What is the optical power of an optical system and why is it a very important index of an optical system? How is the unit of optical power defined?

What are the rules of the composition of ideal optical instruments?

1. What is the ideal optical setting?

An optical system that can form an ideal image is called an ideal optical system or an ideal optical system, or optical system for short.

1. What conditions does an ideal light group need to form a perfect image?

The ideal conditions for perfect imaging of optical groups are as follows: concentric beams in object space can be transformed into concentric beams in image space, that is, the image composed of light from a point in object space is still a point, that is, object space and image space are corresponding; Line-line correspondence; Face to face.

2. What is the difference between the so-called ideal light group in paraxial (Gaussian) optics and the ideal light group in applied optics?

An optical group in paraxial optics (such as a combination of refractive balls or lens groups). ) there is no strict design, and only in the paraxial region can a perfect image be formed, that is, the imaging range and beam width are infinite.

In the actual light group, such as photographic lens imaging, the object always has a certain size, and the beam of each object point also requires a certain width. In applied optics, it is defined as an ideal optical system, and an arbitrarily large beam forms a perfect image in an arbitrarily large range. Although the optical groups in applied optics, such as photographic lenses, have been strictly designed, they still cannot form perfect images. Therefore, the ideal optical system in applied optics is only an approximation of the actual optical group. However, the imaging quality of the actual optical system can be compared and estimated by using the characteristics of the ideal light synthesis image.

In an ideal optical system in a homogeneous medium, the light in both the object space and the image space is a straight line. A little bit in the object space is still a little bit in the image space. Therefore, the positions of objects and images can be determined by light through geometric relations. This geometric relationship of object image is called * * * line imaging (or * * * line transformation and * * * line optics).

3. What is the main content of linear optics theory?

* * * The main contents of linear optics theory are as follows:

(1) Every point in the object space corresponds to a point in the image space, and there is only one point; These two points are called the * * * yoke points of the object-image space;

(2) Every straight line in the object space corresponds to a straight line in the image space, and this pair of corresponding straight lines is called the * * * yoke line in the object-image two space;

(3) If any point in the object space is on a straight line, then the * * * yoke point in the image space must also be on the * * * yoke line of the straight line;

④ Any plane in the object space corresponds to a plane in the image space.

Second, what is the meaning of the ideal light group?

When designing an optical system, the designer should first have a clear idea and put forward specific requirements according to the use conditions. Such as the position of objects and images, magnification, image correction, longitudinal and transverse dimensions of optical system, etc. The above requirements must be calculated according to the ideal light group theory to obtain data.

To study or analyze an existing optical system, such as a photographic lens at hand, it is necessary to apply the theory of ideal optical system, determine the function of each optical element and understand the relationship between each element.

3. What is the basic point of an ideal light group?

We know that the optical theory of * * * line is a point between an object and an image, and a line corresponds to a line, which mainly uses light to determine the position of the object and the image through geometric relations. The geometric relationship between an object and an image is usually formed by several pairs of typical rays with special optical characteristics, and then the position and magnification (horizontal magnification and angular magnification) of the object and the image are determined according to the corner relationship of the figure. There are three pairs of yoke points on the main optical axis of the optical group: focus, principal point and node, which are collectively referred to as base points.

1. What is the focal point and focal plane of the ideal optical group?

No matter whether a light group is simple (such as a refractive ball and a thin lens) or complex (such as a photographic lens composed of multiple lenses), as long as it is regarded as an ideal light group, the * * * yoke relationship of an object image can be determined by some basic points and basic surfaces. As for the detailed plot of the light group, such as the curvature and spacing of the image refraction surface and the optical material that constitutes the lens, it is not necessary to consider, as shown in Figure 2-2 1. Figure 2-2 1(a) shows the positive light group (convergent light group), with the object focus in the object space and the image focus in the image space; Figure (b) shows the negative (divergent) light group, in which the object focus is in the image space and the image focus is in the object space. Various incident rays diverge after passing through the negative light group. Only the first and last two refractive surfaces and the main optical axis in the optical group are shown in the figure. The incident beam parallel to the principal axis (the object point is infinitely far on the principal axis of the object space) and the outgoing beam passing through the optical group intersect with a point F' on the principal axis of the image space, and F' is called the image focus (or secondary focus or back focus) of the optical group. The vertical plane passing through F' is called the square focal plane (second focal plane and back focal plane) of the optical group image; The object point F of the yoke point * * * at infinity is called the object focus (the first focus and the front focus), and the vertical axis plane passing through F is called the object focus plane (the front focus plane and the first focus plane) (as shown in Figures (a) and (b)).

