Tarasov - Laser - Physics - Mir, Notas de estudo de Física

Baixe Tarasov - Laser - Physics - Mir e outras Notas de estudo em PDF para Física, somente na Docsity! L.V. Tarasov Mir Publishers Moscow About the Book The book aims at bringing the reader up-to-date with the latest achievements and trends in the physics of laser processes, as well as at providing sufficient information so that the reader can then independently make use of specialized literature in this field. It describes the present state of affairs in the development of laser technology from the point of view of research as well as applications in various branches of industry. 11. B. TapaCOB ttJJI13V1KA npou.ECCOB B rEHEPATOPAX KOrEPEHTHOrO OnTV1LtECKOrO JI1311YLtEHV1~ HSAaTellbCTBO «PaAHO H CB~Sb») MOCKBa L~ Tarasov Translated from the Russian by Ram S. Wadhwa MIR Publishers MOSCOW First published 1983 Revised from the 1981 Russian edition Ha aH,8J1,uilc1W.~ Sl8blne @ llaAaTeJILCTBO «PaAHO H COHaLI, 1981 © English translation. Mir Publishers, 1983 CONTENTS ·.:IIII.,I,t'.· I. Population Inversion in Active Media . t ,1 Sumo General Aspects . . . . . . . . . . . . . 1.~ Optical Pumping. Solid-state Lasers . . . . . 1.a. Organic Dye Lasers . . . . . . . . . . . . 1.4 Gas Lasers with a Wideband Optical Pumping. ... l.rl Pumping by Self-maintained Electric Discharge in Rarefied Gnses . i,n 1'~)ocLro-ionisation Lasers . . f .7 (;n~HJ ynnmic Lasers (Thermal Pumping) . f ,M Chnrnienl Lasers . . . . . . . . . . . . I. U PI"~ln" Lasers (Recombination Pumping) . flu"tlt'" ~. Iru.·llu,llon of Radiation Field in a Laser Resonator ~~ .1. (."IIOI·ulioll Condition . . . . . . . . . . . . . . . ~,~ •)pt.ieul Hesonator and Laser Radiation (Preliminary Remarks) ~.:' (~"IU'l'"I Hemurks about Open Resonators . . . . . . . . . . :l,.le I.C'II~ ~n ve~uides and Open Resonators (Geometrical Optics "ppl'OX imntlun) . • . • . . . . . . . . • . • . • • • • • ~,h U.,t.lIllill", of an Open Resonator . . . . . . :.!." "1l"ly~i~ or Open Resonators on the Basis of Fox-Li Iterative l\1 •• tluul. l'~quivnl~nt Resonators . ~~.'I (~""Mto'i"ll Bpulns . ~~ H 'I'runsf'ornuulon and l\latching of Gaussian Beams . ~ U (.,,"~~I,," Hcnms in Stable Resonators . . . . . ~~ 10 11,1"1,,.1.1,. ltesonntors . . . . . . . . . . . . . ~, t I 1t','.t'llIetru'.y Sf'lcction . . . . . . . . . . . . . :.! I:.! lIul" hur'lllll~ I~rr~ct and Frequency Pulling Effect . ~.I:' 'l'lu-rtunl f,oll~ .••••..•••••••••• ~~, t ~ \V"V"M'U 1dC' Husonators . . . . . . . . . . . • . . • • • . ~~ Ir. OptiC'" I U"dl"tion in a Thin-film Waveguide. Distributed F.,,'dltllrl( . .•... a. t ("'''''1,,,1 U"IIl"I'k~1 nbout Laser Operating Conditions . . .. :,,~ "ppl'n,I.""l.u 1·~CJunt.lons Describing the Dynamics of Laser I'I'U."''''''''''''I (B"lnll(~p Equations) . . . . . . . . . . . . • . ;, :, I"t',,,, (;"'IC'r'"t.lon, H(~~1I1nr Damped Pulsations of Radiation It.,wI'I· ...........••.••••••••••• :'1 11""Iuhl,' Upl-4onnlor Laser. Undamped Pulsations of Radia- 11,1" 1"'\\"1' . . . . . . . . . • . • . . .• ••••• :, r, ".'1 I v.' ~fOfltll"ll()n or the Resonator Q-factor . . . . . :,,, (U".II 1'1I1P41' (}PIH~rllti()n ill the Case of Active Modulation of IIII' ""~olllltor ()-fnet.or . ;1 i 1I1..".'II"hl., Fllt.H· LIlM(lr~ ••••••••••••• ;, H(.,,,.11. 1'1I1~,· (~prH~rati()fl for Passive Q-modulation of the 1'''~IIIl"lfll' ....,.............. 5 9 9 20 30 36 39 50 54 58 64 73 73 78 88 100 112 116 134 142 151 163 175 183 190 195 200 214 215 229 244 256 262 270 "283 296 8 Contents 3.9 Longitudinal Mode Locking (Ultrashort Light Pulse Genera- tion) . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Measurement of Duration of Ultrashort Light Pulses . . . . 3.11 Analysis of Self-locking of Longitudinal Modes in a Bleachable Filter Laser Based on Fluctuation Concepts . . . . . . ... 3.12 Time-based Description of Active Longitudinal Mode Locking in a Laser with a Uniformly Broadened Gain Line . . . . . Appendices References Index • • • . 304 313 319 328 336 345 354 Chapter One POPULATION INVERSION IN ACTIVE MEDIA 1.1. SOME GENERAL ASPECTS Inversion of Active Medium as a Necessary Condition for Laser Generation. The generation of laser radiation is caused by transi- tions between certain energy levels of active centres. We shall call these levels the laser levels, and denote their population (per unit volume of the active medium) by n1 (lower laser level) and n 2 (up- per laser level) respectively. The difference N == n2 - (g2/g1) n1 (1.1.1) is called the inverse population density of laser levels. Here, gl and g2 represent the degeneracy of the corresponding levels. For the sake of simplicity, we assume that gl == g2' If the condition JV > 0 (1.1.2) is satisfied, an inversion of the active medium is said to take place. Since N is, in general, a function of time and space coordinates, the concept of inversion may be applied to definite time intervals and to specific regions of the active medium. In a medium in thermodynamic equilibrium, N is a negative quantity, i.e, the lower level is more densely populated than the upper one. In order to create an inversion in the active medium, we must bring it to a nonequilibrium state. Inversion of the active medium is a necessary condition for laser generation. It should be recalled that the amplification factor x of a homogeneous medium is given by the expression! Ix=aN I (1.1.3) where a is the cross section of induced transitions between laser levels. It can be seen from Eq. (1.1.3) that the very fact of the ampli- fication of radiation upon its passage through an active medium as- Humes that the condition (1.1.2) is satisfied, i.e. N is a positive quan- t.ity. The need for ensuring an inversion is physically apparent, since only for N > 0 (i.e. for n2 > nt ) can the process of induced emission at the lasing transition dominate over the reverse process nf absorption of radiation. In order to create and sustain inversion, we have to excite (or Jlump) the active medium in some way. Several methods are avail- 1 See, for example, Eqs. (5.6.9) and (5.6.10) in [1]. 12 Ch. 1 Population Inversion in Active Media 'l'l ..., lnvorslon cond it ion (1.1.2) in this case assumes the form y > 1. (1.1 ..8) Sub~t.itHting in Eq. (1.1.8) the expression for y obtained from Eq. (1.1.7), we get the following inequality: R 1 > [ A21+F1 F2+~~~;:+R3) J. (1.1.9) The ratio R 3/(R 3 + R~) is the relative probability that an active centre goes from level 3 to level 2. Consequently, F 3R37(R 3 + R~) is the probability that level 2 is populated via level 3. Since the transition 0 -+ 2 also leads to level 2, the sum F = F 2 + [F3R3/(R 3 + R~)] (1.1.10) is the total probahility of populating the upper laser level. Further, we observe that the sum R = A 21 + R 2 (1.1.11) is apparently the total probability of depopulation (relaxation) of the upper level. Taking into account Eqs. (1.1.10) and (1.1.11) we can rewrite the inversion condition (1.1.9) in the form I (R 1 - AZ1)1R > F 11F I. (1.1.12) General Principles for Creating Inversion. In order to create in- version, the selectivity of population or depopulation of the corres- ponding levels of an active centre is of prime importance. In this connection, let us consider the condition (1.1.12), from which it can be seen that stationary inversion is attained under the assumption that the inequality R1 > A 21 (1.1.13) is satisfied. In other words, the total probability of depopulation of the lower laser level must be higher than the probability of its population due to spontaneous transitions from the upper laser level. It also follows from the inequality (1.1.12) that it is desirable to satisfy the following inequalities in order to create an inversion: F ~ F1 , (1.1.14) R1 ~ R. (1.1.15) These inequalities reflect the above-mentioned selectivity factor necessary for creating an inversion. The inequality (1.1.14) means that the total probability of populating the upper laser level must be considerably higher than the total probability of populating the lower laser level. The second inequality (1.1.15) means that the 1.1 Some General Aspects 13 total probability of relaxation of the lower laser level must be consid- urabl y higher than the total probability of relaxation of the upper luser level. III actual practice, inversion can be created if at least one of these t.wo inequalities is satisfied. In this connection, it should be rem em- bored that inversion can be created not only through a preferential population of the upper laser level (in preference to the lower laser lovel ), but also through a preferential depopulation of the lower laser level. Thus it is necessary to take into consideration the relation hotween the probabilities of population and depopulation mechan- isms in laser levels while considering different methods for pump- illg. Mechanisms for Population of Levels (Excitation Mechanisms). A bsorption of Light (Optical Pumping). Pumping radiation with H specially selected spectrum can lead to a high selectivity of optical «xcitation on account of predominant population of the upper laser luvel: F» Fl. I nelastic collisions with free electrons, where a part of kinetic energy of the electron is transferred to the active centre (electronic excita- I iou). The exciting free electrons may be created by a self-sustained J.rU~ discharge, or by using ionising radiation in combination with an ox Iornal electric field for accelerating electrons (non-self-sustained cI ischarge). The latter case corresponds to electro-ionisation pumping. FoJ' the case of electronic excitation, the inequality F < F} is uxunll y satisfied instead of the inequality (1.1.14)2. I nelastic collisions with auxiliary atoms and molecules with a reson- 1'111, transfer of excitation energy from an atom (or molecule) to an nr t.ive centre. The resonance mechanism of energy transfer assumes " high pumping selectivity due to predominant population of the upper laser level (F ~ F I ) . Chemical pumping corresponds to an excitation of levels through specially selected exothermic reactions. Chemical pumping is char- nrtorised by a highly pronounced selectivity (F ~ F I ) . llecombination pumping means an excitation of levels as a result 01' recombination of electrons and ions in a specially created plasma. 'I'ho inequality (1.1.14) may not be satisfied when this method is u ppl ied for excitation. Thermal pumping is the excitation of levels through heating of the H"I ivo medium. In this case, F < Fl. Mechanisms for Depopulation of Levels (Relaxation Mechanisms). Hadiatioe depopulation by spontaneous emission. The selectivity of this mechanism is determined by the selection rules (a transition na"y be optically allowed or forbidden, depending on the properties 01' tho levels under consideration) and the distance between the level ~ The self-terminating transitions discussed later in this hook are an ex- t I' pHon to this. 14 Ch. 1 Population Inversion in Adive Media being depopulated and the low-lying level at which the active cen- tres arrive as a result of relaxation (in other words, on the energy nro of the spontaneously emitted photon). The higher the value of Iu», the higher the probability of radiative depopulation (subject, of course, to the condition that such a transition is optically allowed). I t should be remembered that Einstein's coefficient A for sponta- neous emission in the case of dipole (i.e. optically allowed) transi- tions is given by the expressions (1.1.16) where (d) is the matrix element of the electric dipole moment of the radiating system, corresponding to a transition. The mechanism of radiative depopulation may be characterised by a high selectivity. If the transition 2 -+ 0 is optically forbidden while the transition 1-+ 0 is optically al lowed' and has a fairly high frequency, then R1 ~ R. I nelastic collisions with free electrons, resulting in an increased kinetic energy of electrons (electronic de-excitation). This mechanism does not possess any special selectivity. Inelastic collisions with auxiliary atoms and molecules, resulting in the energy transfer from an active centre to an atom (or a mole- cule). There are two mechanisms of energy transfer, namely, the resonance mechanism and the gas-kinetic mechanism. In the first case, specific atomic or molecular levels are excited. This mechan- ism has a fairly high selectivity (R 1 ~ R). In the second case, the kinetic energy of the atom (or molecule) increases. The gas-kinetic mechanism of energy transfer is considerably less selective and slower in comparison to the resonance mechanism. As a special case, let us mention the de-excitation of active centres upon collisions with the walls of the reservoir containing an active medium (de-excitation at the walls). De-excitation through adiabatic expansion of a gaseous active me- dium. The excitation energy of the active centres is converted into the kinetic energy of the particles of a rapidly expanding gas. This mechanism allows for a highly selective pumping due to predomi- nant depopulation of the lower laser level (R 1 ~ R). Chemical de-excitation through specially selected chemical reac- tions has a very high degree of selectivity (R 1 ~ R). Transfer of excitation energy to the collective motions in a medium. In particular, this includes a transfer of energy to phonons (increase in the vibrational energy of atoms in a crystal lattice). 3 See, for example, Eq. (2.5.84) in [1]. 4 The labelling of energy levels corresponds to the diagram shown in Fig. 1.1. 1.1 Some General Aspects 17 2 Jnw___~_}exc (b) '1)'=0•.6 Fig. 1.4 o .2 1)'= 0.1 --- ,(a) (1.1.13). A sufficiently fast relaxation of the lower laser level is required in order to fulfil this condition. The inequality (1.1.13) considerably limits the choice of laser transitions which can create a stationary inversion, and hence a continuous-wave generation. Suppose that relaxation of the lower laser level is caused by spontaneous emission. In this case, a high relaxation rate is ensured if the lower laser level is situated fairly high above the ground level, in accordance with (1.1.16). Consequently, a laser transition must take place between high levels, which means a fairly quantum yield 11' ~= Jiw/Eexc of the laser. For a typical atomic gas laser operating in the CW mode, the energy E exc ~ 10-20 eV, and, as n 1'1I1e, 11' does not exceed 0.1. H('sides, the use of high levels mnuns small values of the factors " and f in the expression (1.1.5) fOI' ] aser efficiency, since there is " probabil itv of excitation of addi- I iOHH 1 low-lying levels as well as a prohu hi] ity of relaxation of active ('I'll t I'PS 10 these levels. Hence it is 1101 surpr-ising that the efficiency CtI' H (: \V atomic gas laser is just of tho order of 10-3 • FiM'ure 1.4a shows a characteristic d ill~I'Ulil o] the laser levels under IIIC' cOlld i I ion that the lower level I'~ dc'popll) ill nd by spontaneous "1I1i~~~·doll. 11('1'0. () is the ground level, 1 and 2 are the laser levels. '1'111' 01114'1' 1C'\,('ls shown in the figure are parasitic levels in this case. lur 1,lj~ I ,'eo, 'i' 0.1. In order to improve the efficiency of the Lnlt'l', "'I' ~"unlcl M'O 0\'01' Irum the diagram shown in Fig. 1.4a to II,,, 0'"' III FiM'. 1.st., 1'01' which 11' == 0.6. In other words, we should luu~icl.q' I I'HII~itioll~ hetweon rolatively low laser levels. /\ clC'(,I'c'il~n ill I htl rud iutivo depopulation of the lower laser level 1'-' uuuvuid a hl« ill t his ease, and hence supplementary mechanisms un' Ilt't'P~~H"Y [01' I'pln xat-ion of the lower level. One such supplemen- IIlI'\' nuxhnu ism nuiv hp tho inelastic collisions of active centres with "IO'lIl~ .. lid )nolel'lIl~~s speria lly introduced into the active gaseous nu-rliutn. Such lasers are often called collision lasers. Muhx-ulnr and electro-ionisation lasers generating on vibrational transi- t i()l1~ or a CO2 molecule are examples of collision lasers. The cases shown in Fi~. 