geometrical optics: some applications of the intensity...

16 August 2006SPIE Novel Optics 6289-161

Geometrical Optics:some applications of the intensity law

David L. ShealyDepartment of PhysicsUniversity of Alabama at BirminghamE-mail: [email protected]

16 August 2006SPIE Novel Optics 6289-1616 August 2006

ObjectivesConcepts of rays, wavefronts, energy propagation

Review general integral of eikonal equationRelate principal curvatures of wavefront to Laplacian of eikonalReview intensity law of geometrical optics

Ray optics field - solution of scalar wave equationRelate principal curvatures of wavefront to ray optics field amplitudeRelate intensity of field to propagation distance & principal curvaturesReview generalized ray trace equations

Imaging applications of intensity lawCaustic surfaces and optical design via caustics

Nonimaging applications of intensity lawIllumination engineeringBeam shaping


Concepts of Rays

Vector ray equation can be used to compute trajectory of ray paths r(s):

Propagation vector:Define 3 unit vectors:

Frénet Equations:

( ) ( ) ( )d sd n nds ds

⎡ ⎤= ∇⎢ ⎥

⎣ ⎦

rr r p

b

t

( )0

d sds nk

= =r kt

dds ρ

=t p

= ×b t p

dds

τρ

= − +p t b d

dsτ= −

b p


Using Frénet Equations

Curvature of ray path:

Torsion of ray path can be evaluated in similar way using Frénet Equations

( ) ( ) ( ) ( )d ndn n nds ds ρ

∇ = = +⎡ ⎤⎣ ⎦r pr r t t r

( )1 ln nρ= ∇p ri

pb

t


Concepts of Wavefronts

Eikonal φ(r0,r) = optical distance along ray path (r0,r)Wavefront satisfies φ(r0,r) = const.Propagation vector satisfies

k = k0∇ φ(r0,r)Rays ⊥ to wavefrontPhase of wavefront is k0 φ(r0,r) Eikonal Equation is

[∇φ(r)]2 = n2r0

r

n(r)

t

p

Wavefronts

RayPath


General Integral of eikonalequation

Eikonal equation in Cartesian coordinate

Stavroudis obtained a general integral of the eikonal equation

( ) ( ) ( ) ( )2 2 2

2nx y z

φ φ φ∂ ∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞+ + =⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠

r r rr

( )( )( )

,

u

v

ux vy wz h u v

uz u h w

vz y h w

φ = + + +

= +

= +( )

( ) ( ) ( )

( )

2

2 2 2,

, ,,

,

x y z

x y

u v

pn

u v u v n u v

h u v h u vH u v

u v

p ns h u v uh vh

= −

= + + − −

∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

= − + +

W S H

S e e e

e e

or in Cartesian form

where h(u,v) is arbitrary functionto be determined by boundaryconditions.


Properties of general integral of eikonal equation

Stavroudis has evaluated differential geometry properties of wavefront:

Surface normal:

Principal curvatures in terms of h(u,v), hu, hv

Caustic surfacesFor more details, see ON Stavroudis and RC Fronczek, “Caustic

surfaces and the structure of the geometrical image,” JOSA 66.8, 795 (1976).

( ),u v

u v

u vn

×= = =

×SW WN t

W W


Principal Curvatures of Wavefront

Qualitative approach for evaluating relationship between element of area on wavefront and principal radii of curvatures of wavefront. Assume origin of s-coordinate axis is located at dS1. See Born & Wolf, pp. 116-117. ( )

( ) ( )1 1 2

2 1 2

( )dS

dS s s

ρ δθ ρ δφ

ρ δθ ρ δφ

=

= + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦

Q1Q2

δφδθ

ρ2

ρ1

s

s

dS2

dS1

Principal Sections


Principal Wavefront Curvatures

Taylor expansion of unit tangent vector t

Divergence of tangent vector t = ∇φ/n

( ) ( ) ( )

( )

1 2

0 0

1 21 2

2 1

0 0

0 01 2

01 2

( ) s s

s ss ss s

h hs s

h h

δ

τ τρ ρ

∂ ∂+ = + +

∂ ∂

= + − + −

r r

r r

t r t rt r r t r

e et r e e

t(r0)

t(r0 +δr)

δr

ro

Wavefront

e2

e1

( ) ( ) ( )( )

( ) ( )

2

1 2 1 2

1 1 1 10 0s s n s s

φρ ρ ρ ρ⎡ ⎤ ⎡ ⎤∇

∇ = + = = +⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦

rt ri


Intensity Law

Intensity:

Conservation of Energy for E&M fields requires

Applying Gauss’ Theorem

( ) ( ) ( ) 2S ,I t n A= ∝r r r

( ) ( ) 0I∇ =⎡ ⎤⎣ ⎦r t ri

11 1 2 2 2

2 1

I or IdS /

I dS I dSdS

= =


Ray Optics Field

Assume the ray optics fieldsatisfies scalar wave equation

which requires

Luneburg and Kline asymptotic series in (-ik0)-m

( ) ( ) ( )2 2 20 0u k n u∇ + =r r r

( ) ( )0expA ik φ−⎡ ⎤⎣ ⎦r r

( ) ( ) ( ) ( ) ( ) ( )( )

( )( )

22 2 2 20 0 2 0

A Ak n ik

A Aφ φ φ

⎡ ⎤∇ ∇⎡ ⎤∇ − + − ∇ + ∇ + =⎢ ⎥⎣ ⎦

⎣ ⎦

r rr r r r

r ri


Ray Optics Field Transport

Transport equation (2nd term):

Field amplitude satisfies

Thus

( ) ( ) ( ) ( ) ( ) ( ) ( )2 2 2 22 0A A A A nφ φ ⎡ ⎤∇ + ∇ ∇ = ∇ =⎣ ⎦r r r r r r r ti i

