Fresnel number calculator
Check which diffraction model fits your setup: a 0.001 m aperture radius, 1 m of distance and 6.328e-7 m light give a Fresnel number of 1.58027812895, labeled as the transition zone. Free in your browser.
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Worked example
- Radius
- 0.001
- Distance
- 1
- Wavelength
- 0.0000006328
- Fresnel number
- 1.580278129
- Regime
- transition
The Fresnel number calculator evaluates the dimensionless ratio that connects aperture size, propagation distance, and wavelength in diffraction problems. Enter the radius of a circular aperture, the distance from that aperture to the observation plane, and the wavelength of the light, all in one consistent length unit. The calculator applies N_F = a²/(λz), reports the value with stable rounding, and gives a practical near-field, transition, or far-field label. It is useful for optical experiments, beam propagation checks, imaging layouts, and coursework where you need a quick, reproducible indication of which diffraction approximation is likely to matter.
What the Fresnel number describes
The Fresnel number, commonly written N_F, is a dimensionless measure of how strongly the finite size of an aperture influences a diffraction pattern at a chosen observation distance. A large aperture or a short propagation distance increases the number, while a longer wavelength or a more distant observation plane decreases it. Because every length uses the same unit, the squared length in the numerator cancels the product of two lengths in the denominator.
The result therefore has no unit. Fresnel number is often used as a compact guide to diffraction regime. Values much greater than one usually indicate pronounced near-field, or Fresnel, behavior with pattern structure that depends on propagation distance.
Values much less than one tend toward the far-field, or Fraunhofer, situation. Values around unity occupy a transition where a simple label should not replace a fuller propagation model.
How to enter radius, distance, and wavelength
Provide radius, distance, and wavelength as positive finite numbers. Radius means the distance from the center to the edge of the diffracting aperture, not its diameter. If a specification gives diameter, divide it by two before entering the value; forgetting this step makes the computed Fresnel number four times too large because radius is squared.
Distance is the propagation distance z from the aperture plane to the observation plane. Wavelength is λ for the light in the relevant medium; if you are working outside vacuum, use the wavelength appropriate to that medium when the model requires it. The three values must share one length unit.
Metres are convenient for SI calculations, but millimetres, micrometres, or nanometres also work when every field is converted to that same unit. For example, do not combine a radius entered in millimetres with a wavelength entered in nanometres unless you first convert one of them.
Interpreting the result responsibly
Use the returned number as a scaling parameter and an initial model-selection check. If N_F is far below one, the observation plane may be sufficiently far away for a Fraunhofer approximation, especially when other far-field assumptions are also satisfied. If N_F is far above one, propagation is generally in the Fresnel region and a near-field diffraction treatment is more appropriate.
Around one, neither shorthand should be treated as decisive, so numerical propagation or an exact solution may be warranted. The regime field deliberately uses a conservative transition band from 0.1 through 10 to discourage false precision. It does not certify an experimental layout, account for aberrations, describe aperture transmission, calculate intensity, or replace a wave-optics simulation.
The formula also assumes the supplied radius is the characteristic radius relevant to the aperture geometry; noncircular apertures may use a different characteristic dimension or separate Fresnel numbers by axis. For repeatable automation, store the unit convention beside your inputs even though the output is dimensionless, and compare the raw Fresnel number rather than only its label.
What you can do with it
Choose a diffraction approximation
Estimate whether a circular-aperture setup is clearly near field, clearly far field, or in a transition region before selecting a propagation model.
Check an optical bench layout
Compare aperture sizes and screen distances at a fixed wavelength while documenting the dimensionless Fresnel number for each arrangement.
Validate coursework and simulation inputs
Generate a reproducible N_F value and substitution steps to check hand calculations or establish parameters for a numerical diffraction simulation.
FAQ
What is the formula for Fresnel number?
For this circular-aperture calculation, N_F = a²/(λz), where a is aperture radius, λ is wavelength, and z is observation distance.
Is Fresnel number dimensionless?
Yes. Radius squared and wavelength times distance both have dimensions of length squared, so their units cancel when one consistent length unit is used.
Should I enter aperture radius or diameter?
Enter radius. If you have diameter, divide it by two first. Entering diameter directly would make the result four times too large.
Which units can I use?
Any length unit works if radius, distance, and wavelength all use that same unit. Converting all three to metres is often the safest convention.
How should I interpret a Fresnel number near one?
Treat it as a transition case. The calculator's regime label is a guide, and a full diffraction model may be needed for the accuracy your application requires.
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/fresnel-number \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":0.001,"distance":1,"wavelength":6.328e-7}'const res = await fetch("https://api.kit.forhosting.com/optics/fresnel-number", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 0.001,
"distance": 1,
"wavelength": 6.328e-7
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/optics/fresnel-number",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 0.001,
"distance": 1,
"wavelength": 6.328e-7
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/optics/fresnel-number", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":0.001,"distance":1,"wavelength":6.328e-7}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":0.001,"distance":1,"wavelength":6.328e-7}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/optics/fresnel-number", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"radius": 0.001,
"distance": 1,
"wavelength": 6.328e-7
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "optics.fresnel_number",
"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
What the Fresnel number describes
For a circular aperture in the usual on-axis estimate, it is the aperture radius squared divided by wavelength times propagation distance: N_F = a²/(λz). This calculator labels values below 0.1 as far_field, above 10 as near_field, and the broad interval between them as transition. Those thresholds are practical guides, not universal physical boundaries; geometry, illumination, required accuracy, and the observable being measured can all affect which approximation is acceptable.
How to enter radius, distance, and wavelength
The API also accepts concise aliases: a or aperture_radius for radius, z or observation_distance for distance, and lambda or wl for wavelength. Success returns fresnel_number, a dimensionless flag, the regime guide, the formula, normalized input values, and substitution steps. Numeric strings in ordinary decimal or scientific notation are accepted, while zero, negative, missing, non-finite, or out-of-range values return invalid_input rather than a misleading result.
Interpreting the result responsibly
The same pure solve function runs in the browser and through the API, with no network calls, random values, or clock dependence.