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Chapter 7. Polarization Optics - JonesMatrix

The optics of LCD is complicated by the fact that it isbirefringent as well as electroactive (with a twist).

The simplest approach of modeling LCD optics is to usethe 2x2 matrix. (Jones matrix ). The LC cell ischaracterized by θ(z) and φ(z). Once these two functionsare known, the optical properties of the LCD can becalculated.

7.1 2x2 matrix

The 2x2 Jones matrix is just a simple shorthand way torepresent the polarization state of light. Since LCD isbased on polarization manipulation, the Jones matrix isvery useful.

The polarization state of light is described by a 2x1 vector(Jones vector). Any polarization state can be representedas a sum of two perpendicularly polarized waves withdifferent amplitude and phase:

E = (xa + yb ejδ)Eo ejωt – jkz

where 122 =+ ba , and δ is the phase delay between thex and y components. The Jones vector corresponding tothis wave is

δjbe

a

2

Notice that any common phase in a and b can be takenout and be absorbed by the phase term ejωt-jkz..

Examples of Jones vectors:

αα

sin

cos refers to light polarized at angle α to the x-axis.

j

1

2

1 represents a right-circularly polarized light

In the Jones vector and Jones matrix approach, everyoptical element is represented by a 2x2 matrix.

Jin Jout

M

Then Jout = M Jin

A few examples of Jones matrices:

x-axis polarizer Mx =

00

01

3

Retardation plate with y-axis as the fast axis:

Mδ =

− δ

δ

j

j

e

e

0

02

1

Polarization rotator:

MR(θ) =

−θθθθ

cossin

sincos

Can easily show that a linearly polarized light polarized atα to the x-axis becomes α+θ to the x-axis after passingthrough M3.

=

++

αα

θθαθα

sin

cos)(

sin(

)cos(RM

The Jones matrix is very useful in problems concerningpolarization manipulation, such as the LCD. If there aremore than one polarization manipulation element, wesimply multiply the Jones matrices:

M1 M2 M3 MN

Jin Jout

5

and φ is the rotation angle from (x,y) to (x’,y’) as shown inthe diagram.

Note that Rφ = ΜR−1(φ) = ΜR(-φ)

Also R-φ = Rφ−1 = Rφ

’

Proof :

It is easy to show that

=

=

−'

')(

'

'1y

xM

y

xR

y

xR φφ

Also, any vector J in (x, y) system is related to J’ in (x’, y’)system by a rotation

J = R-φ J’

or J’ = Rφ J

Hence, if K = M J

then K’ = Rφ K = Rφ M J = Rφ M R-φ Rφ J

or K’ = M’ J’

where M’ = Rφ M R-φ

Also the reverse transformation is given by

M = Rφ−1 M’ Rφ

6

For example, rotate x axis by +90o,

−

=01

1090R

A retardation plate with x-axis as fast axis will have aJones matrix of

=

−

−

−

− δ

δ

δ

δ

j

j

j

j

e

e

e

e

0

001

10

0

001

10

The Jones matrix of a retardation plate with the fast axis atφ to the x-axis is given by

−

−=

−−− φφ

φφφφφφ

δ

δφδ

δφ cossin

sincos

0

0cossin

sincos

0

0j

j

j

j

e

eR

e

eR

Note that we are transforming from (x’,y’) to (x,y) in thiscase. The (x’,y’) frame is the principle axes of theretardation plate and the (x,y) axes are the fixedlaboratory frame .

This formula is very important. We shall see later that theLC cell can be regarded as a stack of birefringent plates.

Some more manipulations of Jones matrices:

1. Half wave plate:

The Jone matrix of a half wave plate with c-axis on the x-axis is

−j

j

0

0

4

Then Jout = MN …M3 M2 M1 Jin

7.2 Coordinate transformation

The Jones matrix depends on the definition of thecoordinate system. If the coordinate is rotated by φ, theJones matrix will become different.

x

x’P

y

y’

φ

Rule : If the Jones matrix is M in the (x,y) coordinates andM’ in the (x’,y’) coordinate system, then

M’ = Rφ M Rφ−1

and M = Rφ−1 M’ Rφ

where Rφ is the coordinate transformation matrix

−

=φφφφ

φ cossin

sincosR

7

The Jones matrix of a half wave plate with c-axis at θ tothe x-axis is

−=

−

−

−

φφφφ

φφφφ

φφφφ

2cos2sin

2sin2cos

cossin

sincos

0

0

cossin

sincos

j

j

Check:

=

− φφ

φφφφ

2sin

2cos

0

1

2cos2sin

2sin2cos

which is equivalent to a rotation of the linear polarizationalong x by 2φ. This Jones matrix is not the same as thepolarization rotation matrix since the rotation is dependenton the polarizer angle.

