Image formation Ray tracing
Calculation
Lecture 17.
Lenses Convex
Concave
Optical instruments
Mirrors Convex
Concave
Image formation
Lenses
Convex (converging) lens
Laws of refraction and reflection can be used
to explain how lenses and mirrors operate
Parallel rays (e.g. from the Sun) passing
through a convex lens
Sunlight focused by magnifying glass may
burn hole in paper placed at focal point F.
f
F F: focal point
f: focal length
Example
A farsighted person requires an eyeglass of
strength 2.5 diopters. What is the focal length
of the eyeglass lens?
D =2.5 f = 1 D
=1/2.5 = 0.4 m = 40cm.
F is the focal point.
f is the focal length, an important characteristic
f
F
Image formation
Convex lens
1
f Strength in diopters (D) = ( f is in metres)
Power or strength of the lens
Optic axis
Focal length f = 40 cm.
F
F
s s′
h
h′
Image Formation (ray tracing)
Ray 1 entering lens parallel to optic axis will
exit and pass through the focal point
Ray 2 passing through the focal point will exit
the lens and travel parallel to optic axis
Ray 3 will undergo only a small deviation
(not shown) (thin lens)
3
Rays reversible
Real inverted image formed
Real image (may be projected and displayed
on a screen)
Object at a distance greater than the focal
length from convex
lens
2
1
Optic axis
F F s
s′
h
h′
Image Formation (ray tracing)
Image
• virtual
•upright
•magnified.
Object at a distance less than the focal
length from the lens
Simple magnifier
Convex lens
Concave (diverging) lens
F
Dashed lines indicate the direction from which
the rays appear to come
f
Rays entering lens parallel to axis appear to
originate at focal point
Optic
axis
Concave (diverging) lens
s
s′ F
h
h′
Virtual image always produced by concave lens
Cannot be viewed on screen since rays are
diverging on the right of the lens
However can be viewed with the eye since the
eye converges the rays onto the retina.
Object outside the focal length
Optic
axis
Ray diagrams are useful in sketching the
relationship between object and image
Relationship may also be calculated
F
F
s s′
h
h′
A
B O
C
D
Triangles AOB and
DOC are similar '
'
h h
s s ' 'h s f
h f
1 1 1
's s f
E
Triangles EFO and
DCF are similar
''h s
h s
'
'
hh
f s f
Image Formation---calculation
' 's s f
s f
1 1 1
's s f
Thin lens formula
Object distance s
positive if object is in front of lens
negative if object is behind lens
Image distance s′
positive if image is formed behind the lens (real)
negative if is formed in front of the lens (virtual)
Focal length f
•positive -- convex lens
•negative --concave lens
Image ---calculation
F
s s′
h
h′ B
F
F
s s′
h
h′ B
F
'hM
h
Magnification is defined as
M Negative :
inverted image
M Positive :
upright image
'sM
s or
Magnification
Simple magnifier
Object placed inside focal length of converging
lens;
image viewed
• virtual,
• magnified
• upright
.
F F s
s′
h
h′
Example
An object 0.5 cm in height is placed 8 cm from a
convex lens of focal length 10 cm. Determine the
(a) position, (b) magnification, (c) orientation and
(d) height of the image. 1 1 1
's s f
1 1 1
's f s
1 1 1
' 10 8s s′ =
10 x 8
8-10 = - 40cm.
' 405
8
s cmM
s cm
h′ = M x h = 5 x 0.5cm = 2.5cm 'h
Mh
M +ve
image upright
F F s
s′
h
h′
(a)
(c) (b)
(d)
An object is placed 45 cm from a lens of focal
length -25 cm. Determine the position,
magnification, and orientation of the image.
1 1 1
's s f
1 1 1
's f s
M +ve image upright
Example
1
S’
1
-25 cm
1
45 cm = - S’ = -16.1 cm
M = - S’ S M = -
-16.1 cm 45 cm
= 0.36
s
s′ F
h
h′ Optic
axis
Effective focal length (feff) of combination of a
number of thin lenses close together
1 2
1 1 1......
efff f f
Effective strength (Seff) of combination of a
number of thin lenses close together
1 2 .....effS S S
Combining Lenses
Determine the combined strength of a thin convex lens
and a thin concave lens placed close together if their
respective focal lengths are 10cm and -20cm.
