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Abstract
A design method for the creation of photons with fractional orbital angular momentum (FOAM) is reported through the use of a metal-wedge optical fiber. The main focus was on a limiting zero-degree metal wedge, the case of light beams carrying $\pm \frac {\hbar }{2}$ OAM per photon. We compared this ideal case with a one-degree metal wedge in hopes of encouraging experimentalists to fabricate this type of waveguide. Various intensity and the Poynting vector plots for different modes and varying wedge angles were obtained and analyzed. We found that the Poynting vectors in the transverse direction for the zero and one degree metal wedges were essentially identical. In addition, as the metal wedge angle was increased, the real propagation constant decreased. Furthermore, the metal wedge enables a new method to obtain a local field enhancement in the center of the core of this waveguide, thus providing a wide range of applications in other fields. Finally, we found discrete complex propagation constants leading to discrete leaky modes with potentially an infinite number of these modes.
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1. Introduction
Light beams with an azimuthal component of the form $e^{im\varphi },$ where $m$ is an integer, carry $m\hbar$ OAM per photon [1]. Producing light beams with FOAM has been of great interest and has been previously accomplished using optical devices such as spiral phase steps with a fractional step height [2] along with special holograms [3]. These optical devices operate by inflicting a shift in phase by $e^{i \tilde {m} \varphi }$ where $\tilde {m}$ is not necessarily an integer value [4]. Another method of producing light beams with FOAM was developed by creating a quantum mechanical description of FOAM. This allowed the FOAM to be expressed as a quantum state. These states can be represented as a superposition of light modes with different integer $m$ values [4].
Light beams with integer OAM have many applications in particle guiding and trapping, optical communication, holography and high-precision imaging, and novel light-matter interactions with topological materials [5]. It is also used in quantum key distribution in particular to achieve the transmission of more than one bit of information per photon [6]. Light beams carrying FOAM are of substantial interest for their unique properties and great potential for applications [5], for example, in quantum communications [4]. Furthermore, experiments in the area of two-photon entanglement could benefit greatly from an enhanced propagation distance for the light beams with FOAM [7].
The metal-wedge optical fiber design in this paper provides a different technique for producing light beams with FOAM. We examined plots of the Poynting vector in the transverse and propagation direction for several modes and wedge angles. The main emphasis is to demonstrate the similarity of the Poynting vectors in the transverse direction between the ideal zero-degree metal wedge and the one-degree metal wedge for the $\frac {\hbar }{2}$ FOAM state. There have been previous efforts to enhance the local electric field through the use of a fan-shaped semiconductor waveguide with a sharp dielectric corner which utilizes the corner effect [8]. The enhancement of the light field intensity is useful for various applications such as in electrodynamics [8–10], nonlinear optical effect [8,11], quantum optomechanics [8,12], optical sensors [8,13], and nano-optical tweezers [8,14]. The metal-wedge optical fiber provides a novel way to produce a local field enhancement near the sharp metal edge in the center of the core of this optical fiber.
2. Methods
The model that we will be considering consists of an optical fiber in the azimuthal range $0\leq \varphi \leq \varphi _0$ and an ideal metal wedge in the azimuthal range $\varphi _0 \leq \varphi \leq 2\pi$ as shown in Fig. 1.
