Coupling efficiency

Coupling efficiency $\eta$ is the fraction of the optical power leaving one component that ends up in the intended mode of the next: a laser into a fiber, a fiber into a chip waveguide, one fiber into another. It is quoted either as a fraction or percent, or as a coupling loss in decibels, $-10\log_{10}\eta$. Two well aligned single-mode fibers with matched modes and index-matching gel couple above 99% (under 0.05 dB). A 3 µm diameter waveguide mode butted against the 10.4 µm mode of standard single-mode fiber at 1550 nm couples about 28%, a loss of 5.5 dB, and a uniform silicon grating coupler typically 25–50% (3–6 dB).

Single-mode coupling: the overlap integral

When the receiving component supports a single mode, only the part of the incoming field that has the shape of that mode is captured. The efficiency is the normalized overlap integral of the two transverse fields,

$$\eta = \frac{\left|\int E_1 E_2^{*}\,dA\right|^2}{\int |E_1|^2 dA\,\int |E_2|^2 dA},$$

times any transmission factors for reflections and absorption along the way. Because single-mode fiber modes are close to Gaussian, closed forms cover most cases. For two modes with $1/e^2$ radii $w_1$ and $w_2$ (half the mode field diameters) meeting at their waists,

$$\eta_\text{size} = \left(\frac{2 w_1 w_2}{w_1^2 + w_2^2}\right)^2 .$$

With $w_1$ = 1.5 µm and $w_2$ = 5.2 µm this gives 0.284, or 5.47 dB. For equal radii $w$, a lateral offset $d$ and a tilt $\theta$ give

$$\eta_\text{offset} = \exp\!\left(-\frac{d^2}{w^2}\right),$$ $$\eta_\text{tilt} = \exp\!\left[-\left(\frac{\pi n w \theta}{\lambda}\right)^2\right].$$

Between two 10.4 µm fiber modes, a 1 µm offset keeps 96.4% (0.16 dB) and a 1° tilt at 1550 nm in air keeps 96.7% (0.15 dB). The same 1 µm offset between 3 µm modes keeps only 64% (1.9 dB), which is why small-mode joints need submicrometre alignment. The full set of formulas, including a longitudinal gap and elliptical modes, and tables of alignment tolerances are in the mode mismatch loss entry.

Multimode and incoherent sources

When the receiver accepts many modes, as a multimode fiber does, geometry decides: light is captured if it lands inside the core within the acceptance cone set by the numerical aperture. For an extended, incoherent source the limit is étendue. A Lambertian emitter no larger than the core couples a fraction of about $\text{NA}^2$ of its power into a step-index fiber in air, so a fiber of NA 0.22 collects about 4.8% (13.2 dB below the emitted power). A lens helps only when the emitter is smaller than the core: imaging it larger, up to the core size, narrows its cone and raises the fraction by up to the ratio of core area to emitting area. For an emitter as large as the core or larger, no lens can raise the figure, because a lens that shrinks the spot widens the cone.

Adding up the factors

A practical coupling efficiency is a product of independent factors: mode overlap, lens and window transmission, Fresnel reflection at each uncoated surface, clipping by apertures, and polarization mismatch where the receiver is polarization sensitive. In decibels the losses add. An uncoated silica face ($n$ ≈ 1.444 at 1550 nm) reflects 3.3% of the light (Fresnel reflection), so a link with 80% mode overlap, 95% lens transmission and one bare fiber face has

$$\eta = 0.80 \times 0.95 \times 0.967 = 0.735,$$

or 0.97 + 0.22 + 0.15 = 1.34 dB.

Measurement

Coupling efficiency is measured as the ratio of the power in the receiving mode to the power available from the source, each read on an optical power meter. For a laser into fiber, the reference is the beam power just before the lens, and the coupled power is read after enough fiber, or a tight enough loop, to strip cladding modes. For chip couplers, the total fiber-to-fiber transmission includes two couplers and the waveguide loss, and the per-facet value is extracted by measuring waveguides of several lengths, as described in coupler loss de-embedding. Insertion loss is the related component specification: the drop in power when a part is inserted into a link, which includes the coupling at both of its ports.

Where it matters

Coupling loss enters every optical power budget and often dominates it on photonic chips. An edge coupler with a spot-size converter can be broadband and below 1 dB per facet; a grating coupler is easier to probe on a wafer but narrower in wavelength. The articles fiber-to-chip coupling and free-space to single-mode fiber coupling cover the methods and the bench procedure.

Pitfalls

• Quoting the efficiency into a fiber end without stripping cladding light.
• Comparing values taken at different wavelengths or polarizations; grating couplers are strongly polarization dependent.
• Forgetting that the Gaussian formulas assume both fields are Gaussian; a non-Gaussian waveguide mode loses extra power even with perfect alignment.

Common questions

How do you convert coupling efficiency to dB?

Loss in dB is $-10\log_{10}\eta$: 50% is 3.01 dB, 80% is 0.97 dB, 90% is 0.46 dB.

What is a good fiber coupling efficiency?

For a well collimated single-mode laser into single-mode fiber through a suitable lens, 70–90% is commonly reached on a careful bench setup; fusion splices between matched fibers typically exceed 99% (under 0.05 dB), and good connectors reach about 95 to 98% (0.1 to 0.25 dB).

Is coupling efficiency the same in both directions?

For single-mode to single-mode coupling, yes, because the overlap integral is symmetric. For multimode or incoherent coupling it generally is not: a single-mode fiber couples almost all its light into a multimode fiber, but only a small fraction travels back.

References: H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966); D. Marcuse, "Loss analysis of single-mode fiber splices," Bell System Technical Journal 56, 703 (1977); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).