large numerical aperture, allows more light to be collected. Such a system, with a high numerical aperture is said to be ‘fast’. This terminology has its origins in photography, where the efficient collection of light using wide apertures enabled the use of short exposure times. An alternative convention exists for describing the relative size of the aperture, namely the f-number. For a lens system, the f-number, N, is given as the ratio of the lens focal length to the aperture diameter:
(2.2)
This f-number is actually written as f/N. That is to say, a lens with a focal ratio of 10 is written as f/10. The f-number has an inverse relationship to the numerical aperture and is based on the stop diameter rather than its radius. For small angles, where sinΔ = Δ, then the following relationship between the f-number and numerical aperture applies:
(2.3)
In this narrative, it is assumed that the aperture is a circular aperture, with an entire, unobstructed circular area providing access for the rays. In the majority of cases, this description is entirely accurate. However, in certain cases, this circular aperture may be partly obscured by physical or mechanical hardware supporting the optics or by holes in reflective optics. Such features are referred to as obscurations.
At this stage, it is important to emphasise the tension between fulfilment of the paraxial approximation and collection of more light. A ‘fast’ lens design naturally collects more light, but compromises the paraxial approximation and adds to the burden of complexity in lens and optical design. This inherent contradiction is explored in more detail in subsequent chapters.
Figure 2.2 Location of entrance and exit pupils.
2.3 Entrance Pupil and Exit Pupil
The physical aperture stop may not actually be located conveniently in object space as shown in Figure 2.1. On the other hand, it may be located anywhere within the sequential train of optical components that make up the optical system. An example of this is shown in Figure 2.2, a situation that is true of many camera lenses, where the physical stop is located between lenses.
In the situation described, the entrance pupil is the image of the physical aperture as projected into object space. Correspondingly, the exit pupil is the image of the physical aperture as projected into image space. The exit pupil is located in the conjugate plane to the entrance pupil and may be regarded as the image of the entrance pupil. Along with the cardinal points of a system, the location of the entrance and exit pupils are key parameters that describe an optical system. Most particularly, the numerical aperture in object space is defined by the angle of the marginal ray that intersects the edge of the entrance pupil.
Worked Example 2.1 Cooke Triplet
Figure 2.3 shows a simplified illustration of an early type of camera lens, the Cooke triplet.
By convention, image space is assumed to be on the left-hand side of the illustration. All lenses are assumed to have no tangible thickness (thin lens approximation) and the axial origin lies at the first lens. Positive axial displacement is to the right.
Figure 2.3 Cooke triplet.
1 Position and Size of Exit PupilIt is easiest, first of all, to calculate the position of the exit pupil, as this is the stop imaged by a single lens (the third lens) of focal length 32.8 mm. The position of the aperture stop, the object in this instance, is 6.4 mm to the left of this lens. The distance, v, of the exit pupil from the third lens is therefore given by:Thus, the exit pupil is 7.95 mm to the left of the third lens and 8.05 mm from the origin. The magnification is given by (minus) the ratio of image and object distances and so it is easy to calculate the size of the exit pupil:
2 Cardinal Points of the LensThe distance between the first and second lenses is 6.8 mm and between the second and third lenses is 9.2 mm. By convention, we retrace dummy rays −16 mm back to the origin at the first lens, so that all matrix ray tracing formulae are referred to a common origin. The matrix for the system is given below:To calculate the position of the exit pupil we need to know the focal length of the system and the positions of the two focal points. Following the matrix relations set out in Chapter 1, we can calculate the following:Focal length: 52.3 mmLocation of First Focal Point: −41.2 mmLocation of Second Focal Point: 57.7 mmAll distances are referenced to the axial origin at the first lens. There is, of course, a single effective focal length as both object and image spaces are considered to lie within media of the same refractive index.
3 Position and Size of the Entrance PupilThe imaged pupil or exit pupil lies in image space, 8.05 mm from the origin. This is 49.65 mm to the left of the second focal point. In applying Newton's formula, the second focal distance, x2 is then equal to −49.65. We can now calculate the first focal distance to determine the position of the entrance pupil.The object or entrance pupil therefore lies 55.1 mm to the right of the first focal point and 13.9 mm (−41.2 + 55.1) to the right of the first lens.The location of the entrance pupil expressed as an object distance is 52.3 − 55.1 or −2.8 mm. Similarly the location of the exit pupil expressed as an image distance is equal to −49.65 + 52.3 or +2.65 mm. The magnification (image/object) is, in this instance equal to 2.65/2.8 or 0.946. Therefore we have:The diameter of the entrance pupil is, therefore, 15.1 mmSo, in summary we have:
| Entrance Pupil Axial Location: 13.9 mm | Entrance Pupil Diameter: 15.1 mm |
| Exit Pupil Axial Location: 8.05 mm | Exit Pupil Diameter: 14.3 mm |
Figure 2.4 Optical system with a telecentric output.
2.4 Telecentricity
In the previous example, both entrance and exit pupils were located at finite conjugates. However, a system is said to be telecentric if the exit pupil (or entrance pupil) is located at infinity. In the case of a telecentric output, this will occur where the entrance pupil is located at the first focal point. In this instance, all chief rays will, in image space, be parallel. This is shown in Figure 2.4 which illustrates a telecentric output for two different field positions.
A telecentric output, as represented in Figure 2.4 is characterised by a number of converging ray bundles, each emanating from a specific
