Chapter 23

Geometrical Optics(lecture 1: mirrors)

Dr. Armen Kocharian

Reflection and Refraction at a Plane Surface

The light radiate from a point object

in all directions

The light reflected from a plane mirror

apperas as a point image

The light refracted from a point object

seems closer

Location of the image formed by a plane mirror

it is virtual image

Reflection and Refraction at a Plane Surface

Reversals in a Flat MirrorThe images P1' and P2' are formed by single reflection

Image P3‘ is formed by double reflection

ImagesImages are always located by extending diverging rays back to a point at which they intersectImages are located either at a point from which the rays of light actuallydiverge or at a point from which they appear to diverge

Types of ImagesA real image is formed when light rays pass through and diverge from the image point

Real images can be displayed on screens

A virtual image is formed when light rays do not pass through the image point but only appear to diverge from that point

Virtual images cannot be displayed on screens

Images Formed by Flat Mirrors

Simplest possible mirrorLight rays leave the source and are reflected from the mirrorPoint I is called the image of the object at point OThe image is virtual

Images Formed by Flat Mirrors, 2

A flat mirror always produces a virtual imageGeometry can be used to determine the properties of the imageThere are an infinite number of choices of direction in which light rays could leave each point on the objectTwo rays are needed to determine where an image is formed

Images Formed by Flat Mirrors, 3

One ray starts at point P, travels to Q and reflects back on itselfAnother ray follows the path PR and reflects according to the law of reflectionThe triangles PQRand P’QR are congruent

Images Formed by Flat Mirrors, 4

To observe the image, the observer would trace back the two reflected rays to P’Point P’ is the point where the rays appear to have originatedThe image formed by an object placed in front of a flat mirror is as far behind the mirror as the object is in front of the mirror

|p| = |q|

Lateral MagnificationLateral magnification, M, is defined as

This is the general magnification for any type of mirrorIt is also valid for images formed by lensesMagnification does not always mean bigger, the size can either increase or decrease

M can be less than or greater than 1

''≡ = =

Image heightObject height

h ymh y

Lateral Magnification of a Flat Mirror

The lateral magnification of a flat mirror is 1This means that h' = h for all images

Reversals in a Flat MirrorA flat mirror produces an image that has an apparentleft-right reversal

For example, if you raise your right hand the image you see raises its left hand

Reversals in a Flat Mirror (cont)Formed image is

VirtualErectReversed (switched right and left)

Reversals, cont.The reversal is not actually a left-right reversalThe reversal is actually a front-back reversal

It is caused by the light rays going forward toward the mirror and then reflecting back from it

Properties of the Image Formed by a Flat Mirror – Summary

The image is as far behind the mirror as the object is in front

|p| = |q|The image is unmagnified

The image height is the same as the object heighth’ = h and M = 1

The image is virtualThe image is upright

It has the same orientation as the objectThere is a front-back reversal in the image

Spherical MirrorsA spherical mirror has the shape of a section of a sphereThe mirror focuses incoming parallel rays to a pointA concave spherical mirror has the silvered surface of the mirror on the inner, or concave, side of the curveA convex spherical mirror has the silvered surface of the mirror on the outer, or convex, side of the curve

Concave Mirror, NotationThe mirror has a radius of curvature of RIts center of curvatureis the point CPoint V is the center of the spherical segmentA line drawn from C to V is called the principal axis of the mirror

Focal LengthWhen the object is very far away, then p → ∞and the incoming rays are essentially parallelIn this special case, the image point is called the focal pointThe distance from the mirror to the focal point is called the focal length

The focal length is ½ the radius of curvature

Paraxial RaysWe use only rays that diverge from the object and make a small angle with the principal axisSuch rays are called paraxial raysAll paraxial rays reflect through the image point

Spherical AberrationRays that are far from the principal axis converge to other points on the principal axisThis produces a blurred imageThe effect is called spherical aberration

Image Formed by a Concave Mirror

Geometry can be used to determine the magnification of the image

h’ is negative when the image is inverted with respect to the object

'h qMh p

= = −

Reflection at Spherical Surface(Concave Mirror)

For small angles αall rays from point P will intersect at P'

The eye sees the rays as coming from point P‘

Graphical Methods for Mirrors(Concave and Convex)

Principle-ray diagramsParallel ray to the axes passes trough FA ray through F reflected parallel to the axesA ray along the radius reflected backA ray to the vertex V is reflected forming equal angles with the optic axis

