Pages: 12

The interaction of light with matter results in reflection, transmission and absorption. To describe these phenomena an effective theory can be formulated by averaging over volumes large in comparison with the molecular scale, but small on a macroscopic scale. A material is described by a volume density of electric and magnetic dipoles, the so-called polarization field https://www.w3.org/1998/Math/MathML" display="inline"> P ( r ,   t ) and magnetization field https://www.w3.org/1998/Math/MathML" display="inline"> M ( r ,   t ) . The properties of the polarization field are derived, and the concept of bound charge is introduced. The first macroscopic Maxwell equation, which relates the electric displacement field https://www.w3.org/1998/Math/MathML" display="inline"> D ( r ,   t ) to the free charge density, is derived. Guided Problems treat the calculation of the average electric field produced within a sphere by an arbitrary charge distribution; investigate the electric field inside and outside a uniformly polarized spherical electret; and solve for the field distribution when a dielectric sphere is placed in a uniform external electric field.

Pages: 17

A time-varying bound charge density results in a bound current density, which acts as a new source for the magnetic field https://www.w3.org/1998/Math/MathML" display="inline"> B ( r ,   t ) . The full set of macroscopic Maxwell equations for a polarizable material follows. General materials may also exhibit a bulk magnetization https://www.w3.org/1998/Math/MathML" display="inline"> M ( r ,   t ) . Spatial variations in the magnetization, either at a surface or in the bulk, act as a new type of bound current density. Its inclusion leads to the most general form of the macroscopic Maxwell equations. The concept of electric and magnetic energy density is introduced, along with the Poynting vector https://www.w3.org/1998/Math/MathML" display="inline"> S ( r ,   t ) , which measures the flow of energy. Conservation of energy is described by Poynting's theorem. Material-dependent constitutive relations are introduced to relate the polarization and magnetization to the electric and magnetic fields. The fields inside and outside a permanent spherical magnet are calculated in a Guided Problem. In a second Problem, the static https://www.w3.org/1998/Math/MathML" display="inline"> B and https://www.w3.org/1998/Math/MathML" display="inline"> H fields are calculated for a magnetizable sphere placed in a uniform external magnetic field.

Pages: 16

The propagation of electromagnetic plane waves in a vacuum is reviewed. While all physical quantities in electromagnetic theory are real-valued, the introduction of complex fields simplifies the mathematics. The relationship of complex fields to the underlying real fields is described. The Fourier integral is introduced as a means to represent quasimonochromatic light and wave packets. The relationship between a wave packet and its Fourier transform is explored for light in a vacuum. The concept of dispersion is introduced, along with the distinction between phase velocity and group velocity for a wave packet traveling in a dispersive medium. Guided Problems treat methods for time-averaging products of complex harmonic functions; determine the exact wave-number distribution for a traveling one-dimensional Gaussian wave packet; and analyze the spreading of a Gaussian wave packet in a dispersive medium.

Pages: 13

Since the time-dependent polarization field in a material does not respond instantaneously to changes in the electric field, the associated constitutive relation is frequency-dependent. The Drude-Lorentz model treats the electrons in a metallic material classically by introducing a phenomenological damping force proportional to the electron velocity. The model leads to a frequency-dependent relative permittivity https://www.w3.org/1998/Math/MathML" display="inline"> ε ( ω ) / ε 0 connecting the complex polarization field and electric field. The relative permittivity has both real and imaginary parts, indicating the fields vibrate out of phase. Numerical results are obtained for the relative permittivity and ac conductivity in a model system. The Drude-Lorentz model is extended to dielectric insulators by modeling bound electrons using Hooke's law force, and once again the relative permittivity is calculated for a model system.

