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Third Edition

Adelphi University

^ADDISON-WESLEY

An imprint of Addison Wesley Longman, Inc.

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Library of Congress Cataloging-in-Publication Data

Hecht, Eugene

Optics / Eugene Hecht; — 3rd ed.

p. cm.

Includes index.

ISBN 0-201-83887-7

1. Optics. II. Title.

QC355.2.H42 1998

535—dc20

Copyright © 1998 by Addison Wesley Longman, Inc.

All rights reserved. No part of this publication may be reproduced,

stored in a retrieval system, or transmitted, in any form or by any

means, electronic, mechanical, photocopying, recording, or otherwise,

without the prior written permission of the publisher.

Printed in the United States.

3 4 5 6 7 8 9 0-MA-009998

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8.7 Retarders 349

where m — 0,1,2,..it will function as a half-wave plate (A<p

= Tr, Stt, Stt, etc.).

Although its behavior is simple to visualize, calcite is not

often used to make retardation plates. It is brittle and difficult

to handle in thin slices, but more than that, its birefringence,

the difference between ne and n0, is a bit too large for convenience. On the other hand, quartz with its much smaller birefringence is frequently used, but it has no natural cleavage

planes and must be cut, ground, and polished, making it rather

expensive. The biaxial crystal mica is used most often. Several forms of mica serve the purpose admirably, for example,

fluorophlogopite, biotite, or muscovite. The most commonly

occurring variety is the pale brown muscovite. It is very easily cleaved into strong, flexible, and exceedingly thin largearea sections. Moreover, its two principal axes are almost

exactly parallel to the cleavage planes. Along those axes the

indices are about 1.599 and 1.594 for sodium light, and

although these numbers vary slightly from one sample to the

next, their difference is fairly constant. The minimum thickness of a mica half-wave plate is about 60 microns. Crystalline

quartz, single crystal magnesium fluoride (for the IR range

from 3000 nm to about 6000 nm), and cadmium sulfide (for

the IR range from 6000 nm to about 12,000 nm) are also widely used for wave plates.

Retarders are also made from sheets of polyvinyl alcohol

that have been stretched so as to align their long-chain organic molecules. Because of the evident anisotropy, electrons in

the material do not experience the same binding forces along

and perpendicular to the direction of these molecules. Substances of this sort are therefore permanently birefringent,

even though they are not crystalline.

A rather nice half-wave plate can be made byjust attaching

a strip of old-fashioned glossy cellophane tape over the surface of a microscope slide. (Not all varieties work—the bestis

LePage's "Transparent Tape.") The fast axis, that is, the vibration direction ofthe faster ofthe two waves, corresponds to the

transverse direction across the tape's width, and the slow axis

is along Us length. During its manufacture, cellophane (which

is made from regenerated cellulose extracted from cotton or

wood pulp) is formed into sheets, and in the process its molecules become aligned, leaving it birefringent. If you put your

half-wave plate between crossed linear polarizers, it will show

no effect when its principal axes coincide with those of the

polarizers. If, however, it is set at 45° with respect to the polarizer, the E-field emerging from the tape will be flipped 90°

and will be parallel to the transmission axis of the analyzer.

I

FIGURE 8.46 A hand holding a piece of Scotch tape stuck to a

microscope slide between two crossed polaroids. (E.H.)

Light will pass through the region covered by the tape as if it

were a hole cut in the black background ofthe crossed polarizers (Fig, 8,46). A piece of cellophane wrapping will generally also function as a half-wave plate. See if you can determine

the orientation of each of its principal axes using the tape

retarder and crossed Polaroids. (Notice the fine parallel ridges

on the sheet cellophane.)

The Quarter-Wave Plate

The quarter-wave plate is an optical element that introduces

a relative phase shift of Arp = 7r/2 between the constituent

orthogonal o- and e-components of a wave. It follows once

again from Fig. 8.7 that a phase shift of 90° will convert linear

to elliptical light and vice versa. It should be apparent that linear light incident parallel to either principal axis will be unaffected by any sort of retardation plate. You can't have a

}'('Iativc phase difference without having two components.

With incident natural light, the two constituent .^-stales are

incoherent; that is, their relative phase difference changes randomly and rapidly. The introduction of an additional constant

phase shift by any form ofretarder will still result in a random

phase difference and thus have no noticeable effect. When linear light at 45° to either principal axis is incident on a quarterMicrosoft HoloLens WSOU000003941

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350 Chapter 8 Polarization

wave plate, its o- and e-components have equal amplitudes.

Under these special circumstances, a 90° phase shift converts

the wave into circular light. Similarly, an incoming circular

beam will emerge linearly polarized.

Quarter-wave plates are also usually made of quartz, mica,

or organic polymeric plastic. In any case, the thickness of the

birefringent material must satisfy the expression

d{\n0 - ne\) = (4m + l)Ao/4

You can make a crude quarter-wave plate using household

plastic food wrap, the thin stretchy stuff that comes on rolls.

Like cellophane, it has ridges running in the long direction,

which coincides with a principal axis. Overlap about a half

dozen layers, being careful to keep the ridges parallel. Position

the plastic at 45° to the axes of a polarizer and examine it

through a rotating analyzer. Keep adding one layer at a time

until the irradiance stays roughly constant as the analyzer

turns; at that point you will have circular light and a quarterwave plate. This is easier said than done in white light, but it's

well worth trying.

Commercial wave plates are generally designated by their

linear retardation, which might be, for example, 140 nm for a

quarter-wave plate. This simply means thatthe device has a 90°

retardance only for green light of wavelength 560 nm (i.e., 4 X

140). The linear retardation is usually not given quite that precisely; 140 ± 20 nm is more realistic. The retardation of a wave

plate can be increased or decreased from its specified value by

tilting it somewhat. Ifthe plate is rotated about its fast axis, the

retardation will increase, whereas a rotation aboutthe slow axis

has the opposite effect. In this way a wave plate can be tuned to

a specific frequency in a region about its nominal value.

i

0

FIGURE 8.47 The Fresnel rhomb.

FIGURE 8.48 The Mooney rhomb.

The Fresnel Rhomb

We saw in Chapter 4 that the process of total internal reflection introduced a relative phase difference between the two

orthogonal field components. The components parallel and

perpendicular to the plane-of-incidence were shifted in phase

with respect to each other. In glass (n = 1.51) a shift of 45°

accompanies internal reflection at the particular incident angle

of 54,6s(Fig. 4.43e). The Fresnel rhomb shown in Fig. 8.47

utilizes this effect by causing the beam to be internally reflected twice, thereby imparting a 90° relative phase shift to its

components. If the incoming plane wave is linearly polarized

at 45° to the plane-of-incidence, the field components [£,] and

will initially be equal. Alter the first reflection the wave

within the glass will be elliptically polarized. After the second

reflection it will be circular. Since the retardance is almost

independent of frequency over a large range, the rhomb is

essentially an achromatic 90° retarder. The Mooney rhomb (n

= 1.65) shown in Fig. 8.48 is similar in principle, although its

operating characteristics are different in some respects.

8.7.2 Compensators

A compensator is an optical device that is capable ofimpressing a controllable retardance on a wave. Unlike a wave plate

where Atp is fixed, the relative phase difference arising from a

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