1.3, whore relaxation of the lower level by inelastic collisions plays an huportunt role in cl'pating inversion, also belong to this category of lasers. Advantages of Pulsed Pumping; Generation on Self-terminating Trnnsltlons. For pulsed pumping, inversion is created just over ('(','tain time intervals and hence a fast depopulation of the lower 18 Ch. 1 Population Inversion in Active Media Fig. 1.5 laser level is not significant. Suppose that when an excitation pulse is switched on, the upper laser level is populated at a faster rate than the lower laser level. In this case, the laser operates on account of inversion created at the beginning of the excitation pulse. Such a situation is illustrated in Fig. 1.5 where curves 1 and 2 describe the time variation of the population density of the lower and upper laser levels respectively, and curve 3 shows the excitation pulse. It can be seen from the figure that inversion is created at the begin- ning of the excitation pulse (in time tt). Apparently, the rate of relaxation of the lower laser level in this case is not signifi- cant. It follows from above that in the case of pulsed pumping, genera- tion is possible in a large number of media and for a larger number of transitions in any medium, than for sta- tionary pumping. In particular, genera- tion is possible on the so-called sel/- terminating transitions. Self-terminating transitions are ones for which the first resonance level of the active centre is the upper laser level, while the lower laser level is a metastable one. Metastability of the lower level forb- ~ t ids its depopulation by radiation and thus a stationary inversion is not. pos- sible for such transitions in spite of the fact that the first resonance level, as a rule, has the largest cross section of elec- tronic excitation in a discharge. However, pulsed generation is possible for such transitions. This can happen at the beginning of the excita- tion pulse, when the rate of relaxation of the lower level is not signifi- cant, and the ratio of the population rates of laser levels is signifi- cant. Since the duration of generation is restricted in this ease by the properties of the transition itself, such transitions are called self-terminating transitions. The ratio of population rates of laser levels for self-terminating transitions is especially favourable on account of metastability of the lower laser level. As a matter of fact, the electron excitation cross sections for optically forbidden transitions are usually much lower than for the allowed ones [3]. Hence the probabili.ty of electronic excitation of the first resonance (upper laser) level is considerably higher than the probability of excitation of the metastable (lower laser) level. That is why the gain in lasers working on self-terminat- ing transitions is very high 7• 'lOwing to this circumstance, such lasers may operate in the superlumines- cence regime, i.e. without the resonator mirrors or with just one mirror (see Ch. ~ in [4]). 1.1 Some General Aspects 19 r 2 Fig. 1.6 o Another advantage of pulsed pumping is that from a technical point of view, it must be easier than continuous pumping. A con- I iuuous stable supply of energy to the active medium is, as a rule, more complicated than a pulsed supply. Besides, there is no need 1'01' a forced cooling of the heated active medium in the case of pulsed pumping. .Pulsed pumping leads to different pulsed generation conditions", This helps in obtaining an exceptionally high concentration of the omitted optical energy in space and time. For example, operating .coud it.ions have been attained for generation of high-power ultrashort light pulses of duration 10-11-10-12 sec and a peak power up to 1012 w. I n actual practice, pulse lasers are encountered much more fre- quently than CW-Iasers. Solid-state and I iqu id lasers often operate ill the pulsed regime by employing pulsed or continuous pumping. Gas lasers E ure characterised by a continuous opera- I ion, but even for such lasers, there is ~I'owing interest towards pulsed genera- tion [5]. Noble-gas Molecular Lasers. In con- clusion of this section, we shall describe n fairly effective mechanism for depopu- lating the lower laser level. This mech- nil ism is based on the use of the noble- gas molecules as active centres. Tho ground state of a noble-gas mole- ru le is unstable (repulsive). Such a molecule can exist only in electroni- cally excited states. Upon a transition Irom an excited state into the ground xt alo. it dissociates into atoms. I I, is well known that the atoms of noble Lt'nsps "do not like" to form molecules. III order to create the molecules Ar2 , Kr 2 , or Xe 2 , energy has to be 0\ ponded. Such molecules exist only in the excited state and can I.., treated as disintegrating molecules. Figllre 1.6 shows the electron energy levels of a diatomic noble- ~HS mnlecu le (as functions of the distance between the atomic cen- l n-x). Here, level 1 corresponds to the ground state while 2 and 3 n~pr'PS(HI t excited energy levels. In con trast to state 1, states 2 IIlIcI .1 are stable and the vibrational levels of molecules in these ,~I nips are shown by horizontal lines. The vertical arrows indicate th., transit.ion 2 ~ 1; lu» is the energy of a photon emitted during " Pulsed generation conditions will be described in Ch. 3. It should be re- nu-mbered that pulsed generation is possible even in the case of continuous plIlllping. ~'. 22 Ch. 1 Population Inversion inActive Media ed for the case when there is no generation), the total number of active centres per unit volume is denoted by n'. The balance equations for the population densities of the levels have the form n01wp - n0 2w21 = 0; nOl + n0 2 == n'. Introducing the inverse population density No = n 0 2 - (g2 Igl) nOl, we get (1.2.5) (1.2.6) Using Eqs. (1.2.4) and (1.2.3) (having replaced g by gl and g' by g2' of course), we can rewrite (1.2.5) in the Iol lowingforrn: N.,. - g2' A21o---n gl A21+ (1+ gl/g2) Wp • We introduce the term pumping rate Q and define it as the proba- bility that the active centres arrive at the upper laser level. In this 3 2 , 1 Fig. 1.8 Fig. 1.9 case, Q == wp • Using this term, Eq. (1.2.6) can be rewritten in the form No _ 1?2 A 2 1 (1 2 r-r) 7- -g; A21+(1+g1/g2) Q · • ., Figure 1.8 shows the dependence of N oln' on Q as defined by formula (1.2.7). It can be easily seen that irrespective of the pump- ing rate, the quantity 1J0 remains negative, and so inversion is not created. This shows that a two-level diagram is not suitable for the case of optical pumping. Let us now consider the three-level case shown in Fig. 1.9, where 1-3 is the excitation channel and 1-2 is the generation channel. The balance equations for the population densities of levels in this case have the form - n 02 A21 + n03w32 == 0; } n01Wp-n03(W31+W32) ==0; (1.2.8) not +n02 + n0 3 == n': 1.2 Optical Pumping. Solid-state Lasers 23 Solving this system, we get 120 1 == (n'ID) A 21 (W 31 + W 3 2); 120 2 == (n'ID) W pW 3 2 , (1.2.9) where D is the determinant of the system of equations (1.2.8). From this, it follows that No _ Wp W32-(g2/g1) A 21 (W31 +W32) 11,' - A 21 (wp +W31 +W32) +WpW3 2 • (1.2.10) 'rho pumping rate Q for this system of equations can be written in the form Q == W p W 3 2I(w31 + to32) • ( 1.2.11) lJsing (1.2.11)~ we can rewrite Eq. (1.2.10) in the following form: No Q-(g2/g1) A 21 2 2 --;;;-= A 21+Q(1+A21/ W32) • (1..1 ) ------ Noln ' o~---,::....--~-_..I..--_--~" I I I I I R'l 1-g, \ or. Figure 1.10 shows the dependence of Noln' on Q as given by Iorrnula (1.2.12). Of the two curves shown in this figure, the dashed one corresponds to the special r nse when A 2 1 ~ W 3 2• I t can be easily seen that ~I alionary inversion may be C'Hsily created in the case of a 111I'ee-Ievel diagram. For this, it i~ necessary that the condition Q > (g2 Ig1) A 2 1 (1.2.13) lI'p>' (g2/gl) (W31+ W 32) (A 21/W32)' (1.2.14) i~ satisfied. The quantity Q 1D V == Fig. 1.10 (g2/gl) A 21 is called the inver- sunt threshold pumping rate. The smaller the value of A 2 1 , the luwer the inversion threshold. For a three-level case, when the inequality (1.2.15) i~ satisfied, the magnitude of Qlnv decreases and the limiting value of the ratio N 0/12' increases. 'l'h is leads to the conclusion that it is best to use such diagrams I'or' which the value of A 21 is the lowest. This conclusion, however, i~ not correct. It turns out that an extremely low value of A 21 is nul desirable. This' question will be discussed in Sec. 2.1. 24 Ch. 1 Population Inversion in Active Media Solid-state Lasers. Practical Attainment of Optical Pumping, Op- erating Laser Diagrams 12. Figure i.11a shows a schematic for optic- al pumping in a solid-state laser. Here, 1 is the active element, 2-the source of pumping radiation (for example. a gas discharge flash lamp); 3-a reflector for focussing the pumping radiation on the active element; 4-an optical resonator mirror, and 5-the laser radiation. Figure i.iib shows the laser cross section. The reflector surface in the cross section is represented by an ellipse at whose 5 ~ 4 Fig. 1.11 foci lie the active element and the pumping source. In actual prac- tice, reflectors of different forms and construction can be· applied. Thus, a two-lamp reflector, whose cross section is shown in Fig. L'l lc, can be used. This type of construction increases the output power of radiation. A solid-state active element consists of a matrix (basis) and an activator (active centre) introduced into the matrix in the form of impurity. The matrices employed may be crystalline or amorphous (glass-l ike). The stimulated emission effect has so far been observed in more than 250 dielectric crystals activated by transition group ionic impurities. The most common among these are the oxide crystals having an ordered structure (for example, Al 20 3 crystals w ith Cr 3+ ion impurity, or Y 3Al5012 and YAIO a crystals with Nd 3+ ion impur- ity). We can also mention fluoride crystals having an ordered structure (for example, CaF 2 crystals with Dy2+ ion impurit.ies). The efficiency of optical pumping is determined by t \VO main factors. Firstly, the pumping radiation must be absorbed effect.ively by the active centres and at the same time, it should not be absorbed by the matrix at all. Secondly, the quantum efficiency of pumping 12 General problems concerning the working of solid-state lasers have been discussed in [7-9]. The basic theories of solid-state lasers have been described in [10-121. Spectroscopic properties of activated laser crystals have been system- atised by Herlich et ale [13] and Kaminskii [14]; see also r4]. Specific problems dealing with fabrication of lasers, and construction and properties of light sources have been described in [15]. The thermal conditions in solid-state lasers have been discussed in [16]. Finally, the preparation of solid-state active elements has been considered in [17]. 1.2 Optical Pumping. Solid-state Lasers 27 .. 1. 1.__ Fig. 1.15 200 300 400 T, K A,~m I 0.695 ~ L0.694 ! O 693 ~­. I I (the separation between these two levels is 0.003 urn-1). These levels play the role of upper laser levels. Upon transition from the states 4F1 and 4[/2 to the upper laser level, the chromium ions transfer a part of their energy to the crystal lattice (non-optical transitions are shown in the figure by undulating arrows). The lower laser level is denot.ed by 4A 2. This is also the ground state!'. I t is clear Irorn the figure that ruby lasers may be represented by a three-level dia- gram. Figure 1.14a shows the ruby absorption spectrum for the case when pumping' radiation propagates parallel (curve 1) and perpendicular (curve 2) to the optical axis of the crystal!". In the absorption spectrum, two relatively broad lines with maxima at A ~ 0.41 f.tm. (blue absorption band) and A ~ 0.55 urn (yellow-green absorption band) can be dis- tinguished. These correspond to the excita- tion' of states 4}ll and -IF2 respectively. The dashed curve shows the radiation spectrum of a mercury lamp which is the source of excitation. I t can be easily seen that the spectrum of the mercury lamp coincides fairly wel l with the absorption bands of ruby, thus ensuring a high selectivity of excitation and increasing the fraction of gainfully consumed power of pumping radiation. Figure 1.14b shows the luminescence spectrum of chromium ions in ruby. Two lines can be observed in this spectrum: the R 1-line at 0.6943 urn corresponding to the transition E --.+4A 2, and the R 2- line at 0.6929 f.!m corresponding to the transition 2A --.+ 4A2 • The curve shown in the figure has been obtained at a temperature of 300 1(. On reducing the temperature, both lines shift towards the shorter wavelength region of the spectrum (Fig. 1.15). Nd:YAG Laser. One of the most widely used solid-state lasers at present is that in which Yttrium Aluminium Garnet (Y 3Al r; 012) serves as the matrix, while Nd 3+ ions serve as the activator. This laser is known as Nd : YAG laser. , The Nd : YAG laser has a comparatively low excitation thresh- old and high thermal conductivity, which lead to generation at a high pulse frequency, as well as continuous operation. The ef- ficiency of this laser is comparatively high (of the order of a few percent). The main transitions of a neodymium ion in garnet are shown ill Fig. 1.16. Transitions are accomplished between definite atomic 14 The lower laser level consists of t\VO sublevels separated by 4 X 10-6 flm-1. 15 Ruby.crystal has a trigonal (rhombohedral) symmetry. Its optical axis has a three-fold symmetry. 28 Ch. 1 Population Inversion in Active Media energy leoels'" shown as energy bands in the figure. Each band (each energy level) has a corresponding group of closely spaced energy sublevels appearing as a result of a splitting of an energy level in the electric field of the crystal lattice of garnet (Stark's splitting). As a result of pumping, the neodymium ions go over from the ground state corresponding to the energy level 419/ 2 to three groups J!A, ~m-1 1.5 1.4 2Hg/'l1.3 4S3/2,4F7/2 483/ 2===-4"1 4 4FS/ 2 1.2 _~2,F5/2 4F7/2 1.1 4 F 3 2 1.0 E 0.9 C'J Ll') 0.8 ~to ~ 0 ~ 0.7 ...: 0.80 'A,ltm 0.6 4115/ 2 (a) 0.5 -------l LOu; 0.4 ~lr3J2 0 -- - .,.; 0.3 ---4-~ 0.2 /11/2 4F~41 --~~ I 3/2 11/2 I 0.1 1.04 1.08 1.12 A,J!m 0 '19/2 (b) Fig. 1.16 Fig. 1.17 of energy ,levels A, S, and C. Group A corresponds to the energy levels 4J?7/2 and 48 3/ 2 ; group B to the levels 4F5i 2 and 2H9/ 2 ; and group C to the level 4/l 3/ 2• To these three groups of energy levels correspond three bands in the absorption spectrum of neodymium in garnet. These bands are represented in Fig. 1.17a as A. B, and C bands respectively. The fine structure of the absorption bands, which can be seen clearly in the figure, reflects Stark's splitting of energy levels. The energy level 4F3/2 is the upper laser "level", By omitting energy, the neodymium ions jump down from this level to the energy levels 4111/ 2 , 419/ 2 , 4[13/2' and 4[15/2' The major part of this energy (60%) is omitted during the transition 4F3/2 -+ 4[11/2' Levels cor- responding to the state 4[11/2 are considered as louier laser levels. 16 The notation used for designating atomic levels has been given in [7, 23]. See also Sec. 3.1 in [1]. 