2. Quarterwave plate

The Jones matrix of a quarterwave plate with c-axis alongthe x-axis

+

−=

j

jM

10

01

2

1

If light polarized at 45o to the x-axis passes through it, thenew Jones vector is

−=

+−

=

+

−=

j

j

j

j

j

jJ

1

2

1

1

1

2

1

1

1

10

01

2

1

which is a right circularly polarized wave.

3. Polarizer

8

The Jones matrix of a polarizer with polarizing axis along x

is

00

01. So the Jones matrix of a polarizer with the

polarizing axis at θ is given by

−

−

φφφφ

φφφφ

cossin

sincos

00

01

cossin

sincos

=

φφφφφφ

2

2

sincossin

cossincos

4. Eigenvalues and eigenvectors of the Jones matrix

Any matrix can be diagonalized to find the eigenvaluesand eigenvectors. For the 2x2 Jones matrix, the 2eigenvectors correspond to the Jones vector that canpropagate through the system without any change ofpolarization state.

M J = λ J

Exercise: Find the eigenvectors of the retardation plateand the polarizer Jones matrices.

7.3 LCD Optics Modeling

In the most common model, the LC cell is thought of ascomposed of N retardation plates. N is sufficiently large sothat each slice can be regarded as having constant θ andφ, i.e. constant c-axis orientation. For twist angle smallerthan 1800, N<80 is large enough for LCD modeling. Also N

9

> 20 for accuracy. This is the approach used in allcommercial LCD modeling software.

The LC cell then has a Jones matrix given by

MLC = MN ….M3 M2 M1

where Mn is the Jones matrix of a birefringent plate with c-axis at angle φn to the x-axis and at θn to the z-axis.

φn

θn z

x

y

n

Jones matrix Mn

If the fast axis is at angle φn to the x-axis, then the Jonesmatrix is given by a coordinate transformation:

Mn = Rφn –1 M’ Rφn

where Rφn is the transformation matrix

10

−

=nn

nnnR

φφφφ

φ cossin

sincos

Jones matrix of the general birefringent plate with c-axisalong the principle axis (x’ ) is given by

M’ =

−

δ

δ

j

j

e

e

0

0

whereλθπδ ))(( oe nnd −=

with2

2

2

2

2

sincos

)(

1

eoe nnn

θθθ

+=

The above formulas can be used for the modeling of LCDwith arbitrary φ(z) and θ(z) distribution numerically. Inparticular, we shall work out a simple case of φ(z) and θ(z)below.

7.4 Jones matrix of twisted nematic cells with uniformtilt

The Jones matrix of a T-cell without any voltage appliedcan be obtained analytically.

Assuming no pretilt for simplicity, θ(z) = 0. If there is auniform tilt, then we can simply reduce ∆n for all thebirefringent plates.

11

The twist angle is given by

φ(z) = qz = Φ z/d

where Φ is the total twist angle of the LC cell. All twistangles are measured relative to the x-axis. The aboveequation already assumes that φ(0) = 0, i.e. the inputdirector is parallel to the x-axis.

Assuming N plates, then there are N Jones matrices. Thetwist angle of the nth plate is

φn = n ∆φ

where ∆φ = Φ / Ν

N∆φ∆φ 2∆φ

z

xy

Therefore the LC Jones matrix is given by

MLC = (RN∆φ−1M’RN∆φ)….(R2∆φ

−1M’R2∆φ)(R∆φ-1M’R∆φ)

where

=

−

Nj

Nj

e

eM

/

/

0

0' δ

δ

12

with kdnd ∆=∆=

λπδ

where ∆k =λ

π nkk oe ∆=−2

Now Rn∆φR(n-1)∆φ−1 = R∆φ

So MLC = RΦ−1(M’R∆φ)N

We can now make use of the Chebychev identity tosimplify this matrix further. It can be shown that

−

−=

−

−

1

1

NNN

NNNN

UDUCU

BUUAU

DC

BA

whereΩΩ=

sin

sinNUN

and cos Ω = ½ (A + D)

This is valid for any unitary matrix with AD – BC = 1.

