Model for light scattering and lasing in dye-doped nematic liquid ...

Introduction Scattering in NLC A Laser Active Medium Results Future Development Conclusions

Model for light scattering and lasing in dye-dopednematic liquid crystals

A. Veltri, M. Infusino, S. Ferjani, G. Strangi

LICRYL (Liquid Crystals Laboratory),National Institute for the Physics of Matter (INFM-CNR), Center of Excellence

CEMIF.CAL and Department of Physics - University of Calabria, 87036Arcavacata di Rende (CS), Italy

Alessandro Veltri LICRYL, INFM-CNR, Department of Physics UNICAL

Model for light scattering and lasing in dye-doped nematic liquid crystalselectronic-

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Lasing in Scattering Media

LasersImage from “Tunable photonic materials and random lasers” Stefano Gottardo - PhD Thesis



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Lasing in Scattering Media

All that you need

Random Lasers: A Scattering Medium A Laser-Active Dopant A bit of luck

Image from http://www.martin-klimas.de/

Image from http://www.greenbeans.co.nz/



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NLC as a Scattering Medium

Nematic Molecules and Photons

Light-Nematic InteractionNematic Length L ∼ 100nmLaser Wavelenght λ = [500 ÷ 600]nm

Director FluctuationInvolving ∼ 103 Nematic MoleculesDeformation Length ∼ Laser Wavelenght



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Director Fluctuations and Light Beams





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Director Fluctuations and Light Beams



Interaction Between Light Beams [N-photons] and Elastic Deformations



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Light in the Medium

Light and the Medium: Parameters

Light Beam position r(x , y , z)

polarization α = 1, 2 frequency ω

wave vector k

Nematic dielectric constants

ε⊥ and ε|| elastic constants κ1,2,3

temperature T

Images from http://www.geocities.com/ and http://it.wikipedia.org/wiki/Cristalli_liquidi



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Light in the Medium

Frames of reference

“Laboratory” Frame [x , y , z ]

H. Stark et al. “Light diffusion and diffusing-wavespectroscopy in nematic liquid crystal”J. Opt. Soc. Am. A, 14(1):156, January 1997.

y -axis perpendicular to the glass slabs

planar alignment n0 = (0, 0, 1)



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Light in the Medium

Frames of reference

“Impact” Frame [u1, u2, u3]


u2-axis perpendicular to [n0, k]-plane. u1-axis perpendicular to [u2, n0]-plane. u3 parallel to n0.

u2 =n0 × k|n0 × k|

u1 = u2 × n0

u3 ≡ n0

e1 is the electric field corresponding toextraordinary polarization (α = 1).

e2 is the electric field corresponding toordinary polarization (α = 2).



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Dealing with Impacts

Effect of Fluctuations


Input Beam wavevector k polarizarion α

Output Beam wavevector q polarizarion β



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Dealing with Impacts

Structure Factor Bαβ


Input Beam wavevector k polarizarion α

Output Beam wavevector q polarizarion β

Bαβ(k, q) = ∆ε2kBTω4

c4

2∑δ=1

N(eα, eβ, uδ)

kδ(qS)

by defining kδ(q) = κδq2⊥ + κ3q2

|| and usingL = (κ3c2)/(∆ε2kBTω2) as the normalization lenght:

Bαβ(k, q) =ω2

c2

2∑δ=1

N(eα, eβ, uδ)[κδκ3

q2S⊥ + qS||

]

where N = [(n0 · eβ)(uδ · eα) + (uδ · eβ)(n0 · eβ)]2



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Results of the Model

After a Scattering Event

Path to cover until next scattering event: R Wavevector after the scattering event: q New polarization of the beam: β

R =lα

cos δ ln(

11−xr

) 1lα(k)

=nα(k)

4π

2∑β=1

∫dΩq

4πBαβ(k, q)n3

β(q)

xr is a random number ∈ [0 : 1],

cos δ = 1n2

hsin2 θε2⊥

+ cos2 θε||

iand θ is the angle between k and n0

where




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∫q

Bαβd q ≥ Cαβzr

Cαβ =

Cα1, if β = 1Cα2, if β = 2

Cα1 =nα(k)

4π

∫dΩq

4πBα1(k, q)n3

β(q)

Cα2 =nα(k)

4π

∫dΩq

4πBα2(k, q)n3

β(q)

zr is a random number ∈ [0 : 1]

where




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β =

1, if g ≤ Rαβ

2, if g > Rαβ

Rαβ =Cα1

Cα1 + Cα1

Cα1 =nα(k)

4π

∫dΩq

4πBα1(k, q)n3

β(q)

Cα2 =nα(k)

4π

∫dΩq

4πBα2(k, q)n3

β(q)

g is a random number ∈ [0 : 1]

where




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The Path




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Model for Active Dopant

How many levels you said?