1.2 Optical Pumping. Solid-sfate Lasers 29 Figure 1.'17b shows the luminescence spectrum of Nd in garnet for the transition 4Fsi 2 ~ 4/ 11/ 2 • This spectrum consists of 7 lines, the ones corresponding to 1.0615 f.1ID and 1.0642 f.1m being the brightest of these. Wavelengths for 18 luminescence lines obtained by taking different transitions into consideration have been given in Table 1.1 [14]. The data given in this table have been obtained at a tempera- ture of 300 K. TABLE 1.1 Transitions .I.,Luminescence Relative intensity tilne wavelength. of transi tions, % IJ.rn 0.8910 0.8999 0.9385 0.9460 1.0521 1.0615 1.0642 1.0737 1.1119 1.1158 1.1225 1.3184 1.3331 1.3351 1.3381 1.3533 1.3572 25 60 14 The Nd : YAG laser can be considered in a more simplified way hy using the four-level d iagram!? consisting of the ground "level" 4/9/ 2, the lower laser "level" 4/11/ 2, the upper laser "level" 4F:i ! 2. and the excitation "levels" 4F7i 2 and 4F5i 2. I t should be noted that the F ~ / transitions are forbidden in the dipole approximation (optically forbidden transitions), since the orbital quantum number for a neodymium ion changes by 3 in the case of such a transition. Thus, the states corresponding to F energy levels are metastable. Krypton lamps are often employed for pumping Nd : YAG lasers. In some cases, chromium ions are introduced in the garnet lattice as sensitizers, and xenon lamps are used for pumping. Chromium has t wo broad absorption bands in garnet (corresponding to the wave- lengths 0.43 urn and 0.59 urn), These coincide fairly well with the 17 The four-level diagram is characteristic for solid-state lasers with Nd3 + ions as active centres. 32 Ch. 1 Population Inversion in Active Media are an effective substitute for parametric light generators in the visible and near infrared regions of the spectrum.P? Optical Pumping of Dye Lasers. Most frequently, coherent pump- ing by radiation from solid-state pulsed lasers (Nd : YAG, Nd-glass, ruby. etc.) is used for excitation of dyes. Fundamental frequency as well as harmonics (for example, the second harmonic (A, === 0.53 urn) and the third harmonic (A, === 0.35 urn) of the Nd : YAG laser radia- tion) are used for pumping. The efficiency of dye lasers excited with the help of an auxiliary pulsed laser is of the order of tens percent. For a solution a rhodamine 6G in ethanol, an efficiency of 75% was achieved upon pumping by the second harmonic of a Nd-glass laser. \'2 eft,~ J-----=---- . - - -----=.- =~ 4 5----- Fig. 1.20 Dye lasers employing coherent pumping may function as wide-band amplifiers in the optical region. They can also be used as relatively simple and efficient converters of optical frequencies. Coherent pulsed pumping may be transverse or longitudinal. In the former case, the pumping radiation propagates perpendicular to the direction of dye oscillations, while in the latter case the two are parallel. As an example, one of the alternatives for longitudinal pumping is shown in Fig. 1.20. Here, 1 is the dye, 2-the pumping rad iation. 3-the radiation generated by the dye; 4-a totally reflecting prism. and 5-the resonator mirror (opaque to the pump- ing radiation). Besides pulsed pumping. continuous coherent pumping is also employed. For this purpose, an ionised argon laser is often used. In order to increase the generation threshold of the dye, the pump- ing radiation is focussed on the dye solution over a region of about 10 urn diameter. This is liable to produce thermooptical distortion in the active medium. In order to eliminate this risk, the dye solu- tion is quickly circulated through the generation zone (a complete replacement of the solution in the generation zone is attained over a period of about 1 usee). Circulation of the dye solution is impor- tant from other points of view too, for example, for removal of photo- disintegration products frorn the generation zone. 20 For a detailed description of organic dyes and lasers based on them, soo [25-27], as well as Ch. 18 in [4]. 1.3 Organic Dye Lasers 33 Fig. 1.21 Incoherent optical pumping is also applied to the dye lasers, by using, for example, flash tubes or coaxial flash lamps. Such lasers hnve an efficiency of about 1 % and below. Level Diagram and Basic Transitions. A dye molecule has a com- plox structure, is composed of a large number of atoms, and has a large 1111 rnber of states which are complicated mixtures of electronic, v ihrational , and rotational states. Hence it is impossible to draw n sufficiently accurate energy level (I iagram. While considering the mechanism of creation of inversion S2{~-~ in a dye laser an extremely simple, nnd to a certain extent, arbitrary level diagram is used, reflecting only some main aspects of quantum transitions in a dye molecule. This <:'IIIIU l l l diagram is shown in Fig. 1.21. #J The .energy levels shown by thick and thin horizontal lines in Fig. 1.21 refer to vibrational and rotational states of the molecule respectively. The vibration-rotation states have been grouped together near the electronic states of the mole- cule, denoted in the figure as So, "')1' 8 2, T 1 , T 2 , etc. The state corresponding to 8 0 is the ground ~t.ate. An excitation leads to the transi- tion of one of the electrons of the molecule to a more excited state. If the spin of this electron remains antiparallel to the spin of the rest of the molecule, the electronic state is called a singlet (states 8 1 , 8 2 , as well as the ground state 8 0) . If, however, excitation is accompanied by a spin flipping, rendering it parallel to that of the remaining molecule, the electronic states are called triplets (states T 1 , T 2). The singlet-triplet transitions are associated with a flipping of the electron spin and are therefore less probable than the singlet-singlet or triplet-triplet transitions. Transi- tions are shown in Fig. 1.21 by arrows. The straight arrows indicate optical transitions (laser transition is indicated by a thick arrow), while the undulating arrows show nonoptical transitions. Optical pumping excites a molecule from the electronic state 8 0 to electronic state 8 1 • The excited molecule undergoes' a nonradiative relaxation to the lower vibrational level of the state 8 1 very quickly (over a period of about 10-11 sec), transferring its surplus energy to the solvent. This molecule may then undergo a radiative (laser) transition 8 1 ~ So, or one of the following three transitions compet- .ing with the laser transition: the transition 8 1 -+ 8 2 associated with 3--0436 34 Ch. 1 Population Inversion in Active Media an additional absorption of the pumping radiation; the nonradiat.ive- transition 8 1 -+- 8 0 (internal conversion); or the nonradiative transi- tion 8 1 -+ T 1 (singlet-triplet conversion). The singlet-triplet conver- sion may then be followed by additional absorption of pumping radiation for T 1 -+ T 2 transition (triplet-triplet absorption). Besides. this transition, the nonradiative and radiative T1 -+ 8 0 transitions- are also possible. While considering the creation of inversion, all these transitions must be taken into account. The transitions 8 1 -+ S 2 and T1 -+ T 2- lead to additional expenditure of pumping energy. The nonradiative· transition 8 1 -+ So as well as the transitions 8 1 -+ T1 -+ So decrease- the number of molecules de-excited during the laser transition. The unfavourable influence of the triplet-triplet absorption may be reduced- if steps are taken to increase the probability of T1 --+-So transitions. This prob- ability (and hence the lifetime of the molecule in T1 state, which is of the or- der of 1llsec) depends on the composition of the solvent. In order to increase- this probability, certain special substances (the so-called extinguishers of the- triplet state) are added to the dye solution. For example, addition of paramagnet- ic impurities (like oxygen) facilitates "mixing" of singlet and triplet states and consequently increases the rate of triplet-singlet transitions. Tuning of Generation Wavelength; Selective Resonators. As has been mentioned above, tuning of generation .wavelength is possible in dye lasers. For this we can use the dependence of the position of J 4 Fig. 1.22 dye generation line on the concentration of the dye molecules in the solution, temperature of the solution, and the reflection coefficients of the resonator mirrors. Selective resonators are often employed for- tuning of the generation wavelength. A selective resonator is one which contains a spectral selective element in addition to the active medium. Spectral optical filters,. Fabry-Perot interferometers, dispersion prisms, or diffraction grat- ings can be employed as such elements. Figure 1.22a shows a dye laser with a continuous tuning of the- generation wavelength with the help of a dispersion prism. Here, 1 is the dye cuvette, 2-the pumping radiation (from the auxiliary laser), 3-the output radiation, 4-the output resonator mirror, 5-a prism, and 6-a reflecting mirror rotating with respect to the: prism. The cuvette is oriented in such a way that the perpendicular 1.4 Gas Lasers with a Wideband Optical Pumping 37 r 1 2 Fig. 1.24 1)0 spent uselessly in heating the gas and surrounding objects. Hence widoband optical pumping is not employed for the case of gas lasers. Posslhlltty of Wideband Optical Pumping in Gases; Photodisso- .'Intioll Lasers. Exceptions t<ithe rule are gaseous media containing mulecules which dissociate under the influence of radiation. As a Inn tter of fact, the absorption spectrum due to photodissociation of J.(H~OOUS molecules is continuous and is romparable in width to the absorption E ~IH'('tra for solid and liquid active media. III this connection, we turn to Fig. 1.24. 'l'he figure shows electronic energy lr-vels of a diatomic molecule (curves 1 £3 IIl1d 2). Suppose that the molecule is in 0110 of the vibrational states of level 1 E2o--t,----;..~-~-- (J~'11 is the energy of the molecule). By E1V--~"'T'T'T-MF'P'I"'I"'. nhsorbing the pumping radiation, the iuulecule may jump to the electronic t'florg-y level 2. In accordance with the Eov--~....-.........., Franck-Condon principle'", such transi- I inns are shown in the figure by vertical ,,- 1ll'I'OWS (dotted lines). It is evident from o~-------...... I lip figure that a transition 1 -+ 2 is pos- ~ihle upon absorption of photons whose ouorg y is nw and satisfies the inequali- t it'~ (E 1 - Eo) < !iro < (E s - Eo). If !iw < (E 2 - Eo), a transi- t iou takes place from the initial vibrational state of energy lr-v«l 1 to one of the vibrational states of energy level 2. The P'4p(\ctrum then consists of a number of narrow lines. If, however, 110) > (E 2 - Eo), the transition 1 -+ 2 occurs, leading to the dissociation of the molecule. The absorption spectrum in this rusowilI be continuous and itswidth will be equal to E s - E 2 - Owing to the continuous nature and comparatively large width of the absorption spectrum of photodissociating molecules, a wide- huurl optical pumping can be employed for stimulating the photo- elissociation reaction: AB + n« -+A* + B. At least one of the products of molecular dissocisation is found 10 he in the excited state (A*) and may he used for laser generation. '1' Itus, wideband optical pumping in gas lasers is possible if mulocular photodissociation is employed as the intermediate pro... ...,~~~.I. Such gas lasers are called photodissociation lasers. 'l'here are two types of photodissociation lasers. The first type urnploys one of the products of molecular dissociation as the excited 2:' See, for example, Sec. 3.7 in [1]. :l4 The method of wideband optical pumping for laser generation in gases willa dissociating molecules has been proposed in [30]. 38 Ch. 1 Population Inversion in Active Media active centre (excitation in primary photoprocesses). In the second type, the excited active centres are formed as a result of chemical reactions involving the products of molecular dissociation (excita- tion in secondary chemical reactions). Such lasers are also called photochemical lasers. At present, possibilities are being explored for creating gas lasers based on allowed electronic transitions of molecules with direct wideband optical pumping (without photodissociation of molecules). These possibilities are based on the relatively broad absorption spectra of a number of molecules (especially polyatomic) in the ultraviolet region of spectrum. The small magnitude of vibrational quanta may result in a nearly continuous absorption in this region'", Iodine Photodissociation Laser. As an example of a photodissocia- tion laser, we consider the laser generating on transitions in iodine atoms [32]. Under the action of wideband optical pumping (with wavelength 0.27 um), the CF31 molecules dissociate: CF31 + tiro -+ -+CF3 + 1*. The iodine atoms 1* appear in the excited state 2Pl/2. On the transitions in these atoms to the state 2p3/2' radiation with a wavelength 1.315 um is emitted. In addition to CF31 molecules, CHsI molecules may also be used in this process. . The efficiency of photodissociation lasers is of the order of 0.5 %. This efficiency may be increased in principle by broadening the ab- sorption spectrum of dissociating molecules of a gaseous active medium as well as through chemical reactions corresponding to photodissociation. In this connection, we consider the problem of enlarging the class of active media applicable for use in photodisso- ciation lasers [31]. Direct Conversion of Solar Energy into Laser Radiation. Hol- ger et ale [33] have examined the possibility of creating a photodis- sociation laser excited by the solar radiation, and called the sun- pumped laser. The active medium of such a laser must satisfy a number of requirements: the absorption band must cover a consider- able part of the solar spectrum, the quantum yield should be as high as possible, and the generation must take place for a relatively low-intensity continuous pumping. A gaseous mixture containing cesium and xenon atoms has been proposed as active medium for a sun-pumped laser. The CS2 mole- cules are disintegrated into cesium atoms by absorbing solar radiation in the band 0.44 urn to 0.52 urn approximately. The products of dissociation contain excited cesium atoms in 7281 / 2 state (denoted by Cs**). Laser oscillation may appear on transitions in these atoms into less excited states 62P3/2 and 62P1/ 2 : { 7281/ 2 -+ 62P3/2, Cs**-+Cs* 7281/ 2 -+ 62P 1/2. 26 Widehand optical pumping in gas lasers with or wlthout.photodissocia- tion of molecules has heen described in [31]. 1.5 Pumping by Self-maintained Electric Discharge ;> 39 These transitions correspond to the generation wavelengths 1.47 und 1.36~m respectively. The question of selective depopulation of the lower laser levels of cesium atoms (without deexciting the upper leser level) is of great importance in these lasers. A chemical depopulation mechanism has been proposed. It is based on active combination of Cs* atoms (and not Cs** atoms) with Xe atoms, leading to the creation of excit- ·ed CsXe molecules. These molecules are dissociating molecules and their ground state is unstable. The molecules of CsXe, returning ·to the ground state, dissociate into unexcited Cs and Xe atoms. 