F and f' are not * * * yoke points, because the object point is placed in F, like it is not in F', and vice versa. Like the square focal plane, the yoke plane is also a vertical axis plane located at infinity. Parallel beams from an object at infinity in any direction will converge to a point (secondary focus) on the focal plane of the image after passing through the optical group; The focal plane of the object is yoked to the plane perpendicular to the main axis at infinity, so the light beam emitted from any point on the focal plane of the object will be parallel to the secondary optical axis passing through that point after passing through the optical group. The properties of the focal plane mentioned above are often used when drawing optical circles.

2. What are the principal points and planes of an ideal light group?

① What are principal points and principal planes?

Any ideal optical group has a pair of yoke planes with transverse magnification equal to 1. The main plane of an object belongs to the object, and the point on its axis is called the main point of the object (or the first main point, the front main point); The main plane of an image is called the main plane of the image, and the points on its axis are called the main points of the image. H and H' represent the front principal point and the back principal point respectively. Figs. 2-22(a) and (b) show the principal points and principal planes of the convex lens. The beam emerging from the object focus f is parallel to the main optical axis after twice refraction; The light beam parallel to the main optical axis passes through the image focus after being refracted twice. In the two drawings, each pair of yoke lines extend and intersect, and the locus of these intersections is the longitudinal axis plane, which is the main plane, and their intersections with the main axis are the main points.

(2) Why is the main plane a * * * yoke with horizontal magnification equal to plus one?

In Figure 2-23, H 1 is the intersection of two * * * yoke lines with double arrows on the main plane of the object; H' 1 is the intersection of two yoke lines of a single arrow on the main plane of the image. As can be seen from Figure 2-22, whether it is the light emitted from F or the incident light parallel to the main optical axis. Its incident height (the distance from the intersection of incident light and the main plane of the object to the main axis) is arbitrary; The outgoing light in Figure 2-22(a) is just the incident light in Figure (b); If the incident heights in two pictures are equal, the heights of emergent rays (the distance from the intersection of emergent rays and the main plane of the image to the principal axis) must be equal. Therefore, the situation shown in Figure 2-23 is completely achievable. In this case, H 1 can be regarded as the convergence point of two incident rays-the virtual object point on the main plane of the object, and H' 1 can be regarded as the virtual image point of H 1. This pair of * * * yokes are on the same side of the main shaft and have the same height from the main shaft, so the lateral magnification is positive. Similarly, two line segments H 1H and H' 1H' are * * yokes. If this figure revolves around the axis, the two planes where H 1H and H' 1H' are also * * yokes.

The focal length and object distance of the object are based on the main point H of the object, and the right is positive and the left is negative; The image quantity takes h' as the coordinate origin, which is also right positive and left negative. However, the volume of matter cannot be calculated from H'; Image quantity cannot be calculated from H.

4. What is the relationship between objects and images in an ideal light group?

To find the image of a known object in an ideal light group, graphic (drawing) method and analytic (algebraic) method can be used:

1. What is a graphic method?

According to the properties of the focal point and principal point of the light group and the positions of the points, lines and surfaces in the object space, the positions of the * * * yoke points, lines and surfaces can be found by graphic method, which is called graphic method.

