## Abstract

The generalized eikonal formalism has recently been shown able to describe wave phenomena, which cannot be obtained by classical geometrical optics. However, whether this generalized eikonal equation can be used to describe interference effects due to linear superposition of waves still remains unanswered. It is shown here that the generalized eikonal formalism self-consistently satisfies the superposition principle and allows the interference effect. First, it is shown analytically to comply with the superposition principle. It is also shown numerically that the formalism satisfies the linear superposition of waves. The computed trajectories provide quantitative information on optical wave propagation as well as physical insights in the formation of interference fringes. Thus the generalized eikonal formalism is expected to be readily developed for the investigation of optical beam transformation in much the same manner as in the geometrical optics.

© 2002 Optical Society of America

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### Equations (12)

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(1)
$$(\nabla L{)}^{2}={n}_{0}^{2}+\frac{1}{{k}_{0}^{2}\phi}{\nabla}^{2}\phi ,$$
(2)
$$\frac{\mathrm{d}}{\mathrm{d}s}(\nabla L)=\nabla |\nabla L|,$$
(3)
$$\nabla \xb7({\phi}^{2}\nabla L)=0,$$
(4)
$$\mathbf{H}=\sqrt{\frac{{\epsilon}_{0}}{{\mu}_{0}}}\left[\nabla L\times (\stackrel{\u02c6}{e}\phi )+\frac{1}{{\mathit{ik}}_{0}}\nabla \times (\stackrel{\u02c6}{e}\phi )\right]exp({\mathit{ik}}_{0}L).$$
(5)
$$\mathrm{Re}\{\mathbf{P}\}=\mathrm{Re}\left\{\frac{1}{2}\mathbf{E}\times {\mathbf{H}}^{*}\right\}=\frac{1}{2}\sqrt{\frac{{\epsilon}_{0}}{{\mu}_{0}}}({\phi}^{2}\nabla L).$$
(6)
$$\mathbf{E}=\frac{1}{{\epsilon}_{r}}\left\{(\nabla L{)}^{2}(\stackrel{\u02c6}{e}\phi )-\frac{1}{{k}_{0}^{2}}{\nabla}^{2}(\stackrel{\u02c6}{e}\phi )+\frac{i}{{k}_{0}}[{\nabla}^{2}L(\stackrel{\u02c6}{e}\phi )+2(\nabla L\xb7\nabla )(\stackrel{\u02c6}{e}\phi )]\right\}exp({\mathit{ik}}_{0}L).$$
(7)
$$(\nabla {L}_{\mathrm{1,2}}{)}^{2}={n}_{0}^{2}+\frac{1}{{k}_{0}^{2}{\phi}_{\mathrm{1,2}}}{\nabla}^{2}{\phi}_{\mathrm{1,2}}$$
(8)
$$\nabla \xb7({\phi}_{\mathrm{1,2}}^{2}\nabla {L}_{\mathrm{1,2}})=0.$$
(9)
$${\nabla}^{2}{\mathrm{\Phi}}_{3}={\nabla}^{2}({\mathrm{\Phi}}_{1}+{\mathrm{\Phi}}_{2})=-{n}_{0}^{2}{k}_{0}^{2}{\mathrm{\Phi}}_{3}$$
(10)
$$(\nabla {L}_{3}{)}^{2}={n}_{0}^{2}+\frac{1}{{k}_{0}^{2}{\phi}_{3}}{\nabla}^{2}{\phi}_{3},$$
(11)
$$\nabla \xb7({\phi}_{3}^{2}\nabla {L}_{3})=0.$$
(12)
$${X}_{min}=\frac{1}{4}\frac{\mathrm{\lambda}}{{x}_{0}}z.$$