Lecture 2: Propagation of light waves and pulses Petr Kužel

• Propagation of plane waves in linear homogeneous isotropic (LHI) media: ! refractive index ! intensity ! absorption • Propagation of light pulses: ! group velocity ! group dispersion — chirp ! superluminal effects (?) Wave equation

∂B  ∂2 E N 2 ∂2 E ∇ ∧ E + = 0 ∇2 E − εµ = 0 ∇2 E − = 0 ∂  0 ∂ 2 2 ∂ 2 t  ⇒ t or c t ∂D ∂2 H N 2 ∂2 H ∇ ∧ H − = 0 ∇2 H − εµ = 0 ∇2 H − = 0 ∂t  0 ∂t 2 c2 ∂t 2

2 1 (∇ ∧ ()∇ ∧ E = ∇∇ ⋅ E − ∇ E ) ε = ε ε c = 0 r ε µ 0 0 well-known identity ε = ε′ − iε′′ r r r 2 = εµ 2 = ε N 0c r dielectric constant refractive index

ε′ = 2 − κ2 ε′′ = κ r n , r 2n , = − κ N n i 1 1 n = ( ε′2 + ε′′2 + ε′ , κ = )( ε′2 + ε′′2 − ε′ ) 2 r r r 2 r r r Propagation in the vacuum

1 ∂2 E = i()ωt−k⋅r ∇2 E − = 0 E E0e 2 ∂ 2 ω c t with k ≡ k = 1 ∂2 H () c ∇2 H − = 0 H = H ei ωt−k⋅r c2 ∂t 2 0

• Back to Maxwell’s equations:

∂B 1 ∇ ∧ E + = −ik ∧ E + iωµ H = 0 ()k ∧ E = H ∂ 0 µ ω 0 0 t 0 ∇ ⋅ E = −ik ⋅ E = 0

⊥ ⊥ ⊥ k E0 H0 k s = k k = η−1 ()∧ H0 0 s E0 µ 0 − η = ≈ 377Ω = 1 ()∧ 0 ε B0 c s E0 0 Superpositions of plane waves

•In the k-space: 3-dimensional ()= ()i()ωt−k⋅r E t,r ∫∫∫E0 k e dk ()= ()i()ωt−k⋅r H t,r ∫∫∫H0 k e dk •In the ω-space: 1-dimensional (propagation along z: k = k = 0) ω x y ()ω E()t, z = ∫ E ()ei t−kz dω 0 ω ()= ()i()ωt−kz ω H t, z ∫ ωH0 e d ω ω ω ω • Pulse: central frequency 0; central wave vector k0 = 0/c. ω ω ω ω ()− − ()()()()− − − ()= ()i 0t k0z = i 0t k0z ()i 0 t k k0 z = E t, z E1 t, z e ω e ∫ E0ω e d () − ()− ()− ()ω − = i 0t k0z ()i 0 t z c ω = ()− i 0t k0z e ∫ E0 e d E1 t z c e ω Propagation in the matter

N 2 ∂2 E = i()ωt−k⋅r ∇2 E − = 0 E E0e 2 ∂ 2 ω c t with k ≡ k = N N 2 ∂2 H () c ∇2 H − = 0 H = H ei ωt−k⋅r c2 ∂t 2 0

• Back to Maxwell’s equations: ∂B 1 ∇ ∧ E + = −ik ∧ E + iωµ H = 0 ()k ∧ E = H ∂ 0 µ ω 0 0 t 0 ∇ ⋅ E = −ik ⋅ E = 0

⊥ ⊥ ⊥ k E0 H0 k s = k k N H = ()s ∧ E 0 η 0 η µ 0 η = 0 = 0 N N ε B = ()s ∧ E 0 c 0 Propagation direction (vector k) k = k′ − ik′′ • k' = n ω/c, k'' = κ ω/c • k' and k'' can have different different directions: ! k'∙r = const: constant phase planes ! k''∙r = const: constant amplitude planes • k'': boundary conditions of an absorbing sample

k i α α kr n1

= − κ N2 n2 i 2 z k''t β' k't y x Propagation direction (vector S) • Energy flow (Poynting vector S)

