A cryo-EM image is not a direct picture of the protein — the microscope's lens system multiplies every spatial frequency of the true signal by the contrast transfer function (CTF), which oscillates in sign and rings the image with alternating bright/dark bands (Thon rings) in its power spectrum:
γ(k) = -π·Δf·λ·k² + (π/2)·Cs·λ³·k⁴
CTF(k) = -[√(1-A²)·sin γ(k) + A·cos γ(k)] · exp(-B·k²/4)
k = spatial frequency (1/Å, resolution = 1/k), Δf = defocus, λ = relativistic electron wavelength set by accelerating voltage, Cs = spherical aberration of the objective lens, A ≈ 0.07 is the fixed amplitude-contrast fraction, and the exp(-B·k²/4) envelope damps high frequencies from beam/detector incoherence. This 2D version evaluates CTF(k) independently at every (kx, ky) point of the frequency plane — the concentric rings you see are the actual zero-crossings of that surface, not drawn circles.
- The power-spectrum image shows intensity ∝ CTF(k)², exactly as a real Fourier-transformed micrograph does — dark rings mark exact zero crossings, bright bands mark local extrema.
- The line chart below it is the same CTF(k) plotted as a signed 1D radial profile, so you can read off the sign flips directly.
- Higher defocus packs the rings tighter (more phase contrast at low resolution, but earlier information loss at high resolution).
- Higher voltage shortens λ, which flattens γ(k) and pushes the first zero crossing to higher resolution (smaller Å).
- Turn on ring-fitting mode and click a dark ring in the power spectrum: the engine reads off the spatial frequency k under your click and numerically inverts CTF(k, Δf) = 0 for Δf — the same root-finding step CTFFIND/Gctf perform on a real micrograph — then compares the recovered defocus against the value the slider actually set.