INCREASING THE EFFICIENCY OF LASER RADIATION AMPLIFIERS G. D. Divin, L. V. Ivanushkina, and B. M. Sedov

UDC 621.325

V. I. Korolev,

The efficiency of laser equipment constructed for the purpose of amplifying the radiation of a master laser with amplifying stages is determined mainly by the efficiency of the latter. It is known (e.g., see [i, 2]) that the efficiency of an amplifier, which one can define as the ratio of the energy removed from excited ions W r participating in the amplification process to the total energy W s stored in the active medium, increases as the density of amplified radiation increases. The density of the amplified radiation is limited in turn by the radiation resistance of the surface and volume of the active medium [3, 4], This leads to the result that upon the amplification of Reams which are almost parallel, the energy density Eo at the amplifier input should not exceed the value EI/M , where E l is the limiting permissible energy density, which is dictated by notions of the radiative resistance of the surface of the laser material, and M is the amplification per pass, A method was proposed in [5-7] for amplification of the radiation in diverging beams, and calculations are given for some particular cases of amplification by this method, The idea behind the method is the realization of amplification conditions with conservation of the energy density of the amplified radiation along the entire length of the amplifier at the limiting level by virtue of an increase in the transverse cross-sectional area of the beam. Estimates are made in this paper of the maximum values of the efficiency of laser ampli-fiers based on neodymium glass which one can achieve by increasing the energy density of the amplified radiation to the maximum possible level, Amplification of parallel and diverging beams is discussed for several levels of excitation of the active medium and various distributions of the inversion along the length of the active element. For this purpose, we used a computer to numerically solve the equation which describes the propagation of monochromatic radiation in a medium with an inverted population No (see [7]):

d____.~_

1 No (l __ e_~a~) __ ( ~ +?)e,

dr

a

r

(1)

,

where e(r) is the total number of photons of the amplified radiation which have passed during the pulse time ~n through unit area with coordinate r (the units are cm-2), e is a parameter characterizing the layout of the energy levels of the active medium of the amplifier whereby = 1 when Zp >> T=I and ~ = 2 when rp << T21, where T2~ is the lifetime of the lower level 4I:~/2, o is the transverse cross section of induced transitions, B = 2 for a beam with spherical divergence, ~ = 1 for a cylindrically converging beam, and for ~ = 0 the beam is parallel, ~ and y is the nonactive absorption coefficient of the active medium. The boundary condition is: S/r=ro = eo, where ro is the distance from the center of the divergent wave to the input face of the active element. It ~.~= assumed that the energy is uniformly distributed over beam cross section, the inversion is .zonstant for any cross section with coordinate r, and the effect of luminescence and superluminescence is insignificant. As an example, the curves obtained for the variation of e along the length of the active medium are presented in Fig. 1 for the case in which No = 2.12"10 i~ cm-3~ which corresponds to a stored energy d e n s i t y ~ 0 . 4 J/cm 3. In order to analyze the method of amplifying parallel beams for the case in which the length of the active element is equal to 600 mm, we will introduce the dependences of the amplification M and the efficiency of the amplifier n on the energy density of the input radiation Eo (Fig. 2). It is evident that in the general case it is necessary to increase both Eo and No in order to raise n. For the case ~ = 1 both ways will give the expected effect, and for ~ = 2 it is preferable to increase No, since starting from Eo = 5 J/cm = a further increase of Eo does not lead to an increase of ~. Limitation of the energy density 1980.

