Light and Matter Tim Freegarde School of Physics & Astronomy

Light and Matter Tim Freegarde School of Physics & Astronomy
University of Southampton

Light and Matter refraction absorption absorption scattering thermal
excitation Light and Matter: I love the title of this lecture course! What else is there but light and matter? Well, there’s gravitation, and the weak force that’s important in describing radioactivity, but a very great deal of physics is about light and matter – especially when you realize that, in the modern description, the electromagnetic force is regarded as being propagated by virtual photons of light. To give some examples of how light interacts with matter, let’s consider the processes which allow us to read the printed text of a book. Firstly, we need a light source – a lamp or the sun – where thermal excitation of atoms results in the emission of photons of light. These are scattered by the atmosphere and the walls of the room, to fall upon the printed page. Here, light which is not absorbed by the dark print is scattered and may be imaged by refraction in a lens. If that lens is in the eye, then the image on the fovea causes electronic excitation, which triggers the optic nerves. To create the printed text, we might use a laser printer, in which the light is scanned using reflecting mirrors and modulated by Bragg scattering. Further processes involving photoconductivity transfer the scanned image to the page. reflection Bragg scattering stimulated emission

Light and Matter effect of matter on light: absorption & dispersion
refraction transmission & reflection scattering (Rayleigh, Mie, Bragg) birefringence and optical activity effect of light on light: Pockels & Kerr effects harmonic and parametric generation self-focussing, self-phase-modulation effect of light on matter: dipole force (optical tweezers) scattering force (laser cooling) coherent scattering (Bragg; interferometry)

Technologies effect of matter on light: polarizers lenses & prisms
mirrors and filters liquid crystal displays acousto-optic modulators effect of light on light: electro-optic modulators frequency doubling & mixing laser mode-locking effect of light on matter: optical tweezers laser cooling quantum computing

Programme electromagnetism optics waves polarization
optical properties of materials quantum mechanics atomic response and causality optical control of matter

Programme FOUNDATIONS wave mechanics classical electromagnetism
CONTROLLING LIGHT WITH MATTER classical interaction of light and matter polarization of light CONTROLLING LIGHT WITH LIGHT optical nonlinearity tensor nature of susceptibility CONTROLLING MATTER WITH LIGHT quantum mechanical interaction of light and matter optical forces and wave-particle duality

Wave mechanics Light and Matter Tim Freegarde
School of Physics & Astronomy University of Southampton

Wave mechanics – la Ola

Wave mechanics response to action of neighbour delayed reaction e.g.
waves are bulk motions, in which the displacement is a delayed response to the neighbouring displacements

Gravitational waves delay may be due to propagation speed of force (retarded potentials) vertical component of force

Wave mechanics waves are collective bulk disturbances, whereby the motion at one position is a delayed response to the motion at neighbouring points propagation is defined by differential equations, determined by the physics of the system, relating derivatives with respect to time and position e.g. in certain circumstances, a wave may propagate without distortion: a surface of constant phase, , is known as a wavefront, and propagates with the phase velocity,

Linearity and superpositions
if the system is linear, then the wave equation may be split into separate equations for superposed components; i.e., if y1 and y2 are wave solutions, then so is any superposition of them if sinusoidal solutions are allowed, then the wave shape at any time may be written as a superposition of sinusoidal components Fourier analysis complex coefficients allow waves which are complex exponentials:

Dispersion linear systems may show dispersion – that is, the wave speed varies with frequency if sinusoidal solutions are allowed, then the wave shape may still be written as a superposition of sinusoidal components dispersion causes the components to drift in phase as the wave propagates the wave may no longer be written as

Dispersion 10 sinusoidal components: 2 sinusoidal components:
spreading of wavepacket this illustration corresponds to the wavepacket evolution of a quantum mechanical particle, described by the Schrödinger equation

Plane wave solutions to wave equations
linear, non-dispersive linear, dispersive non-linear solitons:

Alternative solutions
show that spherical waves of the form are valid solutions to the Schrödinger equation of a free particle

Wave mechanical operators
an operator is a recipe for determining an observable from a wave function e.g. an operator could yield the parameter from the wave for convenience, to avoid the observable depending upon the magnitude of the wavefunction, we instead define the general operator i.e. the square brackets are commonly omitted

Wave mechanics waves result when the motion at a given position is a delayed response to the motion at neighbouring points derivatives with respect to time and position are related by the physics of the system, which lets us write differential equations e.g. in certain circumstances, a wave may propagate without distortion: the solutions depend upon whether the system shows linearity or dispersion

Messenger Lecture Richard P. Feynman (1918-1988) Nobel prize 1965
Messenger series of lectures, Cornell University, 1964 Lecture 6: ‘Probability and Uncertainty – the quantum mechanical view of nature’ see the later series of Douglas Robb memorial lectures (1979) online at

Light and Matter next : electromagnetism and electromagnetic waves
for handouts, links and other material, see

Classical electromagnetism
Light and Matter Classical electromagnetism Tim Freegarde for handouts, links and other material, see School of Physics & Astronomy University of Southampton

Programme FOUNDATIONS wave mechanics classical electromagnetism
CONTROLLING LIGHT WITH MATTER classical interaction of light and matter polarization of light CONTROLLING LIGHT WITH LIGHT optical nonlinearity tensor nature of susceptibility CONTROLLING MATTER WITH LIGHT quantum mechanical interaction of light and matter optical forces and wave-particle duality

Electromagnetic waves
electrostatic force acts through vacuum retardation due to finite speed of light , enhanced by inertia of any charged particles net force due to oscillating dipole

James Clerk Maxwell first colour photograph defined nature of gases
expressed fundamental laws of light, electricity and magnetism Listen again to Melvyn Bragg’s In Our Time:

Maxwell’s equations Gauss no monopoles Faraday Ampère
constitutive equations

Constitutive equations
conservation of charge constitutive equations

Maxwell’s equations constitutive equations conservation of charge

Electromagnetic wave equation
use constitutive equations to reduce electric & magnetic fields to single functions differentiate equations to allow electric or magnetic field to be eliminated constitutive equations conservation of charge apply vector relations to produce wave equation

Sinusoidal plane wave solutions

Continuity conditions
transverse waves on a guitar string x T T continuity of y … finite extension conservation of energy continuity of … finite acceleration conservation of momentum

electromagnetic fields parallel components E// conservation of energy 1 2 perpendicular components 1 2 conservation of momentum

Electromagnetic energy density & flow
constitutive equations conservation of charge

Electromagnetic energy density & flow
BBC Radio 4 long wave transmitter, Droitwich frequency: 198 kHz l = 1515 m power: 400 kW MSF clock transmitter, Rugby frequency: 60 kHz l = 5000 m power: 60 kW

Electromagnetic waves
electrostatic force acts through vacuum retardation due to finite speed of light , enhanced by inertia of any charged particles net force due to oscillating dipole

Classical interaction between light and matter
Tim Freegarde School of Physics & Astronomy University of Southampton

