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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
1. Motivation
• 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)
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2. Geometrical Optics
•
Rays (wavefront normals) carry the
information of amplitude and phase
A ray is given by
ei k 
U  U0
J
U0 initial amplitude
 eikonal (optical path)
J divergence of the ray
Flux conservation:
2
2
U 0 da0  U da
1
1
U

J
Rm Rs
Field diverges (U  ) if Rm  0 or Rs  0
Rm  QmAm
Rs  QsAs
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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).
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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: ( 2  k 2 ) U (x)  0
Fresnel-Kirchhoff or Rayleigh-Sommerfeld diffraction integrals:
ik
U ( P)  
2π

A
ei k s
U ( A)
dA
s
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For a spherically aberrated wave with
small angles everywhere we get
U(,z)  I(R,Z)
where R  , Z  z
We introduce the integral I(R,Z) and name it Bessoid integral
I ( R, Z ) 


J 0 ( R ρ1
 ρ2 ρ4 
 i  Z 1  1 
)e  2 4 
ρ1 dρ1
0
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The Bessoid integral
Bessoid Integral I
3-d: R,,Z
Cuspoid catastrophe + ‘hot line’
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Stationary phase and geometrical optics rays
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4. Wave Picture: Matching Geometrical Optics
and Bessoid Integral
Summary and Outlook:
•
•
•
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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”)
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U geometrical (  , z )  U Bessoid ( R, Z )
3
U0

j 1
e
i k j (  , z )
J j ( , z)
 [ A I ( R, Z )  AR I R ( R, Z )  AZ I Z ( R, Z )] ei  ( R,Z )
6 knowns: 1, 2, 3, J1, J2, J3
6 unknowns: R, Z, , A, AR, AZ
And this yields
R = R(j) = R(, z)
Z = Z(j) = Z(, z)
 = (j) = (, z)
A = A(j, Jj) = A(, z)
AR = AR(j, Jj) = AR(, z)
AZ = AZ(j, Jj) = AZ(, z)
Coordinate transformation
Amplitude matching
Matching removes divergences of geometrical optics
Expressions on the axis rather simple
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5. The Sphere
Sphere radius: a = 3.1 µm
Geometrical optics solution:
f 
Wavelength:  = 0.248 µm
Refractive index: n = 1.42
a n
 5.24 µm
2 n 1
Bessoid matching:

f d  f 1 

3 π n (3  n)  1 
  3.93 µm
4 k a n (n  1) 
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
a
large depth of a narrow ‘focus’
(good for processing)
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Illustration
Bessoid-matched
Bessoid integral
solution
Geometrical
optics solution
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Refractive index: n = 1.5
Bessoid calculation
Mie theory
intensity |E|2  k a  a / 
q  k a = 300
a0.248 µm  11.8 µm
q  k a = 100
a0.248 µm  3.9 µm
q  k a = 30
a0.248 µm  1.18 µm
q  k a = 10
a0.248 µm  0.39 µm
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Electric field immediately behind the sphere (z  a) in the x,y-plane
(k a = 100, incident light x-polarized, normalized coordinates)
Bessoid matching
Theory of Mie
SiO2/Ni-foil,  = 248 nm (500 fs)
sphere radius a = 3 µm
linear polarization
D. Bäuerle et al., Proc SPIE (2003)
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Conclusions
•
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•
•
•
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
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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)
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Appendix
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Analytical expressions
for the Bessoid integral
On the axis
(Fresnel sine and cosine functions):
π
I ( R  0,Z ) 
2
Z 2 π
i
e 4
Z iπ 
erfc e 4 
2



Near the axis (Bessel beam)
I ( R,Z )  J 0 ( R  Z ) I (0 ,Z )
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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
T100x100 > 1 hour
2.
Numerical integration along a line in the complex plane (Cauchy theorem)
Integration converges
T100x100  20 minutes
3.
Solving numerically the corresponding differential equation for the Bessoid
integral I (T100100  2 seconds !)
2i IZ  ΔRI  0
(Δ R I ) R  Z I R  i R I  0
paraxial Helmholtz equation
in polar coordinates + some
tricks 
one ordinary differential equation
in R for I (Z as parameter)
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Properties of Bessoid important for applications:
•
•
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near the axis: Bessel beam with slowly varying cross section
smallest width is not in the focus
3 
width from axis to first zero of Bessel function: w0 
8 sin 
(width is smaller than with any lens)
diverges slowly: large depth of focus (good for processing)

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Generalization to Vector Fields
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 the higher-order Bessoid integrals:
Im 


ρ1m1 J m ( R
 ρ2 ρ4 
 i  Z 1  1 
ρ1 ) e  2 4  dρ1
I0  I
0
Geometrical optics terms with -dependence cos(m) or sin(m)
have to be matched with m-th order Bessoid integral Im
Coordinate equations (R, Z, ) remain the same (cuspoid catastrophe)
Amplitude equations (Am, ARm, AZm) are modified systematically
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