(Exercise: Prove the Chebychev identity by induction)

Then it is straight-forward to show that as N -> infinity,

13

(M’R∆φ)N =

∆+Φ−

Φ∆−

dk

iddd

dd

dk

id

ββ

βββ

ββ

ββ

β

sincossin

sinsincos

where β2d2 = Φ2 + ∆k2d2

Finally, MLC can be written as a simple matrix

MLC =

+−−−−ibaidc

idciba

where a = cos Φ cos βd + dβ

Φ sin Φ sin βd

b = βk∆

cos Φ sin βd

c = sin Φ cos βd - dβ

Φ cos Φ sin βd

d = βk∆ sin Φ sin βd

This is an extremely useful result.

Properties:(1) MLC is normalized, i.e. 1=LCM . (Check it.)

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(2) MLC is unitary. i.e. MLC*T = MLC

-1 . (Check it). Recallfrom matrix algebra: the eigenvalues of unitarymatrices have forms ejα (or unit length).

(3) MLC changes if we change the twist sense, i.e.Φ → −Φ. The off-diagonal elements changes sign. Butthe properties remains the same. (e.g. the eigenvaluesand eigenvectors are the same.) Therefore it does notmatter how we define the twist sense, as long as it isconsistent.

(4) If the wave propagates in the opposite direction, i.e. wehave a left-handed coordinate system, then MLC

becomes MLC*. Proof: if z → -z, then M’ → Μ’*. This isa useful result for reflective displays.

(5) It can be shown that

MLC = Rφ−1 Τ−1

−

di

di

e

eβ

β

0

0 Τ

where T =

−

−χχ

χχcossin

sincos

i

i and sin 2χ = φ/βd.

In this form, the LC cell behaves as a “rotatingwaveplate”.

7.5 Eigenvalues and eigenvectors of M LC

The eigenvalues of the LC Jones matrix can be obtainedindirectly. Let us first find the eigenvectors of (MR∆φ)N.Recall that MLC = RΦ

−1(M’R∆φ)N

Write (M’R∆φ)N =

− gh

hg *

15

where g = dk

id ββ

β sincos∆+

and h = dd

ββ

sinΦ

Then the eigenvalues are given by the secular equation:

0*

=

−−

−λ

λgh

hg

The solution is easily derived to be:

λ = e-jβd and ejβd

The corresponding normalized eigenvectors are

v1 =

∆−−

∆+

ββ

ββ

2

2k

j

k

and v2 =

∆+

∆−

ββ

ββ

2

2k

j

k

These are elliptically polarized waves. Note that v1••••v2 = 1which is another property of unitary matrices.

This result is easily obtained if we note that

16

(M’R∆φ)N = Τ−1

−

di

di

e

eβ

β

0

0 Τ

where T =

−

−χχ

χχcossin

sincos

i

i and sin 2χ = φ/βd.

By inspection we know that the eigenvalues are e-jβd andejβd and the eigenvectors are

− χ

χsin

cos

iand

χχ

sin

sin

i

To visualize what is going on inside the LC cell, just take d= z in all the above formulas, and replace Φ by φ(z). Theneverything else is still valid.

Now MLC = RΦ−1(M’R∆φ)N

So MLC vI = RΦ−1(M’R∆φ)N vI = λI RΦ

−1 vI

Hence inside the LC, the eigenvectors of polarization arerotating elliptically polarized waves. The rotation is in thesame sense and pitch as the director twist angle. Thephysical picture is clearer if we take at the 2 limits of highpitch and low pitch.

(1) Low twist large birefringence limit (d ∆∆∆∆k >> φφφφ)

In this case β ∼ ∆k

The 2 eigenvectors are

17

v1 ~

0

1

v2 ~

1

0j

These are linearly polarized light along the x-axis (e-wave)and along the y-axis (o-wave). The waves inside the LCcell are linearly polarized light rotating along as thedirector. This is called the waveguiding limit. It is alsocalled the adiabatic limit or the Mauguin limit.

