1. 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

2. Motivation
Thin film optics
having to deal with multiple reflections
and requiring refractive indices
often only rely on optical measurements.

3. Transfer Matrix Method Multilayer Stack Comparison to Experiment
Contents
Basic Principles
Transfer Matrix Method
Multilayer Stack
Comparison to Experiment
Advanced Features
Surface Roughness
Inhomogeneous Layers
Effective Medium
User Interface
Outlook 3

4. 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:
𝐸 𝑖 + 𝐸 𝑖 − = 1 𝑡𝑖,𝑗 1 𝑟 𝑖,𝑗 𝑟 𝑖,𝑗 𝐸 𝑗 + 𝐸 𝑗 −
𝑟 𝑖,𝑗 =- 𝑟 𝑗,𝑖 = 𝑁𝑖− 𝑁 𝑗 𝑁𝑖+ 𝑁 𝑗 , 𝑡 𝑖,𝑗 = 2𝑁𝑖 𝑁𝑖+ 𝑁 𝑗 , 𝑡 𝑗,𝑖 = 2 𝑁 𝑗 𝑁𝑖+ 𝑁 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:
𝐸 𝑖 + 𝐸 𝑖 − = Φ−1 o o Φ E,i+ E,i− Φ= 𝑒 −𝑖 2𝜋 𝜆 𝑁 𝑖 𝑑
Where d is the thickness of medium, ω is the frequency of the propagating light and c is the speed of light
Propagation through mediums at
normal incidence 4

5. Oblique incidence Within the layer At interface 𝞱 𝑖 𝞱 𝑖
Basic Principles: Transfer Matrix Method Advanced Features User Interface
At interface
Within the layer
𝞱 𝑖
𝞱 𝑖
Medium i
P polarization:
Medium j
𝞱 𝑗
𝑟 𝑖,𝑗 = 𝑁 𝑗 𝑐𝑜𝑠 𝞱 𝑖 − 𝑁 𝑖 𝑐𝑜𝑠 𝞱 𝑗 𝑁 𝑗 𝑐𝑜𝑠 𝞱 𝑖 + 𝑁 𝑖 𝑐𝑜𝑠 𝞱 𝑗
𝑡 𝑖,𝑗 = 2 𝑁 𝑖 𝑐𝑜𝑠 𝞱 𝑖 𝑁 𝑗 𝑐𝑜𝑠 𝞱 𝑖 + 𝑁 𝑖 𝑐𝑜𝑠 𝞱 𝑗
Φ= 𝑒 −𝑖 2𝜋 𝜆 𝑁 𝑚 𝑑/𝑐𝑜𝑠 𝞱 𝑗
S polarization:
𝑟 𝑖,𝑗 = 𝑁 𝑖 𝑐𝑜𝑠 𝞱 𝑖 − 𝑁 𝑗 𝑐𝑜𝑠 𝞱 𝑗 𝑁 𝑖 𝑐𝑜𝑠 𝞱 𝑖 + 𝑁 𝑗 𝑐𝑜𝑠 𝞱 𝑗
𝑡 𝑖,𝑗 = 2 𝑁 𝑖 𝑐𝑜𝑠 𝞱 𝑖 𝑁 𝑖 𝑐𝑜𝑠 𝞱 𝑖 + 𝑁 𝑗 𝑐𝑜𝑠 𝞱 𝑗
Fresnel coefficiencts for oblique incidence 5

6. Coherent Layers – Interference Effects
Basic Principles: Multilayer Stack Advanced Features User Interface
Validity condition:
𝑑 ∆𝜆 𝐶𝑜ℎ𝑒𝑟𝑒𝑛𝑐𝑒 << 1 n θ
Phase difference between transmission orders:
𝛿= 2𝜋 𝜆 2𝑛𝑑 cos 𝜃
When δ=2𝑚𝜋 , 𝑚∈ℕ there will be constructive interference in T 6

