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12/17/12 ZEISS Microscopy Online Campus | Microscopy Basics | Numerical Aperture and Resolution

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Introduction

The numerical aperture of a microscope objective is the measure of itsability to gather light and to resolve fine specimen detail while working at afixed object (or specimen) distance. Image-forming light waves passthrough the specimen and enter the objective in an inverted cone asillustrated in Figure 1(a). White light consists of a wide spectrum ofelectromagnetic waves, the period lengths of which range between 400 and700 nanometers. As a reference, it is important to know that 1 millimeter equals 1000micrometers and 1 micrometer equals 1000 nanometers. Light of green color has a wavelengthrange centered at 550 nanometers, which corresponds to 0.55 micrometers. If small objects(such as a typical stained specimen mounted on a microscope slide) are viewed through themicroscope, the light incident on these minute objects is diffracted so that it deviates from theoriginal direction (Figure 1(a)). The smaller the object, the more pronounced the diffraction ofincident light rays will be. Higher values of numerical aperture permit increasingly oblique rays toenter the objective front lens, which produces a more highly resolved image and allows smallerstructures to be visualized with higher clarity. Illustrated in Figure 1(a) is a simple microscopesystem consisting of an objective and specimen being illuminated by a collimated light beam,which would be the case if no condenser was used. Light diffracted by the specimen ispresented as an inverted cone of half-angle (α), which represents the limits of light that can enterthe objective. In order to increase the effective aperture and resolving power of the microscope, acondenser (Figure 1(b)) is added to generate a ray cone on the illumination side of the specimen.This enables the objective to gather light rays that are the result of larger diffraction angles,increasing the resolution of the microscope system. The sum of the aperture angles of theobjective and the condenser is referred to as the working aperture. If the condenser apertureangle matches the objective, maximum resolution is obtained.

In order to enable two objectives to be compared and to obtain a quantitative handle on

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resolution, the numerical aperture, or the measure of the solid angle covered by an objective isdefined as:

Numerical Aperture (NA) = η • sin(α) (1)

where α equals one-half of the objective's opening angle and η is the refractive index of theimmersion medium used between the objective and the cover slip protecting the specimen (η = 1for air;; η = 1.51 for oil or glass). By examining Equation (1), it is apparent that the refractive indexis the limiting factor in achieving numerical apertures greater than 1.0. Therefore, in order toobtain higher working numerical apertures, the refractive index of the medium between the frontlens of the objective and the specimen cover slip must be increased. The highest angularaperture obtainable with a standard microscope objective would theoretically be 180 degrees,resulting in a value of 90 degrees for the half-angle used in the numerical aperture equation. Thesine of 90 degrees is equal to one, which suggests that numerical aperture is limited not only bythe angular aperture, but also by the imaging medium refractive index. Practically, apertureangles exceeding 70 to 80 degrees are found only in the highest-performance objectives thattypically cost thousands of dollars.

The resolution of an optical microscope is defined as the smallest distance between two pointson a specimen that can still be distinguished as two separate entities. Resolution is directlyrelated to the useful magnification of the microscope and the perception limit of specimen detail,though it is a somewhat subjective value in microscopy because at high magnification, an imagemay appear out of focus but still be resolved to the maximum ability of the objective and assistingoptical components. Due to the wave nature of light and the diffraction associated with thesephenomena, the resolution of a microscope objective is determined by the angle of light wavesthat are able to enter the front lens and the instrument is therefore said to be diffraction limited.This limit is purely theoretical, but even a theoretically ideal objective without any imaging errorshas a finite resolution.

Observers will miss fine nuances in the image if the objective projects details onto theintermediate image plane that are smaller than the resolving power of the human eye (a situationthat is typical at low magnifications and high numerical apertures). The phenomenon of emptymagnification will occur if an image is enlarged beyond the physical resolving power of theimages. For these reasons, the useful magnification to the observer should be optimally above500 times the numerical aperture of the objective, but not higher than 1,000 times the numericalaperture.

Immersion Media

One way of increasing the optical resolving power of the microscope is to use immersion liquidsbetween the front lens of the objective and the cover slip. Most objectives in the magnificationrange between 60x and 100x (and higher) are designed for use with immersion oil. Good resultshave been obtained with oil that has a refractive index of n = 1.51, which has been preciselymatched to the refractive index of glass. All reflections on the path from the object to the objectiveare eliminated in this way. If this trick were not used, reflection would always cause a loss of lightin the cover slip or on the front lens in the case of large angles (Figure 2).

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Fluorescence Microscopy

Polarized Light

Microscope Ergonomics



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The useful numerical aperture of the objective and therefore the resolving power would be

reduced by the reflection described above. The numerical aperture of an objective is also

dependent, to a certain degree, upon the amount of correction for optical aberration. Highly

corrected objectives tend to have much larger numerical apertures for the respective

magnification as illustrated in Table 1.

