Fundamental theoretical framework in geometrical optics
This page provides an overview of the fundamental theories of geometrical optics, omitting the derivation explanations. Detailed derivations are presented on separate pages, or only the names of the laws are shown.
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.
Reflection and refraction of light rays
The laws of reflection and Snell's law, shown below, are fundamental principles that determine how light changes its path at an interface, and are the most fundamental laws that underpin all decisions in optical design. These laws, which mathematically describe the phenomena of reflection and refraction, are the starting point for all calculations, from lens design to optical simulations.
This law is based on the electromagnetic properties of light, and its derivation is shown in EM1-(4)-⑦ and EM1-(5)-⑦ .
・Law of reflection
When a light ray is reflected at a medium interface, the angle of incidence with respect to the normal to the interface is defined as θ, and the angle of reflection is defined as φ.
Figure 1-1
The angular relationship when light rays reflect at a boundary is described by the law of reflection shown below. This law is a fundamental geometric principle that governs the phenomenon of reflection.
・Snell's Law (Law of Refraction)
When light travels from a medium with refractive index n₁ to a medium with refractive index n₂, the light ray changes its path at the interface. If the angle of incidence with respect to the normal to the interface is θ₁ and the angle of refraction is θ₂, then the relationship between the two is uniquely determined by the refractive index of the medium.
Figure 1-2
The phenomenon of light bending is described by the following equation, based on the correspondence between angle and refractive index. This relationship is a fundamental principle that defines how light changes its path at an interface, and is known as Snell's Law.
• Paraxial Theory System of Optical Systems ① – Focal Length and its Properties –
When parallel light rays are focused to a single point on the image side, the distance from the principal point of the optical system to that point is the focal length. The focal length functions as a fundamental quantity that determines the relationship between the field of view and the field of view, as well as the magnification of the optical system.
This chapter discusses the paraxial theory of focal length, which is constructed based on rule B.
Rule B: Definition of the paraxial region
Assuming that the light ray travels at a sufficiently small angle to the optical axis,
tanθ≃θ, sinθ≃θ, cosθ≃1
This approximates the paraxial condition, which greatly simplifies the treatment of light rays and provides a powerful framework for understanding the basic structure of optical systems.
・Definition of focal length
Consider a system in which parallel light is incident on an optical system and the exiting light converges to a point F' on the optical axis. In this case, the point where the intersection of the incident and exiting light rays, extended toward the optical system, is projected perpendicularly to the optical axis is defined as the rear principal point H'.
In the paraxial region, all refraction by the optical system can be considered to occur in the plane orthogonal to the optical axis (rear principal plane) that contains H'.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 2-1a
In this case, the H'F' vector is called the rear focal length (and is denoted by f').
Similarly, consider a system where light rays emitted from the front focus F are incident on the optical system, and the emitted light is parallel. In this case, the point where the intersection of the incident and emitted rays, extended toward the optical system, is projected perpendicularly to the optical axis is defined as the front principal point H. In the paraxial region, all refraction by the optical system can be considered to occur in the plane orthogonal to the optical axis (front principal plane) that contains H.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 2-1b
In this case, the FH vector is called the front focal length (denoted by f).
These can be summarized in the following diagram.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 2-2
Here, each vector in the figure represents the following:
FF: Front Focus
HD: Front Principal Point Distance
HH': Distance between principal points
HD': Rear principal point distance
BF: Back focus
Furthermore, a ray incident at the front principal point at an angle ω with the optical axis exits from the rear principal point at an angle ω'. In this case, the following relationship holds ( (IG)-(1)-② ).
This is closely related to the relationship between the field of view and the focal length, which will be discussed later.
Furthermore, as shown in Figure 2-2, the relationship between each vector can be expressed by the following equation.
Furthermore, the relationship between the front focal length and the rear focal length is expressed as follows (see (IG)-(1)-⑦ for the derivation).
Here, assuming the refractive index of the object medium is 1, the rear focal length can be written as f' = n'f, which gives the impression that the rear focal length f' is proportional to the refractive index n' of the image medium. However, in reality, even a change in either the object refractive index n or the image refractive index n' will change both the front focal length f and the rear focal length f'. The amount of change depends on the design of the optical system and cannot be uniquely expressed as a general formula. Furthermore, the same applies to key parameters of the optical system such as FF, HD, HH', HD', and BF; even a change in either the object refractive index n or the image refractive index n' will change their values depending on the configuration of the optical system.
・The relationship between the field of view and the focal length.
From here, we will describe the specific properties that focal length represents.
Parallel light incident on the optical system at an angle ω with respect to the optical axis, after passing through the optical system, forms an image at the rear focal point F', shifted by y' from the optical axis.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 2-3a
From this, the following equation holds true using trigonometric functions.
Here, the relationship between ω and ω' is obtained from (2)-②a.