Translated from Zhurnal Prikladnoi Spektroskopii, Original article submitted August 2, 1979,

0021-9037/80/3206-0563507.50

Vol, 32, No. 6~ pp. 979-984, June~

9 1980 Plenum Publishing Corporation

563

of the radiation at the output of the active element E o u t ~ E

l (we will further assume~ in

accordance with Refs. 3 and 4, that E l = i0 J/cm 2) results in the fact that Eo has a value corresponding to only the initial section of the n(Eo) dependence, in which ~ does not yet reach the maximum value. One can approximate Eo to E~ by decreasing the length L of the ac~ tire element. An estimate shows that for an active element operating according to a two-level amplification scheme (= = 2) a decrease of L from 600 to I00 mm will lead to an increase of from 29 to 31% (for No = 1.06"i0 le cm-3), from 30 to 35% (for No = 2.12.10 z8 cm~3)~ and from 31 to 37% (for No = 4.24-i0 ze cm-3). The value of D increases from 43 to 50, 37 to 51, and 30 to 52.5%, respectively, for a three-level medium with the same No. Therefore, it is more advantageous in the case of amplification of parallel beams to select an amplification loop composed of a larger number of stages with active elements of shorter length, creating in each of the elements amplification conditions with Eou t = E l, Now we will discuss the amplification of spherically diverging beams in an amplifier with an active element uniformly excited along its length. We will restrict ourselves in estimating the value of n to amplification conditions which are of greatest interest~ namely when Eo = Eou t = ~ . It is evident from Fig. ib that these conditions can occur only for specified va• the parameter set re, No, and L. It is not difficult to convince oneself that for a selected value of No the largest value of ~ out of all possible alternatives of realizing conditions withEo = Eou t = E l corresponds to the case with the minimum dip of E in the central part of the active element. The results of an estimation of the efficiency of amplifiers of diverging beams which has been made for conditions with a minimum dip of c in the volume of the active element are given in Table i, We note that it is necessary in connection with the necessity of creating a multistage amplifier of diverging beams to introduce into its optical layout elements which correct the aperture angle of the beam with the goal of providing identical amplification conditions in each amplifying stage. For example, the optical layout of Fig. 3 permits realizing these conditions. It is known from the literature (e.g., see [9, i0]) that the threshold of volume breakdown of neodymium glass in the case of pulse durations T = 5-50 nsec is attained at energy P densities of the amplified radiation E v which are 2-5 times larger than E~. U s i n g this fact, one can attempt to increase the value of n of an amplifier still more by increasing the energy density of the amplified radiation in the volume of the active element, leaving the value Eou t = E l at its output end face, It is possible to accomplish such conditions by altering the excitation level of the active medium along the length of the active element, This condition can be satisfied, e.g., in active elements of increasing transverse cross section with uniform illumination of their lateral surfaces by optical pumping sources. In this case one should expect in the active element not a uniform inversion distribution along the length but one which decreases according to the law N(r) = No(ro/[ro +L]) 2 Calculation shows that with such an inversion distribution the curves of the variation of e along the element length will have the appearance illustrated in Fig. ic. In this case the fact that the value of N is less for amplification conditions with larger maximum value emax in the volume of the active medium, although one would expect the opposite, is interesting, The cause lies in the competition of the effect of two factors on the value of ~ -- the increase in it when E increases and the decrease in it when No decreases (see Fig. 2b). The second factor is the decisive one in the majority of cases, since a larger emax corresponds to a larger L, ~ d consequently, a larger difference between the values of N at the input and output end faces. A large gain in the efficiency can occur under the condition L << re (short active element) when N differs only insignificantly at the input and output ends. This case occurred for No = 4.24.1018 cm -3 and = = i (see Table i). From this follows the conclusion that it is possible to increase appreciably the efficiency of an amplifier by increasing the energy density of the radiation in the volume of the active element only in the case of an inversion distribution which is constant throughout the volume by selecting amplification conditions with an increase of along the length of the active element (e.g., see the solid curves 2 and 3 and the dashed curve 3 in Fig. lb). But a problem arises in this connection with the emergence of the radiFor this reaation from the output end, since for Eou t > E~ = Eo theend face is destroyed. son it is impossible to realize amplification conditions with Eo = Eou t = E l in the active 564