Objective Numerical Aperture versus Optical Correction

MagnificationPlan Achromat

(NA)

Plan Fluorite

(NA)

Plan Apochromat

(NA)

0.5x 0.025 n/a n/a

1x 0.04 n/a n/a

2x 0.06 0.08 0.10

4x 0.10 0.13 0.20

10x 0.25 0.30 0.45

20x 0.40 0.50 0.75

40x 0.65 0.75 0.95

40x (oil) n/a 1.30 1.40

63x 0.75 0.85 0.95

63x (oil) n/a 1.30 1.40

100x (oil) 1.25 1.30 1.40

Table 1

The Airy Disk and Microscope Resolution

When light from the various points of a specimen passes through the objective and is

reconstituted as an image, the various points of the specimen appear in the image as small

patterns (not points) known as Airy patterns. This phenomenon is caused by diffraction or

scattering of the light as it passes through the minute parts and spaces in the specimen and the

circular rear aperture of the objective. The limit up to which two small objects are still seen as

separate entities is used as a measure of the resolving power of a microscope. The distance

where this limit is reached is known as the effective resolution of the microscope and is denoted

as d0. The resolution is a value that can be derived theoretically given the optical parameters of

the instrument and the average wavelength of illumination.

It is important, first of all, to know that the objective and tube lens do not image a point in the

object (for example, a minute hole in a metal foil) as a bright disk with sharply defined edges, but

as a slightly blurred spot surrounded by diffraction rings, called Airy disks (see Figure 3(a)).

Three-dimensional representations of the diffraction pattern near the intermediate image plane

are known as the point-spread function (Figure 3(b)). An Airy disk is the region enclosed by the

first minimum of the airy pattern and contains approximately 84 percent of the luminous energy,

as depicted in Figure 3(c). The point-spread function is a three-dimensional representation of the

Airy disk.



Resolution can be calculated according to the famous formula introduced by Ernst Abbe in thelate 19th Century, and represents a measure of the image sharpness of a light microscope:

Resolutionx,y = λ / 2[η • sin(α)] (2) Resolutionz = 2λ / [η • sin(α)]2 (3)

where λ is the wavelength of light, η represents the refractive index of the imaging medium asdescribed above, and the combined term η • sin(α) is known as the objective numerical aperture(NA). Objectives commonly used in microscopy have a numerical aperture that is less than 1.5,restricting the term α in Equations (2) and (3) to less than 70 degrees (although new high-performance objectives closely approach this limit). Therefore, the theoretical resolution limit atthe shortest practical wavelength (approximately 400 nanometers) is around 150 nanometers inthe lateral dimension and approaching 400 nanometers in the axial dimension when using anobjective having a numerical aperture of 1.40. Thus, structures that lie closer than this distancecannot be resolved in the lateral plane using a microscope. Due to the central significance of theinterrelationship between the refractive index of the imaging medium and the angular aperture ofthe objective, Abbe introduced the concept of numerical aperture during the course of explainingmicroscope resolution.

The diffraction rings in the Airy disk are caused by the limiting function of the objective aperturesuch that the objective acts as a hole, behind which diffraction rings are found. The higher theaperture of the objective and of the condenser, the smaller d

0 will be. Thus, the higher the

numerical aperture of the total system, the better the resolution. One of the several equationsrelated to the original Abbe formula that have been derived to express the relationship betweennumerical aperture, wavelength, and resolution is:

Resolutionx,y or d0 = 1.22λ / [NAObj + NACon] (4)

Where λ is the imaging wavelength of light, NACon

is the condenser numerical aperture, andNA

Obj equals the objective numerical aperture. The factor 1.22 has been taken from the

calculation for the case illustrated in Figure 4 for the close approach of two Airy disks where theintensity profiles have been superimposed. If the two image points are far away from each other,they are easy to recognize as separate objects. However, when the distance between the Airydisks is increasingly reduced, a limit point is reached when the principal maximum of the secondAiry disk coincides with the first minimum of the first Airy disk. The superimposed profiles displaytwo brightness maxima that are separated by a valley. The intensity in the valley is reduced byapproximately 20 percent compared with the two maxima. This is just sufficient for the humaneye to see two separate points, a limit that is referred to as the Rayleigh criterion.



A comparison may help to make this easier to understand. It is most unlikely that a telephone

cable would be used for the electronic transfer of the delicate sound of a violin, since the

bandwidth of this medium is very restricted. Much better results are obtained if high-quality

microphones and amplifiers are used, the frequency range of which is identical to the human

range of hearing. In music, information is contained in the medium sound frequencies;; however,

the fine nuances of sound are contained in the high overtones. In the microscope, the subtleties

of a structure are coded into the diffracted light. If you want to see them in the imaging space

behind the objective, you must make sure that they are first gathered by the objective. This

becomes easier with a higher aperture angle and thus an increased numerical aperture.

