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1
Foundation of Geometrical Optics
SOLO HERMELIN
Updated: 7.10.06http://www.solohermelin.com
2
SOLO Foundation of Geometrical Optics
Table of Content
Derivation of Eikonal Equation
The light rays and the Intensity Law of Geometrical Optics
Derivation from High Frequencies Assumptions
Derivation from the Wave Equation in a Non-Homogeneous Media
Derivation from Maxwell Equations
Lagrange’s Invariant Integral
The Laws of Reflection and Refraction
Optical Length
Fermat’s Principle
3
SOLO Foundation of Geometrical Optics
Table of Content (continue)
References
Proof of Fermat’s Principle Using Calculus of Variation
Euler-Lagrange Equations
Transversality Conditions
Weierstrass-Erdmann Corner Conditions
Hilbert’s Invariant Integral
Second Order Conditions: Legendre’s Condition for a Weak Local Minimum
Second Order Conditions: Weierstrass’s Condition for a Strong Local Minimum
Hamilton’s Canonical Equations
Hamilton-Jacobi Equations
4
Foundation of Geometrical OpticsSOLO
“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. A physical model of a pencil of rays may be obtained by allowing the light from a source of negligible extension to pass through a very small opening in an opaque screen. The light which reaches the space behind the screen will fill a region the boundary of which (the edge of the pencil) will, at first sight, appear to be sharp. A more careful examination will reveal, however that the light intensity near the boundary varies rapidly but continuously from darkness in the shadow to lightness in the illuminated region, and the variation is not monotonic but is of an oscillatory character, manifested by the appearance of bright and dark bands, called diffraction fringes. The region in which this rapid variation takes place is only on the order of magnitude of the wavelength.…for small wavelengths the field has the same general character as that of a plane wave, moreover, that within the approximation of geometrical optics the laws of refraction and reflection established for plane waves incident upon a plane boundary remain valid under more general conditions.”
From Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3
Return to Table of Content
5
SOLO
DERIVATION OF EIKONAL EQUATION
Foundation of Geometrical Optics
Derivation from Maxwell Equations
Consider a general time-harmonic field:
( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ]tjrHtjrHtjrHaltrH
tjrEtjrEtjrEaltrE
ωωωωωω
ωωωωωω
−+==
−+==
exp,exp,2
1exp,Re,
exp,exp,2
1exp,Re,
*
*
in a non-conducting, far-away from the sources ( )0,0 == eeJ ρ
No assumption of isotropy of the medium are made; i.e.: ( ) ( )( )rr εεµµ == ,
Far from sources, in the High Frequencies we can write using the phasor notation:
( ) ( ) ( ) ( ) ( ) ( )00000 &,&, 00 εµωωω
∆−− === kerHrHerErE rSjkrSjk
Note
The minus sign was chosen to get a progressive wave:
End Note
( ) ( ) ( )[ ] ( ) ( ) ( )[ ]SktjSktj erHaltrHerEaltrE 0000 Re,&Re, −− == ωω
James Clerk Maxwell(1831-1879)
6
SOLO
From those equations we have
Foundation of Geometrical Optics
( )[ ] ( ) ( )[ ]( ) Sjktj
SjkSjktjSjktjtj
eeESjkE
EeeEeeEeerE0
000
000
000,−
−−−
×∇−×∇=
×∇+×∇=×∇=×∇ω
ωωωω
( )[ ] ( ) ( )
Sjk
SjktjSjktjtj
eHjk
eHejeHejerHt
0
00
0
00
0
0
00
000
1
1,
−
−−
=
==∂∂
εµ
εµεµωωω ωωω
from which
( ) 0
00
0000 HjkESjkEFεµ
µ−=×∇−×∇
and
01 0
00
0
00
0
∞→
→×∇=−×∇k
Ejk
HESεµ
µ
DERIVATION OF EIKONAL EQUATION (continue – 2)
Derivation from Maxwell Equations (continue – 2)
7
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,−
−−−
×∇−×∇=
×∇+×∇=×∇=×∇ω
ωωωω
( )[ ] ( ) ( )
Sjk
SjktjSjktjtj
eEjk
eEejeEejerEt
0
00
0
00
0
0
00
000
1
1,
−
−−
=
==∂∂
εµ
εµεµωωω ωωω
( ) 0
00
0000 EjkHSjkHAεµ
ε=×∇−×∇
01 0
00
0
00
0
∞→
→×∇=+×∇k
Hjk
EHSεµ
ε
8
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
µε
εωµω
( )
( )
( )( )
==
==
→∂∂
=
=
+
=⋅∇
=⋅∇
+∂∂=×∇
∂∂−=×∇
∆
0&0
2
0 00
ee
e
e
J
ck
jt
HB
ED
BGM
DGE
Jt
DHA
t
BEF
ρ
λπω
ω
µε
ρ
André-Marie Ampère1775-1836
Michael Faraday1791-1867
Karl Friederich Gauss1777-1855
9
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
EeeEeeEeerEω
ωωω
εεεεεεωε
( ) 00000 =⋅∇−⋅∇+⋅∇ ESjkEEGE εεε
01 0
000
0
∞→
→
⋅∇+⋅∇=⋅∇
k
EEjk
ESεε
We also have
from which
and
( )[ ] ( ) ( ) ( )[ ]( ) 0
,0
000
0000
000
=⋅∇−⋅∇+⋅∇=
⋅∇+⋅∇=⋅∇=⋅∇−
−−−
Sjktj
SjkSjktjSjktjtj
eeHSjkHH
HeeHeeHeerHω
ωωω
µµµµµµωµ
( ) 00000 =⋅∇−⋅∇+⋅∇ HSjkHHGM µµµ
01 0
000
0
∞→
→
⋅∇+⋅∇=⋅∇
k
HHjk
HSµµ
10
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 =−×∇ HESFεµ
µ
( ) 00
00
0 =+×∇ EHSAεµ
ε
( ) 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∇
11
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 6)
Derivation from Maxwell Equations (continue – 6)
( ) 00
00
0 =−×∇ HESFεµ
µ
( ) 00
00
0 =+×∇ EHSAεµ
ε
From the second equation we obtain
000
0 HSE ×∇−=ε
εµ
And by substituting this in the first equation
( ) 00 000
00
00
000 =+×∇×∇→=−
×∇×∇− HHSSHHSS
εµεµ
εµµ
εεµ
But ( ) ( ) ( ) ( )
2
00
02
0
0
00
n
HSHSSSHSHSS
=
∇−=∇⋅∇−∇⋅∇=×∇×∇
εµεµ
12
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 Table of Content
13
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 8)
Derivation from High Frequencies Assumptions
R.K. Luneburg and M. Kline proposed for high frequencies the empirical asymptotic series (see Crispin & Siegel, “Radar Cross-Section”, 1968, pp.21-27, Maffet, “Topics for a Statistical Description of Radar Cross Section”, 1988, § 7.2.
( ) ( ) ( )
( ) ( ) ( )∑
∑∞
=
−
∞
=
−
=
=
0
0
1,
1,
0
0
mmm
rSjk
mmm
rSjk
rHerH
rEerE
ωω
ωω
From those equations we obtain:
( )
( ) ∑∑∑
∑∑∑∞
=
−∞
=
−∞
=
−
∞
=
−∞
=
−∞
=
−
×∇+×∇−=
×∇=×∇
×∇+×∇−=
×∇=×∇
000
0
000
0
111,
111,
000
000
mmm
Sjk
mmm
Sjk
mmm
Sjk
mmm
Sjk
mmm
Sjk
mmm
Sjk
HeHSejkHerH
EeESejkEerE
ωωωω
ωωωω
Let use Faraday (F) and Ampère (A) equations, without sources:
( )( )
=×∇
−=×∇
EjHA
HjEF
εωµω
Rudolf Karl Luneburg1903 - 1949
14
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 9)
Derivation from High Frequencies Assumptions (continue – 1)
or
( )
( )
=×∇+×∇−
−=×∇+×∇−
∑∑
∑∑∞
=
−∞
=
−
∞
=
−∞
=
−
000
000
11
11
00
00
mmm
Sjk
mmmm
Sjk
mmm
Sjk
mmmm
Sjk
EjeHHSjke
HjeEESjke
εω
ωω
µω
ωω
( )
( )
=−×∇+×∇−
=+×∇+×∇−
∑
∑∞
=
∞
=
01
01
000
000
mmmmm
mmmmm
EjHHSj
HjEESj
εωεµωω
µωεµωω
This is true if all the coefficients of the same power of 1/ω are zero
,2,1
01
01
&0
0
100
100
0000
0000 =
=×∇−+×∇
=×∇−−×∇
=+×∇
=−×∇
−
−
m
Hj
EHS
Ej
HES
EHS
HES
mmm
mmm
εεµ
µεµ
εεµ
µεµ
15
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 10)
Derivation from High Frequencies Assumptions (continue – 2)
For high frequencies 1/ω → 0, only the m = 0 term must be considered
,2,1
01
01