(a) Under ideal imaging conditions, the light beams emitted from one point will meet at one point after being refracted by the lamp group. So in order to determine the position of the image point, we only need to find the yoke ray of two typical rays in the image space, and their intersection point is the image point. As shown in Figure 2-24, the key points of the system are H and h'; The positions of the focal points f and F' are both known, and the position and size of the vertical axis object PQ are also known. Find the position and size of its image. First, the light QM parallel to the main optical axis passes through the Q point and the main plane of the object at the M point. According to the nature of the focal point and the main plane, the outgoing ray M'p' of the ray QM refracted by the optical system must pass through the back focal point F'. Then a ray passing through the focus of the object is drawn from point Q, and the front principal plane is at point N, so its yoke ray N'Q' must be parallel to the principal axis. The intersection point Q' of two refraction lines is the image point of Q point; The longitudinal axis segment Q' p' passing through q' is the image of PQ.

(b) If the object point is on the main axis, the above two typical rays overlap to form a ray propagating along the main axis. Therefore, it is necessary to introduce light in any direction, and to determine the direction of light, it is necessary to apply the properties of focal plane. As shown in Figure 2-25(a), find the image point of point A on the axis graphically: if A is regarded as any incident ray AM, it can be considered as one of the parallel beams (oblique beams) emitted by an object point infinitely far from the axis, and then an auxiliary ray is used to be parallel to it after the front focus F, and these two rays form an oblique parallel beam. They should converge at a point on the focal plane of the image. This can be determined by the auxiliary light, because after the auxiliary light exits the system, it should be parallel to the main optical axis and intersect with the back focal plane at B', so the direction of the light can be determined, and the intersection with the main axis A' is the required image point; You can also use the method shown in Figure 2-25(b) to find the image point A'.

2. What is an analytical method?

If the position of the object relative to the optical group is known, the method of calculating the position and size of the image by formula is called analytical method. This method is not as convenient and intuitive as drawing, but it is more accurate. Because of the different origin of coordinates, it can be divided into Newton formula and Gaussian formula. As shown in figure 2-26. X is the focal object distance, with point F as the origin, X' as the focal image distance and point F' as the origin, and the symbols are all negative Zuo Zheng right. All marks in the figure are geometric positions (positive values).

(1) How to derive Newton's formula?

As shown above, there are four triangles: 1, 2, 3, 4. You can have the following formula:

The transverse magnification relative to Newton's formula is

(2) How to deduce the Gaussian formula?

As shown in the above figure, s represents the distance (object distance) from the object point P to the object principal point H, s' represents the distance from the image point P' to the image principal point H', and the symbols of s and s' take the principal points (H and H') as the coordinate origin, which is still negative Zuo Zheng right. As can be seen from the figure: x = s-f; X' = s'-f', substitute Newton's formula, and get Gaussian formula as follows:

In most cases, photographic lenses are used in the same medium. At this time, the formula f =-f=-f' becomes:

Add f' on both sides, then substitute x'+f' = s' and x+f = s, and sort it out:

The above Gauss formula and its transverse magnification are all derived from Newton's formula. Conversely, Newton's formula can also be derived from Gaussian formula; Or all of them are directly derived from the corner relationship in the optical path diagram.

Any type of photographic lens, as long as it is regarded as an ideal light group, can be drawn according to the method in Figure 2-26, and the position of the image after obtaining the base point can be obtained.

5. Why is the magnification of the combination lamp group equal to the product of the magnifications of each part?

An optical system can be composed of one component or several components, and each component can be composed of one lens and several lenses. Each component can be regarded as an optical group separately. For example, the zoom lens of a camera usually consists of four parts: front fixed group, zoom group, compensation group and rear fixed group. The magnification of the zoom lens is equal to the product of four parts. Next, we derive the magnification of the optical system composed of three elements. If the length of the object is y, the imaging heights through the three components are y' 1, y'2 and y'3. Prove:

β=β 1β2β3 (2-24)(b)

Because the image of the first light group is the object of the second light group, that is:

y2=y' 1

The image of the second light group is the object of the third light group, namely:

y3=y'2

The image of the third lamp group is also the image of the combined lamp group, namely:

y'3=y '