Maxwell’s equations constitutive equations conservation of charge

Electromagnetic wave equations
use constitutive equations to reduce electric & magnetic fields to single functions differentiate equations to allow electric or magnetic field to be eliminated constitutive equations conservation of charge apply vector relations to produce wave equation

use constitutive equations to reduce electric & magnetic fields to single functions use constitutive equations to reduce electric & magnetic fields to single functions differentiate equations to allow electric or magnetic field to be eliminated differentiate equations to allow electric or magnetic field to be eliminated apply vector relations to produce wave equation apply vector relations to produce wave equation

apply Newtonian mechanics to determine response of medium to applied field define polarization P and magnetization M governed by properties of the optical medium use result to write (complex) conductivity, dielectric constant etc. vapours, dielectrics, plasmas, metals magnetization usually too slow to have effect at optical frequencies insert into constitutive equations and hence derive wave equation as usual assume (for now) D[E] to be linear and scalar

Electromagnetic waves in isotropic media
atoms and molecules are polarized by applied fields induced polarization alignment of permanent dipole moment polarization modifies field propagation: refractive index; absorption

Vapours and dielectrics
bound or massive nuclei electrons confined in harmonic potential restoring force proportional to displacement Newtonian dynamics frequency dependence (dispersion)

Metals and conductors free charges
diffusion in response to applied field equilibrium velocity characterized by conductivity frequency dependence (dispersion) damped solutions (absorption) dissipation through resistive heating

Plasmas and the ionosphere
independent, free charges inertia in response to applied field Newtonian dynamics frequency dependence (dispersion)

Reflection at metal and dielectric interfaces

transverse waves on a guitar string x T T continuity of y … finite extension conservation of energy continuity of … finite acceleration conservation of momentum

electromagnetic fields parallel components E// conservation of energy 1 2 perpendicular components 1 2 conservation of momentum

electromagnetic fields conservation of energy conservation of momentum E// 2 1 parallel components perpendicular components combine forward and reflected waves to give total fields for each region apply continuity conditions for separate components hence derive fractional transmission and reflection

combine forward and reflected waves to give total fields for each region apply continuity conditions for separate components hence derive fractional transmission and reflection

Reflection at multiple dielectric interfaces
combine forward and reflected waves to give total fields for each region apply continuity conditions for separate components hence derive fractional transmission and reflection

Reflection at multiple dielectric interfaces

Classical interaction between light and matter

How light interacts with matter
atoms and molecules are polarized by applied fields induced polarization modifies field propagation: refractive index; absorption

Lorentz theory of atomic polarization
bound or massive nuclei electrons confined in harmonic potential restoring force proportional to displacement Newtonian dynamics dissipation of motion through frictional force

complex dielectric constant G=0.050 1 freq real part: refractive index imaginary part: (absorption)

complex dielectric constant G=0.325 G=0.350 G=0.300 G=0.275 G=0.375 G=0.400 G=0.500 G=0.475 G=0.450 G=0.425 G=0.250 G=0.200 G=0.225 G=0.050 G=0.100 G=0.075 G=0.125 G=0.150 G=0.175 1 freq real part: refractive index imaginary part: absorption

complex dielectric constant G=0.200 1 freq real part: refractive index imaginary part: absorption ‘stop band’ from to : strong attenuation even for small G

complex dielectric constant G=0.325 G=0.300 G=0.250 G=0.275 G=0.350 G=0.400 G=0.500 G=0.475 G=0.450 G=0.425 G=0.375 G=0.225 G=0.200 G=0.050 G=0.075 G=0.100 G=0.175 G=0.150 G=0.125 1 freq real part: refractive index imaginary part: absorption ‘stop band’ from to : strong attenuation even for small G

Causality and the dispersion relations
causality: effect follows cause E time  causality:  must obey time  E time

the Kramers-Krönig dispersion relations
causality: effect follows cause Kramers-Krönig relations relate the real and imaginary parts of (w) if , then

Implication for all dielectrics
evaluate e1 as w→0 if e1 ≠ 1, there must be frequencies at which e1 ≠ 0 (absorption) dielectrics cannot be transparent at all wavelengths

Application to a single sharp absorption
suppose a single absorption at w = w0 1 Kramers-Krönig then gives freq

Quantum mechanical interaction between light and matter

Quantum description of atomic polarization
spatial part of eigenfunctions given by and energy full time-dependent eigenfunctions therefore any state of the two-level atom may hence be written

write time-dependent Schrödinger equation for two-level atom insert energy of interaction with oscillating electric field reduce to coupled equations for a(t) and b(t) spatial part of eigenfunctions given by and full time-dependent eigenfunctions therefore any state of the two-level atom may hence be written

write time-dependent Schrödinger equation for two-level atom insert energy of interaction with oscillating electric field reduce to coupled equations for a(t) and b(t) full time-dependent eigenfunctions therefore spatial part of eigenfunctions given by and any state of the two-level atom may hence be written

x/a0 x/a0 electron density depends upon relative phase of superposition components

Atomic polarization G=0.325 G=0.300 G=0.250 G=0.350 G=0.275 G=0.400
response of massive electrons to applied electric field G=0.325 G=0.300 G=0.250 G=0.350 G=0.275 G=0.400 G=0.475 G=0.500 G=0.450 G=0.425 G=0.225 G=0.375 G=0.175 G=0.075 G=0.200 G=0.100 G=0.050 G=0.125 G=0.150 resonant frequency due to confining potential of electrons in atom 1 electron displacement leads to atomic polarization freq frequency-dependent amplitude and phase lag of response related by causality Newtonian and quantum mechanical models give same result

Polarization of light Light and Matter Tim Freegarde

Optical polarization light is a transverse wave: perpendicular to
for any wavevector, there are two field components any wave may be written as a superposition of the two polarizations

The Fresnel equation return to derivation of electromagnetic wave equation consider oscillatory waves of definite polarization aω apply vector identity twice and simplify

The Fresnel equation for isotropic media
electromagnetic waves are transverse the Poynting vector is parallel to the wavevector

Characterizing the optical polarization
wavevector insufficient to define electromagnetic wave we must additionally define the polarization vector e.g. linear polarization at angle

Jones vectors normalized polarization vector is known as the Jones vector defines polarization state of any wave of given and real field corresponds to superposition of exponential form and complex conjugate

Categories of optical polarization
linear (plane) polarization coefficients differ only by real factor circular polarization coefficients differ only by factor elliptical polarization all other cases

Polarization notation
circular polarization RCP plane of incidence right- or left-handed rotation when looking towards source traces out right- or left-handed thread perpendicular parallel linear (plane) polarization parallel or perpendicular to plane of incidence plane of incidence contains wavevector and normal to surface

Controlling light with matter
Light and Matter Controlling light with matter Tim Freegarde School of Physics & Astronomy University of Southampton

