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x
m.
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Displacement, velocity and acceleration
in simple harmonic motion
Displacement
a .sin ( ω .t
φ)
t
Velocity
a .ω .cos ( ω .t
φ)
t
Acceleration
1
2
a .ω .sin ( ω .t
φ)
1
0
t
10
Displacement lags behind velocity by π/2 radians, and
lags behind acceleration by π radians.
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Adding phasors
Two phasors, initially φ out of phase, and after
rotating through θ = ωt
θ
φ
time=0, θ= 0
PLUS
θ
Later time, phase change = θ
φ
θ
Resultant - unit initial amplitudes
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Addition of Phasors
A
a
ψ
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50
40
vectors with relative phase φ = 0
30
20
Addition of N = 50
50
40
40
40
30
30
20
20
10
10
10
20
vectors with relative phase φ = 1
50
0
10
20
30
40
vectors with
10 relative phase φ = 2
30
Addition of N = 50
deg
50
50
50
deg
Addition of N = 50
vectors with relative phase φ = 10
50
20
50
40
30
40
30
40
30
20
50
20
10
10
10
0
10
20
30
40
50
50
40
30
20
10
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40
30
20
10
0
10
20
30
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40
50
10
20
30
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10
20
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10
10
20
20
30
30
40
40
50
50
vectors with relative phase φ = 10
deg
500
400
300
200
100
500
400
300
200
100
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100
200
300
400
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100
200
300
400
Beats
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9.9 , 9.8 .. 9.9
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Two equal-amplitude sinusoids combine to give a resultant which is a
product of a carrier and an envelope. The frequencies of these are the
half-sum and half-difference of the original frequencies.
The beat frequency, the frequency at which the amplitude varies, is the
difference between the original frequencies.
10
sin( t .π )
8
sin( 1.3 .t .π )
sin( t .π )
2
sin( 1.3 .t .π )
2 .cos( 0.15 .t .π )
6
6
6
4
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9
2
0
2
10
5
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5
10
t
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t
Two equal-amplitude sinusoids combine to give a resultant which is a
product of a carrier and an envelope. The frequencies of these are the
half-sum and half-difference of the original frequencies.
The beat frequency, the frequency at which the amplitude varies, is the
difference between the original frequencies.
Addition of Phasors Unequal Amplitudes
b
b
R
φ
ψ−
ψ θ
φ
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A General Wave Disturbance
c
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0.6
0.4
0.2
0
1
0
1
2
3
4
5
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Fourier Series
ceil
Consider the Fourier Series for a square wave sq ( x , L )
n
. .
. .
4
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ser ( n , x , L )
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π .( 2 . p 1 )
p=0
( 1)
x
2 .x
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1.5
1
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ser ( 0 , x , 1 ) 0.5
ser ( 1 , x , 1 )
0
ser ( 2 , x , 1 )
ser ( 3 , x , 1 )
0.5
1
1.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
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Close-up:
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1.8
2
1.0001 , 1.0005 .. 1.1
1.5
sq ( x , 1 )
1
ser ( 30 , x , 1 )
ser ( 100 , x , 1 ) 0.5
0
1
1.02
1.04
1.06
1.08
1.1
x
The Gibbs phenomenon is the concentration of the 'overshoot' of
the sudden rise into a region close to the edge of the step.
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Sinusoidal Waves
A single-frequency wave may be written as
f ( x , t ) A . cos ( k .x ω . t )
where
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1
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4
5
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4
5
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Typical variation of refractive index
Frequency
Refractive index
Absorption
Limiting value = 1
Infrared
Visible
Ultraviolet
X-ray
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Two Superposed Waves
a . cos ω 1 . t
y ω 1,ω 2,k 1,k 2,x,t
k 1 .x
a . cos ω 2 . t
k 2 .x
Take the angular frequencies to be 10 .π and 11.5 . π ,
wavevectors to be 10 . π and 12 . π
2
1
0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
1
2
time t=0
2
1
0.2
0
0.2
0.4
0.6
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1.4
1.6
1.8
1
2
time t=0.15
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04/02/99 12:29
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W avelength dependence of selected optical m aterials,
all show ing norm al dispersion
( refactive index decreasing w ith increasing wavelength)
d e no te s visib le re g ion
D en se flin t g la ss
R e fr a c tiv e in d e x
1.7
1.6
L ig h t flin t g lass
C ry sta l q u a rtz
1.5
B o ro silic ate cro w n g la ss
A cry lic p la stic
V itre o u s q u a rtz
1.4
0 .0 0
0 .4 0
0 .8 0
W a v e len gth (m ic r o n s)
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k .γ
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ρ
k .ρ .