Similarly, a light ray originating from a point on the front focal plane at a distance y from the optical axis is emitted as parallel light at an angle ω' with respect to the optical axis after passing through the optical system.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 2-3b
From this, the following equation holds true using trigonometric functions.
Here, the relationship between ω and ω' is obtained from (2)-②a.
Thus, the optical system plays a role in relating the positional information of the object and image sides, y, y', to the angular information ω, ω', which corresponds to parallel light, via the focal length.
・Optical contribution of focal length at finite distances
Next, consider a system in which a light ray emitted from point O passes through the optical system and is imaged at point O'. In this case as well, among the light rays from off-axis object points, the ray passing through the front principal point maintains its tilt angle at the front principal point and is imaged at the corresponding image point via the rear principal point.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 2-4
For this system, the magnification β of the optical system is defined as follows.
The sign of the magnification β is determined by how many times the image is inverted within the optical system. If the number of inversions is odd, the image is inverted and β is negative; if the number of inversions is even, the image is upright and β is positive. If only a virtual image is formed, no inversion occurs, the image is upright, and β is positive.
At this time, the positions of the object point O and image point O' relative to the front focus F and rear focus F' are represented by the following vector. *See (IG)-(1)-①' for derivation.
Therefore, the distance between objects can be expressed as follows:
Here, HH' is a quantity determined by the specific design parameters of the optical system and does not have a direct relationship with primary performance indicators of the system such as focal length or magnification.
In many imaging systems, the refractive index of both the object and the image is 1 (n = n' = 1), so under these conditions, the object-image distance is given by the following equation ( (IG)-(1)-⑫ ).
Furthermore, solving for β yields the following solution.
*Composite refers to both the direct and inverse mappings
• The Paraxial Theory System of Optical Systems ② – The Pupil Surface and its Role –
The pupil plane is the position and size of the "effective aperture" that an optical system presents to the outside. The pupil plane is an important element that directly contributes to the performance of the optical system, such as brightness, depth of field, and resolution.
Here too, we treat it as a paraxial region and continue to apply rule B.
・Definition of the pupil
An optical system is equipped with an aperture to define the angular range of light rays that pass through it. When this aperture is located inside the optical system, its appearance when observed from the outside appears as a virtual image whose position and size have changed due to the image-forming action of the optical system.
Furthermore, the image of this aperture shows different positions and sizes depending on whether it is viewed from the incident side or the exit side.
Therefore, the following definitions are introduced:
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The image of the aperture as seen from the entrance side is called the entrance pupil, its position is P, and its apparent aperture diameter is Dep.
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The image of the aperture as seen from the exit side is called the exit pupil, its position is P', and its apparent aperture diameter is Dxp
Figure 3-1a
Figure 3-1b
・Contribution of the pupil plane and F-number in parallel light incidence
Therefore, if we illustrate Figure 2-3a again, including the pupil surface, it will look like this.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 3-2a
Here, the F-number on the image side is defined as follows:
The F-number is an indicator of the brightness of an optical system. Since the amount of light passing through an optical system is proportional to the area of the entrance pupil, the amount of light is proportional to the square of the entrance pupil diameter. Therefore, if we denote the amount of light passing through the optical system as I and keep the rear focal length f' constant, the following relationship holds between the amount of light and the F-number.
In addition to the F-number, another indicator of the amount of light passing through the optical system is the numerical aperture (NA), defined by the following formula.
Here, in the paraxial region, the focusing angle θ′ can be approximately expressed using the F-number and numerical aperture as follows:
Therefore, the following equation holds true as the paraxial approximation.
Similarly, if we illustrate Figure 2-3b again, including the pupil surface, it will look like this.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 3-2b
Here, we define the F-number on the object side as follows:
The F-number is an indicator of the brightness of the optical system. Since the amount of light passing through the optical system is proportional to the area of the pupil, the amount of light is proportional to the square of the exit pupil diameter. Therefore, if we denote the amount of light passing through the optical system as I and keep the front focal length f constant, the following relationship holds between the amount of light and the F-number.
In addition to the F-number, another indicator of the amount of light passing through the optical system is the numerical aperture (NA), defined by the following formula.
Here, in the paraxial region, the focusing angle θ′ can be approximately expressed using the F-number and numerical aperture as follows.
Therefore, the following equation holds true as the paraxial approximation.
・Contribution of the pupil surface and effective F-number at finite distances
Furthermore, if we re-illustrate Figure 2-4, including the pupil surface, it will look like this.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 3-3
Here too, the horizontal scaling factor is defined as follows.
Furthermore, pupil magnification is defined as follows:
At this time, by using the transverse magnification, pupil magnification, and F-number when parallel light is incident, the image-side F-number, i.e., the effective F-number in a finite imaging system can be expressed as follows. *See (FN)-(1)-⑤' for derivation.
Furthermore, in finite imaging systems, the object-side numerical aperture and the image-side numerical aperture are defined as follows:
At this time, the following relationship holds. This is an approximate expression of the Lagrangian invariant (PX)-(2)-③ in terms of numerical aperture, by applying the paraxial condition (θ≃sinθ).
