spatial resolution optical systems

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This version reflects the comments of the core participants as reviewed and

incorporated in accordance with CORD's FIPSE-supported Curriculum

Morphing Project.

MODULE 10-8

ESPATIAL

RESOLUTION

OF OPTICAL

SYSTEMS

INTRODUCTION

An optical system is normally designed to give information aboutthe object being viewed. Usually, the information appears in theform of an image. The amount and quality of information depends

on whether the object is surrounded by a dark or light background(contrast), the size of the object (spatial frequency), and the qualityof the optical system. The modulation transfer function (MTF) of anoptical system is a measure of the systems imaging capabilities andwill largely determine the amount of fine detail that will be observedin the image. This module examines and explains in some detail theconcept and utility of the modulation transfer function, particularlyas it applies to resolution.

MODULE PREREQUISITES

The student should have completed Module 1-4, "Properties ofLight"; Module 1-8, "Temporal Characteristics of Lasers"; Modules2-8 through 2-11 of Course 2, "Geometrical Optics"; Module 6-1,"Optical Tables and Benches"; Module 6-8, "Lenses"; Module 7-8,"Mechanical and Bleachable Dye Methods"; and Module 9-6,"Power Supply and Calibration of a Photomultiplier." The studentshould also have a basic knowledge of algebra and geometrical andwave optics, and be able to operate a helium-neon laser,photomultiplier, and electrometer.

Course # 10: Module 8: Spatial Resolution of Optical Systems http://cord.org/cm/leot/course10_Mod08/Module10-8.htm

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Upon completion of this module, the student should be able to:

1. Explain how the modulation transfer function of an opticalsystem determines the quality of the optical system.

2. Explain the difference between the square-wave and thesine-wave MTF.

3. Calculate the MTF of an optical system, given the MTFs of theindividual components.

4. Set up the equipment and measure the square-wave MTF of twolenses.

Historically, the earliest measure of performance for an optical

component or instrument was resolving power. Resolving power ofan optical system refers to the ability to separate two closely spacedobjects in the image generated by the optical system. To test thequality of an optical system, it is common to test it with linepatterns. The resolving power is commonly expressed in terms of aspatial frequency (i.e., lines/mm or line pairs/mm).

A type of object frequently used to test the performance of anoptical system consists of a series of alternating light and dark barsof equal width with sharp boundaries, as indicated in Figure 1a.

Fig. 1Typical bar object or target with corresponding intensity plot


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If the pattern of the bars is repeated inXmillimeters (1 period), thenthe pattern has a frequency of 1/Xlines per millimeter (lines/mm). Aplot of the intensity of the light transmitted by the bar target isshown in Figure 1b. When an image is formed by an optical system,each point is imaged as a blurred spot due to aberrations, diffraction,

scattering, and absorption.

The usual procedure is to photograph such a test chart and then pickout the finest pattern in which individual lines can be identified. Thereciprocal of the width of a line-plus adjacent-space is called thelimiting resolving power of the system. As an example, the testchart shown in Figure 2 has been imaged by a 250 mm lens. Thisimage is shown in Figure 3.

Fig. 2 Fig. 3

Official Air Force resolution test chart made Image of USAFtest chart up of a series of progressively smallerpatterns using 250 mm imaging lens

As the bar becomes smaller, it becomes more difficult to clearlyidentify the individual lines. For purposes of calculation, assumethat the finest pattern which is discernible is group 1 element 1. Thelimiting resolving power of the optical system, which in this caseincludes the 250 mm lens and the photographic film, can bedetermined by measuring the period of the bar pattern in group 1element 1 in the test target shown in Figure 2. The period is 0.5 mm.

Thus, the limiting resolution for this hypothetical case is 2lines/mm. The reason the lines are more difficult to see as thepattern becomes smaller is that the apparent change in intensitybetween the dark and white regions is decreasing as one approachesthe limiting resolution of the system. The intensity pattern forseveral spatial frequencies is shown in Figure 4. (The spatialfrequencies vary from low on the top curve to high on the bottomcurve.)


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Fig. 4Intensity pattern for several spatial frequencies

The difference in intensity between the dark and white regions is thesame for all frequencies at the object. But notice how it decreases inthe image as the spatial frequency increases. The dark regionsbecome lighter and the white regions darker, i.e., the contrastdecreases. Also notice how the sharp discontinuities in the objecthave been rounded off in the image. When the contrast in the imageis smaller than the system (e.g., the eye, film, or photodetector) candetect, the pattern can no longer be resolved.

The contrast Cis defined in terms of intensityI, as given byEquation 1.

C=Equation 1

If the background is perfectly black,Imin

= 0, then

C= = 1

Equation 2

which is the highest contrast possible. If the bars and the intervalsare mere shades of gray, the contrast decreases correspondingly.Least contrast will result ifI

max=I

min, in which case C= 0. Thus,

contrast may vary between zero and one. The human eye requires

approximately 5% contrast to resolve an image.


