Numerical analysis of eikonal equation

A. V. Korolkova; D. S. Kulyabov; M. N. Gevorkyan; T. R. Velieva

arxiv: 1906.09467 · v1 · pith:UTT5RHMHnew · submitted 2019-06-22 · ⚛️ physics.comp-ph · cs.NA· math.NA

Numerical analysis of eikonal equation

D. S. Kulyabov , A. V. Korolkova , T. R. Velieva , M. N. Gevorkyan This is my paper

Pith reviewed 2026-05-25 18:08 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cs.NAmath.NA

keywords eikonal equationmethod of characteristicsMaxwell lensLuneburg lensgeometric opticsnumerical analysisray tracing

0 comments

The pith

The eikonal equation reduces to an ODE system via characteristics for ray tracing in Maxwell and Luneburg lenses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper transforms the eikonal equation into a system of ordinary differential equations using the method of characteristics. This system is then specialized for the refractive index profiles of Maxwell and Luneburg lenses. The approach allows numerical computation of light rays in these gradient-index devices under the geometric optics approximation. A reader would care because it bridges wave optics to practical ray calculations for lens design without solving full Maxwell equations.

Core claim

The eikonal equation, obtained from the Helmholtz equation in the short-wavelength limit, is converted to an ODE system by the method of characteristics. This system is explicitly written for the cases of Maxwell and Luneburg lenses, enabling numerical integration for ray paths in these optical devices.

What carries the argument

Method of characteristics applied to the eikonal equation, reducing it to a first-order ODE system for the ray trajectories in inhomogeneous media.

If this is right

Ray paths in Maxwell and Luneburg lenses can be computed by solving the derived ODE system numerically.
The geometric optics approximation connects wave solutions to classical ray tracing for these lenses.
Applications to calculation of optical devices follow directly from the transformed equations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar reductions could apply to other gradient-index optics problems if the refractive index allows characteristic curves.
Numerical stability of the ODE system might depend on the specific lens parameters chosen.
Extensions to time-dependent or anisotropic media would require modifying the characteristic equations.

Load-bearing premise

The short-wavelength approximation underlying geometric optics remains valid for the Maxwell and Luneburg lenses under consideration.

What would settle it

A direct comparison showing that numerical ray paths from the ODE system deviate significantly from exact wave solutions for these lenses at the wavelengths considered.

Figures

Figures reproduced from arXiv: 1906.09467 by A. V. Korolkova, D. S. Kulyabov, M. N. Gevorkyan, T. R. Velieva.

**Figure 3.** Figure 3: The trajectories of the rays in case of Luneburg lens for a point source and [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Figure 4: The wavefronts in case of Luneburg lens for a point source and [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

read the original abstract

The Maxwell equations have a fairly simple form. However, finding solutions of Maxwell's equations is an extremely difficult task. Therefore, various simplifying approaches are often used in optics. One such simplifying approach is to use the approximation of geometric optics. The approximation of geometric optics is constructed with the assumption that the wavelengths are small (short-wavelength approximation). The basis of geometric optics is the eikonal equation. The eikonal equation can be obtained from the wave equation (Helmholtz equation). Thus, the eikonal equation relates the wave and geometric optics. In fact, the eikonal equation is a quasi-classical approximation (the Wentzel-Kramers-Brillouin method) of wave optics. This paper shows the application of geometric methods of electrodynamics to the calculation of optical devices, such as Maxwell and Luneburg lenses. The eikonal equation, which was transformed to the ODE system by the method of characteristics, is considered. The resulting system is written for the case of Maxwell and Luneburg lenses.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper reduces the eikonal equation to an ODE system via characteristics for the Maxwell and Luneburg lenses using a standard technique with no new framework or results.

