laterally, which was demonstrated in an experiment where an OVB was injected into rubidium vapor [25]. Recently the possibility of using $\delta$ to measure a transverse flow $\vec {V}$ in a plasma was discussed and related studies are continuing [26–28]. Indeed, this method may reveal a complicated profile of $\vec {V}$ in the divertor region of a toroidal fusion plasma [29] where energetic ions flow out towards the divertor along open field lines surrounding the toroidal plasma’s last closed flux surface.

In this paper, we aim to study the feasibility of the OVB to directly measure $\vec {V}$ in a plasma. In section2, we present a new way of deriving $\delta$ due to $\vec V$, which is based on conservation of the momentum and energy of the atom that is excited by the OVB, unlike the derivation in the first paper published by Allen, et al. [20]. In section3, by referring to the obtained results for $\delta$, we consider the feasibility of measuring $\vec V$ laterally with an OVB from a PCSEL injected into the plasma. Finally, a summary is given in section 4.

2. Classical way of deriving $\delta$

The mode of an OVB can be obtained from a mathematical solution of the paraxial wave equation in the cylindrical coordinates, and this mode is called the Laguerre-Gaussian (LG) mode [30,31]. Since a rigorous derivation can settle the confusion over the various descriptions regarding the OVBs in recent papers [1,26,32], we show a detailed derivation in Appendix A. Here we define the LG mode as

(1-3)\begin{cases} {} \vec{E} = E_0(r, z) \exp[i \Theta(r, \phi, z)]\vec{e},\\ E_0(r, z) = \sqrt{\frac{2p!}{\pi (p + |l|)!}}\left(\frac{\sqrt{2} r}{w(z)}\right)^{|l|} \exp \left[- \frac{r^2}{w(z)^2}\right] L_p^{|l|} \left[\frac{2r^2}{w(z)^2}\right] \frac{w_0}{w(z)}, \\ \Theta(r, \phi, z) = kz - l\phi - (1+2p+|l|)\tan^{-1}\left(\frac{z}{z_R}\right) + \frac{kr^2}{2R(z)}, \end{cases}

where $\vec {e}$ is the polarization vector, $p$ is a radial index, $l$ is a topological charge, $w(z)$ is the beam waist, $w_0$ is the minimum value of $w(z)$, $L_p^{|l|}$ indicates a Laguerre polynomials, $k = 2 \pi / \lambda$ is the wave number with wavelength $\lambda$, $z_R$ is the Rayleigh length, and $R(z)$ is the radius of curvature. The solution of Eq. (2) gives $I(r)$ of LG modes, as shown in Fig. 1. Thus, $I(0)$ is null and two symmetric peaks appear in the radial profile except for $l = 0$. The peaks shift outward with increasing $|l|$, and the outermost positions where the normalized intensities decrease to $1/e^2$ are located at $r/w(z)$ = 1.5, 1.8, and 2.0 for $l$ = 1, 2, and 3, respectively.

Fig. 1. Calculated $I$ of OVB (a) on the $r\phi$ plane and (b) along the $r$ axis, and (c) the corresponding phase fronts for the cases $l$ = 0, 1, 2, and 3.

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We derive $\delta$ expected in the measurement. Figures 2(a) and (b) show a schematics diagram of the measurement system and how a two-level atom absorbs the light, respectively. The vector of $\vec {V}$ is written as

(4)$$\vec{V} = V_z \vec{e}_z + V_r \vec{e}_r + V_\phi \vec{e}_\phi,$$

where $V_z$, $V_r$ and $V_\phi$ are the velocity components in the axial, radial and azimuthal directions, respectively, and $\vec {e}_z$, $\vec {e}_r$ and $\vec {e}_\phi$ are the unit vectors in these directions. Hereafter, light with angular frequency $\omega$ and momentum $\vec {p}$ is assumed to enter an atom whose absorption frequency and total atomic mass are indicated by $\omega _0$ and $M$, respectively. When an OVB is injected into a plasma, some photons contained in OVB are absorbed by flowing atoms. Then, electrons in the flowing atoms transition to an upper level. As a result, $\vec {V}$ changes correspondingly to $\vec {V'}$ and $\omega$ changes to $\delta$ = $\omega - \omega _0$. From the momentum and energy conservation laws, we have

(5-6)\begin{cases} {} M\vec{V} + \vec{p} = M\vec{V'},\\ E_1 + \frac{1}{2}M\vec{V}^2 + \hbar \omega = E_2 + \frac{1}{2}M\vec{V'}^2,\end{cases}

where $E_1$ and $E_2$ are the energies of the lower and upper levels. Thus, the value of $E_2 - E_1$ equals $\hbar \omega _0$. Then, we obtain $\delta$ as follows:

(7)$$\delta = \frac{1}{\hbar} \left( \vec{p} \cdot \vec{V} + \frac{\vec{p}^2}{2M} \right).$$

Fig. 2. Schematic diagrams of a $\vec {V}$ measurement. (a) The OVB emitted from the PCSEL goes across the plasma flow, and $\delta$ occurs as a result. We measure $\delta$ with a spectrometer after the beam enters an optical fiber. (b) During the interaction between the injected OVB and the plasma, some of the electrons are excited by photons in the OVB.

