Laser Model Final Report

Transcription

1 Laser Model Final Report Dann Caballero, Steve Pollock April 24, 2012 Abstract We modeled a pair of coupled di erential equations that represent a simple mathematical model for a laser. The model is nonlinear, but can be investigated both using analtic and numerical methods. We describe this model in detail connecting each variable in the model to the phsics it controls/describes. We found fied points (i.e., critical points) for this sstem and have determined their stabilit. This sstem appears to have a number of variables that control the stabilit, but b non-dimensionalizing the model, we have found that a certain ratio of these parameters control lasing of the sstem. We have plotted interesting results from the analtics and modeled particular trajectories of the sstem. 1 Introduction 1.1 How do lasers work? Most lasers work on the principle of an inverted population. That is, atoms in the laser sstem a pumped into an ecited state. The resulting deca of those atoms to a lower state produces photons at a particular wavelength. The atoms in a laser sstem are tpicall immersed in a gain medium. The photons that are emitted bounce between two mirrors on either end of the gain medium, causing further stimulated emission which amplifies (increases power) the output. One of the mirrors is partiall transparent so that some photons escape. These photons are of a single wavelength and coherent (roughl, the same phase). Our sample report appears long, but it reall is not. There are three reasons for its apparent length: (1) The margins have been increased to accommodate annotations like these, (2) we worked out a lot of algebra inline ( 3 pages), and (3) figure take up a lot of space ( 4 pages). These annotation boes are meant to help ou de cide what goes into our final report. An abstract or summar is nice, but not necessar. It helps th reader figure out the point of our project It also provides ou with a nice summar of what ou did. In this section, we have tried to eplain/motivate the model that we are go ing to work with. Yo might notice this is the eact tet from our progress report. That s OK! 1.2 The laser model Milonni and Eberl [1] proposed a mathematical model for laser dnamics that takes into account both the number of laser photons (n) and ecited atoms (N): 1 We are simpl presenting the model that we are going to work with. Notice, w have cited where it came from.

2 dn dt dn = GnN kn, (1a) dt = GnN fn + p. (1b) In this model, there are N ecited atoms in the sstem which can produce the n laser photons. The number of laser photons (n) increases when there are more This section of tet is ver important. It ecited atoms (N) and more laser photons (n) available to ecite those atoms. Moreover, the gain coe cient, G, for the medium controls how e ective these photons nation of the model. provides the epla- We are attempting are in eciting more atoms and thus creating more photons. This is the first term in to make sense of the Eq. 1, GnN. The number of photons (n) decreases as photons leave the sstem at model and connect it a rate k; this is proportional to the number of laser photons in the sstem (n). This back to the phsics. Notice, we haven t is the second term in Eq. 1, kn. As more photons are emitted b atoms, fewer performed an calculations et. We are atoms are ecited; the rate at which this happens is proportional to the number of simpl illustrating ecited atoms (N) and the e cienc of the medium (G). This is the first term in that we understand Eq. 2, GnN. The number of ecited atoms (N) also drops as atoms emit photons some of the details of the model that we at a rate f; again, this is also proportional to the number of ecited atoms (N). have chosen to work This is the second term in Eq. 2, fn. Finall, the number of ecited atoms (N) with. increases as the pumping of energ into the sstem increases, p. This is the final term in Eq. 2, p. We can consider the limit when the rate of ecited atom production is much slower than the production of laser photons. This might seem like a counter This section still con intuitive limit, how can ecited atom production be slower than the the production of laser photons. In fact, what we are saing is that laser photons sta in the sstem for a much longer time than atoms are ecited. That is, atoms quickl drop down and release a photon which stas in the cavit of a long time. This limit is k f 1, that its the deca rate of photons due to scattering and mirror transmission is much smaller than the rate of spontaneous emission. This limit is e ectivel taking Ṅ 0 because atoms are not ecited long enough to be counted compared to the photons the produce [2]. In the quasi-static limit, when Ṅ is small compared to ṅ, the model reduces to a single non-linear di erential equation, dn dt =0= GnN fn + p! N = p dn p dt = Gn Gn + f 1.3 Non-dimensionalizing the laser model Gn + f kn (2) tains more sensemaking. We are attempting to motivate asimplermodel,b consider a particular limit. Notice, that w don t actuall take the limit and perform the mathematics unt we have eplained th phsics behind takin this limit. We chose to enumerate onl final equations. Later, ou wil see that we have performed some algebra inline. You can do this, or turn in separate calculations. Equations 1 and 2 are mathematical models of a laser that have eplicit units (e.g., f has units of atoms/second). Often, it makes sense to remove the units from these models in a process called, non-dimenionalization. This produces a mathematical 2 We chose to nondimensionalize our model and this section provides the mo tivation for that work You are certainl fre to maintain dimensions in our model. In that case, ou might discuss tpical parameter values for the model our a working with.

