1. 18. Images 18.1. Images in plane mirrors
Image is a reproduction derived from light. There are two kinds of images:
a virtual image (existing only within the brain, when rays are traced back to perceived location of source.) and a real image (exists whether you are looking or not).
18.1. Images in plane mirrors
Figure from
HRW,8
Formation of virtual image of an extended object in a plane mirror.
It appears that light originates at point I.

2. Spherical mirrors
Spherical mirror is a small section of the surface of a sphere. The central axis extends through the center of curvature of a sphere O and the center of the mirror o.
Concave mirror
Convex mirror
Spherical mirrors:
f = r/2
f – focal length
r - radius of
curvature
r > 0 for concave
r < 0 for convex
In a concave mirror (a) a real focus F is formed on the same side as incident rays and in a convex mirror (b) the light rays seem to diverge from a virtual focus on the side oppsite to the light rays. Parallel incident rays close to the central axis after reflection from the mirror determine the focal point F.

3. 18.3. Images from spherical mirrors
Real images form on the side where the object is located. Virtual images form on the opposite side.
Formation of real image of an extended object in a concave mirror.
Formation of virtual image of an extended object in a convex mirror.
When rays from an object form only small angles with the central axis, the following relation between the object distance x, the image distance y and the focal length f can be written:
spherical mirror formula (18.1)
Focal length can be positive or negative.

4. Proof of spherical mirror formula (eq. 18.1)
From trigonometry theorem it follows
for exterior angle of a triangle PaO
b = a + q
and for a triangle PaI
g = a + 2q, from which one obtains
a + g =2b (18.2)
The angles in radian measures are equal
= ac/cP = ac/x,
b = ac/cO = ac/r
g = ac/cI = ac/y
In effect eq.(18. 2) can be written as
follows :
ac/x +ac/y = 2ac/r or
1/x + 1/y = 2/r and because f = r/2 one obtains finally
Formation of real image of an extended object in a concave mirror.

5. Images from spherical mirrors, cont.
For spherical mirrors one observes the phenomenon called spherical aberration.
One focus point exists only for rays close to the central axis.
One focal point one obtains only for paraboloidal mirrors.
Spherical concave mirror is tangent to embracing paraboloid mirror. Both mirrors have the same curvature radius at the tangent point.
Optical rays are emitted by the source placed in a focus F.
Only the ray „a” reflected from paraboloid mirror is parallel to the central axis. The ray a’ reflected from the spherical mirror is not parallel to the central axis.

6. Images from spherical mirrors, cont.
Magnification
Lateral magnification m of an object reflected in a mirror can be obtained taking into account that triangles abc and dec are similar . In this case one can write:
(18.3)

7. 18.4. Thin lenses
Two refracting surfaces with coinciding central axes form a lens. When light rays initially parallel to the central axis converge, the lens is called converging. If the rays diverge, the lens is called diverging.
Rays parallel to the central axis of
a converging lens give a real focus F.
(b) Magnified top part of the converging lens. Both refractions bend the ray downward.
(c) Rays parallel to the central axis of a diverging lens give a virtual focus F’at intersection of their extensions.
(d) Magnified top part of the diverging lens. Both refractions bend the ray upward.
Converging lens
For a thin lens with refractive index nl
placed in a medium with refractive index nm the focal length f is given by:
Diverging lens

8. Images from thin lenses
The image of a point object is located where the rays passing through the lens intersect. The image of an extending object is obtained by locating the positions of selected points of this object.
(a) Formation of real image in a converging lens. Only two of indicated three special rays are necessary to obtain the image.
(b) Formation of a virtual image in the diverging lens with two concave sides.
Distances of the object x and the image y to the lens are related to each other by equation which is the same as for mirrors:
Similarly the lateral magnification of the lens is given by

9. Two–lens system
Working in steps one constructs the image by the first lens alone ane then taking this image as an object one constructs the image by the second lens. The overall magnification is given by the product of lateral magnifications of both lenses
For a system of two lenses placed very close to each
other the overall focal length can be found from relation or
where Z is the refractive power.
Refractive power is measured in diopters (D) equal to
the reciprocal of the focal length measured in metres.
For a lens with f = 0.5m Z = 2D.
System of two lenses separated by L.
(b) The image I1 produced by lens 1.
(c) The image I2 produced by lens 2 where the image I1 acts as an object O2.
Image I2 is the final image.

10. 18.5. Optical instruments, single magnifying lens
(a) The same object seen at higher angle gives larger image on the eye retina. However the minimum distance for normal eye is d = 25 cm.
(b) Use of magnifying lens enables placing the object close to the eye. If it is inside focal point F of the lens, one obtains the magnified virtual image.
For small angles from fig. (a) one obtains and from fig. (b)
The angular magnification is then equal:
(18.4)

11. Compound microscope
Small object is placed very close to the objective focus, so its distance to the lens is a bit higher than fob . The inverted, enlarged real image forms between the eyepiece lense and its focus F1. This image serves as an object for the eyepiece and the observer sees a final inverted, virtual and enlarged image.
The lateral magnification produced by the objective is (taking into account that the distance between lenses s is much higer than both focal lengths):
Total magnification of the microscope is a product of lateral magnification of the objective and angular magnification of the eyepiece:
where d = 25 cm.

12. Refracting Telescope
Parallel rays from distant object give a real image.
Real image formed at the common focal points F1 and F2 acts as an object for the eyepiece, which produces a virtual final image at a great distance from the observer.
The angular magnification of the telescope is aey/aob. For rays close to the central
axis a ob = h/fob, aey = h/fey what gives

13. Human eye The basic structure of the human eye.
Light is refracted first by the cornea and then by a lens whose shape (and thus the ability to focus the light) is controlled by muscles.
A normal eye can focus the incident light on the retina.
Correction of farsightedness by a converging lens to add refractive power.
Correction of nearsightednes by a diverging lens to reduce refractive power.

14. Fresnel lens
The conventional lens is divided into a set of concentric annular sections and the overall thickness of each section is decreased. As a result the continuous surface of a standard lens is divided into a set of surfaces of the same curvature but with stepwise discontinuities between them.
An ideal Fresnel lens would have infinitely many such sections.
A spherical Fresnel lens gives sharp images, however disturbed by diffraction of light at the edges.
The curved surface can be replaced by a flat surface, but such a lens does not produce sharp images. It can be used only for the focusing of light, e.g. for focusing of sunlight on a solar panel (the concentration can reach a ratio of 500:1).
Comparing to a conventional lens, a lot of volume and mass is reduced in Fresnel lens.
Cross section of a spherical
Fresnel lens.
Cross section of a
conventional plano-convex
lens.