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The first term describes the interaction between the incident light and the atom. The second term is a recoil shift, and is smaller than the first term. Therefore, we can ignore the second term and rewrite Eq. (7) as

(8)$$\delta = \frac{\vec{p} \cdot \vec{V}}{\hbar}.$$

Here, $\vec {p}/\hbar$ can be substituted with $\nabla \Theta (r, \phi , z)$ as derived in Appendix B. Then, Eq. (8) is described as

(9)$$\delta = \nabla \Theta(r, \phi, z) \cdot \vec{V}.$$

Using Eq. (3), Eq. (9) can be written as

(10)$$\begin{aligned} \delta &= \left(\frac{\partial \Theta(r, \phi, z)}{\partial z}\right) V_z + \left(\frac{\partial \Theta(r, \phi, z)}{\partial r}\right) V_r + \left(\frac{1}{r} \frac{\partial \Theta(r, \phi, z)}{\partial \phi}\right) V_\phi\\ &= \left[k - \frac{kr^2}{2(z^2+z_R^2)} \left(\frac{2z^2}{z^2+z_R^2}-1\right) - \frac{(1+2p+|l|)z_R}{z^2+z_R^2}\right] V_z + \frac{kr}{R(z)} V_r - \frac{l}{r} V_\phi.\end{aligned}$$

This derivation is valid for emission from the atom. Then, the emission shift $\delta '$ is written as

(11)$$\delta' = \frac{1}{\hbar} \left(\vec{p} \cdot \vec{V} - \frac{\vec{p}^2}{2M}\right).$$

Since the second term is again much smaller than the first term, $\delta '$ results in equal to $\delta$. The above is a classical approach to deriving $\delta$, which yields the same results as in first paper [20] in which a quantum approach was used for the derivation.

3. Feasibility of measuring $V_\phi$ from $\delta$ of OVB

Here we consider the feasibility of measuring $\vec V$ from $\delta$ of the OVB emitted from the PCSEL. The OVB has a wavelength of $\lambda \approx$940 nm, and the spectral width is smaller than 7 pm. The half divergence angle $\theta _d$ is less than 1 deg [19,33]. When $\vec {V}$ of a flowing plasma is predominantly perpendicular to the beam axis and a well-collimated OVB is used, Eq. (10) is approximated by

(12)$$\delta = -\frac{l}{r} V_\phi.$$

Thus, as $V_\phi$ is increased, the value of $\delta$ also increases.

For an actual $V_\phi$ measurement, the absorption time $t_a$ of photons in the plasma must be shorter than the passing time $t_p$ that it takes for the injected OVB to pass through the plasma. Here, $t_a$ is derived from:

(13)$$t_a =\frac{1}{B\rho(\nu)}= \frac{g_1}{g_2}\frac{16 \pi^2 \hbar}{\lambda^3}\frac{1}{A}\frac{1}{\rho(\nu)}=\frac{g_1}{g_2} \left\{\exp\left(\frac{2 \pi \hbar c}{k_B T \lambda}\right)-1\right\} \frac{1}{A},$$

where $B$ is Einstein’s B coefficient, and $\rho (\nu )$ is the spectral energy density of blackbody radiation. The value of $B$ can be obtained from Einstein’s A coefficient [34]. The symbols $g_1$ and $g_2$ are the degeneracies of the lower and upper levels, respectively, $k_B$ is Boltzmann’s constant, and $T$ is the plasma temperature, which is assumed to be in the range between $10^4$ and $10^8$ K. The OVB with $\lambda \approx$940 nm is absorbed in nitrogen (N) or argon (Ar) plasmas. The $t_a$ values of these plasmas are on the order of $10^{-9}$ s for $T\sim\,10^8$ K.

On the other hand, the minimum value of $t_p$ can be estimated by $t_p \approx D/V_\phi$, where $D$ is the diameter of the electrode of the PCSEL, which determines the envelope of $I(r)$ of the OVB. When $D$ is $\sim$0.10 mm, the maximum value of $V_\phi$ is on the order of $10^5$ m/s, because $t_a$ is on the order of $10^{-9}$ s. Hereafter, we will assume that $V_\phi$ is on the order of $10^5$ m/s [35].

Figure 3(a) shows the dependence of $|\delta _f|$ on $r$, where $\delta _f$ = $\delta /2\pi$, which has the frequency unit. As recognized in Eq. (12), $\delta _f$ is proportional to $l$ but is inversely proportional to $r$. As mentioned in Sec. 2, $I(0)$ = 0, and a pair of symmetric peaks shift outwardly in the $r$ direction as the value of $l$ increases. Therefore, the spectrum of $\delta$ depends on both $l$ and $I(r)$ of the OVB. Figure 3(b) shows the expected wavelength shift $|\Delta \lambda |$, which is obtained from $|\Delta \lambda |$ = $|2\pi c (1/\omega - 1/\omega _0)| \approx |\delta _f|\lambda ^2/c$, where the OVB is assumed to be a line spectrum. Not only does each peak of $|\Delta \lambda |$ shift proportionally to $l$, but the spectral broadening also changes according to $l$. To clearly evaluate the broadening, full widths at half maxima (FWHM) of the spectra for different $l$ in Fig. 3(b) are plotted in Fig. 3(c). Values of FWHM decrease with increasing $l$ because the width between the pair of symmetric peaks of each $I(r)$ increases as $l$ increases.

