Ring Resonator Calculator

A ring resonator is a waveguide closed on itself, beside one or two straight bus waveguides. Light in a bus leaks into the ring at the coupler. At the wavelengths where one round trip holds a whole number of wavelengths, the light in the ring adds up in phase, and the bus transmission dips. The calculator draws the spectrum, phase and group delay at the bus ports, and a result card for the resonance nearest the wavelength you design for.

Everything it computes comes from four inputs:

• the effective index n_eff and the group index n_g, typed, or read across wavelength from a waveguide you saved in the PIC Waveguide Mode Solver;
• the loss per round trip: the propagation loss you enter plus the bend loss;
• κ², the fraction of the power that crosses between bus and ring at each coupler, typed, or imported from a coupler you saved in the PIC Directional Coupler Simulator;
• the ring’s length, the circumference at the radius you enter.

Two configurations: “All-pass: one bus waveguide, and the light that enters the ring returns to it. Add-drop: a second bus picks off the light at resonance and sends it to the drop port.”

Each import is optional. With nothing imported the page computes from the values you type, and every imported value can be detached and edited.

The left panel has four sections. Before any import, the panel shows only what a ring needs: the configuration, the radius, the propagation loss, the bend loss, n_eff and n_g, κ², and the design wavelength. What an import brings appears only after it is applied.

The presets’ n_eff and n_g come from the PIC Waveguide Mode Solver and their κ² from the PIC Directional Coupler Simulator, each on a stated cross-section and gap, and the Preset control’s ⓘ names them. Their radius, loss and design wavelength are design values, chosen to show a kind of ring rather than taken from a device. The LNOI Modulator’s κ² is a design value too, because the coupler does not model lithium niobate.

Ring

Waveguide properties

The propagation loss is always yours: it stays editable after an import. The other three can be filled by an import (“Importing the waveguide”).

Coupling

Simulation

The points set the even grid across the range. Narrow resonances are not left to that grid: the page always adds a dense window of points around every resonance it finds, so a line narrower than the grid spacing is still drawn at its true depth (“The transfer functions and the card”).

With an imported table, the automatic range is cut to the wavelengths the table covers, and the note says so. A range you set yourself is never cut: if it leaves the table, the page refuses it by name (“What the ring refuses”).

The button “Import from Mode Solver” (hover: “Import from the PIC Waveguide Mode Solver”) opens “Import from a waveguide design”, with “Your saved waveguide designs”. Pick a design to see what it carries; nothing on the page changes until you press Apply. The summary lists one line per thing it will set:

• The runs the design holds: “Wavelength sweep, … to … nm” or “Single wavelength, … nm”, and “Bend, R = … µm”.
• “Mode: …”, with a mode select when the run kept more than one guided mode.
• “n_eff and n_g: … wavelengths from … to … nm (Full-vectorial (FDE): saved run)” for a wavelength sweep, or “n_eff at … nm only.” for a run at one wavelength.
• “Bend loss: from the bend solve at R = … µm”, or “Bend loss: none in this design, so it stays as you entered it.”
• “Waveguide: …”: the cross-section the device view draws and the coupler check compares (see “When the coupler’s waveguide differs”).

One visit for both runs. The modal’s footer link, “Open the PIC Waveguide Mode Solver set up for this ring: a wavelength sweep over … to … nm and a bend at R = … µm”, opens the mode solver with this ring’s waveguide, a sweep over the ring’s range and a bend at its radius. Its hover: “Opens in a new tab with this ring's waveguide. Run the sweep and the bend, saving after each; both runs go into one design, and Import here takes both.” A saved design with one of the two runs missing offers “Add the bend run in the PIC Waveguide Mode Solver” or “Add the wavelength sweep in the PIC Waveguide Mode Solver” under its summary once you pick it.

