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

Reflection Coefficients

i r

t

SIGMA Polarization

Er

HrE

H

r12

n1 cos i n2 cos tn1 cos i n2 cos t

12

Reflection Coefficients

Pi Polarization

i r

t

12

Hi

Ei

Hr

Err12

n2 cos i n1 costn2 cos i n1 cost

Ht

Et

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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)