(2) High twist limit small birefringence limit ( φ φ φ φ >> >> >> >> d∆∆∆∆k)

In this case β ∼ φ/d

The 2 eigenvectors are

v1 ~

− j

1

2

1

v2 ~

j

1

2

1

These are circularly polarized waves. Hence the wavesinside the LC cell are rotating circularly polarized waves. Itturns out from more rigorous wave propagation theory thatonly the circularly polarized wave with the same twistsense as the director can propagate. Hence the othercircularly polarized wave will be reflected. This is theprinciple of the cholesteric display.

18

7.6 Parameter space

All the operating modes of a LCD can be shown on theparameter space.

If the LCD is composed of a polarizer at angle α to the x-axis, an LC cell with input director along the x-axis, and anoutput polarizer at angle γ to the x-axis, then thetransmission is given by

T = T(α, γ, φ, d∆n) = ( )2

sin

cossincos

••

αα

γγ LCM

There are only 4 parameters, in the zero volt state. If wefix any 2 of the parameters, T can be plotted as a functionof the other 2 parameters in a 2D contour map. This iscalled the parameter space. In most cases, the polarizersare either parallel of perpendicular. So γ = α or γ = α+π/2.So in fact there are 3 parameters in T(α, φ, d∆n).

Common situation: α = 0, γ = 90o. This is for example thecase for a TN display. The following parameter space canbe obtained:

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Each line represents a constant transmittance contour.The increment is 10% transmittance. The shaded partrepresents T>90%. The series of peaks show the Mauguinmodes. The series at 90o twist is the normal TN display.The series at 270o shows the STN display. The series at180o does not show near 100% transmittance. It is calledthe OMI mode. It has a maximum transmittance of 41%,but has other advantages such as B/W operation and lowdispersion.

This parameter space contains a lot of information. It isalso very easy to understand the similarities anddifferences of TN, HTN (High TN), STN (Supertwistednematic), OMI (Optical mode interference), SBE(Supertwist birefringent effect) and ECB modes.

Nomenclature:

-400 -200 0 200 4000

0.5

1

1.5

2

Twist angle

d∆n

20

LCD Mode Twist Angle Polarizer AngleECB 0o 45o

TN 90o 0o

HTN 120-150o 15o

STN 180-240o 45o

SBE 270o -32.50

OMI 180o 0o

The PS gives the transmittance at no voltage (nonselect).When a voltage is applied, the transmittance will changebecause ∆n decreases. For a first approximation, we canregard this change as a vertical line going towards the x-axis.

Also the dispersion can be visualized easily. A change in λis equivalent to change in ∆n since the parameter ∆n/λappears together in all formulas. Therefore it is equivalentto a vertical axis scaling.

The parameter space above is for the waveguidingsituation, with α = 0. If α = 45o, the ECB modes will beobtained. Here we show a series of parameter spaces toshow the systematic variation of the operating modes ofany LCD.

21

22

The parameter space can also be plotted with α and d∆nas the free parameters. They are useful for designing newLCD operating modes. For example, the following PSshows the 240o twist STN display with cross polarizers.The optimum d∆n of 0.75µm, and optimum polarizer angleof 30o can be obtained easily. This is in agreement withthe best design.

7.7 Gooch and Tarry formulas:

Gooch and Tarry derived the analytical expressions of theoptical properties of the GTN cell. We can derive thoseimportant formulas using the Jones matrix easily.

0 20 40 60 800

0.5

1

1.5

2

2.5

d x

delta

n

Polarizer angle

23

Case 1: α = 0, γ = Φ+π/2.

The transmission of the LCD is given by

( )2

0

1)2/sin()2/cos(

••+Φ+Φ= LCMT ππ

Therefore

T = du

dd

βββ

22

222

2sin

1

1sin

+=Φ

where u = Φ∆=

Φ∆=

Φkdnd

λπδ

This is the original Gooch and Tarry formula. In particular,for the 90o TN cell, the LCD will have parallel polarizersand the transmission is given by

T = dd

ββ

222

2sin

Φ

which can be used to derive the waveguiding modes ofthe TN LCD. This will be discussed in the next Chapter.

Case 2: α = 0, γ = Φ.

