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.

Right spatial collector power

P_R/P_0

$0.50$

Modal transfer length

L_c

$50\%$

Total longitudinal flux

P_{\rm out}/P_0

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Evanescent Directional Coupler

Edge-to-edge gap, not centre separation

g

1.50\,\mu\mathrm m

Physical parallel interaction length

L

1\,\mathrm{mm}

Width of each identical slab core

w

3\,\mu\mathrm m

Vacuum wavelength; assumed fixed indices

\lambda_0

1550\,\mathrm{nm}

Supermode mix: zero even, ninety odd

\theta

45^{\circ}

Input phase of the odd mode relative to the even mode

\varphi

180^{\circ}

Inspection position along the guides

x/L

50\%

Light does not hop between the guides along a diagonal path. Two extended supermodes interfere, so power moves back and forth between the parallel cores while the total flux stays the same. Pull the cores apart, launch only the even or the odd mode, then design and record a half-power output.

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.

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 misconceptionThe familiar sine-squared curve is the exact power that any spatial detector catches.

Better mental modelNo. 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

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.
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.
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.