06. Modulation Transfer Function
Description
This article is from the Photographic Lenses Tutorial, by David Jacobson with contributions by others.
06. Modulation Transfer Function
The modulation transfer function is a measure of the extent to which a lens, film, etc., can reproduce detail in an image. It is the spatial analog of frequency response in an electrical system. The exact definitions of the modulation transfer function and of the related optical transfer function vary slightly amongst different authorities.
The 2-dimensional Fourier transform of the point spread function is known as the optical transfer function (OTF). The value of this function along any radius is the fourier transform of the line spread function in the same direction. The modulation transfer function is the absolute value of the fourier transform of the line spread function.
Equivalently, the modulation transfer function of a lens is the ratio of relative image contrast divided by relative object contrast of an object with sinusoidally varying brightness as a function of spatial frequency (e.g. cycles per mm). Relative contrast is defined as (Imax-Imin)/(Imax+Imin). MTF can also be used for film, but since film has a non-linear characteristic curve, the density is first transformed back to the equivalent intensity by applying the inverse of the characteristic curve.
For a lens, the MTF can vary with almost every conceivable parameter, including f-stop, object distance, distance of the point from the center, direction of modulation, and spectral distribution of the light. The two standard directions are radial (also known as sagittal) and tangential.
The MTF for an an ideal lens (ignoring the unavoidable effect of diffraction) is a constant 1 for spatial frequencies from 0 to infinity at every point and direction. For a practical lens it starts out near 1, and falls off with increasing spatial frequency, with the falloff being worse at the edges than at the center. Adjacency effects in film can make the MTF of film be greater than 1 in certain frequency ranges.
An advantage of the MTF as a measure of performance is that under some circumstances the MTF of the system is the product (frequency by frequency) of the properly scaled MTFs of its components. Such multiplication is always allowed when the phase of the waves is lost at each step. Thus it is legitimate to multiply lens and film MTFs or the MTFs of a two lens system with a diffuser in the middle. However, the MTFs of cascaded ordinary lenses can legitimately be multiplied only when a set of quite restrictive and technical conditions is satisfied.
As an example of some OTF/MTF functions, below are formulas for and plots of the OTFs three cases:
1. Diffraction for an f/22 aperture.
2. A .03mm circle of confusion
3. The combination of these; more precisely, the OTF of an otherwise ideal lens with an f/22 aperture and defocused to produce a .03mm circle of confusion.
Let lambda be the wavelength of the light, and spf the spatial frequency in cycles per mm.
For diffraction the formula is
OTF(lambda,N,spf) = 2/Pi (ArcCos(lambda N spf) -
lambda N spf Sqrt(1-(lambda N spf)^2)) if lambda N spf <=1
= 0 if lambda N spf >=1
Note that for lambda = 555 nm, the OTF is zero at spatial frequencies of 1801/N cycles per mm and beyond.
For a circle of confusion of diameter C,
OTF(C,spf) = 2 J1(Pi C spf)/(Pi C spf)
where, again, J1(x) is the first order Bessel function. The OTF goes negative at certain frequencies. Physically, this would mean that if the test pattern were lighter right on the optical center than nearby, the image would be darker right on the optical center than nearby. Some authorities use the term "spurious resolution" for spatial frequencies beyond the first zero. The MTF is the absolute value of the OTF.
Consider the case where there is a combination of diffraction and focus error dz, the distance the between the film plane and the plane of sharpest focusing. (A focus error of dz by itself would cause a circle of confusion of diameter dz/N.) The OTF for this combination is given by the following formula, which involves an integration that must be done numerically. Let s = lambda N spf, and a = Pi spf dz / N. Then the OTF is given by
OTF = 4/(Pi a) integral y=0 to sqrt(1-s^2) of sin(a(sqrt(1-y^2)-s)) dy
for s < 1
0 for s >= 1
This formula is an approximation that is best at small apertures.
Here is a graph of the OTF of the f/22 diffraction limit, a .03mm circle of confusion, and the combined effect.
OTF
*
1 *****
# +$$*
# +$*
# + *$ $$$$ Diffraction
0.8 # + **$ **** Circle of confusion
# ++ *$$ ++++ Combined diffraction and circle of confusion
# + * $$
# + * $
0.6 # ++* $$
# +* $$
# * $$
# * $$
0.4 # * $$
# *++++ $$
# * +++++ $$$$
# * +++++$$$$
0.2 # * ++++$$
# * +$$$
# * $$$$*****************
#######################**##################**$$$$$$$$$$$$$$$$$******$$$
0 # ** ***** ***
# 20 40 ***** ***** 80 100 120
# **
Spatial Frequency (cycles/mm)
Note how the combination is not the product of each of the effects taken separately.
Some authorities present MTF in a log-log plot.
The classic paper on the MTF for the combination of diffraction and focus error is H.H. Hopkins, "The frequency response of a defocused optical system," Proceedings of the Royal Society A, v. 231, London (1955), pp 91-103. Reprinted in Lionel Baker (ed), _Optical Transfer Function: Foundation and Theory_, SPIE Optical Engineering Press, 1992, pp 143-153.
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