Light Propagation in Generalized Lens-Like Media (1965)

This paper provides a preliminary assessment of electromagnetic wave propagation in focusing media which departs from those previously studied, ideal lenses and continuous media with square-law index variation. New approximate methods are described for obtaining the transverse beam width, phase constant, and ray trajectory in continuous lens-like media, and for determining stability conditions in lens waveguides, where the lenses contain spherical aberration and the continuous media contain fourth-order or higher-order terms of variation in index of refraction. It is proven that only in aberrationless-lens waveguides or in a continuous medium with square-law index variation will the shape of a beam injected off axis or with an angle to the medium's axis remain constant about a beam axis which oscillates about the axis of the medium. In non-square law media the beam will spread, but knowledge of the coefficients describing the medium and the position and angle of the injected beam enables one to specify the maximum radius within which all of the energy will be confined. The following is an example of the type of solution obtained for non-square law media: for a medium characterized by the transverse index variation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n_a (1-{1/2}a_4 x^4)$</tex> and assuming no index variation exists in the direction of propagation, the radius to the 1/e point infield is approximately <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$w_e = 0.666{\lambda^{1/3} \over a_{4}^{1/6}}$</tex> and the phase constant is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\beta = {2\pi \over \lambda} - {0.256(a_4 \lambda)^{1/3}(m + 1)^2 \over \left({m + 2.5 \over 2.5})^2/3}$</tex> where the free-space wavelength <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\lambda_0 = n_{a} \lambda$</tex> and m is the order of the mode, m = 0, 1, 2 … Quite generally, non-square law media show dispersion, and unlike the square-law media the various modes travel with different group velocities. Expressions are given to allow these effects to be evaluated for small perturbations on a square-law medium as well as for higher-order index variations. The transverse beam shape associated with any law of index variation is shown to be as well approximated (in the region of signifteant power density) by a cosine function or Ga1lssian function as is the field for an ideal lens-waveguide approximated by a Gaussian function in the presence of typical diffraction losses. Normal mode shapes are obtained for resonators with fourth-order and eighth-order mirrors by the method of Fox and Li; diffraction 108se8 for a few Fresnel number8 are also given. In a certain range of Fresnel n1lmbers, fourth- and eighth-order mirrors give lower diffraction losse8 than spherical mirrors. An approximate method for solving the paraxial ray equation for rather general (non-square-law) media is outlined. Requiring only reciprocity and symmetry about the medium's axis, it is shown that tlu) radial position of the ray (x) is related to distance (z) along the axis of the medium by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x = \sum b_m cos m\beta z$</tex> where m = 1, 3, 5, 7 …. Moreover, it is shown that this series converges very rapidly, making it possible to get a good approximate representation with only a few terms. For example, for the fourth-order medium described above, an approximate solution is <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$x = x_{0} \{0.959 cos \beta z + 0.041 cos 3 \beta z \}$</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$ \beta = x_{0} \sqrt{1.44a_4}$</tex> . It is characteristic of all non-square law media to have a ray period 2Π/β which is a function of the peak ray displacement, x <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> Lens waveguides with fourth- or higher-order terms in the focusing or index function can of course exhibit increasingly strong focusing for energy departing farther from the medium's axis, and if the lenses are suitably spaced might be useful in reducing the magnitude of beam wander due to imperfections or guide-axis curvature. However, for a given beam spot size, non-square-law lenses must be placed closer together than square-law lenses. Use of non-square law lenses or distributed media in transmission systems will most probably require repeater-system techniques which are operable with multi-mode signals at the receiver input.