Problems and answers in wave optics

118 Pages·2011·1.603 MB·English

by  Ryabukho, Vladimir P

Problems and
Answers in
Wave
Optics
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Library of Congress Cataloging-in-Publication Data

Ryabukho, Vladimir P.
Problems and answers in wave optics / Vladimir P. Ryabukho.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-8194-8971-5 (alk. paper)
1. Wave theory of light. 2. Optics--Mathematics. I. Title.
QC403.R93 2011
535'.2--dc23
2011041769

Published by

SPIE
P.O. Box 10
Bellingham, Washington 98227-0010 USA
Phone: [phone]
Fax: [phone]
Email: [email protected]
Web: http://spie.org

Copyright © 2011 Society of Photo-Optical Instrumentation Engineers
(SPIE)

All rights reserved. No part of this publication may be reproduced or
distributed in any form or by any means without written permission of
the publisher.

The content of this book reflects the work and thoughts of the
author(s). Every effort has been made to publish reliable and
accurate information herein, but the publisher is not responsible for the
validity of the information or for any outcomes resulting from reliance
thereon.

Printed in the United States of America.
First printing

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Preface
This collection presents problems related to a wide scope of
wave phenomena in optics, studied within the framework of
the university course of general physics. The problems cover
the basic topics of wave optics, i.e., electromagnetic waves and
oscillationsofopticalrange,spectralpropertiesofelectromagnetic
waves, polarization of oscillations and waves, reflection and
refraction of light, total internal reflection, optics of anisotropic
media and crystal optics, interference of monochromatic
and mutually coherent waves, temporal and spatial light
coherence, interference of partially coherent light, multiple-beam
interference,diffractionoflight,diffractiongratings,diffractionof
lightbyvolumegratingsandacousticwaves,andopticsofmoving
media.
This book is written as a supplement for students studying
physics,mathematics,andengineering,includingmedicalphysics
and engineering, biomedical optics, and biophotonics. The prob-
lems are chosen such that their solution supports the study and
understanding of the basic concepts of wave optics. This is why
the collection contains a sufficient number of relatively simple
problemsforeachtopic.Pairedwithanypopularopticstextbook,
these problems illustrate the principles learned from lectures and
lab work. The content of some problems allows them to be used
for self-training. Many problems are accompanied by schematic
illustrationsforclarification,sincethestudyofopticalproblemsis
largelyassociatedwithvisual–spatialperception.
What follows is the result of more than 20 years of experience
teaching optics to students at Saratov State University. I greatly
appreciate the cooperation, contribution, and support of all of
v
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vi Preface
my students, postgraduate students, and colleagues from the
university’s Department of Optics and Biophotonics, especially
from the head of the department, Professor Valery Tuchin. I am
indebted to my colleague Vladimir Derbov, a professor in the
Department of Theoretical Physics at Saratov State University,
forhisEnglish-languageassistance.Iappreciatehisfastandhigh-
qualitytranslation,andhisspiritoffruitfulcollaboration.Iwould
also like to express my gratitude to my wife, Anna, and my son,
Peter,fortheirindispensablesupport,understanding,andpatience
duringthewritingofthisbook.
VladimirRyabukho
October2011
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Contents
Preface..................................................................................... v
Chapter 1 Electromagnetic Waves in the Optical
Range....................................................................................... 1
1.1 Equations and parameters of electromagnetic
waves.................................................................................... 1
1.2 Spectralpropertiesofelectromagneticwaves........ 6
1.3 Polarizationofelectromagneticwaves..................... 8
1.4 Energy,power,andintensityoflight......................... 9
1.5 Standingelectromagneticwaves................................. 10
Chapter2 ReflectionandRefractionofLight................... 13
2.1 Fresnelformulae............................................................... 13
2.2 Energyreflectionandtransmissioncoefficients.... 14
2.3 Totalinternalreflection.................................................. 17
Chapter3 OpticsofAnisotropicMedia:CrystalOptics... 21
3.1 Lightrefractionattheboundaryofananisotropic
medium................................................................................ 21
3.2 Linearpolarizersandretarderplates......................... 23
3.3 Interferenceofpolarizedlight...................................... 26
3.4 Polarizationprisms.......................................................... 28
3.5 Inducedopticalanisotropy............................................ 30
Chapter4 CoherenceandInterferenceofLight............... 33
4.1 Interference of monochromatic and mutually
coherentlight..................................................................... 33
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viii Contents
4.2 Amplitudedivisioninterferometers:TheMichel-
sonandMach–Zehnderinterferometers................... 41
4.3 Wavefrontdivisioninterferometers:TheYoung
interferometer.................................................................... 44