3 model which has fundamentall the same dnamics as the model with dimensions. However, a non-dimensional model is often easier to work with because the number of parameters is tpicall reduced (i.e., parameters tend to form dimensionless ratios) [2] The one-dimensional model We begin b non-dimensionalizing equation 2 because it is a single di erential equation. Equation 2 has a single independent variable t and a single dependent variable n. We propose two critical numbers (t c and n c ) which will ield dimensionless variables ( = t/t c and = n/n c ). We plug these definitions into equation 2 and obtain n c d t c d = Gn p c kn c. Gn c + f If we divide this equation b n c and multipl b t c the resulting equation is dimensionless, d d = Gt p c kt c. Gn c + f We choose t c =1/k to simplif the second term to a single non-dimensional variable, d d = G p. k Gn c + f We also divide out an f from the denominator to ensure it is dimensionless and swap for p in the parentheses, d d = Gp. fk Gn c /f +1 We choose n c = f/g to simplif the first term in the denominator to a single non-dimensional variable, d d = Gp. fk +1 Finall, we rewrite the ratio Gp/f k as c, the single parameter which characterizes the sstem. d d = c +1. (3) We can interpret c as the ratio of parameters that contribute to lasing (high gain, G; and high pumping, p) to parameters that can detract from it (high deca 3 This is how ou nondimensionalize most equations, in general As ou might notice, we have worked out the algebra in detail. This is for our benefit; should ou chose to non-dimensionaliz our own model. We have performed some algebra to obtain our dimensionles model; but, more importantl, we have tried to make sense o the resulting dimensionless ratio. We ar tring to make sense of the phsics ever step of the wa.

4 rates, f and k). Indeed, f is a funn parameter because it is possible for it to both contribute to lasing and detract from it. In this case, we interpret is as being unproductive for lasing because higher f with fied G and p will not lead to lasing. So we epect that the laser should operate (i.e., lase) when c>1. We shall see this is the case in the net full section The two-dimensional model Non-dimensionalizing the two-dimensional model, given b equation 1, proceeds the same wa as before. However, in this model, the are now two dependent variables (n and N), hence we propose a third critical number (Ñc) that will remove the dimensions of the second dependent variable (i.e., = N/N c ). We plug the two previous definitions into equation 1 to obtain: ñ c d t c d = Gñ cñc kñ c, Here s another eample of nondimensionalizing a model. There s a bit more algebra in this section, but we have attempted to eplain each step for our benefit. Isolating ẋ and ẏ, we obtain, Ñ c t c d d = Gñ cñc fñc + p. d d = GÑc t c k t c, d d = Gñ c t c f t c + p t c Ñ c. Again, the simplest isolation of a variable comes from setting t c =1/k. This isolates in the first equation, d d = d d = GÑc, k Gñ c k f k + p. kñc The net simplest isolation is a the product () in the first equation. B setting Ñ c = k/g, we obtain, d d =, d d = Gñ c k f k + pg k 2. The final characteristic number can be set to isolate the first term in the second equation. In fact, setting it equal to Ñc makes the most sense (i.e., ñ c = k/g), 4

