Nonlinear Optics - Chapter 1 pptx

[...]... E1 , 0χ P ( 1 + ω2 + ω3 ) = 6 P ( 1 + ω2 − ω3 ) = 6 P ( 1 + ω3 − ω2 ) = 6 P (ω2 + ω3 − 1 ) = 6 P (2 1 + ω2 ) = 3 P (2ω2 + 1 ) = 3 P (2ω3 + 1 ) = 3 P (2 1 − ω2 ) = 3 P (2ω2 − 1 ) = 3 P (2ω3 − 1 ) = 3 2 E1 E2 , (3) 2 0 χ E2 E1 , (3) 2 0 χ E3 E1 , (3) 2 ∗ 0 χ E1 E2 , (3) 2 ∗ 0 χ E2 E1 , (3) 2 ∗ 0 χ E3 E1 , 0χ (3) (3) 3 E2 , 0χ P (3ω3 ) = 0χ (3) 3 E3 , (3) E1 E2 E3 , ∗ 0 χ E1 E2 E3 , (3) ∗ 0 χ E1... 2 1 If we again perform the summation over field frequencies in Eq (1. 3 .12 ), we obtain Pi (ω3 ) = (2) χij k (ω3 , 1 , 1 )Ej ( 1 )Ek ( 1 ) 0 (1. 3 .17 ) jk Again assuming the special case of an input field polarization along the x direction, this result becomes Pi (ω3 ) = (2) 2 0 χixx (ω3 , 1 , 1 )Ex ( 1 ) (1. 3 .18 ) Note that a factor of two appears in Eqs (1. 3 .15 ) and (1. 3 .16 ), which describe sum-frequency... convention, one finds that 1 1 (2) 2 P (2 1 ) = P (2ω2 ) = 0 χ (2) E22 , 0 χ E1 , 2 2 P ( 1 + ω2 ) = 0 χ (2) E1 E2 , P ( 1 − ω2 ) = 0 χ (2) E1 E2∗ , P (0) = 0 χ (2) E1 E1∗ + E2 E2∗ Note that these expressions differ from Eqs (1. 2.7) by factors of 1 2 8 1 ♦ The Nonlinear Optical Susceptibility We see from Eq (1. 2.7) that four different nonzero frequency components are present in the nonlinear polarization... of Nonlinear Optical Processes 13 and the negative of each Again representing the nonlinear polarization as ˜ P (3) (t) = P (ωn )e−iωn t , (1. 2 .16 ) n we can write the complex amplitudes of the nonlinear polarization for each of the positive frequencies as P ( 1 ) = (3) ∗ ∗ ∗ 3E1 E1 + 6E2 E2 + 6E3 E3 E1 , P (ω2 ) = 0χ (3) ∗ ∗ ∗ 6E1 E1 + 3E2 E2 + 6E3 E3 E2 , P (ω3 ) = P (3 1 ) = 0χ 0χ (3) ∗ ∗ ∗ 6E1 E1... = n0 + n2 I, (1. 2 .14 a) F IGURE 1. 2.5 Third-harmonic generation (a) Geometry of the interaction (b) Energy-level description 12 1 ♦ The Nonlinear Optical Susceptibility F IGURE 1. 2.6 Self-focusing of light where n0 is the usual (i.e., linear or low-intensity) refractive index, where n2 = 3 2n2 0 c 0 χ (3) (1. 2 .14 b) is an optical constant that characterizes the strength of the optical nonlinearity,... in Eq (1. 1.2) that the second-order contribution to the nonlinear polarization is of the form ˜ P (2) (t) = 0χ (2) ˜ E(t)2 , (1. 2.4) we find that the nonlinear polarization is given by ˜ P (2) (t) = 0χ (2) 2 2 E1 e−2i 1 t + E2 e−2iω2 t + 2E1 E2 e−i( 1 +ω2 )t ∗ ∗ ∗ + 2E1 E2 e−i( 1 −ω2 )t + c.c + 2 0 χ (2) E1 E1 + E2 E2 (1. 2.5) 1. 2 Descriptions of Nonlinear Optical Processes 7 It is convenient to express... simplicity, we have taken the ˜ ˜ fields P (t) and E(t) to be scalar quantities in writing Eqs (1. 1 .1) and (1. 1.2) In Section 1. 3 we show how to treat the vector nature of the fields; in such a case χ (1) becomes a second-rank tensor, χ (2) becomes a third-rank tensor, and so on In writing Eqs (1. 1 .1) and (1. 1.2) in the forms shown, we have also assumed that the polarization at time t depends only on... ωt) (1. 3.8) is represented by the complex field amplitudes 1 E(ω) = E eik·r , 2 1 E(−ω) = E e−ik·r , 2 (1. 3.9) 1. 3 Formal Definition of the Nonlinear Susceptibility 19 or alternatively, by the slowly varying amplitudes 1 A(−ω) = E 2 1 A(ω) = E , 2 (1. 3 .10 ) In either representation, factors of 1 appear because the physical field ampli2 tude E has been divided equally between the positive- and negative-frequency... Sections 2.2 and 2.4 F IGURE 1. 2.2 Sum-frequency generation (a) Geometry of the interaction (b) Energy-level description 1. 2 Descriptions of Nonlinear Optical Processes 9 1. 2.4 Difference-Frequency Generation The process of difference-frequency generation is described by a nonlinear polarization of the form ∗ P ( 1 − ω2 ) = 2 0 χ (2) E1 E2 (1. 2 .10 ) and is illustrated in Fig 1. 2.3 Here the frequency of... we find that Eat = 5 .14 × 10 11 V/m.∗ We thus expect that under conditions of nonresonant excitation the secondorder susceptibility χ (2) will be of the order of χ (1) /Eat For condensed matter χ (1) is of the order of unity, and we hence expect that χ (2) will be of the order of 1/ Eat , or that χ (2) 1. 94 × 10 12 m/V (1. 1.3) 2 Similarly, we expect χ (3) to be of the order of χ (1) /Eat , which for . 508 References 508 11 . The Electrooptic and Photorefractive Effects 511 11 .1. Introduction to the Electrooptic Effect 511 11 .2. Linear Electrooptic Effect 512 11 .3. Electrooptic Modulators 516 11 .4. Introduction. Parametric Amplification 10 5 2.9. Optical Parametric Oscillators 10 8 2 .10 . Nonlinear Optical Interactions with Focused Gaussian Beams 11 6 2 .11 . Nonlinear Optics at an Interface 12 2 Problems 12 8 References 13 2 3 Equation 5 61 13.3. Interpretation of the Ultrashort-Pulse Propagation Equation 567 13 .4. Intense-Field Nonlinear Optics 5 71 13.5. Motion of a Free Electron in a Laser Field 572 13 .6. High-Harmonic