INTRODUCTION. Amplified spontaneous emission (ASE) has been under extensive theoretical and experimental studies during the past and recent years.
ISSN 1054660X, Laser Physics, 2012, Vol. 22, No. 12, pp. 1861–1873.
PHYSICS OF LASERS
© Pleiades Publishing, Ltd., 2012. Original Text © Astro, Ltd., 2012.
A New Approach for Describing Amplified Spontaneous Emission in a KrF Excimer Laser Using zDependency of GainCoefficient1 A. Hariri and S. Sarikhani Laser and Optics Research Department, Nuclear Science and Technology Research Institute, North Kargar Avenue, P.O. Box 113658486, Tehran, Iran Received April 4, 2012
Abstract—Based on our recent realization concerning the geometrically dependent gain coefficient in self terminating gas lasers, where it is shown that in onedimensional approach it is zdependent, we applied the gain formulation to explain, both numerically and analytically, the behavior of the amplified spontaneous emission (ASE) output energy vs. excitation length of the active medium. As an example, we used experimen tal measurements reported for a KrF excimer laser. In the approach it was realized that it is needed to present a gainprofile slightly lower than the gainprofile deduced from different reports appeared in the literature, where it is also indicating that the contribution of the ASE on the laser output is significant and it is the active medium length dependent. The present analytical presentation of the ASE output energy behavior, also, introduces a generalized formulation compared to that appeared in the literature. With this approach it is pos sible to remove most of the present ambiguities existing on understanding of the ASE behavior. DOI: 10.1134/S1054660X12120109 1
1. INTRODUCTION Amplified spontaneous emission (ASE) has been under extensive theoretical and experimental studies during the past and recent years. Due to importance of the subject, in particular, the presence of ASE in high power lasers, in addition of a great amount of published papers dealing with different aspects of the ASE real ization, it has also been discussed in some text books in some details [1, 2]. The earliest experimental consider ation goes back to 1965, when Leonard used a 200 cm long N2laser oscillator (OSC) operating at 337.1 nm. He showed that by increasing the excitation length of the laser OSC, the output energy starts to grow from a threshold length and increases monotonically by increasing the excitation length of the oscillator [3]. By observing the reduction of the beam divergence upon employing an end mirror, it was concluded that the device is amplifying the spontaneous emission origi nating from one end of the apparatus. The ASE prob lem was further investigated theoretically and experi mentally by Allen and Peters [4–7] by employing long laser tubes of He/Ne lasers operating at 3.39 μm. For explaining the experimental measurements for both He/Ne and N2lasers, as appeared in [3], they used rate equations, and also applied the theoretical conclusions from an earlier published paper by Stenholm and Lamb on the semiclassical theory of gas lasers [8]. In recent years, on the other hand, due to various applications of high power excimer lasers, in particu lar, ebeam pumped KrF lasers and high power solid state lasers in laser fusion, understanding of these laser facilities has become one of the major challenging tasks of the researchers. In this approach, the ASE 1 The article is published in the original.
study has attracted more attention, where this emis sion is especially undesirable in high gain system and it can grow to a level where it leads to depletion of the excited state population of the laser medium. For the past theoretical work we can refer to a number of papers such as: Hunter and Hunter approach for introducing bidirectional amplification [9]; ASE effects in small aspect ratio laser oscillators [10]; intro ducing timedependent code to model ASE in a KrF amplifier [11]; ASE in dye laser oscillator and ampli fier (OSCAMP) [12]; low pressure study of ebeam Kr/F2 laser amplifier [13]; ASE in slab [14] and rod amplifiers [15]; gain depumping due to ASE [16, 17]; developing a 3dimentional code for the ASE in KrF [18], and DF [19] lasers, etc. In addition to study large size ebeam and electric discharge gas lasers, solid state, and dye lasers, ASE has shown its presence in other media such as: conju gated polymer [20], cyano substituted oligo single crystal [21], semiconducting polymer waveguides [22], etc., where they can be categorized as very small size media, (for example in [21] the size of crystal is about 2 mm long). The gain coefficient in these media measured to be very high (i.e., in [21], for example it is ~33 cm–1), which is consistent with our proposed model of geometrical dependency of the gain coefficient. In contrast to large size electric discharge or e beam machines, in smallsize media [like: 20–22] as the pump length of the samples can be easily changed, therefore a large amount of information concerning the ASE behavior in these systems is attainable. For example, in [20] it was shown that output power increases first gradually, and then rises almost rapidly when the pump intensity increases. In this type of
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excitation it was found that the output intensity from one end of stripe is given by the following expression g ( λ )l
e –1 (1) I ( λ ) = A ( λ )I p ⎛ ⎞ , ⎝ g(λ) ⎠ where A(λ) is a constant, Ip is the pump intensity, g(λ) is the small signal gain (or net gain when absorption is con sidered in the medium), and l is the length of the pumped stripe (or the medium excitation length). By varying the excitation length of the medium, the data from ASE out put intensity (or energy) are usually fitted to Eq. (1), and the results of the fittings are rather excellent. Generally, there are a few remarks and comments for applying Eq. (1) to experimental measurements. 1. I(λ) in the measurement is often expressed in an arbitrary unit, or it is normalized to unity, whereas, its value has to be obtained by the elementary informa tion given for any given medium such as: stimulated emission cross section, σstim, frequency of the emitted light, ν, etc. 2. For a given radiation or laser system, there is some length threshold, zth, where it corresponds to the onset of the ASE. Although the zth measurement is hard to be made for very smallsize media or perhaps it becomes an impossible measurement, but for some long channel lasers such as N2laser in [3], it has been shown that it is not small (~40 cm). For KrF laser, where we are using data from Kühnle et al. [23] to explain our present model analysis, it is reported to be 24 cm (it must be cited that this is the length where authors in [23] they could still perform the ASE output energy measurement, and the electrodes length where the ASE is actually goes to zero in their system is evi dently less than the above reported figure). Also, in one of our experiment of using an OSCAMP N2laser system, with an AMP capable to operate with different lengths (lAMP = 2.2–31.0 cm), when an electrode with the length of lAMP = 2.2 cm was used, no ASE was observed in the AMP section, while it was possible to observe the light amplification upon injecting the OSC output signal into the AMP section [24]. 3. Analytically, the ASE intensity has been com monly obtained from the photon density rate equation for the steady state condition under the following assumptions: (a) Intensity, I, or photon density, nph, for the steady state solution is zero at z = 0, where z is the direction of the ASE photons emerging from the other end of the medium. (b) The fraction of light emitted within the solid angle Ω(z), i.e., Ω(z)/4π is considered to be a constant, where it is assumed that amplified spontaneous emission is essentially arising from the emitting element at z = 0 [1]. So, with these assumptions, prediction of a value for zth is not possible. 4. And the last, g0 (small signal gain, or net gain net
