Optical calculations

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Calculations for the technical design exemplified by an objective with 10/0.25 and a condenser with NA 0.9

All methods developed (VBDC, VPDC, VPBC) use a combination of at least two different types of illumination that differ significantly with respect to their light yield. In addition, the apertures and optically effective cross-sectional areas of the contributing objectives also differ depending on their magnification and optical design. In view of these circumstances, I will use an objective/condenser combination laid out in this work which is also very commonly used in practice to carry out calculations on how the respective light transmission areas of the different methods can be designed to achieve an optimum light yield and balanced intensities of the superimposed partial images.

Methodological procedure:

To compare the respectively relevant light intensities and light yields, the manufacturer's specifications and my own measurements were used to calculate the f-values (f-numbers) corresponding to the aperture values commonly used in microscopy, which are regarded as standard in photography for specifying light intensities and image brightnesses.

The sequence of "classical" f-numbers used in photography has defined steps. Each subtracted step halves the light transmission area. Therefore, for a constant image brightness, the exposure time must be increased by the inverse value of the factor by which the transmission area decreases as the f-step increases (Table 1). Successive f-steps differ numerically by a factor of  respectively 1.414. The increment from one f-step to the next can also be given as integer multiples in EV levels (EV = exposure value). In this case, one f-step corresponds to one EV

Table 1: Relationships between aperture setting (f-stop, f-value) respectively light intensity, aperture area (light transmission area), exposure time and numerical aperture (NA), changes of aperture size and exposure time relative to the aperture area for an f-value of 2

Two equations are used to calculate the condenser aperture ratio CAR (f-value, f-number) and the corresponding numerical aperture NA (Henkel, 2003):

CAR = condenser aperture ratio (f-value)
f = focal distance,  NA = numerical aperture
d = diameter of the optical system (aperture breadth)

Corresponding mesaurements of several light outlets are compiled in Table 2.

Table 2: Light-transmission areas and derived parameters calculated for a condenser with
 NA = 0.9 and f = 13 mm, equipped with light stops (light masks on filter slides)

Further processing steps to determine suitable ratios of light-transmission areas for VBDC, VPDC and VPBC

Calculation of the areas of all implemented light-transmission openings (light outlets), and calculation of the diameters (= aperture breadths) of circles with the same area (Table 2

Determination of condenser aperture ratios (CAR, f-values) and NA values of the calculated circular openings having the same area (Table 2)

Determination of the required exposure times for different types of illumination for a given light source intensity and a given object (Table 3). Taking account of the respective transmission areas and exposure times, it can be concluded that an advantageous area ratio of the associated light openings produces a balanced intensity and so a comparable brightness of the partial images for all newly developed methods.

Table 3: Light efficiency (light yield) and effective light-transmission areas, corresponding alterations of exposure time and exposure examples, determined for different modes of illumination.
* The exposure data also apply to axial darkfield

Explanations for Tables 2 and 3:

Measurements were made of the light-transmission areas of the light stops on slides supplied by the manufacturers and used for phase contrast (Leitz, Phaco 1 for objective 10/0.25) and darkfield (original light mask, Leitz), as well as our own models (hand-made prototype) used for the new types of illumination (24 perforations for darkfield, in combination with a three-hole light stop for phase contrast to produce VPDC; axial light transmission for axial darkfield or brightfield, in combination with a standard sized phase-contrast light annulus to produce axial VPDC or axial VPBC; 8 perforations for concentric-peripheral brightfield, in combination with a standard sized phase-contrast light annulus to produce peripheral VPBC).

The relative light yields and image brightnesses of the different types of illumination were estimated using the exposure meter of a microphotographic camera to measure the exposures required respectively in brightfield, normal darkfield and standard phase contrast for suitable typical standard objects. The corresponding exposures were also determined for axial darkfield and concentric-peripheral darkfield based on the hand-made light openings for VPDC in the peripheral and axial darkfield light path.


Optical designs:

The basic optical designs of the phase-contrast objective Phaco 10/0.25 and the associated condenser for the light stop (NA 0.9) are shown in Figures 60 and 61.

Fig. 60: Optical design of the phase-contrast objective Phaco 10/0.25

For the Phaco 10/0.25 objective with NA = 0.25, f= 17 mm and a working distance of 7.6 mm, the following applies (see Fig. 60):

The equation NA= n* sin. (α) was used to determine the maximum angle of incidence α for the light entering the objective, in this case, 14.5°.

An aperture width of 8.5 mm (Fig. 60, right) and f = 17 mm gives a light intensity (f-number) of 2.0, in accordance with the light intensity of the condenser with an adjusted aperture width.

Fig. 61: Optical design of a typical condenser with NA = 0.9.
Ratios for a fully opened aperture diaphragm, respectively full aperture width (a)
and for adjustment of the aperture diaphragm or aperture width to the cross-sectional area of the objective (b)

For a condenser with NA= 0.9 and f =13 mm, the following applies (see Fig. 61):

For a fully opened aperture diaphragm (aperture width: 23.5 mm), the above-defined angle is α 65° (Fig. 61a).

