DOCSLIB.ORG

The Twist in Light's Tail Miles Padgett FRS FRSE

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Twist in Light's Tail Miles Padgett FRS FRSE The Twist in Light’s Tail Miles Padgett FRS FRSE Kelvin Chair of Natural Philosophy follow me on Twitter @MilesPadgett Light beams carry a Linear Momentum • Light carries a linear momentum – Force exerted = Power/c • Laser pointer exerts 10-11 N • Strong enough to move cells and bacterial • Or (with a very large sail!) a spacecraft…. Light’s Angular Momentum in History We now refer to this as the SPIN angular momentum of the photon But light has an extra angular momentum in addition to photon spin (this extra momentum is needed for some rare atomic transitions) Getting started on Orbital Angular Momentum of Light • 1992, Allen, Beijersbergen, Spreeuw and Woerdman Miles Padgett • 1994, Les meets Miles at dinner…... Orbital Angular Momentum from helical phase fronts E E B B S S Sθ = 0 Sθ ≠ 0 Helical phasefronts and orbtial angular momentum • Phase cross-section ℓ= 0 Φ = exp (iℓφ) ℓ = 0, plane wave ℓ = 1, helical wave ℓ = 2, double helix ℓ= 1 ℓ = 3, pasta fusilli etc. 2π ℓ= 2 Phase 0 x-section of helical wave ℓ = 3 Intensity Phase Angular momentum in terms of photons • Spin angular momentum Polarisation σ = +1 Circular – Circular polarisation σ" per photon • Orbital angular momentum – Helical phasefronts σ = -1 ℓ" per photon Helical Phasefronts ℓ = -1 ℓ = 0 ℓ = 1 ℓ = 2 ℓ = 3 A double-start helix (ℓ=2): Home improvments Chambord castle (chateaux de la Loire) Making helical phasefronts • Pass plane-wave through a spiral- phase plate (thickness α φ) – step height= ℓλ/(n-1) Designing helical phase hologram ! Spiral Phase-plate s= ℓλ/(n-1) s 2π • Holographically + = e.g. ℓ = 1 ▹ 0 Making helical phasefronts with diffraction gratings Making OR measuring phasefronts with holograms Make interactive by using SLM Switching time Generate ≈5-20mSec Efficiency ≈50% Light source OR detector Measure Richard Bowman A gift for all the family….. Orbital angular momentum from Skew rays Poynting vector E B S Orbital angular momentum from skew rays Angular Momentum Transfer Spin Angular Momentum Orbital Angular Momentum (circular angular momentum) (helical phasefronts) The OAM in…… OAM in Quantum Optics e.g. Zeilinger OAM in Communication OAM in tweezers OAM in Quatum Optics e.g. Willner e.g. Rubinsztein-Dunlop e.g. Zeilinger OAM in Imaging e.g. Ritsch-Marte OAM in Lasers OAM in NOT optical e.g. Forbes e.g. Verbeeck OAM and SLM control OAM in Phenomenology in micromanipulation OAM in Quantum OAM in Imaging OAM in Communications Optics Fractality and Topology of Light’s darkness Kevin O’Holleran Florian Flossmann Mark Dennis (Bristol) Vortices are ubiquitous in nature • Whenever three (or more) 2π plane waves interfere optical vortices are formed – Charge one vortices occur 2π wherever there is diffraction or scattering 2π 0 Map out the vortex position in different planes • Either numerically or experimentally one can map the vortex positions in different planes The tangled web of speckle • ≈1600 plane waves c.f. Gaussian speckle Quantum entanglement with spatial light modulators Jonathan Leach Barry Jack Sonja Franke-Arnold (Glasgow) Steve Barnett and Alison Yao (Strathclyde) Bob Boyd Anand Jha (Rochester) OAM in second harmonic generation • Poynting vector “cork screws”, azimuthal skew angle is 1 green θ = ℓ/kr photon ℓ =2ℓ • Does this upset a co-linear “cork screwing” 0 phase match? -No Poynting vector • Frequency & ℓ-index both double • “Path” of Poynting vector stays the same – phase matching maintained 