Pages: 26

The effect of a complex relative permittivity https://www.w3.org/1998/Math/MathML" display="inline"> ε ( ω ) / ε 0 in the electromagnetic wave equation is treated in detail. The resulting electromagnetic wave exhibits a complex wave vector, implying a new velocity, as well as the presence of damping due to absorption. The concept of a frequency-dependent skin depth is introduced. The boundary conditions satisfied by https://www.w3.org/1998/Math/MathML" display="inline"> E , https://www.w3.org/1998/Math/MathML" display="inline"> D , https://www.w3.org/1998/Math/MathML" display="inline"> H and https://www.w3.org/1998/Math/MathML" display="inline"> B at the interface between two materials are derived, then used to study reflection and transmission of a plane wave in the special case of normal incidence. The concepts of reflectivity and transmissivity are introduced. The optical properties (reflection, transmission and absorption) of a metal and of a dielectric are analyzed within the Drude-Lorentz model, and numerical results are presented for the same systems treated in Chapter 4. Properties depend on the relationship between the light frequency and the characteristic frequencies of the material, including the plasma frequency. Guided Problems examine plasma oscillations in a metal; determine the form of the damped electromagnetic wave transmitted in a metal for the regime https://www.w3.org/1998/Math/MathML" display="inline"> ω τ ≪ 1 ; apply the Drude-Lorentz model to calculate optical properties of Zn metal; examine the propagation of waves in a metamaterial with a negative refractive index; analyze reflection at a perfectly conducting metal interface by calculating the wave emitted by an oscillating surface current; and investigate interference effects in the flow and absorption of energy at a boundary between materials with complex refractive indices.

Pages: 23

The reflection and transmission of a plane wave at a planar interface is analyzed in detail for an arbitrary angle of incidence and linear polarization. It is assumed absorption can be ignored in both materials, i.e., the indices of refraction are real-valued. The law of reflection and Snell's law follow immediately. The so-called Fresnel equations for the reflection and transmission coefficients are derived, and the flow of energy is analyzed. When light is incident from an optically dense material on an optically sparse material, total internal reflection may result; the wave in the sparse material is then evanescent. Detailed results are obtained for the properties of the reflected and transmitted waves. Guided Problems treat the transmission of light at a dielectric-metal interface, and analyze the so-called Goos-H https://www.w3.org/1998/Math/MathML" display="inline"> a ¨ nchen effect for a narrow beam that undergoes total internal reflection at a boundary between dielectrics.

Pages: 35

The most general polarization for an electromagnetic plane wave is elliptical. At any point in space, the tip of the electric field vector traces out an ellipse in either a clockwise or counterclockwise direction as viewed by an observer looking back toward the source. Special cases are linear and circular polarization. A general elliptically polarized wave is analyzed in detail, and the so-called orientation angle and ellipticity angle are introduced. Three numerical examples are included. The four Stokes parameters for polarized light are defined, then used to construct the Poincar https://www.w3.org/1998/Math/MathML" display="inline"> e ′ sphere. Any two antipodal points on the sphere constitute an orthonormal polarization basis. The Stokes parameters are used to generalize the discussion to include noisy waves that are unpolarized or partially polarized. The polarization of a wave can be manipulated using a polarizer or a wave plate. Polarizers fall into two categories: beam splitters and dichroic materials. Beam-splitting prisms that rely on birefringence are described at length. Wave plates, which are also constructed from birefringent materials, alter the relative phase between component waves with perpendicular polarizations. The effect of polarizers and analyzers can be described using the so-called Jones calculus or Mueller calculus. The Jones calculus applies to fully polarized elliptical waves; the polarization state of the electric field is represented by a two-component vector, and the optical device by a two-by-two matrix. In contrast the Mueller calculus allows the description of partially polarized waves; the polarization state of the intensity is represented by a four-component vector, and the optical device by a four-by-four matrix. The Jones matrices for a number of devices are derived. The method for translating a Jones matrix to a Mueller matrix is described, then used to work on a set of examples. The use of a Fresnel rhomb as a substitute for a quarter-wave plate is examined quantitatively in a Guided Problem. A second Problem analyzes the change in polarization experienced by a linearly polarized plane wave that reflects from a dielectric or metallic surface.