The transmission of the LCD is given by

( )2

0

1sincos

••ΦΦ= LCMT

24

Therefore

T = 21

1

u+u2 + cos2 βd

This is just 1-T of case (1).

Here is a plot for φ = 90ο, 180ο and 270ο. These are thewaveguiding modes.

25

The 90o TN cell with cross polarizers is a special exampleof this case. Here the transmission is given by

T = 1 - dd

ββ

222

2sin

Φ

Case 3: α = 45o, γ = −45ο.

The transmission of the LCD is given by

( ) 222

1

111

4

1cbMT LC +=

••−=

This can be used to analyze the ECB mode displays,which will be presented in the next Chapter. Here is a plotof the transmission as a function of d∆n forφ = 0ο, 90ο, 180ο, 270ο. It can be seen that these are trulyinterference modes.

26

Case 4: α = 0, γ = π/2

Here the polarizers are always crossed. The transmissionis given by

( ) 222

0

110

4

1dcMT LC +=

••=

These examples show the power of the Jones matrix andthe parameter space approach in analyzing LCD optics.

7.7 Wave propagation theory of LC cell

There is another approach to LCD optics – wavepropagation theory. It yields the same results as the Jonesmatrix approach.

We first obtain the differential equation for the Jonesvector.

z

∆zL(z)

Let n = (cos φ, sin φ, 0)

φ = qz

km = (ne + no) π/λ

27

φm = km ∆z

∆k = π∆n/2λ

δ = ∆k ∆z

then as before the Jone matrix of the nth slice of LC cellcan be approximated by a birefringent plate:

φδ

δφ

φ Re

eRezL

j

jmj

=

−−0

0)( 1

Represent the Jones vector at z by J(z), then

J(z+∆z) = L(z) J(z)

Therefore

∆J = J(z+∆z) – J(z) = [ L(z) – I] J(z)

As ∆z 0,

φφ δδ

φ Rj

jRjzL m

+

−−≈ −

10

01)1()( 1

Simplifying, we get

∆

∆−∆

−∆−= − zRkj

kjRIzjkzL m φφ 0

0)1()( 1

Hence the differential equation for J(z) is given by

28

)(0

0)( 1 zJRkj

kjRjk

dz

zdJm

∆−

∆−−= −

φφ

Now we need to get rid of Rφ by making the transformation

Let )()( zJRezJ zmjkR φ=

Then it can be shown that, after a few steps,

)()(

zJkjq

jqkj

dz

zdJR

R

∆

−∆−=

Now we need to diagonalize this equation. Let theeigenvalue be β, then

0=−∆

−−∆−β

βkjq

jqk

which gives

22 qk +∆±=β

which is the same as β2d2 = Φ2 + ∆k2d2

obtained previously in section 7.4.

Now the normalized eigenvectors corresponding to the +and – solutions are

29

v+ =

∆+

∆−

ββ

ββ

2

2k

j

k

and v- =

∆−−

∆+

ββ

ββ

2

2k

j

k

Therefore the solutions for JR(z) are

zjR evzJ β++ =)(

zjR evzJ β−− =)(

[Verify that these are indeed the solutions of thedifferential equation for JR(z)].

Therefore the Jones vectors inside the LC cell is given by

+−−−+ = vRezJ zmkj 1)()( φβ

and −−+−− = vRezJ zmkj 1)()( φβ

These are exactly the same results as in section 7.5. Thewaves inside the LC cell are elliptically polarized waveswith rotating axis guided by the LC director. The resultshere also provide the additional phase factors.

30

The wave propagation approach is especially useful forcholesteric case if we take into consideration continuousreflection inside the cell. It will give the reflection of thecircular polarized light, which we cannot get here.

31

Chapter 8. LCD Optical Modes

LCD can operate in many modes. Every point in theparameter space can be a quiescent operating point forthe LCD. Depending on the polarizer angle, an LCD canbe in either the ECB, or waveguiding or mixed mode ofoperation.

8.1 ECB modes (Interference modes)

Classic ECB: No twist

There are several classic ECB modes (homogeneous cell,hybrid aligned cell, homeotropic cell). They all rely on thebirefringence of the LC cell. The LC cell behaves as aretardation plate with variable retardation.