Fig. 3. Expected values of $|\delta |$ in a plasma with $V_\phi \sim\,10^5$ m/s. (a) Increasing $l$ yields larger $|\delta _f|$. (b) Spectra of the OVB become narrower with increasing $l$. Here, $w(z)$ of the OVB is set at 0.05 mm. Further, the spectral peak shifts to larger values along the $|\Delta \lambda |$ axis as $l$ increases. (c) Values of FWHM become demonstrably smaller with increasing $l$.

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Furthermore, we mention the exact values of $\delta$ in the most feasible case where we injected an OVB with $l$ = 2 and $D$ = 0.05 mm. The value of $D$ of the fabricated PCSEL determined the envelop of $I(r)$ of the emitted OVB (see also Fig. 1(b)). Thus, we can obtain the following relationship between $D$ and $w_0$ for the case of $l$ = 2: $D$ = 2$\times$1.8$w_0$. Thus, $w_0$ is calculated to be 0.014 mm. After being launched from the PCSEL (see also Fig. 2(a)), the OVB starts to expand radially. If the beam is collimated at $z$ = 5 mm by a lens, $w(z)$ could shorten to less than 0.10 mm because $\theta _d$ $\leq$ 1 deg. In this case, the peak $|\Delta \lambda |$ would be 0.94 pm and its FWHM would broaden by 0.87 pm as shown in Figs. 4(b) and (c). These values are as large as 13 % of the spectral width ($\approx$ 7 pm) of the OVB emitted from the PCSEL, which suggests that the beam can be applied to measure $V_\phi \sim 10^5$ m/s in a flowing plasma.

Fig. 4. Summaries of peak values and FWHM for different values of $l$ and $w(z)$. Red, blue, and green markers are the data for $l=$1, 2 and 3, respectively, where $V_\phi \sim 10^5$ m/s, and $\lambda$ is 940 nm. (a) Peak values of $|\delta _f|$. (b) Peak values of $|\Delta \lambda |$. (c) FWHM of $|\Delta \lambda |$.

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Finally, we will briefly mention the prospect of an actual measurement of $\vec V$ using the OVB. First, high-temperature plasmas are confined in a vacuum chamber to which several types of viewing windows are attached. The emitted OVB propagates while being accompanied by some attenuation that occurs in the viewing window. To alleviate the attenuation, the viewing window should be made of quartz glass that is capable of transmitting 940 nm infrared light. Second, when plasmas are produced, some spectral broadening may take place due to mechanical vibration of the whole measurement system. In addition, stray light may infiltrate a detector. Therefore, the measurement system needs to be carefully set-up and used. Third, if the measurement system is applied to a magnetically confined flowing plasma, as this type of plasma never remains in the same place. These plasmas usually fluctuate owing to its finite temperature and are sometimes disrupted, which is called magnetohydrodynamic instability. In this case, the accuracy of the measurement point in the flowing plasma should be double-checked by other instruments.

4. Summary

In this paper, we revisited the derivation of $\delta$ in detail, involving conservation of momentum and energy. The feasibility of measuring $\vec {V}$ using the $\delta$ of an OVB was studied. The value of $\delta$ is proportional to $l$ and is inversely proportional to $r$ from the beam axis. Owing to $I(r)$ of the OVB, the spectrum of $\delta$ is broadened. In a specific case when an OVB with $l = 2$ from a PCSEL having $w(z)$ = 0.10 mm is applied to the measurement, we can measure $|\vec {V}|$ of $10^5$ m/s by the spectral changes: the peak $|\Delta \lambda |$ is $\sim$1 pm, and its spectrum broadens by $\sim$1 pm at FWHM. The OVBs emitted from PCSELs, potentially having both a small beam waist and a small spectral width, are promising for realizing a compact plasma flow measurement system and would be a powerful tool in the fields of plasma diagnostics, albeit with some outstanding concerns in actual experiments. We will address some of these experimental issues and present results in subsequent papers.

A. Derivation of LG mode

Here, we derive the LG mode in detail. In a vacuum, Maxwell’s equations are written as the following set of partial differential equations:

(14-15)\begin{cases} {} \rho = 0, \\ \vec{i} = 0, \label{Maxwellco2} \end{cases}

(16-19)\begin{cases} {} \nabla \cdot \vec{E} = 0, \\ \nabla \times \vec{E} = - \frac{\partial \vec{B}}{\partial t}, &\\ \nabla \cdot \vec{B} = 0, &\\ \nabla \times \vec{B} = \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}, \end{cases}

where $\rho$ is charge density, $\vec {i}$ is current density, $\vec {E}$ is the electric field, and $\vec {B}$ is the magnetic field. Additionally, $\mu _0$ and $\epsilon _0$ satisfy $\mu _0 \epsilon _0 = 1/c^2$, where $c$ is the speed of light.