Fig. 1. Optical fiber with step-index profile and an ideal metal wedge. The variable $a$ represents the core radius and the core is assumed to be a denser medium than the cladding that is $n_1 > n_2.$
We assumed the ideal metal condition, i.e. the electric and magnetic fields are zero inside the metal region. We could introduce a complex form of the solution in the propagating mode as
From Maxwell equations $\mathbf {\nabla } \times \mathbf {E} = - \frac {\partial \mathbf {B}}{\partial t}$ and $\mathbf {\nabla } \times \mathbf {H} = \frac {\partial \mathbf {D}}{\partial t}$ one can assume the independent components in the $z$-direction only by the following:
From the azimuthal component, we write that $\sin \left (\frac {m\pi }{\varphi _0}\varphi \right ) = \frac {1}{2i} \left ( e^{i\varphi \frac {m\pi }{\varphi _0}}-e^{-i\varphi \frac {m\pi }{\varphi _0}}\right )$ which shows that the light beam has photons carrying $\pm \frac {m \pi }{\varphi _0} \hbar$ OAM with equal probability. The other components of electromagnetic fields are uniquely related to the z-components via the Maxwell equation in respective regions [15]. After applying boundary conditions at $\rho = a,$ we obtain for the TM mode a condition for the propagation constant $\beta$ given by
This parameter embodies both the fiber structural parameters and the optical wavelength [16]. The value of $V$ must be larger than a certain critical value $(V_{critical})$ for the existence of a real beta solution for that fundamental mode. To observe the $\frac {\hbar }{2}$ orbital angular momentum state, we plotted the Poynting vector for $m=1$ mode by writing the $\sin (\frac {\varphi }{2})$ (with $m = 1$ and $\varphi _0 = 2\pi$) component from Eq. (2) as $\frac {1}{2i}(e^{i \frac {\varphi }{2}}-e^{-i \frac {\varphi }{2}})$ and taking only electromagnetic fields with the portion $e^{i\frac {\varphi }{2}}.$ In practice, there have been methods developed for the efficient separation of OAM modes through the use of a radial varying phase [17]. It is important to note that the intensity plot in Fig. 2 and the plots of the Poynting vectors in sections 5 and 6 are of relative strength.
Fig. 2. Intensity distribution associated with the solution $\beta = 7040792.65 \, \frac {rad}{m}$ including the one-degree metal wedge located at $259-360$ degrees. The spike at the origin is associated with the metal wedge’s sharp edge.
3. Field enhancement by metal wedge
We present the ideal limiting zero-degree metal wedge optical fiber to produce light beams carrying $\pm \frac {\hbar }{2}$ OAM per photon and to study the local field enhancement near the sharp metal edge. This ideal zero-degree metal wedge gives rise to the singularity behavior of the local field enhancement in the center of the optical fiber. From Fig. 2, we observed the divergence of the intensity at the origin of this waveguide including the presence of a one-degree metal wedge, but the ideal limiting zero-degree case produces a nearly identical plot. At the origin $\rho =0$ resulting in $E_\rho$ and $E_\varphi$ to diverge [15] which indicates the expected singular behavior there due to the presence of the metal edge. This metal wedge optical fiber has the unique feature that it naturally produces photons with both $\frac {\hbar }{2}$ and $-\frac {\hbar }{2}$ OAM with equal probability. The electric field that was used to produce this Fig. 2 has both photons with $\frac {\hbar }{2}$ and $-\frac {\hbar }{2}$ OAM.
4. Metal wedge angle
As the angle of the metal wedge increases, the value of $V_{critical}$ must increase as well since we will need a larger core radius for the existence of a real beta solution. In Fig. 3(a), we can see that the value of $V_{critical}$ increases linearly for small angles for the $m = 1$ mode, and we find it is also the case for the $m = 2$ mode. Moreover, we can examine how the value of the real beta solution changes as we increase the metal wedge angle. From Fig. 3(b), we can see that within small angles, the real solution decreases linearly as the metal wedge angle increases. As we move away from small angles, it was found that the real solution decreases nonlinearly, as expected.
Fig. 3. The metal wedge is located between $2\pi (1-\frac {\theta }{360}) \leq \varphi \leq 2\pi.$ $\mathbf {a})$ For $m = 1$ mode, a positive linear relationship between the $V_{critical}$ value as a function of the metal wedge angle. $\mathbf {b})$ For $m = 1$ mode, negative linear relationship between the real beta solution as a function of the metal wedge angle inside of the small angle range.