t
t
which is given by g 0 = g0 – α L , where α L is the internal absorption losses) has been considered to be a
constant, otherwise, it is not possible to reach a closedform relation, given, for example, by Eq. (1). These assumptions, in general, do not disturb seri ously the validity of Eq. (1) for very smallsize media, as it was applied successfully, for example, in [20–22], and that is because of the following reasons: Firstly, no distinction can be made for the position of zth when IASE is plotted vs. lAMP, and zth ≈ 0 can be considered as a good approximation for the threshold length. Sec ondly, with a very good approximation Ω(z)/4π ≅ Ω(z ≈ 0)/4π = 0.5, so we can be assured for consider ing Ω(z)/4π as a constant. For large scale laser systems, however, these simpli fying assumptions are introducing a severe draw back for understanding the ASE behavior in largescale high power lasers. As the spatial dependency of the gaincoefficient in our recent proposed formulation for the gain coefficient in selftermination gas lasers has been verified experimentally, so this realization deserves to be discussed in more details. The dependency of g0(z) on the zdirection [25, 26] was verified for variety of gas lasers, including N2, + excimers, N 2 , F2 lasers and CVLs [26]. Thus, accord ing to the reported experimental observations and measurements, g0 appeared in the rate equations can not be considered as a constant, and its dependency on the zdirection (or x, y, z directions) must be consid ered for the analysis. In onedimensional approach, the formulation in our g0model is expressed by [26] max
γL – bz = m' + 1 + ( 1 + bz )e , z = l AMP , (2a) z z or, upon expanding the 3rd term in Eq. (2a), we have exact
g0
max
exact
g0
1 + γL max = m' + + bγ L z
∞
i
( – bz ) , ∑ ( i + 1 )! i
(2b)
i=1
max
where z ≡ lAMP is the AMP length, γ L is the maxi mum power loss in the zdirection, b is a constant and has been introduced according to the simulation and can be determined by the g0 measurements. b appeared to be a very small quantity. m' is a constant, and was initially introduced to remove the inequality ∂nph/∂t > 0 to show the photon density above the threshold in a simplified twolevel system. m' can also be determined by experimental measurements of the gain coefficient versus the length of laser medium. By numerical cal culation and gain measurement in a N2laser [25], it was realized that the loss coefficient γL depends on the zdirection and can be expressed approximately by max – bz γL(z) = γ L e , where it was further found to be con sistent with all measurements in self terminating gas + lasers such as; CVLs, N2, N 2 , F2, and excimer lasers. As b is a very small quantity, if we let b ≈ 0 in Eq. (2b) we will obtain a simple equation, given by LASER PHYSICS
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1 + γL = m' + (3) , z = l AMP . z Equation (3) in the form of exp n (4) g 0 = m + , z = l AMP , z was introduced initially for fitting the g0measure ments vs. lAMP [24]. We can further let m' = m, and n = approx
g0
max
1 + γ L , if we compare Eq. (3) with Eq. (4). Equations (2a) or (2b) shows, that for an active medium of very short z we have a very high gain value (for example in the range ~33 cm–1 as given in [21], or even much higher, for xray lasers [1]). For a large scale ebeam pumped KrF having lAMP ~100 cm, and diameter of 27 cm, for example [16, 17], the zdepen dent terms on the righthand side of Eqs. (2a) or (2b) are very small, and lasers of this type are usually exhib iting very small gain values. In this article for understanding the ASE behavior, we used the rate equation for photon density, nph(z), in the steady state condition. In our first approach, the differential equation for the photon density was solved numerically, keeping in mined that both g0(z) and Ω(z)/4π are zdependent. With this approach it was possible to observe the exact numerical solution of the photon density and output energy for the amplified spontaneous emission. For applying our proposed method of the calculation, the ASE experimental data for the KrF laser output energy were adopted from Kuhnel et al. measurements [23]. Our calculation pre dicts the presence of a length threshold, calculated to be ~7 cm. It should be emphasized here that for veri fying this prediction precisely, a very careful measure ment of ASE output energy, εASE, vs. lAMP, in particular at the onset of ASE emission is needed. In this calculation, it was further realized that for explaining the ASE measured energy, different gain max ASE parameters (m', b, γ L ) corresponding to the g 0 must be assigned for the medium. These parameters exp are different from the g 0 gain parameters, obtained experimentally and reported by different researchers ASE for a large number of lasers. Both profiles, i.e., g 0 exp
and g 0 , for z zth and z 0, respectively, are pre dicting very large values for the gain coefficient, while for a large enough z (or amplifying length), both gain ASE coefficients approach to very small values. The g 0 (z) exp
and g 0 (z) are both obeying Eqs. (2a) and (2b). Upon ASE
introducing the g 0 (z), we further observed the effect of the ASE on the KrF laser emission by defining the ASE exp g 0, KrF / g 0, KrF ratio. The plot corresponding to this ratio vs. z is also introduced. LASER PHYSICS
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The next approach was directed toward obtaining ε vs. lAMP, analytically. For this purpose, first, we took some length averaging for the Ω(z)/4π factor appeared in the rate equation to observe the consis tency with the measured εASEvalues, i.e., we still kept the assumption that the Ω(z)/4π factor is a constant [1], but we introduced its average value within a dis charge length of 84 cm. In order to realize the effect of the Ω(z)/4π factor on the analytical calculation, two other values for the Ω(z)/4π were also selected. For the second selection of Ω(z)/4π, we used zth ≅ 7.0 cm, and with a best fit of the ASE energy measurement to the εASE analytical solution, Ω(z)/4π = 0.0056 was obtained. This figure is very much close to 〈Ω(z)/4π〉 = 0.0074 obtained in our first selection. We also applied the commonly used Ω(z = 0)/4π = 0.5 as a third selec tion. By observing the Ω(z)/4π selections on the εASE(lAMP) behavior, it was realized that different Ω(z)/4π values are affecting on the ASE threshold length zth where our intention in this respect was to get mainly the best fit to the experimental measurement. For the g0 formulation, the exact expression for the exact approx exact g 0 (z) and also the g 0 (z) [i.e., g 0 (z) for b = 0] were used. In this respect, we were able to predict the presence of zth ≠ 0, and a closedform formulation for the ASE output energy, εASE, was obtained. With the approximation of b = 0, the εASE formulation was found to be very much close to the expression appeared in the literature. For large laser systems, however, the exact exact expression of g 0 (z) has to be used. ASE
