Model-free extraction of refractive index
from measured optical data
A Tool for Refractive InDex Simulation
Martina Schmid, Guanchao Yin, Phillip Manley
Helmholtz-Zentrum Berlin,
Nanooptical concepts for photovoltaics
Motivation
Thin film optics
having to deal with multiple reflections
and requiring refractive indices
often only rely on optical measurements.
Contents
• Basic Principles
• Transfer Matrix Method
• Multilayer Stack
• Comparison to Experiment
• Advanced Features
• Surface Roughness
• Inhomogeneous Layers
• Effective Medium
• User Interface
• Outlook
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Basic Principles – Transfer Matrix Method
Basic Principles: Transfer Matrix Method
Advanced Features
User Interface
one wave with positive direction(E+)
Superposition of electric field
-
one wave with negative direction(E )
Propagating through an interface:
𝐸𝑖 +
− =
𝐸𝑖
1
𝑡𝑖 𝑗
,
1
𝑟𝑖,𝑗
𝑁 −𝑁
2𝑁𝑗
2𝑁
𝑟𝑖,𝑗 =-𝑟𝑗,𝑖 =𝑁𝑖+𝑁𝑗,
𝑖
𝑟𝑖,𝑗 𝐸𝑗 +
−
1 𝐸𝑗
𝑡𝑖,𝑗 = 𝑁 +𝑁𝑖 , 𝑡𝑗,𝑖 =𝑁 +𝑁 ;
𝑗
𝑖
𝑗
𝑖
𝑗
𝑟, 𝑡 are, respectively, the complex amplitude reflection and
transmission Fresnel coefficients; 𝑁 is the complex
refractive index of the layer
Propagating within a layer:
𝐸𝑖 +
Φ−1
− =
𝐸𝑖
o
Propagation through
mediums at
normal incidence
o
Φ
E,i+
E,i−
Φ=𝑒
−𝑖
2𝜋
𝑁𝑑
𝜆 𝑖
Where d is the thickness of medium, ω is the frequency of
the propagating light and c is the speed of light
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Oblique incidence
Basic Principles: Transfer Matrix Method
Medium i
ᆈ𝑖
Advanced Features
User Interface
At interface
ᆈ𝑖
Within the layer
P polarization:
Medium j
ᆈ𝑗
𝑟𝑖,𝑗 =
𝑡𝑖,𝑗
𝑁𝑗 𝑐𝑜𝑠𝞱𝑖 − 𝑁𝑖 𝑐𝑜𝑠𝞱𝑗
𝑁𝑗 𝑐𝑜𝑠𝞱𝑖 + 𝑁𝑖 𝑐𝑜𝑠𝞱𝑗
2𝑁𝑖 𝑐𝑜𝑠𝞱𝑖
=
𝑁𝑗 𝑐𝑜𝑠𝞱𝑖 + 𝑁𝑖 𝑐𝑜𝑠𝞱𝑗
S polarization:
𝑟𝑖,𝑗
Φ=𝑒
−𝑖
2𝜋
𝑁 𝑑/𝑐𝑜𝑠𝞱𝑗
𝜆 𝑚
𝑁𝑖 𝑐𝑜𝑠𝞱𝑖 − 𝑁𝑗 𝑐𝑜𝑠𝞱𝑗
=
𝑁𝑖 𝑐𝑜𝑠𝞱𝑖 + 𝑁𝑗 𝑐𝑜𝑠𝞱𝑗
𝑡𝑖,𝑗 =
2𝑁𝑖 𝑐𝑜𝑠𝞱𝑖
𝑁𝑖 𝑐𝑜𝑠𝞱𝑖 + 𝑁𝑗 𝑐𝑜𝑠𝞱𝑗
Fresnel coefficiencts for
oblique incidence
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Coherent Layers – Interference Effects
Basic Principles: Multilayer Stack
Advanced Features
User Interface
Validity condition:
𝑑
n
∆𝜆𝐶𝑜ℎ𝑒𝑟𝑒𝑛𝑐𝑒
θ
<< 1
Phase difference between
transmission orders:
𝛿=
2𝜋
2𝑛𝑑 cos 𝜃
𝜆
When δ = 2𝑚𝜋 , 𝑚 ∈ ℕ there will
be constructive interference in T
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Incoherent Layers & Substrate Layers
Basic Principles: Multilayer Stack
Advanced Features
User Interface
To removed coherency, calculate the
Intensity instead of the Electric Field
𝐼 = 𝐸 ∗ 𝐸 = |𝐸0 |2 𝑒 𝑖𝑘𝑛𝑑 𝑒 −𝑖𝑘𝑛𝑑 = |𝐸0 |2
phase information
𝑡
∆𝜆𝐶𝑜ℎ𝑒𝑟𝑒𝑛𝑐𝑒
>> 1
Phase relationships between
interior reflections is destroyed –
therefore there is no interference
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Multilayer Stack
Basic Principles: Multilayer Stack
Advanced Features
User Interface
Coherent Stack
• Includes interference
• Typical thickness 0 ~ 2000 nm
Incoherent Layer
• Interference “turned off”
• Typical thickness 1 mm
• 9 Total layers implemented in RefDex
• Combine coherent and incoherent layers in any order
• For R,T calculation, d, n and k must be known for all layers
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Input Spectrum – R and T
Basic Principles: Comparison to Experiment
Advanced Features
User Interface
Absorbing Region
• Reflection loses
coherency peaks
• Transmission
drops to zero due
to absorption
Transparent Region
• R and T both show
coherency peaks
• R does not drop to
0 due to reflection
from glass
substrate