2 infra red photons ℓ = ℓ0 Correlations in angular momentum +10 ℓs -10 -10 +10 i π ℓ Near perfect (anti) Correlations in angular Δℓ momentum Prob. -20 ℓs -ℓi +20 Correlations in angle +π φs π i π φ +π Near perfect Correlations in angle Δφ Prob. π φs -φi +π Angular EPR Correlations in complimentary basis sets 2 2 2 2 Δ s i Δ φs φi = 0.00475 << 0.25 [ ( ) ] [ ( )] Assuming we believe QM to be complete we have to accept that it is non-local €- > EPR for angular momentum and angles! The OAM communicator • Every OAM state forms an independent channel Rotational Doppler Shifts Martin Lavery Steve Barnett and Fiona Speirits Linear vs. Rotational Doppler shifts Light source Light source - exerts force = P/c - exerts torque = ℓP/ω Ω v Doing work on a light beam changes its energy and hence shifts it’s frequency Doppler shift from a SPINNING surface N photons per second Torque= ℓN"ω/ω Work Done= ΩNℓ" Work per photon Ω≠0 "Δω = Ωℓ" Frequency shift Δω = Ωℓ v=0 Illuminate with OAM at +/- ℓ and measure Δω A B 0 L -5 dB H -10 -15 Power -20 . -25 Rel -30 2200 2250 2300 2350 2400 2450 2500 fmod Hz C 500 H L 450 L 1 400 - s . 350 rad H 300 W 250 200 1200 1400 1600 1800 2000 2200 2400 2600 fmod Hz H L Conclusions • Beyond the field of Orbital Angular Momentum itself, the realisation that the Poynting vector can be engineered over the beam cross- section has triggered research into many beam types, and light-matter interactions in both the classical and quantum regimes. • Better communication, better sensing, better imaging. Richard Bowman Students/RAs, Graham Gibson especially Martin Lavery Jonathan Leach Kevin O’Holleran Me Sonja Franke-Arnold Johannes Courtial Jackie Romero Neil Simpson Collaborators Les Allen Steve Barnett External Bob Boyd Mark Dennis Mervyn Miles Monika Ritsch-Marte Wilson Sibbett !.
Load more

Recommended publications
  • Photons That Travel in Free Space Slower Than the Speed of Light Authors
    Title: Photons that travel in free space slower than the speed of light Authors: Daniel Giovannini1†, Jacquiline Romero1†, Václav Potoček1, Gergely Ferenczi1, Fiona Speirits1, Stephen M. Barnett1, Daniele Faccio2, Miles J. Padgett1* Affiliations: 1 School of Physics and Astronomy, SUPA, University of Glasgow, Glasgow G12 8QQ, UK 2 School of Engineering and Physical Sciences, SUPA, Heriot-Watt University, Edinburgh EH14 4AS, UK † These authors contributed equally to this work. * Correspondence to: [email protected] Abstract: That the speed of light in free space is constant is a cornerstone of modern physics. However, light beams have finite transverse size, which leads to a modification of their wavevectors resulting in a change to their phase and group velocities. We study the group velocity of single photons by measuring a change in their arrival time that results from changing the beam’s transverse spatial structure. Using time-correlated photon pairs we show a reduction of the group velocity of photons in both a Bessel beam and photons in a focused Gaussian beam. In both cases, the delay is several microns over a propagation distance of the order of 1 m. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves. Introducing spatial structure to an optical beam, even for a single photon, reduces the group velocity of the light by a readily measurable amount. One sentence summary: The group velocity of light in free space is reduced by controlling the transverse spatial structure of the light beam. Main text The speed of light is trivially given as �/�, where � is the speed of light in free space and � is the refractive index of the medium.
