From magneto-optics From magneto optics to ultrafast manipulation - PowerPoint PPT Presentation

From magneto-optics From magneto optics to ultrafast manipulation of magnetism Andrei Kirilyuk Radboud University Nijmegen, The Netherlands 1 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Outline of the lecture Light as a probe � linear magneto-optics linear magneto optics � nonlinear (magneto-)optics Example: all-optical FMR Light as an excitation � classification of effects � classification of effects � basics of opto-magnetism � coherent control � local control of spins p can this become too-ultrafast? 2 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Outline of the lecture Light as a probe � linear magneto-optics linear magneto optics � nonlinear (magneto-)optics Example: all-optical FMR Light as an excitation � classification of effects � classification of effects � basics of opto-magnetism � coherent control � local control of spins p can this become too-ultrafast? 3 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Radboud University Nijmegen 1.5 – 3.2 eV Andrei Kirilyuk, Targoviste – August 2011 Optics 4

Why are certain wavelengths “visible”? Transmission through water 1 km 1 km wave UV IR X ray X-ray Microw 1 m Radio 1 mm 1 µm 1 km 1 m 1 mm 1 µm 1 nm wavelength visible spectrum spectrum 5 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Optics 1.5 – 3.2 eV r r r r = ε ˆ ε = χ ˆ ε or D E P E 0 0 r r r r r r r r = ε + ε = + χ ˆ ˆ 1 r r D E P − ω = ( ) i k r t 0 E E e 0 ε = 2 2 n 6 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Anisotropic media ⎛ ⎞ ε ε ε ⎜ ⎜ ⎟ ⎟ xx xy xz ε = ε ε ε ˆ ⎜ ⎟ yx yy yz ⎜ ⎜ ⎟ ⎟ ε ε ε ⎝ ⎝ ⎠ ⎠ r zx zy zz E E y if H = 0 r E E ε ⎛ ⎛ ⎞ ⎞ 0 0 0 0 x ⎜ ⎟ xx ε ε = ε ε ˆ 0 0 0 0 ⎜ ⎜ ⎟ ⎟ yy ⎜ ⎟ ε 0 0 ⎝ ⎠ zz i.e. one could chose such a coordinate system 7 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Driven oscillator model ω sin E t 0 0 ⎛ ⎞ 2 d x ω ⎜ ⎟ + ω = 2 m x F ⎜ ⎟ 0 0 2 ⎝ ⎝ ⎠ ⎠ dt dt = ω sin F eE t 0 2 d x e + + ω ω = = ω ω 2 sin sin x x E E t t 0 0 2 dt m 8 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Driven oscillator model solution in the form: r r eE = ω = ω ( ) sin 0 sin ( ) ( ( ) ) x t x x t t E t 0 ω − ω e 2 2 m 0 amplitude lit d amplitude depends on ω x ω 0 ω 0 ω ω damped oscillator ω ° 0 0 phase p ° − 180 9 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Sum of the two waves: Δ z incoming + outgoing amplitude ( ( ) ) phase phase = = ω 0 0 sin i E E z E E t 0 10 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Phase of the light after transmission Δ z ⎛ ⎞ ( ) ( ) z = ω − ⎜ ⎜ ⎟ ⎟ sin i E E z E E t t 0 ⎝ ⎠ c Δ Δ z z − ) ( 1 extra time because of n : n c Δ Δ ⎛ ⎛ ⎞ ⎞ ( ) z z = ω − − − ⎜ ⎟ sin ( 1 ) E z E t n 0 ⎝ ⎠ c c Δ Δ z ω − ) ( 1 phase delay: n c z z = 0 ( ( ) ) = = ω 0 0 sin i thus if a phase delay occurs thus if a phase delay occurs, E E z E E t t 0 this is equivalent to the refractive index 11 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Absorption en refractive index multiple resonances: amplitude = absorption phase = refractive index γ γ ω − ω 2 2 / 2 Ne Ne α = − = 0 0 1 1 n ε ω − ω + γ ε ω ω − ω + γ 2 2 2 2 2 ( ) ( / 2) 4 ( ) ( / 2) c m m 0 0 0 0 0 e e γ accounts for damping t f d i 12 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Kramers-Kronig relations ε ε = = ε ε + + ε ε ( ) du ( ) ∞ ∞ i i ω − ε 2 1 1 2 ( ) u ∫ ε ω = 1 ~ 2 π π − ω ω 2 2 = = + + u u n n n n ik ik 0 ( ) d ( ) du ∞ ε ε 2 ( ) ( ) u u u u ∫ ∫ ε ω = 2 2 2 2 ε = n − 1 k π − ω 2 2 u 1 0 ε 2 = 2 2 nk k real and imaginary parts are not independent! 13 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Dispersion of glass ω λ 14 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Optical constants of metals Au Au Ni Ni 15 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Interaction of light with magnetic solids How does magnetic field (magnetization) modify dielectric tensor? dielectric tensor? r r r r = εε D E y 0 ⎡ ε ⎤ 0 0 xx xx ⎢ ⎢ ⎥ ⎥ ε = ε = 2 0 0 H =0 n ⎢ ⎥ yy x ⎢ ⎢ zz ⎥ ⎥ ε ε 0 0 0 0 ⎣ ⎣ ⎦ ⎦ if isotropic if isotropic z ⎡ ⎤ ? ? ? r M or H ⎢ ⎥ ε ε = = ? ? ? ? ? ? E E H ≠ 0 H ≠ 0 ⎢ ⎢ ⎥ ⎥ ⎢ ⎥ ? ? ? ⎣ ⎦ 16 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