We are not interested on:

Internal conversion

Inter System crossing

Triplet triplet absorption

Phosporescence

Taking into account that:

Transition d → b is very fast

We can consider a base levelat timeTypical energy spectrum of a dye molecule



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Internal conversion



Phosporescence



We can consider a base levelat timeThree level approximation



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Internal conversion



Phosporescence



We can consider a base levelat timeOur Choice

Two level approximation



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Gain Equation

Let’s consider a beam of N photons σ-polarized propagating in suchmedium(

∂

∂t+ vσ

g · ∇)

Nσ(k) = Aσab(k)

[(Nσ(k) + 1)nb − Nσ(k)

ga

gbna

]Hypothesis:

gbAσab(k) = gaAσ

ba(k)

∂∂t = 0 [Steady State Solution]

Intensity:

I = hω3Nσ/(2π)3c2

Analytic Solution:

Nσ(S) = (N0 + 1/γ)eγAσabS − 1/γ

where γ = 1 − gbna/ganb

Aσab = Aσ

abnb/vσg

A. Aspect, C. Fabre, G. Grynberg “Optique Quantique 1: Lasers”Majeure de Physique (2005).



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Gain on the Path

Fluorescence



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Gain on the Path

Fluorescence

nb > na → γ > 0nb < na → γ < 0



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Gain on the Path

Fluorescence

nb > na → γ > 0nb < na → γ < 0

Nσ(S) = (N0+1/γ)eγAσabS −1/γ



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Wavelength and length of Path

More the Scattering, Longer the Path

Taking into account that free main path R ∝ (κ3λ2)/(∆ε2kBT )

lower λ ⇒ lower R ⇒ higher number of scattering events ⇒ longer the path S

S1 < S2



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S1 < S2

longer path



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S1 < S2

longer path

γ > 0



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S1 < S2

longer path

γ > 0γ < 0



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Monte Carlo Simulation

Initial Conditions

Randomly generatedinitial position r0and direction k

Parameters for the Medium E7 nematic liquid crystal

Dielectric Constants ε⊥ = 2.31526656 ε|| = 3.04921444

Elastic Constants κ1 = 10.3 · 10−12N κ2 = 7.21 · 10−12N κ3 = 16.5 · 10−12N

Temperature T = 300K

Parameters for Light λ ∈ [540 : 620]nm

α =

(1, 50%

2, 50%



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Outputs

Sample Thickness 1mm



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Outputs

Sample Thickness 1mm

Path λ-dependent



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Outputs

Sample Thickness 100µm



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Outputs

Sample Thickness 100µm

Path λ-independent



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Discussing the Results

No Lasing in 100µm Samples?



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No Lasing in 100µm Samples?

"contra facta non est argumentum" [you can’t deny the facts]



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The Rutherford Syndrome

We have to find a different description for the scattering centers distribution



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Hunting Down the “bug”

Fluctuation Spectrum

Calculation of structure factor Bαβ isbased on the fluctuation spectrum:

〈|δn(qS)|2〉 ∼ kBT/κq2S

where qS is the wavevector associatedto director fluctuation

Large fluctuations are statistically favorite



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Fluctuation Spectrum

Calculation of structure factor Bαβ isbased on the fluctuation spectrum:

〈|δn(qS)|2〉 ∼ kBT/κq2S

where qS is the wavevector associatedto director fluctuation

Large fluctuations are statistically favorite

Hypothesis:Fluctuations Spatially uncorrelated



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Non Linear Interactions

Used spectrum describe director fluctuationsinteracting with a pure, uniform alignedmedium



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Fluctuation could also interact eachothers giving rise non-linear terms



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Fluctuation could also interact eachothers giving rise non-linear terms

This could change the slope of the spectrum at some scales



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We are using a dye doped nematicliquid crystal.We should have to take into accountsome kind of Janossy effect.Again non linear terms.



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We are using a dye doped nematicliquid crystal.We should have to take into accountsome kind of Janossy effect.Again non linear terms.

This could change the slope of the spectrum at some scales



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Conclusions

We have explored scattering in nematic liquid crystal as awavelength selector for random lasing in this medium

The idea works well for ideal thick samples, but isinappropriate for the thin ones utilized in experiments

The model represents a good diagnostic tool studying physicalphenomena occurring in this kind of systems

As a future develop, we plan to realize a more complex modeltaking into account spatial correlation for director fluctuationand/or interactions with dye molecules.



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Thank you for your attention!

Presentation made in LATEX



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Model for light scattering and lasing in dye-doped nematic liquid ...

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