Pages: 12

In the small-wavelength limit, the Maxwell equations and electromagnetic wave equation take on a simplified form. The expansion parameter is https://www.w3.org/1998/Math/MathML" display="inline"> λ 0 / L , where https://www.w3.org/1998/Math/MathML" display="inline"> λ 0 is the vacuum wavelength and https://www.w3.org/1998/Math/MathML" display="inline"> L is a typical macroscopic length. To leading order in the expansion parameter, the wave equation reduces to the so-called eikonal equation, which is the basis for geometric optics. Assuming a wavefront is specified at some initial time, the eikonal equation generates the set of wavefronts at all later times; the process can be formulated geometrically using Huygens' principle. A line that cuts each wavefront along the normal is a geometric ray; in a material with a uniform refractive index, the rays are just straight lines. It is shown that the rays in a general inhomogeneous medium satisfy Fermat's principle. The equation that relates the bending of rays in an inhomogeneous material to variations in the refractive index is derived. A Guided Problem applies Fermat's principle to derive Snell's law of refraction at a planar interface.

Pages: 16

As discussed in the preceding chapter, the leading order approximation for the electromagnetic wave equation in the small-wavelength limit is the eikonal equation. The next-to-leading-order approximation supplements the eikonal equation with the so-called transport equation, which is derived in Appendix B. The transport equation specifies the variation in the electric field magnitude and direction as a light ray is traversed. It is shown that the transport equation guarantees the time-averaged Poynting vector has divergence zero, guaranteeing conservation of energy in a nonabsorptive material. Basic ideas from the differential geometry of surfaces are introduced, including the concept of a principal direction and the corresponding center of curvature. Each point on a surface is characterized by two principal directions, which are mutually perpendicular, each with its own center of curvature. In a homogeneous material, the wavefronts constitute a family of surfaces intersected by straight geometric rays. At the points of intersection with a single ray, all wavefronts share the same two centers of curvature. When a wave recedes from a center of curvature, the successive wavefronts become flatter along the corresponding principal direction; in contrast, when a wave approaches a center of curvature, the successive wavefronts become more sharply curved along the corresponding principal direction. The locus of centers of curvature is a surface of two sheets, the so-called caustic surface. As the area of the infinitesimal patch crossed by a pencil of geometric rays at successive wavefronts changes, the intensity at the center of the patch also changes to ensure conservation of energy. When a wavefront intersects a caustic surface, the intensity at the point of intersection is infinite within the geometric-optics approximation. It is demonstrated that the transport equation directly reproduces the same result for the change in light intensity obtained by tracking a pencil of rays and applying conservation of energy. For cases in which the location of points in a ray pencil can be specified analytically in terms of distance along the central ray https://www.w3.org/1998/Math/MathML" display="inline"> s and two auxiliary parameters https://www.w3.org/1998/Math/MathML" display="inline"> α and https://www.w3.org/1998/Math/MathML" display="inline"> β , it is shown the evolution of the light intensity can be calculated using the Jacobian https://www.w3.org/1998/Math/MathML" display="inline"> J of the vector function that expresses the location of points https://www.w3.org/1998/Math/MathML" display="inline"> ( x ,   y ,   z ) in terms of https://www.w3.org/1998/Math/MathML" display="inline"> ( s ,   α ,   β ) . The evolution equations for the electric and magnetic field unit vectors at points on a geometric ray are derived explicitly in a Guided Problem.