= ∧ = 1 { ∧ ∗} S E H 2 Re E H

1 ∗ ∗ 1 ∗ ∗ ∗ ∗ S = {}Re E ∧ ()k ∧ E = Re{}k ()E ⋅ E − (k ⋅ E)E = 2ωµ 2ωµ #"! 0 0 0 1 ∗ − ′′⋅ k′ 2 − ′′⋅ = {}Re ()k′ − ik′′ ()E ⋅ E e 2k r = E e 2k r ωµ 0 0 ωµ 0 2 0 2 0

• Propagation along z (k // z)

− ω 2 − ωκ n 2 −α S = ()2ωµ 1 n E e 2 z c = E e z 0 0 η 0 c 2 0 ωκ πκ α()ω = 2 = 4 c λ Optical pulses: dispersion

• Dispersion relation (non-absorbing medium): ω k = n()ω c • Optical pulse as a linear combination of eigenmodes: ()= ()i()ω()k t−kz = ()ω ()i ωt−k()ω z ω E t, z ∫ E0 k e dk ∫ E0 e d ω ∆ω ω ∆ • Mean frequency 0 and wave vector k0: ‹‹ 0, k ‹‹ k0:  ω   2ω  β ω()= ω + d ()− + 1  d  ()− 2 = ω + − ()+ ()− 2 k 0   k k0   k k0 0 v g k k0 k k0  dk  2 2 2 0  dk 0    2  ψ ()ω = + dk ()ω − ω + 1  d k  ()ω − ω 2 = + −1 ω − ω ()+ ()ω − ω 2 k k0   0   0 k0 v g 0 0  dω  2 dω2 2 0  0

ψ = −β 3 v g … group velocity dispersion (GVD) Group velocity (first order effects)

• Propagation of a pulse ∞ ∞ ()ω − i()v (k−k )t−(k−k )z ()ω − i(v t−z)(k−k ) ()= ()i 0 t k0z g 0 0 = i 0 t k0z () g 0 = E t, z ∫ E0 k e e dk e ∫ E0 k e dk −∞ −∞ ()ω − − ()− == ei 0t k0z E ()z −v t = e ik0 z vt E ()z −v t 1 g #$" $! 1 g ( A) #$" $! (B) •(A): field oscillation with the central frequency ω c v = 0 = wave front velocity: ω k0 n( ) •(B): propagation of envelope without deformation ω = d group velocity v g dk ω 0 ω c k = n()ω d dk ⇒ v = c g n + ωdn dω Optical pulses: energy flow

• Poynting vector µ 1 0 n 2 − ωκ c = , η = S(z −v t) = E (z −v t) e 2 z / c ε µ 0 ε g η 1 g 0 0 0 2 0 • Energy conservation law

ωκ ∂ U ω ∗ 1  d()ωε′ ∂E ∗ ∂E ∗  e2 z / c = ε′′E E +  1 ⋅ E + ε′ 1 ⋅ E  = ∂t 2 1 1 2  dω ∂t 1 ∂t 1 

∗  ω  ∂E ∗ = ω nκ ε E E + n2 + dn2 dω ε 1 ⋅ E 0 1 1  2  0 ∂t 1 #$"$ $!$ n()n+ωdn dω =nc vg

n ∂ 2 − ωκ ∇ ⋅ S = E (z −v t) e 2 z / c = η ∂ 1 g 2 0 z   ωnκ ∗ n ∂E ∗ − ωκ ∂ U = − E ⋅ E + 1 ⋅ E  e 2 z / c = −  η 1 1 η ∂ 1  ∂  0c 0v g t  t Group velocity dispersion GVD (second order effects)