read the letter

The one thing to know is that this paper takes the eikonal equation, applies the method of characteristics to convert it to a system of ODEs, and then specializes that system to the Maxwell and Luneburg lenses. The second thing is that this is a direct, standard procedure with no additional physical assumptions beyond the usual short-wavelength limit for geometric optics. What the paper does well is to carry through the algebra for these two specific index profiles. The radial symmetry simplifies the characteristics, and spelling out the resulting ODEs for each lens gives a concrete starting point for numerical integration if that's the goal. The transformation itself follows the usual steps for the nonlinear first-order PDE |grad S| = n(r). The soft spots are more about what is missing than what is wrong. The title promises numerical analysis, but the abstract stops at writing the system. There is no indication of how they solve the ODEs, what integrators they use, or how the results compare to known analytic solutions for these lenses. Without that, it is difficult to assess accuracy or usefulness. The soundness of the reduction looks fine on its face, as it is a textbook step, but the lack of any verification leaves the practical value unclear. The citation pattern is not an issue here since the claim is just the application. No circular reasoning appears. This paper is for a reader who already knows the eikonal equation and wants the explicit ODE form for these two lenses in one place. It does not offer enough for someone looking for new methods or broad insights in computational optics. I would not recommend sending it for peer review. The work is too narrow and does not advance the state of the art enough to merit referee effort.

Referee Report

2 major / 2 minor

Summary. The manuscript derives the eikonal equation from the Helmholtz equation under the short-wavelength (geometric optics) approximation and applies the method of characteristics to convert the first-order nonlinear PDE |∇S| = n(r) into an equivalent system of ODEs. Explicit forms of this ODE system are presented for the radially symmetric refractive-index profiles of the Maxwell lens and the Luneburg lens, with the goal of enabling numerical ray-tracing analysis in these gradient-index devices.

Significance. The transformation itself is a standard application of Charpit’s method (or equivalent characteristic equations) to the eikonal equation and therefore adds little conceptual novelty. If the manuscript supplied reproducible numerical implementations, error bounds, or comparisons against known analytic ray paths for these lenses, the work could serve as a practical computational template for GRIN optics design; absent such content, its significance remains limited to a routine exercise in classical geometric optics.

major comments (2)

The title announces 'Numerical analysis,' yet the abstract and manuscript description contain no numerical results, discretization scheme, integration method, error estimates, or benchmark comparisons. Without these elements the central claim that the ODE system supports numerical analysis of the lenses cannot be evaluated.
The short-wavelength premise is invoked to justify the eikonal equation, but no quantitative check (e.g., wavelength-to-index-variation scale) is supplied for the Maxwell or Luneburg profiles; this assumption is load-bearing for the validity of the entire geometric-optics reduction.

minor comments (2)

Notation for the refractive index n(r) and the eikonal S should be introduced with explicit functional forms for each lens before the characteristic equations are written.
The manuscript should cite standard references for the method of characteristics applied to the eikonal equation (e.g., Born & Wolf or modern treatments of Charpit’s method) to place the derivation in context.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and the opportunity to clarify the scope of our work. We address the two major comments below.

read point-by-point responses

Referee: The title announces 'Numerical analysis,' yet the abstract and manuscript description contain no numerical results, discretization scheme, integration method, error estimates, or benchmark comparisons. Without these elements the central claim that the ODE system supports numerical analysis of the lenses cannot be evaluated.

Authors: We acknowledge that the manuscript presents the derivation of the characteristic ODE system but does not carry out explicit numerical integrations, error analysis, or benchmark comparisons. The central contribution is the explicit reduction of the eikonal equation to a first-order ODE system for the Maxwell and Luneburg profiles, which is directly amenable to standard numerical ODE solvers. To address this point we will revise the title to 'Characteristic equations for the eikonal equation in gradient-index optics' and add a short section illustrating numerical integration of the system together with a basic error estimate using a standard Runge-Kutta scheme. revision: yes
Referee: The short-wavelength premise is invoked to justify the eikonal equation, but no quantitative check (e.g., wavelength-to-index-variation scale) is supplied for the Maxwell or Luneburg profiles; this assumption is load-bearing for the validity of the entire geometric-optics reduction.

