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 95616 dmkrol@ucdavis.edu 1
Active Optical Properties of Glass 1. Light emission Optical amplification and lasing (fluorescence, luminescence) 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 dmkrol@ucdavis.edu 2 2 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
Linear optics-classical electron oscillator electron Equation of motion for bound electron: m e d 2 x dt 2 = � m e � dx 2 K = m e � 0 dt � Kx � e E 0 e � i � t resonance damping force binding force driving force frequency ion core Solution: electron oscillates at driving frequency x ( t ) = x 0 e � i � t = -e E 0 1 2 � � 2 � i �� e � i � t m � 0 Oscillating dipole: p = � e x ( t ) P ( t ) = � p i = � N e x ( t ) = Ne 2 E 0 1 2 � � 2 � i �� e � i � t Polarization: m � 0 dmkrol@ucdavis.edu 3 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
Frequency dependent dielectric constant electron P ( t ) = � 0 � E ( t ) 2 linear susceptibility K = m e � 0 � o E ( t ) = N e 2 ( ) = P ( t ) 1 resonance � � 2 � � 2 � i �� frequency � o m � 0 ion core � o E ( t ) = N e 2 � D ( � ) = 1 + � ( � ) = P ( t ) 1 2 � � 2 � i �� � o m � 0 ω 0 dmkrol@ucdavis.edu 4 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
Nonlinear optics-anharmonic binding force nonlinear optics linear optics electron F ion core x x F = � Kx F = � Kx � ax 2 binding force (+higher order terms) Equation of motion for bound electron: d 2 x dt 2 = � m e � dx dt � Kx � ax 2 + (.....) � e E 0 e � i � t m e damping force binding force driving force dmkrol@ucdavis.edu 5 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
Nonlinear optics-classical electron oscillator electron Equation of motion for bound electron: d 2 x dt 2 = � m e � dx dt � Kx + ax 2 � e E 0 e � i � t + cc [ ] m e 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 λ E(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 6 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
Nonlinear optics-classical electron oscillator (1) + 2 γ ˙ (1) + ω 0 2 x (1) = − eE ( t )/ m : ˙ ˙ x x 2 = 0 (2) + 2 γ ˙ (2) + ω 0 2 x (2) + a x (1) 2 : [ ] ˙ ˙ x x (3) + 2 γ ˙ (3) + ω 0 2 x (3) + 2 ax (1) x (2) = 0 3 : ˙ ˙ x x x (1) ( t ) = -e E 0 1 2 � � 2 � i �� e � i � t x (1) ( t ) � E ( t ) m � 0 x (2) ( t ) = � ae 2 E 0 2 � � 1 1 2 D (2 � ) D 2 ( � ) e � i 2 � t + x (2) ( t ) � E ( t ) [ ] � � m 2 D (0) D ( � ) D ( � � ) � � oscillates at 2 ω 2 � � j “oscillates” at 0 2 ) � i � j � D ( � j ) = ( � 0 dmkrol@ucdavis.edu 7 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
χ (2) resonances x (2) ( t ) = � ae 2 E 0 2 � � 1 1 2 D (2 � ) D 2 ( � ) e � i 2 � t + x (2) ( t ) � E ( t ) [ ] � � m 2 D (0) D ( � ) D ( � � ) � � oscillates at 2 ω “oscillates” at 0 2nd-order nonlinear polarization P (2) ( t ) = � Nex (2) ( t ) 2nd-order susceptibility (for SHG) � (2) (2 � ) = P (2) (2 � ) E ( � ) 2 = aNe 3 E 0 2 1 ω 0 m 2 D (2 � ) D 2 ( � ) 2 ω ω 2 � � j 2 ) � i � j � D ( � j ) = ( � 0 2nd-order susceptibility is enhanced when either ω or 2 ω are equal to ω 0 dmkrol@ucdavis.edu 8 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
