Wavepackets Outline - Review: Reflection & Refraction - Superposition of Plane Waves - Wavepackets - ΔΔk Δx Relations 1
Sample Midterm (one of these would be Student X s Problem) Q1: Midterm 1 re-mix (Ex: actuators with dielectrics) Q: Lorentz oscillator (absorption / reflection / dielectric constant / index of refraction / phase velocity) Q3: EM Waves (Wavevectors / Poynting / Polarization / Malus' Law / Birefringence /LCDs) Q4: Reflection & Refraction (Snell's Law, Brewster angle, Fresnel Equations) Q5: Interference / Diffraction
Electromagnetic Plane Waves The Wave Equation E y E ω y = ɛμ k = z t c E y = A 1 cos (ωt kz)+a cos (ωt + kz) E A H x = 1 A cos (ωt kz)+ η η cos (ωt + kz) H t 3
H reflected wave E Reflection of EM Waves at Boundaries E 1 (z =0)=E (z =0) incident wave Write traveling wave terms in each region Determine boundary condition transmitted wave Infer relationship of ω 1, & k 1, Solve for E ro (r) and E to (t) Medium 1 Medium E 1 = E i + E r ( =ˆx a Ei0 e jk1z + E r0 e +jk ) 1z ( =ˆx a Ei0 e jk1z + re i0 e +jk 1z ) E = E t =ˆxE a t0 e jk z =ˆxtE a i0 e jk z 4
H reflected wave Reflection of EM Waves at Boundaries E E 1 (z =0)= E (z =0) incident wave H 1 (z =0)= H (z =0) transmitted wave Write traveling wave terms in each region Determine boundary condition Infer relationship of ω 1, & k 1, Solve for E ro (r) and E to (t) Medium 1 Medium E 1 H 1 a =ˆx a =ˆy ( Ei0 e jk1z + E r0 e +jk ) 1z ( ) E i0 E e jk 1z r0 e +jk 1z η 1 η 1 At normal incidence.. 5 E H =ˆxE a t0 e jk z a =ˆy E r E t0 e jk z η r = 0 η η 1 = E0 i η + η 1
E-field polarization perpendicular to the plane of incidence (TE) Oblique Incidence at Dielectric Interface Write traveling wave terms in each region Determine boundary condition Infer relationship of ω 1, & k 1, Solve for E ro (r) and E to (t) E i =â y Ei0 e jk ixx jk iz z E-field polarization parallel to the plane of incidence (TM) x y z H i =ˆyH a i0 e jk ixx jk iz z 6
Oblique Incidence at Dielectric Interface Write traveling wave terms in each region Determine boundary condition E r =â y E r0 e jk rxx+jk rz z Infer relationship of ω 1, & k Solve for E ro (r) and E to (t) 1, E i =â y E i0 e jk ixx jk iz z E t =â y E t0 e jk txx jk tz z x y z 7
Oblique Incidence at Dielectric Interface Write traveling wave terms in each region Determine boundary condition E r =â y E r0 e jk rxx+jk rz z Infer relationship of ω 1, & Solve for E ro (r) and E to (t) k 1, E i =â y E i0 e jk ixx jk iz z E t =â y E t0 e jk txx jk tz z x y z Tangential field is continuous (z=0).. E i0 e jk ixx + E r0 e jk rxx = E t0 e jk txx 8
Tangential E-field is continuous Snell s Law E i0 e jk ixx + E r0 e jk rxx = E t0 e jk txx n 1 n REMINDER: ω k = = v p ωn c x y z k ix = k rx n 1 sin θ i = n 1 sin θ r θ i = θ r k ix = k tx n 1 sin θ i = n sin θ t 9
Reflection of Light (Optics Viewpoint μ 1 = μ ) TE: r = n 1 cos θ i n cos θ t n 1 cos θ i + n cos θ t E r E i H r TE H i Et H t H r Et E r Ht E i TM x Hi y E-field perpendicular to the plane of incidence E-field parallel to the plane of incidence z Reflection Coefficients TM: r = n cos θ i n 1 cos θ t n cos θ i + n 1 cos θ t r θ B θ c r Incidence Angle n 1 =1.44 n =1.00 θ i 10
Electromagnetic Plane Waves The Wave Equation E y E ω y = ɛμ k = z t c E y = A 1 cos (ωt kz)+a cos (ωt + kz) E A H x = 1 A cos (ωt kz)+ η η cos (ωt + kz) H t When did this plane wave turn on? Where is there no plane wave? 11
Superposition Example: Reflection E H incident wave reflected wave transmitted wave Medium 1 Medium How do we get a wavepacket (localized EM waves)? 1
