Waveguide Modes and Photonic Integration

How light gets trapped inside a strip of silicon 220 nm tall, what determines which modes exist, and why different platforms produce such different behavior.

Total internal reflection, but make it wave optics

Light in free space spreads. Point a laser at the Moon and the beam arrives 2 km wide. But trap that same light inside a strip of silicon 220 nm tall and 500 nm wide, and it stays confined for centimeters, bending around corners, splitting into channels, and interfering with copies of itself. That is guided-wave optics, and it is the foundation of every photonic integrated circuit (PIC) shipping today.

You already know the ray-optics explanation: light bouncing between two interfaces at angles steeper than the critical angle stays trapped by total internal reflection (TIR). If you want a refresher, see our Fresnel Equations tutorial. That picture is correct but incomplete. It tells you that light can be trapped, but not which field patterns are allowed.

The wave-optics version adds a constraint. The field inside the core is a standing wave in the transverse direction and a traveling wave along the guide. For the mode to be self-consistent, the round-trip transverse phase must be an integer multiple of 2π, after accounting for the phase shifts at each reflection. Only a discrete set of bounce angles satisfy this condition. Each solution is a mode, labeled TE0, TE1, TM0, TM1, and so on.

The key parameter is the effective index $n_{\text{eff}}$. It is the propagation constant β divided by the free-space wavenumber $k_0$:

For a guided mode, $n_{\text{eff}}$ must sit between the core index and the cladding index:

If $n_{\text{eff}}$ equals $n_{\text{core}}$, the field is entirely inside the core (no evanescent tail). If it drops to $n_{\text{clad}}$, the field extends infinitely into the cladding and the mode is no longer guided.

If you have worked with single-mode fiber, this is the same physics at a different scale. SMF-28 has an index contrast of 0.36% and a core diameter of 8.2 μm. Silicon-on-insulator has an index contrast of 58% and a core width of 500 nm. The 160-fold difference in contrast is why one fits in a cable and the other fits on a chip.

The slab waveguide eigenvalue equation

The simplest waveguide to analyze is the slab: a planar film of thickness h and refractive index $n_{\text{core}}$, sandwiched between a substrate ($n_{\text{sub}}$) and a cover ($n_{\text{cov}}$). The field inside the core oscillates back and forth — that is $k_y$, the transverse wavevector, which measures how fast the field wiggles vertically. Outside the core, the field decays exponentially — that is γ, the evanescent decay rate. The eigenvalue equation says: the oscillation inside the core must match the decay outside, at both interfaces, simultaneously.

For TE modes (electric field polarized parallel to the interfaces), the eigenvalue equation is:

where $m=0,1,2,\dots$ is the mode order, and:

• $k_y=\sqrt{n_{\text{core}}^2{k}_0^2-{\beta }^2}$ is the transverse wavevector in the core
• ${\gamma }_{\text{sub}}=\sqrt{{\beta }^2-n_{\text{sub}}^2{k}_0^2}$ is the evanescent decay constant in the substrate
• ${\gamma }_{\text{cov}}=\sqrt{{\beta }^2-n_{\text{cov}}^2{k}_0^2}$ is the evanescent decay constant in the cover

The left side is the round-trip phase accumulated across the core. The right side is the sum of the phase shifts at the two interfaces (the arctan terms). Each time you increase the mode order m by one, you add another half-oscillation inside the core.

This equation cannot be solved in closed form. You find $n_{\text{eff}}$ by scanning β between $n_{\text{clad}}{k}_0$ and $n_{\text{core}}{k}_0$ and looking for the values where the left and right sides match. Each crossing is a mode.

Worked example: SOI at 1550 nm

Silicon photonics lives on the silicon-on-insulator (SOI) platform: a thin film of crystalline silicon ($n=3.476$ at 1550 nm) sitting on a silica buried oxide ($n=1.444$). Let us solve for the modes of a 220 nm slab with silica on both sides.

First, the V-number:

Since $V<\pi =3.14$, only the fundamental TE and TM modes exist. This is single-mode operation. Solving the eigenvalue equation numerically:

Two things stand out. First, TE0 has a much higher effective index than TM0 (2.848 vs 2.053). The birefringence is 0.795, which is enormous compared to a fiber (where birefringence is typically less than 0.001).

Second, the evanescent penetration depth is about 100 nm for TE0. That means the field extends roughly half the core thickness into the cladding. In a 220 nm SOI slab, the mode is only about 65% confined to the core. The rest of the light rides the evanescent tail — simultaneously a challenge (propagation loss from surface roughness) and a feature (evanescent coupling for sensors and directional couplers).

The V-number and cutoff

The V-number controls how many modes a waveguide supports. For a symmetric slab:

The m-th TE mode appears when $V$ exceeds $m\pi$. So TE0 exists for any $V>0$, TE1 appears at $V=\pi$, TE2 at $V=2\pi$, and so on.

For our SOI example (NA = 3.162 at 1550 nm), the TE1 cutoff thickness is:

So a 220 nm silicon slab is single-mode, but barely. Push to 250 nm and a second TE mode appears. This is why the SOI industry settled on 220 nm: it is the thickest single-mode silicon slab at telecom wavelengths.

