1
OPTICSPart I
SOLO HERMELIN
Updated: 16.01.10http://www.solohermelin.com
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
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 PrincipleDerivation 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
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
OPTICSPart II
5
Table of Content (continue)
SOLO OPTICS
References
Optical Aberration
Monochromatic Seidel Aberrations
Chromatic Aberration
Interference
OpticsPart II
6
Optics SOLO
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
7
Optics SOLO
Hierarchy of Optical Theories
8
MAXWELL’s EQUATIONSSOLO
SYMMETRIC MAXWELL’s EQUATIONS
Magnetic Field Intensity H
1mA
Electric Displacement D 2 msA
Electric Field Intensity E 1mV
Magnetic InductionB 2 msV
Electric Current Density eJ
2mA
Free Electric Charge Distributione 3 msA
Fictious Magnetic Current Density mJ 2mV
Fictious Free Magnetic Charge Distributionm 3 msV
1. AMPÈRE’S CIRCUIT LW (A) eJ
t
DH
2. FARADAY’S INDUCTION LAW (F)mJ
t
BE
3. GAUSS’ LAW – ELECTRIC (GE) eD
4. GAUSS’ LAW – MAGNETIC (GM) mB
Although magnetic sources are not physical they are often introduced as electricalequivalents to facilitate solutions of physical boundary-value problems.
André-Marie Ampère1775-1836
Michael Faraday1791-1867
Karl Friederich Gauss1777-1855
James Clerk Maxwell(1831-1879)
9
SOLO
The Electromagnetic Spectrum
10
SOLO
Visible Spectrum
11
SOLO
The Infrared (IR) Spectrum of Interest
Return to TOC
12
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions
2t
1t
h
2H
1H
1
2
C
CS1P2P
3P
4P
b
21ˆ n
ek
ldtHtHhldtHldtHldHh
C
2211
0
2211ˆˆˆˆ
where are unit vectors along C in region (1) and (2), respectively, and 21ˆ,ˆ tt
2121 ˆˆˆˆ nbtt
- a unit vector normal to the boundary between region (1) and (2)21ˆ n- a unit vector on the boundary and normal to the plane of curve Cb
Using we obtainbaccba
ldbkldbHHnldnbHHldtHH e
ˆˆˆˆˆˆ21212121121
Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have:
b
ekHHn
2121ˆ
S
e
C
Sdt
DJdlH
dlbkbdlht
DJSd
t
DJ e
h
e
S
eˆˆ
0
AMPÈRE’S LAW
1
0lim:
mAht
DJk e
he
13
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 1)
2t
1t
h
2E
1E
1
2
C
CS1P2P
3P
4P
b
21ˆ n
mk
ldtEtEhldtEldtEldEh
C
2211
0
2211ˆˆˆˆ
where are unit vectors along C in region (1) and (2), respectively, and 21ˆ,ˆ tt
2121 ˆˆˆˆ nbtt
- a unit vector normal to the boundary between region (1) and (2)21ˆ n- a unit vector on the boundary and normal to the plane of curve Cb
Using we obtainbaccba
ldbkldbEEnldnbEEldtEE m
ˆˆˆˆˆˆ21212121121
Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have:
b
mkEEn
2121ˆ
S
m
C
Sdt
BJdlE
dlbkbdlht
BJSd
t
BJ m
h
m
S
mˆˆ
0
FARADAY’S LAW
1
0lim:
mVht
BJk m
hm
14
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 2)
h
2D
1D
1
2
21ˆ n
dS
1n
2n
e
SdnDnDhSdnDSdnDSdDh
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)21ˆ 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 2121ˆ
dSdShdv e
h
e
V
e 0
GAUSS’ LAW - ELECTRIC
1
0lim:
msAhe
he
V
e
S
dvSdD
15
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 3)
h
2B
1B
1
2
21ˆ n
dS
1n
2n
m
SdnBnBhSdnBSdnBSdBh
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)21ˆ 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 2121ˆ
dSdShdv m
h
m
V
m 0
GAUSS’ LAW – MAGNETIC
1
0lim:
msVhm
hm
V
m
S
dvSdB
16
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (summary)
2t
1t
h
22 , HE
11, HE
1
2
C
CS1P2P
3P
4P
b
21ˆ n
me kk
,21ˆ n
dS
11, BD
22 , BD
me ,
mkEEn
2121ˆ FARADAY’S LAW
ekHHn
2121ˆ AMPÈRE’S LAW 1
0lim:
mAht
DJk e
he
1
0lim:
mVht
BJk m
hm
eDDn 2121ˆ GAUSS’ LAW
ELECTRIC 1
0lim:
msAhe
he
mBBn 2121ˆ GAUSS’ LAW
MAGNETIC 1
0lim:
msVhm
hm
Return to TOC
17
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
For Homogeneous, Linear and Isotropic Medium
ED
HB
where are constant scalars, we have ,
Jt
EJ
t
DH
t
t
H
t
BE
ED
HB
Since we have also tt
t
J
t
EE
DED
EEE
t
J
t
EE
2
222
2
2
&
18
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS (continue 1)
Define
meme KK
c
KKv
00
11
where
smc /103
1036
1104
11 8
9700
is the velocity of light in free space.
The absolute index of refraction n is
me KKv
cn
0
The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is
t
J
t
E
vE
2
2
22 1
19
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS (continue 2)
In the same way
The Inhomogeneous Wave (Helmholtz) Differential Equation for the Magnetic Field Intensity is
Jt
EJ
t
DH
t
H
t
BE
t
ED
HB
Since are constant andtt
,
J
t
HH
HHB
HHH
Jt
HH
2
222
2
2
0&
Jt
H
vH
2
2
22 1
Return to TOC
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 tjrEjtjt
rEtrEt 00 expexp,
Hence
m
e
m
e
jt
m
e
m
e
B
D
JBjE
JDjH
BGM
DGE
Jt
BEF
Jt
DHA
)(
21
ELECTROMAGNETICSSOLO
Fourier Transform
The Fourier transform of can be written as: trHtrE ,,,
dttjtrHrHdtjrHtrH
dttjtrErEdtjrEtrE
exp,,&exp,2
1,
exp,,&exp,2
1,
This is possible if:
drHdttrH
drEdttrE
22
22
,2
1,
,2
1,
