Photons: Light Waves behaving as Particles

Photons:  Electromagnetic radiation behaves as both waves and particles.  The energy in an electromagnetic wave is carried in units called photons.  The energy E of one photon is proportional to the wave frequency f and inversely proportional to the wavelength \lambda, and is proportional to a universal quantity h called Planck’s constant: E=hf=\frac{hc}{\lambda}.  The momentum of a photon has magnitude E/c: p=\frac Ec=\frac{hf}c=\frac h{\lambda}.

The photo-electric effect:  In the photo-electric effect, a surface can eject an electron by absorbing a photon whose energy hf is greater than or equal to the work function \phi of the material.  The stopping potential V_0 is the voltage required to stop a current of ejected electrons from reaching an anode: eV_0=hf-\phi.

Photon production:  X rays can be produced when electrons accelerated to high kinetic energy across a potential increase V_{AC} strike a target.  The photon model explains why the maximum frequency and minimum wavelength produced are given by the equation: eV_{AC}=hf_{\max}=\frac{hc}{\lambda_{\min}} (bremsstrahlung).


Invariance of physical laws, simultaneity:  All of the fundamental laws of physics have the same form in all inertial frames of reference.  The speed of light in vacuum is the same in all inertial frames and is independent of the motion of the source.  Simultaneity is not an absolute concept; events that are simultaneous in one frame are not necessarily simultaneous in a second frame moving relative to the first.

Time dilation:  If two events occur at the same space point in a particular frame of reference, the time interval \Delta t_0 between the events as measured in that frame is called a proper time interval.  If this frame moves with constant velocity u relative to a second frame, the time interval \Delta t between the events as observed in the second frame is longer than \Delta t_0: \Delta t=\frac{\Delta t_0}{\sqrt{1-\frac{u^2}{c^2}}}=\gamma\Delta t_0, \gamma=\frac1{\sqrt{1-u^2/c^2}}.

Length contraction:  If two points are at rest in a particular frame of reference, the distance l_0 between the points as measured in that frame is called a proper length. If this frame moves with constant velocity u relative to a second frame and the distances are measured parallel to the motion, the distance l between the points as measured in the second frame is shorter than l_0.  l=l_0\sqrt{1-\frac{u^2}{c^2}}=\frac{l_0}{\gamma}.

The Lorentz transformation:  The Lorentz coordinate transformations relate the coordinates and time of an event in an inertial frame S to the coordinates and time of the same event as observed in a second inertial frame S' moving at velocity u relative to the first.  For one-dimensional motion, a particle’s velocities v_x in S and v_x' in S' are related by the Lorentz velocity transformation.  x'=\frac{x-ut}{\sqrt{1-u^2/c^2}}=\gamma(x-ut), y'=y, z'=z, t'=\frac{t-ux/c^2}{\sqrt{1-u^2/c^2}}=\gamma(t-ux/c^2), v_x'=\frac{v_x-u}{1-uv_x/c^2}, v_x=\frac{v_x'+u}{1+uv_x'/c^2}.

The Doppler effect for electromagnetic waves:  The Doppler effect is the frequency shift in light from a source due to the relative motion of source and observer.  For a source moving toward the observer with speed u, the received frequency f in terms of the emitted frequency f_0 is f=\sqrt{\frac{c+u}{c-u}}f_0.

Relativistic momentum and energy:  For a particle of rest mass m moving with velocity \vec{v}, the relativistic momentum is \vec{p}=\frac{m\vec{v}}{\sqrt{1-v^2/c^2}}=\gamma m\vec{v}, the relativistic kinetic energy is K=\frac{mc^2}{\sqrt{1-v^2/c^2}}-mc^2=(\gamma-1)mc^2.  The total energy E is the sum of the kinetic energy and the rest energy mc^2: E=K+mc^2=\frac{mc^2}{\sqrt{1-v^2/c^2}}=\gamma mc^2.  Also, E^2=(mc^2)^2+(pc)^2.


Fresnel and Fraunhofer diffraction:  Diffraction occurs when light passes through an aperture or around an edge.  When the source and the observer are so far away from the obstructing surface that the outgoing rays can be considered parallel, it is called Fraunhofer diffraction.  When the source or the observer is relatively close to the obstructive surface, it is Fresnel diffraction.

Single-slit diffraction:  Monochromatic light sent through a narrow slit of width a produces a diffraction pattern on a distant screen.  The condition for destructive interference (a dark fringe) at a point in the pattern at angle \theta: \sin\theta=\frac{m\lambda}{a}, (m=\pm1,\pm2,\pm3,\ldots).  The intensity in the pattern as a function of \theta: I=I_0\{\frac{\sin[\pi a(\sin\theta)/\lambda]}{\pi a(\sin\theta)/\lambda}\}^2.

