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$: Reconstruction of the radial refractive index profile of ... · Reconstruction of the radial refractive index pro le of optical waveguides from lateral di raction patterns Seminar$
Reconstruction of the radial refractive index pro�le ofoptical waveguides from lateral di�raction patterns

Seminar der AG Computerorientierte Theoretische Physik, 28. Juni 2017

Gunnar Clauÿen1

1Fachbereich IngenieurwissenschaftenJade Hochschule am Standort Wilhelmshaven

OUTLINE

1 Introduction and motivation

2 TheoryMie theory for cylindersInverse problems: IRGN algorithm

3 The research project

4 Results: Diameter of a homogeneous cylinder

5 Work in Progress: Strati�ed cylinders

6 Conclusion and outlook

28.06.2017 Reconstruction of the radial refractive index pro�le... 1

INTRODUCTION AND MOTIVATION (I)

Motivation

Fibres in communication technology (waveguides)

Distinct types: Gradient and step-index �bres

Desired: Measurement and control of parameters (refractive index pro�le)during production, i.e. the drawing process

Current knowledge

Methods exist for the determination of the outer diameter

These methods work well for intransparent cylinders

No algorithm is known for the refractive index pro�le n(d) or {dj , nj}


INTRODUCTION AND MOTIVATION (II)

General approach

Illumination of a cylinder with plane-wave incidence, ζ = 90◦

Refraction on layer boundaries and di�raction on edges

Di�raction pattern is measured through an array of CCD cells

Evaluation of the di�raction pattern for parameter determination

Experimental setup

θ

k

r=d/2

yCCD

y

x

nm

λ

ns

Please note

Setup is the same for strati�edcylinders

Distance yCCD has to be knownaccurately

All e�ects add up to scattering


INTRODUCTION AND MOTIVATION (III)

Theoretical approach: Inverse problem

Direct problem: Parameters known, result unknown

Inverse problem: Result known, parameters unknown

Illustration of an inverse problem in [Bohren & Hu�man 1983]


MIE THEORY FOR CYLINDERS (I)

Light scattering: Basics of Mie theory

Propagation of EM waves is expressed through Maxwell equations

Especially: Vector wave equation ∇2 ~E + k2 ~E = 0 and continuity atboundaries, [ ~E2 − ~E1]× n̂ = 0

Mie theory: Transformation to speci�c coordinates corresponding to geometryallows expansion of scattered �elds into Bessel functions

Scattering on cylinders

Incident wave: ~Ei =∑∞n=−∞En ~N

(1)n

Scattered wave: ~Es =∑∞n=−∞En

[bnI ~M

(3)n + ianI ~N

(3)n

]For normal incidence (ζ = 90◦), only z-components of ‖-polarized

components are relevant: Es‖(z) = −∞∑

n=−∞Enbn‖Nn(z)


MIE THEORY FOR CYLINDERS (II)

Scattering coe�cients

General calculation for homogeneous (r = 1) or strati�ed (r > 1) cylinders:

bn=mrJn(xr)[J′n(mrxr)+Tr−1

n N′n(mrxr)]−J′n(xr)[Jn(mrxr)+Tr−1n Nn(mrxr)]

mrHn(xr)[J′n(mrxr)+Tr−1n N′n(mrxr)]−H′n(xr)[Jn(mrxr)+T

r−1n Nn(mrxr)]

(1)

Tsn=

msJn(xs)[J′n(msxs)+Ts−1n N′n(msxs)]−J′n(xs)[Jn(msxs)+Ts−1

n Jn(msxs)]

msJn(xs)[J′n(msxs)+Ts−1N′n(msxs)]−N′n(xs)[Jn(msxs)+Ts−1n Jn(msxs)]

(2)

Recursive calculation: Layers s = 1...r, start with T0 = {0}Depends only on the relative refractive index ms =

ns

ns+1and the �size

parameter� xs = ks+1 · rs of the s-th layer

Outline of the calculation of the di�raction pattern

1 Calculate scattering coe�cients bn up to nmax = dx+ 4 · 3√x+ 2e

2 Transform CCD array to polar coordinates (x, y) 7→ (ρ, ϕ)

3 Calculate vector-harmonic generating functions Nn(z) for these coordinates

4 Calculate sum Es‖(z) = −∑nmax

−nmaxEnbn‖Nn(z).


MIE THEORY FOR CYLINDERS (III)

...and it works!

Agreement for an intransparent d = 370 µm cylinder, λ = 632.8 nm

x-position [mm]-10 -8 -6 -4 -2 0 2 4 6 8 10

Inte

nsi

ty [

a.u.]

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

MeasurementCalculationDifference


IRGN ALGORITHM (I)

Inverse problem (now formally...)

