STEADY STATES ANALYSIS OF THE SOLID STATE LASER SYSTEMS

Transcription

1 Journal of Pure and Applied Mathematics: Advances and Applications Volume 0, Number, 03, Pages -68 STEADY STATES ANALYSIS OF THE SOLID STATE LASER SYSTEMS Department of Computer Science and Erlangen Graduate School in Advanced Optical Technologies (SAOT) Friedrich-Alexander University Erlangen-Nuernberg Cauerstr., D-9058 Erlangen Germany Abstract The constant output power of a solid state laser is desired in many applications. Therefore, the property of the steady states of a laser is crucial. We consider a three-dimensional (3D) solid state laser model with several Gaussian modes. Discretized version, scalar version, and single-mode version of this model are shown. Using the scalar system, we achieve some stability property of the steady states by linearization techniques. We not only use the discretized system in numerical simulations, but also it to prove the uniqueness of the positive steady state of the single-mode system. Based on the results of the single-mode system, we obtain some theorems concerning the existence, nonexistence, nonuniqueness of the positive steady states of the multimode system. A prior-estimate of the steady state population inversion density and an inequality of the steady state output power are also given. Finally, we perform numerical simulations for the 3D laser model to confirm the analysis and to explore some of the dynamic scenarios. 00 Mathematics Subject Classification: 35B3, 35B40, 35D05. Keywords and phrases: steady states, solid state laser, 3D laser model. Received February, Scientific Advances Publishers

2 . Introduction A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. There are many different types of laser active medium: solid, gas, liquid or semiconductor. Solid state lasers are those whose active medium is a glass or crystal to which a dopant such as neodymium, chromium, or erbium is added. As we all know that atoms or ions in a medium can exist only in certain energy states. The state of the lowest energy is called groundstate; all other states having more energy than the ground-state are called excited-states. Almost all atoms or ions remain in the ground-state unless some external stimulus (e.g., a pumping source such as another laser) is applied to it. That more atoms or ions are in the excited-state than in the ground-state is called population inversion. There are three atomic processes in lasers: absorption, spontaneous emission, and stimulated emission. Absorption means that an atom or ion takes up the energy of the pumping source and jumps from the low energy level to the high energy level. Spontaneous emission is that an atom or ion stays in the high energy level, then it moves from the high energy level to the low energy level and at the same time emits a photon. Stimulated emission denotes that an atom or ion lying in the high energy level interacts with an incoming photon and drops to the low energy level. In this process, a second photon is produced. Thus, the light is strengthened. A photon created in this manner has the same phase, frequency, polarization, and direction as the incident photon. This is in contrast to spontaneous emission, which occurs without regard to the incident photons. To create a laser, three components are necessary an active medium (also called gain medium), a resonator (also called cavity), and a pumping source. The resonator has an input mirror at one end and an output mirror at the other. The two mirrors cause light inside the resonator to reflect back and forth through the active medium. If a certain pumping threshold is achieved, then the three processes in the laser produce more and more photons with each pass (see Figure ).

3 STEADY STATES ANALYSIS OF THE SOLID STATE 3 Figure. An example of laser. Solid state lasers play an important role in the medical, scientific, military, and industrial fields. Since the birth of solid state laser [], various regimes have been studied in numerous papers [,, 4, 6, 7, 8, 9, ]. Recently, Wohlmuth et al. [5] considered several Gaussian modes in a resonator and assumed that each mode oscillates independently and the al photon number is the sum of each photon number: M Φ() t = Φ () t, where M i= i is the al mode number. () Figure. Laser resonator sketch.

4 4 Figure 3. An example of four-level gain medium ( e.g., Nd : YVO4 ). Assuming that = S D [ 0, L] Ω is the 3-D domain of a resonator with length L (see Figure ), Ω g is the 3D domain of the gain medium with length l g, they have shown a multimode laser system to describe the 4-level gain medium laser (see Figure 3): N ( t x) t M, σcair = N( t, x) ( Φi ( t) U g ( x) ) V i eff i= N( t, x) N + Rp( x) τ f N( t, x), N () dφi () t dt σc = V air eff Ω g U gi ( x) N( t, x) dx Φi (), t τc (3) where σ c, V, τ, N, τ > 0; R ( x) > 0, x Ω ;, air eff f c p g