JEAN FOURIER
1768-1830
22
ELECTROMAGNETICSSOLO
NoteThe assumption that can be written as: trHtrE ,,,
tjrHtrHtjrEtrE 00 exp,,exp,
is equivalent to saying that has a Fourier transform; i.e.: trHtrE ,,,
dtjrHtrHdttjtrHrH
dtjrEtrEdttjtrErE
exp,2
1,&exp,,
exp,2
1,&exp,,
This is possible if:
drHdttrH
drEdttrE
22
22
,2
1,
,2
1,
00
0
exp
expexpexp,,
rEdttjrE
dttjtjrEdttjtrErE
End Note
23
ELECTROMAGNETICSSOLO
m
e
m
e
ED
HBm
e
JHjE
JEjH
JHjE
JEjH
JBjE
JDjH
me JJjEkE
2
em JJjHkH
2
22 f
c
c
fk
Using the vector identity AAA
For a Homogeneous, Linear and Isotropic Media:
m
e
ED
HBm
e
H
E
B
D
e
me JJjEkE
22
m
em JJjHkH
22
and
we obtain
Monochromatic Planar Wave Equations (continue)
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 whichconstrkt
25
ELECTROMAGNETICSSOLO
To satisfy the Maxwell equations for a source free media we must have:
Monochromatic Planar Wave Equations (continue)
we haveUsing: 1ˆˆ&ˆˆ sssc
nsk
0
0
H
E
HjE
EjH
0ˆ
0ˆ
ˆ
ˆ
0
0
00
00
Hs
Es
HEs
EHs
sPlanar Wave
0E
0Hr
0
0
0
0
00
00
rkj
rkj
rkjrkj
rkjrkj
ekje
eHkj
eEkj
eHjeEkj
eEjeHkjrkjrkj
0
0
0
0
00
00
Hk
Ek
HEk
EHk
For a Homogeneous, Linear and Isotropic Media:
Return to TOC
26
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
Spherical Waveforms z
x
y
rcosr
,,rP
sinsinr cossinr
The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is
t
J
t
E
vE
2
2
22 1
In spherical coordinates:
cos
sinsin
cossin
rz
ry
rx
2
2
2222
22
sin
1sin
sin
11
rrr
rrr
For a spherical symmetric wave: rErE
,,
Errrr
E
rr
E
r
Er
rrE
2
2
2
22
22 121
27
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
SourceSourceSource
Spherical Waveforms z
x
y
rcosr
,,rP
sinsinr cossinr
The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is assuming no sources
011
2
2
22
2
t
E
vEr
rr
In spherical coordinates:
cos
sinsin
cossin
rz
ry
rx
01
2
2
22
2
Ertv
Err
or:
A general solution is:
waveregressive
waveeprogressiv
tvrFtvrFEr 21
0,0,0,0 meme JJ
r
eEerEtrE
rktjtj
0,,
Assume a progressive monochromatic wave of phase rkt
) a regressive wave has the phase ( rkt
r
eErE
rkj
0, Return to TOC
28
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
Cylindrical Waveforms
z
x
yr
zrP ,,
sinr
cosr
In cylindrical coordinates:
zz
ry
rx
sin
cos
2
2
2
2
22 11
zrrr
rr
For a cylindrical symmetric wave: rEzrE
,,
r
Er
rrE
12
The Inhomogeneous Wave (Helmholtz) Differential Equation for the Electric Field Intensity is assuming no sources
011
2
2
2
t
E
vr
Er
rr
0,0,0,0 meme JJ
29
ELECTROMAGNETICSSOLO
ELECTROMGNETIC WAVE EQUATIONS
Source
Cylindrical Waveforms
z
x
yr
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
011
2
2
2
t
E
vr
Er
rr
0,0,0,0 meme JJ
Assume a progressive monochromatic wave of phase rkt
) a regressive wave has the phase ( rkt
tjerEtrE ,,
0
12
2
2
Evr
E
rr
E
k
The solutions are Bessel functions which for larger approach asymptotically to: rkje
r
ErE 0,
Return to TOC
30
SOLO
Energy and Momentum
Let start from Ampère and Faraday Laws
t
BEH
Jt
DHE e
EJt
DE
t
BHHEEH e
HEHEEH
But
Therefore we obtain
EJt
DE
t
BHHE e
First way
This theorem was discovered by Poynting in 1884 and later in the same year by Heaviside.
ELECTROMAGNETICS
John Henry Poynting1852-1914
Oliver Heaviside1850-1925
31
SOLO
Energy and Momentum (continue -1)
We identify the following quantities
-Power density of the current density EJe
HEDEt
BHt
EJe
2
1
2
1
BHt
pBHw mm
2
1,
2
1
DEt
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 ,
DEtt
DE
ED
2
10
BHtt
BH
HB
2
10
ELECTROMAGNETICS
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 1sm
Electric Field Intensity E 1mV
Magnetic InductionB 2 msV
Hendrik Antoon Lorentz1853-1928
e
Force on the free chargeF
Ne
Second way
ELECTROMAGNETICS
33
SOLO
Energy and Momentum (continue – 4)
The power density of the Lorentz Force the charge
EJBvEvp e
Bvv
Jve
ee
0
or
HEt
BHE
t
D
Et
DHEEH
Et
DHEJp
t
BE
HEHEEH
Jt
DH
e
e
e
ELECTROMAGNETICS
34
SOLO
Energy and Momentum (continue – 5)
HEDEt
BHt
EJe
2
1
2
1
dve
E
B
eJv
,
V
FdF
Fd
Let integrate this equation over a constant volume V
VVVV
e dvSdvDEtd
ddvBH
td
ddvEJ
�
2
1
2
1
If we have sources in V then instead of we must use
E
sourceEE
Use Ohm Law (1826)
sourceee EEJ
VV td
d
t
Georg Simon Ohm1789-1854
sourcee
e
EJE
1
For linear, isotropic electro-magnetic materials HBED
00 ,
ELECTROMAGNETICS
35
SOLO
Energy and Momentum (continue – 6)
VVVR
n
V
sourcee dvSdvDE
td
ddvBH
td
ddRIdvEJ
2
1
2
12
V
FieldMagnetic dvBHtd
dP
2
1
V
FieldElectric dvDEtd
dP
2
1 SV
Radiation SdSdvSP
V
sourceeSource dvEJP
V
sourcee
R
n
V
sourcee
L S eee
V
sourcee
L S eee
V
e
dvEJdRI
dvEJdS
dldSJdSJdvEJldSdJJdvEJ
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 Poynting1852-1914
36
ElectromagnetismSOLO
EM People
John Henry Poynting1852-1914
Oliver Heaviside1850-1925
Nikolay Umov1846-1915
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 byPoynting in 1884 and later in the same year by Heaviside.