Diffraction gratings:  A diffraction grating consists of a large number of thin parallel slits, spaced a distance d apart.  The condition for maximum intensity in the interference pattern is the same as for the two-source pattern, but the maxima for the grating are very sharp and narrow.  d\sin\theta=m\lambda (m=\pm1,\pm2,\pm3,\ldots).

X-ray diffraction:  A crystal serves as a three-dimensional diffraction grating for x rays with wavelengths of the same order of magnitude as the spacing between atoms in the crystal.  For a set of crystal planes spaced a distance d apart, constructive interference occurs when the angles of incidence and scattering (measured from the crystal planes) are equal and when the Bragg condition is satisfied: 2d\sin\theta=m\lambda, (m=1,2,3,\ldots).

Circular apertures and resolving power:  The diffraction pattern from a circular aperture of diameter D consists of a central bright spot, called the Airy disk, and a series of concentric dark and bright rings.  Equation \sin\theta_1=1.22\frac{\lambda}D gives the angular radius \theta_1 of the first dark ring, equal to the angular size of the Airy disk.  Diffraction sets the ultimate limit on resolution (image sharpness) of optical instruments.  According to Rayleigh’s criterion, two point objects are just barely resolved when their angular separation \theta is given by the equation.


Interference and coherent sources:  Monochromatic light is light with a single frequency.  Coherence is a definite, unchanging phase relationship between two waves.  The overlap of waves from two coherent sources of monochromatic light forms an interference pattern.  The principle of superposition states that the total wave disturbance at any point is the sum of the disturbances from the separate waves.

Two-source interference of light:  When two sources are in phase, constructive interference occurs where the difference in path length from the two sources is zero or an integer number of wavelengths; destructive interference occurs where path difference is a half-integer number of wavelengths.  If two sources separated by a distance d are both very far from a point P, and the line from the sources make an angle \theta with the line perpendicular to the line of the sources, then the condition for constructive interference at P is d\sin\theta=m\lambda, (m=0,\pm 1,\pm 2,\ldots).  The condition for destructive interference is d\sin\theta=(m+\frac12)\lambda, (m=0,\pm1,\pm2,\ldots).  When \theta is very small, the position y_m of the mth bright fringe on a screen located at distance R from the sources is: y_m=R\frac{m\lambda}d, (m=0,\pm1,\pm2,\ldots).

Intensity in interference pattern:  When two sinusoidal waves with equal amplitude E and phase difference \phi are superimposed, the resultant amplitude E_P and intensity I are as follows: E_P=2E|\cos\frac{\phi}2|, I=I_0\cos^2\frac{\phi}2.  If the two sources emit in phase, the phase difference \phi at a point P (located a distance r_1 from source 1 and a distance r_2 from source 2) is directly proportional to the path difference r_2-r_1: \phi=\frac{2\pi}{\lambda}(r_2-r_1)=k(r_2-r_1).

Interference in thin films:  When light is reflected from both sides of a thin film of thickness t and no phase shift occurs at either surface, constructive interference between the reflected waves occurs when 2t is equal to an integral number of wavelengths.  If a half-cycle phase shift occurs at one surface, this is the condition for destructive interference.  A half-cycle phase shift occurs during reflection whenever the index of refraction in the second material is greater than in the first.

Michelson interferometer:  The Michelson interferometer uses a monochromatic light source and can be used for high-precision measurements of wavelengths.  Its original purpose was to detect motion of the earth relative to a hypothetical ether, the supposed medium for electromagnetic waves.  The ether has never been detected, and the concept has been abandoned; the speed of light is the same relative to all observers.  This is part of the foundation of the special theory of relativity.

Geometric Optics

Reflection or refraction at a plane surface:  When rays diverge from an object point P and are reflected or refracted, the directions of the outgoing rays are the same as though they had diverged from a point P' called the image point.  If they actually converge at P' and diverge again beyond it, P' is a real image of P; if they only appear to have diverged from P', it is a virtual image.  Images can be either erect or inverted.

Lateral magnification:  The lateral magnification m in any reflecting or refracting situation is defined as the ratio of image height y' to object height y, m=\frac{y'}{y}.  When m is positive, the image is erect; when m is negative, the image is inverted.

Focal point and focal length:  The focal point of a mirror is the point where parallel rays converge after reflection from a concave mirror, or the point from which they appear to diverge after reflection from a convex mirror.  Rays diverging from a focal point of a concave mirror are parallel after reflection; rays converging toward the focal point of a convex mirror are parallel after reflection.  The distance from the focal point to the vertex is called the focal length, denoted as f.  The focal points of a lens are defined similarly.

