Magnifiers and Microscopes

The magnification of a microscope or a magnifier is, like that of a tele-scope, defined as the ratio of the angle subtended by the image to the angle subtended by the object. The difficulty here is that we are con-cerned with an object at a finite distance, and the angle which the object subtends will vary with that distance. The answer to this dilemma is that the object is considered to be viewed at a convention-al distance of 10 in, chosen as the “nearest distance of distinct vision.”

This convention is obviously a compromise, since the eyes of a young person can focus to a distance of a few inches, whereas an older indi-vidual may be unable to focus closer than several feet.

If the object to be examined is placed at the focal point of the micro-scope/magnifier, the image is seen at infinity, and the magnification is simply

MP

(2.16)

This expression is valid for either a simple magnifying glass or a com-pound microscope, where F is the effective focal length of the micro-scope/magnifier. If, as shown in Fig. 2.12, the object is between the lens and the first focal point, then the magnification depends not only on the focal length F, but also on the image distance S′, as well as the distance from the magnifier to the eye R, as follows:

250 mm

F

10 in

F

S'

F

S R

Object

Image F1

Figure 2.12 The optics of a magnifier or simple microscope, showing the object distance S, the image distance S′, and the eye distance R. The object is located at, or within, the first focal point F₁, and the image is virtual and located far enough from the eye that it can be comfortably viewed; this image distance is usually quite large.

MP
(2.17) Equation (2.16) is used to determine the stated power of magnifiers, eyepieces, and microscopes, whereas Eq. (2.17) is useful for determin-ing the magnification of devices such as slide viewers. It should be apparent that if the definition of magnification were changed to use the angle subtended by the object from a distance D instead of 10 in, we would simply substitute D for 10 in in the above equations.

Sample calculations

Lay out a 4 tabletop slide viewer which is to be used at a distance of 20 in from the eye, and which is to provide an image that is 0.67 diopters from the eye. A distance of 0.67 diopters is a distance of (1/0.67)
1.5 m ≈ 60 in. With reference to Fig. 2.12, R
20 in To locate the slide position, we use Eq. (1.4):





S
1.66666 in

For a slide diagonal of 1.6 in, how large must the lens diameter be? The magnified image is 41.6
6.4 in, and at an image distance of 40 in it subtends an angle of 6.4/40
0.16. At the eye distance of 20 in this angle requires a lens diagonal of 200.16
3.2 in. This is a large diameter for a

1

will show; see Sec. 5.5 for sketch techniques). At this point our alterna-tives are: (1) change the initial specifications so that the lens focal length is longer and its required diagonal is smaller; (2) use more than one ele-ment for the lens; (3) make the lens smaller than 3.2 in and force users to shift their heads to see the full slide; or some combination of these.

In the compound microscope, shown in Fig. 2.13, the objective lens forms a magnified image of the object, which is viewed through the eyepiece. The magnification is the product of the objective magnifica-tion (S′/S) and the eyepiece magnification (10 in/Fe), or

MP
^• (2.18)

The microscope magnification can also be determined by calculating the effective focal length of the combination of the objective and eyepiece, using Eq. (1.27) or (1.28). This gives the focal length for the microscope F_m:

F_m

(2.19)

and a magnification of

MP

(2.20)

which yields exactly the same result as does Eq. (2.18).

10 in(F_o S′)

Figure 2.13 The compound microscope consists of an objective lens which forms an enlarged image of the object, and an eyepiece which further magnifies the image and allows the eye to view it comfortably.

Sample calculations

We wish to view an object 40 in away at a magnification of 5. The optical instrument can be 10 in long. A 5 magnifier/microscope has a focal length of 10/MP
2 in, but since MP may be plus or minus, so may the focal length. If the object is placed at the focal point of the microscope (so that the image presented to the eye is at infinity for comfortable viewing), the back focus must be B
40 in. The compo-nent powers will be minimized if the space between compocompo-nents is the maximum allowed, or d
10 in. Using Eqs. (1.31) and (1.32) and fab

This is the galilean version, with a negative eyelens and a positive objective; the image is erect; there is no internal focus. If we use MP

5, fab
2.0 in, and we get

f_a


0.4762

f_b


7.692

This arrangement corresponds to the conventional compound micro-scope with an inverted image and an internal focus. The internal image allows the use of a reticle or crosshair.

The optics of a HUD (or head-up display) and those of an HMD (or head/helmet-mounted display) are basically magnifiers. Typically a HUD projects collimated information from a CRT (cathode ray tube),

40052

The object is placed at or near the focal point of the optics so that the image produced is at infinity or a large distance. Depending on the application, the infinitely distant image may be reflected from a 45°

tilted semireflecting “combiner” mirror, through which the user can also see directly. A military aircraft HUD may provide weapon or air-craft status information and also provide an aiming point for the weapon system. Because the user’s eye is usually located a significant distance from the lens system, the field of view of a HUD is limited by the aperture of the optics. The system must also be large enough to accommodate a certain amount of motion of the eye without vignetting (obscuring) the image. In an HMD the eye is usually much closer to the optics and the function is more like that of an ordinary magnifier. Some HMDs have optical systems which incorporate relay optics and are thus analogous to the compound microscope. This is usually done in order to make the center of gravity of the total HMD assembly roughly coincident with that of the head. A concave mirror, either fully or semireflecting is sometimes used as the collimating/magnifier in conjunction with a 45° semireflecting com-biner mirror. In some applications, such as an HMD for surgery or virtual reality, there is no see-through capability and the view of the outside world, if required, is seen by looking beneath the HMD optics.

2.9 Telephoto and Retrofocus