Foundation of geometrical optics

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


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


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


( )[ ] ( ) ( )[ ]( ) 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


from which

and



( )[ ] ( ) ( )[ ]( ) 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




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


from which

and



( )[ ] ( ) ( ) ( )[ ]( ) 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




( ) 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




( ) 00

00


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




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.


13



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



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




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




=+×∇

=−×∇

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




To summarize, we obtained

=−∇⋅∇

=⋅∇=⋅∇

0

0

0

2

0

0

nSS

ES

HS

The last equation is the Eikonal Equation


18



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



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




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




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




( ) ( ){ 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




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




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


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


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


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



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



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



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



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



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



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



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



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



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



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




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




( ) ( ) ( )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


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


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:


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:


( ) ( ) ( ) 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



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==


We found that the ray propagation vector is


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==


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


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


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


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



2222

2

2 :r

lrnrgzd

rd ρββ −−==


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



2222

2

2 :r

lrnrgzd

rd ρββ −−==


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.


Core axisCladding


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)


2222

2

2 :r

lrnrgzd

rd ρββ −−==


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


( )

≥=<=

=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)


2222

2

2 :r

lrnrgzd

rd ρββ −−==


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



0222 >−−= lng cladding β

55

SOLO Optical Fiber – Ray TheoryStep-index Fiber (continue – 9(


2222

2

2 :r

lrnrgzd

rd ρββ −−==


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


( ) 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


22 β

ρ

−=

cladding

rad

n

lr

0222 >−− lncore β

0222 <−− lncladding β

56

For a step-index core fiber ncore = constant.


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



0222 <−−= lng cladding β

rρ

0=l

claddingcore( )rg

meridional ray

022 <−= βcladdingng

022 >−= βcoreng


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



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


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



57


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

a, 08/17/2005

In fact the Lagrange Invariant is only a special one dimensional case of a general integral invariants discussed by J.H. Poincare in his "Les Methodes Nouvelles de la Mecanique Celeste" (see comment in Max & Born, "Principles of Optics", 6th Ed., p.127

58


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


59


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


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


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 =


62


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


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


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

−=∫≥∫


65


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


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


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


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


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


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


72


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


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



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




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




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




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



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 ,,


81


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



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



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



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


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



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



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



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.


89


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



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



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


92


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



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


94

William Rowan Hamilton (1805-1855) Canonical Equations of Motion 1835

where H is the Hamiltonian defined as:

Hamilton-Jacobi Theory


95



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



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




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




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




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


William Rowan Hamilton1805-1855

Carl Gustav Jacob Jacobi

1804-1851

Hamilton-Jacobi Equation

101



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



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.


103

SOLO

References


[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)


[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)


[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


January 4, 2015 106


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

Foundation of geometrical optics

Science

Transcript of Foundation of geometrical optics