focusing of light in axially symmetric systems within the...

Focusing of Light inAxially Symmetric Systems withinthe Wave Optics Approximation

Diplomarbeit zur Erlangung des akademischen Grades

Diplomingenieur

im Diplomstudium der Technischen Physik

Angefertigt am Institut fur Angewandte Physik

Eingereicht von

Johannes Kofler

Betreuung und Beurteilung:

Dr. Nikita Arnold

o. Univ.-Prof. Dr. Dr. h.c. Dieter Bauerle

Linz, September 2004

Ohne den Glauben daran, daß es grundsatzlich moglich ist, die Wirklichkeitdurch unsere logischen Konstruktionen begreiflich zu machen, ohne den Glaubenan die innere Harmonie unserer Welt konnte es keine Naturwissenschaft geben.Dieser Glaube ist und bleibt das Grundmotiv jedes schopferischen Gedankensin der Naturwissenschaft.

Albert Einstein und Leopold Infeld, ”Die Evolution der Physik”

Eidesstattliche Erklarung

Ich erklare an Eides statt, daß ich die vorliegende Diplomarbeit selbstandig und ohne fremde Hilfe verfaßt,andere als die angegebenen Quellen und Hilfsmittel nicht benutzt respektive die wortlich oder sinngemaßentnommenen Stellen als solche kenntlich gemacht habe.

i

Acknowledgement

First of all, I wish to express my deepest thanks to my parents. Their unconditional support made it possiblefor me to dedicate myself to my studies in far-reaching freedom. I will remember the last five years as awonderful and most important period in my life and I shall always be very grateful for that.

I want to thank Prof. Dieter Bauerle for giving me the opportunity to write a theoretical diploma thesisat his institute, for attracting my attention to the problem of light focusing by microspheres, and for hisrepetitious endeavour to provide an inspiring working environment.

My profound thanks go to Dr. Nikita Arnold for his ceaseless support in the past year. His supervision,in fact, was outstanding both in a scientific and in a personal way. Without his excellent review and hisnumerous suggestions broad parts of this work would not have reached their present form.

Linz, September 2004 Johannes Kofler

ii

Abstract

The present work was motivated by the question of how the electromagnetic light field behind a focusingsphere of micron dimensions can be described. We wanted to find an answer to this question in an illustrativeway and improve the understanding of the physical situation. In particular, we sought to avoid solvingMaxwell’s equations explicitly as it is done in the theory of Mie, for it leads to an uninstructive infinitesummation of Bessel functions and associated Legendre polynomials. However, in order to reach this aim itwas necessary to develop a quite general mathematical apparatus, which is capable of solving a whole familyof problems. Hence, eventually, the field behind a sphere becomes only a specific case to solve.

Starting from the scalar diffraction theory, a paraxial spherically aberrated wave is studied, leading to anintegral representation of the light field. In parallel, we recapitulate the fundamentals of geometrical opticsand apply it to the eikonal of a spherically aberrated wave. The field diverges in the caustic regions, givenby a cuspoid surface and the axis of symmetry up to the focus. The cuspoid forms the border between thelit region, where three geometrical rays arrive at each point, and the shadow, where only one real ray exists.

The diffraction integral for a spherically aberrated wave can be brought into a canonical form of catas-trophe theory. This Bessoid integral is the axially symmetric two-dimensional analog of the well knownPearcey integral which appears in many fields of physics and was studied in detail. However, for the Bessoidintegral a complete investigation is still lacking and thus analytical expressions in special coordinate regimesare presented, especially along the axis of symmetry. Furthermore, differential equations are derived, whichallow fast numerical calculation of the Bessoid integral in the whole space of its arguments.

Subsequently, we describe general non-paraxial focusing by matching an arbitrary solution of geometricaloptics, obeying the cuspoid topology of strong spherical aberration, with a wave field built by the Bessoidintegral and its derivatives. The unknown coordinates and amplitude factors can be found in an algebraicway and the resulting field does not show the divergences of geometrical optics. On and near the axis allexpressions can be widely simplified. By this means one obtains general elucidating results for the positionof the diffraction focus, that is the point of maximum intensity, and for the field on the axis in arbitraryaxially symmetric focusing systems.

Then, as a special case, the problem of a refracting sphere is solved within the picture of geometricaloptics. We apply the Bessoid matching procedure in order to obtain a scalar wave field without causticdivergences. The consideration of an incoming linearly polarized plane wave leads us to the necessity toallow angular dependent vectorial amplitudes of the electric and magnetic field on the spherically aberratedwavefront, breaking the axial symmetry. This requires the introduction of higher-order canonical Bessoidintegrals and new amplitude matching formulas. The Bessoid coordinates, on the other hand, are the samefor all orders, since the cuspoid topology and thus the structure of the ray picture remains unchanged.The comparison with the theory of Mie shows excellent agreement with the vectorial Bessoid matching andreveals the limits of applicability of the approach developed in this work.

iii

Contents

Introduction 2

1 Diffraction Integrals, Geometrical Optics and Spherical Aberration 41.1 Scalar Diffraction Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Geometrical Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 The Bessoid Integral 162.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Off the Caustic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.3 On and Near the Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.4 At the Cuspoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.5 Numerical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Relation between Geometrical and Wave Optics 323.1 Matching with the Bessoid Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.2 Determination of Coordinates and Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 The Matched Solution for a Spherically Aberrated Wave . . . . . . . . . . . . . . . . . . . . . 373.4 General Expressions On and Near the Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 The Sphere 424.1 Geometrical Optics Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 The Scalar Wave Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 On the Axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.4 The Vectorial Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.5 Comparison with the Theory of Mie . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Conclusions 62

References 64

Curriculum Vitae 66

Introduction

In this thesis we develop a theoretical description of the focusing of light by spheres with diameters whichare comparable with or larger than the wavelength. The field enhancement produced by such microlenseshas been used lately for different types of high-throughput laser material processing1 and it also plays animportant role in dry laser cleaning2. Small spherical particles naturally arise in many areas of science,because they have the minimum surface area and surface energy for a given volume.

Strong spherical aberration makes the focusing properties by no means trivial. The only known exactsolution is obtained on the basis of the Mie theory3, which does not give much of a physical insight. It is arigorous electrodynamic treatment of a plane wave incident on a sphere, leading to an infinite summation ofBessel functions and associated Legendre polynomials. But in fact the most prominent features of the fielddistribution and the focusing properties of transparent dielectric spheres originate rather from the pictureof geometrical optics.

In the lowest approximation the sphere acts as an ideal lens. This picture is of course completely inappli-cable, but even classical calculations for weak spherical aberration4 do not yield useful results for the lightfield behind a sphere. The maximum intensity is kept unchanged and its position does not depend on thewavelength.

The present approach5 bases on the canonical integral of catastrophe theory for the cuspoid ray topologyof strong spherical aberration. This Bessoid integral – though arising from the problem of a paraxial wave– is used to describe arbitrary axially symmetric and strongly spherically aberrated focusing by appropriatecoordinate transformations and amplitude equations.

The Bessoid integral is the higher-dimensional generalization of the Pearcey integral6, which plays animportant role in many short wavelength phenomena besides optical caustics, including elastic scattering ofatoms and ions, the semiclassical theory of chemical reactions, collisions of heavy nuclear ions, the asymptoticevaluation of path integrals, the propagation of acoustic, electromagnetic and water waves, scattering fromsurfaces as well as various general features of semiclassical quantum mechanics.

We first investigate a paraxial strongly spherically aberrated scalar wave with the correct axially symmet-ric topology from two points of view. On the one hand, the Fresnel-Kirchhoff diffraction integral leads us tothe wave field corresponding to the cuspoid catastrophe, represented by the Bessoid integral. In geometricaloptics, on the other hand, rays carry the phase and amplitude of the field. For a spherically aberrated waveit illustrates the field divergence along the focal line and on the cuspoid caustic surface. The latter formsthe border between the 3-ray lit region and the 1-ray geometrical shadow. Also the concept of caustic phaseshift is recapitulated.

Subsequently, we investigate the Bessoid integral both analytically and numerically. Analytic approxima-tions can be found by the method of stationary phase for three different regions of its arguments, that is offthe caustic, near the axis of symmetry and near the cuspoid. The numerical evaluation is non-trivial, as theBessoid integrand is highly oscillatory. It is done fastest by solving an ordinary differential equation whichwe derive.

Then the general methods of uniform caustic asymptotics are applied to the axially symmetric cuspoidtopology. Arbitrary non-paraxial focusing with strong spherical aberration is described by matching thesolution of geometrical optics with a wave field which is constructed from the Bessoid integral and itsfirst partial derivatives. By this means the caustic divergences are removed and the resulting field is finiteeverywhere. The equations for the coordinates and amplitudes can be solved in an algebraic way7 andnear the axis significant simplification is possible, yielding general formulas for the on-axis field distributionof arbitrary axially symmetric systems with a cuspoid topology as well as an intuitive condition for thediffraction focus, that is the point of maximum intensity.

1 Bauerle, D. et al., Laser-processing with colloid monolayers, Proc. SPIE 5339 (2004), 20–26, and the references therein.2 Luk’yanchuk, B. (Editor), Laser Cleaning, World Scientific Publishing (2002), p. 103 et seq.3 Mie, G., Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen, Annalen der Physik (4), 25 (1908), 377.4 Born, M. and Wolf, E., Principles of Optics, Cambridge University Press, Seventh Expanded Edition (2002), p. 517 et seq.5 Following Kravtsov, Y. A. and Orlov, Y. I., Caustics, Catastrophes and Wave Fields, Springer Series on Wave Phenomena

(Volume 15), Springer-Verlag, Second Edition (1999), p. 92 et seq.6 Pearcey, T., The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic, Lond. Edinb. Dubl.

Phil. Mag. 37 (1946), 311–317.7 For the Pearcey integral the coordinate transformations can be found in Connor, J. N. and Farrelly, D., Theory of cusped

rainbows in elastic scattering: uniform semiclassical calculations using Pearcey’s Integral, J. Chem. Phys. 75(6) (1981), 2831–2846, whereas both the coordinate and amplitude equations are written in Brekhovskikh, L. M. and Godin, O. A., Acoustics ofLayered Media II, Springer Series on Wave phenomena (Volume 10), Springer-Verlag (1992), p. 213 et seq.

Introduction 3

The case of a plane wave incident on a refracting sphere obeys the cuspoid topology and the derivedapparatus of Bessoid matching can be applied. First, we solve the geometrical optics problem for a scalarwave field. The consideration of incident linearly polarized light requires a modulation of amplitude on thespherically aberrated wavefront, without changing the structure of rays and the type of catastrophe. For thevectorial field it is therefore necessary to generalize the Bessoid integral and the coordinate and amplitudematching procedure to higher orders. The formulas for the coordinate transformations remain the same andthe amplitudes can be modified systematically for the higher orders. The higher-order Bessoid integrals areappropriate for the matching of arbitrary cuspoid focusing where the axial symmetry is broken with respectto the amplitudes only.

The Bessoid matched vectorial electric field behind the sphere is dependent on the azimuthal angle,showing a quite complicated structure of maxima and minima. The magnetic field and the Poynting vectorcan be found in a straightforward way.

Finally, we compare our vectorial results for the sphere with the theory of Mie. There is an excellentagreement within a wide range of parameters. Only for small spheres and at distances to the sphere surfacewithin the order of a wavelength significant deviations are present. This is natural, since geometrical opticsbecomes invalid for such situations.

Chapter 1

Diffraction Integrals, GeometricalOptics and Spherical Aberration

In this chapter the fundamental integrals of scalar diffraction theory are derived. We apply them to thesituation of an incoming paraxial spherically aberrated wave incident on a plane screen with circular aperture.In the last section of the chapter the fundamentals of geometrical optics are developed and applied to aspherically aberrated wave. Thereby, we will be confronted with the divergence of geometrical optics solutionsin the caustic regions leading to infinite fields.

1.1 Scalar Diffraction Theory

Starting1 from the scalar Helmholtz2 equation(∇2 + k2

)U(x) = 0 , (1.1)

where k is the wavevector and U is the scalar wave field, we apply the Gauß3 theorem of vector analysis inits integral representation, i.e. ∫

V

∇A d3x =∫

∂V

An da , (1.2)

with A an arbitrary vector field, n the outward surface normal unity vector, and V and S ≡ ∂V a givenvolume and its closed surface, respectively. By choosing A = φ∇ψ and using the notation

n∇ψ ≡ ∂ψ

∂n, (1.3)

one directly arrives at Green’s4 first identity∫

V

(φ∇2ψ +∇φ∇ψ

)d3x =

∫

∂V

φ∂ψ

∂nda . (1.4)

Exchanging φ and ψ in (1.4) and subtracting from the original equation, we obtain Green’s second identity∫

V

(φ∇2ψ − ψ∇2φ

)d3x =

∫

∂V

(φ

∂ψ

∂n− ψ

∂φ

∂n

)da . (1.5)

The Green’s function G of the Helmholtz equation is defined as the solution of (1.1) with a point source atx′ ≡ (x′, y′, z′) as inhomogeneity:

(∇2 + k2)G(x,x′) = −δ(x− x′) . (1.6)

By setting φ = G and ψ = U (now taking n as the inward normal at the point x′), equation (1.5) becomes

U(x) =∫

∂V

(U(x′)

∂G(x,x′)∂n′

−G(x,x′)∂U(x′)

∂n′

)da′. (1.7)

1 Following the derivation in Jackson, J. D., Classical Electrodynamics, John Wiley and Sons, Second Edition (1975), p. 40et seq. and p. 427 et seq.

2 Hermann Ludwig Ferdinand von Helmholtz (1821–1894).3 Carl Friedrich Gauß (1777–1855).4 George Green (1793–1841).

1.1 Scalar Diffraction Theory 5

This relation holds for all x ≡ (x, y, z) within the volume V . For x outside V , the field vanishes: U(x) = 0.One possible choice of G is the outgoing spherical wave, assuming harmonic time dependence exp(−iω t) –with ω the angular frequency and t the time – in the original time dependent equations:

G(x,x′) =e i k R

4 π R, (1.8)

with R = x− x′. We obtain

U(x) = − 14 π

∫

∂V

e i kR

R

[(i k − 1

R

)U(x′)

RR

n′ +∂U(x′)

∂n′

]da′. (1.9)

One has to relate this formulation to the corresponding physical picture of diffraction theory. The integralcan be split into 2 parts (see figure 1.1), ∂V ≡ S = S1 + S2.

V

S1

A

Q

ar as

P

sr

n

S2

Figure 1.1: Notation

The first (S1) contains a screen and its aperture (A), the second (S2), e.g. a half sphere, tends to infinity.Since the field has come through S1, it can be characterized as a diverging wave near S2. Thus, U obeys theSommerfeld5 radiation condition

U ∝ e i k R

R, (1.10)

1U

∂U

∂R∝ i k − 1

R, (1.11)

such that S2 gives no contribution. As a result we arrive at Kirchhoff’s6 diffraction integral

U(x) = − 14 π

∫

S1

e i k R

R

[(i k − 1

R

)U(x′)

RR

n′ +∂U(x′)

∂n′

]da′. (1.12)

In order to calculate this integral one has to know the values of U and its derivative on the surface S1. Ingeneral, that is not possible and one has to make the fundamental Kirchhoff assumptions7:

(a) U and ∂U/∂n are zero everywhere on the screen S1 except at the openings A,

(b) at the openings A the values of U and ∂U/∂n are the same as if no screen was present.

Mathematically more exact and consistent is the following approach. Starting again from (1.7), we choosethe Green’s functions which satisfy Dirichlet8 (index D) or von Neumann9 (index N) boundary conditions,respectively:

GD(x,x′) = 0 , (1.13)∂GN(x,x′)

∂n′= 0 , (1.14)

5 Arnold Johannes Wilhelm Sommerfeld (1868–1951).6 Gustav Robert Kirchhoff (1824–1887).7 All standard calculations for diffraction problems in classical optics are based on Kirchhoff’s approximation. But one has

to add that it is mathematically inconsistent, since in fact from condition (a) follows that U vanishes everywhere.8 Johann Peter Gustav Lejeune-Dirichlet (1805–1859).9 Janos von Neumann (1903–1957).


with x′ on S1. Hence, from (1.7) the so called generalized Kirchhoff integrals are found:

U1(x) = +∫

S1

U(x′)∂GD(x,x′)

∂n′da′, (1.15)

U2(x) = −∫

S1

∂U(x′)∂n′

GN(x,x′) da′. (1.16)

We will investigate the special case of a plane screen S1 at z′ = 0. The method of mirror images providesGreen’s functions satisfying (1.13) and (1.14), respectively (minus sign for D, plus sign for N):

GD,N(x,x′) =1

4 π

(e i k R+

R+∓ e i k R−

R−

), (1.17)

where the distances from x to the point source and its mirror image with respect to the plane z = 0 areR± = (x− x′)2 + (y − y′)2 + (z ∓ z′)2. If one sets s ≡ R+(z′=0), one can write10

GD(z′=0) = 0 ,∂GD

n′(z′=0) =

e i k s

2 π s

(i k − 1

s

)(− cos αs) ,

GN(z′=0) =e i k s

2 π s,

∂GN

n′(z′=0) = 0 .

(1.18)

This yields the Rayleigh11-Sommerfeld diffraction formulas of first and second kind for the field U at thepoint P ≡ x:

U1(x) = +1

2 π

∫

S1

U(x′)∂

∂z′e i k s

sdx′dy′, (1.19)

U2(x) = − 12 π

∫

S1

∂U(x′)∂z′

e i k s

sdx′dy′. (1.20)

Now we assume an incident spherical monochromatic wave emitted by a point source Q at distance12 rfrom the point of integration on the aperture x′ = (x′, y′, z′=0):

U = Ce i k r

r, (1.21)

∂U

∂z′= C

e i k r

r

(i k − 1

r

)cos αr , (1.22)

with a factor C, being the amplitude at r = 1. The angles αr and αs are measured between the correspondingvector and the inward surface normal (see again figure 1.1). Thus, (1.19) and (1.20) become

U1(x) = − C

2 π

∫

S1

e i k (r+s)

r s

(i k − 1

s

)cosαs dx′dy′, (1.23)

U2(x)= − C

2 π

∫

S1

e i k (r+s)

r s

(i k − 1

r

)cosαr dx′dy′, (1.24)

In this notation and with (1.21) and (1.22), the Kirchhoff integral (1.12) is referred to as Fresnel13-Kirchhoffdiffraction formula and it has the form

U(x) = − C

4 π

∫

S1

e i k (r+s)

r s

[(i k − 1

r

)cosαr +

(i k − 1

s

)cosαs

]dx′dy′. (1.25)

We postulate that the distances from the screen are significantly larger than the wavelength λ = 2 π/k, i.e.k r, k s À 1. Then one can neglect the second terms of the corresponding brackets in (1.25).

Therefore, we may sum up:

U(x) = − i k C

2 π

∫

S1

e i k (r+s)

r sg(αr, αs) dx′dy′, (1.26)

10 The negative sign in front of the cosine appears as a consequence of the chain rule due to the partial derivative of s withrespect to z′.

11 Lord Rayleigh, John William Strutt (1842–1919).12 Notice that r is not the absolute value of x = (x, y, z). The first variable is connected with Q, whereas the latter is with P .13 Augustin Jean Fresnel (1788–1827).


where the inclination factor is

g(αr, αs) =

cos αs Rayleigh-Sommerfeld of first kind, U known on S1,cos αr Rayleigh-Sommerfeld of second kind, ∂U/∂n known on S1,

12 (cos αs + cos αr) Fresnel-Kirchhoff (Kirchhoff-Approximation), S1 → A.

(1.27)

Similarly, in the case of an arbitrary field on the plane aperture one can write

U(x) = − i k2 π

∫

S1

e i k s

sU(x′) g(αr, αs) dx′dy′. (1.28)

Here αr is the angle between the inward normal and the (slowly varying) normal to the wavefront at x′.To describe focusing, we start from equation (1.26) and consider an incoming converging spherical wave.

This implies that one has to change r → −r in the exponent. Moreover, we introduce a Cartesian14 coordinatesystem (x, y, z) with origin in the focus F ≡ (0, 0, 0). At a distance z = −f (where f > 0) there is a planecircular aperture A (with the center O1) with radius a. The positions on it are denoted as (x1, y1). Thus,the whole system is axially symmetric with respect to the z-axis (figure 1.2).

A

s

r

P

a

z

xx1

yy1

O1

F

f

Figure 1.2: Notations

The relevant distances are

r =√

x21 + y2

1 + f2 , (1.29)

s =√

(x− x1)2 + (y − y1)

2 + (z + f)2 . (1.30)

The lowest non-trivial order Taylor15 expansion for the path in the exponent exp[ i k (s− r)] yields

s− r ≈ z − 1f

(xx1 + y y1)− z

2 f2(x2

1 + y21) . (1.31)

Furthermore, the cosines in (1.27) shall be replaced by unity, i.e. g(αr, αs) ≈ 1, because we assume smallangles everywhere (paraxial approximation). In this case the expressions for the Fresnel-Kirchhoff and theRayleigh-Sommerfeld integrals become identical. In the lowest order the product of the inverse distances,1/r s, is replaced by 1/f2, and we redefine C ≡ U0 f in such a way that U0 is the amplitude of the sphericalwave at the aperture (at the distance f). Then, (1.26) reads

U(x, y, z) = − i k U0

2 π f

∫

A

e i k[z− 1

f (x x1 + y y1)− z2 f2 (x2

1 + y21)

]dx1dy1. (1.32)

Having in mind axial symmetry, we introduce cylindrical coordinates (ρ, ϕ, z) in the focal region and(ρ1, ϕ1, z1 = −f) in the aperture:

x = ρ cosϕ, y = ρ sinϕ,x1 = ρ1 cosϕ1, y1 = ρ1 sin ϕ1, dx1dy1 = ρ1 dρ1dϕ1.