What the effective index tells you

The effective index is the phase velocity of the mode expressed as an equivalent refractive index. A mode with $n_{\text{eff}}=2.848$ accumulates phase at the same rate as a plane wave in a hypothetical bulk material with $n=2.848$. Over a length L, the phase is:

A Mach-Zehnder interferometer with arm length difference $\mathrm{\Delta }L$ produces a phase difference of:

The group index $n_g$ plays the analogous role for pulses. A pulse envelope travels at $v_g=c\mathrm{/}{n}_g$, where:

In our 220 nm slab, $n_g=3.58$, so pulses are 26% slower than the phase fronts. This distinction matters for any device that operates on pulse timing: modulators, delay lines, ring resonators.

Mode profiles and confinement

Each mode has a characteristic field profile. TE0 is a half-cosine in the core with exponential tails in the cladding. TE1 has a full cosine (one node). Higher modes have more nodes.

The confinement factor Γ measures what fraction of the mode power lives inside the core:

For the 220 nm SOI slab at 1550 nm, ${\mathrm{\Gamma }}_{\text{TE0}}=0.651$ (65.1%). The confinement factor increases with thickness:

At h = 100 nm, more than half the light is in the cladding. By h = 500 nm, three quarters is in the core — but now you have three TE modes. Waveguide design is always a negotiation between confinement and single-mode operation.

Dispersion: how n_eff changes with wavelength

Even if the core and cladding materials had perfectly flat refractive indices, the effective index would still change with wavelength. This is waveguide dispersion: as the wavelength changes, the mode "sees" a different balance of core and cladding.

At shorter wavelengths, the mode is more tightly confined (higher $n_{\text{eff}}$, closer to $n_{\text{core}}$). At longer wavelengths, the mode spreads into the cladding (lower $n_{\text{eff}}$, approaching $n_{\text{clad}}$).

The slope $d{n}_{\text{eff}}\mathrm{/}d\lambda$ at 1550 nm is about $-4.7\times {10}^5{\text{m}}^{-1}$, giving $n_g=3.58$. A 1550 nm pulse in this waveguide travels at $c\mathrm{/}3.58=\mathrm{83,800}$ km/s, about 25% slower than the phase velocity. This large waveguide dispersion is a distinctive feature of high-contrast platforms like SOI.

From slabs to strips: the effective index method

Real waveguides are not infinite slabs. They are rectangular strips with finite width. The standard SOI strip waveguide is 500 nm wide and 220 nm tall. Solving the full 2D mode equation requires a numerical solver, but there is an elegant approximation that gives surprisingly good results: the effective index method (EIM).

The idea is to separate the 2D problem into two 1D problems:

Step 1: Solve the vertical slab (thickness h = 220 nm, Si core, SiO₂ cladding) to get $n_{\text{slab}}$. For the quasi-TE mode, use the TE slab equation. This gives $n_{\text{slab}}=2.831$ for the asymmetric case (air top, SiO₂ bottom).

Step 2: Solve a horizontal slab of width w, using $n_{\text{slab}}$ as the core index and the lateral cladding index. For the quasi-TE mode, the correct convention is to use the TM slab equation in this lateral step. This "crossed polarization" rule comes from the fact that the dominant electric field component switches its orientation relative to the slab interfaces between the two steps.

For a 500 × 220 nm SOI strip (asymmetric, air top):

Quasi-TE mode:

• Step 1 (vertical, TE): $n_{\text{slab}}$ = 2.831
• Step 2 (lateral, TM): $n_{\text{eff}}$(quasi-TE) = 2.474

Quasi-TM mode:

• Step 1 (vertical, TM): $n_{\text{slab}}$ = 1.891
• Step 2 (lateral, TE): $n_{\text{eff}}$(quasi-TM) = 1.703

The birefringence in the strip (2.474 − 1.703 = 0.77) is nearly as large as in the slab. Most silicon photonic circuits are designed to operate in a single polarization (usually TE) and include polarization splitters and rotators to manage the other.

Platform comparison: Si vs SiN vs LiNbO₃

Not all waveguide platforms are alike. The core-cladding index contrast determines almost everything: how small the waveguide can be, how tight the bends, how many modes, and how sensitive the device is to fabrication errors.

SOI has the highest contrast. Bending radii can be as small as 5 μm, but propagation losses are typically 1–3 dB/cm. Silicon is transparent from 1.1 to 8 μm but absorbs in the visible.

Silicon nitride (Si₃N₄) has moderate contrast. Propagation losses can be as low as 0.1 dB/m in ultra-low-loss platforms. SiN is transparent from visible wavelengths through the near-IR.

Lithium niobate on insulator (LNOI) combines moderate contrast with strong electro-optic and nonlinear optical properties. Modulators on LNOI can operate at speeds exceeding 100 GHz.

The fiber connection

To put all of this in perspective, here is how an SOI strip waveguide compares to the fiber you may already be familiar with:

The V-numbers are similar (both around 2), but the core sizes differ by a factor of 20 and the mode areas by a factor of 800. All of that scaling comes from the index contrast. This is also why coupling light between a fiber and a chip is one of the hardest practical problems in integrated photonics.

References

1. Saleh & Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Chapters 8–9.
2. Okamoto, Fundamentals of Optical Waveguides, 2nd ed. (Academic Press, 2006).
3. Chiang, "Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides," IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
4. Bogaerts et al., "Silicon microring resonators," Laser Photonics Rev. 6, 47–73 (2012).
5. Reed & Knights, Silicon Photonics: An Introduction (Wiley, 2004).

Where to go deeper

Try the calculations yourself in the PIC Waveguide Mode Solver: load the SOI 220 nm preset and verify the numbers in this tutorial. All the refractive indices used here are available in the Material Database.

Design a ring filter with the waveguide you just learned about → Ring Resonators in Photonics