    [Show full text]
  • Arxiv:1603.00726V1 [Physics.Optics] 2 Mar 2016
    Single-pixel 3D imaging with time-based depth resolution Ming-Jie Sun,1, 2, ∗ Matthew. P. Edgar,2 Graham M. Gibson,2 Baoqing Sun,2 Neal Radwell,2 Robert Lamb,3 and Miles J. Padgett2, y 1Department of Opto-electronic Engineering, Beihang University, Beijing, 100191, China 2SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK 3Selex ES, Edinburgh, UK Time-of-flight three dimensional imaging is an important tool for many applications, such as object recognition and remote sensing. Unlike conventional imaging approach using pixelated detector array, single-pixel imaging based on projected patterns, such as Hadamard patterns, utilises an alternative strategy to acquire information with sampling basis. Here we show a modified single-pixel camera using a pulsed illumi- nation source and a high-speed photodiode, capable of reconstructing 128×128 pixel resolution 3D scenes to an accuracy of ∼ 3 mm at a range of ∼ 5 m. Furthermore, we demonstrate continuous real-time 3D video with a frame-rate up to 12 Hz. The sim- plicity of the system hardware could enable low-cost 3D imaging devices for precision ranging at wavelengths beyond the visible spectrum. Introduction Whilst a variety of 3D imaging technologies are suited for different applications, time-of-flight (TOF) systems have set the benchmark for performance with regards to a combination of accuracy and operating range. Time-of-flight imaging is performed by illuminating a scene with a pulsed light source and observing the back-scattered light. Correlating the detection time of the back-scattered light with the time of the illumination pulse allows the distance, d, to objects within the scene to be estimated by d = tc=2, where t is the TOF and c is the propagation speed of light.
    [Show full text]
  • Arxiv:2005.03760V1 [Physics.Class-Ph] 4 May 2020
    Amplification of waves from a rotating body Marion Cromb,1 Graham M. Gibson,1 Ermes Toninelli,1 Miles J. Padgett,1, ∗ Ewan M. Wright,2 and Daniele Faccio1, 2, y 1School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK 2College of Optical Sciences, University of Arizona, Tucson, Arizona 85721, USA (Dated: May 11, 2020) In 1971 Zel'dovich predicted that quantum fluctuations and classical waves reflected from a ro- tating absorbing cylinder will gain energy and be amplified. This key conceptual step towards the understanding that black holes may also amplify quantum fluctuations, has not been verified ex- perimentally due to the challenging experimental requirements on the cylinder rotation rate that must be larger than the incoming wave frequency. Here we experimentally demonstrate that these conditions can be satisfied with acoustic waves. We show that low-frequency acoustic modes with orbital angular momentum are transmitted through an absorbing rotating disk and amplified by up to 30% or more when the disk rotation rate satisfies the Zel'dovich condition. These experiments address an outstanding problem in fundamental physics and have implications for future research into the extraction of energy from rotating systems. Introduction. In 1969, Roger Penrose proposed a an amplifier. Outgoing waves then have an increased am- method to extract the rotational energy of a rotating plitude, therefore extracting energy from the rotational black hole, now known as Penrose superradiance [1]. Pen- energy of the body in the same spirit of Penrose's pro- rose suggested that an advanced civilisation might one posal. day be able to extract energy from a rotating black hole Satisfying the condition in Eq.
    [Show full text]
  • PDF Program Download
    May 17-20, 2021 | Virtual Conference 2021 Please visit our website for more information! CONFERENCE i2mtc2021.ieee-ims.org PROGRAM Sponsors and Organizers SPONSORS AND ORGANIZERS Table of Contents Welcome Message from the General Co-Chairs ........................................................................................................ 3 I2MTC 2021 Organizing Committee ............................................................................................................................... 5 I2MTC Board of Directors ................................................................................................................................................ 6 I2MTC 2021 Associate Technical Program Chairs .................................................................................................... 7 Special Session Organizers ............................................................................................................................................ 8 I2MTC 2021 Reviewers ...................................................................................................................................................... 9 I2MTC 2021 Conference Sponsors .............................................................................................................................. 11 I2MTC 2021 Plenary Speakers ...................................................................................................................................... 12 I2MTC 2021 Plenary Speakers (continued) ..............................................................................................................