(1.33)

Because of xx1 + y y1 = ρ ρ1 cos(ϕ1 − ϕ), expression (1.32) can be rewritten as

U(ρ, z) = − i k U0

2 π fe i k z

a∫

0

2 π∫

0

e− i k

(ρ ρ1 cos(ϕ1−ϕ)

f − z ρ21

2 f2

)

ρ1 dρ1dϕ1. (1.34)

14 Rene Descartes (1596–1650).15 Brook Taylor (1685–1731).


This is simplified using an integral representation for the zero-order Bessel16 function17:

J0(t) =1

2 π

2 π∫

0

e i t cos θ dθ . (1.35)

Since the angular integration in (1.34) is taken over the whole period 2 π, the shift ϕ in the cosine argumentis unimportant. One can discard the minus sign in the argument of the Bessel function for it is symmetric.We therefore obtain the one-dimensional integral18

U(ρ, z) = − i k U0

fe i k z

a∫

0

J0

(k

ρ ρ1

f

)e− i k

z ρ21

2 f2 ρ1 dρ1. (1.36)

In order to consider a spherically aberrated wave one has to disturb the phase in the exponent of thespherical wave by higher order terms19 in ρ1. Due to symmetry these corrections must be of an even power.As a consequence, in the lowest order we have to add a fourth-order perturbation with the strength20 B > 0.Thus, we arrive at:

U(ρ, z) = − i k U0

fe i k z

a∫

0

J0

(k

ρ ρ1

f

)e− i k

z ρ21

2 f2 − i k B ρ41 ρ1 dρ1. (1.37)

To illustrate the physical meaning of the parameter B, we state that k δ is the phase shift (δ is the distance)between the aberrated and the non-aberrated wavefront at the focal distance f and a small angle γ, say atthe edge of the aperture a (figure 1.3).

g

d

F

f

Figure 1.3: Spherical (solid) and spherically aberrated (dashed) wave. Notation

Hence, with sin γ ≈ γ,

B =δ

a4=

δ

(γ f)4(1.38)

For the sake of generality we introduce dimensionless coordinates ρ1, R and Z. This is done by the followingdefinitions and transformations21:

ρ41

4≡ k B ρ4

1 → ρ1 = 4√

4 k B ρ1 ,

Z ρ21

2≡ k

z ρ21

2 f2→ Z =

√k

4 B

z

f2,

R ρ1 ≡ kρ ρ1

f→ R = 4

√k3

4 B

ρ

f.

(1.39)

16 Friedrich Wilhelm Bessel (1784–1846).17 See e.g. http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html.18 This integral can be found in the literature, e.g. in Born, M. and Wolf, E., Principles of Optics, Cambridge University

Press, Seventh Expanded Edition (2002), p. 487.19 It is also possible to directly insert a spherically aberrated wave U(x′) in (1.28).20 To shift the aberrated focus towards the aperture, it is convenient to choose a negative sign for this correction term and

positive B.21 The factors 1/2 and 1/4 on the left hand side of the first two definitions are convention.

1.2 Geometrical Optics 9

Substitution into (1.37) gives22

U(R, Z) = − i k U0

f√

4 k Be i√

4 k B f2 Z

ρ1(a)∫

0

J0(R ρ1) e− i

(Z ρ2

12 +

ρ414

)

ρ1 dρ1. (1.40)

Having in mind future applications, we briefly consider a sphere as the source of the spherically aberratedwave. Its radius be the radius of the aperture, a. If we keep the wavelength λ fixed (k = const) and changethe size of the sphere, then, for constant angle γ, both the focal length f and the aberration path differenceδ are proportional to the sphere radius a. Hence, for the aberration strength from (1.38), B ∝ 1/a3, andfor the field from (1.40), U ∝

√k a. The intensity – the absolute square of the field – will thus have the

functional dependence |U |2 ∝ k a.

1.2 Geometrical Optics

Within the picture of geometrical optics the rays carry the wave field’s information of amplitude and phase.They are the curves whose tangents coincide with the direction of propagation and are normals to thewavefronts, i.e. normals to the surfaces of equal phase. A ray’s field at a point P is determined by

U = U0e i k ψ

√J

, (1.41)

where U0 is the amplitude at some initial wavefront, ψ is the eikonal23 (optical path along the ray from thiswavefront to P ) and J is the generalized geometrical divergence. The overall field in P is given by the sumof all ray fields arriving at that point.

The eikonal equation is24

(∇ψ)2 = n2 , (1.42)

with n the refractive index. It is a non-linear partial differential equation of the Hamilton25-Jacobi26 type.Hence, the theory of geometrical optics is inherently related to the calculus of variations, i.e. to Maupertuis’27

and Hamilton’s principle of least action in classical mechanics and to Fermat’s28 principle of shortest opticalpath or principle of least time29. The latter states that the variation of the eikonal, δψ, vanishes for thepropagation between any two points P1 and P2 along the path30 σ:

δψ = δ

P2∫

P1

n dσ = c δ

P2∫

P1

dt = 0 , (1.43)

with c the velocity of light in vacuum and t the time. The ray equations in optics are therefore similar tothe Euler31-Lagrange32 equations in mechanics. With a convenient definition of the Hamiltonian33

H =p2 − n2(x)

2= 0 , where p = ∇ψ , (1.44)

22 Note that in general (1.40) is not an exact expression for the field. It implicitly contains the Kirchhoff approximation aswell as the restriction of small angles.

23 From the Greek word eikon for image.24 In fact, there are three conditions which must be met that the eikonal equation is valid, namely that the relative variations

(a quantity divided by the absolute value of its gradient) of amplitude, wavenumber and refractive index are all small comparedto the wavelength.

25 Sir William Rowan Hamilton (1805–1865).26 Carl Gustav Jacob Jacobi (1804–1851).27 Pierre Louis Moreau de Maupertuis (1698–1759).28 Pierre de Fermat (1601–1665).29 Strictly speaking, these are extremal principles, where the minimum is physically manifested.30 One assumes that the variation of the integrand vanishes at the borders: δn(P1) = δn(P2) = 0.31 Leonhard Euler (1707–1783).32 Joseph-Louis Lagrange (1736–1813).33 The subsequent review of geometrical optics follows Kravtsov, Y. A. and Orlov, Y. I., Geometrical Optics of Inhomogeneous

Media, Springer Series on Wave Phenomena (Volume 6), Springer-Verlag (1990), p. 3 et seq.


we obtain the ray equations in their canonical form:

dxdτ

=dH

dp= p , (1.45)

dpdτ

= −dH

dx=

12∇n2 . (1.46)

Here x ≡ (x, y, z) is the position and p ≡ (px, py, pz) is the canonical momentum of the ray. The parameterτ is related to the length along the ray, σ, by dτ ≡ dσ/n = c dt/n2.

Under typical conditions the initial field is specified on a certain surface, x0(ξ, η) = x(ξ, η, τ =0), endowedwith the curvilinear ray coordinates ξ and η:

Ux0(ξ, η) = U0 e i k ψ0 , (1.47)

where U0 = U0(ξ, η) and ψ0 = ψ0(ξ, η) are the initial amplitude and eikonal on x0 with τ = 0 (figure 1.4).

x h

x = ( , , )h tx x

x0( , )x h

Figure 1.4: Ray path and coordinates

The eikonal in x = x(ξ, η, τ) is given by

ψ = ψ0 +

τ∫

0

p2 dτ = ψ0 +

σ∫

0

n(x) dσ . (1.48)

The generalized divergence of rays can be calculated from flux conservation along the ray tubes (figure 1.5).

da0

da

Figure 1.5: Conservation of energy flux in a ray tube with area elements da0 and da

In differential form it reads|U0|2 n0 da0 = |U |2 n da , (1.49)

where da (da0) and n (n0) are the area element and the refractive index (on the initial surface). Therefore,from (1.41).

J =|U0|2|U |2 =

n da

n0 da0, (1.50)

The divergence can also be represented as the ratio

J =D(τ)D(0)

, (1.51)

where D(τ) is the determinant of the Jacobian of the transition from ray coordinates (ξ, η, τ) to the Cartesiancoordinates (x, y, z):

D(τ) =∂(x, y, z)∂(ξ, η, τ)

≡∣∣∣∣∣∣

∂x/∂ξ ∂x/∂η ∂x/∂τ∂y/∂ξ ∂y/∂η ∂y/∂τ∂z/∂ξ ∂z/∂η ∂z/∂τ

∣∣∣∣∣∣. (1.52)


In terms of the catastrophe theory34 the projection of the extended space (x, y, z, ξ, η, τ) onto the three-dimensional configuration space (x, y, z) brings about singularities – identified with caustics35 – where theJacobian of the transition to ray coordinates vanishes and the field diverges:

D(τ) = 0 and U ∝ 1√J

=

√D(0)D(τ)

. (1.53)

Caustics correspond to an intersection of infinitesimally close rays (da = 0). They are the envelopes ofray families and the rays are always tangent to the caustic. When one crosses the caustic, an even numberof rays is born or annihilated. This discrete appearance or disappearance gives rise to the name catastrophe,for it is a qualitative change in the number of rays.

In homogeneous space (n = const.) caustics are surfaces where one of the main (principle) radii ofcurvature36 of the initial wavefront becomes zero, i.e. they are the loci of all centers of curvature. Onecan express the divergence in terms of these main radii. For an axially symmetric system these are themeridional radius Rm (in the plane of the ray and the axis of symmetry) and the sagittal radius Rs (in theplane perpendicular to the meridional plane). We define

Rm0 ≡ QmAm0 , Rm ≡ QmAm ,

Rs0 ≡ QsAs0 , Rs ≡ QsAs ,(1.54)

with Qm and Qs the meridional and sagittal centre of curvature, respectively (figure 1.6).

da0

QmQs

Am0

As0

Am

As

da

Figure 1.6: Change of the area element along the ray from the initial element da0 to da

Then the divergence becomes37

J =n Rm Rs

n0 Rm0 Rs0. (1.55)

In the general inhomogeneous case again (n 6= const.) the Jacobian D(τ) – and thus the divergence J –changes its sign (a radius of curvature changes its sign from minus to plus, i.e. from converging to diverging)when one moves along the ray. This happens if a caustic is passed, and the corresponding ray undergoes aphase shift (phase delay) of −π/2:

1√J→ 1√

− |J | =1

i√|J | =

e− iπ2√|J | . (1.56)

In a situation where a ray is tangent to several caustics, the overall caustic phase delay, denoted as Sc, equalsthe sum of the phase shifts the ray acquired. Thus, the field of a ray can be written as

U = U0e i k ψ

√J

= U0e i k ψ + i Sc

√|J | . (1.57)

34 See e.g. Berry, M. V., Phase-space projection identities for diffraction catastrophes, J. Phys. A: Math. Gen. 13 (1980),149–169, and Kravtsov, Y. A. and Orlov, Y. I., Caustics, Catastrophes and Wave Fields, Springer Series on Wave Phenomena(Volume 15), Springer-Verlag, Second Edition (1999), p. 34 et seq.

35 From the Greek word kaustikos which means burning.36 Main radii are maximal and minimal radii possible and lie in planes that are perpendicular to each other.37 Explicitly, da0 = Rm0 Rs0 dαm dαs and da = Rm Rs dαm dαs, where dαm and dαs are the meridional and sagittal angle

elements which cancel upon division.


Up to three caustic delays can be taken into account by choosing the square root√

J properly, though, ifnecessary, by overruling the standard definition of the square root which is a branch cut along the negativereal axis38. If a ray passes more than three caustics, then

√J cannot take into account these delays anymore

and only the expression exp(i Sc)/√|J | is valid.

Let us now consider the particular case of a spherically aberrated wave – propagating in vacuum (n = 1)– on an infinitely large plane screen at z = −f with the amplitude U0 (notation as in figure 1.2). Its initialeikonal is

ψ(0) = −r −B (x21 + y2

1)2, (1.58)

where we used the notation ψ(0) ≡ ψ0. One can approximate

r =√

x21 + y2

1 + f2 ≈ f +x2

1 + y21

2 f. (1.59)

The ray coordinates on the initial surface are Cartesian: ξ = x1 and η = y1. Hence, the initial canonicalmomenta are

px =∂ψ(0)

∂x1≡ ψ(0)

x1= −x1

f− 4 B x1 (x2

1 + y21) ,

py =∂ψ(0)

∂y1≡ ψ(0)

y1= −y1

f− 4 B y1 (x2

1 + y21) , (1.60)

pz =∂ψ(0)

∂z1=

√1− p2

x − p2y .

The last equal sign follows from the eikonal equation39

p2x + p2

y + p2z = 1 . (1.61)

In vacuum the rays are straight lines (the canonical momentum p is conserved), and thus the ray equation(1.45) has the form

x = x1 + s px ,

y = y1 + s py , (1.62)z = −f + s pz ,

where s is the third ray coordinate (τ), indicating the position on the ray. It represents the distance fromthe position on the screen (x1, y1,−f) to the point of observation P = (x, y, z):

s =√

(x− x1)2 + (y − y1)

2 + (z + f)2 (1.63)

The geometrical situation in two dimensions is shown in figure 1.7.Substituting (1.60) into (1.62), we can write

x =(

1− s

f

)x1 − 4 B x1 (x2

1 + y21) s , (1.64)

y =(

1− s

f

)y1 − 4 B y1 (x2

1 + y21) s , (1.65)

Without loss of generality one may set y = 0 and thus define the meridional plane. Then the above equationsyield the points from which the rays arrive at the point (x, y=0, z) in the form of the following expressions:

x31 +

14 B

(1f− 1

s

)x1 +

x

4 B s= 0 , (1.66)

y1 = 0 . (1.67)

38 For example assume an initially converging ray in homogeneous space (both main radii of curvature are negative and Jis positive). After two caustic passings the ray is diverging and J is positive again (both radii of curvature are positive). Thenone has to choose the negative square root

√J = −|√J | because Sc = −π/2− π/2 = −π and exp(−i π) = −1.

39 The right hand side is the square of the refractive index, but we are using n = 1.


Figure 1.7: Rays corresponding to a spherically aberrated wavefront on a plane screen (vertical straight line on theleft) propagate along straight lines. Wavefront after the screen (vertical curved line on the left), z-axis (horizontalline) and focus (small dot on the right) are indicated

The cubic-like equation40 (1.66) has 3 roots, which we denote as x1,j (j = 1, 2, 3), corresponding to threegeometrical rays. We number them in such a way that x1,1 is always real (and negative), the two others, x1,2

and x1,3, are either real (and positive) or complex conjugate. If there are 3 real solutions, ordered in the way

x1,1 < x1,2 < x1,3 , (1.68)

then one is situated in the lit region, whereas in the shadow of geometrical optics there is only one real rayreaching each point. The border between the lit and the shadow region has the form of an axially symmetricthree-dimensional cuspoid41 and its vertex is situated at the focus (origin) F (figure 1.8).

(b)(a)

11

2F

P

P

F

3

Figure 1.8: (a) 3-ray region inside the cuspoid (dashed line). (b) 1-ray region outside

The overall eikonal at the point of observation is

ψj = ψ(0)j + sj , (1.69)

where sj is the distance from (x1,j , 0,−f) to (x, 0, z).The remaining step is to find an expression for the ray amplitudes which are connected with the divergence

J . For this purpose we have to calculate the determinant of the Jacobian

D(s) =∂(x, y, z)

∂(x1, y1, s). (1.70)

The quantities x, y and z are given by (1.62). Here one has to consider s as an independent ray variable andnot as a function of x1 and y1 as it was the case above, where it represented the distance to a given pointP . With the help of the eikonal equation (1.61) one obtains

D(s) =ψ

(0)x1x1ψ

(0)y1y1 − (ψ(0)

x1y1)2

pzs2 +

ψ(0)x1x1 [1− (ψ(0)

y1 )2] + ψ(0)y1y1 [1− (ψ(0)

x1 )2] + 2 ψ(0)x1 ψ

(0)y1 ψ

(0)x1y1

pzs + pz , (1.71)

40 One must keep in mind that s contains x1 due to (1.63).41 A cuspoid is the surface of revolution of a cusp which is its two-dimensional analog.


where multiple indices denote multiple partial derivatives. From (1.60):

ψ(0)x1x1

= − 1f− 4 B (3x2

1 + y21) ,

ψ(0)y1y1

= − 1f− 4 B (x2

1 + 3 y21) , (1.72)

ψ(0)x1y1

= −8 B x1 y1 .

The divergence of the j-th ray is given by the ratio of the local and initial Jacobian, Jj(s) = Dj(s)/Dj(0).Here, the index j in Dj means that the j-th stationary point (x1,j , 0) has to be inserted. Thus, the formalexpression (1.71) simplifies tremendously. The eikonal’s first derivative with respect to y1 – see (1.60) – van-ishes upon substituting y1,j = 0 (no momentum in y-direction due to symmetry, as the point of observationlies in the x,z-plane). Furthermore, the mixed derivative in (1.72) vanishes for the same reason. Hence, with

ψ(0)y1,j = ψ

(0)x1y1,j = 0 , (1.73)

and since D(0) = pz, one finds (omitting the index j)

J =ψ

(0)x1x1ψ

(0)y1y1

p2z

s2 +ψ

(0)x1x1 + ψ

(0)y1y1 p2

z

p2z

s + 1 . (1.74)

We write (1.74) as a product:

J =

(1 + s

ψ(0)x1x1

p2z

)(1 + sψ(0)

y1y1) . (1.75)

The field amplitude is proportional to the inverse square root of the divergence. Equating (1.75) to zero andsubstituting p2

z = 1− (ψ(0)x1 )2, yields the solutions

s1 = −1− (ψ(0)x1 )2

ψ(0)x1x1

, s2 = − 1

ψ(0)y1y1

. (1.76)

This means that for each ray the amplitude Uj diverges twice. On the one hand, it diverges on the cuspoid42

after propagation s1, on the other hand on the axis43 after the path s2. In the present case of a sphericallyaberrated wave, the caustic is given by the three-dimensional cuspoid and the axis up to the focus and eachray is tangent to the cuspoid. It is two rays which disappear (become complex) when moving across thecuspoid from the lit into the shadow region.

The result (1.76) could have been written down right away. Indeed, s1 corresponds to the (negative)meridional radius of curvature of the plane curve ψ(0)(x1, 0) = −f − x2

1/2 f −B x41, and s2 is the (negative)

distance from the point (x1, 0) to the z-axis, which is the radius of curvature of the sagittal plane, as a coneof rays intersects on the axis.

In the lit region – by virtue of figure 1.8a – we expect ray 1 to be shifted by ∆ϕ1 = −π/2−π/2 = −π (ittouched the cuspoid and crossed the focal line and multiple phase shifts are added), ray 2 not to be shiftedat all, i.e. ∆ϕ2 = 0 (it has neither touched the cuspoid nor the focal line yet), and ray 3 by ∆ϕ3 = −π/2 (ithas touched the cuspoid). The phase shifts of the second and third ray are automatically considered correctlydue to positive and negative Jacobian, respectively44. The first ray (∆ϕ1 = −π) must be shifted manuallyby choosing the negative square root45. Hence, we write

√J1 = −

√D1(s)D1(0)

,√

J2 = +

√D2(s)D2(0)

,√

J3 = +

√D3(s)D3(0)

. (1.77)

42 At the cuspoid the real rays 2 and 3 merge and become complex (conjugate) outside.43 On the axis – for negative values of z – there is an infinite number of rays arriving at each point, coming from a whole

circle on the aperture.44 J3 < 0 (ray 3 touched the cuspoid and Rm,3 is thus positive, whereas Rs,3 is still negative for the ray is converging to the

axis in the sagittal plane) and J2 > 0 (both Rm,2 and Rs,2 are negative). The standard definition of the square root makes√J3 purely imaginary with positive imaginary part. Since 1/

√J3 is the important quantity, ray 3 is in fact shifted by 1/i = −i

with respect to ray 2, as√

J2 is real and positive. All these statements are true for the lit (3-ray) region. In the shadow (1-ray)region, ray 2 and 3 are complex conjugate.

45 See discussion below (1.57) and the corresponding footnote.


In the lit region, one might take the absolute value of J and consider the phase shift explicitly:

1√Jj

=e i Sc,j

√|Jj |, where Sc,j =

−π for j = 1 (cuspoid and focal line)0 for j = 2

−π/2 for j = 3 (cuspoid)(1.78)

Thus, in the point of observation – due to axial symmetry, represented by cylindrical coordinates (ρ, z) –the geometrical optics solution for the spherically aberrated eikonal (1.58) is given by

U(ρ, z) = U0

m∑

j=1

e i k ψj + i Sc,j

√|Jj |, (1.79)

where the sum has to be taken over the number of rays. Outside the cuspoid m = 1, since the two complexrays are non-physical. There, Sc,1 = −π still holds, since crossing the upper branch of the cuspoid does notaffect ray 1 at all (figure 1.8b). Inside the cuspoid m = 3. Figure 1.9 shows the absolute square of expression(1.79). It reveals divergence on the cuspoid and the axis (up to the focus).

-3

-1.5

0

1.5

z0.6

0.4

0.2

0

Ρ

0

1500

3000

ÈU È2

-3

-1.5

0z

Figure 1.9: Absolute square of the geometrical optics amplitude U for the initial eikonal ψ(0) of a sphericallyaberrated wave (1.58). Parameters are k = 100, f = 10, B = 2.5×10−4 and U0 = 1. The field diverges at the cuspoidand at the axis up to the focus F = (0, 0)

Chapter 2

The Bessoid Integral

While the geometrical optics field diverges in the caustic regions, scalar diffraction theory for a paraxialspherically aberrated wave led us to a two-dimensional integral where one integration could be carried outanalytically. Correspondingly, we will define this Bessoid integral in a canonical way, compare with its one-dimensional analog, the Pearcey integral, and also derive analytic expressions in different regions, that isfar from the caustic, on the axis, and on the cuspoid. For small wavelengths the correspondence betweengeometrical and wave optics lies in the fact that the rays originate from the points of stationary phase inthe diffraction integrals. Furthermore, we investigate the numerical evaluation of the Bessoid integral andderive an ordinary differential equation, allowing its fast computation.

