The Interaction of Light and Matter: α and n

Transcription

1 The Interaction of Light and Matter: α and n The interaction of light and matter is what makes life interesting. Everything we see is the result of this interaction. Why is light absorbed or transmitted by a particular medium? Light causes matter to vibrate. Matter in turn emits light, which interferes with the original light. Destructive interference means absorption. Mere out-of-phase interference changes the phase velocity of light, or refractive index. Excited atoms & the forced oscillator Complex Lorentzian: 1/(δ iγ) α Refractive index Absorption coefficient n 1 Absorption coefficient, α, and refractive index, n. Frequency, ω

2 Adding complex amplitudes When two waves add together with the same complex exponentials, we add the complex amplitudes, E + E '. Constructive interference: Destructive interference: "Quadrature phase" ±9 interference: = -. + = -.i + = i time time time Laser Absorption Slower phase velocity

3 Light excites atoms, which emit light that adds (or subtracts) with the input light. When light of frequency ω excites an atom with resonant frequency ω : Electric field at atom! E( t) Electron cloud! x ( ) e t On resonance (ω = ω ) Emitted field! + Eʹ( t) = Incident light Emitted light Transmitted light An excited atom vibrates at the frequency of the light that excited it and reemits the energy as light of that frequency. The crucial issue is the relative phase of the incident light and this re-emitted light. For example, if these two waves are ~18 out of phase, the beam will be attenuated. We call this absorption.

4 The Forced Oscillator: The relative phase of the oscillator motion with respect to the input force depends on the frequencies. Let the oscillator s resonant frequency be ω, and the forcing frequency be ω. Below resonance ω << ω On resonance ω = ω Force Oscillator Weak vibration. In Strong vibration. 9 out of We could think of the forcing function as a light electric field and the oscillator as a nucleus of an atom in a molecule. Above resonance ω >> ω Weak vibration. 18 out of

5 The relative phase of an electron cloud s motion with respect to input light depends on the frequency. Below resonance ω << ω Electric field at atom Electron cloud Weak vibration. 18 out of Let the atom s resonant frequency be ω, and the light frequency be ω. The electron charge is negative, so there s a 18 phase shift in all cases (compared to the previous slide s plots). On resonance ω = ω Above resonance ω >> ω Strong vibration. -9 out of Weak vibration. In

6 The relative phase of emitted light with respect to the input light depends on the frequency. The emitted light is 9 phase-shifted with respect to the atom s motion. Below resonance ω << ω On resonance ω = ω Above resonance ω >> ω Electric field at Electron atom cloud Emitted field Weak emission. 9 out of Strong emission. 18 out of Weak emission. -9 out of

7 The Damped Forced Oscillator The undamped oscillator has infinite amplitude on resonance, which is unphysical. We fix this, ad hoc, by using a damped forced oscillator: d xe dxe e e e e m + m γ + m ω x = ee exp( iωt) dt dt The solution is now: ( e / m ) e xe ( t) = E( t) ( ω ω iωγ ) The electron always oscillates at the light frequency but the complexvalued solution introduces a frequency-dependent phase shift, and frequency-dependent amplitude.

8 Why we included the damping factor, γ Atoms spontaneously decay to the ground state after a time. Also, the vibration of a medium is the sum of the vibrations of all the atoms in the medium, and collisions cause the sum to cancel. Collisions "dephase" the vibrations, causing cancellation of the total medium vibration, typically exponentially. (We can use the same argument for the emitted light, too.) time

9 The atom s response is approximately a Complex Lorentzian. Consider: x ( t) e e / m e / me = ω ω ωγ ( ω ω)( ω + ω) iωγ e i Assuming ω ω, this becomes: = = e / me ω ( ω ω) iωγ e / me 1 ω ( ω ω) iγ / In terms of the variables δ = ω ω and Γ = γ/, the function 1/(δ iγ), is a complex Lorentzian. Its real and imaginary parts are: 1 δ iγ = 1 δ + iγ δ iγ δ + iγ = δ δ + Γ + iγ δ + Γ

10 Complex Lorentzian

11 Damped forced oscillator solution for light-driven atoms The forced-oscillator response is sinusoidal, with a frequencydependent strength that's approximately a complex Lorentzian: Here, e <. e 1 1 xe ( t) E( t) E( t) ωm ( ω ω iγ / ) ( ω ω iγ / ) e When ω << ω, the electron vibrates 18 out of phase with the light wave: When ω = ω, the electron vibrates -9 out of phase with the light wave: When ω >> ω, the electron vibrates in phase with the light wave: 1 xe ( t) E( t) E( t) ( ω) 1 xe ( t) E( t) ie( t) ( iγ / ) 1 xe ( t) E( t) E( t) ( ω)

12 The relative phase of an electron cloud s motion with respect to input light depends on the frequency. Below resonance ω << ω Electric field at atom Electron cloud Weak vibration. 18 out of Let the atom s resonant frequency be ω, and the light frequency be ω. The electron charge is negative, so there s a 18 phase shift in all cases (compared to the previous slide s plots). On resonance ω = ω Above resonance ω >> ω Strong vibration. -9 out of Weak vibration. In

13 Okay, so we ve determined what the light wave does to the atom. Now, what does the atom do to the light wave?

14 Re-emitted light from an excited atom interferes with original light beam The re-emitted light may interfere constructively, destructively, or, more generally, somewhat out of phase with the original light wave. We model this process by considering the total electric field, + = z Incident light Emitted light Transmitted light E(z,t) = E original (z,t) + E re-emitted (z,t) Maxwell's Equations will allow us to solve for the total field, E(z,t). The input field will be the initial condition.