1873 - 1884
Return to TOC
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
trPtrtc
tr eee ,1
,1
,0
2
2
22
Heinrich Rudolf Hertz1857-1894
- speed of propagation of the EM wave [m/s]00
1
c
- Polarization Vector Intensity eP 2 msA
- Permitivity of space 2122 mNsA
- Electric Hertz’s Vector Potential (1888)e NsA 11
tA e
000 eV
0
Using the Electric Hertz’s Vector Potential we obtain :
The field vectors are given by ee
tcV
t
AE
2
2
200 1
tAH e
000
1
38
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
Compute (continue-3) ee
tcE
2
2
2
1
We have
32
0
4
0
2
5
0
2
2
2
2 44
3
4
31
rc
rpr
rc
rprpr
r
rprrp
tcE ee
e
230 44 rc
rp
r
rp
tH e
r
ptre
04,
krtjkrtj epedqp 00
Let use spherical coordinates
zyxr rrr 1111 cossinsincossin
111 sincos00 rz krtjkrtj epepp
krtjeprccr
jr
rc
rprrp
rc
rprrp
r
rprrpE
rr
02
0
2
2
0
3
0
32
0
2
4
0
2
5
0
2
4
sin
4
sincos2
4
sincos2
44
3
4
3
11111
r1
1
1
pckpp
pckjpjp222
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
2sin1
4sin
44
krtjkrtj ep
rk
j
r
kcep
rcr
jH
krtjepr
k
r
kj
r
rccrj
rE
r
rr
0
2
23
0
2
0
2
2
0
3
0
111
11111
sinsincos21
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
ck
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
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
102sin
4
tj
FieldNear epr
kcjH
tj
FieldNear epr
E r 03
0
11 sincos24
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.tje
02
r
rk
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
102sin
4
krtj
FieldteIntermedia epr
kcjH
krtj
FieldteIntermedia epr
kj
rE r
023
0
11 sincos21
4
1
Near, Intermediate and Far Fields
• Intermediate (induction) zone: ~rd
• Far (radiation) zone: rd
10
2
sin4
krtj
FieldFar epr
kcH
10
0
2
sin4
krtj
FieldFar epr
kE
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)
12
r
rk
12010
36
11041
:9
7
0
0
1
0
00
c
FieldFar
FieldFar
cH
EZ
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
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
sin4
1
krtjep
cr
j
rH
krtjepcrc
jrc
jrrr
E r
02
2
2
0
11 sin11
cos12
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:
1sincossin4
20
krt
ckrt
rr
pHrealH
11 sinsin
1cos
1cossincos
12
4 2
2
2
0
0 krtrc
krtcr
krtc
krtrrr
pErealE r
1
1
cossincossinsincos1
4
2
sincossinsincos1
42
0
32
2
0
22
2
2
2
0
22
2
0
krtc
krtr
krtc
krtrr
p
krtc
krtr
krtrc
krtcrr
pHES r
The Poynting Vector of the Electric Dipole Field is given by:
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
cossincossinsincos1
4
2
sincossinsincos1
42
0
32
2
0
22
2
2
2
0
22
2
0
krtc
krtr
krtc
krtrr
p
krtc
krtr
krtrc
krtcrr
pHES r
T
TdttS
TS
0
1lim
2
12cos
1lim
2
11lim
2
1cos
1limcos
0
0
1
00
22
T
T
T
T
T
Tdtrkt
Tdt
Tdtrkt
Trkt
2
12cos
1lim
2
11lim
2
1sin
1limsin
0
0
1
00
22
T
T
T
T
T
Tdtrkt
Tdt
Tdtrkt
Trkt
02sin1
lim2
1cossin
1limcossin
0
00
T
T
T
Tdtrkt
Tdtrktrkt
Trktrkt
rrc
pS 12
23
0
2
42
0 sin42
11 cossin
4sin
1
42
22
0
32
2
02
2
2
2
2
2
2
0
22
2
0
rcrcr
p
rccrcr
pS r
we obtain:
or: Radar Equation
Irradiance
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
rrc
pS 12
23
0
2
42
0 sin42
Radar Equation
45 90 135 1800
0
5
10
15
20
25
30
0
45
90
135
180
225
270
315
z
y5.0 0.1
Polar Angle , in degrees
Rel
ativ
e P
ower
, in
db
The Total Average Radiant Power is:
0
22
23
0
2
42
0 sin2sin42
drrc
pdSSP
Arad
22
0
22
120123
0
42
0
3/4
0
3
23
0
42
0 4012
sin16
0
prc
pd
rc
pP
c
c
rad
3
4
3
2
3
2cos
3
1coscoscos1sin
0
30
2
0
3
dd
Return to TOC
46
ELECTROMAGNETICSSOLO
To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations
we haveUsing: 1ˆˆ&ˆˆ0 kkknkkk
0
0
H
E
HjE
EjH
0ˆ
0ˆ
ˆ
ˆ
0
0
00
00
Hk
Ek
HEk
EHk
kˆPlanar Wave
0E
0Hr
0
0
0
0
00
00
rkj
rkj
rkjrkj
rkjrkj
ekje
eHkj
eEkj
eHjeEkj
eEjeHkjrkjrkj
22
22&
2
ˆ
2
ˆHwEwwcn
kwwcn
kS meme
Time Average Poynting Vector of the Planar Wave
Reflections and Refractions Laws Development Using the Electromagnetic Approach
47
SOLO REFLECTION & REFRACTION
iE
iH
rE
rH
ik rk
tH
tE
tk
21n
z
x yi
r
t
Consider an incident monochromatic planar wave
c
nk
eEkH
eEE
iiii
rktjiii
rktjii
ii
ii
1
00
110011
0
0
The monochromatic planar reflected wave from the boundary is
11
1
1
0
0
&n
cv
vc
nk
eEkH
eEE
rrr
rktjrrr
rktjrr
rr
rr
The monochromatic planar refracted wave from the boundary is
22
2
2
0
0
&n
cv
vc
nk
eEkH
eEE
ttt
rktjttt
rktjtt
tt
tt
Reflections and Refractions Laws Development Using the Electromagnetic Approach
48
SOLO REFLECTION & REFRACTION
The Boundary Conditions at z=0 must be satisfied at all pointson the plane at all times, impliesthat the spatial and time variations of
This implies that
iE
iH
rE
rH
ik rk
tH
tE
tk
21n
z
x yi
r
t
Phase-Matching Conditions
yxteEeEeEz
rktjt
z
rktjr
z
rktji
ttrrii ,,,,0
00
00
0
yxtrktrktrktz
ttz
rrz
ii ,,000
ttri
yxrkrkrkz
tz
rz
i ,000
must be the same
Reflections and Refractions Laws Development Using the Electromagnetic Approach
49
SOLO REFLECTION & REFRACTION
tri nnn sinsinsin 211
iE
iH
rE
rH
ik rk
tH
tE
tk
21n
z
x yi
r
t
Phase-Matching Conditions
zyxc
nk
zyxc
nk
ttttttt
irirrrr
ˆcossinˆsinsinˆcos
ˆcossinˆsinsinˆcos
2
1
yyxc
nrk
yxc
nrk
yc
nrk
tttz
t
irrz
r
iz
i
ˆsinsincos
sinsincos
sin
2
0
1
0
1
0
yxrkrkrkz
tz
rz
i ,000
2
tr
ttri
x
y
Coplanar
Snell’s Law
zzyyxxr
zyc
nk iiii
ˆˆˆ
ˆcosˆsin1
Given:
Let find:
Reflections and Refractions Laws Development Using the Electromagnetic Approach
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
zyc
nk iiii
ˆˆˆ
ˆcosˆsin1
zyxc
nk
zyxc
nk
ttttttt
irirrrr
ˆcossinˆsinsinˆcos
ˆcossinˆsinsinˆcos
2
1
ynnync
kkz
ynnync
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
21n
z
x yi
r
t
Reflections and Refractions Laws Development Using the Electromagnetic Approach
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
21n
z
x yi
r
t yxrkrkrk
zt
zr
zi ,
000
yxtrktrktrktz
ttz
rrz
ii ,,000
Vector Notation
ScalarNotation
Reflections and Refractions Laws Development Using the Electromagnetic Approach
52
SOLO REFLECTION & REFRACTION
iE
iH
rErH
ik rk
tH
tE
tk
21n
z
x yi
r
t
i r
ttH
tE
tk
rH
rk
rE
iH
iE
ik
21n
Boundary
ti
ti
i
r
nn
nn
E
Er
coscos
coscos
2
2
1
1
2
2
1
1
0
0
ti
i
i
t
nn
n
E
Et
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
Er
sin
sin21
0
0
ti
it
i
t
E
Et
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
SOLO REFLECTION & REFRACTION
iE
iH
rE
rHik rk
tH
tE
tk
21n
z
x yi
r
t
i r
t
tH
tE
tk
rH
rk
rE
iH
iE
ik
21n
Boundary
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
Er
coscos
coscos
1
1
2
2
1
1
2
2
||0
0||
ti
i
i
t
nn
n
E
Et
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
Er
tan
tan21
||0
0|| titi
it
i
t
E
Et
cossin
cossin221
||0
0||
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Fresnel Equations
See full development in P.P.“Reflection & Refractions”
54
SOLO REFLECTION & REFRACTION
ti
ti
i
r
nn
nn
E
Er
coscos
coscos
1
1
2
2
1
1
2
2
||0
0||
ti
i
i
t
nn
n
E
Et
coscos
cos2
1
1
2
2
1
1
||0
0||
ti
ti
i
r
nn
nn
E
Er
coscos
coscos
2
2
1
1
2
2
1
1
0
0
ti
i
i
t
nn
n
E
Et
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 Lorentz1853-1928
See full development in P.P.“Reflection & Refractions”
Return to TOC
55
IR Radiometric Quantities SOLO
RTA
DA 2cm 2cm
TARGETSOURCE
DETECTORRECEIVER
Radiation Flux Power W
Spectral Radial Power
m
W
Irradiance
2mc
W
AE
Spectral Radiant Emittance
mmc
WMM
2
Radiant Intensity
str
WI
Spectral Radiant Intensity
mstr
WII
Radiance
strmc
W
A
IL
2cos
Spectral Radiance
mstrmc
WLL
2
Radiant Emittance
2mc
W
AM
Spectral Irradiance
mmc
WEE
2
T
TdttS
TS
0
1lim
Irradiance is the time-average of the Poynting vector
Return to TOC
56
Physical Laws of Radiometry SOLO
Plank’s Law
1/exp
1
2
5
1
Tc
cM 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
in eTemperaturAbsolute-
constantBoltzmannsec/103806.1
constantPlanksec106260.6
lightofspeedsec/458.299792
10439.1/
107418.32
23
234
4
2
4242
1
MAXPLANCK
(1858 - 1947)Plank’s Law
57
Physical Laws of Radiometry SOLO
Plank’s Law
1/exp
1
2
5
1
Tc
cM BB
Plank 1900
Plank’s Law applies to blackbodies; i.e. perfect radiators.