2. THEORETICAL APPROACH ASE 2.1. Exact (Numerical) Solution for n ph and εASE For a 4level laser system with ΔN ≅ N2, the rate equation for the photon density in an amplifying medium is given by ∂n ph ( z, t ) ∂n ph ( z, t ) + c = cg 0 n ph ( z, t ) ∂t ∂z (5) N2 n ph ( z, t ) – + γ . τ sp τc For the steady state solution, we can let ∂/∂t = 0, and Eq. (5) simplifies to N ∂n ph ( z ) t (6) = ( g 0 – α L )n ph ( z ) + γ 2 . cτ sp ∂z In Eq. (5) or (6) nph is the photon density correspond ing to photons traveling in the zdirection, τc is the photon lifetime which is related to the total loss coef t t ficient, α L , through the relation 1/τc = cα L . ΔN = N2 – N1 is the population density, and N2 is the popu lation of the upper state level. For our example in KrF laser for the lower population, we can use N1 ≅ 0 and apply ΔN ≅ N2. γ is 1/4π of the solid angle Ω(z), and for
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a gas laser with an electrodes separation of d, it is given by [27, 28] Ω ( z) = 1 1 – z (7) . 2 2 4π 2 z + d /4 In some approximation for z Ⰷ d, Eq. (7) reduces to 2
Ω ( z) 1 d ≅ 2 . 4π 16 z
(8)
Equation (8) for z > 5 cm with a relative error of less than 5% can be applicable for the case we are studying. For our numerical approach, however, we used the exact expression given by Eq. (7). g0 in Eq. (6) is the small signal gaincoefficient of the medium, and in a onedimensional laser system it has a form similar to Eqs. (2a) or (2b), or Eq. (4). (g0 – t
net
t
α L ) is the net gain, i.e., g 0 = g0 – α L . Thus, in a medium with some losses, g0 which is defined as the gross gain, is slightly higher than the experimental measurements. In the present work, the absorption losses are not in our concern, therefore we can use t net α L ≅ 0, and let g 0 ≅ g0. Keeping in mind that the absorption losses must be added to the measured net gain value to have the gross gain of the medium. As an example, we are referring to gain and nonsaturable absorber measurements in the KrF laser [29], where according to the authors’ report upon using a 92 cm long epumped excimer amplifier, the values of their measurements are given as: g0 = 0.12 cm–1, and αL = 0.0045 cm–1. In another example in an ebeam pumped KrF laser with lAMP = 40 cm, and diameter of 7.5 cm, the g0 and αL values at 1000 Torr gas mixture were measured to be 0.16–0.18 cm–1, with the absorp tion losses of 0.0075 and 0.0125 cm–1, respectively. At a higher gas mixture pressure of 2500 Torr, the authors reported, respectively, the values of 0.17–0.19 cm–1 and 0.010–0.015 cm–1 [30]. These figures are indicating t that α L ≅ 0 can be considered as a rather good approx imation, unless some interest is oriented for evaluating the loss coefficient in a specific high power medium. The ASE output energy εASE is obtained using ASE
ε = hνVn ph , (9) where, V is the discharge volume, V = Sz, and S is the discharge crosssection. Referring to Eq. (6), as the two indicated terms on the righthand side of the equation are zdependent, so our first approach is to solve this equation numeri cally. In this case, the second term on the righthand side of the equation is not considered to be a constant and consequently the results give the actual solution for presenting the ASE output energy, εASE. For this purpose, we consider, first, an idealized large laser sys tem, where lAMP and dAMP are large enough that we can set gaincoefficient to be a constant [i.e., in Eqs. (2a)
or (2b) in an onedimensional model, we let z ∞]. z→∞ Then, we have g 0 ≅ m', which is a constant. If we solve Eq. (6) numerically for different constant values of m', where we call it here m 'i , ranging from m 'i = 0.08 to 0.35 cm–1, different profiles for εASE will be obtained. These profiles are indicating that in the ASE experimental energy measurements of εASE vs. lAMP, in a real laser system when lAMP is finite, i.e., different from infinity, we are dealing with different excitation lengths of lAMP, and each excitation length lAMP of the medium is experiencing different m i' value, or a con stant gaincoefficient. In other words, the ASE output energy experimental profile shows that the length dependency of the ASE gaincoefficient in the KrF ASE laser (i.e., g 0, KrF ) is responsible for the measured ASE output energy. The results of this calculation is shown in Fig. 1. With this argument, the KrF laser ASE gain coefficient for each lAMP, was obtained and the results are introduced in Fig. 2. The observed deduced data, as appeared in Fig. 2, were further fitted to a function of the type given by Eq. (2a), and the results of this curve fitting introduce the ASEKrF gain parameters which have also been tabulated in the first column in ASE Table 1. By solving Eq. (6) upon using the g 0 parameters and with a best fit to the experimental KrFASE energy measurement, the results of solution appeared in Fig. 3, as C1profile. It also turns out that zth ≅ 7.0 cm. In Fig. 3 three profiles, C1, C2, C3, are corresponding to the calculated ASEKrF laser output energy measurement for three different cases: when ASE g 0, KrF , calculated from the ASE energy measurement, according to Fig. 2, has been used, (C1); the case when +
all the measured gain values of excimers/ N 2 /F2 lasers exp
have been used to obtain the g 0, excimers , (C2); and only the data from KrF lasers measurements have been used exp to obtain the g 0, KrF , (C3). The calculated ASE energy in C1, C2, and C3 profiles show that ASE initiates at some length threshold, zth, and as it was mentioned earlier zth ≅ 7 cm corresponds to the C1profile. The authors in [23] for zth ~ 24 cm, they still could do the ASE energy measurements, as shown also in their reproduced data in Fig. 3. Thus, as the energy measurement close to the threshold is considered to be a very difficult task, our calculated value for the case when εASE = 0 at z = zth, for obtaining the threshold length is not expected to be very close to the reported rough value given in [23] and actu ally this range needs to be very carefully measured experimentally to verify further the validity of the pro posed geometrical model for the gain coefficient. Also, it was realized that the predicted C2 and C3profiles are getting close to C1profile for short lAMP, while their deviations from C1profile occur at some large lAMP LASER PHYSICS
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ASE output energy, εASE, mJ 80 (0.32) (0.26) (0.35) (0.29)
60
40
20
(0.20)
(0.23)
(0.14) (0.11)
(0.17)
Experimental data from [23] Numerical calculation for different m'values shown in parentheses
Threshold length, zth = 7 cm (0.08)
0
10
20
30
40
50
60
70 80 90 100 Excitation length, lAMP, cm
Fig. 1. Results of the KrFASE output energy numerical calculation vs. AMP length, lAMP, using Eq. (6), and applying different values of constant gainvalues ranging from m 'i = 0.08 cm–1 to m 'i = 0.35 cm–1. m 'i is referring to different constant gainvalues given to Eq. (6). The experimental measurements deduced from [23] for KrF laser. This figure clearly shows that short excitation lengths, lAMP, correspond to high gain values, while for long lAMP the gain coefficient drops to a low value, i.e., the calculated ASE gain coefficient is lAMP dependent.