5 STEADY STATES ANALYSIS OF THE SOLID STATE 5 i =,, M (M is the al mode number); x : ( x, y, z); N: population inversion density of the gain medium (i.e., population density of the gain medium in energy level of Figure 3); Φ i: photon number of the i-th mode in the cavity; N : doping concentration of the gain medium (i.e., the number of ions per unit volume of the gain medium); R p : pumping rate; U g i : the i-th normalized Gaussian mode in the gain medium, which + + U g i satisfies ( x, y, z) dxdy = ; σ : emission cross section of the gain medium; c air : speed of light in vacuum; τ f : fluorescence lifetime of the gain medium; τ c: photon resonator lifetime; [4] τ c = Lopt [ ( ) ] ; cairln r r los L optical path length; L = n l + l ; opt : opt n g : refractive index of the gain medium; g g air r : reflectivity of the input mirror to the generated laser photon; r : reflectivity of the output mirror to the generated laser photon; los : internal loss of the generated laser photon per pass in the cavity; l length of air; l = L ; air : air l g L: length of the cavity (see Figure ); V eff = n g l g + l air.

6 6 Explanation of each term in system () and (3) σc N V air eff M i= ( Φ U ): i g i the decreased ions number of the gain medium in energy level (see Figure 3) caused by stimulated emission in the unit time and unit volume; σcair Φ i U g ( x) N( t, x) dx: V the number of photon concerning the Ω i eff g i-th mode created in the unit time by the whole gain medium; N : the decreased ions number of the gain medium in energy level τ f (see Figure 3) caused by spontaneous emission in the unit time and unit volume; R p N N : N pumping; see Figure 3); pumping force (population inversion is caused by Φi τc : al rate at which the i-th mode photons leave the resonator, Φi τc = Φiln( r ) cair Φiln( r ) cair Φiln( los) + + L L L opt opt opt cair ; (4) Φiln( r ) c L opt air : rate at which the i-th mode photons leave the resonator through the input mirror; Φiln( r ) c L opt air : rate at which the i-th mode photons leave the resonator through the output mirror; Φiln( los) L mirror losses. opt cair : all other the i-th mode photons losses except

7 using STEADY STATES ANALYSIS OF THE SOLID STATE 7 From (4), the output power of the i-th mode P i ( t) can be calculated by Φ i as follows: ln( r ) cair Pi () t = hω Φi (), t (5) L opt where h is Planck s constant, 34 h = J s; ω is the frequency of generated photons; and The al output power is given by h ω is the energy of one photon. P al M () t = P (). t (6) i= In this paper, we consider the 3-D multimode laser system () and (3). The main goal of this paper is to study the property of steady states of the laser systems. The outline of this paper is as follows. In Section, we show a discretized version of system () and (3), which will not only be used in numerical simulations, but also be used to prove the uniqueness of positive steady state of the single-mode system. In Section 3, we show the scalar version of the laser model and derive some stability results. In Section 4, the uniqueness of positive steady state of the single-mode system will be proved. In Section 5, we achieve some theorems concerning the existence, nonexistence, nonuniqueness of the positive steady states of the multimode system. Then, the prior-estimate of the steady state population inversion density and an inequality of the steady state output power are also given. In Section 6, we perform numerical simulations for the 3-D laser model to confirm the analysis and to explore some of the dynamic scenarios. Throughout this paper, we use the following pumping rate: i P p α r Rp ( x) = Rp ( r, θ, z) = ηrηt ( )( ) exp( ) exp( αz), hν p πwp wp (7)

8 8 where η r : radiative efficiency; η t : transfer efficiency; P p: pump power; ν p : frequency of the pumping beam; α : absorption coefficient ( α = σn ); w p : pump radius.. Numerical Discretization In order to realize the 3D numerical simulations of the system () and (3), we apply a finite volume discretization to the gain medium. Let us denote c g I the finite volume discretization cells of the gain medium and vol ( c gi ) is the volume of the cell. Let N I () t cg I N( t, x) d( x) vol( c ) gi, RpI cg I Rp ( x) d( x) vol( c ) gi, U gii cg I U gi ( x) d( x) vol( c ) gi, then c g I N( t, x) R ( x) d( x) p vol( c ) gi N I () t RpI, c g I N( t, x) U ( x) d( x) gi vol( c ) gi N I gii () t U. The discretized multimode version of the laser system is dn I () t dt M σcair = N I () t ( Φi ( t) U g I ) V i eff i= N I () t N N I () t + RpI, τ f N (8)