&0
0
100
100
0000
0000 =
=×∇−+×∇
=×∇−−×∇
=+×∇
=−×∇
−
−
m
Hj
EHS
Ej
HES
EHS
HES
mmm
mmm
εεµ
µεµ
εεµ
µεµ
=+×∇
=−×∇
0
0
0000
0000
EHS
HES
εεµ
µεµ
×∇−=
×∇=
000
0
000
0
HSE
ESH
εεµ
µεµ
We can see that
( )
( )
=×∇⋅∇−=⋅∇
=×∇⋅∇=⋅∇
0
0
000
0
000
0
HSSES
ESSHS
εεµ
µεµ
16
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 11)
Derivation from High Frequencies Assumptions (continue – 3)
=+×∇
=−×∇
0
0
0000
0000
EHS
HES
εεµ
µεµ
×∇−=
×∇=
000
0
000
0
HSE
ESH
εεµ
µεµ
By substituting in the second equation we obtain0H
( ) 00000 =+×∇×∇ EESS ε
µεµ
or
( ) ( ) ( )
( ) ( )[ ] 02
000
000
00000
00
EnSSESS
ESSEESSEESS
−∇⋅∇−=
−∇⋅∇−=
+∇⋅∇−⋅∇∇=+×∇×∇=
εµεµ
εµεµ
εµεµ
17
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 12)
Derivation from High Frequencies Assumptions (continue – 4)
To summarize, we obtained
=−∇⋅∇
=⋅∇=⋅∇
0
0
0
2
0
0
nSS
ES
HS
The last equation is the Eikonal Equation
Return to Table of Content
18
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 13)
Derivation from the Wave Equation in a Non-Homogeneous Media
For non-homogeneous media ε,μ are functions of position
( )
( )
( )( )
=
=+
=⋅∇
=⋅∇
+∂∂=×∇
∂∂−=×∇
HB
ED
BGM
DGE
Jt
DHA
t
BEF
e
e
µε
ρ
0
( )
( )
( ) ( )( ) ( )
=⋅∇
=⋅∇
+∂∂=×∇
∂∂−=×∇
0HGM
EGE
Jt
EHA
t
HEF
e
e
µ
ρε
ε
µ
From those equations we have
( )
( ) ( )
( )[ ] ( )
×∇×∇−∇−⋅∇∇=
=×∇×
∇+×∇×∇=
×∇×∇
∂∂−
∂∂−=×∇
∂∂=
×∇×∇+
∂∂=∇×
∂∂=∇×
EEE
EEE
t
J
t
EH
tE e
Jt
EH
t
HE e
µµµ
µµµ
εµ
εµ
22
2
2
11
111
1 ( ) ( ) ( )( )
( ) ( )[ ]
∇+∇⋅−∇=
+⋅∇−∇=⋅∇∇→
→=⋅∇+⋅∇=⋅∇
∂∂=×∇×∇+⋅∇∇−
∂∂−∇
ερε
ερ
εε
ρεεε
µµµε
ee
e
e
EEE
EEE
t
JEE
t
EE
ln
ln2
22
From which
( )[ ] ( ) ( )
∇+
∂∂
=×∇×∇+∇⋅∇+∂∂−∇
ερµµεµε ee
t
JEE
t
EE
lnln2
22
19
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 14)
Derivation from the Wave Equation in a Non-Homogeneous Media (continue – 1)
Also
or
( )
×∇+
∂∂−=
×∇+×∇
∂∂=
×∇×∇
∂∂
−=∇×+∂∂
=∇×
εµ
εε
µε
et
HE
e
Jt
EH
J
t
HJE
tH
e
2
21
and
( ) ( )
( )[ ] ( ) ( )( ) ( )
⋅−∇=⋅∇−=⋅∇→=⋅∇+⋅∇=⋅∇
×∇×∇−∇−⋅∇∇=
=×∇×
∇+×∇×∇=
×∇×∇
HHHHHH
HHH
HHH
µµµµµµ
εεε
εεε
ln0
ln11
111
2
( ) ( ) ( )
×∇−=×∇×∇+⋅∇∇−
∂∂−∇
εεεµε eJ
HHt
HH
ln2
22
( )( ) ( ) ( )
×∇−=×∇×∇+∇⋅∇+
∂∂−∇
εεεµµε eJ
HHt
HH
lnln2
22
20
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 15)
Derivation from the Wave Equation in a Non-Homogeneous Media (continue – 2)
Far from sources, in the High Frequencies we can write, using the phasor notation
( ) ( ) ( )0000 &, 0 εµωω
∆− == kerErE rSjk
The Wave Equation in a Non-Homogeneous Media, without sources is:
( )[ ] ( ) ( ) 0lnln2
22 =×∇×∇+∇⋅∇+
∂∂−∇ EE
t
EE
µεµε
or in phasor notation
( )[ ] ( ) ( ) µεωµε∆==×∇×∇+∇⋅∇++∇ kEEEkE &0lnln22
21
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 16)
Derivation from the Wave Equation in a Non-Homogeneous Media (continue – 3)
Let compute
( ) ( ) ( )( ) ( ) ( ) ( )( ) 000000, EeeEerErE rSjkrSjkrSjk −−− ∇+∇=∇=∇ ω
( ) SjkSjk eSjke 000
−− ∇−=∇( ) ( ) ( ) ( )[ ]
( ) ( )[ ] Sjk
SjkSjkSjk
eESjkESjkESjkESjkE
EeeEeErE0
000
00000002
002
0002 ,
−
−−−
∇−∇⋅∇−∇⋅∇−∇−∇=
∇+∇⋅∇=∇⋅∇=∇ ω
( ) ( ) ( )( ) Sjk
SjkSjkSjk
eESjkE
EeeEeErE0
000
000
000,−
−−−
×∇−×∇=
×∇+×∇=×∇=×∇ ω
( ) ( )[ ] ( )[ ] ( )[ ] ( ) ( )( )[ ] ( ){ } Sjk
SjkSjkSjk
eESjkE
EeEeeErE0
000
lnln
lnlnlnln,
000
000
−
−−−
∇⋅∇−∇⋅∇=
∇⋅∇+∇⋅∇=∇⋅∇=∇⋅∇
εεεεεεω
( )[ ] ( ) ( ) 0lnln22 =×∇×∇+∇⋅∇++∇ EEEkE
µε
( ) ( ){ 02
00000002
002 EkESjkESjkESjkESjkE +∇−∇⋅∇−∇⋅∇−∇−∇
( )[ ] ( ) ( ) [ ]} SjkeESjkEESjkE 0000000 lnlnln −∇−×∇×∇+∇⋅∇−∇⋅∇+ µεε
Starting from
22
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 17)
Derivation from the Wave Equation in a Non-Homogeneous Media (continue – 4)
( ) ( ){ 02
00000002
002 EkESjkESjkESjkESjkE +∇−∇⋅∇−∇⋅∇−∇−∇
( )[ ] ( ) ( ) [ ]} SjkeESjkEESjkE 0000000 lnlnln −∇−×∇×∇+∇⋅∇−∇⋅∇+ µεε
Since ,by dividing the previous equation bywe obtain
000
00
knk =
== εµ
εµµε
ωµεωSjkek 02
0−
But
( ) ( ) ( ) ( )[ ]
( )( ) ( ) ( )[ ]{ } 0lnln
1
lnln1
02
0020
0002
00
2
=∇−∇⋅∇−∇××∇+
∇⋅∇−×∇×∇−∇−−∇⋅∇−
EEEjk
ESESESjk
ESSn
εµ
εµ
( ) ( ) ( ) ( ) ( )[ ]( )[ ] ( ) ( )[ ] 000
0000
lnln2ln
lnlnlnln
ESSEnES
EESESES
µµεµεµ
∇⋅∇+∇⋅∇−=∇⋅∇+⋅∇+⋅∇−∇=∇⋅∇−×∇×∇−
Hence we obtain
( ) ( )( ) ( )[ ]
( )( ) ( ) ( )[ ]{ } 0lnln
1
ln2ln1
02
0020
002
00
2
=∇−∇⋅∇−∇××∇+
∇⋅∇−∇−∇⋅∇−∇⋅∇−
EEEjk
SEnESSjk
ESSn
εµ
µ
23
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 18)
Derivation from the Wave Equation in a Non-Homogeneous Media (continue – 5)
or
( ) ( )( ) ( )[ ]
( )( ) ( ) ( )[ ]{ } 0lnln
1
ln2ln1
02
0020
002
00
2
=∇−∇⋅∇−∇××∇+
∇⋅∇−∇−∇⋅∇−∇⋅∇−
EEEjk
SEnESSjk
ESSn
εµ
µ
( ) ( )( )
( ) 0,,1
,,,1
,, 020
00
0 =+− µεµ EMjk
nSELjk
nSEK
where
( ) ( )( ) ( )( ) ( )
( ) ( ) ( ) ( )[ ] 02
000
002
0
02
0
lnln,,
ln2ln,,,
,,
EEEEM
SEnESSnSEL
ESSnnSEK
∇−∇⋅∇−∇××∇=
∇⋅∇−∇−∇⋅∇=
∇⋅∇−=
∆
∆
∆
εµµε
µµ
24
SOLO Foundation of Geometrical Optics
DERIVATION OF EIKONAL EQUATION (continue – 19)
Derivation from the Wave Equation in a Non-Homogeneous Media (continue – 6)
In the same way using
( ) ( ) ( ) ( ) 0,,1
,,,1
,, 020
00
0 =+− µεµ HMjk
nSHLjk
nSHK
we obtain
( ) ( )( ) ( )( ) ( )
( ) ( ) ( ) ( )[ ] 02
000
002
0
02
0
lnln,,
ln2ln,,,
,,
HHHHM
SHnHSSnSHL
HSSnnSHK
∇−∇⋅∇−∇××∇=
∇⋅∇−∇−∇⋅∇=
∇⋅∇−=
∆
∆
∆
εµµε
µµ
( ) ( ) ( )0000 &, 0 εµωω
∆== kerHrH rSjk
( ) ( ) 0,, 02
0 =∇⋅∇−= HSSnnSHK
or
For sufficient large (high frequencies) the second and third term may be neglected and the wave equations becomes
000 εµω=k
( ) 22 nS =∇ Eikonal EquationReturn to Table of Content
25
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 εµωωω
∆−− === kerHrHerErE rSjkrSjk
We found the following relations
( ) 00
00
0 =−×∇ HESFεµ
µ
( ) 00
00
0 =+×∇ EHSAεµ
ε
( ) 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
×=
26
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
ωωωωωωε
ωωωωωωωωε
ωωε
ε
But( ) ( )[ ] ( )
( ) ( )[ ] ( )0
2
2exp2exp
2
12exp
1
02
2exp2exp
2
12exp
1
0
0
00
∞→
∞→
→−=−=−
→==
∫
∫
T
TT
T
TT
Tj
Tjtj
Tjdttj
T
Tj
Tjtj
Tjdttj
T
ωωω
ωω
ωωω
ωω
Therefore
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )rErEerEerEdtT
rErEw rSjkrSjkT
e
*00
*00
0
*
22
1,,
200
εεωωε ==∫= −
Let compute the time averages of the electric and magnetic energy densities
27
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
εεωωε ==∫= −
( ) ( ) ( ) ( ) ( ) ( )rHrHdttrHtrHT
dttrBtrHT
wTT
m*00
00 2,,
1,,
1 µµ === ∫∫
Using the relations
( ) 000
0 HSEA ×∇−=ε
εµ
( ) 000
0 ESHF ×∇=µ
εµ
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
µεµµµ
εεµεε
28
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
×∇⋅=⋅+⋅=+= µεThe 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
,,
ωωωω
ωω
ϖϖϖϖ
ϖϖ
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]
( ) ( ) ( ) ( )[ ]ωωωω
ωωωωωωωω ωω
,,,,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
29