• Main effect: pulse broadening in time • This treatment is necessary for ! Ultrashort pulse (sub-50 fs) propagation ! Ultrashort pulse (ps or sub-ps) generation in lasers (multiple passes through dispersive media) ! Short pulse (sub-ns) propagation in fibers (extremely long distances) β ω()k = ω +v k − (k + )()k − k 2 0 g 0 2 0

− ψ k()ω = k +v 1 ω− (ω + )()ω − ω 2 0 g 0 2 0 Intuitive treatment

• Spectral components coming from two ∆ω ends of the spectrum propagate with different velocities dv ∆ = g ∆ω ω z v v (ω ) vg( 2) g dω g 1 • GVD is defined as 2   d k d  1  1 dv g ψ = = = − ω ω2 ω   2 ω 0 ω d d v g  v g d • Spreading due to a thickness L of the dispersive material:

L L dv  ψ L  ∆τ ≈ ∆v = g ∆ω = L ψ ∆ω = 4ln 2  2 g 2 ω  τ  v g v g d  0  for a gaussian pulse shape Exact treatment

• Superposition of the spectral components

∞ ∞ ()ω − ω ω −()i(ω−ω )(t−z v ) iβz (2v3 ) (ω−ω )2 ()= ω ()i t k( )z ω = i 0 t k0z ()ω 0 g g 0 ω E t, z ∫ E0 e d e ∫ E0 e e d −∞ −∞

• One gets for a gaussian pulse for a crystal length L after all the integrations: α()t−z v 2 ()1−iδ − g ()ω − 1 (δ2 + ) 4ln 2 ψL E()t, z = Aei 0 t k0z e 1 with δ = 2αψL = + δ τ2 1 i 0

• Spreading due to a thickness L of the dispersive material: 2  4ln 2 ψL   ψ  τ()L = τ 1+ δ2 = τ 1+   ∆τ ≈ 4ln 2 L  0 0  τ2   τ   0   0  (intuitive treatment) z = 0 z = z 0 z = 2z0 z = 3z0 E

(A)

(B)

(C)

z ∂Φ ()t − z v 4α2 ψL ω()= = ω + g Mean frequency varies in time: z,t 0 ∂t 1+ ()2α ψL 2 ∂2Φ GDD = − = ψL Chirp (group delay dispersion): ∂ω2 Pulse spreading: example

120 Dispersion of a BK7 plate

100

80

60 Thickness: 10 mm Pulse length after the plate (fs) 40 6 mm 3 mm

20 20 40 60 80 100 Bandwidth-limited pulse length (fs) “Superluminal” effects

• What can be superluminal? ! Phase velocity? ! Group velocity? ! Velocity of an information? • Definition of the speed of an information transfer Information speed

Amplification process

The noise can only slow down the information transfer Phase velocity

k = 0, v = ∞ k cosθ, v/cosθ

θ

k, v Group velocity

c v = g dn n + ω dω

• Group velocity can exceed the vacuum speed of the light: ! region of anomalous dispersion " dielectric resonance " gain medium 10 10 Dielectric 8 8 resonance 6 6

4 4 refractive index • group velocity absorption index 2 2 c v = g dn 0 0 n + ω 0 1 2 3 4 dω Frequency (arb. u.)

1.5 3 λ c 1 • group velocity can be 2 0.5 superluminal 0 • penetration depth = 1

fraction of λ -0.5 group velocity normalized to penetration depth normalized to -1 0 0 1 2 3 4 Frequency (arb. u.) Gain-assisted superluminal light propagation L. J. Wang, A. Kuzmich & A. Dogariu NEC Research Institute, 4 Independence Way, Princeton, New Jersey 08540, USA

Gain-assisted Measured pulse advancement Conclusion: anomalous dispersion: corresponds to a negative group • pulse maximum and velocity leading / trailing edge of the pulse are advanced • there is almost (!!) no change in the pulse shape • the very first non-zero signal can never (!!) be advanced