9 STEADY STATES ANALYSIS OF THE SOLID STATE 9 dφi () t dt = σcair V eff I vol( cg ) U g I N I () t Φi (), t I i τc (9) where i =,,, M. Remark. () This discretized system looks like the corresponding continuous system. () When we use this set of notation, the physical units of variables do not change (e.g., the unit of N( t, x ) is density; the unit of N I () t is also density, N I () t is the average density in a cell). Let Num g be the al number of cells of the gain medium discretization, then the above system consists of Num g + M ordinary differential equations. 3. Steady States Analysis of the Scalar System The scalar version of the laser system is dn() t dt = σcairlg Lopt N() t Φ() t N() t N N() t + Rp, τ f N (0) dφ() t dt σcairlg = Lopt N() t Φ(), t τc () where N: population inversion density of the gain medium; Φ : al photon number in the cavity; area g : cross section area of the gain medium. We use U g =, s.t. =. area U g dxdy area g lg =. g S g We suppose that

10 0 Obviously, system (0) and () has only two steady states (, Φ Rp N ) =, 0, and R p + τ f N ( L, ), ( ) opt Lopt N Φ =. + Rp τ σ σ c cairlg cairlg τ f N N If and only if + Rp( ) > 0, τ f N N then 0. Φ > Let Rp critical. () τ f ( ) N N In the case of N < N ( i.e., R > 0), if and only if R p > R pcritical, p critical then 0. Φ > Now we show theorems about the two steady states: Theorem 3. (Property of the steady states). In the case of N < N ( i. e., R > 0), we have p critical 0 = (a) R < R < N < N < N and Φ < Φ 0; p pcritical 0 (b) R > R < N < N < N and 0 = Φ < Φ ; p pcritical 0 (c) R = R < N = N < N and 0 = Φ = Φ. p pcritical In the case of N > N, we have 0 < N < N and Φ < Φ = 0, R > p 0.

11 STEADY STATES ANALYSIS OF THE SOLID STATE Proof. In the case of N < N, of (b) and (c) is similar to that of (a). we only give the proof of (a). Proof Rp Rp < Rp R < < N N < N. critical p ( ) R τ p f + N N τ f N R p < R pcritical L σc opt l air g τ f + R p ( ) < 0 Φ < 0. N N It is obvious that N < N. In the case of N > N, the conclusion holds evidently. Theorem 3. (Stability of the steady states). In the case of N < N ( i. e., R 0), we have p critical > (a) if R p < R pcritical, then ( N, Φ ) is a stable nodal point and ( N ) is a saddle point;, Φ (b) if R p > R pcritical, then ( N, Φ ) is a saddle point; (c) if point; (d) if point, R > and F ( ) 0, then ( N ) is a stable nodal p R pcritical p R pcritical R p, Φ R > and F ( ) < 0, then ( N ) is a stable focal R p, Φ where Rp 4 ( ) ( ) ( ). F Rp + R p N τc τ f N N (3) In the case of N > N, we have ( N ) is a stable nodal point and ( N ) is a saddle point, R > 0., Φ p, Φ

12 Proof. Let σc airlg N N N NΦ + R f p L opt τ f f = N, σ (4) f cairlg N Φ Lopt τc then ( f, ) (, ) f Df N Φ ( N, Φ) = σcairlg Φ Lopt σcairlg Φ Lopt τ f Rp N σcairlg Lopt N σcairlg Lopt N. τc (5) Step. Proof of (a) and (b). Rp σcairlg N (, ) τ f Lopt Df N Φ = N. σcairlg 0 N Lopt τc The eigenvalues of ( Rp Df N, Φ ) are λ = < 0 τ f N λ σc = L l air g opt N τ c. (6) and If R p < R pcritical, then λ < 0. Therefore, ( N, Φ ) is a stable nodal point. If R p > R pcritical, then λ > 0. Therefore, ( N, Φ ) is a saddle point. σcairlg Rp σcairlg Φ N (, Φ ) = L opt τ f Lopt Df N N. σcairlg σ cairlg Φ N Lopt Lopt τc (7)

