9 Electromagnetic Waves 9.1 Waves in One Dimension 9.1.1 The Wave Equation.

9 Electromagnetic Waves 9.1 Waves in One Dimension 9.1.1 The Wave Equation. 9.1.2 Sinusoidal Waves 9.1.3 Boundary Conditions: Reflection and Transmission 9.1.4 Polarization 9.2 Electromagnetic Waves in Vacuum 9.2.1 The Wave Equation for E and B 9.2.2 Monochromatic Plane Waves 9.2.3 Energy and Momentum in Electromagnetic Waves 9.3 Electromagnetic Waves in Matter 9.3.1 Propagation in Linear Media 9.3.2 Reflection and Transmission at Normal Incidence 9.3.3 Reflection and Transmission at Oblique Incidence 9.4 Absorption and Dispersion 9.4.1 Electromagnetic Waves in Conductors, 9.4.2 Reflection at a Conducting Surface 9.4.3 The Frequency Dependence of Permittivity 9.5 Guided Waves 9.5.1 Wave Guides 9.5.2 TE Waves in a Rectangular Wave Guide 9.5.3 The Coaxial Transmission Line Chapter 9: Electromagnetic Waves

Traveling wave 9.1 Waves in One Dimension 9.1.1 The Wave Equation How to describe a wave that propagates with a fixed shape at constant speed mathematically? Let represent the displacement of the string at the point z, at time t. The displacement at point , at the later time , is the same as the displacement a distance to the left (at ), back at time t = 0: Any function that depends only on represents a wave of fixed shape traveling in the direction at constant speed . represent travelling wave at constant speed v. do not represent travelling wave at constant speed v. A and b are constants. Chapter 9: Electromagnetic Waves

The wave equation 9.1 Waves in One Dimension 9.1.1 The Wave Equation Why does a stretched string support wave motion? The net transverse force on the segment A string under tension Assuming these angles are small, the sine can be replaced by the tangent: Newton's second law says is the mass per unit length The wave equation Chapter 9: Electromagnetic Waves

The general solution 9.1 Waves in One Dimension 9.1.1 The Wave Equation The wave equation admits as solutions all functions of the form The wave equation involves the square of , so is another class of solutions. The most general solution to the wave equation is the sum of a wave to the right and a wave to the left: The wave equation is linear: The sum of any two solutions is itself a solution. Chapter 9: Electromagnetic Waves

Formula 9.1 Waves in One Dimension 9.1.2 Sinusoidal Waves Phase constant Amplitude Phase Angular wave number At , the phase is zero; If , the central maximum passes the origin at time ; is the distance by which the central maximum (and therefore the entire wave) is "delayed." The wavelength At any fixed point z, the string vibrates up and down, undergoing one full cycle in a period . The frequency (number of oscillations per unit time) is The angular frequency traveling to the right traveling to the left traveling to the left we could simply switch the sign of k to produce a wave with the same amplitude, phase constant, frequency, and wavelength, traveling in the opposite direction. Chapter 9: Electromagnetic Waves

Euler's formula 9.1 Waves in One Dimension 9.1.2 Sinusoidal Waves Euler's formula denotes the real part of the complex number . The complex wave function The complex amplitude The actual wave function is the real part of The advantage of the complex notation is that exponentials are much easier to manipulate than sines and cosines. Chapter 9: Electromagnetic Waves

Example 9.1 Waves in One Dimension 9.1.2 Sinusoidal Waves Find the amplitude and phase constant of the wave resulted from combing two sinusoidal waves of the same frequency and wave number. Chapter 9: Electromagnetic Waves

Linear combinations of sinusoidal waves 9.1 Waves in One Dimension 9.1.2 Sinusoidal Waves Any wave can be expressed as a linear combination of sinusoidal ones: is a function of . runs through negative values in order to include waves going in both directions. The formula for , in terms of the initial conditions and , can be obtained from the theory of Fourier transforms. Therefore if you know how sinusoidal waves behave, you know in principle how any wave behaves. So from now on we shall confine our attention to sinusoidal waves. Chapter 9: Electromagnetic Waves