5 d =, d d d = f k + pg k 2. We have reduced two non-linear di erential equations with four free parameters to two non-linear di erential equations with two free dimensionless ratios. We set a = f/k and b = pg/k 2 to finall obtain, d = ( 1), (4a) d d = ( + a)+b. (4b) d We can understand the dimensionless ratios as a measure of losses (if a is large and b is small) or gains (if a is small and b is large). Therefore, we epect the laser to operate (i.e., lase) if b>a. This is the same condition as before because c = b/a. We shall this is the case in the net full section. 2 Analtical analsis of the laser model Analtical analsis of equations 3 and 4 can help us make sense of how lasing is possible in each of these models. First, we can look for stead solutions (i.e., fied or critical points). Stead solutions do not change with time and it is possible for such solutions to be stable (attractor) or unstable (repeller). That is, it is possible that given a particular set of parameter values all solutions either approach or run awa from a particular stead solution. To determine which is the case, we must evaluate the stabilit of our obtained stead solutions. For one-dimensional sstems, this is quite simple [3]. For multidimensional sstems, we can evaluate linear stabilit using the Jacboian [2]. 2.1 Stead solutions of the one-dimensional sstems To find stead solutions for the one-dimensional sstem, we set equation 3 to zero and solve for. ẋ =0= c +1 In this case, we have a slightl more complicated nondimensional model, but we tried to make sense of the new dimensionless ratios an how the might influ ence the phsics. Each project must contain some analtical work. For a nonlinear model, finding stead solutions is an appropriate analtical eercise. Your project might lend itself to analtics in some limit or ou might compare it to asimpleranaltical model. 5

6 c +1 = c = 2 + 0= 2 +(1 c) 0= ( +(1 c)) Hence, there are two possible stead solutions 0 = 0 or 0 = c 1. Recall that Again, we worked ou the dimensionless ratio c is a free parameter. The stabilit of these solutions might depend on our choice of this parameter. We postulated that c>1 should produce lasing. To evaluate the stabilit of these solutions we take evaluate the sign of the derivative of right-hand side of equation 3 with respect to at the fied points. This is identical to checking the sign of ẍ, that is, concavit near the fied point. ẋ = F () = F 0 () = F 0 (0) = c 0+1 c +1 c +1 c ( + 1) (0 + 1) 2 1=c 1 the algebra inline. This is not necessar but please attach tha work to our final write-up. F 0 (c 1) = c (c 1) + 1 c(c 1) ((c 1) + 1) 2 1= 1 c If F 0 ( 0 ) > 0, the solution is unstable (repeller), and if F 0 ( 0 ) < 0, the solution is stable (attractor). If c<1, the first stead solution is stable (i.e., 0 = 0 means no lasing) and the second stead solution is unstable ( 0 = c 1). For this situation (c < 1), the second solution is unphsical ( 0 = c 1 < 0); this corresponds to negative photon numbers. If c>1, the first stead solution is unstable and the second stead solution is stable. In this situation (c >1), the second stead solution is the lasing solution ( 0 = c 1 > 0) that we predicted earlier. In particular, the laser photon number is some finite positive value. 2.2 Stead solutions of the two-dimensional sstems To find stead solutions for the two-dimensional sstem, we set both di erential equations in equation 4 to zero and solve for and simultaneousl., 1 The mathematics we performed had a pur pose. We connected our earlier prediction (i.e., lasing occurs when c>1) to the mathematics necessar to prove this. Notice that we have also made sense of th other limit (i.e., that the unstable solution is unphsical). ẋ =0=( 1), ẏ =0= ( + a)+b. 6

