Spin-Orbit Coupling in Quasi-Monochromatic Beams

Abstract

We investigate the concept that the value of the spin-orbit coupling is the energy efficiency of energy transfer between orthogonal components. The energy efficiency changes as the beam propagates through the crystal. For a fundamental Gaussian beam, its value cannot exceed 50%, while the energy efficiency for Hermite–Gaussian and Laguerre–Gaussian beams of higher orders of the complex argument can reach a value close to 100%. For Hermite–Gauss and Laguerre–Gauss beams of higher orders of real argument, the maximum energy efficiency can only slightly exceed 50%. It is shown that zero-order Bessel–Gauss beams are able to achieve an energy efficiency close to 100% when generating an axial vortex in the orthogonal component in both monochromatic and polychromatic light, while for a polychromatic Laguerre–Gauss or Hermite–Gauss beam of a complex argument, the energy efficiency reduced to a value not exceeding 50%. The spin angular momentum is compensated by changing the orbital angular momentum of the entire beam, which occurs as a result of the difference in the topological charge of the orthogonally polarized component by 2 units.

1. Introduction

In 1996, an article by Berry and Klein [1] was published that analyzed the structure of an optical catastrophe of the “assembly” type (spherical aberration), obtained as a result of the diffraction of polychromatic (white) light. The authors accompanied the theoretical description with experimental results obtained as a result of white light diffraction on corrugated glass for bathrooms. They found that near the geometric caustic, a number of regions with a peculiar coloration were formed that border the characteristic optical singularities of the optical vortex type. The point is that any optical catastrophe, with the exception of the simplest one, the “fold” type, is formed due to the regular “stacking” of optical vortices [2]. It is these optical vortices that give the characteristic coloration of the diffraction pattern in the vicinity of the core. Developing this idea, Berry in [3,4] considers the structure of the distribution of colors near the singularity. He found a special saturation of dark shades of various colors concentrated near a very narrow region of the phase singularity. Based on this, he proposed a method of optical chromatoscopy by saturating the area near the singularity with colors during computer data processing. This method consists of finding the maximum brightness at each point in the image by comparing the red, green and blue components of the brightness on the monitor, and then these values are normalized to this maximum value. As a result, the picture near the singularity turns out to be brighter, and the colors in it become easily distinguishable. This method was developed for the analysis of light scattered by corrugated surfaces [5,6,7]. Optical vortices present in scattered light are of a random nature and do not require special devices for their formation. However, the creation of single optical vortices carried by paraxial beams of the Laguerre–Gauss and Bessel–Gauss types encounters serious technical difficulties. The point is that the traditional method of obtaining optical vortices is based on light diffraction either on computer-synthesized holograms [8,9] or on spiral phase plates [10,11,12,13]. These methods are based on strict observance of the diffraction condition in the vicinity of a phase feature on a hologram or a helicoidal transponder, which is critical to the wavelength. Any non-observance of these conditions leads to the disappearance of the characteristic picture of a solitary vortex in the beam [13]. In this case, special mention should be made of optical beams that have the property of Fourier invariance [14].

Nevertheless, in [15,16,17], we managed to avoid such strict requirements imposed on the formation of an optical vortex due to the processes of light propagation in a uniaxial anisotropic medium. We have shown that a circularly polarized beam propagating along the optical axis of the medium is capable of forming optical vortices on the axis with the same localization regardless of the wavelength. Such optical vortices were called “white” vortices, in contrast to colored vortices [3,4,5,6,7], for which the position of the optical vortex is different for different spectral components.