6. What is the power of the combination lamp group?

The optical power of the optical system is a numerical representation of the convergence or divergence of the optical system, and its numerical value is measured by n'/f' or n/f. If j is used to represent the optical power of the optical group, then:

If the lamp group is placed in the air

Ordinary photography is suitable for the situation shown in Formula (2-27). The positive optical power of an optical system means that it is a positive group and has a convergent effect on light; Negative J indicates that the light group is negative and has divergent effect on light. Camera lens is usually a combined optical group composed of light groups with positive and negative powers, and the power of this combined optical group must be positive. The power of the short focal length lens (fisheye lens and wide-angle lens) of the camera is large, which will produce a very large deflection effect of the outgoing beam relative to the incident beam. The telescopic system (afocal system) does not deflect the beam (only changes the aperture of the beam), with zero focal power and infinite focal length.

7. What is the general formula for an ideal light group to image an object of any size with an arbitrarily wide beam?

In Figure 2-27, the ray PM emitted by the object point P on the axis forms a U angle with the optical axis, and the intersecting main plane is at point M. The * * * yoke ray M'P' with the incident height of h..PM intersects the main plane at point M, and the included angle with the optical axis is U'. It consists of right triangles △PMH and △P'M'H'

For an ideal optical group, the above formula is applicable to any value of u (or y) and u' (or y'), and of course it is also applicable when u (or u') tends to zero:

Compared with the lagrange invariant muy' = n' u' y' in formula (2-6), we can get:

If the object and image media are the same:

f=-f '

Replace (2-29) with (2-28):

It is a general formula-LAH formula for optical groups to image objects of any size with arbitrarily wide beams.

Eight, what is the angular magnification?

As shown in Figure 2-27, when passing through a pair of * * * yoke points on the optical axis, take a pair of * * * yoke rays PM and P'M', and their included angles with the optical axis are U and U' respectively. The ratio of the tangents of these two angles is called the angular magnification of a pair of yoke points, namely:

Substitute the relationship between tgu' and tgu into formula (2-28) to get the following result.

If the optical system is in the same medium, then

Obviously, the angular magnification has nothing to do with angles u and u', but only with the position of the object. At the same pair of * * * yokes, the tangent ratio of all * * * yokes to the optical axis is constant.

9. What are the nodes of the optical group?

What are the nodes of the 1. optical group?

On the principal axis of the optical group, there are a pair of * * * yoke points with diagonal magnification equal to+1, which are called the nodes of the optical group. Those belonging to the object side are called object side nodes, and those belonging to the image side are called image side nodes. Represented by k and K' respectively. The angle is equal to plus one, which means that a pair of yoke lines passing through the node are parallel and in the same direction, as shown in Figure 2-28.

2. How to determine the location of the node?

The above results show that the distance between the object node and the object focus is equal to the image focus value, such as f'

If the optical groups are in the same medium, the nodes coincide with the principal points. In ordinary photography, the lens is in the same medium-air. If it is a thin lens, the four points of the front and rear principal points and the front and rear nodes are combined into one called the optical center, which is represented by O, so the O point has the properties of both principal points and nodes.

3. How to draw the optical path diagram with the attributes of nodes?

In the past, we used the properties of focus and principal point to find images graphically. Similarly, attributes that use focus and nodes can also be used to find images graphically. As shown in Figure 2-29, when the camera lens is placed in the air, the principal point coincides with the node, and the positions of the principal point and the focus are known. If the Q point of the object PQ guides a ray through the front node (i.e. the front principal point), then the * * * yoke ray must pass through the rear node (i.e. the rear principal point) and be parallel to the incident ray; Then make a ray parallel to the optical axis or pass through the focus of the object, and the ray from its yoke must pass through the focus of the image or parallel to the main optical axis, and intersect with the ray passing through the node of the image at point Q', that is, the image of point Q, and the longitudinal axis segment P'Q' passing through point Q is the image of object PQ.

