Zone Plate Calculator: Focal Length & Design

A zone plate calculator works out the focal length of a Fresnel zone plate from its diameter, the number of zones, and the wavelength of light (or other radiation) you plan to use. You enter those three values, and the calculator returns the focal length in your chosen unit, along with the width of the outermost zone, which matters for fabrication. This page explains the math behind it, how to choose the right parameters, and common pitfalls when building or simulating zone plates.

Formula

The focal length \(f\) of a zone plate is given by the relation between the radius of the outermost zone, the number of zones, and the wavelength. The standard formula is:

\(f = \frac{r_n^2}{n \lambda} – \frac{\lambda}{4}\)

where:

• \(r_n\) = radius of the outermost zone (mm) – half the plate diameter
• \(n\) = total number of zones (integer)
• \(\lambda\) = wavelength of the radiation (mm)
• \(f\) = focal length (mm)

The term \(-\lambda/4\) is a small correction that comes from the exact path-length difference for a spherical wave. For most practical purposes (especially in visible light), it is negligible, but it matters for X-ray or extreme UV where wavelengths are not tiny compared to zone widths.

The outermost zone width \(\Delta r_n\) is simply the difference between the radii of the last two zones:

\(\Delta r_n = r_n – r_{n-1} = \frac{r_n}{n} – \frac{r_n}{n+1}\)

This value is critical because it sets the minimum feature size you need to fabricate.

Worked Example

Using the default values from the calculator: diameter = 50 mm, zones = 100, wavelength = 500 nm (green light).

1. Radius \(r_n = 50/2 = 25\) mm
2. Wavelength in mm: \(500 \text{ nm} = 500 \times 10^{-6} \text{ mm} = 0.0005 \text{ mm}\)
3. Apply the formula: \(f = (25^2) / (100 \times 0.0005) – 0.0005/4\)
4. \(25^2 = 625\)
5. \(100 \times 0.0005 = 0.05\)
6. \(625 / 0.05 = 12500\) mm
7. Subtract the correction: \(12500 – 0.000125 \approx 12500.00\) mm

So the focal length is about 12,500 mm, or 12.5 meters. That is a very long focal length, which makes sense: a 50 mm plate with only 100 zones is a low-power lens. The outermost zone width is \(25/100 – 25/101 = 0.25 – 0.2475 = 0.0025\) mm, or 2.5 micrometers – easily printable with a photolithography setup.

How a Zone Plate Works

A zone plate is a diffractive optical element that focuses light by alternating transparent and opaque rings (zones). The radii of the zones are chosen so that light passing through each transparent ring arrives at the focal point in phase, constructively interfering. The path difference between adjacent zones is exactly one wavelength.

Unlike a refractive lens, which bends light continuously, a zone plate works by diffraction. This means it is wavelength-dependent: a zone plate designed for 500 nm will focus red light at a different distance, and blue light at yet another. This chromatic aberration is much stronger than in a normal glass lens, but it also makes zone plates useful for monochromatic sources like lasers or synchrotron X-rays.

Zone plates are the standard focusing optics for X-ray microscopy, because no material can refract X-rays efficiently. They are also used in UV lithography, and as cheap, lightweight lenses for radio astronomy or even solar observation (using a simple zone plate made of wire mesh).

Choosing Parameters

When you design a zone plate, you typically start with the required focal length and wavelength, then decide on the diameter and number of zones. The key trade-off is resolution vs. difficulty of fabrication.

• Resolution: The smallest spot a zone plate can focus is roughly equal to the width of the outermost zone. So if you need a 100 nm spot (for X-ray microscopy), you need the outermost zone to be about 100 nm wide.
• Focal length: For a given outermost zone width, the focal length is proportional to the square of the diameter. Doubling the diameter quadruples the focal length (if the number of zones stays the same).
• Number of zones: More zones means a larger numerical aperture (and thus better resolution), but also a much smaller outermost zone width. The number of zones also affects efficiency – more zones concentrate more light into the focal spot.
• Wavelength: Shorter wavelengths (like X-rays) allow finer resolution, but also require much smaller zone widths for the same focal length.

A practical approach is to set the resolution you need (outermost zone width), then choose a diameter that gives a manageable focal length. For visible light, a 1 mm diameter plate with 100 zones gives a focal length of about 5 meters – not very practical. You would need many more zones or a smaller diameter to get a short focal length.

Common Mistakes

The most frequent error is mixing up units. Wavelength is usually given in nanometers (nm) or micrometers (µm), while diameter is in millimeters or centimeters. Always convert everything to the same unit before calculating. Our calculator does that internally, but if you do it by hand, be careful.