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 3)
Using the relations
( ) 000
0 HSEA ×∇−=ε
εµ ( ) 000
0 ESHF ×∇=µ
εµ
( ) ( ) ( ) ( )[ ] ( ) ( )( ) ( )( ) ( )
××∇−×∇×=×+×= rHrHSrESrErHrErHrES 0
*
0
*
0000
0*
0*
00 222
1
4
1 µεεµεµ
we obtain
( ) ( ) ( ) ( )
⋅∇+⋅∇−⋅∇−⋅∇= *
00
0
0*
0
0
0*
0*
0000
22222
1HHSHSHESEEES
µµεεεµεµ
( ) ( ) ( ) ( ) ( ) ( )( )*
0000*
00 22rHSrErHrHrErEwww me
×∇⋅=⋅+⋅=+= µε
we obtain
Using
( ) wSn
cwwSS me ∇=+∇=
200
00 22
1
εµεµ
εµ
00
2
00
&1
εµεµ
εµ== nc
30
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
r
0s
0r
A Bundle of Light Rays
31
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
r
0s
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=
32
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 Ssd
rdn ∇=← ray
( ) ( ) ( ) ( ) ( ) ABBAABBABA
∇⋅+∇⋅++×∇×+×∇×=⋅∇AB
≡↓
( ) ( ) ( ) AAAAAA
∇⋅+×∇×=⋅∇2
1
SA ∇≡↓
( ) ( ) ( ) ( ) SSSSSSSS ∇∇⋅∇=∇∇⋅∇+∇×∇×∇=∇⋅∇∇←
0
2
1( )SSn
∇⋅∇∇=2
1
2nSS =∇⋅∇←( )2
2
1n
n∇=
n∇=
33
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
r
0s
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( )
00
1rayray
ray rrpnsd
rd
==
( ) 0ˆ0 snpnpsd
d=∇=
34
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 8)
Example 1
In a homogeneous medium n is constant and the ray 2nd order differential equation
02
ray2
=sd
rd
has the solution
ssrr 00ray ˆ+=
From this equation we can see that in a homogeneous medium the light rays are straight lines.
where is a constant vector and is a constant unit vector.0r
0s
35
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 9)
Example 2
Earth atmosphere
Assume that because of the change in atmosphere density with the altitude we have
( ) milesRrmileskr
krn 000,4,971
2
2
=>=+=
36
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 10)
Example 2 (continue – 1)
Let compute ( )[ ] ( )snsd
drsn
sd
rdsrnr
sd
dray
rayray ˆˆˆ ×+×=×
Since 0ˆˆˆˆ =×=×⇒= snssnsd
rds
sd
rd rayray
From ( ) nsnsd
d
sd
rdn
sd
d ∇==
ˆray
( ) ( ) ( )0ˆ =×=∇×=×
ray
ray
ray
rayrayrayrayray rd
rnd
r
rrrnrsn
sd
dr
also
( )[ ] ( ) cconstsrnrsrnrsd
drayrayrayray
==×⇒=× ˆ0ˆ
37
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 11)
Example 2 (continue – 2)
From Figure
( )[ ] ( ) cconstsrnrsrnrsd
drayrayrayray
==×⇒=× ˆ0ˆ
θθθθθθ
ˆˆ rayrayray
ray
rayray
ray
rayray
ray
rayray
ray rrd
rd
r
r
d
dr
r
r
d
rd
r
rr
d
d
d
rd+=
+
=
=
2
2
ˆˆ
ˆ
rayray
rayrayray
ray
ray
rd
rd
rrd
rd
d
rd
d
rd
s
+
+==
θ
θθ
θ
θ
( ) ( ) ( )c
rd
rd
rrn
rd
rd
rrd
rd
rnrrsrnr
rayray
rayray
rayray
rayrayray
rayrayrayrayray =
+
=
+
+×=×
2
2
2
2
2
ˆˆ
ˆˆ
θθ
θθ
38
SOLO Foundation of Geometrical Optics
THE LIGHT RAYS AND THE INTENSITY LAW OF GEOMETRICAL OPTICS (continue – 12)
Example 2 (continue – 3)
( ) ( ) ( )c
rd
rd
rrn
rd
rd
rrd
rd
rnrrsrnr
rayray
rayray
rayray
rayrayray
rayrayrayrayray =
+
=
+
+×=×
2
2
2
2
2
ˆˆ
ˆˆ
θθ
θθ
( ) 222 crrnc
r
d
rdrayray
rayray −=θ
From which
Separation of variables will give
( )∫−
=rayr
rayrayray
ray
crrnr
rdc
222θ
Return to Table of Content
39
A step-index cylindrical fiber has a central core of index ncore surrounded bycladding of index ncladding where ncladding < ncore.
SOLO Optical Fiber – Ray Theory
Cladding
Coreaxisθ
0θ
iθ
Core axisCladding
Skew ray in core of fiber
Meridional ray in corewith two reflexions
When a ray of light enters such afiber at an angle θ0 is refracted at anangle θ, and then reflected back at the boundary between core and cladding, if the angle of incidence θi is greater than the critical angle θc.
Two distinct rays can travel inside the fiber in this way:
• meridional rays remain in a plan that contains fiber axis
• skew rays travel in a non-planar zig-zag path and never cross the fiber axis
Example 3
40
For the meridional ray
SOLO Optical Fiber – Ray Theory
Cladding
Coreaxisθ
0θ
iθ
Meridional ray in corewith two reflexions
Snell’s Law at the fiber enter
If the ray is refracted from the core to the cladding than according to Snell’s Law:
222
0 sin1cossinsin claddingcoreicoreicorecore nnnnn −<−=== θθθθ
r
core
cladding
i n
nθθ sinsin =
If there is no tunneling from core to cladding. 1sin:sin ≤=> c
core
cladding
i n
nθθ
Since we have90=+ iθθ
θθ sinsin 0
1
coreair nn =
Therefore total internal reflection will occur if:2
22
0 1sin
−=−<
core
cladding
corecladdingcore n
nnnnθ
41
We consider only two types of optical fibers:
SOLO Optical Fiber – Ray Theory
Skew ray in step-indexcore fiber
Meridional ray in step-indexcore fiber
Core axisCladding
Core axisCladding
zθ
φθφ1
r1z1.constnn corecladding =<
Meridional ray in a grated-index core
Core
axisCladding
Skew ray in a grated-index core of fiber
( )rnncore =
Core axisCladding
zθφθ
r
r1
φ1
• step-index core fiber where the index of refraction in core is constant and changes by a step in the cladding such that
corecladding nn <
• graded-index core fiber where the index of refraction in core changes as function of radius r such that ( )rnncore =
42
For a graded-index core fiber ncore = n ( r ) let develop the ray equation:
SOLO Optical Fiber – Ray Theory
( ) ( ) ( ) rrnrd
drn
sd
rdrn
sd
d1ray =∇=
zzrrr 11ray +=
where:rayr
- ray vector
rayrdsd=
Assuming a cylindrical core fiber we will use cylindrical coordinates
zzddrrrdrd 111ray ++= φφ
Graded-index Fiber
szsd
zd
sd
drr
sd
rd
sd
rd1:111ray =++= φφ
=
−=
=
01
11
11
zd
rdd
drd
φφ
φφ
011111 =−== zsd
dr
sd
d
sd
d
sd
dr
sd
d φφφφ
=
+−=
+=
zz
yx
yxr
11
1cos1sin1
1sin1cos1
φφφ
φφ
to describe the ray vector:
( ) ( ) ( ) ( ) 22222/1zddrrdrdrdsd rayray ++=⋅= φ
ray propagation direction
43
SOLO Optical Fiber – Ray Theory
Skew ray in core of fiber
zθ
φθ
φ1
r1
z1
ρ
Q
P
zrrr zzz 1cos1cossin1sinsin1 ray θθθθθ φφ ++=
ρφθ
CoreQ' axis
Core
axisCladding
zθφθ
r
r1
φ1
ray1r
( )rnncore =
( ) ( ) ( ) rrnrd
drn
sd
rdrn
sd
d1ray =∇=
Graded-index Fiber (continue – 1)
zsd
zd
sd
drr
sd
rd
sd
rd111ray ++= φφ
( )
( ) ( )
( ) ( )
( ) ( )
0
ray
11
11
11
sd
zd
sd
zdrnz
sd
zdrn
sd
d
sd
d
sd
drrn
sd
drrn
sd
d
sd
rd
sd
rdrnr
sd
rdrn
sd
d
sd
rdrn
sd
d
+
+
+
+
+
=
φφφφ
( ) ( ) ( ) ( ) ( ) zsd
zdrn
sd
dr
sd
drnr
sd
drnr
sd
d
sd
d
sd
rdrnr
sd
rdrn
sd
d11111
2
+
−
++
= φφφφφ
( ) ( ) ( ) ( ) ( ) ( ) ( )r
rd
rndz
sd
zdrn
sd
d
sd
d
sd
rdrn
sd
drn
sd
dr
sd
drnr
sd
rdrn
sd
d
sd
rdrn
sd
d11121
2
ray =
+
+
+
−
=
φφφφ
011111 =−== zsd
dr
sd
d
sd
d
sd
dr
sd
d φφφφ
44
SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 2)
( ) ( ) ( ) ( ) ( ) ( ) ( )r
rd
rndz
sd
zdrn
sd
d
sd
d
sd
rdrn
sd
drn
sd
dr
sd
drnr
sd
rdrn
sd
d
sd
rdrn
sd
d11121
2
ray =
+
+
+
−
=
φφφφ
From this equation we obtain the following three equations:
( ) ( ) ( )rd
rnd
sd
drnr
sd
rdrn
sd
d =
−
2
φ
( ) ( )02 =+
sd
d
sd
rd
r
rn
sd
drn
sd
d φφ
( ) 0=
sd
zdrn
