International Conference on Industrial andApplied Mathematics

Zurich, Switzerland, July 2007

Shekhar GuhaUnited States Air Force Research Laboratory

Leo GonzalezGeneral Dynamics Information Technology

Qin ShengBaylor University

Description of light focusing by a

spherical lens

Outline

1. Motivation and Background

2. Some specific examples of light propagation

through a spherical lens

Nanostructures

Moth-eye AR surface

Trapped Atom Beams

IR FPA

Motivation

Enhanced Transmission of Nanohole Arrays

CCD Micro-lenses

High Resolution Imaging

Commercial DVDs

Nanoscale structures are widespread

(Optical elements are inserted near focal plane)

Light Propagation Theories

Rays – Geometrical optics and application of Snell’s Law through surfaces

Waves – Solution of wave propagation equation with initial and boundary conditions known

Hybrid - Combination of ray and wave optics

Fields – FDTD Method solving for EM fields from linear Maxwell’s Equations

“Holy Grail”

We would like to know the exact complex electromagneticfield for every point in space for propagation of light throughlinear and nonlinear optical regions

i. e., solve the wave equation

, 22 EfEk

for all space.

Challenges

For feature sizes close to the wavelength of light, vector nature ofthe E-M fields becomes important

The presence of small features in the beam of light alters the field distributions, so the boundary conditions for propagation equationbecome unknown

For feature sizes much larger than the wavelength of light, diffraction calculations become computationally intensive.

For propagation over small distances, computation also becomeschallenging

A ‘general’ wave-propagation theory describing light propagationthrough commonly used optical elements is not easily available

Goal

To determine the field distribution of light refracted by a lens using full diffraction theory

• Ray picture inadequate• Nijboer-Zernike diffraction theory of aberration assumes small angles

Kirchhoff vector diffraction formalism is used to describe

light propagation, starting from a curved surface

Focusing by small-f-number lenses in the presence of aberrations

S. Guha, Optics Letters 25 (19), pp. 1409-1411 , 2000

Previous work: Ray Theory + Diffraction:

Diffraction Integral Method

Surface 1

Surface 2

022 Ek

Green’s Function Method

s v

dvUVVUdsnUVVU 22ˆ

From Green’s scalar theorem,

where U and V are continuous functions of position, and aresolutions of the wave equation,

• • •

(lots of math, and some assumptions)

we get the complete Rayleigh-Sommerfeld diffraction integral:

S

r’

r

Field values at r can be obtained from the known values on a general surface S

Stratton – Chu Integrals

For a flat surface, the third term goes to zero and the more familiar

terms of the Kirchhoff diffraction theory are obtained

Rayleigh-Sommerfeld Solution

x1 x2

y1 y2

z = 0 z = z

11

ip

1122 dydxip

11

ey,xE

i2

zpy,xE

where

22

12

2

12 zyyxx and n2

p

(Fresnel Approximation)

22 1

22 1

2 2 ( ) ( )x x y y z

e e e

z

ik ikzik

zx x y y

2

2

2

2 12

2 12

[( ) ( ) ]

Paraxial Approximation

f number of a lens =focal length

diameter f

2af# = f / 2a

Paraxial approximation is valid when the longitudinal distance is large compared with the transverse distances:

z x x y y22 1

22 1

2 ( ) ( )

True for slow system, i.e f# >> 1Not true for fast system f# ~ 1

Validity of the Paraxial Approximation

Fourth term

To satisfy Maxwell’s equation at the boundary, a fourth term

is needed:

ldHGi

E

14 )(1

ldEG

iH

14

1

Stratton (1941)

Kottler (1923)

ˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

21

321

321

crcz

bbrby

aarax

ybxa

zcybxa

zcybxar

ˆˆˆ

ˆˆˆˆ

ˆˆˆˆ

33

222

111

13

112

111

sina

coscosa

cossina

13

112

111

cosb

sincosb

sinsinb

12

11

sinc

cosc

xeEE zik ˆ1101

Incident fields:

Transmitted fields:

)ˆˆˆ(ˆ

)ˆˆˆ(

32101

32101

11

11

hhrhzik

eerezik

TbTbTbryeHH

TaTaTareEE

yeHH zik ˆ1101

r

1

z

1 2

1

1

sin

sinTT t

pre 1

1

cos

cos

t

pe TT se TT

11

12

sinn

sinnTT t

srh 11

12

cos

cos

n

nTT t

sh ph Tn

nT

1

2

1211

11

coscos

cos2

tp nn

nT

1112

11

coscos

cos2

ts nn

nT

Conversion from Cartesian to Spherical Coordinates

parameter p2 = 2 n R1 /

Spherical Surface

Normalized coordinates and dimensionless parameters usedto simplify the integrals

