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Ray Propagation and Aberrations in a Plane‑Parallel Plate: A Theoretical Analysis Based on Snell’s Law

Various equations can be derived from the law of refraction of light at material interfaces (Snell's law). Here we introduce some practical ones.

table of contents:

  • Image position of convergent light incident perpendicularly on a plane-parallel plate

The position of the real image relative to the virtual image is determined for two cases: when the real image is inside the plane-parallel plate, and when it is beyond the plane-parallel plate.

A ray diagram of light rays passing through a plane-parallel plate is shown along with a virtual image.

(1) When the real image is in a plane-parallel plate:

平行平面板と結像位置の関係_042.png

*For dimensional notation, see Appendix: Definitions of direction and sign for one‑dimensional vectors and angles

Here, dh, dθ1, and dθ2 are assumed to be minute quantities.

The conditions shown in the figure can be expressed as the following equations:

平行平面板と結像位置の関係_043.png

Here, the second equation is Snell's law. Also, since dθ1 and dθ2 are infinitesimal quantities, the following can be said:

Description of light rays entering a material_003.png

So the original equation can be replaced with:

平行平面板と結像位置の関係_044.png

Solving this leads to the following:

Relationship between parallel plane plate and image position_013.png

(2) When the real image is beyond the plane-parallel plate:

平行平面板と結像位置の関係_045.png

*For dimensional notation, see Appendix:Definitions of direction and sign for one‑dimensional vectors and angles

Here, dh, dθ1, and dθ2 are assumed to be minute quantities.

The conditions shown in the figure can be expressed as the following equations:

平行平面板と結像位置の関係_046.png

Here, the second equation is Snell's law. Also, since dθ1 and dθ2 are infinitesimal quantities, the following can be said:

Description of light rays entering a material_003.png

So the original equation can be replaced with:

平行平面板と結像位置の関係_047.png

Solving this leads to the following:

The difference between the position of a light ray passing through a plane-parallel plate and the position of a virtual image

  • Image formation position and astigmatism when parallel planes are positioned obliquely to the optical axis

Consider a light beam that is obliquely incident on a parallel plane as shown below, and forms an image at the exit point. In this case, the imaging position is determined by separating it into the tangential and sagittal directions, and expressing the relative position of each to the virtual image position (the difference between these is the amount of astigmatism).

・ Derivation of the imaging position of the tangential component

Consider the system shown in the figure below. Here, the small angles dθ1,2 of the tangential component are taken parallel to the screen.

平行平面板斜め入射_022.png

Here, dhT , dθ1 , and dθ2 are assumed to be infinitesimal amounts.

The conditions shown in the diagram can be expressed as follows:

平行平面板斜め入射_023.png

Here, let's discuss the formulation of equation (2)-1-④. Since dθ1 and dθ2 are infinitesimal quantities, if we consider the red line and the green dotted line to be parallel, then the ray width before incidence on the interface is twice the ray width after passing through the interface by cosθ1/cosθ2 times, and we formulated equation (2)-1-④ taking this into account.

平行平面板斜め入射_024.png

Differentiating both sides of (2)-1-③,

平行平面板斜め入射_025.png

From (2)-1-①,②,③',④,⑤, eliminating s, xA, xBT, x0, dhT, dθ1, dθ2,

平行平面板斜め入射_026.png

Here, from (2)-1-①,⑥,

平行平面板斜め入射_027.png

Also, from (2)-1-③,

平行平面板斜め入射_029.png

Therefore, the desired solution can be expressed as follows.

平行平面板斜め入射_030.png

Furthermore, by applying (2)-1-③ to eliminate θ2 ,

平行平面板斜め入射_031.png
平行平面板斜め入射_032.png

・Derivation of the imaging position of the sagittal component

Consider the system shown in the figure below. Here, the small angle dφ1,2 of the sagittal component is taken in the depth direction relative to the screen.

平行平面板斜め入射_033.png

Here, dhS , dφ1 , and dφ2 are assumed to be minute amounts.

