detector chip. Nonetheless the logic followed here still applies. Figure 2.8 shows the general set up.
The two lenses are separated by the nominal tube length, d, and an intermediate image is formed by the objective lens within the tube. The eyepiece then presents the final image to the eye at the infinite conjugate. In other words, the intermediate image is designed to be located at a distance f2 (the eyepiece focal length) from the eyepiece. If the objective lens focal length is f1, then the matrix of the system is:
The entire co-ordinate system is referenced to the position of the objective lens. Of particular relevance here is the first focal length. From the above matrix we have the following equation for the system focal length:
The logic of Eq. (2.9) is that a shorter system focal length can be created than would be reasonably practical with a single lens. Using the same definition as used for the simple magnifying lens, the effective system magnification, Msystem, is given by the ratio of the closest approach distance d0, (250 mm), and the system focal length. The system magnification, is given by:
The bracketed quantity, (s − f1 − f2), i.e. the lens separation minus the sum of the lens focal lengths is known as the optical tube length of the microscope, and this will be denoted as d. Generally, for optical microscopes, this tube length is standardised across many commercial instruments with the standard values being 160 or 200 mm. Equation (2.10) may be rewritten as:
Figure 2.8 Compound microscope.
(2.11)
The above formula gives the total magnification of the instrument as the product of the individual magnifications of the objective lens and eyepiece. In this context, these individual magnifications are defined as in Eqs. (2.12a) and (2.12b):
The equations above establish the standard definitions for microscope lens powers. For example, the magnification of microscope objectives is usually in the range of ×10 to ×100. For a standard tube length, d, of 160 mm, this corresponds to an objective focal length ranging from 16 to 1.6 mm. A typical eyepiece, with a magnification of ×10 has a focal length of 25 mm (d0 = 250 mm). By combining a ×100 objective lens with a ×10 eyepiece, a magnification of ×1000 can be achieved. This illustrates the power of the compound microscope.
The entrance pupil is defined by the aperture of the objective lens. This entrance pupil is re-imaged by the eyepiece to create an exit pupil that is close to the eyepiece. Ideally, this should be co-incident with the pupil of the eye. The distance of the exit pupil from final mechanical surface of the eyepiece is known as the eye relief. Placing the exit pupil further away from the physical eyepiece provides greater comfort for the user, hence the term ‘eye relief’. Objective lens aperture tends to be defined by numerical aperture, rather than f-number and range from 0.1 to 1.3 (for oil immersion microscopes).
Figure 2.9 Basic optical telescope.
2.11.3 Simple Telescope
A classical optical telescope is an example of an afocal system. That is to say, no clearly defined focus is presented either in object or image space. As the name suggests, the telescope views distant objects, nominally at the infinite conjugate and provides a collimated output for ocular viewing in the case of a traditional instrument. As far as the instrument is concerned, both object and image are located at the infinite conjugate. Of course, this narrative does assume that the instrument is designed for ocular viewing as opposed to image formation at a detector or photographic plate. In any case, the design principles are similar. Fundamentally, the telescope provides angular magnification of a distant object, and this angular magnification is a key performance attribute.
The basic layout of a simple telescope is shown in Figure 2.9. Light from the distant object is collected by an objective lens whose focal length is f1 and then collimated by an eyepiece with a focal length of f2. These two lenses are separated by the sum of their focal lengths, thus creating an afocal system with an angular magnification given by the ratio of the lens focal lengths.
The matrix of the telescope is similar to that of the compound microscope, with an objective lens and eyepiece separated by some fixed distance.
The separation, s, is simply the sum of the two focal lengths and the system matrix is given by:
(2.13)
The angular magnification (the D value of the matrix) is simply −f1/f2. It is important to note the sign of the magnification, so that for two positive lenses, then the magnification is negative. In line with the previous discussion with regard to the optical invariant, the linear magnification (given by matrix element A) is the inverse of the angular magnification. Also, the C element of the matrix, attesting to the focal power of the system, is actually zero and is characteristic of an afocal system.
As in the case of the microscope, the objective lens forms the system entrance pupil. The exit pupil is formed by the eyepiece imaging the objective lens. This is located a short distance, approximately f1 from the eyepiece, this distance determining the ‘eye relief’. Ideally, for ocular viewing, the pupil of the eye should be co-incident with the exit pupil. Unlike the compound microscope, the exit pupil of a simple (ocular) telescope is relatively large, about the size of the pupil of the eye. Clearly, if the exit pupil were significantly larger than the pupil of the eye, then any light falling outside the ocular pupil would be wasted. In fact, in a typical telescope, where f1 ≫ f2, the size of the exit pupil is approximately given by the diameter of the objective lens multiplied by the ratio of the focal