4. What are the applications of the physical properties of nodes?

The characteristics of nodes are not only used to draw the optical path diagram, but also used in panorama camera, which is called the turning point. Its principle is to make the camera lens rotate around the axis of the image node and change its negative image on the arc surface with the radius of the image distance, so that you can take pictures of the big scene.

As shown in Figure 2-30, the photographed crowd is on an arc with the image node K' as the center and the sum of the object distance and the distance between two nodes as the radius; The film is placed on the cambered surface with the back node as the center and the image distance as the radius. Ordinary transfer machines are often equipped with slits (exposure windows) before and after the lens. The light from the subject must pass through two slits at the same time (the former is incident light and the latter is outgoing light), so that a small area of the film can be sensitized and a clear image can be formed. When the slit of the exposure window is located in U, the light from the upper left subject can form a clear image in a small area of A film through the lens; When the slit between the photographic lens and the exposure window rotates clockwise around the k' point to the front, the light injected from the front subject can form a clear image in the small area B of the film through the photographic lens; With the rotation of the camera lens, the slit of the exposure window sweeps the whole film in turn, and you can get a bigger picture of the scene.

X. how to determine the base point of a photographic lens?

Although the basic point of a photographic lens is invisible and intangible, it exists objectively and can be measured. We don't involve the problem of how to accurately determine the base point, but only introduce simple and cheap methods.

1. How to determine the focus?

The whole photographic lens is a positive light group, which is equivalent to a thick lens and plays the role of converging light. Let the lens with direct sunlight, as shown in Figure 2-3 1, use a small screen (such as a piece of white paper) to move left and right along the optical axis behind the light group. If you move to A or B, the diameter of the light spot on the screen is larger, except that the light spot at F' is the smallest, which can be approximated as a point (where the paper screen can be burned), and F' is the back focus of the lens; Similarly, the focus f of an object can be measured by switching the lens left and right.

2. How to determine the main point and main node of the lens?

In ordinary photography, the lens is always in the air, and the main point of the object coincides with the node of the object; The main point of the image coincides with the node of the image. Therefore, as long as its nodes are measured, the principal points are naturally measured. It is necessary to measure nodes according to their attributes.

As shown in Figure 2-32: Keep the object and image plane still. Rotate the lens around an axis perpendicular to the paper. When the position of the axis is different, the position of the image point is different. However, when important official passes through the rear node of the lens, the position of the image point remains unchanged. The lens shown in figure (a) is stationary, and the image point of the parallel beam is p'; As shown in figure (b), the lens rotates clockwise around the image point K' by a small angle, and the position of the image point remains unchanged; As shown in figure (c). When important official passes through the rear node, the lens rotates by a small angle, but the position of the image point changes. Because the two nodes are * * * yoke points with angular magnification equal to+1, the direction of incident light in Figure 2-32 is along the X direction of the abscissa axis, so the outgoing light of the rear node must be parallel to the incident light, and its intersection with the image plane is the clear image point of the parallel beam.

Make the camera lens slowly translate along the X-axis and gently rotate around the axis (O) perpendicular to the paper surface until the parallel light beam or the image of the infinite scene does not shift when rotating. At this time, the rotation center o coincides with the back node, so the K' point can be accurately determined; If you turn the lens upside down, you can also measure the front node of the lens.

XI。 How to lay out the basic points of photographic lens?

The structural dimensions of all kinds of cameras are determined according to the use requirements, and their structural contents include the composition of the system, the focal length of each component, the relative position and lateral dimensions of each component. Among them, the layout of the base point of the photographic lens directly affects the axial (length) size of the camera, and the aperture of the lens directly affects the lateral size of the camera.

1. What are the main parts in the horizontal dimension of the photographic lens?

(1) What is the image plane positioning distance?

The distance between the axial positioning end face of the lens barrel matched with the lens holder and the lens focal plane (exposure window plane) is called the image plane positioning distance.

(b) What is the back working distance (back intercept, back vertex focal length, image vertex focal length)? The distance from the back vertex of the last lens of a photographic lens to its image focus is called the back working distance.