15 The Inhomogeneous Wave Equation The induced polarization, E z P! and e is the electron charge, and N = the electron number density. For our vibrating electrons: 1 c E t = µ P t, is due to the medium: where:!! P( t) = Nex ( t) e e! P( t) e 1 xe ( t) E( t) ωm ( ω ω iγ / ) e e 1 P( z, t) Ne E exp[ i( kz ω t)] ωm ( ω ω iγ / ) x We often also write: P E(z,t) or: P( t) = ε χ E( t) P = ε χe where: χ Ne 1 = ε ωm ( ω ω iγ / ) e

16 The Slowly Varying Envelope Approximation For now, we'll assume that the wave won't be dramatically affected by the induced polarization, which itself will not change in time. Let ( ω ) and ( ω ) E( z, t) = E ( z)exp i kz t P( z, t) = P ( z)exp i kz t Specifically, the envelope, E (z), is assumed to vary slowly; the fast variations will all be in the complex exponential, exp i kz Constant in time ( ωt) The time derivatives are easy (as before, they just bring down a ω ) because the envelopes are independent of t. The z-derivatives: E( z, t) E = + ike ( z) exp i kz t z z ( ω ) E( z, t) E E = ik k E + exp i ( kz ωt) z z z Because variations of the envelope, E (z), in space will be slow, we ll neglect the nd z-derivative.

17 The Slowly Varying Envelope Approximation Substituting the derivatives into the inhomogeneous wave equation: E ω ik k E E exp i ( kz ω t) = µ ω P exp i ( kz ωt) z c Now, because k = ω/c, the nd and 3rd terms cancel. And canceling the complex exponentials leaves: ik E z = µ ω P Or: E z = ik P ε If P is constant, the integration is trivial. Usually, however, P = P (E ) and hence P(z), too. Converting to finite differences, the re-emitted field at a given z will be: ik ΔE P Δz ε Note the i, which means that the re-emitted field has a 9 phase shift with respect to the polarization and hence a -9 phase shift with respect to the electron cloud motion.

18 The re-emitted wave leads the electron cloud motion by 9 in Below resonance ω << ω Electric field at Electron atom cloud Emitted field Weak emission. 9 out of This phase shift adds to the potential phase shift of the electron cloud motion with respect to the input light. On resonance ω = ω Strong emission. 18 out of Above resonance ω >> ω Weak emission. -9 out of

19 Solving for the slowly varying envelope P ε χe E Substituting for P, the SVEA becomes: This differential equation has the solution: z = ik P ε Define new quantities for the real and imaginary parts of χ: E k = i χ E z k E ( z) = E ()exp i χ z so that: α = k Im χ 1 ( n 1) = Re χ { α } E ( z) = E ()exp i [ i / + ( n 1) k] z where α is the absorption coefficient and n is the refractive index.

20 The complete electric field in a medium: The electromagnetic wave in the medium becomes (combining the slowly varying envelope with the complex exponential): { α } E( z, t) = E ()exp i[ i / + ( n 1) k] z exp[ i( kz ωt)] Simplifying: E (z) E( z, t) = E () exp[( α / ) z] exp[ i( nkz ωt)] Absorption attenuates the field Refractive index changes the k-vector

21 A light wave entering a medium Vacuum (or air) Medium n = 1 n = λ Absorption depth = 1/α k nk λ /n Wavelength decreases E( z, t) = E () exp[ i( k z ωt)] E () exp[( α / ) z] exp[ i( nk z ωt)] Typically, the speed of light, the wavelength, and the amplitude decrease, but the frequency, ω, doesn t change.

22 Absorption Coefficient and the Irradiance The irradiance is proportional to the (average) square of the field. Since E(z) α exp(-αz/), the irradiance is then: I(z) = I() exp(-α z) where I() is the irradiance at z =, and I(z) is the irradiance at z. Thus, due to absorption, a beam s irradiance exponentially decreases as it propagates through a medium. The 1/e distance, 1/α, is a rough measure of the distance light can propagate into a medium.

23 Refractive index and Absorption coefficient n comes from the real part of χ: (n 1)k = k Re{ χ} = k Re ε ε α comes from the imaginary part of χ: α / = Simplifying: k Im{ χ} = k Im ε ε n 1 = α = Ne / m e ω (ω ω iγ / ) Ne / m e ω (ω ω iγ / ) Ne (ω ω ) 4ε ω m e (ω ω ) + (γ / ) Ne γ / ε c m e (ω ω ) + (γ / ) These results are valid for small values of these quantities.

24 Refractive index and Absorption coefficient ω Frequency, ω Ne γ / Ne ω ω α = = ε ( ω ω) ( γ / ) 4 ε ω ( ω ω) ( γ / ) n 1 c me + me +