5. Ideal zero-degree metal wedge
The ideal zero-degree metal wedge is a limiting case, but for our calculations is defined as $\varphi _0 = 2\pi.$ In Fig. 4(a), we can see the Poynting vector for the $m = 1$ TE mode for a real beta solution inside the core region with most of the energy being concentrated towards the center. The phase that is $\omega t- \beta z$ from Eq. (1) was set to $\frac {\pi }{2}.$ The Poynting vector is zero along the positive $x-$axis due to the ideal zero-degree metal wedge as seen in Fig. 4. Next, we found some complex solutions for various modes and plotted the corresponding Poynting vector for $m = 1$ TE mode in Fig. 4(b). We can observe that the energy is radiating outwards from the center and this is a result of the negative imaginary component from the complex solution. It is important to note that there are multiple complex solutions. In particular, a complex solution exists for both positive and negative imaginary components with perfect symmetry, agreeing with the time-reversal symmetry expected. For each mode $m,$ once the real component of the complex solution is fixed, there appears to be a continuum of complex solutions within a particular range. We suspect, but cannot prove at the moment that there are an infinite number of complex solutions leading to an infinite number of leaky modes. In Fig. 5, we can see plots of the Poynting vector for the $m=1$ TE mode at a phase of $\frac {\pi }{2}$ inside the core region but for the $\frac {\hbar }{2}$ OAM state with a real beta solution in Fig. 5(a) and a complex solution in Fig. 5(b). In Fig. 5(a), the Poynting vector plot is rotating in the clockwise direction with a concentration of energy towards the center of the core similar to Fig. 4(a). In Fig. 5(b), the Poynting vector has a spiral shape and rotates in the clockwise direction. When plotting the $-\frac {\hbar }{2}$ OAM state, we obtain the same plots for both the real and complex solutions but instead rotates in the counterclockwise direction.
Fig. 4. $\mathbf {a})$ Poynting vector in the transverse plane for $m = 1$ TE mode with real beta solution $\beta = 7040810.66 \, \frac {rad}{m}. \, \mathbf {b})$ Poynting vector in the transverse plane for $m = 1$ TE mode with complex beta solution $\beta = 7040865.37-101265.82i \, \frac {rad}{m}.$ In both plots, the phase that is $\omega t- \beta z$ from Eq. (1) is equal to $\frac {\pi }{2}$ and the Poynting vector must be zero along the positive $x$-axis due to the ideal zero-degree metal wedge. The blue circle indicates the core region.
Fig. 5. $\mathbf {a}$) Poynting vector in the transverse plane for $m=1$ TE mode with real beta solution $\beta =7040810.66 \, \frac {rad}{m}$ in the $\frac {\hbar }{2}$ OAM state. $\mathbf {b}$) Poynting vector for m = 1 TE mode with complex beta solution $\beta = 7040865.37-101265.82i \, \frac {rad}{m}$ in the $\frac {\hbar }{2}$ OAM state. In both plots, the phase that is $\omega t- \beta z$ from Eq. (1) is equal to $\frac {\pi }{2}$ and the Poynting vectors are rotating in the clockwise direction.
The intensity of the Poynting vector in the z-direction for the $\frac {\hbar }{2}$ OAM state with a real and complex solution can be seen in Fig. 6. The real solution results in a concentration of intensity towards the core and the complex solution results in a concentration of energy away from the core rotating in a clockwise direction.
Fig. 6. $\mathbf {a})$ Poynting vector in the z-direction for $m = 1$ TE mode with real beta solution $\beta =7040810.66 \, \frac {rad}{m}$ in the $\frac {\hbar }{2}$ OAM state. Note that this plot is scaled differently since there were not many features outside of this region. $\mathbf {b})$ Poynting vector in the z-direction for $m = 1$ TE mode with complex beta solution $\beta = 7040865.37-101265.82i \,\frac {rad}{m}$ in the $\frac {\hbar }{2}$ OAM state which rotates in the clockwise direction. In both plots, the phase that is $\omega t- \beta z$ from Eq. (1) is equal to $\frac {\pi }{2}.$
6. Including a one-degree metal wedge
