Fibre coupling loss calculator
Estimate coupling loss between fibres, waveguides and lasers from mode field mismatch, offset, tilt, gap and reflection.
Analog, RF, photonics and packagingFree to useHow it worksSourcesWhere it helps
Fills the inputs below with typical figures. Replace any of them with your own.
- Total coupling loss
- 0.053 dB
- 98.8% of the power is coupled
- Mode size mismatch
- -0.000 dB
- Offset and tilt
- 0.053 dB
- Together
- Gap
- 0.000 dB
- Rayleigh range of mode 1: 80.5 µm
- Return loss at the first surface
- Not applicable
- When surface reflections are on and the indices differ
- Mode size mismatch-0.000 dB−10 log10 of 4 w1^2 w2^2 / (w1^2 + w2^2)^2 on each axis
- Separation (gap)0.000 dBBeam spreading across the gap, extra to the mismatch
- Lateral offset0.040 dBExtra to the two terms above
- Angular tilt0.013 dBExtra to the terms above
Show the data as a table
| Item | Loss (dB) |
|---|---|
| Mode size mismatch | -0.000 dB |
| Separation (gap) | 0.000 dB |
| Lateral offset | 0.040 dB |
| Angular tilt | 0.013 dB |
Sign in to save your inputs and come back to them later.
| Term | Loss | Basis |
|---|---|---|
| Mode size mismatch | -0.000 dB | −10 log10 of 4 w1^2 w2^2 / (w1^2 + w2^2)^2 on each axis |
| Separation (gap) | 0.000 dB | Beam spreading across the gap, extra to the mismatch |
| Lateral offset | 0.040 dB | Extra to the two terms above |
| Angular tilt | 0.013 dB | Extra to the terms above |
- Offset horizontal, with the gap and tilt entered (solid line, circles)
- Offset horizontal, no gap and no tilt (dashed line, squares)
Show the data as a table
| Lateral offset (µm) | Offset horizontal, with the gap and tilt entered | Offset horizontal, no gap and no tilt |
|---|---|---|
| 0 | 0.0127 | 0 |
| 0.441 | 0.0439 | 0.0313 |
| 0.882 | 0.138 | 0.125 |
| 1.32 | 0.294 | 0.281 |
| 1.76 | 0.513 | 0.5 |
| 2.21 | 0.794 | 0.782 |
| 2.65 | 1.14 | 1.13 |
| 3.09 | 1.54 | 1.53 |
| 3.53 | 2.01 | 2 |
| 3.97 | 2.55 | 2.53 |
| 4.41 | 3.14 | 3.13 |
| 4.85 | 3.8 | 3.78 |
| 5.29 | 4.52 | 4.5 |
| 5.74 | 5.3 | 5.28 |
| 6.18 | 6.14 | 6.13 |
| 6.62 | 7.05 | 7.04 |
| 7.06 | 8.02 | 8 |
| 7.5 | 9.05 | 9.04 |
| 7.94 | 10.1 | 10.1 |
| 8.38 | 11.3 | 11.3 |
| 8.82 | 12.5 | 12.5 |
| Extra loss | Offset, horizontal (µm) | Offset, vertical (µm) | Tilt, horizontal (degrees) | Tilt, vertical (degrees) |
|---|---|---|---|---|
| 0.5 dB | 1.76 | 1.76 | 1.26 | 1.26 |
| 1 dB | 2.5 | 2.5 | 1.78 | 1.78 |
| Separation (µm) | Total coupling loss |
|---|---|
| 0 | 0.053 dB |
| 1 | 0.053 dB |
| 2 | 0.053 dB |
| 5 | 0.057 dB |
| 10 | 0.069 dB |
| 20 | 0.119 dB |
| 50 | 0.451 dB |
| 100 | 1.463 dB |
How it works
Light leaving one fibre, waveguide or laser is a bell-shaped (Gaussian) beam, and the loss at a joint is the share of that beam the other side does not pick up. This tool works out the loss from the two mode sizes, the lateral offset, the angular tilt, the gap and the surface reflections, shows each cost separately, and gives the offset and tilt that cost an extra half and one decibel.
