Afocal System Calculator
Two-lens afocal (telescope and beam expander) systems: afocal spacing, angular magnification, beam size and the ray-transfer matrix, Keplerian or Galilean, and the focal distances when the spacing is off.
Results
System power vs lens spacing
Model
[A B; C D] = L(f₂)·T(t)·L(f₁): A = 1 − t/f₁, B = t, C = −1/f₁ − 1/f₂ + t/(f₁f₂), D = 1 − t/f₂
Power φ = −C = φ₁ + φ₂ − tφ₁φ₂; afocal when C = 0, t = f₁ + f₂
Afocal: M = D = −f₁/f₂, beam size × A = −f₂/f₁
Paraxial thin lenses in air. Keplerian telescopes (both lenses positive) invert the image and have a real intermediate focus where a field stop or spatial filter can sit; Galilean ones (one negative lens) are upright and shorter. As a beam expander, either scales a collimated beam's diameter by |f₂/f₁| and divides its divergence by the same ratio. Away from the afocal spacing the system has power, and the angle a ray leaves at depends on its height, so it has no single angular magnification.
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Paraxial thin lenses in air, from the two-lens ray-transfer matrix. Aberrations, lens thickness and apertures (vignetting) are not included.
- Greivenkamp J. E. (2004). Field Guide to Geometrical Optics. SPIE Press. Two-lens systems, afocal systems and ray-transfer matrices. doi:10.1117/3.547461