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vg
2
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1
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0.001 . m , 0.002 . m .. .2 . m
2 .π
1
λ
0.8
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0.6
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0.4
0.2
0
0
0.05
0.1
0.15
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θ
MassesT m spaced aT apart on a string under tension T
are displaced sideways. The displacement of mass r is
yr, for which the equation of xmotion is
2a
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y r+1
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04/02/99 15:38
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Waves on a beaded string
y( r , t , k)
A .cos ( k .r .a
ω .t )
2 .π
k
10 . a
1
0
1
0
k
2
4
6
8
10
12
14
16
18
20
2
4
6
8
10
12
14
16
18
20
2 .π
6 .a
1
0
1
0
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0
1
0
2
4
6
8
10
12
14
16
18
20
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Derivation of Equation of Motion
for String under Tension
An element of length ds (approximately
equal to dx) is acted upon by the
tension T at an angle θ at x and θ+dθ at
x+dx.
Displacement
y
beadxmp.mcd
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04/02/99 16:27
T
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θ
T
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dx+d ξ
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07/02/99 22:14
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Jet engine
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140
120
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100
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soundsdb.mcd
1
08/02/99 14:25
63Y onP k1
ω.
Z1
ρ 1 .T
ω.
k2
T
Z2
i . ω .t
yi ( x , t)
T
ρ 2 .T
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e
i . ω .t
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i . ω .t
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rr
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Z1
Z2
Z1
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Re ( yr ( x , t) ) , Re ( yt ( x , t) ) )
displacement, y
1
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0
0.5
1
3
2.5
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0
0.5
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i . ω .t
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i . ω .t
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ddy ( x , t )
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1
stringjn.mcd 12/02/99 12:24
1
Re ( dyr ( x , t) ) , Re ( dyt ( x , t) ) )
if ( x < 0 , Re ( ddyi ( x , t) )
Re ( ddyr ( x , t) ) , Re ( ddyt ( x , t) ) )
gradient, dy/dx
5
0
5
10
0.1
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
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position, x
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0
50
100
150
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2
12/02/99
´3µm© nPªÂ
12:25
Losses in a multilens system
•Ã–—m˜ ÄQÅ ‹
‚zÆ –„Ç.¤žj˜ Ç È ž€œ˜GɗmÊ
Assume that 96% of the light energy is transmitted through
each interface:
a lens involves two interfaces (front and back)
n
1 , 2 .. 8
1
0.9
( .96 )
2 .n
0.8
0.7
0.6
0.5
1
2
3
4
5
6
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Untitled 15/02/99 10:45
´3µËÙ ª ×
•Ã–—m˜ Ä%Ú Û
ÜzÆ –„Ç.¤žj˜ Ç È ž€œ˜GɗmÊÝ ÜÜ
Reflection and transmission paths at a matching layer
Medium 1
Medium 2
Medium 3
1
t12
r12
t12r23
t12r23r21
t12r23t21
t12r23r21r23
t12t23
t12r23r21t23
t12r23r21r23r21
t12r23r21r23t21
t12r23r21r23r21r23
t12r23r21r23r21t23
t12r23r21r23r21r23t21
¥ ¦ §$¨+² ³S´3µØ'¶‘ÌAË ·XÞ ¨+¸w¿N¦&º'¸&³1² ³*ßu³*½*¿N¦ ¹$¼+à¦&¼»½¹A»K¿;¦&¼+§'¶\ṿN³S¿;Ðu»K¿¹$¼+¸wϊ¿NÐ+³V»±kÙ
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Note »that
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change in phase with distance travelled through the
media.
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Z2
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Now the range of wavelengths in the visible is about a
factor of 2
0.03
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1
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1.8
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File version: 17/02/99 09:58
E:\1B24\Lectures\figs07\lenslost.mcd/lens
Assume that 99% of the light energy is transmitted through
each interface:
a lens involves two interfaces (front and back)
n
1 , 2 .. 8
1
0.9
( .99 )
2 .n
0.8
0.7
0.6
0.5
1
2
3
4
5
6
7
8
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à ³²;¦ ³à ¹ÀGÑÒ»K¦&²ZÓR§$¸g»à àZÓ»K¦&²ZÔ"¦&¼A¿;³² ÀÕ»K½³à*°Ö»à;à ¨+±¦&¼+§×$×jºL³² ½³¼A¿¿N²;»¼+à;±¦&à à;¦ ¹$¼i»K¿ ³m»K½ZÐ
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lenslost.mcd 17/02/99 11:25
ñ Ãòómô Āõ Û öø÷ ùCúÇ.ûómü Æ Ãý"Ç þ"Éùjü ÃüÇ ÿómÆ Ç3üXö
ûó*Ãü Wavefront curvature decreases
with distance
from
ð !Ì
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Planes of constant phase perpendicular to k,
which is here directed radially outwards.
