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6.5. Strehl ratio
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6.6.1. MTF  aberration compounding, CCD, limitations
► 6.6. MTF  Modulation transfer function
PAGE HIGHLIGHTS Modulation transfer function (MTF) is commonly used to describe the convolution of point spread functions (PSF) and the Gaussian (geometric) image of an object that is a continuous sinusoidal intensity pattern, in effect a continuum of dark and bright lines gradually changing from the maxima (in the middle of the bright line) to minima (middle of the dark line). The convolution integral sums up energy arising from the PSF blotches created at every point of the Gaussian image, and so describes the corresponding diffraction image of the pattern. Changes in the PSF due to aberrations, pupil obstructions and other factors affect the quality of this diffraction image, specifically its contrast level and phase distribution. In general, convolution with the PSF smoothes out, i.e. flattens intensity distribution of the sinusoidal (or any other) pattern, lowering the contrast and acting as a lowpass filter (i.e. imposing limit to resolution)  the larger PSF vs. pattern's frequency, the more so. The MTF is a part of the complex function describing this process, called Optical transfer function (OTF). While OTF is limited to the effect of system PSF on imaging this single form of object  a sinusoidal intensity distribution  it is accepted as a performance indicator for imaging in general. The OTF consists of two components: (1) MTF, and (2) phasetransfer function (PTF). The former is its modulus, or magnitude, expressing the relative change in contrast of the pattern image vs. aberrationfree aperture, and the latter quantifies the linear lateral shift of this pattern caused by the change in PSF intensity distribution due to aberration. That intensity distribution change may be radially symmetric (such as draining of the energy from the central maxima with defocus or spherical aberration), or asymmetric, for example due to coma aberration (FIG. 100, middle). The basic relation between OTF, MTF and PTF can be presented as: OTF = OTFe iPTF = MTFe iPTF implying that the MTF is the absolute value of the OTF (e is the natural logarithm base, 2.718; note that the equality is not present with the the commonly used form of the OTF, MTF, which entirely omits the phase factor). Since OTF contains the imaginary number i, it is a complex function consisting from the real and imaginary part, the latter being the part containing the imaginary number. The imaginary number i=(1)0.5 is convenient for expressing angular shifts, in this case angular shift of the sinusoidal MTF intensity pattern phase at a given Gaussian image point due to the pattern's lateral shift, caused by aberrated PSF. With eiPTF=cos(PTF)+isin(PTF), the two parts of OTF function can be expressed as: OTF = MTF[cos(PTF) + isin(PTF)] = MTFcos(PTF) + MTFisin(PTF), with the two factors at right being the real and imaginary part of the OTF, respectively (noting that the PSF is usually annotated as angle). Either can be plotted separately, the former, being cosine function, originating at 1 for zero spatial frequency, and the latter originating at zero. If PSF retains any form of symmetry around a line splitting it in two halves (i.e. symmetry in its pupil aberration function), the imaginary part vanishes (because in such case the shift can only be zero or π, for contrast reversal), and OTF reduces to OTF= MTFcos(PTF). This is the case with defocus, spherical aberration and astigmatism, for any orientation of the PSF, and with coma only for a single orientation, along the axis of aberration. For all other orientations the coma OTF consists of both, real and imaginary part. It is important to note that, since eiPTF=cos(PTF)+isin(PTF) does not represent a sum, rather coordinates of a complex number, the real part of the OTF does not represent the magnitude of OTF (i.e. MTF), except when PTF equals zero (no phase shift). If we use the common notation for the real and imaginary part, Re and Im, respectively, then the relation between them and the MTF is given by OTF= MTF= (Re2+Im2)0.5. As illustrated below, eiPTF is not a value, rather phase descriptor. Hence OTF magnitude is not affected by it.
