Zeiss Read MTF Curves

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Carl Zeiss Camera Lens Division December 2008

How to Read MTF Curves

by

H. H. Nasse


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Carl Zeiss Camera Lens Division December 2008

Preface

„ The rules of optics are complex and nasty! “

This is a nice, honest sentence from an internet discussionabout ‚How to read MTF curves’, telling us how difficult tounderstand this world of numbers might be forphotographers.

Nevertheless I am going to show you on the followingpages, that things are not that bad, and that you canunderstand the basic relationships without an excursion into

higher mathematics of Fourier-optics.

After reading the paper you will be able to conclude aboutthe character of a lens from MTF data published by makersor testing institutes. You will however also learn about thelimitations of MTF, so that you can read lens reviewscritically.

And those who see too many numbers and curves may beassured that these are not really necessary for goodphotography, since photography is mainly based onexperience. But it is great fun to understand your tools in abetter way, and this what I wish you to have during reading

this first part. In a second part we will show you a number ofillustrating images.


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Carl Zeiss Camera Lens Division 3

Point Spread Function

When photographers want to take a verynatural-looking picture of a subject, theywould like to have an ideal lens on theircamera, one which allows all light rays

emanating from one point on the object tomeet again at exactly one point of theimage. We now know that with real lenseswe can go only part way to achieving thisideal. Image points in the geometric senseof the word do not exist in reality.

Aberrations of the lens systems, productiontolerances, and ultimately the wave-likenature of light as well, are the reasons whythe light originating from one point of theobject is always distributed over an areaaround the ideal image point.

To a certain extent, this area is the‘smallest possible circle of confusion’;however, the light therein is not evenlydistributed, the intensity usually

decreases from the inside to the outsideand the shape is rarely circular. Thiseffect is known as “point spreadfunction”. Its shape and sizecharacterize the image quality of a lens.

If it is possible to compare photographywith painting, the point spread function isthe handwriting, the brushstroke of alens. Just as there are wide, flat, pointedor even bristly brushes, lenses also havevery different styles of handwriting.

But why then is it still not used toquantitatively describe image quality?

There are three reasons for this:

First of all, the shape is sometimes very

complicated and therefore defies a simplenumerical description. This is illustrated inthe following pictures taken with amicroscope. The first six point spreads inthe images 1 to 6 on the next page areexamples of useable, but moderate imagequality and are typical

of high-speed lenses at full aperture,wide-angle lenses on the edge, or slightdefocusing. A small white square has been pastedinto each image for a size comparison; itrepresents an 8.5µm pixel area likethose of a 12 MP, 35mm format full-frame camera. All these point spreadsare thus considerably larger than this(relatively large) pixel area.


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1 2 3 4

5 6 7 8

The point spread Nr. 7 in the second rowabove is an example of outstandingimaging performance. A digital sensor

generally does not see such small pointspreads, however. The image Nr. 8 on thefar right shows the same point spreadbehind a low pass filter that is usuallypositioned in front of the sensor and isintended to suppress the Moiré effect. Theimage quality is therefore artificiallydeteriorated in the low pass filter byincreasing the point spread considerablyby means of several birefringent discs.

The second reason is that you almostnever see such single, isolated point

spreads. For example: Only if you takepictures of stars on a dark night do youachieve the same effect as that here in thelab. Most images are generated in thecamera in a complicated way of combiningthe parts of a large number of single pointspreads.

This is because a small area of theobject consists of many densely packedpoints and these correspond to many

densely packed ideal image points in theimage behind the lens. Since the realpoint spreads cannot be infinitely small,this means that the individual pointspreads overlap: The intensity at onepoint of the image (you could even sayin one pixel) is generated by a two-dimensional integration (summation) ofmany point spreads. There is thus a not-so-easily manageable mathematicalconnection between the ‘brushstroke’and the image that we see.

The third reason is that the entireimaging chain from the lens to the eyecan be much more elegantly describedwith the method that I would now like toexplain.


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Modulation transfer

Since we are primarily interested in howextended objects are imaged, objects

which, unlike stars, comprise an infinitenumber of points, we must find anotherway to quantitatively describe the imagequality. We use a sinusoidal brightnessdistribution to examine how an objectthat looks as simple as possible is imaged.The sinusoidal brightness distribution is apattern of bright and dark stripes in whichthe transition between bright and darkoccurs gradually and continuously, i.e.sinusoidally, just as the electric power inour sockets varies with time.We use the sinusoidal stripe pattern

because the result in the image is onceagain a sinusoidal pattern,

regardless of how complicated the shapeof the point spread may be.

Several of its properties also remainstable or at least have nothing to do withimaging quality: The direction of thestripes does not change and thefrequency – the number of stripes perunit length – only changes according tothe imaging scale.What is no longer identical to the originalis the difference in brightness betweenthe dark and bright stripes. This isbecause the extended point spreadsensure that part of the light falls on a

position that would actually becompletely dark, instead of falling on abright location.

-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90

µm

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Sinusoidal Pattern 20 Lp/mm Image

Point Spread Profile

Point Spread Profile

white

black

This graph shows a sinusoidal stripe

pattern (black curve) as an intensity profile(a cross-section perpendicular to thestripes). It has 20 periods per millimeter,so one period is 50 µm long. The red andblue curves are cross-sections of thebrightness distribution in a point spread.The brightness that would exist with idealimaging at the point of the sine patternmarked in blue is distributed to thesurrounding area according to the bluecurve. You can therefore see that some ofthis light falls into the dark “valleys” at 25µm next to the blue point.

Light also falls there from the red dot on

the flank of the sine pattern. Althoughthe sine pattern on the flank is darker, alarger fraction reaches the point at -25µm, since the red dot is closer to thedark valley. Thus, the intensity in thedark areas of the pattern is the sum ofmany such contributions from theneighboring areas. The result is then theweaker modulated curve labeled“image". The brightness of the darkstripes in the image is raised by theaberrations, while the bright stripes getdarker.


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In optics, the difference between brightand dark is referred to as “contrast”. Seenfrom a more general point of view, thedifference between maximum andminimum for all sinusoidal, periodicallychanging quantities is called“modulation.” If we compare themodulation of the image with themodulation of the object by simply dividingthese two figures by each other, we get asimple figure that provides a statementabout the imaging properties of the lens:The modulation transfer . Thus, we havealready understood the first two letters ofthe term "MTF". It is a number between 0and 1 or between 0 and 100%.

