- 8 3 Show That Equations 8 6 22 Are Consistent With The Fact That The Increases In The Photon Flux At W And W Are Ide 1 (13.5 KiB) Viewed 70 times
- 8 3 Show That Equations 8 6 22 Are Consistent With The Fact That The Increases In The Photon Flux At W And W Are Ide 2 (49.31 KiB) Viewed 70 times
- 8 3 Show That Equations 8 6 22 Are Consistent With The Fact That The Increases In The Photon Flux At W And W Are Ide 3 (49.98 KiB) Viewed 70 times
separation SECOND HARMONIC GENERATION AND PARAM 8 that is filled with a medium whose polarization is given by (8.6-11). Using the This can be done by considering a parallel plate capacitance of area and relations (8.6-12) din) = (0) + p = e(t) the dielectric constante can be written as *(1 + ) + de and the capacitance C = Ns as (8.6-13) (1 + X)A. Ad C 5 If the electric field is given by es -E, sin , the capacitance becomes sin w (8.6-14) cell + x1 _ AdEo 5 . points of view used which is of a form identical to (8.6-4). It follows that the two to describe parametric processes--the one represented by (8.6-4), in which an es ergy-storage parameter is modulated, and that in which the electric (or magnetic) response is nonlinear, as in (8.6-11)—are equivalent. We return now to the basic nonlinear parametric equations (8.2-13) to analyze the case of optical parametric amplification. We find it convenient as in (8.3-22) to introduce a new field variable, defined by RE E 1 = 1,2,3 (8.6-15) so that the power flow per unit area at is given by P, . ER WAP (8.616) A when n; is the index of refraction at . The power flow P/A per unit area is related to the flux N. (photons per square meter per second) by PA 1 Nλω, • Arai (8.6-17) 2 VM so that AP is proportional to the photon flux at w. The equations of motion (8.2-13) for the A, variables become A The electric displacement dit) should not be confused with the nonlinear constant din (8.6-11).
PARAMETRIC AMPLICATION 305 dA - - 214,- KjAxe*** -KAN dA; dz dA dz as, - xli lze Kalea (8.6-18) where Ak=ky - (k: + k) Kd టర్నూలు, ni EO 40 1 = 1, 2, 3 (8.6-19) The advantage of using the A instead of E, is now apparent since, unlike (8.2-13). relations (8.6-18) involve a single coupling parameter We will now use (8.6-18) to solve for the field variables A (2), A_{2), and A3(2) for the case in which three waves with amplitudes ACO), A_CO), and Ag(0) at fre- quencies w. wz, and w, respectively, are incident on a nonlinear crystal at z = 0. We take w, - , + w, a, = az = ax = 0 (no losses), and Ak = ks - ks - k = 0. In addition, we assume that w, A.(2) and Az(2) remain small compared to |A,(0) throughout the interaction region. This last condition, in view of (8.6– 17), is equivalent to assuming that the power drained off the pump" (at w.) by the "signal" (w) and idler (2) is negligible compared to the input power at W. This enables us to view A,(z) as a constant. With the assumptions stated above, equations (8.6-18) become dA 1 A3 dA3 _ig A, (8.6-20) dz dz 2 where ve Eo WW2 8 = KA,(0) DE:(0) (8.6-21) 7112 The solution of the coupled equations (8.6-20) subject to the initial conditions A, (z = 0) = A (0), Az(z = 0) = A20),A,O) = A (0), i.e., g is real, is A(z) = A (0) cosh iA (0) sinh2 2- be Az) = 43(0) cosh 8 + LA,(0) sinh : Z (8.6-22)