Two tied strings 9.1 Waves in One Dimension 9.1.3 Boundary Conditions: Reflection and Transmission Suppose that a string is tied onto a second string. The tension is the same for both stings. The mass per unit length is not the same. The wave velocities and are different. transmitted pulse reflected pulse Incident pulse Knot knot Assume the incident wave is a sinusoidal oscillation that extends all the way back to , and has been doing so for all time. Assume the same goes for and (except , extends to ). The incident wave All parts of the string are oscillating at the same frequency (a frequency determined by the person at , who is shaking the string in the first place). The reflected wave The transmitted wave Chapter 9: Electromagnetic Waves

Boundary conditions 9.1 Waves in One Dimension 9.1.3 Boundary Conditions: Reflection and Transmission With incident and reflected waves of infinite extent traveling on the same piece of string, it's going to be hard to tell them apart. The wavelengths and wave numbers are different. is continuousat : Else there would be a break between the two strings. If the knot itself is of negligible mass, then the derivative of must also be continuous: Otherwise there would be a net force on the knot, and therefore an infinite acceleration. Since the imaginary part of differs from the real part only in the replacement of cosine Chapter 9: Electromagnetic Waves

Phase of reflected waves 9.1 Waves in One Dimension 9.1.3 Boundary Conditions: Reflection and Transmission The second string is heavier than the first The second string is lighter than the first Amplitudes always positives The reflected wave is out of phase by. The reflected wave is inverted. or If the second string is infinitely massive or, if the first string is simply fixed at the end There is no transmitted wave-all of it reflects back. Chapter 9: Electromagnetic Waves

Transverse waves 9.1 Waves in One Dimension 9.1.4 Polarization Wave on a string Transverse waves The displacement is perpendicular to the direction of propagation Longitudinal waves The displacement is along the direction of propagation Slinky compression waves Electromagnetic waves are transverse. For transverse wavesthere are two dimensions perpendicular to any given line of propagation. Transverse waves occur in two independent states of polarization: The polarization vector defines the plane of vibration. Because the waves are transverse, is perpendicular to the direction of propagation: Vertical polarization In terms of the polarization angle , Horizontal polarization Chapter 9: Electromagnetic Waves

The Wave Equation 9.2 Electromagnetic Waves in Vacuum 9.2.1 The Wave Equation for E and B In regions of space where there is no charge or current, Maxwell's equations read In vacuum, each Cartesian component of and satisfies the three-dimensional wave equation, The velocity of light, c. Chapter 9: Electromagnetic Waves

The electromagnetic spectrum 9.2 Electromagnetic Waves in Vacuum 9.2.2 Monochromatic Plane Waves We confine our attention to sinusoidal waves of frequency . Since different frequencies in the visible range correspond to different colors, such waves are called monochromatic. Chapter 9: Electromagnetic Waves

Transverse waves 9.2 Electromagnetic Waves in Vacuum 9.2.2 Monochromatic Plane Waves Suppose that the sinusoidal waves are traveling in the z direction and have no or dependence. These are called plane waves because the fields are uniform over every plane perpendicular to the direction of propagation. Chapter 9: Electromagnetic Waves

Transverse waves 9.2 Electromagnetic Waves in Vacuum 9.2.2 Monochromatic Plane Waves Maxwell's equations impose extra constraints on and Electromagnetic waves are transverse: the electric and magnetic fields are perpendicular to the direction of propagation. E and B are in phase and mutually perpendicular; their (real) amplitudes are related by Chapter 9: Electromagnetic Waves

Example 9.2 Electromagnetic Waves in Vacuum 9.2.2 Monochromatic Plane Waves If points in the x direction, then points in the y direction. Chapter 9: Electromagnetic Waves