13 STEADY STATES ANALYSIS OF THE SOLID STATE 3 The eigenvalue equation of Df ( N, Φ ) is ˆ λ + p λˆ + q = 0. (8) The eigenvalues of Df ( N, Φ ) are ˆλ and λ. ˆ p = σc ˆ ˆ airlg Rp σcairlg Rp λ + λ = Φ + N = < L L opt τ f N opt τc N 0. (9) q = λˆ λˆ = det( Df ( N, Φ )) = + Rp( ). τ c τ f N N (0) If R p < R pcritical, then ˆ ˆ λ > 0, λ < 0. Therefore, ( N, Φ ) is a saddle point. Thus (a) and (b) of the theorem have been proved. Step. Proof of (c) and (d). Let Rp 4 F ( Rp ) p 4q = ( ) + R ( ). p N τc τ f N N If R > and F ( ) 0, then λ ˆ < 0, λˆ 0. Therefore, p R pcritical R p < ( N Φ ) is a stable nodal point., If R > and F ( ) < 0, then λˆ = a + bi, λˆ = a bi, a 0. p R pcritical R p < Therefore, ( N Φ ) is a stable focal point., Step 3. Proof of the conclusion in the case of N > N. λ The eigenvalues of ( Rp Df N, Φ ) are λ = < 0 τ f N σcairlg = Lopt N. τc and

14 4 N L σc τc opt air g > N > N N < τcσcairlg Lopt l 0. Noticing that N < N, we have λ = σcairlg Lopt N < σcairlg Lopt N < τ c 0., Therefore, ( N Φ ) is a stable nodal point, R > 0. p The eigenvalues of Df ( N, Φ ) are ˆλ and λ ˆ. q = λˆ λˆ = det( Df ( N, Φ )) = + Rp( ). τ c τ f N N N q < 0, R > 0 λˆ 0, ˆ > λ > N p < 0. Therefore, ( N Φ ) is a saddle point, R > 0., p (a)

15 STEADY STATES ANALYSIS OF THE SOLID STATE 5 (a) (a3)

16 6 (b) (b)

17 STEADY STATES ANALYSIS OF THE SOLID STATE 7 (b3) (c)

18 8 (c) (c3) Figure 4. Three scenarios of laser: (a) Case (a) of Theorem 3. ( N Φ ) is a stable nodal point; (b) Case (c) of Theorem 3. ( N Φ ) is a stable nodal point; and (c) Case (d) of Theorem 3. ( N Φ ) is a stable focal point.,,,

19 STEADY STATES ANALYSIS OF THE SOLID STATE 9 Remark. Case (d) is when R ( R R ) in Figure 5. p p, p Figure 5. Case (d) is when R ( R, R ). p p p Remark 3. From (), we know that Φ reaches its minimum or maximum value, when N = N. If > N dφ( t) N, then > 0 ( i.e., Φ ). dt If < N dφ() t N, then < 0 ( i.e., Φ ) (see Figure 6). dt In Case (c) of Theorem 3., there exists only finite number of t, s.t. N ( t ) = N ; however in Case (d) of Theorem 3., there exists infinite number of t, s.t. N ( t ) = N.

20 0 (a) (b)

21 STEADY STATES ANALYSIS OF THE SOLID STATE (c) Figure 6. Relationship between N ( t) and Φ ( t): (a) Case (a) of Theorem 3.; (b) Case (c) of Theorem 3.; and (c) Case (d) of Theorem Steady States Analysis of the Single-mode System The single-mode version of the laser system is N ( t x) t σcair = N( t, x) Φ() t U g ( x) V, eff N( t, x) N + Rp( x) τ f N( t, x), N () dφ() t dt = σc V air eff Ω g U g ( x) N( t, x) dx Φ(). t τc () The discretized form of () and () is (see Section ) dn I () t dt = N I σcair () t Φ() t U V eff gi N I () t N N I () t + RpI, τ f N (3)

22 dφ() t dt = σc V air eff I vol( cg ) U g N I () t Φ(), t I I τc (4) where I =,,, m. The above system consists of m + ordinary differential equations. Before we discuss the property of the steady states, let us firstly see a lemma. Let ai fi ( x) =, (5) b x + c F m i m i ( x) f ( x), (6) i i= where cm cm c ai, bi, ci > 0, i =,, m and < < < < 0. b b b m m Lemma 4. (Property of a class of functions). m ai (a) F m( 0) = > 0. c i= i ~ (b) F m ( x) = C has m roots, where C ~ is a positive constant. ~ (c) If F m ( 0) C, then there exists no x ˆ > 0, s.t. F ( xˆ ) C ~ m =. ~ (d) If F m ( 0) > C, then there exists a unique x ˆ > 0, s.t. F ( xˆ ) C ~ m =. Proof. Obviously, we have F m m a ( 0) = i > 0, c i= i lim ( x) = 0+, F m x +

