Categories: Fiber optics, Optics, Physics.

Step-index fiber

As light propagates in the zz-direction through an optical fiber, the transverse profile F(x,y)F(x,y) of the electric field can be shown to obey the Helmholtz equation in 2D:

 ⁣2F+(n2k2β2)F=0\begin{aligned} \nabla_{\!\perp}^2 F + (n^2 k^2 - \beta^2) F = 0 \end{aligned}

With nn being the position-dependent refractive index, k=ω/ck = \omega / c the vacuum wavenumber, and β\beta the mode’s propagation constant (i.e. wavenumber), to be found later. In polar coordinates (r,ϕ)(r,\phi) this can be rewritten as follows:

2Fr2+1rFr+1r22Fϕ2+μF=0\begin{aligned} \pdvn{2}{F}{r} + \frac{1}{r} \pdv{F}{r} + \frac{1}{r^2} \pdvn{2}{F}{\phi} + \mu F = 0 \end{aligned}

Where we have defined μn2k2 ⁣ ⁣β2\mu \equiv n^2 k^2 \!-\! \beta^2 for brevity. From now on, we only consider choices of μ\mu that do not depend on ϕ\phi or zz, but may vary with rr.

This Helmholtz equation can be solved by separation of variables: we assume that there exist two functions R(r)R(r) and Φ(ϕ)\Phi(\phi) such that F(r,ϕ)=R(r)Φ(ϕ)F(r,\phi) = R(r) \, \Phi(\phi). Inserting this ansatz:

RΦ+1rRΦ+1r2RΦ+μRΦ=0\begin{aligned} R'' \Phi + \frac{1}{r} R' \Phi + \frac{1}{r^2} R \Phi'' + \mu R \Phi = 0 \end{aligned}

We rearrange this such that each side only depends on one variable, by dividing by RΦR\Phi (ignoring the fact that it may be zero), and multiplying by r2r^2. Since this equation should hold for all values of rr and ϕ\phi, this means that both sides must equal a constant, which we call 2\ell^2:

r2RR+rRR+μr2=ΦΦ=2\begin{aligned} r^2 \frac{R''}{R} + r \frac{R'}{R} + \mu r^2 = -\frac{\Phi''}{\Phi} = \ell^2 \end{aligned}

This gives an eigenvalue problem for Φ\Phi, and the well-known Bessel equation for RR:

Φ+2Φ=0r2R+rR+(μr2 ⁣ ⁣2)R=0\begin{aligned} \boxed{ \Phi'' + \ell^2 \Phi = 0 } \qquad \qquad \boxed{ r^2 R'' + r R' + (\mu r^2 \!-\! \ell^2) R = 0 } \end{aligned}

We will return to RR later; we start with Φ\Phi, because it has the simplest equation. Since the angle ϕ\phi is limited to [0,2π][0,2\pi], Φ\Phi must be 2π2 \pi-periodic, so:

Φ(0)=Φ(2π)Φ(0)=Φ(2π)\begin{aligned} \Phi(0) = \Phi(2\pi) \qquad \qquad \Phi'(0) = \Phi'(2\pi) \end{aligned}

The above equation for Φ\Phi with these periodic boundary conditions is a Sturm-Liouville problem. Consequently, there are infinitely many allowed values of 2\ell^2, all real, and one of them is lowest, known as the ground state.

To find the eigenvalues 2\ell^2 and their corresponding Φ\Phi, we in turn assume that 2<0\ell^2 < 0, 2=0\ell^2 = 0, or 2>0\ell^2 > 0, and check if we can then arrive at a non-trivial Φ\Phi for each case.

By combining our result for 2=0\ell^2 = 0 and 2>0\ell^2 > 0, we get the following for =0,1,2,...\ell = 0, 1, 2, ...:

Φ(ϕ)=Acos(ϕ)\begin{aligned} \boxed{ \Phi_\ell(\phi) = A \cos(\phi \ell) } \end{aligned}

Here, \ell is called the primary mode index. We exclude <0\ell < 0 because cos(x)cos(x)\cos(x) \propto \cos(-x) and sin(x)sin(x)\sin(x) \propto \sin(-x), and because AA is free to choose thanks to linearity.

Let us now revisit the Bessel equation for the radial function R(r)R(r), which should be continuous and differentiable throughout the fiber:

r2R+rR+μr2R2R=0\begin{aligned} r^2 R'' + r R' + \mu r^2 R - \ell^2 R = 0 \end{aligned}

To continue, we need to specify the refractive index n(r)n(r), contained in μ(r)\mu(r). We choose a step-index fiber, whose cross-section consists of a core with radius aa, surrounded by a cladding that extends to infinity rr \to \infty. In the core r<ar < a, the index nn is a constant nin_i, while in the cladding r>ar > a it is another constant non_o.

