Geometric optics

Geometric Optics

Mirrors, lenses, light, and image formation

Geometric Optics

Understanding images and image formation, ray model of light, laws of reflection and refraction, and some simple geometry and trigonometry

REFLECTION AND REFRACTION AT A PLANE SURFACE

Reflection and refraction on plane mirrors

Key terms

• Anything from which light rays radiate–Object

• Anything from which light rays radiate that has no physical extent– Point object

• Real objects with length, width, and height– Extended objects

Key terms

Specular reflection

Reflection on a plane surface where reflected rays are in the same directions

Diffused reflection

Relfection on a rough surface

Key terms

Virtual image

Image formed if the outgoing rays

don’t actually pass through the

image point

Real image

Image formed if the outgoing rays actually pass through

the image point

Image formation by a Plane mirror


Ray Diagrams

• a diagram that traces the path that light takes in order for a person to view a point on the image of an object

Line of Sight

Principle

• suggests that in order to view an image of an object in a mirror, a person must sight along a line at the image of the object.

Reflection at a Plane Surface


V

θ

θ

θ θ

s s’M M’


• M is the object and M’ is the virtual image

• Ray MV is incident normally to the plane mirror and it returns along its original path

• s= object distance• s’= image distance• s=-s’


• Sign rulesFor the object distance:–When the object is on the same side of the

reflecting or the refracting surface as the incoming light, s is positive

For the image distance:–When the image is on the same side of the

reflecting or the refracting surface as the outgoing light, s’ is positive

Image of an extended object

V’

V

θθ

θ θ

s s’

y

M M’

Q Q’

θ

y’

Image of an extended object

• Lateral magnification–Ratio of image height to object

height–M=y’/y

• Image is erect• m for a plane mirror is always +1• Reversed means front-back

dimension is reversed

REFLECTION AT A SPHERICAL SURFACE

Reflection on Concave and Convex mirrors

Reflection at a Concave Mirror

P P’CV


• Radius of curvatureR• Center of curvature• The center of the sphere

of which the surface is a part

C

• Vertex• The point of the mirror

surfaceV

• Optic axisCV

IMAGE FORMATION ON SPHERICAL MIRRORS

Graphical Methods for Mirrors

Graphical Method

• Consists of finding the point of intersection of a few particular rays that diverge from a point of the object and are reflected by the mirror

• Neglecting aberrations, all rays from this object point that strike the mirror will intersect at the same point

Graphical Method

• For this construction, we always choose an object point that is not on the optic axis• Consists of four rays we can

usually easily draw, called the principal rays

Graphical MethodA ray parallel to the axis,

after reflection passes through F of a concave

mirror or appears to

come from the (virtual) F of a convex mirror

A ray through (or

proceeding toward) F is

reflected parallel to the

axis

A ray along the radius through or away from C intersects the

surface normally and is reflected back

along its original pathA ray to V is

reflected forming equal angles with

the optic axis

Object is at F

Object is between F and Vertex

Object is at C

Object is between C and F

Positions of objects for concave mirrors


If α dec, θi is nearly parallel

Rays nearly parallel or close to R Paraxial rays


If α inc, P’ is close to V Image is smeared out Spherical Aberration


FCV

s at infinity s’= R/2


• All reflected rays converge on the image point• Converging mirror• If R is infinite, the mirror

becomes plane

Reflection at a Concave MirrorThe incident parallel rays converge after reflecting from the

mirror

They converge at a F at a distance

R/2 from V

F is Focal point, where the rays are brought to focus

f is the focal length, distance from the vertex to the focal

point

f= R/2


FCV

s’ at infinity s= R/2


The object is at the focal

points=f=R/2

1/s +1/s’= 2/R 1/s’=0; s’ at infinity

1/s+ 1/s’= 1/fObject image relation, spherical mirror

Image of an Extended Object

m= y’/y Lateral magnification

m= y’/y= -s’/s

Lateral magnification for spherical mirrors

Example• A concave mirror forms an image, on a wall

3.00m from the mirror, of the filament of a headlight lamp 10.0cm in front of the mirror.

a. What is the radius of curvature and focal length of the mirror?

b. What is the height of the image if the height of the object is 5.00mm?

