Optics and photonics an introdution-F.Graham Smith, Terry A.King, Dan Wilkins

Optics and Photonics: An Introduction Second Edition F Graham Smith University of Manchester, UK Terry A King University of Manchester, UK Dan Wilkins University of Nebraska at Omaha, USA Optics and Photonics: An Introduction SECOND EDITION Optics and Photonics: An Introduction Second Edition F Graham Smith University of Manchester, UK Terry A King University of Manchester, UK Dan Wilkins University of Nebraska at Omaha, USA Copyright # 2007 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (þ44) 1243 779777 Email (for orders and customer service enquiries): cs-books@wiley.co.uk Visit our Home Page on www.wileyeurope.com or www.wiley.com 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, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd, 90 Tottenham Court Road, London W1T 4LP, UK, without the permission in writing of the Publisher Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, or emailed to permreq@wiley.co.uk, or faxed to (þ44) 1243 770620 This publication is designed to provide accurate and authoritative information in regard to the subject matter covered It is sold on the understanding that the Publisher is not engaged in rendering professional services If professional advice or other expert assistance is required, the services of a competent professional should be sought Other Wiley Editorial Offices John Wiley & Sons Inc., 111 River Street, Hoboken, NJ 07030, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103-1741, USA Wiley-VCH Verlag GmbH, Boschstr 12, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 42 McDougall Street, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, Clementi Loop #02-01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada Ltd, 6045 Freemont Blvd., Mississauga, Ontario, Canada L5R 4J3 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic books Anniversary Logo Design: Richard J Pacifico Library of Congress Cataloging-in-Publication Data Graham-Smith, Francis, Sir, 1923Optics and photonics : an introduction – 2nd ed / F Graham Smith, Terry A King, Dan Wilkins p cm ISBN 978-0-470-01783-8 – ISBN 978-0-470-01784-5 Optics–Textbooks Photonics–Textbooks I King, Terry A II Wilkins, Dan, 1947III Title QC446.2.G73 2007 535–dc22 2006103070 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 9780470017838 (HB) ISBN: 9780470017845 (PB) Typeset in 10/12pt Times by Thomson Digital Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production Contents PREFACE LIGHT AS WAVES, RAYS AND PHOTONS ix The nature of light Waves and rays Total internal reflection The light wave Electromagnetic waves The electromagnetic spectrum Stimulated emission: the laser Photons and material particles GEOMETRIC OPTICS 19 The thin prism: the ray approach and the wavefront approach The lens as an assembly of prisms Refraction at a spherical surface Two surfaces; the simple lens Imaging in spherical mirrors General properties of imaging systems Separated thin lenses in air Ray tracing by matrices Locating the cardinal points: position of a nodal point, focal point, principal point, focal length, the other cardinal points Perfect imaging Perfect imaging of surfaces Ray and wave aberrations Wave aberration on-axis – spherical aberration Off-axis aberrations The influence of aperture stops The correction of chromatic aberration Achromatism in separated lens systems Adaptive optics OPTICAL INSTRUMENTS 57 The human eye The simple lens magnifier The compound microscope The confocal scanning microscope Resolving power; conventional and near-field microscopes The telescope Advantages of the various types of telescope Binoculars The camera Illumination in optical instruments PERIODIC AND NON-PERIODIC WAVES 83 Simple harmonic waves Positive and negative frequencies Standing waves Beats between oscillators Similarities between beats and standing wave patterns Standing waves at a reflector The Doppler effect Doppler radar Astronomical aberration Fourier series Modulated waves: Fourier transforms Modulation by a non-periodic function Convolution Delta and grating functions Autocorrelation and the power spectrum Wave groups An angular spread of plane waves vi Contents ELECTROMAGNETIC WAVES 115 Maxwell’s equations Transverse waves Reflection and transmission: Fresnel’s equations Total internal reflection: evanescent waves Energy flow Photon momentum and radiation pressure Blackbody radiation FIBRE AND WAVEGUIDE OPTICS 135 The light pipe Guided waves The slab dielectric guide Evanescent fields in fibre optics Cylindrical fibres and waveguides Numerical aperture Materials for optical fibres Dispersion in optical fibres Dispersion compensation Modulation and communications Fibre optical components Hole-array light guide; photonic crystal fibres Optical fibre sensors Fabrication of optical fibres POLARIZATION OF LIGHT 163 Polarization of transverse waves Analysis of elliptically polarized waves Polarizers Liquid crystal displays Birefringence in anisotropic media Birefringent polarizers Generalizing Snell’s law for anisotropic materials Quarter- and