The photographer is used to expressingbright-dark differences in aperture stops,which is also very reasonable as theperception of our eyes follows suchlogarithmic scales. But, what, forexample, does a modulation transfer of50% mean if our pattern of stripesconsists of a difference of 6 aperturestops between the brightest and darkestpoints, i.e. a brightness ratio of 1 : 2

6

= 1 : 64 ? Is the difference in the image 3aperture stops or 1:32, which wouldcorrespond to 5 aperture stops?Both would be wrong. In reality, wewould then still have approximately 1.5aperture stops in the above-mentionedcase. This is because, in optics, the“contrast” parameter is defined asfollows:

Minimum Maximum

Minimum MaximumContrast

+

−=

Therefore, in our example, the contrast ofthe object is 63 divided by 65, or approx.0.97. After imaging with a modulationtransfer of 50%, the contrast in the imageis only half as high, approx. 0.48. Minimumto maximum is then approx. 1:2.9.

(1.9/3.9 = 0.48)

The following graph shows how objectcontrast and image contrast are relatedfor different modulation transfers if theyare measured in aperture stops:

MTF 20 - 97 %

0

1

2

3

4

5

6

1 2 3 4 5 6 7 8 9 10

Object Contrast [Aperture Stops]

I m a g e

C o n

t r a s

t [ A p e r t u r e

S t o p s

]

MTF 97 %

MTF 94 %

MTF 90 %

MTF 81 %

MTF 70 %

MTF 50 %

MTF 20 %


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We can then recognize three importantproperties of MTF here which we shouldremember when reading MTF curves:

1. Small differences in higher MTFvalues are particularly significantat high object contrast levels.

2. On the other hand, weak tonalvalue variations of less than oneaperture stop do not require highMTF values. Differences above70-80% are then hardly relevant.

3. With very low MTF values, itpractically does not matter how

high the object contrast is; theimage contrast is always low.

Incidentally, this is why the datasheetsof films always also gave the resolvingpower for the low contrast of 1:1.6. Theresolution figures for the contrast of1:1000 can only be measured usingcontact exposure. For the fineststructures (i.e. very high spatialfrequencies), no lens in the world iscapable of producing a contrast of tenaperture stops. Estimating the amountof information of film images based onthis higher resolution value is thus toooptimistic.


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Modulation t ransfer function, resolving power

It is obvious that one single stripe patternis not sufficient to characterize the qualityof a lens. A very coarse pattern with largeseparations between bright and dark

stripes could, of course, also be imagedwell by a lens with a relatively large pointspread function. If we decrease theseparation between the stripes, however,so that the separation between bright anddark approaches the size of the pointspread, then a lot of light from the brightzone is radiated into the darker zones ofthe pattern and the image contrastbecomes noticeably lower.

If we want to use the comparison withpainting again: Coarse structures can bepainted well with a thick brush, the pointedfine brush is required for fine details,however.

We therefore need to investigate howthe lens images stripe patterns ofvarious degrees of fineness, i.e. we needto determine a modulation transfer for

every one of these patterns. We thusobtain a whole sequence of numbers,and if we plot them as a function of aparameter which describes the finenessof the stripe pattern, these numbersrepresent a curve, the modulationtransfer function.

The fineness of the stripe pattern can bemeasured by counting how manyperiods of the pattern are contained in adistance of one mm in the image. Aperiod is the separation between twobright or two dark stripes, or the width ofa line pair consisting of one dark andone bright stripe. The number of periodsper millimeter in the image plane is thespatial frequency, given in the unit linepairs per millimeter , abbreviated tolp/mm.

Modulation transfer function of a 50mm lens of 35mm format in the image center,measured at f/ 2 and f/ 5.6, for purposes of comparison the diffraction-limitedtransfer functions for f/5.6 and f/16 are also shown (solid line without circulardots). The diffraction-limited image is the best possible one. On the horizontalaxis we have the spatial frequency in line pairs per mm.

0 10 20 30 40 50 60 70

80 90

100

0 20 40 60 80 100 120 140 160 Spatial frequency lp/mm

M T F

[ % ]

measurement aperture 2 measurement aperture 5.6

diffraction-limited aperture 5.6 diffraction-limited aperture 16


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A diffraction-limited image has an almostperfectly straight MTF curve whichdecreases in proportion to the spatialfrequency. The zero MTF value is reachedat the so-called limit frequency, which is

determined by the f-number and thewavelength of the light.

A rough estimate for mediumwavelengths of visible light is:

The width of the point spread in µmcorresponds to the f-number, and the

limit frequency is about 1500 dividedby the f-number.

Intensity profile of a diffraction-limited point spread

(Airy-disk) at f-number 16

0

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[µm] diameter = f-number

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For real lenses with residual aberrationsthe MTF values initially decrease quicklyand then very slowly approach the zero

line. The curves therefore exhibit apronounced sag. In the above examplethis can clearly be seen for the curve ofaperture 2; for aperture 5.6 the lens is notvery far removed from the optimum whichis physically possible.

The spatial frequency at which the MTFvalue reaches zero or falls below a lowthreshold (e.g. 10%) is the resolvingpower of the lens in air . Periodic stripepatterns can become this fine before theirimage changes to an unstructured gray.

The curve above for aperture 2, inparticular, shows that this resolution limit isdifficult to measure; this is the casebecause the very flat slope at high spatialfrequency values means the resultdepends very sensitively on the minimumcontrast required. The measurement istherefore very imprecise. For this reasonalone the resolving power in air istherefore not a suitable quality criterion forlenses!

Neither should it be confused with theresolving power which is achieved inconjunction with an image sensor . And

this leads to the above-mentioned thirdreason why we describe image qualitywith the modulation function:

We never observe the image of the lensdirectly with our eye, but require furtherlinks in the imaging chain: We alwaysrequire an image sensor, analog ordigital, or maybe a scanner, a printer orprojection optics.