1.5. PUMPING BY SELF-MAINTAINED ELECTRIC DISCHARGE IN RAREFIED GASES Types of Gas Discharge Lasers. Pumping by self-maintained dis- ,eharge in the active medium is often employed for excitation of gas lasers. Such lasers are called gas discharge lasers.t" They operate in fairly rarefied gaseous media, the pressure being of the order of 1- 10 mm Hg. There are three types of gas discharge lasers: those operating on transitions between ionic levels (ion gas lasers or ion lasers); those operating on transitions between neutral atomic levels (neutral .atom lasers, or atomic lasers); and those operating on transitions be- tween molecular levels (molecular lasers). The active centres in these three types of lasers are free ions, free atoms, and free molecules respectively. Ion lasers generate mainly in the visible as well as near ultraviolet region of the spectrum (from 0.3 to 1 um approximately). Atomic lasers generate over a much wider range of wavelength-from 0.4 to ~OO um , the main laser transitions falling in the region 1-20 um, Molecular lasers have the widest range of generation wavelength. Lasers operating on transitions between rotational levels of the same electronic and vibrational state of a molecule (purely rotational transitions) generate in a wide infrared region from 10-20 um up to several hundred micrometres. Molecular lasers operating on transi- tions between vibrational levels of the same electronic state of a molecule (Vibration-rotation transitions) generate infrared radia- tion mainly in the range 5-50 um, Molecular lasers may also generate on transitions hetween electronic states of a molecule. In this case, the generation wavelength is in the visible or ultraviolet region of the .spectrum (nearly 0.2 to 1 ~m)27. 26 Gas discharge lasers have been described in [34-36]. See also [4, 7, 8, 37, 38]. 27 As an example of this kind of las ers, we can mention the molecular lasers on transitions from a stable electron-excited state to the unstable ground state (excimer lasers).. 42 Ch. 1 Population Inversion inActive Media canal is meant for maintaining a uniform pressure throughout the length of the discharge tube. In the absence of such a canal, there is a concentration of gas in the anodic part of the tube soon after the arc discharge is switched on. For high-frequency excitation, the electrodes inside the tube are not used. The discharge may be excited with the help of a high-fre- quency choke mounted on the by-pass canal (Fig. 1.28). The by-pass canal and the choke are shown by 1 and 2 respectively. F· 1 28 T.he method of "squeezing" the discharge19. . from the walls of a discharge tube with the help of a longitudinal magnetic field is fairly widely employed. As a result, the rate of relaxation of excited ionic states decreases on account of collisions with the walls of the tube. Consequently, the output power of the laser radiation increases. Creation of Inversion in the Argon Laser. Figure 1.29 shows the most important transitions for an Ar" ion for creating inversion 16~-~ -----~---+--....... i\=O.488Jlm 0.515 15 1Ix. IIm- '19 7F 18 1 171-------~-+--I----.....~ O'---------L.---:----JL-...- --J 1.5 Pumping by Self-maintained Electric Discharge 43 definite distribution of electrons in the ion over states with different / )rincipal and orbital quantum numbers, nand l respectively), name- y 3p44s, 3p44p, 3p43p, 3p43d, and 3p44d. The group of levels with tho electron configuration 3p44s (four electrons in the state with n d:: 3, l = 1 and one electron in the state with n = 4, l = 0, l.u, four 3p electrons and one 4s electron) corresponds to the lower luser "level" which we shall designate as the "4s-level". The group of lovols with the configuration 3p44p corresponds to the upper laser "Inver' ("4p level"). The group of levels with the configuration 3p&.3p (1.0. with configuration 3p6) corresponds to the ground "level" (u:ip-Ievel"). The term "level" is used within inverted commas here sluce each electron configuration has several energy levels of the Inn (several spectral termsj'". For example, the electron configuration :'1'~4p has 15 terms, hence the "4p-Ieyel" actually consists of 15 energy luvels, It is because of this that each electron configuration has a cor- responding energy band of a definite width. Table 1.3 shows the TABLE 1.3 Electron cH1nf\guration :.'p4:ip :1/,'4, Spectral terms 2p1/22P3/2 2p 1/22PS/24p 1/24PS/24PS/22DS/22D5/22S 1/2 { 2S 1/24S3/22p 1/22P3/24P1/24P3/24P5/2 2l)3/221)5/241)1/24D3/241)S/24D7/22~5/22~7/2 Lines in the argon luminescence spectrum (4p -+ 48 transition): 0.455 um o.see, 0.473, 0.477, 0.488, 0.497, 0.502, and 0.515 J.lID. xpectroscopic energy levels corresponding in this case to the ground, Iower and upper laser "levels". I t should be noted that the energy bands in the level diagram shown in lo'ill' 1.16 (Nd: YAG laser) and Fig. 1 29 (argon laser) have different origins. 1,:,,(',11 band in Fig. 1.16 corresponds to one spectroscopic (atomic) energy level which is subjected to Stark's splitting into several sublevels under the action or the electric field of the crystal lattice. Each band in Fig. 1.29 corresponds to .u'vl'rnl unsplit spectroscopic energy levels. I·~ight lines appear as a result of the 4p -+ 4s transition. Their wuvelengths are shown in Table 1.3. The blue line with wavelength ().I,HH um (relative intensity 45%) and the green line with wavelength O.f,15 um (relative intensity 35%) are the brightest ones. The upper I"Mor "level" ("4p level") is excited as a result of several processes: a,n 'I'he connection between the electron configuration and the spectral terms 4'.'I·I·t~~pondingto this configuration is explained, among others, by Landau and 1.lf,dlitz [40]. See Sec. ~.1 in [1]. 44 Ch. 1 Population Inversion in Active Media (a) the transition 3p -+ 4d ---+ 4p, (b) the transition 3p ---+ 3d -+ 4p, and (c) the transition 3p -+ 4p (the cascade, the step-by-step, and the direct electron excitation respectively). I t is interesting that the rate of electron excitation of the upper laser "level" (denoted by F (4p)) is less than the rate of electron exci- tation of the lower laser "level" (F (48)). It turns out that F (4s)/F (4p) ~ 2. The inversion arises because the lower laser level states relax considerably faster than the upper ones: R (4s)/R (4p) ~ 10. The relaxation of the upper levels takes place mainly on account of spontaneous emission. Significantly, the transition 4p ---+ 3p is forbidden in the dipole approximation (optically forbidden transi- tion), and the laser levels are situated fairly high in comparison with the ground state. Thus, inversion in the argon laser is created not due to the predominant population of the upper laser levels, but rather due to the predominant depopulation of the lower laser levels. Helium-neon Laser; Creation of Inversion. Helium-neon laser is historically the first gas laser [41]. The generation takes place on transitions between levels of neutral neon atoms. Besides neon, the active gaseous medium contains helium as the buffer gas. The gaseous mixture has a pressure of about 1 mm Hg, the partial pressure of helium being about 5-10 times higher than the pressure of neon. The helium-neon laser employs a stationary glow discharge excited by a direct current. Figure 1.30 shows the main transitions in the active medium of a helium-neon laser. The dashed lines show the transitions connect- ed with electron excitation or deexcitation. The solid lines show the optical transitions, while the dotted line shows a transition connect- ed with deexcitation at the walls. The curved arrows indicate the resonant transfer of excitation energy from helium atoms to neon atoms. As in the case of an argon laser, the energy bands shown in this figure for neon atoms correspond to specific electron configura- tions and are consequently composed of several levels (several spec- tral terms). Abbreviated notations of electron configurations (Pas- chen's notations) are used in the figure: is-band corresponds to the 2p 53s configuration, 2s-band to the 2p 54s configuration, 3s-band to the 2p55s configuration, 2p-band to the 2p 53p configuration, and 3p-band to the 2p 54p configuration. Each s-band consists of four levels, while there are ten levels in each p-band. The 3s and 2s bands play the role of upper laser levels, while 3p and 2p are the lower laser levels corresponding to these. The main laser transitions are 38 ---+ 3p (3.39 um), 2s ---+ 2p (1.15 um), and 38 -+ 2p (0.6328 um-i-red line). Inversion in a helium-neon laser is due to the fact that the rate of population of the upper laser levels is considerably higher than 1.5 Pumping by Self-maintained Electric Discharge 47 (b) (a} oc Fig. 1.33 o fif MPvoral components: carbon dioxide, nitrogen, and various addi- II v.'~ Iike helium, water vapours, etc. The active centres are CO2 mole- f'''.~~ radiating on the transitions between vibrational levels of the .,lprlloonic ground state. Nitrogen plays the role of a buffer gas and II~ molecules resonantly transfer the excitation energy to CO2 mole- 1·,d"H. A glow discharge is usually ap- 1'1 ,.,et for exciting CO2 lasers, although H Itulso discharge can also be used. Figure 1.32a shows the initial ver- rdOIl of a CO2 laser. Here, 1 indi- • It I.'H the system of CO2 circulation, 2 '" the nitrogen circulation system, .tJ the region of high-frequency glow tlIMt'hnrge, 4-the lasing volume in which the excited nitrogen molecules ~ W 2 ,tl'l' rnixed with the unexcited CO2 mulucules, 5-the resonator mirror, Hlld 6-the output radiation. In this vorsion of the CO2 laser, the discharge "I.,c·-t.rons excite only the nitrogen urulocules, and the excited nitrogen w3 uiulocules then transfer this energy to I h., active centres in another region. I.nter versions of CO2 laser em- ploy a discharge in the CO2 + N 2 mixture. For this, a d.c.-excited ulow discharge is normally used. The schematic for such a CO2 laser ,~ shown in Fig. 1.32b, in which 1 is the lasing volume, 2 and 3 are I Ito anode and cathode respectively, 4-the resonator mirror, and [)-. the output radiation. Like the previous version, this system is 111~o based on gas circulation, which helps in avoiding an undesirable r hnnge in the chemical composition of the active gaseous mixture, which takes place, in particular, as a result of the reaction ~~( :()2 -+ 2CO + 02' At present, sealed CO2 lasers are frequently employed (without lilly circulation of the gaseous mixture). The life of such tubes may bn of the order of 1000 hours or more. Creation of Inversion in a CO2-Laser. The CO2 molecule has four vibrational degrees of freedom, with three corresponding vibrational modes: the symmetric mode (Fig. 1.33a), the deformation mode (Fig. 1.33b), and the asymmetric mode (Fig. 1.33c)32. The vibrational Iroquencies corresponding to these are denoted by WI' 002 , and Ws respectively, while fiw i == 0.163 eV, fioo 2 == 0.078 eV, fiW3 = 0.276 eV, uud WI ~ 200 2 • 32 The energy of deformation mode is doubly: degenerate (a deformation modo has two vibrational degrees of freedom). .... 48 Ch. 1 Population Inversion inActive Media We shall represent the vibrational states of a CO2 molecule through a set of three vibrational quantum numbers VI' V 2, and V3• These numbers are equal to the excitation factor of the symmetric, defor- mation, and asymmetric modes respectively of the CO2 molecule. For example, the vibrational state (020) (VI == 0, V 2 == 2, V 3 = 0) is a state in which the deformation vibrations are doubly excited while the symmetric and asymmetric vibrations are not excited. The mechanism of creation of inversion in the CO2-laser is illus- trated in Fig. 1.34 which shows the levels corresponding to the three different vibrational modes of 3 CO2 molecule as well as the first 1/A,J.Lm-1 0.25 (000 0.2 I I I 0.15 I I (100) I 0.1 I I (010)1 0.05 I I 0 J~ CO 2 N2 Fig. 1.34 excited vibrational level for an N 2 molecule. The upper laser level is (001), while the levels (020) and (100) form the lower laser level. Laser radiation of wavelength 10.4 um is generated on the transi- tion (001) -+ (100), while on the transition (001) -+ (020), radiation with wavelength 9.4 urn is generated. The excitation of the (001) level takes place as a result of inelastic collisions of CO2 molecules with electrons (electronic excitation) and the excited N 2 molecules (resonant energy transfer) which are excited in collisions with electrons. The relaxation of (020) and (100) levels mainly takes place as a result of resonant energy transfer to the unexcited CO2 molecules, leading to the accumulation of CO2 molecules in the (010) state (see the transition marked in Fig. 1.34 by the letter r): CO2(020) + CO2(OOO) -+ 2C02(010), CO2(100) + CO2(OOO) -+ 2C02(010). The relaxation of the level (010) is the "bottleneck". The main relaxation mechanism for this level is the gas kinetic mechanism of 1.5 Pumping by Self-maintained Electric Discharge .tlllH'~Y transfer to H 20 molecules or He atoms (see the transition uullcnted in Fig. 1.34 by the letter g). Thus, inelastic collisions of (:( 'g molecules (010) with specially introduced additions to the gas uli xtures play an important role in the creation of inversion. It ~la()1I1d be recalled that CO2 lasers belong to the class of collision 1...'t(H'S. I t should be observed that each vibrational level of a CO 2 molecule has a nurrusponding set of rotational sublevels. Hence the laser transitions (001) -. .... ttOO) and (001) -+ (020) must be treated as uibratton-rotatton transitions hav- IIIJl,t.hc corresponding vibration-rotation bands in the laser generation spectrum. R-branch 0.098 0.096 0.094 0.092 1/"-, fLm-1 Fig. 1.35 20 40 V E/P, cm.mm Hg Fig. 1.36 5 10 w 0.8 r-----r--..,..--~--_r__-___, O. 2 I-----I'W---+---~--+--I 0.4 I--r-+---I-lt-----+--...::i~_t_-___t Itetuming to the question of crea- I ion of inversion in a CO2 laser, wo mention the necessity for en- ~11I'illg an optimal EIP ratio (E is I lit' field intensity in the positive .tiM(',hnrge column and P is the prnssure of the gas mixture) for which the cross section of electron "x(',itntion of N 2 molecules and the (001) level of CO2 molecules are MilIfrciently large and the cross sections of excitation of levels (100), (O~O). and (010) of CO2 molecules are relatively small. As an exam- 1'1.\, Fig. 1.36 shows the dependence of the fraction w of energy, 'l'1a.·~m bands correspond to the P and R branches of transitions. These branches huvu been clearly indicated in Fig. 1.35 which shows the generation spectrum of It low-power CO 2 laser (for (001) -+ (100) transition). It should be noted that wll h increasing laser power, the number 01' lines generated simultaneously in P lind fl branches of transition decreases urul rnay come down to just one or two Illl"~ (competition between transitions). III other words, the interaction (competi- t iuu) between transitions may lead to K1'u('rntion mainly on one of the transi- Iloll~ in P and R branches only. Conse- tplf'lltly, high-power CO 2 lasers have a Id"hly coherent radiation. 52 Ch. 1 Population Inversion in Active Media the ionisation of the gas is ensured by ionising radiation, while the electrons simply excite the active centres. Electroionisation CO2 Laser'. 