Step-Index Fibers
This is a continuation from the channel waveguides tutorial.
An optical fiber is basically a cylindrical dielectric waveguide with a circular cross section where a high-index waveguiding core is surrounded by a low-index cladding. Optical fibers are usually made of silica (SiO2) glass. The index step and profile are controlled by the concentration and distribution of dopants. For example, the core can be doped with germania (GeO2) or alumina (Al2O3) or other oxides, such as P2O5 or TiO2, for a slightly higher index than that of a silica cladding. Alternatively, to take advantage of low-loss pure silica, the cladding can be doped with fluorine for a slightly lower index while the core contains undoped pure silica.
Silica fibers are ideal for light transmission in the visible and near-infrared regions because of their low loss and low dispersion in these spectral regions. They are therefore suitable for optical communications and most laser applications in this range of spectrum.
Optical fibers made of other materials are also developed for special applications. For example, low-cost plastic fibers can be used for short-distance interconnections between personal computers and printers in offices. Fibers composed of ZrF4, BaF2, AlF3, LiF3, and other fluorides have a low loss in the range of 2-4 μm in the mid infrared. They can be used for mid-infrared optical communication or medical applications. Fibers for other spectral regions, such as the 10-μm region of CO2 laser wavelengths, are also developed.
Optical fibers have a wide range of applications. Owing to their low losses and large bandwidths, their most important applications are fiber-optic communications and interconnections. Other important applications include fiber sensors, guided optical imaging, remote monitoring, and medical applications.
With active dopants, such as neodymium or erbium, fibers with an optical gain under optical pumping are also used as optical amplifiers and fiber lasers, opening up many new applications.
In addition, because optical fibers provide strong optical confinement over long distances, they also present unique conditions for many interesting nonlinear optical processes, which lead to such applications as optical soliton information and propagation, optical pulse compression, and optical frequency conversion. Within the photonics community, fiber-optic components and systems form a major industry by themselves.
Step-index fibers
A step-index fiber is a nonplanar step-index waveguide that has a circular cross section, as shown in figure 1 below.
The core ha a radius a. The core diameter 2a, typically ranges from a few micrometers for a single-mode fiber to less than 100 μm for a multimode fiber. It is designed for the fiber to support a desired number of guided modes. The outer diameter of a fiber is that of the outside boundary of its cladding, which is typically about 100 μm or somewhat larger. The outer diameter of a fiber is determined by the requirement that the cladding be thicker than the penetration depth of a guide-mode field to prevent the field from reaching the air-cladding boundary and by the consideration of easy handling. The standard outer diameter size for multimode silica fibers is 125 μm.
For a step-index fiber, the waveguide parameter V, also called the V number of the fiber, is defined as
\[\tag{1}V=\frac{2\pi}{\lambda}a\sqrt{n_1^2-n_2^2}=\frac{\omega}{c}a\sqrt{n_1^2-n_2^2}\]
The numerical aperture of the fiber is
\[\tag{2}\text{NA}=\sqrt{n_1^2-n_2^2}=\sin\theta_a\]
which determines the acceptance angle, θa, of an optical fiber. Therefore, the acceptance angle of a circular fiber is simply \(\theta_a=\sin^{-1}(\text{NA})=\sin^{-1}\sqrt{n_1^2-n_2^2}\).
The acceptance angle is the largest incident angle, with respect to the normal of the end surface of a fiber, that allows an optical beam to be coupled into the fiber core. A wave entering the fiber at an incident angle smaller than the acceptance angle will be totally reflected at the core-cladding interface and thus will be guided in the fiber core.
A wave entering at an incident angle larger than θa will be partially transmitted through the core-cladding interface after entering the fiber and will not be guided.
Example
A step-index silica fiber has a core index of 1.452, a cladding index of 1.449, and a core diameter of 8 μm. What are its numerical aperture and acceptance angle? What is the value of its V number at 850 nm wavelength?