Later, other authors developed special methods for generating polychromatic vortices on computer-synthesized holograms [18,19]. In these methods, the chromatic dispersion that occurs during the diffraction of light on a diffraction grating is compensated by a special prism installed after the hologram. However, due to the incomplete compensation of chromatic dispersion, an asymmetric color appears in the vicinity of the optical vortex (in contrast to the “white” vortices obtained not in a crystal). As a result, such vortices are called colored vortices. It is important to note that the detection of a vortex in a circularly polarized component is associated with a number of difficulties, the main being the use of a quarter-wave plate. Since a conventional quarter-wave plate does not allow conversion of circularly polarized light into linearly polarized light in a wide range of wavelengths, in our experiments we used the Fresnel rhombus, which is less sensitive to wavelength. Recently, in the works [20,21,22], the use of special achromatic compensators for this purpose was reported.

Special problems arose with the registration of the position and charge of optical vortices in a polychromatic beam. Thus, the authors of [18] used the chromatoscopic method, and [5,6,7] used both the chromatoscopic method and the interferometry method based on the Mach–Zehnder interferometer. The method [23] for determining the topological charge in a multi-color beam deserves special attention. In [24], the Fresnel biprism was used, which makes it possible to form a color interference pattern in a beam with low temporal coherence.

Recently, a simple method was proposed for determining the magnitude and sign of the topological charge of polychromatic vortices based on express beam analysis based on an astigmatic lens [25].

The method of generating optical vortices in crystals also makes it possible to form special types of beams, the so-called “bottle” beams [21] in an initially circularly polarized component, as well as soliton-like polychromatic beams in a medium with a nonlinearity [26]. However, the method of generation in crystals has its limitations. For example, it does not allow the formation of Bessel–Gauss and Laguerre–Gauss beams, since they have an initial complex phase and amplitude relief. For the first time, polychromatic zero-order Bessel–Gauss beams were formed by the authors of [27], who used for this purpose the light of an argon lamp introduced into the medium of a single-mode optical fiber in order to significantly increase the spatial coherence of the field. Light from the fiber was projected onto a standard axicon, after which the required beam was formed. Recently, significant progress in the formation of special polychromatic beams was achieved through the use of lasers emitting ultrashort pulses (the so-called femtosecond lasers) [28]. In the case of using short pulses, it became possible to generate “flying donuts”, which are of fundamental interest since they will interact with matter in unique ways, including non-trivial field transformations when reflected from interfaces and excitation of a toroidal response and anapole modes in matter, hence, offering opportunities for telecommunications, sensing and spectroscopy [29].

2. Polychromatic Laguerre–Gauss Beams

Let us consider the propagation in a uniaxial crystal of a Laguerre–Gaussian beam formed in white light. As a model of a white light source, we take the radiation of a completely black body with temperature $\mathrm{T}=4800 \mathrm{K}$.

The spectral distribution $f\left(\lambda \right)$ of such a source is shown in Figure 1. In this calculation, the distribution of colors in the picture is as follows. We use the monitor’s RGB brightness components, which are calculated as:

$$\begin{array}{l}R={\left[{\displaystyle \int f\left(\lambda \right)I\left(x,y,z,\lambda \right)\overline{r}\left(\lambda \right)d\lambda }\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\chi $}\right.}\\ G={\left[{\displaystyle \int f\left(\lambda \right)I\left(x,y,z,\lambda \right)\overline{g}\left(\lambda \right)d\lambda }\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\chi $}\right.}\\ B={\left[{\displaystyle \int f\left(\lambda \right)I\left(x,y,z,\lambda \right)\overline{b}\left(\lambda \right)d\lambda }\right]}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\chi $}\right.}\end{array}$$

(1)

where $I\left(x,y,z,\lambda \right)={\left|E\left(x,y,z,\lambda \right)\right|}^2$—intensity distribution; $\overline{r},\overline{g},\overline{b}$—color coordinates in the spectrum, values for the visible range; integration is carried out in the same wavelength range $\lambda \in \left[380,780\right]\times {10}^{-9}\mathcal{M}$. $\chi \approx 1.9$ is the monitor non-linearity coefficient [1]. Since it is given by a table, the integration in (1) is replaced by summation over 81 points with a step of $d\lambda =5 \mathrm{nm}$. If some values of the integrals (1) turn out to be negative, this means that the corresponding colors lie outside the monitor gamut, and we assume these color components (negative) to be equal to zero. For adequate reproduction of colors on the monitor, when calculating (1), the maximum value in the picture was found and all values of the monitor brightness components $R,G,B$ were divided by this value and multiplied by 255 (the maximum brightness of the monitor pixel). In calculations, it was also necessary to take into account the dispersion of the refractive indices of a uniaxial crystal.