33 The electroionisation pumping method is applied in high pressure CO 2 lasers. The mechanism of creation of inversion for an electroionisation CO2 laser is illustrated in Fig. 1.34 which also applies to a gas discharge CO2 laser. The only difference is that the fast electrons exciting the vibrational degrees of freedom of N 2 and CO2 molecules are generated not by the self-sustained discharge, but rather by the ionising radiation and the accelerating external electric field (non-self-sustained discharge). An j IJ I I I 4 2 ~ 2 ~~~~~~=!J 7 5 Fig. 1.38 electron beam from an accelerator (electron energy 100-500 KeV, current density of the beam "",,10-4 A/cm 2) is usually employed as ionising radiation. Figure 1.38 shows the simplified version of an electroionisation CO2 laser. Here, 1 is the lasing volume, 2-the resonator mirrors, a-the electron beam (ionising radiation), 4 and 5-the upper and lower electrons respectively, 6-a metal foil which is transparent to fast electrons, and 7-the laser radiation. It should be noted that the foil 6 is a fairly vulnerable place in an ionisation laser. It separates the evacuated accelerator space from the lasing volume which is under high pressure. An increase in the pressure of the gas mixture leads to an increase in the power of laser radiation per unit volume of the active medium. Let us denote this power as W. Figure 1.39 shows the dependence of W on P. The experimental points A correspond to the gas-dis- charge CO2 laser, while the pointsB correspond to the electroionisa- tion CO2 laser. This dependence is approximately quadratic: W ex: ex P2. In comparison with the gas discharge CO2 laser, the power of laser radiation per unit volume of the active medium is 106 times 'more in an electroionisation laser. 88 The first CO2 lasers with non-self-sustained 'discharge operating at atmos- pheric pressure were 'created in f971 (49; 50]. .. 1.6 Electroionisation Lasers 53 8 Fig. 1.39 10 iO The Choice of an Active Medium. Besides the gas mixtures CO2 + I·N 2 and CO2 + N 2 + He, other active media can also be employed ill electroionisation lasers [48]. For these, we consider the lasing tran- Hi lions not only between vibrational states, but also between electronic 7 W, W/cm 3 ~ ta tes of molecules. iO Electroionisation CO Laser. This laser operates on the transition be- tween higher vibrational levels of a 105 (~() molecule (v ~ 15). The generated wavelength is equal to 5 f1m. Usually, It mixture of gases CO + N 2 or CO + -1-N 2 + He is employed. This laser 102 has an unusually high efficiency, up to 50%. Electroionisation Ar+ N 2 Laser. This luser operates on the transition between .,1 ectronic states of an N 2 molecule. OJ The generated wavelength is equal to 10-2 0.358 urn. Argon plays the role of a buffer gas, resonantly transferring «nergy to nitrogen. E lectroionisation Compressed Xenon Laser, This laser operates on a transi- t.ion between the excited and ground ulectronic states of a disintegrating X 02 molecule. The generated wavelength is equal to 0.172 um, Fnst electrons excite the Xe atoms, which then form the excited .X (~2 molecules as a result of collision with unexcited atoms. Besides pure xenon, a mixture of Ar + Xe is also employed, argon playing the role of a buffer gas. Methods of Ionisation. An electron beam is most frequently employed fol' ionisation. The method of ionisation through ultraviolet radia- tion (photoionisation), obtained, for example, through a spark dis- ch arge [51]34, is under intensive investigation. Also worth considera- tinn is ionisation by the heavy charged particles created in nuclear rr-actions, Laser generation has been obtained in an electroionisation CO2 Iuser placed in the active zone of a stationary nuclear reactor [52]. An active gas mixture CO2 : N 2 : sHe of composition 1 : 4 : 5 was «mployed. The ionisation was caused by the products of the nuclear roaction n + sHe ~ p + T + 0.76 f\IeV. . 34 Electroionisation lasers employing photoionisation are frequently called phototonisation lasers. 54 Ch. 1 Population Inversion in Active Media (where n is a neutron, p-a proton, and T-the tritium nucleus). The reaction was initiated by thermal neutrons created in the moder- ator of the reactor. The density of thermal neutrons was 1014 em -2sec -1. Possibilities are being considered for creating an electroionisa- tion laser by using the energy generated by a nuclear reactor not only for ionisation, but also for accelerating free electrons appearing as a result of ionisation. Such a laser would directly transform nuclear energy into laser radiation. Eo EJ £2 £3 £4 Fig. 1.40 n n (c) 1.7. GASDYNAMIC LASERS (THERMAL PUMPING) Thermal Methods for Creating Inversion. Thermal excitation may be used as a fairly effective means of populating vibrational and rotational levels of a molecule. This involves heating the gas to tempera- tures around 1000-2000 K. Naturally, inversion cannot be created simply by heating a gas, which leads to a decrease in the population density of levels with increasing energy according to the Boltzmann law: nsx: exp (-ElkT). Figures 1.40a and b compare the population of vibrational levels at different temper- atures (at temperatures T 1 and T 2' where T1~ T 2). Curve 1 shows the dependence noc exp (-ElkT1) while curve 2 shows the dependence nix: exp (-E/kT2 ) . It can readily be seen that while an increase in temper- ature leads to a population of the higher levels, it does not ensure a population inversion. Suppose that a gas heated to a tem- perature T 2 is very rapidly cooled to a temperature of T1 , 't being the time over which this ternperature variation takes place. Further, we assume that the rate of relaxation of the first ex- cited level (level E1) is much higher than the relaxation rate of the levels E 2 1 E 3 •••• More precisely, we assume that the relaxation time (171) for the first level E 1 is much shorter than 't, wh ile the relaxation time ('t2 ) for the higher levels is, on the contrary, much longer than 17, i.e, 1'1 ~ l' ~ 't2 o In this case, the population density of the E 1 level will Iollow the temperature varia- tion upon cooling, while the population densities of other levels 1.7 Gasdynamic Lasers (Thermal Pumping) 57 Fig. 1.43 (001) speed of about 1500 m/sec). The energy of translational motion of the- molecules is supplied by the energy of vibrational motion. This- means that the vibrational levels relax very rapidly at the exit from the nozzle. I t is extremely significant that not all vibrational levels. undergo a rapid relaxation, but only such levels for which the relax- nt.ion time is less, i.e, the lower laser levels (020) and (100) of the- <:°2 molecule. As far as the (001) level of the CO2 molecule and the n .(020) vibrational level of the N 2 mole- (~1I1e are concerned, they do not relax to any considerable extent finder a very rapid expansion of the gas. In the laser volume, i.e. under conditions of a sufficiently rarefied gas (the pressure does not exceed 0.1 atm) these levels prac- t.ically do not relax at all, and "freez- ing" of higher vibrational degrees of freedom is observed. Figure 1.43 shows the variation of the population densities of the levels (020) and (001) of a CO2 molecule upon a passage of the gaseous mixture from the combustion chamber (the parameters of the mixture- nre T 2 and P 2) to the cavity (the mixture in the cavity has param- eters T I and PI). In the transition region (the nozzle), there is an almost complete depopulation of the (020) level and only a slight decrease in the population density of the (001) level. In the cavity, the population density of the (001) level practically freezes at values nearly corresponding to the initial temperature T 2. The population inversion of (001) and (020) levels is apparent. 'rhus, the CO2 molecules enter the cavity with nearly unpopulated lower laser levels (more precisely, the population density of these· levels corresponds to the temperature T1 ) . As regards the upper" laser level, it is populated as if the gas continued to be at a tern-- perature T 2. It is also important that the population density of the vibrational level of the N 2 molecules is "frozen". The excited N 2 molecules will resonantly give their excitation energy to CO2 molecules and thus rnaintain the relatively high population density of the (001) level. J f, should be remembered that quantitatively, nitrogen constitutes the main component of the mixture-about 90% in a gasdynamic laser. Hence it may be considered that the laser energy is obtained mainly from the vibrational energy of nitrogen molecules. Thus, the energy accumulated in the vibrational degrees of free- dorn of molecules in the combustion chamber is consumed during t he passage of the gaseous mixture into the cavity through the nozzle nccord ing to the following procedure: part of energy accumulated ~8 Ch. 1 Population Inversion in Active Media -in symmetric deformation vibrations of CO2 molecules is converted into the energy of translational motion of the flow through the nozzle. 'The energy accumulated in asymmetric vibrations of CO2 molecules, .and mainly in the vibrations of N 2 molecules is converted, excluding the losses in the cavity, into laser radiation. Methods of Improving the Efficiency of Gasdynamic Lasers. The .relatively low efficiency of gasdynamic lasers (not more than (1 %) Is due to a number of reasons. Firstly, the energy of the translational motion of the gas flowing from the nozzle is uselessly wasted (after ,deceleration and compression in the diffuser, the gas is found to be at high temperatures). Besides, a certain number of excited N 2 molecules are removed from the cavity by the gas flow before they have been able to convey their excitation energy to CO2 molecules. 'Likewise, a certain number of excited CO2 molecules are removed before they have been able to emit. 39 In order to increase the efficiency of a gasdynamic laser, it is neces- .sary to reduce the losses mentioned above. A reduction of losses -caused by the removal of the excited molecules from the cavity is possible by optimising its parameters or by improving its construc- tion. In order to avert the losses due to heating of the gas in the dif- -fuser, some radical measures have to be taken. For example, it is possible to construct a gasdynamic laser with a closed cycle for the gas mixture. Such a laser must include two additional elements: .a compressor, and a heat-exchanger. These are meant to impart the initial temperature and pressure to the gas mixture. Thus, the energy -of the translational motion of gas flow may be used for bringing the gas into the initial state. In this case the gasdynamic laser is iden- tical to a heat engine (whose working cycle is close to the Carnot cycle). Once such a laser starts generating, the subsequent supply of energy must simply compensate the losses due to the movement of the gas, .losses in the resonator and the radiated light energy. 1.8. CHEMICAL LASERS Chemical compounds are rich accumulators of energy. This energy .may be partially liberated during a rearrangement of chemical bonds (in exoenergic chemical reactions). 'The idea of converting this energy into laser energy is quite tempting. Chemical lasers are based on the realisation of this energy conver- s ion.f" The undiminishing interest to this type of lasers is due both to the practical importance of the problem of direct conversion of 39 A premature removal of a certain number of excited nitrogen and carbon dioxide molecules from the cavity by the gas flow determines the type of losses in the resonator, which are typical of gasdynamic lasers. . .. 40 The possibility of creating a chemical laser was considered by Oraevskii {59] and Tal'roze [60]. Reviews bv Tijoev et ale [61} and Basov et al. [62] have been devoted to chemical lasers. Sec also [81 and Ch , 7 in [4]. 1.8 Chemical Lasers 59 ehnrnical energy into light, as well as to the possibility of using lasers Iur investigating chemical processes, the cross sections of reactions, nnrl their kinetics. 'I'he existing chemical lasers generate on vibrational (more pre- ,·iHnly, vibration-rotation) transitions in molecules. The lower limit of t lu~ generation wavelength range in these lasers is about 2.5 f.Lm. I"vostigations are being carried out to explore the possibilities of .'I't'lll.ing chemical lasers in the visible spectral region (i.e. lasers Jll'unrating on transitions between electronic states of molecules). «:hemical Reactions. Initiation and Acceleration of Reactions. ~1 ost frequently, chemical lasers employ substitution reactions lead- '''Jl to formation of diatomic molecules in excited vibrational states. \Vn shall call these molecules as lasing active centres. They either them- ~I'I yes emit, or cause emission in other molecules to which they trans- t1'1' their excitation energy. 41 The following are the examples of sub- ~f lt.ution reactions used in lasers (the sign * indicates an excited mol- .'.~,a1e): F + H 2 ~ HF* + H; F + D2 -+ DF* + D; H + Cl2 --+ .. IICl* + CI; CI + HI -+ HCI* + I. In order to initiate these reactions, a certain amount of energy has I n he spent on obtaining chemically active reagents (F, H, CI). Be- <1ldns, it is essential to accelerate a chemical process since a chemical I ,,~O(· operates only on fast reactions which ensure a fairly rapid popu- I" I ion of the upper laser level of the emitting molecules. The rate ur It chemical reaction is proportional to the concentrations of the I'na~ents taking part in it. Hence a sufficiently large amount of chem- ~f'nlly active reagents is required in order to accelerate a reaction. Various methods are employed for initiating and accelerating f'hnrnical reactions (in other wodrs, there are several different means 01' obtaining chemically active reagents). In particular, chemically IIf',t.ive reagents are reagents in atomic state, obtained as a result of dissociation of molecules. The most widely used methods are photodis- «ictation; dissociation by electron impingement, and thermal dissociation or molecules. In this context, the following methods for initiation (tH~eelerating) of chemical reactions' are employed: photoinitiation, Initiation by an electron beam or by an electric discharge pulse, and I Iiermal initiation. Quartz flash lamps are often used as sources of radiation for photo- tnitiation of chemical reactions. For pumping at a shorter wavelength (I" ~ 0.2 urn), different types of open discharge (for example, spark d iscliarge) are applied. One of the simplest means of initiating chem- ,.'ul reactions is the self-sustained discharge in a gas. However, this iunthod is applicable only for relatively low pressures and small 11t~i ng volumes. Hence the method of initiation based on the ap- p l ir.af.ion of an electron beam is more interesting. An electron beam 'I In this context, ,ve often speak of direct or indirect creation of inversion III c', II P.mical lasers (see later). 62 Ch. 1 Population Inversion in Active Media Fig. 1.45 pumping. The decisive factor in the creation of inversion is obviously the relationship between the rate of a chemical reaction and the relaxation rates for the levels. In particular, it is important that the inequality R 1 > R 2 be satisfied. In diatomic molecules this inequal- ity is hard to satisfy on account of the fact that there is no process which would lead to a selective depopulation of the lower vibrational levels. On the contrary, the transitions (v + 1) -+ v take place, as a rule, at a: faster rate than the transitions v -+ (v - 1). In. this connection it is expedient to use an indirect inversion. In the indirect creation of inver- sion, the excited diatomic mole- cules appearing as a result of the chemical reaction resonantly transfer their excitation energy to the unex- cited polyatomic (usually tri- atomic) molecules which then radiate in the resonator. In other words, inversion is created on the vibra- tional transitions of "cold" molecules forming the laser medium, as a re- sult of resonant transfer of energy from the "hot"[molecules produced in the chemical reaction. As a rule, CO2 molecules are employed as "cold" molecules while the molecules. of DF, HF, HCI~ etc. serve as "hot" molecules. When energy is trans- ferred to CO2 molecules. the asymmetric vibrations of these molecules (OOn) levels) are most eflect ively excited. DF-C02 Chemical Lasers with Indirect Creation of Inversion [63]. A high output power and a relatively long laser chain have made the D 2 + F 2 + CO2 laser (deuterium. fluorine, and carbon dioxide mixture) as one of the most important chemical lasers. Chemical reactions are initiated in this laser by photolysis, electron beam, or with the help of the reaction NO + F 2 -+ NOF + F. In the latter case, the laser is called a purely chemical DF-C0 2 laser. . A simplified version of such a laser is shown in Fig. 1.45. Here, 1 is the chamber in which the chemical reactions take place, 2 is an optical resonator,3-the resonator mirrors, and 4-laser beam. A mixture of fluorine and helium:" is first introduced into the chamber 1. Carbon dioxide and nitric oxide are then injected into the flow through a number of holes. Atomic fluorine is created as the chem- ically active reagent as a result of the chemical reaction NO + F 2 ~ -+ NOF + F. When the concentration of atomic fluorine is suffi- ciently high, the following chain process takes place as deuterium 41 Helium is introduced in order to prevent the overheating and sponta- neous ignition of the fluorine-hydrogen mixture. Besides, helium facilitates the depopulation of the lower laser level in CO2 molecules. 