Orbital angular momentum due to modes interference

Since the pioneering and interesting work of the orbital angular momentum presented by L. Allen [1], the study of the subject have covered many interesting applications [2]; a compilation overview over the last 25 years, with several theoretical and experimental applications was reported in [3]. The authors in [4] have recently demonstrated that a rotationally arranged nano-antenna can be used to convert the phase information in a twisted light beam into spectral information, which hence can be used to classify the phase state of the twisted light beam. This effect has strong influence on the optical properties of dielectric and plasmonic particles and it is useful for new technological applications. The physical properties of the orbital angular momentum have the potential to improve the performance of optical communication systems in different ways [5], but it is not the aim of this manuscript to explore the full list of possibilities. Exploiting the physical properties of the orbital angular momentum is the subject of an increasing number of research topics nowadays. In this context, here we focus in the use of the scalar formalism for studying the orbital angular momentum for invariant beams; due to its simplicity, this approach has been theoretically [6] and experimentally [7, 8] used and tested.
Recently, the scalar representation has been used to describe scattering problems using partial waves series in the far field approximation, where the impinging source is a structured field [9]. Effects also considering the polarization [10], the extinction [11] and driven acoustic radiation forces [12] have also been considered. Furthermore, the scalar approach is a natural representation for the structured beams family, also known as “non-diffracting beams”. The study of these fields covers different areas, from quantum mechanics to astrophysics [13]. Such fields are the Bessel [14], Mathieu [15] and Weber beams [16], which are constructed by superposition of plane-waves [17], formalism which is called angular spectrum representation [18]. These ideal fields propagate indefinitely without changing their transverse intensity distribution [19], even in the presence of massive phase perturbations and into inhomogeneous media [20]. As an interesting application, among others, this physical effect increases the resolution and contrast to image sub-cellular components and organelles in different microscopy methods [21].
Otherwise, the study of structured invariant beams is of increasing interest in optical physics. Their properties make them particularly attractive for optical design [22], for studying propagation through etalons and crystals [23]. Another interesting example is the use of Bessel beams for driving an optimal single tractor beam for dielectric particles with cylindrical shape [24].
The study of nonparaxial orbital angular momentum was recently revisited in [25] using Bessel beams; in that work, the authors show well defined orbital angular momentum properties, without considering the possibility of contributions due to mixed modes superimposed along their propagation. To the best of our knowledge, this feature has not been fully explored. However, recently, using the scalar potential approach, fundamentals electromagnetic properties, such as the energy density, the Poynting vector, the Maxwell stress tensor for nonparaxial beams were derived [26]. Currently, the study of the Poynting vector has been taken considerable interest due to its features and properties along the propagation.
In the context of structured beams, Novitsky in [27] has shown that Bessel beams possess negative values in the longitudinal and azimuthal components of the Poynting vector, which depends in the superposition of mixed modes and on the phase detuning between the complex amplitudes $c_{TE}$ and $c_{TM}$ of the transversal electric or magnetic part of the beam representation. Since then, many interesting results have been reported, such as the explanation of optical pulling forces [28], effects in meta-materials [29], tractor beams [30] and optical manipulation [31], to name a few of them. Another interesting case was reported for X-Waves in [32], where the propagation direction of their negative Poynting vector could be locally changed using carefully chosen complex amplitudes; however, we showed [26] that the negative behavior can be found independently of the mode interference. Nonetheless, the Poynting vector behavior presented in the Bessel beams and in the X-Waves, mentioned above, has opened a discussion related to tractor beams generation, and with other interesting applications, such as the forces that can be locally oriented in a direction opposite to the propagation wave vector [33].
Then, what is the relationship between mixed modes and the Negative Poynting Vector (NPV)? This effect can be physically defined as an uncommon response produced by the local sign change in the Poynting vector components along its propagation. Recently, we reported in [34] to deep in the understanding of the NPV to study the invariant family beams. The main result of our work was to show the negative local change of the negative Poynting vector, which can be independently obtained without to superposition of mixed modes TE/TM, as it is in the case of Weber beams [34]. Additionally, considerable interest in theoretically and experimentally studies of the vectorial structured fields are driven by the possibility to create a wide variety of exotic optical focal fields with homogeneous and spatially inhomogeneous states of polarization; an interesting review of the wide scope of interest and applications is presented in [35].
In this context, the study of the properties of the orbital angular momentum from the theoretical an experimental points of view can open new engineering technologies. Using the concept of the Poynting vector can be useful, as shown by proposals to measure the orbital angular momentum using superposition of vector mixed modes TE/TM. This was performed and tested for a Laguerre Gauss beams [36]. This approach may open interest for other structured fields, as it was also proposed by using $X$ waves in ultrashort optical pulses [37]. Even, in the case of two electromagnetic plane waves with the same angular frequency and different wave vectors, the superposition fields reveals highly nontrivial structure in the local momentum and spin densities [38], that can be used to enhance the optical manipulations of small particles. For example, the spinning dynamics can be driven by superposing two vortex beams with respective circular and radial polarizations such that the particle spins around a certain optical axis [39]. The authors in [40, 41] have pointed out the importance of the Poynting vector for obtaining orbital angular momentum from spatial superposition of the Poynting vector beams.
This article is organized as follows: in Section 2, we briefly review the theoretical framework based on the scalar potential approach; in Section 3, we present the general negative Poynting vector for the whole invariant beams family; in Section 4, we study the orbital angular momentum in their transversal and longitudinal propagation due to mixed modes and in Section 5, we present our conclusions.