χ (2) frequency terms When the input field has 2 frequency components: 1 e � i � 1 t + E 2 e � i � 2 t E ( t ) = E + C . C the 2nd-order polarization has many frequency components (mixing terms): 2 e � i 2 � 1 t + E 2 2 e � i 2 � 2 t + 2 E 1 E 2 e � i ( � 1 + � 2 ) t + 2 E * e � i ( � 1 � � 2 ) t + c . c ] P (2) ( t ) = � (2) [ E 1 E 1 2 * + E 2 E 2 + 2 � (2) [ E * ] 1 E 1 � P ( � n ) e � i � n t . = n � 1 ) = � (2) E 2 P (2 ( SHG ), 1 Second harmonic generation � 2 ) = � (2) E 2 2 P (2 ( SHG ), * + E 2 E 2 Optical rectification P (0) = 2 � (2) ( E * ) 1 E ( OR ), 1 P ( � 1 + � 2 ) = 2 � (2) E Sum-frequency generation 1 E 2 ( SFG ), Difference-frequency generation P ( � 1 � � 2 ) = 2 � (2) E * 1 E ( DFG ). 2 dmkrol@ucdavis.edu 9 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
Nonlinear optics: general formalism r r r r ( t ) = � (1) � ( t ) + � (2) � ( t ) 2 + � (3) � ( t ) 3 + ��� P E E E linear nonlinear 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 ( � n ) e � i � n t P ( t ) = n The frequency components ω n are determined by the order of the interaction and the input frequencies 2nd-order interaction: input frequencies, E induced polarization frequencies, P One 2 ω 1 , 0 ω 1 Two ω 1 , ω 2 2 ω 1 , 2 ω 2 , 0, ω 1 + ω 2 , ω 1 - ω 2 Three ω 1 , ω 2 , ω 3 2 ω 1 , 2 ω 2 , 2 ω 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 10 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
3rd-order polarization and nonlinear index E = E 0 e � i � t Applied field, one input frequency P (3) ( t ) = � (3) E ( t ) 3 3rd order nonlinear polarization has frequency components: P (3 � ) = � (3) E 0 3 P ( � ) = 3 � (3) E 0 2 E 0 * The total polarization at ω has linear and nonlinear contributions * P ( � ) = � (1) E 0 + 3 � (3) E 0 2 E 0 � ( eff ) = � (1) + 3 � (3) E 0 P ( � ) = ( � (1) + 3 � (3) E 0 E 0 2 * ) E 0 = � ( eff ) E 0 2 � I E 0 n = n 0 + n 2 I n 2 � � (3) intensity n 2 = nonlinear index coefficient dmkrol@ucdavis.edu 11 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
χ (3) - frequency terms With three input fields: ω 1 , ω 2 , ω 3 P (3) has the following frequency components: 2 , 3 3 1, 3 3 2 , 3 1, 2 , 2 1 , 2 1 , 2 2 1 ± 1 ± 3, 2 2 ± 2 ± 3, 2 3 ± 3 ± 2 3 , 2 - 3 , 2 , 3 - 1 + 2 + 1 + 1 + 2 + 1 3 dmkrol@ucdavis.edu 12 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
2nd-order polarization P ( t ) = P (1) ( t ) + P (2) ( t ) = � (1) E ( t ) + � (2) E ( t ) 2 2 ( ) + 0.5sin � t ( ) P ( t ) = sin � t ( ) sin � t linear polarization ( ) � 0.25cos 2 � t nonlinear polarization 0.25 dmkrol@ucdavis.edu 13 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
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 � P (2) ( t ) = � (2) � E ( t ) 2 [ ] P (2) ( t ) = � P (2) ( t ) � � (2) = 0 � (2) = 0 Glass is isotropic dmkrol@ucdavis.edu 14 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
χ (2) tensor P and E are vectors! So χ (n) ’s are tensors (2) ( � n + � m , � n , � m ) E x ( � n ) E x ( � m ) P x ( � n + � m ) = � xxx (2) ( � n + � m , � n , � m ) E x ( � n ) E x ( � m ) P y ( � n + � m ) = � yxx dmkrol@ucdavis.edu 15 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
Nonlinear wave equation Propogation of waves is described by nonlinear wave equation � 2 r � 2 r �� 2 r E + � (1) (1) NL E = � 4 � P c 2 � t 2 c 2 � t 2 which leads to a set of coupled differential equations (example SHG) 2 � z = 4 � id eff � 2 � A 2 � k = k 2 � 2 k 1 2 e i � kz A 1 k 2 c 2 2 � z = 8 � id eff � 1 � A * e � i � kz 1 A 2 A 1 k 1 c 2 Growth of SH wave depends on propagation length and is optimized when Δ k=o (phase matching) dmkrol@ucdavis.edu 16 Advanced Vitreous State - The Properties of Glass: Active Optical Properties of Glass
Recommend
More recommend
