When should I analyze optical problems using ray tracing vs. wavefront analysis?

[TJULICHEN]

TL;DR

This question explores the boundary between geometric and wave optics.

I'm often confused about when it's appropriate to use geometric optics (ray tracing) and when it's necessary to switch to wave optics and analyze the wavefront instead.


For example, in many interference problems, the conclusions derived from ray tracing are often correct—even though interference is inherently a wave phenomenon. However, in some cases that seem well-suited for ray optics, the ray model gives incorrect results.


One example that puzzles me is this:
Consider a light ray incident normally (perpendicularly) onto a flat glass plate, but the refractive index of the plate varies linearly with height. According to standard refraction analysis using Snell's law, the incident angle is zero, so the direction of propagation should remain unchanged. However, if I analyze the wavefront, it becomes clear that the ray will actually bend due to the spatial gradient in refractive index—the glass plate behaves like a prism. So, in this case, wavefront analysis gives the correct physical behavior, while basic ray tracing fails.


This leads to my central question:
How can I determine when it's sufficient to analyze a system using rays and Snell's law, and when I must use wavefront or full wave analysis instead?


I understand that ray optics assumes negligible wavelength effects and is generally valid for large-scale systems, while wave optics accounts for interference, diffraction, and spatial phase variations. But in practice, it's not always clear where to draw the line—especially in systems like graded-index media.


Any insights or guidelines for choosing the appropriate method would be greatly appreciated.

 

[A.T.]

Consider a light ray incident normally (perpendicularly) onto a flat glass plate, but the refractive index of the plate varies linearly with height.

Is the refractive index gradient perpendicular to the surface that the ray enters perpendiculary, and thus parallel to the inital ray direction?

 

[TJULICHEN]

Is the refractive index gradient perpendicular to the surface that the ray enters perpendiculary, and thus parallel to the inital ray direction?

View attachment 363553
Thank you very much for your response. I have sketched a diagram to illustrate the situation. The refractive index of the glass varies along the y-direction, and the light is incident perpendicularly on the end face of the glass, which is a standard rectangular block. The deflection direction of the light ray is roughly drawn and not based on precise calculations.

 

[A.T.]

View attachment 363553
Thank you very much for your response. I have sketched a diagram to illustrate the situation. The refractive index of the glass varies along the y-direction, and the light is incident perpendicularly on the end face of the glass, which is a standard rectangular block. The deflection direction of the light ray is roughly drawn and not based on precise calculations.

OK, that makes sense. In this case rays and Snell's law alone are indeed not helpful.

Not sure if there is a general method how to recognize those cases, other than replacing each single ray, with two parallel pulses close together, considering how they would propagate in the given situation and if the line connecting them (representing the wave front) stays perpendicular to the ray from pure ray-optics.

 

[Andy Resnick]

TL;DR Summary: This question explores the boundary between geometric and wave optics.

I'm often confused about when it's appropriate to use geometric optics (ray tracing) and when it's necessary to switch to wave optics and analyze the wavefront instead.

The branch of optics characterized by the neglect of wavelength (the limiting case of λ→ 0) is geometrical optics, which models the transport of electromagnetic energy as 'light rays'. A light ray is physically interpreted as an infinitesimal packet of radiance (as opposed to intensity or irradiance). The connection between geometrical optics and wave optics is the eikonal equation. Note that geometrical optics also (almost always) ignores polarization.

For example, in many interference problems, the conclusions derived from ray tracing are often correct—even though interference is inherently a wave phenomenon.

Geometrical optics can indeed be used to model interference, aberrations and inhomogeneous and/or anisotropic media.

https://repository.tudelft.nl/record/uuid:f5d05e64-0792-4960-b0fe-5ccbc95da562

However, in some cases that seem well-suited for ray optics, the ray model gives incorrect results.
One example that puzzles me is this:
Consider a light ray incident normally (perpendicularly) onto a flat glass plate, but the refractive index of the plate varies linearly with height. According to standard refraction analysis using Snell's law, the incident angle is zero, so the direction of propagation should remain unchanged. However, if I analyze the wavefront, it becomes clear that the ray will actually bend due to the spatial gradient in refractive index—the glass plate behaves like a prism. So, in this case, wavefront analysis gives the correct physical behavior, while basic ray tracing fails.

I am unfamiliar with this result- can you provide some details? Prism dispersion is well-modeled with geometric optics, but you seem to be describing something totally different.

This leads to my central question:
How can I determine when it's sufficient to analyze a system using rays and Snell's law, and when I must use wavefront or full wave analysis instead?

