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Laser Gain Measurements by Means of Amplified Spontaneous Emission

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Laser Gain Measurements by Means of Amplified Spontaneous Emission Laser gain measurements by means of amplified spontaneous emission G. Marowsky, F. K. Tittel, W. L. Wilson, and E. Frenkel Laser gain determination by means of amplified spontaneous emission is discussed in terms of a rate-equa- tion approach. Numerical solutions show that optical gain can be deduced only for a limited regime of ex- perimental parameters. The theoretical analysis is examined with experimental gain data for organic dyes in the vapor phase and in liquid solution. In addition, the gain of electron-beam-excited Ar-N2 and XeF laser mixtures has been studied. 1. Introduction fects. Finally, this analysis is examined in terms of Optical gain is an essential parameter for the char- actual experimental gain data for optically and elec- acterization of any laser system. Laser threshold con- trically excited organic dyes in liquid solution and in the ditions, output coupling of the resonator, power output, vapor phase and electron-beam-excitedAr-N 2 and XeF efficiency, saturation behavior, and spectral narrowing laser mixtures. In the case of organic dyes, the small- effects are all functions of the gain characteristics of an signal gain as derived from ASE measurements leads active medium. The most direct method for measuring to a determination of the spectral dependence of the optical gain utilizes a laser probe beam of the appro- excited-state absorption (ESA) cross sections. priate wavelength, but frequently no such source exists II. Theoretical Model or is available conveniently, in which case an indirect Experimental details of the ASE technique for mea- technique based on the analysis of amplified sponta- suring gain have been described in Refs. 2-5. 1 The neous emission (ASE) can be employed. ASE tech- technique involves a comparison of the fluorescence niques have been used by several groups 2 4 to compare yields of various lengths of the active medium. An the gain characteristics of organic dyes in liquid solu- analysis of this technique can be made in terms of cou- tion. Initial analysis of ASE was based on an intuitive pled differential laser 2 rate equations that describe the approach and subsequently improved by means of a properties of two oppositely traveling 5 waves I+(x) and rate-equation approach. In this paper, a detailed I-(x) [Fig. 1(a)] in a transversely excited active medium. computer analysis of the ASE technique is described For simplicity, let us consider an ideal four-level system, that utilizes realistic values of typical experimental such as an organic dye of concentration N (total number parameters. Although the model 3 is developed pri- of molecules/cm ), excited-state population density N1 , marily based on the well-known features of the dye laser cross section for stimulated emission ae, cross section gain, the ASE technique can be effectively applied to for pump light p, absorption cross section for reab- other laser systems. The theoretical analysis estab- sorption u'l, and cross section for excited-state ab- lishes well-defined experimental limits of applicability sorption of fluorescenceradiation adz. The propagation of the ASE techniques that are determined by the of the ASE intensities I+(x) and I-x) with uniform stimulated emission cross sections, concentration of transverse excitation with a pump intensity Ip may be active molecules, pump intensity, and geometrical ef- described by the following equations under stationary conditions 6 : a d -L. -Ix) = [g=(x)+ I(x)(1 +k? -k kI - k?1=(x),(1) (eN dxI N G. Marowsky is with Max-Planck-Institut I, + kOI(x) fuer Biophysikalische .Nl(x)=N. Chemie, Gottingen, Federal Republic of Germany, and E. Frenkel is 1 +Ip+ e(l + k)I(x) (2) with Fachhochschule Furtwangen, Furtwangen, Federal Republic I(x) = I+(x) + I(x) of Germany. The other authors are with Rice University, Electrical (3) Engineering Department, Houston, Texas 77001. Equations (1)-(3) have been formulated using dimen- Received 7 September 1979. sionless quantities, and the notation used is the same 0003-6935/80/010138-06$00.50/0. as in Ref. 5. The intensities I, I+, I-, and Ip have been ©1980 Optical Society of America. normalized to their saturation intensities, I, = (1oe)-l 138 APPLIED OPTICS / Vol. 19, No. 1 / 1 January 1980 Ip for the ASE intensities and IPS= (Tlovrap)- for the pump intensity, where r is the first excited-singlet-state decay time. The coefficients k' and k1 have been introduced I (x) to account for losses due to reabsorption (ko =--llu,) I-x) and losses due to excited-state absorption (ki - 1/- - - - so 2r > - - - - - 1 The factors g+(x) denote the probability that a spon- (a) K-13- i- taneously emitted light quantum at point x of the active 9- Ix)' .g+(x) volume penetrates through the end faces of a cylinder Ax of radius 2ri and length Lo. According to Refs. 5 and -X Lo 6, these factors are given by x =xo X=Xj g+(x)=!