4.4 Coherenceoflight............................................................ 47
4.4.1 Temporalcoherenceoflight........................ 47
4.4.2 Transversespatialcoherenceoflight........ 50
4.4.3 Longitudinalspatialcoherenceoflight.... 52
4.5 Interferenceofpartiallycoherentlight..................... 54
4.6 Multibeam interferometry: The Fabry–Pérot
interferometer.................................................................... 60
Chapter5 DiffractionofLight............................................. 63
5.1 Fresnelzonetheoryofdiffraction.............................. 63
5.2 Far-field(Fraunhofer)diffraction............................... 67
5.3 Diffractiondivergenceofopticalbeams:diffrac-
tionlimitofresolutioninopticalsystems................ 69
5.4 Diffractiongratings......................................................... 73
5.5 Lightdiffractionbyvolumegratings:diffraction
byacousticwaves............................................................. 77
Chapter6 OpticsofMovingMedia..................................... 81
6.1 Dopplereffect.................................................................... 81
6.2 Sagnacinterferometer..................................................... 85
Answers................................................................................... 89
1 ElectromagneticWavesintheOpticalRange......... 89
1.1 Equations and parameters of electro-
magneticwaves................................................ 89
1.2 Spectral properties of electromagnetic
waves.................................................................... 90
1.3 Polarizationofelectromagneticwaves..... 90
1.4 Energy,power,andintensityoflight......... 91
1.5 Standingelectromagneticwaves................. 91
2 ReflectionandRefractionofLight............................. 92
2.1 Fresnelformulae.............................................. 92
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Contents ix
2.2 Energy reflection and transmission
coefficients......................................................... 92
2.3 Totalinternalreflection.................................. 93
3 OpticsofAnisotropicMedia:CrystalOptics......... 94
3.1 Refraction of light at the boundary of
ananisotropicmedium................................... 94
3.2 Linearpolarizersandretarderplates......... 94
3.3 Interferenceofpolarizedlight..................... 94
3.4 Polarizationprisms.......................................... 95
3.5 Inducedopticalanisotropy............................ 95
4 CoherenceandInterferenceofLight......................... 96
4.1 Interference of monochromatic and
mutuallycoherentlight.................................. 96
4.2 Amplitude division interferometers:
The Michelson and Mach–Zehnder
interferometers.................................................. 97
4.3 Wavefrontdivisioninterferometers:The
Younginterferometer...................................... 97
4.4 Coherenceoflight........................................... 97
4.4.1 Temporalcoherenceoflight........ 97
4.4.2 Transverse spatial coherence
oflight................................................ 98
4.4.3 Longitudinalspatialcoherence
oflight................................................ 98
4.5 Interferenceofpartiallycoherentlight..... 98
4.6 Multibeam interferometry: The Fabry-
Pérotinterferometer........................................ 99
5 DiffractionofLight......................................................... 100
5.1 Fresnelzonetheoryofdiffraction.............. 100
5.2 Far-field(Fraunhofer)diffraction............... 100
5.3 Diffractiondivergenceofopticalbeams:
diffractionlimitofresolutioninoptical
systems................................................................ 101
5.4 Diffractiongratings......................................... 101
5.5 Light diffraction by volume gratings:
diffractionbyacousticwaves....................... 102
6 OpticsofMovingMedia................................................ 103
6.1 Dopplereffect................................................... 103
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x Contents
6.2 Sagnacinterferometer.................................... 103
References...............................................................................105
Index.........................................................................................107
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Chapter 1
Electromagnetic Waves
in the Optical Range
1.1 Equations and parameters of electromagnetic
waves
1.1 Estimate the phase velocity v of light propagation in a
medium with refractive index n = 1.5. Determine the wave-
length λ of the light in this medium if the frequency of the
lightoscillationsisν = 5·1014 Hz.
1.2 Determine the change ∆v of the phase velocity of light
propagation due to refraction at the interface between glass
(n = 1.5) and water (n = 1.33), and the ratio of wave-
1 2
lengthsλ /λ .
1 2
1.3 Determine the phase velocity v of light propagation in a
medium with relative permittivity ε = 2.5 and permeability
µ = 1.
1.4 Showthattheplane-waveexpressionE(z,t) = E cos(ωt−kz)
0
is a solution of the wave equation and express the phase ve-
locityofthewavevintermsofthewaveparameters,namely,
circular frequency ω and spatial circular frequency (wave
number)k.
1.5 Using the complex form of the expression for a plane har-
monic wave propagating along the z axis, show that this
expressionsatisfiesthewaveequation.
1.6 Using the complex expression for the plane harmonic elec-
tromagneticwaveandMaxwell’sequationsinthedifferential
form,showthatthevectors E(cid:126), H(cid:126) and(cid:126)kofthewaveformthe
1
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