Graphical Methods for Mirrors(Concave and Convex)

Principle-ray diagramsParallel ray to the axes passes trough FA ray through F reflected parallel to the axesA ray along the radius reflected backA ray to the vertex V is reflected forming equal angles with the optic axis

Notation for Mirrors and Lenses

The object distance is the distance from the object to the mirror or lens

Denoted by pThe image distance is the distance from the image to the mirror or lens

Denoted by qThe lateral magnification of the mirror or lens is the ratio of the image height to the object height

Denoted by M

Image Formed by a Concave Mirror

Geometry also shows the relationship between the image and object distances

This is called the mirror equationIf p is much greater than R, then the image point is half-way between the center of curvature and the center point of the mirror

p → ∞ , then 1/p ≈ 0 and q ≈ R/2

1 1 2p q R

+ =

Focal Point, cont.The colored beams are traveling parallel to the principal axisThe mirror reflects all three beams to the focal pointThe focal point is where all the beams intersect

It is the white point

Focal Point and Focal Length, cont.

The focal point is dependent solely on the curvature of the mirror, not on the location of the object

It also does not depend on the material from which the mirror is made

ƒ = R / 2The mirror equation can be expressed as 1 1 1

ƒp q+ =

Focal Length Shown by Parallel Rays

Convex MirrorsA convex mirror is sometimes called a diverging mirror

The light reflects from the outer, convex sideThe rays from any point on the object diverge after reflection as though they were coming from some point behind the mirror The image is virtual because the reflected rays only appear to originate at the image point

Image Formed by a Convex Mirror

In general, the image formed by a convex mirror is upright, virtual, and smaller than the object

Sign ConventionsThese sign conventions apply to both concave and convex mirrorsThe equations used for the concave mirror also apply to the convex mirror

Ray DiagramsA ray diagram can be used to determine the position and size of an imageThey are graphical constructions which reveal the nature of the imageThey can also be used to check the parameters calculated from the mirror and magnification equations

Drawing a Ray DiagramTo draw a ray diagram, you need to know:

The position of the objectThe locations of the focal point and the center of curvature

Three rays are drawnThey all start from the same position on the object

The intersection of any two of the rays at a point locates the image

The third ray serves as a check of the construction

The Rays in a Ray Diagram –Concave Mirrors

Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected through the focal point, FRay 2 is drawn from the top of the object through the focal point and is reflected parallel to the principal axisRay 3 is drawn through the center of curvature, C, and is reflected back on itself

Notes About the RaysThe rays actually go in all directions from the objectThe three rays were chosen for their ease of constructionThe image point obtained by the ray diagram must agree with the value of qcalculated from the mirror equation

Ray Diagram for a Concave Mirror, p < f

The object is between the mirror surface and the focal pointThe image is virtualThe image is uprightThe image is larger than the object (enlarged)

The Rays in a Ray Diagram –Convex Mirrors

Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected away from the focal point, FRay 2 is drawn from the top of the object toward the focal point and is reflected parallel to the principal axisRay 3 is drawn through the center of curvature, C, on the back side of the mirror and is reflected back on itself

Ray Diagram for a Concave Mirror, p > R

The center of curvature is between the object and the concave mirror surfaceThe image is realThe image is invertedThe image is smaller than the object (reduced)

Ray Diagram for a Concave Mirror, p < f

The object is between the mirror surface and the focal pointThe image is virtualThe image is uprightThe image is larger than the object (enlarged)

The Rays in a Ray Diagram –Convex Mirrors

Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected away from the focal point, FRay 2 is drawn from the top of the object toward the focal point and is reflected parallel to the principal axisRay 3 is drawn through the center of curvature, C, on the back side of the mirror and is reflected back on itself

Ray Diagram for a Convex Mirror

The object is in front of a convex mirrorThe image is virtualThe image is uprightThe image is smaller than the object (reduced)

Application – Day and Night Settings on Auto Mirrors

With the daytime setting, the bright beam of reflected light is directed into the driver’s eyesWith the nighttime setting, the dim beam of reflected light is directed into the driver’s eyes, while the bright beam goes elsewhere

Proof of Mirror Equation: Reflection at Spherical Surface(Concave Mirror).