For this fiber, n1 = 1.452, n2 = 1.449, and a = 4 μm. The numerical aperture is
\[\text{NA}=\sqrt{n_1^2-n_2^2}=0.093\]
The acceptance angle is
\[\theta_a=\sin^{-1}0.093=5.34^\circ\]
The V number at λ = 850 nm is
\[V=\frac{2\pi}{\lambda}a\sqrt{n_1^2-n_2^2}=2.758\]
The mode fields of a circular fiber are best described in cylindrical coordinates with
\[\tag{3}\mathbf{E}_{mn}(\mathbf{r},t)=\pmb{\mathcal{E}}_{mn}(\phi,r)\exp(\text{i}\beta_{mn}z-\text{i}\omega t)\]\[\tag{4}\mathbf{H}_{mn}(\mathbf{r},t)=\pmb{\mathcal{H}}_{mn}(\phi,r)\exp(\text{i}\beta_{mn}z-\text{i}\omega t)\]
Note that the first index,m, is associated with the coordinate \(\phi\), while the second index, n, is associated with the coordinate r. This designation of indices will become clear later. The field equations obtained in the optical waveguide field equations tutorial are general equations for waveguides. They can be used for a circular fiber by transforming x and y coordinates to r and \(\phi\) coordinates. For example, (16)-(19) [refer to the field equations tutorial] become
\[\tag{5}(k^2-\beta^2)\mathcal{E}_r=\text{i}\beta\frac{\partial\mathcal{E}_z}{\partial r}+\text{i}\omega\mu_0\frac{1}{r}\frac{\partial\mathcal{H}_z}{\partial\phi}\]\[\tag{6}(k^2-\beta^2)\mathcal{E}_\phi=\text{i}\beta\frac{1}{r}\frac{\partial\mathcal{E}_z}{\partial\phi}-\text{i}\omega\mu_0\frac{\partial\mathcal{H}_z}{\partial r}\]\[\tag{7}(k^2-\beta^2)\mathcal{H}_r=\text{i}\beta\frac{\partial\mathcal{H}_z}{\partial r}-\text{i}\omega\epsilon\frac{1}{r}\frac{\partial\mathcal{E}_z}{\partial\phi}\]\[\tag{8}(k^2-\beta^2)\mathcal{H}_\phi=\text{i}\beta\frac{1}{r}\frac{\partial\mathcal{H}_z}{\partial\phi}+\text{i}\omega\epsilon\frac{\partial\mathcal{E}_z}{\partial r}\]
where \(k^2=\omega^2\mu_0\epsilon(r)=\omega^2n^2(r)/c^2\).
For a step-index fiber, (27) and (28) in the wave equations for optical waveguides tutorial are also valid, but they take the following form in cylindrical coordinates:
\[\tag{9}\frac{\partial^2\mathcal{E}_z}{\partial r^2}+\frac{1}{r}\frac{\partial\mathcal{E}_z}{\partial r}+\frac{1}{r^2}\frac{\partial^2\mathcal{E}_z}{\partial\phi^2}+(k_\text{i}^2-\beta^2)\mathcal{E}_z=0\]\[\tag{10}\frac{\partial^2\mathcal{H}_z}{\partial r^2}+\frac{1}{r}\frac{\partial\mathcal{H}_z}{\partial r}+\frac{1}{r^2}\frac{\partial^2\mathcal{H}_z}{\partial\phi^2}+(k_\text{i}^2-\beta^2)\mathcal{H}_z=0\]
where \(k_1^2=\omega^2n_1^2/c^2\) for the core and \(k_2^2=\omega^2n_2^2/c^2\) for the cladding for i = 1 and 2, respectively. For guided modes, we have \(k_1\gt\beta\gt k_2\) and
\[\tag{11}k_1^2-\beta^2=h^2\]\[\tag{12}\beta^2-k_2^2=\gamma^2\]
In general, fiber modes can be hybrid modes with \(\mathcal{E}_z\ne0\) and \(\mathcal{H}_z\ne0\). Therefore, (9) and (10) have to be solved simultaneously. They can be solved by separation of variables. For example, for \(\mathcal{E}_z\), the solution for \(\phi\) dependence yields