2.1 Definition

We introduce the one-dimensional canonical integral, henceforth denoted as Pearcey integral1:

IP (X, Z) =1√2 π

∞∫

−∞e− i

(X x1 + Z

x212 +

x414

)

dx1. (2.1)

It is closely related to the original Pearcey integral2

P (C1, C2) =

∞∫

−∞e i (C1 t + C2 t2 + t4) dt . (2.2)

One easily finds the connecting expression between (2.1) and (2.2), namely IP (X, Z) = P ∗(√

2 X,Z)/√

π,where P ∗ denotes the complex conjugate of P . Figure 2.1 shows |IP (X,Z)|2.

The integral appearing in (1.40) shall be denoted as I(R,Z). For brevity the tilde symbol in ρ1 shall beomitted in the following (ρ1 → ρ1). Furthermore, we consider an infinitely large aperture3, a → ∞. Thisyields the two-dimensional canonical analog of (2.1) which will henceforth be referred to as Bessoid integral4.Using (1.35), the Bessoid integral can be written in both Cartesian,

I(R, Z) =1

2 π

∞∫

−∞

∞∫

−∞e− i

(R x1 + Z

x21 + y2

12 +

(x21 + y2

1)2

4

)

dx1dy1, (2.3)

1 First computed by Trevor Pearcey (1919–1998): Pearcey, T, The structure of an electromagnetic field in the neighbourhoodof a cusp of a caustic, Lond. Edinb. Dubl. Phil. Mag. 37 (1946), 311–317.

2 We note that the Pearcey integral plays an important role in many short wavelength phenomena besides optical caustics.These include the elastic scattering of atoms and ions, the semiclassical theory of chemical reactions, collisions of heavy nuclearions, the asymptotic evaluation of path integrals, the propagation of acoustic, electromagnetic and water waves, scatteringfrom surfaces as well as various general features of semiclassical quantum mechanics. References to all these fields are listedin Connor, J. N. and Farrelly, D, Theory of cusped rainbows in elastic scattering: Uniform semiclassical calculations usingPearcey’s Integral, J. Chem. Phys. 75(6) (1981), 2831–2846.

3 An infinitely large aperture may seem to stand in contradiction with the previous derivation, where we postulated smallangles. But the main contribution to the integral comes from points within reasonable distances at the aperture as we will see.Besides, and more importantly, we will show that coordinate transformations allow to describe non-paraxial focusing with thehelp of this integral.

4 Kirk, N. P., Connor, J. N. L., Curtis, P. R., and Hobbs, C. A., Theory of axially symmetric cusped focusing: numericalevaluation of a Bessoid integral by an adaptive contour algorithm, J. Phys. A: Math. Gen. 33 (2000), 4797–4808.

2.1 Definition 17

-10

-5

0

5

Z

10

8

6

4

2

0

X

0

1

2

ÈIPÈ2

-10

-5

0Z

Figure 2.1: Absolute square of the Pearcey integral IP (X, Z)

and polar coordinates5,

I(R,Z) =

∞∫

0

ρ1 J0(R ρ1) e− i

(Z

ρ212 +

ρ414

)

dρ1. (2.4)

Due to axial symmetry – without loss of generality – we have set ϕ = 0 (i.e. Y = 0, X = R) in (2.3), whichmeans that the point of observation lies in the X,Z-plane6. Here x1 and y1 are dimensionless integrationvariables similar to ρ1 (i.e. x1 = 4

√4 k B x1 → x1, the same for y1). The dependence of the dimensionless

coordinates R and Z on the original coordinates ρ and z is given by (1.39). In the one-dimensional represen-tation, R has the meaning of a polar coordinate and is consequently restricted to be non-negative. Figure2.2 shows |I(R, Z)|2.

-10

-5

0

5

Z

10

8

6

4

2

0

R

0

2

4

ÈI È2

-10

-5

0Z

Figure 2.2: Absolute square of the Bessoid integral I(R, Z)

The Pearcey and the Bessoid integral correspond to so-called diffraction catastrophes. Their field distri-bution contains caustic zones where the intensity predicted by geometrical optics goes to infinity. While thePearcey integral IP , (2.1), corresponds to a cusp caustic (figure 2.3a), i.e. a one-dimensional manifold in a

5 We find it worth to mention that both the Pearcey and the Bessoid integral are conditionally convergent.6 Originally, the exponent in (2.3) would contain X x1 + Y y1 = R ρ1 cos(ϕ1 − ϕ). With ϕ = 0 this becomes X x1 = R x1 =

R ρ1 cos ϕ1.

2.1 Definition 18

two-dimensional space7, the Bessoid integral I, (2.3) and (2.4), corresponds to a cuspoid caustic, i.e. to anaxially symmetric cuspoid surface of revolution in three dimensions, as well as the caustic focal line up tothe focus F at z = 0 (figure 2.3b).

X X

Y

Z Z

(a) (b)

F F

Figure 2.3: (a) Two-dimensional cusp (Pearcey case). (b) Three-dimensional cuspoid and focal line (Bessoid case)

A caustic is denoted as stable, if it does not change its topology under small perturbations. This is thecase for the cusp catastrophe and the Pearcey integral. The Bessoid integral corresponds to a structurallyunstable caustic, because an infinitely small perturbation will destroy the radial symmetry and the axis willnot be a caustic zone any longer. It is, however, stable on the class of axially symmetric wavefronts. Ingeneral, caustics8 are classified by expressing the corresponding canonical integral as

I(R) =1

(2 π)l/2

∫e i φ(R,t) dlt (2.5)

with the generating function

φ(R, t) = φ0(t) +n∑

p=1

φp(t)Rp , (2.6)

where R = (R1, R2, ..., Rn) are the external parameters and t = (t1, t2, ..., tl) are the internal or statevariables. The number of external parameters, n, is called the codimension of the caustic, and the numberof internal variables, l, is called the corank. The number of corresponding geometrical rays is n + 1.

Continuing with the Pearcey integral9, the position of the cusp can be calculated by combining the pictureof wave and geometrical optics. Since the (large) wavenumber k appears (via X and Z) in the phase

φP (x1) = −X x1 − Zx2

1

2− x4

1

4, (2.7)

the exponent in (2.1) is very rapidly oscillating with x1 in almost the whole region of integration. As a result,the integrand’s contributions for IP will cancel themselves nearly everywhere. Figure 2.4 shows the real partof exp(i φP ).

However, there are regions which give significant contribution to the integral. These regions – correspond-ing to the rays of geometrical optics – are the neighbourhoods of the points of stationary phase10, i.e. thosepoints where the first derivative of the phase (2.7) vanishes:

X + Z x1 + x31 = 0 . (2.8)

This is a cubic equation in normal form. It has 3 solutions for x1 which are given by Cardan’s11 formula:

ξ1 = S + T , S ≡ 3

√−X

2+√

D ,

ξ2 = −12

(S + T ) + i√

32

(S − T ) , where T ≡ 3

√−X

2−√D ,

ξ3 = −12

(S + T )− i√

32

(S − T ) , D ≡ X2

4+

Z3

27.

(2.9)

7 Normally, the term cusp refers to the vertex of the cusp itself. But we use it to denote the two branches in order to keepconsistency with the term cuspoid, that is the surface of revolution of a cusp.

8 Strictly speaking, this is true for so-called simple (null-modal) caustics.9 Here n = 2, R = (X, Z), l = 1, t = x1, φ0 = −x4

1/4, φ1 = −x1 and φ2 = −x21/2.

10 The method of stationary phase was first introduced by Lord Kelvin, William Thomson (1824–1907).11 Girolamo Cardano (1501–1576).

2.1 Definition 19

-4 -2 2 4x1

-1

-0.5

0.5

1

cos ΦP

Figure 2.4: Real part of the exponent in the Pearcey integral for X = 5 and Z = −12. The integrand is highlyoscillatory except in regions around the points of stationary phase (−3.656, 0.423, 3.233), given by (2.8)

For real coefficients (X and Z) we can distinguish between the following cases:

D > 0 one real and two complex conjugated roots,D < 0 three different real roots,D = 0, X 6= 0 two real roots out of which one is twofold,D = 0, X = 0 one threefold real root.

(2.10)

In the case of a real and positive discriminant D ≥ 0 the third roots in (2.9) are clearly defined, namely bytaking the real solution. For a negative discriminant D < 0 (complex radicands) they have to be determinedin such a way that S T = −Z/3. Thus, for computational purposes, it is more convenient to express thesolutions of (2.8) in terms of trigonometric functions, summarized in the following table12:

Z < 0, D ≤ 0 Z < 0, D > 0 Z > 0

P ≡ sgn(X)√|Z| /3 β ≡ 1

3arccos

X

2 P 3β ≡ 1

3arcosh

X

2 P 3β ≡ 1

3arsinh

X

2 P 3

ξ1(X, Z) −2 P cosβ −2 P coshβ −2 P sinhβ

ξ2,3(X, Z) 2 P cos(β ± π

3

)2 P

(cosh β ± i

√3 sinh β

)2 P

(sinhβ ± i

√3 cosh β

)

(2.11)

For X = 0 the solutions of (2.8) are 0, ±√−Z. If Z = 0, one has the three solutions 3√−X. Therefore, ξ1 is

always real (and negative) and it always represents a real ray of geometrical optics, whereas ξ2 and ξ3 areonly real (and positive) if 27 X2 + 4 Z3 < 0, i.e. inside the cusp (lit region). Outside the cusp (geometricalshadow), i.e. for 27X2 + 4 Z3 > 0, they are complex (conjugate) and cannot be used as points of stationaryphase, because the integration variable x1 is a real quantity. The complex solutions correspond to complexrays.

The caustic is the set of points at which two or more rays merge13, i.e. ξ2 = ξ3 ↔ D = 0 for the cusp.The general condition for an arbitrary caustic is the simultaneous vanishing of all first derivatives (conditionfor the existence of a ray) and one vanishing second derivative (condition for merging rays) of the phase.Both approaches yield that the cusp is given by all (X, Z≤0) which lie on the semicubic parabola

27 X2 + 4 Z3 = 0 . (2.12)

Summing up the canonical Pearcey integral (cusp catastrophe):

IP (X, Z) =1√2 π

∞∫

−∞e− i

(X x1 + Z

x212 +

x414

)

dx1,

12 Taken from Bronstein, I. N. and Semendjajew, K. A., Teubner-Taschenbuch der Mathematik (Editor: Zeidler, E.), B. G.Teubner Verlag (1996), p. 649.

13 At the origin, X = Z = 0, all three rays merge.

2.1 Definition 20

27 X2 + 4 Z3 = 0 (1 + 2 merging rays) caustic,27 X2 + 4 Z3 > 0 1 ray (shadow),27 X2 + 4 Z3 < 0 3 rays (lit).

The derivation for the cuspoid caustic14 is similar but a little more complicated. From the phase in (2.3),

φ(x1, y1) = −R x1 − Zx2

1 + y21

2− (x2

1 + y21)2

4, (2.13)

we find the stationary phase condition:

∂φ(x1, y1)∂x1

≡ φ10(x1, y1) = −R− Z x1 − x1(x21 + y2

1) = 0 , (2.14)

∂φ(x1, y1)∂y1

≡ φ01(x1, y1) = −Z y1 − y1(x21 + y2

1) = 0 . (2.15)

This system of equations has 2 sets of solutions. The first is

R + Z x1 + x31 = 0 , (2.16)

y1 = 0 . (2.17)

This is in agreement with the symmetry and the cubic equation is the same as (2.8) with X = R. Therefore,the preceding argumentation holds and the cuspoid surface is given by the same expression (2.12) as thecusp but also includes rotational symmetry15. Points of stationary phase are (ξj , 0), where the ξj are againthose from (2.9) or (2.11). On the other hand, the second set of solutions is

R = 0 , (2.18)

x21 + y2

1 = −Z . (2.19)

This means that to any point on the axis and for negative values of Z, there arrives the axial ray and acontinuum of rays. The latter form a circle of radius −Z on the aperture and a corresponding cone of sagittalrays converging to the axis. Thus, for Z < 0, the axis is a part of the caustic.

Summing up the canonical Bessoid integral (cuspoid catastrophe):

I(R,Z) =1

2 π

∞∫

−∞

∞∫

−∞e− i

(R x1 + Z

x21 + y2

12 +

(x21 + y2

1)2

4

)

dx1dy1

=

∞∫

0

ρ1 J0(R ρ1) e− i

(Z

ρ212 +

ρ414

)

dρ1,

27 R2 + 4 Z3 = 0 (1 + 2 merging rays)R = 0, Z < 0 (1 + ∞ rays)

}caustic,

27 R2 + 4 Z3 > 0 1 ray (shadow),27 R2 + 4 Z3 < 0 and R 6= 0 3 rays (lit).

Later we will also need the first partial derivatives of I(R,Z) with respect to R and Z. We may differ-entiate under the integral sign16 and from (2.4) we obtain

IR(R, Z) ≡ ∂I(R, Z)∂R

= −∞∫

0

ρ21 J1(R ρ1) e

− i

(Z

ρ212 +

ρ414

)

dρ1, (2.20)

where the derivative of the zero-order Bessel function was used:

ddt

J0(t) = J−1(t) = −J1(t) . (2.21)

Expression (2.20) vanishes on the axis due to J1(0) = 0 (figure 2.5).14 Here n = 2, R = (R, Z), l = 2, t = (x1, y1), φ0 = −(x2

1 + y21)2/4, φ1 = −x1 and φ2 = −(x2

1 + y21)/2.

15 This equivalence justifies that henceforth we may sometimes use the term cusp, although, strictly speaking, the cuspoid ismeant.

16 This is allowed, if the partial derivative of the integrand exists and is a continuous function in this variable.

2.2 Off the Caustic 21

-10

-5

0

5

Z

10

8

6

4

2

0

R

0

5

10

ÈIRÈ2

-10

-5

0Z

Figure 2.5: Absolute square of IR(R, Z), the derivative of the Bessoid integral with respect to R

In a similar way one finds

IZ(R,Z) ≡ ∂I(R,Z)∂Z

= − i2

∞∫

0

ρ31 J0(R ρ1) e

− i

(Z

ρ212 +

ρ414

)

dρ1, (2.22)

which is shown in figure 2.6.

-10

-5

0

5

Z

10

8

6

4

2

0

R

0

40

80

ÈIZ È2

-10

-5

0Z

Figure 2.6: Absolute square of IZ(R, Z), the derivative of the Bessoid integral with respect to Z

2.2 Off the Caustic

In the following sections we shall develop asymptotic approximations for the Bessoid integral in variousregions. First, we want to consider I(R,Z) for arguments far away from the caustic (i.e. in regions faraway from the cusp as well as far from the axis). This is extremely important, as it is exactly the regioncorresponding to the range of applicability of geometrical optics. However, it is better to start without thisrestriction and introduce it afterwards.

From (2.14) and (2.15) we found the points of stationary phase (ξj , ηj = 0) with j = 1, . . . ,m, wherem = 1 or 3 is the number of rays and the ξj are determined by (2.9). On the axis, for R = 0 and Z < 0, wehave an infinite number of rays and the stationary points are represented by one axial ray and an infiniteamount of rays, starting from the aperture points (ξj , ηj) where ξ2

j + η2j = −Z.


The phase (2.13) and its second derivatives read

φ(x1, y1) = −R x1 − Zx2

1 + y21

2− (x2

1 + y21)2

4, (2.23)

∂2φ(x1, y1)∂x2

1

≡ φ20(x1, y1) = −Z − 3 x21 − y2

1 , (2.24)

∂2φ(x1, y1)∂y2

1

≡ φ02(x1, y1) = −Z − x21 − 3 y2

1 , (2.25)

∂2φ(x1, y1)∂x1∂y1

≡ φ11(x1, y1) = −2 x1 y1 . (2.26)

In the following, the index j, if not written in brackets or specified differently, shall denote that j-th stationarypoint has to be inserted as argument, e.g. φj ≡ φ(ξj , ηj). A two-dimensional second-order Taylor expansionaround the j-th stationary point yields

φ(j)(x1, y1) = φj +12

φ20,j (x1 − ξj)2 +12

φ02,j (y1 − ηj)2 + φ11,j (x1 − ξj) (y1 − ηj) . (2.27)

The first derivatives vanish because of the stationary phase condition. The integral I(R,Z) can be approxi-mated by the contributions of the stationary points17 due to the method of stationary phase18:

I(R,Z) =1

2 π

∞∫

−∞

∞∫

−∞e i φ dx1dy1

≈ 12 π

m∑

j=1

∞∫

−∞

∞∫

−∞e i φ(j) dx1dy1, (2.28)

with m the number of geometrical optics rays. Introducing19 the Hesse20 matrix

H(x1, y1) ≡(

φ20 φ11

φ11 φ02

)(2.29)

and the vector21

uj ≡(

x1 − ξj

y1 − ηj

), (2.30)

relation (2.27) can be rewritten as:

φ(j)(uj) = φj +12

uTj Hj uj . (2.31)

Thus, (2.28) becomes

I(R, Z) ≈ 12 π

m∑

j=1

e i φj

∞∫

−∞e

i2 uT

j Hj uj duj . (2.32)

The quadratic form in the exponent can be simplified due to the fact that H is real and symmetric. We omitthe index j in the next few lines. Be U the unitary matrix comprised of the orthonormalized eigenvectors ofH which brings the latter into diagonal form. Then, by means of the transformation

u = Uv with v ≡ (v1, v2)T , (2.33)

it follows22

uT Hu = vT U−1 HUv = λ1 v21 + λ2 v2

2 , (2.34)17 If the second derivatives in (2.27) do not vanish, which means well separated rays, (2.28) becomes exact in the limit of

the wavenumber k going to infinity. Formally, this implies that R and Z become infinitely large, but the stationary phaseapproximation works reasonably well even for values of R and Z near 1.

18 See van Kampen, N. G., An asymptotic treatment of diffraction problems, Physica 14 (1949), 575–589, and Focke, J.,Asymptotische Entwicklungen mittels der Methode der stationaren Phase, Ber. Sachs. Akad. 101(3) (1954), 1–48.

19 The following considerations can easily be applied to arbitrary multi-dimensional situations.20 Ludwig Otto Hesse (1811–1874).21 Only in the next lines we are using column and row vectors (and the transpose sign T), whereas otherwise we will not

distinguish between them for notation reasons.22 This normal form is unique due to Sylvester’s inertia law. James Joseph Sylvester (1814–1897).


where λ1 and λ2 are the eigenvalues of H. With23

∞∫

−∞e i c t2 dt =

√π

ci = e i

π4 sgn c

√π

|c| , (2.35)

and writing all indices, equation (2.32) takes on the form

I(R, Z) ≈ 12 π

m∑

j=1

e i φj

√2 π iλ1,j

√2 π iλ2,j

≈m∑

j=1

e i φj + iπ4 (sgnλ1,j + sgnλ2,j)

√|λ1,j λ2,j |. (2.36)

Since the transformation was unitary, the product of the eigenvectors, λ1 λ2, is the determinant of the HessianH itself. We denote the signature24 – the difference of the number of positive and negative eigenvalues λl –of an n× n matrix M with sign M =

∑nl=1 sgnλl. Thus, the Bessoid integral can be approximated by

I(R, Z) ≈m∑

j=1

e i φj + iπ4 signHj

√|detHj |(2.37)

with the determinant in the denominator

detH(x1, y1) = φ20 φ02 − φ211

= (x21 + y2

1 + Z) (3 x21 + 3 y2

1 + Z) , (2.38)

The determinant vanishes on the caustic, i.e. (a) on the cuspoid and (b) on the axis up to the focus. To show(a) one has to insert the (merging) ray(s) (ξ2,3 =− 3

√R/2, η2,3 =0) from (2.9) into (2.38), using the condition

for the cusp (2.12), i.e. 3 x21 = −Z and y1 = 0. To prove (b) it is enough to insert (2.19) into (2.38).

Therefore, the method of stationary phase fails (diverges) on the caustic as we expected from geometricaloptics. This is the case, because two points of stationary phase coalesce (merge) and form a saddle point25.By virtue of (2.35) we see that a vanishing second derivative (c = 0) leads to divergence.

However, for points (R, Z) in regular regions, expression (2.37) simplifies. The only rays (m = 1 outsidethe cusp for 27 R2 + 4 Z3 > 0, or m = 3 inside the cusp for 27 R2 + 4 Z3 < 0) are the (ξj , 0) from (2.9) or(2.11). Due to the vanishing mixed derivative, φ11(ξj , 0) = 0, the Hesse matrix H is already in diagonal formand the second derivatives are the eigenvalues:

detHj = φ20,j φ02,j

= Z2 + 4 ξ2j Z + 3 ξ4

j , (2.39)

signHj = sgn φ20,j + sgn φ02,j

= sgn(−Z − 3 ξ2j ) + sgn(−Z − ξ2

j ) . (2.40)

The signature of the Hessian (2.29) is always an even number ∈ {−2, 0,+2}. It corresponds to the phaseshift of the rays as they touch a caustic and as the determinant changes its sign when going through zero.In general, the phase of a ray field changes by −π/2 upon touching a non-singular caustic (by −π upontouching a three-dimensional degenerate focus), where multiple phase shifts are just added.

Recalling figure 1.8a, we know the rays’ phase shifts, namely ∆ϕ1 = −π, ∆ϕ2 = 0, and ∆ϕ3 = −π/2.This would request the signatures −4, 0 and −2 for the rays 1, 2 and 3, respectively. However, we recall thatthe geometrical optics approach is not represented by the Bessoid integral (2.4) alone but by expression (1.40)for the field, in which an additional factor −i = exp(−i π/2) is present. This recovers the full correspondencewith the caustic phase shifts for all rays:

signHj =

−2 for j = 1

2 for j = 20 for j = 3

and − i e iπ4 signHj =

e− i π for j = 11 for j = 2

e− i π/2 for j = 3(2.41)

23 To prove this statement, it is necessary to introduce an infinitely small imaginary part for c, i.e. c → c + i ε on the lefthand side.

24 The standard definition for the signature is just the number of positive terms in a quadratic form.25 The term saddle point refers to the fact that – considering the phase as a function of x1 – both the first and the second

derivative vanish.

