Ray Propagation Based on the Gaussian Imaging Equation in Ideal Lens Systems …(IG)
This page systematically deals with paraxial ray propagation in an ideal lens system based on the Gaussian imaging equation and demonstrates a method for determining optical system parameters (focal length, principal point position, back focus, etc.) using matrix form. First, the imaging equation that gives the relationship between object distance, image distance, and focal length is derived, and then a matrix representation of geometric ray propagation by an ideal lens based on the Gaussian imaging equation is constructed.
*This page deals with a ray propagation model based on geometrical optics. It does not include the effects of wave-optical diffraction, such as that of Gaussian beams.
The flow of explanation on this page is based on Toshiro Kishikawa's "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 the imaging relationship formula
Here, we start with an illustration based on the definition of focal length and use that concept to derive the imaging relationship at finite distances.
The process by which a light ray emitted from an object point passes through an optical element with focal length f and forms an image at the image point can be illustrated as follows, based on the definition of focal length ( (BS)-(2)-① ).
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
(i) When f > 0, z < 0,
Figure 1-1a
(ii) When f > 0, 0 < z < f,
Figure 1-1b
(iii) When f > 0, f < z,
Figure 1-1c
(iv) When f < 0, z < f
Figure 1-1d
(v) When f < 0, f < z < 0,
Figure 1-1e
(vi) When f < 0, z > 0 ,
Figure 1-1f
Let's elaborate on Figure 1-1. In paraxial optics, ray propagation is linear, so all rays emanating from an object point can be represented as a linear combination of two independent rays. Therefore, if the point where these two rays intersect on the image side is determined, other rays will also reach the same point, and the image point is uniquely determined. Thus, the conjugate relationship between the object plane and the image plane can be defined solely from the intersection point of the two rays, giving us the object plane O and the image plane O'.
In the coding convention of Rule A, (i) to (vi) are all forms of the same model, and if the magnification is β, then from the relationship of similarity of triangles, the following can be said in common for all of (i) to (vi):
Transforming this yields the following:
To intuitively understand the positional relationship of the principal points, let's consider the condition where the horizontal magnification is 1:1 (β = 1). In this case, the object distance and image distance satisfy f = z and f' = −z'. If we draw a ray diagram (Figure 1-2) in this state, we can see that the front principal point coincides with the object plane, and the rear principal point coincides with the image plane. Therefore, rays passing through the front (rear) principal point are projected as rays passing through the rear (front) principal point, and since height is conserved under the condition β = 1, the correspondence between the principal points can be understood as a translation.
Figure 1-2
Here, let n be the refractive index of the medium at the object plane, and n' be the refractive index of the medium at the image plane. When we also draw the light ray passing through the principal point where the principal plane intersects the optical axis, the following relationship holds between each incident ω and the exit angle ω', according to Lagrange's invariant ( (PX)-(2)-③ ) and y=y'.
Next, let's consider the following major rays.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 1-3
From Figure 1-3, the following can be said.
From (1)-①', (1)-②a, and (1)-②b,
Here, for u and u', similar to (1)-①'', the following relationship holds from the Lagrange invariant ( (PX)-(2)-③ ) .
From (1)-④, (1)-②abc, and (1)-①',
From (1)-① and (1)-⑤,
Furthermore, the relationship between the angle of incidence ω and the exit angle ω' to the principal point can be directly read from Figure 1-3, and the result is consistent with (1)-②. This indicates that the geometric derivation using the illustration and the derivation using Lagrangian invariants are consistent with each other.
Next, if we define the vertical 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.
From (1)-①'a,b,c and (1)-⑨a, we obtain a relation that includes the following Newton's equation (or Newton's relation).
Next, by applying (1)-⑨a to (1)-④, the following Gaussian imaging equation is obtained.
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 lateral magnification, substituting n=n'=1 into (1)-⑧ yields the following equation.
・Light propagation based on Gaussian imaging with an ideal lens
1. Relationship between the conversion of incident and emitted ray by optical elements.
Consider a system in which a ray of light passes sequentially from the 0th plane (object plane), the 1st plane, the 2nd plane, ..., the j-th plane, ..., to the nth plane (image plane). In this case, the parameters of the j-th plane are illustrated as follows.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
• Distance from the front focal point to the front principal point of the optical element: fj
Distance from the rear principal point of the optical element to the rear focal point: fj'
• Distance from the front principal point of the optical element to the object-side aerial image: sj
Distance from the rear principal point of the optical element to the image-side aerial image: s'j
• Height of the incident light ray from the object side at the optical element position: hj
• Height of the light rays emitted towards the image side at the optical element position: h'j
- Angle of light rays on the object: uj
• Ray angle on the image side: u'j
Use air as the medium and apply (1)-⑨.
Here, from the diagram,
From (1)-⑪ (Gauss's imaging formula),
From (2)-①a,b,c, (2)-②,
(2)-①a and (2)-②' can be expressed as matrices as follows.
