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Ray propagation in an ideal lens system based on Newton’s formula

First, we derive the equation for the imaging relationship. Then, we show a recurrence model of geometric ray propagation through an ideal lens, applying Newton's Formula. We also show how to construct an imaging optical system by specifying the initial value of the ray and automatically calculating the final value.

This shows the results of ray tracing of an imaging optical system with an ideal lens (focal length: f). The method is to apply Newton's formula (or Newton's relation) to the recurrence formula.

*The content explained here is only a model of geometrical optics ray propagation, and does not take into account the effects of wave optics diffraction. For Gaussian beam propagation, the effects of wave optics diffraction must be taken into account, and the theoretical model is different, so please refer to the link below.

table of contents:

・Derivation of the imaging relationship formula

The manner in which a ray of light from an object point passes through an optical element with a focal length f and is focused onto an image point is illustrated below.

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

(i) When f > 0, z < 0,

理想レンズの配置による光線追跡_156.png

(ii) When f > 0, 0 < z < f,

理想レンズの配置による光線追跡_157.png

(iii) When f > 0, f < z,

理想レンズの配置による光線追跡_158.png

(iv) When f < 0, z < f

理想レンズの配置による光線追跡_159.png

(v) When f < 0, f < z < 0,

理想レンズの配置による光線追跡_160.png

(vi) When f < 0, z > 0 ,

理想レンズの配置による光線追跡_161.png

If the magnification is β, the following can be said for all of (i) to (vi) based on the similarity of triangles.

理想レンズの配置による光線追跡_060.png

This is because (i) to (vi) are all forms of the same model, so we only need to consider one of (i) to (vi).

Next, consider the following:

理想レンズの配置による光線追跡_120.png

From the figure, the following can be said:

理想レンズの配置による光線追跡_124.png

From (1)-①', (1)-②a, and (1)-②b,

理想レンズの配置による光線追跡_123.png

From Lagrange invariant,

理想レンズの配置による光線追跡_125.png

From (1)-④, (1)-②abc, and (1)-①',

理想レンズの配置による光線追跡_126.png

From (1)-① and (1)-⑤,

理想レンズの配置による光線追跡_127.png

Next, if we define the longitudinal magnification α as follows, then from (1)-①, (1)-①', and (1)-⑥,

理想レンズの配置による光線追跡_147.png

It should be noted that this only holds true when z is a small quantity.

*When the medium is air:

Since n=n'=1, the following can be said.

ガウスの結像式
理想レンズの配置による光線追跡_142.png

From (1)-①, (1)-①', and (1)-⑧, we obtain the following equations, including Newton's equations (or Newton's relations).

理想レンズの��配置による光線追跡_148.png

Next, by applying (1)-(3) to (1)-(8), the following Gaussian lens formula is obtained.

理想レンズの配置による光線追跡_149.png
縦倍率

Next, the object-to-image distance will be described.
If the object-image distance is defined as a one-dimensional vector with the object point as the start point and the image point as the end point, it can be expressed as follows:

理想レンズの配置による光線追跡_150.png

Next, for the vertical magnification, by substituting n=n'=1 into (1)-⑦, the following formula is obtained.

理想レンズの配置による光線追跡_151.png

・Recurrence formula model for geometric ray propagation through ideal lenses

1. Definition of parameters of the recurrence formula

Consider a system in which light rays pass through the 0th surface (object surface), the 1st surface, the 2nd surface, ..., the i-th surface, ..., and the n-th surface (image surface) in that order. In this case, the parameters of the i-th surface can be illustrated as follows:

理想レンズの配置による光線追跡_155.png

・Distance from the front focal point to the front principal point of the optical element: fi

・Distance from the rear principal point of the optical element to the rear focal point: fi'

・Distance between principal points of optical elements: HH'i

・Distance from the rear principal point of an optical element to the front principal point of the next optical element: di

・Distance from the object-side aerial image to the front focus of the optical element: zi

・Distance from the back focus of the optical element to the image-side aerial image: z'i

・Ray height at the optical element position: hi

・Image side ray angle: θi

・Distance from the origin position (0th object surface) to the front principal point of the optical element: Li

However, the settings are in blue and the values calculated from the recurrence formula are in black.

2. Derivation of the recurrence formula

Next, to find the recurrence formula, the i-th and (i+1)-th surfaces are illustrated below.

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

理想レンズの配置による光線追跡_169.png

However, here too, the set values are in blue and the values calculated from the recurrence formula are in black.

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

理想レンズの配置による光線追跡_164.png

Here, the second and third equations represent Newton's equations.

Here, the medium is air, and (1)-⑧ is applied.
In addition, θi and θi+1 are assumed to be minute amounts.
In this case, rearranging (2)-① gives the following recurrence formula:

理想レンズの配置による光線追跡_165.png

3. Deriving the first term of the recurrence formula

Next, to obtain the first term of the recurrence formula, we illustrate the 0th and 1st surfaces as follows:

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

理想レンズの配置による光線追跡_168.png

However, here too, the set values are in blue and the values calculated from the recurrence formula are in black.