All these components, even the humaneye, have their own imaging properties,each of which can also be described bya transfer function. And the nice propertyof MTF is that the MTF of the entireimaging chain is (approximately) theproduct of all individual MTFs . Let usconsider a few typical examples:


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0

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0 20 40 60 80 100 120 140 160

Spatial frequency Lp/mm

M T F [ % ]

Lens MTF at f /5.6 MTF of colour f ilm MTF product lens x f ilm

Product of two modulation transfer functions: Very good 35mm format lens andcolor negative film. The product is always smaller than the smallest factor in theimaging chain.

In this case, the total modulation is essentially limited by the film. If one specifiesa minimum of 10% modulation transfer, one must expect a resolving power of 80-100lp/mm. If further elements such as projection optics or the eye are taken intoaccount, the product is even slightly smaller.

0

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0 20 40 60 80 100 120 140 160Spatial frequency Lp/mm

M T F

[ % ]

Lens MTF at f /2 MTF of colour f ilm MTF product lens x f ilm

Product of two modulation transfer functions: 35mm format lens with moderateperformance and color negative film. The product is now determined to an almostequal extent by the lens and by the film.


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If one considers the curve of the product ofonly two transfer functions and takes intoaccount the fact that, in reality, even moretransfer functions are involved, which canonly make the product smaller, then it

becomes clear that it is not necessary toconsider the complete range of the veryhigh spatial frequencies.

Digital sensors with 24 megapixels in the35mm full-frame format or 15 MP in the APS-C format have Nyquist frequencies ofabout 90lp/mm. Their theoretical maximumresolutions are thus roughly comparable tothe color negative film. It is thereforeusually sufficient to consider the spatialfrequencies up to 40lp/mm for theseformats, although with larger numbers of

pixels the 40lp/mm are slightly moreimportant than usual.

Another consideration also suggests thatthis is a reasonable limit: If one looks at an A4-size print from a distance of 25cm, andthus sees the picture width at an angle of60°,

the human eye can resolve1600lp/picture height at most, because itresolves a maximum of 8lp/mm at thisdistance, which is called the ‘leastdistance of distinct vision’. For the 35mm

film format with 24mm picture height thiscorresponds to 66lp/mm. The spatialfrequencies important for the human eyeare therefore also in the range up to40lp/mm.

If one considerably enlarges the imageviewed, however, and neverthelessviews it from a relatively short distance,the eye can use the highest spatialfrequencies of the system and itsuddenly sees weaknesses which wouldnot be noticed during normal viewing of

the image. This is, incidentally, whathappens when one views digital pictureson a large monitor in 100%representation. In this case, the image ofa 12MP camera is more than one meterwide.

Incidentally, a sensor which can also usethe lens performance at higherfrequencies is the low sensitivity black-and-white film:

0

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120

0 20 40 60 80 100 120 140 160

Spatial frequency Lp/mm

M T F

[ % ]

Lens MTF at f /5.6 MTF of Black&White film MTF product lens x f ilm

Good lens combined with high resolution B&W film(this data is from T-Max 100)


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The following images illustrate graphicallythat 40lp/mm is already a spatialfrequency which is quite high, at least forthe 35mm format.

They show a picture of the well-knownSiemens stars, which are used by manyto test cameras. The complete image ofa 12 MP camera in 35mm full-framecontains nine stars:

A greatly enlarged section shows the

center of a Siemens star and how close tothe center the spatial frequency 40lp/mmis:


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Edge definition, image contrast

Maybe we should briefly recapitulate atthis point: We know now why themodulation of sinusoidal stripe patternsdecreases with increasing spatial

frequency in optical imaging and also inthe further stages of the image generationup to the perception. But what do thesenumbers tell us about the quality of realpictures? What is the relationship betweenterms such as definition, brilliance,resolution of detail etc. and thesenumbers?

Our subjects do not contain sinusoidalpatterns, of course. They can only begenerated as an approximation with a lotof effort, even in the laboratory, andinstead one uses other test objects fromwhich the sinusoidal modulation ismathematically deduced.

Stripe patterns with a rectangular intensityprofile, with a sudden change betweenblack and white,

are used on the special test charts forevaluating lenses and cameras and todetermine the effective resolving power.

The modulation transfer for rectangularpatterns is, incidentally, usually a little bitbetter than for sinusoidal patterns of thesame spatial frequency. These preciserectangular shapes are also rarely foundin real photographic subjects, however.

Fine periodic patterns represent only asmall fraction of the subject propertieswhich our eye uses to recognize imagingquality. Most important are really theedges, the borders between two areaswith different brightness or color. Wewould therefore now like to understandwhat the relationship is between MTFand the reproduction of edges, and thisbrings us back again to our startingpoint, the point spread function.

The following images show from left to right:

• Intensity profile of the point spread function, in a logarithmic scaledown to 1/1000 of the maximum intensity in the center. The width of thepoint spread function is given in µm, 1µm is 1/1000mm.

• Intensity profile of two edge images with large and small step inbrightness. The vertical scale is the logarithmic aperture scale familiar tothe photographer: Every graduation signifies a halving of the intensity. Thehorizontal scale is again a measure of the distance in the image in µm.The bright and the dark side of the edge are to the left and rightrespectively.

• The corresponding modulation transfer for five spatial frequencies 5, 10,20, 40 and 80lp/mm is shown as a bar chart.


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0.1

1.0

10.0

100.0

- 6 0

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Edge profile [µm]

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[ E V ]

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7080

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5 10 20 40 80

[Lp/mm]

M T F [ % ]

This is an example of a very good imaging performance in 35mm format; the pointspread is narrow, the transition at an edge from white to black is no wider thanabout 10µm, i.e. very steep. The photographer then says: The image of the edgeis sharp. In the language of modulation transfer, this characteristic is recognizedby the fact that all values at the important spatial frequencies are very high and donot decrease so strongly towards the higher frequencies.

For a lens with such imaging performance, the image quality achieved is usuallylimited by the sensor or by other factors such as focusing accuracy, cameramovements etc.

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Edge profile [µm]

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[ E V ]

010

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5 10 20 40 80

[Lp/mm]

M T F [ % ]

Here, the diameter of the point spread function is significantly larger; the image ofthe edge from white to black is by no means as sharp, the edge profile is flat,because the transition from maximum brightness to black takes between 30 and50µm depending on the size of the brightness step. A deep black is neverthelessachieved after this distance, the contrast between the ends of the above scale istherefore high. The MTF values reveal these characteristics by quickly falling off

towards the higher spatial frequencies, while differing only slightly at the lowestfrequency when compared to the previous example.


