11.6.1 Rayleigh Quarter-Wave Limit14, 47
The Rayleigh limit for image quality indicates that the image will be sensibly perfect
if the emerging wavefront departs from a perfect sphere by no more than one-quarter
of the wavelength of the radiation forming the image. A system which meets this limit
is sometimes descried as “diffraction-limited.”
The amounts of certain aberrations which correspond to a maximum wavefront deformation
of one-quarter wave are as follows:
when the reference point is chosen to minimize the departure of the wavefront from
the reference sphere.
The effect of a quarter-wave of aberration on the diffraction pattern is to shift some
light from the central patch to the rings. A perfect system has 84 percent of the light in
the central patch, 16 percent in the rings. A quarter-wave system has 68 percent in the
central disk, 32 percent in the rings, but the diameter of the disk and rings is essentially
the same. Such a change is detectable, but with difficulty. For most applications a system
with less than a quarter-wave of aberration is a excellent one.
The quarter-wave limit tacitly assumes that the wavefront is relatively smooth.
When this is not the case, the root-mean-squared (rms) wavefront deformation is a better
measure of system quality than the peak-to-valley deformation. An rms deformation
of between a fourteenth (λ/14) and a twentieth (λ/20) wave is approximately the
equivalent of the classical quarter-wave limit.
11.6.2 Strehl Definition8
The Strehl definition is the ratio of the illuminance at the peak of the diffraction pattern
of an aberrated image of a point to that at the peak of an aberration-free image. A
Strehl ratio of 80 percent is the equivalent of the Rayleigh quarter-wave limit. The
Strehl ratio is also equal to the normalized volume under the (three-dimensional) modulation
transfer function (MTF). See Secs. 11.6.3 and 11.6.4.
11.6.3 Optical Transfer Function8,16,42
The optical transfer function OTF(v) is a complex function describing the performance
of an optical system in terms of its imagery of a linear object pattern whose luminance
varies sinusoidally according to the spatial frequency v (in cycles per unit length). The
modulus of the OTF is the modulation transfer MTF(v), which describes the transfer
of contrast (modulation) from object to image; the argument of the OTF is the spatial
phase shift Φ(v) of the sinusoidal image pattern from its nominal position.
For the one-dimensional case the luminance of the sinusoidal object is
where v is the spatial frequency and y is the coordinate in which the luminance varies.
Modulation is defined as the peak-to-valley variation of luminance divided by the sum
of peak and valley. Thus Eq. (11.78) gives for the object modulation
Each line element making up the object is imaged as the line-spread function L(y)of
the optical system (Sec. 11.4.9). Assuming unit magnification and 100 percent transmission
to simplify matters, we can express the illuminance at position y in the image
as the summation of the product of G(y) and L(y):
When normalized by dividing by (11.80)
can be transformed into
F(y) in Eq. (11.81) is the illuminance in the image. Note that it is sinusoidally modulated
at the same frequency as the object, Eq. (11.78), and the pattern is shifted a distance
represented by the phase angle Φ [which is zero if L(y) is a symmetrical function].
The modulation in the image is thus
The modulation transfer function (MTF) is, by definition, the ratio of the modulation
in the image to that in the object
11.6.4 Specific Modulation Transfer Functions4–26
The MTF graphs of this section are plotted as functions of vo, the limiting cutoff frequency.
An optical system is a low-pass filter and will not transmit spatial frequencies
above vo, which is given by
where NA = n sin U, the numerical aperture, λ is the wavelength of the light forming
the image, and f/number is the effective speed or relative aperture at the image.
The limiting frequency can also be expressed as an angular frequency of the object,
subtended from the entrance pupil: vo = D/λ cycles per radian, where D is the diameter
of the entrance pupil.
Perfect System. The MTF at a spatial frequency v for an optical system with no
aberrations and a uniformly illuminated circular aperture is given by
Equation (11.89) is plotted in Fig. 11.27.
For a slit or a rectangular aperture the MTF is
Annular Aperture. In the presence of a central obscuration of the pupil, as in a
Cassegrain mirror system, the MTF of an aberration-free system is reduced greatly at
low frequencies and increased slightly at high frequencies, as shown in Fig. 11.28.