half-wave plates Optical activity Formal descriptions of polarization Induced birefringence INTERFERENCE 185 Interference Young’s experiment Newton’s rings Interference effects with a plane-parallel plate Thin films Michelson’s spectral interferometer Multiple beam interference The Fabry–Pe´rot interferometer Interference filters INTERFEROMETRY: LENGTH, ANGLE AND ROTATION 205 The Rayleigh interferometer Wedge fringes and end gauges The Twyman and Green interferometer The standard of length The Michelson–Morley experiment Detecting gravitational waves by interferometry The Sagnac ring interferometer Optical fibres in interferometers The ring laser gyroscope Measuring angular width The effect of slit width Source size and coherence Michelson’s stellar interferometer Very long baseline interferometry The intensity interferometer 10 DIFFRACTION 231 Diffraction at a single slit The general aperture Rectangular and circular apertures: uniformly illuminated single slit: two infinitesimally narrow slits: two slits with finite width: uniformly illuminated rectangular aperture: uniformly illuminated circular aperture Fraunhofer and Fresnel diffraction Shadow edges – Fresnel diffraction at a straight edge Diffraction of cylindrical wavefronts Fresnel diffraction by slits and strip obstacles Spherical waves and circular apertures: half-period zones Fresnel–Kirchhoff diffraction theory Babinet’s principle The field at the edge of an aperture 11 THE DIFFRACTION GRATING AND ITS APPLICATIONS 259 The diffraction grating Diffraction pattern of the grating The effect of slit width and shape Fourier transforms in grating theory Missing orders and blazed gratings Making gratings Contents vii Concave gratings Blazed, echellette, echelle and echelon gratings Radio antenna arrays: end-fire array shooting equally in both directions: end-fire array shooting in only one direction: the broadside array: two-dimensional broadside arrays X-ray diffraction with a ruled grating Diffraction by a crystal lattice The Talbot effect 12 SPECTRA AND SPECTROMETRY 281 Spectral lines Linewidth and lineshape The prism spectrometer The grating spectrometer Resolution and resolving power Resolving power: the prism spectrometer Resolving power: grating spectrometers The Fabry–Pe´rot spectrometer Twin beam spectrometry; Fourier transform spectrometry Irradiance fluctuation, or photon-counting spectrometry Scattered laser light 13 COHERENCE AND CORRELATION 307 Temporal and spatial coherence Correlation as a measure of coherence Temporal coherence of a wavetrain Fluctuations in irradiance The van Cittert–Zernike theorem Autocorrelation and coherence Two-dimensional angular resolution Irradiance fluctuations: the intensity interferometer Spatial filtering 14 HOLOGRAPHY 329 Reconstructing a plane wave Gabor’s original method Basic holography analysis Holographic recording: off-axis holography Aspect effects Types of hologram Holography in colour The rainbow hologram Holography of moving objects Holographic interferometry Holographic optical elements Holographic data storage 15 LASERS 349 Stimulated emission Pumping: the energy source Absorption and emission of radiation Laser gain Population inversion Threshold gain coefficient Laser resonators Beam irradiance and divergence Examples of important laser systems: gas lasers, solid state lasers, liquid lasers 16 LASER LIGHT 371 Laser linewidth Spatial coherence: laser speckle Temporal coherence and coherence length Laser pulse duration: Q-switching, mode-locking Laser radiance Focusing laser light Photon momentum: optical tweezers and trapping; optical tweezers; laser cooling Non-linear optics 17 SEMICONDUCTORS AND SEMICONDUCTOR LASERS 395 Semiconductors Semiconductor diodes LEDs and semiconductor lasers; heterojunction lasers Semiconductor laser cavities Wavelengths and tuning of semiconductor lasers Modulation Organic semiconductor LEDs and lasers 18 SOURCES OF LIGHT 415 Classical radiation processes: radiation from an accelerated charge; the Hertzian dipole Free– free radiation Cyclotron and synchrotron radiation Free electron lasers Cerenkov radiation 15.2 Pumping: The Energy Source 351 Energy levels Short lifetime Long lifetime Laser transition Population Figure 15.2 Energy levels and the level populations in a three-level laser The ruby laser is an example of a three-level laser in which the active species is the Cr3þ ion rather than a neutral atom One further element is needed to make such an amplifier into a self-excited oscillator; the light must be fed back into the laser material This is achieved by enclosing the lasing material between mirrors, forming a resonant cavity Emission from the device is obtained by arranging that one of the resonator mirrors has a non-zero transmittance 15.2 Pumping: The Energy Source As shown in Figure 15.2 the energy which is converted into laser light is injected, or pumped, into the laser at a higher photon energy hn31 than the laser output photons with energy hn21 The excited atoms (or ions) then lose energy hn32, falling into the intermediate level which has a longer lifetime Atoms accumulate