Chapter 3 Mirrors The most common and familiar optical device
Outline Plane mirrors Spherical mirrors Graphical image construction Two mirrors; The Cassegrain Telescope
Plane mirrors Common household mirrors: a plane glass surface that is silvered on the back surface by the application of metallic coating First-surface mirrors: coated on the front surface, so the reflected light does not pass through the mirror glass
Figure 3.1 A point source S (object) illuminates the mirror surface MM Light striking the mirror from S obeys the law of reflection A person viewing the mirror sees the ray as emanating from the image I and sees the source as being at point I Virtual image, because the light does not actually pass through image point I
Goal Find the relationship between the object and the image in an optical system In this case, s =-s, where s is the image position, and s is the object position (note the negative sign) Distances measured from the right of surface are positive, and those to the left are negative In optics we use Cartesian coordinate system whose origin is placed at the surface under consideration on the line of symmetry of the system The line of symmetry is called the optical axis The exchange of left and right in an image is called image reversion
Example 3.1 How small can a mirror be and still function as a full-length mirror? How much it be positioned to function this way?
Example 3.2 As you are putting on makeup or shaving one morning, a spider suddenly hangs down between you and the mirror. If you are 30 cm from the mirror, and the spider is 15 cm away from the mirror, how far are you from its image?
Multiple plane mirrors Figure 3.2 two plane mirrors set at 90 o to each other can be used to reverse the image
Corner Cube Reflector Constructed of three plane mirrors set at 90o to one another The reflections cause any light incident on the corner cube to return along the same path taken by the source light
Spherical Mirrors Most lens surfaces and many mirrored surfaces are spherical An example: right-hand side-view mirror in many cars convex spherical mirror Play an important role in optics because they can produce both real and virtual images with some magnification
Figure 3.3: a concave mirror surface The light from a point source converges toward another point, forming in this case a real image
Figure 3.4 The real image formed by an extended object in a concave spherical mirror MM
Mirror Equation The techniques allow us to know where the image will be found if we know the shape of the mirror and the position of the object.
Figure 3.5 Line of symmetry OO is the optical axis, its intersection with mirror is the origin Point C is the center of curvature of the mirror Point source S on the optic axis at a distance s illuminates the mirror We will consider the ray that strikes the mirror at R at a height h above the axis The ray makes equal angles with CR and cross the optic axis at I at a distance s
Substitute Eq. (3-5) into Eq. (3-6), we get the mirror equation: This equation fixes the relationship between the object distance s, the image distance s, and the curvature radius R of mirror Note since a plane is a sphere of infinite radius, we can again get Eq. (3-1) s=-s Eq.(3-5). ',, Angles in radian are approximately Eq.(3-4) 2,, s h c R h b s h a b c a i b c i a b (3-6) 2 ' 1 1 R s s
Example 3.3: Figure 3.6 In developing the mirror equation we neglected the small distance x between the based of the vertical h and the mirror surface. If a mirror has a radius of curvature of 15 cm and an aperture of 5 cm, what is the maximum length that is disregarded with respect to the center of curvature of the mirror?
Important Consequences from Mirror Equation Concept of conjugates: for every object point s there is a single corresponding image point s, and this pair of points are called optical conjugates Because of the simple symmetry of Equation (3-6), we have optical reversibility if the source is moved to s, then the image will move to s. If s is infinitely removed from the mirror, 1/s=0, and s =R/2, the focal point. (see Fig. 3.7) 1 s 1 s' 1 f ( f :focal length)
Example 3.4 Assume that optical infinity is 6 m in front of a mirror with a focal length of -25 cm. What would be the error in a measurement of the mirror s focal length?
Figure 3.7 Also illustrate the design of a flashlight or other optical illuminator Placing the source at the focal point results in a parallel beam of light from the mirror
Figure 3.8 Moving the source off the optic axis but still on the focal plane results in a shift in the direction of the parallel beam. Application: automobile headlamps
Example 3.5 A concave spherical mirror with a radius of 15 cm is illuminated by an object 30 cm in front of it. Where is the image formed?
Example 3.6 Can an object and its image be made to lie at the same point on the optic axis?
Example 3.7: virtual image An object is located 5 cm from a concave spherical mirror having a 10 cm focal length. Where is the image located?
Table 3.1 Mirror conjugate positions Object position Image position Image character Infinity to center of curvature Center of curvature to focal point Focal point to mirror Focal point to center of curvature Center of curvature to infinity Behind the mirror Real Real Virtual
Example 3.8: Figure 3.9 A convex mirror with a radius of curvature R=15 cm is illuminated by a source 30 cm to the left of the mirror. Where is the image formed, and is it real or virtual? Any source to the left of a convex mirror will result in a virtual image lying between the mirror and the focal point.
Graphical Image Construction The image of extended object is considerably more interesting than that of point objects Fig. 10.3(a): OO represents an extended object By using 3 selected rays passing through the tip of OO, we can locate the image H (1) the ray parallel to the optic axis must pass out of the system through the focal point f
Figure 3.10(b) (2) The ray passing through the focal point is conjugate with a point at infinity and must pass back to out of the system parallel to the optical axis
Figure 3.10(c) (3) The ray makes equal angles with the optic axis at the mirror surface All three rays intersect at the position of I, the tip of the arrow image, and we can draw the image at that position
Figure 3.10(d) An additional characterization of the image The image is inverted and smaller than the object OO The ratio of the image size to that of the object is called the magnification
Figure 3.11 The triangles OPO and IPI are similar, so the magnification is where the negative sign indicates that the image is inverted m s' s P
Example 3.9 If the object in Example 3.5 is 2 cm tall, how large is the image, and is it erect or inverted?
Figure 3.12: Convex Mirror When viewed from the left of the mirror, the diverging rays will appear to come from the virtual image II.
Example 3.10 If the object in Example 3.8 is 3 cm tall, how tall is the image, and is it erect or inverted?
Two Mirrors; The Cassegrain Telescope Many telescopes use a pair of mirrors as focusing elements Fig. 3.13: a telescope design with the Cassegrain system Objective mirror is a concave mirror with a hole bored through the center Partway along the telescope body a secondary, convex mirror is located between the primary mirror and its focal point f p The object for the secondary mirror is a virtual object, and the secondary mirror forms a real image just behind the primary mirror
Example 3.11 The focal length of the primary mirror of a Cassegrain telescope is 1500 mm. A secondary mirror is set in the telescope tube 1400 mm from the primary, and the image plane is 10 mm behind the primary mirror (see Fig. 3.13). What is the focal length of the secondary mirror?
Example 3.12 A primary mirror in a Cassegrain telescope has a focal length of 2000 cm. The image is to be formed 10 cm in back of the primary mirror, and the magnification is to be 20X. Where is the secondary mirror with a focal length of 150 cm to be set?
Homework Problems 2, 4, 6, 8