7 The first equation above is satisfied if = 0 or = 1. First, we choose =0 and plug this into the second equation to find what must be, 0= (0 + a)+b 0= a + b = b/a Hence, the first stead solution is h 0, 0 i = h0, b/ai. Notice that this corresponds to having no laser photons in the cavit ( 0 = 0). To find the second Here, we have made solution, we allow set = 1 in the second equation and solve for, sense of the first stead solution. 0= 1( + a)+b 0= + b a = b a Hence, the second stead solution is h 0, 0 i = hb a, 1i. This solution can have laser photons in the cavit if b>aas that corresponds to positive 0. This is the lasing solution that we predicted earlier. We can determine the linear stabilit of these stead solutions b evaluating the Jacobian (matri of partial derivatives) of this sstem at these fied points. This methodolog is somewhat beond the scope of the current project (and course!), so we do not detail the method here. Interested readers are directed to Strogatz [2]. But, we sketch out the idea. A particular solution that converges (or runs awa from) to a fied point (stead solution) does so along a particular trajector or path. This path can be simple (such as a straight-line along the or -ais) or comple (a curved path that eventuall is tangent to some direction). The latter is an approach (or run awa) along an eigenvector of the Jacobian evaluated at the fied point. The rate of approach (or run awa) can be determined b that eigenvector s eigenvalue. Hence, the eigenvalues of the Jacobian are ver important in talking about the long time dnamics of the sstem. Do solutions converge to the fied point or run wa from it? The sign of the eigenvalues determine determine whether the fied point is a stable fied point (both eigenvalues negative and real), unstable fied point (both eigenvalues positive and real), or a saddle point stable in one direction but unstable in another (both real, but one positive and one negative). More interesting behaviors is possible (i.e., if the eigenvalues are comple). For the current sstem, the fied point h0, b/ai is linearl stable if b/a < 1 and a saddle point when b/a > 1. That is, lasing appears impossible unless the gains (b) are stronger than the losses (a). The fied point hb a, 1i is linearl stable if b/a > 1 and a saddle (but irrelevant) if b/a < 1. This saddle point is irrelevant because b a<0 is not a phsical solution (i.e., negative photon number!). Here, we have made sense of the second stead solution. This section describes how we probe stabilit for multidimensional sstems. This work is definitel beond the scope of our course, but the description of the method is included for completeness. Yo ma contact us if ou are interested in performing this tpe of investigation. We make sense of the stabilit of both stead solutions and have connected it back to the phsics. 7

8 3 Computational analsis of the laser model We have built some intuition about the laser model. In particular, when losses outweigh gains (regardless of model), lasing is not possible. We present numerical computations to further illustrate these claims. 3.1 Numerical analsis of the one-dimensional sstem In our analtical analsis of the one-dimensional laser model, we have found one phsical solution that occurs at low gains and high losses (c <1). This solution corresponds to no lasing; laser photons in the cavit tend to zero. Below, we plotted the phase space trajector for this one-dimensional sstem and observed that all solutions converge to the no lasing solution in this parameter limit (Figure 1). All solutions in a one-dimensional sstem follow the same phase space trajector [2] This section starts with the numerics. To perform the work in this section, we used NDSolve as wel as various plotting tools. We have, in each figure caption, attempted to make sense of our numerical results and connect the plots to the phsics (a) Low gains (c <1) produce no lasing (b) Two trajectories (the overlap) started with =1and =. Figure 1: Phase space plots of the one-dimensional laser model (c = 0.9). (a) In a one-dimensional sstem, all solutions (blue line) converge to zero photon number (red dot). (b) A series of particular solutions (orange line) follows the same phase space trajector (blue line). Here, we have connected the prediction from the model back to the phsical sstem. We obtained numerical solutions for the model for two choices of initial conditions. After roughl 20, both tend to zero photon number (Figure 2). If c<1, then the sstem has more losses, through mirror transmission and photon scattering, than gains, through pumping. Hence, the sstem is unable to sustain laser photons in the cavit for an appreciable amount of time. The sstem bifurcates and two solutions appear as c is increased above 1. One of the solutions previousl eisted and is unstable (no lasing), but the new solution (lasing) is stable. Below, we plotted the phase space trajector for this one-dimensional sstem and observed that all solutions converge to the lasing solution for c>1 (Figure 3). We obtained numerical solutions for the model for two choices of initial conditions. After roughl, 20, both tend toward (c 1) photon number (Figure 4). If Notice that, again, w c>1, the sstem has more gains, through pumping or choice of medium, than losses, 8 emphasized the connection between the models predictions and the phsics.