in this metastable state, and are available for the stimulated emission process The original ruby laser was pumped by an intense flash of white light, which is selectively absorbed by chromium ions dispersed through the aluminium oxide crystal Only a small part of the energy in the white light is at the right wavelength to be absorbed and produce the population inversion; this is inefficient, which is the reason for the use of an intense source of light Other types of laser use more finely tuned pumping systems; the very common He–Ne gas laser provides a good example The He–Ne laser contains a mixture of the two gases in an electrical discharge tube Both gases are excited and ionized in the discharge The amplifying medium is neon, which is pumped into a state of population inversion by collision with excited helium atoms; these in turn have been energized by electron collisions in the discharge The energy transfer between the two species of gas atoms is very efficient because of a close coincidence between energy levels in the excited helium and the upper levels suited for the laser action in neon Figure 15.3 shows the outline of the He–Ne laser, and the energy levels involved The coincidence is between the two metastable levels 21 S0 and 23 S1 in helium and the two metastable levels 5s and 4s in neon.3 Stimulated emission from the 5s and 4s levels can be The He levels are described by Russell–Saunders coupling, while for Ne the levels are designated by their electron configuration as (1s2 2s2 2p5 )3s, ( )4s, ( )5s, etc.; note that in the older Paschen notation the ( )3s configuration is designated 1s and ( )4s is designated 2s, and so on 352 Chapter 15: Lasers V (a) Energy (in 10 000 cm1 units) 17 16 S1 Infrared lasing 3.39 µm 5s S0 Collisions 4p 4s Green lasing 0.543 µm Red lasing 0.6328 µm Infrared lasing 1.15 µm 15 3p Fast spontaneous decay 14 3s 13 Radiation, collision with walls and electrons 12 Helium S Neon 1s 2s 2p (b) Figure 15.3 The helium–neon laser (a) Laser excited by a d.c electrical discharge, with potential V (b) Simplified energy level diagram through transitions to several different energy levels, allowing laser action at 3.39 mm, 1.15 mm, 632.8 nm and 543.5 nm The familiar red beam of the He–Ne laser is operating on the 632.8 nm transition Figure 15.3 shows the mirrors which enclose the laser, forming a resonator As will appear later, a particular laser wavelength can then be selected by a choice of resonator system 15.3 Absorption and Emission of Radiation We now review the basic theory of the three processes involved in the interaction of radiation and matter, which we introduced briefly in Section 1.7 The processes of absorption, spontaneous emission and stimulated emission are sketched in Figure 15.4 We suppose that the two states, with energies E1 and E2 , are populated with number densities n1 and n2 Absorption occurs when radiation of frequency n ¼ ðE2 À E1 Þ=h is incident on the medium, with excitation from the ground state to the excited state The rate of absorption in which atoms are raised from level to level is   dn1 ¼ ÀB12 n1 uðnÞ ð15:1Þ dt ab 15.3 Absorption and Emission of Radiation 353 2 (a) (b) (c) Figure 15.4 Absorption, spontaneous emission and stimulated emission where n1 is the population per unit volume in level and uðnÞ is the energy density of the incident field (units of energy per unit volume per unit frequency interval, J mÀ3 HzÀ1 Þ uðnÞ is a function of the frequency n of the radiation field B12 is the Einstein absorption coefficient, which is a constant characteristic of the pair of energy levels in the particular type of atom Spontaneous emission of a photon occurs with transition of the electron from the excited level to the ground level with the emitted photon energy hn ¼ E2 À E1 The rate of decrease of the population n2 by spontaneous emission is   dn2 ¼ A21 n2 : ð15:2Þ À dt spon The constant A21 (unit: sÀ1 ) is related to the spontaneous radiative lifetime t of the excited state as A21 ¼ : t ð15:3Þ In stimulated emission atoms in level are stimulated to make a transition to level by the radiation field itself The rate at which the transition occurs is proportional to the number of atoms in level and the energy density of the radiation field:   dn2 ¼ ÀB21 n2 uðnÞ: ð15:4Þ dt stim The constant B21 is the Einstein coefficient for stimulated emission from energy level to level Note that the rate of stimulated emission is proportional to the energy density at the resonant frequency n ¼ ðE1 À E2 Þ=h, so that for high levels of radiation energy density stimulated emission dominates spontaneous emission The rate of change of population in level is the sum of the effects of spontaneous and stimulated transitions given by equations (15.1), (15.2) and (15.4), which yields the rate equation dn2 ¼ ÀB21 uðnÞn2 þ B12 uðnÞn1 À A21 n2 : dt ð15:5Þ Conservation of atoms implies that the ground-state population density obeys dn1 =dt ¼ Àdn2 =dt The relation between the three Einstein coefficients is found by considering an equilibrium situation, where a collection of atoms within a cavity is in thermal