This is a summary of the topics in linear optics we have covered in modules 1 - 7.

We have reviewed geometrical optics.  Geometrical optics is an approximation to linear optics when the wavelengths of the light being studied are very small compared to the dimensions of the equipment used to study the light, and the photon energies are much smaller than the energy sensitivity of the equipment.

We have studied wave propagation in linear, isotropic, homogeneous (lih) media.  In a lih material monochromatic, sinusoidal plane waves with E(r,t) = E₀ exp(i(k×r-wt)) obey the wave equation

with me =m(w)e(w) = (n(w)/c)² = constant, and

When we reconstitute an arbitrary wave as a linear superposition of plane waves with different frequencies the shape of the wave changes as it propagates because of dispersion.

We have developed a microscopic model for the index of refraction in media with and n » 1, 

and confirmed its dependence on w.  We showed that the apparent reduction in the speed of light in a material is a consequence of the interference of the original wave and many scattered waves all traveling with speed c.

We have studied wave propagation in anisotropic, linear, homogeneous (lh) media.  In an anisotropic material D is always perpendicular to k, i.e. D is perpendicular to the normal to the wave front.  This is not necessarily true for E.  D and E are not necessarily parallel vectors.

In coordinate system with its axes along the principal axes, the dielectric tensor e is given by 

and e_iE_i = e₀n²(E_i – (1/k²)k_i(k×E)).  Putting this equation in matrix form and solving for E we find that for a given direction of propagation , there are in general two values for the refractive index, n₁ and n₂.  If n₁² is not equal to n₂², then D₁ and D₂ are perpendicular to each other.

In addition, the direction of the energy flow through the anisotropic material is not along .  When we send a collimated laser beam into an anisotropic medium we, in general, observe two beam propagating in the medium.  There is one vector defining the average direction perpendicular to the wave fronts of the plane waves making up the wave packets that define the beams.  The phase velocity of the plane waves that make up beam i (i = 1,2) is v_pi = c/n_i.  The beam, however, travels with the group velocity v_ri = c/n_icos a in the direction of ¶w/¶k_i.  We measure the ray index or energy index of refraction n_icos a.

We have studied the behavior of plane waves at dielectric-dielectric boundaries.  We derived the Fresnel reflection and transmission coefficients for s- and p-polarization and defined the characteristic wave impedance Z for a plane waves in a medium as the magnitude of ratio of the transverse electric field E_|| to the transverse magnetic induction H_||.  The wave impedance is continuous.  We studied polarization by reflection and total internal reflection.

We briefly looked at reflection and refraction in anisotropic media and found that the simple law of reflection q_i = q_r will not generally hold since the indices of refraction are different for the incident and the reflected waves.

We applied the laws of reflection and refraction to quarter-wave dielectric layers and reflection gratings.  In particular we used the fact that the wave impedance is continuous to find the  index of refraction of a quarter wave layer needed to be to produce an anti-reflection coating. 

We have studied Fresnel diffraction theory.  The scalar field y(x,y,z,t) = y(x,y,z)exp(-iwt) is a solution to the scalar wave equation Ѳy + k²y = 0.  In the paraxial approximation the wave vectors of all plane waves make small angles with respect to the z-axis, so that we can write

k_z = (k² –k_x² –k_y²)^½ » k – (k_x² + k_y²)/2k.

We write y(x,y,z) = u(x,y,z) exp(ikz) and obtain

.

In the Fraunhofer approximation we evaluate the integral for u(x₀,y₀) = 1 if |x₀| < a_x/2, |y₀| < a_y/2 and zero otherwise and obtain the intensity distribution 

.

The far-field (Fraunhofer) diffraction pattern of an image located in the entrance focal plane of a converging lens can be projected by the lens into the exit focal plane of the lens.  The electric field amplitude in the frequency plane is proportional to the Fourier transform of of the field amplitude in the object focal plane of the  lens.  This forms the basis for an optical processor.

We have studied Gaussian beams.  They satisfy the scalar wave equation in the paraxial approximation.  The function  u_mn(xyz)exp(i(kz-wt) represents a beam with a Hermite-Gaussian profile traveling in the positive z-direction.   The set of u_mn(xyz)exp(i(kz-wt) are the Hermite-Gaussian modes.  All modes expand in diameter as they propagate, and all modes have the same radius of curvature and the same basic scale parameter as a function of z.

represents a wave with Gaussian amplitude profile traveling in the +z direction.

We have studied the properties of optical wave guides made from linear, isotropic materials.  We solved Maxwell's equations for modes propagating in a medium with a parabolic index profile and for modes propagating in a slab wave guide. 

In a multi-mode fiber many modes can propagate.  The number of guided modes in a step-index multimode fiber is given by V²/2, and a step index fiber becomes single-mode for a given wavelength when V < 2.405 where V  = k_f a NA.  Even in a single mode fiber a pulse experiences dispersion.  

In a sinlge-mode fiber a pulse envelope satisfies the equation

  ,

which has a Gaussian-pulse solution

.

The pulse is "chirped", d₀ is the chirp parameter.  If d₀ is positive, the front part of the pulse oscillates with a higher frequency than the back part of the pulse.  If d₀ is negative, the back part of the pulse oscillates with a higher frequency than the front part of the pulse.  As the pulse propagates along a lossless fiber the chirp parameter changes.  If d(t) < d₀, the pulse becomes shorter in time.  For d₀ > 0 the high frequency components dominate the front part of the pulse at t = 0.  The low-frequency components dominate the back part of the pulse.  The low frequency components will eventually catch up with the high frequency components and the temporal width of the pulse will decrease.  In a very long fiber, the low frequency components overtake the high frequency components, and the pulse broadens again.

We have studied coupled resonators.  We have considered two resonators characterized by the equations dx₁/dt = -iw₁x₁ + k₁₂x₂ and dx₂/dt = -iw₂x₂ + k₂₁x₁, or by the equations dx₁/dz = ib₁x₁ + k₁₂x₂ and dx₂/dz = ib₂x₂ + k₂₁x₁, where k₁₂ and k₂₁ are the coupling coefficients.  We have solved this problem of two coupled harmonic oscillators and found that the energy is transferred back and forth between the modes.  Full transfer can occur when w₁ = w₂ or b₁ = b₂.  Applications include tunable filters and optical wave guide switches.

We have studied the diffraction of an EM wave by an acoustic wave.  The Bragg condition has to be satisfied for the incident wave, and the diffracted wave has wave vector k_d = k_i + k_s and frequency w_d = w_i + w_s » w_i.  The diffracted wave forms a beam that is deflected from the incident beam.  This beam also interacts with the acoustic wave and is deflected back into the incident wave.  The incident and diffracted wave can act like coupled oscillators, feeding energy from one to the other.  Acousto-optic devices include modulators and beam deflectors.

We have also studied Q-switching and mode locking, and how it can be accomplished with an acousto-optic device.