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

0

1/ 2 20 0

0 01and '2 '

s

s

n s dS sA s A s A s A s ds

n s dS s n sφ⎧ ⎫⎡ ⎤ ∇⎪ ⎪= = −⎨ ⎬⎢ ⎥

⎪ ⎪⎣ ⎦ ⎩ ⎭∫

( ) ( ) ( )( )

( )( )

1/ 2 1/ 2

1 2

1 2

0 00

0 0A s A

s sρ ρ

ρ ρ⎡ ⎤ ⎡ ⎤

= ⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦


Intensity of Ray Optics Field

Intensity along ray path – principal curvatures

Refraction

( ) ( )( )

( )( )

1 22 2

1 2 1 2 1 2 1 2

0 01

I II s

s s s sρ ρ

ρ ρ ρ ρ κ κ κ κ= =

+ + + + + +

( ) ( )/ '

cos ( ) cos ( )

r in n

i r

γγ

γ φ φ

= +Ω

=Ω = − +

A A N


Refraction of ray optics field*Principal and normal curvatures

Intensity – normal curvatures & torsion

General ray trace (GRT) equations

*Burkhard & Shealy, Appl. Opt. 21.18, 3299-3306 (1981)

( ) ( )( ) 2 2

01 ( )

II s

s sκ κ κ κ τ=

+ + + −⊥ ⊥

21 2 1 2 and κ κ κ κ κ κ κ κ τ+ = + = −⊥ ⊥

( ) ( )( ) ( )( ) ( )

2 2cos cos

cos cos

r i

r r i i

r r i i

κ γκ κ

κ γκ κ

τ γτ τ

= +Ω

= +Ω

= +Ω

⊥ ⊥ ⊥


Propagation of refracted field*

*Burkhard & Shealy, Appl. Opt. 21.18, 3299-3306 (1981)


Refracted Field & CausticsNormal to principal curvatures conversion:

Caustic surfaces

*Al-Ahdali & Shealy, Appl. Opt. 29.31, 4551-4559 (1990)

1 1

2 2

C out out

C out out

ρρ

= += +

X X tX X t

2 21

2 22

1 12 21 12 2

κ κ κ κ κ τ

κ κ κ κ κ τ

= ⎡ + ⎤ + ⎡ − ⎤ +⎣ ⎦ ⎣ ⎦

= ⎡ + ⎤ − ⎡ − ⎤ +⎣ ⎦ ⎣ ⎦

⊥ ⊥

⊥ ⊥


Caustics & Aberrations

*Shealy, Appl. Opt. 15.10, 2588-1596 (1976)


Optical Design via Caustics

1. Minimizing separation and spreading of caustics have been used to improve performance of some system

2. Eliminating astigmatism by using GRT equations for rotationally symmetric system (τ=0) and requiring:

which leads to second order differential eq. for z(r):

( ) ( ) ( ) ( )2 or r r r rκ κ κ κ= =⊥ 1

( ) ( )2

2 2

cos 0cos cos

ii ir r

γ κ κ κ κ⎡ ⎤ Ω⎡ ⎤− +Ω − =⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦

⊥ ⊥


Illumination System

Intensity law => differential equation for sag of one surface in system:

Range of solutionshave been studied for ρ(θ)*

*Shealy, “Classical (non-laser) methods” in Laser Beam Shaping – Theory and Techniques, F.M. Dickey and S. C. Holswade, eds, pp. 313-348, Marcel Dekker, New York, 2000.

r

z(r)

z

Mirror

Source Detector

ρ

θ

I(θ) E(R)

( ) ( )( )

( )2 2

2 2

' sin 2 ' cos' cos 2 'sin cos

R rZ R z r

ρ θ ρρ θ ρ

ρ θ ρρ θ ρ θ

+ −−=

− + −

16 August 2006SPIE Novel Optics 6289-1620

Innovations in laser beam shaping

Overview of 40-year cycle from first concepts of using intensity law to devices reaching market


Growth in US patents involving beam shaping

0

200

400

600

800

1000

1200

1400

1600

1976-80 1981-85 1986-90 1991-95 1996-02

US PatentsInvolvingBeam Shaping


Cornwell, “Non-projective transformations in optics,”Ph.D. Dissertation, University of Miami, Coral Gables, 1980

Proc. SPIE 294, 62-72, 1981

Differential Power

Conservation of Energy: Ein=Eout

Magnifications of ray coordinates

OPL conditionDetermine sag z(r) of first surfaceDetermine inverse magnificationDetermine sag Z(R) of second surface

( , ) ( , )in outI x y dxdy I X Y dXdY=

( )( )( )1 2

0

1( )x

xx

X x

a u dum x C C

x A um u

⎡ ⎤= +⎢ ⎥

⎢ ⎥⎣ ⎦∫


Beam Shaping Applications

In a holographic projection processing system featured on 10 January 1999 issue of Applied Optics , a two-lens beam shaping optic increased the quality of micro-optical arrays.

J.A. Hoffnagle & C. M. Jefferson, “Design and performance of a refractive optical system that converts a Gaussian to a flattop beam,” Appl. Opt. 39.30, 5488-5499, 2000.

Cover Graphics for Nov 2003 issue of Optical EngineeringIrradiance of Gaussian beam propagating through beam shaper developed by Hoffnagle & Jefferson, who contributed this graphics for the special section on laser beam shaping.


ConclusionsBeam shaping and illumination engineering are significant outcomes of intensity lawWavefront differential geometry provides full description of image surfaces and offers ways to improve imaging systemsLuneburg and Kline asymptotic series offers a way of extending geometrical optics into the domain of wavelength-dependent phenomena

geometrical optics: some applications of the intensity...

Documents