Since μ\mu is different in the core and cladding, we will get different solutions RiR_i and RoR_o there, so we must demand that the field is continuous at the boundary r=ar = a:

Ri(a)=Ro(a)Ri(a)=Ro(a)\begin{aligned} R_i(a) = R_o(a) \qquad \qquad R_i'(a) = R_o'(a) \end{aligned}

Furthermore, for a physically plausible solution, we require that RiR_i is finite and that RoR_o decays monotonically to zero when rr \to \infty. These constraints will turn out to restrict μ\mu.

Introducing a new coordinate ρrμ\rho \equiv r \sqrt{|\mu|} gives the Bessel equation’s standard form, which has well-known solutions called Bessel functions, shown below. Let ±\pm be the sign of μ\mu:

{0=ρ22Rρ2+ρRρ±ρ2R2Rfor  μ00=r22Rr2+rRr2Rfor  μ=0\begin{aligned} \begin{cases} \displaystyle 0 = \rho^2 \pdvn{2}{R}{\rho} + \rho \pdv{R}{\rho} \pm \rho^2 R - \ell^2 R & \mathrm{for}\; \mu \neq 0 \\ \displaystyle 0 = r^2 \pdvn{2}{R}{r} + r \pdv{R}{r} - \ell^2 R & \mathrm{for}\; \mu = 0 \end{cases} \end{aligned}

First few solutions to Bessel's equation

Looking at these solutions with our constraints for RoR_o in mind, we see that for μ>0\mu > 0 none of the solutions decay monotonically to zero, so we must have μ0\mu \le 0 in the cladding. Of the remaining candidates, ln(r)\ln(r), rr^\ell and I(ρ)I_\ell(\rho) do not decay at all, leading to the following RoR_o:

Ro,(r)={rfor  μ=0  and  =1,2,3,...K(ρ)=K(rμ)for  μ<0  and  =0,1,2,...\begin{aligned} R_{o,\ell}(r) = \begin{cases} r^{-\ell} & \mathrm{for}\; \mu = 0 \;\mathrm{and}\; \ell = 1,2,3,... \\ K_\ell(\rho) = K_\ell(r \sqrt{-\mu}) & \mathrm{for}\; \mu < 0 \;\mathrm{and}\; \ell = 0,1,2,... \end{cases} \end{aligned}

Next, for RiR_i, we see that when μ<0\mu < 0 all solutions are invalid since they diverge at r=0r = 0, and so do ln ⁣(r)\ln\!(r), rr^{-\ell} and Y(ρ)Y_\ell(\rho). Of the remaining candidates, r0r^0 and rr^\ell have a non-negative slope at the boundary r=ar = a, so they can never be continuous with RoR_o'. This leaves J(ρ)J_\ell(\rho) for μ>0\mu > 0:

Ri,(r)=J(ρ)=J(rμ)for  μ>0  and  =0,1,2,...\begin{aligned} R_{i,\ell}(r) = J_\ell(\rho) = J_\ell(r \sqrt{\mu}) \qquad \mathrm{for}\; \mu > 0 \;\mathrm{and}\; \ell = 0,1,2,... \end{aligned}

Putting this all together, we now know what the full solution for FF should look like:

F(r,ϕ)=R(r)Φ(ϕ)={ARi,(r)cos(ϕ)for  raBRo,(r)cos(ϕl)for  ra\begin{aligned} F_\ell(r, \phi) = R_\ell(r) \, \Phi_\ell(\phi) = \begin{cases} A_\ell \: R_{i,\ell}(r) \, \cos(\phi \ell) & \mathrm{for}\; r \le a \\ B_\ell \: R_{o,\ell}(r) \, \cos(\phi l) & \mathrm{for}\; r \ge a \end{cases} \end{aligned}

Where AA_\ell and BB_\ell are constants to be chosen based on the light’s intensity, and to satisfy the continuity condition at r=ar = a.

We found that μ0\mu \le 0 in the cladding and μ>0\mu > 0 in the core. Since μn2k2 ⁣ ⁣β2\mu \equiv n^2 k^2 \!-\! \beta^2 by definition, this discovery places a constraint on the propagation constant β\beta:

ni2k2>β2no2k2\begin{aligned} n_i^2 k^2 > \beta^2 \ge n_o^2 k^2 \end{aligned}

Therefore, ni>non_i > n_o in a step-index fiber, and there is only a limited range of allowed β\beta-values; the fiber is not able to guide the light outside this range.