Wavecurv.mcd
1
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23/02/99 12:11
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24/02/99 09:55
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2
24/02/99 09:56
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24/02/99 09:57
25/03/99 09:12
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2dstndw.mcd
6
25/03/99 09:28
A Guided Wave System
y
x
Reflecting plane
(k x ,-k y )
a
(k x ,k y )
(k x ,k y )
Reflecting plane
B+A%"%$V(+&ut|'*/U‚>0)Bb4)6eAOƒ@J7O:`2C>+.10)29„J]+6M4)AIHB20[D0VE)Og†BW+)&
Guide1.mcd
1
25/03/99 08:48
"%$V({ghR=
A Guided Wave System
n=1
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propagation
n=2
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propagation
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Nodal
planes
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Guide2.mcd
1
25/03/99 08:50
"%$V({ghR‡
kñ jòómôlN‹Œn ñŽqrZjòò‘Jr3ý“’“”•r3ôeþP–˜— %j úVómüqpûjHsý­ô_r3û
ùÁüÿ ™šr3ô_r3ó*úHr3ý~û
Doppler Effect - moving source
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D0FEJ.J23:`^„#7J4IH>!J„A„!JWdB26MA42=:_4)6e:`^N„!4A79)]@Ÿž':AJ2W+JWqx\J^2W~:J&
Waves spread out behind, compressed in front of,
moving source
Doppler.mcd
1
25/03/99 12:14
"%$VT{ghR=œ
em_wave.nb
1
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Radiation from a Dipole Source
Field in the xy plane from a dipole along the z axis
° ±²³!´d›zi)¯'µyt=T'¶¸·b^´d³`¹)W±B¹3º`±»¼¾½@¹3º!º`´J³!¼À¿M³!»Á ¹ÂW±½Š»)ÃB´„A»)³AÄ9´)¶qº`^+´¸W±½Š»Ã´±B„
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Slice along the x axis
1
0
1
Dipole.mcd
0
10
20
30
1
40
50
25/03/99 14:57
·#i)¯{ghRÏ
Fermat's Principle - Reflection
d
x
θ1
a
θ2
b
θ1
θ2
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Path length is
l
a
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x
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x)
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Fermat.mcd
1
25/03/99 12:47
O
mirror
I'
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The rays from the object O are reflected by the mirror.
The mirror forms a virtual image at I', and the eye forms
a real image at I.
Reflect1.mcd
2
·#i)¯{ghRT
25/03/99 12:51
Fermat's Principle - Refraction
refractive index n 1
a
θ1
d-x
x
θ2
b
refractive index n 2
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l n 1. a
2
x
2
Fermat.mcd
n 2. b
2
(d
x)
2
2
25/03/99 12:48
eye
I'
O
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Formation of a virtual image by refraction at an interface
·#i)¯{g‰t=¯
Refract4.mcd
3
25/03/99 13:12
Mirage effects
COLD - large refractive index
HOT - small refractive index
°±²³!´c›zi)¯+µS3t¶e·b^´%ÁH±B³`¹3²´_´9ߊ´JÄ9ºJ]+¿M»³!ÁC´9Wx=Y³!´9¿M³A¹)Ä9ºA±B»¼¹aºb¹ ÃB¹FY´J³D»)¿Ý^»)ºJ]'ôJ„A„
W´9¼„A´ ¹)±³Jµ
Ray passing from more to less optically dense region is
refracted away from the normal.
Mirage effects - star positions
Mirage1.mcd
1
25/03/99 12:49
zenith apparent
direction incident
of star light ray
from star
air layers
of increasing
refractive index
(optical density)
ground level
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Ray passing from less to more optically dense region is
refracted away from the normal.
