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Transcript

ELECTROMAGNETIC FIELDS, the PHYSICS of LIGHT, and APPLICATIONS Lectures given at the Royal University of Phnom Penh February-April 2013 . ~ v 6= 0 ∇×~ . Prof. Dr. Hanspeter Helm http://frhewww.physik.uni-freiburg.de Department of Molecular & Optical Physics, Albert-Ludwigs-University, Freiburg, Germany This script was composed in Phnom Penh, Cambodia, in support of 45 hours of lectures on this topic which I gave in the Master’s Course in Physics, at the Royal University of Phnom Penh, in spring 2013. 1 . . . Prof. Dr. Hanspeter Helm helm@uni-freiburg.de frhewww.physik.uni-freiburg.de Cover page pictures taken from CT scan : http://www.genesishealth.com/services/radiology/ct/ Tarantula thermogram : http://www.sciencephotogallery.com/ Microwave oven : http://www.sciencetablets.blogspot.com/2011/04/how-can-microwaveoven-cook-food-so.html . 1 A picture book on the same topic is available upon request. Contents 1 Introduction 1.1 Repeating units and magnitudes in physics 1.2 History of electromagnetic fields . . . . . . 1.3 Electrical charges . . . . . . . . . . . . . . . 1.4 Electrical forces . . . . . . . . . . . . . . . . 1.5 Principle of superposition . . . . . . . . . . 1.6 Properties of vector fields . . . . . . . . . . 1.7 Images of the laws of electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 2 4 4 6 7 9 2 Static Electric Fields 2.1 Coulomb’s law . . . . . . . . . 2.2 Electric field . . . . . . . . . . . 2.3 Electrical flux . . . . . . . . . . 2.4 Electrostatic potential . . . . . 2.5 A conductor in an electric field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 16 17 20 21 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Current, Capacity & Energy 25 3.1 Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.2 Capacitor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.3 Energy contained in the electric field . . . . . . . . . . . . . . 28 4 Dielectrics 29 4.1 Electrical Dipole . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Dielectrics in an external field . . . . . . . . . . . . . . . . . . 32 4.3 Dielectric polarization . . . . . . . . . . . . . . . . . . . . . . 34 5 Time-dependent fields 35 5.1 Faraday’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 5.2 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 6 Electromagnetic Oscillators 37 6.1 Free oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 6.2 Forced oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 38 6.3 Open oscillator circuit (Hertz dipole) . . . . . . . . . . . . . . 40 I 7 Waves in vacuum 43 7.1 Plane waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.2 Periodic waves . . . . . . . . . . . . . . . . . . . . . . . . . . 44 7.3 Polarization of waves . . . . . . . . . . . . . . . . . . . . . . . 45 8 Waves in resonators 47 8.1 Standing electromagnetic waves . . . . . . . . . . . . . . . . . 47 8.2 Wave guides . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9 Spectrum of electromagnetic waves 49 9.1 Thermal radiation . . . . . . . . . . . . . . . . . . . . . . . . 51 9.2 Energy and momentum . . . . . . . . . . . . . . . . . . . . . 53 9.3 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 10 Atoms and Light 10.1 Transition dipole moment . . . 10.2 Quantum jumps . . . . . . . . 10.3 Classical rate-equation model . 10.4 Interaction with a beam of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 55 57 59 59 11 Application : LASER 61 11.1 Interference of multiple waves . . . . . . . . . . . . . . . . . . 62 11.2 Cavity modes and gain . . . . . . . . . . . . . . . . . . . . . . 64 12 Application : X-ray imaging 65 12.1 Units of Measure and Exposure . . . . . . . . . . . . . . . . . 65 12.2 Sources / detectors of X-Rays . . . . . . . . . . . . . . . . . . 66 12.3 Contrast and Computed Tomography . . . . . . . . . . . . . 66 13 Application : RADAR 13.1 Reflection . . . . . . . 13.2 Doppler Radar . . . . 13.3 Distance Measurement 13.4 Antenna design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 70 70 71 72 14 Application : LIDAR 73 14.1 Scattering and Flourescence . . . . . . . . . . . . . . . . . . . 73 14.2 Differential LIDAR . . . . . . . . . . . . . . . . . . . . . . . . 74 14.3 Coherent Detection . . . . . . . . . . . . . . . . . . . . . . . . 75 15 Application : Thermography 77 15.1 Thermal Energy and Balance . . . . . . . . . . . . . . . . . . 77 15.2 Emissivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 15.3 Infrared Cameras . . . . . . . . . . . . . . . . . . . . . . . . . 80 Chapter 1 Introduction 1.1 Repeating units and magnitudes in physics In order to repeat units used in physics and their meanings we begin by considering the properties of a laser pointer. Laser pointers are made to be eye-safe, so the laser power delivered is below 1 mW (milli-Watt). Laser pointers are typically red, so their wavelength is near λ = 600 nm (nanometers). First we try to answer how many red photons the laser pointer emits per second? The photon energy is Eph = hf where h is Planck’s constant, and f is the frequency of light, f = c/λ, with c being the speed of light. So from 3×108 the wavelength of our laser we calculate its frequency to be f = 6×10 −7 = 12 500×10 Hz. With the frequency we get the photon energy Eph = 3×10−19 Joule which is about 2 eV (electron Volts). Given that 1 Watt corresponds to the delivery of 1 Joule of energy per second we can say that our 1 mW 10−3 15 photons per second. laser delivers 3×10 −19 = 3 × 10 If we describe our laser light in terms of electro-magnetic radiation, we may use the Poynting vector to calculate the electric field strength of the em-wave emerging from our laser pointer. To do so we need to introduce the concept of intensity (the flux energy per unit area and per second, often termed irradiance). In our example we have to specify the cross section of our laser beam in order to do this calculation. Let us assume a cross section area of the laser beam of 1 mm2 = 10−6 m2 . Then the intensity is I = 1 kW/m2 . Now taking the magnitude of Poynting vector (which is equivalent to the intensity) we have I = c0 E02 , with E0 being the electric 1 field amplitude. Remembering that 4π = 10−7 c2 (where 0 is the dielectric 0 constant), we obtain E0 = 600 V/m. Next we use a lens to focus the light from our laser pointer. We may easily focus the light down to a cross section area of 100 µm2 . If we do so the intensity in the focus region increases by a factor 104 . Hence the electric field amplitude now rises to E0 = 100×600 = 60 kV/m, a formidable electric 1 h = 6 × 10−34 Js c = 3 × 108 m/s 2 e = 1.6 × 10−19 EM-Waves & Lasers field. Physicists have learned how to operate lasers in pulsed mode. Typically what happens is that all the energy which is usually delivered in 1 second (in a continuous wave, CW) can be delivered in a short pulse of light, say of 10 ns duration1 (1 ns = 10−9 s). If we operate our laser pointer in√this way the electric field amplitude would magnify by another factor of 108 to E0 ≈ 6 × 108 V/m. The reason of course is that now the same number of photons are present only during this very short duration of time. The power delivered per second has not changed, but for a short instant of time the intensity rises tremendously. When matter is illuminated by light fields of high intensity, the response of matter is typically not linear. Non-linear optics deals with the strong perturbation of matter caused by such fields. In a second direction we may explore the energy required to operate our laser pointer. Usually it is battery operated, say two 1.5 V batteries are used and when running CW the laser draws a current of 60 mA. The CW power drawn from the barry is 1.5 × 0.06 = 0.090 W or 90 mW. The efficiency of conversion of electrical power into light is therefore about 1/90 ≈ 1%. Next we may ask how many electron had to flow from the battery terminal for each photon delivered by the laser pointer. Again we consider CW operation and remember that the electric charge carried by an electron corresponds to the elementary charge e. We recall that the current given in C Ampere refers to the charge (in Coulomb) flowing per second, 1A = 1C/s. From this we conclude that a current of 60 mA corresponds to a flow of 0.06/e ≈ 4 × 10+17 electrons per second. Comparing this number with the number of 3 × 10+15 photons emitted per second, we find that about 130 electrons are required from the battery of our device to generate a single photon. Before we deal in more depth with these any of these subject areas we recall basic facts about electro-dynamics and quantum mechanics, about lasers and optics. 1.2 History of electromagnetic fields In the late 19th century it became clear that electricity and magnetism are not independent phenomena but they are two different aspects of a mathematical concept termed electromagnetic field (em field). In order to explain electrical and magnetic phenomena one had to assume that this em field follows dynamic laws. These laws are collected in Maxwell’s equations. The reality of the electromagnetic field appears in the fact that electromagnetic fields can be observed. Two vectors characterize the ~ and the magnetic field vector B. ~ They may be em field, the electric vector E observed due to the forces they exert on stationary and on moving electric 1 Even femtoseconds (1 fs =10−15 s) and shorter times are achieved nowadays. Introduction 3 charges. Also observable is the electric charge which itself may be viewed as the origin of the electromagnetic field. A surprising observation of the 19th century was that light can be viewed as a temporal and spatial modulation of the electromagnetic field. This phenomenon is predicted by Maxwell’s equations as well. With modulation we refer to a temporal and spatial variation carried by the field vectors, ~ ~ E(x, y, z, t) and B(x, y, z, t), in its simplest form a sinusoidal dependence in ~ 1 spatial dimension E(x, y, z, t) = {E0 , 0, 0} sin (ωt − kz), a plane wave with amplitude E0 , frequency ω = 2πf and wavevektor k and polarization along the x-axis, the wave propagating along the z-axis. Maxwells equations cover a great variety of phenomena of nature. To explore this wide range of topics one typically considers at first, timeindependent (static) fields. This is a refresh of what you have heard in previous lectures. We briefly recall how electrical and magnetic fields appear in observations, how we can generate electrical currents and how electrical currents react to the presence of magnetic fields. Also how currents generate magnetic fields. After the static situation we explore time-dependent cases which lead us to electromagnetic waves and to light. As soon as we combine matter and light we are in the general field of optics. Optics describes the response of matter to an external electro-magnetic field (matter becomes polarized ) and the modification of the external electro-magnetic field by the now polarized matter. An example is the action of a lens which is able to focus light. Matter by itself always gives rise to electromagnetic fields in the form of incoherent blackbody radiation. The law governing blackbody radiation is Planck’s law. The earth and our body (≈ 300 K) emits blackbody radiation at infrared wavelengths, a region we frequently associate with heat. The sun on the other hand, at 20 times higher absolute temperature (5900 K), emits blackbody radiation peaking at visible wavelengths, a region we usually associate with light. Matter would only cease to emit blackbody radiation if it were at absolute temperature limit, 0 K. Suitably prepared matter may generate coherent electromagnetic fields. This is well known from the Hertz dipole driven by an electic current. When considering the situation at higher frequencies (lasers) we need to resort to antennas made of atoms, molecules, or solid-state band-gaps and to quantum jumps as the fundamental origin of radiation. In this context we briefly repeat what you had previously heard in the context of quantum mechanics. Given the high brightness of laser radiation that can be achieved, the electric fields which can be achieved in a focused laser beam easily reach magnitudes of 109 V/m and beyond. The polarization of matter at such high field strength typically shows a non-linear response. This gives rise, among other things, to mixing of waves in matter which in turn leads to the generation of new waves, a topic in the field of non-linear optics. 4 EM-Waves & Lasers 1.3 Electrical charges Electrical charging phenomena are the earliest observation of electricity. While charging-up is a dynamic process, the behaviour of charged objects may be considered in the context of static electric phenomena. From experiments we know today: • There are positive und negative electrical charges. They differ in the direction of force with respect to each other and through their deflection in electric and magnetic fields. • Charges of the same sign repel, of opposite sign attract each other. • Charge is always tied to material object: electrons, protons and objects built from these. Some short lived elementary particles are charged. • The charge q = +e of the proton and q = −e of the electron represent the smallest unit of charge that exists. Whenever charge is observed in nature it comes in multiples of this elementary charge. For a total charge, Q, we always have Q = N e with N being an integer. • The magnitudes of charge +e und −e agree with each other. • Law of conservation: In a closed system the total number of charges stays constant. • In general our environment is electrically “neutral”. However charge may appear suddenly if we separate charges by friction, ionization ..... • We may separate charges in space. e.g. ionization of an atom A by photon impact (photoionization) or in a collision of two atoms A + hf → A+ + e− A + A → A + A+ + e− or mechanical effects (friction or fracture) which separate charges. • Electrical charges can be transported with electrical insulators. 1.4 Electrical forces Below we compare the magnitudes of gravitational and Coulomb force ( m1 , m2 are the mass of the particles, q1 , q2 their electrical charges, r the distance between the two particles, fG , fc are constants).2 2 units will be introduced later. 5 Introduction Strength |F | for 2 protons Sign Gravitational force Coulomb-Force 2 |F~G | = fG m1r·m 2 |F~C | = fc q1r·q2 2 1 5 × 1034 always attractive depends on the sign of the product q1 · q2 r-dependence 1 r2 neutralization impossible 1 r2 macroscopically nearly always Atoms are built from protons and electrons (and the electrically neutral neutrons). Due to the Coulomb law protons and electrons should attract each other. The reason that an atom does not collapse is a quantum effect: Heisenberg’s uncertainty relationship predicts that the mean momentum of the electron ∆p is very high when it is spatially localized to a small area, ∆x . This QM-rule prevents the collapse of the atom. ∆p · ∆x ≥ h̄ (1.1) Nuclei are built from protons (positive charges) und neutrons (neutral). Why do the protons in a heavy atom not fly apart? Reason : nuclear forces exist which are much stronger than the Coulomb interaction, but they act only over a very short range in space. This small range over which nuclear forces act limits the size of nuclei. Uranium with 92 protons is at the border of stability (a small perturbation by an additional neutron suffices for fission). Electron: nobody knows what keeps an electron together, or what size it is. Matter: The combination of electro-magnetic and quantum effects holds matter together and determines its atomic and its macroscopic electric and magnetic properties (e.g. electrical conductor - semiconductor insulator, ferro-, para-, dia-magnetic). Electrical charging: Some materials give up electrons more easily than others. For example rubber molecules retain electrons better than cotton. Thats why by rubbing rubber in a cotton cloth, the rubber gets charged with electrons and the cotton retains the opposite charge (missing electrons). Even a very tiny imbalance of electrical charges (an effective charge of 10−9 C) can lead to huge electrical effects. By comparison the total charge of electrons in 1 gram of copper is 4×104 C but is normally neutralized by the charge of the protons. 