Propagation vector 9.2 Electromagnetic Waves in Vacuum 9.2.2 Monochromatic Plane Waves The propagation (or wave) vector, , points in the direction of propagation and its magnitude is the wave number . The scalar product is the generalization of is the polarization vector. Because is transverse. The actual (real) electric and magnetic fields in a monochromatic plane wave with propagation vector and polarization are Chapter 9: Electromagnetic Waves

Poynting vector 9.2 Electromagnetic Waves in Vacuum 9.2.3 Energy and Momentum in Electromagnetic Waves The energy per unit volume stored in electromagnetic fields is So the electric and magnetic contributions are equal. For a monochromatic plane wave As the wave travels, it carries this energy along with it. The energy flux density (energy per unit area, per unit time) transported by the fields is given by the Poynting vector For monochromatic plane waves propagating in the direction Energy density velocity of the wave In a time , a length passes through area , carrying with it an energy . The energy per unit time, per unit area, transported by the wave is therefore . Chapter 9: Electromagnetic Waves

Momentum 9.2 Electromagnetic Waves in Vacuum 9.2.3 Energy and Momentum in Electromagnetic Waves Electromagnetic fields carry momentum. The momentum density stored in the fields is For monochromatic plane waves In the case of light, the wavelength is so short (), and the period so brief (), that any macroscopic measurement will encompass many cycles. denotes the time average of over a complete cycle. We are interested only in the average value. Another way of finding Chapter 9: Electromagnetic Waves

Intensity 9.2 Electromagnetic Waves in Vacuum 9.2.3 Energy and Momentum in Electromagnetic Waves The intensity is the average power per unit area transported by an electromagnetic wave When light falls on a perfect absorber it delivers its momentum to the surface. In a time the momentum transfer is , so the radiation pressure (average force per unit area) is On a perfect reflector the pressure is twice as great, because the momentum switches direction, instead of simply being absorbed. We can account for this pressure qualitatively, as follows: The electric field drives charges in the direction, and the magnetic field then exerts on them a force in the direction. The net force on all the charges in the surface produces the pressure. Chapter 9: Electromagnetic Waves

Maxwell’s equations 9.3 Electromagnetic Waves in Matter 9.3.1 Propagation in Linear Media Inside matter and in regions where there is no free charge or free current, Maxwell's equations If the medium is linear , In homogeneous medium and do not vary from point to point differ from the vacuum analogs only in the replacement of by Electromagnetic waves propagate through a linear homogeneous medium at a speed is the index of refraction of the material For most materials, is very close to , so where is the dielectric constant Since is almost always greater than 1, light travels more slowly through matter. Chapter 9: Electromagnetic Waves

Boundary conditions 9.3 Electromagnetic Waves in Matter 9.3.1 Propagation in Linear Media All of our previous results carry over, with the simple transcription The energy density The Poynting vector Boundary conditions When a wave passes from one transparent medium into another For monochromatic plane waves Chapter 9: Electromagnetic Waves

Boundary conditions 9.3 Electromagnetic Waves in Matter 9.3.2 Reflection and Transmission at Normal Incidence Suppose the plane forms the boundary between two linear media. A plane wave, traveling in the z direction and polarized in the x direction, approaches the interface from the left It gives rise to a reflected wave travels back to the left in medium and a transmitted wave Chapter 9: Electromagnetic Waves

Reflection and transmission amplitudes 9.3 Electromagnetic Waves in Matter 9.3.2 Reflection and Transmission at Normal Incidence If the permittivitiesare close to their values in vacuum (for most media), Identical to the ones for waves on a string Chapter 9: Electromagnetic Waves

Reflection and transmission coefficients 9.3 Electromagnetic Waves in Matter 9.3.2 Reflection and Transmission at Normal Incidence What fraction of the incident energy is reflected, and what fraction is transmitted? The intensity (average power per unit area) is If , then the ratio of the reflected intensity to the incident intensity is is the reflection coefficient the ratio of the transmitted intensity to the incident intensity is is the transmission coefficient When light passes from air into glass, R = 0.04 T = 0.96 Conservation of energy, of course, requires Chapter 9: Electromagnetic Waves

