Description of Physics of Optics, part I.
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2. 2
Table of Content
SOLO OPTICS
Maxwell’s Equations
Boundary Conditions
Electromagnatic Wave Equations
Monochromatic Planar Wave Equations
Spherical Waveforms
Cylindrical Waveforms
Energy and Momentum
Electrical Dipole (Hertzian Dipole) Radiation
Reflections and Refractions Laws Development
Using the Electromagnetic Approach
IR Radiometric Quantities
Physical Laws of Radiometry
Geometrical Optics
Foundation of Geometrical Optics – Derivation of Eikonal Equation
The Light Rays and the Intensity Law of Geometrical Optics
The Three Laws of Geometrical Optics
Fermat’s Principle (1657)
3. 3
Table of Content (continue)
SOLO OPTICS
Plane-Parallel Plate
Prisms
Lens Definitions
Derivation of Gaussian Formula for a Single Spherical Surface Lens
Using Fermat’s Principle
Derivation of Gaussian Formula for a Single Spherical Surface Lens
Using Snell’s Law
Derivation of Lens Makers’ Formula
First Order, Paraxial or Gaussian Optics
Ray Tracing
Matrix Formulation
4. 4
Table of Content (continue)
SOLO OPTICS
Optical Diffraction
Fresnel – Huygens’ Diffraction Theory
Complementary Apertures. Babinet Principle
Rayleigh-Sommerfeld Diffraction Formula
Extensions of Fresnel-Kirchhoff Diffraction Theory
Phase Approximations – Fresnel (Near-Field) Approximation
Phase Approximations – Fraunhofer (Near-Field) Approximation
Fresnel and Fraunhofer Diffraction Approximations
Fraunhofer Diffraction and the Fourier Transform
Fraunhofer Diffraction Approximations Examples
Resolution of Optical Systems
Optical Transfer Function (OTF)
Point Spread Function (PSF)
Modulation Transfer Function (MTF)
Phase Transfer Function (PTF)
Relations between Wave Aberration, Point Spread Function
and Modulation Transfer Function
Other Metrics that define Image Quality – Srahl Ratio
Other Metrics that define Image Quality - Pickering Scale
Other Metrics that define Image Quality – Atmospheric Turbulence
Fresnel Diffraction Approximations Examples
O
P
T
I
C
S
P
a
r
t
I
I
5. 5
Table of Content (continue)
SOLO OPTICS
References
Optical Aberration
Monochromatic Seidel Aberrations
Chromatic Aberration
Interference
O
p
t
i
c
s
P
a
r
t
I
I
6. 6
OpticsSOLO
Hierarchy of Optical Theories
• Quantum Light as particle (photon)
Emission, absorption, interaction of light and matter
• Electromagnetic Maxwell’s Equations
Reflection/Transmission, polarization
• Scalar Wave Light as wave
Interference and Diffraction
• Geometrical Light as ray
Image-forming optical systems
λ → 0
8. 8
MAXWELL’s EQUATIONSSOLO
SYMMETRIC MAXWELL’s EQUATIONS
Magnetic Field IntensityH
1
mA
Electric DisplacementD
2
msA
Electric Field IntensityE
1
mV
Magnetic InductionB
2
msV
Electric Current DensityeJ
2
mA
Free Electric Charge Distributione 3
msA
Fictious Magnetic Current DensitymJ
2
mV
Fictious Free Magnetic Charge Distributionm
3
msV
1. AMPÈRE’S CIRCUIT LW (A)
eJ
t
D
H
2. FARADAY’S INDUCTION LAW (F)
mJ
t
B
E
3. GAUSS’ LAW – ELECTRIC (GE)
eD
4. GAUSS’ LAW – MAGNETIC (GM) mB
Although magnetic sources are not physical they are often introduced as electrical
equivalents to facilitate solutions of physical boundary-value problems.
André-Marie Ampère
1775-1836
Michael Faraday
1791-1867
Karl Friederich Gauss
1777-1855
James Clerk Maxwell
(1831-1879)
12. 12
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions
2
ˆt
1
ˆt
h
2H
1H
1
2
C
CS1P
2P
3P
4P
bˆ
21
ˆ n
ek
ldtHtHhldtHldtHldH
h
C
2211
0
2211
ˆˆˆˆ
where are unit vectors along C in region (1) and (2), respectively, and21
ˆ,ˆ tt
2121
ˆˆˆˆ nbtt
- a unit vector normal to the boundary between region (1) and (2)21
ˆ n
- a unit vector on the boundary and normal to the plane of curve Cbˆ
Using we obtainbaccba
ldbkldbHHnldnbHHldtHH e
ˆˆˆˆˆˆ 21212121121
Since this must be true for any vector that lies on the boundary between
regions (1) and (2) we must have:
bˆ
ekHHn
2121
ˆ
S
e
C
Sd
t
D
JdlH
dlbkbdlh
t
D
JSd
t
D
J e
h
e
S
e
ˆˆ
0
AMPÈRE’S LAW
1
0
lim:
mAh
t
D
Jk e
h
e
13. 13
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 1)
2
ˆt
1
ˆt
h
2E
1E
1
2
C
CS1P
2P
3P
4P
bˆ
21
ˆ n
mk
ldtEtEhldtEldtEldE
h
C
2211
0
2211
ˆˆˆˆ
where are unit vectors along C in region (1) and (2), respectively, and21
ˆ,ˆ tt
2121
ˆˆˆˆ nbtt
- a unit vector normal to the boundary between region (1) and (2)21
ˆ n
- a unit vector on the boundary and normal to the plane of curve Cbˆ
Using we obtainbaccba
ldbkldbEEnldnbEEldtEE m
ˆˆˆˆˆˆ 21212121121
Since this must be true for any vector that lies on the boundary between
regions (1) and (2) we must have:
bˆ
mkEEn
2121
ˆ
S
m
C
Sd
t
B
JdlE
dlbkbdlh
t
B
JSd
t
B
J m
h
m
S
m
ˆˆ
0
FARADAY’S LAW
1
0
lim:
mVh
t
B
Jk m
h
m
14. 14
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 2)
h
2D
1D
1
2
21
ˆ n
dS
1
ˆn
2
ˆn
e
SdnDnDhSdnDSdnDSdD
h
S
2211
0
2211
ˆˆˆˆ
where are unit vectors normal to boundary pointing in region (1) and (2),
respectively, and
21
ˆ,ˆ nn
2121
ˆˆˆ nnn
- a unit vector normal to the boundary between region (1) and (2)21
ˆ n
SdSdnDDSdnDD e 2121121
ˆˆ
Since this must be true for any dS on the boundary between regions (1) and (2)
we must have:
eDDn 2121
ˆ
dSdShdv e
h
e
V
e
0
GAUSS’ LAW - ELECTRIC
1
0
lim:
msAhe
h
e
V
e
S
dvSdD
15. 15
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 3)
h
2B
1B
1
2
21
ˆ n
dS
1
ˆn
2
ˆn
m
SdnBnBhSdnBSdnBSdB
h
S
2211
0
2211
ˆˆˆˆ
where are unit vectors normal to boundary pointing in region (1) and (2),
respectively, and
21
ˆ,ˆ nn
2121
ˆˆˆ nnn
- a unit vector normal to the boundary between region (1) and (2)21
ˆ n
SdSdnBBSdnBB m 2121121
ˆˆ
Since this must be true for any dS on the boundary between regions (1) and (2)
we must have:
mBBn 2121
ˆ
dSdShdv m
h
m
V
m
0
GAUSS’ LAW – MAGNETIC
1
0
lim:
msVhm
h
m
V
m
S
dvSdB
16. 16
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (summary)
2
ˆt
1
ˆt
h
22 ,HE
11,HE
1
2
C
CS1P
2P
3P
4P
bˆ
21
ˆ n
me kk
,
21
ˆ n
dS
11,BD
22,BD
me
,
mkEEn
2121
ˆ FARADAY’S LAW
ekHHn
2121
ˆ AMPÈRE’S LAW 1
0
lim:
mAh
t
D
Jk e
h
e
1
0
lim:
mVh
t
B
Jk m
h
m
eDDn 2121
ˆ
GAUSS’ LAW
ELECTRIC
1
0
lim:
msAhe
h
e
mBBn 2121
ˆ
GAUSS’ LAW
MAGNETIC
1
0
lim:
msVhm
h
m
Return to TOC
17. 17
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
For Homogeneous, Linear and Isotropic Medium
ED
HB
where are constant scalars, we have,
J
t
E
J
t
D
H
t
t
H
t
B
E
ED
HB
Since we have also
tt
t
J
t
E
E
DED
EEE
t
J
t
E
E
2
2
22
2
2
&
18. 18
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS (continue 1)
Define
meme KK
c
KK
v
00
11
where
smc /103
10
36
1
104
11 8
9700
is the velocity of light in free space.