Expressions for the fields

321321 ,,,,, vvvuuu

are functions of ’ and ’

02

4E

TipA p

tp nn

nT

'cos'cos

'cos2

21

1

Values of the Integrands

n2n1

1. Single spherical surface n2

2. Single spherical lens

Focusing of light by curved surfaces

n

m = 5 deg

m = 20 deg

On-axis Intensities

n = 1.5 p2 = 106

From geometric optics, f = n/(n-1)

p2 = 106 n = 1.5

m = 5 deg

m = 20 deg

p2 = 106

Longitudinal Field Strengths

For large aperture, the longitudinal field can be quite strong

Comparison of the Different Terms

Plane wave incident at an angle

Angle of incidence

Coordinate transformation from x’,y’,z’ to x’’,y’’,z’’ through angle

Integrations performed in’’and’’ coordinates

p2 = 3660

0 deg

7 deg

15 deg

Coma for a small diameter lens

p2 = 100000 deg 7 deg

15 deg

Coma for a larger diameter lens

A Plano-Convex Lens

Newport Catalog

01 8.18m 0

2 8.26m

2

Expressions for the fields

321321 ,,,,, vvvuuu

are functions of ’ and ’

12ipe

12ipe

121 rr

12

2

2 Rnp

5.12 n m 6328.0

Computational challenge arises from large value of p2

52 1048.1 p

R1 = 1 cm

Computational challenge

2

1

z

21

Questions:

1. For a given 2, 2, how many

integration terms are needed on surface 1?

For each 2, 2: Need 6000 x 4000 values of 1, 1, i.e., 40 sec

Need 26500 values of 2, i.e., 40 sec x 26500 x 4 = 50 days

~ 0.5 day using 100 processors

2. How many 2, 2 values are needed?

z3

4 Nodes (512 processors/node) 2048 Processors (2.9 Peak TeraFLOPS) 1 Gigabyte Memory per Compute

Processor (512 GB/Node, 2048 GB/Total) 25.5 Terabyte Workspace MIPS R16000 (700 MHz) Operating System: IRIX UNIX

SGI Origin 3900 (hpc11)

Computational resource

0

0.2x106

0.4x106

0.6x106

0.8x106

1.0x106

1.7 1.8 1.9 2.0 2.1

= 0.6328 m

Single surface

Geometric focus: 1.94

z

21

Fields at the second surface

-10

-5

0

5

10

0 10 20 30

(degrees)

Ex

Ex

-15

-5

5

15

0 10 20 30

(degrees)

Hy

Hy

Real partImaginary part

Highly oscillatory fields (period ~ 0.005 degrees)

On axis intensity after the lens

0

10000

20000

30000

40000

50000

1.9 2.0 2.1 2.2

z3

SIntensity

S32

xE

Geometric focus: 1.7

On

axis

Int

ensi

ty]Re[ *HES

The Finite-Difference Method

z

rr = m hz = n g

m = 0 to m1

n = 0 to n1

(h and g are chosen to be small)

nE

z

E E

gm n m n

, , 1 E

r

E E

hm n m n

1, ,

2

,1,,1

2

2 2

h

EEE

r

E nmnmnm

nmnmnmnm EEEE ,1,,11, (...)(...)(...)

02 11

2 z

EikET

Under cylindrical symmetry:

m

02 11

2 z

EikET

0)z,r(u)z

ik2rrr

1(

2

2

n2

k

5.1n 0.1n

Interface is discontinuous AND curved, so the FD methodcannot be applied directly

Light focusing by a lens

using FD method

z–stretching adaptation on the interface

To handle curved surfaces, one to one stretching coordinatetransformation is used:

),(rr ),(zz

r ZrRRZ

rRRz22

22

Transformation of curved surface to

rectangular

Focusing of light described

through F-D method

The electromagnetic field components of light refracted by a

curved surface were calculated using integral as well as differential method

The calculations were extended to the case of two refracting

surfaces, providing the field distributions for a commercially

available lens, for the first time to our knowledge

Extension to non-spherical rotationally symmetric surfaces is

straightforward

Summary