propagation of partially coherent light in optical systemsgross/... · 2016. 11. 28. ·...
TRANSCRIPT
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www.iap.uni-jena.de
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
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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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