Defocused System. The effects of various amounts of defocus are shown in
Fig. 11.29. The defocus is expressed as a function of sin U so that the graph may be
applied to any optical system. If the defocusing is large, on the order of 4λ/(n sin2U)
or greater, then diffraction effects can be neglected and
Equation (11.91) is the MTF of a uniformly illuminated circular disk image. A system
whose image is a uniformly illuminated slit or band of light, as when the image is
blurred by motion, has an MTF given by
where W is the width of the slit or band.
Aberrated Systems. There is a strong similarity between the MTF curves for systems
afflicated with one-quarter wave of aberration, regardless of the type of aberration.
See Sec. 11.6.1. In general one is fairly safe in assuming that a given amount of wavefront
deformation (from a perfect spherical wave front) caused by any aberration will
produce an MTF characteristic similar to that resulting from the same amount (measured
in wavelengths of deformation) of another aberration; thus Fig. 11.29 is typical
of most aberrations.
11.6.5 Sine Waves and Square Waves13,28
By definition the OTF and MTF apply to the imagery of a sinusoidally modulated target
object. A square-wave target, i.e., a series of alternating light and dark bars, is a
convenient and widely used target for testing the performance of optical systems.
When the MTF (i.e., the sine-wave response) is known, the modulation transfer of a
square-wave target can be calculated by summing the response to the Fourier components
of the square wave:
where S(v) = square-wave target transfer factor and M(v) = sine wave MTF.
If the square-wave factor is known, the MTF can be found from
In general, the modulation transfer factor is higher for a square-wave target than for
a sine-wave target. See Fig. 11.17, for example. The most common form of bar target
for lens testing is the USAF/1951 target which consists of only three dark bars on an
extended white background (or the reverse) for each frequency. If the target frequency
is taken as the reciprocal of the centerline spacing of the bars, the modulation transfer
is much higher for the three-bar target than for an extended sine-wave target. This is
because the frequency content is heavily concentrated in the subharmonics; i.e., a
spectral breakdown (Fourier analysis) of a three-bar target shows much power at frequencies
less than v = 1/(bar spacing).
11.6.6 Aerial Image Modulation Curve
The aerial image modulation (AIM) curve is a plot of the minimum image modulation
required to produce a response in a sensing element, plotted as a function of spatial frequency.
AIM curves are commonly used to describe such detectors as photographic film, image
tubes, and the human eye. A typical AIM curve rises with increasing frequency, indicating
that a higher image modulation is required to produce a response at higher frequencies. If
the AIM curve for a film and the MTF curve of a lens are plotted on the same graph, the
intersection of the two curves indicates the limiting resolution of the combination.
11.6.7 Depth of Focus
Photographic Depth of Focus. Assuming that an arbitrarily selected level of blur (of
diameter B) caused by a defocusing of the optical system can be accepted, the tolerable
depth of focus is then ±B/2NA, if diffraction effects are totally neglected and if the
optical system is free of aberrations. The corresponding depth of field at the object
ranges from Snear to Sfar:
The “hyperfocal distance” is the distance at which the optical system is focused in
order to make Sfar equal to infinity and Snear equal one-half the hyperfocal distance.
Physical Depth of Focus. There is actually no sharp demarcation between being in
focus and out of focus. The image simply deteriorates gradually as the amount of defocus is increased. The wave-front aberration caused by defocusing is given by
Thus a depth of focus “tolerance” corresponding to the Rayleigh quarter-wave criterion is
Note that Eq. (11.99) may be used for both depth of focus (at the image) and depth of
field (at the object) if Um is taken as the slope of the marginal ray at the image or
object, respectively.
11.6.1 Rayleigh Quarter-Wave Limit14, 47
The Rayleigh limit for image quality indicates that the image will be sensibly perfect
if the emerging wavefront departs from a perfect sphere by no more than one-quarter
of the wavelength of the radiation forming the image. A system which meets this limit
is sometimes descried as “diffraction-limited.”
The amounts of certain aberrations which correspond to a maximum wavefront deformation
of one-quarter wave are as follows:
when the reference point is chosen to minimize the departure of the wavefront from
the reference sphere.
The effect of a quarter-wave of aberration on the diffraction pattern is to shift some
light from the central patch to the rings. A perfect system has 84 percent of the light in
the central patch, 16 percent in the rings. A quarter-wave system has 68 percent in the
central disk, 32 percent in the rings, but the diameter of the disk and rings is essentially
the same. Such a change is detectable, but with difficulty. For most applications a system
with less than a quarter-wave of aberration is a excellent one.
The quarter-wave limit...
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