2.3 On and Near the Axis 24

2.3 On and Near the Axis

In this section we find an approximation near the axis for the Bessoid integral (2.4),

I(R,Z) =

∞∫

0

ρ1 J0(R ρ1) e− i

(Z

ρ212 +

ρ414

)

dρ1. (2.42)

After the substitution w ≡ ρ21 one finds

I(R, Z) =12

∞∫

0

J0(R√

w) e− i

(Z

w2 +

w2

4

)

dw . (2.43)

Again the method of stationary phase is applied. This time the argumentation is that near the axis (smallR) the Bessel function is slowly varying compared with the exponent. The integral will have significantcontribution only from the region in which the exponent’s phase is stationary26. Be ϕ the phase of theexponent in (2.43). Then the condition is

∂ϕ

∂w= −1

2(Z + w) = 0 → w = −Z . (2.44)

In the case of Z > 0 the point w = 0 should be taken as a stationary edge point of the integration27.The exponent will vary rapidly everywhere such that the first oscillation (which is the slowest) makesthe strongest contribution. But if there exists a real stationary point, i.e. if Z < 0, we neglect the edgecontribution. Consequently, the single stationary point is given by

ζ = max(−Z, 0) . (2.45)

In a lowest order approximation the Bessel function is considered as constant near the stationary point andcan be pulled out of the integral:

I(R, Z) ≈ 12

J0(R√

ζ)

∞∫

0

e− i

(Z

w2 +

w2

4

)

dw . (2.46)

The phase can be written as a complete quadratic form. With v ≡ (w + Z)/2:

I(R,Z) ≈ J0(R√

ζ) e iZ2

4

∞∫

Z/2

e− i v2dv . (2.47)

This integral is the difference of an integral from 0 to ∞ and an integral from 0 to Z/2. The first can besolved analytically similar to (2.35), the latter can be fit into the Fresnel cosine (C) and sine (S) functionor error function (erf), respectively,

∞∫

0

e− i v2dv =

12

√π

i=√

π

2e− i

π4 , (2.48)

Z/2∫

0

e− i v2dv =

√π

2

[C

(Z√2 π

)− i S

(Z√2 π

)]=√

π

2e− i

π4 erf

(Z

2e i

π4

), (2.49)

where28

C(u) ≡u∫

0

cos(

π t2

2

)dt , S(u) ≡

u∫

0

sin(

π t2

2

)dt , erf(u) ≡ 2√

π

u∫

0

e−t2dt . (2.50)

26 This point of stationary phase cannot be thought as a ray from geometrical optics, because a part of the wave’s phase isin the argument of the Bessel function.

27 In general, the first derivative of the phase does not vanish for edge points of the integration region. This is why they arenamed stationary points of second kind.

28 See e.g. http://mathworld.wolfram.com/FresnelIntegrals.html and http://mathworld.wolfram.com/Erf.html.

2.3 On and Near the Axis 25

Introducing the complementary error function

erfc(u) ≡ 1− erf(u) , (2.51)

a simple expression for the Bessoid integral (2.42) near the axis and for Z < 0 is

I(R, Z) ≈√

π

2J0(R

√−Z) e iZ2−π

4 erfc(

Z

2e i

π4

). (2.52)

In fact, this includes the exact solution on the axis R = 0 as a special case as well as the approximatesolution for R 6= 0 and Z > 0 due to equation (2.45). In both cases the Bessel function becomes J0(0) = 1and consequently (2.52) reads29,

I(R, Z) ≈√

π

2e i

Z2−π4 erfc

(Z

2e i

π4

). (2.53)

The distribution along the axis is shown in figure 2.7.

-10 -8 -6 -4 -2 2 4Z

1

2

3

4

ÈIH0, Z LÈ2

Figure 2.7: Absolute square of I(0, Z), the Bessoid integral on the axis

The maximum of intensity lies at Zm ≈ −3.051 and its value is |I(0, Zm)|2 ≈ 4.305. The width of thefocal line caustic is determined by the argument of the Bessel function, namely R

√−Z. One may define thecaustic width Rw ≡ w0/

√−Z according to the first zero of the Bessel function, w0:

J0(w0) = 0 , w0 ≈ 2.405 . (2.54)

Figure 2.8 compares the exact Bessoid integral with its approximation, I(R,Z) ≈ I(0, Z)J0(R√−Z), for

fixed values of Z, showing the good agreement up to the first minimum and the fact that the caustic widthon the axis becomes narrower towards the aperture.

0.5 1 1.5 2 2.5 3R

0.5

1

1.5

2

2.5

3

ÈIHR,-4LÈ2

0.5 1 1.5 2 2.5 3R

0.5

1

1.5

2

2.5

3

ÈIHR,-9LÈ2

Figure 2.8: Exact Bessoid integral (solid line) and its approximation I(0, Z) J0(R√−Z) (dashed line) at Z = −4

(left) and Z = −9 (right) for moderate values of R

29 If R = 0, one may replace ”≈” by ”=”.

2.4 At the Cuspoid 26

2.4 At the Cuspoid

For the Bessoid integral I(R, Z) we have found an expression for the regular regions off the caustic, (2.37)in combination with (2.39) and (2.40), and an equation valid near the axis, (2.52). We want to complete thepicture by a formula that approximates the integral near the cusp but not near the degenerate focus. Werecall the stationary phase expression (2.37),

I(R, Z) ≈m∑

j=1

e i φj + iπ4 signHj

√|detHj |(2.55)

with m = 1 or 3 and detHj = φ20,j φ02,j . It diverges at the cusp due to the fact that

φ20(x1, y1) = −Z − 3 x21 − y2

1 (2.56)

vanishes for (x1 = ξ2,3 = − 3√

R/2, y1 = η2,3 = 0) and 27 R2 = −4 Z3. The reason for this is that the twostationary points (ξ2 and ξ3) coalesce and form a saddle point. But since the second derivative with respectto y1,

φ02(x1, y1) = −Z − x21 − 3 y2

1 , (2.57)

does not become zero, we will have to modify the Taylor expansion (2.27) only with respect to x1. The firstray, (ξ1, 0), is regular and can be taken from (2.9). The second and third ray, (ξ2, 0) and (ξ3, 0), are notseparated anymore and the standard stationary phase expression (2.55) becomes singular. It is consistentto write just one Taylor expansion for these two merging rays, as they form one stationary region of phasetogether. We take advantage of the vanishing second derivative at the cusp and consider this point, henceforthdenoted as ξ0, as the centre of slowly varying phase, since it lies in between the two merging stationary points.Equating (2.56) to zero and using y1 = 0 we find30

ξ0(R, Z) =

√−Z

3. (2.58)

For negative Z the point ξ0 is real even outside the cusp caustic. There it is an inflection point and itstill represents the point of slowest change in the phase. But in general ξ0 is not a point of stationary phase,because the first derivative does not vanish. Only exactly on the cusp it is a real ray with x1-coordinateξ0 = ξ2,3 = −ξ1/2.

The contribution of the first ray in (2.55) can be kept unchanged and we denote it as Ij=1. The remainingproblem is to find an expression for j = 0 which replaces the contribution of j = 2, 3. From (2.15) we seethat the first derivative with respect to y1 is still zero for the point (ξ0, 0). On the other hand, it is necessaryto expand up to the third order in x1. With u ≡ x1− ξ0, v ≡ y1, and subscript 0 denoting the correspondingargument, i.e. (ξ0, 0), one obtains31

φ(0)(u, v) = φ0 + φ10,0 u +12

φ02,0 v2 +16

φ30,0 u3, (2.59)

where

φ(x1, 0) = −R x1 − Zx2

1

2− x4

1

4, (2.60)

φ10(x1, 0) = −R− Z x1 − x31 ,

φ30(x1, 0) = −6 x1 , φ02(x1, 0) = −Z − x21 .

(2.61)

Then

I(R, Z) ≈ Ij=1 +1

2 πe i φ0

∞∫

−∞

∞∫

−∞e i

(φ10,0 u +

12 φ02,0 v2 +

16 φ30,0 u3

)dudv . (2.62)

Thus, after performing the integration over v:

I(R,Z) ≈ Ij=1 +1√

2 π |φ02,0|e i φ0 + i

π4 sgn φ02,0

∞∫

−∞e i

(φ10,0 u +

16 φ30,0 u3

)du . (2.63)

30 If one does not pay regard to the radial character of R and allows it to be negative, then ξ0 must have the sign of R,because the rays 2 and 3 – forming the cusp – lie at the same side as the point of observation (see figure 1.8).

31 Only the non-zero terms are written.

2.5 Numerical Evaluation 27

In order to determine the remaining integral in (2.63), henceforth denoted by Ic, we substitute w3 ≡ φ30,0 u3/2and obtain

Ic ≡∞∫

−∞e i

(φ10,0 u +

16 φ30,0 u3

)du = 3

√2

φ30,0

w(∞)∫

w(−∞)

ei

(φ10,0

3√

2φ30,0

w +13 w3

)

dw , (2.64)

where the signs of the integral borders depend on the sign of φ30,0. But so does the cubic root in frontof the integral. By taking the absolute value of φ30,0 in this factor and when using the abbreviation χ0 ≡φ10,0

3√

2/φ30,0, one can write

Ic = 3

√2

|φ30,0|

∞∫

−∞e i

(χ0 w +

13 w3

)dw . (2.65)

Due to the integral representation of the Airy32 function33,

Ai(s) =1

2 π

∞∫

−∞e i

(s t +

13 t3

)dt , (2.66)

we obtain

Ic = 2 π 3

√2

|φ30,0| Ai(χ0) . (2.67)

Thus, finally, in a neighbourhood of the cuspoid, the following approximation holds:

I(R, Z) ≈ C0 Ai(χ0) e i ϕ0 + C1 e i ϕ1 , (2.68)

where

C0 ≡√

2 π

|φ02,0|3

√2

|φ30,0| , ϕ0 ≡ φ0 +π

4sgnφ02,0 , χ0 ≡ φ10,0

3

√2

φ30,0,

C1 ≡ 1√|φ20,1 φ02,1|, ϕ1 ≡ φ1 +

π

4(sgnφ20,1 + sgn φ02,1) .

(2.69)

The phase and its derivatives can be found in (2.60) and (2.61), the stationary points from (2.58) and(2.9). Expression (2.68) does not show any singularities as long as one does not approach the axis. Thisis in full agreement with the general theory34, as Airy’s function is a general uniform asymptotic near anon-degenerate caustic where 2 real rays disappear.

Note that this equation is valid only in those regions (R, Z) where there are two regions of stationaryphase, i.e. near the cusp. One is caused by ray 1 (ξ1, 0), for which the standard method works, using secondderivatives only. The other region arises from two coalescing stationary points (ξ2, 0) and (ξ3, 0), which forma saddle point and one slowly varying region of phase. There one constructs an artificial ray (ξ0, 0), givenby the position of the vanishing second derivative with respect to x1. One considers the second derivativefor y1 and the first and third derivative for x1. The former leads to an exponent, the latter to Airy function,showing the field’s oscillatory behaviour in the lit region and its exponential decay in the geometrical shadow.

2.5 Numerical Evaluation

We want to close this chapter with considerations on the numerical evaluation of the Bessoid integral. Thefast numerical computation of (2.4),

I(R, Z) =

∞∫

0

ρ1 J0(R ρ1) e− i

(Z

ρ212 +

ρ414

)

dρ1 ≡∞∫

0

fρ1 dρ1, (2.70)

is by no means trivial, since the integrand fρ1 is highly oscillatory for large ρ1, where the amplitude of theoscillation is proportional to35 √ρ1 (see figure 2.9).

32 George Biddell Airy (1801–1892).33 See e.g. http://mathworld.wolfram.com/AiryFunctions.html.34 Kravtsov, Y. A. and Orlov, Y. I., Caustics, Catastrophes and Wave Fields, Springer Series on Wave Phenomena (Volume

15), Springer-Verlag, Second Edition (1999), p. 21 and p. 77 et seq.35 The asymptotic form of the Bessel function for large arguments (t À 1) is J0(t) =

√2/π t cos(t− π/4).


1 2 3 4 5Ρ1

-1

-0.5

0.5

1

Re fΡ1

Figure 2.9: Real part of the integrand in (2.70) for X = 2, Z = −4

The first approach is to integrate up to a certain border, say a(R, Z), given in such a way that a liesbeyond all slowly varying regions (a is larger than all stationary points) but as small as possible in order notto come too deep into the region of fast oscillation. Then one can correct the missing part (from a to ∞) bythe method of stationary (edge) point of second kind.

However, a more elegant and faster way is to use the main theorem of complex functions theory due toCauchy36. In one formulation it states that for a holomorphic function f of a complex variable z the integralsover all closed Jordan37 curves C vanish: ∮

C

f dz = 0 . (2.71)

Let us consider figure 2.10.

C1

C2

C3

Im r1

Re r1

a

J

Figure 2.10: Paths in the complex plane. Notation

The limit of integration goes to infinity (a → ∞). Since the integrand fρ1 of the Bessoid integral (2.70)is holomorphic38, we can state

∫

C1

fρ1 dρ1 +∫

C2

fρ1 dρ1 +∫

C3

fρ1 dρ1 = 0 , (2.72)

where the first integral along the real line C1 is exactly I(R, Z). The angle ϑ will be chosen below in sucha way that the integrand falls of exponentially39. Hence, the second integral along the arc C2 vanishes.Therefore, with complex ρ1,

I(R, Z) = −∫

C3

fρ1 dρ1 =

∞ eiϑ∫

0

ρ1 J0(R ρ1) e− i

(Z

ρ212 +

ρ414

)

dρ1. (2.73)

Now we definet ≡ ρ1 e− i ϑ, (2.74)

36 Augustin Luis Cauchy (1789–1857).37 Marie Ennemond Camille Jordan (1838–1922).38 The integrand fρ1 is a product of three holomorphic functions (in fact they are even entire functions on the whole complex

plane), i.e. J0, exp and ρ1 itself, and the product of holomorphic functions is holomorphic again.39 We shall add the comment that it is also possible not to integrate along a straight line in the complex plane but to distort

the path of integration in such a way that the integrand falls of as fast as possible. This is denoted as the method of steepestdescent. In this case, the path of integration is a function of the stationary points and thus of the coordinates of the Bessoidintegral. See Kirk, N. P. et al., Theory of axially symmetric cusped focusing: numerical evaluation of a Bessoid integral by anadaptive contour algorithm, J. Phys. A: Math. Gen. 33 (2000), 4797–4808.


and obtain upon substitution

I(R, Z) =

∞∫

0

t e 2 i ϑ J0(R t e i ϑ) e− i e 2 i ϑ

(Z t2

2

)− i e 4 i ϑ

(t4

4

)

dt ≡∞∫

0

ft dt. (2.75)

As we want the integrand ft to fall off exponentially, the factor in front of the highest power (t4) must havea negative real part:

Re(− i e 4 i ϑ) = sin 4 ϑ < 0 . (2.76)

This leads to the alternative restrictions

π/4 < ϑ < π/2 , 3 π/4 < ϑ < π ,−3 π/4 < ϑ < −π/2 , −π/4 < ϑ < 0 .

(2.77)

However, there was the additional condition that the integral along the arc C2 vanishes. If one used e.g.π/4 < ϑ < π/2, then C2 (which lies at ∞) would also extend over the whole region from 0 to π/4, wherethe integrand exponentially grows with ρ1. Consequently, ϑ is bound to fulfil40

−π/4 < ϑ < 0 or 3 π/4 < ϑ < π . (2.78)

The solution −π/4 < ϑ < 0 is more natural and justifies the negative angle in figure 2.10. Experimentally,ϑ = −π/16 is a good choice (see figure 2.11).

1 2 3 4 5t

-0.5

0.5

1

1.5

Re ft

Figure 2.11: Real part of the integrand in (2.75) for X = 2, Z = −4 and ϑ = −π/16

But the computationally fastest is yet another technique. We first illustrate this method for the Pearceyintegral41 and afterwards generalize it to the Bessoid case. All in all, this is simpler and more transparent.

We therefore consider the Pearcey integral

IP (X, Z) =1√2 π

∞∫

−∞e i φP dx1, where φP = −X x1 − Z

x21

2− x4

1

4. (2.79)

The derivatives of the integrand read

∂

∂Xe i φP = −ix1 e i φP ,

∂

∂Ze i φP = −i

x21

2e i φP ,

∂2

∂X2e i φP = −x2

1 e i φP ,

∂3

∂X3e i φP = i x3

1 e i φP .

(2.80)

This leads to the partial differential equation

2 i∂IP

∂Z+

∂2IP

∂X2= 0 , (2.81)

40 The first remaining possibility is obvious, the second is valid due to the fact that it also has the real axis (although negative)as border.

41 This is done in a very general way in Connor, J. N. and Curtis, P. R., Differential equations for the cuspoid canonicalintegrals, J. Math. Phys. 25(10) (1984), 2895–2902. Here the term cuspoid stands for the whole family of canonical catastropheintegrals with corank l = 1, beginning with fold (codimension n = 1), cusp (n = 2), swallowtail (n = 3) and butterfly (n = 4)but does not mean the surface of revolution of a cusp (cuspoid, l = n = 2 ) as we have introduced it.


which is the paraxial Helmholtz equation. Another differential equation can be found by integrating thedifferentiated integrand of (2.79) with respect to x1:

∞∫

−∞

∂

∂x1

e− i

(X x1 + Z

x212 +

x414

) dx1 =

∞∫

−∞−i (X + Z x1 + x3

1) e− i

(X x1 + Z

x212 +

x414

)

dx1 (2.82)

=

e− i

(X x1 + Z

x212 +

x414

)∞

−∞

= 0 . (2.83)

The right hand side of equation (2.82) is just the result of differentiation. The second line represents thefundamental principle of differential and integral calculus due to Newton42 and Leibniz43, which states thatthe integral of the derivative of a function is the difference of the function values taken at the borders. Thevery last equality can be understood if one assumes an infinitely small imaginary part in front of the fourthorder term in the exponent, x4

1 → (1− i ε) x41, ε > 0. Thus, by virtue of (2.82) and (2.80), we can construct

the differential equation∂3IP

∂X3− Z

∂IP

∂X+ i X IP = 0 . (2.84)

Generalizing the one-dimensional cusp problem to the two-dimensional Bessoid cuspoid case, we haveto consider that the coordinates X and Z in IP (X,Z) are Cartesian, whereas R and Z in I(R, Z) arecylindrical44. Intuitively, if we want to transform the differential equations for IP (X,Z) into equations forI(R, Z), we have to use for the differential operators

∂IP

∂X→ ∂I

∂R,

∂IP

∂Z→ ∂I

∂Z,

∂2IP

∂X2→ 1

R

∂

∂R

(R

∂I

∂R

)=

∂2I

∂R2+

1R

∂I

∂R,

∂3IP

∂X3→ ∂

∂R

[1R

∂

∂R

(R

∂I

∂R

)]=

∂3I

∂R3+

1R

∂2I

∂R2− 1

R2

∂I

∂R.

(2.85)

In analogy to (2.81) and (2.84) we write the differential equations for the Bessoid integral:

2 i∂I

∂Z+

∂2I

∂R2+

1R

∂I

∂R= 0 , (2.86)

∂3I

∂R3+

1R

∂2I

∂R2−

(1

R2+ Z

)∂I

∂R+ i R I = 0 . (2.87)

Since the second equation contains only derivatives with respect to R, the Z-coordinate is just a parameter.Introducing the radial Laplacian45 in polar coordinates J ≡ ∆RI, and denoting partial derivatives withindices, we can write (2.87) also as a set of two ordinary differential equations46:

IRR +1R

IR = J ,

JR − Z IR + i R I = 0 .(2.88)

The last step for a numerical evaluation is to find the three necessary boundary conditions. From equation(2.53) the function values on the axis are known:

I(0, Z) =√

π

2e i

Z2−π4 erfc

(Z

2e i

π4

). (2.89)

Furthermore, due to symmetry:IR(0, Z) = 0 . (2.90)

42 Sir Isaac Newton (1643–1727).43 Gottfried Wilhelm von Leibniz (1646–1716).44 An angular coordinate does not appear due to axial symmetry.45 Pierre Simon Laplace (1749–1827).46 In the present work all plots of the Bessoid and its derivatives contain 101×101 mesh points, computed with the software

package Mathematica 5 (Wolfram Research). Direct numerical integration of (2.70) takes more than one hour on a modernpersonal computer. Integration in the complex plane according to (2.75) decreases the amount of time by approximately a factorof 3, depending on the choice of the angle. Solving the ordinary differential equations (2.88), however, lasts only a few seconds.


Finally, by virtue of (2.86):

J(0, Z) = −2 i IZ(0, Z) =√

π Z

2e i

Z2−π4 erfc

(Z

2e i

π4

)+ i

= Z I(0, Z) + i , (2.91)

where we noticed that on the axis the linear relationship holds:

i Z I(0, Z)− 2 IZ(0, Z) = 1 . (2.92)

In the derivation of (2.86) and (2.87) we used an intuitive correspondence between the Pearcey and theBessoid case. We shall briefly explain the strict rigorous derivation. The first equation (2.86) is indeed foundin direct analogy to the Pearcey case, when looking at the derivatives of the Bessoid integrand and using arecurrence relation for consecutive neighbours of Bessel functions:

J1(t) = tJ0(t) + J2(t)

2. (2.93)

The second equation (2.87) is again obtained with the help of the Newton-Leibniz formula. However, onehas to use not the Bessoid integrand itself, but change the order of the Bessel function and go from J0 to J1

and introduce an additional factor i:

∞∫

0

∂

∂ρ1

i ρ1 J1(R ρ1) e

− i

(Z

ρ212 +

ρ414

) dρ1 = 0 . (2.94)

The function within the brackets vanishes at both borders, where for the border at infinity an infinitely smallimaginary part in front of the leading term in the exponent has to be assumed again. Upon performing thedifferentiation on the left hand side of (2.94), one can construct (2.87).

