Optics: Lecture 2 Light in Matter: The response of dielectric materials Dispersive Prism: Dependence of the index of refraction, n(  ), on frequency or wavelength of light. Sir Isaac Newton used prisms to disperse white light into its constituent colors over 300 years ago. When white light passes through a prism, the blue constituent experiences a larger index of refraction than the red component and therefore it deviates at a larger angle, as we shall see.

The effect of introducing a homogenous, isotropic dielectric changes Maxwell’s equations to the extent that  o   and  o  . The phase speed in the medium becomes: The ratio of the speed of an E-M wave in vacuum to that in matter is defined as the index of refraction n: Relative Permittivity and Relative Permeability For most dielectrics of interest that are transparent in the visible, these are essentially non-magnetic and to a good approximation K M  1.

To a good approximation also known as Maxwell’s Relation: K E is presumed to be the static dielectric constant (and works well only for some simple gases, as shown on next slide). In reality, K E and n are actually frequency-dependent, n(  ), known as dispersion. Scattering and Absorption: m n Gas/Solids Excitation energy can be transferred via collisions before a photon is re-emitted. Resonant process h Non-resonant scattering: Energy is lower than the resonant frequencies. E-M field drives the electron cloud into oscillation The oscillating cloud relative to the positive nucleus creates an oscillating dipole that will re-radiate at the same frequency.

Works well Doesn’t work so well

F = -k E x A small displacement x from equilibrium causes a restoring force F. and results in resonant frequency: + - x-axis Light  E(t) and produces a classical forced oscillator. Amplitude ~ m for bright sun light. E(t)E(t)

The result can be modeled like a classical forced oscillator with F E = q e E o cos(  t) = q e E(t). Using Newton’s 2 nd law: Driving Force – Restoring Force = ma, where Rest. Force = -k E x To solve, let x(t) = x o cos  t *Note that the phase of the displacement x depends on  >  o or  <  o which gives x   qE(t).

The electric polarization or density of dipole moments (dipole moment/vol.) is given by: Where N = number electrons per volume. We learn from the dielectric properties of solids that Therefore Since n 2 = K E =  /  o it follows that we obtain the following dispersion equation: *Note that  >  o  n < 1 (above resonance) (Displacement is 180  out-of-phase with driving force.) and  1 (below resonance) (Displacement is in-phase with driving force.)

Consider classically the average power (P av ) delivered by the applied electric field: (See, e.g. Mechanics, Symon, 3 rd Ed.) Phase angle  This average power is analogous to absorption of E-M radiation at the resonant frequency  0.

For light  = ck = 2  c/, we can write the dispersion relation as Thus, if we plot (n 2 -1) -1 versus -2 we should arrive at a straight line.

In reality, there are several transitions in which n > 1 and n < 1 for increasing , i.e., there are several  oi resonant frequencies corresponding to the complexity of the material. Therefore, we generalize the above result for N molecules/vol. with f j different oscillators having natural frequencies  oj, where j = 1, 2, 3.. A quantum mechanical treatment shows further that where f j are weighting factors known as Q.M. oscillator strengths and represent the transition probability for each mode j. The energy is the energy of absorption or emission for a given electronic, atomic, or molecular transition.

When  =  oj then n is discontinuous (and blows up). Actual observations show continuity and finite n. The conclusion is that a damping force which is proportional to the speed m e  dx/dt should generally be included when there are strong interactions occurring between atoms and molecules, such as in liquids and solids. With damping, (1) energy is lost when oscillators re-radiate and (2) heat is generated as a result of friction between neighboring atoms and molecules. The corrected dispersion, including damping effects, is as follows: This expression often works fine for gases.

In a dense solid material, the atoms/molecules may experience an additional field that is induced by the surrounding medium and is given by P(t)/3  o. With this induced field, the dispersion relation becomes We will see that a complex index of refraction will lead to absorption. Presently, we will consider regions of negligible absorption in which n is real and Thus For various glasses, Since oj ~ 100 nm in the ultra-violet (UV).

Note that as    oj, n(  ) gradually increases and the behavior is called “Normal Dispersion.” Again, at  =  oj, n is complex and leads to an absorption band. Also, when dn/d  < 0, the behavior is called “Anomalous Dispersion.”

When white light passes through a glass prism, the blue constituent experiences a larger index of refraction than the red component and therefore it deviates at a larger angle, as seen in the first slide.

Note the rise of n in the UV and the fall of n in the IR, consistent with “Normal Dispersion.” At even lower frequencies in the radio range, the materials become again transparent with n > ~1. Transparency occurs when  >  o. When  ~  o, dissipation, friction and therefore absorption occurs, causing the observed opacity.

Absorption Spectrum of Water: Absorption coefficient: n I is the imaginary part of the index of refraction.

The water vapor absorption bands are related to molecular vibrations involving different combinations of the water molecule's three fundamental vibrational transitions: V1: symmetric stretch mode V2: bending mode V3: asymmetric stretch mode The absorption feature centered near 970 nm is attributed to a 2V1 + V3 combination, the one near 1200 nm to a V1 + V2 + V3 combination, the one near 1450 nm to a V1 + V3 combination, and the one near 1950 nm to a V2 + V3 combination. In liquid water, rotations tend to be restricted by hydrogen bonds, leading to vibrations, or rocking motions. Also stretching is shifted to a lower frequency while the bending frequency increased by hydrogen bonding.

Propagation of Light, Fermat’s Principle (1657) Involves the principle of least time: The path between two points that is taken by a beam of light is the one that is transversed in the least amount of time. To find the path of least time, set dt/dx=0. Since n i = c/v i and n t = c/v t. Snell’s Law of Refraction

Note that where o is the vacuum wavelength. In general, for many layers having different n, we can write Note that if the layers are very thin, we can write = OPL (Optical Path Length)

We can compute t as simply Note that the spatial path length is and for a medium possessing a fixed index n 1, Fermat’s principle can be re-stated: Light in going from S  P traverses the route having the smallest OPL. We will begin next with the E-M approach to light waves incident at an interface and derive the Fresnel Equations describing transmission and reflection.