excitation lengths. In the second column of Table 1, we also introduced zth for C2 and C3profiles. They are 7.0 and 9.5 cm, respectively. They were selected so as to get a consistency with the experimental data for the ampli fying range of lAMP < 40 cm. For producing curves C2 and C3 we have collected most of the reported gain values appeared in the litera ture, in a more complete picture compared to that given in our previous publication [26]. The collected data from excimer lasers experimental measurements are given by Figs. 4 and 5 which are corresponding to deduced exp exp g 0, excimers , and g 0, KrF profiles, respectively. These pro max files are carrying different m', b, γ L parameters, and of course different from the profile appeared in Fig. 2. Comparison of profiles appeared in Fig. 2, with those given by Figs. 4 and 5, and also the figures tabu lated in Table 1, are indicating that for a large lAMP, all g0values appeared in these profiles reduce to small, and somehow constant values, while for a short lAMP, they all increase to high gainparameters. In spite of the fact that our collected data for excimer lasers gain profiles, as given in Figs. 4 and 5, to some extend, in particular in short lAMP excitation lengths, might be able to explain the present theoretical approach for the ASE εASEmeasurements, but by applying the g 0 formu lation, as appeared in Fig. 2, somehow smaller gain coefficient with different gainparameters (i.e., m', b, max γ L ) will be produced which is able to explain the εASEmeasurements in the whole range of the excita tion length, lAMP. This observation, by itself, is carrying enough evidence that some different gaincoefficient, LASER PHYSICS
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according to the geometricalmodel, is responsible for the εASEmeasurements. For applying the present model we used merely the data from Kühnle et al. for the ASE energy measure ments [23]. They used a commercial Lambda Physic EMG150ES, KrF excimer laser with a 2.5 × 1.0 cm cross section and a discharge length of 84 cm. It was filled with 6 mbar F2, 150 mbar KrF, and 1.65 bar He. The system was operating with a voltage of 30 kV. Calculated ASE gaincoefficient –1 for KrF laser, g0,ASE KrF, cm
0.8
Segmented excitation lengths used for calculations
0.6
0.4 m' = 0.001 cm−1 b = 0.066 cm−1
0.2
max
γL
0
20
40 ASE
= 4.504
60 80 100 Excitation length, lAMP, cm
Fig. 2. ASE gain values, g 0, KrF , obtained from the numer ical calculation to be consistent with the experimental ASE output energy measurements, reported in [23] for the KrF laser, deduced from Fig. 1, vs. excitation length lAMP.
1866
HARIRI, SARIKHANI max
Table 1. Gainparameters, m', b, γ L energy
deduced from different approaches, and Ω/4π, zth corresponding to the ASE output max
Ω/4π
zth , cm
0.014
2.40
–
–
0.026
0.015
2.72
Selected to be 7.0
0.003
0.013
4.73
1 – z 1 2 2 2 z +d The same
0.001
0.066
4.50
The same
0.001
0.066
4.50
Descriptions
m', cm–1
b, cm–1
Previous report for the excimer lasers [26]
0.035
Gainparameters for all exci + mer, N 2 , F2 lasers Gainparameters for the KrF laser only Gainparameters, deduced from the numerical solution
γL
Selected to be 9.5 7.0
Referred to Fig. 3, C2 Referred to Fig. 3, C3 Referred to Fig. 3, C1
ASE
for the g 0, KrF Gainparameters from the numerical solution, used for deducing the 2nd column in the analytical solution
ASE
With this realization, and also by defining g 0, KrF = 0 for z ≤ zth, where it refers to the absence of ASE gain ASE
value, g 0 , when no amplified spontaneous emission exp is observed, Fig. 6 is produced. In Fig. 6, g 0, KrF , and ASE
g 0, KrF are depicted, separately. All curves for large excitation lengths approach each others. Large AMP lengths, actually correspond to the cases where most of the ebeam facilities are working, and apparently for analyzing these devices we can keep g0 as a constant. Again, it has to be noted that our present conclusion relies on the reported ASE measurements in a KrF laser. But, in addition to the above observation there are some
Ω 〈 〉 = 0.0074 4π 0.0056 0.5
7.4
Referred to Fig. 8
7.0 18.0
Referred to Fig. 7 Referred to Fig. 8
other existing documented observations in the litera ture, where they deserve to be mentioned here. Theo retically, Hunter, and Hunter introduced two differen tial equation describing the coherent laser fluxes, desig nated by ᏸ± and spontaneous fluxes by ± (for two propagation directions), and then they introduced the total radiative flux, Σ = (ᏸ+ + ᏸ– + + + –) [9]. The same idea is given by Haag et al. [12]. That means that there is no reason for not adopting different gain values to be responsible for the coherent and the amplified spontaneous emission. Also, we had an observation that for lAMP ≤ zth when we did not observe any ASE in the N2laser amplifier section of an oscillatoramplifier laser system, and the measured gaincoefficient had a
ASE output energy, εASE, mJ 80 C3 C2
Experimental data from [23]
C1
C1: Data from numerical calculation
60
C2: Data from all excimer, N2+, F2 lasers gain measurements C3: Data from KrF laser gain measurements
40
20 Plots are based on m', b, γLmax parameters given in Table 1
0
10
20
30
40
50
60
70 80 90 100 Excitation length lAMP, cm
Fig. 3. Results of the ASE output energy calculation vs. KrF excitation length. Curves C1, C2, and C3, respectively, correspond to ASE
exp
exp
max
the cases, when g 0, KrF , g 0, excimers , g 0, KrF with different parameters m', b, and γ L been used for the calculations. The ASE threshold length zth is 7 cm.