After the import

• The button becomes a status line, “Waveguide: “…”, …”, with a link to change the design.
• n_eff and n_g turn read-only and show their value at the design wavelength: “n_eff … at … nm” and “n_g … at … nm”. They follow the design wavelength as you move it.
• Under them, the source: “From “…”: Full-vectorial (FDE): saved run, …–… nm, saved …”. Its ⓘ, “Where these values came from”, gives the full record of the run.
• “Detach” keeps the two values and makes them editable. From its ⓘ: “Keeps the values shown and makes them editable. The ring stops following the imported table, so changing the design wavelength no longer updates them.”

With a bend run in the design

• Under n_eff: “Includes the bend's index shift, …, from the bend solve at R = … µm”. Its ⓘ, “What the bend shift assumes”: “A bent mode's effective index differs from the straight one's. The ring adds this difference, from the same bend run, to the imported table at every wavelength. It is measured at one wavelength and assumed the same across the range; how it changes with wavelength has not been measured.”
• The n_g ⓘ then adds one sentence: “The group index sets the free spectral range. With an imported table it comes from the same run, at the design wavelength. n_g is the straight guide's; the bend shift is applied to n_eff only.”
• Bend loss turns read-only with its source, “From the bend solve at R = … µm in “…””, and the label “Held constant across the range”. Its ⓘ, “Why it is constant”: “The bend loss comes from one bend solve at one wavelength, and the ring uses that value at every wavelength of the range. How it changes with wavelength is not included.”
• Its own “Detach”: “Keeps the value shown and makes it editable. It stops being tied to the bend solve, and a radius change no longer flags it.”
• When the ring’s radius differs from the radius the bend was solved at, a flag says so: “Solved at R = … µm; your ring is … µm. Bend loss changes steeply with radius, so it is not carried from one radius to another.” It offers two actions, “Use R = … µm” and “Solve the bend at … µm”.
• When the solve’s materials absorb, the imported bend loss already includes that absorption: “The imported bend loss already includes the materials' absorption, so enter only scattering loss as yours.”

A design without a bend run imports the table alone, and the page says “No bend run in “…”: n_eff is the straight guide's.” Bend loss then stays as you entered it.

The button “Import from Directional Coupler” (hover: “Import coupling from the PIC Directional Coupler Simulator”) sits under each coupling, one per bus. It opens “Import coupling from a coupler design”, which on an add-drop ring names the bus: “Import coupling from a coupler design for the input bus” or “Import coupling from a coupler design for the drop bus”. The list is “Your saved coupler designs”. The summary of a picked design shows:

• “κ² … at … nm (ring-bus arc)”, naming the kind of region the coupler modeled;
• “Gap … nm”, when the region has one gap;
• the coupling phase, carried with the import and not used by the ring;
• “Excess loss: not modeled by this engine, so the ring treats the coupler as lossless.”

The footer link, “Open the PIC Directional Coupler Simulator with this ring's radius, wavelength and waveguide”, opens the coupler with the ring’s radius, the design wavelength and the imported waveguide, in a new tab.

After the import

The status line reads “Coupler: “…”, ring-bus arc”, and κ² turns read-only. Its source line takes one of three forms, by what the coupler run holds:

• With several wavelengths, κ² is read at every wavelength of the range (see “How κ² is read across wavelength”).
• When κ² comes from exactly two wavelengths, a second line says how closely a straight line between them matched full runs: “κ² between the two runs: within 0.3 %” up to 50 nm apart, “κ² between the two runs: within 2 %” up to 100 nm. For one sweep with two wavelengths the line reads “κ² between the sweep's two wavelengths: within 0.3 %” or “κ² between the sweep's two wavelengths: within 2 %”. The ⓘ, “How this was checked”: “Between two runs the ring draws κ² as a straight line in ln κ². Checked against full runs every 10 to 25 nm on two silicon arcs and one silicon nitride arc at R = 10 and 50 µm, the largest difference was 1.07 % for runs 100 nm apart and 0.29 % for runs 50 nm apart.”
• A run the server refused is named under the source line, one line each: “Not computed at … nm: that run needed more cells than the server allows. The other wavelengths are used.” at an end of the range, or “Not computed at … nm: that run needed more cells than the server allows; the ring does not estimate between … and … nm.” inside it. A run refused for another reason names it: “Not computed at … nm: …” or “Not computed at … nm: …; the ring does not estimate between … and … nm.”.
• “Detach”: “Keeps the value shown and makes it editable. It stops following the coupler run, so changing the radius or the design wavelength no longer re-reads it.”
• The provenance ⓘ, “Where these values came from”: “These values come from the coupler run. Changing the radius or the design wavelength re-reads them where the run allows, and flags them where it does not. Detach to enter κ² by hand.”