\[\tag{13}\mathcal{E}_z(\phi, r)=R(r)e^{\pm\text{i}m\phi},\qquad m=0,1,2,...,\]
where R(r) satisfies
\[\tag{14}\frac{\text{d}^2 R}{\text{d}r^2}+\frac{1}{r}\frac{\text{d}R}{\text{d}r}+\left(h^2-\frac{m^2}{r^2} \right)R=0,\qquad\text{for }r\lt a\]\[\tag{15}\frac{\text{d}^2 R}{\text{d}r^2}+\frac{1}{r}\frac{\text{d}R}{\text{d}r}-\left(\gamma^2+\frac{m^2}{r^2} \right)R=0,\qquad\text{for }r\gt a\]
Equations of the same form define the dependence of \(\mathcal{H}_z\) on \(\phi\) and r. The solution of (14) with the requirement that R(r) be finite at r = 0 is \(J_m(hr)\), the Bessel function of the first kind of order m. Meanwhile, the solution of (15) with the requirement that \(rR^2(r)\rightarrow0\) as \(r\rightarrow\infty\) yields \(K_m(\gamma r)\), the modified Bessel function of the second kind of order m.
Thus the r dependence of \(\mathcal{E}_z\) and \(\mathcal{H}_z\) is found to be \(J_m(hr)\) for r < a and \(K_m(\gamma r)\) for r > a. The leading orders of \(J_m(x)\) and \(K_m(x)\) are plotted in figure 2(a) and 2(b), respectively.
These Bessel functions have the following properties:
\[\tag{16}J_0(0)=1,\qquad J_{m\ne0}(0)=0\]\[\tag{17}K_m(0)=\infty\]
and, for large values of x,
\[\tag{18}J_m(x)\approx\sqrt{\frac{2}{\pi x}}\left[\cos\left(x-\frac{m\pi}{2}-\frac{\pi}{4}\right)-\frac{4m^2-1}{8x}\sin\left(x-\frac{m\pi}{2}-\frac{\pi}{4}\right)\right]\]\[\tag{19}K_m(x)\approx\sqrt{\frac{\pi}{2x}}\left(1+\frac{4m^2-1}{8x}\right)e^{-x}\]
The following identities of the Bessel functions are also found to be useful:
\[\tag{20}J_{-m}=(-1)^mJ_m,\qquad\qquad K_{-m}=K_m\]\[\tag{21}J_m'=\frac{1}{2}(J_{m-1}-J_{m+1}),\qquad K_m'=-\frac{1}{2}(K_{m-1}+K_{m+1})\]\[\tag{22}\frac{m}{x}J_m=\frac{1}{2}(J_{m-1}+J_{m+1}),\qquad\frac{m}{x}K_m=-\frac{1}{2}(K_{m-1}-K_{m+1})\]
Once \(\mathcal{E}_z\) and \(\mathcal{H}_z\) are solved, the other field components can be found using (5)-(8).
The boundary conditions require that the tangential field components, \(\mathcal{E}_z\), \(\mathcal{E}_\phi\), \(\mathcal{H}_z\), and \(\mathcal{H}_\phi\), be continuous at the boundary, r = a, between the core and the cladding. These conditions result in the requirement that the \(\phi\) dependence of \(\mathcal{E}_z\) be 90° out-of-phase with respect to that of \(\mathcal{H}_z\). Therefore, we can choose
\[\tag{23}\mathcal{E}_z(\phi,r)=\left\lbrace\begin{align}&A_mJ_m(hr)\cos{m\phi},\qquad{r\lt a}\\&B_mK_m(\gamma r)\cos{m\phi},\qquad{r\gt a}\end{align}\right.\]\[\tag{24}\mathcal{H}_z(\phi,r)=\left\lbrace\begin{align}&C_mJ_m(hr)\sin{m\phi},\qquad{r\lt a}\\&D_mK_m(\gamma r)\sin{m\phi},\qquad{r\gt a}\end{align}\right.\]
where Am, Bm, Cm, and Dm are constants to be found for a particular fiber mode.
