Optics 083 · Waveguides, structured light, and modern optics

Optical Fiber: Rays to Modes

When does the picture of rays bouncing inside a fiber stop working, so that light must be treated as modes? Once the core is only a few wavelengths across. The V number, set by core radius, wavelength and numerical aperture, says how many guided modes fit. Below a cutoff of about 2.4 only one mode propagates, and a picture of rays bouncing at different angles no longer describes the light.

In a 3D cutaway fiber, change the core radius and compare the ray and wave pictures. Rays escape according to Snell's law, the guided modes are Bessel eigenmodes with tails in the cladding, and near-field samples can be frozen.

Normalized frequency

V

$0.50$

Geometrical numerical aperture

\mathrm{NA}

$50\%$

Selected mode propagation parameter

b

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Optical Fiber: Rays to Modes

Actual core radius, not diameter

a

4.50\,\mu\mathrm m

Assumed core minus cladding index

\Delta n

0.005

Vacuum wavelength; fixed material indices

\lambda_0

1550\,\mathrm{nm}

Independent test ray angle in air

\theta_{\rm air}

5^\circ

Mode family: 0 fundamental, 1 two lobes, 2 radial node

q

0

Selected mode lobe orientation

\phi_0

0^\circ

Shrink the core. The ray still bounces, but a two-lobed wave no longer fits. Solve for the modes that match the boundary conditions, show their tails in the cladding and capture an image of the end face. Separately, steer a test ray outside the acceptance cone and watch it escape.

The model is scalar and weakly guiding, with infinite cladding and assumed nondispersive indices. It inspects three mode families and does not list every mode. The geometric test ray is computed separately. Coupling, bend loss and measured devices are not modeled.

Physics tutorial

Audit the physics of Optical Fiber: Rays to Modes

BackgroundA ray can bounce at many angles, but a wave must match itself all the way around the core boundary. The same radius and index step set $V=\frac{2\pi a}{\lambda_0}\sqrt{n_1^2-n_2^2}$ . Shrink the core and most wave patterns are no longer guided, even though a ray could still be totally reflected.

Why it mattersWhen does the picture of rays bouncing inside a fiber stop working, so that light must be treated as modes?

Start with the essentials

Focus question: When does the picture of rays bouncing inside a fiber stop working, so that light must be treated as modes?
One-sentence intuition: A guided field reaches into the cladding. Its propagation constant lies between the two bulk indices: $0<b=\frac{n_{\rm eff}^2-n_2^2}{n_1^2-n_2^2}<1$ . Near cutoff the tail spreads wide instead of dropping abruptly to zero at the core wall. The three plotted mode families are a chosen sample, never a complete count.

Core mathematical model

The acceptance angle in air and a longer ray path

\begin{aligned}\sin\theta_{\max}&=\mathrm{NA}=\sqrt{n_1^2-n_2^2}\\n_1\sin\theta_{\rm core}&=\sin\theta_{\rm air}\\\delta t_{\rm ray}/L&=\frac{n_1}{c}(\sec\theta_{\rm core}-1)\end{aligned}

Air has an index of one. The test ray enters at the center and is meridional. The delay compares its path in the core with an axial ray in a medium without dispersion. It is only a geometric estimate, and once the ray escapes it says nothing about guiding.

The mode shape comes from matching at the boundary

\begin{aligned}u^2+w^2&=V^2\\u\frac{J_{\ell-1}(u)}{J_\ell(u)}&=-w\frac{K_{\ell-1}(w)}{K_\ell(w)}\\F_\ell(\rho)&=\mathcal N\begin{cases}J_\ell(u\rho),&\rho\le1\\J_\ell(u)\dfrac{K_\ell(w\rho)}{K_\ell(w)},&\rho>1\end{cases}\end{aligned}

The normalized radius is the actual radius divided by the core radius. Ordinary and modified Bessel functions are matched in both field and radial derivative. The displayed field is a real scalar field with a cosine angular dependence; the degenerate sine orientation and the two polarization directions are not counted separately here. The root search is bracketed by published Bessel zeros, and a tiny numerical exclusion right at cutoff is documented.

