Higher-order modes

A higher-order mode is any field pattern a fiber, waveguide or laser resonator supports other than its fundamental mode. The fundamental (LP01 in a fiber, TE0 or TM0 in a slab or chip waveguide, TEM₀₀ in a laser) has a single lobe; each higher-order mode has one or more nodal lines across it, a larger spatial extent, a lower effective index and, in free space, a faster divergence. Whether a guide carries them is set by its V-number: a step-index fiber guides only LP01 below $V = 2.405$, and LP11 joins above it. Standard single-mode fiber has $V \approx 2.04$ at 1550 nm and about 3.7 at 850 nm, where it carries LP11 as well. A 50 µm graded-index multimode fiber with NA 0.20 has $V \approx 37$ at 850 nm and guides about 340 modes.

Cutoff and mode count

In a step-index fiber the LP modes appear at fixed values of $V$: LP11 at 2.405, LP21 and LP02 at 3.832, and further modes at higher zeros of the Bessel functions, as listed in the few-mode fiber entry. Because $V$ scales as $1/\lambda$, each mode has a cutoff wavelength below which it is guided; the cutoff wavelength of a single-mode fiber is that of LP11. For large $V$ the total number of guided modes, counting both polarizations, is approximately

$$M \approx V^2/2 \quad \text{(step index)},$$ $$M \approx V^2/4 \quad \text{(graded index)}.$$

A slab waveguide guides its first-order mode once its thickness exceeds $\lambda/(2\,\mathrm{NA})$, so the single-mode size shrinks as the index contrast rises. The common 500 × 220 nm silicon strip at 1550 nm is slightly wider than the width at which the first-order TE mode starts to be guided, about 450 nm with oxide cladding; that mode is weakly guided, is lost at bends and is not excited by symmetric components, so the strip behaves as single mode in practice.

In a laser resonator with spherical mirrors the modes are the Hermite-Gaussian TEM$_{mn}$ patterns, with $m$ and $n$ nodal lines in the two transverse directions; their structure and resonance frequencies are covered under transverse modes.

Effects

Modal dispersion. In a step-index fiber the spread in arrival time is about $\mathrm{NA}^2/(2 n_1 c)$ per unit length; for NA 0.22 and $n_1 = 1.46$ that is 55 ns/km; graded-index profiles reduce it by a factor of hundreds (see modal dispersion).

Interference and modal noise. Two or more modes with different propagation constants beat along the guide, so the intensity pattern at the output and the power passing any mode-selective element (a splice, a connector, a single-mode component) depend on length, temperature and stress. Single-mode fiber used at 850 nm or 1060 nm, below its cutoff wavelength, shows this as power that fluctuates when the fiber is moved, touched or warmed.

Beam quality. A beam containing higher-order modes has $M^2 > 1$; a pure Hermite-Gaussian TEM$_{mn}$ has $M_x^2 = 2m + 1$ and $M_y^2 = 2n + 1$. Beam quality degrades as power shifts into them, which limits how tightly the beam can be focused.

Laser behavior. In an edge-emitting laser diode, a first-order lateral mode reaching threshold as the current rises causes a change of slope in the L-I curve and a shift or split in the far field, discussed under LIV kinks.

They are also used deliberately: multimode interference couplers rely on the beating of several waveguide modes to form self-images, and mode-division multiplexing sends separate data on separate modes.

Observation and removal

On a camera image of the output, a two-lobed pattern, or one that changes shape when the fiber is moved, indicates LP11 or higher. In lasers, higher-order modes show up in an $M^2$ measurement and as extra beat notes on a fast photodiode. A fiber's LP11 cutoff is measured by comparing transmission versus wavelength through a short fiber with and without a small bend; LP11 is lost in the bend, and the step in the ratio marks cutoff.

Higher-order modes are less tightly bound than the fundamental, so bending removes them first. A few loops of modest radius in a fiber, a short length of single-mode fiber spliced in as a mode filter, a pinhole spatial filter in free space, or a tight bend or adiabatic taper on a chip all serve the purpose. Standard single-mode fiber relies on this effect: its theoretical LP11 cutoff can lie slightly above the shortest wavelength it is used at, because near cutoff LP11 is weakly guided and the bends and length of a real cable strip it within meters, so the cutoff measured on a cabled length is shorter than the theoretical one.

Common questions

Is standard single-mode fiber single mode at 1310 nm?

In practice yes. The theoretical LP11 cutoff of a fiber with $V = 2.42$ at 1310 nm lies just above 1310 nm, but LP11 that close to cutoff is lost to bends within a short length, so the cutoff measured on a cabled length lies below 1310 nm.

What is the LP11 mode?

The first higher-order mode of a weakly guiding fiber: two lobes of opposite phase separated by a nodal line, with two orientations and two polarizations, so four degenerate field patterns in all.

References: A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019); A. E. Siegman, Lasers (University Science Books, 1986); D. Gloge, "Weakly guiding fibers," Applied Optics 10, 2252 (1971).