, given in the first column of Table 1, have
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A NEW APPROACH FOR DESCRIBING AMPLIFIED SPONTANEOUS EMISSION −1 KrF small signal gaincoefficient, gexp 0, KrF, cm 0.40
Excimers small signal gain collected from exp –1 literature, g 0, excimers, cm 0.40 exp max
– bl AMP 1 1 g 0, excimers = m' + + γ L ⎛ b + ⎞ e ⎝ l AMP l AMP⎠
0.35
[16,30,31,33,36,37,38,39,46,49,50,52] [32,41] [34,35,42,43] [39.46] [38,40,44] [45]
0.30 0.25 0.20
1867
KrF F2 KrCl ArF XeF N2+
Experiment measurements
0.35
– bl AMP exp max 1 1 g 0, KrF = m' + + γ L ⎛ b + ⎞ e ⎝ ⎠ l AMP l AMP
0.30 [46]
0.25 0.20
[30] [33]
0.15 0.15 0.10 m' = 0.026 cm−1 −1
0.015 cm 0.05 b =max γL
0.10
m' = 0.003 cm−1
0.05
b = 0.013 cm−1 γLmax = 4.73
[16, 31] [36, 49] [38]
= 2.72
0
20
40
60 80 100 120 Excitation length lAMP, cm
Fig. 4. The experimental small signal gainprofile, exp g 0, excimers
+ excimers/ N 2 /F2 lasers,
collected from the for literature (from Table 2) vs. excitation length. References are given in the brackets.
tendency for not following the gain profile from the case when the ASE was existing [24]. ASE
exp
Also, in Fig. 6, the ratio of g 0, KrF / g 0, KrF is depicted. It is seen that this ratio reduces as lAMP increases, and for short excitation lengths, close to zth, it reaches a maximum value of ~90%. Thus, according to the present analysis, reduction or elimination of the ASE in a large laser system cannot be simply achieved, although different techniques such as reduction of operational gas pressure of Kr and F2 in case of KrF laser have been proposed [13, 31]. Based on the pro posed geometrical model of laser gain coefficient, the operational gas pressure has a small contribution to the gain value in comparison with the laser geometri cal configuration (that is, lAMP and dAMP). Therefore, the pressure dependency of gain coefficient is not ASE exp probably affecting significantly on the g 0 /g 0 ratio and the reduction of gas pressure (or output voltage in a electric discharge system) can lead, only, to reduc tion of the output extraction energy, while this ratio still remains the same. 2.2. Analytical Solution for εASE Although the numerical solution for obtaining εASE, as introduced in section 2.1. is explaining the experimental measurements, however, at this stage it is desirable to look for the εASE analytical expression, where this solution is able to give a considerable amount of information for the ASE behavior, and also it verifies the validity of parameters used for the εASE LASER PHYSICS
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0
20
40
60 80 100 120 Excitation length lAMP, cm
Fig. 5. The same as Fig. 4, but the g0values are given for the KrF laser only. References are given in the brackets.
calculation so as to be consistent with the measure ments. For this purpose we used the rate equation, as given by Eq. (6), with the following assumptions: (i) At the threshold length, zth, the ASE photon density, nph, or ASE output energy, εASE, is zero. (ii) We adopt a constant value for the Ω(z)/4π, instead of using the zdependency of Ω(z)/4π, as given by Eq. (7) or Eq. (8) [1], otherwise, Eq. (6) cannot be solved analytically. For a very large value of the z direction, the Ω(z)/4π reduces drastically; therefore, one choice is to apply some length average of the Ω(z)/4π factor, using Ω ( z) = 1 4π l AMP
l AMP
Ω(z)
dz. ∫ 4π 0
Other selections of the Ω(z)/4π will be also introduced in this paper. (iii) We apply Eq. (2b) with the parameters ASE obtained from the previous section and for the g 0, KrF as tabulated in Table 1. (iv) We ignore the absorption losses, that is, we let t α L ≅ 0. (v) We let g0 = σstimΔN ≅ σstimN2 for the case of the KrF laser. Thus, Eq. (6) can be written as ∂n ph ( z ) ( Ω/4π ) (10) = g 0 ⎛ n ph ( z ) + ⎞ . ⎝ cσ stim τ sp⎠ ∂z If we let Ω⎞ 1 , ψ ( z ) = n ph ( z ) + ⎛ ⎝ 4π⎠ cσ stim τ sp
(11)
1868
HARIRI, SARIKHANI +
Table 2. Summarized gain coefficients measured for excimer, N 2 , F2 laser systems Wave length, nm
Method of excitation
KrCl KrF
248 248 248 193 157 222 248
ebeam ebeam ebeam ebeam elec. discharge 40 kV elec. discharge 30 kV elec. discharge 35 kV
KrCl ArF KrF KrF KrF KrF XeF XeF
222 193 248 248 249 248 352 352
F2
KrCl
157 222
KrCl XeF
222 352
+
428
Type of laser KrF KrF KrF ArF F2
N2
0.16–0.18* 0.17–0.19* 0.126† 0.121† 0.37 0.056, 0.041 0.145 (2 ns)* 0.160 (10 ns)* elec. discharge 10.5 kV 0.016 elec. discharge 45 kV 0.035 elec. discharge 30 kV 0.050* elec. discharge 35 kV 0.043* elec. discharge 30 kV 0.25 double discharge 0.023* double discharge 0.01 elec. discharge 0.241 (at 15 kV) 15, 18 kV 0.232 (at 18 kV) elec. discharge 40 kV 0.032 elec. discharge 45 kV 0.13 Passive absorption ASE technique 45 kV 0.15 elec. discharge 26 kV 0.11 0.13 elec. discharge 15.7 kV 0.10
ArF
193
KrF KrF KrF XeF (B XeF (C KrF KrF
248 248 248 353 448 248 248
ebeam elec. discharge
248 248
elec. discharge ebeam
KrF KrF
X) A)
Gain coefficient, cm–1
fluorescence measure ment elec. discharge 40 kV ebeam ebeam ebeam
0.070 0.042 0.15 0.25–0.27* 0.042–0.054* 0.026* 0.3† 0.05† 0.22 0.038 (at 30 kV)* 0.043 (at 35 kV)* 0.096 0.015†
Dimension, cm lAMP
dAMP
40
– – – –
φAMP
Gas pressure
3 3 1.2
– – – – – – – –
3.5 bar 2 atm 1.7 atm 1.7 atm 1900 torr 1 atm 1.15 atm 870 torr
[30] ″ [29] ″ [32] [35] [33] ″ [34] [39] ″ [36] [37] [38] ″ [40]
80 35
2 2
– –
2 atm 355 pa
[41] [42]
35 50
2 1.5
– –
48
0.8
–
355 pa 2.7 atm 3.7 atm 2.2 atm
[43] [44] ″ [45] ″ ″ [46]
40 92 92 10 40 45 80 80 80 80 14 82 82 30
S = 1 × 0.7 cm2
1.5 S = 1 cm2
1.7 3 4.5 variable S = 2 × 0.5 cm2
7.5 1000 torr 7.5 2500 torr – 750 torr – 963 torr – 2–10 atm – 3 atm – 2 atm
Ref.