Φ = += Φ ++ = Φ2

α θβ θα β

'

=

=

Φ =

hαshβshR

'tan =−hβ

s δtan =

−hα

s δ

'+ =1 1 2s s R

tanΦ =−h

R δ

Focal Point and Focal LengthFocal; length of a spherical mirror

f - focus

Object placed in focal point s=R/2

=2Rf

' ,+ =2 1 2R s R

'⇒ + =1 1 1s s f

' ,+ =∞1 1 2

s R

' = ∞s

f>0 - focus is positive

Image of Extended ObjectLateral magnification

f - focus

Lateral magnificationfor a plane mirror

',=y y⇒ = +1m m

' ' ,= = −1 y sm y s

'= −s s

Image Formation (Ex: 34.1s= 10 cm, s’=300 cm

f - focus (or R)

Lateral magnificationfor a plane mirror

' ' ,.

= = − = − = −1 300cm 30

100cmy s

m y s

+ =1 1 2

10cm 300cm R

Reflection at Spherical Surface(Convex Mirrors)

For small angles αall rays from point P will intersect at P'

The eye sees the rays as coming from point P‘

f<0 - focus is negative

Reflection at Spherical Surface(Convex Mirrors)

For small angles αall rays from point P will intersect at P'

The eye sees the rays as coming from point P‘

'+ =1 1 2s s R

' '= =

y smy s

Notation for Mirrors and Lenses

The object distance is the distance from the object to the mirror or lens

Denoted by pThe image distance is the distance from the image to the mirror or lens

Denoted by qThe lateral magnification of the mirror or lens is the ratio of the image height to the object height

Denoted by M

Spherical MirrorsA spherical mirror has the shape of a section of a sphereThe mirror focuses incoming parallel rays to a pointA concave spherical mirror has the silvered surface of the mirror on the inner, or concave, side of the curveA convex spherical mirror has the silvered surface of the mirror on the outer, or convex, side of the curve

Concave Mirror, NotationThe mirror has a radius of curvature of RIts center of curvatureis the point CPoint V is the center of the spherical segmentA line drawn from C to V is called the principal axis of the mirror

Paraxial RaysWe use only rays that diverge from the object and make a small angle with the principal axisSuch rays are called paraxial raysAll paraxial rays reflect through the image point

Image Formed by a Concave Mirror

Geometry can be used to determine the magnification of the image

h’ is negative when the image is inverted with respect to the object

'h qMh p

= = −

Image Formed by a Concave Mirror

Geometry also shows the relationship between the image and object distances

This is called the mirror equationIf p is much greater than R, then the image point is half-way between the center of curvature and the center point of the mirror

p → ∞ , then 1/p ≈ 0 and q ≈ R/2

1 1 2p q R

+ =

Focal LengthWhen the object is very far away, then p → ∞and the incoming rays are essentially parallelIn this special case, the image point is called the focal pointThe distance from the mirror to the focal point is called the focal length

The focal length is ½ the radius of curvature

Focal Point and Focal Length, cont.

The focal point is dependent solely on the curvature of the mirror, not on the location of the object

It also does not depend on the material from which the mirror is made

ƒ = R / 2The mirror equation can be expressed as 1 1 1

ƒp q+ =

Convex MirrorsA convex mirror is sometimes called a diverging mirror

The light reflects from the outer, convex sideThe rays from any point on the object diverge after reflection as though they were coming from some point behind the mirror The image is virtual because the reflected rays only appear to originate at the image point

Image Formed by a Convex Mirror

In general, the image formed by a convex mirror is upright, virtual, and smaller than the object

Sign ConventionsThese sign conventions apply to both concave and convex mirrorsThe equations used for the concave mirror also apply to the convex mirror

Sign Conventions, Summary Table

Ray DiagramsA ray diagram can be used to determine the position and size of an imageThey are graphical constructions which reveal the nature of the imageThey can also be used to check the parameters calculated from the mirror and magnification equations

Drawing a Ray DiagramTo draw a ray diagram, you need to know:

The position of the objectThe locations of the focal point and the center of curvature

Three rays are drawnThey all start from the same position on the object

The intersection of any two of the rays at a point locates the image

The third ray serves as a check of the construction

The Rays in a Ray Diagram –Concave Mirrors

Ray 1 is drawn from the top of the object parallel to the principal axis and is reflected through the focal point, FRay 2 is drawn from the top of the object through the focal point and is reflected parallel to the principal axisRay 3 is drawn through the center of curvature, C, and is reflected back on itself

Notes About the RaysThe rays actually go in all directions from the objectThe three rays were chosen for their ease of constructionThe image point obtained by the ray diagram must agree with the value of qcalculated from the mirror equation

Sign Conventions, Summary Table