Advanced Vitreous State The Physical Properties of Glass

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1 Advanced Vitreous State The Physical Properties of Glass Active Optical Properties of Glass Lecture 20: Nonlinear Optics in Glass-Fundamentals Denise Krol Department of Applied Science University of California, Davis Davis, CA

2 Active Optical Properties of Glass 1. Light emission (fluorescence, luminescence) Optical amplification and lasing Optical transitions, spontaneous emission, lifetime, line broadening, stimulated emission, population inversion, gain, amplification and lasing, laser materials, role of glass 2. Nonlinear Optical Properties Fundamentals: nonlinear polarization, 2nd-order nonlinearities, 3rd-order nonlinearities Applications: thermal poling, nonlinear index, pulse broadening, stimulated Raman effect, multiphoton ionization Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 2

3 Linear optics-classical electron oscillator electron Equation of motion for bound electron: m e d 2 x dt 2 = "m e# dx dt " Kx " ee 0e "i$t x(t) = x 0 e "i#t = -ee 0 m damping force binding force driving force Solution: electron oscillates at driving frequency 1 # 0 2 "# 2 " i#$ e"i#t ion core K = m e " 0 2 resonance frequency Oscillating dipole: p = "ex(t) Polarization: P(t) = "p i = #Nex(t) = Ne2 E 0 m 1 $ 0 2 #$ 2 # i$% e#i$t dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 3

4 Frequency dependent dielectric constant P(t) = " 0 #E(t) linear susceptibility "(#) = P(t) $ o E(t) = Ne2 $ o m 1 # 0 2 %# 2 % i#& electron ion core K = m e " 0 2 resonance frequency " D (#) =1+ $(#) = P(t) " o E(t) = Ne2 1 " o m # 2 0 %# 2 % i#& ω 0 dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 4

5 Nonlinear optics-anharmonic binding force linear optics electron nonlinear optics F x binding force F = "Kx ion core x F = "Kx " ax 2 (+higher order terms) Equation of motion for bound electron: m e d 2 x dt 2 = "m e# dx dt " Kx " ax 2 + (...) " ee 0 e "i$t damping force binding force driving force dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 5

6 Nonlinear optics-classical electron oscillator Equation of motion for bound electron: electron m e d 2 x dt 2 = "m e# dx dt " Kx + ax 2 " e E 0 e "i$t + cc [ ] damping force binding force driving force ion core This equation has no general solution, we will solve it by using a perturbation expansion. We replace E(t) by λe(t) and seek a solution in the form: x = "x (1) + " 2 x (2) + " 3 x (3) +... The parameter λ is a parameter that characterizes the strength of the perturbation. It ranges continuously between 0 and 1 and we will set it to 1 at the end of the calculation. Substitute expression for x in above equation of motion and group terms with equal powers of λ dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 6

7 Nonlinear optics-classical electron oscillator : x (1) + 2γ x (1) + ω 0 2 x (1) = ee(t)/m 2 : x (2) + 2γ x (2) + ω 2 0 x (2) + a x (1) [ ] 2 = 0 3 : x (3) + 2γ x (3) + ω 2 0 x (3) + 2ax (1) x (2) = 0 x (1) (t) " E(t) x (1) (t) = -ee 0 m 1 " 0 2 #" 2 # i"$ e#i"t x (2) (t) "[ E(t) ] 2 x (2) (t) = "ae2 2 E 0 $ 1 m 2 & D(2#)D 2 (#) e"i2#t + % 1 ' ) D(0)D(#)D("#) ( D(" j ) = (" 0 2 #" j 2 ) # i" j $ oscillates at 2ω oscillates at 0 dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 7

8 χ (2) resonances x (2) (t) "[ E(t) ] 2 x (2) (t) = "ae2 2 E 0 $ 1 m 2 & D(2#)D 2 (#) e"i2#t + % 1 ' ) D(0)D(#)D("#) ( 2nd-order nonlinear polarization P (2) (t) = "Nex (2) (t) oscillates at 2ω oscillates at 0 2nd-order susceptibility (for SHG) " (2) (2#) = P(2) (2#) = ane3 2 E 0 1 E(#) 2 m 2 D(2#)D 2 (#) D(" j ) = (" 0 2 #" j 2 ) # i" j $ ω 2ω ω 0 2nd-order susceptibility is enhanced when either ω or 2ω are equal to ω 0 dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 8