Superposition Example: Interference Incident Light Path Reflected Light Pathways Through Soap Bubbles Constructive Interference (Wavefronts in Step) Reflected Light Paths Incident Light Path Outer Surface of Bubble Destructive Interference (Wavefronts out of Step) Reflected Light Paths Image by Ali T http://www.flickr.com/photos/7768540@n00/7 89338547/ on Flickr. Thin Part of Bubble Inner Surface of Bubble Thick Part of Bubble What if the interfering waves do not have the same frequency (ω, k)? 13
Two waves at different frequencies will constructively interfere and destructively interfere at different times time In Figure above the waves are chosen to have a 10% frequency difference. So when the slower wave goes through 5 full cycles (and is positive again), the faster wave goes through 5.5 cycles (and is negative). 14
Wavepackets: Superpositions Along Travel Direction ( ) E = E e j(ω 1t k 1 z) o + e j(ω t k z) ŷ 4π k + k 1 4π k k 1 REMINDER: k = n ω c SUPERPOSITION OF TWO WAVES OF DIFFERENT FREQUENCIES (hence different k s) 15
Wavepackets: Superpositions Along Travel Direction WHAT WOULD WE GET IF WE SUPERIMPOSED WAVES OF MANY DIFFERENT FREQUENCIES? f(k) + E = E o f (k) e +j(ωt kz) dk LET S SET THE FREQUENCY DISTRIBUTION as GAUSSIAN ( ) 1 (k k f (k o) )= exp πσ k σ k k o σ k 16 k REMINDER: k = n ω c
Reminder: Gaussian Distribution σ 50% of data within ± 0.4 0.3 0..1% 34.1% 34.1% 0.1.1% 0.1% 13.6% 13.6% 0.1% μ 1 f (x) = πσ e (x μ) σ μ specifies the position of the bell curve s central peak σ specifies the half-distance between inflection points FOURIER TRANSFORM OF A GAUSSIAN IS A GAUSSIAN [ ] F x e ax (k) = 17 π e a π k a
Gaussian Wavepacket in Space Re{E n (z,t o )} t = t o f(k) k o Re{E n (z,t o )} n σ k k SUM OF SINUSOIDS = WAVEPA CKET ( ) 1 (k ko) f (k) = exp πσ k σ k WE SET THE FREQUENCY DISTRIBUTION as GAUSSIAN 18
Gaussian Wavepacket in Time ( ) σ E(z,t) =E o exp k (ct z) cos (ω o t k o z) Re{E n (z,t o )} GAUSSIAN ENVELOPE t = t o Re{E n (z,t 1 )} t = t 1 z z k n Re{E n (z,t o )} n k n Re{E n (z,t 1 )} n 19
Gaussian Wavepacket in Space ( ) σ E(z,t) =E o exp k (ct z) cos (ω o t k o z) GAUSSIAN ENVELOPE WAVE PACKET λ o λ o = π k o In free space ω k = c this plot then shows the PROBABILITY OF WHICH k (or frequency) EM WAVES are MOST LIKELY TO BE IN THE WAVEPACKET f(k) Δz = 1 σk σ k k k o 0 ΔkΔz =1/
Gaussian Wavepacket in Time ( σ E(z,t) =Eo exp k (ct z) ) cos (ω o t k o z) WAVE PACKET λ o λ o = π k o Δk Δz Δz = Δk = c Δt n c Δω n UNCERTAINTY RELATIONS ΔkΔz =1/ ΔωΔt =1/ If you want to LOCATE THE WAVEPACKET WITHIN THE SPACE Δz you need to use a set of EM-WAVES THAT SPAN THE WAVENUMBER SPACE OF Δk = 1/(Δz) 1
Wavepacket Reflection E(z,t) E c E c
Wavepackets in -D and 3-D E(x, z)e (x, z) Contours of constant amplitude E(x, z)e (x, z) t = t o -D 3-D Spherical probability distribution for the magnitude of the amplitudes of the waves in the wave packet. 3
In the next few lectures we will start considering the limits of Light Microscopes and how these might affect our understanding of the world we live in incident photon electron Suppose the positions and speeds of all particles in the universe are measured to sufficient accuracy at a particular instant in time It is possible to predict the motions of every particle at any time in the future (or in the past for that matter) 4
Key Takeaways GAUSSIAN WAVEPACKET IN SPACE ( ) σ E(z,t) =E o exp k (ct z) cos (ω o t k o z) WAVE PACKET λ o λ o = π k o GAUSSIAN ENVELOPE UNCERTAINTY ΔkΔz =1/ RELATIONS ΔωΔt =1/ 5
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