1.8 Chemical Lasers 63- is injected into the mixture: F + D2~DF* + D, D + F2-+DF*+F; F+ID2 -+-DF* + D;... The excited DF* molecules transfer the excitation energy to C02- molecules which are then carried by the flow into the resonator- where they radiate. Carbon Disulphide Chemical Lasers (CO Chemical Lasers). The- corrosive and highly toxic nature is the main drawback of the fluorine- hydrogen mixture. Hence other mixtures which are free from this. drawback have been tried. Foremost among these are mixtures- based on carbon' disulphide (CS2) . Let us first consider the mixture CS2 + O2 [64]. Upon irradiating" this mixture with light of wavelength 0.18-0.22 llID, the following: reactions take place in it44 CS2 + nm -+ CS + S (photoinitiation); 8 + 02~ 80 + 0; CS + °-+ co- + S; CS2 + 0 -+ SO + CS. Chemically active reagents CS, 5, and 0 are formed in these reac-- tions. Sulphur monoxide (SO) molecules and excited carbon monox- ide ,(CO*) molecules are the end products of these reactions. The- latter are' used as active centres in the laser medium (direct creation of inversion). The observed emission spectrum is characterised by' a wavelength range from 4.9 to 5.8 !lID. The Iaser radiation is gener- ated on vibrational transitions of CO molecules (v -+ v - 1) where' v = 1, 2, ... , 16. More efficient chemical lasers involving lower losses in photoini-· tiated reactions are C8 2 + 80 3 lasers or CS2 + 0 3 lasers [65, 66]. In the C8 2 + 80 3 (carbon disulphide and sulphur trioxide) Iaser; the second of the above-mentioned reactions is replaced by the reac- tion 80 3 + lu» -+ 80 2 + 0 (the radiation wavelength must be less: than 0.26 urn) , while in the C8 2 + 0 3 (carbon disulphide and ozone) laser, it is replaced by the reaction 0 3 + n«~ O2 + ° (Ap ~ ~ 0.22-0.28 um). . In all chemical lasers based on carbon disulphide, CO molecules' are the active centres of the laser medium. These lasers are often called CO chemical lasers. The main reaction CS + °~ CO* + S· is a distinctly exoenergic reaction; energy of about 80 kcal/mole is liberated in this reaction, and 80 % of this energy is used for excit- ing the vibrations of CO molecules. In order to increase the relaxa- tion rate of lower vibrational levels of a CO molecule, we introduce' agents (like nitrous oxide N20), to the gaseous mixtures of CO chemical lasers. Besides, thermal diluents like sulphur hexafluoride SF6 are also introduced. " The small length of the laser and chemical chain predetermines that the- processes in this laser are not chain processes. Ch. 1 Population Inversion in Active Media Chemical Lasers on Electronic Transitions of Molecules. As men- tioned earlier, investigations are being carried out to create chemical lasers on electronic transitions of molecules (chemical visible lasers). -One of the possibilities of creating such lasers is related to the ex- ·citation of electronic states of a molecule as a result of a combination reaction between an atom and a molecule. In particular, we can use the oxidation reaction [67] Ge + N20 ---+ N2 + GeO (+130 kcal/mole), -or Sn + N20 ---+ N 2 + SnO (+90 kcal/mole) leading to the crea- tion of inversion on electronic transitions of GeO and SnO mole- -cules, For initiating these reactions, we may use the thermal method, employing temperatures around 1000-1500 K. Another method of initiating envisages optical excitation by CO2-laser radiation. This method gives rise to the possibility of an effective conversion of infra- red radiation into visible radiation. 1.9. PLASMA LASERS (RECOMBINATION PUMPING) Recombination Plasma as an Active Laser Medium. The recombi- -nation processes during collisions of free electrons with the positively charged ions in a plasma may lead to a sufficiently effective accumula- tion of atoms (ions) in excited states. In other words, it can be used, in principle, for pumping laser levels. Let Te denote the temperature of the electronic component of the plasma (electronic temperature), and T denote the equilibrium tem- perature at which the degree of ionisation coincides with the given 'value. Suppose that n ; and n+ are the concentrations of electrons and ions respectively. The rate of decrease of n+ on account of recombina- tion electron-ion collisions is given by the equation -dn+/dt = ~n+n~ (1.9.1) 'where ~ is the coefficient of electron-ion recombinat.ion.v" I t increases 'with the decrease in electronic temperature T e according to the rela- tion ~ = DT;9/2 (1.9.2) '(D is a certain constant). It follows from Eq. (1.9.1) that n+ (t) = n+ (0) exp (-t/T:+), (1.9.3) 'where 't' is obtained by taking into consideration Eq. (1.9.2), T+=T;/2/Dn;. (1.9.4) The smaller the value of 't'e, the higher the rate of recombination, and hence the more effective the pumping. It can be easily seen that the recombination plasma must have a sufficiently high electron 45 Here, we consider the recombination by triple collisions, since in a dense plasma the radiative recombination may be neglected [68J. 1.9 Plasma Lasers (Recombination Pumping) 67 The rate of population of level 2 is the rate of recombination of ions. I t should be recalled that 't+ = T:/2/Dn~. Using Eqs. (1.9.6), we get the following expressions for the inverse population density of laser levels: n 2 - n1 = n 2 (1 - n11n2 ) = (n+/R 2't+) (1 - R 21R1) . (1.9.7) In this case, formula (1.1.3) for the amplification factor assumes the form x = o (n 2 - n1) = (an+/R 2't + ) (1 - R 2IR1 ) , (1.9.8) or 2 "J- aD n+ne (1_ R 2 ) (199) r\I- R 2 T9/2 R 1 • • •e As can be seen from Eq. (1.9.9), this simplified model of a plasma laser reflects its main features: the necessity to create high concen- trations n ; and n+ in the plasma, "cooling" of electrons, and ensuring a rapid depopulation of the lower level (in order that the inequality R2 < R 1 is satisfied). F --.......-,.....--2 Fig. 1.46 :{ %j!ij 2 ~ Q.. :::s ~{ i~'Ult ~ i i 0 ....J Fig. 1.47 Depopulation of the Lower Laser Level. The main mechanisms of depopulation of the lower laser level in plasma lasers are: radia- tion, deexcitation by cold free electrons, inelastic collisions with atoms of special additives, and chemical depopulation. Radiative depopulation (depopulation by spontaneous emission) is effective when the lower laser level is quite high, the plasma is not too dense, and the lasing volume is not very large. In the con- trary situation, the depopulation on account of collisions of active centres with the cooled free electrons is more important. The effectiv- ity of this mechanism depends on the electron concentration and the- degree of their cooling. I t should be observed that the cooling of electrons is necessary to ensure recombination (and hence the popula-- tion of the upper laser level) and for accelerating the process of de-- population of the lower laser level. ~. 68 Ch. 1 Population Inversion in Active Media Inversion of electronic deexcitation takes place, in particular, when the levels of the active centre are divided into two groups of closely-spaced levels, the lowest level in the upper group is the upper laser level while the uppermost level in the lower group is the lower laser level (see Fig. 1.47; 1 and 2 are the laser levels). As a matter of fact, the probability of electronic deexcitation between levels of different groups is considerably smaller than between the closely- spaced levels of the same group. Hence collisions with electrons lead, within the limits of each group of levels, to a predominant population of the lower, and depopulation of the upper levels (as indicated by arrows in the figure). Thus, conditions are ensured m n m II r-~- ~ A B A B o A B (a) (b) Fig. 1.48 (c) for creating inversion between the lower level of the upper group and the upper level of the lower group even when the total popula- tion density of the lower group exceeds the population density of the upper group. I nelastic collisions of active centres (let us denote these centres as A) with the atoms of special additives (denoted as B) may also be used for depopulation of lower levels. Figure 1.48 shows the transi- tions corresponding to different collision processes (the ionisation level of the atom is shaded), Fig. 1.48a showing the resonant transjer oj excitation A (m) + B (0) ~ A (0) + B (n). Figure 1.48b shows the ionisation oj an impurity atom upon transfer oj excitation (Penning effect) A (m) + B (0) ~ A (0) + B+ + e; (where e is an electron whose energy is indicated by ~E in the figure). Figure 1.48c shows the resonant charge transjer48 A+ + B (O)~A (m) + B+. By a special choice of the additive (impurity), a predominant de- population of the lower levels can be ensured. This is illustrated 48 It can be easily) seen that 'this process leads not to depopulation, but rather to population of the m-th level of the atom A. 1.9 Plasma Lasers (Recombination Pumping) 69 Fig. 1.49 o n in Fig. 1.49 where the population densities of laser levels n1 and n2 are plotted against the impurity concentration (nimp). I t can be seen that when the impurity concentration exceeds a certain value (the value n in the figure), a population inversion in laser levels takes place. Effective depopulation of lower levels in plasma lasers is also possible through special chemical reactions. Chemical depopulation can lead to inversion even on transitions to the ground state. 49 Pulsed Plasma Lasers. For ionisation of the active medium in these lasers, rapidly decaying pulses of electric discharge with the following characteristic parameters are used: voltage up to 20 kV, current up to 300 A, and pulse duration of about 0.1-1 usee. Pulse generation takes place in an ajterg low discharge. A mixture of an active laser component ionised and an auxiliary (buffer) component which cannot be easily ionised is used as active medium. The latter component is required mainly to ensure a sufficiently rapid cooling of free electrons in plasma. Under the action of a powerful excitation pulse, plasma is gener- ated with a high concentration of free electrons and ions of various mul tiplicities. On termination of the excitation pulse, a rapid cool- ing of free electrons takes place on account of collisions with atoms of the buffer gas. This is followed by an intensive recombination of ions and electrons which effects a pumping of laser levels in active centres consisting of atoms or ions (normally singly charged) of easily ionisable component of the mixture. The electronic deexcita- tion of lower laser levels plays an important role in the creation of inversion. ' Pulsed plasma lasers have been created, in particular, with ionised vapours oj alkaline-earth metals Mg, Ca, Sr, and Ba. As an example, let us consider the 8r-He plasma laser generating on transitions in SrII, i.e. on transitions between the levels of singly charged Sr t ions. Helium plays the role of buffer gas. The excitation pulse gener- ates a plasma with a high concentration of doubly charged stron- tium ions (Sr2+). The recombination processes lead to the formation of excited singly charged Sr" ions which play the role of active centres. Generation is attained on a number of transitions of Sr t ions. The principal lines are at 0.416 urn (6281/ 2~ 52Pl/2 transition) and at 0.431 um (6281/ 2 ~ 52P 3/ 2 transition) [71]. Plasma Lasers Using] Hard Ionising Agents. Reactor Laser. Some of the most promising methods of creating recombination plasma 4' Chemical depopulation will be described in detail later in this book. 72 Ch. 1 Population Inversion in Active Media Plasmachemical Lasers. Unlike chemical lasers where chemical energy is used for exciting upper laser levels, the main function of chemical reactions in plasma lasers is to depopulate the lower laser levels. Investigations of plasmachemicallasers have two basic aspects: a study of chemical reactions which remove atoms (ions) from certain states, and an investigation of media based on dissociating molecules [69]. The first aspect envisages the investigation of kinetics of popula- tion and depopulation of energy levels in the course of some chemical reactions taking place in plasma mixtures. As an example, let us consider the chemical reactions which ensure a rapid removal of hydrogen atoms from the ground state [73]: H + F + HF ~ 2HF; H + CI + HCI ~ 2HCI; H + F + F 2 ~ HF + F 2; H + CI + CI2 ~ HCI + C1 2• By making use of these reactions, generation may be achieved on transitions corresponding to the Lyman series lines in the hydro- gen spectrum. In a plasma laser based on noble-gas molecules, the processes taking place in the recombinating plasma lead to the formation of electron- ically excited dissociating molecules. These molecules may be formed, for example, as a result of recombination of atoms in the ground state [denoted as A (0)] with the atoms in the excited state [denoted as A (1)]: 2A (0) + A (1)~ A 2 (1) -t- A (0). Here A 2 (1) is the dis- sociating molecule in the electronically excited state. Plasma lasers based on electronic transitions oj noble-gas molecules apparently belong to the most promissing type of plasma lasers already in operation, since the problem of depopulation of the lower laser level can be easily solved for them, and also the choice of emit- ting molecules is varied. In particular, the compounds based on heavy inert gases (Ar, Kr, Xe) and the elements having a high affin- ity for electrons (halogens, oxygen) offer bright perspectives. Chapter Two FORMATION OF RADIATION FIELD IN A LASER RESONATOR llesides the active medium and excitation (pumping) system, the ilillical cavity (resonator) is one of the main elements of a laser. To fl'('"l.e laser oscillations, the inversion of the active medium should "ltI~'Il'e a gain which is higher than the total loss in the resonator. By rhnug ing the level and nature of losses in a resonator, one can exer- t'to",.., control over leser processes. Depending on the specific nature ..I' I ho resonator used in a laser, radiation is generated with definite Ap..cl.rum and a definite structure in space and time. The main func- I tOil of the optical resonator in a laser is to create the radiation field "I'tI/. a definite structure. ar, GENERATION, CONDITION ltelatlon Between Initial Gain and Loss Coefficients. It is well l.uuwn that as light propagates in the inverted active medium an ltU'l'l'HSe in the optical flux density is accompanied by the equaliza- II uu of population densities of laser levels. This leads to the effect Il"llnd gain saturation. The gain % (z) at the point