The spectral radial emittance of a blackbody is given by:
MAXPLANCK
(1858 - 1947)Plank’s Law
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 maximumis 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
10670.515
2:
1/exp
1
Kcm
W
hc
k
cm
WTd
Tc
cdMM BBBB
LUDWIG BOLTZMANN(1844 - 1906)
WILHELM WIEN
(1864 - 1928)
Stefan-Boltzmann Law
JOSEFSTEFAN
(1835 – 1893)
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
Physical Laws of Radiometry (Continue -2) SOLO
Emittance of Real Bodies (Gray Bodies)
For real (gray) bodies:
BBMM
- 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
,
BBM
M
61
Physical Laws of Radiometry (Continue -3) SOLO
Kirchhoff’s Law
rM
iE aE
tM
Gustav Robert Kirchhoff1824-1887
- 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
E11
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
Physical Laws of Radiometry (Continue -4) SOLO
Kirchhoff’s Law (Continue – 1)
Gustav Robert Kirchhoff1824-1887
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’stemperature will rise above the surrounding and will be a transfer of energy from acold surrounding to a hot body contradicting the second law of thermodynamics.
TT
222 ,, T2A
111 ,, T1A
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.
2
sin
r
drdrd
A
cosAAn
z
x
y
0
2
cos, L
AL
coscos, 00 IALI
Lambert’s Law
0
2
0
2/
0
00 sincoscos LddLdLA
M
The Radiant Intensity from a Lambertian Surface is
The Radiant Emittance (Exitance) from a Lambertian Surface is
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:
212
2222
211
122
1
cos&
cos R
Add
strcm
W
AL
1A
1dA
1 12R
Radiating(Source)Surface
2A
2dA
ReceivingSurface
2
2d
2
12
2211112
coscos
R
AdAdLd
The total radiant power (radiance) received at surface A2 from A1 is:
2 1
21212
21112
coscos
A A
AdAdR
L
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
ReceivingSurface
2
2d
212
2222
12
111
cos&
cos
R
Add
R
Add
1A
1dA
1 12R
Radiating(Source)Surface
2A
2dA
ReceivingSurface
2
1d
then:
212
2112
12
2211112
coscos
R
AdAdL
R
AdAdLd nn
12121112 dAdLdAdLd nn
The Power is the product of the Radiance, the projected Area, and the Solid Angleusing the other area.
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 Target2, R
AOPTICSTO
ωT,O – solid angle of Target as seen from the Optics2, R
ATARGETOT
ΩD,O – solid angle of Detector as seen from the Optics2, f
ADETECTOROD
ωO,D – solid angle of Optics as seen from the Detector2, f
ADETECTORDO
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
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
Physical Laws of Radiometry (Continue -11) SOLO
Transfer of Radiant Energy (Continue – 5)
Optics (continue – 3)
2
,
R
AAL
AL
TARGETDETECTOR
DTDETECTOROpticsNo
2
,
f
AAL
AL
OPTICSDETECTOR
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
40
44 0#
fAL
f
DAL DETECTOR
Dff
DETECTOR
The Optics increases the energy collected by the Detector
since DTDO ,, 22
#2 4 R
A
ff
A TARGETOPTICS
OpticsNoOpticsWith
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
71
Physical Laws of Radiometry (Continue -13) SOLO
Targets
72
Physical Laws of Radiometry (Continue -14) SOLO
Targets
73
Physical Laws of Radiometry (Continue -15) SOLO
Targets
74
Physical Laws of Radiometry (Continue -16)
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-923660 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:
2876.0207.2397.053 cmWmmM In a tail-on situation the radiant intensity is:
110203660876.0
53
strWA
MmmI NOZZLE
Lambertian
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 molecularvibrations and rotations, only at certain frequencies..
77
Physical Laws of Radiometry (Continue -19) SOLO
Targets (continue – 3)
• The plume (continue – 1)
78
Physical Laws of Radiometry (Continue -20) SOLO
Targets (continue – 4)
• The plume (continue – 2)Breathing engines have exhaust plume temperatures of
K600450 Cruise flight K800600 Maximum Un-augmented Thrust K15001000 Augmented (After burner) Thrust
Rockets have exhaust plume temperatures of
K75002500 Liquid propellant
K35001700 Solid propellant
ExampleAssume:
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:
142
8.27101075.1
55.433.4
strWA
MmmI PLUME
Lambertian
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
11, MachrMachHTT
- air temperature at altitude HTARGET and mach number Mach MachHT ,0
- recovery factor r
vp CC / - specific heat ratio = 1.4 for air
ExampleMach = 2.0, HTARGET = 5000 m
27.0,250.2,50000 KMachmHT
then KT 41422
14.182.01250 2
235
3
1066.1414
cmWdKTMM
assume 215mATARGET
143
3.7910151066.1
53
strWA
MmmI TARGET
Lambertian
80
Physical Laws of Radiometry (Continue -22)
81
Physical Laws of Radiometry (Continue -23)
82
Physical Laws of Radiometry (Continue -24) SOLO
Targets (continue – 6)
• Aerodynamic Heating (continue – 1)
Emissivity Reflectance Absorptance Material
.04 .81 .19 Polished Aluminium
.04 .63 .37 Unpolished Aluminium
.18 .43 .57 Titanium
.05 .60 .40 Polished Stainless Steel
.88 .79 .21 White Paint
.92 .05 .95 Black Paint
.27 .71 .29 Aluminum Paint
83
Physical Laws of Radiometry (Continue -25) SOLO
Sun, Background and Atmosphere
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
mTm 49.0
5900
28982898
This is almost at the middle of the visible spectrum mm 75.040.0
Loss by Scattering
85
Physical Laws of Radiometry (Continue -27) SOLO
Sun, Background and Atmosphere (continue – 2)
Atmosphere
Atmosphere affects electromagnetic radiation by
3.2
11
RkmRR
• 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 μmincludes VIS: 0.4 μm ≤ λ < 0.7 μm
Window # 3 (SWIR): 2.0 μm ≤ λ < 2.5 μm
86
Physical Laws of Radiometry (Continue -28) SOLO
Sun, Background and Atmosphere (continue – 3)
Atmosphere Absorption over Electromagnetic Spectrum
87
Physical Laws of Radiometry (Continue -29) SOLO
Sun, Background and Atmosphere (continue – 4)
Rain Attenuation over Electromagnetic Spectrum
FREQUENCY GHz
ON
E-W
AY
AT
TE
NU
AT
ION
-Db
/KIL
OM
ET
ER
WAVELENGTH
88
Physical Laws of Radiometry (Continue -30) SOLO
Sun, Background and Atmosphere (continue – 3)
Add scanned Figure from McKenzie
Atmosphere (continue – 1)
89
GEOMETRICAL OPTICSSOLO
http://en.wikipedia.org/wiki/Optics
From “Cyclopaedia” or “An Universal Dictionary of Art and Science”Published by Ephraim ChambersIn London in 1728
Return to TOC