Mirage2.mcd
1
25/03/99 12:50
kjH¡¢F£ ¤làn áÝâ§-rG®ã)r%®/r%â£#r 䌪¨«[¬¢F£ ¡qr%âqjH© r%âq«
Young's Slits
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Young's Double Slits
d1
y
k1
S1
d2
r
h
x
k2
S2
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E:\1B24\Lectures\lect21\youngp.mcd/you
Young2.mcd
File version: 03/03/99 09:59
1
25/03/99 16:09
Idealised Young's Slits pattern
Intensity
1
0.5
30
20
10
0
10
20
30
Position on screen (y)
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Realistic Young's Slits pattern
Intensity
1
·#ì'ÓOí‰î=ó
0.5
30
20
10
0
10
20
30
Position on screen (y)
1
youngp.mcd 25/03/99 16:13
Young's Experiment - Alternative Sources
S
Lloyd's single mirror
Fresnel's double mirror
S
Fresnel's biprism
S
Young3.mcd
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1
25/03/99 16:10
·#ì'ÓOí‰îî
10
0
10
x
Add in phase: f(x) and f(x)+f(x-1)
Radiation from a Atomic Sources
1
Pulse train from one atom
e( x , 0 )
10
00
10
20
30
40
Add with1 random
phases:
f(x) and f(x)+f(x-1)
10
0
10
Radiation from a Atomic Sources
x
2
Add in phase: f(x) and f(x)+f(x-1)
1
1
Pulse train from one atom
0
e( x , 0 )
0
E:\1B24\Lectures\lect21\youngp.mcd/you
File version: 03/03/99 09:59
1
1
10
0
10
x
2
10
0
10
30
Add
in phase:
f(x)
and 20f(x)+f(x-1)
10
COHERE.MCD
0
10
20
1
40
30
40
25/03/99 16:05
Add with random phases: f(x) and f(x)+f(x-1)
2
Idealised Young's Slits pattern
1
1
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010
Intensity
0
10
20
30
40
1
Add
with random phases: f(x) and f(x)+f(x-1)
2
2
0.5
10
0
10
20
30
20
10
30
40
0
10
1
COHERE.MCD
0
1
25/03/99 16:05
20
30
Position on screen (y)
1
2
10
0
10
20
30
40
Realistic Young's Slits pattern
1
1
Intensity
COHERE.MCD
25/03/99 16:05
0.5
30
20
10
0
10
20
30
° ±²³!´C›zì'Ó)µSø+Ó)¶[·bé´C¿Ò¹3ê±B¼²d»)¿?ºAé´C¿M³!±B¼²´½@¹aºAº`´9³A¼•¹aº[Ã1¹3³A´J³v¹)¼+²ÃB´9ò ±¼Ôæ5»¼+²Ðyò
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Position on screen (y)
1
youngp.mcd 25/03/99 16:13
·#ì'ÓOí‰î|ï
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Diffraction Grating - Geometry
h
θ
h sin θ
N h sin θ
Gratinh.mcd
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1
25/03/99 17:33
·#ììVí‰î=ø
Diffraction Grating
λ
570 . 10
h
2.3 .10
9
π
sin N . . h . sin ( θ )
λ
F( λ , h , θ , N)
6
sin
N
5
2
π. .
h sin ( θ )
λ
25
20
15
10
Diffraction Grating
5
570 . 10
λ
0
9
π
sin N . . h . sin ( θ )
0.2 λ 0.3
2
° ±²³!´c›zìì'µyøó+¶e·bé´c±¼=º`´J³;¿M´J³!´J¼Ä9´G½@¹3º!º`´J³!¼½³!»'ê+ÄJ´Jêû=ëHÌË´cÃB±B¼+´#ò!»³!ÄJ´Jò9èòAé+»{ö?í
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0.3
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0.1
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N100 5
π. .
h sin ( θ )
λ
25
80
20
60
15
40
10
20
50
0
0.3
0.2
0.1
0
0.3
0.2
0.1
0
Grating.mcd
N
0.1
0.1
0.2
0.2
0.3
0.3
1
25/03/99 17:28
10
100
80
60
40
20
0
0.3
0.2
0.1
0
0.1
0.2
0.3
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Grating.mcd
1
25/03/99 17:28
·#ììVí‰î|œ
Traces at 0.8, 1.0 and 1.2 times
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Rayleigh.mcd
2
25/03/99 17:35
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Rayleigh.mcd
2
25/03/99 17:35
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Huygens's Construction - Plane Wave
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Huygen1.mcd
1
07/03/99 21:33
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Huygens's Construction - Spherical Wave
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Huygen2.mcd
1
07/03/99 21:43
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Huygens's Construction - Reflection and
Refraction
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Incident
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Huygen4.mcd
1
07/03/99 21:54
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Huygens's Construction - Aperture
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Huygen3.mcd
1
07/03/99 21:46
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all these contributions.