9 t (a) Trajector started with = t (b) Trajector started with =1. Figure 2: Plots of dimensionless photon number versus time for the one-dimensional laser model. Regardless of initial conditions, all solutions coverage to zero photon number (a) High gains (c >1) produce lasing (b) Two trajectories (the overlap) started with =0.1 and =. Figure 3: Phase space plots of the one-dimensional laser model (c = 2). (a) In a onedimensional sstem, all solutions (blue line) converge to (c 1) photon number (green dot) and run awa from zero photon number (red dot). (b) A series of particular solutions (orange line) follows the same phase space trajector (blue line). through mirror transmission and scattering. Hence, the sstem is self-sustaining. 3.2 Numerical analsis of the two-dimensional sstem In our analtical analsis of the two-dimensional laser model, we found onl one phsical solution that occurs at high losses and low gains (b/a < 1). This solution corresponds to no lasing; there tend to be no laser photons even though some fied fraction of atoms remain ecited. Below, we plotted the phase space for this twodimensional sstem. The vertical ais in these plots correspond to the number ecited atoms and the horizontal ais correspond to number of laser photons. All phsical solutions converge to the stead solution (Figure 5). We obtained numerical solutions for the model for two choices of initial condi- This section is quite similar to the previous section. Note the sense-making and connections to previous predictions in th section. 9

10 t (a) Trajector started with = t (b) Trajector started with =. Figure 4: Plots of dimensionless photon number versus time for the one-dimensional laser model. Regardless of initial conditions, all solutions coverage to (c 1) photon number. tions. After 20, both tend toward zero photon number and maintain an identical number of ecited atoms b/a (Figure 6). For b/a < 1, the sstem has more losses than gains and, hence, cannot sustain laser photons in the cavit. Interestingl, in the model, atoms are maintain their ecitation and are releasing laser photons, but this is eactl canceled b the losses due to scattering and mirror transmission. The sstem bifurcates and two solutions appear as b/a is increased above 1. One of the solution previousl eisted and is now a saddle point (no lasing), but the new solution (lasing) is stable. Below, we plotted the phase space for the two-dimensional laser model (Figure 7). All solutions (ecept those starting with 0 = 0) appear to converge to the lasing solution. That the previous solution is now a saddle point is interesting. Along all directions, ecept if 0 = 0, the sstem will sustain laser photons in the cavit. That is, there must eist at least a single photon in the sstem to produce lasing. This is generall possible because of thermal fluctuations in the medium. With some probabilit, a few atoms are alread ecited and emitting photons. Hence, the sstem is tpicall primed for lasing. We obtained numerical solutions for the model for two choices of initial conditions. After 20, both tend toward = b a and = 1. If b/a > 1, the sstem has more gains than losses and can, thus, sustain laser photons in the cavit for indefinitel (Figure 8). 4 Conclusions In this paper, we investigated model for a laser that accounts for medium gain, losses through mirrors and scattering, spontaneous emission of atoms, and pumping. The model predicts, in general, if the e ects of pumping, gain, atomic deca outweigh losses through scattering and mirrors, lasing can occur. This is true regardless of initial conditions ecept for the case where there are no laser photons in the cavit to We simpl summariz the results from the model in this section 10