{ _ Lo-x 1 (4) Ip DETECTOR g-(X) = , - (X 2 [(x 2 ++ri") r) 1/2] The factor in Eq. (2) describes the reduction of the inversion level due to the influence of ASE. For e = 0, the excited-state depletion by ASE is neglected. For e = 0, k'I <<Ip, and by neglecting the spatial depen- dence of g'(x), the small-signal gain coefficient go can be derived from Eqs. (1)-(3) upon integration: go = aN[I (1 - k ) - kI/(1 + Ip) Fig. 1. (a) Transverse excitation scheme for generation of the ASE 1 intensities I+(x) and I-(x) and definition of the geometrical factors =uNI*/(1 + Ip) (6) g+(x) and gx). (b) Registration of integrated fluorescence intensity with I1,2 by a detector at a distance D >>Lo. I'P -= Ip(l - k1)I - koI. (6a) On the other hand, a probe beam will experience an effective gain geff, which can also be derived from the rate equations, taking into consideration their spatial dependence with g' = g+(x) and = 1. Then g-ff is given by 1 CLo geff =- ,eNl(x)dx. (7) Lo Jo The dependence of geff on the pump intensity I is shown in Fig. 2. The plots are obtained from a com- puter solution of Eqs. (1)-(3) for different values of eN 1 = 2, 10, and 20 cm- and different E values. It is ap- parent from Fig. 2 that for small pump intensities Ip and 0) small aeN values solutions with e = 0 and e = 1 coincide, Ip - and hence N1 :X Nl(x) or go as defined in Eq. (6) is a good approximation for the small-signal gain. A value LL 10- -E =0.00 1 of OueN = 10 cm- approximately corresponds to the z |deNlOcmflV --e-.1= l.0 characteristics of a 10-4 M solution of an organic dye I -E =1.0 with a cross section for stimulated emission of 10-16 0D 5- 2 w, cm . For even more dilute solutions (N 5. 10-5 M) and Ip as high as unity, the differences between the " realistic case " , approximate solution g(e = 0, Ip) and the noU , , 0.01 0.1 1 l1 20 10o Ip - g(E = 1, Ip) disappear completely. In the case of a broadband dye laser, it is not necessary to use wave- length-integrated cross sections, as discussed in Ref. 5, and to introduce go factors weighted by the quantum efficiency of the total fluorescence spectrum as long as Ip and o-eN are restricted to the values shown in Fig. 2. 0.01 0.1 1 10 20 100 IpP Application of the ASE technique requires alternate PUMP INTENSITY Ip exposures of different volumes of the active medium to the pump. According to Ref. 1, the small-signal gain Fig. 2. Computer evaluation of g(e,Ip) for three different values of is obtained from the ratio of the resultant amplified the experimental gain parameter eN. spontaneous emissions: 1 January 1980 / Vol. 19, No. 1 / APPLIED OPTICS 139 0,- 041~~~~~~~= t 0.01 01 1.0 1 'p _~ Ip _ 1 1.0 0eNlOc !08 to "'04-/ / I, 0, m02- V(b) 0-1 0.01 o.1 1.0 1 Ip Fig. 3. Comparison ofgexp and geff for (a) GeN = 2 cm- and (b) aeN Fig. 4. Comparison gexp and 1 of go for (a) eN =2 cm-l and (b) aeN = 10 cm- for values of a = 4, 10, and 20 where a = D/Lo. = 10 cm-' for values of a = 4, 10, and 20. a minimum detector distance D 2 20Lois required and that the pump power must be close to unity or greater 1 I2(Lo) 1 9ep ~- (8) in order to assure independence of gain data from de- tector position. On the other hand, gexp/geff and gexp/go values deviate considerably from unity if Ip exceeds Lo is the full length of the pumped region, and I2j(Lo) unity and o-eN > 2 cm-' [Figs. 3(b) and 4(b)]. As a and I (Lo/2) are the ASE intensities after exposure to consequence, low o-eN values, Ip values close to 1, and the full length Lo and the half length Lo/2. These in- a detector position D >>Lo should be chosen for a reli- tensities are monitored with a detector of an entrance- able gain measurement when using the ASE technique. aperture diameter 2r2 located at a distance D from the As shown in Figs. 3(b) and 4(b), only in this range of active volume [Fig.1(b)]. The intensities It(Lo/2) and experimental values is it justified to approximate geff 2I(Lo) have been obtained by another integration of Eq.
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