OVERVIEW OF NONLINEAR OPTICS AND HARMONIC GENERATION POLARIZATION AND SUSCEPTIBILITY The interaction of photons with matter is through the susceptibility of the material From the wave equation we have v v E P E 0 0 t t t 2 where the polarization is v v P 0 E and is the complex tensor susceptibility v v 2E 2P E 0 0 2 0 2 t t 2 2 2 n E P 2 0 2 2 c t t 2 The first term is for free space propagation The second term is for the generation absorption of radiation and dispersion The polarization is actually a function of higher orders of the electric field v P 0 n E n 0 1E 2 E 2 3 E 3 n 1 1 is for the absorption and emission of photons as described by the Einstein B coefficients They involve only two waves one in one out 2 3 are for multiphoton processes If the material the electric field propagates through has nonlinear susceptibilities n n 1 then it has the ability to interact with higher powers of the electric field This leads to harmonic generation 1 3 WAVE MIXING Let s assume that we have 3 wave mixing k1 k2 2 1 2 3 energy conservation 1 3 k3 z Ej 1 E j 0 exp i j t k j z cc 2 where cc complex conjugate and 1 E j 0 2 n k E j 0 z Each frequency component must satisfy the wave equation 2 2P E j 0 0 2 E j 0 2 t t 2 and the total electric field amplitude is the sum of all three fields Etotal E1 E 2 E3 The polarization is 2 P 0 1 Etotal 2 Etotal 3 1 0 1 E j 0 exp i j t k j z cc j 1 2 3 1 0 2 E j 0 exp i j t k j z cc j 1 2 3 1 El 0 exp i l t k l z cc l 1 2 When we multiply out the terms for the nonlinear polarization we get terms that correspond to each frequency sums and differences of frequencies We can pick out those terms for 1 2 and 3 For example for 1 3 2 P2 1 0 2 2 E 30 E 20 exp i 3 2 t k 3 k 2 z cc 2 whereas for 3 1 2 P2 3 0 2 2 E10 E20 exp i 1 2 t k1 k 2 z cc where E10E20 is the source term for 3 based on E1 and E2 The right side of the wave equation then becomes for 1 2 E1 2 P1 0 0 0 t 2 t 2 0 0 i 1 2 0 i 12 0 2 1 E10 exp i 1t k1 z cc 2 1 E10 exp i 1t k1 z cc 0 i 12 0 2 E30 E 20 exp i 1 k 3 k 2 z cc NOTE 0 0 12 1 1 0 1 12 12 n12 c 2 k12 k1 0 1 1 2 1 The gradient term to the wave equation is 2 1 E j 2 E j 0 exp i j t k j z cc z 2 2 Since k j 2 E j 0 1 E j0 i 2k j k 2j E j 0 exp i j t k j z cc 2 2 z z E j 0 2 E j0 then z z 2 E j 0 1 2 E j k 2j E j 0 i 2k j exp i j t k j z cc 2 z So if we now group terms having the same frequency exp i j t we get 3 dE10 0 12 ik1 exp ik1 z 0 2 E30 E 20 exp i k 3 k 2 z dz 2 Divide by ik1 exp ik1z 1 dE10 i 2 1 0 0 2 E30 E 20 exp i k 3 k 2 k1 z dz 2 1 1 dE 20 i 2 0 2 0 2 E10 E30 exp i k1 k 3 k 2 z dz 2 2 1 dE30 i 3 0 2 0 2 E10 E 20 exp i k1 k 2 k 3 z dz 2 3 OBSERVATIONS a The growth or decay of each field depends on the other two fields A field at E1 generates a field at E3 b In order that the beating between the waves does not average to zero the wave vectors must sum to zero For E30 this means k1 k2 k3 0 k3 k1 k2 SECOND HARMONIC GENERATION In second harmonic generation 2 1 1 2 3 12 1 E10 E20 Assume that we are perfectly phase matched k3 k1 k2 E 20 Assume E10 is real E10 E 20 E10 4 3 3 2 1 1 2 1 dE10 i 2 1 0 0 2 E30 E 20 dz 2 1 1 dE 20 i 2 0 2 dE 0 2 E10 E30 10 dz 2 1 dz In order for these equations to hold E30 must be imaginary that is 90 out of phase from E1 Defining iE30 E30 E1 E10 E 20 dE1 dE10 dE20 dz dz dz 1 2 dE1 E 1 1 0 0 2 E30 dz 2 1 1 dE30 3 0 2 E12 0 2 dz 2 3 4 The field E1 the sum of E10 and E20 at 1 produces an increase in the field E30 at 2 1 The energy lost from E1 to E30 produces a decrease in the amplitude of E1 5