6 EM-Waves & Lasers Coulomb’s law is only valid for stationary charges. When charges move, the laws are more complicated. A part of the force between moving charges is the magnetic force. From experiments the following principle was determined: The force on a charge q depends only on the position of the charge in the field and on its velocity ~v (Lorentz force) ~ + ~v × B ~ F~L = q · E (1.2) ~ and B ~ give electric and magnetic field strength at the position of Here E the charge q. The effect of all other charges in the universe is collected in ~ and B. ~ The values of E ~ and B ~ may change with time. the two vectors E The general (relativistic) equation of motion for a charge q is d~ p d = dt dt m0~v p 1 − v 2 /c2 ! ~ + ~v × B ~ = F~L = q · E (1.3) ~ and B. ~ Now we only need to figure out the values of E 1.5 Principle of superposition In physics, the superposition principle states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. The superposition principle applies to any linear system. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum. The electric (magnetic) fields generated by two different objects are additive. ~ =E ~1 + E ~2 E ~ =B ~1 + B ~2 B This implies: If we know the field generated by a single electrical charge, we can determine the field of all charges in the world by superposition of their respective fields. The superposition principle is also behind wave interference. When two or more waves are present at the same space, the net amplitude at each point Introduction 7 in space is the sum of amplitudes of the individual waves. In some cases the sum has a smaller amplitude than the individual waves; this phenomenon is called destructive interference.In other cases the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference. 1.6 Properties of vector fields ~ +~v × B). ~ We have defined the field via the force acting on a charge F~L = q (E But fields exist at any point in space, even if there are no charges present. When forces are active on a charged object then something must be present in space, even if the charge is absent. ~ ~ E(x, y, z, t) und B(x, y, z, t) describe the vector fields at a position x, y, z at the time t. We associate each point in space with such a vector. At the time t these fields are the origin of the force on the charge which sits at the position (x, y, z), under the assumption, that moving this charge into this position does not move the charges in the rest of the world. Thereby it does not modify the field which was present at the position (x, y, z) prior to the charge q being placed there. A field describes a physical property whose magnitude depends on space and time. Examples of fields are: • temperature: T (x, y, z, t), a scalar field, • velocity of water molecules in a river: ~v (x, y, z, t), a vectorial field. We may represent a vector field : • abstract as a function: ~ = f (x, y, z, t) E • graphically as a vector, or • as a line of force (tangent along the field vectors). In the following examples we think of a velocity field and a fictitious closed surface in this field, which does however not disturb the flow. The field vector which we are considering describes the direction and magnitude of flow of liquid particles. 8 EM-Waves & Lasers Question 1: How much liquid is lost from our volume (flows out) and how much liquid enters our volume (flows in) per second ? This quantity we term flux ( flux = mean normal component of velocity ) divergence = (mean normal component) times (surface) Question 2: Is the liquid circulating ? Here we think of a loop (in the form of a bicycle inner tube) located inside the liquid (which does not disturb the flow). Now we consider all liquid particles inside our tube and sum over the component of their velocity vectors tangential along the loop. If the result is non-zero a circulation is present and the sign of the result gives us the direction of rotation of the liquid. circulation = (mean tangential component) times (circumference) These two definitions (divergence and circulation) permit us to describe the ~ ( electric field strength laws of electromagnetism in terms of the vectors, E ~ ), and B (magnetic field strength, in most books referred to as magnetic induction). The sign and the magnitude of the two quantities are important. 9 Introduction 1.7 1. Images of the laws of electromagnetism ~ through a closed surface = The flux of E (1/0 )× (sum of all charges inside) 0 is a constant. When no charges are present inside our volume, then the mean normal component summed over the closed surface is zero. 2. ~ along the curve C Circulation of E d ~ through S) = − dt (flux of B C is a closed curve which also forms the boundray of the surface S. The surface S is not closed but may be arbitrarily shaped. 3. ~ through a closed The flux of B surface is always equal to zero. ~ along C) c2 × (Circulation of B 4. = d dt ~ through S) + (flux of E 1 0 ×(flux 5. of electrical current through S) ~ + ~v × B ~ F~L = q · E These 5 laws describe electro-dynamics. In the presence of matter additional rules must be introduced. 10 EM-Waves & Lasers integral form I differential form ~ · dS ~= 1 E 0 S 1 I Z ~ · d~s = − d E dt C 2 I 3 S ~ ·E ~ = 1ρ ∇ 0 ρ dV Z ~ ~ ×E ~ = − ∂B ∇ ∂t ~ · dS ~ B ~ · dS ~=0 B I ~ ·B ~ =0 ∇ Z ~ · d~s = µ0 ~j · dS ~+ 1 d B c2 dt C 4 Z ~ ~ ×B ~ = µ0 ~j + 1 ∂ E ∇ c2 ∂t ~ · dS ~ E c2 = (0 µ0 )−1 Stokes’ law (circulation) : Gauss’ law (flux) : I S ~ · dS ~= E Z V ~ ·E ~ dV ∇ I C ~ · d~s = E Z S ~ ×E ~ · dS ~ ∇ Nabla operator ~ := ∂ , ∂ , ∂ ∇ = (∂x , ∂y , ∂z ) ∂x ∂y ∂z ~ r) describes the scalar distribution of The divergence of a vector field E(~ sources and sinks of the field. ~ and E ~ The divergence has the form of a scalar product between ∇ ~ := ∇ ~ · E(~ ~ r) = ∂x Ex + ∂y Ey + ∂z Ez div E ~ r) describes a vector field. The The rotation or curl of a vector field E(~ ~ und E ~ rotation has the form of a vector product ∇ ~ := ∇ ~ × E(~ ~ r) = rot E ∂y Ez − ∂z Ey , ∂z Ex − ∂x Ez , ∂x Ey − ∂y Ex ! The gradient of a scalar potential φ(~r) is a linear operator, which turns the scalar φ into a vector field. The largest growth in potential appears when we move along the steepest gradient. ~ φ = grad φ := ∇ = (∂x , ∂y , ∂z ) φ unit vector in direction of the maximal growth of φ ! · this maximal growth ! 11 Maxwell equations Qualitative experiments The following experiments show qualitatively the connections which follow from the five equations on page 9. • We send a current trough a wire, which is located above a permanent magnet. Current means that the electrons in the wire travel with some drift velocity ~v . Due to the Lorentz force, ~ F~L = q~v × B, the electrons are deflected from their path and transfer momentum to the atoms in wire. Consequence: the wire moves. • If we do the experiment properly (with high current, thin magnet) we will see that the magnet falls over! Why is this? The 4th law says: a current through ~ the wire means that the circulation of B around the wire is non zero, that is magnetic field due to the current exists. The permanent magnet suddenly sees the magnetic field from the wire. We would argue that the magnetic field which is generated by the current in the wire exerts a force on the permanent magnet. If it is strong enough, the magnet may fall over. • Two wires, each carrying current. Each current (wire) gives rise to a magnetic field at the position of the opposite wire. The wires attract each other, if the currents flow in the same direction. 12 • Currents and magnets generate magnetic fields. A current is the equivalent of a moving charge. If we replace the permanent magnet in the first experiment by a coil through which current flows, we get the same result. • we explore the magnetic field of a straight wire carrying a DC current. ~ along any loop surrounding The 4th law tells us: the circulation of B the wire is a fixed value for a given current. We would say the circulation is identical for whatever loop we draw around the wire. Applying Stokes’ law we can say the integral around the loop is a constant. As a consequence we expect that for a circular loop the tangen~ gets smaller if we increase tial component of B the radius of the loop. As the circumference of ~ must the loop gets bigger the magnitude of B decrease. From this argument we deduce the ~ : r-dependence of |B| I ~ · d~s = const. B 2πr1 B1 = 2πr2 B2 = const. 1 ⇒ B(r) ∝ r • The magnetic field of a permanent magnet has as its origin also in moving charges. What are these currents ? We may think that electrons moving around the nuclei represent a current. In addition, electrons and nuclei (even the neutron) have spins with which a magnetic moment is associated. When the atomic magnetic moments are in disorder there is no net magnetic effect. In a permanent magnet the spins are ordered and a macroscopic magnetization appears. All magnets have their origin in currents. In connection with spins of electrons and nuclei the origin or meaning of this current is however not yet clear. A spinning electron with charge on its outside is not the correct picture for the magnetic moment associated with the electron spin. 13 Maxwell equations ~ =0 • 3rd law: div B There are no magnetic charges (equivalently: there are no magnetic monopoles). If we break a magnet into two pieces, two new magnets emerge. • We charge a capacitor by closing the switch. A current I flows for some time, although the circuit is open (the capacitor plates are separated from each other). Now we imagine a curve C surrounding the wire. This curve encloses an area S1 . On the basis of the 4th law we expect ~ along C) ∝ (flux of I through S1 ) (Zirkulation von B Now we draw a new surface S2 , with the same boundary C. This surface does not cut through the wire but it closes in between the capacitor plates. No regular current flows through this surface, but the ~ along the curve C must be the same. The explanation circulation of B of this puzzle came from Maxwell: When we close the switch, an electric field builds up between the capacitor plates. ~ along C) ∝ (Circulation of B ∂ ∂t ~ through S2 ) (flux of E A temporal change of the electric field gives rise to magnetic effects . 14 • Now we repeat the first experiment but we add a current meter in the circuit. Now we move the wire. Doing this we move the electrons. If the electrons move in a magnetic field of the permanent magnet, they experience the Lorentz force. As a consequence of F~L = ~ we observe a current flowq~v × B ing in the circuit. • Now we move the magnet and again a current flows. If a current flows charges must move. But they are at rest at the beginning of the experiment! What moves the charges ? There must be a force ~ But where does the electric field come from? The closed → F~ = q E. circuit with the ammeter forms the closed curve C and spans a surface S. ~ along C ) ∝ ( Circulation of E ∂ ∂t • Finally we send an AC current through the red wire, I = I0 sin ωt. By combining the second and the fourth law we can explain the generation of em waves: With waves we mean that the ~ and B ~ propagate with the speed fields E of light c away from our antenna. How does the speed of light enter Maxwell’s equations? → c2 = (0 µ0 )−1 ~ through S) (flux of B Chapter 2 Static Electric Fields Electrostatics deals with time-independent problems of stationary charges ~ ~ in the absence of magnetic fields. In the static case (∂ E/∂t = ∂ B/∂t = 0) Maxwell’s equations separate into two pairs : Dynamic ~ ·E ~ = 1ρ ∇ 0 ~ ~ ×E ~ = − ∂B ∇ ∂t ~ ·B ~ =0 ∇ ~ ~ ×B ~ = µ0 ~j + 1 ∂ E ∇ c2 ∂t Static ~ ·E ~ = 1ρ ∇ 0 ~ ×E ~ =0 ∇ ~ ·B ~ =0 ∇ ~ ×B ~ = µ0 ~j ∇ electrostatics magnetostatics ~ and B̄ appear in separate equations. • In the static case, E • Electricity und magnetism are separated phenomena as long as the charges and currents are stationary. Only when the magnetic (electric) field changes with time or if the current changes with time we find a ~ and B̄ . connection between E ~ • stronger argument yet: ∂ E/∂t must be fast in comparison with c2 µ0 ~j, ~ and B̄ to show a significant dependence from each other. in order for E magnetostatics: ~ with divergence=0 vector field B (no magnetic sources) and given rotation (closed B-field lines). electrostatics: ~ with rotation=0 vector field E (no closed E-field lines) and given divergence. 15 16 2.1 CHAPTER 2. STATIC ELECTRIC FIELDS Coulomb’s law Applying Gauss’ law to Maxwell’s first equation we see that the scalar product of the electric field and the surface normal vector (the normal component of the field), when integrated over a closed surface, is proportional to the charge residing inside the volume contained by this surface, I Z Z ρ 1 ~ ~ ~ ~ dV = Q . E · dS = ∇ · E dV = 0 S V V 0 Now we consider a sphere with radius r and place a point-charge Q at the center of a sphere. The field must then be spherically symmetric and be of the same magnitude at each point on the surface of the sphere. The field has only a radial component. Its magnitude |E| on the sphere surface is 1 |E| 4πr2 = Q . (2.1) 0 Now we place a second charge, q, at some position on the sphere surface. This second charge will feel the force ~. F~ = q E (2.2) Inserting for E from Eq. (2.1) we obtain Coulomb’s law 1 qQ |F | = . (2.3) 4π0 r2 The prefactor is defined as : N 1 −7 2 = 10 c . (2.4) 4π0 A2 Here c is the speed of light in vacuum (c = 3 × 108 m/s). The quantity 0 is the dielectric constant 1 N · m2 Volt · m fc = = 8.99 · 109 = 8.99 · 109 and (2.5) 2 4π0 C C 0 = 8.854 · 10−12 A2 · s 4 A·s = 8.854 · 10−12 3 kg · m V·m (2.6) where 1 kg·m2 ·s−2 = 1 N·m = 1 V·A·s. The charge of the electron is q = e = 1.602 · 10−19 C (2.7) Current = amount of charge passing through some surface per second. current charge unit abbreviation ampere coulomb A C conversion 1A=1C/s 1C=1As 17 2.2. ELECTRIC FIELD 2.2 Electric field Coulomb’s law delivers the force acting between two charges. As with gravity one originally assumed that this force action is transmitted instantaneously, however far the distance. According to the theory of relativity no signal propagates faster than the speed of light. With the field concept we have a means to describe the development of the field with time. In the field model the second charge feels the field generated by the first one. The field is the messenger of the Coulomb force. If the first charge moves, the field changes. This change is seen by the second charge at some later time. But for the moment we continue to stay with stationary chrges. A definition for the electric field A charge Q sits at the origin. The force on a ~ is sample charge q at the position R ~ = fc q · Q êR F~ (R) R2 Here êR is the unit vector, pointing to the position of the sample charge q. ~ generated by Q, may be defined The electric field strength at the position R via the force which q experiences in this field ~ = q · E( ~ R) ~ . F~ (R) (2.8) Thereby we obtain a definition of the electric field strength of a point charge, ~ R) ~ = E( 1 Q êR . 