Light and Matter. Thursday, 8/31/2006 Physics 158 Peter Beyersdorf. Document info

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1 Light and Matter Thursday, 8/31/2006 Physics 158 Peter Beyersdorf Document info

2 Class Outline Common materials used in optics Index of refraction absorption Classical model of light absorption Light Molecules Interaction Dispersion

3 Common materials Material n* α* (m -1 ) vacuum 1 air fused silica Al 2 O , BK aluminum silver gold water diamond * at λ=589 nm (the sodium D line)

4 Absorption of Light Quantum mechanical picture of absorption tells us a photon of energy hν is absorbed by an atom or molecule which leaves the atom or molecule in a more energetic state ν E 0 E 1 The Quantum picture does not tell us much about the interaction itself. Thus we consider a classical model for some insight

5 Classical model of light absorption We must model three things to describe absorption Light (sinusoidal electromagnetic field) Matter (simple harmonic oscillator) Interaction (damped oscillator driven by periodic force)

6 Classical model of light Consider only the oscillating electric field (the corresponding magnetic field is implied) Amplitude vector including polarization and spatial dependence (mode) information speed of propagation c=ω/k angular frequency of the light ω=2πf wavevector of the light magnitude is k=2πn/λ 0 direction defines propagation direction of the light

7 Classical model of light For the moment we will consider the 1-dimensional description of a plane wave polarized along x, propagating along z " " " " " Electric field has units of Volts/meter. The intensity is the quantity more commonly measured in the laboratory. It is related to electric field by and has units of W/m

8 Classical model of molecules Treat atoms or molecules as simple harmonic oscillators. For any molecule with an arbitrary potential at equilibrium U =0 and potential can be modeled as a quadratic well (i.e. that of a simple harmonic oscillator) V(r) r=a r

9 Classical model of molecules This model works for arbitrary atoms or molecules covalently bonded atoms have a stable equilibrium distance Electron clouds can be pushed away from atomic nucleus but will be pulled back by a restoring force nuclear spins align with external magnetic field - if they are disturbed from equilibrium they will relax back to aligned state

10 Classical model of interaction Consider the restoring force for the perturbed molecule assuming it is aligned to the light polarization (along x) F = V (x x 0 )î from Newton s 2 nd law so V (x x 0 ) = mẍ where m may need to be replaced by 3. the effective mass of the oscillator 10 10

11 Classical model of interaction V (x x 0 ) = mẍ The equation of motion for the simple harmonic oscillator has solutions of the form x = A cos(ω 0 t + φ) " " " " " " " " " " or with natural frequency

12 Classical model of interaction Now we add a velocity-dependent damping force F = ( V (x x 0 ) γmẋ)î and find the solutions to V (x x 0 ) γmẋ = mẍ

13 Classical model of interaction Solutions for the damped harmonic oscillator are of the form x(t) = Ae (γ/2)t cos (ω t + φ) " " " " " " " " " " " " or with x

14 Classical model of interaction Finally we add a the driving force of the electric field evaluated at r=0 and find the solutions to hint: this is easily done trying the phasor form of a solution

15 Classical model of interaction Solutions for the (complex) amplitude of the damped harmonic oscillator with an external driving force are of the form which means the physical displacement is x(t) = (e/m)e 0 (ω 20 ω2 ) 2 + γ 2 ω 2 cos (ωt β) with tan β = γω ω 2 0 ω2 and E(t) = E 0 cos (ωt)

16 Absorbed power The absorbed power is

17 Absorption Lineshape Absorption is frequency dependent with a Lorentzian lineshape

18 Atomic Susceptibility The macroscopic response of a material to a driving field is called χ, the atomic susceptibility The displacement vector of the material is where the polarization is a function of N, the number density of atoms, and the atomic dipole moment μ=-e x Carefull - This P is the polarization of the material, not the absorbed power

19 Atomic Susceptibility We define the atomic susceptibility to be The index of refraction of the material is then which can be found from the classical electron oscillator model

20 Atomic Susceptibility Complex Index of refraction gives rise to absorption. For a single atom we saw this was For N atoms per unit volume, far away from any atomic resonances this can be expressed as an absorption coefficient that describes the exponential decrease in intensity of light traveling through the material di(ω) dx = α(ω)i(ω)

21 Dispersion We have shown that the index of refraction (and absorption coefficient) id frequency dependent. This frequency dependance has a name: Dispersion normal dispersion in optical materials has a decreasing index with increasing wavelength Dispersion is responsible for the colors seen in this diamond. Different colors see different index of refraction causing them to travel through the jem in different paths

22 Summary The principle optical characteristics of a material are index of refraction and absorption coefficient Interaction of light with matter can be modeled as a driven harmonic oscillator and used to explain frequency (wavelength) dependance of the index of refraction and absorption coefficients The variation of the index of refraction with wavelength is called dispersion. Normally dn/dλ<