# Weakly Guiding Fibers

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This tutorial is a continuation from the step-index fibers tutorial.

Most optical fibers for practical applications are ** weakly guiding fibers** that have a small index step, Δ

*n*, between the core and the cladding:

\[\tag{48}\Delta=\frac{\Delta n}{n_1}=\frac{n_1-n_2}{n_1}\ll1\]

The mathematics for the modes of a weakly guiding fiber can be greatly simplified by taking proper approximations. For example, the cutoff *V*_{c} for the HE* _{mn}* modes with m ≥ 2 of a weakly guiding fiber can be approximated by the

*n*th nonzero root of the equation

\[\tag{49}J_{m-2}(x)=0\]

which is obtained from (45) [refer to the step-index fibers tutorial] under the condition of (48).

Meanwhile, for the modes of a weakly guiding fiber, β^{2}/k_{2}^{2} ≈ 1, and the parameter *η* defined in (39) [refer to the step-index fibers tutorial] has a value of *η* ≈ 1 for HE modes and a value of *η* ≈ -1 for EH modes. Therefore, (35)-(38) [refer to the step-index fibers tutorial] are reduced to a simple form that is useful for obtaining a visualization of the field patterns and intensity distributions of the modes. The resulting approximate transverse electric field components, \(\mathcal{E}_r\) and \(\mathcal{E}_\phi\), and intensity distribution, *I*, for the four types of fiber modes are

\[\tag{50}\left.\begin{align}&TE_{0n}:&&\mathcal{E}_r=0,&&\mathcal{E}_\phi\sim J_1(hr),&&I\sim J_1^2(hr)\\&TM_{0n}:&&\mathcal{E}_r\sim J_1(hr),&&\mathcal{E}_\phi=0,&&I\sim J_1^2(hr)\\&HE_{mn}:&&\mathcal{E_r}\sim J_{m-1}(hr)\cos m\phi,&&\mathcal{E}_\phi\sim -J_{m-1}(hr)\sin m\phi,&&I\sim J_{m-1}^2(hr)\\&EH_{mn}:&&\mathcal{E}_r\sim -J_{m+1}(hr)\cos m\phi,&&\mathcal{E}_\phi\sim-J_{m+1}(hr)\sin m\phi,&&I\sim J_{m+1}^2(hr)\end{align}\right\rbrace\]

Transverse magnetic field components also have a simple form similar to that of transverse electrical field components. Because transverse magnetic field lines are simply orthogonal to transverse electric field lines, the magnetic field components are not spelled out explicitly in (50). The patterns of the field lines and intensity distributions of several leading modes are shown in figure 4 below. Note that the intensity distributions for all four types of modes do not depend on \(\phi\) and have only radial variations.

**Linearly Polarized Modes**

It can be seen that except for the HE_{11} mode, the fields of the fiber modes shown in figure 4 are not plane polarized because the field lines are not straight parallel lines.

However, in the weakly guiding approximation, it is possible to represent the fields in a fiber in terms of ** linearly polarized modes**, called

**. Indeed, all of the HE**

*LP modes*_{1n}modes are very much plane polarized, particularly in weakly guiding fibers.

For other modes, many are nearly degenerate, and plane polarized fields can be formed by linear combinations of these nearly degenerate modes if the weakly guiding approximation leading to (50) is valid. For example, in the weakly guiding limit, the cutoff *V*_{c} determined by (49) for the HE_{21} mode is the same as that of TE_{01} and TM_{01} modes. These three modes are nearly degenerate. Combinations of these nearly degenerate modes result in LP modes.

The discussions above can be demonstrated by considering the *x* and *y* components of the transverse electric field:

\[\tag{51}\mathcal{E}_x=\mathcal{E}_r\cos\phi-\mathcal{E}_\phi\sin\phi\]\[\tag{52}\mathcal{E}_y=\mathcal{E}_r\sin\phi+\mathcal{E}_\phi\cos\phi\]

For any HE_{1n} mode, we have

\[\tag{53}\mathcal{E}_r\sim J_0(hr)\cos\phi\qquad\text{and}\qquad\mathcal{E}_\phi\sim -J_0(hr)\sin\phi\]

from (50).