First, we take the curl of both sides of Eq. (17) as

(20)$$\nabla \times (\nabla \times \vec{E}) = \nabla \times \left(- \frac{\partial \vec{B}}{\partial t} \right).$$

By using $\nabla \times (\nabla \times \vec {E})$=$\nabla (\nabla \cdot \vec {E})- \nabla ^2 \vec {E}$, we obtain

(21)$$\nabla^2 \vec{E} = \frac{\partial}{\partial t}\left(\nabla \times \vec{B}\right) = \frac{\partial}{\partial t}\left(\mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}\right) = \frac{1}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2}.$$

Assuming that the optical beams propagate along the $z$ axis, the primary spatial dependence of $\vec {E}$ may be the $\exp (\pm ikz)$ variation, which has a spatial period equal to the wavelength $\lambda$. Here, the beam propagates along the $+z$ direction. Then, we set $\vec {E}$ as

(22)$$\vec{E} = u(r, \phi, z) \exp i(kz - \omega t)\vec{e},$$

where $u(r, \phi , z)$ is a complex scalar wave amplitude that describes the transverse profile of the beam, $k=2\pi /\lambda$ is the wave number with $\lambda$, $\omega$ is angular frequency, and $\vec {e}$ is the polarization vector. Substituting Eq. (22) into Eq. (21), we obtain

(23)$$(\nabla^2 + k^2) \vec{E} = 0.$$

In cylindrical coordinates, the scalar operator $\nabla ^2$ is written as

(24)$$\nabla^2 = \frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right) +\frac{1}{r^2}\frac{\partial^2}{\partial \phi^2} +\frac{\partial^2}{\partial z^2}.$$

Thus, the wave equation in cylindrical coordinates is given by:

(25)$$\left\{\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)\ +\frac{1}{r^2}\frac{\partial^2}{\partial\phi^2} +\frac{\partial^2}{\partial z^2} + k^2 \right\} \vec{E}\ = 0,$$

Substituting Eq. (22) into the left side of Eq. (25), the reduced equation is

(26)$$\begin{aligned} &\exp i(kz-\omega t)\vec{e}\left\{\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right)+\frac{1}{r^2}\frac{\partial^2}{\partial \phi^2}\right\}u(r, \phi, z)\\ &+\exp i(kz-\omega t)\vec{e} \frac{\partial^2 u(r, \phi, z)}{\partial z^2} + 2ik \exp i(kz-\omega t)\vec{e} \frac{\partial u(r, \phi, z)}{\partial z}\\ &= 0. \end{aligned}$$

The $z$ dependence of $u(r, \phi , z)$ is in general weak compared not only to $\lambda$, as in $\exp i(kz-\omega t)$, but also to the transverse variations owing to the finite width of the beam. This slowly varying dependence of $u(r, \phi , z)$ along the $z$ axis can be expressed by the paraxial approximation as

(27)$$\left|\frac{\partial^2 u(r, \phi, z)}{\partial z^2}\right| \ll \left|2k\frac{\partial u(r, \phi, z)}{\partial z}\right|.$$

Under this approximation, we ignore the $\partial ^2 / \partial z^2$ term in Eq. (26) and eliminate $\exp i(kz-\omega t) \vec {e}$. Then, Eq. (26) can be written as

(28)$$\left\{\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right) +\frac{1}{r^2}\frac{\partial^2}{\partial \phi^2} +2ik \frac{\partial}{\partial z} \right\}u(r, \phi, z) = 0.$$

A fundamental Gaussian mode that satisfies the paraxial wave equation can be written as

(29-32)\begin{cases} {} w(z) = w_0 \sqrt{1 + \left(\frac{z}{z_R} \right)^2}, &\\ R(z) = z + \frac{z_R^2}{z}, &\\ \psi(z) = \tan^{-1} \left(\frac{z}{z_R}\right), &\\ z_R = \frac{\pi w_0^2}{\lambda}, \end{cases}

where $w(z)$ is the beam waist, $w_0$ is the minimum value of $w(z)$, $R(z)$ is the radius of curvature, $\psi (z)$ is the Guoy phase, and $z_R$ is the Rayleigh length. The solution of Eq. (28) is the LG mode. Equation (28) can be easily solved by using a coordinate system $(s,\phi ,\psi )$ derived from the cylindrical coordinate system $(r,\phi ,z)$, when $s$ is given by

(33)$$s = \frac{r}{w(z)}.$$

Since $w(z)$, $R(z)$, and $\psi (z)$ are given by Eqs. (29), (30), and (31), $R(z)$ and $w(z)$ can be rewritten as

(34-35)\begin{cases} {} R(z) = z_R \tan \psi + \frac{z_R^2}{z_R \tan \psi} = \frac{z_R}{\cos \psi \sin\psi}, \\ w(z) = w_0 \sqrt{1 + \tan^2 \psi} = \frac{w_0}{\cos \psi}, \end{cases}

since $s$ is defined in Eq. (33), we can write $z$ and $r$ as

(36-37)\begin{cases}{} z = z_R \tan \psi, \\ r = sw(z) = sw_0 \sqrt{1 + \tan^2 \psi} = \frac{sw_0}{\cos \psi}. \end{cases}