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
1exp,Re,
exp,exp,2
1exp,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 0000 Re,&Re,
James Clerk Maxwell(1831-1879)
See full development in P.P.“Foundation of Geometrical Optics”
91
SOLO
From those equations we have
Foundation of Geometrical Optics
Sjktj
SjkSjktjSjktjtj
eeESjkE
EeeEeeEeerE0
000
000
000,
Sjk
SjktjSjktjtj
eHjk
eHejeHejerHt
0
00
0
00
0
0
00
000
1
1,
from which
0
00
0000 HjkESjkEF
and
01 0
00
0
00
0
k
Ejk
HES
DERIVATION OF EIKONAL EQUATION (continue – 2)
Derivation from Maxwell Equations (continue – 2)
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
HeeHeeHeerH0
000
000
000,
Sjk
SjktjSjktjtj
eEjk
eEejeEejerEt
0
00
0
00
0
0
00
000
1
1,
0
00
0000 EjkHSjkHA
01 0
00
0
00
0
k
Hjk
EHS
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 00
ee
e
e
J
ck
jt
HB
ED
BGM
DGE
Jt
DHA
t
BEF
André-Marie Ampère1775-1836
Michael Faraday1791-1867
Karl Friederich Gauss1777-1855
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
01 0
000
0
k
EEjk
ES
We also have
from which
and
0
,0
000
0000
000
Sjktj
SjkSjktjSjktjtj
eeHSjkHH
HeeHeeHeerH
00000 HSjkHHGM
01 0
000
0
k
HHjk
HS
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
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
000
0 HSE
And by substituting this in the first equation
00 000
00
00
000
HHSSHHSS
But
2
00
02
0
0
00
n
HSHSSSHSHSS
97
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 7)
Derivation from Maxwell Equations (continue – 7)
Finally we obtain
0022 HnS
or
zyxnz
S
y
S
x
SornS ,,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
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
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 1)
T
T
T
TT
e
dttjrErErEtjrET
dttjrEtjrEtjrEtjrET
dttjrEalT
dttrEtrET
dttrDtrET
w
0
2**2
0
**
0
2
00
2exp,,,22exp,4
1
exp,exp,exp,exp,4
1
exp,Re1
,,1
,,1
But
0
2
2exp2exp
2
12exp
1
02
2exp2exp
2
12exp
1
00
00
T
TT
T
TT
Tj
Tjtj
Tjdttj
T
Tj
Tjtj
Tjdttj
T
Therefore
rErEerEerEdtT
rErEw rSjkrSjkT
e
*00
*00
0
*
22
1,,
200
Let compute the time averages of the electric and magnetic energy densities
100
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 2)
In the same way
rErEerEerEdtT
rErEw rSjkrSjkT
e
*00
*00
0
*
22
1,,
200
rHrHdttrHtrHT
dttrBtrHT
wTT
m*
00
00 2,,
1,,
1
Using the relations
000
0 HSEA
000
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
*
000
0*
00
*
00
*
000
0*
00
2
1
2
1
2
1
2
1
22
2
1
22
S
0E
0H
101
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 3)
Therefore *002
1rHSrEww me
Within the accuracy of Geometrical Optics, the time-averaged electric and magnetic energy densities are equal.
*0000*
00 22rHSrErHrHrErEwww me
The total energy will be:
The Poynting vector is defined as: trHtrEtrS ,,:,
T
tjtjtjtj
Ttjtj
T
dterHerHerEerET
dterHerEalT
dttrHtrET
trHtrES
0
**
00
,,2
1,,
2
11
,,Re1
,,1
,,
,,,,4
1
,,,,,,,,4
11
**
0
2****2
rHrErHrE
dterHrErHrErHrEerHrET
Ttjtj
rHrErHrE
erHerEerHerE rSjkrSjkrSjkrSjk
0*
0*
00
)(0
)(*0
)(*0
)(0
4
14
10000
The time average of the Poynting vector is:
John Henry Poynting1852-1914
102
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 3)
Using the relations
000
0 HSEA 0
000 ESHF
rHrHSrESrErHrErHrES 0
*
0
*
0000
0*
0*
00 222
1
4
1
we obtain
*
00
0
0*
0
0
0*
0*
0000
22222
1HHSHSHESEEES
*0000*
00 22rHSrErHrHrErEwww me
we obtain
Using
wSn
cwwSS me
200
00 22
1
00
2
00
&1
nc
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 snSn
S
S
Ss ˆ:ˆ
We have swvwSn
cS
n
cv
ˆ2
1
2 2
s
constS constdSS
s
r0s
0r
A Bundle of Light Rays
104
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 5)
swvwSn
cS
n
cv
ˆ2
1
2 2
s
constS constdSS
s
r0s
0r
From this equation we can see that average Poynting vector is the direction ofthe 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
ss
Suppose that the vector describes the light path, then the unit vector is given by
r
s
sd
rd
rd
rds ray
ray
ray
ˆ
where is the differential of an arc length along the ray pathrayrdsd
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
rds ray
ray
ray
ˆ rayrdsd
Ssd
d
sd
rdn
sd
d
ray
Ssd
rd ray
sd
rdf
sd
zd
zd
fd
sd
yd
yd
fd
sd
xd
xd
fd
sd
zyxfd
,,
SSn
1 S
sd
rdn ray
ABBAABBABA
AB
AAAAAA
2
1
SA
SSSSSSSS 0
2
1 SSn
2
1
2nSS 2
2
1n
n
n
106
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 7)
Therefore we obtained nSsd
d
and
nsd
rdn
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
0s
s
constS constdSS
s
r0s
0r
We can transform the 2nd order differential equation in two 1st order differential equations by the following procedure. Define
Ssnsd
rdnp ˆ: ray
We obtain 0ˆ0 snpnp
sd
d
0ˆ0 snpnpsd
d
Return to TOC
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 instraight 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 thesurface normal is equal to the angle between thereflected ray and the surface normal:
2v
1v
Refracted Ray
21ˆ n
2n
1n
i
t
Reflected Ray
21ˆ n
2n
1n
i r
3. Law of Refraction
An optical disturbance moving from a medium ofrefractive index n1 into a medium of refractive indexn2 will have its incident ray, the surface normal betweenthe 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 thereflected 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
SOLO Foundation of Geometrical Optics
Fermat’s Principle (1657)
1Q
1P
2P
2Q1Q
2Q
1S
SdSS 12
2PS
1PS
2'Q
rd
s
s
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
SOLO
1. The optical path is reflected at the boundary between two regions
0
2121
rd
sd
rdn
sd
rdn rayray
In this case we have and21 nn 0ˆˆ
2121
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 ssn11 sn
21 sn
1121 ˆˆˆ snsn
rd 0ˆˆ 121 rdssn
Reflected Ray
21ˆ 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
SOLO
2. The optical path passes between two regions with different refractive indexes n1 to n2. (continue – 1)
0
2121
rd
sd
rdn
sd
rdn rayray
where is on the boundary between the two regions andrd
sd
rds
sd
rds rayray 2
:ˆ,1
:ˆ 21
rd
22 sn
11 sn
1122 ˆˆˆ snsn
0ˆˆˆ 1122 rdsnsn
Refracted Ray
21ˆ 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 21ˆ 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
SOLO
Plane-Parallel Plate
i
r
ri r
t ld
i
A
C
B
E
2n1n
A single ray traverses a glass plate with parallel surfaces and emerges parallel to itsoriginal direction but with a lateral displacement d.
