Modulation Transfer Function

Diffraction-limited incoherent MTF and cutoff 2NA/λ, and the defocused OTF and Strehl ratio with contrast reversal, paraxial (Hopkins) and with the exact high-NA defocus phase, for a circular pupil.

Model: Textbook approximation

Numerical aperture NAWavelength (nm)Refractive index on the NA side, nDefocus Δz on the NA side (µm)Normalised frequency ν = f/f_c

Results

Cutoff f_c = 2NA/λ1.4545 lp/µm

Cutoff1455 lp/mm

Chosen frequency f = ν·f_c727 lp/mm

In-focus MTF(ν)0.39100

Paraxial W₀₂₀ = Δz·NA²/(2n)0.4364 λ

Defocused OTF(ν), paraxial (Hopkins)0.1343

Defocused OTF(ν), exact phase0.1202

Strehl ratio, paraxial sinc²(kW₀₂₀/2)0.5111

Strehl ratio, exact phase0.4794

Scalar, uniform pupil amplitude in ρ = sin θ/(NA/n): an aplanatic lens weights it by (1 − (NAρ/n)²)^(−1/4), which lowers the defocused OTF further at high NA (by a third at NA 0.95, Δz 0.5 µm); no polarization. The exact phase is 2π(nΔz/λ)√(1 − (NAρ/n)²) (Debye integral); its path difference at the pupil edge, n·Δz(1 − cos θ), is 0.455λ here against the paraxial W₀₂₀ = 0.436λ.

OTF vs spatial frequency

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

Scalar incoherent imaging with a uniform circular pupil (no aplanatic apodization, no polarization). The paraxial OTF uses W₀₂₀ = Δz·NA²/(2n) in Hopkins' pupil-overlap integral. The exact-phase OTF uses the Debye defocus phase 2π(nΔz/λ)√(1 − (NAρ/n)²), integrated over the pupil overlap in 2D; its Strehl ratio is in closed form. At NA 0.95, Δz 0.5 µm and 550 nm the Strehl ratio is 0.27 against a paraxial 0.56. An aplanatic lens would weight the pupil by (1 − (NAρ/n)²)^(−1/4); leaving it out over-states the high-NA defocused OTF (0.043 against 0.028 at ν = 0.5 in that example).

Goodman J. W. (2005). Introduction to Fourier Optics, 3rd ed., §6.3. Roberts & Company. Incoherent cutoff 2NA/λ and the diffraction-limited MTF of a circular pupil.
Hopkins H. H. (1955). The frequency response of a defocused optical system. Proc. R. Soc. Lond. A 231, 91–103. OTF of a defocused circular pupil. doi:10.1098/rspa.1955.0158
Richards B., Wolf E. (1959). Electromagnetic diffraction in optical systems II. Structure of the image field in an aplanatic system. Proc. R. Soc. Lond. A 253, 358–379. Debye integral with the defocus phase k n z cos θ. doi:10.1098/rspa.1959.0200
Sheppard C. J. R. (1988). Aberrations in high aperture conventional and confocal imaging systems. Appl. Opt. 27, 4782–4786. High-aperture defocus as a non-quadratic pupil phase. doi:10.1364/AO.27.004782

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