Appendix A: Derivation and Explanation of the Étendue Invariant, Including the Dynamical Analogy; Derivation of the Skew Invariant
Derivation and Explanation of the Etendue Invariant, Including the Dynamical Analogy; Derivation of the Skew Invariant
А.1 THE GENERALIZED ЁТЕМЭИЕ
In Section 2.7 we introduced the invariant
The meaning of this was as follows: let any ray be traced through an optical system, and let rectangular coordinate axes be set up in the entry and exit spaces in arbitrary orientations. Also, let the ray meet the x, у plane in the entry space in (x, y), and let its direction cosines be (L, M, N), and similarly for the exit space. Then for any nearby ray with coordinates (x + dx, у + dy, L + dL, M + dM) we have
We shall prove in Section A.2 that this result is true for any optical system and for any choice of the directions of the axes. But first we discuss some different ways of interpreting this result. We put p = nL and q = nM, and we treat (x, y, p, q) as coordinates in a four-dimensional space. Let U be any enclosed volume in this space. Then U is given simply by
taken over the volume. The result of Equation (A.2) means that if we let a ray sweep out the boundary of this volume in the four-dimensional entry space, it will sweep out an equal volume in four-dimensional exit space or, clearly, in any intermediate space of the optical system. There is a useful analogy between this space and the multidimensional phase space of Hamiltonian mechanics, so we shall call it simply phase space. Thus, our result is that if a ray sweeps out a certain volume of phase space in the entry region, it sweeps out the same total volume in the exit region, or, more concisely, phase space volume is conserved. Of course, the shape of the volume will change from entry to exit space or even if the origin of coordinates is shifted parallel to the z direction. However, it can easily be shown that rotating the axes about their origin does not change the phase space volume, so it is a physical significant invariant.
Another picture that is particularly valuable for the purposes of this book may be obtained by supposing that an area in the x, у plane (in real space, not phase space) is covered with uniformly closely spaced points and that through each point a large number of rays is drawn. These rays are to be in different directions so that their direction cosines increase by uniformly small increments over certain ranges of L and M. The rays so drawn then represent “ray points” uniformly spaced throughout a certain volume of phase space, and our theorem can then be stated in the following form: the density of rays in phase space is invariant through the optical system. If the four-volume U of Equation (A.3) has been defined by means of physical components in the optical system—that is, an aperture stop and a field stop—it is known as the throughput or etendue of the system.
We can see from the ray-point representation that the etendue is a measure of the power that can be transmitted through the optical system from a uniformly bright source of sufficient extent, in the geometrical optics approximation. Here, of course, in addition to neglecting interference and diffraction effects, we are assuming no losses in the system due to reflections, absorption, or scattering. The concept of etendue is applied in comparing the properties of many different kinds of optical instruments, although usually only a simplified version appropriate to axially symmetric systems is used.
A.2 PROOF OF THE GENERALIZED ETENDUE THEOREM
Equation (A.2) has been proved in many different ways. In particular, the analogy from Hamiltonian mechanics can be used to give a simple proof (see, e.g., Marcuse, 1972; Winston, 1970). However, since we are here concerned with applications to geometrical optics, it seems appropriate to give a proof based directly on optical principles. We shall discuss the mechanical analogies in some detail in Section A.3.
We follow closely the method of Welford (1974), which makes use of the pointei- konal, or characteristic function, of Hamilton. This function is defined as follows. We take arbitrary cartesian coordinate systems in entry and exit spaces of the optical system under discussion. Let P and P' be any two points in the entry and exit spaces, respectively, and let them be in the x, у planes of their respective coordinate system." Then the eikonal V is defined as the optical path length from P to P' along the physically possible ray joining them. In general, one and only one ray passes through P and P but if there is more than one, then V is multivalued. Thus, with the preceding restriction V is a function of x, y, x and y'. Let the direction cosines of the ray in the two spaces be (L, M, N) and (L, M N'): then the fundamental property of the eikonal can be stated as follows: [1]
This property is proved in many texts on geometrical optics (e.g., Born & Wolf, 1975; Welford, 1974). To prove our theorem we differentiate Equation (A.4) again, and we obtain, using the notations of Equation (A.3) and using subscripts for partial derivatives,
We next rearrange these terms and put the equations in matrix form
If we denote the two matrices by В and A and the column vectors by M and M', this equation takes the form
and multiplying through by the inverse of B,
This matrix equation can be expanded, and we get together with three similar equations
It can be seen that the determinant of the matrix B~1 A is the Jacobian
which transforms the differential four-volume dx, dy, dp, dq—that is, we have
Our result will be proven if we can show that the Jacobian has the value unity. But the determinant of matrix В has the value
and that of matrix A has the same value, since Vxy> = Vyx , and so forth. Also, the determinant of the product of two square matrices is the product of their determinants, so that
Thus, det(fT'A) = 1 and Equation (A.13) yields our theorem, Equation (A.2).