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sin
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I( θ , λ , d)
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Diffraction - Single Slit
0
15
10
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λ
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Fraunsli.mcd
Fraunslj.mcd
1
11/03/99 12:52
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11/03/99 13:27
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Rayleigh.mcd
2
25/03/99 17:35
œ”ïVí;ï)ø
Diffraction pattern - slit and circular aperture
s( θ , f)
c ( θ , f)
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θ
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1
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0.8
0.6
0.4
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Gratini.mcd
1
11/03/99 20:26
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N
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10
1
Intensity
0.8
0.6
0.4
0.2
0
4
3
2
1
0
1
2
3
4
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zIxpf*t?y·ygfhƒ}gfhƒ´h~y;wutCwuc›|ph~›´‚t?Kny’Îg‚#ÂMh~f-x?zs´ecewu´a«Iwuhƒy °5¢’h~t?vŠ‘‘gt?ezst?´ ± ce“ghœt?´*wu‘gh
´iKcew»´}Rt?vƒcKygd‡k
2 d sin(theta)/lambda
f
10
b
Slit width = slit spacing/10
Gratini.mcd
2
11/03/99 20:26
1
Intensity
0.8
0.6
0.4
0.2
0
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
2 d sin(theta)/lambda
b cedpÎgfih@ϔC½‡kBÐCևoºa‘gh9‚xO“gce‹gv(tCwcKxpyjxCznw‘gh“‡cersfut?v5wucKx?y}RtCwwh~fiyˆx?zgtœd?futCwcKygdg¢×´i‘gx ± y
zIxpfTt?y Îgy‡fh~}‡fh~´ih~y;wŠt¿wuce|?h~› ´‚t?esy’Îg‚#ÂMh~fTx?zA´ecewu´«Iwh~y °Ø¢‡h~t?vŠ‘xpyghwh~y;wu‘t?´ ± cK“gh
t?´aw‘gh8´Kc›wa´}gt?v~ceygdgk
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Gratini.mcd
3
11/03/99 20:30
Diffraction - Fraunhofer and Fresnel Limits
Fraunhofer
Limit
To e
urc
so
det
To
ect
or
Fresnel
Limit
Source
Dector
bcedpÎgfih8ϔC½‡kl¥?هo*a‘gh\“gcersh~fih~ygvƒh8Âsh5w ± hƒh~y1bRfutCÎgyg‘gx?zIhƒf‰t?yg“!bRfh~´iygh~º“‡cersfut?v5wucKx?y¡k
Huygfre.mcd
1
11/03/99 12:59
œ”?½{íV½‡m
ÚÆÅ'Û1Ü(ÝßÞ'àáÈ
âAžãÌä-åpÌä-̞1݉Ìçæ™èêé#ä@Ü~Å'Ë1Ÿìë¨í/̞/Å'î ̞/é
Interference - the oil film
air
oil
h
water
ïðKñ?Îgòó8Ï{ôpÐnõ¦ö‡÷pøùaúgóˆñ?ó~ûpü'óƒýòiþ ûCÿÿIòið gñpójÿIû?òü Cýuðeû ð
The phase change in the reflection in air from the upper surface of the
oil is π; the phase change on reflection at the oil-water interface is 0.
Constructive interference takes place when the total phase difference
between a ray reflected from the surface of the oil and a ray which has
passed into the oil and been reflected from the water is an integer
multiple of 2π. Given that the phase changes on reflection differ by π for
the two rays, this requires the path length in oil to be an odd half-integer
multiple of the wavelength in the oil.
The film is observed at near-normal incidence, so we may take the
geometrical path length in the oil to be 2h.We therefore seek a
1 . λ
wavelength λ in air which satisfies 2 . h
p
2 n oil
Oilfilm.mcd
1
21/03/99 16:35
ùœôpÐ V½;ô
ûpð eü!õ
Interference - a film at non-normal
incidence
d
n1
n2
A
θ2
θ1
D
B
C
ïðeñpÎgòió%Ï{ôpÐnõ¦ö;ônøùaúgóñpóƒûpüó5ýuò£þ1û?ÿ9ÿIòð ‡ñpóÿIûpòiü CýðKû =’þ
ð ~ð gó ƒó?õ
ùœôpÐ V½
×þ?ó~ò Cý gû gû?òü
Interference Fringes - Newton's Rings
Side view
Radius R
Air film
Lens
P
t
Glass plate
Top view
Lens
P
r
Frnewtn.mcd
ïðKñ?Îgòó8Ï{ôpÐnõ¦ö ‡øùaúgó ióƒý QÎ µû?ÿ '՜ó aýû Žòið gñ aóؾ sóƒòðKü'ó ;ý~õ
1
21/03/99
16:10 ùœôpÐ V½)î
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Newton's Rings -- geometrical results
R
R-t
r
t
ïðKñp·òóÏ{ôpÐnõ¦ö)îgø-ùaú‡ó\ñ?ó~ûpü'óƒýòiþû?ÿ{ýuú‡óˆÕœó aýû Žòið gñ aóؾ sóƒòðKü'ó ;ý~õ
r
Frnewtng.mcd
2
R
2
(R
t)
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2
t .( 2 .R
t)
2
21/03/99 16:26
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Interference Fringes - Wedge
Air film
α
ïðKñp·òóÏ{ôpÐnõ¦ö nø-ùaú‡ó\ÿIûpòiü ¿ýuðKû û?ÿ{ÿIòð ‡ñpó að
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Frwedge.mcd
1
21/03/99 16:33
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Interference - Michelson's Inteferometer
Mirror D
Source S
Beam
splitter
A
Compensating
plate
B
Mirror C
Detector E
ï ðeñpÎgòió=Ï{ô?ö‡õ¦ö ‡ø1ùaú‡ó !ð Šúgó iû ð ;ýó~òiÿIóƒòûpü'óƒýó~òƒø ió~ó=ýúgó=ýó Oý ÿIûpò
Mirrors
gó ƒòð Cnand
ýuðKû ¡õD are silvered on their front faces, and the beam
0
6
1
splitter A consists of a glass plate part-silvered on its rear surface.