4π0 R2 (2.9) This definition is independent of the magnitude of q. In a more general situation the charge Q does not rest at the origin, but it sits at some position ~r1 . ~ is In this case the field at the position R ~ R) ~ = E( 1 Q ~ − ~r1 ) , (R ~ − ~r1 |3 4π0 | R (2.10) where the vector components are X x1 X − x1 ~ − ~r1 = R Y − y1 = Y − y1 . Z z1 Z − z1 18 CHAPTER 2. STATIC ELECTRIC FIELDS Fields of multiple charges : The field of several distributed charges are obtained by vector addition ~ R) ~ = E ~1 + E ~2 + E ~3 + . . . E( X Qi ~ − ~ri ) (R = fc ~ | R − ~ri |3 (2.11) i Below we show the lines of force for the field generated from two charges. In the center figure shows the case of a dipole (Q1 = −Q2 ). Scalar charge density : The scalar charge density ρ(~r) (units C/m3 ) is defined via the total charge in the volume Q= Z ρ(~r) dV . V For a general distribution ρ(~r) we have ~ R) ~ = fc E( Z V ~ − ~r R ρ(~r)dV . ~ − ~r |3 |R (2.12) Surface charge density : The total charge sitting on a thin metallic plate can be defined as the integral over the surface charge density σ sitting on the surface S Q= Z σ dS . (2.13) S The units of σ are C/m2 . Next we search for the field of a homogeneously charged plate at a distance a from the plate. The contribution of a surface element dS to the field strength E at a distance b is given by 19 2.2. ELECTRIC FIELD ~ = fc σ dE dS êb b2 (2.14) Here êb is the unit vector in the direction of b. The surface element is dS = r dr dϕ, b = a/ cos α and r = a tan α. Thus dr a = . dα cos2 α (2.15) For an infinitely large plate the horizontal components cancel to zero. After integration over dϕ we have for the component perpendicular to the surface dEv = fc σ 2π sinα dα . (2.16) Integrating α from 0 to π/2 we obtain Ev = σ/20 . The field is homogeneous and independent of the distance from the plate. ~ = σ êa . E (2.17) 20 ~ depends on the sign of the charge density σ. In the space The direction of E below the plate the field has the same magnitude as above but has opposite sign. Thus we may conclude: The normal component of the electrical field strength jumps by σ/0 when we pass through the plate. Plate capacitor: We consider 2 plates with opposite charges Q1 = −Q2 . When the distance between the plates is small compared to the size of the plates then the field strength between the plates is E = σ/0 . (2.18) This field is a factor of two larger than in the example above as now two plates contribute. For infinitely large plates the field inside is homogeneous. In the space outside the fields ±σ/(20 ) compensate to zero. For finite size plates inhomogeneous fields appear near the boundaries. 20 CHAPTER 2. STATIC ELECTRIC FIELDS We conclude from the above discussion: • Electrical charges modify the empty space. ~ r). • Electrical charges are origin for the vector field E(~ ~ is defined via the force • The strength and direction of E ~ ~ r). on a sample charge q, F (~r) = q E(~ • Lines of force give a graphic representation of the field. The direction of force is given by the tangent along these lines. 2.3 Electrical flux Electrical charges are sources and sinks of the electric field. A measure of the spatial density of the lines of force is the differential flux ~ · dS ~. dΦel = E The total flux through a surface S is Φel = Z S ~ · dS ~. E We consider a sphere of radius r, at its center is the charge Q, hence I r̂ ~ · dS 2 S r Z π Z 2π 1 2 = fc Q r sin θ dθ dϕ r2 o 0 Q = f c Q 4π = . 0 Φel = fc Q The electrical flux through a closed surface depends only on the net charge contained in this volume. It is independent of the area of the surface. 21 2.4. ELECTROSTATIC POTENTIAL 2.4 Electrostatic potential We define the work required to move a charge q ~ from position a to position b : in the field E W = Z b a Z b F~ (r) · d~r = q ~ E(r) d~r . a In the field of a point charge Q the work required is W = fc qQ Z r2 d~r r1 1 1 − = fc qQ 2 r r1 r2 . Energy is gained (W > 0) if charges of the same sign move away from each other (r2 > r1 ). The electric field is conservative, the work done is independent of the path we take, it depends only on the positions of the points a and b. The work done on a closed path is zero1 . For such fields can we define a potential. The electrostatic potential φ at position P is defined via the energy q·φ which is required to carry a sample charge q from P to infinity φ(P ) = Z ∞ P ~ E(r) · d~r . (2.19) The potential difference between two points φ(P1 ) − φ(P2 ) = Z P2 P1 ~ E(r) · d~r (2.20) is often referred to as electrical voltage (sometimes called tension) U = φ(P1 ) − φ(P2 ) . (2.21) A charge passing through a potential difference U changes its potential energy by the amount ∆Epot = −q · U . (2.22) Since total energy is conserved we require ∆Ekin = −∆Epot = q · U . 1 (2.23) Note that this is only true as long as the charge is moved with very low speed. 22 CHAPTER 2. STATIC ELECTRIC FIELDS Definition of voltage (tension) [U ] = [energy] kg · m2 · s−2 N·m = = = Volt . [charge] A·s C (2.24) Now we define the electric field in terms of the gradient of the potential ~ = −grad φ = −∇ ~ φ = − ∂φ , ∂φ , ∂φ E ∂x ∂y ∂z ! . (2.25) The electric field points in the direction of the steepest decrease of the potential. With Maxwell’s first equation we obtain ~ ·E ~ = −∇ ~ · (grad φ) = −∆φ = ρ ∇ (2.26) 0 ~ 2 is the Laplace operator, which is in Cartesian coordinates where ∆ = ∇ ∆= ∂2 ∂2 ∂2 + + . ∂x2 ∂y 2 ∂z 2 If we can solve Poisson’s equation ρ ∆φ = − 0 (2.27) (2.28) we may obtain directly the potential distribution from a given set of charges. In the absence of charges Eq. 2.28 becomes the Laplace equation ∆φ = 0 . (2.29) You may ask whether such a simple equation is good for any prediction, after all this thing is just zero! Answer: in the absence of charges the potential field in space is established by conditions at the boundary of this space. Potential of a point charge: with the definition of φ(R → ∞) = 0 we have φ(~r) = fc Q |~r| Equi-potential surfaces : are surfaces along which φ(~r) is constant. Equi-potential surfaces are orthogonal to the lines of force. No work is done when moving a charge along an equipotential surface. This statement is only true if we move the charge slowly. Field of a charged hollow sphere : (2.30) 2.5. A CONDUCTOR IN AN ELECTRIC FIELD 23 We assume a sphere carrying the homogeneous surface charge density σ. The total charge is Q = 4πR2 σ. For a position r > R we have Φel = I ~ · dS ~ = |E| ~ 4πr2 = Q , E 0 S (2.31) hence we have for r > R Q ~ E(r) = fc 2 êR . r The field outside the sphere (r > R) is like that of a point charge Q located at the sphere center. The sphere surface is an equi-potential surface. The potential at a distance r from the center is φ(r) = fc Q . r (2.32) For the field inside a different argument can be made. Any closed surface inside the hollow sphere does not contain charges. Hence the field strength inside the sphere must be zero ~ < R) = 0 . E(r (2.33) The potential inside is constant, and equal that at the surface of the sphere. The field jumps on the surface from |E| = 0 inside to |E| = fc Q/R2 = σ/0 on the surface. This corresponds to the surface charge density on the sphere, σ = Q/(4πR2 ) (see also page 19). 2.5 A conductor in an electric field In a conductor we have free movable ~ moves the charges. The force F~ = q E charges until an opposing field forms inside the conductor, which compensates the external field. If there is no current flowing, the electric field inside the metal is zero and the charges sit on the surface of the sphere, the concept of Faraday’s cage. The appearance of charges on a metal surface due to an external electric field is termed influence. . charged conductor uncharged conductor 24 Experiments to separate and to transport charges: • Two metal plates (with insulated handles) touch each other inside a capacitor • After separation of the two plates we have a field-free zone between the plates. We have separated charges. The field due the capacitor action of the two plates compensates the field from the outer capacitor. • Cup-electrometer: When transporting charge with a metallic spoon to the outside of the cup we can maximally reach the potential of the charge source used to charge the spoon (left). Charges distribute over cup and spoon according to their capacity to carry charges. • If on the other hand we move the charged spoon inside the cup (second figure) the electrometer shows a response, even though we do not touch the cup, a consequence of influence caused by the charge on the spoon. • If the spoon touches the inside of the cup its charges repel to move to the outside of the cup (third figure). The inside of the cup stays field free. If we now remove the spoon we find the spoon free of charge. • This is the principle of a Van-de-Graff generator (fourth picture). The finite resistance of the insulator holding the cup limits the maximal tension one reaches by successive adding of charges inside the cup. Tensions of many Megavolts can be built up using a source of only 1kV. . Chapter 3 Current, Capacity & Energy 3.1 Current Current describes transport of charge per second 1 A = 1 Ampere = 1 C/s. I= dQ = dt Z ~ ~j · dS (3.1) Carriers of charge are electrons or charged ions. In a microscopic model we think of a density of n particles per unit volume, each with the charge q, which move (drift) with the velocity ~v . With this we may define the current density ~j = nq ~v . (3.2) If there are charge carriers of both signs we would write ~j = n− q − ~v − + n+ q + ~v + . (3.3) Drude model for drifting electrons. Electrons in the conduction band move in a lattice of positive ions with thermal speeds, vth ≈ 106 m/s. In collisions with the ions they are deflected from their path. Without external field there is no directed motion of electrons, hvi = 0. - Let the mean time between collisions with the lattice be τ . If there is an external field, electrons will be accelerated in this free time ~ between collisions : ~a = F~ /m = q E/m. As a result they attain the mean ~ drift-speed: h~v i = τ F /m, (typically < 1 mm/s). - With this relationship we may define the conductivity σ ~ ~ ~j = nqh~v i = nq 2 τ E/m = σE (3.4) The unit of conductivity is [σ] = [A/(V m)]. - From this relationship we obtain Ohm’s law. With I = j · A und E = U/L we have for a homogeneous conductor 25 26 CHAPTER 3. CURRENT, CAPACITY & ENERGY I = σAU/L = U/R (3.5) - Resistivity is defined as R= L L = ρs · σA A (3.6) where ρs is the specific resistivity. ρs = [Ω · m]. - The units are [R] = [V/A] = [Ω], ~ 6= 0 inside the conductor. - If a current flows we have E Potential drop: If current flows through a homogeneneous conductor, there is a linear potential drop along the conductor x U (x) = φ1 − φx = R · I . L As a consequence we may pick up potential differences along the conductor U1 = U0 x L Joule’s heat und U2 = U0 L−x . L appears due to the work done by moving charges, W = Q U . The power (Unit [Watt]=[V·A] ) is given by P = dW dQ U2 =U = U I = I2 R = . dt dt R (3.7) Conservation of charge allows to formulate an equation of continuity. ~ The flow of The flow of charge through a surface per time unit is ~j · dS. charge through a closed surface correpsonds to the loss of charge from the volume V enclosed by the surface S, I ~=−dQ=−d ~j · dS dt dt S Using Gauss’ law I S ~= ~j · dS Z V Z ρ dV . (3.8) V ~ · ~j) dV (∇ we may define the equation of continuity ~ · ~j = − ∂ρ . ∇ ∂t For a stationary current we have ~ · ~j = 0 . ∇ (3.9) (3.10) 27 3.2. CAPACITOR 3.2 Capacitor A capacitor is typically formed by two metal plates which are placed a short distance from each other. Frequently a dielectric is placed between the plates to increase the capacity (see section 4). If we turn on circuit a) the lamp glows and gets dimmer and extinguishes when the capacitor is charged. If we now replace the current source by a short circuit (b) we observe the same phenomenon again: The lamp glows for a short time,until the capacitor is discharged. Initially there is no potential difference between the capacitor plates. This is why the lamp glows when we switch on the battery. The current flowing through the lamp leads to charging of the capacitor plates. When the full battery voltage appears as a potential difference between the capacitor plates no more current flows. Now a field has built up between the plates. This field stores energy. We may say the charged capacitor carries the charge +Q and −Q on the plates. The two plates are at the potential φ1 and φ2 , with U = φ1 − φ2 . The surface charge density on the two plates is σ = Q/A. The electric field is proportional to Q, and inverse proportional to the surface of the plates A (see p. 19) E= Q σ = 0 A 0 (3.11) The capacity of this device is defined as C= Q . U (3.12) The unit of capacity is [Farad]=[Coulomb/Volt] . Typical capacity values are in the range of pico-, nano- and microfarad. 28 Field in a planar plate capacitor : We consider two plates at x = 0 and at x = d, which carry the charges ±Q. No charges are between the plates. Hence we may use the Laplace equation (see page 22) in one dimension d2 φ=0 dx2 ⇒ φ(x) = ax + b (3.13) with the boundary conditions φx=0 = φ1 = b and φx=d = φ2 = ad + φ1 . With a = (φ2 − φ1 )/d = U/d we obtain for the potential distribution φ(x) = U x/d + φ1 (3.14) and for the field ~ = −∇φ ~ = −U x̂/d . E (3.15) For a plate of area A we obtain |E| = U σ Q = = 0 0 A d (3.16) and for the capacity Q = C · U , C = 0 A/d . (3.17) Example : for A = 1 cm2 and d = 1 mm we have a capacity C = 0.9 pF. 3.3 Energy contained in the electric field A capacitor contains the energy Wel = U Q/2. Wel = 1 Q2 1 = CU 2 2 C 2 (3.18) We use this expression for our plate capacitor with C = 0 A/d and the tension U = E · d. The volume between the plates is A · d = V . The energy stored in the electric field in this volume is Wel = 1 1 C U 2 = 0 E 2 V . 2 2 (3.19) This expression is valid for arbitrary configurations of the electric field in vacuum. With it we may define the energy density of the electric field, 1 wel = Wel /V = 0 E 2 . 2 [wel ] = [J/m3 ] (3.20) Chapter 4 Dielectrics 4.1 Electrical Dipole An electrical dipole with dipole moment p~ = Q d~ (4.1) is formed by two opposing charges Q placed at a distance d. The dipole potential is φd = fc Q 1 ~ ~ − d/2| |R − 1 ~ ~ + d/2| |R (4.2) The potential distribution shows a singularity at the position of the charges. For R d we have 1 1 q = ~ ~ ~ d~ R |R ± d/2| 1 ± R· R2 + 1 φd ≈ fc ~ · p~ p cosθ R = fc R3 R2 d2 4R2 1 1 q ≈ R 1± (4.3) The potential decreases with 1/R2 at (R d) . The top row gives the exact equi-potential lines from (4.2). The bottom row gives the approximate result (4.3). The two charges are located at the positions x = y = 0, z = ±1. Far from these positions Eq. (4.2) and (4.3) are practically identical. 29 ~ d~ R· R2 1 ≈ R ! ~ · d~ 1R 1∓ ± .... 