Wave numbers 9.3 Electromagnetic Waves in Matter 9.3.3 Reflection and Transmission at Oblique Incidence Suppose a monochromatic plane wave approaches from the left giving rise to a reflected wave All three waves have the same frequency that is determined once and for all at the source and a transmitted wave The three wave numbers are related Chapter 9: Electromagnetic Waves

At the boundary, phases are the same 9.3 Electromagnetic Waves in Matter 9.3.3 Reflection and Transmission at Oblique Incidence The combined fields in medium (1), and , must be joined to the fields and in medium (2), using the boundary conditions. These all share the generic structure notice is that the x, y, and t dependence is confined to the exponents. Because the boundary conditions must hold at all points on the plane, and for all times, these exponential factors must be equal. This is another way to show that the transmitted and reflected frequencies must match the incident one When When for all and all . This can only hold if the components are separately equal, for if , for if , Chapter 9: Electromagnetic Waves

Fundamental laws of geometrical optics 9.3 Electromagnetic Waves in Matter 9.3.3 Reflection and Transmission at Oblique Incidence We may orient our axes so that lies in the plane. Thus Hence and must lie in the plane. First Law: The incident, reflected, and transmitted wave vectors form a plane (called the plane of incidence), which also includes the normal to the surface(here, the z axis). the angle of incidence the angle of reflection the angle of transmission, or the angle of refraction Second Law: The angle of incidence is equal to the angle of reflection, The law of reflection. All of them measured with respect to the normal The law of refraction, or Snell's law. Third Law: All we used was “the phases are same at the boundary”. Therefore, any other waves (water waves or sound waves) can be expected to obey the same "optical“ laws when they pass from one medium into another. Chapter 9: Electromagnetic Waves

polarization parallel to the plane 9.3 Electromagnetic Waves in Matter 9.3.3 Reflection and Transmission at Oblique Incidence 2 1 Suppose that the polarization of the incident wave is parallel to the plane of incidence ( plane) • Boundary conditions represent pairs of equations, one for the x-component and one for the y-component. Chapter 9: Electromagnetic Waves

Fresnel's equations 9.3 Electromagnetic Waves in Matter 9.3.3 Reflection and Transmission at Oblique Incidence 2 1 Fresnel's equations The reflected and the incident waves are either in phase, if , or 180° out of phase, if The polarization of the incident wave is parallel to the plane of incidence The transmitted wave is always in phase with the incident one. Chapter 9: Electromagnetic Waves

Brewster's angle 9.3 Electromagnetic Waves in Matter 9.3.3 Reflection and Transmission at Oblique Incidence Fresnel's equations 2 1 At Brewster's angle, the reflected wave is completely extinguished. Air Glass out of phase Chapter 9: Electromagnetic Waves

Reflection and transmission coefficients 9.3 Electromagnetic Waves in Matter 9.3.3 Reflection and Transmission at Oblique Incidence The power per unit area striking the interface is The average power per unit area of interface, and the interface is at an angle to the wave front. The incident intensity is The reflected intensity is 2 1 The transmitted intensity is The reflection coefficient Air Glass The polarization of the incident wave is parallel to the plane of incidence The transmissioncoefficient Chapter 9: Electromagnetic Waves

Transient behavior of free charge 9.4 Absorption and Dispersion 9.4.1 Electromagnetic Waves in Conductors In the case of conductors is not zero. The continuity equation for free charge Ohm's law, the free current density in a conductor is proportional to the electric field. Using Ohm's law For a homogeneous linear medium Using Gauss's law Any initial free charge density dissipates in a characteristic time . If you put some free charge on a conductor, it will flow out to the edges. For a "perfect" conductor and 0. For a "good" conductor, is much less than the other relevant times in the problem (in oscillatory systems, For a "poor" conductor, is greater than the characteristic times in the problem (). We're not interested in this transient behavior. We'll wait for any accumulated free charge to disappear. From then on . Chapter 9: Electromagnetic Waves