When the coupler’s waveguide is not the ring’s

κ² holds for the waveguide the coupler was designed with. When it differs from the waveguide imported above, the coupling row shows an amber flag: “This coupler was designed with …; the ring's waveguide is …. Its κ² holds for its own waveguide, so the two may not describe one device.” Its action: “Open the coupler with the ring's waveguide”. It warns and does not block: two runs can describe different process corners on purpose. With no waveguide imported, a gray line asks for one: “This coupler was designed with …. Import the ring's waveguide above to check that they match.” Which fields count is in “When the coupler’s waveguide differs”.

When an arc coupler’s radius differs from the ring’s by more than 0.005 µm, the summary says so: “Coupler ring radius … µm; your ring is … µm. The coupling depends on the radius, so it is not carried from one to the other.” Apply keeps the coupler with the ring but does not use its κ²: the ring goes on with the κ² you entered. Choosing “Use R = … µm” sets the ring’s radius to the coupler’s, and only then is the imported κ² applied, read-only, with Detach. A later import of the same bus is flagged when its gap differs by more than 0.5 nm.

The card is titled “At the resonance nearest … nm”, and its first row, “Resonance at … nm”, names the resonance it describes. n_eff and n_g are shown at the design wavelength; every other row is computed at that resonance.

• The coupling regime line under the rows says whether the ring is under-coupled, critically coupled or over-coupled: whether the bus takes out less light per round trip than the ring loses, about the same, or more.
• When anything is imported, the line at the bottom of the card names it, for example “n_eff and n_g from “…”; κ² from “…””.
• When the design wavelength falls where κ² is not estimated (a refused run inside the range), the card shows “No result at … nm: it falls between … and … nm, where the ring does not estimate κ². The spectrum shows the rest of the range.”

The ring is drawn three ways, from the inputs on the page: 3D, Top and Cross-section, the same views the PIC Waveguide Mode Solver and the PIC Directional Coupler Simulator use. The page opens on 3D.

• Top draws the ring at its radius and each bus beside it. The gap is drawn between the facing edges of ring and bus, never between their center lines, and a scale bar gives true length.
• A waveguide is far narrower than a ring is wide, so the drawing can stretch the narrow dimensions: at × N, every waveguide width and gap is drawn N times its true size about its own center line, while the radius stays true. Auto picks N so the gap is visible, and you can choose another value; the largest offered keeps the ring’s inner edge at least half the radius from its center. In 3D the heights are stretched by the same factor. Cross-section always keeps true proportions.
• 3D draws the same outlines through the layer stack.
• In 3D, drag to rotate, scroll to zoom and right-drag to pan; in Top and Cross-section, scroll to zoom and drag to pan. Reset view fits the drawing again; on Top and Cross-section, so does a double-click.
• Before any import, the drawing is estimated from your values: the single-mode width of a 220 nm silicon strip, with oxide below and air above, whose mode index matches your n_eff, and the widest gap at which a ring-bus arc of that width, at your radius, couples your κ². They are for the drawing only. Import a waveguide and a coupler to draw their true dimensions.
• Until a waveguide or coupler is imported, the drawing carries no width, gap or thickness numbers and no dimension lines: its geometry is an estimate. A waveguide import brings the waveguide’s dimensions, and a coupler import brings its own waveguide’s width and the gap; the gap’s number shows only once a coupler is imported. The radius and the scale bar are true from the start.
• Cross-section draws the imported waveguide with the mode solver’s own labels. Before an import it shows “Import a waveguide to see its cross-section.” and draws nothing.