A.3 THE MECHANICAL ANALOGIES AND LIOUVILLE'S THEOREM
In this section we shall indicate the analogies used to identify our theorem of the invariance of U, the etendue, with Liouville’s theorem in statistical mechanics.
Fermat’s principle, on which all of geometrical optics can be based, can be stated in the form
where ds is an element of the ray path from P, to P2. This can be written in the form
where
and the dots denote differentiation with respect to z. Also, we define
The analogy is to regard L as the Lagrangian function of a mechanical system in which x and у are two generalized coordinates, p and q are the corresponding generalized momenta, and z corresponds to the time axis. On this basis the ordinary development of mechanics can be carried out, such as by Luneburg (1964, Article 18), by solving the variational problem of Equation (A.15). The Hamiltonian is found to have the value
In other words, it is -nN where N is the .-direction cosine, and, of course, p and q as just defined are respectively equal to nL and nM. The phase space for this system has the four coordinates (x, y,p, q), and Liouville’s theorem in statistical mechanics can be invoked immediately to state that phase space volume is conserved.
However, the meaning of Liouville’s theorem in mechanics is rather different from the theorem of conservation of etendue. Liouville’s theorem is essentially statistical in nature, and it refers to the evolution in time of an ensemble of mechanical systems of identical properties but with different initial conditions. Each system is represented by a single point in phase space, and the theorem states that the average density of points in phase space is constant in time. An example would be the molecules of a perfect classical gas in equilibrium in a container. Each point in phase space, which in this example has 2N dimensions, where N is the number of molecules, represents one of an ensemble of identical containers, an ensemble large enough to permit taking a statistical average of the density of representative points. Liouville’s theorem states that if all the containers remain in equilibrium, the average density of points remains constant.
Another example would be focused beams of charged particles, as in a particle accelerator. Here we can regard one pulse of particles as constituting one realization of the ensemble and therefore one point in phase space. The statistical averaging is carried out over the random positions and momenta of the particles entering the focusing system from pulse to pulse.
The theorem has been applied to many different physical systems but the essential point is that it makes a statistical statement about the average density of points in phase space, whereas the throughput or etendue theorem is deterministic in nature. Thus, although for convenience we may call the throughput theorem “Liouville’s theorem,” it is desirable to remember that there is a fundamental difference in meaning.
A.4 CONVENTIONAL PHOTOMETRY AND THE ETENDUE
In discussing the action of concentrators we use the notion of a beam of radiation in which a certain cross section dx dy is uniformly filled with rays covering uniformly the direction cosine solid angle dL dM. Note the useful relation dL dM = N dQ.. Then an ideal concentrator takes all the rays from a source of finite area within a certain range of direction cosines and delivers them at the exit aperture to emerge over a solid angle 2л.
These ideas relate easily to particle beams, but it may not be quite clear how they relate to classical photometric ideas. In classical photometry we have a particular kind of ideal source called Lambertian for which the flux or power radiated per unit solid angle per unit projected area of the source is constant over all directions. Many physical sources approximate closely to the Lambertian condition and an ideal blackbody radiator must be Lambertian, since the radiation in a blackbody cavity is isotropic. It is easy to show that if a source radiates the same flux per unit area of the source and per unit element of direction cosine space, then it is Lambertian. Thus, we see than an ideal concentrator will take radiation from a Lambertian source over a certain solid angle and deliver it so as to make the exit aperture appear to be a complete Lambertian radiator over solid angle 2л.
REFERENCES
Born, M., and Wolf, E. (1975). Principles of Optics. 5th Ed. Pergamon, Oxford.
Marcuse, D. (1972). Light Transmission Optics. Van Nostrand-Reinhold, Princeton, NJ. Welford, W. T. (1974). Aberrations of the Symmetrical Optical System. Academic Press, New York.
Winston, R. (1970). Light collection within the framework of geometrical optics. J. Opt. Soc. Am. 60, 245-247.
- [1] This is in no way essential to the definition of the eikonal; it merely simplifies the present calculation.