Circular fringes will be observed at the detector if the reflection of
D in the beam splitter is parallel to C.
The glass in the beam splitter will be dispersive, but it is traversed
three times by (SADAE) but only once by (SACAE). This results
in a wave-length dependent optical path difference. If the
compensator plate is identical (apart from any silvering) to the beam
splitter, and is aligned parallel to it, it will introduce an extra path
length into the second path which is exactly equivalent to the two
extra traverses of the beam splitter in the first path.
Michel1.mcd
1
21/03/99 18:39
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góƒý ?ð eó
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5 Interference - Michelson's Inteferometer
Image of C in A
d
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Source S
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splitter
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plate
B
Mirror C
Detector E
Michelson's Inteferometer - parallel mirrors
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ð ýó~òiü aû?ÿ{ýuú‡ó\ðeü Cñp.ój. û?d ÿ{. ýuúgó8ü'û π 2 Kójü'ðKòiòûpòTð ýuúgó Mó ?ü eðeýýó~òƒõ
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: ;
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Michel1a.mcd
1
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ð ’ýó~ò£ÿIó~òû?üó5ýuó~ò ‰ðeýuú Mû?ýuúšü'ðKòiòûpò Mó~ò só gð Còýuû•ýuú‡ó só ?ü
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~ð gñû?ÿ÷×ÙpÙ ?ó Kó gñ?ýú ~õ
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Michel1b.mcd
21/03/99 18:40
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21/03/99 18:52
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Doublet Fringes in the Michelson Interferometer
I( θ , λ , d)
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Two waves differing in frequency by 5%
Doublet Fringes in the Michelson Interferometer
I( θ , λ , d)
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25
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ïðeñ gòió {ô?önõlö nøµùaú‡ó ?òð Cýuðeû ^û?ÿÿIòið gñ?óµð ;ýuó ðeýQþ ‰ð›ýuú gð £ý ~óÿIûpò Iýuû
pó eó gñCýuú
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Two waves differing in frequency by 5%
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ïðeñ gòió {ô?ö‡õ ‡ø eû ó ûCÿºýuúgó ?òið CýðKû û?ÿºÿIòð gñpójð ’ýó iðeýQþ ‰ðeýú gð £ý ƒóÿIûpò
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pó
eó gñ?ýú
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ý ?ð gð gñ Mû?ýuú ?ó eó gñ?ýú Cý %ü'ðKòòiûpò ƒð gñ'û?ÿ÷
pó eó gñCýuú ƒõ
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ïðeñ gòió {ô?ö‡õ ;ônø*ùaú‡ó ?òið CýðKû û?ÿAÿIòð gñpójð ’ýó iðeýQþ ‰ðeýuú gñ KóŽÿIûpò Iýû
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ý ?ð gð gñ Mû?ýuú ?ó eó gñ?ýú Cý %ü'ðKòòiûpò ƒð gñ'û?ÿ÷ ‡õ›÷ ?ó eó gñ?ýú ~õ
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ƒðKó ’ý
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D
Fabry2.mcd
O
BP ^P
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A
9
gD
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21/03/99 22:37
ùœô n÷
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Interference - Fabry-Perot Inteferometer
etalon
Source
Collimating
lens
Focusing
lens
Screen in
focal plane
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Fabry1.mcd
1
21/03/99 22:36
Reflection - Concave Mirror
θ1
α1
φ
C
A1
θ2
α2
A2
l2
V
r
I1
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Reflect2.mcd
1
œ¨§]7n§
25/03/99 12:55
Reflection - Concave Mirror
Focal plane and focal length
V
f
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Reflect3.mcd
1
25/03/99 12:57
Imaging -- Convex Surface
θ2
θ1
α2
α1
V
A1
φ
A2
C
r
I2
l1
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Reflect2.mcd
2
25/03/99 12:55
Images - Concave Mirror
B1
h1
A2
V
h2
A1
B2
l2
l1
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Reflect6.mcd
1
25/03/99 13:01
Concave Make-up/Shaving Mirror
C
A1
A2
V
250 mm
z=-400
z=-x z = 0
z=250-x
Magnification - upright image
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Reflect7.mcd
1
25/03/99 13:03
Imaging -- Spherical aberration
C
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Reflect3.mcd
2
25/03/99 12:57
What counts as a small angle?