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0.1

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Edge profile [µm]

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5060

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[Lp/mm]

M T F [ %

]

A wide, box-shaped line image naturally gives rise to poorer edge definition. TheMTF values for the lower and medium spatial frequencies up to 20lp/mm are

normal, even at 60lp/mm there is still an acceptable modulation transfer. If onewere to look only at these frequencies, one would have the impression of a quiterespectable imaging performance.

But: There is no contrast at 40lp/mm! The curve of the modulation transfer candrop to zero and then increase again. This is then called “spurious resolution”,which is a somewhat unfortunate expression because the structure with 60lp/mmis reproduced with a clear resolution. One usually does not notice that black andwhite are interchanged (except with the Siemens stars) and the next zero pointwould come again at 80lp/mm and then a resolution again where even black andwhite are in the correct position again. The term ‘spurious resolution’ is intendedto express that the isolated measurement of a high resolution at a single,coincidentally favorable spatial frequency can simulate an image quality which isnot even present. You will not find this type of imaging in published MTF curves,but it is of practical importance with respect to focus errors and motion-inducedblurring.

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[ E V ]

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4050

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5 10 20 40 80

[Lp/mm]

M T F

[ % ]

Here the point spread function is about as narrow as in the first example, but issurrounded by a weak halo. The edge definition is high in parts, but at the sametime a broad, bright fringe stretches into the dark zone. The photographer would

say that the lens exhibits flare. The contrast near the edge is low.


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The MTF values of this 4th type arecharacterized by the fact that theydecrease only slowly with increasingspatial frequency, as in the first example.But the values of the low spatial

frequencies of 5 and 10lp/mm areconspicuously low.The imaging properties of a lens with thischaracter can be somewhat inconsistentand are judged differently depending onthe image content.Edges with low to medium contrast arereproduced with the same degree ofsharpness, particularly if the exposure israther brief. Fine structures with a lot ofcontrast appear a little bit flat, however,edges with a lot of contrast and lights evenshow flare or appear broadened at

generous exposure.Most of the high-speed standard lensesfrom the sixties were corrected like this atlarge apertures. At 10lp/mm they had only60-70% MTF, whereas nowadays 80-90%is usual here.

At the time, one said the lens was‘optimized for resolution’, which is notexactly correct, because they simply hada good edge definition, but the resolvingpower for fine periodic structures was no

better than in lenses with a differentdesign.When black-and-white photography wasstill dominant, one could compensate thelow contrast reproduction of these lensesby enlarging on paper with hardgradation. Colour photography with itsless flexible laboratory processing laterdemanded a change in the correction tobetter contrast rendition.

Lenses with this imaging character arehowever favourable tools for particular

subjects. One should always be carefulwith judgements about lenses.For instance the famous soft portray lens‘IMAGON’ has a modulation transferfunction like this:

Rodenstock Imagon 200mm, centre image

0

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40

60

80

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0 10 20 30 40

Spatial frequency [Lp/mm]

M T F [ % ] normal lens

soft lensImagon

Incidentally, it is not the case that, whendesigning a lens, a decision has to bemade between high resolving power andgood contrast rendition; both are possible

for lenses with good correction.

But what does ‘contrast rendition’ actuallymean? We must not forget that when wetalk about ‘contrast’ we always meanmicro contrast, i.e. structures, which wecan just about see or just cannot see withthe naked eye, for example on a slide. Butif we photograph a chessboard, forexample, so that it fills the format, thecontrast between the black and the whitesquares has nothing to do with this.

MTF measurements say nothing aboutthis macro contrast. They gauge onlythe correction of the lens, i.e. the smalldeviations of the light beams, while the

macro contrast depends on the veilingglare of the lens, i.e. on the largedeviations.These result from undesirable reflectionsbetween the optical surfaces and fromlight scattering at the interior barrelcomponents, so that they usually reachthe image plane a long way from theoriginal target. All these characteristicsare often mixed up with each other in theterm ‘brilliance of the image’. Good MTFvalues at low spatial frequencies arenecessary, but they are no guarantee forbrilliant images.


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Enlargements of format-filling images of a chess board, left with perfect imagingquality, in the middle with low micro contrast, right with high degree of veilingglare.

The characteristics of the images above are also illustrated by their histograms: Inthe picture of the lens with poor micro contrast (middle) the right peak in particularis broadened towards the left, because the white at the edges with glare lights upthe areas which are in reality black. The separation of the two peaks on the grayscale is the same as with the good picture on the left, however.

In the picture on the right with the high level of veiling glare, the lower peak of thehistogram is shifted upwards because the black is lit up by the veiling glare in thewhole area.

The four basic types of point spreadfunction shown above and thecorresponding MTF curves can be foundin all lens data, not always in theexemplary forms shown here, of course,but usually as mixtures and combinationsthereof.

From these examples we also learn that

one must always consider the MTF atseveral spatial frequencies. The meaningof a value of 75% at 10lp/mm would onlybe completely unambiguous for the imageof a sinusoidal pattern. In real images, italways also depends on what the valuesat 20 and 40lp/mm are.

If they are very high, the lens exhibitsglare at edges rich in contrast and withlights, as in our fourth example. And ifthey are lower than normal, the lens issimply less sharp, maybe a little out offocus, but it is free of glare.

Test procedures which measure onlyone point of the modulation transfer

function, for example the resolution orthe spatial frequency where 50% MTF isachieved, are of little value! This appliesto optics just as it does to a HIFI system:If I know at which frequency theloudspeakers will have their maximumtransmission or how loud 440Hz are,I still don’t know how music sounds.


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Edge definition in digital images

When image data is processed digitally,the transfer function of the camera can bestrongly influenced. Edge enhancement

involves making the bright side of an edgea little bit brighter, and the darker side alittle bit darker. This improves the microcontrast and the edge steepness, and thesubjective impression of sharpness issignificantly improved, without significantlyincreasing the resolution of detail. This isconvincing proof that definition andresolution are not the same thing.

