Depth of Field
Microscope depth of field from Inoué's formula, λ₀n/NA² plus the detector term n·e/(M·NA), with the non-paraxial wave term for high NA.
Results
Depth of field vs NA
Model
d = λ₀n/NA² + n·e/(M·NA) (Inoué & Spring)
Non-paraxial wave term: λ₀/(2(n − √(n² − NA²)))
The wave term is the full depth over which defocus adds at most a quarter wave of path at the aperture edge (paraxial), half the distance from focus to the first axial zero, 2nλ₀/NA². The detector term is the defocus whose geometric blur spans the detector's resolution e referred to the object, e/M. Take e as the pixel pitch, or twice it for Nyquist sampling. n is the index between the cover glass and the front lens, so d is the range of focus travel; inside a specimen of index n_s the depth is about d·n_s/n (paraxial). At high NA the paraxial wave term overstates the depth: at NA 1.4 in oil it reads 0.43 µm against 0.29 µm without the approximation. The detector term n·e/(M·NA) is also paraxial; the exact geometry e·√(n² − NA²)/(M·NA) is smaller at high NA (0.027 vs 0.070 µm at NA 1.4 oil, 100×, 6.5 µm).
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
Inoué's paraxial sum of a quarter-wave defocus term and a geometric detector term, n the immersion index. At high NA both terms overstate the depth (the detector term exactly is e·√(n² − NA²)/(M·NA)); the page also shows the wave term without the paraxial approximation.
- Inoué S., Spring K. R. (1997). Microscope Image Formation. In: Video Microscopy, 2nd ed., ch. 2, pp. 13–117. Plenum. Total depth of field d = λ₀n/NA² + n·e/(M·NA). doi:10.1007/978-1-4615-5859-0_2
- Nikon MicroscopyU. Depth of Field and Depth of Focus. The same formula, with e the smallest distance a detector in the image plane resolves. microscopyu.com