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

理想レンズの配置による光線追跡_170.png

Here, the second equation represents Newton's equation.

Here, the medium is air, and (1)-⑧ is applied.
Also, θ0 and θ1 are assumed to be small quantities.
In this case, rearranging (2)-③ gives the first term of the recurrence formula as follows:

理想レンズの配置による光線追跡_167.png

Therefore, by inputting values for the settings of each surface and performing calculations (2)-② and (2)-④ in order from the 0th surface to the nth surface (image surface), ray tracing can be performed.

・Setting the initial state of the light

Next, let us consider how to set the setting values h0 and θ0 for the 0th surface.

There are various possible methods, but here we will show a method that uses the entrance pupil position and the object-side numerical aperture.

By setting the object height y0 , the distance to the entrance pupil zp , and the object-side numerical aperture NAo, we can obtain the five light rays (ray1, ray2, ..., ray5) shown in the figure below.

Ray tracing with ideal lens arrangement_072.png

Here, D is the entrance pupil diameter. If the angle φ is assumed to be infinitesimal, NAo can be expressed as follows:

Ray tracing with ideal lens arrangement_069.png

Therefore, the setting values h0 and θ0 of the 0th surface, which represent each of the rays ray1, ray2, ..., ray5, can be calculated as follows.

Ray tracing with ideal lens arrangement_071.png

・Calculating the closing value of a ray

Finally, the parameters of the optical system and image plane are automatically calculated.

For ray1 (or ray2), calculate dn-1 so that the position where the ray intersects with the central axis after passing through the optical element at the (n-1)-th surface becomes the image surface (the n-th surface) .

In this case, if the light ray for ray1 from the (n-1)-th surface to the n-th surface (image surface) is illustrated, it will look like this:

Ray tracing with ideal lens arrangement_118.png

The conditions shown in the figure for ray1 can be expressed as follows.

Ray tracing with ideal lens arrangement_112.png

Here, if we assume that the medium is air and apply (1)-⑧, we get the following:

理想レンズの配置による光線追跡_153.png

Then, by using the ray height hn(ray3) of ray3 at the image plane position , the magnification β of the entire optical system can be found by calculating the following:

Ray tracing with ideal lens arrangement_110.png

Furthermore, the image-side numerical aperture NAi can be calculated as follows:

Ray tracing with ideal lens arrangement_083.png

For ray 4, the position where the ray intersects with the central axis after passing through the optical element at the (n-1)-th surface is the exit pupil position. The distance from the rear principal point of the (n-1)-th surface to the exit pupil is denoted by dn-1(p) , and the distance from the image plane to the exit pupil is denoted by zp'.

In this case, if the light ray 4 from the (n-1)-th surface to the exit pupil surface is illustrated, it will look like this:

Ray tracing with ideal lens arrangement_119.png

The conditions shown in the figure for ray4 can be expressed as follows.

Ray tracing with ideal lens arrangement_107.png

Here, if we assume that the medium is air and apply (1)-⑧, we get the following:

理想レンズの配置による光線追跡_154.png

Therefore, the exit pupil position zp' can be calculated as follows.

Ray tracing with ideal lens arrangement_114.png

・Creating a format for ray tracing

We will now show an example of how to apply the recurrence formula shown above to create a format in Excel for arranging an ideal lens and tracing rays.

・Input section:

Enter the values for each setting item in the black box below. There are up to 9 surfaces, but it is not necessary to enter all of them. For the remaining surfaces, the focal length should be set to a sufficiently large value equivalent to infinity, and the principal point distance and surface distance should be set to zero.

Ray tracing with ideal lens arrangement_101.png

・Calculation section:

Ray tracing calculations are performed separately for each of rays 1 to 5. The red and blue boxes refer to the input values.

Ray tracing with ideal lens arrangement_100.png

・Ray tracing data section:

The following is data for plotting the results of ray tracing. The position in the optical axis direction is given for each surface as L and L+HH'. Also, lines are left blank to avoid drawing rays between principal points. For rays 3, 4, and 5, new rays 3*, 4*, and 5* have been added, which are inverted with respect to the optical axis.

Ray tracing with ideal lens arrangement_099.png

・Ray tracing result display section:

The results of the ray tracing are shown in the figure below. The calculation results of the optical system and image plane parameters are also shown.

Ray tracing with ideal lens arrangement_098.png

In this way, you can use Excel to place an ideal lens and perform ray tracing. The format is shown below.

  • Related materials

[A] Kishikawa, Toshiro. Yūzā Enjinia no Tame no Kōgaku Nyūmon (An Introduction to Optics for User Engineers). Optronics Co.

  • Update History

  • 2026-01: Comprehensive revision

  • 2025-10: Comprehensive revision

  • 2025-06: Newly released

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