The absolute index of refraction n is
me KK
v
c
n
0
The Inhomogeneous Wave (Helmholtz) Differential Equation for the
Electric Field Intensity is
t
J
t
E
v
E
2
2
2
2 1
19. 19
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS (continue 2)
In the same way
The Inhomogeneous Wave (Helmholtz) Differential Equation for the
Magnetic Field Intensity is
J
t
E
J
t
D
H
t
H
t
B
E
t
ED
HB
Since are constant and
tt
,
J
t
H
H
HHB
HHH
J
t
H
H
2
2
22
2
2
0&
J
t
H
v
H
2
2
2
2 1
Return to TOC
20. 20
ELECTROMAGNETICSSOLO
Monochromatic Planar Wave Equations
Let assume that can be written as: trHtrE ,,,
tjrHtrHtjrEtrE 00 exp,,exp,
where are phasor (complex)
vectors.
rHjrHrHrEjrErE
ImRe,ImRe
We have tjrEjtj
t
rEtrE
t
00 expexp,
Hence
m
e
m
e
j
t
m
e
m
e
B
D
JBjE
JDjH
BGM
DGE
J
t
B
EF
J
t
D
HA
)(
23. 23
ELECTROMAGNETICSSOLO
m
e
m
e
ED
HB
m
e
JHjE
JEjH
JHjE
JEjH
JBjE
JDjH
me JJjEkE
2
em JJjHkH
2
22 f
c
c
f
k
Using the vector identity AAA
For a Homogeneous, Linear and Isotropic Media:
m
e
ED
HB
m
e
H
E
B
D
e
me JJjEkE
22
m
em JJjHkH
22
and
we obtain
Monochromatic Planar Wave Equations (continue)
24. 24
ELECTROMAGNETICSSOLO
Assume no sources:
we have
Monochromatic Planar Wave Equations (continue)
0,0,0,0 meme JJ
022
EkE
022
HkH
nkk
n
k
0
00
00
0
rktjtj
rktjtj
eHerHtrH
eEerEtrE
0
0
,,
,,
022
rkj
rkjrkjrkjrkj
ek
ekkeekje
Helmholtz Wave Equations
satisfy the Helmholtz wave equations ,,, rHrE
rkj
rkj
eHrH
eErE
0
0
,
,
Assume a progressive wave of phase rkt
(a regressive wave has the phase ) rkt
For a Homogeneous, Linear and Isotropic Media
k
0E
0H
r
t
k
Planes for which
constrkt
25. 25
ELECTROMAGNETICSSOLO
To satisfy the Maxwell equations for a source free media we must have:
Monochromatic Planar Wave Equations (continue)
we haveUsing: 1ˆˆ&ˆˆ sss
c
n
sk
0
0
H
E
HjE
EjH
0ˆ
0ˆ
ˆ
ˆ
0
0
00
00
Hs
Es
HEs
EHs
sˆ
Planar Wave
0E
0H
r
0
0
0
0
00
00
rkj
rkj
rkjrkj
rkjrkj
ekje
eHkj
eEkj
eHjeEkj
eEjeHkj
rkjrkj
0
0
0
0
00
00
Hk
Ek
HEk
EHk
For a Homogeneous, Linear and Isotropic Media:
Return to TOC
26. 26
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
Spherical Waveforms z
x
y
r
cosr
,,rP
sinsinr
cossinr
The Inhomogeneous Wave (Helmholtz) Differential
Equation for the Electric Field Intensity is
t
J
t
E
v
E
2
2
2
2 1
In spherical coordinates:
cos
sinsin
cossin
rz
ry
rx
2
2
222
2
2
2
sin
1
sin
sin
11
rrr
r
rr
For a spherical symmetric wave: rErE
,,
Er
rrr
E
rr
E
r
E
r
rr
E
2
2
2
2
2
2
2 121
27. 27
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
SourceSourceSource
Spherical Waveforms z
x
y
r
cosr
,,rP
sinsinr
cossinr
The Inhomogeneous Wave (Helmholtz)
Differential Equation for the
Electric Field Intensity is assuming no sources
0
11
2
2
22
2
t
E
v
Er
rr
In spherical coordinates:
cos
sinsin
cossin
rz
ry
rx
0
1
2
2
22
2
Er
tv
Er
r
or:
A general solution is:
wave
regressive
wave
eprogressiv
tvrFtvrFEr 21
0,0,0,0 meme JJ
r
e
EerEtrE
rktj
tj
0,,
Assume a progressive monochromatic wave of phase
rkt
(a regressive wave has the phase ) rkt
r
e
ErE
rkj
0, Return to TOC
28. 28
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
Cylindrical Waveforms
z
x
y
r
zrP ,,
sinr
cosr
In cylindrical coordinates:
zz
ry
rx
sin
cos
2
2
2
2
2
2 11
zrr
r
rr
For a cylindrical symmetric wave: rEzrE
,,
r
E
r
rr
E
12
The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field
Intensity is assuming no sources
0
11
2
2
2
t
E
vr
E
r
rr
0,0,0,0 meme JJ
29. 29
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
Source
Cylindrical Waveforms
z
x
y
r
zrP ,,
sinr
cosr
SourceSource
In cylindrical coordinates:
zz
ry
rx
sin
cos
The Inhomogeneous Wave (Helmholtz)
Differential Equation for the
Electric Field Intensity is assuming no sources
0
11
2
2
2
t
E
vr
E
r
rr
0,0,0,0 meme JJ
Assume a progressive monochromatic wave of phase
rkt
(a regressive wave has the phase ) rkt
tj
erEtrE
,,
0
1
2
2
2
E
vr
E
rr
E
k
The solutions are Bessel functions which for large
r approach asymptotically to: rkj
e
r
E
rE
0
,
Return to TOC
30. 30
SOLO
Energy and Momentum
Let start from Ampère and Faraday Laws
t
B
EH
J
t
D
HE e
EJ
t
D
E
t
B
HHEEH e
HEHEEH
But
Therefore we obtain
EJ
t
D
E
t
B
HHE e
First way
This theorem was discovered by Poynting in 1884 and later in the same year by Heaviside.
ELECTROMAGNETICS
John Henry Poynting
1852-1914
Oliver Heaviside
1850-1925
31. 31
SOLO
Energy and Momentum (continue -1)
We identify the following quantities
- Power density of the current densityEJe
HEDE
t
BH
t
EJe
2
1
2
1
BH
t
pBHw mm
2
1
,
2
1
DE
t
pDEw ee
2
1
,
2
1
HEpR
eJ
- Magnetic energy and power densities, respectively
- Electric energy and power densities, respectively
- Radiation power density
For linear, isotropic electro-magnetic materials we can write HBED
00 ,
DE
tt
D
E
ED
2
10
BH
tt
B
H
HB
2
10
ELECTROMAGNETICS
32. 32
SOLO
Energy and Momentum (continue – 3)
Let start from the Lorentz Force Equation (1892) on the free charge
BvEF e
Free Electric Chargee 3
msA
Velocity of the chargev
1
sm
Electric Field IntensityE
1
mV
Magnetic InductionB
2
msV
Hendrik Antoon Lorentz
1853-1928
e
Force on the free chargeF
Ne
Second way
ELECTROMAGNETICS
33. 33
SOLO
Energy and Momentum (continue – 4)
The power density of the Lorentz Force the charge
EJBvEvp e
Bvv
Jv
e
ee
0
or
HE
t
B
HE
t
D
E
t
D
HEEH
E
t
D
HEJp
t
B
E
HEHEEH
J
t
D
H
e
e
e
ELECTROMAGNETICS
34. 34
SOLO
Energy and Momentum (continue – 5)
HEDE
t
BH
t
EJe
2
1
2
1
dve
E
B
eJv
,
V
FdF
Fd
Let integrate this equation over a constant volume V
VVVV
e dvSdvDE
td
d
dvBH
td
d
dvEJ
2
1
2
1
If we have sources in V then instead of
we must use
E
source
EE
Use Ohm Law (1826)
source
ee EEJ
VV
td
d
t
Georg Simon Ohm
1789-1854
source
e
e
EJE
1
For linear, isotropic electro-magnetic materials HBED
00 ,
ELECTROMAGNETICS
35. 35
SOLO
Energy and Momentum (continue – 6)
VVVR
n
V
source
e dvSdvDE
td
d
dvBH
td
d
dRIdvEJ
2
1
2
12
V
FieldMagnetic dvBH
td
d
P
2
1
V
FieldElectric dvDE
td
d
P
2
1
SV
Radiation SdSdvSP
V
source
eSource dvEJP
V
source
e
R
n
V
source
e
L S e
ee
V
source
e
L S e
ee
V
e
dvEJdRI
dvEJ
dS
dl
dSJdSJdvEJldSdJJdvEJ
2
11
R
nJoule dRIP
2
RadiationFieldMagneticFieldElectricJouleSource PPPPP
For linear, isotropic electro-magnetic materials HBED
00 ,
R – Electric Resistance
Define the Umov-Poynting vector: 2
/ mwattHES
The Umov-Poynting vector was discovered by Umov in 1873, and rediscovered by
Poynting in 1884 and later in the same year by Heaviside.
ELECTROMAGNETICS
John Henry Poynting
1852-1914
36. 36
ElectromagnetismSOLO
EM People
John Henry Poynting
1852-1914
Oliver Heaviside
1850-1925
Nikolay Umov
1846-1915
1873
“Theory of interaction on final
distances and its exhibit to
conclusion of electrostatic and
electrodynamic laws”
1884 1884
Umov-Poynting vector
HES
The Umov-Poynting vector was discovered by Umov in 1873, and rediscovered by
Poynting in 1884 and later in the same year by Heaviside.
1873 - 1884
Return to TOC
37. 37
Note:
Since there are not
magnetic sources the
Magnetic Hertz’s
Vector Potential is :
0
m
Electrical Dipole (Hertzian Dipole) RadiationSOLO
Given a dipole monochromatic of electric charges defined by the Polarization Vector Intensity
tq
tq
d
r
dqP
dr
tdqdeqaltP tj
e
cosRe 00
we want to find the radiation properties.