20
S = 2 × 0.5 cm2
–
1.7 atm 1.25 atm ~3.7 bar
20 100 100 6
S = 2 × 0.5 cm2
– – –
– 27 27 –
~3.6 bar – ~200 torr 400 kpa
15 80
– 4.5
– –
– 1.7 atm
″ [16] [31] [47] ″ [50] [49]
85 150
1.8 S = 1.5 × 1.5 m2
– –
– ≤200 torr
[52] [13]
† Not included in the analysis in Fig. 4. exp
* Used in Fig. 5 for producing g 0, KrF .
Eq. (10) reduces to ∂ψ ( z) = g ψ ( z ), 0 ∂z
(12)
where, we can consider that Ω(z)/4π to be a constant so as ∂ we can let ( Ω ( z )/4π ) ≈ 0. Eq. (12) can be readily inte ∂z
grated to obtain ψ(z), and then by using Eq. (11), we get: G0 ( z ) ( Ω/4π ) (13) n ph ( z ) = Ae – . cσ stim τ sp In Eq. (13) A is a constant, and for simplicity we have defined G0 ( z ) =
∫ g ( z ) dz,
LASER PHYSICS
(14)
0
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2012
1.00
1869
1.00 exp g 0, KrF ASE g 0, KrF ASE g 0, KrF
0.75
⁄ gexp 0, KrF
exp
0.75
g 0, KrF ⁄ g 0, KrF
0.50
0.50
exp ASE g 0, KrF ⁄ g 0, KrF
exp
g 0, KrF
0.25
ASE
KrF laser small signal gain, g0, KrF, cm–1
A NEW APPROACH FOR DESCRIBING AMPLIFIED SPONTANEOUS EMISSION
0.25
ASE
g 0, KrF
0
0 zth
0 100
20 40 60 80 Excitation length lAMP, cm ASE
ASE
Fig. 6. Plots of gainprofiles corresponding to: theoretical calculation, g 0, KrF ( g 0, KrF is zero for z < zth); experimental measure exp
ASE
exp
ments for the KrF lasers only, g 0, KrF . The plot of g 0, KrF / g 0, KrF ratio vs. lAMP is also given to show the variation of this ratio for different laser excitation lengths.
ASE
where upon using the expression for the g 0 as given by Eqs. (2b), (14) introduces an analytical expression, max denoted by G0(z; m', b, γ L ),
If we use the expression for the ASE output energy εASE, given by Eq. (9); G0(z), and G0(zth) by the use of Eq. (15), we will arrive at the following expression for the εASE(z),
max
G 0 ( z ) ≡ G 0 ( z; m', b, γ L ) max
= m'z + ln z
1 + γL
∞
max
– γL
∑
i=1
i+1
i ( – bz ) . ( i + 1 ) ( i + 1 )!
(15)
The integration constant in Eq. (14) is lumped into A factor appeared in Eq. (13). The negative sign appeared on the righthand side of the third term in Eq. (15) shows that by applying Eq. (15) we get an increase (for the case, when b = 0; secondorder, when i = 2, that is, when two terms are added, etc.) or decrease (for i = 1, 3, …) for the εASE(z) calculation. But, as shown in Fig. 7, by using the higher order terms, the calculated value for the εASE approaches to the numerical solution, as appeared in Subsection 2.1, as well as, the experimental measure ments. But, we get a constant value of Ω/4π = 0.0056, when zth ≅ 7 cm is used. By introducing zth to be different from zero, Eq. (15) gets some finite value (in other words, if we do not introduce zth, infact, at z = 0, G0(z) approaches to infinity). The coefficient in Eq. (13) can be evaluated using the boundary condition of n ph ( z ) at z = zth = 0, where it gives our desired analytical expression for the nph(z) and εASE(z), ( Ω/4π ) G0 ( z ) – G0 ( zth ) n ph ( z ) = ( e – 1 ), (16) cσ stim τ sp G0(zth) is the value of G0(z) given by Eq. (15) at the threshold length, z = zth. LASER PHYSICS
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hνS z ⎧ ⎛ z ⎞ 1 + γL e m' ( z – zth ) ( z ) = ( Ω/4π ) ⎨ ⎝ ⎠ cσ stim τ sp ⎩ z th max
2012
ε
ASE
(17) i+1 i+1 ⎫ – ( – bz ) ( – bz ) i th – 1 ⎬. (i + 1) ( i + 1 )! ⎭ i=1 ∞
× exp
max –γL
∑
It is interesting to observe the behavior of the solution when b = 0. In this case, Eq. (17) reduces to a some how familiar expression, given by ε
ASE
( z, b = 0 )
⎫ z ⎞ 1 + γL m' ( z – zth ) Ω⎞ hνS z ⎧ ⎛ = ⎛ e – 1 ⎬, ⎨ ⎝ 4π⎠ cσ stim τ sp ⎝ z th⎠ ⎩ ⎭ max
(18)
where, by multiplying both the numerator and denominator of the equation by the upper state life time, τu, and using definition of saturation intensity given by Is = hν/σstimτu, and the medium quantum yield, φ = τu/τsp, the ASE intensity, IASE(z) = cρASE = cεASE/V, can be obtained. ρASE is the ASE energy den sity, where for a traveling wave we can apply I = cρ. max
I
ASE
Ω z 1 + γL m' ( z – zth ) ( z, b = 0 ) = ⎛ ⎞ I s φ ⎛ ⎞ e –1 , ⎝ 4π⎠ ⎝ z th⎠ (19) z th ≠ 0.
1870
HARIRI, SARIKHANI ASE output energy, εASE, mJ 80 Experiment data deduced from [23]
60
40
20
(2)
(4)
(6)
(10)
(12)
(14)
Calculation with all terms in Eq. (17) (15)
m' = 0.001 cm−1 b = 0.066 cm−1 Parameters given to Eq. (17) γLmax = 4.50 zth = 7.0 cm Ω/4π = 0.0056 Deduced from curve fitting
(5)
0
(8)
10
20
30
40
(9)
(7)
50
60
(13)
(11)
70 80 90 100 Excitation length lAMP, cm
ASE
Fig. 7. The calculated ASE output energy, ε KrF using Eq. (17). On the figure, (0) means calculation with b = 0; (1), (2), … mean we let i = 1, 2, … in Eq. (17); that is, we use 1 term, 2 terms, etc. The solid line corresponds to Eq. (6) when γ = Ω(z)/4π is kept max
as a constant and all terms in Eq. (17) are used. zth = 7 cm is used from Subsection 2.1. m', b, and γ L in Table 1. The data points are reproduced from [23]. Ω/4π for the best fit is 0.0056.