z (the flux propagates "lolI~ z-axis) is expressed in terms of the optical flux density S (z) 'It I he following form x (z) = 1+ (a7~; S (z) , (2.1.1) wlu-ro v is the velocity of light in the active medium; ex is the param- t~' 1'1' of nonlinearity of the laser transition; and %0 is the initial lfUill 1. The nonlinearity parameter characterises the rate of equaliza- t htll of the population densities of laser levels upon increasing the 01'1 leal flux density. The initial gain is defined by the inverse popu- I" I i011 density N 0 of the 1aser levels in the absence of generation, I (', for S = 0: %0 = (B 21/iW/V) [n 0 2 - (g2/gl) nOll = oN o- (2.1.2) 11t\1'0, co is the frequency of the generated radiation; B 21 is the Eins- t'''I1'~ coefficient; a = B 2 l n w/v is the cross section of induced emis- ~Ioll in the laser transition; and No == n 0 2 - (g2/gl) nOl (we have uJn'udy encountered this quantity in Sec. 1.2). It should be observed 'It"I, a substitution of (2.1.2) in (2.1.1) leads to the result (1.1.3): No Nx (z)=a 1+(a/v)S (z) a (z).. • Sec. 5.6 in [1]. (2.1.5) 74 Ch. 2 Formation of Radiation Field ina Laser Resonator The optical power generated by the active centres in a unit volume is given by the expression- W (z) = x (z) S (z). (2.1.3) Eliminating ,C) (z) from Eqs. (2.1.1) and (2.1.3), we get W (z) = (vIa,) [xo - x (z)l. (2.1.4) We denote by Wgen the optical power generated in the total volume of the active medium of the laser: L Wgen = S ~ W (z)dz o where s is the area of cross section of the active medium and L is the length of the resonator (for the sake of simplicity, we assume that the active medium completely fills the resonator and that physical quantities do not depend on transverse coordinates). Substituting Eq. (2.1.4) in Eq. (2.1.5), we get W gen = (vLsla,) [xo - (x)], (2.1.6) L where (x) = 1rx (z) dz is the gain averaged over the length of o the resonator .. For the case of stationary generation, the following relation is valid": (x) = 111 + 112, (2.1.7) where 111 and 112 are linear coefficients of attenuation of optical flux corresponding to the unfavourable and favourable losses respective- ly4. The favourable losses are due to removal of a part of energy from the active medium in the form of laser radiation. The coefficient of favourable losses 112 is given by the expression 112 = (1/L) In V1/R1R2 , (2.1.8) where R 1 and R 2 are the reflection coefficients of the resonator mir- rors of the laser. The uniavourable losses may be caused by absorp- tion of radiation by atoms (molecules) which are not active centres, by scattering of radiation through the lateral face of the active me- dium, by diffraction effects, or due to other reasons. By using Eq. (2.1.7), we can rewrite (2.1.6) in the following form W gen = (vLsla,) [xo - (111 + 112)]. (2.1.9) 2 See, for example, Eq. (4.1.17) in [1]. 3 See, for example, Eq. (5.6.23) in [1]. , In order to explain the physical sense of these coefficients, one has to con- sider Bouguer's differential law: dS = - (rh + 112) S (z) ds, 2.1 Generation Condition 77 the x-axis determines the inversion threshold Qlnv, while the inter- section with the straight line AA determines the generation threshold 'lgen. I t can be easily seen that Q A Q -A bA 21+ (Tl l + Tl 2) . n >ninv = 2t; gen - 21bA21-(fh +Tl2) , ~'gen ~'inv· If Q »A 21 , the magnitude of the initial gain approaches its lim- iting value "'0 11m = bA 21· (2.1.14) Let us compare a few curves %0 vs, Q for different values of A 21 (considering that the inequality (1.2.15), which assumes the metasta- 1 "0 Xotim ---------~--~-~-----J 2 ~ ... w 32t / ... p W51 WJ2 W 2 ,J , w Fig. 2.1 Fig. 2.2 Fig. 2.3 hility of level 2, is satisfied). Figure 2.3 shows four curves, numbereed in order of decreasing value of A 21• It can be seen..Jrom this figure that the lower the value of A 21 (the higher the lifetime of level 2), the easier the creation of in- version. However, this results in n decrease in the limiting value of the initial gain. This reflects A-t----......~~---.,.---A the negative aspect of the meta- stahility of the upper laser level IlS mentioned above. The straight line AA in Fig. 2.3 (Iefines the loss level. I t can be seen that for a value of A 21 cor- responding to the curve 4, gene- ration is not possible at all (an oxcessive decrease in the value of A 21 has brought the limiting value of gain below the loss level). Thus, the lower limit of values of A 21 is determined by the amount of loss. In accordance with (2.1.14) and (2.1.10), the following condition must be satis- Hod: (2.1.15) 78 Ch. 2 Formation of Radiation Field in a Laser Resonator Fig. 2.4 Frequency Dependence of the Initial Gain. In order to consider the frequency dependence of the initial gain, we start with Eq. (2.1.2), replacing B 21 by Einstein's spectral coefficient B 21 (00). I t should be recalled that B 21 (00) = B 21F (00) where F (00) is a function defin- ing the shape of emission spectrum line/: Thus, Xo (00) = B 21F (00) liooNolv. (2.1.16) The factor B 2 1 appearing in (2.1.16) is independent 01 frequency for dipole and magnetic dipole transitions, while B 21 ex 00 2 for quadrupole transitions (see Table 2.1). Consequently, X o (00) ex: ex ooF (00) for dipole and magnetic dipole transitions, while X o (00) ex ex: w3F (to) for quadrupole transi- tions. The qualitative form of the dependence X o «(U) is determined by the function F (co), i.e. by the shape of the emission spectrum A line for the active centre. If, for example, the spectral line is Lorent- zian in shape (uniformly broadened O-lr----:~--<>----O---~Ca) line) and if the dipole transitions are considered, we get xo(w),....,.[w~/[(W-O)O)2+~2],(2.1.17) A--+-------4J. where Wo is the 'frequency corres- ponding to the peak, and ~ is the halfwidth of the line at the middle of its maximum height. Figure 2.4 shows the dependence X o (0) as described by formula (2.1.17). The straight line AA corresponds to the loss level. Only the shaded part of the function X o (oi), called the gain line, is operative during generation. The frequency of radiation generated on a given transi- tion lies within the limits of the gain line width: 0)1 < 00 < 002. We) have mentioned in the preceding chapter the selective resona- tors with a redistribution of the radiation frequency within the limits of the luminescence line width. As a modification, it may be observed that the range of redistribution is determined not by the luminescence line width, but by the gain line width (the width of the luminescence line at the loss level). 2.2. OPTICAL RESONATOR AND LASER RADIATION (PRELIMINARY REMARKS) Optimisation of Favourable Losses. It follows from (2.1.9) that in order to increase W gen. it is sufficient to reduce the losses in a resonator (reduce the sum 111 + 112). In particular, the favourable 6 Einstein's spectral coefficients and the shape of a spectral line have been described by Tarasov (Sec. 4.7 in [1]). 2.2 Optical Resonator and Laser Radiation 79 loss level can be decreased. For this purpose, it is necessary to increase Il tR2 in accordance with (2.1.8). However, an excessive increase III the reflection coefficient of resonator mirrors is not desirable, since " reduced 112 means a decrease in the fraction of generated radiation converted into laser radiation. This fraction is defined as the ratio 'tI 2/(1')1 + 112)· Taking Eq. (2.1.9) into account, the laser radiation flower is given by the expression W - 1)2 W - vLs [~ - ( + )] 112 (2 2 1) - 1)1 + 112 gen - ex. "'0 f)1 f)2 111 +112 • · • A decrease in 'll2 means a decrease in 112/(1"11 + f)2) but an increase in the factor [x o - ('ll1 + 1')2)]. Consequently, there should be an op- tlmal value 'rI2oPt for the coefficient S of favourable losses, which gives tho maximum output power. Usually, one considers not the output power tV, but rather the rlonsity S of the light flux emerging Irorn the resonator. The quantities ~V and S are connected by the rulation l'Jzopt Xo-1), '12 S = W/s (2.2.2) Fig. 2.5 W11 ere s is the area of cross-section of the light beam (for the sake of simplicity, we assume it to ho equal to the area of cross sect ion of the active medium). In view of the above, we can rewrite Eq. (2.2.1) in the form S = (vL/a) [xo - ('ll1 + 'll2)] fl2/('ll1 + fl2). (2.2.3) Figure 2.5 shows the S vs. "12 dependence defined by the expression (~.2.3). The threshold value of the coefficient of favourable losses iM equal tox, - 'rI1. In order to find the value of 112oPt, let us differ- ont.iate the function S ('ll2) and equate the derivative to zero. This Uiv cs (2.2.4) l lonce (2.2.5) Thus, for given values of )(0 and 1"h, there is an optimal value for t hn coefficient of favourable losses, which is given by the relation (~.2.4). The maximum density of the emerging optical flux corre- "ponding to it is given by the formula (2.2.5). A further increase in ,\i tI, fl X requires an increase in )(0 and a decrease in 111. Suppose that the laser radiation emerges from one of the reso- "" lor mirrors, while the other mirror is totally reflecting. From 82 Ch. 2 Formation of Radiation Field in a Laser Resonator Putting wo/n ~ 1015 sec:", Ln ~ 10 cm , we get Q> 106 from (2.2.15). The values obtained in practice for the Q-factor of laser resonators usually satisfy the condition (2.2.15). Under actual conditions, additional factors crop up, influencing the reso- lution of narrow spectral lines. One of such factors is the non-uniformity of the refractive index of solid-state active media. By using (2.2.11), it can be easily shown that the fluctuation 6n in the refractive index causes a fluctuation 600 ~ ~ w6n/n in the resonance frequency. For resolution of narrow lines in the fre- quency spectrum, it is necessary that 600 < 600'. Consequently, besides condi- tion (2.2.15), the condition 6n/n < ncl sol:n must be satisfied. It follows from this that 6n/n < 10-6 • This inequality is not always satisfied in solid-state lasers. Vibrational Modes of an Optical Resonator. As has been shown above, the resonator of a laser has a considerable influence on the power of Iaser radiation as well as on its spectral propert ies. In actual practice, the influence of the resonator on the properties of laser radiation is even more significant and fundamental. As a matter of fact, the resonator forms definite states in the radiation field. These are called vibrational modes of the resonator. An individual mode is designated as TEl\tIm n q' TEl\tl stands for transverse electromagnetic modes (the electrogm agnetic vibrations in an optical resonator may be considered as transverse). m and n are the transverse indices of the mode and q is the longitudinal one. Each mode is characterised by a definite spatial field configuration (a definite amplitude and phase distribution) in a direction perpen- dicular to the resonator axis, in particular at the mirror surfaces of the resonator. The specific nature of this configuration is determined by the transverse mode indices m and n. Besides, each mode is charac- terised by a definite phase shift at the resonator axis upon a round- trip in the resonator. This phase shift is equal to 2 seq, where q is the longitudinal mode index. I t should be observed that the phase shift is considered at the resonator axis due to the fact that, strictly speaking, a mode is not a plane wave. Clearly, such a restraint would be unnecessary for a plane wave since the distance from the resonator axis of any point in a plane wave front is the same. For each definite combination of indices m and n reflecting a definite transverse field configuration in a resonator, there is a number of corresponding modes with differ- ent values of q. These are called longitudinal modes (also called axial modes). For each one of them, there is a corresponding narrow line in the generation spectrum. The set of longitudinal modes with a given combination of indices m and n is termed a transcerse mode. Obviously, the transverse mode is characterised by only 1ransverse indices (and is denoted by TEl\11n n ) . Each type of transverse mode has a definite structure of the Ium i- nous spot on the resonator mirror. Figure 2.7 a shows the structure of intensity pattern obt ained in a circular mirror for the simplest 2.2 Optical Resonator and Laser Radiation 83 (lowest) transverse modes. The corresponding nature of change in I he sign of amplitude of the field on the mirror surface is shown in Fig. 2.7b. It is evident from the figure that the index m indicates I he number of times the field amplitude changes sign in the radial direction, while n indicates the number of times it changes sign upon It rotation by 1800 around the centre of the mirror. The transverse mode TElVloo is called the lowest-order mode. II, is characterised by the simplest structure of the spot. I t is apparent from Fig. 2.7 a that the smaller the value of transverse indices. the more concentrated is the mode field near the centre of the mirror. OCDE90@) @(j(j@jQj TEM 10 TEM lI TEM J2 TEM 10 TEM1J (a) (b) Fig. 2.7 An intensity pattern, as observed under actual conditions, is often n superposition of several transverse modes (multimode generation). 'l'he spectrum of the generated radiation usually contains several narrow lines (multifrequency generation}!". . The Role of an Optical Resonator in a Laser. According to Ger- rnrd et al. [4], the main function of an optical resonator is to impart It definite shape and defin ite mode struct ure to optical waves radiated hy a laser. The mode pattern of a laser radiation field is formed in I he resonator during the process of successive reflections of the radia- I ion at the resonator mirrors. In som e resonators the field configura- t ion is formed after a large Dumber of passages from one mirror to I he other (multipass resonators), while in some other resonators this process takes place after just a jew passes."! I n order to understand the role of a resonator in a laser, it is con- vr-nient to use the quantum concept, since it takes into account the nature of the laser generation process in the optical spectral region III the best possible way. In particular, the fact that optical radiation 10 The mode patterns for solid-state lasers may be distorted on account of uonuniforrnity of the refractive index of the solid active medium. 