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Mode Properties and Propagation Effects of Optical Orbital Angular Momentum (OAM) Modes in a Ring Fiber
Figure 1
Comparison of different schemes for multiplexing multiple optical spatial modes for fiber transmission. (a) Higher order LP modes are composed of two fiber eigenmodes 
$(\hbox{LP}_{2, 1} \!= \hbox{HE}_{3, 1} + \hbox{EH}_{1, 1})$ having different propagation constants. The two fiber eigenmodes walk off as they propagate along the fiber. OAM modes are composed of two fiber eigenmodes with same propagation constant 
$(\hbox{OAM}_{0, 2} = \hbox{HE}_{2, 1}^{\rm even} + i\times \hbox{HE}_{2, 1}^{\rm odd})$, and thus, there is no walk-off after propagation. (b) To multiplex multiple OAM modes into multimode step-index fiber, a small change of launching condition can excite radially higher order modes and results in the crosstalk. With proper design, single-ring fiber can support only radially fundamental modes with reduced crosstalk. (c) 
$j$ OAM modes with different azimuthal phase order can be multiplexed into the single ring fiber. Using a multiple-ring fiber with 
$k$ rings can increase the multiplexed mode number with another factor of 
$k$. This can potentially transmit 
$k \times j$ OAM modes in a single fiber.
$(\hbox{LP}_{2, 1} \!= \hbox{HE}_{3, 1} + \hbox{EH}_{1, 1})$ having different propagation constants. The two fiber eigenmodes walk off as they propagate along the fiber. OAM modes are composed of two fiber eigenmodes with same propagation constant $(\hbox{OAM}_{0, 2} = \hbox{HE}_{2, 1}^{\rm even} + i\times \hbox{HE}_{2, 1}^{\rm odd})$, and thus, there is no walk-off after propagation. (b) To multiplex multiple OAM modes into multimode step-index fiber, a small change of launching condition can excite radially higher order modes and results in the crosstalk. With proper design, single-ring fiber can support only radially fundamental modes with reduced crosstalk. (c) $j$ OAM modes with different azimuthal phase order can be multiplexed into the single ring fiber. Using a multiple-ring fiber with $k$ rings can increase the multiplexed mode number with another factor of $k$. This can potentially transmit $k \times j$ OAM modes in a single fiber.
Figure 2
(a) OAM mode number supported in the single-ring fiber as a function of the ring outer radius 
$(r_{2})$ with different ring-cladding index difference 
$(\Delta n)$. (b) OAM mode number as a function of the wavelength with different ring-cladding index difference 
$(\Delta n)$.
$(r_{2})$ with different ring-cladding index difference $(\Delta n)$. (b) OAM mode number as a function of the wavelength with different ring-cladding index difference $(\Delta n)$.
Figure 3
Effective refractive index difference of 
$\hbox{TE}_{0, 1}$ and 
$\hbox{HE}_{2, 1}$ modes as a function of (a) the refractive index in the fiber ring region with different ring outer radius 
$(r_{2})$ and (b) wavelength with different ring-cladding index difference 
$(\Delta n)$.
$\hbox{TE}_{0, 1}$ and $\hbox{HE}_{2, 1}$ modes as a function of (a) the refractive index in the fiber ring region with different ring outer radius $(r_{2})$ and (b) wavelength with different ring-cladding index difference $(\Delta n)$.
Figure 4
Intensity and phase distribution of the supported 
$\hbox{OAM}_{0, m}$ modes 
$(\hbox{HE}_{m, 1}^{\rm even} + i\times \hbox{HE}_{m, 1}^{\rm odd}, m = 1 \sim 5)$ in the ring fiber (
$r_{1} = 4\ \mu\hbox{m}$, 
$r_{2} = 5\ \mu\hbox{m}$, and 
$\Delta n = 0.05$).
$\hbox{OAM}_{0, m}$ modes $(\hbox{HE}_{m, 1}^{\rm even} + i\times \hbox{HE}_{m, 1}^{\rm odd}, m = 1 \sim 5)$ in the ring fiber ($r_{1} = 4\ \mu\hbox{m}$, $r_{2} = 5\ \mu\hbox{m}$, and $\Delta n = 0.05$).
Figure 5
Intensity and phase variation of 
$\hbox{OAM}_{0, 3} \ (\hbox{HE}_{3, 1}^{\rm even} + e^{i\theta}\times \hbox{HE}_{3, 1}^{\rm odd})$ modes azimuthally along the center of ring region with different mode walk-off 
$(\theta)$ (
$\theta = 90^{\circ}$ provides a perfect OAM mode). (a) Normalized intensity variation with different mode walk-off and azimuth. (b) Azimuthally normalized intensity variation for 
$\theta = 90^{\circ}$, 120 °, 150°, and 180°. (c) Phase variation with different mode walk-off and azimuth. (d) Azimuthal phase variation for 
$\theta = 90^{\circ}$, 120°, 150 °, and 180°.
$\hbox{OAM}_{0, 3} \ (\hbox{HE}_{3, 1}^{\rm even} + e^{i\theta}\times \hbox{HE}_{3, 1}^{\rm odd})$ modes azimuthally along the center of ring region with different mode walk-off $(\theta)$ ($\theta = 90^{\circ}$ provides a perfect OAM mode). (a) Normalized intensity variation with different mode walk-off and azimuth. (b) Azimuthally normalized intensity variation for $\theta = 90^{\circ}$, 120 °, 150°, and 180°. (c) Phase variation with different mode walk-off and azimuth. (d) Azimuthal phase variation for $\theta = 90^{\circ}$, 120°, 150 °, and 180°.
Figure 6
Standard deviation of (a) intensity, and (b) phase azimuthally along the center of ring region as a function of fiber ellipticity for different OAM modes.
Figure 7
$2\pi$ and 10-ps walk-off length as a function of fiber ellipticity for different OAM modes.
Figure 8
Charge weight of (a)–(h) different well-aligned OAM modes in a 
$\varepsilon = 1\%$ fiber. (i)–(p) 
$\hbox{OAM}_{0, 3}$ modes after 0, 5, 10, and 15-m propagation in a 
$\varepsilon = 1\%$ fiber.
$\varepsilon = 1\%$ fiber. (i)–(p) $\hbox{OAM}_{0, 3}$ modes after 0, 5, 10, and 15-m propagation in a $\varepsilon = 1\%$ fiber.
Figure 9
Demultiplexing efficiency of 
$\hbox{OAM}_{0, 3}$ mode as a function of the propagation length in the ring fiber with different 
$fiber\ ellipticity$ (periodic versus 
$2\pi$ walk-off length).
$\hbox{OAM}_{0, 3}$ mode as a function of the propagation length in the ring fiber with different $fiber\ ellipticity$ (periodic versus $2\pi$ walk-off length).
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