Propagation constant (β)

For a guided optical mode propagating along $z$ with angular frequency $\omega$, the field has the form

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

where $\beta$ is the propagation constant. The phase advances by $\beta$ radians per unit length of propagation.

$\beta$ is the in-plane component of the wavevector, with magnitude bounded by the cladding and core wavenumbers:

$$n_\text{clad} \, k_0 \;<\; \beta \;<\; n_\text{core} \, k_0, \qquad k_0 = 2\pi / \lambda_0 = \omega / c.$$

Modes with $\beta < n_\text{clad} k_0$ are radiating (continuum) rather than guided; modes with $\beta > n_\text{core} k_0$ are evanescent in both core and cladding (unphysical for steady-state guided modes).

Relation to effective and group index.

Effective index:

$$n_\text{eff} \;=\; \beta / k_0 \;=\; \beta c / \omega.$$

Group index:

$$n_g \;=\; c \frac{d \beta}{d \omega} \;=\; n_\text{eff} - \lambda_0 \frac{d n_\text{eff}}{d \lambda_0}.$$

Group velocity dispersion. The second derivative of $\beta$ with respect to frequency defines group velocity dispersion (GVD):

$$\beta_2 \;=\; \frac{d^2 \beta}{d \omega^2} \quad [\text{s}^2 / \text{m}].$$

The conventional dispersion parameter $D$ used in telecom is related by

$$D \;=\; -\frac{2\pi c}{\lambda_0^2} \beta_2 \quad [\text{ps} / (\text{nm} \cdot \text{km})].$$

Higher-order dispersion ($\beta_3, \beta_4, \ldots$) becomes important for short pulses or wide spectra (ultrafast laser pulses, supercontinuum generation, comb spectroscopy).

Phase matching. For nonlinear processes that involve multiple optical fields (sum-frequency generation, four-wave mixing, parametric oscillation, etc.), efficient energy transfer requires the total propagation constants of input and output fields to match:

$$\Delta \beta \;=\; \sum_\text{out} \beta_\text{out} - \sum_\text{in} \beta_\text{in} \;\approx\; 0.$$

Engineering of $\beta(\omega)$ by waveguide design (dispersion engineering) is central to integrated-photonic nonlinear devices.

Field decay in absorbing media. For a lossy waveguide, $\beta$ is complex:

$$\beta \;=\; \beta_\text{real} + i \alpha / 2,$$

where $\alpha$ is the power loss coefficient (per unit length). Field magnitude decays as $e^{-\alpha z / 2}$; intensity decays as $e^{-\alpha z}$.

Higher-order modes. Multi-mode waveguides support several discrete values of $\beta$ at any wavelength, one per supported mode. Each mode has its own effective index, group index, and group velocity. Inter-modal beating and mode-dependent losses are the dominant effects in mode-division multiplexed transmission.

Extraction. $\beta$ at a given operating point is computed by waveguide mode-solving (finite-difference, finite-element, or beam-propagation methods) and verified experimentally by ring-resonator FSR measurement or by interferometric techniques. For canonical step-index fibers, $\beta$ is implicit in the LP mode solutions of the Helmholtz equation.