The drawing sits above the spectrum and the result card below them, and the panel scrolls. The divider under the drawing trades height between the drawing and the spectrum: drag it to give either more room.

A wavelength sweep from the mode solver carries n_eff and n_g at each swept wavelength, for the one mode the sweep tracks. The ring reads the whole table. n_g is read as the run served it: the solver forms it with a central stencil on a grid padded beyond the table’s ends, and forming it again from the table would give a second number for the same quantity and lose the ends.

Between the swept wavelengths

Between two swept wavelengths, n_eff is a cubic whose values are the two served n_eff and whose slopes come from the served n_g, by the definition of the group index:

At a swept wavelength the page uses the served values exactly. The phase the resonances depend on therefore follows the waveguide’s own dispersion between the samples.

Measured on a served sweep of Si 440 × 220 nm, Air above, SiO₂ below, 1500 to 1600 nm in 11 wavelengths: keeping one wavelength every 20 nm and comparing with the 5 wavelengths left out,

Gaps and edges

• A swept wavelength the run marked is not used: where the mode stopped being guided, where n_eff showed a step, where the sweep lost track of the mode, or where the group index was withheld.
• The usable table is the longest run of consecutive good wavelengths that contains the design wavelength. The cubic never bridges a gap.
• The table is never extrapolated. Holding n_eff at the table’s edge would freeze the dispersion and misplace every resonance beyond it by an amount nothing on the page would show, so a wavelength outside the table is refused instead (“What the ring refuses”).

Without a table

With values typed, or a run at one wavelength imported, the phase index is the first-order model about that wavelength λ₀:

For typed values λ₀ is the design wavelength; for an imported run at one wavelength it is that run’s wavelength. A ring with nothing imported therefore places its resonances where the first-order model puts them.

The bend’s index shift in the phase

In a ring the light travels in the bent mode, whose effective index differs from the straight guide’s. When the design carries a bend run, the ring adds the difference to the table at every wavelength:

• n_straight is the bend run’s own straight index, solved with the same absorbing boundary as the bent mode, not a separate straight solve. Both indices come from one run on one grid with one boundary, so the solver’s own offset cancels in the difference.
• Δn_bend is measured at the bend run’s wavelength and assumed the same across the range. How it changes with wavelength has not been measured.
• n_g is not corrected for the bend. The bend moves n_g by a similar order, and that is not modeled; the free spectral range and the resonance search use the straight guide’s n_g.

The bend loss

The bend run gives the loss per quarter turn, which the ring converts to a loss per length of ring:

• A single bend solve is used only when its radius matches the ring’s within 0.005 µm. Otherwise the import is kept but not used, and bend loss stays as you entered it, until you choose “Use R = … µm”, which sets the ring’s radius to the solved one. A bend sweep over radius is interpolated between the two solved radii around the ring’s, with the logarithm of the loss linear in radius, and never beyond them.
• The value is held constant across the range. Bend radiation depends on wavelength; how much it changes across the range has not been measured here and is not included.

The round-trip amplitude combines both losses over the ring’s length L = 2πR:

From a saved coupler run the ring reads κ² = |S₄₁|², the power that crosses from the bus input (port 1) into the ring (port 4), in the coupler’s own port numbering. Three things are left out on purpose:

• The coupling phase is carried for the record and not used: the ring’s model does not need it.
• The bus’s own phase through the coupler is never imported. The ring already counts the phase along its whole length, and adding the coupler’s would count the coupling region twice.
• The coupler is treated as lossless, because no engine that serves these runs models its excess loss.

Across wavelength

• A run at one wavelength gives one κ², held constant across the range.
• Several wavelengths (full runs of a ring-bus arc, or a coupler’s wavelength sweep) give κ² exactly at each of them. Between them, κ² is the exponential of the quadratic in ln κ² through the three nearest:

κ² is never read outside the first and last run wavelengths. A range beyond them is refused by name.