9 χ (2) frequency terms When the input field has 2 frequency components: E(t) = E 1 e "i# 1 t + E 2 e "i# 2 t + C.C the 2nd-order polarization has many frequency components (mixing terms): P (2) (t) = " (2) [E 2 1 e #i 2$ 1 t + E 2 2 e #i 2$ 2 t + 2E 1 E 2 e #i ($ 1 +$ 2 )t + 2E 1 E * 2 e #i ($ 1 #$ 2 )t + c.c] + 2" (2) [E 1 E * 1 + E 2 E * 2 ] = P($ n )e #i$ nt %. n P(2" 1 ) = # (2) 2 E 1 P(2" 2 ) = # (2) 2 E 2 P(0) = 2# (2) (E 1 E * 1 + E 2 E * 2 ) P(" 1 +" 2 ) = 2# (2) E 1 E 2 P(" 1 $" 2 ) = 2# (2) * E 1 E 2 (SHG), (SHG), (OR), (SFG), (DFG). Second harmonic generation Optical rectification Sum-frequency generation Difference-frequency generation dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 9

10 Nonlinear optics: general formalism r P (t) = " (1) # r E (t)+ " (2) # r E (t) 2 + " (3) # r E (t) 3 +### The em field is expressed as sum of frequency components, for example E(t) = E 1 e "i# 1 t + E 2 e "i# 2 t + c.c. The induced polarization can be written as: P(t) = $ n P(" n )e #i" n t The frequency components ω n are determined by the order of the interaction and the input frequencies 2nd-order interaction: linear nonlinear input frequencies, E induced polarization frequencies, P One ω 1 2ω 1, 0 Two ω 1, ω 2 2ω 1, 2ω 2, 0, ω 1 +ω 2, ω 1 -ω 2 Three ω 1, ω 2, ω 3 2ω 1, 2ω 2, 2ω 3,0, ω 1 +ω 2, ω 1 +ω 3, ω 2 +ω 3, ω 1 -ω 2, ω 1 -ω 3, ω 2 -ω 3 dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 10

11 3rd-order polarization and nonlinear index Applied field, one input frequency 3rd order nonlinear polarization E = E 0 e "i#t P (3) (t) = " (3) E(t) 3 has frequency components: P(3") = # (3) 3 E 0 P(") = 3# (3) E 2 * 0 E 0 The total polarization at ω has linear and nonlinear contributions P(") = # (1) E 0 + 3# (3) E 2 * 0 E 0 P(") = (# (1) + 3# (3) E 0 E * 0 )E 0 = # (eff ) E 0 " (eff ) = " (1) + 3" (3) 2 E 0 n = n 0 + n 2 I n 2 " # (3) n 2 = nonlinear index coefficient E 0 2 " I intensity dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 11

12 χ (3) - frequency terms With three input fields: ω 1, ω 2, ω 3 P (3) has the following frequency components: 3 1, 3 2, 3 3 1, 2, ± 2, 2 1 ± 3, 2 2 ± 1, 2 2 ± 3, 2 3 ± 1, 2 3 ± , , , dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 12

13 2nd-order polarization P(t) = P (1) (t) + P (2) (t) = " (1) E(t) + " (2) E(t) 2 P(t) = sin ("t) + 0.5sin ("t) 2 linear polarization sin ("t) nonlinear polarization "0.25cos( 2#t) 0.25 dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 13

14 χ (2) requires lack of inversion symmetry P(t) = P (1) (t) + P (2) (t) = " (1) E(t) + " (2) E(t) 2 P (2) (t) = " (2) E(t) 2 In material with inversion symmetry [ ] 2 "P (2) (t) = # (2) "E(t) P (2) (t) = "P (2) (t) # $ (2) = 0 Glass is isotropic " (2) = 0 dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 14

15 χ (2) tensor P and E are vectors! So χ (n) s are tensors P x (" n + " m ) = # (2) xxx (" n + " m," n," m )E x (" n )E x (" m ) P y (" n + " m ) = # (2) yxx (" n + " m," n," m )E x (" n )E x (" m ) dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 15

16 Nonlinear wave equation Propogation of waves is described by nonlinear wave equation "# r 2 E + $ r (1) % 2 E (1) = " 4& c 2 %t 2 "A 2 "z = 4#id eff$ 2 A k 2 c e i%kz "A 1 "z = 8#id eff$ 1 A k 1 c 2 2 A 1 c 2 % 2 r P NL %t 2 which leads to a set of coupled differential equations (example SHG) 2 * e %i&kz "k = k 2 # 2k 1 Growth of SH wave depends on propagation length and is optimized when Δk=o (phase matching) dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 16

17 Nonlinear optics in glass 2nd-order nonlinearities In normal glasses χ (2) =0 3nd-order nonlinearities All materials, including glasses, have a χ (3) In glass there are only three independent χ (3) tensor elements χ (3) processes involve the interaction of 3 input waves to generate a polarization (4th wave) at a mixing frequency -with 3 different input frequencies there are many possible output frequencies Strength of generated signal depends on propagation length -optical fibers! Phase matching: Δk=k 4 -k 3 -k 2 -k 1 =0 dmkrol@ucdavis.edu Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass 17