Glossary entry

Modulation transfer function (MTF)

How much of an object's contrast an optical system passes to the image, as a function of spatial frequency in line pairs per millimetre. For a diffraction-limited circular aperture it falls from 1 to zero at the cutoff 1/(λN): 227 lp/mm at 550 nm and f/8.

Optics & beams Optics fundamentalsUpdated September 2026

An imaging system blurs every point of the object into its point spread function, so a pattern of fine lines comes through with reduced contrast. The modulation transfer function describes that reduction frequency by frequency. A sinusoidal pattern of spatial frequency $\nu$ has a contrast, or modulation,

M = \frac{I_{max} - I_{min}}{I_{max} + I_{min}},

and is imaged as a sinusoid of the same frequency with lower modulation; the ratio of image to object modulation is $\mathrm{MTF}(\nu) = M_i/M_o$ . Formally it is the magnitude of the optical transfer function, the Fourier transform of the point spread function normalized to one at zero frequency, and it describes incoherent imaging; coherent illumination has a different transfer function.

Diffraction limit

For an aberration-free system with a circular pupil and incoherent light, the MTF is

\mathrm{MTF}(\nu) = \frac{2}{\pi}\left[\phi - \cos\phi\,\sin\phi\right],

with

\phi = \arccos\left(\frac{\nu}{\nu_c}\right),

where the cutoff frequency $\nu_c = 1/(\lambda N) = 2\,\mathrm{NA}/\lambda$ depends only on the wavelength and the f-number. Above the cutoff no contrast is transmitted at all. The curve falls to 0.87 at a tenth of the cutoff, 0.69 at a quarter, 0.39 at half and 0.14 at three quarters. At 550 nm and f/8 the cutoff is 227 lp/mm, and a camera sensor with 5 µm pixels, whose Nyquist frequency is 100 lp/mm, samples that lens where its diffraction-limited MTF is 0.46.

Use

The MTF is the standard specification for imaging lenses, usually plotted up to some frequency of interest for several field positions, and separately for sagittal and tangential lines because off-axis aberrations differ between them. It is more informative than a single resolution limit such as the Rayleigh criterion, because it shows how contrast degrades before detail disappears. Aberrations lower the curve below the diffraction limit at mid frequencies; the Strehl ratio summarizes the same loss as a single number. The MTFs of cascaded parts, the lens, the sensor's pixel aperture and any motion blur, multiply to give the system MTF.

Measurement

The slanted-edge method of ISO 12233 images a sharp edge tilted a few degrees from the pixel grid, builds an oversampled edge profile, differentiates it to a line spread function and Fourier-transforms that to the MTF; it needs only one target and is standard for cameras. Bar or sinusoidal targets at known frequencies give points on the curve directly, with the caveat that square-wave bars measure the contrast transfer function, which differs from the MTF. Lens test benches image a pinhole or slit and compute the MTF from the measured spread function; interferometers compute it from the measured wavefront.

References: J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017), Ch. 7; ISO 12233:2017, Photography: Electronic still picture imaging: Resolution and spatial frequency responses; G. D. Boreman, Modulation Transfer Function in Optical and Electro-Optical Systems (SPIE Press, 2001).