alt="equation"/>
The optical invariant, O, is given by:
(2.4)
The optical invariant is, in the paraxial approximation, preserved on passage through an optical system. That is to say:
n′, h′, θ′, etc. are ray parameters following propagation.
Derivation of the above invariant is straightforward using matrix analysis.
Hence:
From (1.23) we know that the determinant of the matrix is given by the ratio of the refractive indices in the relevant media, so:
Finally we arrive at Eq. (2.5)
The optical invariant is a generalised constraint that relates system lateral and angular magnification and applies to any arbitrary pair of rays. A very specific descriptor is created when the ray pair consists of the chief ray and the marginal ray. This special case of the optical invariant is known as the Lagrange invariant. The Lagrange invariant, H is given by:
(2.6)
If we now simply evaluate H at the entrance and exit pupils where, by definition, hchief is zero, then the product nhmarginalθchief is constant. The Lagrange invariant then simply articulates the fact that the angular and lateral magnifications are inversely related. In fact, the Lagrange invariant captures a more fundamental constraint to an optical system. If the object plane is uniformly illuminated, then the total light flux emanating from the plane is proportional to the square of the maximum field angle. The proportion of that flux that is admitted by the entrance pupil is itself proportional to the square of the marginal ray height. Therefore, the total flux passing through an optical system is proportional to the square of the Lagrange invariant, H2. Thus the Lagrange invariant is an expression of the conservation of energy as light propagates through an optical system. This will become of paramount significance when, in later chapters, we consider source brightness or radiance and the impact of the optical system on optical flux flowing through it.
2.10 Eccentricity Variable
The eccentricity variable, E, is a measure of how far an axial location in an optical system is from the stop. It is expressed in terms of the chief ray to marginal ray height at that particular location. Of course, at the pupil (entrance or exit) itself, the eccentricity variable will be zero. The eccentricity variable is defined as:
(2.7)
E is of course infinite at the focal point of a system. The variable is of great significance in the analysis of optical imperfections or aberrations where the distance of a component from the aperture stop is of critical importance.
2.11 Image Formation in Simple Optical Systems
These introductory chapters provide a complete description of ideal optical systems. That is to say, in the paraxial approximation, where imaging imperfections, or aberrations may be ignored, the analysis presented is substantially complete. Some very simple optical instruments are introduced at this point; their deficiencies are discussed later.
2.11.1 Magnifying Glass or Eye Loupe
The magnifying glass or eye loupe is perhaps the simplest optical system conceivable, in that is consists of a single lens that is intended to be used with the eye to magnify close objects. Our ability to resolve small, close objects is limited by our ability to focus at close quarters. Typically, although this varies with age and other factors, a comfortable distance for viewing near objects is about 250 mm. If the eye can resolve an angle of 1 arcminute, then this corresponds to a resolution of somewhat under 0.1 mm. Addition of a simple lens allows the eye to view objects at a much shorter distance. This is shown in Figure 2.7.
For the two cases illustrated in Figure 2.7, the eye's focussing power remains the same. Therefore, addition of a lens of focal length f will change the closest approach distance, d0, to:
Figure 2.7 Simple magnifying lens.
If the magnification, M, provided by the lens is defined as the ratio of the final image sizes in the two scenarios, the magnification is given by:
(2.8)
In describing magnifying lenses, as suggested earlier, d0 is defined to be 250 mm. Thus, a lens with a focal length of 250 mm would have a magnification of ×2 and a lens with a focal length of 50 mm would have a magnification of ×6. In practice, simple lenses are only useful up to a magnification of ×10. This is partly because of the introduction of unacceptable aberrations, but also because of the impractical short working distances introduced by lenses with a focal length of a few mm. For higher magnifications, the compound microscope must be used.
Naturally, the pupil of this simple system is defined by the pupil of the eye itself. The size of the eye's pupil varies from about 3 mm in bright light, to about 7 mm under dim lighting conditions, although this varies with individuals.
2.11.2 The Compound Microscope
In the preceding subsection, the limitations of a simple magnifying lens were made clear. Overall, its functionality, in delivering a high magnification, is to convey an intermediate image, located at the infinite conjugate, to the human eye. Furthermore, to provide maximum magnification, the focal length of this lens must be as small as possible. For practical reasons, a magnification of greater than ×10 cannot be delivered. This difficulty is solved by the compound microscope where a two-lens system is used to provide a system focal length that is considerably shorter than would be afforded by a single lens. In essence, a compound microscope consists of two lenses, or lens groups. The first lens is the objective lens that lies close to the object and the second lens, in the traditional microscope, is the eyepiece. Of course, in many modern instruments, the eye is replaced