Alternatively, we can choose
\[\tag{25}\mathcal{E}_z(\phi,r)=\left\lbrace\begin{align}&A_m'J_m(hr)\sin{m\phi},\qquad{r\lt a}\\&B_m'K_m(\gamma r)\sin{m\phi},\qquad{r\gt a}\end{align}\right.\]\[\tag{26}\mathcal{H}_z(\phi,r)=\left\lbrace\begin{align}&C_m'J_m(hr)\cos{m\phi},\qquad{r\lt a}\\&D_m'K_m(\gamma r)\cos{m\phi},\qquad{r\gt a}\end{align}\right.\]
where A'm, B'm, C'm, and D'm are also constants for a particular fiber mode.
For \(m\ne0\), these two sets of choices are degenerate because one can be transformed into the other by a change of reference of the angle \(\phi\) for one or the other, which has no physical significance in a circular fiber. However, for \(m=0\), they represent distinctly different sets of modes, as discussed below.
Application of the boundary conditions for a nontrivial solution of Am, Bm, Cm, and Dm for (23) and (24) or that of A'm, B'm, C'm, and D'm for (25) and (26) yields the following eigenvalue equation for the allowed values of h and γ for the guided modes:
\[\tag{27}\begin{align}\left[\frac{J'_m(ha)}{haJ_m(ha)}+\frac{K'_m(\gamma a)}{\gamma aK_m(\gamma a)}\right]\left[\frac{n_1^2J'_m(ha)}{haJ_m(ha)}+\frac{n_2^2K'_m(\gamma a)}{\gamma aK_m(\gamma a)}\right]\\=m^2\frac{c^2\beta^2}{\omega^2}\left(\frac{1}{h^2a^2}+\frac{1}{\gamma^2a^2}\right)^2\end{align}\]
where \(J'_m\) and \(K'_m\) are the derivatives of the Bessel functions.
Recall that each mode in a circular fiber is characterized by two mode indices m and n. As seen above, the first index m refers to the angular dependence \(\cos{m\phi}\) or \(\sin{m\phi}\). The second index n refers to the order of the allowed solutions for eigenvalues h or, equivalently, γ. Therefore, m is called the azimuthal mode index, or the angular mode index, while n is called the radial mode index.
In general, (27) has to be solved numerically.
Fiber Modes
It can be seen that when m = 0, the first set of solutions for the longitudinal components of the mode fields given in (23) and (24) results in the TM fields with \(\mathcal{H}_z\)=0, while the second set of solutions given in (25) and (26) results in the TE fields with \(\mathcal{E}_z\)=0. Therefore, for m = 0, the guided modes are either TE or TM modes, and (27) becomes two separate eigenvalue equations:
\[\tag{28}\frac{J_1(ha)}{haJ_0(ha)}+\frac{K_1(\gamma a)}{\gamma aK_0(\gamma a)}=0,\qquad\text{for TE modes}\]
and
\[\tag{29}\frac{n_1^2J_1(ha)}{haJ_0(ha)}+\frac{n_2^2K_1(\gamma a)}{\gamma aK_0(\gamma a)}=0,\qquad\text{for TM modes}\]
where the relations \(J'_0=-J_1\) and \(K'_0=-K_1\) are used.