Authors: The short-wavelength (geometric-optics) limit is the standard justification for the eikonal equation and requires that the wavelength be much smaller than the spatial scale over which the refractive index varies. For the lenses considered here that scale is set by the lens radius. We will add a brief paragraph in the revised manuscript stating this condition explicitly and noting that it is satisfied for typical optical wavelengths (hundreds of nanometers) relative to lens sizes of centimeters or larger. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct standard transformation

full rationale

The paper's central step is the application of the method of characteristics (Charpit's method) to convert the first-order nonlinear PDE known as the eikonal equation |∇S| = n(x) into an equivalent ODE system, then specializing the resulting system to the radially symmetric refractive indices of the Maxwell and Luneburg lenses. This is a purely algebraic reduction that follows directly from the definition of the eikonal equation and the standard theory of first-order PDEs; it does not rely on any fitted parameters, self-citations that justify uniqueness, or ansatzes imported from prior work by the same authors. The short-wavelength premise is invoked only to motivate adoption of the eikonal equation itself, not to close any loop within the characteristic reduction. No load-bearing self-citation chains or renamings of known results appear in the abstract or described derivation. The work is therefore self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard short-wavelength assumption of geometric optics and the applicability of the method of characteristics; no free parameters or invented entities are mentioned.

axioms (1)

domain assumption Short-wavelength approximation holds for the optical devices considered
Explicitly stated in the abstract as the basis for geometric optics and the eikonal equation.

pith-pipeline@v0.9.0 · 5726 in / 1109 out tokens · 22904 ms · 2026-05-25T18:08:20.847622+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 1 internal anchor

[1]

= 2𝑝1 𝜕𝑝1 𝜕𝑥 + 2𝑝2 𝜕𝑝2 𝜕𝑥 = 2𝑛 𝜕𝑛 𝜕𝑥 , 𝜕 𝜕𝑦 (𝑝2 1 + 𝑝2

work page
[2]

RUDN University Program 5-100

= 2𝑝1 𝜕𝑝1 𝜕𝑦 + 2𝑝2 𝜕𝑝2 𝜕𝑦 = 2𝑛 𝜕𝑛 𝜕𝑦 . After that the following system of equations is obtained: 𝑝1 𝜕𝑝1 𝜕𝑥 + 𝑝2 𝜕𝑝2 𝜕𝑥 = 𝑛 𝜕𝑛 𝜕𝑥 , 𝑝1 𝜕𝑝1 𝜕𝑦 + 𝑝2 𝜕𝑝2 𝜕𝑦 = 𝑛 𝜕𝑛 𝜕𝑦 , =⇒ (︂ p, 𝜕p 𝜕𝑥 )︂ = 𝑛 𝜕𝑛 𝜕𝑥 , (︂ p, 𝜕p 𝜕𝑦 )︂ = 𝑛 𝜕𝑛 𝜕𝑦 . Since 𝜕𝑝1 𝜕𝑦 = 𝜕2𝑢(𝑥, 𝑦) 𝜕𝑦𝜕𝑥 = 𝜕2𝑢(𝑥, 𝑦) 𝜕𝑥𝜕𝑦 = 𝜕𝑝2 𝜕𝑥 , then 𝜕𝑝1 𝜕𝑦 = 𝜕𝑝2 𝜕𝑥 . Using this equality our expressions may be converted i...

work page
[3]

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th Edition, Cambridge University Press, Cambridge, 1999

work page 1999
[4]

J. A. Stratton, Electromagnetic Theory, MGH, 1941

work page 1941
[5]

L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, 4th Edition, Course of Theoretical Physics. Vol. 2, Butterworth- Heinemann, 1975

work page 1975
[6]

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Edition, Course of Theoretical Physics. Vol. 8, Butterworth-Heinemann, 1984

work page 1984
[7]

