Geometrical ray propagation in ideal lens systems based on lens 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.

Recurrence formula model for Gaussian beam propagation through ideal lenses

・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: Direction and sign of one-dimensional vectors and angles

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

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

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

(iv) When f < 0, z < f

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

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

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

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:

From the figure, the following can be said:

From (1)-①', (1)-②a, and (1)-②b,

From Lagrange invariant,

From (1)-④, (1)-②abc, and (1)-①',

From (1)-① and (1)-⑤,

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

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

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

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

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:

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

・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:

・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: Direction and sign of one-dimentional vectors and angles

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.

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:

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: Direction and sign of one-dimentional vectors and angles

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:

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:

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:

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:

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.