Eq. (28) can be easily solved by using a coordinate system $(s,\phi ,\psi )$ derived from the cylindrical coordinate system $(r,\phi ,z)$ as

(38-39)\begin{cases}{} \frac{\partial}{\partial r} = \frac{\cos \psi}{w_0} \frac{\partial}{\partial s}, &\\ \frac{\partial}{\partial z} = \frac{\cos^2 \psi}{z_R} \left(\frac{\partial}{\partial \psi} - \frac{sw_0 \sin \psi}{\cos^2 \psi} \frac{\partial}{\partial r}\right)\nonumber \\ =\frac{\cos^2 \psi}{z_R} \frac{\partial}{\partial \psi} - \frac{s \cdot \sin \psi \cos \psi}{z_R} \frac{\partial}{\partial s}. \end{cases}

Additionally, we rewrite Eq. (28) in the $(s,\phi ,\psi )$-coordinates as

(40)$$\begin{aligned} &\left\{\frac{1}{r}\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right) + \frac{1}{r^2}\frac{\partial^2}{\partial \phi^2} +2ik \frac{\partial}{\partial z} \right\}u(r, \phi, z)\\ &= \left\{\frac{1}{s} \frac{\partial}{\partial s} s \frac{\partial}{\partial s} +\frac{1}{s^2} \frac{\partial^2}{\partial \phi^2} + 2i \frac{w_0^2 k}{z_R} \left( \frac{\partial}{\partial \psi} - \tan \psi \cdot s \cdot \frac{\partial}{\partial s}\right)\right\}u(r, \phi, z)=0. \end{aligned}$$

Then, we can calculate $w_0^2 k/z_R$ as

(41)$$\frac{w_0^2 k}{z_R} = \frac{w_0^2 k \lambda}{\pi w_0^2} = \frac{k \lambda}{\pi} = \frac{2 \pi \lambda}{\lambda \pi} = 2.$$

Therefore, we can rewrite Eq. (28) as

(42)$$\left\{\frac{1}{s} \frac{\partial}{\partial s} s \frac{\partial}{\partial s}+\frac{1}{s^2} \frac{\partial^2}{\partial \phi^2} + 4i \left( \frac{\partial}{\partial \psi} - \tan \psi \cdot s \cdot \frac{\partial}{\partial s} \right) \right\} u(r, \phi, z) = 0.$$

Assuming a trial solution of the paraxial wave equation, a higher-order LG solution of the $(s, \phi , \psi )$-coordinates wave equation can be derived as

(43)$$u(r, \phi, z) = v(s,\phi,\psi) \exp\left\{\left(\frac{ik r^2}{2R(z)}\right)\right\} = v(s,\phi,\psi) \exp(is^2 \tan \psi).$$

Next, we substitute Eq. (43) into the first term of Eq. (42)) as

(44)$$\begin{aligned} &\frac{1}{s} \frac{\partial}{\partial s} s \frac{\partial}{\partial s}u(r, \phi,z) =\frac{1}{s} \frac{\partial}{\partial s} s \frac{\partial}{\partial s} v(s,\phi,\psi) \exp(is^2 \tan \psi)\\ &=\exp(is^2 \tan \psi) \left\{4i \tan \psi - 4s^2 \tan^2 \psi + \left( 4is \tan \psi + \frac{1}{s}\right)\frac{\partial}{\partial s} + \frac{\partial^2}{\partial s^2} \right\}v(s,\phi,\psi). \end{aligned}$$

In the same way, the second term is rewritten as

(45)$$\frac{1}{s^2} \frac{\partial^2}{\partial \phi^2} u(r, \phi,z) =\frac{1}{s^2} \exp(is^2 \tan \psi) \frac{\partial^2}{\partial \phi^2}v(s,\phi,\psi).$$

The third term is also rewritten as

(46)$$\begin{aligned}&4i \left( \frac{\partial}{\partial \psi} - s \tan \psi \frac{\partial}{\partial s} \right)u(r, \phi,z)\\ &= 4i \exp(is^2 \tan \psi)\left\{ \left(\frac{is^2}{\cos^2 \psi} + \frac{\partial}{\partial \psi} \right) - s\tan \psi \left( 2is \tan \psi + \frac{\partial}{\partial s} \right)\right\}v(s,\phi,\psi). \end{aligned}$$

As a result, we rewrite Eq. (43) as

(47)$$\begin{aligned} &\exp(is^2 \tan \psi) \left\{4i \tan \psi - 4s^2 \tan^2 \psi + \left( 4is \tan \psi + \frac{1}{s} \right)\frac{\partial}{\partial s} + \frac{\partial^2}{\partial s^2} \right\}v(s,\phi,\psi)\\ &+ \frac{1}{s^2} \exp(is^2 \tan \psi) \frac{\partial^2}{\partial \phi^2}v(s,\phi,\psi)\\ &+ 4i \exp(is^2 \tan \psi)\left\{ \left(\frac{is^2}{\cos^2 \psi} + \frac{\partial}{\partial \psi} \right) - s\tan \psi \left( 2is \tan \psi + \frac{\partial}{\partial s} \right)\right\}v(s,\phi,\psi) = 0. \end{aligned}$$