Pages: 28

The geometry of caustic surfaces is explored for general wavefronts in a homogeneous medium and for specific cases that can be worked out analytically. It is shown that when a caustic is nondegenerate (i.e., a true surface, rather than a line or a point), each geometric ray intersects the caustic tangentially. For cases in which a Jacobian https://www.w3.org/1998/Math/MathML" display="inline"> J ( s ,   α ,   β ) can be calculated analytically, the caustic surface can be calculated by solving for the points at which https://www.w3.org/1998/Math/MathML" display="inline"> J vanishes. Reflection caustics are calculated for light that strikes the interior surface of (i) a right cylinder and (ii) a sphere; in both cases, the light is assumed incident from an appropriately positioned source at infinity. As an additional example, the reflection caustic is calculated for light from a source at infinity that strikes the interior surface of a parabolic cylinder at an arbitrary angle. In this case, the caustic is a cylinder formed by translation of the curve called a Tschirnhausen cubic. A method from multivariable calculus allows the determination of the envelope of a family of planar curves. It is demonstrated that this method is equivalent to the Jacobian method for determining the caustic surface for a family of cylindrical wavefronts. Guided Problems provide an example of tracking the evolution of a model wavefront; examine the formation of a reflection caustic by a plane wave incident on a spherical mirror; determine the reflection caustic formed by a spherical wave that originates on the boundary of a spherical mirror; and analyze the caustic of the primary rainbow.

Pages: 50

As shown in the preceding chapter, the reflection caustic formed by light from a source at infinity on the symmetry axis of a paraboloidal mirror is a single point, the so-called focal point of the mirror. The imaging of the source point is said to be stigmatic. It is in fact not possible to create an optical instrument that provides stigmatic and geometrically similar imaging of object points in an extended region; however, it is possible to produce images that are nearly stigmatic and nearly similar. A set of differential equations is derived to find the position of points on a geometric ray that passes through a general optical instrument with cylindrical symmetry about a central axis, the so-called optical axis. The paraxial approximation consists of simplifying the ray equations by retaining only terms linear in transverse displacements and direction cosines. It is shown that the transverse position and direction of a ray at any displacement https://www.w3.org/1998/Math/MathML" display="inline"> z along the optical axis can be obtained using an approach based on two-by-two transfer matrices. The transfer matrix for a spherical refracting surface is obtained within the paraxial approximation, and it is shown that any object point is mapped to an image point. The concepts of image and object focal length are introduced, and an equation relating the object position https://www.w3.org/1998/Math/MathML" display="inline"> z and image position is derived. The magnification is expressed in terms of https://www.w3.org/1998/Math/MathML" display="inline"> z , and the refractive indices of the object and image space. A geometric method for locating the image using principal rays is introduced. The manipulations for a refracting surface are repeated for a spherical mirror and a spherical lens. After the completion of calculations within a Cartesian coordinate system, a set of physically intuitive sign conventions is introduced. A set of numerical examples is worked out for the refracting surface, mirror and lens. In more complex problems, it may be convenient to reduce a sequence of lenses and mirrors to a single element using the process called Gaussian reduction. The Newtonian form of the imaging equation for a general focal system is worked out by first considering the example of a refracting surface. The concept of focal power is introduced, and the six cardinal points of a focal system are introduced. These are the focal points, unit points and nodal points of the object and image space. The equations that allow Gaussian reduction of two focal systems to a single element are worked out, then explored through multiple examples. Guided Problems examine the formation of stigmatic images by a Maxwell fisheye; provide practice with the use of transfer matrices in lens and mirror problems; investigate image formation in large-angle lens systems, including a derivation of the Abbe sine condition; apply Gaussian reduction to locate the image formed by a thick lens; and treat the Gaussian reduction of lens and mirror systems that include an afocal element.

Pages: 22

Spherical electromagnetic waves are introduced, first within the eikonal approximation, then as an exact solution of the macroscopic Maxwell equations. When an electric or magnetic point-dipole oscillates with frequency https://www.w3.org/1998/Math/MathML" display="inline"> ω , spherical waves of the same frequency appear in the electric and magnetic fields. Augustin-Jean Fresnel extended Huygens' principle by formulating a quantitative theory based on the emission of spherical wavelets from every point on an initial wavefront. The Huygens-Fresnel integral is constructed and compared with a formally exact vector theory based on the emission of electromagnetic wavelets from effective point-dipoles on the wavefront. The vector theory is explored in depth in Appendices G, H and I. The stationary phase approximation is used to evaluate the free-space Huygens-Fresnel integral in the short-wavelength limit. When the scalar field in the integral is identified with the amplitude of the electric field in the eikonal approximation, the free-space results obtained in Chapter 9 using the transport equation are reproduced exactly. This verifies that the Huygens-Fresnel theory reduces to geometric optics in the short-wavelength limit. A step-by-step analysis of the emission of spherical waves by an oscillating electric point-dipole is provided in a Guided Problem.

