Lecture Notes (Geometric Optics)

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1 Lecture Notes (Geometric Optics) Intro: - plane mirrors are flat, smooth surfaces from which light is reflected by regular reflection - light rays are reflected with equal angles of incidence and reflection Plane Mirror Images: - when you look into a plane mirror, you see an image of yourself that has four properties: 1) The image is upright. 2) The image is the same size you are. 3) The image is located as far behind the mirror as you are in front of it. 4) The image has left-right reversal. That is, if you wave your right-hand it is the left hand of the image that waves back. - light rays leave each point on an object, but we will simplify this process by looking at only two rays - let's draw a diagram of an object in front of a plane mirror

2 - the two rays leave the object; these rays reflect from the mirror and enter the eye; to the eye, it appears that the rays originate from behind the mirror, back along the dashed lines - although rays of light seem to come from the image, it is evident that no light emanates from behind the plane mirror - because the rays of light do not actually emanate from the image in the mirror, it is called a virtual image; in the text, the light rays that appear to come from a virtual image are represented by dashed lines - images in plane mirrors are located as far behind the mirror as the object is in front of it - the object distance is given the symbol d 0 and the image distance is d i - the object height is the same as the image height in plane mirror reflection; the object height is symbolized as h 0 and the image height is h i - if the object and the image are pointing in the same direction, the image is called an erect image Curved Mirrors: - unlike plane mirrors, which only produce virtual images, curved mirrors can produce real images

- real images are those which can be seen on a piece of paper or projected onto a screen, because

spherical mirror; a spherical mirror has the shape of a section from the surface of a sphere - the

convex mirror - for spherical mirrors, the center of curvature is located at point C and the radius

which is called the vertex, V - it is found experimentally that rays striking a concave mirror

3 - real images are those which can be seen on a piece of paper or projected onto a screen, because rays actually converge and pass through the image - the most common type of curved mirror is a spherical mirror; a spherical mirror has the shape of a section from the surface of a sphere - the inner surface of a hollow sphere will produce a concave mirror; the outer surface will produce a convex mirror - for spherical mirrors, the center of curvature is located at point C and the radius is R; the principal axis of the mirror is a straight line through C and the midpoint of the mirror which is called the vertex, V - it is found experimentally that rays striking a concave mirror parallel to its principal axis, and not too far away from this axis, are reflected by the mirror such that they all pass

4 through the same point, F, on the principal axis; this point, which is lies between the center of curvature and the vertex, is called the focal point, or focus, of the mirror - the distance along the principal axis from the focus to the vertex is called the focal length of the mirror, and is denoted, f - the focal point, F, lies halfway between the center of curvature, C, and the vertex, V; therefore, the focal length, f, is equal to one-half the radius, R f = ½ R - rays that lie close to the principle axis are called paraxial rays; and the above equation is only valid for such rays - rays that are far away from the principal axis do not converge to a single point after reflection from the mirror - the result is a blurred image; the fact that a spherical mirror cannot bring all rays parallel to the axis to a single image point is known as spherical aberration Parabolic Mirrors: - you can eliminate spherical aberration by either covering the outer edges of a spherical mirror, or by using a mirror of a different shape, such as a parabolic mirror

- all rays from a point source of light at the focus, which fall on the mirror, are reflected parallel to the axis; the mirror keeps them together and thus acts like a search light - rays of light

purposes; they use long rows of concave parabolic mirrors that reflect the sun's rays at an oilfilled pipe located at the focal point - the sun heated oil is used to generate steam; the steam is used

5 - all rays from a point source of light at the focus, which fall on the mirror, are reflected parallel to the axis; the mirror keeps them together and thus acts like a search light - rays of light from a distant source of light coming to the mirror parallel to the axis, pass on reflection through the focus - parabolic mirrors are used in a method of capturing solar energy for commercial purposes; they use long rows of concave parabolic mirrors that reflect the sun's rays at an oilfilled pipe located at the focal point - the sun heated oil is used to generate steam; the steam is used to drive a turbine connected to an electric generator Convex Mirrors: - in convex mirrors, rays will diverge after being reflected - if the incident parallel rays are paraxial, then the rays will seem to come from a single point behind the mirror; this point is the focal point, F, of the convex mirror and its distance from the vertex, V, is the focal length, f - we assign a negative value to the focal length of the convex mirror; f = -½ R

6 Image Formation: A. Concave Mirrors: - as we have seen earlier, some of the light rays emitted from an object in front of a mirror strike the mirror, reflect from it, and form an image - we will draw three rays to form an image; this process is called ray tracing Ray 1: this ray is initially parallel to the principle axis and therefore passes through the focal point, F, upon reflection from the mirror Ray 2: this ray passes through the focal point, F, and is reflected parallel to the principle axis Ray 3: this ray travels along a line that passes through the center of curvature; this ray strikes the mirror perpendicularly and reflects back upon itself

7 Ray Tracing Example: (Concave Mirrors) - you may have difficulty understanding how an entire image of an object can be deduced once a single point on the image has been determined - in theory, it would be necessary to pick each point on the object and draw a separate ray diagram to determine the location of the image of that point; that would require a lot of ray diagrams

8 - for our purposes, we will only deal with the simpler situations in which the object is a vertical line which has its bottom located upon the principal axis - for such simplified situations, the image is a vertical line with the lower extremity located upon the principal axis More Ray Tracing Examples: (Concave Mirrors)

9 Convex Image Formation: - for determining the image, location and size of a convex mirror, it is similar to a concave mirror, but the focal point and center of curvature of a convex mirror lie behind the mirror, not in front of it - the method of drawing ray diagrams for convex mirrors is described below Step 1: Pick a point on the top of the object and draw two incident rays traveling towards the mirror. Step 2: Once these incident rays strike the mirror, reflect them according to the two rules of reflection for convex mirrors. Step 3: Locate and mark the image of the top of the object.