Generally, a operator F on a parameter set r creates (maps to) a far-�eld

pattern u∞, i.e. a simple operator equation:

F : r 7→ u∞ or F(r) = u∞ (3)

Direct problem: F and r are known, u∞ is to be determined

Inverse problem: F and u∞ are known, r is sought

Solution of the inverse scattering problem

Solution through the iteratively regularized Gauss-Newton (IRGN) algorithm:

||F ′[rn]hn + F(rn)− uδ∞||2 + αn||hn + rδn − r0||2!= min (4)

In each iteration n: Calculate alteration hn of the parameter set rn

Iteration terminates after N steps, if ||F(rN )− uδ∞||2 ≤ τδ


IRGN ALGORITHM (II)

Regularisation

Special treatment for noisy data, weighting of far-�eld pattern

The step update hn of parameters rn is calculated according to

hn = −F′[rδn]

∗(F (rδn)− uδ∞) + αn(rδn − r0)

F ′[rδn]∗F ′[rδn] + αnI

(5)

with regularisation parameter αn = αn0 , α0 = 1/2

Notes

Derivative F ′[rn] to the parameters rn has to be known

For homogeneous cylinders: Analytcally...

For strati�ed cylinders: Numerically...

Analytical calculation does not pay o� via a considerable time advantage!


THE RESEARCH PROJECT (I)

General properties

Research project at Jade Hochschule

Applied and carried out by Prof. Dr. Werner Blohm since 2014

Aim: Improvement of the precision of di�raction-optics methods

Doctoral research is embedded in the project through Jade2Pro

Several related �milestones� have been proposed for both the applicant (Prof.Blohm) and the doctoral researcher (me)

�Applicant� side ended in 2016, but research will be supported further


THE RESEARCH PROJECT (II)

Structogram of the Project


THE RESEARCH PROJECT (III)

Formal development of the doctorate

Cooperation with Zentrum für Technomathematik (ZeTeM), Fachbereich 3,Bremen University

Supervision by Prof. Dr. Armin Lechleiter, head of AG Inverse Probleme

Guest in Bremen in WS 2015/16, seminar talk

Beginning of WS 2016/17: Ph.D. student at Bremen

In WS 2016/17: Recognition of the research topic by FB3

Complete Title: �Bestimmung des radialen Brechungsindexpro�ls optischer

Lichtwellenleiter aus lateralen Streulichtverteilungen�

In WS 2016/17: Project successfully evaluated by Jade2Pro

Project-related �publicity� so far

Talks at 2nd, 3rd und 5th Jade2Pro-Kolloquium, Oldenburg

Poster presentation at the DPG Spring Meeting 2017 of the Atomic,Molecular, Plasma Physics and Quantum Optics Section (SAMOP), Mainz


RESULTS: CYLINDER (I)

The actual inverse scattering problem

Operator F : Mie theory for cylinder scattering

Parameters r: Only diameter d, everything else is �xed (ns, nm, yCCD, λ)

Far-�eld u∞: Pattern on CCD array at distance yCCD

Fréchet derivative F ′ for d is analytically known

(d-dwahr

)/λ-10 -8 -6 -4 -2 0 2 4 6 8 10

||F(

d)-

F(d

wah

r)||

2

0

0.5

1

1.5

2

2.5

λ = 523 nmλ = 633 nmλ = 780 nmλ = 1046 nm

Encountered �problem�

Over d, the patterns resemble each other

An IRGN algoritum iteratively minimizes theresidual ||F(d)− u∞||2The residual, when compared to a referencedreal, exhibits local minima...

Ill-posed problem (i.e. non-bijective)

Danger of false convergence


RESULTS: CYLINDER (II)

�Solution�: Modi�cation of the IRGN algorithm

But: Minima are spaced periodically by δmin,s ≈ 2.18λ

This property may be utilized...⇒ Detect local convergence and then �jump� by ±2.18λ

Example for such a way of the algorithm

-18 -16 -14 -12 -10 -8 -6 -4 -2 0 20

0.2

0.4

0.6

0.8

1

1.2

local search

jumps and local searches afterwards

global minimum

resi

dual

||f

(d)-

f(d

real)|

| 2

distance (d-dreal)/λ


RESULTS: CYLINDER (III)

Where does the periodicity δmin,s ≈ 2.18 stem from?

In the present case, ns = 1.458 ≡ ms =ns

nm...

Research: Compare di�raction patterns for a range of d with a reference dreffor several values of ms, measure minima spacing...

Dependence of minima spacing

0.0

2.0

4.0

6.0

8.0

10.0

1.0 1.5 2.0 2.5 3.0

Abst

and

der M

inim

a de

s Res

iduu

ms

||F(

d)-F

(dw

ahr)||

2

Brechungsindex ns

Daten

Fitfunktion f(ns) = a+b/(ns-1)c

Funktion 1/(ns-1)

-20.0

0.0

20.0

40.0

60.0

80.0

100.0

1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10

Abst

and

der M

inim

a de

s Res

iduu

ms

||F(

d)-F

(dw

ahr)||

2

Relativer Brechungsindex ms

Daten: 2 Schichten, Variation von Schicht 1

Fitfunktion f(ns) = 2/3*(ms-1)-1 Result

δmin,s =1

ms − 1

...thus:1

1.458−1 = 2.183...


RESULTS: CYLINDER (IV)

Ideas for further improvement

Improvement of initial values: Sample for a few values of d around d0, choosevalue of least residual

Approximation through intransparent cylinders:

Subtraction of the 3rd sinusoid component from the di�raction pattern

approximates the pattern for a intransparent cylinder of the same dFor intransparency, only a single global minimum exists...

