Tool · Optical alignment

Fiber Coupling Efficiency Calculator

How much of a Gaussian beam’s power enters a single-mode fiber, and how quickly does it fall off with lateral offset, tilt and a gap? The tool computes the exact overlap of the beam with the fiber’s Gaussian mode at the fiber face, including the growth and wavefront curvature of the beam when the face sits away from the focus, and gives the 1 dB and 3 dB alignment tolerances. Background: mode field diameter, Gaussian beams, and mode mismatch loss.

Beam and fiber

Wavelength λ (nm)

980 nm pump diodes, 1060 nm Yb fiber, O-band 1310 nm, C-band 1550 nm, L-band 1625 nm. At 850 nm the model applies only to fiber that is single-mode at 850 nm; SMF-28 is multimode there.

Beam waist w₀ (µm, 1/e² radius)

Lensed fiber: typically 2–6 µm. A beam matched to SMF-28 has w₀ = MFD/2: 4.6 µm at 1310 nm, 5.2 µm at 1550 nm.

Fiber MFD (µm, full width)

SMF-28: 9.2 µm at 1310 nm, 10.4 µm at 1550 nm. HI-1060: 6.2 µm at 1060 nm.

Gap index n

The medium between beam focus and fiber face: 1 for air; an index-matching gel or adhesive is close to silica, 1.444 at 1550 nm. The tilt below is measured in this medium.

Alignment

Lateral offset d (µm)

where the beam axis crosses the fiber face, measured from the fiber axis

Gap z (µm from the focus)

positive: the fiber face lies beyond the beam focus; negative: before it

Tilt θ (degrees)

angle between the beam axis and the fiber axis, in the gap medium

Presets

Scalar, paraxial overlap of two Gaussian fields. Fresnel reflection at the fiber face (3.3 %, or 0.15 dB, for bare silica in air at 1550 nm), lens aberrations and polarization are not included, so the result is an upper bound on what an alignment can reach.

Coupling efficiency

–

Efficiency against lateral offset

at the current gap and tilt1 dB window3 dB window

Efficiency against gap

at the current offset and tilton axis, no tilt

Checked against

These checks run in your browser on every load. The closed forms are compared with hand-computed values from Joyce and DeLoach (1984) and Marcuse (1977), with the worked example below the tool, and with a direct numerical integration of the overlap that this page performs itself.

Check	Expected	Computed	Tolerance

The table sets each value this page computes beside a reference: the size, gap and offset formulas of Joyce and DeLoach and of Marcuse evaluated by hand, the tilt formula for equal modes, the analytic tolerances w√ln 2 and w√(0.1 ln 10), a Simpson integration of the overlap run in the page, and optima root-found with scipy on a quadrature of the overlap. The tolerance is the largest difference from Expected, in the same units, that still passes.

The model

The beam is a Gaussian with waist radius $w_{b}$ at its focus, and the fiber mode is taken as a Gaussian of radius $w_{f}$ , half the mode field diameter, with a flat phase front at the fiber face. When the face sits a distance $z$ from the focus, in a gap medium of index $n$ (1 for air), the beam arrives with radius $w (z)$ and wavefront radius $R (z)$ :

z_{R} = π n w_{b}^{2} / λ

w (z)^{2} = w_{b}^{2} [1 + (z / z_{R})^{2}]

\frac{1}{R (z)} = \frac{z}{z^{2} + z_{R}^{2}}

Writing the two fields as $e^{- a_{1} r^{2}}$ and $e^{- a_{2} r^{2}}$ , with $a_{1} = 1 / w (z)^{2} + i k / 2 R (z)$ , $a_{2} = 1 / w_{f}^{2}$ and $k = 2 π n / λ$ , and adding the phase $e^{- i k θ x}$ of a beam tilted by $θ$ , the normalized power overlap for a lateral offset $x$ is, with $s = a_{1} + a_{2}$ ,