sd
d
( ) ( ) 022 =+
sd
d
sd
rdrrn
sd
drn
sd
dr
φφ2r×
( ) 02 =
sd
drnr
sd
d φ
( ) constsd
zdrn == β ( ) .2 constl
sd
drnr == ρφ
Integration
Integration
where:
l,β - dimensionless constants (ray invariants) to be defined
ρ - radius of the boundary between core and cladding
By integrating the last two equation we obtain:
(1)
(2)
(3)
(3’) (2’)
45
( ) ( ) zrnsd
zdrn θβ cos==
SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 3)
We found that the ray propagation vector is
Skew ray in core of fiber
φ
φ1
r1 z1
Q
P
zrs zzz 1cos1cossin1sinsin1 θφθθθθ φφ ++=Core
Q' axis
Core
axis
Cladding
zθ
ssd
rd1:ray =
φ
φ1
φθ
r
r1
φ1innercaustic
outercaustic
s1
z1
zθ
( )rnncore =
szsd
zd
sd
drr
sd
rd
sd
rd1111ray =++= φφ
( )rnsd
zd β= ( )rnrl
sd
d2
ρφ =
( ) ( ) sd
rdz
rnrnr
lr
sd
rds ray1111
=++= βφρ
( )sd
rdzrs zz
ray1cos1cos1sinsin1
=++= θφθθθ φφ
Let write also as a function of two geometric parameterss1 φθθ ,z
φθ - skew angle
zθ - angle between ands1 z1
( )rnrl
z
ρθθ φ =cossin ( ) φθθρ
cossin zrnr
l =
(3’) (2’)
46
( ) ( ) zrnsd
zdrn θβ cos==
SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 4)
We found
φθ
r
r1
φ1innercaustic
intesectsray path
outercaustic
intersectsray path
0=φθ
0=φθ
The skew rays take a helical path, as seen from the cross-section figure.
( ) φθθρ
cossin zrnr
l =
( ) ( ) ( ) ( ) 22222 cossincos
β
ρ
θ
ρθ
ρθ φ−
=−
==rn
l
rrnrn
l
rrn
l
rz
z
( ) ( ) 0== ocic rr φφ θθ
A particular family of skew ray will not come closer to the fiber axis than the inner caustic cylindrical surface of radius ric and further from the axis than the outer caustic cylindrical surface of radius roc. From the figure we can see that at the intersection of ray path with the caustic surface
Therefore the caustic radiuses can be found by solving:( )
( ) 10cos22
===−
φθβ
ρ
rn
l
r
or ( ) ( ) 0:2
2222 =−−=r
lrnrgρβ ( ) ( ) 0== ocic rgrg
47
SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 5)
We obtained:
( )rnsd
zd β= ( )rnrl
sd
d2
ρφ =
( ) zd
d
rnzd
d
sd
zd
sd
d β==
( ) ( ) ( ) ( ) ( )( ) ( )rnrd
rnd
rnr
lrnr
zd
rd
rnrn
zd
d
rn×=
−
2
2
ρββ
( )2
2
3
22
2
22
2
1
rd
rnd
rl
zd
rd =− ρβ
Define:zd
rdr =:'
rd
rdr
zd
rd
rd
d
zd
rd
zd
rd
zd
d
zd
rd ''
2
2
=
=
=
( )2
2
3
222
2
1'
rd
rnd
rl
rd
drr =− ρβ Integration ( ) constrn
rl
zd
rd +=+
2
2
22
2
2
2
1
2
1
2
1 ρβ
( ) constsd
zdrn == β(3’) ( ) .2 constl
sd
drnr == ρφ
(2’)
( ) ( ) ( )rd
rnd
sd
drnr
sd
rdrn
sd
d =
−
2
φ(1)
( )( )
2
2
2222
2
2 2 βρββ +⋅+−−=
const
rlrn
zd
rd
rg
48
SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 6)
We obtained: ( )( )
2
2
2222
2
2 2 βρββ +⋅+−−=
const
rlrn
zd
rd
rg
φθ
r
r1
φ1innercaustic
intesectsray path
outercaustic
intersectsray path
0=φθ
0=φθ
To determine the constant we use the fact that at
the caustic we havetherefore
( ) ( ) 0&02
2222 =−−==r
lrnrgzd
rd ρβ
02 2 =+⋅ βconst
Finally we obtain the ray path equation:
( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
Since a ray path exists only in the regions where0
2
2 ≥
zd
rdβ ( ) 0>rg
49
SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 7)
Analysis of: ( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
A ray path exists only in the regions where ( ) 0>rg
1. Bounded rays
The rays are bounded in the core region iff:
g (r)>0 for ric<r < roc and g (r)<0 for r ≥ ρ
rρ
ocricr
2
22
rl
ρ
cladding
core
0≠l
( )rg
skew ray
β<claddingn( ) ociccore rrrrn ≤≤> β
( ) ociccorecladding rrrrnn ≤≤<< β
rρ
ocr
0=l
cladding
core( )rg
meridional ray
50
SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 8)
Analysis of: ( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
A ray path exists only in the regions where ( ) 0>rg
2. Refracted rays
The rays are refracted from the core in the cladding region iff:
g (r)>0 for r ≥ ρ
rρicr
2
22
rl
ρ
cladding
core
0≠l
( )rg
skew ray
222 lncladding +> β
51
SOLO Optical Fiber – Ray TheoryGraded-index Fiber (continue – 9)
Analysis of: ( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
A ray path exists only in the regions where ( ) 0>rg
3. Tunneling rays
The rays escape in the cladding region iff:
g (r)<0 for ρ <r<rrad and g (r)>0 for r ≥ rrad
222 lncladding +< β
rρ
ocr
icr
2
22
rl
ρ
cladding
core
0≠l
( )rg
skew ray
radr
β>claddingn
22 lncladding +<< ββ
( ) 02
2222 =−−=
rad
claddingrader
lnrgρβ
22 β
ρ
−=
cladding
rad
n
lr
The energy leaks from the core tothe cladding region.
52
For a step-index core fiber ncore = constant.
SOLO Optical Fiber – Ray Theory
Core axisCladding
Skew ray in core of fiber
zθ
φθ
s1
φ1
r1
z1
ρ
Q
P
zrrs zzz 1cos1cossin1sinsin1 θθθθθ φφ ++=
ρφθ
Core
PQ
Q' axis
P Q'ρ
φθρ sin2' =PQ
φθ
φθ
icr
φθρ cos=icr
φθ
innercaustic
.constnn corecladding =<
Step-index Fiber
( ) ( ) zrnsd
zdrn θβ cos==
( ) φθθρ
cossin zrnr
l =
( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
( )
≥=<=
=ρρ
rconstn
rconstnrn
cladding
core
2
1
53
SOLO Optical Fiber – Ray TheoryStep-index Fiber (continue – 7)
Analysis of: ( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
A ray path exists only in the regions where ( ) 0>rg
1. Bounded rays
The rays are bounded in the core region iff:
g (r)>0 for r = ρ- ε and g (r)<0 for r = ρ+ε
β<claddingnβ>coren
corecladding nn << β
rρ22 β
ρ
−=
core
ic
n
lr
2
22
rl
ρ
claddingcore
0≠l
( )rg
skew ray
22 β−coren
22 β−claddingn
corenn = claddingnn =
0222 >−−= lng core β
0222 <−−= lng cladding β
rρ
0=l
claddingcore( )rg
meridional ray
022 <−= βcladdingng
022 >−= βcoreng
corenn = claddingnn =
( )
≥=<=
=ρρ
rconstn
rconstnrn
cladding
core
2
1
( ) 0=icrg φ
θθρ
θβθρ
β
ρ φ
coscossin
cos22
zcore
zcore
nl
n
core
ic
n
lr
=
==
−=
P Q'ρ
φθρ sin2' =PQ
φθ
φθ
icr
φθρ cos=icr
φθ
innercaustic
54
SOLO Optical Fiber – Ray TheoryStep-index Fiber (continue – 8)
Analysis of: ( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
A ray path exists only in the regions where ( ) 0>rg
2. Refracted rays
The rays are refracted from the core in the cladding region iff:
g (r)>0 for r ≥ ρ
22 lncladding +> β
( )
≥=<=
=ρρ
rconstn
rconstnrn
cladding
core
2
1
rρ22 β
ρ
−=
core
ic
n
lr
2
22
rl
ρ
claddingcore
0≠l
( )rg
skew ray
22 β−coren
22 β−claddingn
corenn = claddingnn =
0222 >−−= lng core β
0222 >−−= lng cladding β
55
SOLO Optical Fiber – Ray TheoryStep-index Fiber (continue – 9(
Analysis of: ( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
A ray path exists only in the regions where ( ) 0>rg
3. Tunneling rays
The rays escape in the cladding region iff:
g (r)<0 for ρ <r<rrad and g (r)>0 for r ≥ rrad
222 lncladding +< β β>claddingn
22 lncladding +<< ββ
( ) 02
2222 =−−=
rad
claddingrader
lnrgρβ
22 β
ρ
−=
cladding
rad
n
lr
The energy leaks from the core tothe cladding region.