Pages: 13

A mathematically rigorous justification for the Huygens-Fresnel integral is provided by the Helmholtz-Kirchhoff identity for solutions of the scalar Helmholtz equation. The identity is derived in Appendix E for three-dimensional waves and in Appendix H for cylindrical (or two-dimensional) waves. The so-called Fresnel-Kirchhoff integral for a general surface https://www.w3.org/1998/Math/MathML" display="inline"> S reduces to the Huygens-Fresnel integral when https://www.w3.org/1998/Math/MathML" display="inline"> S is a wavefront. It is demonstrated in Appendix F that the more general integral reproduces the predictions of geometric optics. Diffraction occurs when a wave encounters a small aperture or opaque barrier with a scale comparable to the wavelength. The Fresnel-Kirchhoff integral can be used to calculate the diffracted wave once an approximation is made for the field in the vicinity of the aperture or barrier. The Kirchhoff approximation is introduced and used to calculate the diffracted field on the symmetry axis beyond an annular aperture in an opaque screen. The alternate approximations introduced by Rayleigh and Sommerfeld are derived in Appendix E. The distinction between the near-field (Fresnel) and far-field (Fraunhofer) diffraction zones depends on the size of the Fresnel number https://www.w3.org/1998/Math/MathML" display="inline"> F = a 2 / λ L , where https://www.w3.org/1998/Math/MathML" display="inline"> a is the characteristic length of the aperture or barrier, https://www.w3.org/1998/Math/MathML" display="inline"> λ is the light wavelength and https://www.w3.org/1998/Math/MathML" display="inline"> L is the distance from the diffraction screen to the measurement point. For https://www.w3.org/1998/Math/MathML" display="inline"> F larger than unity, the measurement is in the near-field zone, and for https://www.w3.org/1998/Math/MathML" display="inline"> F smaller than unity in the far-field zone. A Guided Problem treats diffraction by a Fresnel zone plate.

Pages: 16

General approximations for the Fresnel-Kirchhoff integral are derived to treat both the far-field and near-field limits. Expressions are obtained for both three-dimensional and two-dimensional geometries. Simplified expressions apply when a paraxial approximation is allowed, i.e., when all relevant light rays are close to the optical axis. Fourier transforms appear in the far-field integrals, allowing the application of methods first introduced in Chapter 3. It is demonstrated that a far-field (Fraunhofer) diffraction pattern is obtained when a converging lens is placed in the path of outgoing light rays and measurements are made in the lens focal plane. A Guided Problem provides an introduction to computed tomography, emphasizing the application of Fourier transforms.

Pages: 18

In the far-field limit, the general properties of Fourier transforms guarantee that energy is conserved in the diffraction of light by a general aperture (three-dimensional geometry) or system of slits (two-dimensional geometry for cylindrical waves). The derivation relies on the Parseval-Plancherel theorem. The diffraction pattern produced by a single slit (two-dimensional geometry) is analyzed for both the far-field and near-field limits assuming the paraxial approximation. In the near-field limit, the solution requires the introduction of Fresnel integrals. In addition diffraction of an obliquely incident wave is analyzed without assuming the paraxial approximation; it is demonstrated that a change of coordinates (i.e., a rotated paraxial approximation) allows a simpler comparison with previous results. The single-slit analysis is extended to treat diffraction by a rectangular slit of finite length. A Guided Problem examines the diffraction of a plane wave by a knife-edge barrier.