    [Show full text]
  • A Mind-Bending Technology Goes Mainstream Electronic Line Calling Get Closer to the Hawk-Eye Innovations, UK UK’S Game-Changing Technology
    Taiwan and the one-China fiction IS up against the wall in Mosul In praise of quinoa, millet and teff Is there a bubble in the markets? MARCH 11TH–17TH 2017 Quantum leaps A mind-bending technology goes mainstream Electronic Line Calling Get closer to the Hawk-Eye Innovations, UK UK’s game-changing technology. Millimetres separate victory from defeat in any sport. Now used by 20 sports in over 80 tournaments worldwide, Hawk-Eye’s electronic line calling service is making sport fairer, smarter and more spectacular. It’s just one example of the ingenuity that the UK’s 5.5 million companies can offer your business. Find your ideal trade partner at great.gov.uk Contents The Economist March 11th 2017 3 5 The world this week 32 WikiLeaks, again The spy who came in for the code Leaders 33 Chicago 7 Subatomic opportunities This American carnage Quantum leaps 33 Campus free speech 8 Britain’s budget Blue on blue Spreadsheets v politics 34 Lexington 8 Stockmarkets Safety politics Bubble-spotting 9 Geopolitics The Americas Ryancare The House proposal One China, many meanings 35 Brazil’s president to amend Obamacare may 10 Food snobbery and Accidental, consequential break parts of America’s economics health-insurance market, In praise of quinoa 36 Race in the Caribbean On the cover Curry cultures page 30 After a century stuck in 38 Bello textbooks, mind-bending Letters Stealing Venezuela quantum effects are about to 11 On renewable energy power mainstream innovation: leader, page 7. Middle East and Africa Big firms and startups are Briefing 39 Defeating
    [Show full text]
  • Directory 2016/17 the Royal Society of Edinburgh
    cover_cover2013 19/04/2016 16:52 Page 1 The Royal Society of Edinburgh T h e R o Directory 2016/17 y a l S o c i e t y o f E d i n b u r g h D i r e c t o r y 2 0 1 6 / 1 7 Printed in Great Britain by Henry Ling Limited, Dorchester, DT1 1HD ISSN 1476-4334 THE ROYAL SOCIETY OF EDINBURGH DIRECTORY 2016/2017 PUBLISHED BY THE RSE SCOTLAND FOUNDATION ISSN 1476-4334 The Royal Society of Edinburgh 22-26 George Street Edinburgh EH2 2PQ Telephone : 0131 240 5000 Fax : 0131 240 5024 email: [email protected] web: www.royalsoced.org.uk Scottish Charity No. SC 000470 Printed in Great Britain by Henry Ling Limited CONTENTS THE ORIGINS AND DEVELOPMENT OF THE ROYAL SOCIETY OF EDINBURGH .....................................................3 COUNCIL OF THE SOCIETY ..............................................................5 EXECUTIVE COMMITTEE ..................................................................6 THE RSE SCOTLAND FOUNDATION ..................................................7 THE RSE SCOTLAND SCIO ................................................................8 RSE STAFF ........................................................................................9 LAWS OF THE SOCIETY (revised October 2014) ..............................13 STANDING COMMITTEES OF COUNCIL ..........................................27 SECTIONAL COMMITTEES AND THE ELECTORAL PROCESS ............37 DEATHS REPORTED 26 March 2014 - 06 April 2016 .....................................................43 FELLOWS ELECTED March 2015 ...................................................................................45
    [Show full text]
  • Academic, Academic Related and Honorary Sta¡ List