OTF can also be defined in terms of PSF, as OTF(ν)=∫PSF(x)eiνx dx, where x is the spatial coordinate in the image plane. It implies that OTF is the Fourier transform of PSF. It is also evident from the fact that OTF can be obtained by selfconvolution of the pupil function (same as autocorrelation for symmetrical functions) which  according to the convolution theorem stating that the Fourier transform of a convolution of two functions equals the product of their individual Fourier transforms  implies that the Fourier transform of OTF is the squared modulus of the coherent (point) spread function (which is the Fourier transform of aberration free pupil function), i.e. PSF. Or, more directly, just as the Fourier transform of aberrationfree pupil function (i.e. Fourier transform of the square pulse) is sinc function, so is the Fourier transform of triangular function  which is the form of aberrationfree 1D OTF for square pupil, with the OTF form for circular pupil being only slightly different  the sinc function squared, or PSF. In other words, the PSF is proportional in form to the continuous spectrum of frequencies contained as a sum of harmonics in the triangular (or neartriangular) function form. It is important to note that the OTF relations above are, by their basic concept, valid only for incoherent light. Formal derivation states that this type of pattern in coherent light would be reproduced without any distortions in amplitude or phase, i.e. with 100% contrast transfer, with the sudden drop to zero at the cutoff frequency 2λ/D (coherent transfer function). As above relations show, MTF does not include possible lateral shift of intensity pattern in any way. This is not by choice, rather by the inherent property of MTF defined as the modulus (magnitude) of contrast transfer as a function of spatial frequency, with every spatial frequency treated as an isolated contributor to the function. Thus, lateral shift of the pattern is no factor, only the contrast level is recorded. MTF alone shows the efficiency of contrast transfer from the object to its image with zero intensity phase shift, for a single orientation in the aberrated image, normally that along the axis of aberration. However, 180° (π radians) phase shift, which can be caused by large amounts of aberrations, will result in contrast reversal, i.e. switch of bright and dark areas in the image vs. object. For instance, such image of F, in addition to it being distorted by the large level of aberration, could make it unrecognizable, even if contrast needed for detection is there. Hence, the OTF phase factor is potentially important factor in predicting image quality, but only in the presence of large aberrations, well above the level found near axis in optical telescopes (for instance, contrast reversal due to primary spherical occurs at near 2λ PV wavefront error). Similar to MTF is the contrast transfer function (CTF), the difference being that the latter describes impulse response (i.e. contrast change in the image vs. object) based on an object that is a continuous square wave intensity distribution, in effect a continuum of bright and dark lines with even intensity distribution over them. Some additional basic aspects of the OTF/MTF are given below. INSET A illustrates characteristic form of the MTF, showing contrast drop as a function of spatial frequency ν for brightly illuminated object with high inherent contrast, as well as the effects of change in illumination/contrast level. For comparison, graph also shows the CTF. INSET B illustrates the effect of phase shift, INSET C shows OTF expanded to include radial PSF dimension and its effect, and INSET D illustrates polychromatic MTF.
INSET A:
TOP LEFT: Standardized MTF object form
is a continuum of parallel, equally spaced
bright contrasty lines on dark background, with irradiance
modulation defined as infinite sinusoidal distribution. The line width, equaling that
of the dark/light pair, is
inversely proportional to the spatial frequency
ν,
normalized to 1 at the cutoff frequency D/λ cycles per radian (the
inverse of the angular limit to resolution λ/D
in radians), where image contrast drops to zero (linear cutoff
frequency is the inverse of the linear limit to resolution
λF,
or 1/λF in lines per mm, for λ in mm). This form of
intensity distribution can be represented as periodic sinusoidal wave,
with the contrast determined by the relative intensity of its maxima
vs. minima. Image of this pattern has the relative intensity and
contrast dampened due to the effect of
diffraction, whether without
or with aberrations present, At any given frequency, MTF is a
ratio of the output (image) to input (object) modulation amplitude.
Mathematically, it is Fourier transform of the system's
PSF (more specifically, it is
an integrated sum of the PSFs for every point over its intensity
distribution profile, i.e. convolution of object's Gaussian image
and aperture's PSF). At
the limit of resolution for aberrationfree aperture, the line width
is nearly equal to the aperture's
FWHM.