The literature covers the calculation of the Pearcey integral in a far-reaching way, that is by solvingdifferential equations47, as a series representation48 and by analytical calculation of the first terms of asymp-totic expansion49. The Bessoid integral was expressed in terms of parabolic cylinder functions50 and againas a series representation51. The latter work gives reference to an unpublished work of Pearcey52 (motivatedby the design of optical instruments and of highly directional microwave antennas), stating that differen-tial equations for the Bessoid integral were employed there. Apart from that, no further indication for theexistence of such equations, let alone their boundary conditions, could be found.

47 Connor, J. N. and Curtis, P. R., Differential equations for the cuspoid canonical integrals, J. Math. Phys. 25(10) (1984),2895–2902.

48 Connor, J. N., Semiclassical theory of molecular collisions: Three nearly coincident classical trajectories, Mol. Phys. 26(1973), 1217–1231.

49 Stamnes, J. J. and Spjelkavik, B., Evaluation of the field near a cusp of a caustic, Optica Acta 30 (1983), 1331–1358.50 Janssen, A. J., On the asymptotics of some Pearcey-type integrals, J. Phys. A: Math. Gen. 25 (1992), L823–L831.51 Kirk, N. P. et al., Theory of axially symmetric cusped focusing: numerical evaluation of a Bessoid integral by an adaptive

contour algorithm, J. Phys. A: Math. Gen. 33 (2000), 4797–4808.52 Pearcey, T. and Hill, G. W., Spherical aberrations of second order: the effect of aberrations upon the optical focus,

Melbourne: Commonwealth Scientific and Industrial Research Organization (1963), 71–95.

Chapter 3

Relation between Geometrical andWave Optics

The geometrical optics solution of any optical problem with a 3-ray cuspoid topology can be found bycalculating the phases and divergences of the rays. However, the resulting field shows singularities at thecaustic, especially on the axis, which is the most interesting region for applications. On the other hand, theproblem of a paraxial spherically aberrated wave has led us to the Bessoid integral, being finite everywhere.We therefore seek to describe arbitrary axially symmetric focusing – in particular non-paraxial focusingwith strong spherical aberration and large angles – by matching the solution of geometrical optics with awave field constructed from the Bessoid integral and its derivatives. The unknown arguments and amplitudefactors can be found in an algebraic way and thereby the divergences of geometrical optics are removed.Finally, compact analytic expressions for the field on the axis are derived.

3.1 Matching with the Bessoid Integral

Let us assume that we know only the geometrical optics solution U for an axially symmetric problem with(strong) spherical aberration. In dimensional coordinates r ≡ (ρ, z):

U(r) =3∑

j=1

Uj(r) e i k ψj(r) . (3.1)

According to (1.69) and (1.79) and the corresponding discussion, the amplitudes and phases have the form

Uj(r) =U0,j e i Sc,j

√|Jj |, (3.2)

ψj(r) = ψ(0)j + sj . (3.3)

Here we have generalized to the case that the amplitudes U0,j can be different for different rays, taking intoaccount everything except the geometrical divergence, e.g. radially dependent transmission coefficients of ascreen. Now we want to find a solution of this problem in terms of the Bessoid (wave optics) integral (2.3)and, by this means, remove the caustic divergences (of geometrical optics). This is done by the method ofuniform caustic asymptotics based on general standard integrals1.

The arguments of the Bessoid integral are first unknown at this stage and we denote them as2 R and Z.With R ≡ (R, Z) and r1 ≡ (x1, y1):

I(R) =1

2 π

∞∫

−∞e i φ(R,r1)dr1, (3.4)

φ(R, r1) = −R x1 − Zx2

1 + y21

2− (x2

1 + y21)2

4. (3.5)

1 Following Kravtsov, Y. A. and Orlov, Y. I., Caustics, Catastrophes and Wave Fields, Springer Series on Wave Phenomena(Volume 15), Springer-Verlag, Second Edition (1999), p. 92 et seq.

2 In general, the functions R and Z differ from the dimensionless coordinates in (1.39), since now the search for generalrelations between the Bessoid coordinates R and Z on the one hand, and real space coordinates ρ and z on the other, is exactlythe task of this section.

3.1 Matching with the Bessoid Integral 33

Since we have 6 known quantities (Uj , ψj), we are free to choose 4 functions besides the coordinates R andZ. We make the Ansatz

U(r) =(

A(r) I(R) +1i

AR(r)∂I(R)

∂R+

1i

AZ(r)∂I(R)

∂Z

)e i χ(r) . (3.6)

Thus, we are seeking a solution in terms of the Bessoid integral and its first derivatives. This requires the 3amplitude factors3 A, AR and AZ as well as the phase function χ.

The physical meaning of (3.6) is multiple. First, we construct the Bessoid-like solution which asymp-totically coincides with the geometrical optics in the regions where the latter works. The resulting solutionis close to the three-dimensional wave equation everywhere. The transformation of coordinates allows todescribe even non-paraxial situations, although the Bessoid integral originated from a paraxial problem.Finally, the slowly varying amplitudes account for different intensities on the rays and non-paraxiality.

One can write (3.6) as

U(r) =1

2 π

∞∫

−∞g(r, r1) e i f(r,r1) dr1, (3.7)

where

g(r, r1) = A(r) + AR(r)∂φ(R, r1)

∂R+ AZ(r)

∂φ(R, r1)∂Z

, (3.8)

f(r, r1) = χ(r) + φ(R, r1) . (3.9)

The integral (3.7) can be solved by the method of stationary phase where everything is considered as slowlyvarying compared with the exponent. From (3.9) we see that the stationary points r1 of f are the same asfor φ, given by the condition

∂f(r, r1)∂r1

=∂φ(R, r1)

∂r1= 0 , (3.10)

since χ and R do not depend on r1. The points of stationary phase be denoted as r1,j ≡ tj ≡ (tj , 0) wherethe tj are given by Cardan’s solution of

R + Z t + t3 = 0 . (3.11)

They can be found in (2.9) or (2.11), where one has to replace ξj by tj . Hence, (3.7) has the asymptoticrepresentation

U(r) ≈3∑

j=1

Wj(r) e i fj(r) , (3.12)

where

Wj(r) = g(r, tj)e i

π4 signHj

√|detHj |=

(A(r) + AR(r)

∂φ(R, tj)∂R

+ AZ(r)∂φ(R, tj)

∂Z

)e i

π4 signHj

√|detHj |, (3.13)

fj(r) = χ(r) + φj(R) . (3.14)

Here fj(r) ≡ f(r, tj), φj(r) ≡ φ(R, tj) and the Hessian

Hj ≡(

φ20(R, tj) φ11(R, tj)φ11(R, tj) φ02(R, tj)

). (3.15)

We match the asymptotic representation (3.12) with the geometrical optics solution (3.1). This impliesmatching of amplitudes and phases:

Wj = Uj , (3.16)fj = k ψj . (3.17)

By virtue of (3.13) and (3.14) one obtains 6 equations (as j = 1, 2, 3), from which 6 quantities – R, Z, χ, A,AR and AZ – have to be determined as functions of r:

(A(r) + AR(r)

∂φ(R, tj)∂R

+ AZ(r)∂φ(R, tj)

∂Z

)e i

π4 signHj

√|detHj |= Uj(r) , (3.18)

χ(r) + φj(R) = k ψj(r) . (3.19)

3 The indices R and Z in the amplitude factors do not indicate derivatives.

3.2 Determination of Coordinates and Amplitudes 34

For the sake of transparency and completeness, we shall explicitly draw attention to the following functionaldependences: stationary points4 r1,j ≡ tj = tj(R) and coordinates R = R(r).

3.2 Determination of Coordinates and Amplitudes

Now we return to (3.2) and (3.3) and insert the solution of geometrical optics into (3.18) and (3.19), respec-tively.

First, we want to consider the equation for the phases (3.19):

χ−R tj − 12

Z t2j −14

t4j = k ψj . (3.20)

From (3.11) we see upon multiplication with tj that

t4j = −R tj − Z t2j , (3.21)

and thus, (3.20) can be written as

χ− 34

R tj − 14

Z t2j = k ψj . (3.22)

It is appropriate5 not to solve these three equations directly but to go to quantities that are permutationallyinvariant with respect to the roots tj . Thus we write the sum, the sum of the squares, and the sum of thecubes of the three equations (3.22). But first we state that the points of stationary phase, tj , obey Vieta’s6

formulas

t1 + t2 + t3 = 0 ,

t1 t2 + t2 t3 + t3 t1 = Z , (3.23)t1 t2 t3 = −R .

We introduce the quantity

an ≡3∑

j=1

tnj , (3.24)

and immediately obtain a0 = 3, a1 = 0 and a2 = −2 Z (the former is trivial, the next is just Vieta’s firstformula, the last is obtained by expressing the sum of squares by the square of the sum and using the firstand second Vieta formula). In general, a recurrence relation can be found directly from the stationary phasecondition (3.11) upon multiplying by powers of tj and summation:

an+1 = −R an−2 − Z an−1 n = 2, 3, . . . (3.25)

One can now calculate a3 = −3 R, a4 = 2 Z2, a5 = 5 R Z and a6 = 3 R2− 2 Z3, which is the last that will beneeded. We start with the sum of the three equations (3.22). With the help of (3.24) this can be written as

3 χ− 34

R a1 − 14

Z a2 =3∑

j=1

k ψj . (3.26)

Substituting a1 = 0 and a2 = −2 Z, we find

χ +16

Z2 = b1 , where b1 ≡ 13

3∑

j=1

k ψj . (3.27)

The next task is the sum of squares of (3.22). We express χ from (3.27) and transfer b1 to the right handside and then square and take the sum:

3∑

j=1

(−3

4R tj − 1

4Z t2j −

16

Z2

)2

=3∑

j=1

(k ψj − b1)2 . (3.28)

4 The partial derivatives with respect to R and Z in (3.13) and (3.18) must be evaluated in such a way as the tj were heldconstant, although they are functions of R themselves.

5 Following Brekhovskikh, L. M. and Godin, O. A., Acoustics of Layered Media II, Springer Series on Wave phenomena(Volume 10), Springer-Verlag (1992), p. 213 et seq.

6 Francois Viete (1540–1603).


Calculating the left hand side and substituting an, one obtains

−94

R2 Z +124

Z4 = b2 , where b2 ≡3∑

j=1

(k ψj − b1)2 . (3.29)

Similarly, the sum of the cubes requires coefficients up to a6 and the solution reads

8164

R4 − 1532

R2 Z3 − 1288

Z6 = b3 , where b3 ≡3∑

j=1

(k ψj − b1)3 . (3.30)

Since the bl (l = 1, 2, 3) are known functions of the geometrical optics phases, (3.20) has been reduced tothe following set of equations:

χ +16

Z2 = b1 , (3.31)

−94

R2 Z +124

Z4 = b2 , (3.32)

8164

R4 − 1532

R2 Z3 − 1288

Z6 = b3 . (3.33)

Expressing R2 by means of the second equation and inserting it into the third, yields the polynomial equation

Z8 − 16 b2 Z4 +256 b3

3Z2 − 64 b2

2

3= 0 . (3.34)

With the redefinitionv ≡ Z2 (3.35)

(3.34) takes on the standard form of a quartic (biquadratic) equation

v4 − 16 b2 v2 +256 b3

3v − 64 b2

2

3= 0 . (3.36)

There are general algebraic solutions named due to Ferrari7, but they are rather cumbersome. It is moreconvenient to solve this quartic equation ourselves and also extract the physical meaning. Starting from(3.36), we express this polynomial in the form

(v2 + K)2 − (Lv + M)2 = 0 . (3.37)

Equating like coefficients in (3.36) and (3.37),

v0 : −64 b22

3= K2 −M2 ,

v1 :256 b3

3= −2 LM , (3.38)

v2 : −16 b2 = 2 K − L2 ,

and introducing the quantity

q ≡ 3

√6 b2

3 − b32 , (3.39)

we find

K =83

(2 q − b2) ,

L = ± 4

√23

(q + b2) , (3.40)

M = ± 163

√q2 − q b2 + b2

2 .

On the other hand, (3.37) is a binomial expression:

(v2 + K)2 − (Lv + M)2 = (v2 + K + Lv + M) (v2 + K − L v −M) = 0 . (3.41)7 Lodovico Ferrari (1522–1565).


Its roots are determined by the roots of the factors, given by the quadratic equations

v2 + K ± Lv ±M = 0 , (3.42)

Since Z should be a real coordinate, and due to v ≡ Z2, only the real and non-negative solutions for v arephysical. From (3.29) we find that b2 is always non-negative but, by virtue of (3.30), b3 can be both positiveand negative8. One can write the real non-negative solution as

v = −2 sgn(b3)

√23

(b2 + q) + 2

√23

(2 b2 − q + 2

√b22 − b2 q + q2

), (3.43)

and hence

Z = ±√−2 sgn(b3)

√23

(b2 + q) + 2√

23

(2 b2 − q + 2

√b22 − b2 q + q2

). (3.44)

where by virtue of (3.32) the sign of Z has to be chosen in such a way that

sgn(Z) = sgn(Z4 − 24 b2) , (3.45)

since this provides R being real:

R =

√Z3

54− 4 b2

9 Z. (3.46)

Here the positive solution was chosen (in accordance with the radial character of the coordinate R). Finally,(3.31) yields

χ = b1 − 16

Z2 . (3.47)

To clarify the meaning of the developed formulas, we express the quantity q from (3.39) in the followingways:

q3 = 6 b23 − b3

2

=1

211R2 (27 R2 + 4 Z3)3

= −2 k6 (ψ1 − ψ2)2 (ψ2 − ψ3)2 (ψ3 − ψ1)2 . (3.48)

The first equal-sign was the original definition, the second shows that q vanishes at the caustic in Bessoidcoordinates R and Z, i.e. for R = 0 (axis) as well as for 27 R2 + 4 Z3 = 0 (cusp). The third line (3.48) is thepermutationally invariant product of phase differences9. It becomes zero, if two or more phases are equal.This happens exactly at the caustic line (ψ1 = ψ3) and at the cusp (ψ2 = ψ3) in real space coordinates ρand z.

Next, we solve the equations for the amplitudes. Omitting the arguments, (3.18) reads(

A− tj AR − 12

t2j AZ

)e i

π4 signHj

√|detHj |=

U0,j e i Sc,j

√|Jj |. (3.49)

It is clear that the transformation R = R(r) keeps the correspondence between the caustic phase shifts inreal r- and Bessoid R-coordinates. In other words, signHj is in accord with Sc,j , given by (2.41) and (1.78),which implies

e i Sc,j e− iπ4 signHj = −i . (3.50)

However, it turns out that this is only true in the lit region (inside the cusp). In the shadow (outside thecusp) one has to deal with complex phases in a more rigorous way. The second equal-sign in (2.35) and thus(2.36) are no longer valid for complex eigenvalues and we have to go back one step and write

e iπ4 signHj

√|detHj |→

√i

φ20,j

√i

φ02,j≡ 1√

Hj

, (3.51)

8 This is true in the lit region. In the shadow, b2 is still real but not necessarily non-negative, because of the imaginary partsof the complex conjugate eikonals ψ2 and ψ3. The quantity b3 becomes purely imaginary. By this means q is real in both thelit and the shadow region.

9 Note that these phases do not include the caustic phase shift. Rather they are the eikonal (optical path) multiplied by thewavenumber.

3.3 The Matched Solution for a Spherically Aberrated Wave 37

with complex Hj . Furthermore, we again use complex geometrical amplitudes, which include phase shiftsand are valid everywhere,

e i Sc,j

√|Jj |→ 1√

Jj

. (3.52)

Therefore (3.49) becomesA− tj AR − 1

2 t2j AZ√Hj

=U0,j√

Jj

, (3.53)

and the solutions are10

A = −U0,1

√H1√J1

t2 t3(t3 − t1) (t1 − t2)

− U0,2

√H2√J2

t3 t1(t1 − t2) (t2 − t3)

− U0,3

√H3√J3

t1 t2(t2 − t3) (t3 − t1)

,

AR = U0,1

√H1√J1

t1(t3 − t1) (t1 − t2)

+ U0,2

√H2√J2

t2(t1 − t2) (t2 − t3)

+ U0,3

√H3√J3

t3(t2 − t3) (t3 − t1)

,

AZ = 2 U0,1

√H1√J1

1(t3 − t1) (t1 − t2)

+ 2 U0,2

√H2√J2

1(t1 − t2) (t2 − t3)

+ 2 U0,3

√H3√J3

1(t2 − t3) (t3 − t1)

,

(3.54)where the tj are functions of R and Z, given by (2.9). The square roots have to be chosen in such a way thatthe amplitudes are smooth functions11. The divergences of geometrical optics are removed in the expressionsabove. Let us, for example, briefly consider the case of merging rays 2 and 3 at the cusp. Then t2 → t3,J2, J3 → 0, H2,H3 → 0 but A,AR, AZ are finite12 due to compensation of the diverging second and thirdterm in each formula13.

The three expressions (3.44), (3.46) and (3.47) are the solutions of matching the three geometrical phaseswith the Bessoid phase at its three stationary points. The functions bl (l = 1, 2, 3) are given by (3.27), (3.29)and (3.30). On the other hand, (3.54) shows the correspondence between the geometrical optics amplitudesand the amplitude factors in front of the Bessoid integral and its derivatives, respectively. Therefore, we havefound all unknown functions for our original Ansatz (3.6).

We may sum up in the following way. The Bessoid integral and its first derivatives have the right topologyfor any 3-ray problem with spherical aberration and axial symmetry. The solution of geometrical optics(eikonals ψj and amplitudes Uj) has to be matched with the Bessoid coordinates, phase and amplitudes (R,Z, χ, A, AR and AZ) in the regions where the former is valid. By this means the geometrical optics solutionis mapped onto a combination of Bessoid wave integrals. Thereby all divergences are removed.

It is interesting that this method involves all solutions of geometrical optics, including the complex rayswhich have been designated as non-physical before. It turns out that both real and complex rays provide thegeometrical skeleton of the wave flesh14.

3.3 The Matched Solution for a Spherically Aberrated Wave

We apply the mathematical apparatus developed in the previous sections to the problem of the sphericallyaberrated wave with initial eikonal ψ(0), given by (1.58), and with U0,j = U0 for all three rays. The geometricaloptics solution diverges at the caustic (see figure 1.9). Matching with the Bessoid approach (3.6),

U(ρ, z) =(

A I +1i

AR IR +1i

AZ IZ

)e i χ , (3.55)

removes the divergences and results in figure 3.1.In particular, we want to note that the maximum of intensity is not in the paraxial focus (ρ=0, z =0),

but is shifted towards the aperture due to spherical aberration.10 We used the first Vieta formula to simplify the numerators of AR in (3.54), i.e. t2 + t3 → −t1 and so on.11 For instance, when crossing the cusp from the lit into the shadow region,

√J2 and

√J3 become complex conjugate (ray

2 and 3 become complex) and due to the usual convention of square roots it may be necessary to choose the signs of theirimaginary parts manually – i.e. interchange

√J2 and

√J3 in the shadow to make

√H2,3/

√J2,3 continuous – if otherwise the

amplitudes are not smooth. But this is a technical problem and no analytical peculiarity.12 In fact, A, AR and AZ are bounded purely imaginary functions.13 Only the combination of diverging terms remains finite, not the second and third term on their own.14 Kravtsov, Y. A. and Orlov, Y. I., Caustics, Catastrophes and Wave Fields, p. 86.

3.4 General Expressions On and Near the Axis 38

-3

-1.5

0

1.5

z0.6

0.4

0.2

0

Ρ

0

1500

3000

ÈU È2

-3

-1.5

0z

Figure 3.1: Absolute square of the field U , calculated by matching the geometrical optics solution of the sphericallyaberrated wave with the Bessoid integral and its derivatives. Parameters are the same as in figure 1.9

3.4 General Expressions On and Near the Axis

In general, the expressions for the Bessoid coordinates and amplitudes are rather complicated. However, itis possible to simplify the situation, if one is close to the axis. We introduce the phases

ϕj ≡ k ψj , (3.56)

and consider first the special case where

ρ = 0 and z < 0 , (3.57)

that is on the axis and inside the cuspoid (in real space coordinates). The first and the third ray have thesame eikonal. We may write

ϕ1 = ϕ3 , ϕ1 > ϕ2 . (3.58)

According to (3.48),q = 0 , (3.59)

and expression (3.44) takes on the simple form

Z = ±√

4− 2 sgn(b3)4

√23

b2 . (3.60)

The quantities b2 and b3 are

b2 =23

(ϕ1 − ϕ2)2 , b3 = −29

(ϕ1 − ϕ2)3 . (3.61)

Due to ϕ1 > ϕ2, sgn(b3) = −1. The sign of Z is given by (3.45), which is zero on the axis, but approachingfrom negative real values:

limρ→0+, z<0

(Z4 − 24 b2) = 0− . (3.62)

Hence15, sgn(Z) = −1. This leads toZ = −2

√ϕ1 − ϕ2 , (3.63)

and from (3.46) one can confirm that R = 0.The Bessoid integral has its global maximum at Zm ≈ −3.051. Then equation (3.63) yields the phase

difference

ϕ1 − ϕ2 =Z2

m

4≈ 2.327 , (3.64)

which is close to 3 π/4 ≈ 2.356. This has a transparent physical meaning, because ray 1 and 3 are shifted by−π/2 as they touch the cusp. Besides, they are shifted again by −π/2 when crossing the focal line, but on

15 Or, in other words, sgn(Z) = sgn(z).


the axis exactly half of this delay has occurred. As a result, the overall phase delay is ∆ϕ1 = ∆ϕ3 = −3 π/4,while ∆ϕ2 = 0. Consequently, (3.64) is the condition for the (first) constructive interference of the axial ray(ray 2) and the cone of non-paraxial rays (ray 1 and 3 in the meridional plane), where the latter are, ofcourse, all in phase.