Characterizing the optical polarization
wavevector insufficient to define electromagnetic wave we must additionally define normalized polarization vector, known as the Jones vector,

linear (plane) polarization coefficients differ only by real factor circular polarization coefficients differ only by factor elliptical polarization all other cases

complex electric field given by real electric field corresponds to superposition with complex conjugate for monochromatic fields, Jones vector is constant

Polarization of time-varying fields
complex polychromatic electric field given by beating between frequencies causes field to vary with time even stabilized lasers have linewidth in the MHz range Jones vector may therefore vary on a microsecond timescale – or faster

Stokes parameters with polychromatic light, the Jones vector varies
we therefore describe polarization through averages and correlations: are the instantaneous field components is their relative phase STOKES PARAMETERS

Stokes parameters total intensity, I1
related to horizontally polarized component, I2 … component polarized at +45º to horizontal, I3 … right circularly polarized component, I4

Unpolarized (randomly polarized) light
average horizontal component = average vertical component average +45º component = average -45º component average RCP component = average LCP component … = half total intensity orthogonal polarizations are uncorrelated

Degree of polarization
for partially polarized light, the quantity represents the degree of polarization, where unpolarized (randomly polarized) completely polarized

Completely polarized light
constant Jones vector Stokes parameters given by when simply defining the polarization state, it is common to drop the intensity factor I1

The Poincaré sphere (a) plot the Stokes vector (f)
right elliptically polarized (a) right circularly polarized [0, 0, 1] (d) (b) left circularly polarized [0, 0,-1] (c) horizontally polarized [1, 0, 0] (g) (d) vertically polarized [-1, 0, 0] (c) (e) (e) polarized at +45º [0, 1, 0] left elliptically polarized (f) elliptically polarized [δ1,δ2,δ3]/δ0 (g) unpolarized [0, 0, 0] (b)

Polarizers many optical elements restrict or modify the polarization state of light polarization-dependent transmission/reflection sheet polarizers (Polaroid) Nicol, Wollaston prisms etc polarizers, polarizing filters, analyzers polarization-dependent refractive index waveplates, retarders four categories of physical phenomena polarization-sensitive absorption (dichroism) polarization-sensitive dispersion (birefringence, optical activity) reflection at interfaces scattering

Polarizers plane of incidence
each mechanism may discriminate between either linear or circular polarizations mechanisms depend upon an asymmetry in the device or medium perpendicular parallel

Linear polarization upon reflection
for normal incidence, no distinction between horizontal and vertical polarizations if wavevector makes angle with interface normal, s- and p-polarizations affected differently we consider here the reflection of p-polarized light; s-polarized beams may be treated similarly we resolve the electric field into components parallel and normal to the interface all magnetic field components are parallel to the interface

Linear polarization upon reflection
for normal incidence, no distinction between horizontal and vertical polarizations if wavevector makes angle with interface normal, s- and p-polarizations affected differently we consider here the reflection of p-polarized light; s-polarized beams may be treated similarly we resolve the electric field into components parallel and normal to the interface all magnetic field components are parallel to the interface combine forward and reflected waves to give total fields for each region apply continuity conditions for separate components hence derive fractional transmission and reflection

Fresnel equations p-polarization s-polarization
combine forward and reflected waves to give total fields for each region p-polarization apply continuity conditions for separate components s-polarization hence derive fractional transmission and reflection

Light and Matter next: polarizing crystals and devices

Polarizers many optical elements restrict or modify the polarization state of light polarization-dependent transmission/reflection sheet polarizers (Polaroid) Nicol, Wollaston prisms etc polarizers, polarizing filters, analyzers polarization-dependent refractive index waveplates, retarders four categories of physical phenomena polarization-sensitive absorption (dichroism) polarization-sensitive dispersion (birefringence, optical activity) reflection at interfaces scattering

Polarizers plane of incidence
each mechanism may discriminate between either linear or circular polarizations polarization dependence corresponds to an asymmetry in the device or medium perpendicular parallel

Linear dichroism conductivity of wire grid depends upon field polarization electric fields perpendicular to the wires are transmitted fields parallel to the wires are absorbed WIRE GRID POLARIZER

Linear dichroism crystals may similarly show absorption which depends upon linear polarization absorption also depends upon wavelength polarization therefore determines crystal colour pleochroism, dichroism, trichroism TOURMALINE

Circular dichroism absorption may also depend upon circular polarization SCARAB BEETLE LEFT CIRCULAR POLARIZED LIGHT RIGHT CIRCULAR POLARIZED LIGHT the scarab beetle has polarization-sensitive vision, which it uses for navigation the beetle’s own colour depends upon the circular polarization

Polarization in nature
CUTTLEFISH (sepia officinalis) the European cuttlefish also has polarization-sensitive vision … and can change its colour and polarization! MAN’S VIEW CUTTLEFISH VIEW (red = horizontal polarization)

Birefringence asymmetry in crystal structure causes polarization dependent refractive index opposite polarizations follow different paths through crystal birefringence, double refraction

Linear polarizers (analyzers)
o-ray birefringence results in different angles of refraction and total internal reflection 38.5º e-ray many different designs, offering different geometries and acceptance angles e-ray o-ray s-ray a similar function results from multiple reflection p-ray

Waveplates (retarders)
at normal incidence, a birefringent material retards one polarization relative to the other linearly polarized light becomes elliptically polarized WAVEPLATE

Compensators a variable waveplate uses two wedges to provide a variable thickness of birefringent crystal adjust a further crystal, oriented with the fast and slow axes interchanged, allows the retardation to be adjusted around zero variable fixed with a single, fixed first section, this is a ‘single order’ (or ‘zero order’) waveplate for small constant retardation SOLEIL COMPENSATOR

Optical activity (circular birefringence)
optical activity is birefringence for circular polarizations CH2 CH3 H CH2 CH3 H an asymmetry between right and left allows opposing circular polarizations to have differing refractive indices l-limonene (orange) r-limonene (lemon) optical activity rotates the polarization plane of linearly polarized light CHIRAL MOLECULES may be observed in vapours, liquids and solids

Jones vector calculus if the polarization state may be represented by a Jones vector then the action of an optical element may be described by a matrix JONES MATRIX

Jones vector calculus transmission by horizontal polarizer if the polarization state may be represented by a Jones vector then the action of an optical element may be described by a matrix retardation by waveplate JONES MATRIX projection onto rotated axes

Müller calculus field averages and correlations following optical element depend linearly upon parameters describing incident beam Müller matrix elements may be written in terms of Jones matrix elements, e.g. MŰLLER MATRIX

Müller calculus the actions of optical materials can be represented by geometrical transformations of the Stokes vector in the Poincaré sphere right elliptically polarized optical activity: rotation about a vertical axis left elliptically polarized

Müller calculus the actions of optical materials can be represented by geometrical transformations of the Stokes vector in the Poincaré sphere right elliptically polarized optical activity: rotation about a vertical axis birefringence: rotation about a horizontal axis left elliptically polarized

Light and Matter next : problem sheet 2

Müller calculus field averages and correlations following optical element depend linearly upon parameters describing incident beam Müller matrix elements may be written in terms of Jones matrix elements, e.g. MŰLLER MATRIX

Problem sheet 2

Atoms with discrete spectra
2.1 The absorption spectrum of an atomic vapour consists of a series of narrow lines at frequencies , , etc., each of which may be represented by a Dirac -function, multiplied by a constant to account for the line strength: Use the Kramers-Krönig relation to find the real part of the permittivity

The stop band 2.2 By considering the propagation of plane electromagnetic waves of the form where , derive and sketch the dispersion curve of vs. for the atomic vapour of problem 2.1 and show that there exist stop bands in which, for real , only evanescent waves may propagate. Determine the frequencies of the edges of the stop bands and sketch the propagation (attenuation) coefficients of evanescent waves within the bands.