For \(m\ge1\), the guided modes in a circular fiber are hybrid modes. Both \(\mathcal{E}_z\) and \(\mathcal{H}_z\) exist in these modes. As a result, all six field components exist. In this case, the solution given in (23) and (24) is degenerate with that given in (25) and (26) with
\[\tag{30}\frac{A_m}{C_m}=-\frac{A'_m}{C'_m}\qquad\text{and}\qquad\frac{B_m}{D_m}=-\frac{B'_m}{D'_m}\]
Therefore, for a hybrid mode, we only have to consider the solutions given by, say, (23) and (24). The hybrid modes can be classified into two groups. Those with Am and Cm having the same sign are called HE modes, while those with Am and Cm having opposite signs are called EH modes. For each given \(m\ge1\), the eigenvalue equation in (27) yields two sets of solutions, one for HE modes and another for EH modes.
Using (5)-(8), all field components can be found from \(\mathcal{E}_z\) and \(\mathcal{H}_z\). For the fields in the core region, the resulting field expression can be simplified by using the identities in (21) and (22) for \(J_m(x)\).
1. For TE0n modes, \(\mathcal{E}_z=\mathcal{E}_r=\mathcal{H}_\phi=0\) and
\[\tag{31}\mathcal{H}_z=J_0(hr),\qquad\mathcal{E}_\phi=\frac{\text{i}\omega\mu_0}{h}J_1(hr),\qquad\mathcal{H}_r=-\frac{\text{i}\beta}{h}J_1(hr) \]
2. For TM0n modes, \(\mathcal{H}_z=\mathcal{E}_\phi=\mathcal{H}_r=0\) and
\[\tag{32}\mathcal{E}_z=J_0(hr),\qquad\mathcal{E}_r=-\frac{\text{i}\beta}{h}J_1(hr),\qquad\mathcal{H}_\phi=-\frac{\text{i}\omega\epsilon_1}{h}J_1(hr)\]
3. For HEmn and EHmn modes, all six field components exist and are given by
\[\tag{33}\mathcal{E}_z=J_m(hr)\cos{m\phi}\]\[\tag{34}\mathcal{H}_z=\frac{\beta}{\omega\mu_0}\eta J_m(hr)\sin m\phi\]\[\tag{35}\mathcal{E}_r=\frac{\text{i}\beta}{h}\left[\frac{1+\eta}{2}J_{m-1}(hr)-\frac{1-\eta}{2}J_{m+1}(hr)\right]\cos m\phi\]\[\tag{36}\mathcal{E}_\phi=-\frac{\text{i}\beta}{h}\left[\frac{1+\eta}{2}J_{m-1}(hr)+\frac{1-\eta}{2}J_{m+1}(hr)\right]\sin m\phi\]\[\tag{37}\mathcal{H}_r=\frac{\text{i}\omega\epsilon_1}{h}\left[\frac{1+\eta\beta^2/k_1^2}{2}J_{m-1}(hr)+\frac{1-\eta\beta^2/k_1^2}{2}J_{m+1}(hr)\right]\sin m\phi\]\[\tag{38}\mathcal{H}_\phi=\frac{\text{i}\omega\epsilon_1}{h}\left[\frac{1+\eta\beta^2/k_1^2}{2}J_{m-1}(hr)-\frac{1-\eta\beta^2/k_1^2}{2}J_{m+1}(hr)\right]\cos m\phi\]
where
\[\tag{39}\eta=\frac{\omega\mu_0C_m}{\beta A_m}\]
The value of the constant η is a characteristic of a particular HE or EH mode and is determined by the boundary conditions through solution of (27). For η > 0, (33)-(38) represent the field components of the HEmn mode. For η < 0, they represent the field components of the EHmn mode.
Note that a multiplicative constant common to all of the field components in a mode is omitted in the above representation. Thus, these mode fields are not normalized.