- Describe each mode as a Gaussian. A mode field diameter (MFD) is twice the radius w at which the field has fallen to 1/e of its peak. Silicon waveguides, laser diodes and gratings are often elliptical, so each mode has a radius on each axis (x and y), and the coupling is the product of the two axes. If you enter the waist radius instead of the diameter, the tool uses it directly.
w = MFD / 2; coupling efficiency = efficiency on x x efficiency on y; loss (dB) = −10 x log10(efficiency) - Mode size mismatch. With the two waists at the same place and no misalignment, the efficiency on one axis depends only on the ratio of the radii. For circular modes it is the square of that axis value.
efficiency on one axis = 2 w1 w2 / (w1^2 + w2^2); circular modes: loss = −10 x log10( 4 w1^2 w2^2 / (w1^2 + w2^2)^2 ) - Lateral offset and angular tilt. An offset d between the axes, and a tilt of angle theta (in the gap medium of index n at wavelength lam), each multiply the efficiency by an exponential that falls quickly as it grows. The offset term is the one that dominates single-mode joints. These are the expressions given by Marcuse for single-mode splices and later extended to collimators.
offset factor = exp( −2 d^2 / (w1^2 + w2^2) ); tilt factor = exp( −2 (pi n w1 w2 theta / lam)^2 / (w1^2 + w2^2) ) - Longitudinal gap. A gap z lets the first beam spread before it meets the second. For circular modes with no other misalignment the efficiency has a closed form; with offset and tilt present, the tool keeps the beam's curvature as well as its growth and evaluates the overlap of the two Gaussians exactly, which reduces to the forms above at zero gap.
gap and mismatch, circular modes: efficiency = 4 / ( (w1/w2 + w2/w1)^2 + (lam z / (pi n w1 w2))^2 ); Rayleigh range z_R = pi n w^2 / lam - Surface reflections. Where the light passes from glass or semiconductor into the gap material, part is reflected. At normal incidence each surface reflects a fraction R. With one pass, the transmitted power is the product of the two surface transmissions; if the reflections between the surfaces are added as power, the transmission is a little higher. Coherent cavity effects (fringes) are not modelled.
R = ((n_a − n_b) / (n_a + n_b))^2; one pass: T = (1 − R1) x (1 − R2); multiple reflections as power: T = (1 − R1) x (1 − R2) / (1 − R1 x R2); return loss = −10 x log10(R1) - Tolerances. Because the offset and tilt terms are exponentials in d^2 and theta^2, the value that costs a given extra loss follows directly. Each is the effect of that one error alone, with the gap as entered; combined errors cost the sum of their losses.
offset for extra loss L (dB) at zero gap: d = sqrt( L x ln(10) x (w1^2 + w2^2) / 20 ); tilt: theta = ( lam / (pi n w1 w2) ) x sqrt( L x ln(10) x (w1^2 + w2^2) / 20 )
Assumptions
- Both modes are Gaussian, with the waist of each at its own surface, and the beams are paraxial (small angles).
- The tilt is the angle between the two axes measured in the gap medium; the offset is the displacement of the axes at the second surface.
- The losses of mismatch, gap, offset and tilt are shown one after the other, each extra to the ones before it; together they are the full overlap of the two Gaussians (the tilt is applied as a separate factor).
- The surfaces are flat and perpendicular to the axes, and the Fresnel loss is for normal incidence without coatings.
- The preset mode sizes and fixed losses are editorial figures for a typical case; replace them with your own design or measured data.
Limits and warnings
- Multimode fibre is outside the model: its loss depends on the launch, the core and the numerical aperture, not on a single Gaussian.
- Real modes are not exactly Gaussian. Fibre modes are close to Gaussian over their central part; silicon waveguides, gratings and laser facets can differ in shape and add their own losses, which the fixed loss input is for.
- Large tilts, offsets of more than a mode diameter and gaps of many Rayleigh ranges leave the range where the paraxial Gaussian expression is reliable.
- Tilt and gap are treated with the beam at the second waist's plane; a tilt about a pivot away from the surface adds some offset, which you can enter as offset.
- Wavelength dependence of the mode size is not computed: enter the mode size at the wavelength you use.
Sources
The methods and default figures rest on these public sources. The wording here is our own; read the source for its full text.