Refractive index of atomic helium
2.3 Given that the first P-states of atomic helium lie about 21.2 eV above the ground S-state, and assuming that the helium atom may be modelled as two independent electrons bound as harmonic oscillators, estimate the refractive index of helium at standard temperature and pressure for visible wavelengths.

Polarization states 2.4 Describe completely the state of polarization of each of the following waves: (a) (b) (c) (d) In each case, calculate the associated Jones vector, and indicate the corresponding position on the Poincaré sphere.

The linear polarizer 2.5 (a) A beam of linearly polarized light with its electric field vertical impinges perpendicularly upon an ideal linear polarizer with a vertical transmission axis. If the incoming beam has an intensity of 200 W.m-2, what is the intensity of the transmitted beam? (b) Given that 300 W.m-2 of light from an ordinary tungsten lamp bulb arrives at an ideal linear polarizer, what is the intensity upon emerging? (c) A beam of vertically plane-polarized light is perpendicularly incident upon an ideal linear polarizer. Show that, if its transmission axis lies at 60º to the vertical, only 25% of the incident intensity will be transmitted.

The linear polarizer 2.5 cont’d
(d) If light that is initially unpolarized and of intensity I passes through two sheet polarizers whose transmission axes are parallel, what will be the intensity of the emerging beam? (e) What will be the intensity of the transmitted beam if the second polarizing sheet of the previous problem is rotated through 30º?

The Pockels cell 2.6 If a Pockels cell modulator is illuminated by light of intensity , it will transmit a beam of intensity such that where the retardance is given in terms of the voltage by Make a plot of the retardance and transmission versus applied voltage. What is the significance of the voltage that corresponds to maximum transmission? What is the lowest voltage above zero that will cause to be zero for ADP? Calculate the half-wave voltage

Jones matrices 2.7 Identify the two linear optical filters which have Jones matrices and

Stokes vectors 2.8 Two incoherent light beams represented by (1, 1, 0, 0) and (3, 0, 0, 3) are superimposed. (a) Describe in detail the polarization states of each of these. (b) Determine the resulting Stokes parameters of the combined beam and describe its polarization state. (c) What is its degree of polarization? (d) What is the reslting light produced by overlapping the incoherent beams (1, 1, 0, 0) and (1, -1, 0, 0)? Explain. (e) Sketch a Poincaré sphere, showing all of the polarization states examined.

The quarter-wave plate
2.9 Confirm that the matrix will serve as a Müller matrix for a quarter-wave plate with its fast axis at +45º. Shine linear light polarized at 45º through it. What happens? What emerges when a horizontal polarization state enters the device? Derive the Müller matrix for a quarter-wave plate with its fast axis at -45º. Check that this matrix effectively cancels that given above, so that a beam passing through the two wave plates successively remains unaltered.

The variable wave plate
2.10 Determine the Jones and Müller matrices for a variable wave plate, such as the Soleil compensator, which introduces an adjustable phase between the two polarization components.

Circular polarization with linear polarizers
2.11 By calculating the Jones and Müller matrices, show that a linear polarizer may be combined with a quarter wave plate to form a circular polarizer, and confirm the orientation of the wave plate axes relative to those of the polarizer for which this occurs.

S- and P-polarizations
2.12 Referring to a German dictionary, look up the words senkrecht and parallel. 'senk·recht (Adj.) 1 im Winkel von 90º zu einer Ebene od. Geraden stehend; beim rechten Winkel stehen die beiden Schenkel ̴ aufeinander 1.1 im Winkel von 90º zur Erdoberfläche (d. h., in Richtung des Lotes) stehend; Sy vertikal, lotrecht; zu beiden Seiten des Pfades stürzten die Felsen fast ̴ ab bleib ̴ ! (umg.; scherzh.) fall nicht hin! das ̴ e Lot am Faden aufgehängtes Gewicht, Lot zur Bestimmung der Senkrechten 2 immer schön ̴ bleiben! (fig.; umg.) Haltung, Fassung bewahren 3 das ist das einzig Senkrechte (fig.; umg.) das Richtige par·al'lel (Adj.) in der Parallele, in gleicher Richtung u. gleichbleibendem Abstand zueinander verlaufend; ̴ e Linien; die Straßen laufen ̴ (miteinander); der Weg läuft ̴ zum Fluß

Brewster’s angle 2.13 A beam of light is reflected off the surface of some unknown liquid, and the light is examined with a linear sheet polarizer. It is found that when the central axis of the polarizer (that is, the perpendicular to the plane of the sheet) is tilted down from the vertical at an angle of 54.30º, the reflected light is completely passed, provided that the transmission axis is parallel to the plane of the interface. From this information, compute the index of refraction of the liquid. At what angle will the reflection of the sky coming off the surface of a pond (η = 1.33) completely vanish when seen through a Polaroid filter?

White light polarimetry
2.14 Explain the principles of operation of a linear polarizer and a quarter wave plate. Explain how one could produce circularly polarized light from an unpolarized beam. Describe a method to determine whether a beam is right- or left-circularly polarized. A beam of white light is incident on a system consisting of a linear polarizer followed by a sheet of birefringent material, followed by a second identical polarizer. The polarizers have parallel transmission axes at 45º to the fast axis of the sheet. Assume that the ratio of the fast and slow velocities in the sheet is independent of wavelength and that the light in the incident beam covers the range from 360 nm to 720 nm with uniform intensity. The sheet acts as a half wave plate at the wavelength of 720 nm. Determine the relative intensities at 360 nm, 480 nm and 720 nm in the emerging beam.

Interference 2.15 Two monochromatic plane wave light beams propagate with their wavevectors at right angles in a horizontal plane, and strike a vertical screen which makes an angle of 45º to each wavevector. Determine the intensity and polarization of the interference pattern in the plane of the screen if (a) the beams are both vertically polarized (b) the beams are both horizontally polarized (c) one beam is polarized horizontally and the other vertically.

Light and Matter next : the tensor nature of susceptibility

The tensor nature of susceptibility
Light and Matter The tensor nature of susceptibility Tim Freegarde School of Physics & Astronomy University of Southampton

Birefringence asymmetry in crystal structure causes polarization dependent refractive index ray splits into orthogonally polarized components, which follow different paths through crystal note that polarization axes are not related to plane of incidence

Anisotropic media difference in refractive index (birefringence) or absorption coefficient (dichroism) depending upon polarization recall that where difference in or implies difference in susceptibility cannot be a simple scalar

Linear dichroism fields normal to the conducting wires are transmitted
fields parallel to the conducting wires are attenuated current flow is not parallel to field susceptibility not a simple scalar WIRE GRID POLARIZER

Birefringence – mechanical model
springs attach electron cloud to fixed ion different spring constants for x, y, z axes polarization easiest along axis of weakest spring polarization therefore not parallel to field susceptibility not a simple scalar

Susceptibility tensor
the Jones matrix can map JONES MATRIX similarly, a tensor can describe the susceptibility of anisotropic media e.g. SUSCEPTIBILITY TENSOR

Diagonizing the susceptibility tensor
if the polarization axes are aligned with the principal axes of birefringent crystals, rays propagate as single beams the susceptibility tensor is then diagonal a matrix may be diagonalized if symmetrical: DIAGONAL SUSCEPTIBILITY TENSOR the optical activity tensor is not symmetrical; it cannot be diagonalized to reveal principal axes

The Fresnel ellipsoid surface mapped out by electric field vector for a given energy density symmetry axes x’,y’,z’ are principal axes semi-axes are FRESNEL ELLIPSOID

The Fresnel ellipsoid allows fast and slow axes to be determined:
establish ray direction through Poynting vector electric field must lie in normal plane fast and slow axes are axes of elliptical cross-section axis lengths are FRESNEL ELLIPSOID

The optic axis if the cross-section is circular, the refractive index is independent of polarization the Poynting vector then defines an optic axis if , the single optic axis lies along (uniaxial crystals) if , there are two, inclined, optic axes (biaxial crystals) FRESNEL ELLIPSOID

Uniaxial crystals the single optic axis lies along
one polarization is inevitably perpendicular to the optic axis (ordinary polarization) the second polarization will be orthogonal to both the ordinary polarization and the Poynting vector (extraordinary polarization) positive uniaxial: FRESNEL ELLIPSOID negative uniaxial:

Poynting vector walk-off
Poynting vector obeys wavevector vector obeys in anisotropic media, and are not necessarily parallel wavevector the Poynting vector and wavevector may therefore diverge Poynting vector Fresnel ellipsoid

Light and Matter next : optical nonlinearity

The tensor nature of susceptibility
Light and Matter The tensor nature of susceptibility Tim Freegarde School of Physics & Astronomy University of Southampton

How light interacts with matter
atoms are polarized by applied fields Lorentz model: harmonically bound classical particles

energy harmonic oscillator two-level atom weak electric field electron density depends upon relative phase of superposition components

energy harmonic oscillator two-level atom weak electric field x/a0 x/a0 electron density depends upon relative phase of superposition components

Optical nonlinearity potential is anharmonic for large displacements
polarization consequently varies nonlinearly with field in quantum description, uneven level spacing distortion of eigenfunctions higher terms in perturbation

Electro-optic effect exploit the nonlinear susceptibility
nonlinearity mixes static and oscillatory fields susceptibility at hence controlled by Pockels effect; Kerr effect

Second harmonic generation
again exploit the nonlinear susceptibility distortion introduces overtones (harmonics) where

Nonlinear tensor susceptibilities
nonlinear contributions to the polarization depend upon products of electric field components e.g. each product corresponds to a different susceptibility coefficient the induced polarization has three components (i =x,y,z): terms in the susceptibility expansion are therefore tensors of increasing rank

Nonlinear tensor susceptibilities
the susceptibility depends upon the frequencies of the field and polarization components e.g. if , any susceptibility unless

Symmetry in susceptibility
the susceptibility tensor may be invariant under certain symmetry operations e.g. rotation reflection inversion the symmetries of the susceptibility must include – but are not limited to – those of the crystal point group optically active materials fall outside the point group description (nonlocality) materials showing inversion symmetry have identically zero terms of even rank

Properties of susceptibility
depends upon frequency dispersion and absorption in material response depends upon field orientation anisotropy in crystal and molecular structure tensor nature of susceptibility depends upon field strength anharmonicity of binding potential hence nonlinearity series expansion of susceptibility

Pockels (linear electro-optic) effect
nonlinearity mixes static and oscillatory fields applying intrinsic permutation symmetry, in non-centrosymmetric materials, dominates

Kerr (quadratic electro-optic) effect
nonlinearity mixes static and oscillatory fields applying intrinsic permutation symmetry, in centrosymmetric materials,

Pockels cell voltage applied to crystal controls birefringence and hence retardance mounted between crossed linear polarizers longitudinal and transverse geometries for modulation field polarizer allows fast intensity modulation and beam switching polarizer modulation voltage

The Pockels cell 2.6 If a Pockels cell modulator is illuminated by light of intensity , it will transmit a beam of intensity such that where the retardance is given in terms of the voltage by Make a plot of the retardance and transmission versus applied voltage. What is the significance of the voltage that corresponds to maximum transmission? What is the lowest voltage above zero that will cause to be zero for ADP? Calculate the half-wave voltage

Light and Matter next : the Faraday effect and harmonic generation

Controlling light with light
Light and Matter Controlling light with light Tim Freegarde School of Physics & Astronomy University of Southampton

Optical nonlinearity potential is anharmonic for large displacements
restoring force is nonlinear function of displacement polarization consequently varies nonlinearly with field is a tensor of rank

Electro-optic effect exploit the nonlinear susceptibility
consider with nonlinearity mixes static and oscillatory fields Pockels (linear) and Kerr (quadratic) effects

again exploit the nonlinear susceptibility consider strong field distortion introduces overtones (harmonics) where

incident field: fundamental constant component: optical rectification new frequency: second harmonic generated intensities depend upon square of fundamental intensity focussed and pulsed beams give higher conversion efficiencies non-centrosymmetric materials required

if the fundamental field contains differently polarized components then the harmonic field contains their products the harmonic polarization need not be parallel to ,

Sum and difference frequency generation
if the fundamental field contains different frequency components then the harmonic field contains their products where

Sideband generation bias Pockels cell to
add r.f. field to modulate transmitted intensity transmitted field contains sum and difference frequency sidebands transmission

Harmonic generation fields may be at optical, radio or quasistatic frequencies energy combines in pairs, to produce sums and differences higher terms in susceptibility may combine more frequencies frequency tripling, quadrupling high harmonic generation e.g. total photon energy conserved:

Phase matching transit time through crystal
harmonic beam is superposition of contributions from all positions in crystal for contributions to emerge in phase, choose opposite polarizations for and use birefringence to offset dispersion conservation of photon momentum

Light and Matter next : Dr Danny Segal, Imperial College

Controlling light with light
Light and Matter Controlling light with light Tim Freegarde School of Physics & Astronomy University of Southampton