At right, shown is the effect of 1 wave PV of
defocus on the image of an object that is not the standard MTF
pattern, but one in which bar width is oriented horizontally, with
the line width changing continuously according to the spatial
frequency. Top image (A) is aberrationfree, and bottom (B)
defocused one. If the aberration is large enough  in the case of
defocus ~0.9 wave PV or larger  image of such object will show
contrast reversal  where bright lines become dark and vice versa  indicated by the OTF, but not the MTF. The reason
was immediately apparent from the PSF intensity distribution for 1
wave of defocus, developing dark circle (minima) at the center of
the pattern. Hence, MTF does provide information on contrast
transfer, but not on the image itself. To reconstruct the image, we
need OTF. This, however. may have significance only with large
aberrations, generally about 1 wave PV and larger (a few more
examples at ). OTF for smaller aberrations does not fall below zero,
thus there is no contrast reversal to affect image.
INSET
C: In addition to phase shift,
if PSF intensity is varying radially it will also cause different contrast transfer for different
orientations of the PSF relative to MTF pattern. A complete MTF integral has
radial component, thus it integrates for all orientations of the PSF
relative to the line pattern. Graphically, it is
represented by a volume figure, formed by continuous change of the
integration angle (i.e. pattern orientation) through a full circle,
or 2π radians
(right). This volume may be looked at as a total contrast, transfer.
For aberrationfree clear aperture, the MTF is given by MTF=(2/π)[cos1νν(1ν2)0.5]. It can be also expressed as:
with the angle α
in degrees found from cos(α)=ν. Graphically,
MTF
contrast transfer equals the relative overlapping area of two
identical circles, in units of the circle area (FIG. 62, top
right), with the circle diameter normalized to 1, and the center
separation s=ν
varying from 0 when the circles are coinciding (ν=0),
to 1 when only touching (cutoff frequency ν=1).
Similarly, the normalized MTF for reduced aperture (still of unit diameter, with v=0 when the two circles are touching, smaller inside the larger), equals the overlapping area with the smaller circle appropriately reduced in diameter, with overlapping area being in units of the smaller circle area. The actual range of resolvable frequencies of a smaller aperture is in proportion to the aperture reduction factor. In terms of MTF, CTF is given as CTF=(4/π)[MTF(ν)(MTF(3ν)/3)+(MTF(5ν)/5)...]. It is important to understand that the MTF graph, such as the one above, does not set absolute values for the contrast drop, or limit to resolution. Both are strictly applicable only to the particular MTF object form used for its calculation: a pattern of bright lines on dark background, λF/2ν wide linearly, F being the focal ratio ƒ/D (i.e. linear width of the bright line at resolution limit is λF/2, or nearly one fifth of Airy disc diameter). Actual contrast dropoff and limiting resolution will vary with the specific properties of details observed, background, and peculiarities of eye perception, or detector properties. For noncontinuous patterns, sinusoidal, squarewave, or others, contrast transfer will generally increase with the reduction in the number of lines (FIG. 103). One example is the resolution threshold for lowcontrast MTFlike planetary details which is, according to the LCB threshold level in FIG. 100, approximately half of that for brightly illuminated contrasty object. Another is a dark line on light background, which can be detected at an angular width several times smaller than the angular diffraction resolution limit for pointimages of ~λ/D radians, because it is the Edge Spread Function, not the PSF, that is applicable in determining its diffraction intensity distribution. An actual object that comes very close to the standardized MTF scenario is a pair of nearly equally bright stars at the optimum brightness level. Resolutionwise, the MTF limiting resolution (cutoff frequency) is nearly identical to the empirical Dawes' limit in double star observing. However, for pairs farther from the optimum brightness level, or, especially, pairs with significant difference in brightness, the resolution limit is lower, or much lower. Still, despite the MTF being standardized to a single object form sample and brightness level, it is considered to be a reliable general indicator of the effect of wavefront aberrations  or any other factor affecting wave interference in the focal zone  on image quality. As mentioned, given relatively low RMS wavefront level of any aberration will result in nearidentical overall contrast loss, but the specifics will vary somewhat. FIG. 101 illustrates variations in the aberrated PSF (left) and MTF (right) for common wavefront aberrations of 1/13.4 and 1/6.7 wave RMS (graphs generated by Aberrator freeware, Cor Berrevoets).