c. R=19.4cm; f= 9.7cm; m= -30.0; y’= 150mm

Example

• An object, 1cm high, is 20cm from the vertex of a concave mirror whose radius of curvature is 50cm. Compute the position and size of the image. Is it real or virtual? Upright or inverted?• s’=-100cm; y’=5cm, m is pos., erect,

virtual

Reflection at a Convex Mirror

F C

s or s’ at infinitys’ or s= R/2

Image formation on spherical mirrors

• Sign rulesFor the object distance:–When the object is on the same side

of the reflecting or the refracting surface as the incoming light, s is positive; otherwise, it is negative


• Sign rulesFor the image distance:–When the image is on the same side

of the reflecting or the refracting surface as the outgoing light, s’ is positive; otherwise, it is negative


• Sign rules:For the radius of curvature of a

spherical surface:–When the center of curvature C is

on the same side as the outgoing light, the radius of curvature is positive, otherwise negative

Reflection at a Convex Mirror

• The convex side of the spherical mirror faces the incident light

• C is at the opposite side of the outgoing rays, so R is neg.

• All reflected rays diverge from the same point

• Diverging mirror

Reflection at a Convex MirrorIncoming rays are

parallel to the optic axis and are not

reflected through F

Incoming rays diverge, as though they had come from point F behind the mirror

F is a virtual focal point

s is positive, s’ is negative

Example • Santa checks himself for soot, using his

reflection in a shiny silvered Christmas tree ornament 0.750m away. The diameter of the ornament is 7.20cm. Standard reference work state that he is a “right jolly old elf,” so we estimate his height to be 1.6m. Where and how tall is the image of Santa formed by the ornament? Is it erect or inverted?

• s’= -1.76cm; m= 2.34x10-2 ; y’= 3.8cm

REFRACTION AT A SPHERICAL SURFACE

Refraction at spherical interface

Refraction at a Spherical Surface

VC

Refraction at a Spherical Surface

na/s + nb/s’= (nb-na)/R

Object-image relation, spherical refracting surface

m=y’/y= -(nas’/nbs)

Lateral magnification, spherical refracting

surface

na/s + nb/s’=0At a plane refracting

surface

Example

A cylindrical glass rod in air has an index of refraction 1.52. and one end is ground to a hemispherical surface with radius R=2.00cm. a.) find the image distance of a small object on the axis of the rod, 8cm to the left of the vertex. b.) find the lateral magnification.

n=1.00

s’= 11.3cm

m= -0.929

ExampleA ray of light in air makes an angle of incidence of 45° at the surface of a sheet of ice. The ray is refracted within the ice at an angle of 30°A. What is the critical angle for the ice?B. A speck of dirt is embedded 3/4in below the surface of the ice. What is it’s apparent depth when viewed at normal incidence?

A. θcrit= 45°B. s’= -0.53in or 0.53in below

GRAPHICAL METHOD FOR LENSES

Biconcave and biconvex thin lenses

Lenses

LensesBiconvex lens; converging

Biconcave lens; diverging

LensesOnly F is

needed for the ray

diagram

Chief ray through the

center is undeviatedRay parallel is

refracted in such a way that it goes through

F on transmission

through the lensFocal ray is parallel to the axis of

transmission

For concave lens, the rays

appear to have passed through F on the object’s side of the lens

Lenses

An object is placed 30cm from a biconcave lens with a focal length of 10cm. Determine the image characteristics graphically.

ANALYTICAL METHOD FOR THIN LENSES

Lens maker's equation

Equations for thin lenses

1/s + 1/s’= 1/f Object-image relation, thin lenses

m=y’/y= -s’/sLateral

magnification, thin lenses

1/f=(n-1) [(1/R1)- (1/R2)]Lensmaker’s equation

Example• A biconvex lens is made of glass with n

= 1.65 and has radii of curvature R=R1=R2= 42 cm. Determine its focal length.

• f=32cm• A biconcave lens is made of glass with

n = 1.65 and has radii of curvature R=R1=R2= 42 cm. What is its focal length?

• f=-32cm

Example

An object is placed 30cm from a biconcave lens with a focal length of 10cm. Determine the image characteristics analytically.

s’=-7.5cmm=0.25

Virtual, upright, half the size of the object

Geometric optics

Education

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