In the transfer function this manipulationcan be seen by the fact that the normaldecrease with increasing spatial

frequency is partially or completelycancelled, as happens with lenses withhigh edge definition. In digital imageprocessing it is even possible toexaggerate this edge enhancement andto generate a transfer function whichincreases with increasing spatialfrequency. In the language of transfertheory it then has a partial high-passcharacter – and such systems showmarked artifacts at edges.

0

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Lp/Image Height

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Lp/mm

Modulation in the picture of a DSLR in the 35mm format, 24 MP, with variousparameters of edge enhancement of the camera JPEG processing. The curveswith a flat slope up to about 50lp/mm belong to pictures with very high edgedefinition.

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Modulation in the picture of a 2/3’’ camera, with minimum, medium and maximumedge enhancement. Significant artifacts must be expected with the buckled curve,

additional bright lines usually appear next to the edges of dark areas.


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Tangential and sagittal

Up to now, we have been concerned with

the question of the relationship betweenthe modulation transfer and the pointspread function. We have seen how theshape of the point spread function and thedistribution of the light intensity within itscomplete area influence the modulationtransfer at various spatial frequencies. Forthis, we have plotted the MTF as afunction of the spatial frequencyparameter.Such a curve is valid only for one singlespot in the image, however, and even forthis spot we really require several curves,

because we have seen from our point

spread examples that they are notnecessarily circular. Some can be bettercompared with a flat brush with whichone can draw fine lines in one directiononly. If we rotate the orientation of thestripe pattern, we must expect differentMTF curves depending on whether theshorter or longer elongation of the pointspread function is perpendicular to thestripe pattern.

The main orientations, i.e. the shortest andlongest elongations of the point spreadfunctions, are always parallel orperpendicular to the radius of the picturecircle, because lenses are rotationallysymmetric. Stripe patterns where thelongitudinal orientation of the stripes istoward the center are therefore calledradial or sagittal (sagitta = Latin for‘arrow’) in optics. This direction usuallyhas the better modulation transfer.

Stripes perpendicular to this have, ofcourse, the same orientation as atangent to a circle around the imagecenter. This orientation of the stripes istherefore called tangential or alsomeridional.

sagittaltangential

lower MTF

higher MTF

non-circular point spread


20/33


MTF curves for lenses

Since the imaging quality of lensesgenerally changes from the center to theedge and since precisely these differencesare of special interest to us, we naturally

need more curves than the two fortangential and sagittal orientation. Half adozen or so measuring points between thecenter and the corner are required in orderto be able to describe the spatial changesof the imaging properties with sufficientprecision. This would be 12 curves in total – not very clear and legible.

The MTF curves which we have got toknow so far, where we have plotted themodulation transfer on the vertical axisand the spatial frequency on the horizontalaxis, are really only suitable for sensorswhere there are no spatial variations. Thisrepresentation is not so suitable forlenses.

Since the MTF curves as a function ofthe spatial frequency always slope tothe right and fall off more or lessrapidly, it is sufficient to read off only

three numerical values from eachcurve, i.e. from three sufficientlydifferent spatial frequencies, usually 10,20 and 40lp/mm. If one shows howthese MTF values change in the imagearea for three frequencies, one obtainsa graphical representation which ismuch better suited to lenses.

This is why you will find MTF curves inour datasheets, where the modulationtransfer is plotted on the y-axis and theimage height, the separation from theoptical axis, on the x-axis. The diagramcontains six curves, i.e. the tangential(broken lines) and sagittal values (solidlines) in each case for three spatialfrequencies. The top curve of the sixalways relates to the lowest spatialfrequency, of course, and the bottomone to the highest.

f-number 1.4

0

20

40

60

80

100

0 5 10 15 20

u' (mm)

MTF (%)

f-number 5.6

0

20

40

60

80

100

0 5 10 15 20

u' (mm)

MTF (%)

MTF curves for the Planar 1.4/50 ZF lens, at 10, 20 and 40lp/mm, white light anddistance to object on infinity.


21/33



22/33


In the center, this lens achieves more than80% MTF at 10lp/mm even at full apertureand decreases to just below 40% at40lp/mm. This means good contrastrendition and medium definition, which has

a slightly soft effect only after the picturehas been greatly enlarged.

Away from the center the MTFdecreases at 10lp/mm to 70%, thetendency to produce flare at edges richin contrast therefore increases.In the corner of the image the sagittal

curves, in particular, are very closetogether at low level, we must thereforeexpect significant glare with open lightsources at the corners.

If the lens is stopped down, all MTF valuesincrease greatly; the curves are very closeto each other at a high value. The MTFvalues therefore decrease only relativelyslowly with increasing spatial frequency.This means excellent edge definition and

very good micro contrast right down to thefinest structures which can be reproducedby the sensor or the film.

In the corner of the picture, all the curvestail off somewhat, those for 10lp/mm alittle, those for the higher frequenciesmore. This indicates that the very goodflatness of the visual field stretches toabout 18mm picture height and that thereis a defocusing in the corner of the picture

due to the field curvature which suddenlyappears.

One should not take the small variationsof the curves for 40lp/mm too seriously,they are visible only at extreme

enlargements of the picture and whentaking photographs of flat objects, inmost pictures they are thereforeinvisible. They are caused by fieldcurvature and shifting of the focus. I willexplain why these lead to such variationsin the section on the three-dimensionalcharacteristics of MTF.


23/33


f-number k = 5.6 f = 85 mm

0

20

40

60

80

100

0 5 10 15 20

u (mm)

MTF (%)


0

20

40

60

80

100

0 5 10 15 20

u (mm)

MTF (%)

A comparison of short Tele focal length for 35mm format, stopped down to f/5.6. On the left ahigh quality prime lens (Planar 1.4/85 ZF), on the right an inexpensive 5x zoom lens. Imagequality with the prime is in the whole frame practically limited by the sensor and allows highest

magnifications. The zoom lens is quite good in the centre, but drops continuously to the edges.Except from the corners a good overall contrast may be expected, but the lens lacks a ‚biting’sharpness, since the MTF values drop rapidly at higher spatial frequencies. The lens couldonly be recommended for moderate enlargements.


