REFRACTION OF LIGHT, REFRACTOMETRY AND INTERFEROMETRY 



of ]Mach. This lends support to a remark where abouts of the rays ultimately emerging 



made by Destouches (21) that "the essential, from the optical system considered. This is 



in order to build a theory, is to possess simple given by Malus' theorem, according to which 



and schematic ideas, of origin rather intui- all the rays issued from a point source *Sr, 



tive than purely experimental, and a long after undergoing multiple reflections and/or 



work of the mind is required." refraction, remain normal to a family of 



de Fermat's principle is often expressed parallel surfaces E, each point of which is 



mathematically as a "stationary time in- equidistant from S (conjugate) by the same 



tegral": optical length L. The surface defines the 



"wave-front surface" or surface of equal 



/ 



n ds = {2) phase. It is that which is reached by light 



after a unique time T (equations 3 and 4) 

 \vhere n is the cartesian refraction index, ds from all the possible trajectories so that: 



T = L/C = S(Z/F) (-^) 



/ 



is the geometric path length, and the quan- 

 tity nds defines the optical path length. 



It is interesting to note that a very similar where I is the geometric path length. This is 



principle of optimum path length was al- in strict conformity with de Fermat's prin- 



ready proposed in the third century by Hero ciple. A bundle of light rays satisfying such 



of Alexandria who implicitly admitted a conditions is said to form a congruence of 



finite light velocity (22). The principle is normals when their direction of propagation 



sometimes more useful in the equivalent is defined by a series of parameters (X, Y, Z) 



form : which depend upon at least two independent 



variables (such as V, U, etc.). In the limited 



ds/d'K = {3) case of geometric optics (n = Cte) with only 



one variable, one has a totality of curves 



In all its forms, the principle states that orthogonal to a family of wave surfaces in- 

 the sum of all optical path lengths followed stead of a congruence. Huygens construction 

 by the light rays through any succession of (1690) is merely a convenient geometric 

 isotropic, homogeneous transparent media demonstration of this theorem, 

 separated by stigmatic surfaces, is station- It is perhaps clear to those conversant with 

 ary, that is, two such adjacent rays differ in Huygens' work that the ideal stigmatic sur- 

 optical length only by an infinitesimal quan- face of a given optical system may differ 

 tity at least of second order. The principle from the actual physical surface of its diop- 

 never states whether this length is a maxi- tic elements. Thus, three situations may 

 mum, a minimimi, or an inflexional quantity, arise, according to which the actual surface 

 but only that it is constant. The writer, is either tangent, internal, or secant to the 

 among others, has cautioned about the in- stigmatic surface. Accordingly, the optimum 

 discriminate interpretation of this principle path length of de Fermat is either a maxi- 

 (23). The argument can be summarized as mum, a minimum, or an inflectional quantity, 

 follows. In the relatively simple case of a respectively, and in the order given above, 

 beam of parallel rays of light, the algebraic The implications of de Fermat's principle 

 sum of all of the partial path lengths of in- pervade much of contemporary physical- 

 cident plus reflected rays remains constant chemistry. Almost simultaneously with its 

 regardless of the angle of incidence on a publication, Pierre Maupertuis showed, in- 

 given surface, be it an ellipse, an hyper- dependently, that when a "material point in 

 boloid or a parabola. action" is analyzed, the sum of all the ener- 



It is obviously important to know the gies involved is stationary, the representa- 



496 