equilibrium with a radiation field Then the populations of the two levels n1 and n2 are constant dn2 dn1 ¼ ¼ 0: dt dt ð15:6Þ 354 Chapter 15: Lasers In thermal equilibrium, when there is detailed balancing between the processes acting to populate and depopulate the energy levels, setting dn2 =dt ¼ we obtain A21 n2 þ B21 uðnÞn2 ¼ B12 uðnÞn1 ð15:7Þ giving the relation between the values of n1 ; n2 and uðnÞ at thermal equilibrium We now use two fundamental laws relating uðnÞ and the relative populations n1 ; n2 to the temperature T These are Planck’s radiation law for cavity radiation (Section 5.7) uðnÞ ¼ 8phn3 c3 expðhn=kTÞ À and the Boltzmann distribution of atoms between the two energy levels:     n2 g2 E2 À E1 g2 hn ¼ exp À : ¼ exp À kT n1 g1 kT g1 ð15:8Þ ð15:9Þ Here we have allowed for the possibility that either level is degenerate, i.e that for the jth level, there are gj ð¼ 1; 2; 3:::Þ quantum states with the same energy (gj ¼ is the non-degenerate case) From equations (15.7) and (15.9) A21 B12 ðn1 =n2 Þ À B21 ð15:10Þ A21 : B12 ðg1 =g2 Þ expðhn=kTÞ À B21 ð15:11Þ uðnÞ ¼ or uðnÞ ¼ This equation may be combined with equation (15.8) to give   A21 8phn3 ¼ : expðhn=kTÞ À ðg1 =g2 ÞB12 expðhn=kTÞ À B21 c3 ð15:12Þ Equation (15.12) is satisfied when A21 8phn3 ¼ B21 c3 g1 B12 ¼ g2 B21 : (15.13) (15.14) These are the required relations between the three Einstein coefficients (see also Problem 15.3) These equations and the concept of transition probability are fundamental to the theory of exchange of energy between matter and radiation The crucial factor for lasers is the ratio between the rates of stimulated and spontaneous emission: rate of stimulated emissions B21 uðnÞ ¼ : ¼ rate of spontaneous emissions A21 ðexpðhn=kTÞ À 1Þ ð15:15Þ For lasing to be feasible, this ratio should be much greater than In that case, stimulated emission dominates spontaneous emission, and the latter is less able to erode away a population inversion before lasing can occur 15.4 Laser Gain 355 Example Assuming thermal equilibrium at room temperature (T ¼ 300 K), evaluate the ratio of equation (15.15) for l ¼ 600 nm (visible) and l ¼ cm (microwave) Solution In general, if T ¼ 300 K, and we measure l ½expð48=lmm Þ À 1ŠÀ1 Hence stimulated rate % expðÀ80Þ ’ 10À35 spontaneous rate in mm, ½expðhn=kTÞ À 1ŠÀ1 ¼ ðl ¼ 0:6 mmÞ ð15:16Þ and 200 at l ¼ cm The factor exp ðhn=kTÞ shows us that in thermal equilibrium stimulated emission is very unlikely at optical frequencies, and explains why the first successful device was the maser, operating at a much lower radio frequency It is not surprising then to learn that all lasers developed until now operate with radiation that is far from thermal equilibrium We also note that since the stimulated emission rate is n2 B21 uðnÞ we may increase the rate by increasing uðnÞ, which is achieved in a resonant cavity, and by increasing n2 in the population inversion resulting from pumping 15.4 Laser Gain We can now consider the growth of a light wave as it passes through an active laser medium, and find the conditions for the wave to grow by stimulated emission The resulting fractional rate of growth, in equation (15.25) below, depends on four factors: the population inversion, the spectral lineshape, the frequency and the transition probability A21 We start by finding the emission and absorption in a small element dz of the path through the laser medium, and then integrating over the whole path, which may involve many to-and-fro reflections in a resonator First we look at the attenuation of an absorbing medium in which a plane wave of monochromatic radiation is travelling as illustrated in Figure 15.5 The reduction in irradiance (power flow across unit area) as the wave travels from position z to z þ dz for a uniform medium is proportional to the magnitude of the irradiance and the distance travelled: dIðzÞ ¼ Iðz þ dzÞ À IðzÞ ¼ ÀaIðzÞdz: I(z) I(z+dz) z Figure 15.5 z+dz Attenuation of a wave in a slab of material ð15:17Þ 356 Chapter 15: Lasers Here a is the absorption coefficient Hence dIðzÞ ¼ ÀaI: dz ð15:18Þ IðzÞ ¼ I0 expðÀazÞ; ð15:19Þ On integration where I0 is the irradiance of the incident beam This represents exponential attenuation If the number of stimulated emissions exceeds the number of absorptions, rather than being attenuated the wave will grow The number of stimulated emissions depends on the energy density uðnÞ The irradiance is the product of the energy density and the velocity, so that in free space or a thin gas I uðnÞ ¼ : c ð15:20Þ The change in irradiance dI of the wave in travelling a distance dz is now proportional to the difference between the numbers of stimulated emissions and absorptions:   I I dI ¼ n2 B21 gðnÞ À n1 B12 gðnÞ hndz: ð15:21Þ c c Here we have introduced the normalized spectral function, or lineshape gðnÞ for the transition, which describes the frequency spectrum of the spontaneously emitted radiation The lineshape is dependent on the mechanism determining the broadening of the transition, as described in Chapter 12 and Appendix In gas lasers inhomogeneous broadening usually dominates due to the thermal