Graded-azimuthal-index fiber as a control element for radial-index and orbital-angular-momentum modes of Laguerre-Gaussian beams

We propose an azimuthally varying graded-index fiber that can efficiently create and control a definite set of radial-index and orbital-angular-momentum (OAM) modes of Laguerre-Gaussian beams and their superpositions. A rigorous coupled-wave analysis formalism is developed, which provides a ready analysis for the azimuthal-refractive-index variation in the Laguerre-Gaussian basis. In particular, we present effective control in the generation and coupling of the various OAM modes, including radial-index modes of the Laguerre-Gaussian beam within the fiber, wherein the propagation length of the fiber and the wavelength act as the control parameters. We further present the possibility of switching every alternate channel in a wavelength division multiplexing protocol to a higher-order spatial mode within the optical fiber to substantially reduce interchannel crosstalk. The graded azimuthal index fiber can be utilized in high-speed fiber-optic communications as a versatile control element for OAM modes and the radial-index mode of the Laguerre-Gaussian beam transformation.

Figure 1

(a) Typical GRIN fiber relative permittivity contour lines in the $x\text{−}y$ plane. The proposed GRAIN fiber contour lines are for (b) $\overline{\varepsilon }\left(\varphi \right)=hcos\left(\varphi \right)$ and (c) $\overline{\varepsilon }\left(\varphi \right)=h\varphi$ including the radial parabolic refractive index. (d) Notation involving the GRAIN fiber element

Figure 2

Dependence of the normalized effective refractive index of the LG basis modes on the $V$ parameter. We are working with the $V$ parameter such that a few lower-order modes are controlled within the GRAIN fiber element.

Figure 3

(a) Input Gaussian mode: $(l,p)=(0,0)$ switches to $(l=\pm 1,p=0)$ at a propagation length of 900 $\mu \text{m}$ and $\lambda =1.5\mu \text{m}$. The energy exchange happens periodically between the modes. The transverse intensity profiles of the modes (b) $l=0$ and (c) $l=\pm 1$ are shown below, with a core radius $a=5.627\mu \text{m}$ (shown as a white circle) and a beam waist ${\rho }_0=2.516\mu \text{m}$.

Figure 4

(a) Input Gaussian mode: $(l,p)=(0,0)$ switches to the radial-index mode $(l=0,p=1)$ of the LG beam at a propagation length of 1600 $\mu \text{m}$ and $\lambda =1.5\mu \text{m}$ for $V=7$ and $m_0=2$. Also shown are the transverse intensity profiles: (b) an equal magnitude superposition state of four modes $({\text{LG}}_{2,0}$, ${\text{LG}}_{-2,0}$, ${\text{LG}}_{0,0}$, and ${\text{LG}}_{0,1})$ at the propagation length of 750 $\mu \text{m}$ and (c) radial-index mode ${\text{LG}}_{0,1}$ with a core radius $a=7.878\mu \text{m}$ (shown as a white circle) and a beam waist ${\rho }_0=2.977\mu \text{m}$.

Figure 5

Total of ten WDM channels of OAM mode switching and six WDM channels of radial-index mode switching accessible for the GRAIN fiber of propagation length 50 mm for (a) $m_0=1$ and $V=5$ and (b) $m_0=2$ and $V=7$. The adjacent wavelength channels are converted from (a) $(l=0,p=0)$ to $(l=\pm 1,p=0)$ and (b) $(l=0,p=0)$ to $(l=0,p=1)$, which helps in reducing the crosstalk between adjacent wavelength channels.

Figure 6

Linearly varying GRAIN fiber element: (a) input Gaussian mode $(l,p)=(0,0)$ switches to $(l=1,p=0)$ and $(l=-1,p=0)$ at propagation lengths of 500 and 1200 $\mu \text{m}$, respectively, at $\lambda =1.5\mu \text{m}$ for $V=5$ (corresponds to a core radius of 5.627 $\mu m$ and a beam waist ${\rho }_0=2.516\mu m$), and (b) input Gaussian mode $(l,p)=(0,0)$ switches to $(l=1,p=0)$ and $(l=2,p=0)$, $(l=-2,p=0)$, and $(l=-1,p=0)$ at the propagation lengths of 400, 800, 1500, and 1900 $\mu \text{m}$, respectively, at $\lambda =1.5\mu \text{m}$ for $V=7$ (corresponds to a core radius of 7.878 $\mu \text{m}$ and a beam waist ${\rho }_0=2.977\mu \text{m}$)

Figure 7

Wavelength switching in the linear GRAIN fiber. (a) Each handedness of the OAM mode is accessed at different wavelengths with an efficiency of about $90\%$ for $V=5$. (b) Inclusion of the higher spatial modes with the efficiency of access about $50\%$ for $V=7$.

Figure 8

Dispersion curves of the linear GRAIN fiber. The above curves are for nondegenerate modes only. One can anticipate splitting the curves among the degenerate modes due to the azimuthal coupling. Solid dispersion curves indicate a slight change in the cutoff criteria of spatial modes compared with the unperturbed case. The number of spatial modes supported for a given $V$ parameter remains the same as the $h=\mathrm{\Delta }=0.01$ case. When $h=5\mathrm{\Delta }=0.05$ (dashed dispersion curves), the number of spatial modes supported by the GRAIN fiber changes for a given $V$ parameter.

Figure 9

Superposition state propagation in the GRAIN fiber, showing periodic revival of the input state $({\text{LG}}_{2,0}-{\text{LG}}_{0,1}+{\text{LG}}_{-2,0})/\sqrt{3}$ between the Gaussian modes.