Optics
irriri lld cossincossinsin
r
tl
cos
r
iritd
cos
cossinsin
ir nn sinsin 0Snell’s Law
n
ntd
r
ii
0
cos
cos1sin
For small anglesi
n
ntd i
01
112
SOLO
Plane-Parallel Plate (continue – 1)
t
r
ii n
ntd
cos
cos1sin
1
2
1n
2n
i
r
r
i
i n
nt
dl
cos
cos1
sin 1
2
l
Two rays traverse a glass plate with parallel surfaces and emerge parallel to theiroriginal direction but with a lateral displacement l.
Optics
irriri lld cossincossinsin
r
tl
cos
r
iritd
cos
cossinsin
ir nn sinsin 0Snell’s Law
n
ntd
r
ii
0
cos
cos1sin
r
i
i n
nt
dl
cos
cos1
sin0 For small anglesi
n
ntl 01
Return to TOC
113
SOLO
Prisms
2i1i1t
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
Optics SOLO
Dispersive Prisms
2i1i1t
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
1sin 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
115
Optics SOLO
Prisms
2i1i1t
11 ti
2t 22 it
1
2/1
1
221
1 sincossinsinsin iii n
116
Optics SOLO
Prisms
2i1i1t
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 equations1
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
11
2 i
t
d
d
21 it
12
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
Optics SOLO
Prisms
2i1i1t
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.
2i1i
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
Optics SOLO
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.
2i1i
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
mn
Find the angle θi1 for which the deviation angle δ is minimal; i.e. δm (continue – 2).
119
Optics SOLO
Prisms
The refractive index of transparent substances varies with the wavelength λ.
1
2/1
1
221
1 sincossinsinsin iii n
2i1i1t
11 ti
2t 22 it
120
Optics SOLO
http://physics.nad.ru/Physics/English/index.htm
Prisms
υ [THz] λ0 (nm) Color
384 – 482482 – 503503 – 520520 – 610610 – 659659 - 769
780 - 622622 - 597597 - 577577 - 492492 - 455455 - 390
RedOrangeYellowGreenBlueViolet
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 thatthe white light consisted of a mixture of various colors and the diffraction was color dependent.
Isaac Newton1542 - 1727
121
SOLO
Dispersing PrismsPellin-Broca Prism
Abbe Prism
Ernst KarlAbbe
1840-1905
At Pellin-Broca Prism an incident ray of wavelength λ passes the prism at a dispersing angle of 90°. Because the dispersing angleis a function of wavelengththe ray at other wavelengthsexit at different angles.By rotating the prism aroundan axis normal to the pagedifferent rays will exit at
the 90°.
At Abbe Prism the dispersing
angle is 60°.
Optics - Prisms
122
SOLO
Dispersing Prisms (continue – 1)Amici Prism
Optics - Prisms
123
SOLO
Reflecting Prisms
2i
1i
1t
2t
E
B D
G
A
F C
BED 180
360 ABEBEDADE
190 iABE
290 tADE
3609090 12 it BED
12180 itBED
21180 tiBED
The bottom of the prism is a reflecting mirror
Since the ray BC is reflected to CD
DCGBCF Also
CGDBFC CDGFBC
FBCt 901CDGi 90221 it
202 sinsin ti nn Snell’s Law
Snell’s Law 110 sinsin ti nn 21 ti 12 i
CDGFBC ~
Optics - Prisms
124
SOLO
Reflecting Prisms
Porro Prism Porro-Abbe Prism
Schmidt-Pechan Prism
Penta Prism
Optics - Prisms
Roof Penta Prism
125
SOLO
Reflecting Prisms
Abbe-Koenig Prism
Dove Prism
Amici-roof Prism
Optics - Prisms
126
SOLO
http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html
Polarization can be achieved with crystalline materials which have a different index ofrefraction 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
SOLO
Polarizing Prisms
A Glan-Foucault prism deflects polarized lighttransmitting the s-polarized component. The optical axis of the prism material isperpendicular to the plane of the diagram.
A Glan-Taylor prism reflects polarized lightat an internal air-gap, transmitting onlythe 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 areperpendicular to the plane of the diagram.
Optics - Prisms
Return to TOC
128
Optics SOLO
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 FocalPoint
Second FocalPoint
Principal Planes
Second Principal Point
First Principal Point
Light Rays from Left
EFLBFL
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
Optics SOLO
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 beseen 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 theelements preceding the Aperture Stop.
Exit Pupil: the image of the Aperture Stop as seen from an axial point on theimage plane.
A.S. F.S.
I
Aperture and Field Stops
Entrancepupil
Exitpupil
A.S.
I
xpEnpE
ChiefRay
Entrance and Exit pupils
EntrancepupilExit
pupil
A.S. I
xpE
npE
ChiefRay
130
Optics SOLO
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 beseen 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 theelements preceding the Aperture Stop.
Exit Pupil: the image of the Aperture Stop as seen from an axial point on theimage plane.
Entrancepupil
Exitpupil
A.S.
I
ChiefRay
MarginalRay
Exp Enp
131
Optics SOLO
Definitions (continue – 3)
Principal Planes: the two planes defined by the intersection of the parallel incident raysentering an optical system with the rays converging to the focal pointsafter passing through the optical system.
FFL
First FocalPoint
Second FocalPoint
Principal Planes
Second Principal Point
First Principal Point
Light Rays from Left
EFLBFL
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
Optics SOLO
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 bundlesmultiplied 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
1sin
ff
DuNA
EFL
Du
Last Principal Plane of theOptical System (Spherical)
133
Optics SOLO
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 ImageSystemOptical
Object ImageSystemOptical
Object ImageSystemOptical
134
Optics SOLO
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
Optics SOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle
Karl Friederich Gauss1777-1855
s 's
n 'n
h
l 'l
'
M
T
CAM’
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 Tmoves on the spherical surface we must have
0d
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 obtainR
nn
s
n
s
n
'
'
'
Gaussian Formula for a Single Spherical Surface
Return to TOC
136
Optics SOLO
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 raysMT and TM’ are paraxial theangles 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
hnn
s
hn
s
hn '
''
or
r
nn
s
n
s
n
'
'
'
Gaussian Formula for a Single Spherical SurfaceKarl Friederich Gauss
1777-1855
Willebrord van Roijen Snell
1580-1626
s 's
n 'n
h
l 'l
'
M
T
CAM’
r
137
Optics SOLO
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Snell’s Law
for s → ∞ the incoming rays are parallel to opticalaxis and they will refract passing trough a commonpoint called the focus F’.
r
nn
s
n
s
n
'
'
'
s '' fs n 'n
h
'l
'
T
CA
F’
R
fs 's
n 'n
h
l
F
T
CA
R
'
r
nn
f
nn
'
'
'r
nn
nf
'
''
for s’ → ∞ the refracting rays are parallel to opticalaxis and therefore the incoming rays passes trough a common point called the focus F.
r
nnn
f
n
'' rnn
nf
'
'' n
n
f
f
Return to TOC
138
Optics SOLO
Derivation of Lens Makers’ Formula
We have a lens made of twospherical surfaces of radiuses r1
and r2 and a refractive index n’,separating two media havingrefraction indices n a and n”. Ray MT1 is refracted by the firstspherical surface (if no secondsurface 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 objectfor 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
A2C1
1rC2 F’1F’’2
M’’
F’2F1
''2f'2f'1s
'2s''2s
2r
n 'n ''n
139
Optics SOLO
Derivation of Lens Makers’ Formula (continue – 1)
M”
M
f
s
AxisA1
A2C1
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
A2C1
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” ands”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
'"'
"
"
140
Optics SOLO
Derivation of Lens Makers’ Formula (continue – 2)
M”
M
f
s
AxisA1
A2C1
1rC2
F”
F
''f
''s
2r
n 'n n
If the media on both sides of the lens is the same n = n”.