The conditions shown in the diagram can be expressed as follows:

平行平面板斜め入射_034.png

From (2)-2-①,②,④,⑤,⑦, Eliminating s, xA, xBS, x0, dhS, dφ1, dφ2,

平行平面板斜め入射_035.png

Here, from (2)-2-①,⑥,

平行平面板斜め入射_036.png

Also, from (2)-2-③,

平行平面板斜め入射_037.png

Therefore, the desired solution can be expressed as follows.

平行平面板斜め入射_038.png

Furthermore, by applying (2)-2-③ to eliminate θ2 ,

平行平面板斜め入射_039.png
平行平面板斜め入射_040.png

・Derivation of astigmatism

From (2)-1-⑤'' and (2)-2-⑤'', the amount of astigmatism can be calculated using the following formula.

平行平面板斜め入射_041.png

*This derivation is based on our own approach, but similar results are shown in standard literature [A] .

  • Image position when a parallel plate is placed at an angle to the optical axis

Consider the following optical arrangement:

This shows the spherical aberration that occurs in a plane-parallel plate.

dθ1 and dθ2 are assumed to be minute amounts, and the longitudinal spherical aberration δs and the horizontal spherical aberration δx are calculated.

The conditions shown in the figure can be expressed as the following equations:

Relationship between parallel plane plate and image position_029.png

① and ③ represent Snell's law, and ⑤ represents the numerical aperture.

From ①, ②, and ⑤,

Relationship between parallel plane plate and image position_030.png

From ③ and ④,

Relationship between parallel plane plate and image position_031.png

From ②' and ④', the longitudinal spherical aberration is as follows:

Amount of longitudinal spherical aberration generated by a plane-parallel plate

From ⑥ and ⑦, the lateral spherical aberration is as follows:

Transverse spherical aberration generated by a plane-parallel plate

The amount of spherical aberration is shown in the diagram below: As can be seen from the diagram, for the same plate thickness, the amount of spherical aberration peaks when the relative refractive index is around 1.6.

Refractive index dependence of longitudinal spherical aberration generated by a plane-parallel plate
Refractive index dependence of transverse spherical aberration generated by a plane-parallel plate

Here, the influence of aberration will be considered.

The point image resolution (Airy disk radius) due to wave optics diffraction broadening is expressed by the following formula:

Amount of blurring of spots due to wave optics diffraction broadening

As a rough guideline, when δx is much smaller than δd , δd dominates the resolution, so the effect of δx is small. On the other hand, when δx is much larger than δd , δx dominates the resolution.

A graph comparing δx and δd is shown below. While this is only a rough guideline, at a wavelength of 532 nm, the effects of spherical aberration become dominant over the point image resolution due to diffraction spread when using cover glass (approximately 0.15 mm thick) with an NA of 0.32 or higher, and when using glass with a thickness of 1.5 mm with an NA of 0.18 or higher. At a wavelength of 1064 nm, the effects of spherical aberration become dominant over the point image resolution due to diffraction spread when using cover glass (approximately 0.15 mm thick) with an NA of 0.38 or higher, and when using glass with a thickness of 1.5 mm with an NA of 0.22 or higher.

Comparison of spot blur due to lateral spherical aberration and diffraction spread when NA is a variable

Therefore, in actual microscopes, spherical aberration can occur when observing through glass, degrading the image. This effect is particularly pronounced with objective lenses that have a large numerical aperture (NA). For this reason, with high NA objective lenses, aberration-correcting objectives are used that are designed to cancel out the spherical aberration caused by the glass, for specific applications such as observation through a cover slip [1] .

  • References

[1] Cover glass and correction ring | Basic knowledge (advanced) | Nikon Solutions Inc.

  • Related literature

[A] Warren J. Smith, Modern Optical Engineering, 4th Ed., McGraw-Hill Education (2008). Chapter 13.10 "The Plane Parallel Plate" / Section 13.5 "Astigmatism"

  • Update History

  • 2026-04: "Image formation position and astigmatism when parallel planes are positioned obliquely to the optical axis": Added derivation of sagittal component and astigmatism amount.

  • 2025-10: "Image formation position and astigmatism when parallel planes are positioned obliquely to the optical axis": Corrected errors in the process and results of deriving the tangential component.

  • 2025-10: Added "Spherical aberration occurring in parallel plane plates".

  • 2025-08: Newly released

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