Lasers with Nonlinear Parameters in .NET Development GTIN - 13 in .NET Lasers with Nonlinear Parameters .NET barcode

Lasers with Nonlinear Parameters using barcode drawer for none control to generate, create none image in none applications.ean 13 creating ~ ~ ~comb i none for none ~ F (n0 + n ) + F +q (n0 + n +q ) f f +q f k +q Gk = Fk . ~ fk 4 ,q 1 + iq / . gs1 databar (7.91). Assuming that t none for none he medium is sufficiently slow in its relaxation ~ ( / = / . >> 1 ) a nd passing over to the real amplitudes and phases ( f k = Fk exp i k ) we write instead of Eq. (7.91).

~ 2n + n + n + q Fk + q F F + q ~ Re Gkcomb cos kq , (7.92) 0 q Fk 4 ,q where (7.93) Ba sed on the assumption that phased mode locking occurs, i.e.

, kq = 0 , the laser spectrum has a rectangular form, Fk F , and, in addition, n kq n , we have. kq = k + q k + q + . ~ n ( A 1) 1 2k / N 2k ~ Re Gkcomb + . ln 2 1 + 2k / N N (7.94). The enumeration of the modes is such that the spectrum boundaries correspond to the indices k = ( N 1) / 2 . The function F ( k / N ) in the brackets is different from a linear one and, therefore, the expression (7.94) does not satisfy the phase equations (7.

90). Thus, we infer that mode locking due to the combination tone interaction in a slowly relaxing laser medium is not possible. The assumption that all the quantities nkq are equal is reasonable only in the case of uniform saturation of the laser medium.

Undoubtedly, such an assumption is valid for an unidirectional ring laser. In a Fabry Perot cavity, a situation similar to that occurs when many modes are excited or when the laser threshold is only slightly exceeded. Moreover, the laser medium should uniformly fill the cavity or be concentrated in a relatively thin layer near the mirror.

Any other arrangement of the laser rod changes the situation with n , but this fact is not essential to the conclusion that the phase self-mode-locking is not possible. In steady state the form of the spectrum is typically parabolic rather than rectangular. However, this is not very important in this context.

Only the finite width of the spectrum matters. If the laser frequency band is ~ infinite, then the condition kq = 0 leads to Re Gkcomb 0 making the selfmode-locking (spontaneous formation of -pulses) feasible. ~ Assuming that the laser medium is fast, i.

e., / << 1 Eq. (7.

91) transform to. Fundamentals of Laser Dynamics f k +q f f +q i ~ ~ Gkcomb = Fk ( 2n 0 + n + n + q ) . fk 4 ,q (7.95). Disregarding the nonequidistancy of the cavity eigenfrequences Eqs. (4.78) are written as Fk + q F F + none none q d k 1 = G (2n0 + n + n +q ) sin kq . (7.96) Fk d 4 ,q These equations are satisfied by the steady states kq = 0 , which correspond to phased mode locking, and the problem is reduced to studying the stability of such a solution.

Complete analysis of the laser stability of the mode-locked state is rather time consuming. However, it is relatively easy to investigate the laser stability with respect to a particular form of perturbation involving only the mode phases. Assume that the phase deviation of the k th mode, k = k k , is small, and linearize Eqs.

(7.96) near kq = 0 . The linearized equations written as.

Fk + q F F + q 1 d( k ) 1 = G ( 2n0 + n + n + q ) , Fk k d 4 ,q (7.97). have a positive right-hand side. This is an indication that the steady-state solution is unstable. This result means that combination tone mode-mode coupling in an inertialess laser medium does not lead to phase locking either.

Meanwhile, it is seen from Eq. (7.97) that a sign reversal of the population difference changes the situation.

Thus, we have arrived at the idea of passive mode locking in a laser with a saturable absorber.. 7.4.2 Threshold Conditions of Passive Mode Locking In the presence of a saturable absorber the phase equations (4.78) transform to d k 1 1 ~ = G ck ~ k + G (Re Gk + Re Gak ) . d 2 (7.98). Consider the mo none for none st important practical case where the transition frequencies and the active and the absorption media coincide ( 0 = a ) and the absorber is fast ( / . a << 1 ). If the lasing modes occupy the frequency band ( N << a ), then for the saturable absorber the expression
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