( )
≥=<=
=ρρ
rconstn
rconstnrn
cladding
core
2
1
rρ22 βρ
−=
core
ic
n
lr
2
22
rl
ρ
claddingcore
0≠l
( )rg
skew ray
22 β−coren
22 β−claddingn
corenn = claddingnn =
22 β
ρ
−=
cladding
rad
n
lr
0222 >−− lncore β
0222 <−− lncladding β
56
For a step-index core fiber ncore = constant.
SOLO Optical Fiber – Ray Theory
P Q'ρ
φθρ sin2' =PQ
φθ
φθ
icr
φθρ cos=icr
φθ
innercaustic
Step-index Fiber
( ) ( )2
2222
2
2 :r
lrnrgzd
rd ρββ −−==
( )
≥=<=
=ρρ
rconstn
rconstnrn
cladding
core
2
1
rρ22 β
ρ
−=
core
ic
n
lr
2
22
rl
ρ
claddingcore
0≠l
( )rg
skew ray
22 β−coren
22 β−claddingn
corenn = claddingnn =
0222 >−−= lng core β
0222 <−−= lng cladding β
rρ
0=l
claddingcore( )rg
meridional ray
022 <−= βcladdingng
022 >−= βcoreng
corenn = claddingnn =
corecladding nn << β
rρ22 β
ρ
−=
core
ic
n
lr
2
22
rl
ρ
claddingcore
0≠l
( )rg
skew ray
22 β−coren
22 β−claddingn
corenn = claddingnn =
0222 >−−= lng core β
0222 >−−= lng cladding β
rρ22 β
ρ
−=
core
ic
n
lr
2
22
rl
ρ
claddingcore
0≠l
( )rg
skew ray
22 β−coren
22 β−claddingn
corenn = claddingnn =
22 β
ρ
−=
cladding
rad
n
lr
0222 >−− lncore β
0222 <−− lncladding β
1. Bounded rays
2. Refracted rays
222 lncladding +> β
3. Tunneling rays
22 lncladding +<< ββ
Return to Table of Content
57
SOLO Foundation of Geometrical Optics
Lagrange’s Invariant Integral
Joseph Louis Lagrange1736-1813
From , since S is only a function of the position relative to the light source, we have
snS ˆ=∇
rdsnrdSSd ⋅=⋅∇= ˆ
We can see that for any two points and and for any curve that joint them
( )1111 ,, zyxP ( )2222 ,, zyxP
( ) ( )111222 ,,,,ˆ2
1
2
1
zyxSzyxSSdrdsnP
P
P
P
−=∫=∫ ⋅
Lagrange’s Invariant Integral orPoincaré’s Invariant
Jules Henri Poincaré1854-1912
58
SOLO Foundation of Geometrical Optics
Lagrange’s Invariant Integral (continue – 1)
To the same result we could arrive by tacking the curl
Using Stoke’s Theorem when performing the following integral on any closed path C that encloses a single connected region of area SC.
( ) 0ˆ =∇×∇=×∇ Ssn
( ) 0ˆˆ =⋅×∇=⋅ ∫∫∫∫CS
C
Stokes
C
Sdsnrdsn
We can see that for any two points and on C
( )1111 ,, zyxP ( )2222 ,, zyxP
( ) ( )111222 ,,,,ˆ2
1
2
1
zyxSzyxSSdrdsnP
P
P
P
−=∫=∫ ⋅
George Gabriel Stokes1819-1903
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59
SOLO Foundation of Geometrical Optics
The Laws of Refraction and Reflection
Consider two regions with different refractive indices n1 and n2.
Consider first a ray that is reflected from region (1( to region (2(.
Let take any plane normal to the boundary, and in this plane a closed curve C (P1, P2, P3, P4) that is closed to the boundary and passes through two regions (P1 and P2 in (1( and P3, P4 in (2( are parallel to the boundary, and P2, P3 and P1, P4 normal to the boundary) and defines an area SC (see Figure).
This can be developed to give
( ) ( ) 0ˆˆˆˆˆˆˆˆˆ 222111
0
222111 =⋅+⋅=Θ+⋅+⋅=∫ ⋅→
ldtsntsnhldtsnldtsnrdsnh
C
60
SOLO Foundation of Geometrical Optics
The Laws of Refraction and Reflection (continue – 1)
( ) ( ) 0ˆˆˆˆˆˆˆˆˆ 222111
0
222111 =⋅+⋅=Θ+⋅+⋅=∫ ⋅→
ldtsntsnhldtsnldtsnrdsnh
C
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 ⋅×≡×⋅
( ) ( ) ( )[ ] 0ˆˆˆˆˆˆˆˆˆˆˆ 22112121221112211 =⋅−×=×⋅−=⋅− −− ldbsnsnnldnbsnsnldtsnsn
Since this must be true for any vector that lies on the boundary between regions (1( and (2( we must have:
b
( ) 0ˆˆˆ 221121 =−×− snsnn
61
SOLO Foundation of Geometrical Optics
The Laws of Refraction and Reflection (continue – 2)
( ) 0ˆˆˆ 221121 =−×− snsnn
This is Snell’s Law
1. are in the same plane
2121 ˆ,ˆ,ˆ ssn −
2. if is the angle between and , and is the angle between and , than
1θ 1s 21ˆ −n 2θ21ˆ −n
2s
Willebrord van Roijen Snell1580-1626
For the reflected ray we can use the same reasoning, and by taking n1 = n2, we obtain
2211 sinsin θθ nn =
( ) 0ˆˆˆ 2121 =−×− ssn
21 θθ =
Return to Table of Content
62
SOLO Foundation of Geometrical Optics
Optical Length
Let define a new integral along any curve C that joints the points P1 and P2.
∫2
1
C
P
P
dsn
To see the difference between this integral and the Lagrange’s Integral
( ) ( )111222 ,,,,ˆ2
1
2
1
zyxSzyxSSdrdsnP
P
P
P
−=∫=∫ ⋅
let use two paths of integration, along an optical ray P1 Q1 Q2 P2, and not along an optical ray P1 Q1 Q2 P2
63
SOLO Foundation of Geometrical Optics
Optical Length (continue – 1)
Assume that through the points P1 and P2 passes one and only one optical ray P1 Q1 Q2 P2,
A neighborhood where the optical rays don’t intersect is called a regular neighborhood.
Using the Lagrange’s Invariant Integral we obtain ∫ ⋅=∫ ⋅
22112211
ˆˆPQQPPQQP
rdsnrdsn
Along the optical ray P1 Q1 Q2 P2, we have the optical ray unit vector that is collinear with the path differential , therefore and
sd
rds ray
=ˆ
rd
sdsdsd
rd
sd
rdrds rayray
ray =
⋅
=⋅
ˆ
( ) ( )1222112211
ˆ PSPSsdnrdsnPQQPPQQP
−=∫=∫ ⋅
Along the path P1 Q1 Q2 P2, we have the optical ray unit vector that is not collinear with the path differential , therefore and
sd
rds ray
=ˆ
rd
sdsdsd
rd
sd
rd ray ≤
⋅
≤
1
∫≤∫ ⋅22112211
ˆPQQPPQQP
sdnrdsn
64
SOLO Foundation of Geometrical Optics
Optical Length (continue – 2)
( ) ( )1222112211
ˆ PSPSsdnrdsnPQQPPQQP
−=∫=∫ ⋅
The equality sign occurs only if along the path P1 Q1 Q2 P2, and are collinear, but this is not possible since we assumed that along P1 and P2 passes one and only one optical ray.
sd
rds ray
=ˆ rd
∫≤∫ ⋅22112211
ˆPQQPPQQP
sdnrdsn
From those two integrals we obtain:
This is the Fermat’s Principle proved using the Geometrical Optics.
( ) ( )1222112211
PSPSsdnsdnPQQPPQQP
−=∫≥∫
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65
SOLO Foundation of Geometrical Optics
Fermat’s Principle (1657)
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.
66
SOLO Foundation of Geometrical Optics
Fermat’s Principle (continue - 1)
If the regularity condition (optical ray not intersecting( doesn’t hold, the optical ray may not be a minimum, as we can see from the Figure, where the optical ray reflected from the planar mirror and reaches the point P2 (P1MP2) is longer than the direct ray from P1 to P2.
67
SOLO Foundation of Geometrical Optics
Fermat’s Principle (continue - 2)
On other example is given in Figure bellow on the rays from a point source refracted by a lens. The refracted rays form an envelope called caustic. The point P’2 where the refracted ray touches the caustic is called a conjugate point. From the Figure we can see that this point is reached by, at least, two rays with different optical paths.
68
SOLO Foundation of Geometrical Optics
Fermat’s Principle (continue - 3) Example of the stationarity of the Fermat’s Principle
Suppose that we have a elliptical mirror and a point source locate at one of it’s foci P1.
The elliptical mirror has the following properties:
1. The sum of the distances from the two foci to any point R on the ellipse is constant.
2121 PRRPPRRP EE +=+ 2. The normal at any point R on the ellipse bisects the angle P1RP2.
2P1P
Point Source
Elliptic Mirror
RER
RnERs
Rs
According to Snell’s Law, all the rays originated at the focus P1 will be reflected by the elliptical mirror and intersect at the second foci P2.