The zfunctional dependency of Eq. (19), is somehow close to the expression which is commonly applied in the literature (for example, in [1]), and it is also given by Eq. (1). But, since the approach to find this equa tion is based on the experimental realization of the z functional dependency of the gaincoefficient, and also the g0geometrical model, it has many advantages to be used in data analysis in many experimental mea surements for small media. To show this similarity it is sufficient to let z – zth = l, where for a small medium, 1+
2/(πΔν0), where Δν0 is the bandwidth of the emission lineshape. Thus, from Eq. (20) we get 2
c0 (20a) σ stim ( ν ≈ ν 0 )τ sp ≅ 2 , 2 π 8πn ν 2 Δν 0 and the factor in front of the large bracket in Eq. (18) turns out to be ⎛ 8πn 3 ν 2 ⎞ π Ω⎞ ⎛ Δν nηSz ⎜ ⎟. 0 2 ⎝ 4π⎠ ⎝ c3 ⎠
max γL
≅ 1. In this case, we have l ≅ 0, and (z/zth ) Eq. (19) gives an exact expression given by Eq. (1), except for the factor g(λ) which it is now a constant (given by m') and appeared in em'l term. For large size media, however, Eqs. (18) and (19) or Eq. (1) cannot be used to explain the experimental observations, in other words, by pushing Eq. (18), such as applying sat uration effects etc. in this equation to follow Eq. (17) as shown their distinction clearly in Fig. 7. In addition, Eqs. (17) and (18) and also Eq. (19), show the depen dency of the εASE or IASE on the medium emission fre quency, ν, and other medium parameters which will be introduced here. Considering the expression for σstim, given by [1] 2
λ0 σ stim = ᒄ ( ν – ν 0 ), 2 8πn τ sp
(20)
where ᒄ(ν – ν0) is the medium emission lineshape and n = c0/c is the medium index of refraction, c0 and c are the velocity of light in vacuum and in the medium, respectively. At the peak of the emission of a Lorentzian lineshape, we can use ᒄ(ν = ν0) =
ASE
for the g 0, KrF are given
(20b)
0
The expression inside the second parenthesis in Eq. (20b) gives exactly the total number of modes per unit volume within the emission bandwidth Δν0 in black body radiation [1], i.e., Δᏺ(ν)/V. Thus, Eq. (18) can also be expressed by the following expression ε
ASE
Ω⎞ Δ ᏺ ( ν ) ( z, b = 0 ) = π ⎛ hνSz 2 ⎝ 4π⎠ V max
z⎞ 1 + γL e m' ( z – zth ) – 1 . × ⎛ ⎝ z th⎠
(21)
Equation (21) [or similarly for the case when b ≠ 0 i.e., Eq. (17)], shows clearly that the ASE output energy is proportional to number of modes per unit volume 1 within a bandwidth Δν0, discharge volume Sz, of 4π solid angle, i.e., ΔΩ(z)/4π, and the ASE functional z dependency which appeared inside the larger bracket. max A plot of εASE(z; m', b, γ L ) for a number of terms upon using Eq. (17), as explained earlier, is depicted in max Fig. 7. The numerical values of m', b, γ L for the KrF LASER PHYSICS
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2012
A NEW APPROACH FOR DESCRIBING AMPLIFIED SPONTANEOUS EMISSION
amplified spontaneous emission have been used according to Table 1. As it is indicated in this figure, by increasing the numbers of terms appeared in Eq. (17), the calculated εASE appears to approach to the exact numerical solution. For the product of the effective stimulated emission cross section and the spontaneous lifetime in the KrF laser, i.e., σstimτsp, Hawryluk et al. reported that it is (1.75 ± 0.25) × 10–24 cm2 s [48]. This figure has also been reported by Watanabe et al. [49]. In [48], they also used the calculated value for the lifetime τsp = 6.5 ns. On the other hand, in [31], the stimulated cross section is given to be 2.6 × 10–16 cm2, while this value in [50] was reported to be (1.8 ± 1.1) × 10–17 cm2. An experimentally measured value of τsp ~ 9 ns has also been reported [51]. For our present calculation we used σstimτsp = 1.75 × 10–24 cm2 s [48, 49]. For the Ω(z)/4π term appeared in the given equa tions, 3 different selections have been applied and the results have been compared with each other. For the first selection, an average value of the (Ω(z)/4π) was used, where we considered lAMP = 84 cm [23]. In this case, the calculated value of 〈Ω(z)/4π〉 is 0.0074. This figure corresponds to zth ≅ 7.4 cm, when the best fit to ASE ε KrF
the measurement was made. We also used the Ω(z)/4π, obtained from Fig. 7, when zth ≅ 7.0 cm was used to get the best fit of the ASE energy measurement to the εASE analytical solution. In this case Ω(z)/4π = 0.0056 was obtained. And the last, Ω(z = 0)/4π = 0.5 was also used to see its effect on the length threshold. In this latter case, zth ≅ 18 cm was deduced. This figure is higher than the results obtained numerically, where it indicates that although Ω(z = 0)/4π = 0.5, which cor responds to the assumption that amplified spontaneous emission initiates at z = 0, is rather a good approxima tion, however, this assumption fails due to its defi ciency for the zth prediction in large scale systems. In the 2nd column of Table 1, we have also com pared different selections of the Ω(z)/4π and their cor responding zthvalues. In Fig. 8 for the analytical cal ASE
culation of ε KrF vs. lAMP, three profiles corresponding to different Ω(z)/4π have been depicted, with the intention of comparing these 3 profiles with each other. In general, there is no any pronounced discrim ination between these profiles, except in the region close to the threshold length, zth. It is also useful to introduce some higher order terms appeared in the calculation. If in Eq. (17) we define, Ω hνS C ( ν ) = ⎛ ⎞ , ⎝ 4π⎠ cσ stim τ sp
(22)
for the 1st, 2nd, and 3rd orders of approximation, we have, respectively LASER PHYSICS
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2012
⎧ z 1 + γL m' ( z – zth ) e = C ( ν )z ⎨ ⎛ ⎞ ⎝ ⎠ ⎩ z ph
1871
max
ASE ε 1st ( z )
(23)
⎫ max b 2 2 × exp – γ L ( z – z th ) – 1 ⎬, 4 ⎭ 2
⎧ z 1 + γL m' ( z – zth ) e = C ( ν )z ⎨ ⎛ ⎞ ⎝ ⎠ ⎩ z th max
ASE ε 2nd ( z )
(24)
⎫ max b 2 max b 3 2 3 × exp – γ L ( z – z th ) + γ L ( z – z th ) – 1 ⎬, 4 9 ⎭ 2
3
and ⎧ z 1 + γL m' ( z – zth ) e = C ( ν )z ⎨ ⎛ ⎞ ⎝ ⎠ ⎩ z th max
ASE ε 3rd ( z )
2
3
max b 2 max b 3 2 3 × exp – γ L ( z – z th ) + γ L ( z – z th ) 4 9
(25)
4 ⎫ 4 max b 4 – γ L ( z – z th ) – 1 ⎬. 32 ⎭ These terms as given by Eqs. (23)–(25), and Fig. 7 show that when the amplifying length, z ≡ lAMP, is small, it is not necessary to use higher order terms, while for a large z ≡ lAMP, some higher order terms have to be applied for explaining the experimental measurements.
