Mathematical Fundamentals of Modern Linear Optics

Figure 1

Illustration of the path integral applied to a layer of an inhomogeneous medium. The path integral contains three elements. (1) A collection of virtual paths connecting an input point (𝑥,𝑦) with an output point (𝑥^′,𝑦^′) can be labelled with a parameter 𝑢∶𝛾∶(𝑧,𝑢)→(𝑥,𝑦;𝑧)∈𝑅³, that is, 𝛾_𝑢(0)=(𝑥,𝑦;0)=(𝑥,𝑦) and 𝛾_𝑢(𝑍)=(𝑥,𝑦;𝑍)=(𝑥^′,𝑦^′). (2) All virtual paths going through the optical inhomogeneous medium 𝑛(𝑥,𝑦,𝑧) from the input point (𝑥,𝑦) to the output point (𝑥^′,𝑦^′) have the optical length 𝑆∶𝑢→∫_𝛾𝑢𝑛(𝑥,𝑦,𝑧)1+𝑥̇²+𝑦²̇𝑑𝑧. (3) The path integral 𝐾_𝑆(𝑥,𝑦;𝑥^′,𝑦^′)=∫^∞_−∞𝐴⋅exp[𝑖𝑘_𝑜𝑆_𝑢(𝑥,𝑦;𝑥^′,𝑦^′)]𝑑𝑢 is the optical propagator. The phase function 𝑆_𝑢(𝑥,𝑦;𝑥^′,𝑦^′) in the neighbourhood of the stationary point  𝑢, where 𝜕S_𝑢/𝜕𝑢|_𝑢=𝑢=0 can be approximated by a parabola with the vertex at point (𝑢,𝑆_𝑢). In this case the optical propagator can be approximated by the function 𝐾_𝑆(𝑥,𝑦;𝑥^′,𝑦^′)=𝐶⋅|𝐾_𝑆(𝑥,𝑦;𝑥^′,𝑦^′)|⋅exp[𝑖𝑘_𝑜𝑆_𝑢(𝑥,𝑦;𝑥^′,𝑦^′)], where 𝑆_𝑢(𝑥^′,𝑦^′;𝑥,𝑦) is a point eikonal, that is, the optical length of the path which satisfies Fermat’s principle.