However, not all β\beta in this range are created equal for all kk. To investigate further, let us define the quantities ξi\xi_i and ξo\xi_o like so, assuming nin_i and non_o do not depend on kk:

ξi(k)ni2k2β2(k)ξo(k)β2(k)no2k2\begin{aligned} \xi_i(k) \equiv \sqrt{ n_i^2 k^2 - \beta^2(k) } \qquad \qquad \xi_o(k) \equiv \sqrt{ \beta^2(k) - n_o^2 k^2 } \end{aligned}

It is important to note that the sum of their squares is constant with respect to β\beta:

ξi2+ξo2=(NA)2k2\begin{aligned} \xi_i^2 + \xi_o^2 = (\mathrm{NA})^2 k^2 \end{aligned}

Where NA\mathrm{NA} is the so-called numerical aperture, often mentioned in papers and datasheets as one of a fiber’s key parameters. It is defined as:

NAni2no2\begin{aligned} \boxed{ \mathrm{NA} \equiv \sqrt{n_i^2 - n_o^2} } \end{aligned}

From this, we define a new fiber parameter: the VV-number, which is very useful:

Vaξi2+ξo2=akNA\begin{aligned} \boxed{ V \equiv a \sqrt{\xi_i^2 + \xi_o^2} = a k \: \mathrm{NA} } \end{aligned}

Now, the allowed values of β\beta are found by fulfilling the boundary conditions (for μ0\mu \neq 0):

AJ(aξi)=BK(aξo)AξiJ(aξi)=BξoK(aξo)\begin{aligned} A_\ell J_\ell(a \xi_i) &= B_\ell K_\ell(a \xi_o) \\ A_\ell \xi_i J_\ell'(a \xi_i) &= B_\ell \xi_o K_\ell'(a \xi_o) \end{aligned}

To remove AA_\ell and BB_\ell, we divide the latter equation by the former, meanwhile defining XaξiX \equiv a \xi_i and YaξoY \equiv a \xi_o for convenience, such that X2+Y2=V2X^2 + Y^2 = V^2:

XJ(X)J(X)=YK(Y)K(Y)\begin{aligned} X \frac{J_\ell'(X)}{J_\ell(X)} = Y \frac{K_\ell'(Y)}{K_\ell(Y)} \end{aligned}

We can turn this result into something a bit nicer by using the following identities:

J(x)=J(x)x+J1(x)K(x)=K(x)xK1(x)\begin{aligned} J_\ell'(x) = - \ell \frac{J_\ell(x)}{x} + J_{\ell-1}(x) \qquad \qquad K_\ell'(x) = - \ell \frac{K_\ell(x)}{x} - K_{\ell-1}(x) \end{aligned}

With this, the transcendental equation for β\beta takes this convenient form:

XJ1(X)J(X)=YK1(Y)K(Y)\begin{aligned} \boxed{ X \frac{J_{\ell-1}(X)}{J_\ell(X)} = - Y \frac{K_{\ell-1}(Y)}{K_\ell(Y)} } \end{aligned}

All β\beta that satisfy this indicate the existence of a linearly polarized mode, each labelled LPm\mathrm{LP}_{\ell m}, where \ell is the primary (azimuthal) mode index, and mm the secondary (radial) mode index, which is needed because multiple β\beta may exist for a single \ell.

An example graphical solution of the transcendental equation is illustrated below for a fiber with V=5V = 5, where red and blue respectively denote the left and right-hand side:

Graphical solution of transcendental equation

For the ground state the light is well-confined in the core, but for higher modes it increasingly leaks into the cladding, thereby reducing the wavenumber β\beta (because ni>non_i > n_o) until the fiber can no longer guide the light β<nok\beta < n_o k, and the modes thus stop existing. Therefore, there is a mode cutoff when β=nok\beta = n_o k or equivalently when ξo=0\xi_o = 0. With this is mind, consider a slightly rearranged version of the above transcendental equation:

XJ1(X)K(Y)=YK1(Y)J(X)\begin{aligned} X J_{\ell-1}(X) K_\ell(Y) = - Y K_{\ell-1}(Y) J_\ell(X) \end{aligned}

Because YaξoY \equiv a \xi_o and X2=V2Y2X^2 = V^2 - Y^2, if ξo=0\xi_o = 0 then Y=0Y = 0 and X=VX = V, and the right-hand side is zero; for it to be satisfiable (i.e. for the mode to exist), the other side should also vanish. Therefore, all LPm\mathrm{LP}_{\ell m} have cutoffs VmV_{\ell m} equal to the mmth roots of J1(Vm)=0J_{\ell-1}(V_{\ell m}) = 0, so if V>VmV > V_{\ell m} then LPm\mathrm{LP}_{\ell m} exists, as long as β\beta stays in the allowed range. In the above figure, they are V01=0V_{01} = 0, V11=2.405V_{11} = 2.405, and V02=V21=3.832V_{02} = V_{21} = 3.832.

All differential equations have been linear, so a linear combination of these solutions is also valid. Therefore, the fiber modes represent independent “channels” of light. However, in practice, they can interact nonlinearly, and light can scatter between them, and between polarizations.


  1. O. Bang, Applied mathematics for physicists: lecture notes, 2019, unpublished.
  2. B.E.A. Saleh, M.C. Teich, Fundamentals of photonics, 1st edition, 1991, Wiley.