7. 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= | 𝐸 0 | 2 𝑒 𝑖𝑘 𝑛 𝑑 𝑒 −𝑖𝑘 𝑛 𝑑 = | 𝐸 0 | 2
phase information
𝑡 ∆𝜆 𝐶𝑜ℎ𝑒𝑟𝑒𝑛𝑐𝑒 >> 1
Phase relationships between interior reflections is destroyed – therefore there is no interference 7

8. 9 Total layers implemented in RefDex
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 8

9. Input Spectrum – R and T Absorbing Region
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 9

10. Comparison to Experiment
Basic Principles: Comparison to Experiment Advanced Features User Interface
An example:
thin film
on substrate 10

11. Problem of Uniqueness 𝑅 𝑐𝑎𝑙 𝑛 λ ,𝑘 λ − 𝑅 𝑒𝑥𝑝 λ =0
Basic Principles: Comparison to Experiment Advanced Features User Interface
𝑅 𝑐𝑎𝑙 𝑛 λ ,𝑘 λ − 𝑅 𝑒𝑥𝑝 λ =0
𝑇 𝑐𝑎𝑙 𝑛 λ ,𝑘 λ − 𝑇 𝑒𝑥𝑝 λ =0
Choose the n, k values which minimise the difference between our model and experiment
Adding these equations together we get a function 𝐹 which takes n and k as input
𝐹 𝑛,𝑘 = 𝑅 𝑐𝑎𝑙 𝑛 λ ,𝑘 λ − 𝑅 𝑒𝑥𝑝 λ
+ 𝑇 𝑐𝑎𝑙 𝑛 λ ,𝑘 λ − 𝑇 𝑒𝑥𝑝 λ
Problems arise because two different n,k input pairs can both equal zero!
𝐹 𝑛′,𝑘′ =𝐹 𝑛 ′ ′,𝑘′′ =0
One Physically Meaningful Solution
Many Unphysical Solutions 11

12. 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! 12

13. 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 13

14. 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 14

15. Modified Transfer Matrix Method – Scalar Scattering Theory
Basic Principles Advanced Features: Surface Roughness User Interface
Scalar Scattering Theory
Rough Interface
𝑟 𝑖,𝑗 ′ = 𝑟 𝑖,𝑗 𝑒𝑥𝑝 −2 (2𝜋𝜎/𝜆) 2 𝑛 𝑖 2
Medium a
𝑟 𝑗,𝑖 ′ = 𝑟 𝑗,𝑖 𝑒𝑥𝑝 −2 (2𝜋𝜎/𝜆) 2 𝑛 𝑗 2
Modified
Fresnel
coefficients
𝑡 𝑖,𝑗 ′ = 𝑡 𝑖,𝑗 𝑒𝑥𝑝 − 2𝜋𝜎 𝜆 2 ( 𝑛 𝑖 −𝑛 𝑗 ) 2 /2
Medium b
𝑡 𝑗,𝑖 ′ = 𝑡 𝑗,𝑖 𝑒𝑥𝑝 − 2𝜋𝜎 𝜆 2 ( 𝑛 𝑗 −𝑛 𝑖 ) 2 /2
𝜎 is the interface roughness
Gives us the loss of specular beam intensity due to interface roughness 15

16. Modified Transfer Matrix Method - Examples
Basic Principles Advanced Features: Surface Roughness User Interface
Determination of optical constants
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, (2013)
σ = 9nm
σ = 20nm 16

17. 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 17

18. Inhomogeneous Layers – Coherent / Incoherent Decomposition
Basic Principles Advanced Features: Inhomogeneous Layers User Interface a) b) e) 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
This reduces the number of transfer matrix calculations to 4 18