motion of the atoms o+ ions Inhomogeneous broadening also applies to transitions in doped glasses where variations in the sites of the doped ions lead to a distribution of centre frequencies A typical inhomogeneously broadened lineshape is shown in Figure 15.6 The normalization of the function gðnÞ is such that Z gðnÞdn ¼ 1: ð15:22Þ g(v) FWHM v0 Frequency v Figure 15.6 A typical inhomogeneously broadened Gaussian lineshape function gðnÞ showing the full width at half maximum (FWHM) 15.4 Laser Gain 357 From equation (15.21), and the Einstein relations (15.13) and (15.14),   dI g2 c A21 ¼ n2 À n1 gðnÞI: dz g1 8pn2 ð15:23Þ Integrating gives an exponential dependence on distance z I ¼ I0 expðgðnÞzÞ ð15:24Þ where I0 is the irradiance at z = 0, and gðnÞ is the gain coefficient: gðnÞ ¼   g2 c A21 n2 À n1 gðnÞ: g1 8pn2 ð15:25Þ If n2 > ðg2 =g1 Þn1 , representing population inversion, then gðnÞ > 0, and the irradiance grows exponentially with distance in the medium The gain coefficient depends, as expected, on the transition probability A21 and on the lineshape Note, however, that the frequency dependence ðnÀ2 ) indicates it is more difficult to make lasers for ultraviolet light than for infrared In comparing the suitability of different laser media it is convenient to specify a stimulated emission cross-section parameter sðnÞ, which is related to the gain coefficient gðnÞ by   g2 gðnÞ ¼ n2 À n1 sðnÞ: ð15:26Þ g1 From equation (15.25) sðnÞ ¼ c2 A21 gðnÞ : 8pn2 ð15:27Þ Since the lineshape gðnÞ is normalized (equation (15.22)), the central height of the line gðn0 Þ is inversely proportional to the linewidth,4 and to a useful approximation gðn0 Þ $ 1=Án For the lineshape of homogeneous broadening (see Section 12.2 and Appendix 4) gðn0 Þ ¼ : pÁn ð15:28Þ Then the cross-section parameter at the peak frequency becomes s0 ¼ sðn0 Þ ¼ c2 A21 : 4p2 n20 Án ð15:29Þ This shows that the stimulated emission cross-section for a homogeneously broadened transition is proportional to the ratio A21 =Án, the spontaneous transition rate over the linewidth (In liquid and solid state lasers the higher refractive index n of the medium compared with a gas means that the light speed c should be replaced by c=n.) The linewidth here is the full width at half maximum, or FWHM 358 Chapter 15: E2 E2 E1 E1 n2 n1 Lasers n2 n1 Population Population (a) (b) Figure 15.7 Population inversion The normal Boltzmann distribution (a) of population in two energy levels is shown inverted in (b) (Here we assume g2 =g1 ¼ 1) 15.5 Population Inversion The population inversion condition n2 > ðg2 =g1 Þn1 derived in Section 15.4 is a necessary condition for the gain coefficient to be positive The two cases of thermal equilibrium and population inversion are shown in Figure 15.7 To create population inversion, energy is required to be put selectively into the laser medium such that the population of level is increased over level to form a non-equilibrium distribution Excitation of the laser medium by pumping may be achieved in several ways In gases at normal pressures, the absorption lines have a narrow bandwidth, which limits their ability to absorb light, and pumping is usually by electron collisions in an electrical discharge Solid state crystals and glasses doped with an active ion have broader absorption lines than gases and are usually excited optically by absorption of energy from a lamp or from another laser In semiconductor lasers (Chapter 17), the relevant energy levels correspond to the conduction and valence bands, which are comparatively very broad Here pumping is achieved by applying an electric field across the semiconductor junction Lasers may conveniently be divided into three- and four-level systems depending on the number of levels active in their operation This is illustrated in Figure 15.8 which shows the inverted population at the laser transition 15.6 Threshold Gain Coefficient Laser oscillation is initiated in a system with population inversion by the spontaneous emission of a photon along the axis of the laser For the laser to sustain oscillation the gain in the laser medium must be greater than the losses in the cavity The losses arise from transmission at the cavity mirrors (in order to provide the laser output, a typical transmission is 5% for continuous laser operation) Other losses arise from absorption and scattering by the mirrors and in the laser medium, and diffraction out of the sides of the cavity The threshold for laser oscillation will occur when the gain is equal to the losses To calculate this threshold gain we combine all the sources of loss into one 15.6 Threshold Gain Coefficient 359 Energy Energy short long short long short Population Population (a) (b) Figure 15.8 Three- and four-level laser schemes lumped loss coefficient k At threshold the irradiance neither decreases nor increases; it stays constant Consider a cavity made up of mirrors M1 and M2 with reflectances R1 and R2 and spaced by a distance L A beam of irradiance I0 starting at M1 on reaching M2 has become I1 ¼ I0 exp½ðg À kÞLŠ, where g and k are the gain and loss coefficients On reflection from