11 (a) Low gains and high losses (b/a < 1) produce no lasing (b) Two trajectories started with h 0, 0i = h0.1, 0i and h 0, 0i = h2, 1i. Figure 5: Phase space plots of the two-dimensional laser model (b/a = 0.9). (a) In a two-dimensional sstem, all solutions (blue line) converge to zero photon number (red dot). (b) A series of particular solutions (orange lines) follow the di erent phase space trajectories, but converge to the same solution (red dot). start. However, such a situation is generall unphsical given thermal fluctuations in the cavit. References [1] P.W. Milonni and J.H. Eberl, Lasers. Wile, [2] S.H. Strogatz, Nonlinear dnamics and chaos: With applications to phsics, biolog, chemistr, and engineering. Westview Pr, 1st Edition, [3] J.R. Talor, Classical mechanics. Universit Science Books, 1st Edition,

12 t t (a) Trajectories started with h 0, 0i = h0.1, 0i. (b) Trajectories started with h 0, 0i = h2, 1i. Figure 6: Plots of dimensionless photon number () and dimensionless ecited atom number () versus time for the two-dimensional laser model. Regardless of initial conditions, all solutions converge to zero photon number (no lasing). Contributions to the work Dann did most of the work including the coding and the writeup. Steve o ered some suggestions, but not man were included in the main manuscript. He did go over final draft and o ered helpful suggestions, including catching an algebraic error. We both agree that Dann should receive a higher grade on the assignment. This section will help us decide how to award credit. Make sure that ou work together to avoid our outcome. 12

13 (a) Trajectories started with h 0, 0i = h0.1, 0i (b) Trajectories started with h 0, 0i = h2, 1i. Figure 7: Phase space plots of the two-dimensional laser model (b/a =). (a) In a two-dimensional sstem, almost all solutions (blue lines) converge to (b a) photon number (green dot) and run awa from zero photon number (red dot). (b) A series of particular solutions (orange lines) follow di erent phase space trajectories. One trajector (no photons in cavit to begin with) converges to zero photon number. Out[137]= Out[138]= t t (a) Trajectories started with h 0, 0i = h0.1, 0i. (b) Trajectories started with h 0, 0i = h2, 1i. Figure 8: Plots of dimensionless photon number () and dimensionless ecited atom number () versus time for the two-dimensional laser model. Regardless of initial conditions, all solutions converge to (b a) photon number (no lasing). 13

14 2 Laser_Model_SP12_2.nb << PlotLegends6` a = 1; b = 0.9; tf = 20; H*v=StreamPlot@8H-1L,-*-a*+b<,8,0,2<,8,0,2<, AesLabelØ8,<,StreamPointsØFine,VectorPointsØ10,VectorStleØRed, FrameLabelØ8,<,FrameStleØDirective@FontSizeØ28DD;*L v = StreamPlot@8 H - 1L, - * - a * + b<, 8, 0, 2<, 8, 0, 2<, StreamPoints Ø Fine, FrameLabel Ø 8"", ""<, FrameStle Ø Directive@Thick, FontSize Ø 28D, ImageSize Ø 8500, 500<, ImageMargins Ø 0D; s = NDSolve@8'@tD H@tD - 1L, '@td ã -@td - a + ã ã 0<, 8, <, 8t, 0, tf<d; s2 = NDSolve@8'@tD H@tD - 1L, '@td ã -@td - a + ã ã 1<, 8, <, 8t, 0, tf<d; Plot@8@tD ê. ê. s<, 8t, 0, tf<, AesLabel Ø 8"t"<, PlotRange Ø 880, tf<, 80, 2<<, AesStle Ø Directive@Thick, LargeD, PlotStle Ø 8Thick<, ImageSize Ø 8500, 500<, ImageMargins Ø 0, PlotLegend Ø 8"", ""<, LegendTetStle Ø FontSize Ø 28D Plot@8@tD ê. ê. s2<, 8t, 0, tf<, AesLabel Ø 8"t"<, PlotRange Ø 880, tf<, 80, 2<<, AesStle Ø Directive@Thick, LargeD, PlotStle Ø 8Thick<, ImageSize Ø 8500, 500<, ImageMargins Ø 0, PlotLegend Ø 8"", ""<, LegendTetStle Ø FontSize Ø 28D p = ê. s<, 8t, 0, tf<, PlotRange Ø All, PlotStle Ø 8Orange, Dashed, Thick<, AesLabel Ø 8"", ""<, AesStle Ø Directive@Thick, FontSize Ø 28DD; p2 = ê. s2<, 8t, 0, tf<, PlotRange Ø All, PlotStle Ø 8Orange, Dashed, Thick<, AesLabel Ø 8"", ""<, AesStle Ø Directive@Thick, FontSize Ø 28DD; pts = Graphics@8PointSize@LargeD, Red, Point@880, b ê a<, 8b - a, 1<<D<D; Show@v, ptsd Show@v, p, pts, p2d t Laser_Model_SP12_2.nb 3 4 Laser_Model_SP12_2.nb t