η = \frac{4}{w (z)^{2} w_{f}^{2} ∣ s ∣^{2}} e^{- 2 β x^{2} + 2 g x - A θ^{2}}

β = Re \frac{a_{1} a_{2}}{s}, g = k θ Im \frac{a_{1}}{s}, A = \frac{k^{2}}{2} Re \frac{1}{s}

On axis this reduces to the closed form of Joyce and DeLoach for two waists separated by $z$ ,

η_{0} = \frac{4}{(w_{b} / w_{f} + w_{f} / w_{b})^{2} + (λ z / π n w_{b} w_{f})^{2}}

and at $z = 0$ the lateral term becomes $\exp [- 2 x^{2} / (w_{b}^{2} + w_{f}^{2})]$ , the Gaussian offset loss given by Marcuse for fiber splices; for equal modes the tilt term becomes $\exp [- (π n w θ / λ)^{2}]$ . The cross term $2 g x$ is nonzero only when a tilt and a curved wavefront occur together: the best lateral offset then moves to $x^{*} = g / 2 β$ , where the local wavefront at the fiber center is closer to square to the fiber axis. The insertion loss shown is $- 10 \log_{10} η$ .

Because $\ln η$ is exactly quadratic in $x$ , the offsets at which the coupling has fallen by a given factor from its best value follow in closed form, with no sweep: $x^{*} \pm \sqrt{D / 2 β}$ , where $D = 0.1 \ln 10$ for 1 dB and $D = \ln 2$ for 3 dB (half power). For equal modes at $z = 0$ these are $w \sqrt{0.1 \ln 10} = 0.480 w$ and $w \sqrt{\ln 2} = 0.833 w$ . The tilt tolerance follows the same way from the quadratic in $θ$ , and the on-axis gap for 1 dB from $η_{0}$ .

The calculation is scalar and paraxial. It treats the fiber mode as an exact Gaussian, which is a good approximation for standard single-mode fiber near its operating wavelength and a poorer one far above cutoff or for fibers with strongly non-step profiles. The tilt is taken in the gap medium and must stay small (paraxial). Aberrations of the focusing lens, polarization mismatch, and the Fresnel reflection at an uncoated fiber face (3.3 %, or 0.15 dB, for silica at 1550 nm) are not included, so the result is an upper bound on what an alignment can reach.

Worked example, from the default inputs (beam waist 5.2 µm, MFD 10.4 µm, 1550 nm), which are mode-matched and give 100 %. A lateral offset of 2 µm gives $\exp (- 2 \cdot 4 / 54.08) = 0.862$ , which the tool shows as 86.2 % and 0.64 dB. Returning the lateral offset to zero and moving the fiber 50 µm from the focus gives $λ z / π w_{b} w_{f} = 77.5 / 84.95 = 0.912$ and $η_{0} = 4 / (4 + 0.832) = 0.828$ , shown as 82.8 % and 0.82 dB. With both offsets applied together the tool gives 73.2 % (1.35 dB) rather than the 1.46 dB sum of the two separate losses, because the expanded beam at the face is less sensitive to lateral offset. At the defaults the tool also gives a lateral tolerance of ±2.50 µm for 1 dB and ±4.33 µm for 3 dB ( $5.2 \sqrt{\ln 2} = 4.33$ ), a 1 dB tilt tolerance of ±2.61°, and a 1 dB gap of 55.8 µm. A tilt of 1° alone gives $\exp [- (π \cdot 5.2 \cdot 0.01745 / 1.55)^{2}] = 0.967$ , shown as 96.7 % and 0.15 dB.

Related: the article on fiber array attach to photonic chips applies these alignment tolerances to V-groove arrays and works through the loss budget per channel.

References: W. B. Joyce and B. C. DeLoach, “Alignment of Gaussian beams,” Appl. Opt. 23, 4187 (1984). D. Marcuse, “Loss analysis of single-mode fiber splices,” Bell Syst. Tech. J. 56, 703 (1977). A. E. Siegman, Lasers (University Science Books, 1986).