Complex wave number 9.4 Absorption and Dispersion 9.4.1 Electromagnetic Waves in Conductors In the case of conductors is not zero. Ohm's law These equations still admit plane-wave solutions "wave number" is complex Chapter 9: Electromagnetic Waves

Real and imaginary parts of 9.4 Absorption and Dispersion 9.4.1 Electromagnetic Waves in Conductors "wave number" is complex Chapter 9: Electromagnetic Waves

Skin depth 9.4 Absorption and Dispersion 9.4.1 Electromagnetic Waves in Conductors plane-wave solutions "wave number" is complex The skin depth is the distance it takes to reduce the amplitude by a factor of . It is a measure of how far the wave penetrates into the conductor. , the real part of , determines the wavelength, the propagation speed, and the index of refraction. Chapter 9: Electromagnetic Waves

Transverse waves 9.4 Absorption and Dispersion 9.4.1 Electromagnetic Waves in Conductors No new constraints from Maxwell's equations impose extra constraints on and No components: the fields are transverse We may orient our axes so that is polarized along the direction: E and B are mutually perpendicular Chapter 9: Electromagnetic Waves

Magnetic and electric fields are not in phase 9.4 Absorption and Dispersion 9.4.1 Electromagnetic Waves in Conductors The electric and magnetic fields are no longer in phase. The magnetic field lags behind the electric field. Chapter 9: Electromagnetic Waves

Excellent reflectors 9.4 Absorption and Dispersion 9.4.2 Reflection at a Conducting Surface a complex number For a perfect conductor, nonconducting linear medium conductor 2 1 • Boundary conditions Since on both sides Excellent conductors make good mirrors. Since on both sides The skin depth in silver at optical frequencies is on the order of 100. is the free surface charge For ohmicconductors () there can be no free surface current, since this would require an infinite electric field at the boundary. is the free surface current density is a unit vector perpendicular to the surface, pointing from medium (2) into medium (1) Chapter 9: Electromagnetic Waves

Dispersion 9.4 Absorption and Dispersion 9.4.3 The Frequency Dependence of Permittivity The permittivity , the permeability , and the conductivity depends on the frequency of the waves. a typical glass The index of refraction in a transparent mediumn is a function of wavelength A prism bends blue light more sharply than red. This phenomenon is called dispersion. The medium is called dispersive if the speed of a wave depends on its frequency. In a dispersive medium, a wave form that incorporates a range of frequencies will change shape as it propagates. A sharply peaked wave typically flattens out. Each sinusoidal component travels at the ordinary wave (or phase) velocity, The packet as a whole (the "envelope") travels at the group velocity The energy carried by a wave packet in a dispersive medium travels at the group velocity, not the phase velocity. It is ok that at some circumstances the phase velocity becomes greater than c. We will stick to monochromatic waves. Chapter 9: Electromagnetic Waves

Model 9.4 Absorption and Dispersion 9.4.3 The Frequency Dependence of Permittivity We want to account for the frequency dependence of in nonconductors, using a simplified model for the behavior of electrons in dielectrics. The electrons in a nonconductor are bound to specific molecules. Picture each electron as attached to the end of an imaginary spring, with force constant Any binding force can be approximated by a spring force for sufficiently small displacements from equilibrium. is displacement from equilibrium. is the electron's mass. is the natural oscillation frequency . Newton's second law gives There will be some damping force on the electron The driving force due to an electromagnetic wave of frequency , polarized in the direction Our model describes the electron as a damped harmonic oscillator, driven at frequency . is the charge of the electron. is the amplitude of the wave. We assume that the much more massive nuclei remain at rest. Chapter 9: Electromagnetic Waves

Dipole moment 9.4 Absorption and Dispersion 9.4.3 The Frequency Dependence of Permittivity A damped harmonic oscillator, driven at frequency . It is easier to handle if we regard it as the real part of a complex equation: In the steady state, the system oscillates at the driving frequency The dipole moment is the real part of The imaginary term in the denominator means that is out of phase with -laggingbehind by an angle that is very small when and rises to when Chapter 9: Electromagnetic Waves