11 For a sufficiently large gain, the inversion of the active medium can be n-moved in a single pass. Such lasers may work without mirrors in the superlu- m l nrscence regime. •• 84 Ch. 2 Formation of Radiation Field in a Laser Resonator is generated as a result of a large number of acts of emission by atoms or molecules performing quantum transitions from one state into another is very well reflected by using the quantum concept. In a certain sense, an individual emitting atom (molecule) may be compared with a certain "resonator", or a certain oscillatory system. In the words of Academician L. I. Mandelshtam [51, "vVe imagine that there is an optical resonator of a certain frequency in each gas molecule. The luminescence of the gas is due to the fact that the resonators oscillate at the same frequency and the gas ern its ligh t of this frequency". Obviously, in order to get coherent beams of light, it is essential that the acts of emission by a large number of resonators are mutually correlated, not only in the radiation frequency, but also in the direc- tion of its propagation and in polarisation. In other words, it is essen- tial that the principle of self-regulation or, in the language of elec- tronics, the principle of positive feedback be followed. The possibility of such a correlation (the possibility of attaining a positive feedback) is inherent in the very nature of induced emission: the secondary pho- ton is in the same state as the primary photon which induces the crea- tion of the secondary photon. In order to realise this in actual prac- tice, a selective population of photon states'" must be ensured in the course of induced transitions. In other words, certain states must be isolated in actual practice. In these states, photons are accumulated while in all the remaining states spontaneously created photons must be eliminated as qu ickly as possible. The selectivity of population of photon states is also ensured by the optical resonator in a laser. "The possibility of a positive feedback, inherent in the induced emission process, is realised in a laser with the help of an optical resonator" (6]. Primarily, a res- onator isolates a certain direction in space in which generation pre- dominantly takes place. Besides, the resonator also performs a fre- quency and polarisation selection of the radiation. The parallel plates used frequently in resonators are oriented to the resonator axis at Brewster's angle, and thus ensure the selectivity of the generated photons, according to their polarisation. I t may be said that the main function of an optical resonator is to isolate certain photon states in which predominant laser generation takes place. The coher- ence properties of laser radiation (directionality, monochromatism, degree of polarisation) depend on how well the resonator ensures the selectivity of population of photon states. Considering the question as to how a resonator ensures selectivity of the population of photon states, it should be emphasized that this is attained on acconnt of selective losses for different states. These losses should be small for some states wh ile for the other states these 12 Photon states are characterised by a definite energy, momentum, and polarisation of the photon. 2.2 Optical Resonator and Laser Radiation 87 ul~o be used for solving a large number of problems concerning nrtive resonators. At the same time, it should be remembered that an amp lifying "c'l j ve medium imparts certain properties to the process of field Iormation in a resonator (see, for example, [9]). To begin with, we run mention the mode competition, leading to a reallocation of the ilnnerated power from some modes to others. This reallocation may t"l{e place on the frequency scale (between longitudinal modes) as wul l as in space (between transverse modes). The active medium C',U uses mode competition on account of the nonlinear optical effect of gain saturation. Gain saturation for certain frequencies may lead'0 I he appearance of "holes" in the profile of gain curve (hole burning ,1!Ject). Another consequence of gain saturation is that higher trans- verse modes (modes with comparatively large values. of transverse indices) rather than lower ones may have a higher Q. Let us con- -ider this in greater detail. rna passive resonator, it is the lower transverse modes, and above nil the lowest-order mode TE~loo, which have the lowest loss char- nC"· Ieristics as a rule. These modes are the ones to be excited in the 1'I.I'~t place in the case of an amplification. The field intensity in l huse modes will rise and may attain saturation. This means that II,p gain w ill be m in imum in the part of the active medium that is .d'foctively covered by lower modes (i ,e, in the vicinity of the reso- untor axis!"), Consequently, the inverse density near the resonator 1.1 \ is may turn out to be lower than in the periphery. In this case lito higher order transverse modes which occupy large volumes in Iltn active medium (covering, among others, the peripheral region H~ well) get excited first. A reallocation of power from the lower modes to the higher ones may lead to a significant change in the Hold configuration in a resonator. Besides the gain saturation effect, there are other factors which Influence the formation of radiation field in an active resonator. For example, a dispersion of the refractive index of the active medi- 'lin may lead to the so-called effect of frequency pulling [10], because 01" which the resonance frequency spectrum is no longer equidistant: rusonance frequencies are more closely spaced near the centre of the umpl ificat ion line. Heating of the active medium through absorption 01' the pumping radiation leads to a change in its refractive index. i\~ a result the so-called thermal lens effect takes place: the active uloment works as a converging or a diverging lens for the radiation Insid e a resonator (see, for example, [11]). rna resonator filled with solid (crystal, glass) active medium, supplementaru modes may be excited. Their emergence is due to the f.'Hoet of total internal reflection at the lateral face of the active 13 In accordance with Eq. (2.1.'1), we get Xmln = xol [1 + (alv) Ssatl, where Ssat is the saturation flux density. 88 Ch. 2 Formation of Radiation Field in a Laser Resonator element. Denoting by e the angle between the direction of propa- gation of a mode and the optical axis of a resonator, we can bifurcate these modes into two groups. The modes with e == 'Jt/2 circulate in the cross section of the active element (circular modes), see Fig. 2.8a. Modes with e =1= n/2 shift in the direction of optical axis as shown in Fig. 2.8b. The modes of the first group are parasitic modes. Modes of the second group may be DEed for creating icaceguide resonators [12]. Supplementary modes in the filled resonators have been considered by Stepanov et ale [3] and Mikaelyan et ale [13]. In actual practice, an active resonator usually contains a certain set of elements (including the active element) which are separated from one another. Consequently, while considering such a resonator one has to take into account a whole set of reflecting surfaces which (a) Fig. 2.8 (b) include, besides the resonator mirrors, the end faces of the elements in the resonator (including the end faces of the active element). In this case, the resonator is equivalent to several coupled resonators. The interference effects arising in a system of coupled resonators may significantly influence the resonance frequency spectrum. In this chapter, main attention will be devoted to passive resonat- ors. The questions dealing with the effects of the active medium. on the generation spectrum (hole burn ing effect and frequency pulling effect) and on the configuration of radiation field (thermal lens effect) will be considered separately. Waveguide resonators and thin-film lasers will also be described later. I t should be emphasized that field formation in active resonators is closely related to the dynamics of the processes in generating lasers. This is an important and fundamental problem, and the third chapter of this book will be devoted to its analysis. 2.3. GENERAL REMARKS ABOUT OPEN RESONATORS Resonators used in lasers belong to the class of open resonators!". They are considerably different from the cavity resonators used in the SHF band. 14 The waveguide resonators and thin-film lasers with a distributed feed- back are important examples of this class of resonators (see Sees. 2.14 and 2.15). 2.3 General Remarks about Open Resonators 89 The theory of open resonators has been described by a number of authors [4, 9, 13-15]. See also [3, 7, 16, 17] and Chs. 22 and ~:\ in [2]. Impossibility of Using Cavity Resonators in Optical Band. Let liS consider a rectangular cavity resonator with conducting walls, shown schematically in Fig. 2.9a. Here Land D are linear d imen- ... ions of the resonator. We shall represent the resonator modes simply ill the form of plane waves satisfying certain boundary conditions k Fig. 2.9 Ill, the cavity walls. Using the boundary conditions for a standing wave, we find that the component modes of the wavevector are of I he form k x == stmlI) ; k y == stnlI); k , == siqlI», (2.3.1) \vhere m, n, and q are integers (mode indices). The relations (2.3.1) lead to the following expression for the resonance wavelength 'Am ll q : 1 / ( m)2 (n)2 I ( q )2 Am nq == V 2D + 2D -r 2L . (2.3.2) 'The mode with indices 0, n, q propagates in the yz-plane (for this mode, k x == 0) as an angle e to the resonator axis, which is de- termined by the numbers nand q (Fig. 2.9b) . Accord ing to (2.3.1), k sin e == stnll) and k cos 8 == scqlL: Consequently, sin 8 == n'A/2D; cos 8 == q'A/2L. (2.3.3) Further, it should be recalled that for the volume V of the reso- untor , the number of plane waves, taken over the frequency interval between ffi and co + ~ffi, is proportional to ffi2: 111 oc V ffi2 ~(t). (2.3.4) lt should also be recalled that the Q-factor of a resonator, doterrn ined hy the Joule losses in the conducting walls, is proportional to V ffi: Q oc Vw. (2.3.5) 92 Ch. 2 Formation of Radiation Field ina Laser Resonator or Q= wUI(- dd~ ) • (2.3.12) Thus, the Q-factor of a passive resonator 111ay be defined as the prod uct of the radiation frequency and the ratio of energy stored in the resonator to the energy lost by the resona tor per un it t ime. The ratio (2.3.12) reflects quite well the relation between the Q-fac- tor of a resonator and the losses inherent in it. Losses characteristic for an open resonator can be divided into three groups: (i) losses d lie lo transmission through the oul put mirror of the resonator, (ii) d ifiraction al losses dne to finite apertures of the mirrors and all elements inside the resonator, and (ii i) losses due to partial absorption of radiation inside the resonator by its mirrors, as well as due to scattering of radiation through the lateral surface of the active element. In Sees. 2.1 and 2.2, losses were divided in Lo two ca Legories, the favourable and the unfavourable ones. Apparently ~ the losses pertaining to the first of the above-men tioned three groups belong to the favourable losses (relation (2.1.8) for the coefficient of favourable losses describes losses connected with the transmission throngh resonator mirrors). Losses corresponding to the third group are apparently unfavourable. As regards the diffractional losses, they may he favourable in some cases. and un- favourable in some others. We observe that in the so-called unstable resonators (which will be consid- ered later), the diffractional losses cause an extraction of laser radiation from the resonator and hence must be treated as favourable losses. In the case of stable resonators, the diffraetional losses are, as a rule, unfavourable. However, even in this case, there may be situations when diffractional losses may be consid- ered, at least partially, as favourable losses (for example, in resonators where a reflecting mirror with a hole in the centre is used as the output mirror) ° Let 11S denote the linear coefficient of losses as '11 without specifying the nature of these losses. We shall find a relation between the loss coefficient 11 and the Q-factor of a resona tor. For this purpose. ]et us consider the dependence of the field energy U on the longitudinal coordinate z instead of time (this is identical to the process of at- tenuation of light flux propagating in the absorbing medium): u (z) == U (0) exp (-llZ). (2.3.13) Here, 1/11 is the length over which the quantity U decreases to 1le of its value. The attenuation time corresponding to this is given by 't == 1/11V == nhlC, (2.3.14) where n is the refractive index of the medium filling the resonator. Substituting (2.3.14) in (2.3.10), we get Q = wn/l1c = 2n/'llA. (2.3.15) 2.3 Ger:eral Remarks about Open Resonators 93 Let Qi denote the Q-factor of a resonator, connected with the i-th t ype of losses, and let Q denote the total Q-factor. If different types 01' losses can he considered independently of one another, we get - dU/dt == ~ (- dU/dt)i- i 'raking (2.3.11) into account, this gives 1/Q === ~ 1/Qi. i (2.3.16) 'l'ho result (2.3.15) is in accordance with (2.3.16): 1 _ 'iv _ '" ~ 'n. -== ~ A1li _ ~ 1 Q-2Jt 1l - 2n LJ ·Il L..J 2n - L..J Qt- iii Q-factor of a Resonator due to Transmission Through the Output ~lirrol"_ The process of attenuation of field energy in a resonator on urcount of transmission through the ontput mirror may be con- veniently considered by observing the propagation of radiation u u __ :=.--==¥{Z)(I-R) UlO) z 0 Z2L (0) (b) Fig. 2.10 inside the resonator from one mirror to another. We shall assume I hat only one resonator mirror is the output mirror (with reflection C'oefficient R), and the other totally reflects the radiation. If the rnd iat ion traverses a path of length ~z inside a resonator of length L, it means that it is reflected ~z/2L times at the output mirror. On r-nch such reflection, the resonator loses a part of the energy, equal I () 1 -R. Consequently, in traversing the path Az. the loss of energy hy the radiation is given by -~U = (U6.z/2L) (1 - R). (2.3.17) 'I'Iie dependence of U on z is shown in Fig. 2.10a. This dependence IH step-shaped, the height of a single step being equal to U (z) (t - R). Fell' the sake of comparison. the continuous U vs z curve correspond- illg to the relation (2.3.13) is shown in Fig. 2.10b. In order that the rr-l at ion (2.3.13) describes approximately the ease shown in 9-! Ch. 2 Formation of Radiation Field in a Laser Resonator Fig. 2.10a, it is necessary to assume that the difference '1 - R defining the height of a "step" is small. This means that the output mirror must have a reflection coefficient fairly close to unity. We can write R = 1 -~, where ~ ~ 1. (2.3.18) In this case, we can revert from (2.3.17) to a relation of the type (2.3.13), which will have the form U (z) = U (0) exp [- (1 - R) z/2L]. (2.3.19) I t follows from this that the coefficient of favourable losses due to transmission through the output mirror of the resonator may be expressed as '11 = (1 - R)/2L. (2.3.20) Subst.itut.ing (2.3.20) in (2.3.15), we get Q = 2wLnic (1 - R) = 4:rtL/A (1 - R). (2.3.21) We have given earlier the expression (2.1.8) for the coelficient of favourable losses. Considering that the resonator has only one output mirror, this expression may be rewritten in the form 11 = (1/2L) In (1IR). (2.3.22) Using (2.3.18), we can write In (1IR) = In [1/(1 - ~)l ~ In (1 + ~) ~ ~ = 1 - R. Consequently, the expression (2.3.22) turns into the expression (2.3.20) obtained above. Q-factor and Modes of an Open Resonator. So far, we have been considering the Q-factor of a resonator without taking into account the mode structure of radiation. However, as mentioned earlier, the losses in an open resonator may strongly vary from one mode to another. Hence, strictly speaking, one should consider the Q-factor not of an entire resonator, but of a given mode in a given resonator. I t should be emphasized that it is the difference in losses for different modes that leads to the formation of laser radiation field. This point has been mentioned in Sec. 2.2 while discussingLhe role of a reso- nator in a laser. Let us somehow excite different modes in a passive resonator and observe the process of their gradual attenuation with time. Different losses for different modes lead to different damping rates for them, and so the mode structure of a field changes with time. The most dominating modes in this structure are characterised by the lowest losses. Thus, a resonator is similar to some sort of a "filter" ~ isolating from any radiation field such components that correspond to modes with the lowest losses. In the case of an amplifying active med ium , high-Q modes of a passive resonator will be characterised apparently by the largest difference (x o - 11). These modes will be the first to be amplified.