Consider the formulas derived:
(3.8)
Let us compare formulas (3.7 and 3.8), it is obvious that we can write the following expression relating the optical characteristics of the lens (focal lengths) and the distances characterizing the location of objects and their images:
, (3,9)
where F is the focal length of the lens; D is the optical power of the lens; d is the distance from the object to the center of the lens; f is the distance from the center of the lens to the image. Lens inverse focal length 
called optical power.
This formula is called the thin lens formula. It applies only with the rule of signs: Distances are considered positive if they are counted in the direction of the light beam, and negative if these distances are counted against the course of the beam.
Consider the following figure.
The ratio of the height of the image to the height of the object is called a linear increase in the lens.
If we look at similar HLW and OAB triangles (Fig.3.3), then the linear magnification given by the lens can be found as follows:
, (3.10)
where АВ - image height; AB - the height of the subject.
For high-quality image acquisition, lens and mirror systems are used. When working with systems of lenses and mirrors, it is important that the system is centered, i.e. The optical centers of all the bodies that make up this system lay on one straight line, the main optical axis of the system. When building an image in the system, the principle of consistency is used: an image is built in the first lens (mirror), then this image is the subject of the next lens (mirror) and the image is rebuilt, etc.
In addition to the focal length, the optical characteristic of lenses and mirrors is the optical power; this is the inverse of the focal length:
(3,11)
The optical power of the optical system is always equal to the algebraic sum of the optical forces that make up this optical system of lenses and mirrors. It is important to remember that the optical power of the scattering system is negative.
(3.12)
The optical power is measured in diopters D = m -1 = 1dptr, i.e., one diopter is equal to the optical power of the lens with a focal length of 1m.
Examples of building images using side axes.
Since the luminous point S is located on the main optical axis, all three beams used to build the image are the same and go along the main optical axis, and to build an image you need at least two beams. The stroke of the second beam is determined with the help of additional construction, which is performed as follows: 1) build a focal plane, 2) choose any beam coming from point S;
Optical aberrations
The aberrations of optical systems and methods for reducing or eliminating them are described.
Aberrations are a common name for image errors arising from the use of lenses and mirrors. Aberrations (from the Latin. "Aberration" - deviation), which appear only in non-monochromatic light, are called chromatic. All other types of aberrations are monochromatic, since their manifestation is not associated with the complex spectral composition of real light.
Sources of aberration. The definition of an image contains the requirement that all rays emanating from some point of an object converge at the same point in the image plane and that all points of the object are displayed with the same magnification in the same plane.
For paraxial rays, the mapping conditions without distortion are met with great accuracy, but not absolutely. Therefore, the first source of aberration is that lenses bounded by spherical surfaces refract broad beams not quite "as it is accepted in the paraxial approximation. For example, the foci for rays incident on the lens at different distances from the optical axis of the lens are different and etc. Such aberrations are called geometric.
a) Spherical aberration - monochromatic aberration, due to the fact that the extreme (peripheral) parts of the lens more strongly deflect the rays going from a point on the axis than its central part. As a result, the image of a point on the screen is obtained as a bright spot, fig. 3.5
This type of aberration is eliminated by using systems consisting of concave and convex lenses.
b) Astigmatism - monochromatic aberration, consisting in the fact that the image of a point has the form of a spot of elliptical shape, which at some positions of the image plane degenerates into a segment.
Astigmatism of oblique beams appears when a beam of rays emanating from a point falls on the optical system and makes some angle with its optical axis. In fig. The 3.6a point source is located on the secondary optical axis. In this case, two images appear in the form of straight line segments located perpendicular to each other in the I and P. planes. The image of the source can be obtained only as a blurred spot between the I and P. planes.
Astigmatism due to the asymmetry of the optical system. This kind of astigmatism occurs when the symmetry of the optical system with respect to the light beam is broken due to the structure of the system itself. With such an aberration, the lenses create an image in which the contours and lines, oriented in different directions, have different sharpness. it
observed in cylindrical lenses, fig. 3.6
Fig. 3.6. Astigmatism: oblique rays (a); conditional
cylindrical lens (b)
A cylindrical lens forms a linear image of a point object.
In the eye, astigmatism is formed with asymmetry in the curvature of the lens and cornea systems. To correct astigmatism are glasses that have different curvature in different directions.
directions.
c) Distortion (distortion). When the rays sent by an object make a large angle with the optical axis, another type of aberration is detected - distortion. In this case, the geometric similarity between the object and the image is violated. The reason is that in reality the linear magnification given by the lens depends on the angle of incidence of the rays. As a result, the image of a square grid takes on either a cushion or barrel view, fig. 3.7
Fig. 3.7 Distortion: a) pincushion, b) barrel-shaped
To combat distortion, a lens system with opposite distortion is selected.
The second source of aberration is associated with the dispersion of light. Since the refractive index depends on the frequency, then the focal length and other characteristics of the system depend on the frequency. Therefore, the rays corresponding to the radiation of different frequencies emanating from one point of the object do not converge at one point of the image plane even when the rays corresponding to each frequency carry out an ideal display of the object. Such aberrations are called chromatic, i.e. chromatic aberration lies in the fact that a beam of white light emanating from a point gives its image as a rainbow circle, the violet rays are located closer to the lens than the red ones, fig. 3.8
Fig. 3.8. Chromatic aberration
To correct this aberration in optics, lenses made from glasses with different dispersions are used: achromates,
Laboratory work number 13
Determination of the focal length of the diffusing lens
and its optical power "
Purpose: learn to determine the focal length of the diffusing lens and its optical power, knowing the focal length of the collecting lens.