The height of the ray remains unchanged before and after the front and rear principal points; 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 optical elements
Next, we will describe the propagation of light rays between the j-th and (j+1)-th planes. The parameters for this propagation are shown in the following diagram.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
• Distance between the principal points of the optical element: HH'j
• Distance from the rear principal point of one optical element to the front principal point of the next optical element: dj
• Distance from the origin (first object plane) to the front principal point of the optical element: Lj
At this point, the following can be said from the diagram.
(2)-④b and c can be expressed as matrices as follows:
A ray passing through the rear principal point of the j-th plane propagates while maintaining its angle until it reaches the front principal point of the next (j+1)-th plane, with only its height changing. The matrix representing this propagation process 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.
The product of the interface matrix ((2)-③) and the spatial propagation matrix ((2)-⑤) is defined as follows.
Applying this, (2)-③ and (2)-⑤ can be expressed as follows.
By using the recurrence relation (2)-⑤', we can trace the propagation of light rays from the first surface to the final surface.
Furthermore, the inverse matrices of (2)-③a, (2)-⑤a, and (2)-⑥a are as follows:
Applying this, reverse ray propagation can be represented as follows:
From the above, we have shown that the propagation of light rays in both the forward and reverse directions can be described uniformly using matrix form. Based on these representations, the next chapter will describe a method for evaluating the characteristics of the entire optical system.
・Derivation of optical system parameters in light ray propagation using an ideal lens
For an optical system consisting of k ideal lenses, the optical system parameters are determined using the tracking results of each ray shown in the figure below. Here again, the medium is air, and we treat f=f' and ω=ω' ( (1)-⑨ ).
* 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 angle between the object height and the principal point is normalized to -1, enabling 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 (2)-③,⑤', (2)-③',⑤'',
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From (2)-①a,b,c,
Parameters without subscripts are considered system-specific parameters that characterize the entire optical system.
Furthermore, the total lens length TL in the figure can be expressed as follows using Lj defined in (2)-④a.
(i) Parallel light incident from the front
Here, we determine the combined focal length f, the distance of the rear principal point HD', and the image length BFinf for parallel light incidence.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
From (3)-①a,b,
Since uF',1 = 0 and hF',1 = 1,
From (3)-②a, the following image distance for parallel light incidence is obtained.
Therefore, by applying (3)-④a' and (3)-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 combined focal length f, the front principal point distance HD, and the object distance FFinf for parallel light incidence.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
From (3)-①a',b',
From u'F,k = 0 and h'F,k = -1,
From (3)-②a, the object distance for parallel light incidence is obtained as follows.
Therefore, by applying (3)-⑤a' and (3)-⑤b to the relationship obtained from the figure, we can obtain the focal length f and the front principal point distance HD.
Next, we calculate the distance HH' between the principal points 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 distance between principal points HH' can be calculated as follows using TL, HD, and HD' obtained so far.
(iii) On-axis paraxial tracking
Here, the image distance BF and magnification β are determined using the object distance FF and the front principal point distance HD.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
From the diagram,
Applying these, from (3)-①a,b,
Therefore, by applying (3)-⑥a,b,c to the following equation obtained from (3)-②a,b, we can obtain the image distance BF and magnification β.
(iv) Primal ray paraxial stop
Here, the exit pupil position XP is determined using the object distance FF, the front principal point position HD, and the entrance pupil position EP.
* Apply Rule A (Signing Convention), Rule B (Definition of Paraxial Region), and Rule C (Color Distinction of Parameter).
From the diagram,
Applying these, from (3)-①a,b,
Therefore, by applying (3)-⑦a,b,c to the following equation obtained from (3)-②a, the exit pupil position XP can be obtained.
Furthermore, although the magnification β was found in (3)-⑥e, another method for deriving that magnification β is shown below.
First, the following can be said from the diagram.
Therefore, the magnification factor β can be calculated as follows.
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Calculation format
As an example of implementing the optical system parameter derivation method described above, a corresponding Excel format is provided as an appendix. This format is designed to allow systematic calculation of various parameters (focal length, principal point position, back focus, etc.) based on matrix optics, and can be used for both theoretical studies and practical analyses.
This format also includes the calculations for pupil magnification and effective F-number derived on a separate page (FN) .
・Specific example: Derivation of the combined focal length
This section derives the combined focal length when two optical elements with focal lengths f1 and f2 are placed at a distance d apart. When this system is represented using a paraxial ray propagation model with an ideal lens, it takes the configuration shown in 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 ( d1 = d ) shown in the figure to (3)-4a', we obtain the following:
Calculating (4)-① yields the following result.
Therefore, applying (4)-①' to the following equation obtained from (3)-④b and c, we obtain the following relationship between the image distance BFinf and the combined focal length f.
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Related Topics
For wave-optical propagation models, including diffraction, such as Gaussian beams, please refer to the following link.
We have discussed ray propagation in an ideal lens system, but for its relationship to ray propagation in an actual spherical lens, please refer to the paraxial calculation theory shown below.
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: Comprehensive revision
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2026-01: Comprehensive revision
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2025-10: Comprehensive revision
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2025-06: Newly released