Figure 2 shows such dependence in the visible range for a lithium niobate crystal [30], while we did not take into account the effect of photorefraction.

Since it is difficult to experimentally create a Laguerre–Gaussian beam in a polychromatic beam, let us first consider a circularly polarized white light beam with a Gaussian envelope incident on a uniaxial crystal. During the propagation of this beam, the intensity distribution in the right-hand circular components can be obtained using (1) and [31]. These patterns as the beam propagates in the crystal are shown in Figure 3.

An initially circularly polarized polychromatic beam with a Gaussian envelope propagates in the crystal. After the crystal, we place an achromatic quarter-wave plate and a polarizer. The beam in front of the quarter-wave plate passes a collimating lens that makes the beam parallel. The action of the achromatic $\lambda /4$ plate on the circularly polarized components of the beam is reduced to the transformation of the right-circularly polarized component of the beam into a linear plate oriented at an angle $-{45}^{\circ }$ to the “o” axis with a phase difference $\delta =\pi /2$ introduced by the $\lambda /4$ plate.

As can be seen from the figure, at short crystal lengths $z\approx 1 \mathrm{mm}$, an optical vortex with a double topological charge is generated in the orthogonally polarized component of the beam, which has practically no color. As the beam propagates further along the crystal axis, not only the left-handed circular component of the beam is colored but also the component with the initial polarization, which at small lengths, is due to the dispersion of the material.

It is worth noting that unlike colored vortices obtained by other methods, as mentioned earlier, this optical vortex has a color symmetrical about its axis, and it has a double topological charge, as illustrated in Figure 4 for length $z=5 \mathrm{mm}$. At large lengths, colored conoscopic rings appear in both components, illustrating the interference in white light of the ordinary and extraordinary beams. Similar patterns were obtained experimentally in [16,17,24].

Since, by the method described above, we can obtain an uncolored vortex with a double topological charge at short crystal lengths in an orthogonally polarized component, the sign of which is opposite to the sign of the circulation of this component, we assume that such a beam with right-hand circular polarization and a negative topological charge is incident on the crystal. As such a beam propagates in the crystal in the orthogonally polarized component, a Laguerre–Gauss beam with a complex argument $n=1, l=0$ will appear, as shown in Figure 5.

Figure 6 shows the interference pattern for the circularly polarized components of this beam.

It can be seen from the figures that at small lengths of the crystal in the left-hand circular component, there is a ring dislocation, which manifests itself in a shift of the interference fringes in Figure 6 ($z=1 \mathrm{mm}$). Further propagation of the beam leads to a smoothing of the ring in the orthogonal component and the appearance of a minimum in the center of the pattern. However, as the interference pattern shows, this minimum does not contain dislocations. In the right-circularly polarized component, the vortex with a double topological charge does not decay, which is also illustrated by the interference.

Similarly, it is possible to obtain Laguerre–Gauss beams of a higher order complex argument in an experiment using several thin crystals and circular polarizers between them. However, it must be taken into account that the intensity of the resulting beam will drop by several tens of times compared to the initial beam.

It is interesting to consider the case of propagation of a linearly polarized beam with a Gaussian envelope through a uniaxial crystal. The field in the crystal in this case is represented by a superposition of fields (see [31]) taken with equal weights. The linearly polarized components of such a beam are shown in Figure 7.