The intensity of a mode has to be calculated using equation (37) of the wave equations for optical waveguides tutorial. For the modes of a circular fiber, it is reduce to
\[\tag{40}I=2(\mathcal{E}_r\mathcal{H}_\phi^*-\mathcal{E}_\phi\mathcal{H}_r^*)\]
The power in a mode is obtained by integrating the intensity over the fiber cross section:
\[\tag{41}P=2\displaystyle\int\limits_0^\infty\int\limits_0^{2\pi}(\mathcal{E}_r\mathcal{H}_\phi^*-\mathcal{E}_\phi\mathcal{H}_r^*)r\text{d}\phi\text{d}r\]
In accordance with the discussions in wave equations for optical waveguides tutorial, it can be shown that equation (41) is equivalent to (38) [in the wave equation tutorial] for a TE mode and is equivalent to (39) [in the wave equation tutorial] for a TM mode.
For HE and EH hybrid modes, (40) and (41) cannot be reduce to the form of only an electric field or that of only a magnetic field.
Cutoff Conditions
The cutoff for a particular guided mode of an optical fiber is determined by the condition γ = 0, at which instant the guided mode ceases to be guided. This is the same condition as that for a guided mode of a planar waveguide discussed in the step-index planar waveguides tutorial. At cutoff, we have
\[\tag{42}V_c=ha\]
which has a form similar to that of (70) [refer to the step-index planar waveguides tutorial]. The equation for finding the cutoff value Vc depends on the type of mode:
1. For TE0n and TM0n modes, Vc is the nth root of the equation
\[\tag{43}J_0(x)=0\]
2. For HE1n modes, Vc is the nth root of the equation
\[\tag{44}J_1(x)=0\]
the first of which being x = 0. Therefore, Vc = 0 for the HE11 mode. For HEmn modes with m ≥ 2, Vc is the nth nonzero root of the equation
\[\tag{45}J_{m-2}(x)+\frac{n_1^2-n_2^2}{n_1^2+n_2^2}J_m(x)=0\]
Because \(J_{-1}(x)=-J_1(x)\), (45) reduces to (44) for the HE1n modes when m = 1. Note that the values of Vc for HEmn modes with m ≥ 2 depend on the specific values of the refractive indices n1 and n2.
3. For all EHmn modes, m ≥ 1, and Vc is the nth nonzero root of the equation
\[\tag{46}J_m(x)=0\]
Figure 3 shows the graphic solution of Vc for some leading modes.
As we can see from figure 3(a), the fundamental mode of a circular fiber is the HE11 mode, which has no cutoff.
The first high-order modes are the TE01 and TM01 modes, which have the same cutoff value of Vc = 2.405. Note that although the TE01 and TM01 modes have the same cutoff Vc, they are not degenerate because they have different β defined by different eigenvalue equations in (28) and (29), respectively, when they are above cutoff. This is also true for other modes that have the same cutoff Vc, such as the HE12 and EH11 modes.
A fiber that has a waveguide parameter
\[\tag{47}V=\frac{2\pi}{\lambda}a\sqrt{n_1^2-n_2^2}\lt2.405\]
supports only the fundamental HE11 mode and is called a single-mode fiber. A fiber with V > 2.405 can support more than just the HE11 mode and is called a multimode fiber. Clearly, whether a fiber is single moded or multimoded depends not only on its index step and core radius, but also on the optical wavelength being considered. For a given fiber, V = 2.405 determines its cutoff wavelength, λc, for its single-mode characteristics. The fiber is single mode for λ > λc, but is multimoded for λ < λc.
Example
Is the silica fiber described in the previous example single moded at 850 nm wavelength? What is the cutoff wavelength for its single-mode operation?
As found in the previous example, V = 2.758 > 2.405 for the fiber at λ = 850 nm. Therefore, this fiber is not not a single-mode fiber at 850 nm wavelength. The cutoff wavelength corresponds to Vc = 2.405. It is found as
\[\lambda_c=\frac{2\pi}{V_c}a\sqrt{n_1^2-n_2^2}=975\,\text{nm}\]
This fiber is single moded at wavelengths longer than 975 nm but is multimoded at shorter wavelengths. For example, it is a single-mode fiber at 1.3 μm wavelength.
The next part continues with the Weakly Guiding Fibers tutorial.