total internal reflections in liquid crystals optics and photonics presented in partial fulfillment...
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Total Internal Reflections in Total Internal Reflections in Liquid CrystalsLiquid Crystals
Optics and PhotonicsOptics and PhotonicsPresented in Partial Fulfillment of the
Second Midterm
Clinton Braganza
Liquid Crystal Institute, K.S.U.4/4/2004
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Reflection Coefficients
i r
t
SIGMA Polarization
Er
HrE
H
r12
n1 cos i n2 cos tn1 cos i n2 cos t
12
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Reflection Coefficients
Pi Polarization
i r
t
12
Hi
Ei
Hr
Err12
n2 cos i n1 costn2 cos i n1 cost
Ht
Et
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Total Internal Reflection
SIGMA polarizationWriting t explicitly using Snell’s law
If n1 > n2, all the incident power is reflected if the incident angle is greater than arcsin (n2 / n1).
r12
n1 cos i n2 1n1
2
n22
sin2 i
n1 cos i n2 1n1
2
n22
sin2 i
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TIR in Liquid Crystals:Glass to LC.
Note that liquid crystals are birefringent, therefore reflections will depend of the orientation of the liquid crystal with respect to the direction of light propagation.
Let us consider this liquid crystal:
ne = 1.7no = 1.5nglass = 1.7
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TIR in LC’s: Orientation
We will consider the following configurations:
A homeotropic cell With sigma and pi polarized light incident on the
cell surface
A planar cell– With director parallel to y-axis
and sigma and pi polarized light incident on the cell surface
– With director parralel to x-axis And sigma and pi polarized light incident on the
cell surface.
y
z
y
z
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TIR in LC’s
Homeotropic Cells
This encounters no, therefore when the incident angle is greater than arcsin (no/ng) = 61.9º , all the light is reflected.
Ei, S - polarizationE
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TIR in LC’s
Homeotropic Cells
This encounters
This increase from no to ne, which is the same as glass, therefore TIR does not take place.
Ei, P - polarization
neff neno
ne2 cos2 no
2 sin2 n
k
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TIR in LC’s
Planar Cell : director parallel to y-axis
Here the electric field always encounters no, therefore if the incident angle is greater than the critical angle we have 100% reflectance.
Ei, S - polarization
y
z
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TIR in LC’s
Planar Cell : director parallel to y-axis
As the incidence angle is increase, the refractive index decreases from ne to no as the electric field becomes parallel to the director, therefore TIR happens here.
Ei, Pi - polarization
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TIR in LC’s
Planar Cell : Director parallel to x-axis
Here the electric field is always parallel to ne, therefore we do not have TIR
Ei, S - polarization
y
z
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TIR in LC’s
Planar Cell : Director parallel to x-axis
Here the electric field always encounters no, therefore TIR occurs at incident angles greater than the critical angle.
Ei, Pi - polarization
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TIR in LC’s: LC to Glass
Let us consider a different liquid crystalno =1.5
ne =1.8
Therefore the critical angle is arcsin(nglass/ne) = 70.81º
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TIR in LC’s : LC to Glass
Homeotropic cell
In the liquid crystal the light encounters no, which is less than nglass, therefore no TIR occurs here.
Ei, S - polarizationE
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TIR in LC’s : LC to Glass
Homeotropic cell
In the liquid crystal the light encounters neff, which increases from no to ne. Therefore TIR occurs.
Ei, p - polarization
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TIR in LC’s: LC to Glass
Planar Cell: director parallel to y-axis
In the liquid crystal the light encounters no, therefore no TIR occurs
Ei, S - polarization
y
z
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TIR in LC’s: LC to Glass
Planar Cell: director parallel to y-axis
In the liquid crystal the light encounters neff, which decreases to no as the incident angle increase, therefore no TIR occurs
Ei, p - polarization
y
z
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TIR in LC’s: LC to Glass
Planar Cell: director parallel to x-axis
In the liquid crystal the light encounters ne, therefore TIR occurs
In the liquid crystal the light encounters no, therefore no TIR occurs
Ei, S - polarization
Ei, p - polarization
y
z
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TIR in ChLC’s: Glass to LC
Knowing the orientation of the liquid crystal at the boundary we treat the planar texture as the previous nematic cases.
Focal Conic TextureI expect a periodic behavior here, for example, for s- polarization:
If director is parallel to cell normal we have TIR
if director is parallel to polarization of light we will have no TIR.
Planar Texture
Focal Conic Texture
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Some applications
Switchable fiber optic cables –too expensive A more economical use would be for optical switches.
– Shown below is a telecom optical switch designed by Baker, that can switch light to two different positions without changing the polarization.
ITOITO
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Conclusions
Total internal reflection was solved by carefully analyzing the orientation of the liquid crystal director with respect to light propagation.
It would be nice to get a general solution for TIR in LC’s, without first knowing the director orientation.
For the case of cholesterics this problem would involve studying the effect of the evanescent wave from one chiral layer to another.
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References
Yang, D-K, J. Opt. A: Pure and Appl. Opt, 5(2003) 402-408
Baker, A. P., 1998 Liquid Crystal Optical Switch Having Reduced Crosstalk, USA Patent # 4,720,171
Xianyu, H., et al, Optics Letters, 28 10 (2003)
Boiko, Y., et al, Optics Letters, 27 19 (2002)