In the component with the initial linear polarization, four “colored” optical vortices with a deformed core are formed near the axis [32], as can be seen from Figure 7c,d. The chromatoscopic image shows that at the center of the vortex, the distribution of colors forms a blue–yellow dipole, similar to the case considered by Berry [3,4]; however, moving away from the center of the vortex, a green color appears, in contrast to the Berry dipole.

3. Polychromatic Bessel–Gauss Beams

In [27], a zero-order Bessel–Gauss beam was experimentally obtained in a polychromatic beam on an axicon. Let us consider the possibility of obtaining a Bessel–Gauss beam of the 2nd order from it. The argument under the Bessel function for a beam obtained on an axicon depends on the wavelength [27]: $\frac{K}{z_0\sigma }r=k{\gamma }_{a}r$, where $K$ is complex constant, ${\gamma }_a$—axicon angle, $\sigma =1-i z/z_0^{}$. We assume that the Bessel–Gauss beam waist falls on the input plane of a uniaxial crystal. Then we express the parameter $K_o$ in terms of the parameter for green light: $K_o\left(\lambda \right)=K_o\left({\lambda }_g\right)\frac{z_o\left(\lambda \right)k_o\left({\lambda }_g\right)}{z_o\left({\lambda }_g\right)k_o\left(\lambda \right)}$, where $z_o\left(\lambda \right)$ is Rayleigh length, ${\lambda }_g$—wavelength for green light, $k_o\left({\lambda }_g\right)$ radial wave vector component. Figure 8 shows the distribution of colors in the circularly polarized components of the Bessel–Gauss beam as it propagates along the crystal axis. In contrast to the Laguerre–Gaussian beam, such a beam is colored immediately on the input plane of the crystal.

When propagating in the beam components, the pattern is washed out, which leads to the formation of a weakly colored almost white beam over a certain length (in our case, $z=3 \mathrm{mm}$). Moreover, in the right-circularly polarized component of the beam, a ring with an intensity minimum appears near the axis, which surrounds a weak maximum on the axis. With further propagation, this maximum disappears and at the center of the right-circularly polarized component, there is an intensity minimum but without an optical vortex (see Figure 9a). In the left-circularly polarized component, there is an optical vortex on the axis with a double topological charge (Figure 9b).

An increase in the length of the crystal leads to uneven coloring of the beam components, and colored rings of the conoscopic pattern are formed at sufficiently large lengths. It should be noted that, both in monochromatic and polychromatic light, as the crystal length increases in the circularly polarized components, the concentration of intensity occurs inside the ring, and the dimensions of this ring weakly depend on the number $l$.

4. Spin-Orbit Coupling in Polychromatic Beams

Let us consider spin-orbit coupling in the case of a quasi-monochromatic beam. We note right away that we do not plan to consider hybrid modes (the so-called Hermite–Laguerre–Gauss beams) [33]. Let us consider the spin-orbit coupling of a quasi-monochromatic beam that has passed through an anisotropic medium. The spin and orbital moments of a quasi-monochromatic beam, including a polychromatic beam, are rigidly related to each other by the law of conservation of the total flux of angular momentum along the crystal axis, just as in the case of monochromatic light. The main distinguishing feature of a quasi-monochromatic beam from a monochromatic one is that its state is characterized by off-diagonal elements of the coherence matrix $\langle E_x{E}_{{}_y}^{*}\rangle$, where angle brackets denote time averaging, which is related to frequency averaging by the Fourier transform [34]. Such averaging is justified by the following physical reasons: off-diagonal elements of the coherence matrix characterize the third Stokes parameter $S_3$, which is nothing but the spin moment of the beam. At the same time $S_3$, a polychromatic beam obeys the additive principle, i.e., its value for incoherent light is defined as the sum of individual monochromatic beams with appropriate weighting factors. Thus, knowing the spectral density $f\left(\lambda \right)$, one can determine the integral spin moment and, using the expression for the energy efficiency of spin-orbit coupling for monochromatic beams, find the corresponding energy efficiency for a polychromatic beam. For simplicity, we assume that the spectral function of a monochromatic beam has a Gaussian distribution:

$$f\left(\lambda \right)=\exp \left\{-{\left(\frac{\lambda -{\lambda }_s}{\Delta \lambda }\right)}^2\right\}.$$