Since the rays travel in the same media and the geometrical paths are equal, the optical paths will be equal also.
( ) ( )2121 PRRPnPRRPn EE +=+ Since all the optical paths reflected by the mirror reach the point P2, we call P2 the conjugate point to P1.
69
SOLO Foundation of Geometrical Optics
Fermat’s Principle (continue - 4) Example of the stationarity of the Fermat’s Principle (continue – 1)
Now replace the elliptical mirror with a planar one normal to at the point R.Rn
2P1P
Point Source
Planar Mirror
Elliptic Mirror
RER
PR
RnERs
PRs
Rs
For this reason the ray will be reflected at R and reach the point P2, in the same way as for the elliptical mirror.
RP1
From the Figure we can see that:
( ) ( ) 222121 PRRRPRPRRRPnPRRPn PPEEPPE +<←+<+
In this case the Fermat’s Principle will give a minimum for the optical path
( )21 PRRPn +
70
SOLO Foundation of Geometrical Optics
Fermat’s Principle (continue - 6) Example of the stationarity of the Fermat’s Principle (continue – 3)
Now replace the elliptical mirror with a circular one normal to at the point R(The mirror diameter is smaller than the maximum axis of the ellipse).
Rn
For this reason the ray will be reflected at R and reach the point P2, in the same way as for the elliptical mirror.
RP1
From the Figure we can see that:
In this case the Fermat’s Principle will give a maximum for the optical path
( )21 PRRPn +
2P1P
Point Source
Elliptic Mirror
CircularMirror
R
CR
ER
RnCRs
ERsPRs
Rs
( ) ( ) 222121 PRRRPRPRRPnPRRPn EECCCC +<←+>+
71
SOLO Foundation of Geometrical OpticsFermat’s Principle (continue - 7)
Karl Friederich Gauss1777-1855
The optical path connecting points M, T, M’ is'' lnlnpathOptical ⋅+⋅=
Applying cosine theorem in triangles MTC and M’TC we obtain:
( ) ( )[ ] 2/122 cos2 βRsRRsRl +−++=
( ) ( )[ ] 2/122 cos'2'' βRsRRsRl −+−+=
( ) ( )[ ] ( ) ( )[ ] 2/1222/122 cos'2''cos2 ββ RsRRsRnRsRRsRnpathOptical −+−+⋅++−++⋅=Therefore
According to Fermat’s Principle when the point Tmoves on the spherical surface we must have ( )
0=βd
pathOpticald
( ) ( ) ( )0
'
sin''sin =−⋅−+⋅=l
RsRn
l
RsRn
d
pathOpticald βββ
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
Derivation of Gaussian Formula for a Single Spherical Surface Lens Using Fermat’s Principle
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72
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations
We have:
constS =constdSS =+
s
∫2
1
P
P
dsn
1P
2P
( ) ( ) ( )∫∫∫∫ =
+
+===
2
1
2
1
2
1
,,,,1
1,,1
,,1
0
22
00
P
P
P
P
P
P
xdzyzyxFc
xdxd
zd
xd
ydzyxn
cdszyxn
ctdJ
Let find the stationarity conditions of the Optical Path using the Calculus of Variations
( ) ( ) ( ) xdxd
zd
xd
ydzdydxdds
22
222 1
+
+=++=
Define:
xd
zdz
xd
ydy == &:
( ) ( ) ( ) 22
22
1,,1,,,,,, zyzyxnxd
zd
xd
ydzyxnzyzyxF ++=
+
+=
73
Leonhard Euler (1707-1783) generalized the brothers Bernoulli methods in“Me tho dus inve nie nd i line a s c urva s m a x im i m inim ive p ro p rie ta te g a ude nte s s ive s o lutio p ro ble m a tis is o p e rim e tric i la tis s im o s e ns u a c c e p ti” (“Me tho d fo r find ing p la ne c urve s tha t s ho w s o m e p ro p e rty o f m a x im a a nd m inim a ”) published in 1744. Euler solved the G e o d e s ic Pro ble m ,i.e. the curves of minimum length constrained to lie on a given surface.
Joseph-Louis Lagrange (1736-1813) gave the first analytic methods of Calculus of Variations in "Essay on a new method of determining the maxima and minima of indefinite integral formulas" published in 1760. Euler-Lagrange Equation:
SOLOCALCULUS OF VARIATIONS
74
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 1)
Necessary Conditions for Stationarity (Euler-Lagrange Equations)
( ) ( ) ( ) 22
22
1,,1,,,,,, zyzyxnxd
zd
xd
ydzyxnzyzyxF ++=
+
+=
0=∂∂−
∂∂
y
F
y
F
dx
d
( )[ ] 2/1221
,,
zy
yzyxn
y
F
++=
∂∂ [ ] ( )
y
zyxnzy
y
F
∂∂++=
∂∂ ,,
1 2/122
( )[ ] [ ] 011
,, 2/122
2/122=
∂∂++−
++ y
nzy
zy
yzyxn
xd
d
0=∂∂−
∂∂
z
F
z
F
dx
d
[ ] [ ] 011
2/1222/122=
∂∂
−
++++ y
n
zy
yn
xdzy
d
75
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 2)
Necessary Conditions for Stationarity (continue - 1)
We have
[ ] 01
2/122=
∂∂−
++ y
n
zy
yn
sd
d
y
n
sd
ydn
sd
d
∂∂=
In the same way
[ ] 01
2/122=
∂∂−
++ z
n
zy
zn
sd
d
z
n
sd
zdn
sd
d
∂∂=
76
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 3)
Necessary Conditions for Stationarity (continue - 2)
Using ( ) ( ) ( ) xdxd
zd
xd
ydzdydxdds
22
222 1
+
+=++=
we obtain 1222
=
+
+
sd
zd
sd
yd
sd
xd
Differentiate this equation with respect to s and multiply by n
sd
d
0=
+
+
sd
zd
sd
dn
sd
zd
sd
yd
sd
dn
sd
yd
sd
xd
sd
dn
sd
xd
sd
nd
sd
zd
sd
nd
sd
yd
sd
nd
sd
xd
sd
nd =
+
+
222
sd
nd
and
sd
nd
sd
zdn
sd
d
sd
zd
sd
ydn
sd
d
sd
yd
sd
xdn
sd
d
sd
xd =
+
+
add those two equations
77
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 4)
Necessary Conditions for Stationarity (continue - 3)
sd
nd
sd
zdn
sd
d
sd
zd
sd
ydn
sd
d
sd
yd
sd
xdn
sd
d
sd
xd =
+
+
Multiply this by and use the fact that to obtainxd
sd
cd
ad
cd
bd
bd
ad =
xd
nd
sd
zdn
sd
d
xd
zd
sd
ydn
sd
d
xd
yd
sd
xdn
sd
d =
+
+
Substitute and in this equation to obtainy
n
sd
ydn
sd
d
∂∂=
z
n
sd
zdn
sd
d
∂∂=
xd
zd
z
n
xd
yd
y
n
xd
nd
sd
xdn
sd
d
∂∂−
∂∂−=
Since n is a function of x, y, zx
n
xd
zd
z
n
xd
yd
y
n
xd
ndzd
z
nyd
y
nxd
x
nnd
∂∂=
∂∂−
∂∂−→
∂∂+
∂∂+
∂∂=
and the previous equation becomes
x
n
sd
xdn
sd
d
∂∂=
78
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 5)
Necessary Conditions for Stationarity (continue - 4)
We obtained the Euler-Lagrange Equations:
x
n
sd
xdn
sd
d
∂∂=
y
n
sd
ydn
sd
d
∂∂=
z
n
sd
zdn
sd
d
∂∂=
ksd
zdj
sd
ydi
sd
xd
sd
rd
kzjyixr
ˆˆˆ
ˆˆˆ
++=
++=
Define the unit vectors in the x, y, z directionskji ˆ,ˆ,ˆ
The Euler-Lagrange Equations can be written as:
nsd
rdn
sd
d ∇=
We recovered the Eikonal Equation.Return to Table of Content
79
CALCULUS OF VARIATIONS
We obtain also:
Transversality Conditions
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) fitxdtxtxtFdttxtxtxtFtxtxtF iiiiT
xiiiii
T
xiii ,00,,,,,, ==
+
−
••••
••
SOLO
( )( )000 , txtA
( )( )fff txtB ,
( )tx*x
t
( )ε,1 tx
( )ε,2 tx
80
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 6)
Transversality Conditions of Calculus of Variations
Assume that the initial and final boundary are defined by the surfaces and , respectively.
