Cauchy Dispersion Refractive Index Calculator
Evaluates the Cauchy equation for one wavelength: with coefficients of 1.5046 and 0.0042 µm² at 0.55 µm, the refractive index is 1.518484297521. Units are never guessed, so convert nanometres to micrometres first. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- A
- 1.5046
- B
- 0.0042
- Wavelength um
- 0.55
- Refractive index
- 1.5184842975
This Cauchy dispersion refractive index calculator evaluates the two-term equation n(λ) = A + B/λ² for one vacuum wavelength. Enter the dimensionless A coefficient, the B coefficient in square micrometres, and wavelength in micrometres to obtain a deterministic refractive-index estimate. It is useful for optical design checks, material-data validation, classroom calculations, and reproducible engineering workflows where the coefficients and their unit convention are already known. The output also repeats every accepted input for clear calculation records.
Use coefficients with the correct wavelength unit
The two-term Cauchy equation is simple, but its coefficients are tied to a unit convention. This calculator expects the wavelength in micrometres and B in square micrometres, while A is dimensionless. If a datasheet gives wavelength in nanometres, divide that value by 1,000 before entering it.
For example, 550 nm becomes 0.55 µm. Do not enter 550 while keeping a B coefficient fitted for micrometres, because the arithmetic will be valid but the physical result will be wrong. Likewise, coefficients from a source that defines wavelength in another unit must be converted or refitted before use.
Record the source, spectral range, temperature, and unit convention alongside any result that enters an engineering report. The calculator deliberately does not guess units or silently rescale values. That strictness makes each result reproducible and prevents a plausible-looking index from hiding a thousandfold wavelength conversion mistake.
Before calculating, confirm that A and B belong to the same material, fit, and stated wavelength convention.
Understand the calculation and its limits
For the supplied wavelength λ, the calculator squares λ, divides B by that squared value, and adds the result to A. In the usual positive-B form, the dispersion contribution becomes larger toward shorter wavelengths and smaller toward longer wavelengths. The equation is empirical: it summarizes measured behavior over a fitted spectral interval rather than deriving optical response from first principles.
A numerically valid answer therefore does not prove that the model is accurate outside the range used to obtain the coefficients. Absorption bands, strong anomalous dispersion, temperature changes, material composition, and manufacturing variation can all make a two-term fit inadequate. Consult the coefficient source before extrapolating into ultraviolet or infrared regions.
For precision optical work, compare the estimate with measured data or a more complete model such as a multi-term Cauchy or Sellmeier equation. Use this result as the value predicted by the supplied two-term fit, not as an unconditional material constant.
Build a reliable optical workflow
Keep wavelength values in vacuum unless the coefficient source explicitly states another convention. Preserve A, B, wavelength, and the returned index together so another person can reproduce the calculation without searching for hidden defaults. When comparing materials, use coefficients fitted over compatible spectral regions and under comparable conditions.
A good validation workflow calculates one wavelength printed in the original source and confirms that the returned value agrees before processing a larger set. It also rejects missing, non-finite, or physically unsuitable inputs early; this capability requires A above zero, B at least zero, and wavelength above zero.
What you can do with it
Check an optical design wavelength
Estimate the refractive index of a material at a laser or imaging wavelength using published A and B coefficients.
Validate a coefficient table
Recalculate a documented reference point to confirm that coefficients, wavelength units, and stored material metadata agree.
Create reproducible dispersion samples
Evaluate selected wavelengths in an automated workflow and retain each input beside its deterministic refractive-index result.
FAQ
Which Cauchy equation does this calculator use?
It uses the two-term form n(λ) = A + B/λ². It does not include additional Cauchy terms.
What wavelength unit should I enter?
Enter vacuum wavelength in micrometres. The B coefficient must therefore be expressed in square micrometres.
Can I enter a wavelength in nanometres?
Convert it first by dividing by 1,000. A wavelength of 550 nm must be entered as 0.55 µm when B uses square micrometres.
Is the equation accurate at every wavelength?
No. Accuracy depends on the material fit and its stated spectral range. Extrapolation beyond that range may be unreliable.
Why must B be zero or positive?
This capability targets the conventional transparent-region two-term Cauchy model with normal dispersion. Use a model appropriate to the source data if its fit requires different behavior.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/optics/cauchy-dispersion-index \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"a":1.5046,"b":0.0042,"wavelength_um":0.55}'const res = await fetch("https://api.kit.forhosting.com/optics/cauchy-dispersion-index", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"a": 1.5046,
"b": 0.0042,
"wavelength_um": 0.55
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/cauchy-dispersion-index",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"a": 1.5046,
"b": 0.0042,
"wavelength_um": 0.55
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/cauchy-dispersion-index", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"a":1.5046,"b":0.0042,"wavelength_um":0.55}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"a":1.5046,"b":0.0042,"wavelength_um":0.55}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/cauchy-dispersion-index", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"a": 1.5046,
"b": 0.0042,
"wavelength_um": 0.55
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.cauchy_dispersion_index",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Build a reliable optical workflow
A single calculation is helpful for checking a design wavelength, while repeated API calls can evaluate a controlled list of wavelengths for a lens, coating, fiber, or laboratory sample. The browser version is convenient for interactive checks, while automation costs $0.002 per request through the API.
Technical questions
What does an API calculation cost?
Each API request costs $0.002. The same deterministic calculation can also run in the browser.