Pages: 28

Diffraction by a circular aperture is analyzed for both the far-field and near-field limits assuming the paraxial approximation. The diffraction effects present in imaging by circular mirrors and lenses are considered; in particular, it is shown that the image of a point object is an Airy disk, rather than a point. This results in a resolution limit for telescopes and microscopes; the Rayleigh criterion for resolution is introduced. The principal methods for microscope illumination are treated, and the concept of numerical aperture is introduced for cases in which a paraxial approximation does not apply. A Bessel beam is a particular cylindrical wave form that is sometimes characterized as “diffraction-free” over a limited propagation distance. A Bessel beam can be produced by passing an incident plane wave through a thin annular aperture in the incoming focal plane of a converging lens. The propagation of the Bessel beam is analyzed by numerical evaluation of the corresponding Fresnel-Kirchhoff integral. Guided Problems include a treatment of image formation as a diffraction process and introduce the technique of phase contrast microscopy.

Pages: 21

Periodic gratings have applications in multiple areas of applied optics, including spectrometry. An infinite double slit (two-dimensional geometry) is first examined in the far-field limit. The treatment is generalized by the introduction of transmission functions and extended to gratings with https://www.w3.org/1998/Math/MathML" display="inline"> N repeated elements. A Rayleigh criterion is specified for the resolution of closely spaced spectral lines formed by a grating. Guided Problems solve for the diffraction pattern produced by a blazed reflection grating and provide an introduction to holography.

Pages: 9

In the field of signal processing, many key techniques rely on the use of Fourier transforms based on time and frequency variables. In the manipulation of light waves, it is natural to apply concepts from signal processing, again using Fourier transforms, but now based on space and “spatial frequency” (or wave-number) variables. This general approach is known as Fourier optics. Any solution of the scalar Helmholtz equation for wave number https://www.w3.org/1998/Math/MathML" display="inline"> k has a so-called angular-spectrum representation, i.e., the solution can be expressed as a two-dimensional integral over plane waves of the form https://www.w3.org/1998/Math/MathML" display="block"> e i ( k x x + k y y ) e i K z z , where https://www.w3.org/1998/Math/MathML" display="block"> k x 2   +   k y 2   +   K z 2     =     k 2 . If a wave's angular-spectrum representation is known on the plane https://www.w3.org/1998/Math/MathML" display="inline"> z = 0 , the form of the wave can be calculated for all https://www.w3.org/1998/Math/MathML" display="inline"> z > 0 . It is demonstrated explicitly that the two Rayleigh-Sommerfeld solutions for a wave diffracted by a planar screen are equivalent to the angular-spectrum solutions based on knowledge of either (i) the wave or (ii) its normal derivative at the screen plane. The angular-spectrum solution based on knowledge of the wave at the screen plane is evaluated in the far-field limit using the stationary phase approximation. The result is identical to that obtained in Chapter 14 for the far-field limit assuming the inclination factor has the Rayleigh-Sommerfeld form. An application of the angular-spectrum approach is included as a Guided Problem in Chapter 23.

Pages: 32

Interference patterns can be obtained by dividing an electromagnetic wave into two secondary waves that travel different paths before being reunited at a measurement point. Any device that employs this technique is said to produce interference by division of amplitude. The difference in maximum and minimum intensities in a pattern determines the so-called fringe visibility. The visibility depends on the relative intensities of the two interfering waves. It is shown that when a wave is divided at an absorption-free beam splitter, the magnitudes and phases of the reflected and transmitted waves are constrained to obey a set of general conditions that follow from energy conservation and symmetry considerations. Three geometries for division of amplitude are investigated in detail: (i) a dielectric plate or film with parallel surfaces; (ii) a thin dielectric film of varying thickness; and (iii) two widely separated planar surfaces that reflect waves generated by a beam splitter (e.g., a Michelson interferometer). The conditions for constructive and destructive interference at the output ports of a Michelson interferometer are analyzed, and the patterns produced by an incident plane wave and by light from an extended source are discussed. Guided Problems examine the form of the time-averaged Poynting vector in a region with both incident and reflected plane waves; the use of symmetries to generate new solutions of the Maxwell equations from a known solution; the effect of unmeasured phases in the analysis of a Michelson interferometer; the nature of patterns formed by a Mach-Zehnder interferometer; and the factors that determine fringe visibility and localization in a general optical system.