23 STEADY STATES ANALYSIS OF THE SOLID STATE 3 lim F m ( x) = 0, x lim ( x) = +, F c m x i + bi lim F ( x) =, c m x i bi m a ibi F m( x) = ( b x + c ) i= i i < 0, m aibi ( bi x + ci ) Fm ( x) =, 4 ( b x + c ) i= i i cm Fm ( x) < 0, Fm ( x) < 0 when x <, b c F ( ) 0, ( ) 0 when m x > Fm x > x >. b Thus, we obtain the pictures of F ( x) F ( ), and F 3 ( x) (see Figure 7)., x m The conclusion of this lemma is obvious.

24 4 ~ ( a) F ( 0) < C ~ ( a) F ( 0) > C

25 STEADY STATES ANALYSIS OF THE SOLID STATE 5 ~ ( b) F ( 0) < C ~ ( b) F ( 0) > C

26 6 ~ ( c) F3 ( 0) < C ~ ( c) F3 ( 0) > C Figure 7. F ( x) F ( ), and F ( ): (a) F ( ); (b) F ( ); and (c) F ( )., x 3 x x x 3 x

27 STEADY STATES ANALYSIS OF THE SOLID STATE 7 ci c j Remark 4. If i < j m, s.t. =, then the roots of b b ~ F m ( x) = C are less than m. But (a), (c), and (d) of the lemma still hold. Before we give the theorem about system () and (), let us see a theorem about the discretized single-mode laser system (3) and (4). Let ( Φ N, ) be the steady states of system (3) and (4). I Obviously, from (3), we have i j N I = σcair Veff U gi RpI Φ + τ f + RpI N. (7) Let fi A ( Φ ) vol( c ) I g U g N I, A,, > 0, I BI CI (8) I I B Φ + C (4) leads to Therefore, we define I m I i= I F ( Φ ) f ( Φ ), (9) m Fm Veff ( Φ ) =. (30) σc τ ~ Veff C. σcairτ (3) c For mathematical simplicity, we suppose that air c C B m m < C B m m < < C B < 0.

28 8 Theorem 4. (Steady states of the discretized single-mode laser system). m m A R (a) ( ) I pi Fm 0 = = vol( cg ) U > 0. I g C I I = I I = RpI + τ f N (b) System (3) and (4) has m + steady states. ~ (c) If F m ( 0) C, then there exists no Φ > 0, s.t. ( N, Φ ) is a steady state (i.e., all Φ 0 ). ~ (d) If F m ( 0) > C, then there exists a unique Φ > 0, s.t. ( Φ N, ) is a steady state (i.e., all other Φ 0 ). Proof. This theorem can be deduced from Lemma 4.. I I Notice that ( Φ RpI N I, ) =, 0 RpI + τ f N system (3) and (4), (b) holds. is always a steady state of Remark 5. In fact, when m =, it is just the scalar system. Then A Rp Rp F ( 0) = = U g ( x) dx = lg, C Ω Rp R g p + + τ f N τ f N ~ Rp Lopt F ( 0) > C > N > N Rp > Rp. critical Rp τcσcairlg + τ f N The conclusion is the same as in the section Steady States Analysis of the Scalar System. Now, we consider the steady states of system () and ().

29 STEADY STATES ANALYSIS OF THE SOLID STATE 9 Let ( N ( x ), Φ ) be the steady states of system () and (). Obviously, from (), we have N ( x) = σc V air eff U g ( x) R ( x) p Φ + τ f + R ( x). p N (3) Let F ( Φ ) U ( x) N ( x) dx, (33) Ωg g () leads to Veff F ( Φ ) =. (34) σc τ air c Thus, we still define ~ Veff C. σc τ (35) air Theorem 4. (Steady states of the continuous single-mode laser system). Rp( x) (a) F ( 0) = U g ( x) dx. Ω Rp( x) g + τ f N (b) System () and () have infinite steady states. c ~ (c) If F ( 0) C, then there exists no Φ > 0, s.t. ( N ( x ), Φ ) is a steady state (i.e., all Φ 0 ). ~ (d) If F ( 0) > C, then there exists a unique Φ > 0, s.t. ( N ( x ), Φ ) is a steady state (i.e., all other Φ 0 ). Proof. Suppose that the gain medium is divided into infinite cells, whose diameters are infinitely small, then lim F m ( Φ ) = F ( Φ ), m + (36)