We start with the Helmholtz Non-homogeneous
Differential Equation of the Electric Hertz’s
Vector Potential : te
trPtr
tc
tr eee ,
1
,
1
,
0
2
2
2
2
Heinrich Rudolf Hertz
1857-1894
- speed of propagation of the EM wave [m/s]
00
1
c
- Polarization Vector IntensityeP
2
msA
- Permitivity of space 2122
mNsA
- Electric Hertz’s Vector Potential (1888)e
NsA 11
t
A e
000 eV
0
Using the Electric Hertz’s
Vector Potential we obtain :
The field vectors are given by e
e
tc
V
t
A
E
2
2
20
0 1
t
AH e
00
0
1
39. 39
SOLO Electric Dipole Radiation
tq
tq
d
r
zSS rrdqP 10
dr
sinr
cosr
zyx
r
r
rr
111
1
cossinsincossin
r1
1
1
x1
y1
z1
Using we can write
11 0
2
0
2
2
sin1
4
sin
44
krtjkrtj
ep
rk
j
r
kc
ep
rcr
j
H
krtj
ep
r
k
r
kj
r
rccr
j
r
E
r
rr
0
2
23
0
2
0
2
2
0
3
0
111
11111
sinsincos2
1
4
1
4
sin
4
sincos2
4
sincos2
We can divide the zones around the source, as function of the relation between
dipole size d and wavelength λ, in three zones:
Near, Intermediate and Far Fields
22
:
c
f
c
k
The Magnetic Field Intensity is transverse to the propagation direction at all ranges,
but the Electric Field Intensity has components parallel and perpendicular to .r1
r1
E
However and are perpendicular to each other.H
• Near (static) zone: rd
• Intermediate (induction) zone: ~rd
• Far (radiation) zone: rd
40. 40
SOLO Electric Dipole Radiation
tq
tq
d
r
zSS rrdqP 10
dr
sinr
cosr
zyx
r
r
rr
111
1
cossinsincossin
r1
1
1
x1
y1
z1
102
sin
4
tj
FieldNear
ep
r
kc
jH
tj
FieldNear
ep
r
E r
03
0
11 sincos2
4
1
Near, Intermediate and Far Fields (continue – 1)
• Near (static) zone: rd
In the near zone the fields have the character of the static fields. The near fields are
quasi-stationary, oscillating harmonically as , but otherwise static in character.tj
e
0
2
r
rk
41. 41
SOLO Electric Dipole Radiation
tq
tq
d
r
zSS rrdqP 10
dr
sinr
cosr
zyx
r
r
rr
111
1
cossinsincossin
r1
1
1
x1
y1
z1
102
sin
4
krtj
FieldteIntermedia
ep
r
kc
jH
krtj
FieldteIntermedia
ep
r
kj
r
E r
023
0
11 sincos2
1
4
1
Near, Intermediate and Far Fields
• Intermediate (induction) zone: ~rd
• Far (radiation) zone: rd
10
2
sin
4
krtj
FieldFar
ep
r
kc
H
10
0
2
sin
4
krtj
FieldFar
ep
r
k
E
r1
FieldFarE
FieldFarH
At Far ranges are orthogonal; i.e. we have
a transversal wave.
rHE 1,,
In the Radiation Zone the Field Intensities behave like a spherical wave (amplitude
falls off as r-1)
1
2
r
rk
120
10
36
1
1041
:
9
7
0
0
1
0
00
c
FieldFar
FieldFar
cH
E
Z
42. 42
SOLO Electric Dipole Radiation
http://dept.physics.upenn.edu/courses/gladney/phys151/lectures/lecture_apr_07_2003.shtml#tth_sEc12.1 http://www.falstad.com/mathphysics.html
Electric Field Lines of Force
43. 43
SOLO Electric Dipole Radiation
tq
tq
d
r
zSS rrdqP 10
dr
sinr
cosr
zyx
r
r
rr
111
1
cossinsincossin
r1
1
1
x1
y1
z1
The phasors of the Magnetic and Electric Field
Intensities are:
10
2
sin
4
1
krtj
ep
cr
j
r
H
krtj
ep
crc
j
rc
j
rrr
E r
02
2
2
0
11 sin
11
cos
12
4
1
Poynting Vector of the Electric Dipole Field
The Poynting Vector of the Electric Dipole Field is
The Magnetic and Electric Field Intensities are:
1sincossin
4
2
0
krt
c
krt
rr
p
HrealH
11 sinsin
1
cos
1
cossincos
12
4 2
2
2
0
0
krt
rc
krt
cr
krt
c
krt
rrr
p
ErealE r
1
1
cossincossinsincos
1
4
2
sincossinsincos
1
4
2
0
32
2
0
2
2
2
2
2
0
22
2
0
krt
c
krt
r
krt
c
krt
rr
p
krt
c
krt
r
krt
rc
krt
crr
p
HES r
The Poynting Vector of the Electric Dipole Field is given by:
44. 44
SOLO Electric Dipole Radiation
tq
tq
d
r
zSS rrdqP 10
dr
sinr
cosr
zyx
r
r
rr
111
1
cossinsincossin
r1
1
1
x1
y1
z1
Let compute the time average < > of the Poynting vector:
Poynting Vector of the Electric Dipole Field
Using the fact that:
1
1
cossincossinsincos
1
4
2
sincossinsincos
1
4
2
0
32
2
0
2
2
2
2
2
0
22
2
0
krt
c
krt
r
krt
c
krt
rr
p
krt
c
krt
r
krt
rc
krt
crr
p
HES r
T
T
dttS
T
S
0
1
lim
2
1
2cos
1
lim
2
11
lim
2
1
cos
1
limcos
0
0
1
00
22
T
T
T
T
T
T
dtrkt
T
dt
T
dtrkt
T
rkt
2
1
2cos
1
lim
2
11
lim
2
1
sin
1
limsin
0
0
1
00
22
T
T
T
T
T
T
dtrkt
T
dt
T
dtrkt
T
rkt
02sin
1
lim
2
1
cossin
1
limcossin
0
00
T
T
T
T
dtrkt
T
dtrktrkt
T
rktrkt
r
rc
p
S 1
2
23
0
2
42
0
sin
42
11 cossin
4
sin
1
42
22
0
32
2
02
2
2
2
2
2
2
0
22
2
0
rcrcr
p
rccrcr
p
S r
we obtain:
or: Radar Equation
Irradiance
45. 45
SOLO Electric Dipole Radiation
tq
tq
d
r
zSS rrdqP 10
dr
sinr
cosr
zyx
r
r
rr
111
1
cossinsincossin
r1
1
1
x1
y1
z1
Poynting Vector of the Electric Dipole Field
r
rc
p
S 1
2
23
0
2
42
0
sin
42
Radar Equation
45 90 135 1800
0
5
10
15
20
25
30
0
45
90
135
180
225
270
315
z
y
5.0 0.1
Polar Angle , in degrees
RelativePower,indb
The Total Average Radiant Power is:
0
22
23
0
2
42
0
sin2sin
42
dr
rc
p
dSSP
A
rad
22
0
22
120
123
0
42
0
3/4
0
3
23
0
42
0
40
12
sin
16 0
p
rc
p
d
rc
p
P
c
c
rad
3
4
3
2
3
2
cos
3
1
coscoscos1sin
0
3
0
2
0
3
dd
Return to TOC
46. 46
ELECTROMAGNETICSSOLO
To satisfy the Maxwell equations for a source free media we must have:
Monochromatic Planar Wave Equations
we haveUsing: 1ˆˆ&ˆˆ
0 kkknkkk
0
0
H
E
HjE
EjH
0ˆ
0ˆ
ˆ
ˆ
0
0
00
00
Hk
Ek
HEk
EHk
kˆ
Planar Wave
0E
0H
r
0
0
0
0
00
00
rkj
rkj
rkjrkj
rkjrkj
ekje
eHkj
eEkj
eHjeEkj
eEjeHkj
rkjrkj
22
22
&
2
ˆ
2
ˆ
HwEwwcn
k
wwcn
k
S meme
Time Average
Poynting Vector of
the Planar Wave
Reflections and Refractions Laws Development Using the Electromagnetic Approach
47. 47
SOLO REFLECTION & REFRACTION
iE
iH
rE
rH
ik
rk
tH
tE
tk
Boundary
21
ˆ n
z
x y
i
r
t
Plan of
incidence
Consider an incident
monochromatic planar
wave
c
n
k
eEkH
eEE
iiii
rktj
iii
rktj
ii
ii
ii
1
00
11
0011
0
0
The monochromatic planar
reflected wave from the boundary is
1
1
1
1
0
0
&
n
c
v
vc
n
k
eEkH
eEE
r
rr
rktj
rrr
rktj
rr
rr
rr
The monochromatic planar
refracted wave from the boundary is
2
2
2
2
0
0
&
n
c
v
vc
n
k
eEkH
eEE
t
tt
rktj
ttt
rktj
tt
tt
tt
Reflections and Refractions Laws Development Using the Electromagnetic Approach
48. 48
SOLO REFLECTION & REFRACTION
The Boundary Conditions at
z=0 must be satisfied at all points
on the plane at all times, implies
that the spatial and time
variations of
This implies that
iE
iH
rE
rH
ik
rk
tH
tE
tk
Boundary
21
ˆ n
z
x y
i
r
t
Plan of
incidence
Phase-Matching Conditions
yxteEeEeE
z
rktj
t
z
rktj
r
z
rktj
i
ttrrii
,,,,
0
0
0
0
0
0
yxtrktrktrkt
z
tt
z
rr
z
ii ,,
000
ttri
yxrkrkrk
z
t
z
r
z
i ,
000
must be the same
Reflections and Refractions Laws Development Using the Electromagnetic Approach
49. 49
SOLO REFLECTION & REFRACTION
tri nnn sinsinsin 211
iE
iH
rE
rH
ik
rk
tH
tE
tk
Boundary
21
ˆ n
z
x y
i
r
t
Plan of
incidence
Phase-Matching Conditions
zyx
c
n
k
zyx
c
n
k
ttttttt
irirrrr
ˆcossinˆsinsinˆcos
ˆcossinˆsinsinˆcos
2
1
yyx
c
n
rk
yx
c
n
rk
y
c
n
rk
ttt
z
t
irr
z
r
i
z
i
ˆsinsincos
sinsincos
sin
2
0
1
0
1
0
yxrkrkrk
z
t
z
r
z
i ,
000
2
tr
ttri
x
y
Coplanar
Snell’s Law
zzyyxxr
zy
c
n
k iiii
ˆˆˆ
ˆcosˆsin1
Given:
Let find:
Reflections and Refractions Laws Development Using the Electromagnetic Approach
50. 50
SOLO REFLECTION & REFRACTION
Second way of writing phase-matching equations
ri
11
22
2
1
1
2
sin
sin
v
v
n
n
t
iRefraction
Law
Reflection
Law
Phase-Matching Conditions
zzyyxxr
zy
c
n
k iiii
ˆˆˆ
ˆcosˆsin1
zyx
c
n
k
zyx
c
n
k
ttttttt
irirrrr
ˆcossinˆsinsinˆcos
ˆcossinˆsinsinˆcos
2
1
ynnyn
c
kkz
ynnyn
c
kkz
ittrti
irrrri
ˆsinsinsinˆcosˆ
ˆsinsinsinˆcosˆ
122
111
ttri
We can see that
tri
tiri kkzkkz 0ˆˆ
tri
tri
tr
nnn sinsinsin
2/
211
iE
iH
rE
rH
ik
rk
tH
tE
tk
Boundary
21
ˆ n
z
x y
i
r
t
Plan of
incidence
Reflections and Refractions Laws Development Using the Electromagnetic Approach
51. 51
SOLO REFLECTION & REFRACTION
ri
11
22
2
1
1
2
sin
sin
v
v
n
n
t
iRefraction
Law
Reflection
Law
Phase-Matching Conditions (Summary)
ttri
tri
tiri kkzkkz 0ˆˆ
tri
tri
tr
nnn sinsinsin
2/
211
iE
iH
rE
rH
ik
rk
tH
tE
tk
Boundary
21
ˆ n
z
x y
i
r
t
Plan of
incidence
yxrkrkrk
z
t
z
r
z
i ,
000
yxtrktrktrkt
z
tt
z
rr
z
ii ,,
000
Vector
Notation
Scalar
Notation
Reflections and Refractions Laws Development Using the Electromagnetic Approach
52. 52
SOLO REFLECTION & REFRACTION
iE
iH
rE
rH
ik
rk
tH
tE
tk
Boundary
21
ˆ n
z
x y
i
r
t
i r
t
tH
tE
tk
rH
rk
rE
iH
iE
ik
21
ˆ n
Boundary
Plan of
incidence
ti
ti
i
r
nn
nn
E
E
r
coscos
coscos
2
2
1
1
2
2
1
1
0
0
ti
i
i
t
nn
n
E
E
t
coscos
cos2
2
2
1
1
1
1
0
0
For most of media μ1= μ2 ,
and using refraction law: 1
2
sin
sin
n
n
t
i
ti
ti
i
r
E
E
r
sin
sin21
0
0
ti
it
i
t
E
E
t
sin
cossin221
0
0
Assume is normal to plan of incidence
(normal polarization)
E
xEExEExEE ttrrii
ˆ&ˆ&ˆ 000000
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Fresnel Equations
See full development in P.P.
