International Conference on Industrial and Applied Mathematics Zurich, Switzerland, July 2007 Shekhar Guha United States Air Force Research Laboratory Leo Gonzalez General Dynamics Information Technology Qin Sheng Baylor 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 electromagnetic field for every point in space for propagation of light through linear and nonlinear optical regions i. e., solve the wave equation for all space.

Challenges For feature sizes close to the wavelength of light, vector nature of the 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 equation become unknown For feature sizes much larger than the wavelength of light, diffraction calculations become computationally intensive. For propagation over small distances, computation also becomes challenging A ‘general’ wave-propagation theory describing light propagation through 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 , 2000 Previous work: Ray Theory + Diffraction:

Diffraction Integral Method Surface 1 Surface 2

Green’s Function Method From Green’s scalar theorem, where U and V are continuous functions of position, and are solutions 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 x1x1 x2x2 y1y1 y2y2 z = 0 z = z where

(Fresnel Approximation) Paraxial Approximation

f number of a lens = focal length diameter f 2a f# = f / 2a Paraxial approximation is valid when the longitudinal distance is large compared with the transverse distances: True for slow system, i.e f# >> 1 Not 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: Stratton (1941) Kottler (1923)

Incident fields: Transmitted fields: r      1 z 1 2 Conversion from Cartesian to Spherical Coordinates

parameter p 2 = 2  n R 1 / Spherical Surface Normalized coordinates and dimensionless parameters used to simplify the integrals

Expressions for the fields are functions of  ’ and  ’

Values of the Integrands

n2n2 n1n1 1.Single spherical surface n2n2 2. Single spherical lens Focusing of light by curved surfaces n

 m = 5 deg  m = 20 deg On-axis Intensities n = 1.5 p 2 = 10 6 From geometric optics, f = n/(n-1) p 2 = 10 6 n = 1.5

 m = 5 deg  m = 20 deg p 2 = 10 6 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

p 2 = deg 7 deg 15 deg Coma for a small diameter lens

p 2 = deg 7 deg 15 deg Coma for a larger diameter lens

A Plano-Convex Lens Newport Catalog

2 Expressions for the fields are functions of  ’ and  ’ Computational challenge arises from large value of p 2 R 1 = 1 cm

Computational challenge    2    1 z 2 1 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 values of  2, i.e., 40 sec x x 4 = 50 days ~ 0.5 day using 100 processors 2. How many  2,  2 values are needed? z3z3

 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

 =  m Single surface Geometric focus: 1.94 z 2 1

Fields at the second surface Real part Imaginary part Highly oscillatory fields (period ~ degrees)

On axis intensity after the lens S3S3 Geometric focus: 1.7 On axis Intensity

The Finite-Difference Method z r r = m h z = n g m = 0 to m 1 n = 0 to n 1 (h and g are chosen to be small) n Under cylindrical symmetry: m

Interface is discontinuous AND curved, so the FD method cannot 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 coordinate transformation is used:

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