Bruns, Das Eikonal, Vol

H. Bruns, Das Eikonal, Vol. 35, S. Hirzel, Leipzig, 1895

work page
[8]

Klein,¨Uber das Brunssche Eikonal, Zeitscrift f¨ ur Mathematik und Physik 46 (1901) 372–375

F. Klein,¨Uber das Brunssche Eikonal, Zeitscrift f¨ ur Mathematik und Physik 46 (1901) 372–375

work page 1901
[9]

Jeong, R

W. Jeong, R. Whitaker, A Fast Eikonal Equation Solver for Parallel Systems, SIAM conference on ... 84112 (2007) 1–4

work page 2007
[10]

A "missing" family of classical orthogonal polynomials

R. Kimmel, J. A. Sethian, Computing Geodesic Paths on Manifolds, Proceedings of the National Academy of Sciences 95 (15) (1998) 8431–8435.arXiv:arXiv:1011.1669v3, doi:10.1073/pnas.95.15.8431

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1073/pnas.95.15.8431 1998
[11]

Zhao, A Fast Sweeping Method for Eikonal Equations, Mathematics of Computation 74 (250) (2004) 603–628.doi: 10.1090/S0025-5718-04-01678-3

H. Zhao, A Fast Sweeping Method for Eikonal Equations, Mathematics of Computation 74 (250) (2004) 603–628.doi: 10.1090/S0025-5718-04-01678-3

work page doi:10.1090/s0025-5718-04-01678-3 2004
[12]

Beliakov, Numerical Evaluation of the Luneburg Integral and Ray Tracing, Applied Optics 35 (7) (1996) 1011–1014

G. Beliakov, Numerical Evaluation of the Luneburg Integral and Ray Tracing, Applied Optics 35 (7) (1996) 1011–1014. doi:10.1364/AO.35.001011

work page doi:10.1364/ao.35.001011 1996
[13]

P. A. Gremaud, C. M. Kuster, Computational Study of Fast Methods for the Eikonal Equation, SIAM Journal on Scientific Computing 27 (6) (2006) 1803–1816.doi:10.1137/040605655

work page doi:10.1137/040605655 2006
[14]

S. Bak, J. McLaughlin, D. Renzi, Some Improvements for the Fast Sweeping Method, SIAM Journal on Scientific Computing 32 (5) (2010) 2853–2874.doi:10.1137/090749645

work page doi:10.1137/090749645 2010
[15]

D. S. Kulyabov, A. V. Korolkova, L. A. Sevastianov, M. N. Gevorkyan, A. V. Demidova, Geometrization of Maxwell’s Equations in the Construction of Optical Devices, in: V. L. Derbov, D. E. Postnov (Eds.), Proceedings of SPIE. Saratov Fall Meeting 2016: Laser Physics and Photonics XVII and Computational Biophysics and Analysis of Biomedical Data III, Vol. 10...

work page doi:10.1117/12.2267959 2016
[16]

D. S. Kulyabov, A. V. Korolkova, L. A. Sevastianov, M. N. Gevorkyan, A. V. Demidova, Algorithm for Lens Calculations in the Geometrized Maxwell Theory, in: V. L. Derbov, D. E. Postnov (Eds.), Saratov Fall Meeting 2017: Laser Physics and Photonics XVIII; and Computational Biophysics and Analysis of Biomedical Data IV, Vol. 10717 of Proceedings of SPIE, SPI...

work page doi:10.1117/12.2315066 2017
[17]

R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley & Los Angeles, 1964

work page 1964
[18]

S. P. Morgan, General Solution of the Luneberg Lens Problem, Journal of Applied Physics 29 (9) (1958) 1358.doi: 10.1063/1.1723441

work page doi:10.1063/1.1723441 1958
[19]

J. A. Lock, Scattering of an Electromagnetic Plane Wave by a Luneburg Lens I Ray Theory, Journal of the Optical Society of America A 25 (12) (2008) 2971.doi:10.1364/JOSAA.25.002971

work page doi:10.1364/josaa.25.002971 2008
[20]

