OPTICAl METROlOGY

Handbook of

OPTICAl METROlOGY Principles and Applications Edited bv

Toru Yoshizawa

o ~Y~~F;'~~~~"P

Boca Raton London New York

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2008037129

13

Polarization Michael Shribak

CONTENTS 13.1 13.2 13.3 13.4 13.5 13.6

Introduction ......................................................................................................................... 339 Degenerate Polarization States ........................................................................................... 340 Polarization Ellipse ............................................................................................................. 340 Polarization Ellipse Transformations .................................................................................. 343 Linear Retarder between Crossed Linear Polarizer and Analyzer ..................................... 345 Complete Polarization State Generator with Rotatable Linear Polarizer and Quarter-Wave Plate ....................................................................................................... 347 13.7 Complete Polarization State Generator with Rotatable Linear Polarizer and Variable Retarder ......................................................................................................... 347 13.8 Complete Polarization State Generator with Linear Polarizer and Two Variable Retarders ................................................................................................ 348 Acknowledgment ........................................................................................................................... 349 References ...................................................................................................................................... 349

13.1

INTRODUCTION

For all electromagnetic radiation the oscillating components of the electric and magnetic fields are directed at right angle to each other and to the propagation direction. Because a magnetic field vector of the radiation is unambiguously determined by its electric field vector, the polarization analysis considers only the last one. Here, we shall assume a right-handed Cartesian coordinate system XYZ, with the Z-axis pointing along the direction of propagation. Let us assume that we have a plane harmonic wave of angular frequency OJ, which is traveling with velocity c in the direction Z. The angular speed is OJ = 2n

I'c where A is the wavelength. The electric

field vector has two components Ex (z, t) and Ey (z, t), which can be written as

and (13.1) where Eox and Eov are the wave amplitudes Ox and Oy are the arbitrary phases t is the time The phase difference between two components can be denoted as 0 = Oy - ox' where 0 .;:; 0< 2n. 339

Handbook of Optical Metrology: Principles and Applications

340

13.2

DEGENERATE POLARIZATION STATES

Particular cases of propagation of these two components are shown in Figure 13.1. The central part illustrates a distribution of electric field strength along the Z-axis at some moment of time t = O. Phase of the y-component Oy = 0 and Ox = -0, correspondently. The unit of the axes Ex and Ey is volts per meter. The unit length along Z-axis is one wavelength. The wave is moving forward, and the tip of the electric strength vector generates a Lissajous figure in the Ex Ey plane. The shape traced out by the electric vector, as a plane wave passes over it, is a description of the polarization state. The right part in Figure 13.1 represents the corresponding loci of the tip of the oscillating electric field vector. The shown cases are called degenerate polarization states: (1) linearly horizontal polarized (LHP) light, EOy = 0; (2) linearly vertical polarized (LVP) light, EOx = 0; (3) linearly + 45 0 polarized (L + 45P) light, EOx =EOY =Eo, 0 =0; (4) right circularly polarized (RCP) light, EOx = EOY = Eo, 0 = nl2; (5) linearly -45 0 polarized (L - 45P) light, Eox = Eoy = Eo, 0 = 7r; and (6) left circularly polarized (LCP) light, Eox = EOy = Eo, 0 = 3n12. The RCP light rotates clockwise and the LCP light rotates counterclockwise when propagating toward the observer. The clockwise sense of the circle describing right circular polarization is consistent with the definition involving a right-handed helix: if a right-handed helix is moved bodily toward the observer (without rotation) through a fixed transverse reference plane, the point of intersection of helix and plane executes a clockwise circle. This classical convention of right and left polarizations is accepted by most authors [1-4]. However, some authors do not agree, and they use the opposite definition of polarization handedness [5-7]. The latter refer to the quantum-mechanical properties oflight. In this description, a photon has an intrinsic (or spin) angular momentum of either negative -h/21C or positive h/21C, where h = 6.626 X 10-34 J s is the Planck's constant. The angular momentum is parallel to the direction of the photon propagation. In the classical convention, the positive and negative momentums correspond to LCP and RCP photons, respectively. The alternative approach suggests to designate the positive momentum to the right polarized light. Here, we follow the classical convention.

13.3

POLARIZATION ELLIPSE

In a general case, the Lissajous figure of two perpendicular sinusoidal oscillations with same frequencies and stable phase difference is an ellipse. Thus, both linear and circular lights may be considered as special cases of elliptically polarized light. In order to find an equation of the corresponding locus, we have to eliminate the time-space term m(t-~) + Ox in Equation 13.1. Expand the expression for Ey (z, t) and combine it with Ex (z, t)/Eox to yield sin[m(t

-!:...) + Ox] sin 0 = c

Ex cos 0 _ Ey . Eox EOY

(13.2)

Then we multiply both parts of Equation 13.1 by sin 0: (13.3) Finally, by squaring and adding Equations 13.2 and 13.3, we have ( Ex Eox

)2 + ( Ey )2 _2 Ex EOY

Ey cos 0 = sin 2D. Eox EOY

(13.4)

Polarization

341

LHP:

LVP:

L+45P:

RCP:

EOx =Eoy =Eo,

O=~ 2

L-45P:

EOx =EOy =Eo, o=1r

LCP:

EOx =Eoy =Eo,

0= 31r 2

FIGURE 13.1

Degenerate polarization states.