“Reflection & Refractions”
53. 53
SOLO REFLECTION & REFRACTION
iE
iH
rE
rH
ik
rk
tH
tE
tk
Boundary
21
ˆ n
z
x y
i
r
t
i r
t
tH
tE
tk
rH
rk
rE
iH
iE
ik
21
ˆ n
Boundary
Plan of
incidence
Assume is parallel to plan of incidence
(parallel polarization)
E
zyEE
zyEE
zyEE
tttt
rrrr
iiii
ˆsinˆcos
ˆsinˆcos
ˆsinˆcos
0||0
0||0
0||0
ti
ti
i
r
nn
nn
E
E
r
coscos
coscos
1
1
2
2
1
1
2
2
||0
0
||
ti
i
i
t
nn
n
E
E
t
coscos
cos2
1
1
2
2
1
1
||0
0
||
For most of media μ1= μ2 ,
and using refraction law: 1
2
sin
sin
n
n
t
i
ti
ti
i
r
E
E
r
tan
tan21
||0
0
||
titi
it
i
t
E
E
t
cossin
cossin221
||0
0
||
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Fresnel Equations
See full development in P.P.
“Reflection & Refractions”
54. 54
SOLO REFLECTION & REFRACTION
ti
ti
i
r
nn
nn
E
E
r
coscos
coscos
1
1
2
2
1
1
2
2
||0
0
||
ti
i
i
t
nn
n
E
E
t
coscos
cos2
1
1
2
2
1
1
||0
0
||
ti
ti
i
r
nn
nn
E
E
r
coscos
coscos
2
2
1
1
2
2
1
1
0
0
ti
i
i
t
nn
n
E
E
t
coscos
cos2
2
2
1
1
1
1
0
0
The equations of reflection and refraction ratio
are called Fresnel Equations, that first
developed them in a slightly less general form in
1823, using the elastic theory of light.
Augustin Jean Fresnel
1788-1827
The use of electromagnetic approach to
prove those relations, as described above, is
due to H.A. Lorentz (1875)
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Hendrik Antoon Lorentz
1853-1928
See full development in P.P.
“Reflection & Refractions”
Return to TOC
55. 55
IR Radiometric QuantitiesSOLO
RTA DA
2
cm 2
cm
TARGET
SOURCE
DETECTOR
RECEIVER
Radiation Flux Power W
Spectral Radial Power
m
W
Irradiance
2
mc
W
A
E
Spectral Radiant
Emittance
mmc
WM
M
2
Radiant Intensity
str
W
I
Spectral Radiant
Intensity
mstr
WI
I
Radiance
strmc
W
A
I
L 2
cos
Spectral Radiance
mstrmc
WL
L
2
Radiant Emittance
2
mc
W
A
M
Spectral
Irradiance
mmc
WE
E
2
T
T
dttS
T
S
0
1
lim
Irradiance is the time-
average of the Poynting
vector
Return to TOC
56. 56
Physical Laws of RadiometrySOLO
Plank’s Law
1/exp
1
2
5
1
Tc
c
M
BB
Plank 1900
Plank’s Law applies to blackbodies; i.e. perfect radiators.
The spectral radial emittance of a blackbody is given by:
KT
KWk
Wh
kmc
Kmkhcc
mcmWchc
ineTemperaturAbsolute-
constantBoltzmannsec/103806.1
constantPlanksec106260.6
lightofspeedsec/458.299792
10439.1/
107418.32
23
234
4
2
4242
1
MAX
PLANCK
(1858 - 1947)
Plank’s Law
57. 57
Physical Laws of RadiometrySOLO
Plank’s Law
1/exp
1
2
5
1
Tc
c
M
BB
Plank 1900
Plank’s Law applies to blackbodies; i.e. perfect radiators.
The spectral radial emittance of a blackbody is given by:
MAX
PLANCK
(1858 - 1947)
Plank’s Law
58. 58
Physical Laws of Radiometry (Continue -1)SOLO
Wien’s Displacement Law
0
d
Md
BB
Wien 1893
from which:
The wavelength for which the spectral emittance of a blackbody reaches the maximum
is given by:
m
KmTm
2898 Wien’s Displacement Law
Stefan-Boltzmann Law
Stefan – 1879 Empirical - fourth power law
Boltzmann – 1884 Theoretical - fourth power law
For a blackbody:
42
12
32
45
2
4
0 2
5
1
0
10670.5
15
2
:
1/exp
1
Kcm
W
hc
k
cm
W
Td
Tc
c
dMM
BBBB
LUDWIG
BOLTZMANN
(1844 - 1906)
WILHELM
WIEN
(1864 - 1928)
Stefan-Boltzmann Law
JOSEF
STEFAN
(1835 – 1893)
59. 59
Physical Laws of Radiometry (Continue -1a)SOLO
Black Body Emittance M [W/m2]
M (300ºK) 5.86 121
M (301ºK) - M (300ºK) 0.22 2
M (600ºK) 1,719 1,555
M (601ºK) - M (600ºK) 17 7
3 – 5 µm 8 - 12 µm
60. 60
Physical Laws of Radiometry (Continue -2)SOLO
Emittance of Real Bodies (Gray Bodies)
For real (gray) bodies:
BB
MM
- Directional spectral emissivity is a measure of how closely the flux
radiated from a given temperature radiator approaches that from a
blackbody at the same temperature
,
BB
M
M
61. 61
Physical Laws of Radiometry (Continue -3)SOLO
Kirchhoff’s Law
rM
iE aE
tM
Gustav Robert
Kirchhoff
1824-1887
- Incident IrradianceiE
- Absorbed IrradianceaE
- Reflected Radiant ExcitancerM
- Transmitted Radiant ExcitancetM
Law of Conservation of Energy: trai MMEE
i
t
i
r
i
a
E
M
E
M
E
E
11
i
a
E
E
: - fraction of absorbed energy (absorptivity)
i
r
E
M
: - fraction of reflected energy (reflectivity)
i
t
E
M
: - fraction of transmitted energy (transmissivity)
Opaque body (no transmission): 01
Blackbody (no reflection or transmission): 0&01
Sharp boundary (no absorption): 01
62. 62
Physical Laws of Radiometry (Continue -4)SOLO
Kirchhoff’s Law (Continue – 1)
Gustav Robert
Kirchhoff
1824-1887
Kirchhoff’s Law (1860) states that, for any temperature and any
wavelength, the emissivity of an opaque body in an isothermal
enclosure is equal to it’s absorptivity.
This is because if the body will radiate to the surrounding less than it absorbs it’s
temperature will rise above the surrounding and will be a transfer of energy from a
cold surrounding to a hot body contradicting the second law of thermodynamics.