Using (51) and (52), this results in

\[\tag{54}\mathcal{E}_x\sim J_0(hr)\qquad\text{and}\qquad\mathcal{E}_y=0\]

Therefore, the transverse electric fields of all of the HE_{1n} modes given in the form of (53) are plane polarized in the *x* direction. They are designated as LP_{0n} modes. The LP_{01} mode is simply the HE_{11} mode and is the fundamental LP mode. There is two-fold degeneracy in LP_{0n} modes because all HE_{1n} modes are two-fold degenerate.

Before we proceed further, we have to note that each of the HE and EH mode has two-fold degeneracy, whereas TE and TM modes have no degeneracy. This is because the field patterns of the HE and EH modes are functions of \(\phi\), but those of the TE and TM modes are independent of \(\phi\).

An orthogonal field pattern can be generated by rotating the field pattern of any HE_{mn} or EH_{mn} mode by an angle of \(\pi/2m\) in \(\phi\). For example, an HE_{1n} mode given by the form in (50), such as the HE_{11} mode shown in figure 4, has its field lines parallel to the *x* direction, as is demonstrated above. Its degenerate orthogonal mode pattern is one with the field lines parallel to the *y* direction.

For the HE_{21} mode given by (50) and shown in figure 4, its degenerate orthogonal mode pattern HE'_{21} can be obtained by substituting \(\phi\) in (50) with \(\phi+\pi/4\) for *m* = 2. Thus we have

\[\tag{55}\begin{align}&HE_{21}:&&\mathcal{E}_r\sim J_1(hr)\cos2\phi,&&\mathcal{E}_\phi\sim -J_1(hr)\sin2\phi\\&HE_{21}^{'}:&&\mathcal{E}_r\sim-J_1(hr)\sin2\phi,&&\mathcal{E}_\phi\sim-J_1(hr)\cos2\phi\end{align}\]

The TE_{01} and TM_{01} modes have no degeneracy. Their \(\mathcal{E}_r\) and \(\mathcal{E}_\phi\) field components are simply those given by (50). Using (51) and (52), it can be shown that

\[\tag{56}\begin{align}&TE_{01}+HE_{21}^{'}:&&\mathcal{E}_x\sim-2J_1(hr)\sin\phi,&&\mathcal{E}_y=0\\&TE_{01}-HE_{21}^{'}:&&\mathcal{E}_x=0,&&\mathcal{E}_y\sim2J_1(hr)\cos\phi\\&TM_{01}+HE_{21}:&&\mathcal{E}_x\sim2J_1(hr)\cos\phi,&&\mathcal{E}_y=0\\&TM_{01}-HE_{21}:&&\mathcal{E}_x=0,&&\mathcal{E}_y\sim2J_1(hr)\sin\phi\end{align}\]

These are plane polarized fields. They are designated as the LP_{11} mode. There is four-fold degeneracy in the LP_{11} mode because it contains four nearly degenerate modes, TE_{01}, TM_{01}, HE_{21}, and HE'_{21}. The LP_{11} mode is the first high-order LP mode above the fundamental mode.

The discussions above can be extended to other LP modes. Except for LP_{0n} modes, which are just HE_{1n} modes, all other LP modes can be constructed from linear combinations of different basic fiber modes. Their relationships are summarized in table 1 below.

The eigenvalue equation and the equation defining the cutoff conditions of the LP modes, as well as their field and intensity patterns, are much simplified. These characteristics are summarized below.

1. **Eigenvalue equation**.

The eigenvalue equation for all LP* _{mn}* modes can be written as

\[\tag{57}\frac{haJ_{m-1}(ha)}{J_m(ha)}=-\frac{\gamma aK_{m-1}(\gamma a)}{K_m(\gamma a)}\]

For *m* = 0, the relations \(J_{-1}(x)=-J_1(x)\) and \(K_{-1}(x)=K_1(x)\) from (20) [refer to the step-index fibers tutorial] can be used. Note that (57) reduces to (28) [refer to the step-index fibers tutorial] for *m* = 1 because the eigenvalue of the LP_{1n} mode is approximately that of the TE_{0n} mode.