Following the formalism proposed by Stratton [42], we write the electromagnetic fields as

	$\displaystyle\vec{E}$	$\displaystyle=c_{\text{TE}}\vec{M}(\vec{r})+c_{\text{TM}}\vec{N}(\vec{r}),$		(1a)
	$\displaystyle\vec{H}$	$\displaystyle=-i\sqrt{\frac{\varepsilon}{\mu}}\left[c_{\text{TE}}\vec{N}(\vec{r})+c_{\text{TM}}\vec{M}(\vec{r})\right],$		(1b)

where $\vec{M}(\vec{r})$ and $\vec{N}(\vec{r})$ are vector fields proposed as

$$\vec{M}(\vec{r})=\nabla\times[\hat{a}\psi(\vec{r})],$$

(2)

and

$$\vec{N}(\vec{r})=\frac{1}{k}\nabla\times\vec{M}(\vec{r}),$$

(3)

being $\psi$ a scalar field, $\hat{a}$ an arbitrary unit vector that determines the direction of propagation (which we will choose as the $Z$ axis, so $\hat{a}=\hat{e}_{3}$), $k$ the magnitude of the wave vector, $\varepsilon$ the electric permittivity, $\mu$ the magnetic permeability, and $c_{\text{TE}}$ and $c_{\text{TM}}$ two arbitrary complex numbers. This approach has been successfully used to study the properties of the family of invariant beams, theoretically and experimentally [6, 7, 8].
It is straightforward to verify that if a scalar field $\psi(\vec{r})$ satisfies the Helmholtz equation

$$\nabla^{2}\psi+k^{2}\psi=0,$$

(4)

then the fields (1a) and (1b) satisfy the Maxwell equations; so, the scalar field $\psi(\vec{r})$ will be named scalar potential. Note that the vector fields, $\vec{M}$ and $\vec{N}$, are orthogonal, that is $\vec{M}\cdot\vec{N}=0$, and solenoidal, i.e. $\nabla\cdot\vec{M}=0$ and $\nabla\cdot\vec{N}=0$.
On the other hand, for any invariant beam the spatial evolution of the scalar potential $\psi$ can be described by its transverse and longitudinal parts [43]. The transverse part $\varphi(u_{1},u_{2})$ will depend only on the transverse coordinates, $u_{1},u_{2}$, and the longitudinal part $Z(z)$ will depend on the longitudinal coordinate $z$ (as we selected $\hat{a}=\hat{e}_{3}$), physically the propagation axis; i.e., we can write

$$\psi(u_{1},u_{2},z)=\varphi(u_{1},u_{2})Z(z).$$

(5)