TT
222
,, T
2A
111
,, T
1A
63. 63
Physical Laws of Radiometry (Continue -5)SOLO
Lambert’s Law
Johann Heinrich
Lambert
1728 - 1777
http://www-groups.dcs.st-andrews.ac.uk/~history/Biographies/Lambert.html
A Lambertian Surface is defined as a surface from which the radiance
L [W/(cm2 str)] is independent of the direction of radiation.
dr
d
rsin
2
sin
r
drdr
d
A
cosAAn
z
x
y
0
2
cos
, L
A
L
coscos, 00 IALI
Lambert’s Law
0
2
0
2/
0
00 sincoscos LddLdL
A
M
The Radiant Intensity from a Lambertian Surface is
The Radiant Emittance (Exitance) from
a Lambertian Surface is
64. 64
Physical Laws of Radiometry (Continue -6)SOLO
Transfer of Radiant Energy
We have two bodies 1 and 2.
The radiant power (radiance) transmitted from 1 to 2 is:
2
12
22
22
211
12
2
1
cos
&
cos R
Ad
d
strcm
W
A
L
1A
1dA
1 12R
Radiating
(Source)
Surface
2A
2dA
Receiving
Surface
2
2d
2
12
22111
12
coscos
R
AdAdL
d
The total radiant power (radiance) received at surface A2 from A1 is:
2 1
212
12
211
12
coscos
A A
AdAd
R
L
65. 65
Physical Laws of Radiometry (Continue -7)SOLO
Transfer of Radiant Energy (Continue – 1)
Define the projected areas:
and the solid angles:
222111 cos&cos AdAdAdAd nn
1A
1dA
1 12R
Radiating
(Source)
Surface
2A
2dA
Receiving
Surface
2
2d
2
12
22
22
12
11
1
cos
&
cos
R
Ad
d
R
Ad
d
1A
1dA
1 12R
Radiating
(Source)
Surface
2A
2dA
Receiving
Surface
2
1d
then:
2
12
211
2
12
22111
12
coscos
R
AdAdL
R
AdAdL
d nn
12121112 dAdLdAdLd nn
The Power is the product of the Radiance, the projected Area, and the Solid Angle
using the other area.
66. 66
Physical Laws of Radiometry (Continue -8)SOLO
Transfer of Radiant Energy (Continue – 2)
Optics
R f
ATARGET
ADETECTOR
AOPTICS
TO, OD,OT , DO,
For an Optical System define:
ATARGET – Target Area
ADETECTOR – Detector Area
AOPTICS – Optics Area
R – Range from Target to Optics
f – Focal Length (from Optics to Detector)
ΩO,T – solid angle of Optics as seen from the Target
2,
R
AOPTICS
TO
ωT,O – solid angle of Target as seen from the Optics
2,
R
ATARGET
OT
ΩD,O – solid angle of Detector as seen from the Optics 2,
f
ADETECTOR
OD
ωO,D – solid angle of Optics as seen from the Detector 2,
f
ADETECTOR
DO
67. 67
Physical Laws of Radiometry (Continue -9)SOLO
Transfer of Radiant Energy (Continue – 3)
Optics (continue – 1)
R f
ATARGET
ADETECTOR
AOPTICS
TO, OD,OT , DO,
For the Figure we can see that:
ODOT ,,
22
f
A
R
A DETECTORTARGET
Also we found that:
DODETECTOROTOPTICS
ODOPTICSTOTARGET
ALAL
ALAL
OTOD
,,
,,
,,
68. 68
Physical Laws of Radiometry (Continue -10)SOLO
Transfer of Radiant Energy (Continue – 4)
Optics (continue – 2)
R f
ATARGET
ADETECTOR
AOPTICS
TO, OD,OT , DO,
R f
ATARGET
ADETECTOR
AOPTICS
TO, OD,OT , DO,
R f
ATARGET
ADETECTOR
AOPTICS
TO, OD,OT , DO,
R f
ATARGET
ADETECTOR
AOPTICS
TO, OD,OT , DO,
TOTARGETAL ,
ODOPTICSAL ,
OTOPTICSAL ,
DODETECTORAL ,
69. 69
Physical Laws of Radiometry (Continue -11)SOLO
Transfer of Radiant Energy (Continue – 5)
Optics (continue – 3)
2
,
R
A
AL
AL
TARGET
DETECTOR
DTDETECTOROpticsNo
2
,
f
A
AL
AL
OPTICS
DETECTOR
DODETECTOROpticsWith
R f
ATARGET
ADETECTOR
AOPTICS
TO, OD,OT , DO,
R
ATARGET
ADETECTOR
TD,
DT ,
• IR Detector without Optics
• IR Detector with Optics
2
#
/
2
4
0
4
4 0#
f
AL
f
D
AL DETECTOR
Dff
DETECTOR
The Optics increases the energy collected by the Detector
since DTDO ,,
22
#
2
4 R
A
ff
A TARGETOPTICS
OpticsNoOpticsWith
70. 70
Physical Laws of Radiometry (Continue -12)SOLO
Targets
The parts of the aircraft that are especially hot are:
• The exhaust nozzle of the jet engine
• The hot exhaust gas area, or the plume
• The areas in which aerodynamic heating is the highest
75. 75
Physical Laws of Radiometry (Continue -17)SOLO
Targets (continue – 1)
• The exhaust nozzle of the jet engine
The exhaust nozzle can be regarded as a gray body with ε = 0.9.
Example: Turbojet Engine 4-P&W JT4A-9
2
3660 cmANOZZLE
rafterburnewithCT
538
24124
207.22735381067.59.0
cmWTM
We are interested only in the band 3 μm ≤ λ ≤ 5 μm.
By numerically integration or using infrared radiation
calculators we obtain: 397.0
811
4
5
3
KT
BB
T
dM
Hence:
2
876.0207.2397.053
cmWmmM
In a tail-on situation the radiant intensity is:
1
1020
3660876.0
53
strWA
M
mmI NOZZLE
Lambertian
76. 76
Physical Laws of Radiometry (Continue -18)SOLO
Targets (continue – 2)
• The plume
The plume is characterized by the radiant emittance of the hot gases that are expanding
into the atmosphere after passing through the exhaust nozzle.
The products of combustion are H2O, CO2, some times CO (incomplete combustion),
OH, HF, HCl.
The infrared emission is produced by changes in the energy contained in the molecular
vibrations and rotations, only at certain frequencies..
77. 77
Physical Laws of Radiometry (Continue -19)SOLO
Targets (continue – 3)
• The plume (continue – 1)
78. 78
Physical Laws of Radiometry (Continue -20)SOLO
Targets (continue – 4)
• The plume (continue – 2)
Breathing engines have exhaust plume temperatures of
K
600450 Cruise flight
K
800600 Maximum Un-augmented Thrust
K
15001000 Augmented (After burner) Thrust
Rockets have exhaust plume temperatures of
K
75002500 Liquid propellant
K
35001700 Solid propellant
Example
Assume:
mm 55.433.45.0
KCCTPLUME
643273370
then:
2222
55.4
33.4
1075.1105.35.0
cmWcmWdMM
For a plume surface of APLUME = 10000 cm2 = 1 m2 the Radiant Intensity is:
1
42
8.27
101075.1
55.433.4
strWA
M
mmI PLUME
Lambertian
79. 79
Physical Laws of Radiometry (Continue -21)SOLO
Targets (continue – 5)
• Aerodynamic Heating
The Target body is heated by the compression and friction of the air against it’s
surface and by friction. Assuming a negligible friction effect and an adiabatic
compression the Target skin temperature is given by:
2
0
2
1
1, MachrMachHTT
- air temperature at altitude HTARGET and mach number Mach MachHT ,0
- recovery factorr
vp CC / - specific heat ratio = 1.4 for air
Example
Mach = 2.0, HTARGET = 5000 m
27.0,250.2,50000 KMachmHT
then KT
4142
2
14.1
82.01250 2
23
5
3
1066.1414
cmWdKTMM
assume 2
15mATARGET
1
43
3.79
10151066.1
53
strWA
M
mmI TARGET
Lambertian
83. 83
Physical Laws of Radiometry (Continue -25)SOLO
Sun, Background and Atmosphere
84. 84
Physical Laws of Radiometry (Continue -26)SOLO
Sun, Background and Atmosphere (continue – 1)
The spectrum distribution of the sun radiation is like a black body with a temperature of
T = 5900 °K
From Wien’s Law the maximum of Mλ is at
m
T
m 49.0
5900
28982898
This is almost at the middle of the visible spectrum mm 75.040.0
Loss by Scattering
85. 85
Physical Laws of Radiometry (Continue -27)SOLO
Sun, Background and Atmosphere (continue – 2)
Atmosphere
Atmosphere affects electromagnetic radiation by
3.2
1
1
R
kmRR
• Absorption
• Scattering
• Emission
• Turbulence
Atmospheric Windows:
Window # 2: 1.5 μm ≤ λ < 1.8 μm
Window # 4 (MWIR): 3 μm ≤ λ < 5 μm
Window # 5 (LWIR): 8 μm ≤ λ < 14 μm
For fast computations we may use the transmittance equation:
R in kilometers.
Window # 1: 0.2 μm ≤ λ < 1.4 μm
includes VIS: 0.4 μm ≤ λ < 0.7 μm
Window # 3 (SWIR): 2.0 μm ≤ λ < 2.5 μm
86. 86
Physical Laws of Radiometry (Continue -28)SOLO
Sun, Background and Atmosphere (continue – 3)
Atmosphere Absorption over Electromagnetic Spectrum
87. 87
Physical Laws of Radiometry (Continue -29)SOLO
Sun, Background and Atmosphere (continue – 4)
Rain Attenuation over Electromagnetic Spectrum
FREQUENCY GHz
ONE-WAYATTENUATION-Db/KILOMETER
WAVELENGTH
88. 88
Physical Laws of Radiometry (Continue -30)SOLO
Sun, Background and Atmosphere (continue – 3)
Add scanned Figure from McKenzie
Atmosphere (continue – 1)
90. 90
SOLO
DERIVATION OF EIKONAL EQUATION
Foundation of Geometrical Optics
Derivation from Maxwell Equations
Consider a general time-harmonic field:
tjrHtjrHtjrHaltrH
tjrEtjrEtjrEaltrE
exp,exp,
2
1
exp,Re,
exp,exp,
2
1
exp,Re,
*
*
in a non-conducting, far-away from the sources 0,0 eeJ
No assumption of isotropy of the medium are made; i.e.: rr ,
Far from sources, in the High Frequencies we can write using the phasor notation:
00000 &,&, 00
kerHrHerErE rSjkrSjk
Note
The minus sign was chosen to get a progressive wave:
End Note
SktjSktj
erHaltrHerEaltrE 00
00 Re,&Re,
James Clerk Maxwell
(1831-1879)
See full development in P.P.