J. A. Lock, Scattering of an Electromagnetic Plane Wave by a Luneburg Lens II Wave Theory, Journal of the Optical Society of America A 25 (12) (2008) 2980.doi:10.1364/JOSAA.25.002980

work page doi:10.1364/josaa.25.002980 2008
[21]

J. C. Maxwell, Solutions of Problems (prob. 3, vol. VIII, p. 188), The Cambridge and Dublin mathematical journal 9 (1854) 9–11

work page
[22]

Joshi, R

A. Joshi, R. Lakhanpal, Learning Julia, Packt Publishing, 2017

work page 2017

[1] [1]

= 2𝑝1 𝜕𝑝1 𝜕𝑥 + 2𝑝2 𝜕𝑝2 𝜕𝑥 = 2𝑛 𝜕𝑛 𝜕𝑥 , 𝜕 𝜕𝑦 (𝑝2 1 + 𝑝2

work page

[2] [2]

RUDN University Program 5-100

= 2𝑝1 𝜕𝑝1 𝜕𝑦 + 2𝑝2 𝜕𝑝2 𝜕𝑦 = 2𝑛 𝜕𝑛 𝜕𝑦 . After that the following system of equations is obtained: 𝑝1 𝜕𝑝1 𝜕𝑥 + 𝑝2 𝜕𝑝2 𝜕𝑥 = 𝑛 𝜕𝑛 𝜕𝑥 , 𝑝1 𝜕𝑝1 𝜕𝑦 + 𝑝2 𝜕𝑝2 𝜕𝑦 = 𝑛 𝜕𝑛 𝜕𝑦 , =⇒ (︂ p, 𝜕p 𝜕𝑥 )︂ = 𝑛 𝜕𝑛 𝜕𝑥 , (︂ p, 𝜕p 𝜕𝑦 )︂ = 𝑛 𝜕𝑛 𝜕𝑦 . Since 𝜕𝑝1 𝜕𝑦 = 𝜕2𝑢(𝑥, 𝑦) 𝜕𝑦𝜕𝑥 = 𝜕2𝑢(𝑥, 𝑦) 𝜕𝑥𝜕𝑦 = 𝜕𝑝2 𝜕𝑥 , then 𝜕𝑝1 𝜕𝑦 = 𝜕𝑝2 𝜕𝑥 . Using this equality our expressions may be converted i...

work page

[3] [3]

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 7th Edition, Cambridge University Press, Cambridge, 1999

work page 1999

[4] [4]

J. A. Stratton, Electromagnetic Theory, MGH, 1941

work page 1941

[5] [5]

L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, 4th Edition, Course of Theoretical Physics. Vol. 2, Butterworth- Heinemann, 1975

work page 1975

[6] [6]

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd Edition, Course of Theoretical Physics. Vol. 8, Butterworth-Heinemann, 1984

work page 1984

[7] [7]

Bruns, Das Eikonal, Vol

H. Bruns, Das Eikonal, Vol. 35, S. Hirzel, Leipzig, 1895

work page

[8] [8]

Klein,¨Uber das Brunssche Eikonal, Zeitscrift f¨ ur Mathematik und Physik 46 (1901) 372–375

F. Klein,¨Uber das Brunssche Eikonal, Zeitscrift f¨ ur Mathematik und Physik 46 (1901) 372–375

work page 1901

[9] [9]

Jeong, R

W. Jeong, R. Whitaker, A Fast Eikonal Equation Solver for Parallel Systems, SIAM conference on ... 84112 (2007) 1–4

work page 2007

[10] [10]

A "missing" family of classical orthogonal polynomials

R. Kimmel, J. A. Sethian, Computing Geodesic Paths on Manifolds, Proceedings of the National Academy of Sciences 95 (15) (1998) 8431–8435.arXiv:arXiv:1011.1669v3, doi:10.1073/pnas.95.15.8431