Polarizer AnalyzerLC cell

The polarizer and the input director are always at 45o toeach other. The polarizers can be cross (as shown) orparallel.

For a cross polarizer geometry, the transmission is givenby

32

T = sin2 ∫d

0dz

zn

λπ )(∆

where d = film thickness∆n = birefringenceλ = wavelength

For a parallel-parallel polarizer geometry

T= 1 - sin2 ∫d

0dz

zn

λπ )(∆

Example: for a homogeneous cell, ∆n(z) = constant

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

d ∆ n

Tra

nsm

issi

on

33

As the retardation depends on V, so the transmission willalso depend on V. The shape of the curve depends on theinitial retardation value. For example, d∆n(0)=2 µm:

This was calculated using DIMOS, and is an example ofan electro-optic curve.

General ECB modes:

The general ECB mode can be analyzed using theparameter space. For a LC cell with any twist, theinterference mode or ECB mode can be obtained byplacing the input director at 45o to the input director. Theoutput polarizer is at 90o to the input polarizer. In thiscase, it can be shown (homework exercise) easily that

T = b2 + c2

ECB cell

0.00

0.10

0.20

0.30

0.40

0.50

0.00 1.00 2.00 3.00 4.00

Voltage

Tra

nsm

ittan

ce

34

In particular, if Φ = 0, it can be easily shown that

T = sin 2 δ

which is what we have derived for a single birefringentplate with no twist. So the above formula is just anextension of the ECB mode to twisted nematic cells.

It should noted that the STN and SBE modes with near45o polarizer angles, are actually general ECB modes.They are not waveguiding and are dispersive (colored).

The following curves show examples of ECB modes for90o twist and 180o twist cells.

For the interference modes, the transmission is alwaysperiodic in d∆n. There is no waveguiding limit.

00.10.20.30.40.50.60.70.80.9

1

0 2 4 6 8 10 12

Retardation

Tra

nsm

issi

on

35

For ECB mode, there can be several maxima and severalminima. If the initial retardation value is reduced, it canhave just one peak.

Dispersion:

Because of the dependence of T on wavelength, ECBdisplays are intrinsically dispersive, meaning that there isstrong coloring of the transmitted light. This is bad formany applications but is good for some applicationsrequiring color contrast.

For classic ECB, the spectrum can be calculated easily. If∆n = constant independent of z, then the peakscorresponds to

2

1+=∆M

nd

oλfor M = 1, 2, 3,…

where λο is the peak wavelength.

So λπλoMT )

2

1(sin2 −= for the ON states

Similarly,

λπλoMT 2sin= for the OFF state

Note that the ECB cell is designed to be either normally onor normally off, but not both, obviously.

36

An ECB display does not have a true dark state. At any V,there is some ∆n, and that must correspond to peaktransmission of some color. So the color of the displaychanges as V changes.

In the ECB mode, the input polarizer is always at 45o tothe input director.

(Homework) Calculate the spectrum for the general ECBmode (i.e. with a twist).

8.2 Waveguiding (Mauguin) modes

In the waveguiding mode, the input polarizer is at 0o or 90o

to the input director of the LC cell. In this case, thepolarization of the light rotates in the same way as the LCdirector of the LC cell. Therefore if the LC director twist is

400 450 500 550 600 650 7000

0.2

0.4

0.6

0.8

1

tran

smitt

ance

wavelength (nm)

M=1 M=2

37

90o, the polarization also twist by 90o. This rotation issupposedly independent of wavelength.

Typical configuration of a TN LCD:

Let us apply the LC Jones matrix to analyze thewaveguiding modes. Here the input polarizer is parallel tothe input director and the output polarizer is parallel to theoutput director.

x

y

DIn, Pin

Dout, Pout

38

We have shown above that the transmission is given bythe famous Gooch and Tarry formula.

T = 21

1

u+u2 + cos2 βd

Let us examine the case of a 90o twist TN cell. In thiscase, the transmission is given by

T = 1 - dd

ββ

222

2sin

Φ

where

22

2

∆+

=

λππβ nd

d

A plot of T vs λnd∆

shows several peaks.