( )000 ,, zyxA( )fff zyxB ,,
The transversality conditions at the boundaries i = 0,f are defined by:( ) ( ) ( )[ ]
( ) ( ) 0,,,,,,,,
,,,,,,,,,,,,
=++
−−
iziy
izy
dzzyzyxFdyzyzyxF
dxzyzyxFzzyzyxFyzyzyxF
( ) [ ] [ ] [ ]
[ ] ( )sd
xdzyxn
zy
n
zy
znz
zy
ynyzynFzFyzyzyxF zy
,,1
111,,,,
2/122
2/1222/122
2/122
=++
=
++−
++−++=−−
( )[ ] ( )
( )[ ] ( )
sd
zdzyxn
zy
zzyxn
z
FF
sd
ydzyxn
zy
yzyxn
y
FF
z
y
,,1
,,
,,1
,,
2/122
2/122
=++
=∂∂=
=++
=∂∂=
Using those equations we obtain: firdsd
rdn i
iray,00 ==⋅
We can see that the rays are normal (transversal( to the boundary surfaces.
where are tangent to the boundary surfaces andfird i ,0= ( )000 ,, zyxA ( )fff zyxB ,,
Return to Table of Content
81
CALCULUS OF VARIATIONS
The necessary conditions for extremes at the corners (discontinuities in ( are:( )tx
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) 0,,,,
,,,,
,,,,
00
000
000
=
−
+
+
+
−
−
−
+
•
−
•
+
•
+
•
+
•
−
•
−
•
−
•
••
•
•
ccccT
xccc
T
x
cccccT
xccc
ccccT
xccc
txdtxtxtFtxtxtF
dttxtxtxtFtxtxtF
txtxtxtFtxtxtF
Weierstrass-Erdmann Corner ConditionsDeveloped independently by Weierstrass and Erdmann in 1877.
cornerpoints( )( )000 , txtA
( )( )fff txtB ,
( )txx
t
SOLO
Karl Theodor Wilhelm Weierstrass1815-1897
82
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 7)
Weierstrass-Erdmann Corner Conditions of Calculus of Variations Let examine the following two cases:
1. The optical path passes between two regions with different refractive indexes n1 to n2.
In region (1): ( ) ( ) 2211 1,,,,,, zyzyxnzyzyxF ++=
In region (2): ( ) ( ) 2222 1,,,,,, zyzyxnzyzyxF ++=
The Weierstrass-Erdmann Corner Conditions( ) ( ) ( )[ ]{
( ) ( ) ( )[ ]}( ) ( )[ ]( ) ( )[ ] 0,,,,,,,,
,,,,,,,,
,,,,,,,,,,,,
,,,,,,,,,,,,
222111
222111
22222222222
11111111111
=−+
−+
−−−
−−
dzzyzyxFzyzyxF
dyzyzyxFzyzyxF
dxzyzyxFzzyzyxFyzyzyxF
zyzyxFzzyzyxFyzyzyxF
zz
yy
zy
zy
where dx, dy, dz are on the boundary between the two regions. Substituting and we obtain( )zyzyxF ,,,,2( )zyzyxF ,,,,1
( ) ( )0
2121 =⋅
− rd
sd
rdn
sd
rdn rayray
where is on the boundary between the two regions andrd
83
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 8)
Weierstrass-Erdmann Corner Conditions of Calculus of Variations (continue – 1)
1. 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
==
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
84
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 9)
Weierstrass-Erdmann Corner Conditions of Calculus of Variations (continue – 2)
2. 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 =−×− ssn
Return to Table of Content
85
David Hilbert (1862 – 1943) For a field of extremals of the integral
is invariant of .
is the field slope and is the path C slope at the point of C.
SOLO CALCULUS OF VARIATIONS
86
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 10)
Hilbert’s Integral of Calculus of Variations
In a region with no conjugate points (no intersecting optical rays) the following integral is invariant to the path of integration
( ) ( )( ) ( ) ( )[ ] ( ) ( )( ){( )
( )
( ) ( )[ ] ( ) ( )( )} xdzyxzzyxyzyxFzyxZzyxz
zyxzzyxyzyxFzyxYzyxyzyxzzyxyzyxF
z
zyxP
zyxPyC
ffff
,,,,,,,,,,,,
,,,,,,,,,,,,,,,,,,,,,,
,, 0000
−−
∫ −−
where is on an extremal trajectory and on the integration path C.
( ) ( )zyxzzyxy ,,,,, ( ) ( )zyxZzyxY ,,,,,
This is called the Hilbert’s Invariant Integral because it is invariant to the path of integration C as long this curve remains in the field of unique extremal solutions.
87
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 11)
Hilbert’s Integral of Calculus of Variations (continue – 1)
( ) ( ) ( ) ( )zyxx
zzyxzzyx
x
yzyxy ,,,,,,,,,
∂∂=
∂∂=
is the field slope and
( ) ( )CC
x
zzyxZ
x
yzyxY
∂∂=
∂∂= :,,,:,,
is the path C slope at the point (x,y,z) of C.
( ) ( ) dxx
zdxzyxZzddx
x
ydxzyxYyd
CC
CC ∂
∂==∂∂== ,,,,,
( ) ( )( ) ( ) ( )[ ] ( ) ( )( ){( )
( )
( ) ( )[ ] ( ) ( )( )} xdzyxzzyxyzyxFzyxZzyxz
zyxzzyxyzyxFzyxYzyxyzyxzzyxyzyxF
z
zyxP
zyxPyC
ffff
,,,,,,,,,,,,
,,,,,,,,,,,,,,,,,,,,,,
,, 0000
−−
∫ −−
Hilbert’s Invariant Integral
becomes
( ) ( ) ( )[ ]{( )
( )
( ) ( ) }zdzyzyxFydzyzyxF
xdzyzyxFzzyzyxFyzyzyxF
zy
zyxP
zyxPzyC
ffff
,,,,,,,,
,,,,,,,,,,,,,,
,, 0000
−−
∫ −−
88
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 12)
Hilbert’s Integral of Calculus of Variations (continue – 2)
( ) ( ) ( )[ ]{( )
( )
( ) ( ) }zdzyzyxFydzyzyxF
xdzyzyxFzzyzyxFyzyzyxF
zy
zyxP
zyxPzyC
ffff
,,,,,,,,
,,,,,,,,,,,,,,
,, 0000
−−
∫ −−
We found that
( ) [ ] [ ] [ ]
[ ] ( )sd
xdzyxn
zy
n
zy
znz
zy
ynyzynFzFyzyzyxF zy
,,1
111,,,,
2/122
2/1222/122
2/122
=++
=
++−
++−++=−−
( )[ ] ( )
sd
ydzyxn
zy
yzyxn
y
FFy ,,
1
,,2/122
=++
=∂∂=
( )[ ] ( )
sd
zdzyxn
zy
zzyxn
z
FFz ,,
1
,,2/122
=++
=∂∂=
Therefore, we can write the Hilbert’s Invariant Integral as
( )
( )
( )
( )∫ ⋅=∫ ⋅
ffffffff zyxP
zyxP
zyxP
zyxP
ray rdsnrdsd
rdn
,,
,,
,,
,, 1000010000
ˆ
The Hilbert’s Invariant Integral of Calculus of Variation gives the same result as the Lagrange’s Invariant Integral of Geometrical Optics.
Return to Table of Content
89
CALCULUS OF VARIATIONS
Second Order Conditions
Legendre’s Necessary Conditions (1786) for a Weak Minimum (Maximum)
A Path and its “Weak” and “Strong” Neighbors
Using the Second Variation:
where:
TTxxxx PFFP ===: T
xxxxFFQ •• ==: TT
xxxx RFFR === :
Legendre’s Necessary Condition for a Weak (Neighbor) Minimum (Maximum)
The Matrix must be Positive (Negative) Definite along a Weak Minimum (Maximum) Optimal Trajectory.
xxFR =
SOLO
( )( )
( )02
0
2
2
00
00
2 ≥∫ ++===
==
ε
εδδδδδδδ
fii
ii
t
t
TTTxdxd
tddtdtxRxxQxxPxJ
90
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 13)
Second Order Conditions: Legendre’s Condition for a Weak Minimum
Adrien-Marie Legendre1752-1833
From( )
[ ] ( )sd
ydzyxn
zy
yzyxn
y
FFy ,,
1
,,2/122
=++
=∂∂=
we obtain
( )[ ] [ ] [ ]
( )[ ] 2/322
2
2/322
2
2/1222/1222
2
1
1
111
,,
zy
zn
zy
yn
zy
n
zy
yzyxn
yy
F
++
+=
++−
++=
++∂∂
=∂∂
From
we obtain
( )[ ] [ ] 2/3222/122
2
11
,,
zy
zyn
zy
zzyxn
yzy
F
++−=
++∂∂=
∂∂∂
( )[ ] ( )
sd
zdzyxn
zy
zzyxn
z
FFz ,,
1
,,2/122
=++
=∂∂=
( )[ ]
( )[ ] 2/322
2
2/1222
2
1
1
1
,,
zy
yn
zy
zzyxn
zz
F
++
+=
++∂∂=
∂∂
From those equations we obtain
( )[ ]
( )( )
+−
−+
++=
∂∂
∂∂∂
∂∂∂
∂∂
=2
2
2/322
2
22
2
2
2
''1''
1
1
,,
yyx
zyz
zy
zyxn
z
F
yz
F
zy
F
y
F
F XX
91
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 14)
Second Order Conditions: Legendre’s Condition for a Weak Minimum (continue – 1)
( )[ ]
( )( )
+−−+
++=
∂∂
∂∂∂
∂∂∂
∂∂
=2
2
2/322
2
22
2
2
2
''1''
1
1
,,
yyx
zyz
zy
zyxn
z
F
yz
F
zy
F
y
F
F XX
Let use Sylvester’s Theorem to check if/when is positive definite (i.e. check that the determinants of all principal minors are positive)
''XXF
James Joseph Sylvester
1814-1897[ ]
( )( )
+−−+
++=
2
2
2/322''1''
1det
1det
yyx
zyz
zy
nF XX
[ ] ( )( )[ ] [ ] 01
111
2/122
2222
2/322>
++=−++
++=
zy
nzyyz
zy
n
( ) 01 2 >+ z
According to Sylvester’s Theorem is positive definite''XXF ( )0'' >XXF
According to Legendre’s Condition ,if the Jacobi’s Condition is satisfied (no conjugate points between P1 and P2), every extremal is a weak minimum.
( )0'' >XXF
Return to Table of Content
92
CALCULUS OF VARIATIONS
Second Order Conditions
Weierstrass Necessary Condition for a Strong Minimum (Maximum) - 1879
Weierstrass defined the function:
( ) ( ) ( ) ( ) ( )xXxxtFxxtFXxtFXxxtE x
−−−= ,,,,,,:,,,
Weierstrass Necessary Condition for a Strong Minimum (Maximum) is:
( ) ( )00,,, ≤≥XxxtE for every admissible set ( )Xxt ,,
Since the Weierstrass condition directly concerns minimality, rather thanstationarity as did Euler-Lagrange condition, it entails no further supportingstatements analogous to the Legendre and Jacobi conditions that support theEuler-Lagrange stationarity condition.