How does magnetic field modify conductivity? g y y y σ ⎡ ⎤ 0 0 r r xx xx ⎢ ⎢ ⎥ ⎥ H 0 H=0 E j σ = σ = σ 0 0 ⎢ ⎥ j E yy ⎢ ⎥ σ 0 0 ⎣ ⎣ ⎦ ⎦ x x zz zz y H E j F L e[ V H ] F L =e[ V × H ] Lorentz force Lorentz force x E y → j x E H E x → j y j σ σ ⎡ ⎤ 0 j H xx xy ⎢ ⎢ ⎥ ⎥ σ = σ σ 0 0 ⎢ ⎥ yx yy E ⎢ ⎥ σ 0 0 ⎣ ⎦ zz 17 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Onsager principle: symmetry of kinetic coefficients symmetry of kinetic coefficients ε ε ⎡ 0 ⎤ ε ⎡ ⎤ 0 0 xx xy ⎢ ⎢ ⎥ ⎥ xx ⎢ ⎢ ⎥ ⎥ ε = ε ε 0 ε = ε H ≠ 0 0 0 H =0 ⎢ ⎥ ⎢ ⎥ yx yy yy ⎢ ⎥ ε ⎢ ⎥ ε 0 0 0 0 ⎣ ⎦ ⎣ ⎦ zz zz S = S ∂ ∂ ij ji W X X = = X S S f f f f i ij j ∂ ∂ t t If S is a function of magnetic field X - response S (H) S ij (H)=-S ji (H)=S ji (-H) S (H) S ( H) f - stimulus W - energy ∂ ∂ W D = ε Onsager principle = ε ij (H)=- ε ji (H)= ε ji (-H) D E E ∂ ∂ is applicable to ε i ij j t t ε ij = ε ji * in non-absorbing media L Landau & Lifshitz, Theoretical Physics, vv. 5 and 8. d & Lif hi Th i l Ph i 5 d 8 18 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Faraday effect – 1 Isotropic medium in a magnetic field: ε − ε ⎛ ⎛ ⎞ ⎞ ε ∝ 0 i M ⎜ ⎟ 0 xy xy z ε = ε ε ˆ 0 ⎜ ⎜ ⎟ ⎟ i 0 0 xy xy ⎜ ⎜ ⎟ ⎟ ε 0 + ε ε ∝ 2 0 0 ⎝ ⎠ M zz zz z r r r = + r r r E i E j E in x y H = = r H H H E E z out E in r = ??? E x out z y 19 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Faraday effect – 2 To find the eigenvalues of the problem: ε − ε ⎛ ⎛ ⎞ ⎞ ⎛ ⎛ ⎞ ⎞ 0 0 i i E E ε − ε ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ 0 ⎟ ⎜ ⎟ xy x r r r i E E ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎟ = xy x 2 x 2 = ε = ε ε = ˆ 0 ⎜ ⎟ ⎜ ⎟ n ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ D E i E n E ε ε 0 xy y i E E ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ ⎝ ⎝ ⎠ ⎠ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ 0 0 xy xy y y y y ε 0 0 0 ⎝ ⎠ ⎝ ⎠ 0 2 = ε ± ε ε − 2 − ε ⇒ n n i 0 0 = xy xy 0 ε ε ε ε − 2 1 ε ε i i n n 1 − − 0 xy − ε 3 4 ≅ ε ± xy ~ 10 10 n 0 xy ε 2 0 ⎛ 1 ⎞ ⎛ 1 ⎞ = ± ± ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ eigenmodes g , or , or and and E E iE iE ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ x x y y ⎝− i ⎝ i ⎠ ⎠ 20 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Faraday effect – 3 = ± E iE x y Two circularly polarized waves 1 ε xy 1 with different refractive indices: ith diff t f ti i di ≅ ε ± xy n ± 0 ε 2 0 r r r l E E E E E E out − + − + in α F F = = + + ε ε π l π 2 2 l α = xy Faraday rotation: λ ε F 0 M. Faraday, On the magnetization of light and the illumination of magnetic lines of force , Phil. Trans. R. Soc. Lond. 136 , 104 (1846). ( ) f f 21 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Kerr effect: various geometries θ θ θ θ θ r r r M M M r H θ >> θ >> θ ≈ o o o 0 0 0 polar longitudinal transverse 22 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011

Magnetic linear birefringence light propagates along x axis, so that r r r r r = + E j E k E in y z H z ε − ε ⎛ ⎞ 0 i x ⎜ 0 ⎟ xy ε ε = = ε ε ε ε ˆ 0 0 ⎜ ⎜ ⎟ ⎟ y y i i 0 xy ⎜ ⎟ ε + ε 0 0 ⎝ ⎠ 0 zz ε ε + ε ε zz ∝ 2 2 , Eigenvalues M 0 0 zz r r , Eigenmodes E y E z 23 Radboud University Nijmegen Andrei Kirilyuk, Targoviste – August 2011