2. **Cutoff conditions**.

Except for the LP_{0n} mode, the cutoff *V*_{c} value for the LP_{mn} mode is the *n*th *nonzero* root of the equation

\[\tag{58}J_{m-1}(x)=0\]

This condition can be obtained by considering the cutoff conditions for the TE, TM, HE, and EH modes discussed in the step-index fiber tutorial in the weakly guiding limit of (48). It can also be obtained by directly applying the cutoff condition of *γ* = 0 to the eigenvalue equation in (57) for the LP modes. For the LP_{0n} mode, *m* = 0 and (58) becomes

\[\tag{59}J_1(x)=0\]

The first root, *x* = 0, counts even though is is a trivial root. The LP_{01} mode, which is simply the HE_{11} mode, has no cutoff, as discussed earlier. Therefore, the cutoff *V*_{c} for the LP_{0n} mode is the *n*th root of the (59), counting *x* = 0 as the first one.

3. **Number of modes**.

For a multimode fiber with a large *V* number, the number of modes supported by the fiber can be estimated. Since the cutoff \(V_{mn}^c\) for the LP_{mn} mode is the *n*th nonzero root of (58), we have

\[\tag{60}V_{mn}^c=\left(m+2n-\frac{3}{2}\right)\frac{\pi}{2}\approx(m+2n)\frac{\pi}{2}\]

from (18) [refer to the step-index fibers tutorial].

This means that for a given large value of *V*, the maximum value of *m* is \(m_{max}\approx2V/\pi\), while the maximum value of *n* for a given *m* is \(n_{max}=V/\pi-m/2\). Since there is a four-fold degeneracy for each LP_{mn} mode with \(m\ne0\), the total number of modes is approximately

\[\tag{61}M\approx4\sum_{m=0}^{2V/\pi}\,\sum_{n=1}^{V/\pi-m/2}1=\frac{4V^2}{\pi^2}+\frac{2V}{\pi}\approx\frac{4V^2}{\pi^2}\]

4. **Field patterns**.

The fields of the LP modes are plane polarized. Because of the degeneracy in each LP mode, there are two possible polarizations for an LP_{0n} mode and four possible combinations of polarizations and angular distributions for an LP_{mn} mode with *m* ≥ 1. This characteristics is discussed above for the LP_{01} and LP_{11} modes and can be seen in (56) for the LP_{11} mode.

For simplicity, we consider the field to be polarized in the *y* direction and the azimuthal angular distribution to be such that \(\mathcal{E}_y\) has a maximum at \(\phi=0\). Then, for any LP_{mn} mode, the field pattern is simply

\[\tag{62}\mathcal{E}_y\sim\left\lbrace\begin{align}&\frac{1}{J_m(ha)}J_m(hr)\cos m\phi,&&r\lt a\\&\frac{1}{K_m(\gamma a)}K_m(\gamma r)\cos m\phi,&&r\gt a\end{align}\right.\]

and \(\mathcal{E}_x=0\). Note that the boundary conditions for a circular fiber do not require \(\mathcal{E}_y\) to be continuous at *r* = *a*. Rather, they require \(\mathcal{E}_\phi\) and \(\mathcal{H}_\phi\) to be continuous at *r* = *a*. Because (62) does not satisfy the boundary conditions exactly, it is only an approximation under the weakly guiding condition of (48).

5. **Intensity distributions**.

The intensity distribution of the LP_{mn} mode has the following pattern:

\[\tag{63}I(\phi, r)\sim\left\lbrace\begin{align}&\frac{1}{J_m^2(ha)}J_m^2(hr)\cos^2m\phi,&&r\lt a\\&\frac{1}{K_m^2(\gamma a)}K_m^2(\gamma r)\cos^2m\phi,&&r\gt a\end{align}\right.\]

This characteristic is also summarized in table 1. Figure 5 below shows the intensity profiles of a few LP modes.

6. **Confinement factors**.