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spherical surface
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LENS2.MCD
l1
1
22/03/99 15:59
l2
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LENS2.MCD
2
22/03/99 16:00
Refraction - principal rays
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prinray.mcd
1
23/03/99 10:53
䨧F½3ñ8³F½
Refraction
-- the
thin
lens
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so as to pass directly or by projection through the second
focal point
F2
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Lens3.mcd
1
23/03/99 22:51
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axis
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Lens4.mcd
1
ñ8³%ÿ
23/03/99 22:53
F2
F2
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first focal point, emerges from the lens parallel to optical
axis
F1
F1
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ßJÝæ Ý the
ç˜
ß6ç,centre
èAêgìßaç.é5åÝæ à æã[ß6ç,è>Ýæ
A
ray passing through
of the lens
é6åàçÀ²Ý¨çæ¸à3²Ý@é5ÝæFÞ'é5åßJì°ÞåLé5åàJàìæ¸ßJÝæ Ý çlìÝæ%é,á
without deviation or lateral displacement (in
the thin lens limit)
Lens4.mcd
1
23/03/99 22:53
%ÿGá Ì Fã<ä[åàß6ÝæÝ ç /ß5ç,è
ÜÝÞ ßJàË
àæ
>êgìßmç.é6åÝæ
Lenses
- Chromatic
Spherical
é6åß6ìFÞå
é5åàà æ%é5ß6àìand
ê˜é5åà
àæ á
Aberrations
ã[ß5ç,è
ݸà æÔé
ç!JÝæÞ"Fæ#à$GÝçBé5à#
White
Red rays
Red focus
Violet focus
Violet rays
Linear
chromatic
aberration
%ÿá%
ܯÝÞ ßJàË
Lens4.mcd
'&WåFß6ìîçBé5ÝEç(làßJß5çBé6Ýìæ)(%èç'é5åÝæ*àæ
ÌÛFã
2
23/03/99 22:53
%ÿ +ñ ä
Lens5.mcd
1
23/03/99 22:54
à"é5ì
çß6ç
á
Achromatic Doublet
Lenses - Chromatic and Spherical
Aberrations
White
light
Red and violet
Common
focus
Red rays
Red focus
Large refractive
index,
Violet focus
moderate
Violet
rays distortion
Moderate refractive
Linear
index, large dispersion
chromatic
aberration
rays
White
%ÿá% ÌFã,&Wìß6ßJà
ÜÝÞ ßJà)Ë
(à
-SåßJìîçÀé5Ý'ç(fà
é6Ýìæ·ìê
.(%è·çæçSåßJìîçÀé5Ý/#ìñ
ß6ß6çBé5Ýìæ
éá
Lens9.mcd
1
23/03/99 23:13
Image formation by a Convex Lens
With Cartesian sign convention,
1
23/03/99 22:54
ÜÝÞß6àEË%ÿGáã10åà ß6Ý,ç/ç(làßJß5çBé6ÝìæÝæç2çß6Þà3ñ8çfà
f = 100
mm, u = -150 mm,
v = 300 mm,
M = -2
Lens5.mcd
3 ,àæ
ßJé Fß6à
á
u
v
object
h
O
C
I
F2
F1
h'
image
%ÿá%7ÿãeä[åàEêgìßJîçÀé5Ýìæ_ìêçæ&ÝîçBÞà,(%èLçìæ4$àß6ÞÝæÞ5à æ á
ÜÝÞ ßJàEË
%ÿ +ñ â
ä
Lens6.mcd
1
23/03/99 22:56
Image formation by a Concave Lens
With Cartesian sign convention,
f = -100 mm, u = -150 mm,
v = -60 mm,
M = 0.4
object
u
v
h
h'
O
F1
C
I
F2
Image formation by a Concave Lens
image
with a virtual object
With Cartesian sign convention,
ß6àamm,
Ë6ÿáu
7Fã=e+80
ä[åàEmm,