10 Step 4: Repeat the process for the bottom of the object. Mirror Equation: - ray diagrams drawn to scale are useful for determining the location and size of the image formed by a mirror, but for accurate descriptions of an image, more analytical techniques are needed - the mirror equation gives the image distance if the object distance and the focal length of the mirror are known - the mirror equation is: d d f o i Magnification: - the magnification (m) of a mirror is the ratio of the image height (h i ) to the object height (h o ) - if the image height is less than the object height, (m) is less than one image height h d m = = = - object height h d - the value of m is positive if the image is upright and negative if the image is inverted i o i o

11 Summary of Sign Conventions: Lenses: Object Distance: - d o is positive if the object is in front of the mirror (real object) - d o is negative if the object is behind the mirror (virtual object) Image Distance: - d i is positive if the image is in front of the mirror (real image) - d i is negative if the image is behind the mirror (virtual image) Focal Length: - f is positive for a concave mirror - f is negative for a convex mirror Magnification: - m is positive for an image that is upright with respect to the object - m is negative for an image that is inverted with respect to the object - lenses are the most widely used optical devices; this is true not even considering the fact that we see the world through a pair of lenses in our eyes - human made lenses date back to antiquity where the use of "burning-glasses" to start fires was common

Rock Crystal Lenses - one of the first direct refernces to lenses was by a Greek playwright named Aristophanes in a play he

- eyeglasses were made from lenses as early as the 1200's; in the 1700's, lenses were used in combination to create a

telescope famous through his ability to construct quality telescopes and also through his astronomical observations - the

12 Rock Crystal Lenses - one of the first direct refernces to lenses was by a Greek playwright named Aristophanes in a play he wrote in 423 B.C. - eyeglasses were made from lenses as early as the 1200's; in the 1700's, lenses were used in combination to create a telescope - contrary to popular belief, Galileo did not invent the telescope, although he was the first person to make the telescope famous through his ability to construct quality telescopes and also through his astronomical observations - the telescope was invented in 1608 by Hans Lippershey; Lippershey was refused a patent and Galileo copied Lippershey's idea a year later - lenses have continued to be used in modern day equipment such as cameras, camcorders, and binoculars

$Types of Lenses: - a lens is a refracting device made of transparent material such as glass or plastic that reconfigures an$ $that is thicker at its center is called a convex lens - convex lenses are also called converging lenses because they refract$

13 Types of Lenses: - a lens is a refracting device made of transparent material such as glass or plastic that reconfigures an incoming energy distribution - a lens has two faces; the faces may be part of a sphere (concave and convex), or flat - a lens that is thicker at its center is called a convex lens - convex lenses are also called converging lenses because they refract light rays that travel parallel to the principle axis toward the axis - concave lenses are thinner in the middle than at the edges

$- concave lenses are also called diverging lenses because they cause incident parallel rays to diverge after exiting the lens Convex Lenses: - when light passes through a lens, refraction occurs at$

14 - concave lenses are also called diverging lenses because they cause incident parallel rays to diverge after exiting the lens Convex Lenses: - when light passes through a lens, refraction occurs at the two lens surfaces - in order to simplify our calculations, we will consider all refraction to occur at only one plane that passes through the center of the lens - this is known as the "thin-lens" model

15 - as we did for mirrors, we will define a point called the focal point (F); rays parallel to the principal axis will pass through the focal point after being converged by a convex lens - the distance from the focal point to the lens is called the focal length (f) - you can also say that the focal length is the image distance that corresponds to an infinite object distance - lenses have two focal points because light rays can pass through either side of the lens - to find the location of an image, we use the lens/mirror equation: d d f - in order to find the magnification of an image, we use the magnification equation: image height hi d m = = = - i object height h d - we use the following lens/mirror equation conventions: o i Object Distance: - d o is positive if the object is in front of the mirror (real object); positive on the object side of the lens - d o is negative if the object is behind the mirror (virtual object); negative on the image side of the lens Image Distance: - d i is positive if the image is in front of the mirror (real image); positive on the image side of the lens - d i is negative if the image is behind the mirror (virtual image); negative on the object side o o

16 Focal Length: - f is positive for a concave mirror; positive for convex lenses - f is negative for a convex mirror; negative for concave lenses Magnification: - m is positive for an image that is upright with respect to the object - m is negative for an image that is inverted with respect to the object Ray Diagrams for Thin-Lenses: - ray diagrams are used to determine the image formed by a single lens or a sytem of lenses - they are also used to clarify the above sign conventions - there are three steps in drawing ray diagrams for lenses - the first step is to draw a ray parallel to the principle axis; after being refracted by the lens, this ray passes through one of the focal points

known as diverging lenses - the focal length of concave lenses are negative - the images created by diverging lenses are virtual, erect, and

17 - the second step is to draw a ray through the other focal point upon which it emerges in a direction parallel to the principal axis - the third ray is drawn through the center of the lens; this ray continues in a straight line Concave Lenses: - remeber, concave lenses are also known as diverging lenses - the focal length of concave lenses are negative - the images created by diverging lenses are virtual, erect, and reduced in size - ray tracings for concave lenses follow the same general rules as for convex lenses but you need to remeber to diverge instead of converge

Lecture Notes (Reflection & Mirrors)

Lecture Notes (Reflection & Mirrors) Intro: - plane mirrors are flat, smooth surfaces from which light is reflected by regular reflection - light rays are reflected with equal angles of incidence and reflection