I'm not sure there is a 'bright line' separating the two approaches. Geometrical optics is valid within the constraints of the eikonal equation, which is derived using several simplifying assumptions- for example, that spatial variations of both the fields and the optical path are much larger than the wavelength. This is why geometrical optics fails when one or more of those quantities varies rapidly, like at a boundary or discontinuity.

The primary strength of geometrical optics is simplicity, based on the assumption that locally the field behaves as a plane wave.

Born and Wolf's text is the standard source of this discussion.

 

[DaveE]

https://courses.washington.edu/me557/reading/geometric_optics_1.pdf

 

[TJULICHEN]

The branch of optics characterized by the neglect of wavelength (the limiting case of λ→ 0) is geometrical optics, which models the transport of electromagnetic energy as 'light rays'. A light ray is physically interpreted as an infinitesimal packet of radiance (as opposed to intensity or irradiance). The connection between geometrical optics and wave optics is the eikonal equation. Note that geometrical optics also (almost always) ignores polarization.



Geometrical optics can indeed be used to model interference, aberrations and inhomogeneous and/or anisotropic media.

https://repository.tudelft.nl/record/uuid:f5d05e64-0792-4960-b0fe-5ccbc95da562


I am unfamiliar with this result- can you provide some details? Prism dispersion is well-modeled with geometric optics, but you seem to be describing something totally different.



I'm not sure there is a 'bright line' separating the two approaches. Geometrical optics is valid within the constraints of the eikonal equation, which is derived using several simplifying assumptions- for example, that spatial variations of both the fields and the optical path are much larger than the wavelength. This is why geometrical optics fails when one or more of those quantities varies rapidly, like at a boundary or discontinuity.

The primary strength of geometrical optics is simplicity, based on the assumption that locally the field behaves as a plane wave.

Born and Wolf's text is the standard source of this discussion.

View attachment 363553
Thank you very much for your response. I have sketched a diagram to illustrate the situation. The refractive index of the glass varies along the y-direction, and the light is incident perpendicularly on the end face of the glass, which is a standard rectangular block and is isotropic. The deflection direction of the light ray is roughly drawn and not based on precise calculations.

 

[TJULICHEN]

https://courses.washington.edu/me557/reading/geometric_optics_1.pdf

Thank you so much for your prompt and precise reply. I truly appreciate it!

 

[Andy Resnick]

Thank you very much for your response. I have sketched a diagram to illustrate the situation. The refractive index of the glass varies along the y-direction, and the light is incident perpendicularly on the end face of the glass, which is a standard rectangular block and is isotropic. The deflection direction of the light ray is roughly drawn and not based on precise calculations.

Thanks, but the diagram you provided isn't helpful. For all I can tell, you just made that up using your imagination. Where is the analysis?

 

[A.T.]

Thanks, but the diagram you provided isn't helpful. For all I can tell, you just made that up using your imagination.

Seems like this is what the OP has in mind:
https://en.wikipedia.org/wiki/Gradient-index_optics

 

[TJULICHEN]

Thanks, but the diagram you provided isn't helpful. For all I can tell, you just made that up using your imagination. Where is the analysis?

For this problem, I analyzed it using the concept of a phase mask (screen function). When light passes through a medium with a gradually varying refractive index, a phase mask function can be defined as

P(y) = \exp[i\,k\,n(y)\,L]

where k = 2\pi/\lambda is the wavenumber, n(y) is the refractive index as a function of y, and L is the thickness.

If the refractive index varies linearly with y, the phase mask becomes a linear phase delay:

P(y) = \exp[i\,k\,(a + y)]

Through Fourier analysis, the spatial frequency corresponding to this phase function directly determines the deflection direction of the light beam. Therefore, from the perspective of wave optics, when the refractive index varies along the propagation direction, the direction of light propagation will inevitably change. This principle is also applied in GRIN prisms.

The derivation may not be very rigorous, as it assumes that the light propagates horizontally through the glass. However, the conclusion can still qualitatively describe the physical phenomenon.

 

[A.T.]

when the refractive index varies along the propagation direction

Did you mean perpendiculary to the propagation direction? That is at least how I understood your diagram.

 

[Andy Resnick]

The derivation may not be very rigorous, as it assumes that the light propagates horizontally through the glass. However, the conclusion can still qualitatively describe the physical phenomenon.

Thanks, that's helpful. I guess I don't understand why you think ray tracing does not give valid results; ray tracing through GRIN lenses is fairly standard and often based on Hamiltonian optics and/or the eikonal equation:

https://opg.optica.org/ao/abstract.cfm?uri=ao-26-15-2943
https://pure.tue.nl/ws/portalfiles/portal/344323264/josaa-41-9-1656.pdf
https://opg.optica.org/ao/abstract.cfm?uri=ao-53-19-4343