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Equation 1 is used to solve a problem in Example A.

Example A: Calculation of the Contrast

Given:

An intensity pattern such as the oneshown in Figure 4 has a maximumintensityI

max= 0.85 and a minimum

intensityImin

= 0.35.

Find: The contrast of the intensity pattern.

C=

(Equation 1)

Solution:

C=

=

C= 0.417

To evaluate an optical system, more information than the limitingresolution is required. To determine how much contrast can betransferred from the object to the image at all spatial frequencies,the modulation transfer function (MTF) is needed. The MTF is theratio of the modulation in the image to that in the object as afunction of spatial frequency. Thus, if one plots the contrast (image-to-object) as a function of spatial frequency, one obtains a curve (theMTF) for the particular optical component or system. Two suchcurves for two different imaging systems are shown in Figure 5.

Notice that both systems have the same limiting resolutionfrequency. However, the system represented byA will produce asuperior image because the greater modulation at lower frequencieswill produce crisper, more contrasting images. Unfortunately, thetype of choice that one is usually faced with in choosing a system isnot as clear as that implied by the curves in Figure 5.


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Fig. 5MTFs for two optical systems having

identical limiting resolution frequencies

Consider the two systems shown in Figure 6, where one systemshows limiting resolution (B) and the other shows high contrast at

low spatial frequencies (A). In situations of this kind, the decisionmust be based on the relative importance of contrast versusresolution in the specifically intended function of the system.

Fig. 6MTFs for two optical systems having

different limiting resolution frequencies

The preceding discussion has been based on square-wave intensitypatterns and the square-wave MTFs. However, if the object patternis in the form of a sine wave, the intensity distribution in the imageis also described by a sine wave. If the MTF is not indicated as

square-wave MTF, it is generally assumed to be a sine-wave MTF.

Sine-wave targets are quite difficult to obtain, so generally it iseasier to measure the square-wave MTF and then convert thesquare-wave MTF to the sine-wave MTF mathematically. There aresystems that measure the sine-wave MTF of systems, but they arevery costly compared to those capable of measuring square-waveMTFs. However, the square-wave MTF can be converted to thesine-wave MTF using the following equation:

Equation 3


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MTF (f1)

sine wave =

where:

Bk

= (1)M(1)(k 1)/2 ift=M

Bk

= 0 ift


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Find: Bkfor k= 9 and k= 11.

Solution:

For k= 9, the number of different primefactors is one. The prime factor is 3.3. Thus, t= 1 andM= 2. Since t


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sine-wave MTF of 0.4 0.8 = 0.32. If the object photographedwith this camera has a contrast of 0.2, then the image will have a

contrast of 0.32 0.2 = 0.064. However, note that the MTF of asystem does not equal the product of the MTFs of the individualcomponents if the components are not directly connected; that is,the lenses are not separated by diffusers. Aberrations of onecomponent may compensate for the aberrations in other componentsin a system of lenses and, thus, produce an image quality which issuperior to that of either component. Any "corrected" optical systemillustrates this point.

Figure 8 shows the MTF for a correctly focused f/4 lens system freeof aberrations, transmitting quasimonochromatic light of a meanwavelength l = 500 nm. The MTFs of two typical photographic

emulsions are also shown.

Fig. 8Typical MTFs

Helium-neon laser (1-5 milliwatts)

Beam-expanding telescope

Photometer

Translator (micron resolution)

Piece of diffusing glass

Set of spatial frequency targets

One-micron slit

6328 filter

Iris


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High-quality 25 cm focal length lens

Poor-quality 25 cm focal length lens

Isolation table

Before beginning, familiarize yourself with and heed all appropriatesafety rules concerning the use of laser systems. Avoid the hazardsof high-voltage electrical systems. The following tasks will beperformed:

1. Set up the equipment necessary to measure the square-wave MTFof lenses.

2. Measure the square-wave MTF of the poor-quality lens.

3. Measure the square-wave MTF of the poor-quality lens when thediameter of the lens is half that used in Task 2.

4. Measure the square-wave MTF of the high-quality lens.

The experimental arrangement as shown in Figure 9 should first be

constructed. To minimize vibrations, it is recommended that theentire experimental apparatus be mounted on an isolation table.

Fig. 9Experimental arrangement for measuring

the square-wave MTF of a lens

The expanded helium-neon laser beam impinges on a diffusing glassto eliminate the coherence of the laser beam and fully fill the testlens aperture. The spatial frequency target is mounted just behind

the diffusing glass in a mount so that various spatial frequencytargets can be placed in identical positions with the bars oriented


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vertically. The distance between the target and the lens to be testedshould be approximately twice the focal length of the lens. A 632.8nm filter should be mounted in a light-tight manner directly in frontof the photometer to eliminate spurious room light from getting intothe detector. This eliminates the necessity of performing theexperiment in a darkened room. The one-micron slit is placed

directly in front of the 632.8 nm filter, absolutely parallel to the barsof the spatial target, and the entire assembly (slit, filter, anddetector) is mounted on the translator so that the slit can be movedacross the image of the spatial target, as shown in Figure 10. Thistotal assembly should be mounted perpendicular to the optical axisof the laser-lens-target system.