342


The above formula describes an ellipse, which is called the polarization ellipse. The standard form equation of an ellipse with semi-major axis a and semi-minor axis b is the following:

(Ea ')2 + (Eb ,)2 =1. _x_

_y_

(13.5)

Equation 13.5 can be transformed into Equation 13.4 by a coordinate rotation on angle lfI:

Ex' = Ex cos lfI + Ey sin lfI E/ = -Ex sin lfI + Ey cos lfI.

(13.6)

Thus, the equation of ellipse with the major axis at angle lfIbecomes (13.7)

Equation 13.5 can be written in the similar way: (13.8) Comparing coefficients in the quadratic Equations 13.7 and 13.8 yields

E Ox 2 = a 2 cos 2 lf1 + b 2 sin 2 lf1 ' 2EoxEoy cos 8 =

(a 2 -

b2 )sin

(13.9)

2lf1, and

±EoxEOY sin 8 = abo Subtracting the first equation of Equation 13.9 from second, we obtain

EOx 2 - EOy 2 = (a 2 -b 2 )cos 2lf1.

(13.10)

The major axis orientation lfI of the polarization ellipse can be found after dividing the third equation of Equation 13.9 by Equation 13.10: (13.11) The orientation is the angle between the major axis of the ellipse and the positive direction of the X-axis. All physically distinguishable orientations can be obtained by limiting lfIto the range 1r

--~

2

1r

lfI ----:--'----'1- tan 2lf11 tan 2qJ 1+ tan 2 2qJ cos Ll

(13.24)

sin 2£2 = sin 2qJ sin Ll Intensity of the beam 1 passed through the linear analyzer with transmission axis at 90° can be found from the ratio

(13.25)

where 10 is intensity of the beam after the polarizer. The ratio angle of the polarization ellipse at the front of the polarizer first equation of Equation 13.21:

0"2

is computed from the

In the considered case, after some straightforward trigonometric transformations, we obtain a simple analytical expression for the ratio angle:

2

cos 20"2

sin 2 2£2 2 l+tan 2lf12

= cos 22£2 cos 22lf12 = 1-

(1- sin 2 2qJ sin 2 Ll) (1 Han 2 2qJ cos Ll/

J

[(1- cos Ll )tan 2qJ + (1 + tan 2 2qJ cos Ll)2 (13.26)

Finally, intensity of the transmitted beam 1 is reduced to

Polarization

347

I

Ll

(13.27)

·22 .2 = 10 Sill cp Sill "2.

Equation 13.27 can also be derived using Jones or Muller matrix computation [8]. Advantage of the computation with the polarization ellipse is that the technique does not require employing matrix algebra and complex numbers, and it explicitly shows a transformation of the polarization by each optical element. Of course, not every optical system can be described by a simple analytical expression like Equation 13.27. In this case, the ellipse transformation formulas allow to create a numerical mathematical model. 13.6

COMPLETE POLARIZATION STATE GENERATOR WITH ROTATABLE LINEAR POLARIZER AND QUARTER-WAVE PLATE

A polarization generator consists of a source, optical elements, and polarization elements to produce a beam of known polarization state. An example of the polarization generator is shown in Figure 13.4. It includes a rotatable linear polarizer P with transmission axis at angle eand rotatable quarter-wave plate R with the slow axis oriented at angle cp. This arrangement is called Senarmont compensator. The polarization states at the front and rear faces of the quarter-wave plate are marked as E and E'. It is convenient to consider the setup in the coordinate system in which X'-axis is parallel to the slow axis of the quarter-wave plate. To obtain parameters of the output polarization ellipse E', we substitute angle difference cp - e for angle e, and 7Cl2 for Ll in Equation 13.23:

{

Iff' = 0 £'

(13.28)

= cp - e·

As it follows from this result, the principal axis of the ellipse is oriented along the slow axis of the quarter-wave plate, and the ellipticity angle is a difference between azimuths of quarter-wave plate and polarizer. Thus, we can obtain any orientation of the polarization ellipse by rotating the entire setup and any ellipticity by turning the polarizer relatively to the quarter-wave plate. For example, the difference cp - e has to be 45° for the right circular polarization (If = 7Cl4). Since the generator allows to produce a beam with any polarization state, it is called the complete polarization state generator. 13.7

COMPLETE POLARIZATION STATE GENERATOR WITH ROTATABLE LINEAR POLARIZER AND VARIABLE RETARDER

Another optical configuration of polarization state generator is presented in Figure 13.5. Linear polarizer P and variable retarder R are mounted in the rotatable body. The angle between transmission

P( 8)

R(7r!2, cp)

E

FIGURE 13.4

E'

Complete polarization state generator with rotatable linear polarizer and quarter-wave plate.


348

P(O)

Rotatable body

R(~.