(2)

The flux of spin angular momentum for a quasi-monochromatic beam can be written as:

$$S_{z,\lambda }=\frac{{\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(\lambda \right)\left[{\Im }_{+}\left(\lambda \right)-{\Im }_{-}\left(\lambda \right)\right]d\lambda }}{{\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(\lambda \right)\left[{\Im }_{+}\left(\lambda \right)+{\Im }_{-}\left(\lambda \right)\right]d\lambda }}.$$

(3)

where ${\Im }_{+}, {\Im }_{-}$—total intensity of right- and left-circularly polarized beam components. Then, the expression for the energy efficiency of a quasi-monochromatic beam with an initial right-handed circular polarization can be written as

$${\eta }_{\lambda }=\frac{{\Im }_{\mp }}{{\Im }_{+}+{\Im }_{-}}=\frac{1-S_{z,\lambda }}{2}.$$

(4)

where $S_{z,\lambda }$ spin angular momentum flux in a uniaxial crystal along the axis, ${\Im }_{\mp }$—intensity in the orthogonally polarized component. Using the expressions for Laguerre–Gauss beams of the complex argument together with (3) and (4), we find the energy efficiency of the spin-orbit coupling. Figure 10 shows the dependence curves depending on the length of the crystal.

They describe the energy efficiency for beams with different indices $n,l$ and different spectral line half-widths. For a Gaussian beam, this dependence is described by smooth curves. If the half-width of the spectral line is $\Delta \lambda ={10}^{-7}\mathrm{m}$, then ${\eta }_{\lambda }\left(z\right)$ such a beam differs little from the analogous curve of a monochromatic beam. An increase in the half-width of the spectral line leads to a rapid change in the energy efficiency, so that over a crystal length of 2 cm, the energy efficiency reaches 45% for the half-width $\Delta \lambda ={10}^{-5}\mathrm{m}$. Quasi-monochromatic Laguerre–Gauss beams of higher orders behave somewhat differently. Thus, for beams with $l=2, n=10$, the dependence curve ${\eta }_{\lambda }\left(z\right)$ for a source with a spectral half-width $\Delta \lambda ={10}^{-7}\mathrm{m}$ experiences rapid oscillations, reaching 98% at the maximum, and then tends toward 50% as the crystal length increases.

These oscillations decay very quickly with increasing line half-width, so that the value ${\eta }_{\lambda }\to 50\%$ at a length of 0.5 mm if $\Delta \lambda ={10}^{-5} \mathrm{m}$. Similar results can be obtained for quasi-monochromatic Hermite–Gauss beams of complex argument.

Interesting results were obtained in the study of the energy efficiency of spin-orbit coupling for Bessel–Gaussian beams. The curves ${\eta }_{\lambda }\left(z\right)$ shown in Figure 11 experience rapid oscillations, reaching a maximum of 94%. In this case, the width of the spectral line $\Delta \lambda =300 \mathrm{nm}$ covers the entire visible range. At the same time, the energy efficiency for all other types of beams decays rapidly, and the oscillation amplitude is very small. This unusual behavior of Bessel–Gaussian beams should be explained. A typical Bessel beam is characterized by a circle of radius $k_{\perp }$, where $k_{\perp }$ is the transverse wave number. At the same time, the Bessel–Gaussian beam in the space of wave numbers is characterized by a ring, and the distribution of vectors k [35] is described by a Gaussian envelope (see Figure 12).

Such a spectrum of plane waves, propagating through the crystal, forms in a certain plane an annular intensity distribution in both circular components of the beam, as mentioned above. When the first annular dip is observed in the intensity distribution in the component $E_{+}$, the value of the spin-orbit coupling is maximum (see Figure 13). In this case, the intensity $E_{+}$ sharply decreases, and the components $E_{-}$, on the contrary, increase.

