Transcript ppt

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
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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
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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 geometrica l (  , z )  U Bessoid ( R, Z )
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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:
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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
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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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