While aberration plots give the optical designer a good indication of the quality of a lens, it is widely accepted that a measurable, objective performance criterion is highly desirable, especially for users or evaluators of the lens. MTF is the most widely used criterion, as it applies to common imaging situations with incoherent illumination and where image contrast, or subjective “sharpness,” is important. In fact, many modern lenses are specified and measured directly in terms of their MTF performance.
As shown in Fig. 4.23, we consider an optical system that is imaging a grating with a sinusoidal variation of intensity. It can be shown that with incoherent illumination, the image will also have a sinusoidal variation of intensity, but with reduced contrast.
As stated below, the MTF is the ratio of the image contrast to the object contrast; it is of course a function of the spatial frequency, and the use of plots of MTF against frequency (usually in cycles/mm) is very common in lens design.
We shall see many examples in later chapters.
The optical transfer function is defined as Image contrast
Figure 4.23. Object and image contrast.
Strictly, we should distinguish between the optical transfer function (OTF) and the modulation transfer function (MTF). The OTF is a vector quantity that takes into account any variation of phase in the image whose amplitude is the MTF.
In practice, MTF is what lens designers are most concerned about. The phase term, sometimes called the phase transfer function (PTF), represents the displacement of the sinusoidal image from its ideal position expressed as a phase angle as a function of spatial frequency. If this exceeds 180 deg, then it is possible for the MTF to become effectively negative. This represents a phase reversal, which means that the image has reversed contrast. Many lenses actually show this behaviour at high spatial frequencies.
4.5.1 Theory
For a system that is used with incoherent illumination, the MTF is given by the Fourier transform of the line spread function. MTF is usually computed by this method, but there are other methods that are often faster.
We use the auto-correlation integral to calculate the true, or “diffraction”
OTF:
( ) ( )
( )
{ }
1 2
( ) exp / 2, / 2, ,
S
D s i W x s y W x s y dx dy
A
æ π ö
=
òò
çè λ ÷ø + − − × (4.28)where D(s) = OTF A = pupil area
S = area common to two sheared pupils, as shown in Fig. 4.24 W = wavefront aberration
s = reduced spatial frequency, equal to fλ/(NA) where NA is the
numerical aperture.
Figure 4.24. S = area of integration for the calculation of MTF.
4.5.2 The geometrical approximation
In the geometrical optics approximation, we assume that λ approaches zero, and it can then be shown that the MTF (in the tangential case) is given by
( )
exp 2 .
A
i f y dA
− π × × δ ×
òò
(4.29)This is integrated over the whole pupil, and in practice this can be approximated conveniently by a simple summation.
4.5.3 Practical calculation
Since we almost always need to know the MTF in both the sagittal and tangential azimuths, we calculate
MTF=
å
exp 2(− π × × δi f x) (4.30) for the sagittal MTF, and the corresponding expressionMTF=
å
exp 2(− π × × δi f y) (4.31) for the tangential MTF.It is convenient to simply trace a relatively large number of rays (typically over 100), and of course we include in the summation only the rays that actually pass through the system. We can also trace rays in several different wavelengths, and include them in the summations with appropriate spectral weighting factors.
It is also convenient to apply a correction factor, by multiplying the geometrical MTF by the diffraction-based MTF for a perfect system. The result of this is that the geometrical results will be correct when we have large aberrations and also when we have very small aberrations. In between, the geometrical MTF will usually give a pessimistic result.
We have to choose how many rings of rays to use. The ray pattern is chosen as shown in Fig. 4.25, where it can be seen that each segment of the semi-circle has an equal area. It should be clear that, with n rings of rays, we will have n2 rays. In order to save computing time, when the optical system has plane symmetry (about the meridian, or tangential, y-plane), the program need not trace rays with negative values of x.
Figure 4.25. Ray pattern for calculating geometric MTF and spot diagrams.
4.5.4 The diffraction limit
From Eq. (4.28) it can be shown that the highest limiting spatial frequency allowed by diffraction of light from the grating object into the finite-sized lens pupil is given by the following equation:
lim 2NA. f =
λ (4.32)
For example, with an f/2 camera lens, the numerical aperture, NA, is 0.25, and for a wavelength of λ = 0.0005 mm, the cut-off frequency will be:
flim =2 × 0.25 /0.0005 = 1000 cycles / mm.
This corresponds to a grating with a spacing of 0.001 mm = 1 µm, but note that at this frequency the MTF is actually zero; at 500 cycles/mm (2-µm spacing) the MTF of a perfect lens would be about 40%, so this might be a more realistic indication of the usefulness of an f/2 lens. However, you should note that very few detectors can resolve 500 or 1000 cycles/mm, so it should be clear that many very useful lenses are not diffraction-limited.
References
1. W. T. Welford, Aberrations of Optical Systems, Adam Hilger (1986).
2. M. Born and E. Wolf, Principles of Optics, Cambridge University Press (1999).
3. H. H. Hopkins, Wave Theory of Aberrations, Oxford University Press (1950).
4. M. J. Kidger, “The calculation of the optical transfer function using Gaussian quadrature,” Optica Acta, Vol. 25, No. 8, p. 665-680 (1978).
Refractive Index