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Propagation of partially coherent light

in optical systems

Minyi Zhong a, Herbert Gross a,b

a Institute of Applied Physics, University of Jena

b Fraunhofer Institute for Applied Optics and Precision Engineering IOF, Jena

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Optical communications

Optical tomography

Photolithography

Figures from apcmag.com; lickr.com; elprocus.com

Applications of partially coherent light

Microscopy

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Goal of our work: to investigate modeling methods to propagate

partially coherent light through various optical components.

Contents of this talk:

1. Introduction

2. Propagation method 1: phase space

3. Propagation method 2: modal expansion

4. Conclusions

Contents

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Introduction to partial coherence

Complete

incoherence:

Complete

coherence: He-Ne laser

Patial coherence

in space:

Correlation function: 𝛤 𝑟1, 𝑟2 = 𝐸∗ 𝑟1 𝐸(𝑟2)

Normalized 𝛤 between 0 and 1: degree of coherence

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Method 1: phase space – introduction

Pointsource:

Partially coherent source:

x

z

Rays

Diffraction

x

u

Δu

x

I

local Poynting vector 𝑆

Integration over u axis

x

z

Δu

x(position)

u (angle)

x u

x

u

Real space: Phase space:

xo

xo

Definition:

𝑊 𝑥, 𝑢 = 𝛤 𝑥 +∆𝑥

2, 𝑥 −

∆𝑥

2exp −𝑖

2𝜋

λ𝑢∆𝑥 d∆𝑥

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Negative values indicate destructive interference, thus useful for analyzing diffractive

components.

Method 1: phase space – Wigner function

Double

slits:Phase

space:

x

u

Paraxial propagation – coordinate transform:

𝑥𝑜𝑢𝑡𝑢𝑜𝑢𝑡

=𝐴 𝐵𝐶 𝐷

𝑥𝑖𝑛𝑢𝑖𝑛

𝑊𝑜𝑢𝑡 𝑥𝑜𝑢𝑡 , 𝑢𝑜𝑢𝑡 = 𝑊𝑖𝑛 𝐴𝑥𝑖𝑛 + 𝐵𝑢𝑖𝑛, 𝐶𝑥𝑖𝑛 + 𝐷𝑢𝑖𝑛

(Reference: Testorf, Phase-Space Optics, 2006)

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Method 1: phase space – diffractive elements

x (mm)

∆𝑧

∆𝜑 = 𝑛 − 1∆𝑧

λ

a) Surfaces with discontinuity in space:

ii. Two steps:

∆𝑧

Δ𝜑 = 5.5

i. Step phase:

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sin u = mλ/d

𝑑

∆𝑧

iii. Grating: iv. Kinoform lens:

Δ𝜑 = 5.5

Method 1: phase space – diffractive elements

a) Surfaces with discontinuity in space:

Phase space right behind the element:

Phase space right behind the element:

Δ𝜑 = 1.4

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Method 1: phase space – diffractive elements

b) Surfaces with discontinuity in slope:

i. Axicon: ii. Lens array:

Phase space at 𝑧 = 0 𝑚𝑚: Phase space at plane A:Phase space at plane B:Phase space at plane C:

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Method 1: phase space – summary

Advantages: 1) Visualizing light in positions and angles,

2) Paraxial propagation with ABCD matrices,

3) Combining ray optics and wave optics, diffraction effects

included.

Disadvantages: 1) Non-paraxial propagation is more complicated,

2) For light with two transverse dimensions, we need a 4D

Wigner function,

3) components beyond thin element approximation, e.g.

waveguides.

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Method 2: modal expansion – introduction

• A partially coherent beam = an incoherent sum of modes.

• Each mode is a coherent beam.

• No interference between every two modes.

• 𝛤 𝑟1, 𝑟2 = 𝑛 λ𝑛𝜙𝑛∗ 𝑟1 𝜙𝑛 (𝑟2)

Orthogonal modes: Non-orthogonal modes:

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Method 2: modal expansion – waveguide

a) Slab waveguide with a step-index profile:

• Gibbs phenomenon: flat-top profile unachievable with finite modes, even at a larger propagation distance.

• Waveguide: always finite modes.

Transverse intensity @ z = 2000 mm

Phase space at 𝑧 = 1 𝑚𝑚 Phase space at 𝑧 = 6 𝑚𝑚

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b) waveguide with a parabolic-index profile:

Method 2: modal expansion – waveguide

Ray paths in Zemax: (𝑧 = 70 𝑚𝑚, Gaussian apodization)

Influence of partial coherence on FWHM of the focus:

Gaussian-Schell beam (waist 150 µm, correlation length 50 µm)

Coherent Gaussian beam:FWHM 18 µm at focus

𝑧 = 15.7 𝑚𝑚

𝑧 = 15.7 𝑚𝑚

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Conclusions about two modeling methods

Phase space Modal expansion

Optical effects Visualization of ray optics and wave

optics, especially diffraction effects.

Direct solution to the complex fields

of each mode.

Propagation Paraxial: ABCD matrix,

Nonparaxial: Helmholtz wave fields.

Wave propagation for each mode,

paraxially or non-paraxially.

Optical

components

Convenient with thin element

approximation, e.g. discontinuous

surfaces.

Suitable for components beyond

thin-element approximation, e.g.

waveguides

Computer memory 4D data for two transverse

dimensions: 𝑊(𝑥, 𝑦, 𝑢, 𝑣).

3D data for two transverse

dimensions: 𝜙𝑛(𝑥, 𝑦).

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