êgìßJîçBé6ÝìæìêeçæLÝîçÞà,(%èLç5#FÝ$à
f Ü=ÝÞ-50
v = -133 mm,
M = -1.66
v
à æ á
ß6ÞÝæÞ
u
object
h
I
Lens7.mcd
1
F2
h'
C
23/03/99 22:57
F1
O
image
8%ÿGáâãWä[åàêgìß6îçBé6ÝìæLìê`çBæµÝîçÞàìêhç5$GÝßé3 ç ˜ì( 9Jà3éaÝæµç ìæ,ç:$à
àæ á
ÜÝÞ ßJà
The Magnifying Glass
h'
h
Lens8.mcd
1
23/03/99 22:58
l
l'
86ÿá<;ãhä[åà=6Ýî à,àæJà#&çAçiîçÞæFÝê èGÝæÞjÞ9ç
ÜÝÞ ßJà
%ÿ +ñ 4;
ä
Lens11.mcd
1
23/03/99 22:44
á
Thick Lens System Principal Planes and Principal Points
A1
K1
Q1
Q2
G1
F1
V1
K1
G2
H1
H2
V2
8%ÿá%7>ãhä[åà=?à
ÜÝÞ ß6à
Lens10.mcd
A2
F2
3 @?Aà æ Jè
èêgàçBé Fß6à ¨ìêç)é5åÝ
1
23/03/99 22:42
%ÿ +ñ >
ä
Jé6àîá
òRó.ôKõ,ö
B2B ø CÐôµúeõöAù/DMþE.F¯úeý'G.H
Two-Lens Systems General Treatment
IE_úJFïþþ
ü
B
A
C
h1
h2
V1
V2
H2
F2
G
s
d
f1
f
KMLNO!P,86 Q<RSUTWVP=NPXY P[Z3O]\"X^U_Za`-XbcPd
Twolens.mcd
1
]\ Z3PYQ
24/03/99 00:39
T<b+4R
Imaging by a Two-Lens System
1
2
h1
I1= O 2
O1
I2
4
3
f1
f2
v1
u1
d
u2
v2
KMLNand
O!P,862ee are
Q7 used
SUfdPhgto
_Y construct
ijPIX^J_k]\k]Z3the
PYl`image
LZ3V*Za`-X jPdk PkQ
Lines 1
I1 of O1 in lens 1; lines 3 and 4 are used to
construct the image I2 of I1, treated as the
object of lens 2.
v1 = 300 mm
u2 = - 150 mm
v2 = 150mm
M=2
Example: f1 = 100 mm
Two-Lens Systems
f
=
75
mm
Compound Microscope
2
u1 = - 150 mm
d = 450 mm
Lens13.mcd
1
23/03/99 22:46
α2
h'
h
fE
fO
fO
fE
L
d
LNm O!Pneat
e Q%oinfinity,
o7p SMTWVPqangular
XY5irXm ds*Y LqO!Xk!qXirPQ
With the final KMimage
magnification is
- α2/(h/standard near point)
= - (h'/fE)/(h/250 mm)
= - (L/fO)(250mm/fE)
Tee<b+tt
Note that many manufacturers standardize the
distance from the second focus of the objective
to the first focus of the eyepiece at L=160 mm.
Lens16.mcd
1
23/03/99 22:50
Two-Lens Systems Two-Lens
Systems
Astronomical
Telescopes
Astronomical Telescopes
h
h
h"
h"
fE
fE
fO
fO
d
d
magnification
=Lqf_Ejr/fZ!OPjPk!qXiwPS'jLdP:_OxYA_NdLyzq_Z3LXd{Q
KMLNLinear
m O!Pnee Q%o<uv SMTWVP/_k Z3O XdXY5
Linear magnification = fE/fO
α2
α2
α1
α1
fO
fO
fE
fE
d
d
Angular magnification = fO/fE
Angular
KMLNm O!PInee Q%o<Systems
uoSUmagnification
TWVP_k Z!O!-XdXY Lq:_=jwZ3fPOjP/fk qEXiwPS1_dNmj|_OxYA_NdLyzq_Z3LXd{Q
Two-Lens
Lens14.mcd
23/03/99 22:47
Galilean Telescopes1
Lens14.mcd
1
23/03/99 22:47
h
h"
fE
fE
fO
d
Linear
KMLNm O!P
nee Q%o<uu magnification
SMTWVP}=_jLjP:_dAZ!P=jPk!fE
qX/fiwOPS'jLdP:_OxYA_NdLyzq_Z3LXd{Q
Tee<b+tp
α2
α1
fE
fE
fO
d
Angular magnification = fO/fE
Two-Lens Systems Galilean Telescopes
h
h"
fE
fE
fO
d
Linear magnification = fE/fO
α2
α1
fE
fE
fO
d
KMLNmAngular
O!Pnee Q%o<uemagnification
SUTWVP/}=_jLjP:_d5Z3P=jPfk O
qX/fiwEPS1_dNmj|_OWY _Nd Lyzq_Z3LXd{Q
Lens15.mcd
1
23/03/99 22:48
Tee<b+pv