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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, Δn, 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 Vc for the HEmn modes with m ≥ 2 of a weakly guiding fiber can be approximated by the nth 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, β2/k22 ≈ 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.

Figure 4  Field line patterns and intensity distributions for several leading modes of a circular fiber. The dark curves in the field patterns are the electric field lines, and the gray curves are the magnetic field lines. The thin circle in each profile locates the core boundary of a step-index fiber.

 

Linearly Polarized Modes

It can be seen that except for the HE11 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 LP modes. Indeed, all of the HE1n 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 Vc determined by (49) for the HE21 mode is the same as that of TE01 and TM01 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 HE1n 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 HE1n modes given in the form of (53) are plane polarized in the x direction. They are designated as LP0n modes. The LP01 mode is simply the HE11 mode and is the fundamental LP mode. There is two-fold degeneracy in LP0n modes because all HE1n 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 HEmn or EHmn mode by an angle of \(\pi/2m\) in \(\phi\). For example, an HE1n mode given by the form in (50), such as the HE11 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 HE21 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 TE01 and TM01 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 LP11 mode. There is four-fold degeneracy in the LP11 mode because it contains four nearly degenerate modes, TE01, TM01, HE21, and HE'21. The LP11 mode is the first high-order LP mode above the fundamental mode.

The discussions above can be extended to other LP modes. Except for LP0n modes, which are just HE1n modes, all other LP modes can be constructed from linear combinations of different basic fiber modes. Their relationships are summarized in table 1 below.

Table 1  Fiber Modes

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 LPmn 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 LP1n mode is approximately that of the TE0n mode.

2. Cutoff conditions.

Except for the LP0n mode, the cutoff Vc value for the LPmn mode is the nth 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 LP0n 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 LP01 mode, which is simply the HE11 mode, has no cutoff, as discussed earlier. Therefore, the cutoff Vc for the LP0n mode is the nth 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 LPmn mode is the nth 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 LPmn 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 LP0n mode and four possible combinations of polarizations and angular distributions for an LPmn mode with m ≥ 1. This characteristics is discussed above for the LP01 and LP11 modes and can be seen in (56) for the LP11 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 LPmn 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 LPmn 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.

Figure 5  Intensity profiles of a few LP modes. The intensity pattern of the LPmn mode consists of m node lines intersecting at the center and n intensity peaks counted radially out from the center. The thin circle in each profile locates the core boundary of a step-index fiber.

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 LPmn 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 LP01 mode has a confinement factor \(\Gamma_{01}\approx0.84\) at the cutoff point of V = 2.405 for the LP11 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.

Figure 6  Confinement factors of leading LP modes as a function of the fiber V number

 

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 n1 = 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 Δn = 0.025 for n2 = 1.455, which corresponds to Δ = 1.69%. With n2 = 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 LP0n modes, which are simply HE1n 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 LPmn mode with m ≥ 1 is a linear combination of some nearly, but not exactly, degenerate modes. Consider the combination LP11 =TM01 + HE21 given in (56). Because the TM01 and HE21 modes are not exactly degenerate, there is a slight difference, Δβ, in their propagation constants. As the LP11 field propagates over a long enough distance, this small Δβ eventually causes the phase relation between the TM01 and HE21 fields, which together make up the LP11 field, to change. As a result, the combined field will not always be plane polarized in the same direction. Therefore, the LP11 mode is not a true normal mode because it is not truly invariant in propagation.

However, as can be expected, Δβ decreases with Δn 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.


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