(c) What is the front zenith focal length?

The distance from the front vertex of the front lens of the photographic lens to the focus of the object is called the front vertex focal length. Its value determines the distance between the front lens and the object plane.

(d) What is optical length?

The distance from the front vertex of the first lens to the back vertex of the last lens is called the optical path of the photographic lens.

2. Photographic lenses are divided into several categories according to focal length?

Photographic lenses can be classified according to different standards. The focal length can be divided into three categories: standard lens, wide-angle lens and telephoto lens.

(a) What is a standard lens?

In the lens series used in cameras, the photographic lens with the focal length close to the diagonal of the frame is usually called the standard photographic lens. For example, Canon 135 single-lens reflex camera has a frame size of 24 mm×36 mm, and its photographic lens series * * * is equipped with 55 kinds of photographic lenses with different focal lengths and performances. A photographic lens with a focal length of 50 mm (diagonal length of the frame is 43. 27 mm) is called the standard lens of 135 camera.

(b) What are long focal length and short focal length photographic lenses? Usually, a lens with a longer focal length than a standard lens is called a long focal length lens; A lens with a shorter focal length than a standard lens is called a short focal length photographic lens.

3. What are the basic point layouts of photographic lenses?

Ordinary photographic lens is placed in the air, and the nodes coincide with the main points, so only the distribution of lens focus and main points is studied. The distribution of basic points is varied, and we only give a few examples to illustrate it.

Example: 1: a Kirk lens of 120, with a frame size of 56 mm×56 mm and a diagonal size of 79.20 mm, where -lF is the focal length of the front ceiling, lF is the working distance of the rear ceiling, and the distribution of the object (image) focus and the object (image) principal point is shown in Figure 2-33. Figure 2-34 shows the base point distribution of Nico 50 mm lens. F and f' are the focus of objects and images; H and h' are the main points of objects and images; -f is the focal length of the object; F' is the focal length of the image; LF is the reverse working distance; -lF is the front working distance; △ is the optical path; L is the length of the lens barrel. For ordinary photographic lenses, f is usually located before the front vertex.

For Example 2, the base point distribution of telephoto lens is shown in Figure 2-35 (a) and (b). In order to shoot the distant view and make the distant objects form a larger image on the image plane, a long focal length lens must be used. The longer the focal length, the larger the camera structure. In order to shorten the length of the cylinder, the positive and negative groups are often separated and the positive group is in front. Fig. (a) shows a Kirk telephoto objective with a focal length of100 mm to 500 mm; The visual field is 20 ~ 40; The relative aperture is from 1: 8 to 1: 3.5, which is the most basic long focal length photographic lens. This telephoto structure makes the main surface push horizontally into the object space, and the length (χ) of the tube is smaller than the focal length (f'), which can generally be shortened by one third.

Fig. 2-35 (b) is the optical structure diagram of Nico, Q-Auto 400 mm, 1: 4.5 long focal length photographic lens, and the base point distribution is shown in fig. (b). It can be seen that the distribution of base points is not exactly the same for the same telephoto lens. Figure (a)H' is located outside the front vertex (left); Figure (b)H' is located in the middle of the lens group behind the front vertex.

Example 3: The basic point distribution of short focal length (reflective telephoto lens) is shown in Figure 2-35 (c). In general photography or cinematography, a wide-angle lens with short focal length should be adopted in order to obtain images with large field of view and rich stereoscopic impression. Because it is necessary to place a light splitting element or a light reflecting element between the objective lens and the negative film, it is desirable for the lens to have a long back working distance. Therefore, the anti-telephoto structure shown in 2-35 (c) should be adopted, so that the back working distance larger than the focal length can be obtained.

In short, the positions of principal points H and H' of different photographic lenses are different with respect to the lens barrel: some are located near the diaphragm blade, some are located in front of the lens barrel, some are located behind the lens barrel, and some are located outside the photographic lens. Usually h is located in the object side and H' is located in the image side. %A