15 Laser_Model_SP12_2.nb 5 6 Laser_Model_SP12_2.nb a = 1; b = ; tf = 20; H*v=StreamPlot@8H-1L,-*-a*+b<,8,0,2<,8,0,2<, AesLabelØ8,<,StreamPointsØFine,VectorPointsØ10,VectorStleØRed, FrameLabelØ8,<,FrameStleØDirective@FontSizeØ28DD;*L v = StreamPlot@8 H - 1L, - * - a * + b<, 8, 0, 2<, 8, 0, 2<, StreamPoints Ø Fine, FrameLabel Ø 8"", ""<, FrameStle Ø Directive@Thick, FontSize Ø 28D, ImageSize Ø 8500, 500<, ImageMargins Ø 0D; Show@v, pts, pts2d s = NDSolve@8'@tD H@tD - 1L, '@td ã -@td - a + ã ã 0<, 8, <, 8t, 0, tf<d; s2 = NDSolve@8'@tD H@tD - 1L, '@td ã -@td - a + ã ã 1<, 8, <, 8t, 0, tf<d; s3 = NDSolve@8'@tD H@tD - 1L, '@td ã -@td - a + ã ã.2<, 8, <, 8t, 0, tf<d; p3 = Plot@8@tD ê. ê. s<, 8t, 0, tf<, AesLabel Ø 8"t"<, PlotRange Ø 880, tf<, 80, 2<<, AesStle Ø Directive@Thick, LargeD, PlotStle Ø 8Thick<, ImageSize Ø 8500, 500<, ImageMargins Ø 0, PlotLegend Ø 8"", ""<, LegendTetStle Ø FontSize Ø 28D p4 = Plot@8@tD ê. ê. s2<, 8t, 0, tf<, AesLabel Ø 8"t"<, PlotRange Ø 880, tf<, 80, 2<<, AesStle Ø Directive@Thick, LargeD, PlotStle Ø 8Thick<, ImageSize Ø 8500, 500<, ImageMargins Ø 0, PlotLegend Ø 8"", ""<, LegendTetStle Ø FontSize Ø 28D p = ê. s<, 8t, 0, tf<, PlotRange Ø All, PlotStle Ø 8Orange, Dashed, Thick<, AesLabel Ø 8"", ""<, AesStle Ø Directive@Thick, FontSize Ø 28DD; p2 = ê. s2<, 8t, 0, tf<, PlotRange Ø All, PlotStle Ø 8Orange, Dashed, Thick<, AesLabel Ø 8"", ""<, AesStle Ø Directive@Thick, FontSize Ø 28DD; p5 = ê. s3<, 8t, 0, tf<, PlotRange Ø All, PlotStle Ø 8Orange, Dashed, Thick<, AesLabel Ø 8"", ""<, AesStle Ø Directive@Thick, FontSize Ø 28DD; pts = Graphics@8PointSize@LargeD, Red, Point@880, b ê a<<d<d; pts2 = Graphics@8PointSize@LargeD, Green, Point@88b - a, 1<<D<D; Show@v, p, pts, p2, pts2, p5d Laser_Model_SP12_2.nb 7 8 Laser_Model_SP12_2.nb t