21
111
'
"
11
rrn
n
ss
21
111
'1
"
1
rrn
n
ff
Therefore
"
11
"
11
ffss
Lens Makers’ Formula
141
Optics SOLO
First Order, Paraxial or Gaussian Optics
In 1841 Gauss gave an exposition in “Dioptrische Untersuchungen”for thin lenses, for the rays arriving at shallow angles with respect toOptical axis (paraxial).
Karl Friederich Gauss1777-1855
Derivation of Lens Formula
'y
s 's
M’A F’
M
T
F
'ffx 'x
Q
Q’'y
y
S
Axisy From the similarity of the triangles
and using the convention:
''
''~'
f
y
s
yyTAFTSQ
Lens Formula in Gaussian form
f
y
s
yyFASQTS
''~
0' y
Sum of the equations:
'
'
'
''
f
y
f
y
s
yy
s
yy
since f = f’ fss
1
'
11
Return to TOC
142
Optics SOLO
First Order, Paraxial or Gaussian Optics (continue – 1)
Gauss explanation can be extended to the first order approximationto any optical system.
Karl Friederich Gauss1777-1855
'y
s 's
M’P1 F’
M
T
F'ffx 'x
Q
Q’'y
yAxis
y P2First Focal
Point
First PrincipalPoint
Second FocalPointSecond Principal
Point
Optical SystemObject
Image
Lens Formula in Gaussian form
'y
s 's
M’A F’
M
T
F
'ffx 'x
Q
Q’'y
y
S
Axisy
fss
1
'
11
s – object distance (from the first principal point to the object).
s’ – image distance (from the second principal point to the image).
f – EFL (distance between a focal point to the closest principal plane).
143
Optics SOLO
Derivation of Lens Formula (continue)
'y
s 's
M’A F’
M
T
F
'ffx 'x
Q
Q’'y
y
S
Axisy
From the similarity of the trianglesand using the convention:
f
y
x
yFASQMF
'~
Lens Formula in Newton’s form
f
y
x
yQMFTAF
'
''''~'
0' y
Multiplication of the equations:
2
'
'
'
f
yy
xx
yy
or 2' fxx
Isaac Newton1643-1727
First Order, Paraxial or Gaussian Optics (continue – 2)
Published by Newton in “Opticks” 1710
144
Optics SOLO
Derivation of Lens Formula (continue)
'h
s 's
M’A F’
M
T
F
'ffx 'x
Q
Q’'h
h
S
Axish
First Order, Paraxial or Gaussian Optics (continue – 3)
Lateral or Transverse Magnification
f
x
x
f
s
s
h
hmT
'''
(-) sign (+) sign Quantityvirtual object real object s
virtual image real image s’
diverging lens converging lens f
inverted object erect object h
inverted image erect image h’
inverted image erect image mT
145
Optics SOLO
ConcaveSpherical
ConvexSpherical
Paraboloidal
ConicEllipsoidal
GeneralAspherical
Plane
Converging : General use
Diverging : General use
Accurately focuses a parallel beamor produces a parallel beam froma point source
Refocuses a diverging bundle atanother point (P) displaced fromthe point of origin (O)
Change the direction of beam
Used mostly in combination systems of twoor more components
BASIC MIRRORS FORMS
146
Optics SOLO
Convex
PlanoConvex
Meniscus
Concave
PlanoConcave
Meniscus
Doublet
Multi-Element
Aspheric
Converging: General Use, Magnification
Converging: Used often in opposed doubles to reduce spherical aberration
Converging: reduced spherical aberration
Diverging: General Use, Demagnification
Diverging: Used in multi-element combinations
Diverging: reduced spherical aberration
Corrected for chromatic aberration
High order of aberration correction used incomplex systems
Corrected for spherical aberrationused in condenser systems
BASIC LENS FORMS
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147
Optics SOLO
Ray Tracing
F CO
I
Object VirtualImage
ConvexMirror
R/2 R/2R
F
F’CO
I
Object
RealImage
ConvergingLens
FCO
I
Object
RealImage
ConcaveMirror
F
F’CO I
Object
VirtualImage
DivergingLens
Ray Tracing is a graphically implementation of paralax ray analysis. The constructiondoesn’t take into consideration the nonideal behavior, or aberration of real lens.
The image of an off-axis point can be located by the intersection of any two of thefollowing three rays:
1. A ray parallel to the axis that isreflected through F’.
2. A ray through F that is reflectedparallel to the axis.
3. A ray through the center C of thelens that remains undeviated andundisplaced (for thin lens).
148
Optics SOLO
InfinityPrincipalfocus
SUMMARY OF SIMPLE IMAGING LENSES
f f2f2 f 0
's
'ss
fs 2 fsf 2'
fs 2 fs 2'
fsf 2 fs 2'
's
's
s
s
fs 's
s
s's
fs fs '
s's
fsf 2 fs '
Real, invertedsmall
Telescope
Real, invertedsmaller
Camera
Real, invertedsame size
Photocopier
Real, invertedlarger
Projector
No image Searchlight
Virtual, erectlarger
Microscope
Virtual, erectsmaller
Various
Figure ObjectLocation
ImageLocation
ImageProperties Example
L.J. Pinson, “Electro-Optics”, John Wiley & Sons, 1985, pg.54
Return to TOC
149
Optics SOLO
Matrix Formulation
The Matrix Formulation of the Ray Tracing method for the paraxial assumption was proposed at the beginning of nineteen-thirties by T.Smith.
Assuming a paraxial ray entering at some input plane of an optical system at the distancer1 from the symmetry axis and with a slope r1’ and exiting at some output plane at the distancer2 from the symmetry axis and with a slope r2’, than the following linear (matrix) relationapplies:
PrincipalPlanesInput
planeOutputplane
Ray path
1h2h
1r2r
'1r
'2r
Symmetryaxis
''' 1
1
1
1
2
2
r
rM
r
r
DC
BA
r
r
DC
BAMwhere ray transfer matrix
When the media to the left of the input planeand to the right of the output plane have thesame refractive index, we have:
1det CBDAM
150
Optics SOLO
Matrix Formulation (continue -1)
Uniform Optical Medium
In an Uniform Optical Medium of length d no change in ray angles occurs:
Ray path
d
1r2r
'1r
'2r
Symmetryaxis
1 2
''
'
12
112
rr
rdrr
10
1 dM
MediumOpticalUniform
Planar Interface Between Two Different Media
Ray path
1r 2r
'1r '2r
Symmetryaxis
1 2
1n 2n
12 rr
'' 1
2
12
12
rn
nr
rr
Apply Snell’s Law: 2211 sinsin nn
paraxial assumption: tan'sin r
From Snell’s Law: '' 1
2
12 r
n
nr
21 /0
01
nnM
InterfacePlanar
1det2
1 n
nM
InterfacePlanar
1det MediumOpticalUniformM
The focal length of this system is infinite and it hasnot specific principal planes.