Why the logarithm: across 1500 to 1600 nm, κ² of the measured arcs grew by a factor of 2.7 to 3.1, close to exponentially in wavelength, so a curve in ln κ² follows it far better than one in κ². Measured on two ring-bus arcs run in full every 10 nm (air above) and every 25 nm (oxide above), R = 10 µm, gap 200 nm, 20 nm mesh: the largest relative difference from the full runs at the wavelengths left out (%):

How the runs are placed: three when the ring’s range is 100 nm or less (both ends and the middle), otherwise equally spaced and no more than 50 nm apart, both ends included. Each is a full run; the runs share one solve window, sized for their most demanding wavelength.

From exactly two wavelengths

Two wavelengths do not define a quadratic, so the ring draws a straight line in ln κ² between them. Measured on the same two silicon arcs and on one silicon nitride arc, against the full runs between the two ends (%):

• The page prints the bound as the ceiling of the largest measured difference at one significant figure: within 0.3 % for wavelengths up to 50 nm apart, and within 2 % up to 100 nm.
• Two wavelengths more than 100 nm apart are not interpolated: the line was not checked that far.
• Under the placement rule above, two runs remain only when an end run of a three-run range was refused, so they are at most 50 nm apart. A wavelength sweep with two points can be wider.
• The bound was measured on two silicon arcs at R = 10 µm and one silicon nitride arc at R = 50 µm. It is printed for every coupler, and its ⓘ states that scope.
• Three or more wavelengths keep the quadratic rule; measured on the same three arcs, it stays within 0.03 % of the full runs.

A refused run inside the range

When the server refuses one of the runs inside the range, the stretch between its two neighbors has no run in it, and the ring does not estimate κ² there. A curve through the next three runs would span that stretch with no data, which is the extrapolation the rule forbids. The spectrum is drawn on the rest of the range, resonances inside the stretch are not reported, and a refused end run shortens the range to the last computed one.

At every wavelength λ of the grid, with n(λ) and κⱼ²(λ) read as above:

These are the calculator’s transfer functions, unchanged; the group delays keep their closed forms.

Finding the resonances

A resonance is a wavelength where the ring holds a whole number m of wavelengths, n(λ)·L/λ = m. The page solves it by Newton steps, started from the first-order model’s closed form:

A dense window of points, scaled to each resonance’s linewidth and spacing, is placed around every resonance found in the range and added to the even grid.

The card

At the resonance nearest the design wavelength, λ_res, with κ² read there:

• The card’s Q is the loaded Q: its linewidth includes the light the bus couples out. The intrinsic Q keeps only the ring’s own loss.
• The finesse above is its high-finesse form.
• The coupling regime compares r₁ with a (all-pass) or with r₂a (add-drop): under-coupled when r₁ is larger, over-coupled when it is smaller, and critically coupled inside the band

The coupler’s ring-side waveguide is compared with the waveguide imported from the mode solver. The bus waveguide may legitimately differ from the ring’s and is not compared.

The tolerances are display precision, not physics: a difference too small to see in either tool’s inputs is not flagged. How fast κ² moves with the ring guide’s width at a fixed gap has not been measured for this comparison, so no physics tolerance is claimed. The wavelength is not a waveguide field; each import’s wavelength coverage is checked on its own.

Where the inputs cannot be read, the page says so by name rather than estimate. No refusal is a clamp: a value is never held at an edge to fill a gap.

• How the bend’s index shift changes with wavelength: it is measured at one wavelength and held across the range.
• The bend’s effect on the group index: n_g is the straight guide’s.
• How the bend loss changes with wavelength: one value is held across the range.
• The coupler’s excess loss: the coupler is treated as lossless.
• The coupler’s phase: carried with the import, not used.
• A physics tolerance for the waveguide comparison: the comparison is at display precision.
• Reflections at the couplers and backscattering into the counter-propagating ring mode, which splits resonances: the transfer functions assume light travels one way around the ring.
• Racetrack rings: the ring is a circle.