2 R2 30 CHAPTER 4. DIELECTRICS Nabla operator in polar co-ordinates : To calculate the electric field distribution around the dipole using (4.3) E = −∇Φ we need the nabla operator in polar coordinates. In Cartesian coordinates we have ~ = êx ∂ , êy ∂ , êz ∂ ∇ ∂x ∂y ∂z . (4.4) In any ortho-normal basis we have : ê1 ×ê2 = ê3 , ê2 ×ê3 = ê1 , ê3 ×ê1 = ê2 . (4.5) The unit vectors in polar coordinates are in the Cartesian basis: êR = { sin θ cos ϕ, sin θ sin ϕ, cos θ } radial (4.6) tangential to circle of longitude (4.7) tangential to circle of latitude (4.8) êθ = { cos θ cos ϕ, cos θ sin ϕ, − sin θ } êϕ = { − sin ϕ, cos ϕ, 0 } We now think of infinitesimal travels on the surface of the (earth) sphere : • walking east we only vary ϕ our path is = R sin θ dϕ • walking south we only vary θ our path is = R dθ • traveling upwards we vary R our path is = dR Hence we have for nabla in polar coordinates ~ = ∇ ∂ 1 ∂ 1 ∂ , êθ , êϕ êR ∂R R ∂θ R sin θ ∂ϕ . (4.9) With it we calculate the electric field strength in polar coordinates ~ = −∇φ ~ d=− E ∂ 1 ∂ 1 ∂ , , φdipol . ∂R R ∂θ R sin θ ∂ϕ (4.10) where we place the dipole axis along the z-axis (~ p || ~z ). Using the approximate expression (4.3) for the dipole potential, we obtain for the rotationally symmetric field (rotationally symmetric around z) 31 4.1. ELECTRICAL DIPOLE the field components 2p cosθ R3 p sinθ = fc R3 ER = fc (4.11) Eθ (4.12) Eϕ = 0 (4.13) Potential energy of a dipole in an external field An external field has the potential values φ1 and φ2 at the positions of the two dipole charges. The potential energy of the dipole is therefore Wpot = Q(φ1 −φ2 ) ~ = Q d~ · ∇φ Wpot ~ = −~ p·E (4.14) ~ ~ Wpot is maximal for −~ ~ =0 if p~ ⊥ E. Wpot is minimal for p~ || E. p || E. Force on a dipole The forces on the individual charges are ~ 1 und F~2 = −Q E ~ 2. F~1 = +Q E For a homogeneneous field we have ~1 = E ~ 2 = E, ~ a resulting force equal to zero, E but we have a torque ~ = p~ × E ~. D In an inhomogeneneous field the dipole feels a torque and a force : h ~ − E(~ ~ r + d) ~ r) F~1 + F~2 = Q E(~ = p~ ~ dE d~r i where the vector gradient is required, at the example of Fx , Fx = Q Ex+ −Ex− = p~ · grad Ex ∂Ex ∂Ex ∂Ex + py + pz = px ∂x ∂y ∂z 32 CHAPTER 4. DIELECTRICS Molecular dipoles Molecules with a permanent dipole moment, e.g. H2 O can be oriented in an external field. The macroscopic result is called orientational polarization. Non-polar molecules, e.g. CO2 carry no permanent dipole moment, but in an external electric field a dipole moment can be induced in them. The induced dipole moment in a single atom (or molecule) is characterized ~ In an electric field which is oriented by its polarizability α where p~ = α E. along z, the centers of charge separate by |~z| and p~ = q ~z. In a molecule the polarizability will in general be dependent on the direction of the electric field relative to the molecular axes. In this case α is a tensor P ~i . and we have p~i = j αi,j E 4.2 Dielectrics in an external field A conductor has freely movable charges. Dielectrics are insulators, its charges are not freely movable, but the they may be polarized on the microscopic (atomic, or molecular) scale. With polarization we mean the displacement of the positive and negative charge centers in a molecule (atom) by a tiny fraction of the size of the molecule (atom). Faraday observed that the voltage on a capacitor decreases when an insulator (or equally a conductor) is brought into the field of the capacitor. We had for the capacity of a parallel plate capacitor (distance d, surface A) C = 0 A d with Q = C U (4.15) being the charge on each plate. As the charge has not changed when we entered the insultor the capacity must increase when the voltage decreases. Hence the integral U= Z ~ · d~s E (4.16) between the plates has decreased. In vacuum we had U= 1 Q Q = d C 0 A (4.17) 4.2. DIELECTRICS IN AN EXTERNAL FIELD With a metallic plate of thickness b inside we have (the surface of the conductor must be an equipotential surface) U= 1 Q (d − b) 0 A (4.18) The capacity of this arrangement is C = 0 1 A d 1 − b/d (4.19) where b/d is the fraction of the volume filled by the conductor. If we fill the capacitor volume completely with an insulator then C = 0 A d (4.20) where is a characteristic number for the material which we inserted between the capacitor plates and > 1. How is the field reduced? How is the capacity increased? Model : The dielectric consists of small polarizable spheres. In an external field charges appear on the surface of these spheres due to polarization. Residual (uncompensated) charges appear on the surface of the insulator. The electric field of the surface charges compensates part of the external field inside the insulator and therefore reduces the voltage across the capacitor. We consider the green surface S. ~ through this surface is The flux of E ~ ·E ~ = ρ/0 . ∇ It is smaller when the dielectric is present. Hence the field strength inside the insulator must be smaller. 33 34 CHAPTER 4. DIELECTRICS 4.3 Dielectric polarization signifies the separation of charges q in an atom by some distance. If we have n atoms per unit volume, then the dipole moment per unit volume is P~ = n q ~z = n p~ (4.21) Inside the insulator charges compensate each other. What remains is a surface charge density σpol , for example on the top σpol = Qpol nqzA = = n · |~ p| = |P~ | A A where P~ is the polarization induced by the external field. This quantity is independent of the volume of the dielectric. The direction of the polarization vector is defined by p~, the magnitude of P~ is equal to the surface charge density σpol . For small fields the linear approximation that z in (4.21) is ~ is valid. For isotropic materials E ~ points in the direction proportional to E : ~ diel . P~ = −χ 0 E (4.22) Here χ is the dielectric susceptibility χ = n α /0 and α is the polarizibility of a single unit (atom or molecule). In an empty capacitor we have Evac = σfree 0 In a dielectric the field is reduced Ediel = σfree + σpol σfree + P = 0 0 Hence we have ~ ~ diel = E ~ vac + P = E ~ vac − χ E ~ diel E 0 (4.23) ~ vac = (1 + χ) E ~ diel = E ~ diel E (4.24) and where = 1 + χ is the dielectric constant of the material. The dielectric constant is dimensionless. Typical values for are air ≈ 1, H2 O ≈ 80, SrT iO3 ≈ 12000. If the boundary of the dielectric and vacuum is perpendicular to the external electric field the field jumps from Evac to Ediel = Evac /. This jump corresponds to the surface charge density at the interface to the dielectric σpol Evac Evac − Ediel = Evac − = . 0 Chapter 5 Time-dependent fields From the second law we see that a timechanging magnetic field is origin for a ~ (closed lines of force of circulation of E the electric field). Analogous but with opposite sign is the last term in equation (4). A timechanging electric field gives rise to a cir~ culation of the B-field. 5.1 ~ ·E ~ = 1ρ (1) ∇ 0 ~ ~ ×E ~ = − ∂B (2) ∇ ∂t ~ ~ (3) ∇ · B = 0 ~ ~ ×B ~ = µ0 ~j + 1 ∂ E (4) ∇ c2 ∂t Faraday’s law Observations with a conducting loop which lead to a flow of current in the circuit (see experiment on page 14): when a permanent magnet is moved into or out of the loop, or when the loop is moved into or out of the field. Conclusion : The temporal change of the magnetic flux produces an electric field along the conducting loop. A potential difference appears across the length of the conductor. If we break the conductor loop we may extract this inductance-voltage. Using Stokes’ law and the second Maxwell equation we obtain Faraday’s law Z Z ~ ×E ~ · dS ~ = −∂ ~ · dS ~ ∇ B ∂t S S I Z ∂ ~ ~ · dS ~ = − ∂ Φm . Uind = E · d~s = − B ∂t S ∂t C Φm is the magnetic flux and Uind is the induced voltage. Note that the area S is bounded by the curve C. The meaning of the minus sign in (2): the induced voltage generates a magnetic field which counteracts the external change in magnetic field flux. This is called the rule of Lenz. The conducting loop attempts to keep the magnetic flux constant. 35 36 CHAPTER 5. TIME-DEPENDENT FIELDS The inductance voltage leads to a flow of current I = Uind /R which uses the energy P = I 2 R. This energy comes from the kinetic energy of motion of the magnet or that of the conducting loop. Aluminum ring on an electromagnet: When turning on the electromagnet, a current is induced in the ring. The magnetic field due to this current is opposite to the field of the coil and the ring is accelerated. (no action if the ring is cut open) Waltenhofen pendulum is decelerated due to the induced magnetic fields which are induced by the eddy currents. 5.2 Inductance The minus sign in Maxwells second equation limits the rise of current in a coil. If the coil has N windings, a changing magnetic field flux inside the coil (for example due to the external AC signal applied to the coil) gives rise to an inductance voltage dΦm Uind = −N . (5.1) dt The magnetic flux inside the coil (SP1) is proportional to the current flowing through the coil: N Φm = L I , (5.2) where L is the inductance of the coil. The magnitude of L depends on the geometry of the coil Uind = −N dΦm dI = −L . dt dt (5.3) The inductance of an arbitrary conducting system is defined as Uind L=− . (5.4) dI/dt The magnitude of L for a simple coil with N windings can be estimated as follows: The field inside the coil is B = µ0 w I, where w = N/` is the number of windings per meter and ` is the length of the coil. In this case the magnetic flux is Φm = B A and using (5.2) and the volume of the coil, V = ` A, we have L = µ0 w2 V . The units of inductance are [L] =Vs/A= Wb/A = Henry. Chapter 6 Electromagnetic Oscillators 6.1 Free oscillator A simple electrical oscillator circuit is built from a capacitor and an inductance. The capacitor C is discharged and charged periodically. In parallel the magnetic field in the coil with inductance L appears and disappears. This is analogous to a mechanical oscillator. The potential energy of the mass M corresponds to the field energy in the capacitor 1 Wel = CU 2 2 The kinetic energy of the mass M corresponds to the magnetic energy 1 Wmag = LI 2 2 A resistor R in an electric circuit is analogous to the friction term in the mechanical oscillator. For the mechanical oscillator with the restoring force Fx = −kx x and the friction term −αẋ we had the equation of motion mẍ + αẋ + kx x = 0 (6.1) The amplitude of oscillation of the mass of the damped HO is equivalent to the current in an L-C-R circuit. For the voltages along the circuit we have Q LI˙ + RI + = 0. C Taking the time derivative (Q̇ = I) we have LI¨ + RI˙ + I/C = 0. Using the ansatz I = Aeλt , where A and λ may be complex, we obtain 37 38 CHAPTER 6. ELECTROMAGNETIC OSCILLATORS λ2 + R 1 λ+ =0 L LC (6.2) with the solutions λ1,2 R =− ± 2L s 1 R2 − = −α ± β 2 4L LC The general solution is therefore I = A1 e−(α−β)t + A2 e−(α+β)t For the case R2 < 4L/C we have β imaginary. Setting β = iω and α = R/(2L) we have the general solution I = e−αt A1 e+iωt + A2 e−iωt . With A1,2 = a ± ib a real solution is h I = e−αt a e+iωt + e−iωt + ib e+iωt + e−iωt i = |A|e−αt cos (ωt + ϕ) √ where |A| = 2 a2 + b2 and tan ϕ = b/a. The initial conditions determine |A| and ϕ. The eigenfrequency of the circuit is ω= s R2 1 − . LC 4L2 (6.3) For R = 0 we obtain Thomson’s formula, ω0 = √ 1 , LC (6.4) The eigenfrequency rises, when L and C get smaller. 6.2 Forced oscillator If we apply an external periodic voltage U = U0 cos ωt, at some arbitrary frequency ω (which is independent of the frequency definition in (6.3)), we enforce a damped harmonic oscillation in the circuit. The sum of external voltage and the inductance voltage must be equal to the voltage drop on R and C: Q C I = LI¨ + RI˙ + C U + Uind = +RI + U̇ 39 6.2. FORCED OSCILLATOR With the complex ansatz U = U0 eiωt and I = I0 ei(ωt−ϕ) iωU = −Lω 2 + iωR + 1 C I The complex resistence is defined as Z= U 1 = R + i ωL − I ωC and the impedance |Z| , (Z = |Z| · eiϕ ) |Z| = s R2 2 1 + ωL − ωC The time dependent current is I= U0 cos (ωt − ϕ) |Z| tan ϕ = with ωL − Im{Z} = Re{Z} R 1 ωC . The current maximizes for ωL−1/(ωC) = 0, that is ω = ω0 . In this case the phase angle ϕ = 0 and the current is in phase with the external AC. The power dissipated in R is P = I2 R = U02 cos2 ωt R. |Z|2 The voltage on the capacitor is UC = = R Q Idt = C C U0 eiωt I = . iω Z C iωC Phase relationships are ( −i = e−i π/2 ) : 40 6.3 CHAPTER 6. ELECTROMAGNETIC OSCILLATORS Open oscillator circuit (Hertz dipole) How can we reduce inductance and capacitance? A continuous transition to a single wire is possible, thus modifying the LC-circuit into a Hertz dipole, a linear dipole, whose ends are alternatively charged positive and negative. Its eigenfrequency is determined by the length of the dipole. If we feed the antenna at the center (at z = 0) with an AC-signal, the current will be I(z, t) = I0 (z) sin ωt . The current has to be zero I(z = ±`/2) = 0 at the ends of the wire. This is the origin of the resonance feature of the antenna. Resonances appear if the length of the dipole is a multiple of half the wavelength, ` = nλ/2. The wavelength connects with the phase velocity c via ω = 2πν = 2πc/λ. A periodic current at the lowest resonance, (n = 1), gives as the spatial distribution of current I(z) = I0 cos (πz/`), and voltage U (z) = U0 sin (πz/`). The standing wave can be observed with light bulbs. Bright are the lamps across which the voltage drop, and hence current is high. For ` = λ/2 = 6 cm, we have ν = 2.5 GHz. At high frequency the Hertz dipole is strongly damped, due to radiation of an electromagnetic wave (not due to its ohmic resistance). To understand the radiative loss we explore the propagation of an em-pulse along two parallel wires, connected at one end to a high-frequency AC-source : Snapshots of the potential distribution along the conductor at various times, A) applying a B) applying a C) applying DC voltage voltage pulse an AC-voltage Only in case C) (temporally periodic) we obtain a spatial periodicity, a wave, and only because of the finite speed of light. Heinrich Hertz (1898) 6.3. OPEN OSCILLATOR CIRCUIT (HERTZ DIPOLE) 41 found that electrical signals propagate along conductors with v ∼ = c. From parallel wires we can form a dipole antenna by bending it open. Now we see how em waves are formed: at an AC frequency of 100 MHz the polarity changes every 5 ns (for 2.5 GHz every 200 ps). The electric field due to the charges on the wire propagates with the speed ~ of light. When the charges meet at the center of the antenna the E-field lines separate from the dipole and propagate away from it in the form of closed loops. Analogously for the magnetic field, which is generated by the current flowing through the antenna. Near radio transmitters we may detect the orientation and magnitude of the field using a glow lamp or a light bulb with differently shaped antennas: ~ induces an AC dipole antenna: a periodic field E current in a straight wire segment. The lamp will glow at positions of maximal E-field, if it is directed along the electric field vector. ~ induces an AC current loop: a periodic field B current and the lamp will glow when the loop is oriented perpendicular to the field. Hertz grid: series of parallel wires with distance smaller than wavelength. • The grid blocks radiation, if the grid wires are parallel to dipole axis. • It is transparent for radiation if the wires are perpendicular to the dipole axis (no current flows in the grid wires). • Under the angle α relative to the orientation of the dipole axis, radiation is transmitted at a reduced intensity, I ∝ cos2 α. 42 CHAPTER 6. ELECTROMAGNETIC OSCILLATORS Mathematical description of Hertz dipole ~ of the antenna at time t0 The electrical dipole moment P ~ 0 ) = Q ~z(t0 ) = Q z0 sin ωt0 êz = p(t0 ) êz P(t (6.5) describes the oscillation of electrons in the wire where the total charge of movable charges is Q. If we use for the retarded time t0 = t−r/c (6.6) we obtain after a lengthy derivation (easiest by using the vector potential) the magnetic field at time t at the position r, ϑ in the far field 1 sin ϑ µ0 p̈(t0 ) (6.7) |B(r, ϑ, t)| = 4π c r Using (|B| = |E|/c) we get for the electric field strength µ0 sin ϑ p̈(t0 ) (6.8) 4π r The radiative power is given from the energy density of both fields 1 wem = 0 E 2 + c2 B 2 = 0 E 2 (6.9) 2 as the flux of energy density |E(r, ϑ, t)| = sem = c wem = c 0 E 2 . (6.10) Radiation preferentially occurs perpendicular to the dipole axis ∝ sin2 ϑ. Due to the periodic time dependence of (6.5) we have ṗ ∝ ω and p̈ ∝ ω 2 . E2 H The total power emitted from the dipole is Pem = sem dS, with the surface element dS = r2 sinϑ dϑ dϕ. The mean power emitted is µ0 hPem i = Q2 ω 4 z02 . (6.11) 12πc The loss due to radiation increases with ω 4 . We see from (6.7) und (6.8) that the radiation of proportional to the acceleration of charges. The power radiated is proportional to the square of the acceleration. Acceleration / deceleration of electrons gives rise to Synchrotron radiation and to Röntgen-Bremsstrahlung. Chapter 7 Waves in vacuum In vacuum where ρ = 0 and ~j = 0 Maxwells equations appear symmetric: ~ ~ ×E ~ = − ∂B ∇ ∂t ~ ~ ×B ~ = 1 ∂E ∇ c2 ∂t ~ ·E ~ =0 ∇ ~ ·B ~ =0 ∇ In 1860 Maxwell discovered that the speed of light connects the electromagnetic fields with light. To see this we differentiate the second equation with respect to time 2~ ~ ∂ ~ ~ =∇ ~ × ∂E = − ∂ B ∇×E ∂t ∂t ∂t2 (7.1) ~ an insert the derivative ∂ E/∂t from the fourth equation : ~ ∂2B ∂t2 ~ 1 ∂2B − 2 2 c ∂t − ~ ~ × ∂ E = c2 ∇ ~ ×∇ ~ ×B ~ = ∇ ∂t ~ ∇ ~ ·B ~ − ∇ ~ ·∇ ~ B ~ = ∇ ~ = ∇ 0 ~ = −∆B ~ 2B ~ −∇ Here we have used the product rule, ~a × (~b × ~c) = ~b(~a · ~c) − (~a · ~b)~c. From ~ and analogously for E ~ this we obtain the wave equation for B ~ = ∆B ~ 1 ∂ 2B c2 ∂t2 ~ = ∆E 43 ~ 1 ∂ 2E c2 ∂t2 44 7.1 CHAPTER 7. WAVES IN VACUUM Plane waves ~ depends on only a single spatial co-ordinate E ~ = E(z). ~ For plane waves E ~ ~ Hence ∂ E/∂x = ∂ E/∂y = 0 and the wave equation simplifies to : ~ 1 ∂2E ∂2 ~ E = ∂z 2 c2 ∂t2 (7.2) ~ ~ ~ ~ ·E ~ = ∂ Ex + ∂ Ey + ∂ Ez = 0 ∇ ∂x ∂y ∂z (7.3) Since ~ z /∂z = 0, hence Ez = const. We choose Ez = 0. Then the we also have ∂ E ~ = {Ex , Ey , 0}. A general solution would be field has the components E Ei = fi (z−ct) + gi (z+ct) (7.4) where i = x, y. We demand that the functions f, g can be differentiated with respect to z ± ct. Since the argument for all points in a plane z = const. is identical, at a given time, (plane of constant phase), one speaks of plane waves. The function fi (z −ct) describes a wave (or its phase fronts), which move along the positive z-direction, the wave gi (z+ct) describes a wave (or its phase fronts), which move along the negative z-direction. To determine the speed of the phase front we differentiate the argument with respect to time : d(z − ct)/dt = dz/dt − c = 0 → dz/dt = +c Plane waves need not be periodic. 7.2 Periodic waves A special solution for a periodic wave results from the condition fi (z − ct) = fi (z + λ − ct) . (7.5) This function has the identical value after a spatial period of λ, the wavelength. We call k the wave number and T the period (time) k= 2π λ c=νλ= ω λ 2π ω = kc T = 1 2π = ν ω (7.6) For a harmonic plane wave along the positive z-direction we have ~ = E ~ 0 f (z − ct) E ~ 0 sin k(z − ct) = E ~ 0 sin(kz − ωt) = E (7.7) 45 7.3. POLARIZATION OF WAVES ~ = 0, t = 0) or more general with a phase φ , which value depends on E(z ~ E = ~ 0 sin(kz − ωt + φ) E (7.8) It is often easier to deal with the complex notation ~1 = E ~ 0 ei(kz−ωt) E ~2 = E ~ 0 e−i(kz−ωt) E ~ 0 ei(ωt−kz) = E ~ 1∗ = E (7.9) From addition of these we can construct the real solutions ( e±iϕ = cos ϕ ± i sin ϕ) n ~1 + E ~2 <e E n ~1 − E ~2 =m E o o ~ 0 cos (kz − ωt) = 2E ~ 0 sin (kz − ωt) = 2E (7.10) For the general case of a plane wave propagating along an arbitrary wave vector k we have ~k = {kx , ky , kz } (7.11) where |k| = 2π/λ and ~ =E ~ 0 ei(ωt−~k·~r) E 7.3 (7.12) Polarization of waves ~ Linearly polarized is a wave with an E-vector which oscillates in a plane ~ t) = [E0x êx + E0y êy ] ei(ωt−kz) E(z, (7.13) The components along x- and y oscillate in phase. ~ For a circularly polarized wave the arrow of the E-vector rotates along 46 CHAPTER 7. WAVES IN VACUUM a circular spiral in space and time. This is the case when E0x = E0y and when the two components oscillate out of phase by 90◦ . ~ t) = E(z, h i E0x êx + E0y eiπ/2 êy ei(ωt−kz) i h = E0 êx + i êy ei(ωt−kz) (7.14) or written in its components ( Ex Ey ) = E0 ( 1 i ) ei(ωt−kz) ForE0x 6= E0y the wave is elliptically polarized. Using Jones Vectors we have : ( linearly polarized along x linearly polarized under an angle θ with x ( right circular √1 2 left circular √1 2 1 0 ) Ex = cos (ωt − kz) Ey = 0 cos θ sin θ ) ( ) √ Ex = (1/√2) cos (ωt − kz) Ey = (1/ 2) cos (ωt − kz + π/2) ) √ Ex = (1/√2) cos (ωt − kz) Ey = (1/ 2) cos (ωt − kz − π/2) ( 1 i 1 −i Ex = cos θ cos (ωt − kz) Ey = sin θ cos (ωt − kz) Chapter 8 Waves in resonators When an electric wave is reflected in a conducting plane a phase jump of 180◦ occurs. We may envision this as follows. The wave travels perpen~ is parallel to the surface. The surface dicular on to a metal surface, hence E ~ is assumed to be a perfect conductor (σel = ∞ und ~j = σel E). ~ at the surface is not zero, If E an infinitely high current flows. ~ to vanish we reIn order for E quire a secondary wave such, that the field of the primary wave is compensated. Inside the metal the field compensate, outside a standing wave appears. 8.1 Standing electromagnetic waves A linearly polarized plane wave Ex (z) = E0 cos(ωt − kz) propagates along the positive z-direction and is reflected by an ideal conducting metal surface ~ at the wall has to be zero at z=0. The tangential component of E Ex (z = 0) = E0in + E0ref = 0 ⇒ E0in = −E0ref (8.1) and we have Ex (z, t) = E0in cos(ωt − kz) + E0ref cos(ωt + kz) = E0in [cos(ωt − kz) − cos(ωt + kz)] = −2E0in sinωt sinkz (8.2) Since ∂By /∂t = +∂Ex /∂z we have ∂By /∂t = 2k E0 sinωt coskz (8.3) 47 48 CHAPTER 8. WAVES IN RESONATORS After integration over time we obtain By (z, t) = −2 k E0 cosωt coskz = − +2B0 cosωt coskz ω (8.4) ~ and B ~ are spatially separated by λ/4 and they are temThe maxima of E ~ and B ~ are in phase. porally separated by T /4. In a running wave E 8.2 Wave guides inside a metallic box standing waves in 3 dimensions may appear. A waveguide is a resonator with an open end face. Two typical forms of solutions appear ~ = {Ex , Ey , 0}. • TE -waves: E ~ ⊥ to the direction of propagation z E ~ = {Bx , By , 0}. • TM-waves: B ~ ⊥ to the direction of propagation z B These transverse electromagnetic waves are characterized TEnm or TMnm , where (n, m) give the number of nodal planes in the x, y-directions. A wave ~ = {0, Ey , 0} and B ~ = {Bx , 0, Bz }. guide excited in the TE10 mode has E For this case the fields in a rectangular wave guide are shown below. The electric field is shown in red (left), the magnetic field in blue (right): One couples into such a resonator with a small dipole antenna at the position of maximal electric field strength or via a circular antenna at the position of maximal magnetic field. Chapter 9 Spectrum of electromagnetic waves All electromagnetic waves (radio waves, light waves, X-rays ...) propagate in vacuum with the speed of light. What differs is the frequency of oscillation, or the wavelength, or the size of the antenna, or the type of detector. In vacuum we have λ = c/ν. Only a tiny portion of the sun’s spectrum is at visible wavelengths. A substantial fraction of the sun’s spectrum is absorbed by the atmosphere before reaching ground level, in particular the high-energy portion (ultraviolet) and selected regions in the infrared, where water molecules and CO2 absorb. frequency ν Hz 3 × 1020 . 3 × 1016 type of radiation Gamma wavelength energy of photons1 1 pm 1 MeV 100 Å 100 eV 500 nm 2.0 eV 100 µm 10 meV 1m 1 µeV X-rays 6 × 1014 ultraviolett visible infrared 3 × 1012 microwaves 3 × 108 radiowaves 1 In order for the photon energies to be exactly the value given by wavelength and frequency we need to multiply every number in this column by a factor of 1.18 . From quantum theory we know: • the energy contained in the field is quantized. Photons appear as packets of energy with E = hν = h̄ω. • the momentum of an individual photon is p = h̄k. Before we deal with this we discuss various concepts of generation of radiation. 49 50 CHAPTER 9. SPECTRUM OF ELECTROMAGNETIC WAVES Sources of radiation and appropriate detectors differ greatly in different frequency ranges. √ Radiowaves: LC-circuit, Thomson’s relationship, ω0 = 1/ L C. Microwaves: In a Clystron pulses of electrons run through small resonators in which they induce a periodic magnetic field. In a Magnetron electron packets circulate with the cyclotron frequency and pass by appropriately shaped resonators. Infrared: Rotational transitions in molecules and molecular vibrations, (only in molecules with a permanent electrical dipole moment, e.g. OH or HF ...). Optically active lattice vibrations also can absorb and generate infrared radiation. Visible light and near UV: electronic transitions in the outer shell of atoms and molecules are primarily respnsible for radiation in this range. UV, X-ray range: electronic transitions in inner shells (higher Z numbers allow shorter wavelengths). X-rays, γ-range: nuclear transitions, Bremsstrahlung (synchrotron, wiggler-magnet radiation) Sun: Over a wide frequency range the sun is a nearly perfect black-body radiator at a temperature if T ≈ 5900 K. The solar constant at the surface of the earth is about 1 kW/m2 , at the top of the earth’s atmosphere it is about 1.4 kW/m2 . Bandwidth of radiation: • discrete radiation (monoenergetic): Emission from free atoms and molecules (low pressure), but also from impurities in solid lattices. ( line spectra ) • continuous radiation: Gases at high pressure, solid state band structure, liquid dyes, Bremsstrahlung, radiation during molecular interactions. 9.1. THERMAL RADIATION 9.1 51 Thermal radiation The electromagnetic field (thermal radiation) permits the energy exchange between bodies until there is thermal equilibrium. This happens without physical contact between the bodies. Even if thermal equilibrium is reached, the thermal radiation field still exists. It pervades all space and the energy contained in the thermal radiation field reflects the temperature of matter present in the radiation field. Black-body radiation A hollow container with inner surface A and a tiny opening ∆A A has the property that radiation, which enters the container through this aperture, undergoes many reflexions inside. As a portion of the radiation is absorbed upon each reflection, we may argue that the radiation which enters, never comes back. We may claim that the probability of absorption of radiation entering through ∆A is practically equal to one. In this idealized framework this is true for any wavelength. Such a body is called black body. Note that the small opening ∆A is called the black body, not the container. If this were the complete story, our container would get hotter and hotter as time goes on, as all the radiation entering through the small opening is absorbed inside, increasing the temperature of the container. A balance occurs in that the matter making up the inner surface A of the container itself emits electromagnetic radiation, according to its temperature. For example the thermal motion of lattice atoms of the container walls is one of the origins of this radiation. Electromagnetic radiation from the container walls fills the inside volume of the container with a certain energy density (which strongly depends on the temperature). Part of this radiation which zaps around inside the container, bouncing from wall to wall, also being partially absorbed and re-emitted by the walls, emerges through the small opening ∆A. One can actually see this radiation when one heats the container. At room temperature the small opening appears black to our eye, but it becomes progressively brighter, from dark red going to bright red, and eventually to white color, if we heat the container. Note that the container can be made of any material (of course we do not want the container to go up in flames, this would be a different story of light involving chemistry). The radiation emitted through the small opening of our container is called black-body radiation. To correctly describe its spectral distribution was one of the achievements of Max Planck at the end of the 19th century. Counting the modes per unit volume, which can be supported by a resonator, Planck found in 1900 that agreement with the observed spectrum of radiation can be achieved if one demands that the energy contained in an electromagnetic wave at frequency ν cannot take any arbitrary value, but can only be increased or decreased by adding discrete units of energy ∆E = hν. This is the so called quantum hypothesis . The quantity h is called Planck’s constant, h = 6.626 · 10−34 J s. 52 CHAPTER 9. SPECTRUM OF ELECTROMAGNETIC WAVES Any mode (any electromagnetic wave) of the radiation field can take up arbitrary many quanta (meaning arbitrary many photons). In the case of the black-body radiation the number of quanta, which are present in a specific mode, rises with temperature and the spectral energy density of radiation of the black body is given by Planck’s law of black-body radiation, w(ν, T ) dν = 1 8πhν 3 dν . 3 hν/kT c e −1 (9.1) The spectral energy density of radiation w(ν, T ) [J m−3 Hz−1 ] gives the energy which is present in the frequency range between ν and ν + dν per unit volume. R∞ The spectral integrated energy density is w(T ) = 0 w(ν, T ) dν [J m−3 ]. Using the relationship λ = c/ν and dν = c · dλ/λ2 , Planck’s law can be written in terms of wavelength w(λ, T ) dλ = 1 8πhc dλ . 5 hc/λkT λ e −1 (9.2) The units of spectral energy density of radiation are [w(λ, T )]=[J m−3 m−1 ]. 1.0 1.0 6000 K 0.8 5000 0.6 4000 0.4 0.2 0.0 200 400 600 800 1000 1200 Frequenz n HTHzL wHlL H10+5 Wêm2 nm-1 L wHnL H10+5 Wêm2 THz-1 L 1.2 6000 K 0.8 0.6 5000 0.4 0.2 4000 0.0 200 400 600 800 1000 1200 Wellenlänge l HnmL From Planck’s equation two famous other laws, that were known previously from experiment) could be explained. Wien’s law describes the position of maximum intensity of radiation as a function of temperature. The maximum of the spectral distribution shifts towards smaller wavelength, when the temperature is raised. λmax · T = const = 2.9 [mm K] . (9.3) For example the maximum intensity of radiation is at 545 nm for T = 5000 K while it is near 3 µm for T = 1000 K, and near 9 µm for T = 300 K. The law of Stefan-Boltzmann describes the integrated radiation energy (integrated over all frequencies). This quantity rises with the fourth power of the temperature Z ∞ w(T ) = w(ν, T ) dν ∝ T 4 . (9.4) ν=0 53 9.2. ENERGY AND MOMENTUM 9.2 Energy and momentum An electromagnetic wave leaving an antenna obviously carries away energy. The radiation field has the energy density 1 0 E 2 + c2 B 2 = 0 E 2 . 2 wem = (9.5) This energy density propagates with the speed of light in direction of the wave vector ~k. The energy which passes through a plane, perpendicular to ~k per second is called intensity or energy flux density I = c wem = c 0 E 2 . (9.6) For a linearly polarized wave E(z = 0, t) = E0 sin(ωt) the intensity is I(t) = c 0 E02 sin2 (ωt) (9.7) appears periodic. The temporal mean is hIit = 1 c 0 E02 . 2 (9.8) In vacuum the intensity of an electromagnetic wave is equal to the magnitude of Poynting’s vector ~ = c2 0 E ~ ×B ~ . S (9.9) An electromagnetic wave not only carries energy, but also momentum. The momentum per unit volume is πem = I wem = 2. c c (9.10) The units of πem are [πem ] → kg m/s energy s momentum = = m3 m m3 m3 The units of wem are equivalent to that of pressure 2 [wem ] → energy kg m/s force = = = pressure 3 2 m m m2 The pressure which is exerted by radiation on a mirror can be measured with a precise balance (Lebedev in Moskow 1901). At full daylight it corresponds to a gravitational force of 0.5 mg on a surface of 1 m2 . This radiation pressure is the origin for the comet’s tails (pointing away from the sun, not along the path of the comet). Efforts are underway to use the radiation pressure in a solar sail for cheap interstellar travel.2 The radiation pressure reduced to the action of a single photon corresponds to the momentum carried by the photon, h̄k. Proper and controlled transfer of photon momentum to atoms is the basis for laser cooling of atoms. 2 http://solarsails.jpl.nasa.gov http://www.planetary.org/solarsail/ 54 CHAPTER 9. SPECTRUM OF ELECTROMAGNETIC WAVES 9.3 Photons The nature of light was origin for controversy over many years. Newton postulated the concept of light particles, Huygen pointed out that waves were required to explain interference. After the discovery of electromagnetic waves by Hertz the wave picture appeared to have won for good. But a problem appeared in that no sound explanation of the spectrum of thermal black-body radiation was possible on the basis of the wave picture. Planck’s formula and experimental findings such as the photo-effect led Einstein to the conclusion that the energy contained in light comes in small packages, called quanta or photons. The two concepts (waves and particles) are not in contradiction, rather there are different phenomena in which light can appear, as wave (interference) and in particle form (photo-effect, Taylorexperiment). Note that this duality is also true for material particles (particles carrying mass). Photons are the messengers of the electromagnetic interaction. Photons have zero mass, they propagate in vacuum with the speed of light c0 = 3 · 108 m/s and they transport the energy W = hν (9.11) They carry the momentum p = hν/c = h̄k. Momentum transfer by photons follows along the wave vector ~k, p~ = h̄~k (9.12) In the photon picture the intensity of an electromagnetic wave is given by the number of photons N which pass through the unit of area per second I = N hν (9.13) In the wave picture the intensity is I = c0 E 2 hence the dependence of the electric field strength on photon number is r hν N E= c0 (9.14) (9.15) The classical description of light is appropriate in the limit of large photon numbers. The experiment of Taylor clearly shows the photon structure of light : A spherical wave is emitted from the point source at the center of a sphere. We have a number of detectors, sitting on the surface of the sphere with radius R. At high intensity of the source, all detectors receive the same flux of energy. However, when decreasing the intensity the statistical feature appears. The detectors randomly receive photons. In a similar fashion double-slit interference builds up after many single photon impacts. Chapter 10 Atoms and Light In a first approximation, any atom or any molecule can be treated as a two level system, a quantum system with two realizations, an excited state |ei and a ground state |gi. The two states are separated in energy by ∆E = h̄ω0 . We call ω0 the eigenfrequency (resonance frequency) of the system. If the parity of the state |ei is opposite to the parity of the state |gi, electromagnetic dipole radiation can emerge in a transition from |ei → |gi. Likewise, electromagnetic dipole radiation can be absorbed by the atom in a transition going from |gi → |ei. 10.1 Transition dipole moment The different parity of the two states implies that the expectation value of the dipole operator in the transition matrix element connecting the two states, µeg = hg| qz |ei , (10.1) is not zero, as the dipole operator qz has odd parity. We say the dipole operator qz connects the two states, a transition between them can be enforced by electric dipole radiation. This transition may happen in absorption or emission. Examples for electric dipole transitions between |gi and |ei are: Electronic excitation. An example is H(1s) → H(2p). Here ∆E = R(1 − 1/4), corresponding to ω0 = 2π × 3 · 1014 Hz, a transition in the ultraviolet.1 For the state vector of the electronic ground state we write R1s Y00 where Rn ` (r) is the radial wavefunction for hydrogenic states, which only depend on r, the separation between the proton and the electron, and the Y`m (θ, φ) are spherical harmonics which only depend on the angles θ and φ of the vector ~r. Likewise for the electronically excited state we write R2p Y1m . If we pick the case where the excited state has n = 2, ` = 1, and m = 0, the transition matrix element (10.1) integrates out to be √ hg| qz |ei = q × 128 2/243 a0 = q × 0.74 a0 (10.2) where a0 signifies the length in units of one Bohr, (a0 = 0.5 × 10−10 m). The charge involved is the elementary charge of the electron (it changes its average position) 1 R is the Rydberg constant, 13.6 eV. 55 56 and hence we may say that this transition is associated with a dipole moment of 8 × 10−30 C·m. The squared magnitude of this dipole moment is connected to the probability that a transition will occur. The figure below (left) shows the radial wavefunctions for the 1s and 2p states, together with the effective potentials. The center and right figures gives harmonic oscillator wavefunctions and the probability distribution in the lowest HO-states. r R and energy 2p 1s 0 5 10 15 r Vibrational excitation. Example of the HF molecule, HF(n = 0) → HF(n = 1). Here n gives the vibrational quantum number and ∆E = h̄ω0 , where ω0 is the vibrational frequency of HF (about 2π × 1014 Hz), a transition in the infrared (λ = 3 µm). This molecule has a permanent dipole moment. This dipole moment changes when the distance between the two atoms changes. If we set the equilibrium distance to be z0 , a first approximation for the dependence of the dipole moment on separation between the H atom and the F atom is p(z) = p0 1 + α(z0 − z) (10.3) where α ≈ 10+10 m−1 and p0 = 2 × 10−29 C·m. We approximate the binding potential between the two atoms as harmonic oscillator with the vibrational wave functions ϕn (z), the expression hϕ0 | p(z) |ϕ1 i (10.4) describes the transition moment associated with a vibrational transition. The term which contributes to the integral is the term depending on z in (10.3). Due to the orthogonality of HO functions the contribution hϕ0 | p0 |ϕ1 i from Eq. (10.3) √ vanishes. Expressed in HO-length units we have2 hϕ0 | z |ϕ1 i = 1/ 2 . The dipole moment which is associated with a transition between (n = 0) ↔ (n = 1) is √ p0 hϕ0 | αz |ϕ1 i = 2 × 10−30 C · m . (10.5) A harmonic oscillator transition in which the vibrational quantum number changes by one is associated with a change in dipole moment, provided the molecule has an intrinsic dipole moment which depends on internuclear distance. Molecules without intrinsic dipole moment are barred from optically allowed vibrational transitions. Lattice vibrations in a crystal. The argument made above for the diatomic molecule can also be made for more complex molecules. It can also be made for lattice vibrations (phonons) in a crystal. If the motion of crystal atoms is related to a change in dipole moment, phonons (optical phonons) give rise to emission and absorption of electromagnetic radiation. 2 For HF the HO-length is about 10−11 m. 57 10.2 Quantum jumps We see from the examples above that a quantum jump (that is a transition |gi ↔ |ei) in an atom or a molecule can be associated with a change in electrical dipole moment. Such a quantum jump happens instantly and gives rise to emission or absorption of a photon of the energy E = h̄ω0 . If many such quantum jumps occur, many such photons are generated. If we observe light, at a level of P = 1 mW emerging from a box containing 1 atmosphere of hydrogen atoms and of the size of 1 cm3 , we may argue that we see 10−3 P = 6 × 1014 = h̄ω0 (3/4)13.6 × 1.6 · 10−19 photons per second (10.6) emerging from the box. As there are about 2 × 1019 atoms in the box we may say that, on average, every one of the hydrogen atoms in the box makes 3 quantum jumps in a time period of 1000 seconds! Or equivalently we may say: one out of 333 atoms in the box makes one jump per second. The word quantum jump refers to the fact that no precise time can be given at which such a transition might occur. Rather there is a probability for making this transition. If the atom is left all by itself in the excited state, it will to this transition according to its so called natural lifetime, τ , and the probabilty to find the atom in the excited state after some time t is taken as ∝ e−t/τ . If we have a sample of Ne atoms in the excited state at time t = 0 we will find that after a time t, the number of atoms in the excited state which is still left is Ne e−t/τ . (10.7) Equally we might say that the rate at which excited atoms decay is given by dNe 1 = − Ne = −A Ne , dt τ (10.8) where A, the so called Einstein A-coefficient, describes spontaneous emission. The time when energy is absorbed from the radiation field and when the atom makes a transition from the ground state |gi to the excited state |ei can also not be specified precisely. Only a probability argument can be made that a quantum jump occurs. Obviously this probability depends on the amount of radiation present at the correct frequency. In the presence of radiation density w(ν) [J·s/m3 ], the ground state atoms make a transition to the excited state at the rate dNg = −Beg w(ν) Ng , dt (10.9) where Beg describes stimulated absorption, the so called Einstein B-coefficient. In order to explain the shape of the black-body spectrum and its dependence on temperature, Einstein found that a third process had to be invoked. This process is called stimulated emission and it may be viewed as follows. If an atom is in the excited state and radiation is present at the correct frequency, then the atom can be stimulated to make a transition back to the ground state, the rate being dNg = +Bge w(ν) Ne . dt (10.10) In this process the energy contained as internal energy of the atom (E = h̄ω0 ) is added to the external field. 58 The coefficients, A and Beg = Bge are related to the transition dipole moment3 , h J · s m3 i 8πh̄ν 3 −1 B with dimension : [s ] = (10.11) A= eg c3 m3 Js2 where (see Eq. 10.1) h i 2 Beg = Bge ∝ µeg with dimension : m3 J−1 s2 (10.12) How do we think of a photon in this framework : As long as the atom is still in the excited state, the photon has not yet emerged ! This rather unclear situation develops all by itself and irreversibly into the final state, when the atom is in the ground state and and the photon is released (this statement is correct with very high probability, provided t τ ). In the mean time, when the photon emission has not yet occurred, the atom is neither in the excited state nor in the ground state, both are simultaneously allowed ! Conclusion: A photon is a spatially extended object, similar to a classical dipole wave, it extends in the direction of propagation over a length of c·τ . It is this large size which permits the interference properties known for light. A photon can interfere with itself (and only with itself). However: the energy of the photon is NOT distributed over this large size. In any measurement the photon energy is found in its entirety localized at the detector. Natural linewidth The absorption profile of a stationary atom is given by the Lorentz-curve L(ω) = L(ω0 ) Γ2 Γ2 + 4(ω − ω0 )2 (10.13) where ω = 2πν and L(ω0 ) is the amplitude at the central frequency, ω0 . This profile reflects the finite lifetime of the excited state, τ , where 1/τ = Γ is related to the Einstein A coefficient for spontaneous emission. The full-width-at-half-maximum (FWHM) of this curve is given by ∆ω = 2π∆ν = Γ. If we introduce the detuning δ = ω − ω0 and normalize the curve to peak height equal one, we obtain L(δ) = 1 2 1 + (2δ/Γ) . In general the linewidth observed in an experiment is always larger than the natural linewidth. Reasons are a variety of broadening mechanisms. For one the motion of atoms introduces a Doppler effect leading to Doppler broadening which can be substantially larger than the natural linewidth. In this case the absorption line profile of the atom appears in a Gaussian distribution, the width being proportional to the temperature of the atoms. 3 For simplicity I have omitted statistical weight factors which often are used in this expression, but which are of no consequence for the picture developed here 59 10.3 Classical rate-equation model We first consider the situation of thermodynamic equilibrium where N individual atoms are inside a box at the temperature T . In this case the number of atoms in the excited state, Ne and the number of atoms in the ground state Ng are stationary, dNe dNg = =0 dt dt (10.14) and we have N = Ng + Ne . Hence we may write for the detailed balance dNg = +Ne w(ν)Bge − Ng w(ν)Beg + Ne A = 0 . dt (10.15) Using the expression of A from (10.11) we may solve this expression for w(ν). w(ν) = Ne 1 8πh̄ν 3 8πh̄ν 3 = . 3 c Ng − Ne c3 Ng /Ne − 1 (10.16) In thermal equilibrium the population in the ground and excited state is distributed according to the Boltzmann factor Ne = Ng e−∆E/kT . (10.17) Inserting this condition of thermal equilibrium into (10.16) we obtain Planck’s law for radiation with ∆E = hν w(ν, T ) = 8πh̄ν 3 1 . 3 hν/kT c e −1 (10.18) We see that the detailed balance of atoms, which interact with the thermal background radiation field, requires three processes to be taken into account4 • spontaneous emission • stimulated emission • stimulated absorption These three processes are required for any kind of matter interacting with the electromagnetic radiation field, at all frequencies, and also for radiation fields which are not in thermal equilibrium as we discuss next. 10.4 Interaction with a beam of light We consider a beam of light with the spectral intensity I(ν) = 4πcw(ν) propagating 2 along the z-axis. spectrally integrated R ∞The dimension of I(ν) is [W·s/m ]. The intensity, I = 0 4πcw(ν) dν, has the dimension [W/m2 ]. When such a beam travels through a medium containing atoms in the ground and excited states, it may suffer absorption or amplification. If the medium is homogeneous and thin, and if the light level is low, we may argue that, after travelling over a small distance ∆z, the beam intensity has changed in a linear way ∆I(z) = I(z + ∆z) − I(z) . (10.19) 4 This was first recognized by Einstein, thus the naming of the three coefficients A, Beg , and Beg as Einstein coefficients. 60 Introducing a coefficient of absorption α we may define 5 ∆I(z) = −α I(z) ∆z , (10.20) and write for the differential expression dI(z) = −α I(z) . dz (10.21) After integration (α is a constant) we obtain Beer’s law of absorption I(z) = I(0) e−αz . (10.22) We consider this case for a sample of thermal atoms. Thermal implies thermodynamic equilibrium, that is there are fewer atoms in the excited state than in the ground state. In this case the change in beam intensity is given as I(z) dI(z) = + ne − ng Beg h̄ω0 , dz 4πc (10.23) where ne and ng are now densities of atoms homogeneously distributed across the profile of the light beam and h̄ω0 = ∆E = Ee − Eg . By comparison with (10.20) we can write for the absorption coefficient B eg h̄ω0 with dimension : [m−1 ] . (10.24) α = ng − ne 4πc We see that the beam is attenuated when ng > ne , which is the thermal case. However when a situation is created such that ng < ne , the change in beam intensity can be positive and we may define a small-signal gain coefficient, β = −α, which describes an exponential growth of beam intensity, I(z) = I(0) eβz . A situation where β is positive is the key to laser operation. 5 A positive sign of α implies loss of radiation. (10.25) Chapter 11 Application : LASER A requirement for achieving laser oscillation (Light Amplification through Stimulated Emission of Radiation) is the presence of small signal gain, β > 0, B eg . (11.1) β = ne − ng 4πc which only appears for population inversion ne > ng . This is typically a manmade situation, brought about by prudent optical pumping of population into the excited state. Different lasers differ by their method of generating this population inversion, but also other criteria are of issue such as continuous operation (CW1 ) or pulsed operation, the bandwidth, the brightness, and the beam shape of the electromagnetic radiation field and the form of the gain medium, whether it is gaseous, liquid or solid. Further criteria concern the quantum yield and overall efficiency of the laser. All lasers require a gain medium, sometimes referred to as the active medium, an efficient pumping process with which inversion of population is achieved, and nearly always an optical feedback mechanism in the form of an optical resonator which also defines the spatial profile of the laser-light beam produced. Creating population inversion A classification of lasers is frequently made by counting the number of energy levels which are actively used in the lasing cycle. Two-level schemes for laser operationare only possible if a spatial separation of excited and ground state atoms is possible, the classical examples being the ammonia maser and the hydrogen maser. Typically three-level and four-level schemes find use in lasing at optical frequencies. In a three level scheme a pump mechanism transfers ground state atoms to an excited state at energy E3 . If this state rapidly decays into the state at energy E2 , and if this state is sufficiently long lived (compared to the rate at which the ground state level, at energy E1 is pumped to E3 ) a population inversion between E2 and E1 can appear. This lasing action can continue until the population inversion disappears. Such lasers typically operate pulsed. CW lasing is achieved with four levels. The ground state at energy E0 is pumped to the level at E3 which decays rapidly into the state at energy E2 . Optical transitions from this level end in a state at energy E1 , this will be the lasing transition. If one can achieve the 1 The abbreviation CW stands for continuous wave. 61 62 situation that the state at E1 rapidly decays to the ground state a continuous population inversion between E2 and E1 may be sustained, enabling CW-operation. E3 E3 fast decay fast decay E2 E2 Laser transition pump Laser transition pump E1 fast decay n E1 n E0 Optical resonator For many lasers an optical resonator is essential as the small signal gain per unit length is small and light has to be sent through the gain medium repeatedly to pick up sufficient amplification. An optical resonator is built from high-quality mirrors, forming a cavity for electromagnetic radiation. This setup is analogous to that of a Fabry-Perot interferometer. A laser beam exit is provided at one end of the resonator by using a mirror which is only partially reflective. This allows both, the recycling of light in the cavity and the coupling-out of a useful amount of laser radiation from the cavity. 11.1 Interference of multiple waves We consider the interference of M√waves, um , each wave with equal amplitude, I0 , and of equal phase difference. To calculate the total intensity we add individual amplitudes p um = I0 ei(m−1)ϕ (11.2) where m = 1, 2, ...M . With the abbreviation h = eiϕ the sum of amplitudes becomes X m um = p p 1 − hM I0 1 + h + h2 + . . . = I0 1−h from which we obtain the total intensity (square of sum of amplitudes) I = I0 sin2 M ϕ/2 sin2 ϕ/2 63 This occurs for example in Bragg reflection where M parallel planes of atoms each reflect a tiny portion of the primary radiation, such that the intensity which is scattered in each plane is practically constant. The phase shift, ϕ, is controlled by the distance between the Bragg planes, d, together with the angle of incidence. Interference of waves of progressively smaller intensity Now we consider the case that an infinite number of waves of constant phase difference interfere, each with a slightly smaller amplitude. We have h = r eiϕ where r < 1. The sum gives X um = m p I0 p 1 1 = I0 1−h 1 − r eiϕ and for the total intensity I= I0 = I0 S(ν) . 1 + (2F/π)2 sin2 ϕ/2 √ where the finesse F is defined as F = π r/(1 − r). This situation applies to the Fabry-Perot interferometer or to a laser resonator (laser cavity) in which case r is the reflectivity of the mirrors which form the cavity. The reflectivity of the mirror is in the range 0 ≤ r ≤ 1. The higher the reflectivity, the higher is the finesse, the more restrictive is the frequency range in which waves can enter/exit the resonator. Note that ϕ is related to the number of half-wavelengths which (almost) fit inside the resonator. If we denote by L the length of the resonator we have L = nλn /2+ϕ. A perfect resonator (r = 1) supports the modes λn = 2L/n and neighboring modes are spaced by νn − νn−1 = ∆ν = c/(2L). ∆ν is called the cavity mode spacing. Transmission SHΝL 1.0 0.8 r=0.2 0.6 0.4 0.5 0.2 0.7 0.0 0.9 Ν Ν+DΝ Ν+2DΝ Ν+3DΝ Ν+4DΝ Anti-reflection coatings. The reflectivity of any optical substrate can be controlled by appropriate thin-film coatings. Typically multiple layers of thin coatings of materials of different refractive index are used as shown in the graph below. We consider the case that the substrate has a refractive index n3 and a single dielectric coating, of thickness d, with refractive index n2 on top of it. A beam of light incident on the thin layer will partially be reflected by it, while part of the beam will be reflected at the interface between substrate and coating. The optical path difference between the two reflected waves is ∆s = 2n2 d and destructive interference can diminish the reflection at both interfaces when : 64 ∆ϕ = 2π ∆s = (2m + 1)π λ hence n2 d = λ 4 Nearly full suppression of reflection can be achieved if one manages to make the amplitudes of the two reflected beams identical. Here one also has to consider that infinitely many reflected beams emerge from the interface between substrate and the thin coating. The optimum √ suppression is reached if n2 = n1 n3 . For a glass substrate, n3 = 1.5 and the transition from air, n1 = 1, the optimal refractive index is found to be n2 = 1.225 . 11.2 Cavity modes and gain We now consider an active medium inside a laser resonator with cavity mode spacing ∆ν = c/(2L). We assume that the medium is pumped, such that inversion of population exists and we consider a beam of light trapped between the cavity mirrors. For simplicity we assume that the length of the active medium equals that of the laser cavity, L. One of the mirrors has the reflectivity r = 1, the other (called the output coupler) has r < 1. If the medium has a small signal gain β, a beam of light inside the resonator may experience amplification. Scattering and other losses will also be present, these we denote by a loss coefficient γ. In one round trip through the cavity the beam visits the output coupler once. There it looses the fraction 1 − r. The gain over one round trip, G, may be written as G = r exp 2(β − γ)L . If G can be made larger than one, G > 1, the threshold for laser oscillation is reached. The frequencies which appear as output from the laser are determined by the interplay between the frequency dependent gain of the active medium, β(ν), and the electromagnetic spectrum which is supported by the optical cavity, S(ν), see page 63 and Eq. (11.1). In the figure to the right we show S(ν) for the case r = 0.95. The frequency dependent gain, β(ν), depends on the spectral width of the optical transition. Typically this width is larger than the natural linewidth. We assume here that the spectral profile is Gaussian, β(ν) = β0 e−(ν−ν0 )/(2Γ) . Folding this profile with the cavity mode profile tells us which modes are supported for laser action, those for which G(ν) > 1. Chapter 12 Application : X-ray imaging X-rays are electromagnetic waves with a wavelength in the range from 0.01 to 10 nm, corresponding to frequencies in the range ≈ 1016 − 1019 Hz, and energies in the range 100 eV to 100 keV. X-rays are shorter in wavelength than UV but longer than gamma rays. Sometimes called Röntgen radiation, after Wilhelm Conrad Röntgen, who discovered it. Soft X-rays (100 eV to 10 keV) and hard X-rays (10 keV-100 keV) differ in penetrating ability. (Soft X-rays hardly penetrate matter at all; the attenuation length of 600 eV X-rays in water is less than 1 micrometer.) X-rays are emitted by electrons outside the nucleus, while gamma rays are emitted by the nucleus. Common use of X-rays: take images of the inside of objects in diagnostic radiography and crystallography. 1.2. X-ray Imaging Hertz Kirchhoff Röntgen 12.1 Units of Measure and Exposure X-ray frequency : f = c/λ, photon energy : E = h f . Exposure : a measure of the ionizing ability of X-rays. - Coulomb per kilogram (C/kg) is the unit of ionizing radiation exposure. The amount of radiation required to create one Coulomb of charge of each polarity in one kilogram of matter.1 The effect of ionizing radiation on living tissue is more 1 The roentgen (R) is a traditional unit of exposure, which represented the amount of radiation required to create one electrostatic unit of charge of each polarity in one cubic centimeter of dry air. 1 roentgen = 2.58 × 104 C/kg. 65 66 closely related to the amount of energy deposited rather than charge generated. Absorbed dose: a measure of energy absorbed. - The gray (Gy), which has units of (joules/kilogram). Equivalent dose : a measure of the biological effect of radiation on human tissue. For X-rays it is equal to the absorbed dose. - The sievert (Sv) is the unit of equivalent dose, which for X-rays is numerically equal to the gray (Gy). Medical X-rays are a significant source of man-made radiation exposure. Most exposure is natural (≈80%), medical X-rays account for <20% (typical American). 12.2 Sources / detectors of X-Rays X-ray fluorescence : If the electron has enough energy it can knock an orbital electron out of the inner electron shell of a metal atom, and as a result electrons from higher energy levels then fill up the vacancy and X-ray photons are emitted. This process produces an emission spectrum of X-rays at a few discrete frequencies. The spectral lines generated depend on the target (anode) element used and thus are called characteristic lines. Usually these are transitions from upper shells into K shell (called K lines), into L shell (called L lines) and so on. Bremsstrahlung : This is radiation given off by the electrons as they are scattered by the strong electric field near the high-Z (proton number) nuclei. These X-rays have a continuous spectrum. The intensity of the X-rays increases linearly with decreasing frequency, from zero at the energy of the incident electrons, the voltage on the X-ray tube. Detectors : Photographic film (crystals of AgBr in a gelatin film, turned into small crystals of Ag by radiation), phosphor screens, Geiger counter, scintillators, semiconducting detector (the sensor layer is a semiconductor, such as GaAs or CdTe in which the incident radiation generates an electron/hole cloud.) 12.3 Contrast and Computed Tomography Beer’s attenuation law in practice I(z) = I0 e−(µ+α)z (12.1) The attenuation coefficient, α, generally increases with atomic weight. Contrast agents help to mark structures and functions, for example bariumsulphate for gastro-intestinal tract or iodine containing media. Scattering coefficient µ. CT (computed tomography) uses X-ray equipment to produce three-dimensional representations of body or industrial components by computed reconstruction. The mathematical tool for tomographic imaging was laid down by Johann Radon to obtain cross-sectional images of patients. In X-ray CT, the line integral represents the total attenuation of the beam of X-rays as it travels in a straight line through the object, the signal corresponds to an integrated attenuation coefficient. To obtain a 2D image µ(x, y), this scan is repeated for various angles, thereby producing a volume of data that can be manipulated through inverse Radon transfer, to visualize the various bodily structures based on their ability to block 67 the X-ray beam. To explain how CT works we attempt the reconstruction of an image of an unknown object viewed from 2 perpendicular angles. Here we only record the outline of the objects projected along the two perpendicular axes: We see that a mere recording of the outline of the object does not give us conclusive information as to the real 3D shape of the object. However if we record the integrated density of the object the situation is better. The bottom traces of the figure to the right shows profiles of integrated density in plane z = 0, the center of the vertical axis. We see that the integrated density contains more information about the actual shape of the object. This is exploited in CT where one records the integrated density under many angles, ranging from −π/2 to π/2. Then a mathematical proceedure called Inverse Radon transformation can be used to estimate the actual shape of the object. A simple example is shown below where we view a solid disk with a square hole inside. By rotating the object by an angle θ one obtains the sequence of integrated intensity shown in the center column. The Radon transform is shown at the right. 68 1.2. X-ray Imaging CT scans CT: computed tomography Figure 1: Parallel beam geometry. Each projection is made up of the set of line integrals through the object. The projection of an object at a given angle θ is made up of a set of line integrals. I = I0 exp ⇣ p(r, ✓) = ln(I/I0 ) = http://en.wikipedia.org/wiki/X-ray_computed_tomography Z ⌘ µ(x, y) ds Z µ(x, y) ds Solve for µ(r, ✓) by inverse Radon Transform. Chapter 13 Application : RADAR Microwave generation, range and speed measurements Use of radio- or micro-waves to determine the range, altitude, direction, or speed of objects. Can detect aircraft, ships, motor vehicles, weather formations, and terrain. Developed during World War II, radio detection and ranging. Antenna (radar dish) transmits pulses of electromagnetic waves which bounce off a object. Directing radio waves towards objects is called illumination, regardless of the fact that radio waves are invisible to the human eye. Only a weak signal is reflected back to the antenna. Materials of good electrical conductivity (metals = mirror), but also water and rain drops reflect radar waves. The weak reflected radar signals are captured by the receiving antenna and can be amplified and analyzed. Radar receivers are often at the same location as the transmitter. Certain frequencies are absorbed or scattered well by water vapor, raindrops, or atmospheric gases (oxygen) hence the choice of frequency is important. Radio frequencies are typically less attenuated in air than visible or infrared light. Therefore radar can detect objects at relatively long ranges. Fog, clouds, rain and falling snow block visible light but can be almost transparent to radio waves. Sophisticated methods of signal processing are used in order to analyze the weak return signal. 69 70 13.1 Reflection Boundary conditions of electric and magnetic fields at interfaces, reflection, refraction, diffraction Electromagnetic waves reflect (scatter) whenever a change of the dielectric constant occurs. Hence solid objects in air or any significant change in atomic density between objects will scatter radar waves. This is particularly true for electrically conductive materials, such as metal and carbon fiber, making radar well suited to detect aircraft and ships. If the wavelength is much shorter than the object size, the wave will bounce off in a way similar to light when reflected by a mirror. If the wavelength is much longer than the size of the target, the target may not be visible because of poor reflection. This is described by Rayleigh scattering, an effect that creates the Earth’s blue sky and red sunsets. When the two length scales are comparable, there may be resonances. Modern systems use short wavelength (a few centimeters) to image objects. The most reflective targets for short wavelengths have 90o angles between the reflective surfaces. A structure consisting of three flat surfaces meeting at a single corner, like the corner on a box, will reflect waves entering its opening directly back at the source. These so-called corner reflectors are commonly used as radar reflectors to make difficult-to-detect objects easier to locate (corner reflectors on a boat or buoy). For similar reasons, objects attempting to avoid detection will angle their surfaces in a way to eliminate inside corners and avoid surfaces and edges perpendicular to likely detection directions, which leads to ”odd” looking stealth aircraft. These precautions do not completely eliminate reflection because of diffraction, especially at longer wavelengths. Half wavelength long wires or strips of conducting material, such as chaff, are very reflective but do not direct the scattered energy back toward the source. The extent to which an object reflects or scatters radio waves is called its radar cross section. The power Pr returning to the receiving antenna is given by the equation: Pr = Pe σ 16π 2 R4 (13.1) where Pe is the emitted power, σ is the reflection factor, R is the distance to the object(surface of a sphere with radius R is 4πR2 ). Very small returning power! 13.2 Doppler Radar Our radar sends out a signal with frequency f0 . Due to the Doppler effect an object moving towards or away from the emitter with velocity v, records a frequency f which is given by ( c is the speed of light) v f = 1+ f0 . (13.2) c When the object is moving towards the emitter (v > 0) the frequency received is higher. This higher frequency is reflected back to the emitter station which sees a 71 returned signal with frequency v v v 1+ f0 ≈ 1 + 2 f0 . fret = 1 + c c c The beat frequency of the two signals is v ∆f = fret − f0 = 2 f0 c For f0 = 3 GHz and v = 36 km/h we have ∆f = 200 Hz. 13.3 (13.3) (13.4) Distance Measurement Transit time using pulsed radar: The round-trip time for the radar pulse to get to the target and return is measured. The distance is proportional to this time (factor of two !). Since radio waves travel at the speed of light, accurate distance measurement requires high-performance electronics. Continuous wave (CW) radar with frequency modulation : Another form of distance measuring radar is based on frequency modulation. Frequency comparison between two signals is much more accurate, than timing the signal. By measuring the frequency of the returned signal and comparing that with the original, the difference can be easily measured. Practical example for this are radar altimeters in aircraft : A ”carrier” radar signal is frequency modulated with a sine wave or sawtooth pattern at audio frequencies. The signal is then sent out from one antenna and received on another, typically located on the bottom of the aircraft, and the signal can be continuously compared using a simple beat frequency modulator that produces an audio frequency tone from the returned signal and a portion of the transmitted signal. Since the signal frequency is changing, by the time the signal returns to the aircraft the transmit frequency has changed. The amount of frequency shift is proportional to the distance. 