151
Optics SOLO
Matrix Formulation (continue -2)
A Parallel-Sided Slab of refractive index n bounded on both sides with media of refractive index n1 = 1
Ray path
d
21 rr 43 rr
'1r '4r
Symmetryaxis
'2r
'3r
nn 211 n 11 n
We have three regions:• on the right of the slab (exit of ray):
'/0
01
' 3
3
124
4
r
r
nnr
r
• in the slab:
'10
1
' 2
2
3
3
r
rd
r
r
• on the left of the slab (entrance of ray):
'/0
01
' 1
1
212
2
r
r
nnr
r
Therefore:
'/0
01
10
1
/0
01
' 1
1
21124
4
r
r
nn
d
nnr
r
21
21
122112 /0
/1
/0
01
/0
01
10
1
/0
01
nn
nnd
nnnn
d
nnM
mediaentranceslabmediaexit
S labSidedParallel
10
/1 21 nndM
SlabSidedParallel
1det SlabSidedParallelM
152
Optics SOLO
Matrix Formulation (continue -3)
Spherical Interface Between Two Different Media
Ray path
21 rr '1r
'2r
Symmetry axis
1n2n
i r
1
12 rr
Apply Snell’s Law: rnin sinsin 21
paraxial assumption: rrii sin&sin
From Snell’s Law: rnin 21
2
1
2
1
2
1
12
21
0101
n
n
n
D
n
n
Rn
nnMInterfaceSpherical 1det
2
1 n
nM
InterfaceSpherical
12
11
'
'
rr
ri From the Figure:
122111 '' rnrn
111 / Rr
12
121
2
112
''
Rn
rnn
n
rnr
1
12
11
1122
12
''
n
rn
Rn
rnnr
rr
1
121 :
R
nnD
where: Power of the surface If R1 is given in meters D1 gives diopters
153
Optics SOLO
Matrix Formulation (continue -4)
Thick Lens
21 rr
43 rr
'1r
i
2 1
'2r '3r
r
2R
1R
f
'4r1C2F IO 1F
2C
Principal planes
2n
1n
s 's
d
We have three regions:• on the right of the slab (exit of ray):
'
01
' 3
3
1
2
1
2
4
4
r
r
n
n
n
Dr
r
• in the slab:
'10
1
' 2
2
3
3
r
rd
r
r
• on the left of the slab (entrance of ray):
'
01
' 1
1
2
1
2
1
2
2
r
r
n
n
n
Dr
r
Therefore:
'
101
'
01
10
101
' 1
1
2
1
2
1
2
1
2
1
1
2
1
2
1
1
2
1
2
1
1
2
1
2
4
4
r
r
n
n
n
D
n
nd
n
Dd
n
n
n
Dr
r
n
n
n
Dd
n
n
n
Dr
r
2
2
21
21
1
21
2
1
2
1
1
1
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
MLensThick
2
212 R
nnD
1
121 :
R
nnD
2
1
21
21
1
21
2
1
2
2
1
1
1
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
MLensThick
1det LensThickM
or21 DD
154
Optics SOLO
Matrix Formulation (continue -5)Thick Lens (continue -1)
21 rr
43 rr
'1r
i
2 1
'2r '3r
r
2R
1R
f
'4r1C2F IO 1F
2C
Principal planes
2n
1n
2R
1R
2f
1C 2F I
O
1F
2C
Principal planes
2n
1n
1h2h
s
s 's
's
d
Ray 2
Ray 1
1f
Let use the second Figure where Ray 2 is parallelto Symmetry Axis of the Optical System that is refractedtrough the Second Focal Point.
'1
1
' 1
1
2
2
21
21
1
21
2
1
2
1
4
4
r
r
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
r
r We found:
2141 /'&0' frrr Ray 2:
By substituting Ray2 parameters we obtain:
1
2
1
21
21
1
214
1' r
fr
nn
DDd
n
DDr
1
21
21
1
212
nn
DDd
n
DDf
frrr /'&0' 414 Ray 1:
We found:
'1
1
' 4
4
2
1
21
21
1
21
2
1
2
2
1
1
r
r
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
r
r
4
1
4
21
21
1
211
1' r
fr
nn
DDd
n
DDr
2
1
21
21
1
211 f
nn
DDd
n
DDf
155
Optics SOLO
Matrix Formulation (continue -6)
Thin Lens
21 rr
43 rr
'1r
i
2 1
'2r '3r
r
2R
1R
f
'4r1C2F IO 1F
2C
Principal planes
2n
1n
s 's
d
For thick lens we found
2
2
21
21
1
21
2
1
2
1
1
1
n
Dd
nn
DDd
n
DD
n
nd
n
Dd
MLensThick
21
21
1
211
nn
DDd
n
DD
f
For thin lens we can assume d = 0 and obtain
11
01
f
MLensThin
1
211
n
DD
f
2
212 R
nnD
1
121 :
R
nnD
211
2
1
21 111
1
RRn
n
n
DD
f
21 rr
43 rr
2R
1R
f
'4r1C2F IO 1F
2C
Principal planes
2n
1n
s 's
'1r
156
Optics SOLO
Matrix Formulation (continue -7)
Thin Lens (continue – 1)
For a biconvex lens we have R2 negative
211
2 111
1
RRn
n
f
For a biconcave lens we have R1 negative
211
2 111
1
RRn
n
f
11
01
f
MLensThin
157
Optics SOLO
Matrix Formulation (continue -8)
A Length of Uniform Medium Plus a Thin Lens
f
d
f
dd
f
MMMMediumUniform
LensThin
LensThinMediumUniform 1
1
1
10
1
11
01
21 rr
43 rr
2R
1R
f
'4r1C2F IO 1F
2C
Principal planes
2n
1n
s 's
'1r
d
Combination of Two Thin Lenses
2n
1d
1f
2d
2f
2n
21
21
2
2
2
1
1
1
21
2
21
1
2121
2
2
1
1
1
1
22
2
111
1
11
1
11
1
1122
ff
dd
f
d
f
d
f
d
ff
d
ff
f
dddd
f
d
f
d
f
d
f
d
f
d
MMMMMdMedium
UniformfLens
ThindMedium
UniformfLens
Thin
LensesThinTwo
The Focal Length of the Combination of Two Thin Lenses is:
21
2
21
111
ff
d
fff
158
Optics SOLO
Matrix Formulation (continue -9)
Mirrors r
Spherical Mirror
i
i
ii i
i
iy
RSpherical MirrorCenter of Curvature
r
Ryiii /tan
Consider a Spherical Mirror of radius R.From the geometry:
For small angles:
Ryiii /
also: iri 2 2/rii Ryiri /2
Define by n the index of reflexion of the medium:
Rynnn
yy
iir
ir
/2
i
i
r
r
n
y
Rnn
y
1/2
01
Therefore:
1/
01
1/2
01
fnRnM
MirrorSpherical
159
Optics SOLO
Matrix Formulation (continue -10)
Cavity of two Mirrors
d
12M
21M2MirrorM
1MirrorM
Spherical Mirror M1
Radius R1
Spherical Mirror M2
Radius R2
O
10
1
1/2
01
10
1
1/2
01
121221
12
d
Rn
d
RnMMMMM
MMirrorSpherical
MMirrorSphericalCavity
Figure shows two spherical mirrorsfacing each other forming an opticalcavity.
Light leaves point O, traverse the gapin the positive direction, is reflected byMirror M1, retraces the gap in thenegative direction, and is reflected byMirror M2. The System Matrix is:
21221
221
12
1
1122 /21/21/2/4/2/2
/22/21
/21/2
1
/21/2
1
RdnRdnRdnRRdnRnRn
RdndRdn
RdnRn
d
RdnRn
d
2122
21212
21
12
1
/4/4/21/4/2/2
/22/21
RRdnRdnRdnRRdnRnRn
RdndRdnM Cavity
April 13, 2023 161
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA