Propagation constant (β)

The propagation constant $\beta$ of a guided mode is the rate at which its phase advances along the guide, in radians per unit length. It is the waveguide counterpart of the free-space wavenumber $k_0=2\pi \mathrm{/}{\lambda }_0$, scaled by the mode's effective index: $\beta =n_{\text{eff}}{k}_0$. At 1550 nm, $k_0=4.054$ rad/μm; a silicon-on-insulator strip mode with $n_{\text{eff}}=2.44$ has $\beta =9.89$ rad/μm, which is 1574 full cycles of phase per millimeter. Its frequency derivatives give the group index and dispersion, and its imaginary part gives loss.

Definition

A mode traveling along $z$ has the form

$$E=E_{\text{m}}(x,y)e^{i(\beta z-\omega t)},$$

where the transverse profile $E_{\text{m}}$ does not change with $z$. For a guided mode, $\beta$ lies between the cladding and core wavenumbers:

$$n_{\text{clad}}{k}_0<\beta <n_{\text{core}}{k}_0\mathrm{.}$$

Below $n_{\text{clad}}{k}_0$ the field is oscillatory in the cladding and radiates, forming the continuum of radiation modes. A value above $n_{\text{core}}{k}_0$ would require a field that is evanescent in every region, which has no bounded solution. Fiber texts often normalize $\beta$ as

$$b=\frac{n_{\text{eff}}^2-n_{\text{clad}}^2}{n_{\text{core}}^2-n_{\text{clad}}^2},$$

which runs from 0 at cutoff to 1 far above it and, for a step-index guide, depends only on the V number. For standard single-mode fiber at 1550 nm, $V\approx 2.04$ for the equivalent step-index core and $b\approx 0.43$; the datasheet NA of 0.14 would give $V=2.33$, but it overstates the step-index value, as the V number entry explains.

Derived quantities

Taylor-expanding $\beta (\omega )$ about the carrier frequency ${\omega }_0$ organizes most of linear pulse propagation:

$$\beta (\omega )={\beta }_0+{\beta }_1\mathrm{\Delta }\omega +\frac{1}{2}{\beta }_2\mathrm{\Delta }{\omega }^2+\dots$$

• ${\beta }_0=n_{\text{eff}}{\omega }_0\mathrm{/}c$ sets the phase.
• ${\beta }_1=d\beta \mathrm{/}d\omega =n_g\mathrm{/}c$ is the inverse group velocity, with

$$n_g=n_{\text{eff}}-{\lambda }_0\frac{d{n}_{\text{eff}}}{d{\lambda }_0}\mathrm{.}$$

• ${\beta }_2=d^2\beta \mathrm{/}d{\omega }^2$ is the group velocity dispersion parameter, in s²/m. Telecom practice uses

$$D=-\frac{2\pi c}{{\lambda }_0^2}{\beta }_2,$$

in ps/(nm·km). Standard single-mode fiber has $D=17$ ps/(nm·km) at 1550 nm, which corresponds to ${\beta }_2=-21.7$ ps²/km. The sign difference between $D$ and ${\beta }_2$ causes frequent confusion: positive $D$ is anomalous dispersion.

• ${\beta }_3$ and higher terms matter for femtosecond pulses, supercontinuum generation and microcomb design, and near a zero-dispersion wavelength where ${\beta }_2$ vanishes.

Loss

In a lossy guide $\beta$ is complex. With the $e^{i(\beta z-\omega t)}$ convention used here,

$$\beta ={\beta }_r+i\alpha \mathrm{/}2,$$

where $\alpha$ is the power attenuation coefficient: the field falls as $e^{-\alpha z\mathrm{/}2}$ and the power as $e^{-\alpha z}$. A propagation loss of 2 dB/cm corresponds to $\alpha =0.461$ cm⁻¹ (46.1 m⁻¹), so $\mathrm{I}\mathrm{m}\beta =0.230$ cm⁻¹. Engineering texts that write fields as $e^{j(\omega t-\beta z)}$ put the loss in with the opposite sign, $\beta -j\alpha \mathrm{/}2$; mixing the two conventions in a mode solver or transfer-matrix code produces gain where loss was intended. Confusing the field and power coefficients gives a factor-of-two error.

Phase matching

Nonlinear and coupled-mode processes transfer power efficiently only when the propagation constants balance. For a generic process,

$$\mathrm{\Delta }\beta =\sum {\beta }_{\text{out}}-\sum {\beta }_{\text{in}}\approx 0.$$

The coherence length $L_c=\pi \mathrm{/}\mathrm{\Delta }\beta$ is the distance over which the generated wave stays in phase with its drive. An effective-index mismatch of 0.01 at 1550 nm gives $L_c=77.5$ μm. Phase matching by birefringence, quasi-phase-matching or waveguide dispersion engineering pushes $\mathrm{\Delta }\beta$ to zero. In a directional coupler, the same quantity between the two guides limits the maximum power transfer.

How it is obtained

Calculation. $\beta$ is an eigenvalue of the waveguide's Helmholtz equation. For step-index fibers it follows from the LP-mode characteristic equation; for integrated waveguides, finite-difference or finite-element mode solvers return $n_{\text{eff}}$ and hence $\beta$ at each wavelength.

Measurement. The absolute value of $\beta$ is rarely measured directly, because a phase of 1574 cycles per millimeter is only known modulo $2\pi$. Its derivatives are measured instead. The free spectral range of a ring resonator gives $n_g$ and therefore ${\beta }_1$; the wavelength dependence of the FSR gives ${\beta }_2$. Loss, the imaginary part, comes from ring linewidths or the cutback method. Grating couplers and prism couplers, which excite a mode only at the angle that matches $\beta$, do give $n_{\text{eff}}$ directly.

Multimode guides. A multimode waveguide has one discrete $\beta$ per mode at each wavelength. The differences set the beat length between modes, $L_B=2\pi \mathrm{/}({\beta }_a-{\beta }_b)$ for modes $a$ and $b$, and the modal delay spread in multimode fiber.

Common questions

What is the difference between $\beta$ and $k$?

$k=n{k}_0$ is the wavenumber of a plane wave in a uniform medium; $\beta$ is the axial wavenumber of a mode, lying between the cladding and core values of $k$.

What units is it given in?

rad/m or rad/μm for the real part; loss is usually quoted separately as $\alpha$ in dB/cm or dB/km.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019); Agrawal, Nonlinear Fiber Optics (6th ed., 2019), Ch. 1 and 2; Yariv & Yeh, Photonics (6th ed., 2007); Snyder & Love, Optical Waveguide Theory (Chapman and Hall, 1983).