The confinement factor for a mode is the fractional power in the core region and is given by

\[\tag{64}\Gamma_{mode}=\frac{P_{core}}{P_{mode}}=\frac{\displaystyle\int\limits_0^a\int\limits_0^{2\pi}I(\phi,r)r\text{d}r\text{d}\phi}{\displaystyle\int\limits_0^{\infty}\int\limits_0^{2\pi}I(\phi,r)r\text{d}r\text{d}\phi}\]

For the LP_{mn} mode, the integrals in (64) can be calculated using the intensity distribution given in (63), resulting in

\[\tag{65}\Gamma_{mn}=1-\frac{h^2a^2}{V^2}\left[1-\frac{K_m^2(\gamma a)}{K_{m-1}(\gamma a)K_{m+1}(\gamma a)}\right]\]

This expression has to be evaluated numerically. An approximate expression is

\[\tag{66}\Gamma_{mn}=1-\frac{h^2a^2}{V^2}\frac{1}{\sqrt{\gamma^2a^2+m^2+1}}\]

The confinement factors for some leading LP modes are shown as a function of the fiber *V* number in figure 6 below. We see that the fundamental LP_{01} mode has a confinement factor \(\Gamma_{01}\approx0.84\) at the cutoff point of *V* = 2.405 for the LP_{11} mode.

Note that as cutoff is approached, the power for a mode with *m* = 0 or *m* = 1 moves away from the core to the cladding so that \(\Gamma_{mn}\rightarrow0\). However, for LP modes with *m* ≥ 2, a large fraction of power remains in the core at cutoff. For a mode with large *m*, the power remains primarily in the core.

**Example**

A multimode silica fiber has a core index of 1.48 and a core diameter of 50 μm. Find the index step needed for it to support at least 1000 guided modes at 850 nm wavelength. How many modes does this fiber support at 1.3 μm wavelength if dispersion can be ignored?

According to (61), the *V* number needs to be \(V=\pi\sqrt{M}/2\gt49.67\) so that *M* > 1000 for the fiber to support at least 1000 modes. Using *n*_{1} = 1.48, *a* = 50 μm/2 = 25 μm, and λ = 850 nm as given, we find that

\[n_2\approx\sqrt{n_1^2-\left(\frac{V\lambda}{2\pi a}\right)^2}\lt1.4554\]

Therefore, we can choose an index step Δ*n* = 0.025 for *n*_{2} = 1.455, which corresponds to Δ = 1.69%. With *n*_{2} = 1.455, we find that *V* = 50.05 > 49.67 and *M* = 1015 > 1000, as required.

Because \(M\propto V^2\propto\lambda^{-2}\), we can find the number of modes at 1.3 μm directly from that at 850 nm if dispersion is ignored. Therefore, the number of modes at 1.3 μm is

\[M=\frac{0.85^2}{1.3^2}\times1015\approx434\]

It has to be noted that although eigenvalue equations and cutoff conditions are written for the LP modes, they are approximations valid only in the weakly guiding limit.

Except for LP_{0n} modes, which are simply HE_{1n} normal modes, ** the LP modes are not the exact solutions of Maxwell's equations for a fiber and thus are not true normal modes of a fiber**.

This concept can be understood from the fact that an LP_{mn} mode with m ≥ 1 is a linear combination of some *nearly, but not exactly, degenerate* modes. Consider the combination LP_{11} =TM_{01} + HE_{21} given in (56). Because the TM_{01} and HE_{21} modes are not exactly degenerate, there is a slight difference, Δ*β*, in their propagation constants. As the LP_{11} field propagates over a long enough distance, this small Δ*β* eventually causes the phase relation between the TM_{01} and HE_{21} fields, which together make up the LP_{11} field, to change. As a result, the combined field will not always be plane polarized in the same direction. Therefore, the LP_{11} mode is not a true normal mode because it is not truly invariant in propagation.

However, as can be expected, Δ*β* decreases with Δ*n* and becomes insignificant for most practical applications, except for vey-long-distance propagation of the mode. For practical applications, because the true modes that make up an LP mode are very nearly degenerate, they can be excited simultaneously if they are above cutoff.

Consequently, if a plane polarized optical wave in free space is coupled into a fiber, it usually results in the excitation of an LP mode. The mode patterns shown in figure 5 are those usually seen at the output of a fiber.

The next part continues with the graded-index fibers tutorial.