In geometric optics, the so-called ideal optical system is an optical system that can form perfect and ideal images for each point in a large enough space with a wide enough beam.The ideal optical system converts the concentric wide beam in the object space to the concentric beam in the image space. This transformation from one space to another can be mathematically reduced to the problem of "collinear transformation" or "collinear imaging". The theory of this coaxial ideal optical system is based onGaussianTherefore, people also call the ideal optical system theory Gaussian optics.
The theory of ideal optical system was proposed by Gauss in 1841, so the theory of ideal optical system is also called "Gauss optics".In isotropic homogeneous media, the object image relationship of an ideal optical system should have the following characteristics:
Figure 1
1. Point to point image: that is, for each point in the object space, there must be one point corresponding to it in the image space, and only one point corresponding to it. Such two corresponding points are called conjugate points in the object image space (as point A and point A 'in Figure 1).
2. Line forming image: that is, for each line in the object space, there must be a line corresponding to it in the image space, and only one line corresponding to it. Such two corresponding lines are called conjugate lines in the object image space (as shown in Figure 1 BC and B ′ C ′).
3. Plane image formation: that is, every plane in the object space must have a plane corresponding to it in the image space, and only one plane corresponds to it. Such two corresponding planes are called conjugate planes of the object image space (such as PQ plane and P ′ Q ′ plane in Figure 1).
Thus, if any point D in the object space is located on the line BC, then its conjugate point D 'in the image space must also be located on the conjugate line B ′ C ′.Similarly, a concentric beam in the object space must correspond to another concentric beam in the image space.The above-mentioned point to point, line to line and plane to plane imaging is called collinear imaging.
The collinear imaging theory is the basic theory of the ideal optical system, which is only a basic assumption. In fact, there is no such ideal optical system.Obviously, the ideal optical system is the direction of the actual optical system, so it is beneficial to find out the basic characteristics of the ideal optical system for seeking the actual system that is close to the ideal optical system in some aspects.When designing the actual optical system, people often use some optical properties and formulas abstracted from the ideal optical system to carry out the initial calculation of the actual optical system, so as to make the design of the actual optical system possible, simplify the calculation and improve the quality.
In the paraxial area of the actual optical system, the collinear imaging theory can be satisfied. Therefore, when designing an optical system, the quality of the system is often measured by the imaging properties of its paraxial area.[1]
Base point and base plane
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According to the characteristics of the ideal optical system, if a ray parallel to the optical axis of the optical system enters the ideal optical system in the object space, there must be a ray conjugate with it in the image space.
Figure 2
As shown in Figure 2, OoneAnd OkThe two points are the vertices of the first and last side of the ideal optical system, FOoneOkF 'is the optical axis.A straight line AE parallel to the optical axis in the object spaceoneAfter being refracted by the optical system, the refracted ray GkF 'intersects the optical axis at F', and the other object light FOoneCoincident with the optical axis, whichRefraction rayOkF 'is still emitted along the optical axis without refraction.Because of the image square GkF′、OkF ′ and the object side AE respectivelyone、FOonePhase conjugation, so the intersection F ′ is AEoneAnd FOoneThe conjugate point of the intersection point (located on the optical axis at the infinity of the object side), so F 'is the image of the point on the infinity axis of the object side. All other incident rays parallel to the optical axis converge at the point F', which is called the image focus of the optical system (or the back focus, the second focus).Obviously, the image focus is the conjugate point of the point on the infinite axis of the object.
Similarly, point F is called the object focus (or front focus, first focus) of the optical system, and it is conjugate with the point on the infinite axis of the image square.After any incident light passing through the F point is refracted by the ideal optical system, the outgoing light must be parallel to the optical axis.The plane passing through the image square focus F 'and perpendicular to the optical axis is called the image square focal plane (image square focal plane);The plane passing through the object focus F and perpendicular to the optical axis is called the object focal plane (object focal plane).Obviously, the conjugate plane of the image square focal plane is at infinity, and the beam emitted from any object point on the image square focal plane must be a parallel beam after being emitted by an ideal optical system;Similarly, the conjugate plane of the object square focal plane is also at infinity. Any incident parallel light, refracted by an ideal optical system, must converge at a point on the image square focal plane.It must be pointed out that the focus and focal plane are a pair of special points and planes of an ideal optical system.The object focus F and the image focus F 'are not conjugate with each other. Similarly, the object focus plane and the image focus plane are not conjugate.