0

20

40

60

80

100

0 5 10 15 20

u (mm)

T (%)


0

20

40

60

80

100

0 5 10 15 20

u (mm)

T (%)

A comparison of two super-wide lenses, which are much more difficult to make, stoppeddown to f/ 5.6. On the left data measured with a Distagon 2.8/21 ZF, on the right a lens,where the lateral chromatic aberration is corrected less good. Its sagittal MTF showssome focus shift, but is otherwise not too bad. But the tangential MTF is very low towards

the edges of the frame; and what this means can be seen in the following two thumbnails(200x200 pixels from a 12MP-image, image height about 12 mm ):


24/33


Three-dimensional characteristics

It is, of course, a truism that imagedefinition also depends on whether thelens is focused correctly. It should thusalso be possible to describe this with MTFcurves; and therefore you are now beingintroduced to a third type of MTF curvewhich is not so commonly known.

Here the MTF values are not plotted asfunctions of the spatial frequency or thepicture height, but of a focus parameter.For this we measure how the MTFchanges in the longitudinal direction onthe image side of the lens and thusobtain the following curves:

Lens focused in the centre at f/1.4

0

10

20

30

40

50

60

70

80

90

100

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Lens > Pressure plate

M T F [ % ]

1 0

, 2 0

, 4 0

L p

/ m m

The MTF for 10, 20 and 40lp/mm is againplotted on the vertical axis. The zero pointon the horizontal axis corresponds to thebest focus: The MTF value for themedium frequency 20lp/mm is a maximumthere, so this is where the sensor or thefilm should be positioned, as symbolizedby the yellow line. To the left we are closerto the lens, to the right we are behind thesensor.

Incidentally, you can see that the tolerancerange for using the best MTF values at thisaperture is only a few hundredths of amillimeter. The two black triangles showthe depth of focus on the image side,calculated purely geometrically for a circleof confusion of 0.03mm diameter.

Incidentally, with this criterion for thedepth of focus, the MTF values at thelimit of the area which counts as focusedare around 20% at 40lp/mm.

Incidentally, it is quite possible for themaxima for different spatial frequenciesto be at different positions. And thecurves are often skewed, which meansthat the type of blurring is different infront of and behind the focus.

So what can happen when one stopsdown the lens aperture? We decreasethe aperture of this lens by three stopsand repeat the measurement, but do notchange our focusing scale, i.e. the zerostill means:MTF maximum in the center of thepicture at 20lp/mm and f-stop 1.4.


25/33


Lens stopped down to f/4, image centre

0

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30

40

50

60

70

80

90

100

-0.3 -0.2 -0.1 0 0.1 0.2 0.3


M T F [ % ]

1 0

, 2 0

, 4 0 L p

/ m m

2.0

2.5

3.0

3.5

4.0

4.5

O b j e c

t d i s t a n c e

[ m ]

The maximum values of the MTF curvesincrease significantly, of course, becausestopping down greatly reduces theresidual aberrations. At the same time,however, we see a shift of the curves tothe right, i.e. further away from the lens.The lens is now not at all optimallyfocused onto our sensor position (yellow

line), the MTF increase does not take fulleffect at that position. The geometricallycalculated depth of focus on the imageside is quite wrong, its length is still alright,but the position is wrong.

This phenomenon is called ‘focus shift’; itis usually more marked with extremelyhigh-speed lenses and is connected to thespherical aberration, because this meansthat beams of light which pass through theaperture area at different distances fromthe optical axis have a different focus.

The focus shift here is about 0.05mm.The black dots in the above graphshow how this shift in the image spaceis connected to the distances in theobject space in front of the camera(scale on the right hand side). If thelens was originally focused withaperture 1.4 at 3m distance, for

example, the best focus has nowmoved to 3.25m if the lens settings arenot altered.

Should this focus shift be correctedwhen taking a photograph? Not really,unless the best possible performancereally has to be in the center of thepicture. But 0.05mm is about 20% ofthe distance between the markings ofthe depth of focus scale for aperture 4,not so easy to control, therefore. Andanyway, it is a completely differentstory in the other parts of the frame.We therefore again measure the MTFin the longitudinal direction, not in thecenter of the picture, but at a distanceof 10mm.


26/33


10 mm image height, f/4, .... tangential, ___ sagittal

0

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30

40

50

60

7080

90

100

-0.3 -0.2 -0.1 0 0.1 0.2 0.3


M T F [ % ]

4 0

L p

/ m m

2.0

2.5

3.0

3.5

4.0

4.5

O b j e c

t d i s t a n c e

[ m ]

Away from the center of the picture we stillhave to distinguish between tangential andsagittal orientation; in order to keep thegraph clear, the curves for 10 and20lp/mm have now been left out.

We can now see that both curves areshifted less and even to the left. Theposition of the maximum thus moveswhen we move in the image area.


0

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60

70

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-0.3 -0.2 -0.1 0 0.1 0.2 0.3


M T F

[ % ]

4 0

L p

/ m m

2.0

2.5

3.0

3.5

4.0

4.5

O b j e c

t d i s t a n c e

[ m ]


27/33



0

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50

60

7080

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100

-0.3 -0.2 -0.1 0 0.1 0.2 0.3


M T F [ % ]

4 0

L p

/ m m

2.0

2.5

3.0

3.5

4.0

4.5

O b j e c

t d i s t a n c e

[ m ]

Here we have reached the edge of theimage at a distance of 18mm from thecenter of the image and we can see thatthe sagittal maximum has now returnedprecisely to the zero of our focusing scale.The field curvature should therefore not beimagined as a uniform curvature of theimage area, but there are reversal points.

This combination of residual fieldcurvature and focus shift leads in anycase to the fact that the MTF curves forthe same lens can look completelydifferent if we do not focus onto the localmaximum for every picture height, butmeasure strictly in a fixed plane:

f-number k = 4

0

20

40

60

80

100

0 5 10 15 20

u' [mm]

MTF [%]

Focus [µm]: axis 0

f-number k = 4

0

20

40

60

80

100

0 5 10 15 20

u' [mm]

MTF [%]

Focus [µm]: axis 0.05 mm

So these two graphs do not mean: “Theleft lens is a little worse in the center thanin the field, the right lens by contrast isvery good in the center of the picture, buthas a significantly deteriorating picturedefinition in the zone around 15mm pictureheight.“

Both measurements are from the samelens, but were taken with a slightlydifferent focus. This difference of0.05mm is of the same order asconventional mechanical cameratolerances such as adjusting the AF andthe focusing screen.