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Comparison to Experiment
Basic Principles: Comparison to Experiment
Advanced Features
User Interface
An example:
thin film
on substrate
10
Problem of Uniqueness
Basic Principles: Comparison to Experiment
Advanced Features
𝑅𝑐𝑎𝑙 𝑛 λ , 𝑘 λ
− 𝑅𝑒𝑥𝑝 λ
=0
𝑇𝑐𝑎𝑙 𝑛 λ , 𝑘 λ
− 𝑇𝑒𝑥𝑝 λ
=0
𝐹 𝑛, 𝑘 = 𝑅𝑐𝑎𝑙 𝑛 λ , 𝑘 λ
+ 𝑇𝑐𝑎𝑙 𝑛 λ , 𝑘 λ
One Physically
Meaningful Solution
Choose the n, k values
which minimise the
difference between our
model and experiment
− 𝑅𝑒𝑥𝑝 λ
− 𝑇𝑒𝑥𝑝 λ
𝐹 𝑛′, 𝑘′ = 𝐹 𝑛′ ′, 𝑘′′ = 0
User Interface
Adding these equations
together we get a function 𝐹
which takes n and k as input
Problems arise because two
different n,k input pairs can both
equal zero!
Many Unphysical
Solutions
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Problem of Uniqueness – Physical Picture
Basic Principles: Comparison to Experiment
Advanced Features
User Interface
Spurius solution branches
Physically meaningful solution
Results need to be
interpreted – More
on this Later!
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Determination of optical constants in multiple-layer configuration
Basic Principles: Comparison to Experiment
Advanced Features
User Interface
Take the configuration of CIGSe/TCO/glass substrate as an example:
G. Yin et al., Influence of substrate and its temperature on the optical constants
of CuIn1-xGaxSe2 thin films, accepted for Journal of Physics D: Applied Physics
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Surface Roughness – Effect on R and T
Basic Principles
Advanced Features: Surface Roughness
User Interface
Absorbing Region
• Reflection Strongly
Reduced
• Transmission Slightly
Reduced
Transparent Region
• R and T reduced
prefferentially at
coherency peaks
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Modified Transfer Matrix Method – Scalar Scattering Theory
Basic Principles
Advanced Features: Surface Roughness
Scalar Scattering Theory
Rough Interface
′
𝑟𝑖,𝑗
= 𝑟𝑖,𝑗 𝑒𝑥𝑝 −2(2𝜋𝜎/𝜆)2 𝑛𝑖 2
Medium a
Modified
Fresnel
coefficients
Medium b
User Interface
′
𝑟𝑗,𝑖
= 𝑟𝑗,𝑖 𝑒𝑥𝑝 −2(2𝜋𝜎/𝜆)2 𝑛𝑗 2
2
′
𝑡𝑖,𝑗
2𝜋𝜎
= 𝑡𝑖,𝑗 𝑒𝑥𝑝 −
𝜆
2
′
𝑡𝑗,𝑖
2𝜋𝜎
= 𝑡𝑗,𝑖 𝑒𝑥𝑝 −
𝜆
(𝑛𝑖 − 𝑛𝑗 )2 /2
(𝑛𝑗 − 𝑛𝑖 )2 /2
• 𝜎 is the interface roughness
• Gives us the loss of specular beam
intensity due to interface roughness
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Modified Transfer Matrix Method - Examples
Basic Principles
Advanced Features: Surface Roughness
Determination of
optical constants
User Interface
G. Yin et al.,The effect of surface
roughness on the determination of
optical constants of CuInSe2 and
CuGaSe2 thin films, J. Appl. Phys.,
133, 213510 (2013)
σ = 9nm
σ = 20nm
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Inhomogeneous Layers – Effect on R and T
Basic Principles
Advanced Features: Inhomogeneous Layers
User Interface
Absorbing Region
• Small reduction in
R and T
Transparent Region
• Coherency
reduced for both R
and T
• Transmission
strongly reduced
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Inhomogeneous Layers – Coherent / Incoherent Decomposition
Basic Principles
Advanced Features: Inhomogeneous Layers
User Interface
a)
b)
e)
c)
d)
a)
b)
c)
d)
2D slice through the 3D inhomogeneous film
Overlay a rectangular grid
The resulting discretised representation of the film
Layers containing voids can be modelled incoherently allowing the use of
average layer thicknesses
e) This reduces the number of transfer matrix calculations to 4