Figure 3
As shown in Figure 3, extend the incident light AEoneAnd outgoing ray GkF′，The intersection point Q 'is obtained;Similarly, extend the incident light BEkAnd GoneF，The intersection point Q can be obtained.Set ray AEoneAnd BEkThe incident height of is the same, and they are all in the meridian plane.Obviously, point Q and point Q 'are a pair of conjugate points.Point Q is ray AEone"Virtual object point" intersected with FQ;Point Q 'is ray BEkAnd GkThe "virtual image point" intersected by F '.Make planes QH and Q ′ H ′ perpendicular to the optical axis through point Q and point Q ′, then these two planes are also conjugate with each other.It can be seen from Figure 3 that the conjugate line segments QH and Q ′ H ′ in these two planes have the same height and are on the same side of the optical axis, so their vertical axis magnification β=+1.We call this pair of conjugate planes with a vertical axis magnification of+1 as the main plane, where QH is called the main plane of the object side (or the front main plane, the first main plane), and Q ′ H ′ is called the main plane of the image side (or the back main plane, the second main plane).The intersection point H between the object's main plane QH and the optical axis is called the object's main point, and the intersection point H 'between the image's main plane Q ′ H ′ and the optical axis is called the image's main point.
The main point and the main plane are also a pair of special points and surfaces of an ideal optical system.The main plane of the object side and the main plane of the image side, the main point of the object side and the main point of the image side are conjugate with each other.
The distance from the main point H of the object to the focus F of the object is called the object focal length (or front focal length, first focal length), which is expressed by f;The distance from the main point H 'of the image square to the focus F' of the image square is called the image square focal length (or the back focal length, the second focal length), which is represented by f '.The positive and negative focal lengths are determined by taking the corresponding main point as the origin.If the direction from the main point to the corresponding focus is the same as the ray propagation direction, the focal length is positive;Otherwise, it is negative.The case shown in Figure 3 is f<0, f '>0.
From △ FQH and △ F ′ Q ′ H ′, the expressions of object focus and image focus are
Figure 4
A pair of main points and a pair of focal points constitute the base point of the optical system, a pair of main planes and a pair of focal planes constitute the base plane of the optical system, and they constitute the basic mode of an optical system (as shown in Figure 4).For an ideal optical system, regardless of its structure (r, d, n), as long as its focal length value and the position of the focus or the main point are known, its properties are determined.[2]
Object image relation
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Given the position of the base point and the base plane of an ideal optical system, when the position and size of the object are fixed, the position and size of the image can be calculated analytically.In order to derive the relevant image position formula, it is necessary to determine the relevant parameters of the image.
Newton formula
Figure 5
As shown in Figure 5, the size of the image A ′ B ′ of the object AB with the size of y after passing through the ideal optical system is y ′.The positions of F, F ′, H, H ′ in the system are known.
In Newton's formula, the object distance x of object AB takes the object focus F as the origin, and the positive and negative signs of the object distance x are determined according to the following rules: if the direction from object focus F to object point A is consistent with the direction of light propagation, the object distance x is positive;Otherwise, it is negative.The object distance x in Figure 5 is negative.Similarly, the image distance x 'is from the image focus F' to the image point A '. If it is consistent with the propagation direction of the light, the image distance is positive, otherwise it is negative.The image distance x 'in Figure 5 is positive.
It can be obtained from similar △ BAF and △ RHF
Similarly, in △ Q ′ H ′ F ′ and △ B ′ A ′ F ′
From this we can get
This is the image position formula with the focus as the origin, called Newton's formula.
Gauss formula
The object image position of Gauss formula is determined relative to the main point of the ideal optical system.As shown in Figure 5
It represents the distance from object point A to object main point H, with
It refers to the distance from image point A 'to image main point H'.The direction is specified to take the main point as the origin. If the direction from H to A or from H 'to A' is consistent with the propagation direction of light, it is positive;Otherwise, it is negative.It can be seen from Figure 5
Substituted into Newton's formula
Divide both sides simultaneously
, Yes
This is the object image position formula with the main point as the origin, which is called Gauss formula.[3]