28/33


Limits to the significance of MTF curves

The relationships presented in the lastsection are a suitable transition to now

move on to talk about the limits of thisworld of numbers. If the shape of thecurves is so sensitive to small changes inthe focusing, one cannot, of course,expect to recognize the curves in everypicture if the subject is three-dimensional,i.e. when even the different distancescause some details to be focused well andothers not so well.

The measurement conditions of MTFcurves are comparable to reproductionphotography, where one plane is strictly

imaged onto another plane. The only othermember of this category is thephotography of subjects which are verydistant with short focal lengths.

The scales of the measured quantityMTF do not do justice to our perception.

Some experience is needed to be able totranslate the graph of the curves into aprediction of the subjective perception ofthe picture. One must take the viewingconditions into account; there is adifference whether one looks at an A4print or at a significantly larger 100%representation on a large monitor fromthe same distance.

The graphic appearance of the MTFcurves usually leads to the significanceof the 40lp curve for normal picture sizes

being overrated and the significance ofthe 10lp curve being underrated. If onelooks at a projected picture, for example,from about the projector distance forconventional projection focal length, thenthe normal human eye can resolve about20lp/mm at best from a 35mm format.

The reasons why the scale does notmatch our perception are partly that theMTF curves of lenses describe only thefirst link of the imaging chain, of course,and do not take into account those thatfollow. Sensor, scanner, projectors, theeye, in short everything afterwards alsoalways has a transfer function whichdecreases towards high spatialfrequencies. And thus they lead to a

decrease in the variations of the lens athigh spatial frequencies because alltransfer functions are of course multiplied.Let us take slide projection as an example:The eye does not see what the 40lp/mmcurve is doing when one is sitting behindthe projector. A further cause is that oureye’s logarithmic perception of brightnessis not taken into account.

Thus there have already been manyinvestigations regarding the question ofhow to translate MTF measurement datainto a scale which is related to ourperception, including the Heynachernumbers used at Zeiss, and othervalues based on psycho-physical factorssuch as SQF (subjective quality factor),MTFA (modulation transfer area), SQRI (square root integral). Their common

feature is that they all compute areasbelow the ‘modulation over spatialfrequency’ curve. A further common feature is thereforethat they all attempt to describe thequality at one image point by one singlenumber. As we have seen above, this is,of course, sometimes an unduesimplification of the data.Unfortunately, the scope of this articleprecludes me from going any deeper intothis.


29/33


Phase transfer function

The desire for simplification is also thereason why, until now, I have withheldsomething from the reader: The MTF

values are nowhere near the whole truthabout the correction state of a lens. Butno-one should really be surprised that asystem as complex as a lens cannot becompletely described by only these fewnumbers. The performance data of a lens,irrespective of whether it is calculated bycomputer or measured in the laboratory,fill a small file.

Simplification is necessary in order tomake it digestible and clear, but one hasthen to put up with the fact that the

precision of the description suffers.

So much to the introduction, and now tomore concrete details again: It ispossible that two lenses which have thesame MTF data produce quite differentimages of one detail of the subject, notrandomly, but systematically – here is anexample:

Images of two high-speed wide angle lenses at full aperture, details near theedge.

The picture shows the roof of a house anda tree in front of a bright sky, i.e. a typicalpicture of a horizon which is rich incontrast. The MTF at low spatialfrequencies is particularly important at theedges of the dark foreground objects,because it determines the amount of glareat these edges. In the picture on the left,the roof exhibits no glare, but the treedoes, in the picture on the right it is theother way round. If there were no tree inthis picture, one would judge the pictureon the left to be the better one (in black-and-white at any rate). At these edgepicture heights both lenses have the sameMTF values for all spatial frequencies,however.

The MTF does not tell us anything aboutthis difference, because it does not yetcompletely describe the characteristics ofthe point spread function.

The really complete optical transferfunction OTF also has a second part,the phase transfer function PTF, whichis usually neglected. It has something todo with the symmetry of the point spreadfunction.

We did take into consideration the factthat the point spread functions can beelongated, that they therefore havedifferent spreads in the tangential andthe sagittal directions. We thereforemeasure two MTF curves for each imagepoint.In the previous examples, however, wehave tacitly assumed that the brightnessdistribution is symmetric in one cross-sectional direction of the point spreadfunction. In reality, this is often not thecase, however.


30/33


Point spread functions can be as skewed as inthe following example. The most frequentcauses are coma errors

which produce point spread functionswith a tail in the radial direction.

0.1

1.0

10.0

100.0

- 6 0 - 4 0 - 2 0 0 2 0 4 0 6 0

µm

R e

l a t i v e

i n t e n s

i t

0

1

2

3

4

5

6

7

- 3 0

- 2 5

- 2 0

- 1 5

- 1 0

- 5 0 5

1 0

1 5

2 0

2 5

3 0

Edge profile [µm]

I n t e n s

i t y

[ E V ]

0

10

20

30

40

50

60

70

80

90

100

5 10 20 40 80 [Lp/mm]

M T F [ % ]

0.1

1.0

10.0

100.0

- 6 0

- 4 0

- 2 0 0

2 0

4 0

6 0

µm

R e

l a t i v e

i n t e n s

i t

0

1

2

3

4

5

6

7

- 3 0

- 2 5

- 2 0

- 1 5

- 1 0 - 5 0 5 1

0 1 5

2 0

2 5

3 0

Edge profile [µm]

I n t e n s

i t y

[ E V ]

0

1020

30

40

50

60

70

80

90

100

5 10 20 40 80

[Lp/mm]

M T F [ % ]

The orientation of an edge is veryimportant for such a skewed point spreadfunction intensity profile, of course. This

point spread function has a halo of 1% ofthe maximum intensity on the left, on theright it stops suddenly. If the bright side ofthe edge is to the right, it will produceglare to the left (bottom). If, however, theopposite is the case and the left side of theedge is bright (top), then the contrast ofthe edge image is high because the pointspread function only extends a shortdistance to the right.