A normalized field and a finite image

\begin{aligned}\int_{\mathbb R^2}|\psi|^2\,d^2\rho&=1\\b&=w^2/V^2\\n_{\rm eff}^2&=n_2^2+b(n_1^2-n_2^2)\\P_{\rm crop}/P&\approx\sum_j |\psi_j|^2\Delta A_\rho\end{aligned}

In weak guidance, the scalar norm stands in for power. Normalization uses Bessel integrals over the core and over the infinite tail. Recorded images keep their original samples and settings, and the power inside the crop is summed from those pixels. Near cutoff, a broad tail can extend past the displayed square. The usual high-frequency estimate of the mode count is not used as an exact count.

Group delay belongs to the mode, not its test ray

\begin{aligned}\eta_m&=P_{{\rm core},m}/P_m\\n_{g,m}&=\frac{n_2^2+\eta_m(n_1^2-n_2^2)}{n_{{\rm eff},m}}\\\tau_{g,m}/L&=n_{g,m}/c\end{aligned}

When the material indices have no dispersion, differentiating the scalar Helmholtz eigenvalue gives the mode’s mean squared index divided by its effective index. The table subtracts propagation in bulk cladding and reports the spread only across the mode families that are resolved and inspected. Near cutoff, the core power fraction and the group delay change together. This is not material dispersion, a pulse calculation over the full mode spectrum, or a delay recovered from an intensity image.

Common difficulties

Does the fundamental mode disappear at the famous cutoff?

Typical misconceptionBelow the first cutoff, the fiber cannot guide light.

Better mental modelThe first higher-order LP family cuts off at the first zero of the ordinary Bessel function. In the ideal model with infinite cladding, the fundamental mode has no positive cutoff. In a real fiber with finite cladding, weak confinement can still cause large bending or leakage losses; this model does not predict those losses.

Run the experiment

01
Scene 1: Fundamental survives
Start with the thin core. Select the first higher-order family and see the empty end face, then return to the fundamental, which is still guided.
What to observe: Laboratory X is the straight fiber axis. Y is vertical, and Z completes the transverse plane. Radius means the core radius, not the diameter. The vacuum wavelength is entered in nanometres and the radius in micrometres. The core index is fixed at 1.46, and the cladding index is lower by the chosen difference. These are assumed teaching values with no dispersion, not a Sellmeier fit for silica.
02
Scene 2: Two-lobed guided mode
Increase the core radius and watch the cylindrical interface and the eigenvalue change together. Select a two-lobed mode, then rotate its lobe axis; the propagation constant does not change.
What to observe: The input angle in air is measured from the fiber axis; the incidence angle for total reflection is measured from the normal to the core wall. The visible fiber spans five hundred micrometres, drawn with different axial and transverse scales. The outer cladding surface is only a cutaway crop. When a ray hits the wall without total reflection, only the transmitted geometric direction is followed to the edge of the crop; partial Fresnel reflection is left out.
03
Scene 3: Radial node and ray escape
Acquire an end-face image, change the wavelength, then compare the frozen and live images. Separately, move the amber test ray outside the acceptance cone: it refracts into the cladding instead of bouncing forever.
What to observe: Each mode is set up ideally, independent of the test ray, and normalized over infinite cladding. The end-face images sample the squared field over each pixel area; they do not infer how a ray couples into a mode. The cropped physical end face and the enlarged data image show the same field, and the enlarged square reaches farther into the ideal cladding. The display uses a fixed square-root curve, with no auto-exposure. The geometric excess delay is neither a mode group delay nor a simulation of pulse spreading.

Optical Fiber: Rays to Modes

Audit the physics of Optical Fiber: Rays to Modes

Core mathematical model

The acceptance angle in air and a longer ray path

The mode shape comes from matching at the boundary

A normalized field and a finite image

Group delay belongs to the mode, not its test ray

Common difficulties

Does the fundamental mode disappear at the famous cutoff?

Run the experiment

Scene 1: Fundamental survives

Scene 2: Two-lobed guided mode

Scene 3: Radial node and ray escape

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