Electro-optic effect we exploit the nonlinear susceptibility
consider with nonlinearity mixes static and oscillatory fields Pockels (linear) and Kerr (quadratic) effects

Faraday (magneto-optic) effect
optical properties may also be influenced by magnetic fields consider effect of longitudinal field upon bound electrons induced circular birefringence, characterized by Verdet constant B magneto-optical glass non-reciprocal

x/a0 x/a0 electron density depends upon relative phase of superposition components

Faraday optical isolator
45º rotation in permanent magnetic field optical ‘diode’ passes incident light but rejects reflection B polarizer magneto-optical glass polarizer

Light and Matter next : controlling matter with light

Controlling matter with light
Light and Matter Controlling matter with light Tim Freegarde School of Physics & Astronomy University of Southampton

Mechanical effect of the photon
electromagnetic waves carry momentum emission momentum flux (Maxwell stress tensor) defined by absorption photons carry momentum

electromagnetic waves carry momentum emission emission momentum flux (Maxwell stress tensor) defined by absorption absorption photons carry momentum

Optical scattering force
each absorption results in a well-defined impulse emission emission isotropic spontaneous emission causes no average recoil average scattering force is therefore absorption absorption where is photon absorption rate

photons carry energy visible photon photons carry momentum visible photon momentum flux sunlight Cosmos 1, due for launch early 2004 © Michael Carroll, The Planetary Society

Solar sails and comet tails
photons carry energy visible photon photons carry momentum visible photon momentum flux Comet Hale-Bopp, 1997 sunlight Cosmos 1, due for launch early 2004 © Malcolm Ellis © Michael Carroll, The Planetary Society

Acousto-optic modulation
Fraunhofer diffraction condition crystal phonon Bragg diffraction condition Doppler shift transducer energy and momentum are conserved

Optical dipole force high force is gradient of dipole potential
towards high intensity depends upon real part of susceptibility freq 1 G=0.050 towards low intensity low

Optical dipole force recoil k-Dk k+Dk dipole interaction scatters photon between initial and refracted beams Dw maximum recoil momentum k

Optical tweezers Controlled rotation of small glass rod
Trapping and rotation of microscopic silica spheres © Kishan Dholakia, University of St Andrews

Diffracting atoms E M Rasel et al, Phys Rev Lett (1995)

The perplexed passenger
München why is first bus to arrive usually heading in the wrong direction?

Light and Matter next : laser cooling, trapping and atom interferometry for handouts, links and other material, see

Controlling matter with light
Light and Matter Controlling matter with light Tim Freegarde School of Physics & Astronomy University of Southampton

Optical scattering force
electromagnetic waves carry momentum emission photon absorption gives a well-defined impulse isotropic spontaneous emission causes no average recoil absorption average scattering force is therefore where is photon absorption rate maximum absorption rate is

Optical forces electromagnetic waves carry momentum
emission absorption forces therefore accompany radiative interactions position-dependent interaction gives position-dependent force TRAPPING

Optical forces electromagnetic waves carry momentum
forces therefore accompany radiative interactions position-dependent interaction gives position-dependent force TRAPPING velocity-dependent interaction gives velocity-dependent force COOLING

Optical forces POSITION VELOCITY continuous wave magneto-optic dipole
Sisyphus dynamical (cavity) Doppler VSCPT modulated c.w. stochastic adiabatic pulsed time-of-arrival Raman interferometric TRAPPING COOLING

Doppler cooling use the Doppler effect to provide a velocity-dependent absorption photon absorption gives a well-defined impulse red-detuned photon reduces momentum spontaneous emission gives no average impulse momentum k

Doppler cooling use the Doppler effect to provide a velocity-dependent absorption photon absorption gives a well-defined impulse red-detuned photon reduces momentum spontaneous emission gives no average impulse illuminate from both (all) directions momentum sweep wavelength to cool whole distribution k

Zeeman slowing opposite circular polarizations see opposite shifts in transition frequency in presence of longitudinal magnetic field ZEEMAN EFFECT Zeeman / Faraday effect red-detuned (s-) laser beam atomic beam tapered solenoids

Optical ion speed limiter
red-detuned laser beam accelerating ions electrostatic acceleration cancelled by radiation pressure deceleration

Magneto-optical trap LCP RCP RCP RCP RCP RCP RCP LCP
anti-Helmholtz coils RCP RCP LCP

Magneto-optical trap LCP Zeeman tuning in inhomogeneous magnetic field provides position-dependent absorption red-detuned laser beams also produce Doppler cooling RCP RCP RCP sweep frequency towards resonance for coldest trapped sample RCP anti-Helmholtz coils typical values: 107 atoms, 10μK LCP

spatial part of eigenfunctions given by and energy full time-dependent eigenfunctions therefore any state of the two-level atom may hence be written

write time-dependent Schrödinger equation for two-level atom insert energy of interaction with oscillating electric field reduce to coupled equations for a(t) and b(t) spatial part of eigenfunctions given by and full time-dependent eigenfunctions therefore any state of the two-level atom may hence be written

write time-dependent Schrödinger equation for two-level atom insert energy of interaction with oscillating electric field reduce to coupled equations for a(t) and b(t) full time-dependent eigenfunctions therefore spatial part of eigenfunctions given by and any state of the two-level atom may hence be written

Rabi oscillations solve for initial condition that, at , solutions are
write time-dependent Schrödinger equation for two-level atom insert energy of interaction with oscillating electric field reduce to coupled equations for a(t) and b(t) solve for initial condition that, at , solutions are where is the Rabi frequency

Rabi oscillations solve for initial condition that, at , solutions are
where is the Rabi frequency

Pi-pulses coherent emission as well as absorption
RABI OSCILLATION time half-cycle of Rabi oscillation provides complete population transfer between two states

Coherent deflection two photon impulses p
atom returned to initial state experiences opposite impulse p Kazantsev, Sov Phys JETP (1974)

Amplification of cooling
spontaneous emission pz p p p velocity selective excitation t

Stimulated scattering: focussing and trapping
München

München Garching plane of coincidence first bus is more likely to be heading towards plane of coincidence

plane of coincidence first pulse excites …………………. photon absorbed second pulse stimulates decay… photon emitted coherent process – can be repeated many times spontaneous emission only in overlap region

plane of coincidence Freegarde et al, Opt Commun (1995) rectangular Sech2 Gaussian FORCE HEATING

EXPERIMENTAL DEMONSTRATION 852 nm transition in Cs 30 ps, 80 MHz sech2 pulses from Tsunami stimulated force ~10x max spontaneous force Freegarde et al, Opt Commun (1995) Goepfert et al, Phys Rev A 56 R3354 (1997) rectangular Sech2 Gaussian FORCE HEATING

Atom interferometry p/2 pulses quarter Rabi cycles
RABI OSCILLATION p/2 pulses time quarter Rabi cycles atomic beam-splitters pure states become