・Relationship between effective F-number and depth of field
Incidentally, as previously shown, the F-number is a quantity related to the amount of light passing through the optical system, but it is also a quantity related to the depth of focus. This will be explained below.
* Apply Rule A (Symbol Convention) and Rule B (Definition of Paraxial Region).
Figure 3-4
In the figure above, let δ be the amount of blur allowed on the image plane. In this case, the relationship between the effective F-number and the image-side depth of focus di is given by the following equation from the geometric relationship of trigonometric functions.
Furthermore, from the vertical magnification relationship ( (IG)-(1)-⑫ ), the depth of focus on the subject side (object side), i.e., the depth of field do , is expressed by the following formula.
Therefore, the depth of field can be expressed as follows, using the effective F-number, horizontal magnification β, and acceptable circle of confusion δ.
• Theoretical framework of aplanatic optical systems in the non-paraxial region
・Properties of the principal and pupil surfaces in aplanatic optical systems
In the paraxial region as shown so far, the principal plane is treated as a plane perpendicular to the optical axis containing the principal point. However, in the non-paraxial region, and in an aplanatic optical system where spherical aberration and coma aberration are sufficiently corrected, Abbe's sine condition causes the principal plane to becomes a spherical surface with a radius of curvature equal to the focal length, as shown in Figure 4-1.
* Rule A (Symbol Convention) applies
Figure 4-1
Here, as with the paraxial region, the image-side F-number is defined as follows using the entrance pupil diameter and the rear focal length.
However, this is distinguished from the F-number defined paraxially and is positioned as the effective F-number. In this case, the principal surface is a sphere with radius of curvature f', so the entrance pupil diameter for the focusing angle θ' is expressed by the following equation.
Furthermore, the image-side numerical aperture is defined in the non-paraxial region as well, similar to the case of paraxial optics, and this definition more directly expresses the physical meaning of the numerical aperture.
Then, in the paraxial region, the following relationship, which holds under the approximation θ ≃ sinθ ≃ tanθ, also holds in the non-paraxial region without approximation, by substituting sinθ for tanθ, if the optical system satisfies the aplanatic condition (Abbe's sine condition).
・Characteristics and diffraction-limited resolution of aplanatic finite optical systems
Next, consider an aplanatic system in which a light ray originating from point O passes through an optical system and is imaged at point O'. In this case, Abbe's sine condition is satisfied, and the entrance pupil is given as a sphere with radius OP, and the exit pupil is given as a sphere with radius P'O'.
* Rule A (Symbol Convention) applies
Figure 4-2
Here, as with (3)-③d, the object-side numerical aperture and the image-side numerical aperture are defined as follows.
In this case, the following relationship holds for object height y and image height y'.
Although this equation is formally the same as (3)-③f, the assumptions under which the two equations hold are fundamentally different. (3)-③f is an approximate equation derived under the paraxial condition, based on the Lagrangian invariant and the approximation relation θ ≃ sinθ.
On the other hand, (4)-②b is a relationship derived from Abbe's sine condition and the etendue conservation law in an optical system that satisfies the aplanatic condition (zero spherical aberration and coma aberration). This equation is derived without using the angular approximation (θ≃sinθ≃tanθ) and therefore does not depend on the paraxial approximation. However, it should be noted that its validity in the non-paraxial region is an approximation near the aplanatic point and is affected by higher-order aberrations of the optical system.
・Resolution limits and depth of field of the aplanatic system, governed by the wave nature of light.
In an optical system with perfectly corrected aberrations, the resolution limit δ is not determined by geometrical optics but by the diffraction limit due to the wave nature of light. The resolution limit δ depends on the wavelength λ and the numerical aperture NAi , and for practical evaluation, two definitions, Abbe's and Rayleigh's, are used depending on the application.
・Rayleigh's resolution limit
As an empirical resolution limit that gives the minimum distance at which two points appear separated,
The distance from the central peak of the point image formed by a circular aperture to the first dark ring, i.e., the Airy disk radius, is used as an indicator. This is shown in (AR)-(2)-28 .
・Abbe's resolution limit
This defines the theoretical resolution limit where the periodic structure can no longer be completely identified.
Furthermore, the depth of focus (on one side) is expressed by the following formula, using the wavelength of light, the image-side numerical aperture, and the refractive index of the image-side medium.
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Related Topics
The laws of reflection and refraction (Snell's Law) are derived as results that appear in steps (4)-⑦-2 and (5)-⑦-2 during the process of deriving Fresnel's formula below.
The laws concerning focal length are derived as follows.
Based on the rules governing pupils, the effective F-number is 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.
[2] Tomowaki Takahashi, "Lens Design: From Aberration Coefficients to Automated Design," Tokai University Press.
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Update History
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2026-08: Newly released