Pupil Matching in Microscopy
Exit pupil D = 2·(250 mm)·NA/(M_obj·M_eye). Between 0.5 and 1 mm the visual magnification is in the useful range 500–1000 × NA.
Results
Exit pupil between 0.5 and 1 mm: the useful range.
Exit pupil vs objective NA
Model
Back aperture 2·f_obj·NA, imaged by f_eye/f_TL: D = 2·f_obj·NA·f_eye/f_TL
With M_obj = f_TL/f_obj and M_eye = 250 mm/f_eye: D = 500 mm · NA / (M_obj · M_eye)
The tube lens focal length cancels, so the marked magnifications and the NA fix the exit pupil. It is usually far smaller than the eye pupil (0.5–1 mm in the useful range against 2–8 mm); at low magnification (2×/0.1 with a 10× eyepiece: 2.5 mm) it can exceed a daylight eye pupil, which then clips it. While the exit pupil is the smaller, retinal image brightness scales as (NA/M)², its square. An exit pupil above about 1 mm wastes resolution because the eye, not the objective, limits it.
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
First-order optics: the objective's back aperture 2·f_obj·NA (Abbe sine condition) imaged to the eyepoint by f_eye/f_TL, with eyepiece power 250 mm/f_eye, so the tube lens cancels.
- Greivenkamp J. E. (2004). Field Guide to Geometrical Optics. SPIE Press. Microscopes: objective magnification, eyepiece power 250 mm/f, pupils. doi:10.1117/3.547461
- Nikon MicroscopyU. Useful Magnification Range. Useful total magnification 500 to 1000 × NA. microscopyu.com