ABCD Matrix Calculator
Build an optical system from sequential elements and compute the ray transfer matrix.
System ABCD matrix (B in mm, C in 1/mm)
det M = 1.000000 ✓ (expected nin/nout = 1.000000)
Output ray
Height: -2.0000 mm
Angle: -10.0000 mrad
Rear focal length f′ = −1/C: 100.000 mm
Imaging condition (B = 0): magnification A = -2.0000
Real ray angles: a flat interface multiplies the angle by n₁/n₂, so det M = nin/nout. Mirrors are unfolded along the reflected axis; a concave mirror (R > 0) acts as a lens of focal length R/2.
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 ray matrices with real ray angles, so a flat interface gives det = n₁/n₂. Thin lenses, unfolded mirrors and no aberrations; valid for small angles and heights.
- Kogelnik H., Li T. (1966). Laser beams and resonators. Appl. Opt. 5, 1550–1567. doi:10.1364/ao.5.001550
- Gerrard A., Burch J. M. (1975). Introduction to Matrix Methods in Optics. Wiley.