GN iterations allow an approximation of d0 to dreal

Exact determination of dreal is however not possible this way

�Plausibility check�: Are minima also present at dN for further observingangles θ? Otherwise, it's certainly no reliable result...


RESULTS: CYLINDER (V)

Precision of the algorithm: Numerical evaluation

n = 1.4, N = 2910 random dreal = 110...150 µm and d0 ∈ dwahr ± 25λ

The actual diameter dreal is met precisely (i.e. |dN − dreal| < 0.1 µm,de�nition of a �success�) for 99.62% of the cases

The calculation time exhibits no visible dependence on the distance betweeninitial and actual values of d


RESULTS: CYLINDER (VI)

Precision in presence of noise

Noise level [%]0 1 2 3 4 5 6 7 8 9 10

Succ

ess

rate

[%

]

20

30

40

50

60

70

80

90

100

Med

ian d

evia

tion

|d-d

real|

[nm

]

0

1.25

2.5

3.75

5

6.25

7.5

8.75

10

Noise: Multiplicative, normally distributed

�Noise level�: Scaling factor for the width(variance) of the distribution

Deviation of the result calculated only for�successful� cases

Note: �Failures� still �nish within localminima

Summary

Method for diameter determination is very precise

Precision drops linearly with noise level

Improvements lead to constant calculation times


STRATIFIED CYLINDER (I)

Basics

Aim: Extension of the IRGN algorithm for strati�ed cylinders

Calculation of di�raction patterns: Only di�erent scattering coe�cients

Experience from homogeneous cylinders: Periodic local minima

Expected di�culties

Parameters to be found: Layer diameters dj

More complex operation, more parameters must be optimised

Especially: How can the �jumps� of the modi�ed IRGN algorithm beimplemented?


STRATIFIED CYLINDER (II)

Example: J = 2 layers, dref = [50, 120] µm, n = [1.55, 1.5]

Periodic behaviour and a global minimum at d ≡ dref exist

ResidualsDiagonal ADiagonal B

-20-15

-10-5

0 5

10 15

20

Distance (d1-d1,ref)/λ

-2-1.5

-1-0.5

0 0.5

1 1.5

2

Distance (d2-d2,ref)/λ

0 0.2 0.4 0.6 0.8

1


STRATIFIED CYLINDER (III)

Further �ndings

Periodicity of minima for layer j < J : δmin,s =2

3

1

ms − 1

Remark: Refractive indices di�er only marginally between layers,|nj − nj+1| = 0.001...0.02⇒ nj+1 = 1.5 : mj ≈ 1.0006...1.013

Thus: For inner layers �large� periodicity can be expected,δmin,s ≈ 50....1000λ

For the outermost layer j = J the jump from nJ to nm is much larger ⇒ cf.homogeneous cylinder


STRATIFIED CYLINDER (IV)

Even more further �ndings

The residuals along the diagonals in parameter space (d1, d2):

0

0.2

0.4

0.6

0.8

1

1.2

1.4

-4 -3 -2 -1 0 1 2 3 4 5

Res

idua

l ||f

(d)-

f(d ref

)|| 2

Factor (d-dref)/(lambda*δj,min)

Diagonal ADiagonal B


STRATIFIED CYLINDER (V)

Ideas for layer diameter determination

IRGN steps lead into a �trench�

The �trench� is linear and described through the periodicities δj,min

Jumps should lead from one �trench� to another (diagonal A)

Motion within one �trench�: local Minima (diagonal B)?

Summary: A combination of IRGN, jumping and sampling methods seemspromising for determing both diameters d1 and d2

...and how can that be applied to J > 2?


FURTHER OUTLOOK

Aims to be reached

Milestone: Algorithm for strati�ed cylinders is to be completed

Experimental veri�cation for homogeneous cylinder (experiment exists)

Optionally: Generalized Lorenz-Mie theory for spherical waves

Temporal constraints

Current position is running until April 2018

Probably extended by one year from remaining funds

Current di�culty: A. Lechleiter out of o�ce until further notice


REFERENCES

References

Bohren, C.F.; Hu�man, D.R.: �Absorption and Scattering of Light by Small Particles�,

John Wiley & Sons, New York, 1983

Gerlinski, V.: �Bestimmung geometrischer Parameter von Glasfasern aus

Streulichtmustern�, Master thesis, Universität Bremen, 2016

Quinten, M.: �Optical Properties of Nanoparticle Systems. Mie and Beyond�, Viley-VCH,

Weinheim, 2011

Hohage, T.; Schormann, C.: �A Newton-type method for a transmission problem in

inverse scattering�, Inverse Problems 14, 1207�1227, 1998

Schäfer, J.-P.: �Implementierung und Anwendung analytischer und numerischer Verfahren

zur Losung der Maxwellgleichungen für die Untersuchung der Lichtausbreitung in

biologischem Gewebe�, Ph.D. thesis, Ulm, 2011


Thanks for your attention!

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