19. Replace propagation operator inside inhomogeneous layer with:
Inhomogeneous Layers – Coherent / Incoherent Decomposition
Basic Principles Advanced Features: Inhomogeneous Layers User Interface
𝑅 𝑐𝑎𝑙𝑐 𝑛,𝑘 = 𝑤 𝑐 𝑅 𝐶 + 𝑤 𝐼 𝑅 𝐼
(Same equations for T not shown here)
Standard Calculation
Replace propagation operator inside inhomogeneous layer with:
𝑷 𝑖 = 𝑚=1 𝑀 𝑷 𝑚 𝑛 𝑚 𝑫 𝑚,𝑚−1 𝑛 𝑚 ,𝑛 𝑚− 𝑷 0 𝑛(0) ,
𝑛 𝑚 = 𝑛 0 , 𝑚=𝑒𝑣𝑒𝑛, 𝑚≠0 ¬𝑛 0 , 𝑚=𝑜𝑑𝑑
𝑛 0 = 𝑛 𝑖 ∗ or
𝑛 0 = 𝑛 𝑣 ∗
Void scattering as from a rough surface. (Slide 13)
Requires statistical knowledge of 3D void distribution as input 19

20. 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 20

21. 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) 21

22. Effective Medium Approximation - Background
Basic Principles Advanced Features: Effective Medium User Interface
Volume Fraction Approximation
Direct mixing of the two materials via the volume fraction
Does not consider polarisation effects arrising due to mixing
𝑛 𝑒𝑓𝑓 = 𝑤 ℎ 𝑛 ℎ + 𝑤 𝑖 𝑛 𝑖
𝑘 𝑒𝑓𝑓 = 𝑤 ℎ 𝑘 ℎ + 𝑤 𝑖 𝑘 𝑖
Maxwell Garnett Approximation
Based on elementary electrostatics
Assumes spatially separated polarisable particles
𝜀 𝑒𝑓𝑓 − 𝜀 ℎ 𝜀 𝑒𝑓𝑓 + 2𝜀 ℎ = 𝑤 𝑖 𝜀 𝑖 − 𝜀 ℎ 𝜀 𝑖 + 2𝜀 ℎ
Bruggeman Approximation
Assumes two kinds of spherical particles randomly arranged.
Spatial separation between particles should be small (i.e. 𝑤 𝑖 is large)
𝑤 ℎ 𝜀 ℎ − 𝜀 𝑒𝑓𝑓 𝜀 ℎ + 2𝜀 𝑒𝑓𝑓 = −𝑤 𝑖 𝜀 𝑖 − 𝜀 𝑒𝑓𝑓 𝜀 𝑖 + 2𝜀 𝑒𝑓𝑓 22

23. 𝒓 𝒑 𝒓 𝒔 = 𝐭𝐚𝐧𝚿 𝒆 𝒊∆ Ellipsometry ModE 𝒓 𝒑 𝒓 𝒔 = 𝐭𝐚𝐧𝚿 𝒆 𝒊∆
𝒓 𝒑 𝒓 𝒔 = 𝐭𝐚𝐧𝚿 𝒆 𝒊∆
Type equation here.
Ellipsometric parameters Ψ and Δ simulated by RefDex
Useful for highly absorbing substrates
Currently incompatable with roughness and inhomogeneity advanced features
𝒓 𝒑 𝒓 𝒔 = 𝐭𝐚𝐧𝚿 𝒆 𝒊∆ 23

24. n k Data from Ellipsometry – Example of Mo film24

25. Main Interface
Basic Principles Advanced Features User Interface 25

26. Advanced options
Basic Principles Advanced Features User Interface 26

27. 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 27

28. 𝒓 𝒑 𝒓 𝒔 = 𝐭𝐚𝐧𝚿 𝒆 𝒊∆ Summary and Outlook RefDex
calculates T, R (n,k) for a multilayer stack
→ extracts n,k from T, R
considers surface roughness
applies to inhomogeneous layers
has also basic features for ellipsometry
. . .
is freely available from
𝒓 𝒑 𝒓 𝒔 = 𝐭𝐚𝐧𝚿 𝒆 𝒊∆
Type equation here.
Impulse and Networking Fond: VH-NG-928 28