Second, we want to derive similar expressions which are valid also in a neighbourhood of the focal line,that is for small (positive) ρ. We make the following approach – up to the first order in ρ – for the phases:

ϕ1 = ϕnp + k ρ sin β ,

ϕ2 = ϕp , (3.65)ϕ3 = ϕnp − k ρ sin β ,

where ϕnp and ϕp denote the eikonals of the non-paraxial rays and the (par)axial ray, respectively (to becalculated with ρ = 0). The angle with the axis of the non-paraxial ray corresponding to ray 3 is β > 0(figure 3.2).

1

rb

np

np

p

2

3

Figure 3.2: Notation

Inserting these phases into the exact expressions (3.44) and (3.46), Taylor expanding in ρ and resubsti-tuting ϕnp = (ϕ1 + ϕ3)/2, ϕp = ϕ2 and k ρ sin β = (ϕ1 − ϕ3)/2 from (3.65), yields

Z = −2

√ϕ1 + ϕ3

2− ϕ2 = −2

√ϕnp − ϕp , (3.66)

R =(ϕ1 − ϕ3)/2√

2 4√

(ϕ1 + ϕ3)/2− ϕ2

=k ρ sin β√−Z

. (3.67)

In the limiting case ρ = 0 expression (3.66) becomes (3.63) and (3.67) becomes R = 0 again. Therefore, nearthe axis we dispose of simple expressions for the Bessoid coordinates R and Z in terms of the phases ϕj ofgeometrical optics.

We want to remind of (2.52) and that the caustic width is determined by the argument of the Besselfunction J0, i.e. R

√−Z. From (3.66) and (3.67) one finds

R√−Z =

ϕ1 − ϕ3

2= k ρ sin β . (3.68)

The width of the Bessoid focal line caustic, ρw, is defined by the first zero of the Bessel function, w0, givenin (2.54):

ρw =w0

k sin β≈ 0.383

λ

sin β. (3.69)

In the purely geometrical picture the first minimum is reached when ray 1 and 3 interfere destructively thefirst time, i.e. when the phase difference becomes π. This is represented by the condition

ϕ1 − ϕ3 = π +π

2, (3.70)

where the term π/2 takes account of the caustic phase shift of ray 1. This yields the geometrical approximationof (3.69):

ρw =ϕ1 − ϕ3

2 k sin β=

38

λ

sin β= 0.375

λ

sin β. (3.71)


Finally, we derive on-axis expressions for the amplitudes A, B1 and B2. For ρ = 0 (R = 0) the stationarypoints, given by (3.11), are

t1 = −√−Z , t2 = 0 , t3 = −t1 . (3.72)

Then, the amplitude A in (3.54) simplifies to

A = U0,2

√H2√J2

, (3.73)

because on the axis the ratios√

H1,3/√

J1,3 are both finite and the corresponding terms disappear uponmultiplication with t2 = 0. If we restrict ourselves to the lit region, i.e. Z < 0, we have

1√Hj

=e i

π4 signHj

√|detHj |. (3.74)

By virtue of (2.39), i.e. detH2 = Z2, and (2.40), i.e. signH2 = 2, one finds16

√H2 = i Z , (3.75)

and thus:A = i

U0,2 Z√J2

. (3.76)

This approximation for the amplitude A is valid up to the focus (Z = 0). The second ray diverges like theinverse distance from the focus and therefore

√J2 approaches zero as fast as Z → 0−.

The amplitude AR in (3.54) vanishes due to t3 = −t1 and the fact that17

√H1√J1

=√

H3√J3

, (3.77)

which means that ray 1 and 3 have same strength. Consequently,

AR = 0 . (3.78)

With (3.72) and (3.77) the amplitude AZ reads

AZ =2Z

(U0,1

√H1√J1

− U0,2

√H2√J2

). (3.79)

The first term is non-trivial. Both√

H1 and√

J1 are zero on the axis, but their ratio is finite and welldefined. In order to understand this, we have to study the behaviour of t1 when R → 0+. Taylor expandingt1, i.e. the expression −2 P cos β in (2.11), yields in the first order in R:

t1 = −√−Z +

R

2 Z. (3.80)

Therefore, again in first order in R:detH1 = 2 R

√−Z . (3.81)

Due to signH1 = −2 we obtain √H1 = i

√2 R

√−Z , (3.82)

and with (3.75):

AZ = 2 iU0,1

√2 R

√−Z

Z√

J1

− 2 iU0,2√

J2

. (3.83)

This approximation for AZ holds for small values of R. It is finite, since√

J1 approaches zero as fast as thefunction18

√R for R → 0+.

16 It is worth mentioning that√|Z2| = −Z, since Z < 0.

17 This is true, since (3.50) holds in the lit region and the absolute values of H1 and H3 as well as those of J1 and J3 are thesame on the axis.

18 This is clear from equation (1.55).


Using (3.63) and (3.68) for the right hand sides, we can sum up:

A = iU0,2 Z√

J2

= −2 iU0,2

√ϕ1 − ϕ2√J2

,

AR = 0 , (3.84)

AZ = 2 iU0,1

√2 R

√−Z

Z√

J1

− 2 iU0,2√

J2

= − iU0,1

√2 k ρ sinβ√

ϕ1 − ϕ2

√J1

− 2 iU0,2√

J2

.

However, further simplification is possible. We insert these expressions in the original equation for the field(3.6):

U =(

AI +1i

AR IR +1i

AZ IZ

)e i χ

=(

U0,2√J2

(i Z I − 2 IZ) +2 U0,1

√2 k ρ sin β

Z√

J1

IZ

)e i χ . (3.85)

On the axis the phase coordinate becomesχ = ϕ2 , (3.86)

which results from substituting ϕ1 = ϕ3 and Z = −2√

ϕ1 − ϕ2 into (3.47). Furthermore, we use the linearrelationship between the Bessoid and its Z-derivative (2.92). And thus we end up with

U =[

U0,1

√2 k ρ sin β√J1

(i I − 1

Z

)+

U0,2√J2

]e i ϕ2 . (3.87)

Chapter 4

The Sphere

We start with solving the problem of a refracting sphere in the picture of geometrical optics. Then the Bessoidmatching procedure is applied in order to obtain a scalar wave field without divergences. As a subsequentstep an incident linearly polarized plane wave is considered, which leads us to the necessity to allow angulardependent vectorial amplitudes of the electric and magnetic field, breaking the axial symmetry. This requiresthe introduction of higher-order canonical Bessoid integrals. The coordinate transformations are the samefor all orders, since the cuspoid topology and hence the structure of the ray picture remains unchanged, andthe amplitudes of the higher-order Bessoid integrals can be found by a modification of the original equations.Finally, we shall compare these results of vectorial Bessoid matching with the theory of Mie.

The investigation of the field behind a sphere was motivated by experiments on laser processing, usingregular two-dimensional lattices of microspheres as an array of microlenses. Such hexagonal arrays are formedby self-assembly processes, e.g. from colloidal suspensions. This technique permits one to produce on asubstrate surface millions of submicron features with a single laser shot. Regular ablation patterns, thegrowth of circular cones, surface patterning by reactive etching and the deposition of arrays of dots by laserinduced forward transfer were demonstrated1. Processing with arrays of colloidal microspheres combines theadvantages of laser material removal, deposition and transformation techniques with the high throughput oflarge-area parallel processing2.

4.1 Geometrical Optics Solution

We consider the problem of a plane wave falling onto a refracting sphere. The figures 4.1 and 4.2 show thegeometrical situation in two and three dimensions. The first illustrates the formation of the cusp by the raysof geometrical optics. The latter demonstrates the axial symmetry in three-dimensional space.

Figure 4.1: Parallel rays are refracted by a sphere. Two-dimensional cross section containing the axis of symmetry

Let a be the sphere radius and n > 1 its refractive index. The centre M is located at the origin (0, 0) ofan axially symmetric cylindrical coordinate system (ρ, z), and we consider an incident plane wave with rays

1 Bauerle, D. et al., Laser-processing with colloid monolayers, Proc. SPIE 5339 (2004), 20–26, and the references therein.2 Bauerle, D., Laser Processing and Chemistry, Springer-Verlag, Third Edition (2000), p. 185 et seq., p. 335 et seq. and p.

497 et seq.

4.1 Geometrical Optics Solution 43

Figure 4.2: Refraction in three dimensions

parallel to the z-axis. The geometrical optics focus, formed by the paraxial rays, is located at (0, f) with3

f =a

2n

n− 1. (4.1)

Since the surrounding medium is vacuum (refractive index 1), Snell’s4 law reads

sin θi = n sin θt . (4.2)

A ray passes the point Q, is first refracted at Q1, a second time at Q2 and propagates to P . Figure 4.3 isdrawn in the meridional plane, containing the point P and the axis and therefore the ray itself.

qi

a b

qi

qt

qt

M

a

F

P

Q1Q

r

z

Q2

Figure 4.3: Refraction by a sphere (all indicated angles are positive)

One can easily find the two angles

α = 2 θt − θi , (4.3)β = 2 θi − 2 θt . (4.4)

Writing α as a function of θi only and equating to zero the derivative with respect to θi, dα/dθi = 0, yieldsthe condition for the angle θi,c, which is the incident angle of a ray that leaves the sphere exactly at thecusp (see 4.1):

sin θ2i,c =

4− n2

3. (4.5)

The radius (measured from the axis) of the caustic ring on the sphere surface can be shown to be

ρc = a sin αc = a(4− n2)3/2

3√

3 n2. (4.6)

3 See e.g. Bergmann, L. and Schafer, C., Lehrbuch der Experimentalphysik (Band 3), Optik (Herausgeber: Gobrecht, H.),Walter de Gruyter, Siebente Auflage (1978), p. 57 et seq.

4 Willebrord van Roijen Snell (1580–1626).


By virtue of (4.1) and (4.6), the value n = 2 is special, because for this refractive index the geometrical focus(vertex of the cuspoid) lies on the sphere surface, which means f = a and ρc = 0. Another special valueis n =

√2 for which the border rays (θi = π/2) leave the sphere exactly on the axis and then propagate

perpendicular to it.Let us now introduce the distance

s ≡ Q2P . (4.7)

This enables us to write down the ray equation, i.e. the position of the point P ≡ (ρ, z):

ρ = a sin α− s sin β , (4.8)z = a cos α + s cosβ . (4.9)

Writing the position in plane polar coordinates l and θ,

ρ = l sin θ , (4.10)z = l cos θ , (4.11)

we can eliminate s and obtain a single equation

l sin(θ + β) = a sin(α + β) , (4.12)

or, in terms of θi and θt,l sin(θ + 2 θi − 2 θt) = a sin θi . (4.13)

Using Snell’s law (4.2), one can eliminate θt. The resulting expression is

2 θi + θ = arcsina sin θi

l+ 2 arcsin

sin θi

n. (4.14)

This is a transcendent cubic-like equation5 which has three roots everywhere6. Either all three are real orone is real and two are complex conjugate. We denote these solutions as θi,j (j = 1, 2, 3) and choose theirorder in a way consistent with all the previous notations. Therefore, θi,1 is always real and negative, whereasθi,2 and θi,3 are either real and positive with θi,2 < θi,3 or complex conjugate with Im(θi,2) > 0. When θi isknown, we find the θt,j from (4.2) and the αj and βj from (4.3) and (4.4). The three ray coordinates sj canbe written by virtue of (4.8) or (4.9):

sj =l cos θ − a cos αj

cosβj. (4.15)

The eikonal is the optical path7 from Q (on the dashed vertical line in figure 4.3 all rays are still in phase)up to the point of observation, P . Thus, omitting the index j,

ψ = QQ1 + n Q1Q2 + Q2P − a

= a (2n cos θt − cos θi) + s , (4.16)

where we have subtracted the sphere radius a in the first line from the path contributions. By this meansthe eikonal is zero in the center M if no sphere were present8.

The next step is to calculate the geometrical optics amplitudes. The generalized divergence of rays, J , caneither be determined by means of the Jacobian transformation matrix from real space into ray coordinatesas in (1.51) or by flux conservation along the ray tubes, given by (1.49) and illustrated in figure 1.6. We shalluse the latter9:

J =Rm Rs

Rm0 Rs0. (4.17)

There exist general formulas for the refraction on an arbitrary surface with arbitrary orientation of the mainradii with respect to the plane of incidence10. But they are cumbersome, and for clarity we derive herethe solution for the sphere ourselves. Let us consider a point source G and start with the derivation of themeridional radius of curvature (figure 4.4).

5 We found it convenient for numerical root finding to write (4.14) as a polynomial in the variable v ≡ sin θi by applyingtrigonometric transformations and squaring the whole equation three times. This polynomial is of order 10 and the three wantedsolutions are confirmed as they fulfill (4.14).

6 For n ≥ √2 this is true for z ≥ a. If n <

√2, the 3-ray region does not start until some distance behind the sphere.

7 In a homogeneous medium this is the real space path multiplied with the refractive index. For that reason, the lengthQ1Q2 = 2 a cos θt inside the sphere must be multiplied with n.

8 One can add arbitrary constants in the definition of the eikonal, for there is freedom in choosing the plane of ψ = 0.9 The refractive indices in (1.50) are both equal to 1 now.

10 Cerveny, V., Seismic Ray Theory, Cambridge University Press (2001), p. 289 et seq., Kravtsov, Y. A. and Orlov, Y. I.,Geometrical Optics of Inhomogeneous Media, Springer Series on Wave Phenomena (Volume 6), Springer-Verlag (1990), p. 121et seq.


M

E

E’

g

a

de

G

D

Rm0

g

de

Rm

Figure 4.4: Meridional cross section for the determination of the meridional radius of curvature, Rm. M is the centerof the sphere

The initial radius of curvature is Rm0 ≡ GE, the one after refraction is Rm ≡ ED. The infinitesimallyneighboured beam (γ ¿ 1) which is refracted in E′ (the angles of incidence and transmission in E are θi andθt, in E′ they be denoted θ′i and θ′t) also propagates to D. The normals onto Rm0 through E and onto Rm

through E′ are g and d, respectively. In the necessary order the length of the arc EE′ can be approximatedby the distance e ≡ EE′. Thus, because all angles are small,

g = γ Rm0 , (4.18)d = δ Rm , (4.19)e = ε a . (4.20)

On the other hand, we find from the infinitesimal triangles:

g = e cos θi , (4.21)d = e cos θt . (4.22)

This leads toRm = −γ Rm0 cos θt

δ cos θi, (4.23)

where we have introduced a minus sign because the wave is converging after the refraction. The remainingproblem is the angle δ in the denominator. The relations between angles and primed angles are

θ′i = θi + γ + ε , (4.24)θ′t = θt − δ + ε , (4.25)

and with Snell’s law

θt − θ′t = arcsinsin θi

n− arcsin

sin(θi + γ + ε)n

, (4.26)

a first order Taylor expansion in the small angle (γ + ε) yields

θt − θ′t = − (γ + ε) cos θi

n cos θt, (4.27)

where we have applied Snell’s law in the denominator. We express δ from equation (4.25), substitute into(4.23) and finally obtain

Rm =na Rm0 cos2 θt

a cos2 θi + Rm0 (cos θi − n cos θt). (4.28)

For the sagittal radius of curvature we consider figure 4.5.A ray which emerged from G is refracted in E (incident angle θi and transmitted angle θt). The distance

from E to the intersection H of the ray with the line passing through G and the sphere centre M is thesagittal radius of curvature, as a neighboured ray, also emerging from G and hitting the sphere not in Ebut infinitesimally shifted perpendicular to the meridional plane, will also propagate to H due to symmetryaround the line GM . We have Rs0 ≡ GE and Rs ≡ EH. The tangent theorem states11

tanν − µ

2=

a−Rs0

a + Rs0cot

π − θi

2, (4.29)

11 All angles are in general large now.


M

E

a

G

H

Rs0

n

h

m

Rs

Figure 4.5: Meridional cross section for the determination of the sagittal radius of curvature, Rs

where π − θi is just the third angle in the triangle GME. Due to

ν + µ = θi (4.30)

we find

µ =θi

2− arctan

(a−Rs0

a + Rs0cot

π − θi

2

). (4.31)

In the triangle MHE the sine theorem states

−Rs

a=

sin(π − µ)sin η

, (4.32)

where we have again introduced a minus sign for the radius of curvature to take into account convergenceof the refracted wave. Some trigonometric manipulation finally gives the result

Rs =na Rs0

a + Rs0 (cos θi − n cos θt). (4.33)

Let us return to the original problem of a plane wave incident onto a sphere. This is the limiting case ofboth initial radii going to infinity,

Rm0, Rs0 →∞ . (4.34)

The radii of curvature immediately after the first refraction, i.e. in Q1 inside the sphere, are by virtue of(4.28) and (4.33):

Rm,Q1 =na cos2 θt

cos θi − n cos θt, (4.35)

Rs,Q1 =na

cos θi − n cos θt. (4.36)

The radii of curvature in Q2 (still inside the sphere) are obtained by adding the distance Q1Q2:

Rm,Q1 + 2 a cos θt , (4.37)Rs,Q1 + 2 a cos θt . (4.38)

These two radii have to be used as initial radii of curvature for the second refraction in Q2. But it is necessaryto exchange the initial and resulting radii in (4.28) and (4.33), for the ray is leaving the sphere now12. Amore algorithmic approach is to use the formulas (4.28) and (4.33) with a → −a, n → 1/n, and θi ↔ θt,where the new and old radii are Rm → Rm,Q2 , Rm0 → Rm,Q1 + 2 a cos θt (analogous for Rs). Immediatelyafter the second refraction, i.e. in Q2 outside the sphere, we therefore have

Rm,Q2 =a (Rm,Q1 + 2 a cos θt) cos2 θi

n a cos2 θt + (Rm,Q1 + 2 a cos θt) (cos θi − n cos θt), (4.39)

Rs,Q2 =a (Rs,Q1 + 2 a cos θt)

n a + (Rs,Q1 + 2 a cos θt) (cos θi − n cos θt). (4.40)

12 One must be careful with the signs. Equation (4.28) can be applied when using Rm → −(Rm,Q1 + 2 a cos θt), Rm0 →−Rm,Q2 and analogous for (4.33).


Consequently, the radii of curvature in the point P are given by

Rm,Q2 + s , (4.41)Rs,Q2 + s . (4.42)

Eliminating the refractive index with Snell’s law (n = sin θi/ sin θt), we can also write the radii (4.35), (4.36),(4.39) and (4.40) in the compact form13

Rm,Q1 = −asin θi cos2 θt

sin(θi − θt), (4.43)

Rs,Q1 = −asin θi

sin(θi − θt), (4.44)

Rm,Q2 = −acos θi

2

(cos θi sin θt

sin(θi − θt)− 1

), (4.45)

Rs,Q2 = −asin(2 θt − θi)

sin(2 θi − 2 θt). (4.46)

A ray’s overall geometrical divergence after both refractions reads (index j omitted)

1√J

=

√Rm,Q1 Rs,Q1√

(Rm,Q1 + 2 a cos θt) (Rs,Q1 + 2 a cos θt)

√Rm,Q2 Rs,Q2√

(Rm,Q2 + s) (Rs,Q2 + s). (4.47)

Note that ray 1 has a negative angle θi. Besides, it has to be shifted manually with a minus sign as it wasthe case in (1.77). The caustic shifts of the rays 2 and 3 are taken into account automatically by the squareroots.

Finally, the geometrical optics solution for the sphere is

U(ρ, z) = U0

m∑

j=1

e i k ψj

√Jj

, (4.48)

with m = 1 (lit region) or 3 (shadow), where the eikonal ψ is given by (4.16). The main equation, determiningthe three rays, is (4.14). Figure 4.6 shows the absolute square of (4.48) and its caustic divergences for a typicalset of parameters.

1

1.25

1.5

1.75

z0.2

0.1

0

Ρ

0

400

800ÈU È2

1

1.25

1.5z

Figure 4.6: Geometrical optics intensity distribution behind a sphere of radius a = 1 and with refractive indexn = 1.5, located at M = (0, 0). The initial amplitude is U0 = 1 and the vacuum wavenumber is k = 100. Thegeometrical focus is at z = f = a n/2 (n− 1) = 1.5

13 If we express R2,Q2 in terms of α and β, i.e. R2,Q2 = −a sin α/ sin β, one can immediately see by virtue of figure 4.3 thatit is the distance from Q2 to the axis, which precisely is the sagittal radius of curvature.

4.2 The Scalar Wave Field 48

4.2 The Scalar Wave Field

The geometrical optics field (4.48) has axial symmetry and a cuspoid topology. We can apply the apparatusof Bessoid matching to this solution in order to remove its divergences and fit it into the wave picture. Theresult is depicted in figure 4.7.