Another mistake is using the diameter instead of the radius in the formula. The formula uses the radius of the outermost zone, which is half the diameter. Forgetting to divide by 2 will give a focal length four times too large.

People also confuse the number of zones with the number of rings. A zone plate with 100 zones has 100 transparent rings (and 100 opaque ones, if you count the central zone as zone 1). The total number of rings is 2n, but the formula uses n, the total number of zones (transparent + opaque).

Finally, the \(-\lambda/4\) correction: for visible light it is tiny (less than a micrometer), but for X-rays with a wavelength of 1 nm and a focal length of 1 mm, it is 0.25 nm – still negligible. Only for very short focal lengths and long wavelengths does it matter, but it is good practice to include it.

Fabrication Limits

The outermost zone width is the limiting factor for manufacturing. Standard photolithography can easily produce features down to about 1 µm. Electron-beam lithography can go down to 10 nm or less, but it is slow and expensive. For X-ray zone plates, the outermost zone width is typically 30–100 nm, which requires e-beam writing.

If you are making a zone plate for visible light, you can even print one on a transparency film using a high-resolution printer (1200 dpi or more). A 1200 dpi printer has a dot size of about 21 µm, so your outermost zone width must be larger than that – say 50 µm or more. That limits you to a low-resolution zone plate, but it can still work for simple demonstrations.

For radio waves (wavelength in centimeters), you can make a zone plate out of wire mesh or even cardboard with conductive tape. The zone widths are large enough to cut with scissors.

Alternatives to Zone Plates

If you need focusing but cannot use a zone plate, consider these alternatives:

• Refractive lenses: For visible light, glass lenses are simpler and more efficient. Zone plates are only chosen when a refractive lens is impossible (e.g., for X-rays) or when weight is critical.
• Reflective mirrors: For X-rays, grazing-incidence mirrors (like Kirkpatrick-Baez pairs) can focus without the chromatic aberration of zone plates, but they are harder to align.
• Fresnel lenses: These are refractive, not diffractive, but they are also thin and lightweight. They work only for visible/infrared, not X-rays.
• Diffractive gratings: A grating disperses light, but a curved grating can focus as well as disperse – similar to a zone plate but with a different geometry.

For most hobbyist projects, a simple converging lens is the easiest solution. Zone plates are mainly for specialized applications where no other optic works.

Frequently Asked Questions

What is the difference between a zone plate and a Fresnel lens?

A Fresnel lens is a refractive optic – it bends light through a series of concentric prism-like grooves. A zone plate is diffractive – it uses alternating transparent and opaque rings to create constructive interference. Fresnel lenses work for broad-spectrum light (though with some chromatic aberration), while zone plates are highly wavelength-specific.

Can I use a zone plate with white light?

Yes, but the image will be very blurry because different wavelengths focus at different distances. You will see a series of colored spots along the optical axis. For white light, a zone plate acts like a very strong chromatic lens, so it is only practical with a monochromatic source or a narrowband filter.

Why is the outermost zone width important?

The outermost zone width determines the numerical aperture and thus the resolution of the zone plate. The smallest resolvable feature is approximately equal to this width. It also dictates the fabrication method – if the width is below 1 µm, you need e-beam lithography.

How do I choose the number of zones?

More zones give a sharper focus and higher efficiency, but also a smaller outermost zone width. Start with the resolution you need (outermost zone width), then choose the number of zones that gives a reasonable diameter. For a given diameter, the number of zones is limited by the minimum feature size you can fabricate.

What is the efficiency of a zone plate?

A simple binary zone plate (opaque/transparent) has a maximum diffraction efficiency of about 10% in the first order. That means only 10% of the incident light goes into the focal spot. Phase zone plates (which shift phase instead of blocking) can reach up to 40% efficiency. This is much lower than a refractive lens, which is why zone plates are only used when nothing else works.

Can I make a zone plate at home?

Yes, for visible light you can print one on a transparency using a high-resolution printer. Use the calculator to get the zone radii, then generate a black-and-white image with those radii. The main limitation is printer resolution – the outermost zone width should be at least 3–4 times the printer dot size. For radio waves, you can cut zones from cardboard and cover them with aluminum foil.

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in d | Plate diameter | mm | 50 | 0 | 1000
in n | Number of zones | - | 100 | 1 | 10000
in wl | Wavelength | nm | 500 | 1 | 1000000
out f | Focal length | mm | (d/2)^2 / (n * wl / 1e6) - wl / 1e6 / 4 | 2
out dr | Outermost zone width | mm | (d/2) / n - (d/2) / (n + 1) | 4