    Academic, Academic Related and Honorary Sta¡ List The following list details Academic, Academic Related and Honorary sta¡ in the University as at May 2001. Changes and new appointments applicable in the academic year 2001^02 are included where available. To update personal details please consult www.gla.ac.uk/humanresources/persdata.htm Information Section Human Resources Abbreviations: * Head of Department y Dean + Head of Division HCL Honorary Clinical Lecturer HCSL Honorary Clinical Senior Lecturer HCT Honorary Clinical Teacher HL Honorary Lecturer HP Honorary Professor HRA Honorary Research Assistant HRF Honorary Research Fellow HSL Honorary Senior Lecturer HSRF Honorary Senior Research Fellow PE Professor Emeritus PV Visiting Professor Y Honorary Member of sta¡ appointed by the Medical Faculty a¤liated to a department outwith the Faculty (eg Pre-Medical). For hospitals: DH Dental Hospital GNG Gartnavel General GRI Glasgow Royal In¢rmary HMH Hairmyers Hospital MLH Monklands Hospital RAH Royal Alexandra Hospital SGH Southern General Hospital SHH Stobhill Hospital VI Victoria In¢rmary WGH Wishaw General Hospital WI Western In¢rmary YKH Yorkhill Hospital S.2 Academic and Academic-Related Sta¡ Faculty of Arts Dean Professor John M Caughie MA Administrative Aileen M O'Neil MA Clerk to the Faculty Jacqueline A Keltie BSc MSc Faculty Planning and Resources O¤cer ARCHAEOLOGY Professor * William S Hanson BA PhD FSA FSAScot Roman Archaeology A Bernard Knapp BA MA PhD Mediterranean Archaeology Senior Lecturer Stephen T Driscoll BA MSc PhD Allan
    [Show full text]
  • Electron Vortices : Beams with Orbital Angular Momentum
    This is a repository copy of Electron vortices : Beams with orbital angular momentum. White Rose Research Online URL for this paper: https://eprints.whiterose.ac.uk/133764/ Version: Accepted Version Article: Lloyd, S. M., Babiker, M. orcid.org/0000-0003-0659-5247, Thirunavukkarasu, G. orcid.org/0000-0002-8978-5304 et al. (1 more author) (2017) Electron vortices : Beams with orbital angular momentum. Reviews of Modern Physics. 035004. ISSN 0034-6861 https://doi.org/10.1103/RevModPhys.89.035004 Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ Electron Vortices - Beams with Orbital Angular Momentum S. M. Lloyd, M. Babiker, G. Thirunavukkarasu and J. Yuan Department of Physics, University of York, Heslington, York, YO10 5DD, UK⇤ (Dated: 3rd May 2017) The recent prediction and subsequent creation of electron vortex beams in a number of laboratories occurred after almost 20 years had elapsed since the recognition of the phys- ical significance and potential for applications of the orbital angular momentum carried by optical vortex beams.
    [Show full text]
  • Arxiv:2004.12788V2 [Quant-Ph] 16 Nov 2020
    Fourier Transform of the Orbital Angular Momentum of a Single Photon Jaroslav Kysela,1, 2 Xiaoqin Gao,3, 1, 2, ∗ and Borivoje Daki´c1, 2, y 1Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria. 2Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria. 3National Mobile Communications Research Laboratory, Quantum Information Research Center, Southeast University, Sipailou 2, 210096 Nanjing, China. (Dated: November 17, 2020) Optical networks implementing single-qudit quantum gates may exhibit superior properties to those for qubits as each optical element in the network can work in parallel on many optical modes simultaneously. We present an important class of such networks that implements in a deterministic and efficient way the quantum Fourier transform (QFT) in an arbitrarily high dimension. These networks redistribute the initial quantum state into the orbital angular momentum (OAM) and path degrees of freedom and offer two modes of operation. Either the OAM-only QFT can be implemented, which uses the path as an internal auxiliary degree of freedom, or the path-only QFT is implemented, which uses the OAM as the auxiliary degree of freedom. The resources for both schemes scale linearly O(d) with the dimension d of the system, beating the best known bounds for the path-encoded QFT. While the QFT of the orbital angular momentum states of single photons has been applied in a multitude of experiments, these schemes require specially designed elements with non-trivial phase profiles. In contrast, we propose a different approach that utilizes only conventional optical elements.