“Foundation of Geometrical Optics”
91. 91
SOLO
From those equations we have
Foundation of Geometrical Optics
Sjktj
SjkSjktjSjktjtj
eeESjkE
EeeEeeEeerE
0
000
000
000,
Sjk
SjktjSjktjtj
eHjk
eHejeHejerH
t
0
00
0
00
0
0
00
000
1
1
,
from which
0
00
0000 HjkESjkEF
and
0
1 0
0
0
0
00
0
k
E
jk
HES
DERIVATION OF EIKONAL EQUATION (continue – 2)
Derivation from Maxwell Equations (continue – 2)
92. 92
SOLO
From Maxwell equations we also have
Foundation of Geometrical Optics
from which
and
DERIVATION OF EIKONAL EQUATION (continue – 3)
Derivation from Maxwell Equations (continue – 3)
Sjktj
SjkSjktjSjktjtj
eeHSjkH
HeeHeeHeerH
0
000
000
000,
Sjk
SjktjSjktjtj
eEjk
eEejeEejerE
t
0
00
0
00
0
0
00
000
1
1
,
0
00
0000 EjkHSjkHA
0
1 0
0
0
0
00
0
k
H
jk
EHS
93. 93
SOLO
DERIVATION OF EIKONAL EQUATION (continue – 4)
Foundation of Geometrical Optics
Derivation from Maxwell Equations (continue – 4)
We have Faradey (F), Ampére (A), Gauss Electric (GE), Gauss Magnetic (GM)
equations:
0
0
HGM
EGE
EjHA
HjEF
0&0
2
0 0
0
ee
e
e
J
c
k
j
t
HB
ED
BGM
DGE
J
t
D
HA
t
B
EF
André-Marie Ampère
1775-1836
Michael Faraday
1791-1867
Karl Friederich Gauss
1777-1855
94. 94
SOLO
From Maxwell equations we also have
Foundation of Geometrical Optics
from which
and
DERIVATION OF EIKONAL EQUATION (continue – 4)
Derivation from Maxwell Equations (continue – 4)
0
,
0
000
0000
000
Sjktj
SjkSjktjSjktjtj
eeESjkEE
EeeEeeEeerE
00000 ESjkEEGE
0
1 0
00
0
0
k
EE
jk
ES
We also have
from which
and
0
,
0
000
0000
000
Sjktj
SjkSjktjSjktjtj
eeHSjkHH
HeeHeeHeerH
00000 HSjkHHGM
0
1 0
00
0
0
k
HH
jk
HS
95. 95
SOLO
To summarize, from k0 → ∞ we have
Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 5)
Derivation from Maxwell Equations (continue – 5)
00
00
0 HESF
00
00
0 EHSA
00 ESGE
00 HSGM
We will use only the first two equations, because the last two may be obtained from
the previous two by multiplying them (scalar product) by .S
96. 96
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 6)
Derivation from Maxwell Equations (continue – 6)
00
00
0 HESF
00
00
0 EHSA
From the second equation we obtain
0
00
0 HSE
And by substituting this in the first equation
00 0
00
00
00
0
00
HHSSHHSS
But
2
00
0
2
0
0
00
n
HSHSSSHSHSS
97. 97
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 7)
Derivation from Maxwell Equations (continue – 7)
Finally we obtain
00
22
HnS
or
zyxn
z
S
y
S
x
S
ornS ,,0 2
222
22
S is called the eikonal (from Greek έίκων = eikon → image) and the equation is
called Eikonal Equation.
Return to TOC
98. 98
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS
From Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3
00000 &,&, 00
kerHrHerErE rSjkrSjk
We found the following relations
00
00
0 HESF
00
00
0 EHSA
00 ESGE
00 HSGM
We can see that the vectors are perpendicular in the same way as the
vectors for the planar waves (where is the Poynting vector).
SHE ,, 00
SHE
,, 00 00 HES
S
0E
0H
99. 99
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS
(continue – 1)
T
T
T
TT
e
dttjrErErEtjrE
T
dttjrEtjrEtjrEtjrE
T
dttjrEal
T
dttrEtrE
T
dttrDtrE
T
w
0
2**2
0
**
0
2
00
2exp,,,22exp,
4
1
exp,exp,exp,exp,
4
1
exp,Re
1
,,
1
,,
1
But
0
2
2exp
2exp
2
1
2exp
1
0
2
2exp
2exp
2
1
2exp
1
0
0
0
0
T
T
T
T
T
T
Tj
Tj
tj
Tj
dttj
T
Tj
Tj
tj
Tj
dttj
T
Therefore
rErEerEerEdt
T
rErEw rSjkrSjk
T
e
*
00
*
00
0
*
22
1
,,
2
00
Let compute the time averages of the electric and magnetic energy densities
100. 100
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS
(continue – 2)
In the same way
rErEerEerEdt
T
rErEw rSjkrSjk
T
e
*
00
*
00
0
*
22
1
,,
2
00
rHrHdttrHtrH
T
dttrBtrH
T
w
TT
m
*
00
00
2
,,
1
,,
1
Using the relations
0
00
0 HSEA
0
00
0 ESHF
since and are real values , where * is the
complex conjugate, we obtain
S )**,( SS
e
m
e
wrHSrErHSrErHSrE
rESrHrESrHrHrHw
rHSrErHSrErErEw
*
00
*
0
*
00
**
0
*
00
*
0
00
0
*
00
*
00
*
0
00
0
*
00
2
1
2
1
2
1
2
1
22
2
1
22
S
0E
0H
101. 101
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS
(continue – 3)
Therefore
*
00
2
1
rHSrEww me
Within the accuracy of Geometrical Optics, the time-averaged electric and
magnetic energy densities are equal.
*
0000
*
00
22
rHSrErHrHrErEwww me
The total energy will be:
The Poynting vector is defined as: trHtrEtrS ,,:,
T
tjtjtjtj
T
tjtj
T
dterHerHerEerE
T
dterHerEal
T
dttrHtrE
T
trHtrES
0
**
00
,,
2
1
,,
2
11
,,Re
1
,,
1
,,
,,,,
4
1
,,,,,,,,
4
11
**
0
2****2
rHrErHrE
dterHrErHrErHrEerHrE
T
T
tjtj
rHrErHrE
erHerEerHerE rSjkrSjkrSjkrSjk
0
*
0
*
00
)(
0
)(*
0
)(*
0
)(
0
4
1
4
1 0000
The time average of the Poynting vector is:
John Henry Poynting
1852-1914
103. 103
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS
(continue – 4)
Using 22
nS Eikonal Equation
we obtain nS
Define snS
n
S
S
S
s ˆ:ˆ
We have swvwS
n
c
S
n
c
v
ˆ
2
1
2 2
sˆ
constS
constdSS
sˆ
r
0
ˆs
0r
A Bundle of Light Rays
104. 104
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS
(continue – 5)
swvwS
n
c
S
n
c
v
ˆ
2
1
2 2
sˆ
constS
constdSS
sˆ
r
0
ˆs
0r
From this equation we can see that average Poynting vector is the direction of
the normal to the geometrical wave-front , and its magnitude is proportional to the
product of light velocity v and the average energy density, therefore we say that
defines the direction of the light ray.
S
sˆ
sˆ
Suppose that the vector describes the light path, then the unit vector
is given by
r
sˆ
sd
rd
rd
rd
s
ray
ray
ray
ˆ
where is the differential of an arc length along the ray pathrayrdsd
105. 105
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS
(continue – 6)
Let substitute in and differentiate it with respect to s.
sd
rd
rd
rd
s
ray
ray
ray
ˆ rayrdsd
S
sd
d
sd
rd
n
sd
d
ray
S
sd
rd
ray
sd
rd
f
sd
zd
zd
fd
sd
yd
yd
fd
sd
xd
xd
fd
sd
zyxfd
,,
SS
n
1 S
sd
rd
n
ray
ABBAABBABA
AB
AAAAAA
2
1
SA
SSSSSSSS
0
2
1 SS
n
2
1
2
nSS 2
2
1
n
n
n
106. 106
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS
(continue – 7)
Therefore we obtained
nS
sd
d
and
n
sd
rd
n
sd
d
ray
We obtained a ordinary differential equation of 2nd order that enables to find the
trajectory of an optical ray , giving the relative index and the initial
position and direction of the desired ray.
srray
zyxn ,,
00 rrray
0
ˆs
sˆ
constS
constdSS
sˆ
r
0
ˆs
0r
We can transform the 2nd order differential equation in two 1st order
differential equations by the following procedure. Define
Ssn
sd
rd
np ˆ:
ray
We obtain
0
ˆ0 snpnp
sd
d
0
ˆ0 snpnp
sd
d
Return to TOC
107. 107
SOLO
The Three Laws of Geometrical Optics
1. Law of Rectilinear Propagation
In an uniform homogeneous medium the propagation of an optical disturbance is in
straight lines.