Dividing both sides of Eq. (47) by $\exp (is^2 \tan \psi )$ and rearranging the equation, we can rewrite Eq. (47) as

(48)$$\left(4i \tan \psi - 4s^2 + \frac{1}{s} \frac{\partial}{\partial s} + \frac{\partial^2}{\partial s^2} x + \frac{1}{s^2}\frac{\partial^2}{\partial \phi^2} + 4i \frac{\partial}{\partial \psi}\right)v(s,\phi,\psi) = 0.$$

In the $(s,\phi ,\psi )$-coordinates, the elementary solution can be separated into products of identical solutions in the $s$, $\phi$ and $\psi$ directions as

(49)$$v(s,\phi,\psi) = f(s)g(\phi)h(\psi),$$

where $f(s)$, $g(\phi )$ and $h(\psi )$ have the same mathematical form. We can therefore find the solution in only one $(s,\phi ,\psi )$-coordinates. Then, we substitute Eq. (49) into the Eq. (48) and obtain

(50)$$\frac{1}{s} \frac{f'(s)}{f(s)} + \frac{f^{\prime\prime}(s)}{f(s)} + \frac{1}{s^2} \frac{g^{\prime\prime}(\phi)}{g(\phi)} - 4s^2 + 4i \frac{h'(\psi) + h(\psi)\tan \psi}{h(\psi)} = 0,$$

where $f'(s)$ and $f''(s)$ are the first and second derivatives of $f(s)$ with respect to its total arguments: $f'(s) = \partial f(s)/\partial s$ and $f''(s) = \partial ^2 f(s)/\partial s^2$. Then, we solve Eq. (50) in terms of $h(\psi )$. Since $\psi$ and other variables have been separated, we can use separation of variables as

(51)$$i \frac{h'(\psi) + h(\psi) \tan \psi}{h(\psi)} = C,$$

where $C$ is a constant. Equation (51) gives $h(\psi )$, which is

(52)$$h(\psi) = C_1 \cos \psi \exp(-iC\psi),$$

where $C_1$ is constant. Then, substituting $h(\psi )$ into Eq. (50) and multiplying both sides by $s^2$, we obtain

(53)$$s \frac{f'(s)}{f(s)} + s^2 \frac{f^{\prime\prime}(s)}{f(s)} + \frac{g^{\prime\prime}(\phi)}{g(\phi)} - 4s^4 + 4Cs^2 = 0.$$

In the same way, we solve it in terms of $g(\phi )$. Furthermore, the third term in Eq. (53) can be written as

(54)$$\frac{g^{\prime\prime}(\phi)}{g(\phi)} = -l^2.$$

Here, $l$ is defined as positive when the beam possesses a right-handed phase rotation, that is, when the phase moves clockwise along the circumference of the beam cross-section. Then, we can write $g(\phi )$ as

(55)$$g(\phi) = C_2 \exp(-il\phi).$$

Then, we substitute Eq. (55) into Eq. (53) and divide both sides by $s^2$, yielding

(56)$$\frac{f^{\prime\prime}(s) + \frac{1}{s} f'(s) -4s^2f(s)}{f(s)} - \frac{l^2}{s^2} + 4C = 0.$$

Finally, in the same way, we solve it in terms of $f(s)$. Equation (56) is written as

(57)$$\left\{\frac{1}{s} \frac{\partial}{\partial s} s \frac{\partial }{\partial s} + \left(-4s^2 -\frac{l^2}{s^2} + 4C \right)\right\}f(s) = 0.$$

When we replace $s$ by $\sigma$ to solve Eq. (57) easily, as

(58-59)\begin{cases} {} \sigma = 2s^2, \\ f(s) = f(\sigma) = \sigma^{\frac{|l|}{2}} \exp\left(-\frac{\sigma}{2}\right) f_0(\sigma).\label{fs0} \end{cases}

Thus, Eq. (59) is written as

(60)$$\left\{\sigma \frac{\partial^2}{\partial \sigma^2} +\left(|l| - \sigma + 1 \right)\frac{\partial }{\partial \sigma} +\left( - \frac{|l|}{2} -\frac{1}{2} + \frac{C}{2}\right)\right\}f_0(\sigma)= 0.$$

Substituting

(61)$$p = - \frac{|l|}{2} -\frac{1}{2} + \frac{C}{2} $$

into Eq. (60), we thus obtain

(62)$$\left\{\sigma \frac{\partial^2}{\partial \sigma^2} +\left(|l| - \sigma + 1 \right)\frac{\partial }{\partial \sigma} +p\right\}f_0(\sigma) = 0. $$

Equation (62) is similar to the standard differential equation for the Laguerre polynomials $L$, which has the form of