    [Show full text]
  • The Royal Society of Edinburgh Issue 3 • Winter 2001
    news THE ROYAL SOCIETY OF EDINBURGH ISSUE 3 • WINTER 2001 RESOURCE THE NEWSLETTER OF SCOTLAND’S NATIONAL ACADEMY A Science Strategy for Scotland Science policy in Scotland is to be shaped by an independent, expert group, to be formed by The Royal Society of Edinburgh (RSE). Minister for Enterprise & Lifelong Learning, Wendy Alexander, announced the creation of the Scottish Science Advisory Committee, a key part of the first ever Scottish Science Strategy, at its launch in the Glasgow Science Centre in August. This is a central element of the science policy framework which sets out a new way forward to foster science in Scotland. The Scottish Science Advisory Committee, which is being set up under the auspices of the RSE will identify priorities, inform policy in science and technology, and advise Scottish Executive Ministers on issues of scientific importance. It will monitor progress and advise the Executive on how the Scottish Science Strategy is being implemented across departments. This remit has been agreed following consultation between RSE President, Sir William Stewart, and the Scottish Executive, which will fund the work of the Committee. The Society will be responsible for selecting the eminent men and women of the Committee and advertisements for positions on it, will appear shortly in the National Media. The Royal Society of Edinburgh will appoint a Chairperson to the Committee and it is expected that the Chairperson will also act as chief adviser on science to the Executive. Once appointed, the new Committee will be housed in its own part of the George Street Ashley Coombes, The ATOM Photographic offices of RSE.
    [Show full text]
  • The Blackett Laboratory
    IMPERIAL COLLEGE OF SCIENCE, TECHNOLOGY AND MEDICINE Faculty of Physical Sciences The Blackett Laboratory Department of Physics Review 2001 Introduction The key event of the past year was the outcome of the Gr oup. An initiative to rei n f o r ce this area over the coming 2001 Research Assessment Exercise on which we had all years has been developed to start in 2002. Initiatives been working so hard. The Department gained the top started in previous years have come to fruition: Prof grade of a 5*A (the A meaning that 95-100% of staff were Steve Cowley and Dr Bill Dorland joined us to lead new entered), a score that was only achieved by three other di r ections in the Plasma Physics Group, and Prof Michael Physics Departments. This is a well-deserved recogni- Thompson joined our Space and Atmospheric Physics tion of the excellent work in the Department. We will Group. Dr John Tisch was appointed as a lecturer in the build on this strong base and continue our policy of Quantum Optics and Laser Science Group, and Dr Yvo n n e renewal and reinforcement of activities. Un r uh in the As t r ophysics Group. The new MSc in Optics and Photonics started in October 2001 supported by a Many of the department's research laboratories (about 20 major EPSRC training grant. per cent of our total floor area) are being upgraded through our 6 million pound share of the Strategic Members of the Department received recognition for Re s e a r ch Infrastruc t u r e Fund's award to Imperial College.
    [Show full text]
  • Light in a Twist: Orbital Angular Momentum Miles Padgett
    Light in a Twist: Orbital Angular Momentum Miles Padgett Kelvin Chair of Natural Philosophy The talk today • Orbital Angular Momentum, what is it? • What has been done with OAM • A couple of example of what we have done and doing! A question • A photon carries a spin angular momentum of • So how does a multi-pole transition (ΔJ > ) conserve angular momentum? Linear momentum at a radius exerts a torque k r k x r -> multipole transition Providing the lever is long enough, a fixed linear momentum can exert an arbitrary high torque Getting started on Orbital Angular Momentum of Light • 1992, Allen, Beijersbergen, Spreeuw and Woerdman Miles Padgett • 1994, Les meets Miles at dinner…... Orbital Angular Momentum from helical phase fronts r" E" z" p! B" p E" z" B" p = 0 θ pθ ≠ 0 Angular momentum in terms of photons • Spin angular momentum – Circular polarisation σ = +1! σ per photon • Orbital angular momentum – Helical phasefronts σ = -1! per photon = 0! = 1! = 2! = 3! etc! Optical vortices, Helical phasefronts , Angular momentum • Intensity, I ≥0 = 0! • Phase, 2π ≥ φ ≥0 = 0, plane wave = 1, helical wave = 1! = 2, double helix Interference+/- = 3, pasta fusilli etc. = 2! # = 3! = vortex charge! I! φ& I! Orbital angular momentum from Skew rays Poynting vector Making helical phasefronts with holograms Making OR measuring phasefronts with holograms Make interactive by using SLM Switching time Generate ≈5-20mSec Efficiency ≈50% Light source OR detector Measure Richard Bowman A gift for all the family….. And the point of shaping the spot is…… A double-start helix (=2) Chambord castle (chateaux de la Loire) OAM in optical manipulation He et al.
    [Show full text]