- Loss Analysis of Single-Mode Fiber Splices (opens in a new tab), D. Marcuse; Bell System Technical Journal, vol. 56, no. 5, pp. 703 to 718, May 1977. The Gaussian expressions for mode mismatch, offset and tilt of single-mode splices. Citation details confirmed from several independent references; the paper's full text was not opened, and the combined overlap used here was derived from the Gaussian beam overlap integral.
- General formula for coupling-loss characterization of single-mode fiber collimators by use of gradient-index rod lenses (opens in a new tab), S. Yuan and N. A. Riza; Applied Optics, vol. 38, no. 15, pp. 3214 to 3222, May 1999. A general Gaussian-field formula with separation, lateral offset, angular tilt and spot-size mismatch together, and the single-misalignment special cases. Abstract and citation confirmed; the formula details were not read.
- Coupling strategies for silicon photonics integrated chips [Invited] (opens in a new tab), R. Marchetti, C. Lacava, L. Carroll, K. Gradkowski and P. Minzioni; Photonics Research, vol. 7, issue 2, p. 201, 18 February 2019. A review of in-plane (edge) and out-of-plane (grating) couplers; the preset fixed losses are editorial figures in the ranges it discusses. Abstract and metadata confirmed; the full text was not read.
- SMF-28 Ultra optical fiber, Product Information PI 1424 (opens in a new tab), Corning Incorporated, February 2024. Gives a 9.2 micrometre mode field diameter for the SMF-28 Ultra fibre. The 10.4 micrometre figure used at 1550 nm is a typical value that was not confirmed from this sheet; check the data sheet of your fibre.
Where it helps
The lifecycle stages this tool serves: 2 stages in 1 lifecycle.
More analog, rf, photonics and packaging tools
Method notev1Formula, ranges, sources and limits
Estimates the loss where light passes between two single-mode fibres, waveguides or lasers from their mode field sizes, lateral offset, angular tilt, gap and surface reflections, with each cost shown separately.
Formula
w = MFD / 2; coupling efficiency = efficiency on x x efficiency on y; loss (dB) = −10 x log10(efficiency); efficiency on one axis = 2 w1 w2 / (w1^2 + w2^2); circular modes: loss = −10 x log10( 4 w1^2 w2^2 / (w1^2 + w2^2)^2 ); offset factor = exp( −2 d^2 / (w1^2 + w2^2) ); tilt factor = exp( −2 (pi n w1 w2 theta / lam)^2 / (w1^2 + w2^2) ); gap and mismatch, circular modes: efficiency = 4 / ( (w1/w2 + w2/w1)^2 + (lam z / (pi n w1 w2))^2 ); Rayleigh range z_R = pi n w^2 / lam; R = ((n_a − n_b) / (n_a + n_b))^2; one pass: T = (1 − R1) x (1 − R2); multiple reflections as power: T = (1 − R1) x (1 − R2) / (1 − R1 x R2); return loss = −10 x log10(R1); offset for extra loss L (dB) at zero gap: d = sqrt( L x ln(10) x (w1^2 + w2^2) / 20 ); tilt: theta = ( lam / (pi n w1 w2) ) x sqrt( L x ln(10) x (w1^2 + w2^2) / 20 )
| Input | Unit | Range it holds for | Comes from |
|---|---|---|---|
| Sizes entered as | choice | mfd, waist | Your figure |
| Size unit | choice | um, nm | Your figure |
| First mode size, x | um or nm | 0.001 to 100 | Your figure |
| Second mode size, x | um or nm | 0.001 to 100 | Your figure |
| Wavelength | nm | 400 to 2500 | Your figure |
| Offset unit | choice | um, nm | Your figure |
| Lateral offset, x | um or nm | from 0 | Your figure |
| Lateral offset, y | um or nm | from 0 | Your figure |
| Tilt unit | choice | deg, mrad | Your figure |
| Tilt, x direction | degrees or mrad | from 0 | Your figure |
| Tilt, y direction | degrees or mrad | from 0 | Your figure |
| Separation unit | choice | um, nm, mm | Your figure |
| Separation | um, nm or mm | from 0 | Your figure |
| Refractive index of the gap medium | index | 1 to 4.5 | Your figure |
| Surface reflections | choice | off, single, multiple | Your figure |
| Index of the first mode's medium | index | 1 to 4.5 | Your figure |
| Index of the second mode's medium | index | 1 to 4.5 | Your figure |
| Other fixed loss | dB | 0 to 30 | Your figure |
Outputs: Total coupling loss (dB); Mode size mismatch loss (dB); Offset loss (dB); Tilt loss (dB); Gap loss (dB); Coupling efficiency (fraction).