M2 and travelling in return through the medium and undergoing reflection at M1 , the irradiance becomes I2 ¼ I0 R1 R2 exp½2ðg À kÞLŠ The round-trip gain, G, is defined as I2 =I0 Then G ¼ I2 =I0 ¼ R1 R2 exp½2ðg À kÞLŠ: ð15:30Þ The threshold condition for laser oscillation is G ¼ 1, giving R1 R2 exp½2ðgth À kÞLŠ ¼ where gth is the threshold gain coefficient, at which the laser will begin to oscillate From equation (15.31) we find   1 gth ¼ k þ ln : 2L R R2 ð15:31Þ ð15:32Þ The first term is the loss within the cavity, and the second term is the loss due to the mirror transmission (or absorption), i.e including that leading to the useful laser output Continuously operating lasers are called CW lasers, standing for continuous wave Once a CW laser is operating in a steady state, the gain stabilizes at the threshold value, since if the gain were greater or less than unity the irradiance would increase or decrease The level at which the irradiance stabilizes depends on the pump power 360 15.7 Chapter 15: Lasers Laser Resonators Most lasers require a long path through the active medium to obtain sufficient overall gain This is achieved by multiple reflections in an optical resonator, often referred to as a resonant cavity An optical resonator both increases laser action and defines the frequency at which it occurs Optical feedback is provided by the optical resonator which retains photons inside the cavity, reflecting them back and forth through the laser medium The simplest basic optical resonator is a pair of shaped mirrors at each end of the laser medium, as in a Fabry–Pe´rot interferometer There are various configurations using plane and curved mirrors used in optical Fabry–Pe´rot resonators; some of these are shown in Figure 15.9 Not all configurations of mirror curvatures and spacings will give stable operation Usually one of the mirrors is arranged to be practically 100% reflecting at the laser wavelength; the other mirror (the output mirror) has a finite transmission, so that light will be transmitted out of the optical cavity to provide the laser output The optical Fabry–Pe´rot resonator made up of two plane-parallel mirrors is similar to the Fabry–Pe´rot etalon or interferometer described in Chapter The resonance condition for waves at normal incidence, along the axis of a cavity with optical length L, as for standing waves, is m l ¼L ð15:33Þ where m is an integer Then the resonant frequency nm for each longitudinal mode of the cavity is nm ¼ m c : 2L ð15:34Þ This equation is important in defining the resonant frequencies at which the laser will oscillate, as it will if they fall within the gain profile of the laser transition, as illustrated in Figure 15.10 The possible oscillating frequencies are termed the longitudinal modes of the laser; they are spaced by c Án ¼ : ð15:35Þ 2L Each of these frequencies may, however, be broken into a more narrowly spaced set; these are due to transverse modes, in which the field pattern may have different structures transverse to the beam M1 M2 r1 = r2 = ∞ Plane parallel Long radius r1 = r2 >> L Confocal r1 = r2 = L r1 = ∞, r2 = L Hemispherical L Figure 15.9 Common laser resonator configurations r1 and r2 are the radii of curvature of mirrors M1 and M2 15.7 Laser Resonators 361 Gain profile (a) Gain Loss v (b) (c) vm v ∆v Figure 15.10 Gain profile and resonant frequencies in a cavity laser: (a) gain profile of the laser transition; (b) allowed resonances of the Fabry–Pe´rot cavity; (c) oscillating laser frequencies direction A transverse mode is an electric and magnetic field configuration at some position in the laser cavity which, on propagating one round trip in the cavity, returns to that position with the same pattern; some of these field patterns are shown in Figure 15.11 The laser output is at one or more frequencies from this set of modes When only one longitudinal and transverse mode is selected, in a single mode laser (Chapter 16), the bandwidth of the laser light is almost unbelievably small For comparison, light from a single line of a low-pressure gas discharge lamp has a spectral width of about 1000 MHz Non-pulsed laser light in contrast typically has a bandwidth of less than MHz and may, with careful design, have a bandwidth of less than 10 Hz As can be seen in Figure 15.11, the transverse modes can have polar (or circular)5 symmetry or Cartesian (rectangular) symmetry; these are known respectively as Laguerre–Gaussian modes and HG1,0 LG HG5,0 LG HG3,1 LG HG3,3 3 LG Figure 15.11 Distribution of irradiance for various transverse modes: Hermite–Gaussian (HG), where the double subscript refers to the number of nodes in the x and y directions, and the corresponding Laguerre– Gaussian (LG), where the superscript and subscript refer to cycles of azimuthal phase and the number of radial nodes respectively In three dimensions, the LG modes are actually helical and carry angular momentum 362 Chapter 15: Lasers Hermite–Gaussian modes Although most lasers are constructed with circular symmetry, the modes with Cartesian symmetry are most common; this arises when some element in the laser cavity