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Inhomogeneous Layers – Coherent / Incoherent Decomposition
Basic Principles
Advanced Features: Inhomogeneous Layers
𝑅𝑐𝑎𝑙𝑐 𝑛, 𝑘 = 𝑤𝑐 𝑅𝐶 + 𝑤𝐼 𝑅𝐼
User Interface
(Same equations for T
not shown here)
Standard Calculation
Replace propagation operator inside inhomogeneous layer with:
𝑷𝑖 =
𝑛 𝑚
𝑀
𝑚=1 𝑷𝑚
𝑛 𝑚 ,𝑛 𝑚−1
𝑫𝑚,𝑚−1
𝑛(0)
𝑷0
, 𝑛 𝑚 =
𝑛 0 , 𝑚 = 𝑒𝑣𝑒𝑛, 𝑚 ≠ 0
¬𝑛 0 , 𝑚 = 𝑜𝑑𝑑
𝑛 0 = 𝑛𝑖∗
or
𝑛 0 = 𝑛𝑣∗
• Void scattering as from a rough surface. (Slide 13)
• Requires statistical knowledge of 3D void distribution as input
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Inhomogeneous Layers – Modelling Distribution of Voids
Basic Principles
Advanced Features: Inhomogeneous Layers
User Interface
• Measurement of real 2D surface
used to generate 3D distribution
• From 3D distribution we obtain
inputs for the RefDex calculation
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Inhomogeneous Layers – Recalculating n and k
Basic Principles
Advanced Features: Inhomogeneous Layers
User Interface
n k data from an inhomogeneous CISe2 film is in good agreement to the n k data
from a homogeneous film using the inhomogeneous layer feature.
P. Manley et al.,A method for calculating the complex refractive index of inhomogeneous thin films,
(submitted)
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Effective Medium Approximation - Background
Basic Principles
Advanced Features: Effective Medium
𝑛𝑒𝑓𝑓 = 𝑤ℎ 𝑛ℎ + 𝑤𝑖 𝑛𝑖
𝑘𝑒𝑓𝑓 = 𝑤ℎ 𝑘ℎ + 𝑤𝑖 𝑘𝑖
𝜀𝑒𝑓𝑓 − 𝜀ℎ
𝜀𝑖 − 𝜀ℎ
= 𝑤𝑖
𝜀𝑒𝑓𝑓 + 2𝜀ℎ
𝜀𝑖 + 2𝜀ℎ
𝑤ℎ
𝜀ℎ − 𝜀𝑒𝑓𝑓
𝜀𝑖 − 𝜀𝑒𝑓𝑓
= −𝑤𝑖
𝜀ℎ + 2𝜀𝑒𝑓𝑓
𝜀𝑖 + 2𝜀𝑒𝑓𝑓
User Interface
Volume Fraction Approximation
• Direct mixing of the two materials via
the volume fraction
• Does not consider polarisation
effects arrising due to mixing
Maxwell Garnett Approximation
• Based on elementary electrostatics
• Assumes spatially separated
polarisable particles
Bruggeman Approximation
• Assumes two kinds of spherical
particles randomly arranged.
• Spatial separation between
particles should be small (i.e. 𝑤𝑖 is
large)
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ELLIPSOMETRY MODE
𝒓𝒑
= 𝐭𝐚𝐧𝚿 𝒆𝒊∆
𝒓𝒔
Type equation here.
𝒓𝒑
= 𝐭𝐚𝐧𝚿 𝒆𝒊∆
𝒓𝒔
• Ellipsometric parameters Ψ and Δ simulated by RefDex
• Useful for highly absorbing substrates
• Currently incompatable with roughness and
inhomogeneity advanced features
23
n k Data from Ellipsometry – Example of Mo film
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Main Interface
Basic Principles
Advanced Features
User Interface
25
Advanced options
Basic Principles
Advanced Features
User Interface
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Data Extraction Process
Basic Principles
Advanced Features
User Interface
• Interactive fitting
process
• Place nodes which
are automatically
connected by a
smooth function
• User selects
physically meaningful
solutions from multiply
degenerate solution
space
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Summary and Outlook
RefDex
•
𝒓
calculates T, R (n,k) for a multilayer stack 𝒑 = 𝐭𝐚𝐧𝚿 𝒆𝒊∆
𝒓𝒔
→ extracts n,k from T, R
•
considers surface roughness
•
applies to inhomogeneous layers
•
has also basic features for ellipsometry
Type equation here.
...
•
is freely available from
http://www.helmholtz-berlin.de/forschung/oe/enma/nanooptix/index_en.html
Impulse and Networking Fond: VH-NG-928
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