(63)$$xL^{\prime\prime} + \left(\alpha -x + 1\right)L' + nL = 0, $$

where

(64)$$L = L_n^{(\alpha)}(x) = \frac{\exp(x) x^{-\alpha}}{h!} \frac{d^n}{dx^n} \left(\exp(-x) x^{n+\alpha}\right) = \sum_{r=0}^n (-1)^r \left(\frac{n+\alpha}{n-r}\right)\frac{x^r}{r!}.$$

Here, $f_0(\sigma )$ and $L$ are identical when the following three conditions are simultaneously satisfied:

(65-67)\begin{cases} {} \alpha = |l|, \\x = \sigma, \\ n = p.\end{cases}

In this case, $f_0(\sigma )$ is written as

(68)$$f_0(\sigma) = L_p^{|l|}(\sigma).$$

Substituting this into Eq. (59), we obtain

(69)$$f(\sigma) = \sigma^{\frac{|l|}{2}} \exp\left(-\frac{\sigma}{2}\right) f_0(\sigma) = \sigma^{\frac{|l|}{2}} \exp\left(-\frac{\sigma}{2}\right) L_p^{|l|}(\sigma).$$

Then, $f(s)$ can be written as

(70)$$f(s) = (\sqrt{2s^2})^{|l|}\exp(-s^2) L_p^{|l|}(2s^2).$$

Therefore, we can write the solutions of $f(s)$, $g(\phi )$ and $h(\psi )$ as

(71-73)\begin{cases} {} f(s) = \left(\sqrt{2s^2}\right)^{|l|} \exp(-s^2) L_p^{|l|}(2s^2), \\ g(\phi) = C_2 \exp(-il\phi), \\ h(\psi) = C_1 \cos \psi \exp(-iC\psi), \end{cases}

In addition, we can write the LG mode (Eq. (43)) as

(74)$$\begin{aligned} &u(r, \phi, z)\\ &= C_3 (\sqrt{2s^2})^{|l|} \exp(-s^2) L_p^{|l|}(2s^2) \exp(-il\phi) \cos \psi \exp(-iC\psi) \exp(is^2 \tan \psi). \end{aligned}$$

Here, we set $C_3=C_1C_2$. In the cylindrical coordinates, Eq. (74) can be rewritten as

(75-77)\begin{cases} {} \vec{E} = E_0(r, z) \exp[i \Theta(r, \phi, z)]\vec{e}\\E_0(r, z)= C_3\left(\frac{\sqrt{2} r}{w(z)}\right)^{|l|} \exp \left[- \frac{r^2}{w(z)^2}\right] L_p^{|l|} \left[\frac{2r^2}{w(z)^2}\right] \frac{w_0}{w(z)}, \\ \Theta(r, \phi, z) = kz - l\phi - (1+2p+|l|)\tan^{-1}\left(\frac{z}{z_R}\right) + \frac{kr^2}{2R(z)}.\end{cases}

In the case where

(78)$$\int_0^{2\pi} \int_0^\infty E_0(r, z)^2 r dr d\phi = w_0^2,$$

is satisfied, we can exactly calculate $C_3$. Further, Laguerre polynomials become orthogonal as

(79)$$\int_0^\infty L^{\alpha}_n(x)^2 \exp(-x) x^\alpha dx = \frac{(n+\alpha)!}{n!}.$$

Assuming $x=2r^2/w(z)^2$ and substituting Eqs. (66) and (67) into Eq. (79), Eq. (79) can be rewritten as

(80)$$\begin{aligned} &\int_0^\infty L^{|l|}_p\left[\frac{2r^2}{w(z)^2}\right]^2 \exp\left[-\frac{2r^2}{w(z)^2}\right] \left(\frac{2r^2}{w(z)^2}\right)^{|l|}\frac{d}{dr} \left(\frac{2r^2}{w(z)^2}\right) dr\\ &=\int_0^\infty E_0(r, z)^2 \frac{1}{C_3^2} \frac{4}{w_0^2} r dr\\ &= \frac{(p+|l|)!}{p!}. \end{aligned}$$

Integrating the left side of Eq. (80) with $\phi$, that is,

(81)$$\int_0^{2\pi} \int_0^\infty E_0(r, z)^2 \frac{1}{C_3^2} \frac{4}{w_0^2} r dr d\phi = \frac{2\pi (p+|l|)!}{p!},$$

we obtain $C_3 = \sqrt {2p!/\pi (p+|l|)!}$ from Eqs. (78) and (81). Then, we can write the LG mode as

(82)$$\begin{aligned} &u(r, \phi, z)\\ &= \sqrt{\frac{2p!}{\pi (p + |l|)!}}\left(\frac{\sqrt{2} r}{w(z)}\right)^{|l|} \exp \left[- \frac{r^2}{w(z)^2}\right] L_p^{|l|}\left[\frac{2r^2}{w(z)^2}\right]\\ &\times \exp(-il\phi) \frac{w_0}{w(z)} \exp\left[-i(1+2p+|l|)\tan^{-1}\left(\frac{z}{z_R}\right) \right] \exp \left[i \frac{kr^2}{2R(z)}\right]. \end{aligned}$$

As a result, $\vec {E}$ can be written as Eqs. (1), (2) and (3).