30 30 lim F m ( 0) = F ( 0). m + This theorem is a corollary of Theorem 4.. (37) Remark 6. From (a) of Theorem 4., we know that F ( 0) = U g ( x ) Ω g τ f + R ( x) p N dx. (38) Notice Rp ( x) + τ R ( x) f p N, x Ω g, we have () for fixed pump radius, increasing the value of pump power ( P p ) ~ can make F ( 0 ) > C happen ( i.e., Φ > 0); () considering the mathematical expression of g U for the first mode TEM 00 and Equation (38), for fixed P p, small pump radius is ~ more likely to make F ( 0 ) > C happen ( i.e., Φ > 0). 5. Steady States Analysis of the Multimode System Let ( N ( x ), ) Φ i be any steady state of system () and (3). If Φ = 0, i =,, M, then N ( x ) is unique. So system () and (3) i has a trivial steady state N ( x) = Rp + τ f ( x) R ( x), p N (39) Φi = 0, i =,, M. (40)

31 STEADY STATES ANALYSIS OF THE SOLID STATE 3 Theorem 5. (Existence of the nontrivial steady state). System () and (3) has infinite steady states. Proof. Considering the conclusion of Theorem 4.(b), this theorem holds obviously. In the following theorems, we are interested in the case of Φi 0, i =,, M M Φi i= > 0. Theorem 5. (Existence of the positive steady state). If j {,,, M} s.t. ( x) U g j Ω g ( x) dx R ( x) Rp p + τ f N then there exists a steady state, which satisfies the condition ~ ~ Veff > C, C σcairτc, Φi 0, i =,, M M Φi i= > 0. Proof. Considering the conclusion of Theorem 4.(d), we know that the following steady state exists: Φ j > 0, Φk = 0, k j. Theorem 5.3 (Nonuniqueness of the positive steady state, M > ). (a) If U ( x) Ωg g j ~ ~ Veff > C, C, σcairτc Rp ( x) ~ dx > C Rp( x) + τ f N j < j M, Ωg U g j ( x) Rp( x) dx Rp ( x) + τ f N then the steady states, which satisfy the condition 0, Φ i i M =,, M Φ > 0 i= i are not unique.

32 3 i 0 j j M (b) If Φ, i =,, M Φ > 0, Φ > 0, j < j, then the steady states, which satisfy the condition 0, M i =,, M Φ > 0 i= i are not unique. Proof. Step. Proof of (a). Considering the conclusion of Theorem 4.(d), we know that there exists at least the following two steady states: Step. Proof of (b). From (3), we know Φ j > 0, Φk = 0, k j; Φ j > 0, Φk = 0, k j. Φ i Φ j > 0 U g Ω j g Ωg Ω g U g U Φ j g j ~ ( x) N ( x) dx = C ( x) ( x) σcair Veff Rp ( x) M ( Φi U g ( x) ) + i= j τ f ( x) dx R ( x) Rp p + τ f N j > 0 U g j Ω g ( x) > C ~. ( x) dx R ( x) Rp p + τ f N dx Rp( x) + N > C ~. ~ = C (4) (4) (4), (4), and conclusion of (a) lead to conclusion of (b).

33 STEADY STATES ANALYSIS OF THE SOLID STATE 33 Theorem 5.4 (Condition on which the positive steady state does not Rp( x) exist). If U ( ) ~ g i x dx < C, i =,,, M, Ω Rp( x) g + τ f N then the steady state, which satisfies the condition Φi 0, i =,, M M Φ i= i > 0 does not exist. Proof. Suppose Φ j > 0, then Ωg U g j ( x) σcair Veff Rp ( x) M ( Φi U g ( x) ) + i= j τ f + dx Rp( x) N = C ~. Therefore, we have Ω g U g j ( x) ( x) dx R ( x) Rp p + τ f N which is a contradiction to U ( x) Ω g Theorem 5.5 (Prior-estimate of N ( x )). g i > C ~, Rp( x) dx < C ~, i =,,, M. Rp( x) + τ f N In the case of Φi 0, i =,, M Φ > 0, i= i we have M 0 < min x Ω g Veff N ( x) < < max N ( x) < N. σcairτclg x Ωg (43)