3. DISCUSSION Base on the gaingeometrical model we managed to explain, as an example, the experimental measurements of the ASE output energy reported for a KrF laser of 84 cm in length [23]. For this purpose two different approaches have been used: numerical calculations, and analytical solution. Both approaches were able to predict the ASE output energy correctly. By applying the model, it was found that introducing an ASE gaincoefficient, ASE g 0 , for explaining the ASE output energy in a KrF laser is unavoidable. In this discussion there are some points that can be realized, accordingly: (i) The geometrical dependency of the gain coeffi cient removes most of the ambiguities in the measure ments appeared in the literature. For clarifying this statement, we are referring to a number of examples appeared in the literature: in [52] the authors upon using a 85 cm long KrF laser, they experimentally g0 l
observed that e = 3500. They made an attempt to verify their measurement with a 45 cm discharge length laser given by Banic et al. measurement [33], where the later authors reported the value of 0.145 cm–1 for their system. In [52], the authors concluded that
1872
HARIRI, SARIKHANI ASE output energy, εASE, mJ 80 Experimental data deduced from [23] 〈Ω/4π〉 = 0.0074, gives zth = 7.4 cm
60
Ω/4π = 0.0056, zth = 7.0 cm from Fig. 7 Ω(z = 0)/4π = 0.5, gives zth = 18.0 cm
40 m' = 0.001cm−1
20
b = 0.066 cm−1 γLmax = 4.50
0
10
20
30
40
50
60
70 80 90 100 Excitation length, lAMP, cm
Fig. 8. The calculated ASE output energy for: 〈Ω/4π〉 = 0.0074, gives zth = 7.4 cm for the best fit to the experimental measure ments (dotted line); Ω/4π = 0.0056 for zth = 7.0 cm has been accepted from Fig. 7 (solid line); and Ω(z = 0)/4π = 0.5. For this last case the best fit gives zth = 18.0 cm (dashed line). Different selections of Ω(z)/4π is affecting the threshold lengths. The mea surements shown by dark circles deduced from [23].
this large discrepancy is most likely caused by satura tion of the gain by the strong ASE output. We used their data, and fitted to Frantz and Nodvik equation [53] and we found that g0 = 0.097 cm–1, and Es ≅ 3.33 mJ/cm2. The value of g0 ≅ 0.1 cm–1 is very much consistent with our collected data appeared in Figs. 4 and 5. Gower and Edwards [37] with lAMP = 14 cm, they found g0 = 0.25 cm–1, where they compared their data with [33, 49, 54], and they concluded that in their work the wide variation may be due to different types of devices. Their measurements are also in agreement with the proposed g0geometrical model. The gain geometrical dependency has also been realized in work of Okuda et al., where they were using a large aperture amplifier. They reported that in a large aperture ampli fier, the steadystate gaincoefficient is smaller than those of small aperture amplifiers [55]. This last state ment is also consistent with our observation with N2 laser, in our proposed 2dimensional laser system, where it was shown that by increasing the discharge gas separation, dAMP, or laser diameter φAMP (in CVLs), the gain value drops to a small value [26]. (ii) By referring to Fig. 6 for large scale systems, the exp
ASE
gaincoefficient for the g 0, KrF and g 0, KrF are very close to each other. Thus, in these devices, g0 can be con sider as nearly a constant carrying small gain values. (iii) By referring to Fig. 3 or Fig. 7, according to the experimental measurement, when lAMP increases, the εASE increases accordingly. As the present available data were applied to verify our gain formulation and no reduction of the output ASE energy measurement due to gain saturation was reported in [23], so, in this work
the gain saturation has not been considered in the analysis. (iv) For evaluation of the ASE output energy, we calculated Δν0 = 8.94 × 1012 Hz (Δλ = 18.40 Å) from Eq. (20a), and using σstimτsp = 1.75 × 10–24 cm2 s. The deduced εASE formulation [for example, Eq. (21)], shows that εASE is proportional to the total number of modes per unit volume, Δᏺ/V, which is 1.22 × 1013 cm–3 for λ = 248.5 nm of the KrF laser. The pres ence of Δᏺ/V factor is also showing clearly the spon taneous nature of the ASE presenting in the output laser beam. This fact for the presence of spontaneous emission has also appeared in other equations such as Eqs. (17)–(19) where the m'parameter, which was introduced initially for the presence of the spontane ous nature of the emission, appeared also in the z functional dependency of the εASE. REFERENCES 1. O. Svelto, Principles of Lasers, 5th ed. (Springer, 2010). 2. A. E. Siegman, Lasers (University Science Books, Mill Valley, California, 1986). 3. D. A. Leonard, Appl. Phys. Lett. 7, 4 (1965). 4. L. Allen, and G. I. Peters, Phys. Lett. A 31, 95 (1970). 5. G. I. Peters, and L. Allen, J. Phys. A: Gen. Phys. 4, 238 (1971). 6. L. Allen, and G. I. Peters, J. Phys. A: Gen. Phys. 4, 377 (1971). 7. L. Allen, and G. I. Peters, J. Phys. A: Gen. Phys. 4, 564 (1971). 8. S. Stenholm and W. E. Lamb Jr., Phys. Rev. 181, 181 (1969). 9. A. M. Hunter II and R. O. Hunter, Jr. IEEE J. Quan tum Electron. QE17, 1879 (1981). LASER PHYSICS
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