FIGURE 101: PSF and MTF plots for clear aberrationfree aperture (black), and aberrated (red) at the 0.80 Strehl level, and that wavefront error doubled.
DEFOCUS: 1/4 and 1/2 wave PV. Doubling the error halves the peak
PSF intensity (Strehl), but the average contrast loss is three times
greater (20% vs. 60%, given as a ratio number by the differential
between the Strehl and 1). Note that the Strehl for 1/4 wave WFE of
defocus is slightly higher than 0.80.
PRIMARY SPHERICAL ABERRATION: 1/4 wave and 1/2 wave PV errors have
nearly identical effect on the PSF maximum as with defocus, but the
more widely spread energy causes more of contrast loss at the lower
frequencies, while the smaller central maxima results in somewhat
better transfer at midtohigh frequencies. The transfer is again
inferior near the cutoff due to the smaller bright core with
defocus. Being close to zero, the cutoff frequency for 1/2 wave PV
is about 20% lower in field conditions. PRIMARY ASTIGMATISM: The effect of 0.37 and 0.74 wave PV is similar to 1/4 and 1/2 wave of defocus in that the energy spread out of the Airy disc is mostly contained closer to central maxima, resulting in less of contrast loss at low frequencies. Unlike defocus, does not change radial symmetry of PSF, astigmatism gives to it crosslike shape which, similarly to coma's asymmetry, gives the best contrast transfer when cross orientation is identical to that of the bars (green) and the worst when it is at 45° angle. TURNED EDGE: 2.5 waves PV is needed that turned edge starting at 95% the radius to lower the Strehl to 0.80. Lost energy is evenly spread wide out (the abrupt initial contrast loss ending at about 1/40 frequency indicates the radius of this spread as 40 times the cutoff frequency, or 40λF  over 30 times the Airy disc radius). Odd but expected TE property  due to the relatively small affected area (the RMS for 0.80 Strehl is whooping 0.28 wave)  is that increasing the aberration does almost no additional damage. ROUGHNESS: ~1/14 and ~1/7 wave RMS of roughness producing asymmetric PSF pattern, thus having contrast transfer vary with its orientation relative to MTF bars. Due to the random nature of the aberration, the PV wavefront error can vary significantly for any given RMS/Strehl level. IHere, the PV error ranges from 0.169 (blue speck on the wavefront map) to 0.374, and 345 to 0.745, respectively. RMS error also may not be as reliably related to the Strehl as with most other aberrations.
PINCHING: 0.37 and 0.74 wave PV of wavefront
deformation caused by edge pinching at a three 120°
separated points has the typical SEEING ERROR: Averaged PSF/MTF effect of ~1/14 and ~1/7 wave RMS of atmospheric turbulence. The PV error can vary substantially (here is 0.43 and 0.86 wave, respectively). Seeing error fluctuates constantly around the average, as do the corresponding image contrast and resolution. Larger seeing errors (1/7 wave RMS is rather common with medium size apertures) reduces contrast level to that similar to twice smaller aperture for lowtomid frequencies, with some highfrequency transfer (iincluding limiting resolution) advantage remaining.
MTF contrast transfer generalized to the RMS
wavefront error alone, regardless of the type of aberration is, as
shown by Bob Shannon, given by a product of the Aberration Transfer
Function [ATF=1(ω/0.18)0.5(14(ν0.5)0.5,
where ω is the RMS wavefront error in units of
wavelength, and v is the
normalized spatial frequency; not to confuse with the
Amplitude Transfer Function  this
is the only place where the Aberration Transfer Function is mentioned] for any given level of the RMS error,
and aberrationfree