The numerical aperture of objectives increases with the magnification up to about 40x (see

Tables 1 and 2), but levels off between 1.30 and 1.40 (depending upon the degree of aberration

correction) for oil immersion versions. Presented in Table 2 are the calculated values for the

resolution of objectives typically used in research and teaching laboratories. The point-to-point

resolution at the specimen, d0, is listed in the table along with the magnified size of the image

(D0) in the intermediate eyepiece plane (using green light of 550 nanometer wavelength). Also in

the table, the value n represents the number of resolved pixels if they are organized in a linear

array along the field diameter of 20 millimeters (20 millimeters/D0).

Resolution for Selected Objectives

Objective/NAd0(μm)

D0(μm)

n

0.5x / 0.15 2.2 11.2 1786

10x / 0.30 1.1 11.2 1786

20x / 0.50 0.7 13.4 1493

40x / 0.75 0.45 17.9 1117

40x / 1.30 (oil) 0.26 10.3 1942

63x / 1.40 (oil) 0.24 15.1 1325

100x / 1.30 (oil) 0.26 25.8 775

Table 2

You should not try to increase the overall magnification of a microscope by using eyepieces

providing a high additional magnification (for example, 16x, 20x or 25x) or other optical after-

burners if the objective does not supply enough pixels at a low numerical aperture. On the other

hand, you will miss subtle nuances if the objective projects very fine details onto the intermediate

image, and you are using an eyepiece with a low magnification. In order to observe fine

specimen detail in the optical microscope, the minute features present in the specimen must be

of sufficient contrast and project an intermediate image at an angle that is somewhat larger than

the angular resolving power of the human eye. As previously mentioned, the overall combined

magnification (objective and eyepiece) of a microscope should be higher than 500x, but less

than 1000x the objective aperture. This value is known as the range of the useful

magnification.

In day-to-day routine observations, many microscopists do not attempt to achieve the highest

image resolution that is possible with their equipment. The resolving power of a microscope is

the most important feature of the optical system and influences the ability to distinguish between

fine details of a particular specimen. The primary factor in determining resolution is the objective

numerical aperture, but resolution is also dependent upon the type of specimen, coherence of

illumination, degree of aberration correction, and other factors such as contrast-enhancing



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methodology either in the optical system of the microscope or in the specimen itself.

Practical Hints to Increase Resolving Power

Modern microscope objectives permit the theoretical resolving power to be realized in practice

provided that suitable specimens are being observed. However, there are several hints that can

be followed to ensure success. These are listed below.

Are the objective and specimen clean? A fingerprint on the front lens of an air objective

may be sufficient to affect the high-contrast image reproduction of a specimen as a result

of unwanted scattered light. The same caveats apply to immersion objectives that are

soiled with residues of resin or emulsions (such as oil and water). In these cases, careful

cleaning of the objective front lens element using a soft cloth and lens cleaner or pure

ethanol should alleviate problems.

Do the cover slips have the correct thickness? It is of critical importance that cover

slips matched to objectives of high aperture (greater than 0.65) have the standard

thickness of 170 micrometers due to the fact that cover slip thickness is taken into

consideration when the objectives are designed. Therefore, if a cover slip of different

thickness (less than 165 or more than 175 micrometers) is used, the quality of the optical

image visibly suffers. In general, the rule of thumb for objectives having a numerical

aperture of 0.7 or higher is that they can tolerate a variation of 10 micrometers, whereas

lower numerical aperture objectives (0.3 to 0.7) can tolerate a higher level of deviation,

usually up to 30 micrometers.

Are you using the correct immersion oil? All objectives having a numerical aperture

greater than 0.95 are designed for use with immersion media (usually oil of refractive

index 1.515). Immersion oil is free of polychlorinated biphenyls and exhibits hardly any

autofluorescence. The image will be markedly impaired if air bubbles are introduced into

the immersion layer, so oil must be carefully applied to avoid bubbles. Air bubbles can be

readily visualized by removing the eyepiece and examining the objective rear focal plane

through the microscope observation tubes (or using a Bertrand lens). If air bubbles are

observed, the objective and specimen slide should be cleaned and the oil carefully re-

applied.

Contributing Authors

Rudi Rottenfusser - Zeiss Microscopy Consultant, 46 Landfall, Falmouth, Massachusetts, 02540.

Erin E. Wilson and Michael W. Davidson - National High Magnetic Field Laboratory, 1800 East Paul

Dirac Dr., The Florida State University, Tallahassee, Florida, 32310.

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