Complex dielectric constant 9.4 Absorption and Dispersion 9.4.3 The Frequency Dependence of Permittivity Linear medium In general, differently situated electrons within a given molecule experience different natural frequencies and damping coefficients. Let's say there are electrons with frequency and damping in each molecule. If there are N molecules per unit volume, the polarization is given by the real part of is the electric susceptibility This is not a linear medium since is not proportional to because of the difference in phase. The complex polarization is proportional to the complex field is complex susceptibility is the complex permittivity The complex dielectric constant Ordinarily, the imaginary term is negligible; however, when is very close to one of the resonant frequencies () it plays an important role. Chapter 9: Electromagnetic Waves

Absorption coefficient 9.4 Absorption and Dispersion 9.4.3 The Frequency Dependence of Permittivity Linear homogenous In a dispersive medium the wave equation for a given frequency is Non-magnetic It admits plane wave solutions The complex wave number The wave is attenuated , since the damping absorbs energy. Because the intensity is proportional to and hence to , is called the absorption coefficient. Here and have nothing to do with conductivity. They are determined by the parameters of the damped harmonic oscillator. The wave velocity is The index of refraction is Chapter 9: Electromagnetic Waves

Absorption coefficient for gases 9.4 Absorption and Dispersion 9.4.3 The Frequency Dependence of Permittivity For gases, thisis small 1 Most of the time the index of refraction rises gradually with increasing frequency, consistent with our experience from optics. The index of refraction In the neighborhood of a resonance the index of refraction drops sharply. It is called anomalous dispersion. The region of anomalous dispersion coincides with the region of maximum absorption. The absorption coefficient The amplitude of electrons oscillation is relatively large, and hence a large amount of energy is dissipated by the damping mechanism. n runs below 1 above the resonance, suggesting that the wave speed exceeds c. This is ok, since energy does not travel at the wave velocity but rather at the group velocity. Chapter 9: Electromagnetic Waves

Absorption coefficient for gases 9.4 Absorption and Dispersion 9.4.3 The Frequency Dependence of Permittivity The index of refraction Away from the resonances, the damping can be ignored, For most substances the natural frequencies are scattered all over the spectrum. For transparent materials, the nearest significant resonances typically lie in the ultraviolet, In terms of the wavelength in vacuum A is the coefficient of refraction Cauchy's formula B is the coefficient of dispersion It applies reasonably well to most gases, in the optical region. Chapter 9: Electromagnetic Waves

Boundary conditions 9.5 Guided Waves 9.5.1 Wave Guides We consider electromagnetic waves confined to the interior of a hollow pipe, or wave guide. We'll assume the wave guide is a perfect conductor. • Boundary conditions inside the material Boundary conditions at the inner wall Free charges and currents will be induced on the surface in such a way as to enforce these constraints. For monochromatic waves that propagate down the tube: Confined waves are not in general transverse. We need to include longitudinal components and . To avoid cumbersome notation, the subscript 0 and the tilde are left off the individual components. Chapter 9: Electromagnetic Waves

Wave equations 9.5 Guided Waves 9.5.1 Wave Guides v ①② It suffices to determine the longitudinal components and ; if we knew those, we could calculate all the others. Chapter 9: Electromagnetic Waves

No TEM waves in a hollow wave guide 9.5 Guided Waves 9.5.1 Wave Guides TE ("transverse electric") waves. TM ("transverse magnetic") waves. and TEM waves. TEMwaves cannot occur in a hollow wave guide. Proof: Gauss's law ①② Faraday's law Since Laplace's equation admits no local maxima or minima, and since, on the boundary, , this means that throughout. The same argument holds for . Chapter 9: Electromagnetic Waves

9 Electromagnetic Waves 9.1 Waves in One Dimension 9.1.1 The Wave Equation.