imposes a preferred direction on the transverse electric and magnetic field vectors The lowest order transverse electromagnetic mode (HG00 or LG00 ) is labelled TEM00 This is the fundamental mode with the largest scale pattern across the laser beam The zero subscripts indicate that there are no nodes in the x and y directions, transverse to the direction of the laser beam The cavity mirrors are, of course, required to reflect at the laser wavelength in order to make the cavity resonant Typically one mirror has a reflectivity as close to 100% as possible and one is arranged to have a carefully selected transmission, chosen to produce the optimum laser output power; this necessarily means that the transmission must be less than the overall laser gain For efficient operation the deviations of the mirrors from their ideal shapes are required to be within a small fraction of the laser wavelength (usually $ l=20) 15.8 Beam Irradiance and Divergence The beam of light leaving the laser is coherent in relation to both its narrow spectral linewidth and its spatial coherence over its emitted wavefront As it leaves the laser, the beam will spread into a narrow angle by diffraction, the width depending on the field distribution across the beam This can be viewed as the beam’s cross-section acting as its own diffraction aperture The simplest mode (the TEM00 mode), which has the narrowest beam, has a Gaussian radial dependence of irradiance Iðr; zÞ with peak irradiance along the axis, so that at radial distance r from the axis   À2r Iðr; zÞ ¼ I0 exp : ð15:36Þ w ðzÞ The radial width parameter w is referred to as a spot size and varies with distance along the axis (For r ¼ w the amplitude is 1=e of the amplitude on-axis, but for convenience this is often referred to as the edge of the beam.) The spot size is smallest within the laser cavity, where there is a beam waist Here the width w0 (Figure 15.12) is related to the length L of the resonator and the wavelength l as  1=2 lL w0 ¼ : ð15:37Þ 2p This applies for both the cavity with two plane mirrors and the symmetric confocal cavity As we discuss below, the cavity mirrors must be significantly larger than this spot size to avoid diffraction loss 2w0 2w L z Figure 15.12 The beamwidth w0 at the waist and at a distance z from the waist 15.8 Beam Irradiance and Divergence 363 The laser beam spreads by diffraction (Figure 15.12) both inside and outside the resonator Analysis of the Gaussian beam solutions of the paraxial wave equation leads to the width wðzÞ of the beam at distance z from the beam waist: "   #1=2 lz wðzÞ ¼ w0 þ ð15:38Þ pw20 which approximates to wðzÞ ’ lz pw0 for z) pw20 : l ð15:39Þ Note that the larger the beam waist, the smaller the angle of spread of the beam For the TEM00 mode, which has a Gaussian spatial profile, the half angle y of the divergence cone for the propagating beam is y¼ l : pw0 ð15:40Þ As expected from Fraunhofer diffraction, the angular width is of order l=w0 For example, an He–Ne laser with l ¼ 632:8 nm operating with a symmetric confocal resonator of length L ¼ 30 cm has  1=2 lL minimum spot radius w0 ¼ ¼ 0:17 mm 2p l ’ 1:2 mrad ¼ 0:066 : divergence angle y ’ pw0 ð15:41Þ ð15:42Þ Note that the beamwidth w0 is determined by the length and not the width of the laser There is, however, a need for the resonator mirrors to be sufficiently wide, so that the beam is not lost by diffraction at each reflection For example, consider the diffraction broadening of a beam that arrives at mirror M2 after it reflects off M1 Assuming initially that the beam fills mirror M1 , the diffraction half angle at mirror M1 is $ l=d1 where d1 is the diameter of the mirror M1 and also of the beam at M1 If d2 is the diameter of M2 , low loss requires d2 ! d1 þ 2Ll=d1 , or approximately d1 d > 1: lL ð15:43Þ This is known as the Fresnel condition For a symmetrical arrangement where d1 ¼ d2 ¼ d the condition is d2 =lL > 1; the quantity d2 =lL is known as the Fresnel number of the optical arrangement (Note the close relationship to the Rayleigh distance (Section 10.4), which defines the boundary between Fraunhofer and Fresnel diffraction.) The beam remains almost parallel for some distance from the laser In equation (15.38) the width is almost constant for distances z 12 z0 , where z0 ¼ pw20 =l defines the Rayleigh range, i.e the distance over which a laser beam is effectively collimated For example, a red-light beam from a laser with mm aperture remains parallel for about m A longer but wider parallel beam can be achieved by using a beam expander, which is a telescope system used in reverse (Figure 16.3) This effectively gives a larger coherent wavefront than the laser aperture alone A survey theodolite with a 25 mm aperture would have a parallel beam over a distance of km Over longer distances the beam 364 Chapter 15: Lasers expander achieves a smaller angular spread than the laser alone Given an optical system accurate to a fraction of a wavelength, and in the absence of atmospheric turbulence, a very narrow beam can be generated A telescope with m diameter aperture can transmit a laser beam with a