Ray Propagation in Spherical Lens Systems Based on Paraxial Approximation …(PX)
A ray in the paraxial region has a very small angle θ with respect to the optical axis of the optical system, and all of its paths pass very close to the optical axis. In this case, the following approximation holds for θ:
sin θ ≃ θ, tan θ ≃ θ, cos θ ≃ 1
Therefore, limiting the area to the paraxial region makes ray tracing calculations extremely simple and is useful for understanding the structure of the optical system. Here, we will show how to derive the calculation formula.
The flow of explanation on this page is based on Toshiro Kishikawa's "Yūzā Enjinia no Tame no Kōgaku Nyūmon (An Introduction to Optics for User Engineers)" [1] . However, the derivations and descriptions have been independently restructured by the author.
table of contents:
This page will proceed with the discussion based on the following rules.
Rule A: Dimensions and angle notation
Dimensions and angles are indicated in Appendix: Following the definitions of dimensions and angles on this site , a single-sided arrow with direction is considered positive for vertical directions (upward), rightward for horizontal directions, and counterclockwise for rotation (counterclockwise is negative). Quantities indicated by double-sided arrows without direction are always treated as positive values.
Rule B: Definition of the paraxial region
On this page, we express angles in radians and use a first-order paraxial approximation that ignores infinitesimal terms of order three or higher. That is, when the angle is θ, we apply the following approximation:
sin θ ≃ θ, tan θ ≃ θ, cos θ ≃ 1
Rule C: Parameter notation (color coding)
For the color coding of each parameter, quantities given by the settings (in the recurrence relation model) are shown in blue, and quantities given by the calculation formula are shown in black.
・Derivation of Abbe's invariants
For the case where paraxial light rays are transmitted through a spherical boundary with refractive index n < n', the typical ray behavior under different incident conditions is illustrated in several cases.
(i) When r > 0, u > 0, u' > 0,
Figure 1-1
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
(ii) When r > 0, u > 0, u' < 0,
Figure 1-2
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
(iii) When r < 0, u < 0, u' > 0,
Figure 1-3
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
Each parameter in the diagram above is defined as follows:
n : Refractive index of the material before incidence on the boundary surface
n' : Refractive index of the material after exiting the boundary surface
r : Radius of curvature of the boundary surface
u: The angle between the incident ray on the boundary surface and the central axis
u': Angle between the exit ray from the boundary surface and the central axis
i: The angle between the incident light on the boundary surface and the normal to the boundary surface
i': The angle between the light emitted from the boundary surface and the normal to the boundary surface
φ: The angle between the normal at the incident position of the ray on the boundary surface and the central axis
h: Height of the light ray incident on the interface relative to its central axis
h': Height of the light ray emitting from the interface relative to its central axis
s: The distance from the boundary surface to the point where the incident light intersects with the central axis
s': Distance from the boundary surface to the point where the light emitted from the boundary surface intersects with the central axis
At this time, the conditions shown in Figure 1-1 can be expressed as follows:
Furthermore, the conditions shown in Figure 1-2 can be expressed as follows:
Furthermore, the conditions shown in Figure 1-3 can be expressed as follows:
(1)-①-1,2,3 all converge to the following identical result.
Other combinations of signs for u and u' exist besides (i), (ii), and (iii), but according to the sign convention of Rule A, all cases converge to the same resulting expression.
Therefore, it can be said that it is sufficient to consider a model of the diagram based on any one combination of signs.
Furthermore, according to Snell's Law , the following can be said in the paraxial region.
From (1)-①'a,b,c,d' and (1)-②, the following Abbe invariants are derived.
(1)-③ (Abbe's invariant) is the relationship between a ray traveling from point P to point P', and it can be seen that the quantities intrinsic to the ray, u, u', and h, do not appear in this equation. Furthermore, if s and s' are fixed as constants, u, u', and h can take any value as long as the refraction condition at the interface (1)-①'b is satisfied. Therefore, in the paraxial region, as shown in Figure 1-3, any ray passing through point P will always travel toward point P'. This means that points P and P' are conjugate points.
Figure 1-3
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
・Derivation of Lagrange invariant
When the entire system is rotated around point O (the center of the sphere on the boundary surface), point P moves to point Q and point P' moves to point Q'. Points Q and Q' are also conjugate, but since the boundary surface remains completely unchanged before and after rotating the entire system, it can be said that points Q and Q' were also conjugate before rotating the entire system.