It is evident that the ideal zero-degree metal wedge is a limiting case that is not achievable in a laboratory. We examined plots of the Poynting vector including the tiny metal wedge of one degree to illustrate the close resemblance with the ideal zero-degree metal wedge solution aforementioned. The one-degree metal wedge in our calculations is defined as $\varphi _0 = 2\pi (1-\frac {1}{360})$, that is the metal wedge is located between $2\pi (1-\frac {1}{360})<\varphi < 2\pi$. In Fig. 7(a), the Poynting vector of the real solution still maintains a concentration in the center of the core region and has the same general patterns as seen in Fig. 4(a). In Fig. 7(b), the Poynting vector for the real beta solution in the $\frac {\hbar }{2}$ OAM state has the same general shape and rotates in a clockwise direction similar to that of Fig. 5(a). The main observable difference is that the Poynting vector in Fig. 7(b) is zero for angles $2\pi (1-\frac {1}{360}) \leq \varphi \leq 2\pi$ due to the presence of the one-degree metal wedge. In Fig. 8(a), the Poynting vector of the complex solution is radiating outwards which is similar to that of Fig. 4(b). In Fig. 8(b), the Poynting vector with a spiral shape has a concentration towards the edges of the core region and rotates clockwise for the $\frac {\hbar }{2}$ OAM state which is similar to that of Fig. 5(b). When including the one-degree metal wedge, we can see from Fig. 7 and Fig. 8 that the resulting Poynting vectors in the $\frac {\hbar }{2}$ OAM state for both a real and complex solutions are almost identical to the ideal zero-degree metal wedge case and preserve close feature resemblances. Therefore, this should encourage experimentalists to fabricate this type of waveguide. This means that the FOAM states that result from an ideal case of a zero-degree metal wedge that is impossible to produce in reality can be achieved within a good approximation with a small one-degree metal wedge.
Fig. 7. $\mathbf {a})$ Poynting vector viewed in the transverse plane for $m = 1$ TE mode with real beta solution $\beta = 7040792.65 \, \frac {rad}{m}$ including the one-degree metal wedge located at $359-360$ degrees. $\mathbf {b}$) Poynting vector viewed in the transverse plane for $m = 1$ TE mode with real beta solution $\beta = 7040792.65 \, \frac {rad}{m}$ including the one-degree metal wedge located at $359-360$ degrees in $\frac {\hbar }{2}$ OAM state which rotates in the clockwise direction. In both plots, the phase that is $\omega t- \beta z$ from Eq. (1) is equal to $\frac {\pi }{2}$ and the Poynting vector must be zero for angles $2\pi (1-\frac {1}{360}) \leq \varphi \leq 2\pi$ due to the one-degree metal wedge.
Fig. 8. $\mathbf {a})$ Poynting vector in the transverse plane for $m = 1$ TE mode with complex beta solution $\beta = 7040792.67-159215.92i \, \frac {rad}{m}$ including the one-degree metal wedge located at $359-360$ degrees. $\mathbf {b})$ Poynting vector in the transverse plane for $m = 1$ TE mode with complex beta solution $\beta = 7040792.67-159215.92i \, \frac {rad}{m}$ including the one-degree metal wedge located at $359-360$ degrees in $\frac {\hbar }{2}$ OAM state which rotates in the clockwise direction. In both plots, the phase that is $\omega t- \beta z$ from Eq. (1) is equal to $\frac {\pi }{2}$ and the Poynting vector must be zero for angles $2\pi (1-\frac {1}{360}) \leq \varphi \leq 2\pi$ due to the one-degree metal wedge.
7. Conclusion
In conclusion, we found that the Poynting vector of the ideal zero-degree metal wedge waveguide and the one-degree metal wedge waveguide were almost identical for the $\frac {\hbar }{2}$ OAM state for both real and complex solutions. This should encourage experimentalists to produce optical fibers with the inclusion of the one-degree metal wedge as a new approach to obtaining light beams with FOAM. Multiple discrete complex propagation constants were found for each mode index $m$, but more research is necessary to prove that there are an infinite number of these complex solutions. Finally, we demonstrated that the sharp metal edge provided a new method to induce a local field enhancement near the center of the core of the optical fiber and can produce larger optical nonlinearity that is mostly desired in nonlinear optical waveguides. Therefore, there exists a wide range of potential new physical applications in electrodynamics, nonlinear optical effect, quantum optomechanics, optical sensors, and nano-optical tweezers once this type of metal-wedge optical fiber is developed in a laboratory.
Funding
Natural Sciences and Engineering Research Council of Canada.
Acknowledgments
Liang Chen would like to acknowledge the funding from the Natural Sciences and Engineering Research Council of Canada (RGPIN-2020-05774).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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