Stimulated scattering: interferometry
excitation probability depends on ψ ‘spin echo’, Ramsey spectroscopy Dψ p/2 p/2

Stimulated scattering: interferometric cooling
p p coherent sequence of operations on atomic/molecular sample short pulses  spectral insensitivity pulses form mirrors of atom/molecule interferometer velocity-dependent phase: p/2 impulses add or cancel M Weitz, T W Hänsch, Europhys Lett (2000) p/2 p/2 z t

Stimulated scattering: interferometric cooling
VELOCITY-DEPENDENT PHASE variation of phase with kinetic energy: where ,  Dψ hence velocity-dependent impulse and cooling…

Light and Matter next : Monday 5 Jan: Q & A
Thursday 9 Jan: problem sheet 3 for handouts, links and other material, see

Light and Matter Quantum computation Tim Freegarde

Binary computing elements
any computer can be built from 2-bit logic gates A B C e.g. half-adder circuit A B C 1 1 gates are not reversible: output does not define input 1 NAND 1 A B C A B C 1 1 XOR 1 D A B C D A B 1 1 C 1 carry HALF-ADDER sum

Reversible binary computing elements
any computer can be built from 2-bit logic gates A B C e.g. half-adder circuit A B C 1 1 gates are not reversible: output does not define input 1 NAND 1 for reversible gates, additional outputs needed A A B C A 1 A B C 1 1 XOR 1 D A B C D A 1 A B 1 1 C 1 carry HALF-ADDER sum

Reversible binary computing elements
any computer can be built from 2-bit logic gates A B C e.g. half-adder circuit A B C 1 1 gates are not reversible: output does not define input 1 NAND 1 for reversible gates, additional outputs needed A A B C A 1 A B C 1 1 XOR 1 A B A D C A B C D A 1 A B D CNOT 1 1 CCNOT (Toffoli) C 1 carry HALF-ADDER sum

Thermodynamics of computation
thermodynamic quantities are associated with any physical storage of information 1 e.g. entropy setting a binary bit reduces entropy by hence energy consumption reversible logic does not change ; no energy consumed if change is slow note that conventional logic gates consume

Quantum computing each data bit corresponds to a single quantum property electronic or nuclear spin of atom or molecule electronic state of atom or molecule polarization state of single photon 1 E D C B A vibrational or rotational quantum number e.g. electron spins in magnetic field gradient electromagnetic interactions between trapped ions lift degeneracies in radiative transitions 11 10 01 00 operations carried out as Rabi -pulses B A evolution described by Schrödinger’s equation CNOT

Quantum computing tiny, reversible quantum bits (qubits) for small, fast, low power computers complex wavefunctions may be superposed: 1 E D C B A parallel processing: result is classical read-out: probabilistic results 11 10 01 00 limited algorithms: B A factorization (encryption security) CNOT parallel searches (data processing)

Quantum computing extension of computing from real, binary numbers to complex, continuous values extension of error-correction algorithms from digital computers to analogue computers 1 E D C B A link between numerical and physical manipulation is quantum mechanics part of computation, or computation part of quantum mechanics? extension of quantum mechanics to increasingly complex ensembles 11 10 01 00 statistical properties (the measurement problem) B A CNOT

Quantum information processing
Kepler 1571 Newton 1642 A C Clarke 1917 Galileo 1564 H G Wells 1866 classical mechanics Fraunhofer 1787 Einstein 1879 Planck 1858 Balmer 1825 Townes 1915 Schawlow 1921 quantum optics observe describe understand predict exploit Schrödinger 1887 Heisenberg 1901 Feynman 1918 Compton 1892 De Broglie 1892 Hertz 1887 quantum mechanics

Further reading R P Feynman, Feynman Lectures on Computation, Addison-Wesley (1996) A Turing, Proc Lond Math Soc ser (1936) D Deutsch, “Quantum theory, the Church-Turing principle and the universal quantum computer,” Proc Roy Soc Lond A (1985) D P DiVicenzo, “Two-bit gates are universal for quantum computation,” Phys Rev A (1995) C H Bennett, P A Benioff, T J Toffoli, C E Shannon

Light and Matter next : problem sheet 3

Coherent control INTRAMOLECULAR VIBRATIONAL ENERGY
REDISTRIBUTION (IVR) non-radiative coupling of vibrational modes rapid decoherence population loss from manipulated levels photochemistry energy nuclear coordinate

Coherent control: the pianoforte
soundboard b keyboard hammer strings bridge escapement action strong coupling  rapid decay time logP weak coupling  slow decay MODES OF COUPLED MOTION logP time “PROMPT” “AFTERSOUND” strike strings together to produce almost pure s strike one string - una corda - to produce equal superposition strike strings together from opposite sides to produce almost pure a reinforced “aftersound” is audible even when played softly FORTE PIANO G Weinreich, Sci Am (Jan 1979) pp94-102

Coherent control INTRAMOLECULAR VIBRATIONAL ENERGY
REDISTRIBUTION (IVR) non-radiative coupling of vibrational modes rapid decoherence population loss from manipulated levels photochemistry STRINGS energy a SOUNDBOARD b STRATEGY FOR MOLECULAR MANIPULATION IVR spectrally tailored pulses excite coherent superposition with reduced coupling to unwanted states shown to reduce IVR to 1% dark states, induced transparency, VSCPT nuclear coordinate

Light and Matter Problem sheet 3 Tim Freegarde

The Fresnel ellipsoid 3.1 Show, by writing eqn. (6.6) in terms of components of and alone that, for a lossless and non-optically active medium, the gradient of the energy density in electric field space , is parallel to the electric displacement . As the gradient of a function is perpendicular to surfaces of constant value, this demonstrates that is normal to the Fresnel ellipsoid.

The Fresnel equation in anisotropic media
3.2 Retrace the argument of sections and 3.7, relaxing the implicit limitation to isotropic media (in which the and fields are parallel), and hence show that plane electromagnetic waves must more generally satisfy the condition

Dusty comet tails 3.3 Calculate the radiation pressure force (optical scattering force) exerted on a smooth particle of reflectivity by sunlight of intensity , if (a) the particle is a disc of radius , facing towards the sun (b) the particle is a disc of radius , orientated at an angle to the sun (b) the particle is a sphere of radius . Hence calculate the acceleration of a spherical particle or radius , density and reflectivity , in sunlight of intensity .

Dusty comet tails 3.3 cont’d Given that the intensity of sunlight approaching the earth is and that the mean radius of the earth’s orbit around the sun is , show that, if the dust particles of a comet’s tail are sufficiently small, then the radiation force will overcome the gravitational pull of the sun. Calculate the critical particle size for completely reflecting particles, and show that it is independent of the distance from the sun. You may assume the earth’s annual orbit around the sun to be circular.