A weak variation is included in the strong variation, therefore a conditionthat is necessary for a weak local minimum (maximum) is also necessary
for a strong local minimum (maximum).
SOLO
93
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 15)
Second Order Conditions: Weierstrass’ Condition for a Strong Local Minimum
Karl Theodor Wilhelm Weierstrass1815-1897
According to Weierstrass’ Condition ,if the Jacobi’s Condition is satisfied (no conjugate points between P1 and P2), every extremal is a strong local minimum.
( )( )0',',',',',',,, ≥ZYXzyxzyxE
Weierstrass’ E function is defined as
( )( ) ( ) ( ) ( ) ( ) ( )zyzyxFzZzyzyxFyYzyzyxFZYzyzyxF
ZYzyzyxE
zy
,,,,,,,,,,,,,,,,,,:
,,,,,,
−−−−−=
[ ] [ ] ( ) [ ] ( ) [ ] 2/1222/122
2/1222/122
1''
111
zy
znzZ
zy
ynyYzynZYn
++−−
++−−++−++=
[ ] [ ] ( ) ( )[ ]zzZyyYzyzy
nZYn
−+−+++
++−++= 22
2/122
2/122 11
1
[ ] [ ] [ ] ( )InequalitySchwarzzyZY
zZyYZYn 0
11
111 2/1222/122
2/122 ≥
++++++−++=
[ ] [ ] ( )zZyYzy
nZYn
++
++−++= 1
11 2/122
2/122
Return to Table of Content
94
William Rowan Hamilton (1805-1855) Canonical Equations of Motion 1835
where H is the Hamiltonian defined as:
Hamilton-Jacobi Theory
SOLO CALCULUS OF VARIATIONS
95
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 16)
Hamilton’s Canonical Equations
Define ( )[ ] ( )
( )[ ] ( )
sd
zdzyxn
zy
zzyxn
z
Fp
sd
ydzyxn
zy
yzyxn
y
Fp
z
y
,,1
,,:
,,1
,,:
2/122
2/122
=++
=∂∂=
=++
=∂∂=
( ) ( ) ( )2222222 1 zynzypp zy +=+++Adding the square of twose two equations gives
( ) ( )2
222
2221
=
+−=++
xd
sd
ppn
nzy
zy
from which
Substituting in ( ) ( ) ( ) 22
22
1,,1,,,,,, zyzyxnxd
zd
xd
ydzyxnzyzyxF ++=
+
+=
gives( ) ( )222
2
,,,,zy
zy
ppn
nppzyxF
+−=
96
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 17)
Hamilton’s Canonical Equations (continue – 1)
From ( )[ ] ( )
( )[ ] ( )
sd
zdzyxn
zy
zzyxn
z
Fp
sd
ydzyxn
zy
yzyxn
y
Fp
z
y
,,1
,,:
,,1
,,:
2/122
2/122
=++
=∂∂=
=++
=∂∂=
solve for
( ) ( )222
2
,,,,zy
zy
ppn
nppzyxF
+−=and
( )
( )222
222
zy
z
zy
y
ppn
pz
ppn
py
+−=
+−=
Define the Hamiltonian( ) ( )
( ) ( ) ( )( ) ( ) ( )
sd
xdzyxnppzyxn
ppn
p
ppn
p
ppn
n
zpypppzyxFppzyxH
zy
zy
z
zy
y
zy
zyzyzy
,,,,
,,,,:,,,,
222
222
2
222
2
222
2
−=+−−=
+−+
+−+
+−−=
++−=
97
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 18)
Hamilton’s Canonical Equations (continue – 2)
From
We obtain the Hamilton’s Canonical Equations
( ) ( ) ( ) ( )sd
xdzyxnppzyxnppzyxH zyzy ,,,,,,,, 222 −=+−−=
( )
( )222
222
zy
z
z
zy
y
y
ppn
p
p
H
xd
zdz
ppn
p
p
H
xd
ydy
+−=
∂∂==
+−=
∂∂==
( )
( )222
222
zy
z
zy
y
ppn
z
nn
z
H
xd
pd
ppn
y
nn
y
H
xd
pd
+−
∂∂
−=∂∂−=
+−
∂∂
−=∂∂−=
98
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 19)
Hamilton’s Canonical Equations (continue – 3)
From( ) ( ) ( ) ( )sd
xdzyxnppzyxnppzyxH zyzy ,,,,,,,, 222 −=+−−= ( )222
zy ppn
n
sd
xd
+−=
By similarity with ( )sd
ydzyxnpy ,,=
define ( ) ( ) ( ) ( )222 ,,,,,,,,: zyzyx ppzyxnppzyxHsd
xdzyxnp +−=−==
Let differentiate px with respect to x ( ) x
H
xd
Hd
ppn
x
nn
xd
pd
zy
x
∂∂−=−=
+−
∂∂
=222
Let compute
( )( )
x
n
n
ppn
ppn
x
nn
sd
xd
xd
pd
sd
pd zy
zy
xx
∂∂=
+−
+−
∂∂
==222
222
99
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 20)
Hamilton’s Canonical Equations (continue – 4)
and
( )( )
x
n
n
ppn
ppn
xn
n
sd
xd
xd
pd
sd
pd zy
zy
xx
∂∂=
+−
+−
∂∂
==222
222
( )( )
y
n
n
ppn
ppn
yn
n
sd
xd
xd
pd
sd
pd zy
zy
yy
∂∂=
+−
+−
∂∂
==222
222
( )( )
z
n
n
ppn
ppn
zn
n
sd
xd
xd
pd
sd
pd zy
zy
zz
∂∂=
+−
+−
∂∂
==222
222
nsd
pd ∇=
xpnsd
xd 1=
( )( )
y
zy
zy
y pnn
ppn
ppn
p
sd
xd
xd
yd
sd
yd 1222
222=
+−
+−==
( )( )
z
zy
zy
z pnn
ppn
ppn
p
sd
xd
xd
zd
sd
zd 1222
222=
+−
+−==
pnsd
rd ray
1=
We recover the result from Geometrical Optics Return to Table of Content
100
William Rowan Hamilton (1805-1855) Canonical Equations of Motion 1835
where H is the Hamiltonian defined as:
Hamilton-Jacobi Theory
SOLO CALCULUS OF VARIATIONS
William Rowan Hamilton1805-1855
Carl Gustav Jacob Jacobi
1804-1851
Hamilton-Jacobi Equation
101
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 21)
Hamilton-Jacobi Equation (continue – 1)
( ) ( ) ( ) [ ]( )
[ ]( )
[ ]( )
[ ] ( )sd
xdzyxn
zy
zyxn
zy
zzyxnz
zy
yzyxny
zyzyxnFzFyzyzyxFzyxS zyx
,,1
,,
1
,,
1
,,
1,,,,,,,,
2/1222/1222/122
2/122
=++
=++
−++
−
++=−−=
( ) ( )[ ] ( )
sd
ydzyxnp
zy
yzyxn
y
FzyxS yy ,,:
1
,,,,
2/122==
++=
∂∂=
( ) ( )[ ] ( )
sd
zdzyxnp
zy
zzyxn
z
FzyxS zz ,,:
1
,,,,
2/122==
++=
∂∂=
We obtain
snsd
rdnS ray ˆ==∇
From this
22 ˆˆ nssnSS =⋅=∇⋅∇
We recovered again the Eikonal Equation
102
SOLO Foundation of Geometrical Optics
Proof of Fermat’s Principle Using Calculus of Variations (continue – 22)
Hamilton-Jacobi Equation (continue – 2)
( ) ( ) sdzyxnsdsd
zd
sd
yd
sd
xdzyxn
sdsd
zdSsd
sd
ydSsd
sd
xdSzdSydSxdSSd zyxzyx
,,,,
1
222
=
+
+
=
++=++=
( ) ( ) ( )∫=− sdzyxnzyxSzyxS ,,,,,, 000Along an extremal trajectory we obtain
( ) ( )sd
ydzyxnzyxS y ,,,, =
( ) ( )sd
zdzyxnzyxS z ,,,, =
( ) ( )2
222
2221
=
+−=++
xd
sd
ppn
nzy
zy
( ) ( ) ( )[ ] ( )
sd
xdzyxn
zy
zyxnFzFyzyzyxFzyxS zyx ,,
1
,,,,,,,, 2/122
=++
=−−=
( ) ( )[ ] ( )[ ]
( )
zy ppzyxH
zyx ppnzy
zyxnzyxS
,,,,
2/12222/1221
,,,,
−
+−=++
=
( ) ( ) 0,,,,,, ≡+ zyx ppzyxHzyxS
Hamilton-Jacobi Equation is an identity.
Return to Table of Content
103
SOLO
References
Foundation of Geometrical Optics
[1] M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation, Interference and Diffraction of Light”, 6th Ed., Pergamon Press, 1980, Ch. 3 and App. 1
[2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996, pp. 387-395, 692-694
104
SOLO
References (continue)
Foundation of Geometrical Optics
[3] W.C. Elmore, M.A. Heald, “Physics of Waves”, Dover Publication, 1969, pp. 315-322
[4] H. Goldstein, “Classical Mechanics”, 2nd Ed., Addison-Wesley, 19??, pp. 484-492
105
SOLO
References (continue)
Foundation of Geometrical Optics
[5] I.M. Gelfand, S.V. Fomin, “Calculus of Variations”, Prentice-Hall, 1963, pp. 34, 36-37, 66, 89, 210, 213
[6] H. Sagan, “Introduction to Calculus of Variations”, Dover Publications, 1969, pp. 67, 73, 106, 230, 277
Return to Table of Content
January 4, 2015 106
SOLO Foundation of Geometrical Optics
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
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