0+45°)

~\ E

FIGURE 13.5

E'

Complete polarization state generator with rotatable linear polarizer and variable retarder.

axis of the polarizer and slow axis of the retarder is 45°. As a variable retarder, one can use BabinetSoleil compensator l3], Berek compensator [3], liquid crystal cell [7], or electro-optic crystal [2], for instance. A phase difference ~ of the retarder can be adjusted. The variable retarder is turned together with the polarizer. Orientation of the polarizer transmission axis is 8. The retarder slowaxis orientation is 8 + 45°, accordingly. The polarization states at the front and rear faces of the variable retarder are shown as E and E'. For analysis of the generator, we use the coordinate system in which X' -axis is parallel to the transmission axis of the polarizer. Parameters of the output polarization ellipse E' are determined from Equations 13.24 at q> = 45°:

lI" = 0 {

(13.29)

~.

£'=-

2

As one can see, the ellipticity angle of the output beam equals a half of retardance, and the principal axis of the ellipse is oriented along the transmission axis of the polarizer. For example, if the retardance is quarter wave (~ = n/2), then the output beam is Rep. 13.8

COMPLETE POLARIZATION STATE GENERATOR WITH LINEAR POLARIZER AND TWO VARIABLE RETARDERS

The polarization generators described above have two or one rotatable components. Arrangement of the complete polarization state generator without any mechanical rotation was proposed by Yamaguchi and Hasunuma [2,9]. A schematic of the Yamaguchi-Hasunuma compensator, which consists of two variable retarders, is shown in Figure 13.6. Two liquid crystal cells LCA and LCB are used

P(OO)

LCB(,B,OO)

LCA(a.45°)

Light

E; FIGURE 13.6

-

E2

E'2

Complete polarization state generator with linear polarizer and two variable retarders.

Polarization

349

as variable waveplate whose retardations a and [3 are controlled by voltages. The slow axes of LCA and LCB are chosen at 45° and 0° (parallel) orientations from the transmission axis of the polarizer P. The polarization states at the front and rear faces of the liquid crystal cells are noted as EI' E}', E2, and E/ Instead of liquid crystal cells, one can use other kinds of variable retarders as mentioned above. Shape and orientation of the polarization ellipse E2 can be obtained by substitution ({) =45° and replacing L1 with a in Equation 13.24:

(13.30)

The corresponding polarization components

and

0"2

~

are obtained with Equation 13.21:

{:':J 2

(13.31)

2

The second variable retarder introduces phase difference [3. Then the output beam E; will have the following ratio angle and phase difference:

{O";

8;

= al2 =

(13.32)

[3 + n/2 .

The relative amplitude of the X- and Y-polarization components of the output light is controlled by a, whereas the phase difference of the same components is controlled by [3. Therefore, by controlling the voltages applied to LCA and LCB, any states of polarization for the output beam can be generated. The corresponding ellipticity and orientation angles of the polarization ellipse can be computed from Equation 13.20: {

tan21j1; = -tana sin [3 sin 2£; = sin a cos [3

.

(13.33)

We have also derived Equation 13.33 by mUltiplication of Jones matrices followed by a complex vector conjugation [10]. This publication describes techniques for fast and sensitive differential measurements of two-dimensional birefringence distribution, which utilize the complete polarization state generator with two variable retarders. The corresponding systems for quantitative birefringence imaging are currently manufactured by CRI, Inc. (www.cri-inc.com). The polarization generator is mentioned as a precisions liquid crystal universal compensator in the company's Web site.

ACKNOWLEDGMENT This work was funded by the National Institute of Health grant ROI EB0057l0.

REFERENCES 1. Shurcliff, w.A., Polarized Light: Production and Use, Harvard University Press, Cambridge, MA,

1962. 2. Azzam, R.M.A. and Bashara, N.M., Ellipsometry and Polarized Light, Elsevier Science, Amsterdam, the Netherlands, 1987.

350


3. Born, M. and Wolf, E., Principles of Optics, 7th ed., Cambridge University Press, New York, 1999. 4. Ramachandran, G.N. and Ramaseshan, S., Crystal optics, in Handbuch der Physik, Vol. 25/1, S. Flugge (Ed.), Springer, Berlin, Germany, 1961. 5. Yariv, A. and Yeh, P., Optical Waves in Crystals: Propagation and Control of Laser Radiation, John Wiley & Sons, New York, 1984. 6. Yeh, P., Optical Waves in Layered Media, John Wiley & Sons, New York, 1988. 7. Robinson, M., Chen, J., and Sharp, G.D., Polarization Engineering for LCD Projection, John Wiley & Sons, New York, 2005. 8. Gerrard, A. and Burch, lM., Introduction to Matrix Methods in Optics, John Wiley & Sons, New York, 1975. 9. Yamaguchi, T. and Hasunuma, H., A quick response recording ellipsometer, Science of Light, 16(1): 64-71, 1967. 10. Shribak, M. and Oldenbourg, R., Techniques for fast and sensitive measurements of two-dimensional birefringence distributions, Applied Optics, 42: 3009-3017,2003.