Pages: 21

When waves are incident on a parallel plate with high-reflectivity surfaces, the treatment of multiple reflections becomes necessary. The fields inside and outside the plate are determined assuming a general angle of incidence, and the reflectivity and transmissivity are calculated. The parallel-plate geometry is the basis of the Fabry-Perot interferometer, which has applications in spectrometry and is an essential part of the optical system in a gravitational-wave detector. When a Fabry-Perot interferometer is used with an extended source, the interference pattern is a system of circular fringes centered on the optical axis. The pattern is analyzed, and the resolution limit for closely spaced spectral lines is calculated in terms of the interferometer's so-called finesse. The basic principles of gravitational-wave detection at a facility such as the Laser Interferometer Gravitational-Wave Observatory (LIGO) are outlined, and the enhancement of the signal by Fabry-Perot cavities in the detector arms is treated quantitatively. The cavities act as light storage devices that dramatically increase the time an electromagnetic wave spends in the presence of a gravitational wave.

Pages: 26

Mutually coherent beams of light exhibit interference, while incoherent beams do not. The subject of coherence is analyzed in both the time and frequency domains using concepts from the field of stochastic processes. The use of complex-valued functions to calculate the properties of real-valued monochromatic fields was introduced in Chapter 3. In order to extend the use of complex-valued functions to noisy fields, it is necessary to employ a formalism developed by Dennis Gabor. The formalism is developed step by step and illustrated with examples. The power spectrum and autocorrelation function of a noisy real-valued field are defined, and their relationship to experimentally measured quantities is established. The Wiener-Khinchin theorem, which relates the power spectrum to the autocorrelation function, is derived, and an example using the autocorrelation function for light from a gaseous source with collision broadening is investigated in detail. The use of convolution to obtain the power spectrum in a system with multiple sources of broadening is illustrated. In addition the power spectrum and autocorrelation function are calculated for a system dominated by Doppler broadening. The correlation between fields at different spatial points is explored with the aid of the cross-correlation, or mutual coherence, function. The coherence properties of light from a distant incoherent source are established by deriving the so-called van Cittert-Zernike theorem. Finally, a section is included to clarify the translation between distribution functions expressed in terms of angular frequency https://www.w3.org/1998/Math/MathML" display="inline"> ω and frequency https://www.w3.org/1998/Math/MathML" display="inline"> ν .

Pages: 19

Many materials used for the manipulation of light have a glassy atomic structure that guarantees the propagation of electromagnetic waves is isotropic. In contrast when a material has a regular crystalline structure, the atomic symmetry may lead to anisotropic wave propagation. On the scale of visible light waves, the relevant material property is the relative permittivity, which behaves as a rotational tensor, rather than as a scalar. In general the fields https://www.w3.org/1998/Math/MathML" display="inline"> E and https://www.w3.org/1998/Math/MathML" display="inline"> D are no longer parallel, and as a result the system becomes birefringent: Two different refractive indices appear for each specified direction of wave propagation. In addition the direction of power flow need not be parallel to the wave vector https://www.w3.org/1998/Math/MathML" display="inline"> k . The general properties of light in an anisotropic material are analyzed using the relative permittivity tensor. The equations that determine the so-called index surface and ray surface are derived, and the relationship between the two surfaces is investigated. In addition the so-called index ellipsoid and ray ellipsoid are defined. The application of these general concepts to the special cases of a uniaxial crystal and a biaxial crystal is worked out in detail. A Guided Problem explores the phenomenon of conical refraction in a biaxial crystal.

