Coupling coefficient (κ)

The coupling coefficient $\kappa$ is the quantity in coupled-mode theory that measures how strongly a perturbation links two modes: the fraction of one mode's field converted into the other per unit length. It has units of inverse length. In a directional coupler, where the perturbation is the neighboring waveguide, silicon devices have $\kappa$ of order 0.1 µm⁻¹ (the 5–10 µm lengths of typical 50:50 couplers correspond to 0.08–0.16 µm⁻¹), and $\kappa$ = 0.1 µm⁻¹ transfers all the power in 15.7 µm; fused fiber couplers, with coupling regions millimeters long, have far smaller values. In a Bragg grating, where the perturbation is the periodic index modulation and the two modes are the forward and backward waves, $\kappa$ is quoted in cm⁻¹: about 2 cm⁻¹ for a fiber grating with an index modulation of $10^{-4}$ and tens of cm⁻¹ for an etched semiconductor grating. The product $\kappa L$ with the device length sets how much power is exchanged.

Definition

For two co-directional modes with amplitudes $A$ and $B$ and propagation constants $\beta_1$ and $\beta_2$, with $\delta = (\beta_1 - \beta_2)/2$, the coupled-mode equations are

$$\frac{dA}{dz} = -i\kappa B\,e^{i2\delta z},$$ $$\frac{dB}{dz} = -i\kappa^{*} A\,e^{-i2\delta z}.$$

With both modes normalized to unit power, $\kappa$ is an overlap integral of the relative permittivity perturbation $\Delta\varepsilon_r$ with the two mode fields:

$$\kappa = \frac{\omega\varepsilon_0}{4}\iint \Delta\varepsilon_r\,\mathbf{E}_1^{*}\!\cdot\mathbf{E}_2\,dx\,dy.$$

The integral is why $\kappa$ grows when the fields overlap more, either because the guides are closer or because the grating sits where the mode is strong. In a coupler it falls roughly exponentially with the gap, at the decay rate of the evanescent field.

Directional couplers

For two identical lossless guides, light launched into one has transferred a fraction $\sin^2(\kappa z)$ to the other after a length $z$, and all of it at the coupling length

$$L_c = \frac{\pi}{2\kappa}.$$

Worked: $\kappa$ = 0.1 µm⁻¹ gives $L_c$ = 15.7 µm, a 50:50 split at 7.85 µm and a 1% tap at 1.0 µm. In the supermode picture $\kappa = \pi(n_e - n_o)/\lambda$, so this coupler has an even–odd supermode index difference of 0.049 at 1550 nm, matching the 0.05 example in the directional coupler entry. When the guides differ, the transfer is capped at $\kappa^2/(\kappa^2 + \delta^2)$, and $\delta = \kappa$ limits it to 50%; the supermode beating behind all of this is described under beat length.

Bragg gratings

A uniform grating of length $L$ reflects, at the Bragg wavelength,

$$R = \tanh^2(\kappa L).$$

For a sinusoidal index modulation of amplitude $\Delta n$ that fills the mode, $\kappa = \pi\Delta n/\lambda$, so a fiber Bragg grating at 1550 nm with $\Delta n = 10^{-4}$ has $\kappa$ = 2.03 cm⁻¹ and needs 14.8 mm to reach $\kappa L$ = 3. For a square-wave step $\Delta n$ with equal widths, as in a quarter-wave stack, $\kappa = 2\Delta n/\lambda$. The coefficient also sets the stop band: its width in wavelength is about $\lambda_B^2\kappa/(\pi n_g)$, so a stronger grating reflects over a wider band.

DFB and DBR lasers

In a DFB laser the grating runs along the gain section and $\kappa L$ fixes the feedback. Practical devices use $\kappa L$ of about 1 to 2, often kept near 1 to 1.5: a weaker grating gives a high effective mirror loss, and a stronger one concentrates the light at the center of the cavity, where longitudinal spatial hole burning degrades the side-mode suppression at high power. A 300 µm DFB with $\kappa L$ = 2 has $\kappa$ = 66.7 cm⁻¹. A DBR laser's rear grating with $\kappa$ = 50 cm⁻¹ and 300 µm length ($\kappa L$ = 1.5) reflects 82%.

Measurement

For couplers, a series of devices of different lengths is measured and the cross-port fraction fitted to $\sin^2[\kappa(L + z_0)]$, where $z_0$ absorbs the coupling in the bends. For gratings, $\kappa L$ follows from the depth of the transmission dip, $\kappa L = \operatorname{artanh}\sqrt{1 - T_\text{min}}$: a dip to 7% transmission gives $\kappa L$ = 2.0. For DFB lasers, the spacing of the two edge modes in the subthreshold spectrum gives $\kappa$ through a coupled-mode fit for the grating's length; for $\kappa L$ near 2 that spacing is about 1.7 times the infinite-grating width above, so applying that formula directly overstates $\kappa$.

Pitfalls

The symbol and the name carry several conventions. In ring resonator analysis, $\kappa$ is often the dimensionless field cross-coupling of the whole coupling region, with $t^2 + \kappa^2 = 1$, instead of a rate per length. Some texts define the coupled-mode equations with $\kappa$ for intensity instead of field, or include a factor of 2 in the mismatch, which shifts formulas by factors of two. Gain-coupled and complex-coupled DFB gratings have complex $\kappa$. Elsewhere on this site $\kappa$ also denotes the extinction coefficient of the complex refractive index and the gain-compression constant in laser chirp. The fiber-to-chip quantity is coupling efficiency, a power ratio, which is a different thing.

Common questions

What is κL in a DFB laser?

The product of the grating's coupling coefficient and the cavity length, a dimensionless measure of the total feedback. Values of about 1 to 2 are standard; the reflectance of the corresponding passive grating, $\tanh^2(\kappa L)$, runs from 58% to 93%.

How is the coupling length related to κ?

For identical waveguides, full transfer happens at $L_c = \pi/(2\kappa)$, and a 3 dB split at half that length.

References: A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE Journal of Quantum Electronics 9, 919 (1973); H. Kogelnik and C. V. Shank, "Coupled-wave theory of distributed feedback lasers," Journal of Applied Physics 43, 2327 (1972); K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006); L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012); R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Academic Press, 2010).