Adjustment of the aperture diaphragm to the optically effective objective cross-section gives an aperture diaphragm diameter of 6.5 mm, according to
 NAobj.= NAcond.=0.25 and α =14,5° (Fig. 61b)   

In the example given above, the cross-sectional area of the objective (diameter: 8.5 mm) is optically congruent with an aperture diaphragm area having a diameter of 6.5 mm. Thus the aperture diaphragm (maximum diameter: 23.5 mm) can be reduced to a diameter of 6.5 mm without affecting the optically effective cross-sectional area of the objective. Consequently, for the 10/0.25 objective, the maximum effective diameter of the aperture diaphragm that can be used for brightfield illumination, is 6.5 mm. The observed brightfield image thus shows no visible changes on gradually narrowing the aperture diaphragm as long as its diameter is larger than 6.5 mm. As soon as the diameter of the aperture diaphragm drops below this value, the well-known changes in the brightfield image caused by closing the aperture diaphragm occur with the objective used (increased contrast and depth of field, more diffraction phenomena, reduced lateral resolution).

For the objective shown here, darkfield illumination can already be achieved by using a light annulus in the condenser close to the aperture diaphragm whose internal diameter is slightly larger than 6.5 mm. For VBDC and peripheral VPBC, on the other hand, this internal diameter must be slightly smaller than 6.5 mm.

The light yield for such an individual adjustment of the light annulus to the objective cross-section can be optimised because the darkfield ring in this calculated example can be designed to be significantly broadened for a constant outer diameter compared to the universally used light annulus for darkfield, whose internal diameter is generally significantly larger (e.g. 21 mm for the microscope Leitz HM-Lux 3).

Conclusions for the ratios of illuminating areas and a comparison with the hand-made prototypes

VBDC (variable bright-darkfield contrast):

The light-transmission area producing the darkfield is about 5.5x larger than that producing brightfield (184:33, see Tab. 2 and 3). In addition, to obtain the same image brightnesses for both partial images, the exposure time of the darkfield image must be 16x to 32x longer, depending on the object. Thus, to adjust the exposure times (equalising the brightnesses), the area of the existing darkfield-producing light annulus (184 mm2) must be enlarged by a factor of 16, respectively, 32 relative to the brightfield-producing transmission area (33 mm2).

To obtain a well-balanced brightness of the partial images, the ratio of the brightfield- and darkfield-producing light transmission areas must lie between

Peripheral VPDC (variable phase-darkfield-contrast)

The area of the light annulus for phase-contrast illumination (4 mm2) is 46x smaller than that for darkfield illumination (184 mm2). To achieve identical exposure times and brightnesses of both partial images, the light annulus provided for darkfield illumination must be enlarged still more by about 2x to 4x, depending on the object.

To obtain a well-balanced brightness of the partial images, the ratio of the phase-contrast- and darkfield-producing light transmission areas must lie between

The three stamped holes in the hand-made prototype of the light stop for phase-contrast illumination have an area of 0.6 mm2, whereas the perforations for the darkfield have a total area of 75 mm2 (see Table 2). This empirically determined ratio of 1:125 corresponds to the calculated design.

Axial VPDC (variable phase-darkfield-contrast:

The exposure times determined for peripheral darkfield with the Leitz light stop (184 mm2) also apply to axial darkfield produced with central light transmission (0.8 mm2). Correspondingly, the exposure times for axial darkfield to produce the same image brightnesses must also be 2x to 4x longer than those for the phase-contrast image produced with the conventional light annulus (4 mm2).

To obtain a well-balanced brightness of the partial images, the ratio of the axial darkfield and phase-contrast-producing light transmission areas must lie between

For the hand-made light stop, the axial light opening (0.8 mm2) was combined with a conventional light annulus for phase-contrast illumination (4 mm2) (area ratio 1:5). This allows variable transitions from phase contrast to darkfield because dominance of the phase contrast partial image can be continuously lowered using the aperture diaphragm. For a light mask, providing the maximum light yield , the ratios calculated  above can be used as a reference to achieve balanced illumination.

Peripheral and axial VPBC (variable phase-brightfield-contrast)

For simultaneous brightfield and phase contrast illumination, the exposure times and image brightnesses of the contributing partial images are the same if both light openings have the same area because the ratios of the transmission areas and exposure times are the same (Table 3).

To obtain a well-balanced brightness of the partial images, the ratio of the brightfield- and phase-contrast-producing light transmission areas must be 1 : 1.

For the hand-made prototype of a light stop for axial VPDC, the ratio of the light openings for brightfield and phase contrast is 0.8:4 or 1:5, respectively (Table 2 ). This means that there are modulation reserves here as well with respect to the intensity of the phase contrast partial image. A ratio of 1:1 for the illuminating areas is very promising for fabricating a prototype providing maximum image brightness and balanced results without closing the aperture diaphragm .

The hand-made light stop for peripheral VPBC was designed on this basis . Eight circularly arranged perforations, each 0.6 mm2 (total area: 4.8 mm2) form the outer lying light-transmission area for concentric-peripheral brightfield. They are combined with a conventional light annulus for phase contrast (4 mm2). Consequently, even minimal closure of the aperture diaphragm produces well-balanced image brightnesses. Further narrowing of this diaphragm leads to gradual domination of phase contrast.

In summary, the technical designs derived on the basis of theoretical optics can be used to obtain well-balanced image exposures with all my own fabricated prototypes.

According to my practical experience so far, which of the methods presented here and which technical design lead to the best results in each individual case ultimately depend on the type of object and the aim of the investigation.

Last Update: August 10th, 2012
Copyright: Timm Piper, 2012