Relating object and image distances:
Plane mirror: s=-s', m=1.
Spherical mirror: \frac1s+\frac1{s'}=\frac2R=\frac1f, m=-\frac{s'}s.
Plane refracting surface: \frac{n_a}s+\frac{n_b}{s'}=0, m=\frac{n_as'}{n_bs}=1,
Spherical refracting surface: \frac{n_a}s+\frac{n_b}{s'}=\frac{n_b-n_a}R, m=\frac{n_as'}{n_bs}.

These object–image relationships are valid only for rays close to and nearly parallel to the optical axis; these are called paraxial rays.  Non-paraxial rays do not converge precisely to an image point.  This effect is called spherical aberration.

Thin lenses:  The lensmaker’s equation relates the focal length of a lens to its index of refraction and the radii of curvature of its surfaces.  \frac1s+\frac1{s'}=\frac1f, \frac1f=(n-1)(\frac1{R_1}-\frac1{R_2}).

Sign rules:  The following sign rules are used with all plane and spherical reflecting and refracting surfaces:

  • s>0 when the object is on the incoming side of the surface (a real object); s<0 otherwise.
  • s'>0 when the image is on the outgoing side of the surface (a real image); s'<0 otherwise.
  • R>0 when the center of curvature is on the outgoing side of the surface; R<0 otherwise.
  • m>0 when the image is erect; m<0 when inverted.

Cameras:  A camera forms a real, inverted, reduced image of the object being photographed on a light-sensitive surface.  The amount of light striking this surface is controlled by the shutter speed and the aperture.  The intensity of this light is inversely proportional to the square of the f-number of the lens: f-number=\frac{\mathrm{Focal\ length}}{\mathrm{Aperture\ diameter}}=\frac{f}{D}.

The eye:  In the eye, refraction at the surface of the cornea forms a real image on the retina.  Adjustment for various object distances is made by squeezing the lens, thereby making it bulge and decreasing its focal length.  A nearsighted eye is too long for its lens; a farsighted eye is too short.  The power of a corrective lens, in diopters, is the reciprocal of the focal length in meters.

The simple magnifier:  The simple magnifier creates a virtual image whose angular size \theta' is larger than the angular size \theta of the object itself at a distance of 25 cm, the nominal closest distance for comfortable viewing.  The angular magnification M of a simple magnifier is the ratio of the angular size of the virtual image to that of the object at this distance.  M=\frac{\theta'}{\theta}=\frac{25\ \mathrm{cm}}f.

Microscopes and telescopes:  In a compound microscope, the objective lens forms a first image in the barrel of the instrument, and the eyepiece forms a final virtual image, often at infinity, of the first image.  The telescope operates on the same principle, but the object is far away.  In a reflecting telescope, the objective lens is replaced by a concave mirror, which eliminates chromatic aberrations.

The Nature and Propagation of Light

Light and its properties.  Light is an electromagnetic wave.  When emitted or absorbed, it also shows particle properties.  It is emitted by accelerated electric charges.

A wave front is a surface of constant phase; wave fronts move with a speed equal to the propagation speed of the wave.  A ray is a line along the direction of propagation, perpendicular to the wave fronts.

When light is transmitted from one material to another, the frequency of the light is unchanged, but the wavelength and the wave speed can change.  The index of refraction of a material n=\frac cv, \lambda=\frac{\lambda_0}n.

Reflection and refraction.  \theta_r=\theta_a (law of reflection), n_a\sin\theta_a=n_b\sin\theta_b (law of refraction).

Total internal reflection.  When a ray travels in a material of index of refraction n_a toward a material of index n_b<n_a, total internal reflection occurs at the interface when the angle of incidence equals or exceeds a critical angle \theta_{\mathrm{crit}}, \sin\theta_{\mathrm{crit}}=\frac{n_b}{n_a}.

Polarization of light.  The direction of polarization of a linearly polarized electromagnetic wave is the direction of the \vec{E} field.

Malus’s law.  When polarized light of intensity I_{\max} is incident on a polarizing filter used as an analyzer, I=I_{\max}\cos^2\phi, I is intensity of the light transmitted through the analyzer, \phi is the angle between the polarization direction of the incident light and the polarizing axis of the analyzer.

Polarization by reflection.  When unpolarized light strikes an interface between two materials, Brewster’s law states that the reflected light is completely polarized perpendicular to the plane of incidence (parallel to the interface) if the angle of incidence is \theta_p=\arctan\frac{n_b}{n_a}.

Huygens’s principle.  If the position of a wave front at one instant is known, then the position of the front at a later time can be constructed by imagining the front as a source of secondary wavelets.