8 '10 -7S e m "2

8

b

I

8 #

2~// 1

8 ~ 7/

2

I

I

50

~O

}

I

700

q

2

'l

200

790

200L~em

Fig. i. Variation of r in proportion to the propagation of the amplified radiation in the active medium; a = 2,10 -2~ cm=~ T = 3.10 -s cm-~, No = 2.12"1018 cm -2, ~ = 1 (solid curves) and 2 (dashed); (a) parallel and (b, c) spherically divering beams; (I') ~o = 5.3, (2') 26.5, and (3', b, c) 53.1018 cm =, i,e,, ( i ' ) Eo = i, (2'), 5, and (3', b, c) i0 J/cm 2, (i) ro = 25, (2)75, and (3) 200 cm; b) the inversion is constant along the length of the active element and (c) it is distributed according to the law N(r) = No(ro/[ro+L]) =

a

M

5

80

8

J

7 60

5

5 -~

5

L ,~'--~-----~

Y --~

~

,

qO

/ 5 q l~ ~ - - ~ - ' , ~ o - ~ u

f

~oO

L ,

9

20

./ I

I

6 810 |

I

I

I

2

l

l

l

l

q 6 ~ E~ I ~

Fig. 2

Fig. 3

Fig. 2. (a) Dependence of amplification per pass and (b) amplifier efficiency on the energy density of the radiation at the amplifier input: ~ = 1 (solid curves) and 2 (dashed curves), (i) No = 1.06, (2) 2.12, and (3) 4.24.10 Is cm -3. Fig. 3. Optical layout of a multistage amplifier of diverging laser beams: 1-3) active elements; 4) optical pumping system; and 5) input end faces of the elements, 9 which are correcting elements providing an identical value of ro. TABLE i. Maximum Values of the Efficiency of Amplifiers of Parallel and Diverging Beams, % No.10 ~s, cm-3 1,06

Ampl. m e t h o d s

2,12

0;=I

in parallel beams

.~ ~

ro

2

~I

4,24 CC-2

~I

GC~2

50

31

51

35

5-9, 5

37

49

28

54

36

59

40

50

35

60

39

565

elements of previous design [7, ii], which are uniformly excited throughout the volume, with the condition that E > Eo = Eou t in the volume of the element. The designs of the active elements in Table 2 permit realizing such amplification condi~ tions. The first of the proposed designs differs from those used earlier in that the output end face of the active element is designed in the form of a plane inclined towards its central axis. This permits increasing the area S of this end face, and consequently Eout, by a factor m, where m=

Vl--tg~.tg2~ cos

~ (1

-

-

tg ~. tg ~

"

(2)

Here ~ is the aperture angle of the active element, and ~ is the angle of incidence of the radiation on the plane of the output face. The surface of the output end face is coated to reduce the reflection losses. If one amplifies polarized radiation, it is possible to use active elements wiLn output end faces tapered at the Brewster angle ~B" One can obtain a uniform inversion distribution in such an active element by increasing the energy liberated in the optical pumping sources in the direction away from the input end face of the active elements towards the output end face according to the law:

W(L)---IV~ r~ )2,

(3)

