Mode mismatch loss

Mode mismatch loss is the loss that occurs when light passes from one guided mode into another whose field differs in size, shape or position. Only the part of the incoming field that matches the receiving mode is coupled into it; the rest goes into radiation or other modes. The coupled fraction $\eta$ is the overlap integral of the two mode fields, and the loss is $-10{\log }_{10}\eta$ in decibels. For two Gaussian modes of radii $w_1$ and $w_2$ meeting at their waists, $\eta =[2w_1{w}_2\mathrm{/}(w_1^2+w_2^2){]}^2$: a 10% difference in mode size costs 0.04 dB, a factor of two 1.94 dB, and standard single-mode fiber, whose mode is 10.4 µm across at 1550 nm, loses 6.85 dB when butted against a 2.5 µm spot. Even with the sizes matched, a lateral offset of 1 µm costs 2.8 dB between 2.5 µm modes but only 0.16 dB between 10.4 µm ones. Mode mismatch is the largest loss in most fiber-to-chip, laser-to-fiber and fiber-to-fiber joints.

Mode mismatch loss formula

For single-mode fibers and waveguides the coupled fraction is the normalized overlap integral of the two transverse fields,

$$\eta =\frac{{\mid \int E_1{E}_2^{\ast }dA\mid }^2}{\int \mathrm{\mid }{E}_1{\mathrm{\mid }}^{2}dA\int \mathrm{\mid }{E}_2{\mathrm{\mid }}^{2}dA},$$

and the loss is $-10{\log }_{10}\eta$ in decibels. This form, with the transverse electric fields alone, is accurate for weakly guiding fibers and for modes that have spread into a uniform cladding; for strongly guiding waveguides the numerator uses the electric field of one mode and the magnetic field of the other. The integral is symmetric in the two fields, so the loss is the same in both directions. It accounts for the shape and position of the fields only: reflection at the joint is a separate loss.

Gaussian overlap integral

The fundamental mode of a single-mode fiber is close to a Gaussian. Computed for a step-index model of standard fiber at 1550 nm, with a core radius of 4.1 µm and a numerical aperture of 0.12 to 0.14, the exact mode overlaps its best-fitting Gaussian by 99.2 to 99.6%, so the closed-form results for Gaussian beams apply to fiber joints almost exactly. For two modes with $1\mathrm{/}{e}^2$ intensity radii $w_1$ and $w_2$ (half the mode field diameters) meeting at their waists, in a medium of index $n$:

$$\begin{array}{rl}{\displaystyle {\eta }_{\text{size}}}& {\displaystyle ={\left(\frac{2w_1{w}_2}{w_1^2+w_2^2}\right)}^2,}\\ {\displaystyle {\eta }_{\text{offset}}}& {\displaystyle =\exp \text{\hspace{0.17em}⁣}(-\frac{2d^2}{w_1^2+w_2^2}),}\end{array}$$

for a size mismatch and for a lateral offset $d$. For two equal modes of radius $w$, a tilt $\theta$ about the joint and a longitudinal gap $z$ give

$$\begin{array}{rl}{\displaystyle {\eta }_{\text{tilt}}}& {\displaystyle =\exp \text{\hspace{0.17em}⁣}[-{\left(\frac{\pi nw\theta }{\lambda }\right)}^2],}\\ {\displaystyle {\eta }_{\text{gap}}}& {\displaystyle =\frac{1}{1+{\left(\lambda z\mathrm{/}2\pi n{w}^2\right)}^2}\mathrm{.}}\end{array}$$

The product ${\eta }_{\text{size}}{\eta }_{\text{offset}}$ is exact for Gaussian modes; the product of all four is a good approximation while each loss is small. In terms of the Rayleigh range of each mode, $z_R=\pi n{w}^2\mathrm{/}\lambda$, the gap term is $1\mathrm{/}[1+(z\mathrm{/}2z_R{)}^2]$. For modes of different sizes separated by a gap, the on-axis result is $\eta =4\mathrm{/}[(w_1\mathrm{/}{w}_2+w_2\mathrm{/}{w}_1{)}^2+(\lambda z\mathrm{/}\pi n{w}_1{w}_2{)}^2]$ (Joyce and DeLoach).

Loss versus mode size ratio

The size term alone:

A 10% difference in mode size costs only 0.04 dB, which is why splices between standard single-mode fibers of slightly different design lose little to mismatch: 0.07 dB between fibers of 10.4 µm and 9.2 µm mode field diameter, as in the fusion splicing entry. A factor of 4.2, as between standard fiber (10.4 µm) and the 2.5 µm spot of a lensed fiber, costs 6.85 dB, and a factor of twenty costs 20 dB between Gaussian modes. The actual mode of a bare 500 × 220 nm silicon strip waveguide, which is not Gaussian, loses 20.1 dB to standard fiber, computed from its full-vectorial mode (inverse taper); its slowly decaying tails gain it about 1 dB over a Gaussian of its best-fit 0.46 µm diameter.