1
f Strength S, in diopters (D) = ( f is in metres)
1 2
1 1 1eff
eff
Sf f f
1 1
.1 .2effS
m m
10 5 5effS diopters
Mirrors Flat Mirror
Concave Mirror
Convex Mirror
Curved mirrors are analogous to lenses
Ray tracing and thin lens equation also valid.
real and virtual images are also formed
Image, upright, virtual
Object and image distance equal
Flat Mirror
d d
object image
Object and image same size
Spherical Mirrors
Spherical mirror
C
R
Hollow sphere
Spherical mirror is a section of hollow sphere
Principal or optic axis
Radius of curvature R = 2f
Curved Mirrors (Spherical)
F
f
f F
Concave mirror
(converging)
Convex mirror
(diverging)
Thin lens formula may be used to determine
object and image distances and focal lengths etc
Real image : inverted (h′ negative),
positive image distance s′
Virtual image: upright (h′ positive),
negative image distance s′
Positive focal length
Negative focal length
Lenses and mirrors
Mirrors
Concave shaving/makeup mirrors
C
F
Image is virtual, upright and enlarged.
Object placed at distance < f from mirror
Application: searchlight
C is the centre of curvature
Mirrors Example
An object is positioned 5 cm in front of a
concave mirror of focal length 10 cm.
Determine the location of its image and its
characteristics.
1 1 1
's s f
s = 5 cm
f = 10 cm 1 1 1
's f s
1 1 1
' 10 5s cm cm
1 1 2 1
' 10 10 10s cm cm cm S’ = -10cm
Characteristics.
•Image virtual
•Located behind mirror
'sM
s
10
5
2
cmM
cm
M
Optical instruments
System may have many optical elements
(example, lenses and mirrors)
Microscopes, telescopes, cameras etc
Thin lens formula or ray tracing may be
used to analyse behaviour of such systems
Simple compound microscope
two convex lenses
Fo
h′′
Fe
Objective
lens
h′
Fo
Object (height h)
Final image
Fe
eyepiece
image formed by objective lens is
inside focal length of eyepiece lens.
Optical instruments
Dental loupes Important
Characteristics
•Resolution*
•Field width *
•Field depth
•magnification
Resolution
•ability to see fine detail
Field width Size of operating site when viewed through loupe
Function of lens system diameter and magnification
Field depth Depth or range of focus
Depends on, available light, optical design,
magnification and accommodation
Magnification,
important but not at the expense of resolution.
Large fuzzy image of little use.
Multiple lenses
Optical instruments
Dental loupes Galilean Design
objective
eyepiece Operating
site
Typically Magnification
m ≈ 2.5 → 4.5 Optical design allows
observer focus at infinity
thereby relieving eyestrain
Typical
working distance
28-38 cm
Simple refracting telescope
Fe
Objective lens eyepiece
Fo,
Optical instruments
Virtual image at infinity,
Magnified and inverted
Objective lens forms image (real, inverted) at
focal point F0 which is also the focal point Fe of the
eyepiece; virtual image is then formed at infinity.
0
e
fM
f
f0 fe
Example
Eyepiece
lens
Objective,
concave mirror
Flat mirror
Simple reflecting telescope
Optical instruments
Effective focal length of the objective in the
Hubble telescope is 57.8 m. What focal length
eyepiece is required to give a magnification of
-8.0 x 103.
0
e
fM
f fe = -Mf0 = -(-8.0 x 10
3) x 57.8 m
=7.23 x10-3 m
An object of height 3 cm is positioned 40 cm from
a concave lens with a focal length of -20 cm.
Determine the position of the image, its
magnification, height, and orientation.
1 1 1
's s f
'sM
s
1 1 1
's f s
1 1 1 3
' 20 40 40s
13.33 1
40 3M
S =40 cm (object in front of lens)
f = -20cm (concave lens)
S′ = -13.33 cm
(Image upright)
'
1' 3.0 1.0
3
hM
h
h Mh cm cm
Question 2.
s
s′ F
h
h′
Virtual image always produced by a concave lens
Object outside the focal length
Optic
axis
An object of height 3 cm is positioned 40 cm from
a concave lens with a focal length of -20 cm.
Determine the position of the image, its
magnification, height, and orientation.
Question 2.
Camera
Optical instruments
CCD array
Real, inverted image formed on CCD array
Lens translated to change image distance S’ to
adjust for different object distances S.
Focal length of lens is fixed.
aperature
1 1 1
's s f
Endoscope for medical investigations—
inserted through small incision or orifice to
inspect and facilitate operation on interior parts
of the body
flexible shaft includes:
•light source to illuminate area,
•image channel to view area under investigation,
•air or water conduit to clear debris,
•instrument conduit
Endoscope
Typical endoscope eyepiece
Transmits light, air , water
Flexible shaft
Instrument entry
Optical instruments
Optical fibre: Total Internal Reflection
Optical instruments
Optical fibre: Total Internal Reflection
Applications
Optical communications:
Image transport, Coherent fibre bundle
Optical fibres used to transmit modulated
laser beams; carrying information
Telephone and internet communications
Rate at which information can be transported
proportional to frequencyof light
Single fibre: many millions of phone
conversations simultaneously.
Cable has many fibres