The solar sail 3.4 See how many errors you can spot in the short article by Prof. T. Gold, FRS,

Optical tweezers 3.5 By considering the recoil accompanying the deflection of a beam of photons, estimate the transverse optical dipole force exerted on a small sphere of glass, of radius and refractive index , in a laser beam of wavelength whose intensity gradient is Hence estimate the energy which must be supplied to remove a silica sphere of radius from the trap formed by a laser beam of wavelength and peak intensity The refractive index of silica may be taken to be Express this energy as a temperature.

Doppler cooling 3.6 (a) Calculate the recoil velocity of an atom of rubidium (atomic mass 85 amu) absorbing a single photon at the wavelength of the laser cooling transition in rubidium, 780 nm. (b) Compare this with the average speed of a rubidium atom at room temperature. How many photons must be absorbed by each atom to cool a room temperature vapour of rubidium down to a final temperature of ? (c) Doppler cooling can only change the atomic momentum by an integral number of photon impulses, and thus cannot reduce the average momentum below half a photon impulse. Calculate the temperature corresponding to this recoil limit in rubidium.

The Kerr lens 3.7 Show that, if the third-order susceptibility is non-zero, then the refractive index for light of frequency depends upon the intensity of the light itself. A lens made from a material with positive is used to focus short pulses of light from a mode-locked laser. Slightly before the focal plane, the focussed beam meets a pinhole which is centred on the beam axis. Explain why the transmission of the pinhole depends upon the instantaneous intensity of the laser pulse, and hence show that such an arrangement can lead to a shortening of the pulse. This process, acting repeatedly upon the pulses circulating within a laser cavity, provides the mechanism of Kerr lens mode-locking.

Dispersion revisited 3.8

The blue-detuned dipole force
3.9 Using your answers to the previous problem, identify the ranges of laser frequencies for which sodium atoms would be repelled from the centre of a focussed laser beam.

Frequency modulation 3.10 Read around the subject of modulation, and show that a laser beam which is modulated in frequency appears, for , to have sidebands at , and compare your result with that for amplitude modulation (examined in our analysis of the Pockels cell modulator). What happens to the spectrum of the modulated laser beam as the modulation index is increased?

Frequency modulation 3.10 cont’d What happens to the spectrum of the modulated laser beam as the modulation index is increased?

Light and Matter next : Examination! (Thursday January 22nd)

Polarization spectroscopy
laser qwp pol vapour cell pol det

Momentum state quantum computer
absorption & emission change momentum by ±k  ladder of momentum states Q0 corresponds to electronic state assume (for now) integer values binary representation {Q3 Q2 Q1 Q0} provides qu-bits 4k 3k 2k k -k -2k -3k 1 Q0 Q1 Q2 Q3 label according to momentum/k p/2 p/2 z t p p Dψ p/2 p/2 pz t

Pockels cell

Problem sheet 1 Light and Matter Tim Freegarde

Spherical waves 1.1 Show that wave motions of the form
are valid solutions to the three dimensional wave equation This expression describes spherical waves, with an intensity decaying according to the inverse-square law, radiating outwards from a point source.

Wave solutions 1.2 Determine whether
are valid solutions to the following wave equations ……… Derive expressions for the phase and group velocities for each of these examples.

Wave solutions 1.2 cont’d Show that the specific form is a solution to
and find an expression for .

Maxwell’s 1865 article 1.3 Read Maxwell’s original article, ‘A dynamical theory of the electromagnetic field,’ and work out the correspondence between today’s nomenclature and the terminology adopted by Maxwell.

Energy flow 1.4 Determine the approximate electric and magnetic field strengths for (a) sunlight reaching the earth with an intensity of 1 kW.m-2 (b) the 1 mm diameter beam from a 1 mW laser pointer (c) the 10 um diameter focus of a pulsed laser delivering 10 mJ pulses with a duration of 10 fs (d) the signal from a 2 W cellular telephone upon reaching a base station 5 km away

Waves in dielectrics with a damped resonance
1.5 Determine the dielectric constant (relative permittivity) for a material which may be considered to be composed of electrons per unit volume, each of which is bound in a damped harmonic oscillator with natural frequency and is subject to a frictional force

Mirror reflectivities
1.6 Calculate the reflectivities in air ( , ) of (a) the gold coating of an infrared mirror, for light of wavelength = 1064 nm, given that ( , , ) (b) a block of borosilica glass ( , )

Skin depth 1.7 The electric field in a homogeneous isotropic medium with conductivity , relative permittivity and relative permeability , satisfies the equation Explain how the second term in the equation arises. Show that such a medium supports the propagation of electromagnetic waves, but with amplitude attenuation. For simplicity, consider a plane wave propagating in the z-direction. For a good conductor, show that the attenuation length, , may be approximated by

Skin depth 1.7 The electric field in a homogeneous isotropic medium with conductivity , relative permittivity and relative permeability , satisfies the equation For an electrically screened laboratory, it is desired that electromagnetic radiation in the frequency range 105 – 106 Hz should be attenuated by at least 70 dB and in the range 106 – 109 Hz by at least 100 dB. Screening will be provided by lining the walls with thin copper sheeting. What is the minimum thickness of copper required? [The conductivity of copper is and should be taken to be unity. An amplitude is (decibels) weaker than amplitude if ]

Multilayer dielectric coatings
1.8 Starting from Maxwell’s equations, derive wave equations for the electric ( ) and magnetic ( ) fields appropriate to an isotropic, linear dielectric medium with permittivity and permeability . Use Maxwell’s equations to obtain a set of relations between the vectors , and for plane waves, where is the wavevector. Show that the magnitudes of and are related by , where is the impedance of the medium, and give an expression for the value of .

Multilayer dielectric coatings
Light is incident normally from medium 1 with impedance , through a layer of medium 2 of uniform thickness and impedance , into a medium 3 of impedance Obtain an expression for the total reflected intensity of light when the thickness corresponds to a quarter wavelength of the incident light in the medium a half wavelength of the incident light in the medium. Show that in case (a) the condition for zero reflection corresponds to Comment on the possible uses of single and multiple layer coatings in the fabrication of optical components for the transmission and reflection of light at particular wavelengths.

Measuring the ionosphere
1.9 Show that, in the absence of damping, the dispersion relation for electromagnetic waves in a plasma of density electrons per m3 may be written in the form where is the modulus of the wavevector and is the associated angular frequency. Find an expression for in terms of and the mass and charge of the electron. Show that electromagnetic waves with impinging on the plasma from free space are reflected.

Measuring the ionosphere
1.9 cont’d In an experiment to investigate the ionosphere, radio signals are transmitted vertically upwards and the reflected signal strengths and time delays measured. It is found that strong reflected signals are observed by the receiver at frequencies of 1.3 MHz and 4.0 MHz with time delays of 6.67 x 10-4 s and 2.0 x 10-3 s, respectively. For frequencies greater than 7.0 MHz, no significant reflected signal is observed. Explain these observations and deduce as much as you can about the ionosphere.

Light and Matter next : how light interacts with matter

Light and Matter Tim Freegarde School of Physics & Astronomy

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