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1073/pnas.95.15.8431 1998

[11] [11]

Zhao, A Fast Sweeping Method for Eikonal Equations, Mathematics of Computation 74 (250) (2004) 603–628.doi: 10.1090/S0025-5718-04-01678-3

H. Zhao, A Fast Sweeping Method for Eikonal Equations, Mathematics of Computation 74 (250) (2004) 603–628.doi: 10.1090/S0025-5718-04-01678-3

work page doi:10.1090/s0025-5718-04-01678-3 2004

[12] [12]

Beliakov, Numerical Evaluation of the Luneburg Integral and Ray Tracing, Applied Optics 35 (7) (1996) 1011–1014

G. Beliakov, Numerical Evaluation of the Luneburg Integral and Ray Tracing, Applied Optics 35 (7) (1996) 1011–1014. doi:10.1364/AO.35.001011

work page doi:10.1364/ao.35.001011 1996

[13] [13]

P. A. Gremaud, C. M. Kuster, Computational Study of Fast Methods for the Eikonal Equation, SIAM Journal on Scientific Computing 27 (6) (2006) 1803–1816.doi:10.1137/040605655

work page doi:10.1137/040605655 2006

[14] [14]

S. Bak, J. McLaughlin, D. Renzi, Some Improvements for the Fast Sweeping Method, SIAM Journal on Scientific Computing 32 (5) (2010) 2853–2874.doi:10.1137/090749645

work page doi:10.1137/090749645 2010

[15] [15]

D. S. Kulyabov, A. V. Korolkova, L. A. Sevastianov, M. N. Gevorkyan, A. V. Demidova, Geometrization of Maxwell’s Equations in the Construction of Optical Devices, in: V. L. Derbov, D. E. Postnov (Eds.), Proceedings of SPIE. Saratov Fall Meeting 2016: Laser Physics and Photonics XVII and Computational Biophysics and Analysis of Biomedical Data III, Vol. 10...

work page doi:10.1117/12.2267959 2016

[16] [16]

D. S. Kulyabov, A. V. Korolkova, L. A. Sevastianov, M. N. Gevorkyan, A. V. Demidova, Algorithm for Lens Calculations in the Geometrized Maxwell Theory, in: V. L. Derbov, D. E. Postnov (Eds.), Saratov Fall Meeting 2017: Laser Physics and Photonics XVIII; and Computational Biophysics and Analysis of Biomedical Data IV, Vol. 10717 of Proceedings of SPIE, SPI...

work page doi:10.1117/12.2315066 2017

[17] [17]

R. K. Luneburg, Mathematical Theory of Optics, University of California Press, Berkeley & Los Angeles, 1964

work page 1964

[18] [18]

S. P. Morgan, General Solution of the Luneberg Lens Problem, Journal of Applied Physics 29 (9) (1958) 1358.doi: 10.1063/1.1723441

work page doi:10.1063/1.1723441 1958

[19] [19]

J. A. Lock, Scattering of an Electromagnetic Plane Wave by a Luneburg Lens I Ray Theory, Journal of the Optical Society of America A 25 (12) (2008) 2971.doi:10.1364/JOSAA.25.002971

work page doi:10.1364/josaa.25.002971 2008

[20] [20]

J. A. Lock, Scattering of an Electromagnetic Plane Wave by a Luneburg Lens II Wave Theory, Journal of the Optical Society of America A 25 (12) (2008) 2980.doi:10.1364/JOSAA.25.002980

work page doi:10.1364/josaa.25.002980 2008

[21] [21]

J. C. Maxwell, Solutions of Problems (prob. 3, vol. VIII, p. 188), The Cambridge and Dublin mathematical journal 9 (1854) 9–11

work page

[22] [22]

Joshi, R

A. Joshi, R. Lakhanpal, Learning Julia, Packt Publishing, 2017

work page 2017