A sharp decrease in intensity $E_{+}$ corresponds to such a direction of the average wave vector of the Bessel–Gauss beam in which the phase difference between the ordinary and extraordinary waves is equal $\pi$ to: ${\delta }_s=\frac{2\pi }{{\lambda }_s}\left[n_o\left({\lambda }_s\right)-n_3\left({\lambda }_s\right)\right]z=\pi$. Let the beam wavelength differ from the average by the value, then the phase difference will be equal to $\delta =\frac{2\pi }{{\lambda }_s+\Delta \lambda }\left[n_o\left({\lambda }_s\right)-n_3\left({\lambda }_s\right)\right]z=\frac{2\pi }{{\lambda }_s}\left[n_o\left({\lambda }_s\right)-n_3\left({\lambda }_s\right)\right]z\left(1-\frac{\Delta \lambda }{{\lambda }_s+\Delta \lambda }\right)$ and the additional phase $\Delta \delta \approx \frac{2\pi \Delta \lambda }{{\lambda }_s^2}\left[n_o\left({\lambda }_s\right)-n_3\left({\lambda }_s\right)\right]z$ difference in the case $\Delta \lambda /{\lambda }_s=1$ will be negligible, and the spin-orbit exchange condition at half-width $\Delta \lambda$ will be approximately the same.

Thus, the efficiency of spin-orbit exchange for broadband Hermite–Gauss and Laguerre–Gauss beams does not exceed 50%, while for such Bessel–Gauss beams it is possible to achieve an energy efficiency of spin-orbit coupling close to 100%.

The method described in the article can be applied by other researchers to analyze the properties of spin and orbital moments in free space [36], anisotropic media [37], as well as to study the properties of topological charges [38].

5. Conclusions

The physical cause of the transformation of the topological charge of the beams is the spin-orbit interaction, in which the total flux of angular momentum, which is the sum of the fluxes of spin and orbital angular momentum along the crystal axis, is a constant value. For example, if a fundamental circularly polarized Gaussian beam that does not carry optical vortices is incident on the input of a crystal, then in the process of propagation in the beam cross section, inhomogeneous polarization occurs, so that in the asymptotic case of an infinitely long crystal length, the degree of polarization of the entire beam is equal to zero. The total spin angular momentum then disappears. The spin angular momentum is compensated by changing the orbital angular momentum of the entire beam, which occurs as a result of the difference in the topological charge of the orthogonally polarized component by 2 units.

It is shown that the value of the spin-orbit coupling is the energy efficiency of energy transfer between orthogonal components. As the beam propagates in the crystal, the energy efficiency changes. For a fundamental Gaussian beam, its value cannot exceed 50%, while the energy efficiency for Hermite–Gaussian and Laguerre–Gaussian beams of higher orders of the complex argument can reach a value close to 100%. For Hermite–Gauss and Laguerre–Gauss beams of higher orders of real argument, the maximum energy efficiency can only slightly exceed 50%. It is shown that zero-order Bessel–Gauss beams are able to achieve an energy efficiency close to 100% when generating an axial vortex in the orthogonal component in both monochromatic and polychromatic light, while for a polychromatic Laguerre–Gauss or Hermite–Gauss beam of a complex argument, the energy efficiency reduced to a value not exceeding 50%.

Experiments were carried out on the generation of polychromatic optical vortices carrying their own orbital angular momentum. However, the theoretical analysis of such states and their properties remains difficult [33]. Here we provide such an analysis of spin-orbit coupling of quasi-monochromatic vortex beams that has passed through an anisotropic medium. Our analysis of spin-orbit coupling of a quasi-monochromatic beam can easily be extended to spatiotemporal vortex impulses of a different nature. For example, quasi-monochromatic vortex beams can be generated in sound waves in fluids or gases, etc.