1

1.25

1.5

1.75

z0.2

0.1

0

Ρ

0

400

800ÈU È2

1

1.25

1.5z

Figure 4.7: Bessoid matched intensity after a sphere. Parameters are the same as in figure 4.6. The global maximum

|U |2 ≈ 907 is located at approximately z ≈ 1.247

Now we want to incorporate Fresnel transmission coefficients. We assume that the incident light is linearlypolarized in x-direction, i.e. the incident electric field vector is

E0 = E0 ex , (4.49)

with ex the unit vector in x-direction and E0 ≡ U0. Since axial symmetry is removed, we extend ourcoordinate system from (ρ, z) to (ρ, ϕ, z), where ϕ is measured from x to y (standard convention). Anarbitrary point of observation P ≡ (ρ, ϕ, z) will be reached by three rays (two may be complex) and theirthree angles θi,j are still determined by (4.14), for all three rays lie in the meridional plane, containing Pand the z-axis (figure 4.8a).

x

(a) (b)

x

z

E0

E0,p

E0,s

Q

Q1

Q2

P

y

y

z

b

j

j

Figure 4.8: (a) One ray propagating from Q to P (refracted in Q1 and Q2) in the meridional plane. (b) Decompositionof the initial electric field vector with length E0 into its π- and σ-component, being parallel and perpendicular to themeridional plane, respectively

The initial polarization in the meridional plane depends on ϕ:

E0,π = E0 cosϕ , (4.50)E0,σ = E0 sin ϕ , (4.51)

where E0,π (E0,σ) is the π-polarized (σ-polarized) component of the incident electric field vector, lying in(being perpendicular to) the meridional plane (figure 4.8b). Both refractions take place in the meridional

4.2 The Scalar Wave Field 49

plane and consequently, the Fresnel coefficients are just multiplied. Neglecting absorption and omitting theindex for the polarization, the amplitude changes upon two refractions (vacuum-sphere and sphere-vacuum)by the factor

t12 t21 = 1− r12 r12 , (4.52)

where tab (rab) is the amplitude transmission (reflection) coefficient from the medium a into (at) the mediumb and index 1 denotes the vacuum and 2 denotes the sphere. The Fresnel reflection coefficients r12 for thetwo polarizations are14

r12,π =n2 cos θi −

√n2 − sin2 θi

n2 cos θi +√

n2 − sin2 θi

, (4.53)

r12,σ =cos θi −

√n2 − sin2 θi

cos θi +√

n2 − sin2 θi

. (4.54)

We define the overall transmission coefficients

Tπ ≡ 1− r212,π , (4.55)

Tσ ≡ 1− r212,σ . (4.56)

Behind the sphere (after the two refractions) the projection onto the original Cartesian system15 (x, y, z)yields the transmission vectors Tπ and Tσ for each ray (omitting the index j):

E0 Tπ = E0,π Tπ

cos β cosϕcos β sin ϕ

sin β

, E0 Tσ = E0,σ Tσ

sinϕ− cosϕ

0

, (4.57)

where E0,π and E0,σ are written in (4.50) and (4.51). Hence, the overall transmission vector is given by

T ≡ Tπ + Tσ = Tπ

cos β cos2 ϕcos β sin ϕ cos ϕ

sin β cosϕ

+ Tσ

sin2 ϕ− sinϕ cosϕ

0

. (4.58)

In the lowest order the scalar approach can be maintained, if we average over the angle ϕ, i.e.

〈T〉 =Tπ

2

cosβ00

+

Tσ

2

100

. (4.59)

One can see that only the x-component remains. This is the consistent scalar solution including Fresnelcoefficients. It is in fact exact on the axis where ρ = 0 (for ρ > 0 the electric field must be ϕ-dependent).Let us match this improved scalar geometrical optics solution with the Bessoid integral and its derivatives.The three rays are16

Uj =U0,j e i k ψj

√Jj

, (4.60)

whereU0,j = U0 Tj , with Tj ≡ 〈Tx,j〉 =

Tπ,j cosβj + Tσ,j

2. (4.61)

1/√

J is written in (4.47) and ψ in (4.16). A decrease of intensity along the axis towards the sphere isobserved, for the non-paraxial rays (1 and 3) have large angles and small transmission coefficients. Figure4.9 shows the resulting intensity distribution.

Figure 4.10 compares the intensity distributions – without (figure 4.7) and with Fresnel coefficients (figure4.9) – along the axis.

14 Here the magnetic permeability of the sphere is assumed to be µ = 1 and the ray coordinate system is used in a way thatr12,π(θi =0) = −r12,σ(θi =0).

15 We used cylindrical coordinates for P .16 On the axis, one could substitute the letter E for U everywhere, because the scalar field U equals the x-component of the

electric field, the y- and z-component being zero for symmetry reasons.

4.3 On the Axis 50

1

1.25

1.5

1.75

z0.2

0.1

0

Ρ

0

400

800ÈU È2

1

1.25

1.5z

Figure 4.9: Absolute square of the averaged x-component of the electric field vector after Bessoid matching. Param-

eters and axis scale (for the sake of comparability) are the same as in figure 4.7. The maximum |U |2 ≈ 692 is locatedat z ≈ 1.258

1.1 1.2 1.3 1.4 1.5 1.6 1.7z

200

400

600

800

ÈU È2

Figure 4.10: Intensity distribution along the axis of figure 4.7 (without Fresnel coefficients, dashed line) and offigure 4.9 (with Fresnel coefficients, solid line)

4.3 On the Axis

In this section we want to investigate the electric field on the axis (ρ = 0) in more detail. The field is givenby the analytical expression (3.87), that is

E = E0

[T1

√2 k ρ sin β√

J1

(i I − 1

Z

)+

T2√J2

]e i ϕ2 , (4.62)

where the Tj are written in (4.61). By virtue of (4.47) and with θi,2 = θt,2 = 0 (the second ray propagatesalong the z-axis) we find

1√J2

=1

1− z/f, (4.63)

where we had to use s2 = z − a for the distance of the second ray behind the sphere. Expression (4.63)diverges not before the focus of geometrical optics z = f . The transmission for normal incidence simplifiesto

T2 =Tπ,2 cos β2 + Tσ,2

2=

4 n

(n + 1)2. (4.64)

Due to (4.16) and s1 = a sin α1/ sinβ1 the phases are

ϕ1 = k [ a (2n cos θt,1 − cos θi,1) + s1 ] = k a

(2 n cos θt,1 − cos θi,1 +

sinα1

sin β1

), (4.65)

ϕ2 = k [ a (2n cos θt,2 − cos θi,2) + s2 ] = k a (2n− 2) + k z . (4.66)

4.3 On the Axis 51

The (positive) phase difference between the first and the third ray at an infinitely small distance from theaxis (ρ → 0+) reads17

ϕ1 − ϕ3 = 2 k ρ sin β . (4.67)

On the other hand, the (positive and vanishing) first ray’s sagittal radius of curvature in (4.47) is

(Rs,Q2)1 + s1 =ρ

sin β. (4.68)

Therefore, we find and define the first ray’s compensated divergence18

D1 ≡√

ϕ1 − ϕ3√J1

= −√

(Rm,Q1)1 (Rs,Q1)1√[(Rm,Q1)1 + 2 a cos θt,1] [(Rs,Q1)1 + 2 a cos θt,1]

√(Rm,Q2)1 (Rs,Q2)1√

(Rm,Q2)1 + s1

√2 k sin β ,

(4.69)which manifestly has no singularity anymore19 until the geometrical focus where (Rm,Q2)1 + s1 → 0+.Altogether the overall field (4.62) is

E = E0

[T1 D1

(i I − 1

Z

)+

T2

1− z/f

]e i ϕ2 . (4.70)

This is valid for z < f (Z < 0) up to values of z which are close to the focus, since the diverging terms D1/Zand (1 − z/f)−1 almost cancel. For z → f , however, the divergence of D1 itself becomes important, as thenon-paraxial ray 1 becomes axial20.

In figure 4.11 we present the position and the value of the maximum electric field’s absolute square as afunction of the refractive index and the dimensionless product k a. The z-coordinate of this global maximumis denoted with fd (diffraction focus) and the intensity there is |E(fd)|2.

50 100 150 200 250 300k a

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

n

fd �a

50 100 150 200 250 300k a

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

n

ÈEH fd LÈ2�E0

2

1.1 2000

Figure 4.11: Left: Diffraction focus in units of the sphere radius as a function of n and k a (contour lines from top tobottom go from 1.1 to 3.0 in steps of 0.1). Right: Normalized maximum of the electric field’s absolute square (contourlines from bottom left to top right are 20, 50, 100, 200, 500, 1000 and 2000)

The main contribution in (4.70) stems from the Bessoid integral, that is from the term ∝ T1 D1 I. Thus,the position of the maximum can be estimated from the condition (3.64), i.e. ϕ1 − ϕ2 ≈ 3 π/4. If the phasedifference ϕ1−ϕ2 from (4.65) and (4.66) is expressed in the only variable θi,1, Taylor expanded and equatedto 3 π/4, then one can use the resulting angle θi to calculate fd from (see figure 4.3)

z = a cosα + asin α

tan β= a

sin θi

sin β. (4.71)

17 The angle β ist the (positive) angle of the non-paraxial ray (figure 3.2). On the axis β = −β1 = β3.18 The minus sign on the right hand side comes from the manual phase shift which has to be applied to the first ray.19 In principle, the structure of this equation is general and valid not only for the sphere. In axially symmetric systems D1

is always finite on the axis, since both the sagittal radius of curvature and the phase difference ϕ1 −ϕ3 are proportional to thedistance ρ.

20 For z > f (Z > 0), on the other hand, one is situated in the shadow and initial assumptions – beginning with (3.74) – areviolated.

4.4 The Vectorial Electromagnetic Field 52

In the lowest non-trivial order of the inverse product k a this yields21

fd ≈ a

2n

n− 1

(1−

√3 π

4 k a

n (3− n)− 1n (n− 1)

). (4.72)

Hence, in the limit of small wavelengths or large spheres the diffraction focus approaches the focus ofgeometrical optics (4.1):

limk a→∞

fd = f (4.73)

4.4 The Vectorial Electromagnetic Field

Let us rewrite the transmission vector (4.58) into T = (Tx, Ty, Tz), using trigonometric formulas:

Tx =Tπ cosβ + Tσ

2+

Tπ cosβ − Tσ

2cos 2 ϕ ,

Ty =Tπ cosβ − Tσ

2sin 2ϕ ,

Tz = Tπ sinβ cos ϕ .

(4.74)

One could match each of these three components – with eikonals and geometrical amplitudes – with theBessoid integral I and its derivatives IR and IZ . However, the result is not a smooth function on the axis, butthe field at R = 0 depends non-physically on the azimuthal angle ϕ. In a consistent picture the ϕ-dependentparts of the electric field (second term in Tx as well as Ty and Tz) have to vanish on the axis22. Therefore,the Bessoid integral I is not the appropriate function for these parts.

This is why we have to introduce new canonical functions for the matching of the terms containingcosϕ, sin 2 ϕ, and cos 2 ϕ. In the general case, new functions are needed to represent arbitrary angulardependence of the field amplitude without changing the initial wavefront, the ray structure, and thus thetype of catastrophe. By this means axial symmetry is lost with respect to the amplitude but not with respectto the phases and divergences of the rays.

The most natural and straightforward generalization of the Bessoid integral

I(R, Z) =

∞∫

0

ρ1 J0(R ρ1) e− i

(Z

ρ212 +

ρ414

)

dρ1 (4.75)

are the canonical higher-order Bessoid integrals23 with the non-negative integer m:

Im(R,Z) =

∞∫

0

ρm+11 Jm(R ρ1) e

− i

(Z

ρ212 +

ρ414

)

dρ1. (4.76)

Such integrals appear naturally if one expands an arbitrary initial field amplitude on the aperture in aFourier24 series:

U0(ρ1, ϕ1) =∞∑

m=0

[ am(ρ1) cos(m ϕ1) + bm(ρ1) sin(m ϕ1) ] . (4.77)

The form of the coefficients am and bm can be seen from a two-dimensional Taylor expansion in Cartesiancoordinates around the point (0, 0),

U0(x1, y1) =∞∑

m=0

m∑n=0

1m!

(m

n

)xm−n

1 yn1

∂mU0(x′1, y′1)

∂x′m−n1 ∂y′n1

∣∣∣∣x′1=0, y′1=0

. (4.78)

21 The geometrical factor 3 π/4 ≈ 2.356 under the square root can be replaced by the more exact Bessoid value 2.327 due to(3.64). Expression (4.72) approximates the position of the maximum of (4.70) within an error of about < 5% for k a > 100 andvalues of the refractive index near n ≈ 1.5.

22 This is precisely reflected by the averaging which resulted in (4.59).23 Asymptotics of these integrals in terms of cylinder functions are investigated in Janssen, A. J., On the asymptotics of

some Pearcey-type integrals, J. Phys. A: Math. Gen. 25 (1992), L823–L831.24 Jean Baptiste Joseph Fourier (1768–1830).


We rewrite this into polar coordinates:

U0(ρ1, ϕ1) =∞∑

m=0

m∑n=0

cmn ρm1 cosm−nϕ1 sinnϕ1 , (4.79)

where

cmn ≡ 1m!

(m

n

)∂mU0(x′1, y

′1)

∂x′m−n1 ∂y′n1

∣∣∣∣x′1=0, y′1=0

. (4.80)

Thus, it becomes clear that ρm1 is the lowest possible power of ρ1 which can be found in one term with

exp(imϕ1). Angular integration over ϕ1 of the two-dimensional representation of the original Bessoid integralin polar coordinates – see (1.34), though in dimensional coordinates – with its phase term exp(−i R ρ1 cos ϕ1)and with the contribution ρm

1 exp(i mϕ1) from the arbitrary field amplitude (4.79) yields the higher-orderBessoids (4.76), where the appearing negative sign in the argument of Jm is omitted by definition25. Hereone has to use the integral representation of the higher-oder Bessel functions26

Jm(t) =1

2 π im

2 π∫

0

e i t cos θ + i m θ dθ . (4.81)

Except I0 ≡ I, all Bessoid integrals vanish on the axis due to Jm(0) = 0 for all m > 0:

Im(0, Z) = 0 for m = 1, 2, 3, ... (4.82)

If we define the functionsIm ≡ Im e i m ϕ , (4.83)

we find that they satisfy the paraxial Helmholtz equation

2 i Im,Z + Im,RR +1R

Im,R +1

R2Im,ϕϕ = 0 , (4.84)

where Im,ϕϕ = −m2 Im. The integral Im is canonical for the field components with the angular dependencesinm ϕ or cos mϕ. Therefore, the first-order Bessoid I1 should be used for Ez, which contains the factorcosϕ, whereas I2 is appropriate for Ey and the second term in Ex, containing sin 2ϕ and cos 2 ϕ, respectively.

It is necessary to express the higher-order Bessoid integrals in terms of I and its derivatives. For I1 wealready know – see equation (2.20) – that the relationship is

I1 = −IR . (4.85)

Hence, the absolute square of the first-order Bessoid integral I1 looks precisely the same as the R-derivativeof I (see figure 4.12 and compare with figure 2.5).

The differentiation of I1 with respect to R yields

I1,R(R, Z) =12

∞∫

0

ρ31 J0(R ρ1) e

− i

(Z

ρ212 +

ρ414

)

dρ1 − 12

∞∫

0

ρ31 J2(R ρ1) e

− i

(Z

ρ212 +

ρ414

)

dρ1. (4.86)

With the recurrence relation (2.93) this simplifies to

I2 = −I1,R +1R

I1 = IRR − 1R

IR , (4.87)

where for the second equal sign (4.85) was used. The absolute square of I2 is shown in figure 4.13.In general, with the help of the recursion relation for Bessel functions,

Jm(t) =t

m

Jm−1(t) + Jm+1(t)2

, (4.88)

we find the recursive relation for the Bessoid integrals

Im+1 = −Im,R +m

RIm . (4.89)

25 The transformation from Cartesian to polar coordinates (dx1dy1 = ρ1 dρ1dϕ1) increases the power of ρ1 by 1.26 See e.g. http://mathworld.wolfram.com/BesselFunctionoftheFirstKind.html.


-10

-5

0

5

Z

10

8

6

4

2

0

R

0

5

10

ÈI1È2

-10

-5

0Z

Figure 4.12: Absolute square of the first-order Bessoid I1

-10

-5

0

5

Z

10

8

6

4

2

0

R

0

40

80

ÈI2È2

-10

-5

0Z

Figure 4.13: Absolute square of the second-order Bessoid I2

For the sake of completeness we give an analytic expression for the Im near the axis. Using (4.89) and (4.88)as well as expression (2.52) for I, one obtains

Im(R,Z) ≈√

π (−Z)m/2

2Jm(R

√−Z) e i

Z2−π4 erfc

(Z

2e i

π4

). (4.90)

Now we can match the vectorial geometrical optics solution for the sphere. Writing the ray index jexplicitly, we shall first introduce abbreviations for the transmission vector (4.74), indicating the order ofϕ-dependence with the superscript (m):

Tx,j = T(0)j + T

(2)j cos 2 ϕ , T

(0)j ≡ Tπ,j cos βj + Tσ,j

2,

Ty,j = T(2)j sin 2ϕ , where T

(1)j ≡ Tπ,j sin βj ,

Tz,j = T(1)j cosϕ , T

(2)j ≡ Tπ,j cos βj − Tσ,j

2.

(4.91)

Hence, the geometrical optics solution – including eikonal ψ and divergence J – reads

Ex,j = E(0)j + E

(2)j cos 2 ϕ ,

Ey,j = E(2)j sin 2 ϕ , where E

(m)j ≡ E0

T(m)j e i k ψj

√Jj

.

Ez,j = E(1)j cos ϕ ,

(4.92)


Each term E(m) is always the sum of the contribution of three rays:

E(m) =3∑

j=1

E(m)j . (4.93)

It is consistent to match each E(m) in analogy to (3.6):

E(m) =(

Am Im(Rm, Zm) +1i

AmR Im,R(Rm, Zm) +1i

AmZ Im,Z(Rm, Zm))

e i χm , (4.94)

where R0 ≡ R, A0 ≡ A and so on27. For m = 0 we know its solutions A, AR, AZ , R, Z and χ, given by(3.54), (3.44), (3.46) and (3.47) respectively. For m ≥ 1 it represents the matching with the higher-orderBessoid integrals to allow for ϕ-dependent parts of geometrical optics.

Since each higher-oder Bessoid integral can be written in terms of the original Bessoid integral, thepoints of stationary phase, tj , remain the same. Therefore, the matching of phases and thus the coordinatetransformations do not change:

Rm = R , Zm = Z , χm = χ . (4.95)

This was expected from the physical point of view. The generalization to the higher-order Bessoid integralsresulted from the need to allow for a variation of amplitude on the initial wavefront, say the front rightbehind the sphere. The shape of the wavefront is not changed in the vectorial approach and neither are thegeometrical divergences and phases of the rays.

However, for m ≥ 2 the derivation of the amplitudes requires additional insight. Let us consider thecase m = 2. The asymptotic behaviour of I2 = IRR − IR/R far from the caustic regions28 where R À 1is determined by the term IRR only, because the second term is small there. Thus – although we needthe second-order Bessoid integral I2 on and near the axis, where it vanishes – we shall use its asymptoticstationary phase expressions far from the axis for the derivation of coordinates and amplitudes. In this regionit is equivalent to the asymptotic of IRR.

In fact, we may generalize this statement to arbitrary order. Due to (4.89) the leading term for thestationary phase calculation is always

Im →(− ∂

∂R

)m

I . (4.96)

Therefore, the equations for the amplitudes do change. They are

(i tj)mAm − tj AmR − 1

2 t2j AmZ√Hj

=E0 T

(m)j√Jj

, (4.97)

which can be seen from the Bessoid integral’s Cartesian representation with the phase (3.5):(− ∂

∂R

)m

I →(−i

∂φ

∂R

)m

= (i x1)m . (4.98)

The equations (4.97) have the same form as (3.53) except an additional factor (i tj)m on the left hand sideand thus the amplitudes (3.54) become

Am = −U0,1

√H1√J1

t2 t3(i t1)m (t3 − t1) (t1 − t2)

− ...− ... (cyclic) ,

AmR = U0,1

√H1√J1

t1(i t1)m (t3 − t1) (t1 − t2)

+ ... + ... (cyclic) , (4.99)

AmZ = 2 U0,1

√H1√J1

1(i t1)m (t3 − t1) (t1 − t2)

+ ... + ... (cyclic) ,

where each of the two cyclic terms arises from the first term via permutation of indices numbering the rays:(1, 2, 3) → (2, 3, 1) → (3, 1, 2).

27 All (higher-order) Bessoid coordinates R, Z, R1, Z1, R2 and Z2 as well as all amplitude factors and the phases are functionsof the real space coordinates ρ and z. Furthermore, we shall mention that (afterwards) one might find it more convenient tochoose different coefficients for the higher-order equations with m > 0, but we decide to keep the factors 1, 1/i and 1/i forconsistency.

28 Note that for the matching procedure we need exactly this asymptotic representation and in the non-caustic regions only.


Finally, the electric field is given by

E(ρ, ϕ, z) =

Ex

Ey

Ez

= E(0)

100

+ E(1)

00

cos ϕ

+ E(2)

cos 2 ϕsin 2 ϕ

0

, (4.100)

where the E(m) are found from Bessoid matching

E(m) =(

Am Im +1i

AmR Im,R +1i

AmZ Im,Z

)e i χ . (4.101)

Figure 4.14 illustrates the absolute square of the electric field29 for ϕ = 0 (x,z-plane) and ϕ = π/2(y,z-plane). One can see the ϕ-dependence even better, if we plot the field as contour plots (figure 4.15).

1

1.25

1.5

1.75

z0.2

0.1

0

Ρ

0

300

600ÈEÈ2

1

1.25

1.5z

1

1.25

1.5

1.75

z0.2

0.1

0

Ρ

0

300

600ÈEÈ2

1

1.25

1.5z

Figure 4.14: |E|2 for ϕ = 0 (left) and ϕ = π/2 (right). Parameters are the same as in figure 4.6, i.e. sphere radiusa = 1, refractive index n = 1.5, vacuum wavenumber k = 100 and initial field amplitude E0 = 1 (polarized inx-direction). The sphere is located at the origin

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7z

-0.2

-0.1

0

0.1

0.2

x

ÈEÈ2

1 1.1 1.2 1.3 1.4 1.5 1.6 1.7z

-0.2

-0.1

0

0.1

0.2

y

ÈEÈ2

Figure 4.15: |E|2 in the x,z-plane (left) and in the y,z-plane (right). Contour shadings go from white (low values)to black (high values)

Figure 4.16 shows |E|2 at the plane z = 1.02 a (immediately behind the sphere30 there are surface effects,which will be discussed in the next section) as a function of the real space coordinates x and y. We alsopresent the absolute square of each component (figure 4.17) in order to demonstrate that the two high peaksof |E|2 in the center (along the x-axis) stem from the z-component.

The distance of these two pronounced maxima from the axis, ρm = xm, can be estimated from the global(first) maximum of the first-order Bessel function J1(R

√−Z) due to (4.90):

J1(w1) → max , w1 ≈ 1.8412 . (4.102)

29 Strictly speaking, it is the square of the vector’s norm, ‖E‖2 = EE∗.30 The radius of the caustic ring on the sphere – calculated from the geometrical optics expression (4.6) – is ρc ≈ 0.198.