After substituting (5) in the Helmholtz equation, we easily obtain that $\varphi(u_{1},u_{2})$ satisfies the two dimensional transverse Helmholtz equation

$$\nabla_{T}^{2}\varphi+k^{2}_{T}\varphi=0,$$

(6)

where $\nabla_{T}^{2}$ is the Laplacian transversal operator, which has a specific form in each coordinate system, and the longitudinal part is $Z(z)=e^{ik_{z}z}$ with the dispersion relation $k^{2}=k_{T}^{2}+k_{z}^{2}$. The two dimensional transverse Helmholtz equation (6) can be separated in Cartesian, cylindrical, parabolic cylindrical and elliptical cylindrical coordinates [43], and that gives origin to plane waves, Bessel beams, Weber beams and Mathieu beams, respectively. Then, we can write the vector operator fields, Eqs. (2) and (3), as follows

$$\vec{M}=-e^{ik_{z}z}\nabla_{T}^{\bot}\varphi,$$

(7)

where

$$\nabla_{T}^{\bot}=-\hat{e}_{1}\frac{1}{h_{2}}\frac{\partial}{\partial u_{2}}+\hat{e}_{2}\frac{1}{h_{1}}\frac{\partial}{\partial u_{1}},$$

(8)

$\hat{e}_{1}$ and $\hat{e}_{2}$ are the base unit vectors corresponding to the transversal direction, and $h_{1}$ and $h_{2}$ are the corresponding scale factors. We note that in the four coordinate systems, in which equation (6) can be separated, the scale factor $h_{3}$ is equal to 1. It is also easy to verify that

$$\vec{N}=\frac{e^{ik_{z}z}}{k}\left(ik_{z}\nabla_{T}+\hat{e}_{3}k_{T}^{2}\right)\varphi,$$

(9)

where $\nabla_{T}$ is the transversal part of the $\nabla$ operators, i.e.

$$\nabla_{T}=\hat{e}_{1}\frac{1}{h_{1}}\frac{\partial}{\partial u_{1}}+\hat{e}_{2}\frac{1}{h_{2}}\frac{\partial}{\partial u_{2}}.$$

(10)

It is worth to notice that the transversal vector operators are related as follows $\nabla_{T}^{\perp}=\hat{e}_{3}\times\nabla_{T}$.

The Poynting vector represents the directional power flux per unit area of an electromagnetic field. For harmonic electromagnetic fields the time average of the Poynting vector is given by [44]

$$\left\langle\vec{S}\right\rangle=\frac{1}{2}\mathrm{Re}\left(\vec{E}\times\vec{H}^{*}\right).$$

(11)

Substituting expressions (1a) and (1b) for the electromagnetic fields into (11), a generalized expression for the Poynting vector of any invariant beam is obtained [26],

$\displaystyle\left\langle\vec{S}\right\rangle=\left|c_{\text{TE}}\right|^{2}\left\langle\vec{S}_{\text{TE}}\right\rangle+\left|c_{\text{TM}}\right|^{2}\left\langle\vec{S}_{\text{TM}}\right\rangle+\left\langle\vec{S}_{\text{TE/TM}}\right\rangle,$

(12)

where

$$\left\langle\vec{S}_{\text{TE}}\right\rangle=\frac{1}{2k}\sqrt{\frac{\varepsilon}{\mu}}\mathbf{Re}\left[\left(\nabla_{T}\varphi\cdot\nabla_{T}\varphi^{\ast}\right)k_{z}\hat{e}_{3}-ik_{T}^{2}\,\varphi^{*}\,\nabla_{T}\varphi\right]$$

(13)

is the transversal electric part,

$$\left\langle\vec{S}_{\text{TM}}\right\rangle=\frac{1}{2k}\sqrt{\frac{\varepsilon}{\mu}}\mathbf{Re}\left[\left(\nabla_{T}\varphi\cdot\nabla_{T}\varphi^{\ast}\right)k_{z}\hat{e}_{3}+ik_{T}^{2}\,\varphi^{*}\,\nabla_{T}\varphi\right]$$

(14)

is the transversal magnetic part, and the interference modes TE/TM

$\displaystyle\left\langle\vec{S}_{\text{TE/TM}}\right\rangle=$

$\displaystyle\frac{1}{2k^{2}}\sqrt{\frac{\varepsilon}{\mu}}\mathbf{Re}\Big{[}i\left(c_{\text{TE}}\,c_{\text{TM}}^{*}k^{2}+c_{\text{TE}}^{*}\,c_{\text{TM}}k^{2}_{z}\right)\left(\nabla_{T}\varphi^{*}\times\nabla_{T}\varphi\right)\cdot\hat{e}_{3}+c_{\text{TE}}^{*}\,c_{\text{TM}}k_{z}k_{T}^{2}\,\hat{e}_{3}\times\nabla_{T}\left(\varphi^{*}\varphi\right)\Big{]}.$