Assumptions
- Both modes are Gaussian, with the waist of each at its own surface, and the beams are paraxial (small angles).
- The tilt is the angle between the two axes measured in the gap medium; the offset is the displacement of the axes at the second surface.
- The losses of mismatch, gap, offset and tilt are shown one after the other, each extra to the ones before it; together they are the full overlap of the two Gaussians (the tilt is applied as a separate factor).
- The surfaces are flat and perpendicular to the axes, and the Fresnel loss is for normal incidence without coatings.
- The preset mode sizes and fixed losses are editorial figures for a typical case; replace them with your own design or measured data.
Uncertainty: The Gaussian overlap is exact for Gaussian modes; real modes differ a little in shape, so expect the offset and tilt losses of a real fibre joint to differ by a few hundredths of a dB and the fixed losses of gratings and edge couplers to dominate the uncertainty. The mode sizes enter squared, so a 5% error in a diameter moves the mismatch term noticeably when the sizes are far apart.
Known limits
- Multimode fibre is outside the model: its loss depends on the launch, the core and the numerical aperture, not on a single Gaussian.
- Real modes are not exactly Gaussian. Fibre modes are close to Gaussian over their central part; silicon waveguides, gratings and laser facets can differ in shape and add their own losses, which the fixed loss input is for.
- Large tilts, offsets of more than a mode diameter and gaps of many Rayleigh ranges leave the range where the paraxial Gaussian expression is reliable.
- Tilt and gap are treated with the beam at the second waist's plane; a tilt about a pivot away from the surface adds some offset, which you can enter as offset.
- Wavelength dependence of the mode size is not computed: enter the mode size at the wavelength you use.
Where the method comes from
Follows Marcuse, Loss Analysis of Single-Mode Fiber Splices (Bell System Technical Journal, 1977) (opens in a new tab). The platform's version departs from it as follows.
- Gap, mismatch and offset are evaluated from the exact overlap of two Gaussian beams with complex beam parameters, and tilt is added as a separate factor Why: It keeps the beam's curvature across the gap, so the terms combine for any separation instead of only in the near field.
- Each mode can be elliptical, with a size on each axis Why: Silicon waveguides, laser diodes and gratings are not round, and the product of the two axes handles them.
- Surface reflections and a fixed loss are added to the Gaussian result Why: Fibre-to-chip joints have glass-to-air surfaces and device-specific losses that the splice analysis leaves out.
Reference example, checked by test: Worked example in this method note: standard single-mode fibre (9.2 micrometre MFD) joined to a 10.4 micrometre MFD fibre at 1550 nm, 1 micrometre offset and 0.5 degree tilt in glass of index 1.468 with no gap, hand-worked from the closed-form mismatch, offset and tilt expressions in the tool's unit test. Mode size mismatch loss 0.065117; Offset loss 0.180205; Tilt loss 0.069514; Total coupling loss 0.314837; Gap loss 0.
Versions
- v1 (2026-10-25): First version, built from the tool idea.
Terms on this page
- Waveguide
- A patterned strip of high-index material, typically silicon or silicon nitride surrounded by oxide, that confines and guides light by total internal reflection.
- dB
- A logarithmic ratio unit, ten times the log of a power ratio or twenty times the log of a voltage ratio.
- Silicon
- The group IV element that carries almost all commercial semiconductor volume, valued for its abundance, its stable native oxide and a manufacturing base refined over sixty years.
- Semiconductor
- A material whose electrical conductivity sits between that of a conductor and an insulator, and can be controlled by doping, temperature, light or an applied electric field.
- NA
- A dimensionless measure of the angular range of light a projection lens can collect and focus, equal to the refractive index times the sine of the maximum acceptance angle.
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