Fig. 10Arrangement of slit and spatial target

Move the translator with the mounted detector, filter, and slit so theimage of the spatial target falls exactly on the slit. This adjustmentis very critical because the resultant MTF will be considerably less

if the spatial target is not focused properly on the slit, thus giving amisleading result.

The apparatus is now ready to measure the MTF of the lens. Withthe 0.25 line/mm spatial target in place, adjust the translator for aminimum reading (R

1) and record. Next, adjust the translator for a

maximum reading (R2) and record. From these two readings, the

contrast of the image is calculated with the aid of Equation 5.

Cimage

=

Equation 5

The Cobject

is equal to one because the bars are perfectly black.

Thus,


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= Cimage

=Equation 6

To measure the square-wave MTFs at other frequencies, remove the0.25 line/mm spatial target and replace it with the other spatialtargets, and repeat the measurements. It should not be necessary torefocus the target onto the slit if care is taken to replace the spatialtargets in the holder in identical orientation. In fact, at higher spatialfrequencies (100 lines/mm or higher), it is impossible to see the barswithout a microscope. Therefore, refocusing the lens is verydifficult.

To perform Task 3, mount an iris directly in front of the lens andadjust the diameter so that it is half the size of the lens diameter.

Now measure the square-wave MTF of the lens. Do you notice anydifference in the square-wave MTFs of the masked and unmaskedlens? What led to this result? Is it reasonable?

In Task 4, remove the iris and replace the poor lensA with thehigh-quality lensB. Place the 0.25 line/mm spatial frequency targetin the holder. Move the translator with attached equipment until asharp image is on the slit. Repeat the measurements as before toobtain the square-wave MTF.

Now by comparing the square-wave MTFs, the lens that would bebest for a particular situation can be chosen. If this lens is going tobe used in an optical system, the square-wave MTF of the lens hasto be converted to a sine-wave MTF before the system MTF can becalculated.

1. What is the spatial frequency of the pattern of bars shown below?

2. A film negative of a farm house also shows a picket fence. Theindividual stakes are barely visible. A measurement shows that the

width of one stake is 0.1 mm and the space between adjacent stakesis 0.2 mm.


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a. What is the limiting resolving power of the cameraand film?

b. If the MTF of the lens is 1.0 at 3 lines/mm, what is theMTF of the film used? (Recall from the test that thehuman eye requires approximately 5% contrast toresolve an image.)

3. Calculate the factorBk

for k= 12 and k= 13.

4. Assume a lens has a square-wave MTF which is one if thefrequency is less than 100 lines/mm and is zero for frequencies over100 lines/mm. Calculate the sine-wave MTF for frequenciesbetween 1 and 100 lines/mm.

5. Assume the film to be used with the lens above has a sine-wave

MTF as shown below. Calculate the system MTF.

6. Describe the probable result that would be obtained if the slit in

front of the photomultiplier in the procedure was canted at an anglewith respect to the target test pattern.

7. The square-wave MTF for an optical system is:

Frequency (lines/mm) Square-wave MTF

10 1

20 1

30 1

40 1

50 1

60 1

70 1

80 0.8

90 0.7


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100 0.6

110 0.4

120 0.3

130 0.2

140 0.1

150 0

Calculate the corresponding sine-wave MTF.

Hecht and Zajac. Optics. Reading, MA: Addison-Wesley Publishing Co., 1974.

Hopkins and Slaymaker. "Knife Edge Testing of OTF in Optical Systems,"Electro-OpticalSystems Design, 4 (13) Dec. 1972. pp. 27-29.

Jensen, N. Optical and Photographic Reconnaissance Systems. New York: John Wiley andSons, Inc., 1968.

Meyer-Arendt, J. R. Classical and Modern Optics. Englewood Cliffs, NJ: Prentice-Hall, Inc.,1972.

Modern Applications of Physical Optics. Wiley-Interscience, 1973.

Nussbaum and Phillips. Contemporary Optics for Scientists and Engineers. Englewood Cliffs,NJ: Prentice-Hall, Inc., 1976.

Shulman, A. R. Optical Data Processing. New York: John Wiley and Sons, 1970.

Smith, David. "OFTQuantitative Image Analysis,"Electro-Optical Systems Design. Dec.

1979. p. 39.

Smith, W. J.Modern Optical Engineering. New York: McGraw-Hill, 1966.

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