TE and TM polarization

Transverse electric (TE) and transverse magnetic (TM) are the two polarizations into which light divides at a flat interface and in a planar waveguide. The names come from microwave waveguides, where a TE mode has no electric field along the direction of propagation and a TM mode no magnetic field along it. For light meeting an interface, the reference is the plane of incidence, which contains the direction of travel and the normal to the surface: a TE wave has its electric field perpendicular to that plane, and so parallel to the surface, while a TM wave has its magnetic field perpendicular to it and its electric field in the plane of incidence, with a component normal to the surface. The two meet different boundary conditions at every interface, so they reflect by different amounts, travel at different speeds in a waveguide and couple differently to components.

TE, TM, s and p

Perpendicular and parallel refer to the plane of incidence, not to the surface: p-polarized light is the one whose electric field has a component normal to the surface. The two reflect differently at oblique incidence (Fresnel equations). TM reflection falls to zero at Brewster's angle, 56.3° for glass of index 1.5, while TE reflection rises steadily with angle, which is why light reflected from water or a road is partly polarized, mostly TE, and why a Brewster window passes TM light without reflection loss. At normal incidence the plane of incidence is undefined and the two behave alike.

In a slab waveguide

A slab waveguide confines light in one direction only, and its modes divide exactly into TE and TM. With the layers normal to $y$ and propagation along $z$, a TE mode has a single electric field component, $E_x$, parallel to the layers, and a TM mode a single magnetic component, $H_x$, with its electric field in $E_y$ and $E_z$. In the ray picture the light zig-zags in the $y$–$z$ plane, its plane of incidence at the layer boundaries, so TE modes are built from s-polarized rays and TM modes from p-polarized ones. For a symmetric slab of thickness $d$, core index $n_1$ and cladding index $n_2$, the even modes satisfy

$$\tan \frac{\kappa d}{2}=\frac{\gamma }{\kappa }\text{ (TE)},\tan \frac{\kappa d}{2}=\frac{n_1^2}{n_2^2}\frac{\gamma }{\kappa }\text{ (TM)},$$

with $\kappa =k_0\sqrt{n_1^2-n_{\text{eff}}^2}$ and $\gamma =k_0\sqrt{n_{\text{eff}}^2-n_2^2}$.

The extra factor $n_1^2\mathrm{/}{n}_2^2$ comes from the TM mode's electric field normal to the layers. The normal component of $\epsilon E$ is continuous at an interface, so $E_y$ jumps by the ratio of the permittivities and is larger just outside the core than just inside it. For a low index contrast the factor is close to 1 and the two polarizations nearly coincide, as in a fiber or a silica waveguide. For silicon in silica it is $(3.476\mathrm{/}1.444{)}^2$ = 5.80, the TM mode reaches much further into the cladding, and its effective index is far lower. Computed for a 220 nm silicon slab in silica at 1550 nm:

The TM mode has the lower effective index but the higher group index, because its effective index falls faster with wavelength.

In silicon and other channel waveguides

A waveguide that confines light in both transverse directions, such as a silicon strip, has no exact TE or TM modes: each mode has all six field components. Its modes are still close to one or the other and are named after the dominant electric field, TE-like (quasi-TE) when it lies mainly along $x$, parallel to the substrate, and TM-like (quasi-TM) when it lies mainly along $y$, normal to the substrate. Marcatili's notation for rectangular guides calls them $E_{pq}^x$ and $E_{pq}^y$, and the "-like" is usually dropped. In a square core within a uniform cladding the two fundamental modes have the same effective index by symmetry; a rectangular core breaks the symmetry and makes the waveguide birefringent. Computed for the standard 500 × 220 nm silicon strip in oxide at 1550 nm:

The difference of 0.675 in effective index is about 1,350 times the birefringence of polarization-maintaining fiber: the two polarizations slip one wavelength apart every 2.3 µm, against 3.1 mm in that fiber. Everything that depends on the index differs between them. A ring resonator of 10 µm radius has a free spectral range of 9.1 nm for TE and 10.2 nm for TM, with its TE and TM resonances at unrelated wavelengths, and an MMI or directional coupler designed for one polarization images or couples the other at the wrong length. The TM mode's field extends further from the core, so it needs larger bends: computed in a uniform oxide cladding, a 90° bend of 5 µm radius loses about 0.06 dB for TM and one of 3 µm about 0.5 dB, while the TE loss stays negligible down to 2 µm. The TE mode's field peaks at the etched sidewalls, the roughest surfaces of the waveguide, and the TM mode's at the smoother top and bottom surfaces, so the two also suffer different scattering losses.

In optical fibers

A round fiber has exact TE and TM modes, but neither is its fundamental mode. The fundamental mode, HE₁₁, is a hybrid mode that comes in two orthogonal polarizations with the same propagation constant in an ideal fiber; the TE₀₁ and TM₀₁ modes are two of the four modes of the LP₁₁ group, guided in few-mode fibers and in standard single-mode fiber at wavelengths below its cutoff. Small asymmetries in a real fiber split the two polarizations of HE₁₁, which causes polarization mode dispersion. Polarization-maintaining fiber makes the split large on purpose and names the two polarizations after its slow and fast axes rather than TE and TM.

Devices that treat TE and TM differently

Working with both polarizations

Light from a standard fiber arrives in an unknown polarization that drifts with temperature and handling, so a circuit that works in one polarization needs one of three arrangements. A polarization controller aligns the input, which suits a laboratory bench but not a deployed link. A polarization-diversity circuit splits the input in two, either turning the TM part into TE with a polarization splitter-rotator or coupling the two fiber polarizations into two TE waveguides with a two-dimensional grating coupler, and processes both halves in identical TE circuits; coherent receivers are built this way as a matter of course, because the signal carries data in both polarizations. Otherwise the waveguide itself is made independent of polarization, with a square core or a low index contrast, which works only while fabrication keeps the geometry within tolerance; what remains appears as polarization-dependent loss and as a shift in wavelength between the two polarizations.

Common questions

What is the difference between TE and TM polarization?

In TE light the electric field is perpendicular to the plane of incidence, and so parallel to the interface or to the layers of a waveguide; in TM light the magnetic field is, and the electric field has a component normal to the interface. In a waveguide, a TE mode has no electric field along the direction of propagation and a TM mode no magnetic field along it.

Is TE the same as s-polarization?

Yes. For reflection and refraction at an interface, TE is s-polarization and TM is p-polarization, both named relative to the plane of incidence. The Fresnel coefficients are also written with ⊥ for s and ∥ for p.

Why is the TM effective index lower in a silicon waveguide?

The TM mode's main electric field is normal to the top and bottom of the thin silicon layer, and the boundary condition makes that field jump by the ratio of the permittivities, 5.8 between silicon and silica, so more of the mode lies in the cladding. In a 500 × 220 nm strip at 1550 nm the TE effective index is 2.446 and the TM 1.771.

Why do silicon photonic circuits mostly use TE?

The TE mode is better confined in the thin silicon layer, so it has the higher effective index, tolerates tighter bends and is less sensitive to the cladding and the substrate, and grating couplers and most component libraries are designed for it. The edge-emitting lasers that feed such circuits also usually emit TE.

Does an optical fiber have TE and TM modes?

Yes, but not as its fundamental mode. The fundamental mode of a round fiber, HE₁₁, is a hybrid mode with two polarizations of equal propagation constant; TE₀₁ and TM₀₁ belong to the next group of modes, LP₁₁, which a single-mode fiber guides only at wavelengths below its cutoff.

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