where Wo is the energy liberated in sources located at the input end. This is easily accomplished with transverse placement of the pumping lamps relative to the element axis. The values of the efficiencies of amplifiers with such active elements when ~ = ~ B (m = 1.87 for the case in question) are given in Table 2. The second of the proposed designs of the active element differs ~n that the active element is made up out of at least two parts with different aperture angles which are joined to each other into a single unit by means of an optical element, e.g., a lens. The aperture angle of the posterior part is larger than that of the preceding part, and the optical element converts radiation with the aperture angle of the preceding part into radiation with the aperture angle of the posterior part. The role of such an optical element can be fulfilled, for example, by a liquid with nli q = n g positioned between the surfaces of the element parts to be joined which are finished with the appropriate radii of curvature, One can also join together the parts of an active element s each of which is made out of glass with different refractive indices, by the method of sintering them, having finished in advance the surfaces to be sintered as spheres with the necessary radii of curvature and so on. One can provide by the choice of the values of rol and LI ~see Table 2) in the first part of this element amplification conditions for which the value of E increases to a value E v towards the point of placement of the optical element. Then one creates by the choice of ro2 (it is provided for by the optical element) and L2 in the second part amplification conditions for which E decreases towards the output end face from E = E v to Eou t = E l . The calculated values of the efficiency coefficients of an amplifier with such active elements are given in Table 2. It was assumed that E v exceeds the value of E l by a factor of two. And so in order to increase the efficiency coefficient of laser radiation amplifiers, it is necessary to excite an active medium to the highest possible level and use laser pulses with a large energy density for the amplification. When the restrictions on the value of the radiation energy density are taken into account, it is possible to attain the largest values for the efficiency of amplifiers of parallel beams by using active elements of small length (5-10 cm). In this case the value of Eo differs insignificantly from Eou t = El, which determines the limiting value for the efficiency of the method of amplification of beams which are almost parallel. Using the method of amplification in diverging beams, one can realize the same amplification conditions in active elements of appreciable length (50-200 cm), which considerably simplifies the optical layout of the amplifier. One can perform optical pumping of elements with increasing transverse cross section with the help of effective illuminators used to pump cylindrical rods. The nonuniformity which arises in the excitation of the active medium along the element axis does not lead, as was shown above, to a decrease in the value of ~. The proposed designs of the elements permit increasing the efficiency of the

566

TABLE 2. Values of the Efficiency of Amplifiers with Active Elements of Altered Design (in %) No.101. , c m - S 4,24

2,12

Design of the elements

~2

oout ~

67,5

38

64

35,5

~l

77, 5

~2

46

"~ ~

42,5

amplifiers by a factor of 1.3-1.4, The difficulties in the production of an excitation of active elements of increasing cross section which is uniform along the length can be circumvented by using them in the shape of truncated rectangular pyramids and employing a pumping system with transverse positioning of the sources along the longer edges of the pyramids. Besides the basic advantage of amplifiers of diverging beams with such active elements -- high efficiency -- one should include ~mong their merits the comparatively low level of superluminescence and parasitic generation [~] and also the possibility of eliminating defects of the active medium associated with self-focusing [12] and the smaller effect of nonlinearity of the amplification, which is expressed in a reduction of the duration of the pulse and a shift of its maximum towards the leading edge [7], All this permits considering the application of amplifiers of diverging beams with the proposed designs of the active elements to be promising in laser systems with high output energy of the radiation, LITERATURE CITED i, 2. 3. 4. 5. 6. 7. 8. 9. i0. ii. 12.

A. A. Mak, B. G. Malinin, V, A. Novikov, et al., Zh, Tekh, Fiz., 39, No, i0, 1886 (1969), P, G, Kryukov and V,S,Letokhov, Usp. Fiz, Nauk, 99_, No, 2, 169 (I~9), N. G. Basov, V. S~ Zuev, P. G, Kryukov, e t a ! . , Zh. Eksp. Teor. Fiz., 54, No. 3, 767 (1968), N. G. Basov, Ch. Kertes, P. G.Kryukov, Yu. A. Matveets, etal., Zh, Eksp. Teor. Fiz., 60, No. 2, 533 (1970), J. M.Jego, Appl, Opt., 9, i (1970). V, M. Ovchinnikov, Zh. Tekh. Fiz., 43, No. 7, 1543 (1973), N. B. Baranova, Yu, V. Senatskii, E. A. Tyurin, and V. A. Shcheglov, Kvantovaya Elektron, 17, No. 5, 57 (1973), L . M.Frants and J, S. Nodvic, J. Appl, Phys., 34, 2346 (1965), C. G. Yong, Proc. IEEE, 57, No. 7, 1267 (1969), Yu. V. Senatskii, AuthoreSs Abstract of Candidate's Dissertation, Moscow (1971), YU. V, Senatskii, Kvantovaya Elektron., No. 5, 109 (1971), P. G.Kryukov and Yu. V. Senatskii, Preprint of the P. N. Lebedev Institute of Physics, Academy of Sciences of the USSR, No. 51 (1971),

567