Alignment tolerances at 1550 nm

The lateral tolerance grows in proportion to the mode size, the tolerance to a gap grows as its square (with the Rayleigh range), and the tolerance to tilt shrinks as the mode grows. Between equal modes a lateral offset of $0.48w$ costs 1 dB and one of $0.83w$ costs 3 dB, and a gap of $1.02z_R$ costs 1 dB. Computed for two matched Gaussian modes at 1550 nm in air, the misalignments that cost 0.5 dB and 1 dB are:

A lensed fiber on a small-mode edge coupler therefore has to be held to a few tenths of a micrometer, while two cleaved fibers tolerate a few micrometers of offset and tens of micrometers of gap. Expanded-beam connectors use the same trade: enlarging the beam relaxes the lateral and gap tolerances and tightens the angular one.

Non-Gaussian and elliptical modes

For modes that are not Gaussian the overlap is computed numerically from the two mode fields, usually taken from a mode solver. The mode at the tip of a silicon inverse taper is an example: computed for tip widths from 160 to 200 nm, it overlaps its best-matching Gaussian by only about 91%, because a small core in a uniform cladding produces a sharply peaked field with slowly decaying tails. Even a perfectly sized lensed fiber loses about 0.4 dB to that difference in shape before any misalignment. The same tails work the other way against a larger mode: the mode of a 180 nm tip loses 5.0 dB to standard fiber, where a Gaussian of the same 2.56 µm diameter would lose 6.7 dB.

Elliptical modes with separate radii in $x$ and $y$, such as the near field of an edge-emitting laser, couple with

$$\eta =\frac{2w_{x1}{w}_{x2}}{w_{x1}^2+w_{x2}^2}\cdot \frac{2w_{y1}{w}_{y2}}{w_{y1}^2+w_{y2}^2},$$

so a 2 × 4 µm mode meeting a round 3 µm mode couples 0.89 of its power (0.52 dB).

Reflection and other losses at a joint

The overlap accounts for the shape of the fields only. Each uncoated glass-air surface also reflects about 3.3% of the power, 0.15 dB, for silica of index 1.444 (Fresnel equations), and the two surfaces of a gap form a weak cavity whose fringes make the loss vary with wavelength; an anti-reflection coating or index-matching gel removes both. Polarization matters for waveguides whose modes differ between TE and TM, since the overlap is then computed separately for each polarization.

Measuring mode mismatch loss

When coupling is measured into a short fiber, light launched into cladding modes can still reach the detector and make the coupling look better than it is; a cladding-mode stripper or a longer fiber removes it. An OTDR looking at a splice between fibers of different mode sizes reads a different loss from each end, even an apparent gain from one of them, because the backscatter it measures is stronger in the fiber with the smaller mode; the average of the two readings is the true loss.

Reducing mode mismatch loss

The loss is reduced by making the two modes the same size and shape and holding them in place: a lens or a lensed fiber to shrink the fiber mode, an inverse taper or spot-size converter to enlarge a chip mode, and a tapered or thermally expanded fiber section between fibers of different mode sizes. The Fiber Coupling Efficiency Calculator computes size mismatch, lateral offset, tilt and gap together, including the wavefront curvature that a gap introduces and the index of the medium filling it, and Fiber-to-chip coupling for silicon photonics compares the methods used on photonic chips.

Common questions

What is mode mismatch loss?

The part of the power lost at a joint between two waveguides or fibers because their mode fields differ in size, shape or position. It is $-10{\log }_{10}$ of the overlap integral of the two normalized fields, and it is usually the largest loss in fiber-to-chip and laser-to-fiber coupling: standard single-mode fiber butted against a 2.5 µm spot loses 6.85 dB to it.

How do you calculate the coupling efficiency between two Gaussian modes?

For modes of radii $w_1$ and $w_2$ at the same plane, $\eta =[2w_1{w}_2\mathrm{/}(w_1^2+w_2^2){]}^2$, multiplied by $\exp [-2d^2\mathrm{/}(w_1^2+w_2^2)]$ for a lateral offset $d$. A 10.4 µm fiber mode coupled to a 2.5 µm mode gives 0.21, or 6.85 dB.

How much loss does a 1 µm offset cause?

It depends on the mode size, since between equal modes the offset loss is $4.34(d\mathrm{/}w{)}^2$ dB. A 1 µm offset costs 0.16 dB between 10.4 µm modes, 1.9 dB between 3 µm modes and 2.8 dB between 2.5 µm modes.

Is mode mismatch loss the same in both directions?

Yes. The overlap integral is symmetric in the two fields, so light going from a small mode into a large one loses the same fraction as light going the other way. An OTDR can still read different splice losses from the two ends, because the backscatter it measures depends on the mode size; the average of the two readings is the true loss.

How can mode mismatch loss be reduced?

By matching the two modes in size and shape and holding them aligned. A lens or lensed fiber shrinks the fiber mode and an inverse taper or spot-size converter enlarges the chip mode; between fibers of different mode sizes, a tapered or thermally expanded section changes one mode gradually into the other.

References: D. Marcuse, "Loss analysis of single-mode fiber splices," Bell System Technical Journal 56, 703 (1977). H. Kogelnik and T. Li, "Laser beams and resonators," Applied Optics 5, 1550 (1966). W. B. Joyce and B. C. DeLoach, "Alignment of Gaussian beams," Applied Optics 23, 4187 (1984). K. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006). A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983). Corning, SMF-28 Ultra product information, for the 10.4 µm mode field diameter at 1550 nm.