-0.2-0.1

00.1

0.2

x-0.2

-0.1

0

0.1

0.2

y

0

60

120

ÈEÈ2

-0.2-0.1

00.1

0.2

x

Figure 4.16: |E|2 at the plane z = 1.02 a

-0.2

0.2x

-0.2

0.2

y0

120

-0.2

x

-0.2

0.2x

-0.2

0.2

y0

120

-0.2

x

-0.2

0.2x

-0.2

0.2

y0

120

-0.2

x

ÈExÈ2 ÈEyÈ

2 ÈEzÈ2

Figure 4.17: |Ex|2, |Ey|2 and |Ez|2 for z = 1.02 a

From (3.68) the condition reads

ρm =w1

k sinβ≈ 0.293

λ

sin β. (4.103)

This can be understood in terms of the geometrical optics phases. On the axis both the y- and thez-component of the electric field vanish (figure 4.18).

b

13

1’

3’

13

2

(b)(a)

x

zy

j

Figure 4.18: Electric field on the axis. (a) Rays and their corresponding electric field vectors (dashed arrows) inthe plane with arbitrary azimuthal angle ϕ. The sum of the z-components of the three rays vanishes. (b) In orderto see that the y-component of the electric field vector vanishes, it is necessary to consider also the rays 1′ and 3′,corresponding to the angle −ϕ (initial electric field vectors and their components of the rays 3 and 3′ are indicated)

But as soon as the phase difference is such that the z-components of ray 1 and 3 add, there will be alarge electric field31. The condition is

ϕ1 − ϕ3 = π +π

2. (4.104)

31 With increasing β the z-components of the electric field vectors of ray 1 and 3 become larger. On the other hand, thereare small transmission coefficients for such rays with large incident angles θi.


The right hand side is due to the fact that for purely geometrical reasons (figure 4.18a) the z-components ofthe electric field of ray 1 and 3 originally point into opposite directions on the axis (π) and ray 1 undergoesa phase delay on the axis (π/2). With (3.68) this yields the geometrical approximation of (4.103):

ρm =ϕ1 − ϕ3

2 k sin β=

38

λ

sinβ= 0.375

λ

sin β. (4.105)

For the calculation of the magnetic field H we have to start with the initial field vector H0. The incidentplane wave must obey Maxwell’s32 equations without source terms, and thus E0 = E0 ex, H0 and the vacuumwavevector k = k ez form a right handed tripod33:

H0 = ez×E0 = H0 ey , where H0 = E0 . (4.106)

Behind the sphere (after two refractions) a ray’s transmission vector for the magnetic field, Vj , is given bythe cross product of the unit vector in the direction of propagation, ekj = kj/k, and the transmission vectorof the electric field, Tj :

Vj = ekj×Tj . (4.107)

With

ekj =

− sin βj cosϕ− sin βj sin ϕ

cosβj

, Tj =

T(0)j + T

(2)j cos 2 ϕ

T(2)j sin 2 ϕ

T(1)j cosϕ

, (4.108)

where Tj was taken from (4.91), we obtain Vj = (Vx,j , Vy,j , Vz,j):

Vx,j = V(2)j sin 2 ϕ , V

(0)j ≡ Tσ,j cos βj + Tπ,j

2,

Vy,j = V(0)j − V

(2)j cos 2 ϕ , where V

(1)j ≡ Tσ,j sin βj ,

Vz,j = V(1)j sin ϕ , V

(2)j ≡ Tσ,j cos βj − Tπ,j

2.

(4.109)

The V(m)j are equal to the T

(m)j if one exchanges Tπ,j and Tσ,j . This is exactly what one expected, because

the π(σ)-component of the magnetic field is transmitted like the σ(π)-component of the electric field. Hence,the geometrical optics solution for the magnetic field is

Hx,j = H(2)j sin 2 ϕ ,

Hy,j = H(0)j −H

(2)j cos 2 ϕ , where H

(m)j ≡ H0

V(m)j e i k ψj

√Jj

.

Hz,j = H(1)j sinϕ ,

(4.110)

In analogy with (4.100), the magnetic field after Bessoid matching has the form

H(ρ, ϕ, z) =

Hx

Hy

Hz

= H(0)

010

+ H(1)

00

sin ϕ

+ H(2)

sin 2 ϕ− cos 2 ϕ

0

, (4.111)

where the H(m) are found from Bessoid matching of the geometrical optics solution (4.110).The flux density of electromagnetic energy is represented by the Poynting34 vector. In Gaussian cgs units

– assuming harmonic time dependence ∝ exp(−i ω t) with ω the angular frequency and t the time – it hasthe form35

S1 =c

4 πRe(E e− i ω t)×Re(H e− i ω t) , (4.112)

where c is the velocity of light in vacuum. The time average of the Poynting vectors before and after thesphere are

〈S0〉 =c

8 πRe(E0×H∗

0) =c

8 πE0 H0 ez and 〈S1〉 =

c

8 πRe(E×H∗) , (4.113)

respectively. We are interested in the flux enhancement normalized to the incident plane wave. Consequently,we define

S ≡ 〈S1〉|〈S0〉| =

Re(E×H∗)E0 H0

. (4.114)

32 James Clerk Maxwell (1831–1879).33 We remind that both the refractive index and the magnetic permeability of the ambient medium (vacuum) are equal to 1.34 John Henry Poynting (1852–1914).35 This is the standard definition. One could add arbitrary rotations of vector fields on the right hand side, for only the

divergence of the Poynting vector enters its origin, namely the equation of continuity.

4.5 Comparison with the Theory of Mie 59

4.5 Comparison with the Theory of Mie

We developed a general apparatus for matching geometrical optics solutions with the Bessoid integral. Themethod can be applied to any axially symmetric system with cuspoid topology of spherical aberration.

For the sphere we are able to compare the results of our approximation with the exact results obtainedfrom the theory of Mie36. On the basis of electromagnetic theory it is a rigorous solution for the diffractionof a plane monochromatic wave by a homogenous sphere in a homogenous medium. One solves Maxwell’sequations with boundary conditions in spherical polar coordinates, which leads to an infinite summation ofBessel functions and associated Legendre37 polynomials. For details concerning the theory of Mie we referto the original article38 and to the literature39.

A main quantity characterizing the sphere is the dimensionless Mie parameter

q ≡ k a , (4.115)

i.e. the product of sphere radius and wavenumber in the surrounding medium (in our case the vacuum).We want to remind of equation (1.40) and the subsequent discussion, where we stated that the expectedintensity enhancement would be proportional to k a. Figure 4.19 compares the absolute square of the electricfield on the axis obtained from the Mie theory40 with the result of Bessoid matched geometrical optics. Theparameters are as in figure 4.6 (a = 1, n = 1.5, E0 = 1) and the Mie parameter41 is q = 300, 100, 30 and 10.

1.1 1.2 1.3 1.4 1.5 1.6 1.7z

25

50

75

100

125

150

ÈEÈ2HcL

1.1 1.2 1.3 1.4 1.5 1.6 1.7z

10

20

30

40

50

ÈEÈ2HdL

1.1 1.2 1.3 1.4 1.5 1.6 1.7z

500

1000

1500

2000

2500

ÈEÈ2HaL

1.1 1.2 1.3 1.4 1.5 1.6 1.7z

200

400

600

800

ÈEÈ2HbL

Figure 4.19: |E|2 on the axis. Dashed lines represent the Mie theory, solid lines are the results of Bessoid matching.The cases (a), (b), (c) and (d) correspond to q = 300, 100, 30 and 10, respectively

We see that down to q ≈ 30 (i.e. a/λ ≈ 4.8) there is very good agreement between Mie theory andBessoid matching. In the case q = 10 (i.e. a/λ ≈ 1.6) the asymptotic behaviour far from the sphere surfaceis still correct. However, for small Mie parameters the characteristic distance – the sphere radius a – is nolonger large compared to the wavelength λ = 2 π/k. In this regime geometrical optics becomes invalid.

36 Gustav Mie (1868–1957).37 Adrien-Marie Legendre (1752–1833).38 Mie, G., Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen, Annalen der Physik (4), 25 (1908), 377.39 See for instance Born, M. and Wolf, E., Principles of Optics, Cambridge University Press, seventh expanded edition (2002),

p. 759 et seq.40 The author would like to thank Prof. B. Luk’yanchuk and Dr. Z. B. Wang (both at the Data Storage Institute, Singapore)

for the program used for the Mie calculations.41 Since a = 1 and E0 = 1, one could also think of varying q = k, and the abscissa and ordinate being in units of z/a and

|E|2 /E20 , respectively.


Next, we compare the electric and magnetic field as well as the z-component of the Poynting vector(4.114) obtained from Bessoid matching and Mie theory. Right behind the sphere (z = a) the agreement isnot perfect (see again figure 4.19), most probably due to surface effects, e.g. evanescent contributions, whichare taken into account in the Mie theory. As a rule of thumb we assume that they disappear at a distance ofabout λ/2 from the surface. Thus, we expect very good agreement starting from the distance z ≥ a + λ/2,i.e.

z

a≥ 1 +

π

q. (4.116)

For q = 100 this results in z/a & 1.03. The following sections are made at z = 1.02 a, showing very goodcorrespondence (figures 4.20, 4.21 and 4.22). For z & 1.05 a the pictures become indistinguishable and evenat z = a all qualitative features are represented in a correct way.

-0.2-0.1

00.1

0.2x

-0.2

-0.1

0

0.1

0.2

y

0

80

160ÈEÈ2

-0.2-0.1

00.1

0.2x

-0.2-0.1

00.1

0.2x

-0.2

-0.1

0

0.1

0.2

y

0

80

160ÈEÈ2

-0.2-0.1

00.1

0.2x

Figure 4.20: |E|2 for z = 1.02 a calculated with Bessoid matching (left) and with the Mie theory (right)

-0.2-0.1

00.1

0.2x

-0.2

-0.1

0

0.1

0.2

y

0

60

120ÈH È2

-0.2-0.1

00.1

0.2x

-0.2-0.1

00.1

0.2x

-0.2

-0.1

0

0.1

0.2

y

0

60

120ÈH È2

-0.2-0.1

00.1

0.2x

Figure 4.21: |H|2 for z = 1.02 a calculated with Bessoid matching (left) and with the Mie theory (right)

-0.2-0.1

00.1

0.2x

-0.2

-0.1

0

0.1

0.2

y

0

30

60Sz

-0.2-0.1

00.1

0.2x

-0.2-0.1

00.1

0.2x

-0.2

-0.1

0

0.1

0.2

y

0

30

60Sz

-0.2-0.1

00.1

0.2x

Figure 4.22: Sz for z = 1.02 a calculated with Bessoid matching (left) and with the Mie theory (right)

Finally, the off axis correspondence shall be investigated by comparing the one-dimensional distributionof |E|2 at z = 1.02 a as a function of x for y = 0 and different Mie parameters (figure 4.23). Again we observe


that for large q the pictures are almost the same, whereas small values of q result in deviations of the Bessoidmatching procedure from the theory of Mie.

-0.2 -0.1 0.1 0.2x

20

40

60

80

100

120

ÈEÈ2HcL

-0.2 -0.1 0.1 0.2x

10

20

30

40

50

ÈEÈ2HdL

-0.2 -0.1 0.1 0.2x

100

200

300

400

500

ÈEÈ2HaL

-0.2 -0.1 0.1 0.2x

20406080

100120140

ÈEÈ2HbL

Figure 4.23: |E|2 at z = 1.02 a for y = 0. Solid (dashed) lines represent Bessoid matching (Mie theory). The cases(a), (b), (c) and (d) correspond to q = 300, 100, 30 and 10, respectively

Conclusions

In the present work we developed a theoretical description of light focusing by microspheres. Our approach1

was to write the canonical integral describing the field for the given ray topology and to adapt it to arbitraryaxially symmetric and strongly spherically aberrated focusing by appropriate coordinate transformationsand amplitude equations.

Therefore, we started with the investigation of a paraxial spherically aberrated wave, which has thecorrect axially symmetric cuspoid topology. The paraxial Fresnel-Kirchhoff diffraction integral (1.26) ledus to a canonical integral of catastrophe theory, the Bessoid integral (2.4), shown in figure 2.2. It is thetwo-dimensional generalization of the well known Pearcey integral (2.1), which appears in many fields ofphysics.

For spherical aberration geometrical optics reveals the field divergence on the cuspoid caustic surface andalong the focal line. In section 1.2 we illustrate this with the example of a paraxial spherically aberratedwave (figure 1.9) and simultaneously discuss the concept of caustic phase shift within the picture of rays,carrying phase and field amplitude.

In chapter 2 the properties of the Bessoid integral needed for the further development are described.It is investigated in the regions off caustic by the method of stationary phase, resulting in the asymptoticapproximation (2.37). Its phase is a fourth-order polynomial with three stationary points. They correspondto the rays of geometrical optics and yield the only significant contributions to this highly oscillatory integral.We also demonstrated the Airy-type behaviour of the Bessoid integral near the caustic cone (2.68).

Near the axis a simple approximation (2.52) can be given in terms of the complimentary error functionwith complex argument (or Fresnel integrals) and the Bessel function. This Bessel-beam type radial depen-dence allows one to estimate the variable width of the caustic focal line. The expression becomes exact onthe axis (2.53).

The numerical evaluation of the Bessoid integral is discussed in section 2.5. As it is a highly oscillatoryintegral, the computation is by no means a trivial task and of large practical importance. Direct numericalintegration along the real axis and the method of steepest descent in the complex plane both have theirdisadvantages. By far the fastest technique is based on the solution of an ordinary differential equation(2.88) which we derived.

Chapter 3 applies the general methods of uniform caustic asymptotics to axially symmetric sphericalaberration with a cuspoid 3-ray topology. We describe arbitrary non-paraxial strongly spherically aberratedfocusing by matching the solution of geometrical optics with a wave field built by the Bessoid integral andits derivatives in the region where the rays are well separated and the stationary phase approximation ofthe Bessoid integral works. By this means the caustic divergences are removed and the result is a light fieldwhich is finite everywhere. The matching is expressed by the Ansatz (3.6) and the deduced equations for theamplitudes and phases are (3.18) and (3.19). The matching procedure was first applied to the scalar case. Itresults in expressions for the Bessoid coordinates and amplitudes in terms of real space coordinates, (3.44),(3.46), (3.47) and (3.54), respectively. These general formulas can be significantly simplified on and near theaxis, (3.66), (3.67), (3.86) and (3.84).

The central part of the Bessoid integral is essentially a Bessel beam with variable cross section due tothe variable angle of the non-paraxial rays. The width of such a scalar beam is given by (3.69). It showsthat the local lateral confinement of the Bessoid integral is always smaller than in the focus of an ideal lenswith the same numerical aperture. Besides, the largest possible apertures can be physically realized withinspherical aberration, which is hardly possible with ideal, non-aberrated and non-paraxial lenses. All this isachieved at the expense of longitudinal confinement.

The field on the axis can be expressed in a general and compact form (3.87), illustrating the relativecontribution of the paraxial ray and the non-paraxial cone of rays. Note that this expression holds even forvectorial problems. The universal result (3.64) gives the position of the diffraction focus for arbitrary axiallysymmetric systems in a very simple and intuitive way. These questions are further illustrated for the caseof a focusing sphere, where the concept of compensated sagittal divergence is introduced and discussed bymeans of equation (4.69).

The original problem of focusing by a refracting sphere is a particular case implying an axially symmetriccuspoid topology. First, we derived compact formulas for the transformation of the meridional and sagittalradii of curvature of the rays undergoing refraction on a sphere, (4.28) and (4.33), respectively. Using these

1 Following Kravtsov, Y. A. and Orlov, Y. I., Caustics, Catastrophes and Wave Fields, Springer Series on Wave Phenomena(Volume 15), Springer-Verlag, Second Edition (1999), p. 92 et seq.

Conclusions 63

results, we computed the geometrical optics solution for an incident plane wave (with linear polarization)with and without Fresnel transmission coefficients, (4.48) and (4.60). The comparison between these twocases demonstrates that the Fresnel coefficients are responsible for a decrease of intensity along the axis fromthe focus towards the sphere surface (figure 4.19). The oscillations in intensity are due to the interferencebetween the paraxial ray and the cone of non-paraxial rays. On the axis the electric field coincides withthe direction of polarization and the scalar expressions are exact. The field can be written in a simple form(4.70), allowing the fast numerical determination of the maximum intensity and its position behind thesphere over a broad range of parameters (figure 4.11). The approximate physical condition for the diffractionfocus is still given by (3.64). Taylor expansion in the inverse Mie parameter 1/k a results in approximation(4.72), assuming small angles. In contrast to classical results for weak spherical aberration, this expressionillustrates the change of the diffraction focus with the wavelength and the size of the sphere.

The off-axis vectorial electric field is dependent on the azimuthal angle and not axially symmetric any-more. But the linear polarization breaks the symmetry only with respect to amplitudes, whereas the pictureof rays and thus the cuspoid type of catastrophe remains unchanged. This amplitude modulation requiredthe introduction of canonical higher-order Bessoid integrals (4.76), for which we found a recurrence relation(4.88) and near-axis approximations (4.90). The equations for the Bessoid coordinates do not change for thehigher orders, while the amplitude equations are modified in a systematic way (4.99). The developed ap-proach can be directly applied to arbitrary non-paraxial vectorial focusing of axially symmetric wavefronts,even if the amplitude of the focused field does not have axial symmetry at all.

The vectorial electric field behind the sphere, given by (4.100), shows maxima and minima depending onthe azimuthal angle (figure 4.15). The magnetic field and the Poynting vector are calculated in a straight-forward way, (4.111) and (4.114). The electric field distribution immediately after the sphere reveals twostrong maxima (figure 4.16). Our analysis clearly shows that they are unrelated to near field effects, mainlyrelated to the axial field component and stem from purely geometrical considerations. This permits simpleestimations for the distance of the maxima from the axis (4.103).

Finally, the results of the Bessoid matching procedure were compared with the theory of Mie (figuresin section 4.5). There is very good agreement for large wavenumbers, that is for Mie parameters k a > 30.The agreement is not perfect for distances behind the sphere of the order of the wavelength (4.116), mostprobably due to evanescent surface contributions, which are taken into account in the rigorous Mie theory.When the ratio of sphere radius to wavelength decreases, the geometrical optics approximation breaks downand deviations increase everywhere. Here, still the agreement is worse close to the sphere and remains betterat large distances.

We derived our expressions first for the scalar case of spherical aberration and adapted them to thevectorial electromagnetic problem originating in optics. But we want to emphasize that the whole approachand in fact the majority of the developed formulas can be applied in other areas of physics where axiallysymmetric focusing is of importance, e.g. radio wave propagation, acoustics, scattering theory, semiclassicalquantum mechanics, etc.

Concluding, let us briefly enumerate several possibilities to extend and refine the developed model forthe sphere. Weak absorption can be incorporated easily, for it just changes the amplitudes along the raysand the transmission coefficients. Strong absorption additionally modifies Snell’s law of refraction, preservingthe axial symmetry of course. One can consider incoming radially or azimuthally polarized beams, whichare known to produce better resolution than linear polarization2. One might also want to calculate theinterference of the diffracted light with the original incident wave or the interference of the light refractedby several spheres or arrays of spheres. The latter yields interesting secondary patterns3 related to the socalled Talbot effect4. Finally, the influence of a finite aperture could be studied. This means, for example, achange in the ray structure and the field distribution right behind the sphere for refractive indices n <

√2.

2 Dorn, R. et al., Sharper focus for a radially polarized light beam, Phys. Rev. Lett. 91(23) (2003), 233901-1–4.3 Experimental achievements are reported in Bauerle, D. et al., Laser-induced surface patterning by means of microspheres,

Lambda Physik 60 (2002), 1–3.4 See Berry, M. V. et al., Quantum carpets, carpets of light, Physics World (June 2001), 39–44.

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World Wide Web

Wolfram Research, http://mathworld.wolfram.com and http://functions.wolfram.com

Curriculum Vitae

Personal History

Name Johannes KoflerProfessional Address Institute for Applied Physics

Johannes Kepler University of LinzAltenbergerstraße 69, 4040 Linz, AustriaTelephone: ++43 732 2468 9257Email: [email protected]

Home Address J.W.-Kleinstraße 36/8, 4040 Linz, AustriaTelephone: ++43 732 250879

Date of Birth June 16, 1980Place of Birth LinzCitizenship AustrianLanguages German (native), English, LatinMarital Status SingleParents Margit (nee Klambauer) and o. Univ.-Prof. Dr. Herbert Kofler

Education

September 1990 – June 1998 Grammar school (1990/1991: Bundesrealgymnasium Auhof, Linz; 1991–1998: Bundesrealgymnasium Viktring/Klagenfurt)

June 1998 Matura diploma (with highest distinction)July 1998 – March 1999 Obligatory military service in Villach and KlagenfurtOctober 1999 Enrolment in the diploma studies of Technical Physics at the Johannes

Kepler University Linz (JKU), Faculty of Engineering and Natural Sci-ences (Technisch-Naturwissenschaftliche Fakultat, TNF)

July 2001 First diploma certificate (with highest distinction)October 2003 – September 2004 Diploma thesis at the Institute for Applied Physics

Professional Experience

August 2002 Traineeship at the Institute for Applied PhysicsOctober 2002 – January 2003 Tutorial for the Theoretical Physics course I (Mechanics) at the Institute

for Theoretical Physics (Prof. Eckhard Krotscheck)March – June 2004 Scientific co-worker at the Institute for Applied Physics, supporting the

Advanced Laser Physics course (Prof. Dieter Bauerle)Scientific co-worker at the Institute for Theoretical Physics, support-ing the Theoretical Physics course IV (Thermodynamics and StatisticalPhysics, Prof. Harald Iro) and V (Advanced Quantum Mechanics, Prof.Franz Schwabl, guest Professor from Munich)

Honours

November 2001 Merit scholarship (JKU, TNF) for the first diploma period (1999–2001)November 2002 Merit scholarship (JKU, TNF) for the year 2001/02November 2003 Merit scholarship (JKU, TNF) for the year 2002/03

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