Chromatic Aberration Simulator — Axial CA & Abbe Number
Visualize the axial chromatic aberration of a singlet lens with the lensmaker's equation and Abbe number. Adjust n_d, V_d, R_1 and R_2 to learn why focal length varies with wavelength.
Parameters
d-line index n_d
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Abbe number V_d
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Radius R_1
mm
Radius R_2
mm
R uses the optical-axis sign convention (left to right). Convex surfaces positive, concave negative. Defaults approximate a BK7 singlet.
Results
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d-line focal length f_d
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Abbe number V_d
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Axial CA Δf = f_C − f_F
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Relative dispersion Δf/f_d
Singlet lens and wavelength-dependent foci
Blue = F-line (486 nm), green = d-line (588 nm), red = C-line (656 nm). The arrow shows axial CA Δf.
Focal length f(λ) vs wavelength λ
Horizontal axis: wavelength (400-700 nm). Vertical axis: focal length f(λ) (mm). C-, d- and F-line positions are marked.
Theory & Key Formulas
The refractive index n of a lens material depends on wavelength λ. This makes the focal length vary with wavelength and produces axial chromatic aberration.
Lensmaker's equation for a thin lens at wavelength λ. n(λ) is the refractive index and R_1, R_2 are the two surface radii:
Abbe number V_d measures how little a glass disperses light. n_d, n_F and n_C are the indices at the d-, F- and C-lines:
$$V_d = \frac{n_d - 1}{n_F - n_C}$$
Approximation for the axial chromatic aberration Δf (C-line focus minus F-line focus) of a singlet:
$$\Delta f = f_C - f_F \approx \frac{f_d}{V_d}$$
A larger Δf means the focus of white light is more spread out by color, producing colored fringes in the image. A larger V_d gives smaller chromatic aberration.
What is the chromatic aberration simulator?
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When I look at stars through an old telescope I sometimes see colored fringes around them. What's that?
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Those fringes are chromatic aberration. Roughly speaking, the refractive index of glass is slightly different for each color (wavelength). Plug that into the lensmaker's equation $1/f = (n-1)(1/R_1 - 1/R_2)$ and the focal length comes out different for blue and red light. So they focus at different points and a colored halo appears around the star. Look at the "singlet lens" canvas above — the blue (F), green (d) and red (C) foci sit at clearly different positions.
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How much do they shift? Enough to see?
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With the defaults (a BK7 singlet, f_d ≈ 129 mm) Δf is about 2.0 mm — roughly 1.56% of the focal length. In photography or microscopy that becomes an obvious smear of color. For a singlet you can estimate it with the super simple formula $\Delta f \approx f_d / V_d$. The larger the Abbe number V_d, the smaller the chromatic aberration.
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So just crank up V_d, right? Except real glasses have a limit, don't they?
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Nice catch. For ordinary optical glasses V_d is realistically about 20 to 95. BK7 (crown) sits at 64, SF11 (flint) at 25, and fluorite (CaF₂) or ED glasses can reach about 95. The catch is that pushing the refractive index up for higher-performance lenses usually pushes V_d down — that's the basic glass-selection trade-off.
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But modern camera lenses hardly show any color fringing. How do they fix it?
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That's where the "achromat" comes in: cement a low-V_d flint concave element to a high-V_d crown convex element so two wavelengths (C and F) share a focus. Push it further to three or more wavelengths and you get an "apochromat", which also kills the secondary spectrum. Telescopes and high-end camera lenses are almost always apochromatic. Once you slide R_1 and R_2 around and feel how badly a singlet behaves, you understand why every real lens has multiple elements.
Frequently Asked Questions
They are Fraunhofer absorption lines in sunlight: hydrogen C at 656.3 nm (red), helium d at 587.6 nm (yellow-green) and hydrogen F at 486.1 nm (blue-green). They span the high-sensitivity region of the visible spectrum and appear as sharp, reproducible absorption lines, so they have been the standard reference for glass dispersion since the 19th century. The e-line (546.1 nm) based V_e is also used today, but this simulator uses the traditional V_d system.
An achromat is a color-corrected design that brings two wavelengths (C and F) to the same focus, leaving a small d-line residual called the secondary spectrum. An apochromat uses special low-dispersion glasses such as fluorite, FK glass or ED glass to bring three or more wavelengths to a common focus and essentially eliminate the secondary spectrum. It is essential for high-magnification microscopes, astronomical telescopes and premium camera lenses, and typically costs several times as much as an achromat.
Axial chromatic aberration is a shift of the focal position along the optical axis — the Δf = f_C − f_F that this simulator computes. It appears as uniform colored blur across the whole image. Lateral (transverse) chromatic aberration is a wavelength-dependent change in magnification, which shows up as colored edges near the corners of the image. Stopping down generally does not remove axial CA, and lateral CA does not appear at the image center. The two are independent aberrations and are corrected separately in optical design.
A single spherical singlet cannot, in principle. Δf ≈ f_d / V_d is always positive and vanishes only in the unphysical limit V_d → ∞. Combining a singlet with a diffractive optical element (DOE) gives some correction, but in practice virtually all designs use two or more elements. Aspherization helps with spherical aberration but not with chromatic aberration, because chromatic aberration is a material property, not a shape problem.
Real-world applications
Camera and telescope lens design: Every high-quality optical system is designed from the start to correct chromatic aberration. A typical camera lens starts from an achromatic core and adds one or two elements of special low-dispersion glass (ED, FLD, UD, etc.) to reach apochromatic-class performance. Telephoto lenses show CA most strongly, so super-telephotos have traditionally used fluorite elements.
Microscope objectives: Microscope objectives are graded by how well they correct chromatic aberration — "achromat", "semi-apochromat (fluorite)" and "apochromat" form a clear performance and price ladder. For high-magnification fluorescence work or quantitative imaging, an apochromat that brings three or more wavelengths to a common focus is essential.
Spectrometers and monochromators: Spectrometers do the opposite — they deliberately exploit chromatic aberration by spatially separating wavelengths with a prism or grating. The detector-side optics are then designed to give the best image quality in a specific wavelength band, or reflective optics (which have zero CA) are used when broadband performance matters.
Laser optics: A pure single-wavelength laser system has, in principle, zero chromatic aberration. But mixed systems that share an axis between a working laser and other wavelengths (a visible aiming beam, a metrology source) need CA correction, and confocal Z-scan systems are also sensitive to wavelength-dependent focal shifts.
Common misconceptions and pitfalls
The most common misconception is that "stopping down removes chromatic aberration". Closing the aperture deepens the depth of field and softens the appearance of axial CA, but the aberration itself does not disappear. Lateral CA in particular still survives near the image corners. The real fix is material selection and adding lens elements; you cannot solve it by shooting technique alone.
Another common error is to think that higher-index glasses are always more chromatic. What controls chromatic aberration is the wavelength variation of the refractive index — the Abbe number V_d — not the absolute index. High-index glasses with high V_d do exist (for example lanthanum-based LAK glasses), but on average higher index trends toward lower V_d (the well-known downward slope of the Abbe diagram). This simulator lets you sweep n_d and V_d independently because they are, in principle, independent physical quantities.
Finally, remember that the Δf shown here is a singlet approximation. $\Delta f \approx f_d / V_d$ is a first-order thin-lens estimate; real lenses also exhibit spherical, coma, astigmatism, field curvature, distortion and lateral chromatic aberration. In a multi-element system the CA contributions of individual elements add or cancel depending on the design, which this tool does not model. Real-world lens design requires rigorous ray tracing in dedicated software such as Zemax or CODE V.