Focusing of Light in Axially Symmetric Systems within the Wave Optics Approximation Johannes Kofler Institute for Applied Physics Johannes Kepler University Linz Diploma Examination November 18th, 2004

2 Goal: Intensity distribution behind a focusing sphere -as analytical as possible -fast to compute -improve physical understanding -interpret and predict experimental results Wave field behind a focusing system is hard to calculate -geometrical optics intensity:  in the focal regions -diffraction wave integrals: finite but hard to calculate (integrands highly oscillatory) -available standard optics solutions (ideal lens, weak aberration): inapplicable -theory of Mie: complicated and un-instructive (only spheres) 1. Motivation

3 2. Geometrical Optics A ray is given by U 0 initial amplitude  eikonal (optical path) J divergence of the ray Flux conservation: Field diverges (U   ) if R m  0 or R s  0 Rays (wavefront normals) carry the information of amplitude and phase Rm  QmAmRs  QsAsRm  QmAmRs  QsAs

4 Caustics Caustics (Greek: ‘burning’): Regions where the field of geometrical optics diverges (i.e. where at least one radius of curvature is zero and the density of rays is infinitely high).

5 3. Diffraction Integrals Wave field in a point P behind a screen A: Summing up contributions from all virtual point sources on the screen (with corresponding phases and amplitudes). Scalar Helmholtz equation: Fresnel-Kirchhoff or Rayleigh-Sommerfeld diffraction integrals:

6 For a spherically aberrated wave with small angles everywhere we get We introduce the integral I(R,Z) and name it Bessoid integral where R  , Z  z U( ,z)  I(R,Z)

7 Bessoid Integral I 3-d: R, ,Z Cuspoid catastrophe + ‘hot line’ The Bessoid integral

8 Stationary phase and geometrical optics rays

9 4. Wave Picture: Matching Geometrical Optics and Bessoid Integral Summary and Outlook: Wave optics are hard to calculate Geometrical optics solution can be “easily” calculated in many cases Paraxial case of a spherically aberrated wave  Bessoid integral I(R,Z) I(R,Z) has the correct cuspoid topology of any axially symmetric 3-ray problem Describe arbitrary non-paraxial focusing by matching the geometrical solution with the Bessoid (and its derivatives) where geometrical optics works (uniform caustic asymptotics, Kravtsov-Orlov: “Caustics, Catastrophes and Wave Fields”)

10 6 knowns:  1,  2,  3, J 1, J 2, J 3 6 unknowns: R, Z, , A, A R, A Z And this yields R = R(  j ) = R( , z) Z = Z(  j ) = Z( , z)  =  (  j ) =  ( , z) A = A(  j, J j ) = A( , z) A R = A R (  j, J j ) = A R ( , z) A Z = A Z (  j, J j ) = A Z ( , z) Coordinate transformation Amplitude matching Matching removes divergences of geometrical optics Expressions on the axis rather simple

11 5. The Sphere Sphere radius: a = 3.1 µm Refractive index: n = 1.42 Wavelength: = µm Geometrical optics solution:Bessoid matching:

12 a large depth of a narrow ‘focus’ (good for processing)

13 Bessoid integralBessoid-matched solution Geometrical optics solution Illustration

14 q  k a = 300 a µm  11.8 µm Refractive index: n = 1.5 Bessoid calculation Mie theory intensity |E| 2  k a  a / q  k a = 100 a µm  3.9 µm q  k a = 30 a µm  1.18 µm q  k a = 10 a µm  0.39 µm

15 Bessoid matching Theory of Mie Electric field immediately behind the sphere (z  a) in the x,y-plane (k a = 100, incident light x-polarized, normalized coordinates) SiO 2 /Ni-foil,  = 248 nm (500 fs) sphere radius a = 3 µm linear polarization D. Bäuerle et al., Proc SPIE (2003)

16 Conclusions Axially symmetric focusing leads to a generalized standard integral (Bessoid integral) with cuspoid and focal line caustic Every geometrical optics problem with axial symmetry and strong spherical aberration (cuspoid topology) can be matched with a Bessoid wave field Divergences of geometrical optics are removed thereby Simple expressions on the axis (analytical and fast) Generalization to non axially symmetric (vectorial) amplitudes via higher-order Bessoid integrals For spheres: Good agreement with the Mie theory down to Mie parameters q  20 (a /  3) Cuspoid focusing is important in many fields of physics: -scattering theory of atoms -chemical reactions -propagation of acoustic, electromagnetic and water waves -semiclassical quantum mechanics

17 Acknowledgements Prof. Dieter Bäuerle Dr. Nikita Arnold Dr. Klaus Piglmayer, Dr. Lars Landström, DI Richard Denk, Johannes Klimstein and Gregor Langer Prof. B. Luk’yanchuk, Dr. Z. B. Wang (DSI Singapore)

18 Appendix

19 On the axis (Fresnel sine and cosine functions): Near the axis (Bessel beam) Analytical expressions for the Bessoid integral

20 Numerical Computation of the Bessoid integral 1.Direct numerical integration along the real axis Integrand is highly oscillatory, integration is slow and has to be aborted T 100x100 > 1 hour 2.Numerical integration along a line in the complex plane (Cauchy theorem) Integration converges T 100x100  20 minutes 3.Solving numerically the corresponding differential equation for the Bessoid integral I (T 100  100  2 seconds !) paraxial Helmholtz equation in polar coordinates + some tricks  one ordinary differential equation in R for I (Z as parameter)

21 Properties of Bessoid important for applications: near the axis: Bessel beam with slowly varying cross section smallest width is not in the focus width from axis to first zero of Bessel function: (width is smaller than with any lens) diverges slowly: large depth of focus (good for processing) 

22 Consider (e.g.) linear polarization of incident light: Modulation of the initial (vectorial) amplitude on the spherically aberrated wavefront  axial symmetry is broken Generalization to Vector Fields Coordinate equations (R, Z,  ) remain the same (cuspoid catastrophe) Amplitude equations (A m, A Rm, A Zm ) are modified systematically Generalization to the higher-order Bessoid integrals: I0  II0  I Geometrical optics terms with  -dependence cos(m  ) or sin(m  ) have to be matched with m-th order Bessoid integral I m