---
title: "Evanescent Directional Coupler Interactive Lab | Sandbox Physics"
description: "How can light cross from one waveguide into a nearby one that it never touches? Each guide's mode has a faint evanescent tail outside its core."
canonical_url: "https://sandboxphysics.com/labs/evanescent-directional-coupler/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/evanescent-directional-coupler/"
last_modified: "2026-10-27"
---

Optics 085 · Waveguides, structured light, and modern optics

# Evanescent Directional Coupler

**How can light cross from one waveguide into a nearby one that it never touches?** Each guide's mode has a faint evanescent tail outside its core. When a neighbor sits inside that tail, light leaks across, and power swaps back and forth between the two guides along their length. A wider gap weakens the coupling and slows the swap. The interaction length decides how much has crossed when the guides part; stop at the right point and you have a fifty-fifty splitter.

Solve the TE supermodes of two planar guides by boundary matching, then set the edge-to-edge gap and the guide width. Watch power beat coherently between the guides, measure it with spatial Poynting collectors, design a half-power output and freeze the complex field.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The model is an ideal planar slab with exactly two supermodes, fixed assumed indices, and infinite cladding and height. It does not represent a fabricated ridge or fiber coupler. Mode ports and spatial collectors are separate measurements, and both the mode preparation and the collectors are ideal.

Physics tutorial

## Audit the physics of Evanescent Directional Coupler

**Background** Both eigenfields reach across the gap. Their relative phase reverses over a length $L_c=\pi/(\beta_e-\beta_o)$. Change the edge gap and watch this distance change.

**Why it matters** How can light cross from one waveguide into a nearby one that it never touches?

Start with the essentials

- **Focus question**: How can light cross from one waveguide into a nearby one that it never touches?
- **One-sentence intuition**: Power moves back and forth because two extended modes interfere, not because a photon jumps across empty space. The modal transfer length is $L_c=\pi/(\beta_e-\beta_o)$, and a full return takes twice that distance.

### Core mathematical model

#### The boundary-value eigenproblem for the guides

$$
f''+(k_0^2n^2-\beta^2)f=0,\qquad [f]=[f']=0
$$

Light propagates along lab X, the transverse coordinate is Z and the electric field points along Y. Each core has width w, and the gap between core edges is g. The displayed height is a crop of a direction that is infinite and uniform. Outside the cores the fields decay, and the TE electric field and its normal derivative are continuous.

#### One coherent forward field

$$
E_Y=\frac{\cos\theta}{\sqrt{\beta_e}}f_e e^{i\beta_e X}+\frac{e^{i\varphi}\sin\theta}{\sqrt{\beta_o}}f_o e^{i\beta_o X}
$$

The eigenfunctions are normalized to one across the transverse direction and are positive on the right core. Mixing them at forty-five degrees with opposite input phase gives the pair concentrated in the left guide. Its evanescent tails still reach into the other half-space. The input is set directly, not computed from a taper.

#### Longitudinal power flux

$$
S_X=|c_e|^2 f_e^2+|c_o|^2f_o^2+\frac{\beta_e+\beta_o}{\sqrt{\beta_e\beta_o}}\Re(c_ec_o^*e^{i(\beta_e-\beta_o)X})f_ef_o
$$

The longitudinal Poynting flux depends on the two unequal propagation constants. Integrating it over each half-space gives the solid power curves. Because the modes are orthogonal, the total is exactly one. The plotted curves approximate the integral with finite samples, and the readouts add the outer tails analytically.

#### Ideal modal ports versus spatial collectors

$$
P_R/P_0=\tfrac12+R I_{eo}\sin(2\theta)\cos[(\beta_e-\beta_o)X-\varphi],\quad R=\frac{\beta_e+\beta_o}{2\sqrt{\beta_e\beta_o}}
$$

The half-space overlap is the integral of the even mode times the odd mode over positive Z. For widely separated cores it approaches one half, and the familiar sine-squared modal result comes back. When the guides are close, the spatial power on one side never drops exactly to zero. The half-power button picks the first positive distance where the modes are in quadrature, rounded to the step of the length control. A pure supermode already splits the power evenly, so the button is disabled for it.

### Common difficulties

#### Does full modal transfer mean no field on the input side?

**Typical misconception** The familiar sine-squared curve is the exact power that any spatial detector catches.

**Better mental model** No. Ideal orthogonal modal ports, each localized on one guide, are not the same as collectors that take in a whole half-space, and the evanescent tails are still there. The solid spatial curves come from integrating the Poynting flux independently; the dashed modal reference is labelled separately.

### Run the experiment

1. 01

   #### Scene 1: Supermode beating

   Use the first preset. Click the half-power button, then halve or double that length. Read the spatial collector and the dashed reference for ideal modal ports separately.

   **What to observe:** The straight, transparent cores do not show a ray bouncing between guides. The Lab solves for a coherent field, and the moving tint does not show the group velocity.

2. 02

   #### Scene 2: Widen the gap

   Prepare only the even mode, then only the odd mode. Both split the power equally between left and right at every distance, but only the odd mode's electric field has a node in the centre.

   **What to observe:** The model is planar, TE only, with two prepared modes; it is not a full three-dimensional coupler as fabricated. The fixed indices are assumptions, not measured silica dispersion.

3. 03

   #### Scene 3: Even mode: no beating

   Change the width and the edge gap and watch the apparatus change with them. Freeze the output samples, change the wavelength, and check that the record stays the same.

   **What to observe:** The data row sits outside the apparatus. Collector brightness only sketches the power; for quantitative results, read the linear numbers and the sampled flux.

Physical conclusion

The gap sets how far the two supermodes split, and the splitting sets the beating. What a collector records depends on the basis it measures in.

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)