2. Law of Reflection
An optical disturbance reflected by a surface has the
property that the incident ray, the surface normal,
and the reflected ray all lie in a plane,
and the angle between the incident ray and the
surface normal is equal to the angle between the
reflected ray and the surface normal:
2v
1v
Refracted Ray
21
ˆ n
2n
1n
i
t
Reflected Ray
21
ˆ n
2n
1n
i r
3. Law of Refraction
An optical disturbance moving from a medium of
refractive index n1 into a medium of refractive index
n2 will have its incident ray, the surface normal between
the media , and the reflected ray in a plane,
and the relationship between angle between the incident
ray and the surface normal θi and the angle between the
reflected ray and the surface normal θt given by
Snell’s Law: ti nn sinsin 21
ri
“The branch of optics that addresses the limiting case λ0 → 0, is known as Geometrical Optics, since in
this approximation the optical laws may be formulated in the language of geometry.”
Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3
Foundation of Geometrical Optics
Return to TOC
108. 108
SOLO Foundation of Geometrical Optics
Fermat’s Principle (1657)
1Q
1P
2P
2Q
1Q
2Q
1S
SdSS 12
2PS
1PS
2'Q
rd
sˆ
sˆ
The Principle of Fermat (principle of the shortest optical path) asserts that the optical
length
of an actual ray between any two points is shorter than the optical ray of any other
curve that joints these two points and which is in a certai neighborhood of it.
An other formulation of the Fermat’s Principle requires only Stationarity (instead of
minimal length).
2
1
P
P
dsn
An other form of the Fermat’s Principle is:
Princple of Least Time
The path following by a ray in going from one point in
space to another is the path that makes the time of transit of
the associated wave stationary (usually a minimum).
The idea that the light travels in the shortest path was first put
forward by Hero of Alexandria in his work “Catoptrics”,
cc 100B.C.-150 A.C. Hero showed by a geometrical method
that the actual path taken by a ray of light reflected from plane
mirror is shorter than any other reflected path that might be
drawn between the source and point of observation.
109. 109
SOLO
1. The optical path is reflected at the boundary between two regions
0
21
21
rd
sd
rd
n
sd
rd
n
rayray
In this case we have and21 nn
0ˆˆ
21
21
rdssrd
sd
rd
sd
rd rayray
We can write the previous equation as:
i.e. is normal to , i.e. to the
boundary where the reflection occurs.
21
ˆˆ ss rd
0ˆˆˆ 2121 ssn
11
ˆsn
21
ˆsn
1121
ˆˆˆ snsn
rd
0ˆˆ 121 rdssn
Reflected Ray
21
ˆ n
1n
i r
REFLECTION & REFRACTION
Reflection Laws Development Using Fermat Principle
ri Incident ray and Reflected ray are in the
same plane normal to the boundary.
This is equivalent with:
&
110. 110
SOLO
2. The optical path passes between two regions with different refractive indexes
n1 to n2. (continue – 1)
0
21
21
rd
sd
rd
n
sd
rd
n
rayray
where is on the boundary between the two regions andrd
sd
rd
s
sd
rd
s
rayray 2
:ˆ,
1
:ˆ 21
rd
22
ˆsn
11
ˆsn
1122
ˆˆˆ snsn
0ˆˆˆ 1122 rdsnsn
Refracted Ray
21
ˆ n
2n
1n i
t
Therefore is normal to .2211
ˆˆ snsn rd
Since can be in any direction on the
boundary between the two regions
is parallel to the unit
vector normal to the boundary surface,
and we have
rd
2211
ˆˆ snsn 21
ˆ n
0ˆˆˆ 221121 snsnn
We recovered the Snell’s Law from
Geometrical Optics
REFLECTION & REFRACTION
Refraction Laws Development Using Fermat Principle
ti nn sinsin 21
Incident ray and Refracted ray are in the
same plane normal to the boundary.
&
Return to TOC
111. 111
SOLO
Plane-Parallel Plate
i
r
ri r
t l
d
i
A
C
B
E
2n
1n
A single ray traverses a glass plate with parallel surfaces and emerges parallel to its
original direction but with a lateral displacement d.
Optics
irriri lld cossincossinsin
r
t
l
cos
r
i
ritd
cos
cos
sinsin
ir nn sinsin 0Snell’s Law
n
n
td
r
i
i
0
cos
cos
1sin
For small anglesi
n
n
td i
0
1
112. 112
SOLO
Plane-Parallel Plate (continue – 1)
t
r
i
i
n
n
td
cos
cos
1sin
1
2
1n
2n
i
r
r
i
i n
n
t
d
l
cos
cos
1
sin 1
2
l
Two rays traverse a glass plate with parallel surfaces and emerge parallel to their
original direction but with a lateral displacement l.
Optics
irriri lld cossincossinsin
r
t
l
cos
r
i
ritd
cos
cos
sinsin
ir nn sinsin 0Snell’s Law
n
n
td
r
i
i
0
cos
cos
1sin
r
i
i n
n
t
d
l
cos
cos
1
sin
0
For small anglesi
n
n
tl 0
1
Return to TOC
113. 113
SOLO
Prisms
2i1i
1t
11 ti
2t
22 it
Type of prisms:
A prism is an optical device that refract, reflect or disperse light into its spectral
components. They are also used to polarize light by prisms from birefringent media.
Optics - Prisms
2. Reflective
1. Dispersive
3. Polarizing
114. 114
OpticsSOLO
Dispersive Prisms
2i1i
1t
11 ti
2t
22 it
2211 itti
21 it
21 ti
202 sinsin ti nn Snell’s Law
10 n
1
1
2
1
2
sinsinsinsin tit
nn
11
21
11
1
2 sincossin1sinsinsincoscossinsin ttttt nn
Snell’s Law 110 sinsin ti nn
11 sin
1
sin it
n
1
2/1
1
221
2
sincossinsinsin iit
n
1
2/1
1
221
1
sincossinsinsin iii
n
The ray deviation angle is
10 n
116. 116
OpticsSOLO
Prisms
2i1i
1t
11 ti
2t
22 it
1
2/1
1
221
1
sincossinsinsin iii
n
21 ti
Let find the angle θi1 for which the deviation angle δ is minimal; i.e. δm.
This happens when
01
0
11
2
1
ii
t
i d
d
d
d
d
d
Taking the differentials
of Snell’s Law equations
22 sinsin tin
11 sinsin ti n
2222 coscos iitt dnd
1111 coscos ttii dnd
Dividing the equations
1
2
1
2
1
1
2
1
2
1
cos
cos
cos
cos
i
t
i
t
t
i
t
i
d
d
d
d
2
22
1
22
2
2
2
2
1
2
2
2
1
2
2
2
1
2
sin
sin
/sin1
/sin1
sin1
sin1
sin1
sin1
t
i
t
i
i
t
t
i
n
n
n
n
1
1
2
i
t
d
d
21 it
1
2
1
i
t
d
d
2
2
1
2
2
2
1
2
cos
cos
cos
cos
i
t
t
i
21 ti
1n
117. 117
OpticsSOLO
Prisms
2i1i
1t
11 ti
2t
22 it
1
2/1
1
221
1
sincossinsinsin iii
n
We found that if the angle θi1 = θt2 the deviation angle δ is minimal; i.e. δm.
Using the Snell’s Law
equations
22 sinsin tin
11 sinsin ti n 21 ti
21 it
This means that the ray for which the deviation angle δ is minimum passes through
the prism parallel to it’s base.
2i
1i
1t
m
11 ti
2t
22 it
21 ti 21 it
Find the angle θi1 for
which the deviation
angle δ is minimal; i.e.
δm (continue – 1).
118. 118
OpticsSOLO
Prisms
1
2/1
1
221
1
sincossinsinsin iii
n
Using the Snell’s Law 11 sinsin ti n
21 it
This equation is used for determining the refractive index of transparent substances.
2i
1i
1t
m
11 ti
2t
22 it
21 ti 21 it
21 it
21 ti
21 ti
m
2/1 t
12 im
2/1 mi
2/sin
2/sin
m
n
Find the angle θi1 for
which the deviation
angle δ is minimal; i.e.
δm (continue – 2).
119. 119
OpticsSOLO
Prisms
The refractive index of transparent substances varies with the wavelength λ.
1
2/1
1
221
1
sincossinsinsin iii
n
2i1i
1t
11 ti
2t
22 it
120. 120
OpticsSOLO
http://physics.nad.ru/Physics/English/index.htm
Prisms
υ [THz]λ0 (nm)Color
384 – 482
482 – 503
503 – 520
520 – 610
610 – 659
659 - 769
780 - 622
622 - 597
597 - 577
577 - 492
492 - 455
455 - 390
Red
Orange
Yellow
Green
Blue
Violet
1 nm = 10-9m, 1 THz = 1012 Hz
1
2/1
1
221
1
sincossinsinsin iii
n
In 1672 Newton wrote “A New Theory about Light and Colors” in which he said that
the white light consisted of a mixture of various colors and the diffraction was color
dependent.
Isaac Newton
1542 - 1727
121. 121
SOLO
Dispersing Prisms
Pellin-Broca Prism
Abbe Prism
Ernst Karl
Abbe
1840-1905
At Pellin-Broca Prism an
incident ray of wavelength
λ passes the prism at a
dispersing angle of 90°.
Because the dispersing angle
is a function of wavelength
the ray at other wavelengths
exit at different angles.
By rotating the prism around
an axis normal to the page
different rays will exit at
the 90°.
At Abbe Prism the dispersing
angle is 60°.
Optics - Prisms
123. 123
SOLO
Reflecting Prisms
2i
1i
1t
2t
E
B D
G
A
F C
BED
180
360 ABEBEDADE
1
90 i
ABE
2
90 t
ADE
3609090 12
it
BED
12
180 it
BED
21
180 ti
BED
The bottom of the prism is a reflecting mirror
Since the ray BC is reflected to CD
DCGBCF
Also
CGDBFC
CDGFBC
FBCt
901
CDGi
902
21 it
202 sinsin ti nn Snell’s Law
Snell’s Law 110 sinsin ti nn 21 ti 12 i
CDGFBC ~
Optics - Prisms
126. 126
SOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Polarization can be achieved with crystalline materials which have a different index of
refraction in different planes. Such materials are said to be birefringent or doubly refracting.