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

d ∆ n

Tra

nsm

issi

on

39

The transmission is 100% when

βd = Nπ for N = 1,2,3…

So the transmission peaks are at

142

1 2 −=∆N

nd

λ

These are called the Mauguin minima . (It will be minimumtransmission if the polarizers are parallel). In thewaveguiding limit, the transmission is 100% always,because the solution is a rotating linearly polarized wave.It occurs at large d∆n. For finite d∆n, the waves areslightly elliptical. The ellipticity parameter is defined as theratio of the major to the minor axis.

Definitions: If Pin is parallel to Din, the wave inside the cellis an e-wave. This is called an e-mode TN cell. If Pin ⊥ Din,then the wave inside the cell is an o-wave. This is o-modeoperation. Whether an o-mode or an e-mode is useddepends also on the viewing angle requirements.

Most TN LCDs operate either in the first or secondminimum. Here are their values

nd∆

First minimum 0.475 µm

Second minimum 1.075 µm

40

Recall from the discussion on refractive index of LC thatmost LC have ∆n of 0.07-0.2. Therefore one can choosethe right combination of cell gap and ∆n to make first orsecond minimum cells.

Sometimes, the choice of the first or second minimum alsohas to do with viewing angle.

We shall show that the description of waveguiding modesfor the Mauguin minima is correct only at large d – thewaveguiding limit. There is some degree of birefringencein the normal first minimum or second minimum operationfor TN LCD.

Dispersion: the waveguiding effect is λ independent.Therefore the waveguiding modes are true black andwhite – no dispersion.

8.3. Mixed modes

If the polarizer angle is at angles other than 0, 45o, or 90o

to the input director of the LC cell, then we have a mixedmode situation. The LCD operates somewhere betweenthe ECB and the waveguiding limit.

In transmittive LCD, mixed modes are not used. Theoptimized STN may have a little bit of mixed modebehavior because the polarizer and analyzer angles maynot be 45o to the directors exactly, and also they are notparallel or perpendicular to each other.

41

8.4 Reflective modes: MTB (mixed TN andbirefringence mode)

A reflective display is different from a transflective display.There is only one polarizer.

For such a truly reflective LCD without the rear polarizer,one has to use the MTB mode or the ECB mode. Theanalysis is greatly simplified by use of the Jones matrix.

Mirror

LC cell

Pol

For a reflective LCD with only one polarizer, thereflectance is given by

( )2

1*sin

cossincos

••= −ΦΦ α

ααα LCLC MRMRR

where M* means opposite twist sence for the LC.

The following figures are parameter space for this displaywith various polarizer angles.

42

The minima in reflectance are called the TN-ECB modes,the MTN modes for α = 0. For nonzero α they are calledMTB modes.

The MTB modes are newly discovered by HKUST. Thereare many new uses of such modes for low power PDAand fancy applications.

HFE

RTN

RSTN

TN-ECB-1

TN-ECB-2

-360 -240 -120 0 120 240 3600

0.3

0.6

0.9

1.2

1.5

Twist angle

d∆n

α=0♣

TN-ECB-2

-360 -240 -120 0 120 240 3600

0.3

0.6

0.9

1.2

1.5

Twist angle

d∆n

α=15♣

MTN

RSTN

-360 -240 -120 0 120 240 3600

0.3

0.6

0.9

1.2

1.5

Twist angle

d∆n

α=30♣

SCTN

-360 -240 -120 0 120 240 3600

0.3

0.6

0.9

1.2

1.5

Twist angle

d∆n

α=45♣

43

There are a lot of interesting physics contained in theseparameter space diagrams. For the case of α = 45o. the zerotwist modes are exactly the ECB modes. For the case of α = o,the TN-ECB modes can be derived exactly.

Given a x-polarized input wave, the ellipticity of the outputthrough the LC cell is given by the Gooch Tarry formula:

+Φ+

= − 222

1 1sin1

2sin

2

1tan u

u

uχ

Therefore, for the TN-ECB mode, the reflected wave should becross polarized. i.e. the LC cell should behave as aquarterwave plate. For this to happen, χ has to be 1.

Then it can be derived easily (homework exercise) that thisleads to the solution

22)12(

π−=Φ N where N=1,2,3…

and the corresponding retardation is given by

d∆n = λΦ/π

For example, at λ=550nm, the first 2 TN-ECB modes are (63o.0.18µm), (189o, 0.54µm). They corresponds exactly to thesolutions depicted in the PS.