72 Terrestrial radar uses low-power FM signals that cover a large frequency range. The multiple reflections are analyzed mathematically for pattern changes with multiple passes creating a computerized synthetic image. Doppler effects are used which allows slow moving objects to be detected as well as largely eliminating ”noise” from the surfaces of bodies of water. 13.4 Antenna design Radio signals broadcast from a single antenna will spread out in all directions, and likewise a single antenna will receive signals equally from all directions. Modern radar systems use a steerable parabolic ”dish” to create a tight broadcast beam, typically using the same dish as the receiver. Most 2D surveillance radars use a spoiled parabolic antenna with a narrow azimuthal beamwidth and wide vertical beamwidth. This beam configuration allows the radar operator to detect ships or aircraft at a specific azimuth but at an indeterminate height. Frequency bands1 name L-band S-band C-band X-band Ku -band K-band Ka -band Range 1-2 GHz 2-4 GHz 4-8 GHz 8-12 GHz 12-18 GHz 18-27 GHz 27-40 GHz typical use long range traffic control weather radar short range tracking marine radar, missile guidance high resolution mapping, satellite altimetry not used (H2 O-absorption) very high resolution mapping Jamming and Cloaking Attempt to generate radio frequency signals originating from sources outside the radar, but at the radar’s frequency to mask targets of interest (electronic warfare tactic). Since the reflected radar signal is typically weak, jammers can be much less powerful than their jammed radars to effectively mask targets. Radar absorbing material, containing resistive and sometimes magnetic substances, is used on military vehicles to reduce radar reflection. This is the radio equivalent of painting something a dark color so that it cannot be seen through normal means. 1 Source: AIAA, American Institute of Aeronautics and Astronautics. Chapter 14 Application : LIDAR LIDAR (Light Detection And Ranging) is an optical remote sensing technology to measure the distance to, or other properties of a target by illuminating the target with light, often using pulses from a laser. LIDAR may use ultraviolet, visible, or near infrared light to image objects, depending on the target to be detected. This technique can be used with a wide range of targets, including nonmetallic objects, rocks, rain, aerosols, clouds, even specific chemical compounds. A narrow laser beam can be used to map physical features with very high resolution. Downward-looking LIDAR instruments fitted to aircraft and satellites and are used for surveying and mapping. They may also enable autonomous height control of a plane or precision safe landing. Typically light is reflected via backscattering. Different types of scattering are used for different LIDAR applications; most common are Rayleigh scattering, Mie scattering, Raman scattering, Thomson scattering as well as fluorescence. Of course regular reflection is the most efficient may of returning light to the sender. Suitable combinations of wavelengths allow remote mapping of atmospheric contents by looking for wavelength-dependent changes in the intensity of the returned signal. 14.1 Scattering and Flourescence Rayleigh scattering is the elastic scattering of light by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. Rayleigh scattering is a function of the electric polarizability of the particles. The Rayleigh scattering model breaks down when the particle size becomes larger than around 10% of the wavelength of the incident radiation. The strength of Rayleigh scattering varies with the fourth power of the frequency (∝ ω 4 ). Rayleigh scattering of sunlight in the atmosphere causes diffuse sky radiation, which is the reason for the blue color of the sky. Mie scattering Scattering by particles similar to or larger than the wavelength of light is typically treated by the Mie scattering. The Mie solution to Maxwell’s equations describes the scattering of electromagnetic radiation by a sphere or elongated object. The solution takes the form of an analytical infinite series. Mie cross 73 74 section scales as R4 where R is the radius of the object, but saturates when the object is much larger than the frequency. Raman scattering is he inelastic scattering of light associated with vibrational transitions in molecules was first reported in 1922 by C. V. Raman and K. S. Krishnan. Today it is a significant tool for analyzing the composition of liquids, gases, and solids. Stokes and anti-Stokes scattering. Fluorescence Fluorescence is is a form of luminescence, the emission of light by a molecule that has absorbed light of a different wavelength. The emitted light usually has longer wavelength (lower energy). However, when the absorbed electromagnetic radiation is intense, it is possible for one electron to absorb two photons; this two-photon absorption can lead to emission of radiation having a shorter wavelength than the absorbed radiation. The most striking examples of fluorescence occur when the absorbed radiation is in the ultraviolet region of the spectrum, and thus invisible to the human eye, and the emitted light is in the visible region. Fluorescence has many practical applications, fluorescence spectroscopy, fluorescent labelling, dyes, biological detectors, and fluorescent lamps. Degrees of Freedom of a Molecule, Molecular Absorption Spectra Quantum Yield, Lifetime, Photochemistry 14.2 Differential LIDAR cross section In atmospheric physics, LIDAR is used as a remote detection instrument to measure densities of certain constituents of the middle and upper atmosphere, such as ozone. But it can equally be used to monitor humidity or the presence of poisonous gases in the atmosphere. In the DIAL technique, the gas concentration profile is determined by analyzing the lidar backscatter signals for laser wavelengths tuned ’on’ and ’off’ an absorption line of the gas of interest. In this technique, the wavelength separation between the on and off wavelengths is minimized to avoid differences in the atmospheric backscattering at the two wavelengths. rA sA on R2 off wavelength I0 PH R 1L DI PH R R1 2L AL 75 The value of the average gas density ρA (cm−3 ) in the range interval from R1 to R2 can be determined from the ratio of the lidar signals at the on and off wavelengths, where σA = σon − σoff is the difference between the absorption crosssections at the on and off wavelengths, and P (on, Ri ) and P (off, Ri ) are the signal powers received from range Ri at the on and off wavelengths, respectively. P (on, R1 ) P (off, R2 ) 1 ln · (14.1) ρA = (R2 − R1 )2σA P (off, R1 ) P (on, R2 ) The location of this cell (it extends from R1 to R2 ) can be moved in space by choosing the appropriate time of the lidar returns, providing the profile of the gas along the line-of-sight of the DIAL system. The magnitude of backscattering in the above example is primarily controlled by the aerosols and clouds in the path. Their distribution can be obtained at the same time from the return signal at the off wavelength. To remotely measure ozone concentrations, the DIAL technique is used in the Hartley-Huggins band near 300 nm, and due to the nature of the ozone absorption, the nearest appropriate wavelength that can be used as the off wavelength is at the edge of the absorption band at 310 nm. Relevant absorption regions for water vapor are in the IR near 727 or 815 nm. This is a sample of the vertical ozone distribution published in the Australia State of the Environment Report 2001. In general measurements on molecules can be used to calculate local temperatures. With phase shift analysis one may also determine the local wind speed. 14.3 Coherent Detection The detection scheme used in DIAL is typically ”incoherent” or direct energy detection. An alternative is to use ”coherent” detection which allows doppler, or phase sensitive measurements such as outlined in the following. A Laser Doppler Vibrometer can measure the vibration of an object without contact, by employing optical heterodyne detection. By FM-demodulating the difference in frequency between reference and measured light, one can can detect the speed and frequency of an object’s vibration, rendering micro-vibration measurement possible. Surface metrology systems are used to quantify surface properties with high precision (3D surface profilometers). Applications in Physics and Astronomy A worldwide network of observatories uses lidars to measure (with mm precision) the distance to reflectors placed on the moon. The Mars Orbiting Laser Altimeter, used a LIDAR instrument in a Mars-orbiting satellite (NASA Mars Global Surveyor) to produce a precise global topography of the red planet. In a similar way NASA’s Phoenix Lander used LIDAR to detect snow in the atmosphere of Mars. 76 In nuclear fusion research, LIDAR Thomson Scattering is used to determine electron density and temperature profiles of the plasma. (Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, the lowenergy limit of Compton scattering: when the particle kinetic energy and photon frequency do not change significantly. This limit is valid when the photon energy is much than the mass energy, mc2 , of the particle). Chapter 15 Application : Thermography Thermal imaging or infrared thermography uses infrared imaging cameras to detect radiation in the infrared range of the electromagnetic spectrum ( 914 µm). Infrared radiation is emitted by all objects according to the black body radiation law. Thermography makes it possible to see the environment with or without visible illumination. The amount of radiation emitted by an object increases with temperature. Therefore thermography allows the visualization of variations in temperature. When viewed through a thermal imaging camera a warm object stands out well against a cooler background. Thus humans become easily visible against the environment at day or night. The use of thermography has increased dramatically with the development of modern infared CCD cameras. 15.1 Thermal Energy and Balance The spectral energy density of radiation in equilibrium with a black body at the temperature T is given by Planck’s law of radiation w(ν, T ) dν = 500 200 100 frequency H THz L 50 (15.1) 40 30 5900 K HsunL 10000 energy H arb. units L 8πhν 3 1 dν , c3 ehν/kB T − 1 1000 100 10 1 0.1 320 K 0.01 300 K 0.001 0 2 4 6 wavelength H mm L 77 8 10 78 The spectral distribution with frequency changes drastically with temperature. The frequency-integrated result gives the total radiation power contained in the field at all frequencies. It scales with T 4 (Stefan-Boltzmann law). Near room temperature the radiation field is primarily in the infrared region. To obtain thermal images (thermograms) requires a camera that senses infrared radiation. These recordings can be turned into a false-color visual displays of the amount of infrared energy emitted by an object. model false-color visual display with Mathematica, temperature color map A contrast will appear in such a picture whenever amount of the radiation emitted and reflected by adjacent objects differs. Generally it is the emitted energy that one intends to measure (for example warm body against a cold landscape, or an object with nonuniform temperature). Applications : A thermogram may show a fault of an electrical circuit (thinner wires get warmer). A thermal imaging camera may visualize the areas of a house which radiate at higher temperature and therefore remove thermal energy from the house, thereby helping to improve the efficiency of heating and air-conditioning units. Airport personnel used thermography to detect suspected swine flu cases during the 2009 pandemic. Firefighters use thermography to see through smoke, to find persons or to localize the base of a fire. Flash thermography :Faults in a forged steel object can be revealed by the nonuniformity in heat propagation, after rapidly heating one end of the object. In clinical diagnostics thermal changes in human beings can be monitored with thermal imaging (cancer detection). A cold blooded spider is readily seen to have different temperature than the human hand holding it: 15.2 Emissivity The ability of an object to emit energy by radiation is characterized by its emissivity, E. It is the ratio of energy radiated by a particular material to energy radiated by a black body at the same temperature. Any real object in thermal equilibrium has E < 1. The absorption coefficient is defined as the ratio of the absorbed power over the incident power. A black body is defined as having an absorption coefficient A = 1. 79 PHnL Ab HnL AHnL Pb HnL Intensity H Wêm 2 L 4 3 integrated power between l = 2 and 4 mm 2 1 0 260 280 300 320 Temperature H K L 340 If we bring two bodies, the green one and the black one, into a closed room (perfect mirrors on the wall) and the two bodies are in thermal equilibrium, then the power radiated by the black body at the frequency ν, Pb (ν), must be absorbed by the green body and vice versa (otherwise the temperature of one of the two objects would change). Thus we must have Pb (ν)A(ν) = P (ν)Ab (ν) at each frequency. Because the power radiated by an object is proportional to its emissivity, P (ν) ∝ E(ν) we have in thermal equilibrium E(ν) Eb (ν) = = K(ν, T ) Ab (ν) A(ν) (15.2) where the function K depends only on frequency and temperature and is actually related to Eq. 15.1. Each material has a different emissivity, depending on the wavelength and on the state of the surface of the object. E can range from practically zero (completely not-emitting) to a theoretical 1 (completely emitting); The more reflective a material is, the lower its emissivity. Highly reflective bodies absorb little radiation and are therefore bad radiators (low E). An example of a substance with low emissivity (in the visible and IR region) would be silver, which has E = 0.02. An example of a substance with high emissivity is asphalt, with an emissivity near 1, E = 0.98. In general, the duller and blacker a material is, the closer its emissivity is to 1. Here is a map showing the emissivity of the earth in the wavelength region between λ = 2 − 3 µm. Not the scale of emissivity shown ranges from 0.85 to 1. The desert areas appear blue and green, which the snow appears white E ≈ 1. 80 15.3 Infrared Cameras Typical CCD and CMOS sensors1 have spectral sensitivity at wavelengths of visible light. By utilizing their spectral sensitivity in the near-infrared (NIR), and by using a suitable filter to block visible light it is possible to obtain true thermal image of objects at temperatures above 300o C. Specialized thermal imaging cameras sensitive beyond λ ≈ 2 µm use focal plane arrays which use InSb, InGaAs, HgCdTe. These cameras are much more expensive than their visible-spectrum counterparts. Some require refrigerators or liquid nitrogen for cooling. A new development are low-cost, uncooled micro-bolometers, but their resolution is considerably lower than that of optical cameras, only of the order of 320x240 pixels. Night Vision Many animals have better night vision than humans do, the result of one or more differences of their eyes: These differences include having a larger eyeball, a larger lens, a larger optical aperture (the pupils may expand to the physical limit of the eyelids), more rods than cones (or rods exclusively, or specialized rods) in the retina, and a tapetum lucidum (reflector at the back of retina, the reason for the shining eyes of certain animals at night). Enhanced intensity range is achieved for humans through the use of an image intensifier, coupled to a CCD, or a high-sensitivity array of photodetectors. 1 CCD stands for charge-coupled device, CMOS stands for complementary metal-oxidesemiconductor.