Here, in the paraxial region, arcs PQ and P'Q' can be considered as straight lines perpendicular to the central axis, so the vectors PQ and P'Q' can be treated as vectors perpendicular to the central axis.
Figure 2-1
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
For each parameter in the above diagram, those shown in Figures 1-1 and 1-2 will be retained, while new ones will be defined as follows.
y: object height
y': image height
ω: Angle of the ray from the object height y toward the intersection of the boundary surface and the optical axis
ω': Angle of the ray from the intersection of the boundary surface and the optical axis toward image height y'
In this case, the conditions shown in the figure can be expressed as the following equations:
Furthermore, according to Snell's law, the following can be said in the paraxial region:
From (2)-①a,b,c and (2)-②, the following relationship can be obtained.
This relation is called the Lagrange invariant or the Helmholtz-Lagrange invariant.
・Recurrence relations and matrix notation for paraxial calculations
1. Conversion relationship between incident and exiting light at the interface.
Consider a system in which a ray of light passes sequentially from the 0th plane (object plane), the 1st plane, the 2nd plane, ..., the jth plane, ..., to the kth plane (image plane). Here, each plane is given to be spherical (planes are given to have an infinite radius of curvature). In this case, the ray of light in the paraxial region before and after passing through the jth plane is illustrated as follows.
Figure 3-1
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
The subscript j in each parameter indicates the j-th boundary surface.
The parameters in the above diagram follow those shown in Figures 1-1 and 1-2 .
First, from (1)-①a,b, the following can be said about the j-th plane.
Furthermore, from (1)-③ (Abbe's invariant), the following can also be said for the j-th plane.
From (3)-①,②
Here, we set it as follows:
Applying this, (3)-②' can be expressed as follows:
(3)-①a and (3)-②'' can be expressed using matrices as follows.
At an interface, the height of the light ray does not change; only the angle changes due to refraction. The matrix representing this refraction relationship is denoted by Sj and is called the interface matrix.
2. Theoretical description of light ray propagation between interfaces
Next, the relationship between the j-th face and the (j+1)-th face can be illustrated as follows.
Figure 3-2
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
For each parameter in the above diagram, those shown in Figures 1-1, 1-2, and 3-1 will be retained, while new ones will be defined as follows.
Lj : Distance from the 1st face to the jth face
dj : Distance from the j-th surface to the (j+1)-th surface
Furthermore, the following can be said from Figure 3-2.
Applying (3)-③b to (3)-⑤a, we get the following:
(3)-⑤b, (3)-③a,b, n'j = nj+1 ,
(3)-⑤a' and (3)-⑤b' can be expressed using matrices as follows.
A ray passing through the j-th plane maintains its angle while only its height changes due to propagation until it reaches the next (j+1)-th plane. The matrix representing this propagation is denoted by Tj and is called the spatial propagation matrix.
3. Matrix representation of ray propagation
The propagation of light rays is tracked using the height and angle of the ray as state variables and described in matrix form. This allows for a unified and concise treatment of the behavior of the optical system.
Here, we define the product of the interface matrix ((3)- ④a) and the spatial propagation matrix ((3)-⑥a) as follows.
Applying this, (3)-④ and (3)-⑥ can be expressed as follows.
Furthermore, the inverse matrices of (3)-④a, (3)-⑥a, and (3)-⑦a are as follows.
Using this, the propagation of light rays in the reverse direction can be expressed as follows.
4. Universality of Lagrangian invariants in optical systems
Next, from (2)-③ (Lagrange invariants), the j-th plane is as follows:
From Figure 3-1, we have n'j = nj+1 , u'j = uj+1 , and y'j = yj+1 , so we can also say the following.
From (3)-⑦ and ⑧, we can conclude that Lagrange's invariants are quantities that hold true for all planes between the object plane and the image plane.
・Deriving optical system parameters through paraxial calculations
The optical system parameters are determined by tracing each ray, as shown in the figure below.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
In paraxial optics, ray height and angle are inherently treated as infinitesimal quantities. However, because the paraxial approximation is linear and homogeneous, the results remain unchanged even if these quantities are normalized to an arbitrary scale during calculations. Of course, this does not mean that actual rays can deviate from the paraxial condition; it is merely a "normalization for computational convenience."