Nicol Prism
The Nicol Prism is made up from
two prisms of calcite cemented
with Canada balsam. The
ordinary ray can be made to
totally reflect off the prism
boundary, leving only the
extraordinary ray..
Polarizing Prisms
Optics - Prisms
127. 127
SOLO
Polarizing Prisms
A Glan-Foucault prism deflects polarized light
transmitting the s-polarized component.
The optical axis of the prism material is
perpendicular to the plane of the diagram.
A Glan-Taylor prism reflects polarized light
at an internal air-gap, transmitting only
the p-polarized component.
The optical axes are vertical in the plane of
the diagram.
A Glan-Thompson prism deflects the p-polarized
ordinary ray whilst transmitting the s-polarized
extraordinary ray.
The two halves of the prism are joined with
Optical cement, and the crystal axis are
perpendicular to the plane of the diagram.
Optics - Prisms
Return to TOC
128. 128
OpticsSOLO
Lens Definitions
Optical Axis: the common axis of symmetry of an optical system; a line that connects all
centers of curvature of the optical surfaces.
FFL
First Focal
Point
Second Focal
Point
Principal Planes
Second Principal Point
First Principal Point
Light Rays from Left
EFL
BFL
Optical System
Optical Axis
Lateral Magnification: the ratio between the size of an image measured perpendicular
to the optical axis and the size of the conjugate object.
Longitudinal Magnification: the ratio between the lengthof an image measured along
the optical axis and the length of the conjugate object.
First (Front) Focal Point: the point on the optical axis on the left of the optical system
(FFP) to which parallel rays on it’s right converge.
Second (Back) Focal Point: the point on the optical axis on the right of the optical system
(BFP) to which parallel rays on it’s left converge.
129. 129
OpticsSOLO
Definitions (continue – 1)
Aperture Stop (AS): the physical diameter which limits the size of the cone of radiation
which the optical system will accept from an axial point on the object.
Field Stop (FS): the physical diameter which limits the angular field of view of an
optical system. The Field Stop limit the size of the object that can be
seen by the optical system in order to control the quality of the image.
Entrance Pupil: the image of the Aperture Stop as seen from the object through the
elements preceding the Aperture Stop.
Exit Pupil: the image of the Aperture Stop as seen from an axial point on the
image plane.
A.S. F.S.
I
Aperture and Field Stops
Entrance
pupil
Exit
pupil
A.S.
I
xpE
npE
Chief
Ray
Entrance and Exit pupils
Entrance
pupilExit
pupil
A.S. I
xpE
npE
Chief
Ray
130. 130
OpticsSOLO
Definitions (continue – 2)
Aperture Stop (AS): the physical diameter which limits the size of the cone of radiation
which the optical system will accept from an axial point on the object.
Field Stop (FS): the physical diameter which limits the angular field of view of an
optical system. The Field Stop limit the size of the object that can be
seen by the optical system in order to control the quality of the image.
Entrance Pupil: the image of the Aperture Stop as seen from the object through the
elements preceding the Aperture Stop.
Exit Pupil: the image of the Aperture Stop as seen from an axial point on the
image plane.
Entrance
pupil
Exit
pupil
A.S.
I
Chief
Ray
Marginal
Ray
Exp Enp
131. 131
OpticsSOLO
Definitions (continue – 3)
Principal Planes: the two planes defined by the intersection of the parallel incident rays
entering an optical system with the rays converging to the focal points
after passing through the optical system.
FFL
First Focal
Point
Second Focal
Point
Principal Planes
Second Principal Point
First Principal Point
Light Rays from Left
EFL
BFL
Optical System
Optical Axis
Principal Points: the intersection of the principal planes with the optical axes.
Nodal Points: two axial points of an optical system, so located that an oblique ray
directed toward the first appears to emerge from the second, parallel
to the original direction. For systems in air, the Nodal Points coincide
with the Principal Points.
Cardinal Points: the Focal Points, Principal Points and the Nodal Points.
132. 132
OpticsSOLO
Definitions (continue – 4)
Relative Aperture (f# ): the ratio between the effective focal length (EFL) f to Entrance
Pupil diameter D.
Numerical Aperture (NA): sine of the half cone angle u of the image forming ray bundles
multiplied by the final index n of the optical system.
If the object is at infinity and assuming n = 1 (air):
Dff /:#
unNA sin:
#
1
2
1
2
1
sin
ff
D
uNA
EFL
EFL
D
u
Last Principal Plane of the
Optical System (Spherical)
133. 133
OpticsSOLO
Perfect Imaging System
• All rays originating at one object point reconverge to one image point after passing
through the optical system.
• All of the objects points lying on one plane normal to the optical axis are imaging
onto one plane normal to the axis.
• The image is geometrically similar to the object.
Object Image
SystemOptical
Object Image
SystemOptical
Object Image
SystemOptical
134. 134
OpticsSOLO
Lens
Convention of Signs
1. All Figures are drawn with the light traveling from left to right.
2. All object distances are considered positive when they are measured to the left of the
vertex and negative when they are measured to the right.
3. All image distances are considered positive when they are measured to the right of the
vertex and negative when they are measured to the left.
4. Both focal length are positive for a converging system and negative for a diverging
system.
5. Object and Image dimensions are positive when measured upward from the axis and
negative when measured downward.
6. All convex surfaces are taken as having a positive radius, and all concave surfaces
are taken as having a negative radius.
Return to TOC
135. 135
OpticsSOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle
Karl Friederich Gauss
1777-1855
s 's
n 'n
h
l 'l
'
M
T
CA
M’
R
The optical path connecting points M, T, M’ is
''lnlnpathOptical
Applying cosine theorem in triangles MTC and M’TC
we obtain:
2/122
cos2 RsRRsRl
2/122
cos'2'' RsRRsRl
2/1222/122
cos'2''cos2 RsRRsRnRsRRsRnpathOptical
Therefore
According to Fermat’s Principle when the point T
moves on the spherical surface we must have 0
d
pathOpticald
0
'
sin''sin
l
RsRn
l
RsRn
d
pathOpticald
from which we obtain
l
sn
l
sn
Rl
n
l
n
'
''1
'
'
For small α and β we have ''& slsl
and we obtain
R
nn
s
n
s
n
'
'
'
Gaussian Formula for a Single Spherical Surface
Return to TOC
136. 136
OpticsSOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law
Apply Snell’s Law: 'sin'sin nn
If the incident and refracted rays
MT and TM’ are paraxial the
angles and are small and we can
write Snell’s Law:
'
From the Figure '
'' nn
nnnnnn '''
For paraxial rays α, β, γ are small angles, therefore '/// shrhsh
r
h
nn
s
h
n
s
h
n '
'
'
or
r
nn
s
n
s
n
'
'
'
Gaussian Formula for a Single Spherical Surface
Karl Friederich Gauss
1777-1855
Willebrord van Roijen
Snell
1580-1626
s 's
n 'n
h
l 'l
'
M
T
CA
M’
r
137. 137
OpticsSOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law
for s → ∞ the incoming rays are parallel to optical
axis and they will refract passing trough a common
point called the focus F’.
r
nn
s
n
s
n
'
'
'
s '' fs
n 'n
h
'l
'
T
C
A
F’
R
fs 's
n 'n
h
l
F
T
CA
R
'
r
nn
f
nn
'
'
'
r
nn
n
f
'
'
'
for s’ → ∞ the refracting rays are parallel to optical
axis and therefore the incoming rays passes trough
a common point called the focus F.
r
nnn
f
n
'' r
nn
n
f
'
'' n
n
f
f
Return to TOC
138. 138
OpticsSOLO
Derivation of Lens Makers’ Formula
We have a lens made of two
spherical surfaces of radiuses r1
and r2 and a refractive index n’,
separating two media having
refraction indices n a and n”.
Ray MT1 is refracted by the first
spherical surface (if no second
surface exists) to T1M’.
111
'
'
'
r
nn
s
n
s
n
11111 ''& sMAsTA
Ray T1T2 is refracted by the second spherical surface to T2M”. 2222 ""&'' sMAsMA
222
'"
"
"
'
'
r
nn
s
n
s
n
Assuming negligible lens thickness we have , and since M’ is a virtual object
for the second surface (negative sign) we have
21 '' ss
21 '' ss
221
'"
"
"
'
'
r
nn
s
n
s
n
M’
M
'1f1f
1s
Axis
T1 T2
A1
A2
C1
1rC2 F’1F’’2
M’’
F’2F1
''2f'2f
'1s
'2s
''2s
2r
n 'n ''n
139. 139
OpticsSOLO
Derivation of Lens Makers’ Formula (continue – 1)
M”
M
f
s
Axis
A1
A2
C1
1rC2
F”
F
''f
''s
2r
n 'n ''n
111
'
'
'
r
nn
s
n
s
n
Add those equations
221
'"
"
"
'
'
r
nn
s
n
s
n
2121
'"'
"
"
r
nn
r
nn
s
n
s
n
M’
M
'1f1f
1s
Axis
T1 T2
A1
A2
C1
1rC2 F’1F’’2
M’’
F’2F1
''2f'2f
'1s
'2s
''2s
2r
n 'n ''n
The focal lengths are defined by
tacking s1 → ∞ to obtain f” and
s”2 → ∞ to obtain f
f
n
r
nn
r
nn
f
n
212
'"'
"
"
Let define s1 as s and s”2 as s”
to obtain
21
'"'
"
"
r
nn
r
nn
s
n
s
n
f
n
r
nn
r
nn
f
n
21
'"'
"
"