(15)

The time averaged Poynting vector is independent of the $z$ coordinate and it satisfies $\nabla\cdot\left\langle\vec{S}\right\rangle=0$, as was proven in [45]. It is important to remark that the interference term in the Poynting vector, expressed in Eq. (15), is different from zero for any invariant beam, of course, whenever at least one of the two constants $c_{\text{TE}}$ and $c_{\text{TM}}$ is not null. Physically this confirms the negative behavior trough the propagation; these results were recently reported in [34] using the Weber beams. The study of its effect on the orbital angular momentum is done in the following sections.

For harmonic electromagnetic fields, the time averaged linear momentum per unit volume carried is $\left\langle\vec{p}\right\rangle=\left\langle\vec{S}\right\rangle/c^{2}$ [44, 50], and then, the time averaged orbital angular momentum density is $\left\langle\vec{j}\right\rangle=\vec{r}\times\left\langle\vec{p}\right\rangle=\vec{r}\times\left\langle\vec{S}\right\rangle/c^{2}$; using Eqs. (13), (14) and (15), we can find explicitly the contributions from the transverse electric, from the transverse magnetic and from the transverse electric/transverse magnetic TE/TM mixed modes. The transversal electric TE part is

$\displaystyle\left\langle\vec{j}^{\,\,\text{TE}}\right\rangle=\frac{\varepsilon}{2\omega}\left|c_{\text{TE}}\right|^{2}\mathbf{Re}\left[k_{z}\left(\nabla_{T}\varphi\cdot\nabla_{T}\varphi^{*}\right)\vec{r}\times\hat{e}_{3}-ik_{T}^{2}\varphi^{*}\,\vec{r}\times\nabla_{T}\varphi\right],$

(16)

whereas the transversal magnetic TM contribution has the form

$\displaystyle\left\langle\vec{j}^{\,\,\text{TM}}\right\rangle=\frac{\varepsilon}{2\omega}\left|c_{\text{TM}}\right|^{2}\mathbf{Re}\left[k_{z}\left(\nabla_{T}\varphi\cdot\nabla_{T}\varphi^{*}\right)\vec{r}\times\hat{e}_{3}+ik_{T}^{2}\varphi\,\vec{r}\times\nabla_{T}\varphi^{*}\right],$

(17)

and the interference mixed modes TE/TM part is given by

$\displaystyle\left\langle\vec{j}^{\;\;\text{TE/TM}}\right\rangle=\frac{c\varepsilon}{2\omega^{2}}\mathbf{Re}\Biggl{\{}i\left(k^{2}c_{\text{TE}}c^{*}_{\text{TM}}+c^{*}_{\text{TE}}c_{\text{TM}}k_{z}^{2}\right)\left(\nabla_{T}\varphi^{*}\times\nabla_{T}\varphi\right)\cdot(\vec{r}\times\hat{e}_{3})+c_{\text{TE}}^{*}\,c_{\text{TM}}k_{z}k_{T}^{2}\,\vec{r}\times[\hat{e}_{3}\times\nabla_{T}\left(\varphi^{*}\varphi\right)]\Biggr{\}}.$

(18)

The set of equations (16), (17) and (18) provide a useful simple recipe to calculate the orbital angular momentum of any invariant beam in terms of scalar potentials. It is important to remark that Eq. (18) physically represents the orbital angular momentum propagation due to the superposition of modes TE/TM.

We have investigated the transversal and longitudinal propagation of orbital angular momentum for invariant beams using a single scalar potential. We have proved that the invariant beams satisfy Maxwell equations and possess well-defined orbital angular momentum [25]. We have shown that the superposition of non-paraxial vectorial beams with axial symmetry can provide a well defined orbital angular momentum. These results exhibit how the modes superposition affects the angular momentum flux density and causes reverse propagation in the case of the fractional Bessel beams [48, 49]. In [59], the authors have studied the importance to handle the amplitude, phase and polarization in order to design structured fields to study the spin-orbit interactions in Bessel beams. Adopting the single scalar potential approach, presented here, may be useful to find interesting applications, as the electromagnetic spin and canonical momentum for paraxial and nonparaxial beams recently reported in [60, 61, 62].