Therefore, in on-axis paraxial tracking, the ray height at the principal point is normalized to 1, and in principal ray paraxial tracking, the angles between the object height, image height, and principal point are normalized to -1/ n1 and -1/n'k, respectively, to enable a simple and unified analysis.
Here, we will distinguish each ray by changing the subscript of each parameter (the part marked with "*" below).
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From (3)-④, (3)-⑥', (3)-④', (3)-⑥'',
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From (3)-① and (3)-③,
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From (3)-③ and (3)-⑩,
Furthermore, the total lens length TL in the figure can be expressed as follows using Lj defined in (3)-⑤c .
(i) Parallel light incident from the front
Here, we determine the focal length f', the distance from the rear principal point HD', and the back focus BFinf for parallel light incidence .
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
From (4)-① a,b,
Since uF',1 = 0 and hF',1 = 1,
From (4)-②c, the following back focus is obtained for parallel light incidence.
Therefore, by applying (4)-④a' and (4)-④b to the relationship obtained from the figure, we can obtain the focal length f' and the rear principal point distance HD'.
(ii) Parallel light incident from the rear (reverse direction)
Here, we determine the focal length f, the front principal point distance HD, and the front focus FFinf for parallel light incidence.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter Types).
From (4)-① a',b',
From u'F,k = 0 and h'F,k = -1,
From (4)-②c, the following front focus is obtained for parallel light incidence.
Therefore, by applying (4)-⑤a' and (4)-⑤b to the relationship obtained from the figure, the focal length f and the front principal point distance HD can be obtained.
Next, we calculate the principal distance HH' as defined in the figure below.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
From the figure, the principal distance HH' can be calculated as follows using TL, HD, and HD' obtained so far.
(iii) On-axis paraxial tracking
Here, we determine the back focus (BF) and magnification (β).
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
From the diagram,
Applying these, from (4)-①a,b,
Therefore, by applying (4)-⑥a,b,c to the following equation obtained from (4)-②c,d, we can obtain the back focus BF and magnification β.
(iv) Primal ray paraxial stop
Here, we determine the magnification β and the exit pupil position XP.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
From the diagram,
Applying these, from (4)-①a,b,
Therefore, by applying (4)-⑦a,b,c to the following equation obtained from (4)-②c, the exit pupil position XP can be obtained.
Furthermore, although the magnification β was found in (4)-⑥e, another method for deriving that magnification β is shown below.
First, the following can be said from the diagram.
From these, the magnification β can be calculated as follows.
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Calculation format
The Excel format for calculating optical system parameters using paraxial ray tracing, as shown above, is provided below.
This format also includes the calculations for pupil magnification and effective F-number derived on a separate page (FN) .
・Specific example: Formula for calculating the focal length of a single lens (Lens-Maker's formula)
Here, we derive the formula for calculating the focal length of a single lens from the paraxial calculation formula. Let n be the refractive index of the material, r1 and r2 be the radii of curvature of the first and second surfaces respectively, and d be the central thickness. When these are applied to the paraxial calculation model, we get the figure below.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
Applying the parameter values shown in the figure ( n1 = n'2 = 1, n'1 = n2 = n , d1 = d ) to (4)-4a', we obtain the following:
Calculating (5)-① yields the following result.
Therefore, applying (5)-② and n'2 = 1 to the following equation obtained from (4)-④b and c, we obtain the following relationship about focal length and back focus.
The resulting (5)-③a is known as the lens manufacturer's formula.
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Related Topics
Snell's Law is derived as a result that appears in steps (4)-⑦-2 and (5)-⑦-2 during the process of deriving Fresnel's formula, which is shown below.
We have discussed light propagation in spherical lens systems, but for its correspondence with ideal lens systems, please refer to the following.
Based on the optical system presented here, the pupil magnification and effective F-number are derived as follows.
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References
[1] Kishikawa, Toshiro. Yūzā Enjinia no Tame no Kōgaku Nyūmon (An Introduction to Optics for User Engineers). Optronics Co.
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Update History
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2026-07: Significantly expand the latter half of the content.
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2026-04: The flow of the first half has been significantly reorganized.
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2025-10: Newly released