16 Laser_Model_SP12_2.nb 9 10 Laser_Model_SP12_2.nb t Laser_Model_SP12_2.nb Laser_Model_SP12_2.nb a1 = Plot@t, 8t, 0, 5<, AspectRatio Ø 1, AesLabel Ø 8"a", "b"<, Filling Ø Bottom, FillingStle Ø Lighter@PurpleD, AesStle Ø Directive@Thick, FontSize Ø 28D, ImageSize Ø 8500, 500<, ImageMargins Ø 0D; a2 = Graphics@Tet@Stle@"No Lasing\nLow\nPumpingêGain", FontSize Ø 28D, 83.5, <DD; a3 = Graphics@Tet@Stle@"Lasing\nHigh\nPumpingêGain", FontSize Ø 28D, 8, 3.5<DD; Show@a1, a2, a3d b Lasing High PumpingêGain No Lasing Low PumpingêGain a H* A case with lasing*l c = 2; r = Plot@c ê H + 1L -, 8, 0, c<, PlotStle Ø 8Thick<, AesLabel Ø 8"", OverDot@""D<, AesStle Ø Directive@Thick, FontSize Ø 28D, PlotRange Ø 880, 1<, 80, 0.2<<, Ticks Ø 880,, 1<, 80, 0.1, 0.2<<D; pts2 = Graphics@8PointSize@0.05D, Red, Point@80, 0<D<D; pts3 = Graphics@8PointSize@0.05D, Green, Point@8c - 1, 0<D<D; Show@r, pts2, pts3d Clear@, tfd tf = 20; sol = NDSolve@8'@tD ã ê H@tD + ã <,, 8t, 0, tf<d; Plot@@tD ê. sol, 8t, 0, tf<, AesStle Ø Directive@Thick, FontSize Ø 28D, AesLabel Ø 8"t", ""<, PlotStle Ø 8Thick<, PlotRange Ø 880, tf<, 80, 1<<, Ticks Ø 880, tf ê 2, tf<, 80,, 1<<D p = ParametricPlot@88@tD, '@td< ê. sol<, 8t, 0, tf<, PlotRange Ø All, PlotStle Ø 8Orange, Dashed, Thick<, AesLabel Ø 8"", OverDot@""D<, AesStle Ø Directive@Thick, FontSize Ø 28DD; Show@ r, pts2, p, pts3d t

17 Laser_Model_SP12_2.nb Laser_Model_SP12_2.nb H* A case with no lasing*l c = 0.9; r = Plot@c ê H + 1L -, 8, 0, c<, PlotStle Ø 8Thick<, AesLabel Ø 8"", OverDot@""D<, AesStle Ø Directive@Thick, FontSize Ø 28D, PlotRange Ø 880, 1<, 8-0.8, 0<<, Ticks Ø 880,, 1<, 8-.8, -0.4, 0<<D; r2 = Plot@c ê H + 1L -, 8, 0, c<, PlotStle Ø 8Thick<, AesLabel Ø 8"", OverDot@""D<, AesStle Ø Directive@Thick, FontSize Ø 28DD; pts2 = Graphics@8PointSize@0.05D, Red, Point@880, 0<, 8c - 1, 0<<D<D; Show@r, pts2d Clear@, tfd tf = 20; sol = NDSolve@8'@tD ã ê H@tD + ã 0.1<,, 8t, 0, tf<d; Plot@@tD ê. sol, 8t, 0, tf<, AesStle Ø Directive@Thick, FontSize Ø 28D, AesLabel Ø 8"t", ""<, PlotStle Ø 8Thick<, PlotRange Ø 880, tf<, 80, 1<<, Ticks Ø 880, tf ê 2, tf<, 80,, 1<<D p = ParametricPlot@88@tD, '@td< ê. sol<, 8t, 0, tf<, PlotRange Ø All, PlotStle Ø 8Orange, Dashed, Thick<, AesLabel Ø 8"", OverDot@""D<, AesStle Ø Directive@Thick, FontSize Ø 28DD; Show@ r, pts2, pd t