Pages: 19

The treatment of laser light is simplified by the introduction of the paraxial wave equation, an approximation to the scalar Helmholtz equation. The paraxial wave equation has as an exact solution the wave form called a paraxial spherical wave, or PSW. The paraxial form of the Fresnel-Kirchhoff integral exactly propagates a PSW forward from an initial plane to a final plane. The PSW can be mathematically modified by replacing the real-valued source point https://www.w3.org/1998/Math/MathML" display="inline"> z 0 with a complex-valued constant https://www.w3.org/1998/Math/MathML" display="inline"> z 0 + I z R , , where https://www.w3.org/1998/Math/MathML" display="inline"> z     R   is the so-called Rayleigh length. The new function, which still obeys the paraxial wave equation exactly, is called a Gaussian beam. The Gaussian beam has all the necessary properties to serve as a model wave function for laser light. The properties of the Gaussian beam are investigated in detail. A Guided Problem provides an alternate derivation of the Gaussian beam wave function using the angular-spectrum representation introduced in Chapter 18.

Pages: 21

The problem of focusing a Gaussian beam using a lens is treated in detail, and simplifying limits are considered. Laser light is typically created inside a linear or ring-shaped resonant cavity. The necessary conditions satisfied by a cavity mode are derived, and the so-called resonator g-parameters are introduced. The laser-cavity stability region is derived, and the standard cavity varieties are described. Discussion is limited to the simplest possible transverse wave structure. The operation of a beam expander is treated in a Guided Problem. The delivery of maximum laser power to a fixed target beyond a lens of variable focal length is treated in a second Problem.

Pages: 42

The diffraction of plane waves by a perfectly conducting knife edge, or half-plane, is a problem first solved by Arnold Sommerfeld using methods of contour integration in the complex plane. The problem is tractable because the geometry is two-dimensional. A general solution can be obtained by considering two cases separately: (i) incident light with https://www.w3.org/1998/Math/MathML" display="inline"> E parallel to the knife edge (E-polarization); and (ii) incident light with https://www.w3.org/1998/Math/MathML" display="inline"> H parallel to the knife edge (H-polarization). An alternate solution that does not require knowledge of complex analysis can be obtained using a method based on image sources. The method is first illustrated to obtain a solution of an electrostatic problem, a line charge placed in the vicinity of a conducting knife edge. The method is then extended to find a solution of the wave equation for a monochromatic line source in the presence of the knife-edge. The solution for an incident plane wave is obtained by taking appropriate limits as the source recedes to infinity. The plane-wave solution is analyzed in detail, and an extensive set of numerical results is obtained for the electric and magnetic fields and the associated Poynting vector. The exact results are compared with results from scalar-field diffraction theory, and excellent agreement is obtained for the regime in which the scalar-field approximations are justified. A Guided Problem considers the solution of the electrostatics problem in greater detail.

Pages: 44

The diffraction of an incident plane wave by an infinite slit in a perfectly conducting plane is a second problem that can be solved exactly. The geometry is two-dimensional, and the cases of E-polarization and H-polarization can be treated separately. The solutions for the fields take the form of an infinite series of so-called Mathieu functions; the series cannot be reduced to a simple closed form. Perturbative studies of the solution were carried out in the 1930s by Morse, Stratton and their collaborators. In more recent years, high-speed computers have enabled numerical solutions to arbitrary accuracy. As a preliminary, the exact form of Babinet's principle for diffraction by complementary planar screens is derived. This result connects the solutions for diffraction by an infinite metallic strip in an open plane and by an infinite slit in a solid plane. The two-dimensional Helmholtz equation is then solved in elliptic coordinates; the solution requires angular and radial Mathieu functions, which are treated in detail. By imposing the required boundary conditions in the metallic-strip problem, a full solution is obtained, then transformed using Babinet's principle to solve the infinite-slit problem. Results are obtained for the electric and magnetic fields, the angular intensity distribution at fixed radial distance, and the total transmitted power. Numerical results are analyzed, and comparisons are made with results from scalar-field diffraction. As in the preceding chapter, the scalar-field results show excellent agreement with exact results in the parameter ranges where approximations are justified.