divergence less than 1/2 arcsecond; this would illuminate a spot only km across on the Moon 15.9 Examples of Important Laser Systems 15.9.1 Gas Lasers Gas lasers may be divided into several types, depending on the active amplifying species in the gas and the excitation mechanism The wide range of gas lasers is summarized in Table 15.1 The wavelengths of gas lasers cover a very broad range from the vacuum UV to the far IR, in continuous wave and pulsed operation, and with some lasers operating up to high powers A mixture of gases is often used in gas lasers to enable excitation by energy transfer between the components or to enhance their operation There are many different pumping mechanisms, including continuous, pulsed or radio frequency electrical discharges, optical pumping, chemical reactions and intense excitation in plasmas The laser emission may be from electronic transitions in neutral atoms (e.g the He–Ne laser) or ionized atoms (e.g Arþ or Krþ ), electronic transitions in molecules (e.g F2 or N2 ), electronic Table 15.1 Examples of gas lasers Laser type Neutral atom He–Ne Cu Ion Arþ Krþ He–Cd Molecular CO2 N2 F2 HCN CH3 F Excimer ArF KrF XeCl XeF Chemical HF I Plasma Se24þ , Ar8þ, etc Typical power or pulse energy Pulsed or CW 632.8 511, 578 1–50 mW 20 mW CW Pulsed 488, 515 647 441.6, 325.0 2–20 W 1W 50–200 mW CW CW CW 10.6 mm 337.1 157 336.8 mm 496 mm 102 –104 W 10 mJ 10 mJ mW mW CW, pulsed Pulsed Pulsed CW CW 193 248 308 351, 353 mJ, mJ, mJ, mJ, Pulsed Pulsed Pulsed Pulsed 2.6–3.3 mm 1.3 mm CW to kW CW to kW Pulsed mJ to J CW, pulsed CW, pulsed 3.5–47 nJ to mJ, ns Pulsed Wavelength (nm) kHz kHz kHz kHz 15.9 Examples of Important Laser Systems 365 transitions in transient excited dimer molecules (termed excimers, e.g KrF or ArF), and vibrational or rotational transitions in molecules (e.g CO2 , CH3 F) Generally gas lasers are excited by an electrical discharge in which excitation of the gas atoms, ions or molecules is by collision with energetic electrons Optical excitation of a gas is usually inappropriate since the absorption lines of gases are very narrow (in contrast to solids) The He–Ne laser described briefly in Section 15.2 was the first gas laser to be operated (in 1960), and was the first continuously operating laser It is still one of the most common lasers, operating on the 632.8 nm wavelength, and is used in many applications requiring a relatively low-power, visible, continuous and stable beam The CO2 gas laser provides large power outputs at the infrared wavelength of 10.6 mm The laser action involves four vibrational energy levels, as in the scheme of Figure 15.13 The broad highest level is closely equal to an excited level in nitrogen, which is an essential added gas component The upper level of the CO2 molecule is populated from this state by collisions with nitrogen molecules The excitation of the nitrogen molecules is by electron collisions, and the electrons are produced in an electric or radio frequency discharge within the laser tube The gas also contains helium, which assists the depletion of the lower levels by collisional de-excitation and stabilizes the plasma temperature Large continuous power outputs, up to some tens of kilowatts, are obtainable; pulsed operation can give pulse energies of joules in microsecond pulses As the gas densities are usually comparatively low, high-powered CO2 lasers must be relatively large to contain a sufficient number of molecules Regarding the rare optical pumping of gas lasers, two exceptions of interest are the atomic iodine photodissociation laser and the neutral atomic mercury laser The iodine laser is pumped by an intense flashlamp, whose light dissociates a molecule such as CF3 I to produce iodine atoms in the first electronic excited state, and stimulated emission is on the magnetic dipole transition P1=2 –2 P3=2 at 1.3 mm The iodine 1.3 mm laser may also be pumped by a chemical reaction in which excited molecular oxygen, formed in a reaction between hydrogen peroxide and chlorine, transfers energy to atomic iodine The mercury laser operates continuously on the strong Hg 546.1 nm transition pumped by a powerful mercury lamp Gain at X-ray wavelengths over to 47 nm has been demonstrated from highly ionized atoms These pulsed lasers operate in a dense plasma pumped by nanosecond laser pulses or electrical discharges Nanosecond X-ray pulses of up to mJ energy (equivalent to megawatt powers) have been produced Figure 15.13 Vibrational energy levels in the CO2 laser ... Congress Cataloging-in-Publication Data Graham -Smith, Francis, Sir, 1923Optics and photonics : an introduction – 2nd ed / F Graham Smith, Terry A King, Dan Wilkins p cm ISBN 978-0-470-01783-8 – ISBN... attributing a rectangular Optics and Photonics: An Introduction, Second Edition F Graham Smith, Terry A King and Dan Wilkins # 2007 John Wiley & Sons, Ltd Chapter 1: Light as Waves, Rays and Photons... Introduction Second Edition F Graham Smith University of Manchester, UK Terry A King University of Manchester, UK Dan Wilkins University of Nebraska at Omaha, USA Optics and Photonics: An Introduction