B. Derivation of $\vec {p}$

The definition of $\vec {p}$ is written with the Poynting vector $\vec {S}$ as

(83)$$\vec{p} = \int\int\int \frac{1}{c^2} \vec{S} r dr d\phi dz.$$

Considering a typical case in which the frequency of a photon is sufficiently larger than that of the atom, and the relaxation time of the atom is sufficiently longer than the period of the light, $\vec {S}$ can be substituted with a time averaged $\langle{\vec {S}}\rangle$ as

(84)$$\vec{S} \approx \langle{\vec{S}}\rangle = \frac{1}{2 \mu_0}Re[\vec{E} \times \vec{B^*}],$$

where $\vec {B^*}$ is the complex conjugate of $\vec {B}$. From Eqs. (16) and (17), when $1/kw(z) << 1$, we can write $\vec {E}$ and $\vec {B}$ as

(85)$$\vec{E} = \left( \begin{array}{c} E_0(r, z)\exp [i(\Theta(r, \phi, z))] \\ 0 \\ \frac{i}{k}\frac{\partial}{\partial x} E_0(r, z)\exp [i(\Theta(r, \phi, z))] \end{array} \right),$$

and

(86)$$\vec{B} = \frac{1}{c}\left( \begin{array}{c} 0 \\ E_0(r, z)\exp [i\Theta(r, \phi, z)] \\ \frac{i}{k}\frac{\partial}{\partial y}E_0(r, z)\exp [i\Theta(r, \phi, z)] \end{array} \right).$$

Then, $\vec {E} \times \vec {B^*}$ can be calculated as

(87)$$\begin{aligned} \vec{E} \times \vec{B^*} &= \frac{1}{c}\left( \begin{array}{c} E_0(r, z)^2 \frac{1}{k}\frac{\partial}{\partial x} \Theta(r, \phi, z)\\ E_0(r, z)^2 \frac{1}{k}\frac{\partial}{\partial y} \Theta(r, \phi, z)\\ -E_0(r, z) \exp[-i\Theta(r, \phi, z)] \frac{i}{k} \frac{\partial}{\partial z} E_0(r, z) \exp[i\Theta(r, \phi, z)] \end{array} \right)\\ &= \frac{1}{c}\left( \begin{array}{c} E_0(r, z)^2 \frac{1}{k}\frac{\partial}{\partial x} \Theta(r, \phi, z)\\ E_0(r, z)^2 \frac{1}{k}\frac{\partial}{\partial y} \Theta(r, \phi, z)\\ E_0(r, z)^2 \frac{1}{k}\frac{\partial}{\partial z} \Theta(r, \phi, z) \end{array} \right)\\ &= \frac{E_0(r, z)^2}{c}\frac{1}{k} \nabla \Theta(r, \phi, z). \end{aligned}$$

Therefore, $\langle{\vec {S}}\rangle$ in Eq. (84) is written as

(88)$$\begin{aligned} \langle{\vec{S}}\rangle &= \frac{1}{2\mu_0} Re[\vec{E} \times \vec{B^*}] = \frac{1}{2\mu_0} \frac{E_0(r, z)^2}{c}\frac{1}{k} \nabla \Theta(r, \phi, z)\\ &=\frac{c^2 \epsilon_0}{2} \frac{E_0(r, z)^2}{c}\frac{1}{k} \nabla \Theta(r, \phi, z) = \frac{c \epsilon_0 E_0(r, z)^2}{2} \frac{1}{k} \nabla \Theta(r, \phi, z). \end{aligned}$$

Here, $\epsilon _0 E_0(r, z)^2/2$ is the density of energy, which is defined by the number of photons per unit volume $N$ as

(89)$$\frac{\epsilon_0 E_0(r, z)^2}{2} = N \hbar c k.$$

Considering one photon and substituting Eq. (88) into Eq. (83), we have

(90)$$\vec{p} = \int\int\int N \hbar \nabla \Theta(r, \phi, z) r dr d\phi dz = \hbar \nabla \Theta(r, \phi, z).$$

Substituting Eq. (90) into Eq. (8), we obtain

(91)$$\delta =\frac{\hbar \nabla \Theta(r, \phi, z) \cdot \vec{V}}{\hbar} = \nabla \Theta(r, \phi, z) \cdot \vec{V}.$$

Thus, the result is Eq. (9).

Funding

Institute of Advanced Energy, Kyoto University (ZE Research Program, IAE (ZE2020A-04)).

Acknowledgments

The authors would like to thank H. Kitagawa, for valuable discussions.

Disclosures

The authors declare no conflicts of interest.

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Revisit: principle of transverse flow measurement by using an optical vortex beam in cylindrical coordinates

Abstract

1. Introduction

2. Classical way of deriving $\delta$

3. Feasibility of measuring $V_\phi$ from $\delta$ of OVB

4. Summary

A. Derivation of LG mode

B. Derivation of $\vec {p}$

Funding

Acknowledgments

Disclosures

References

Cited By

Figures (4)

Equations (71)

OSA Continuum