The MTF values do not take account ofthis dependence on orientation. This iscontained in the phase transfer function,

which differs depending on theorientation of the “tail” of the pointspread function. The name stems fromthe fact that such a skewed point spreadfunction shifts the phase, i.e. the positionof its maxima and minima, of thesinusoidal pattern sideways.


31/33


Color correction

The fact that the optical characteristics ofglass depend on the wavelength of thelight can also be seen in our pictures:

Lenses have color aberrations. While it istrue that each lens has a sophisticatedcompensation system which uses acombination of different types of glass, sothat this type of aberration is usually nolonger critical, some residual aberration isstill present.

There are lenses where the coloraberrations are more critical; this is mainlyat long focal lengths, where only veryrecently has it been possible tosignificantly improve the image quality

thanks to the development of completelynew types of glass.

Long telephoto lenses without thesetypes of glass having extremely lowdispersion or anomalous partial

dispersion have only mediocre MTFvalues. It is nevertheless possible toachieve astonishingly good imagingresults for many subjects.

This is because the MTF of these lensesis strongly dependent on the spectralcomposition of the light. If themeasurement is done with green lightinstead of the conventional white light,where all wavelengths of the visiblespectrum are present with a certainweighting, the MTF curves are

dramatically different:

f-number k = 4 f = 300 mm

0

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40

60

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0 5 10 15 20

u' [mm]

MTF [%]

f-number k = 4 f = 300 mm

0

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0 5 10 15 20

u' [mm]

MTF [%]

MTF curves of a 300mm telephoto lens,on the left measured with white light, onthe right with green light, 100nmbandwidth.

This is why green filters were an important

accessory in the time of black-and-whitephotography. This same effect can also beachieved in color photography if thesubject is predominantly monochrome(nature photographs, red roofs). This is yetanother reason why the imagingperformance is not completely representedby MTF curves.

But it is not always the case that MTFcurves gauge a lens too pessimistically.On the contrary, it is possible that aweakness in the color correction is notvisible in the MTF data for white light. Inother words: MTF says little about color

fringes.


32/33


Comparing MTF in white and in colouredlight helps to understand the reasons ofcolour fringes in images of high contrastedges and highlights.

The following curves illustrate thelongitudinal chromatic aberration of ahigh speed short telephoto lens bymeasuring MTF as a function of focus:

0

10

20

30

40

50

60

70

80

90100

-0.1 -0.05 0 0.05 0.1

Lens > pressure plate

M T F [ % ]

2 0

L p

/ m m

4.5

4.6

4.7

4.8

4.9

5.0

5.1

5.2

5.3

5.45.5

S u

b j e c

t d i s t a n c e

[ m ]

Focus MTF of the wide open Planar 1.4/85 ZA in white light (black curve) and in blue,green and red light. The cross symbols connect position on the image side (horizontalscale) to the subject distance (vertical scale on the right), the lens has been focused inwhite light to a distance of 5m.

MTF values in coloured light are higherthan in white light, but at the same timethe maximum is at different positions, theydon’t have a coincident focus. In the bestfocus for white light (position 0) the redlight MTF is he lowest of all. From thatfollows, that the red line spread has thelargest diameter; the image then shows aslight reddish fringe. This is getting evenstronger, when the subject is at slightlyshorter distance, where the green MTF isat its maximum. Thus this kind of fastlenses produces fringes at highlight detailswhich are red or purple, if the detail is infront of the focal plane, and which isgreen,

if the detail is behind the focal plane. Thesaturation of these colours, calledsecondary spectrum, depends on thedistance of the MTF peak positions andon the slope of the focus MTF curves. Iflenses exhibit more monochromaticaberrations (like old lenses), the curvesare more flat and the colours look pale.Just modern, highly corrected fast lensestend to show more saturated colours.Sine the distances between the peakpositions can’t be made infinitely small,the only chance to make the fringesdisappear is stopping down, since thenthe dept of focus is large compared tothe longitudinal colour aberration, andcoloured MTF differences are gettingsmall.

0

10

20

30

40

50

60

70

80

90

100

-0.4 -0.2 0 0.2 0.4

Lens > pressure plate

M T F [ % ]

2 0

L p

/ m m

4.0

4.5

5.0

5.5

6.0

6.5

7.0

S u

b j e c

t d i s t a n c e

[ m ]

Focus MTF of the Planar 1.4/85 ZA at f/ 5.6


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Bokeh

Curves where the tangential and sagittalvalues are nearly identical in the wholevisual field are often called ideal MTF

curves because, in these cases, the“bokeh”, i.e. the representation of themarkedly defocused background, isparticularly good.

Such statements should be regarded withcaution. MTF only makes statementsabout the focal plane or its immediatesurroundings. And in that case, a circularpoint spread function is indeed anadvantage, because it reproduces smalldetails in a way which is as faithful to theoriginal as possible, with the best trueness

of shape. This is important for the legibilityof writing, for example.

It is not possible to use MTF data todraw conclusions about the brightnessdistribution of the strongly defocused

point spreads, however. There arelenses with nicely parallel tangential andsagittal MTF curves but which arespherically strongly overcorrected. Thiscorrection state causes annulardefocused point spread functions, whichare visible as rim-lights and as pairs oflines and produce a restless-lookingbackground.

This unpleasant characteristic cannot bededuced from the MTF data.

Comparability of MTF data

MTF data is published in manypublications, by manufacturers and nowalso in many independent tests.Unfortunately, one has to be very cautious

when comparing this data, because themeasurement conditions can vary greatly.

The least of the problems would be tooverlook the fact that the spatialfrequencies are different. And, likewise,different spectral weightings of the visiblelight can also leave a comparison wanting.There are also manufacturers who are notafraid to publish data which are better thandiffraction limitations allow, i.e. it isphysically impossible.

This tells you that these values aremerely computed and that this was doneby taking only geometrical optics intoaccount, without considering the wave

nature of light. If the lenses haveexcellent correction, the values stick tothe 100% line. But, please, don’t believethat these numbers are realistic. Reallenses are always a little worse than thecalculation of the optical design program.

MTF data published by Zeiss alwaysoriginates from measured lenses.

Date post:	07-Aug-2018
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