Instruments and equipment:
1. Laboratory optical complex LKO-1.
2. Condenser (module 5) (f = 12 mm).
3. Lens (module 6).
4. Cassette with holder (module 8).
5. Microprojector (module 3).
6. Object number 14.
Theoretical information
Lens - transparent body, bounded by two curved surfaces.
Curved surfaces can be spherical, cylindrical, parabolic, flat (for which the radius of curvature tends to infinity).
Lenses are convex and concave. Their appearance may be as follows:
Bulging
Concave
The lens, whose edges are thinner than the middle, is convex, and if the middle is thinner than the edges, it is concave.
Depending on the refractive index of the lens n l and the refractive index of the medium n cf in which it is located, the lens can be collecting or diffusing:
The beam of light passing through the optical center of the lens does not change its direction of propagation.
About 1 About 2 About 1 About 2
Paraxial rays are rays parallel to the main optical axis.
The main focus is the point at which the paraxial rays intersect or continue after they pass through the lens.
so we know the further course of the rays after the lens:
a) the beam going through the optical center does not change its direction of propagation;
b) the beam going to the lens parallel to the main optical axis after the lens goes through the focus (or out of focus - for a diffusing lens);
c) the beam going through the focus after passing the collecting lens goes parallel to the main optical axis.
These rays are used to build images in the lenses.
To build an image, we run an AS / VO beam, after passing the lens, they will intersect in focal plane (tP), and the point of intersection of the main optical axis and the CM beam gives an image of T.A ".
The distance of the object from the lens OA we denote d, and the image OA "denote f.
Consider the triangles: HLW and B "A" O, they are similar, therefore:
; or . (one)
The triangles of COF and B "A" F are also similar.
From equation (1) and (2) we get:
The last equation is multiplied by:
; from where (3)
The value is called the optical power of the lens and is measured in diopters (diopters).
The formula of the lens, taking into account the refractive index of the material and the radius of curvature of the surface, where R 1 and R 2 are the radii of curvature of the surfaces. For convex surfaces R\u003e 0 for concave surfaces R< 0, для плоской поверхности .
Lens magnification:.
Completing of the work
1. To perform the work it is necessary to assemble the installation according to scheme 1.
Moving the collecting lens (object 6) we achieve a clear image of the light source with the help of a micro-projector (3) on the screen.
2. Measuring the distances a 1 and 1 and using the thin lens formula, we determine the focal length of the collecting lens.
3. We assemble the installation according to scheme 2
M5 M6 M8 M3
In cassette 8 is the object number 14 (diffusing lens).
4. Moving the cassettes 6 and 8 we get a clear image. glowing dots on the screen, and we measure a 2, knowing F c, we find the distance of 2 at which the image should be obtained with the help of a collecting lens (position t.).
5. Determine a p = (in 2 - l) the distance at which is the T. relative to the diffusing lens. In relation to the diffusing lens t. Is the subject. Measuring the distance in p, we determine the focal length of the diffusing lens by the formula:.
6. To record the results of measurements and calculations in the table:
| Item number | a 1 | in 1 | F with | a 2 | at 2 | l a p | in p | F p | ε |
| 1. | |||||||||
| 2. | |||||||||
| 3. | |||||||||
| Average |
For a thin lens, it would be nice to have a formula that will link all of its basic parameters. The focal length F, the distance from the lens to the object d and the distance from the lens to the image f.
First, we construct the image of the object in a thin collecting lens. Consider the following figure.
picture
The image of the object in the lens
Direct from the point A the beam is parallel to the main optical axis. As already known, after refraction, it will pass through the focus of the lens. Next, we construct the ray of AO. As it passes through the optical center of the lens, it will not be refracted. These two rays intersect at point A1. This will be the image of point A in the collecting thin lens.
In principle, we could choose another beam, for example, the one that passes through the focus and build it. This is AD beam. Since it passes through the focus of the lens, after refraction it will be directed parallel to the main optical axis. As you can see, it intersects with other rays at point A1.
Connect point A1 and the main optical axis with a segment. This will be the image of the subject AB in a thin lens.
Thin lens formula
The triangles AOB and A1B1O are similar. Therefore, the following equality will be between their parties:
BO / OB1 = AB / A1B1.
Triangles COF and FA1B1 are similar too. Therefore, the following equality will be between their parties:
CO / A1B1 = OF / FB1.
AB = CO. Consequently,
AB / A1B1 = OF / FB1.
BO / OB1 = OF / FB1.
If we write in terms of the above notation:
By the property of proportion we have:
F * f = F * d = f * d.
We divide each term of this equality by the product f * d * F and get:
This equation is called the thin lens formula. In this formula, the values of f, F, d can be of any sign, both positive and negative. Applying the formula, it is necessary to put signs before the items according to the following rule.
If the lens is collecting, then in front of 1 / F put a plus sign. If the lens is diffusing, then a minus sign is placed in front of 1 / F. If a real image is obtained using a lens, then a plus sign should be put in front of the 1 / f member. If an imaginary image is received, then the 1 / f member needs to put a minus sign.
Before the member 1 / d put a plus sign if the dot is really glowing. If the dot is imaginary, then a minus sign is placed in front of 1 / d. We will use these rules without further proof.
If the values of f, F, d are unknown, first, everywhere they put a plus sign. Then make calculations. If any negative value is obtained, it means that the focus, image or source will be imaginary.
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