34 34 Proof. Step. Proof of 0 < min N ( x). x Ω g If xˆ Ω, s.t. N ( ˆ) 0, then considering Rp ( xˆ ) > 0 and g x Φi 0, i =,, M, we have M σc ( ˆ air N x) ( Φi U g ( xˆ ) ) V i eff i= ( ˆ N x) N N ( xˆ ) + Rp ( xˆ ) τ f N > 0, which is a contradiction to the steady state. Veff Step. Proof of min N ( x) < < max N ( x). x Ω σc τ l x Ω From (3), we have g air c g g Veff σcairτc = > Ω g min x Ω U g g i ( x) N ( x) dx N ( x) U g ( x) dx = min N ( x) lg, i Ω x Ωg g (44) Veff σcairτc = < Ω g U g i ( x) N ( x) dx max N ( x) U g ( x) dx = max N ( x) lg. x Ω i Ω x Ω g g g Step 3. Proof of max N ( x) < N. x Ωg (45) Φi If xˆ Ω, s.t. N ( xˆ ) N, g 0, i =,, M, we have then considering R p ( xˆ ) > 0 and

35 STEADY STATES ANALYSIS OF THE SOLID STATE 35 M σc ( ˆ air N x) ( Φi U g ( xˆ ) ) V i eff i= ( ˆ N x) N N ( xˆ ) + Rp( xˆ ) τ f N < 0, which is a contradiction to the steady state. Theorem 5.6 (CW output power inequality). CW Pout αl < η ( g rηtpp e ), (46) where lg 0 CW Pout denotes the positive steady state output power, i.e., Proof. Step. Preparation π 0 ln( r ) cair CW Pout = hω Φ. i (47) L opt M i= 0 + P l p g αz Rp( r, θ, z) rdrdθdz = ηrηt ( ) α e dz 0 hν 0 p = Pp αl η ( )( g rηt e ), hν p (48) Lopt τ c = [ ( ) ], (49) cair ln r r los ln( r ) cair τ c <, (50) L opt ω ν p <, (5) Ωg lg π + Rp( x) d( x) < Rp ( r, θ, z) rdrdθdz (5) From (3), we have Ω U g i g Veff ( x) N ( x) dx =, if Φi > 0. σc τ air c (53)

36 36 Step. Let N ( t, x ) = 0, t then integrate () on both sides over Ω g. Notice (53), we have τ c M Φi = Ω i= 0 g N ( x) N + Rp ( x) τ f N ( x) d( x). N (54) Therefore, τc M Φ i i= 0 < R p Ωg ( x) d( x). (55) (47), (48), (50), (5), (5), (55) lead to (46). Remark 7. () Theorem 5.: Considering the expression R p ( x) 00 in (7), TEM is most likely to satisfy the condition ( x) Rp ( x) dx Rp( x) + τ f N > C ~. U Ω g j g () Theorem 5.4: In fact, it is the condition on which (39) and (40) is the stable steady state (i.e., in this case, laser can not be excited). (3) Theorem 5.6: In fact, the gain medium. e αl g describes the light absorption of 6. Numerical Results In this section, we turn to numerics to confirm the analysis and to explore some of the dynamic scenarios. An optimized-bdfs-based method [5] is applied to the discretized system (8) and (9). In these simulations, Nd : YVO4 works as the 4-level gain medium.

37 STEADY STATES ANALYSIS OF THE SOLID STATE 37 Simulation and Simulation share the same parameters except mode number M. Simulation, Simulation 3, and Simulation 4 share the same parameters except pump radius. In Simulation 3 and Simulation 4, the pump radius is smaller than that in Simulation, therefore the output power of mode TEM 0, mode TEM 0, and mode TEM approaches zero quickly. However, the dynamics in these two simulations are different. In Simulation 5, pump power ( P p ) is not large enough, the al output power approaches zero quickly. Simulation. Pump radius = 5mm, mode number M = 4 (a)

38 38 (b) (b)

39 STEADY STATES ANALYSIS OF THE SOLID STATE 39 (b) (b3)

40 40 (b4) (c)

41 STEADY STATES ANALYSIS OF THE SOLID STATE 4 (c) (a)

42 4 (al) (a)

43 STEADY STATES ANALYSIS OF THE SOLID STATE 43 (a) (a3)

44 44 (a3) (a4)

45 STEADY STATES ANALYSIS OF THE SOLID STATE 45 (a4) (a5)

46 46 (a5) (a6)

47 STEADY STATES ANALYSIS OF THE SOLID STATE 47 (a6) Figure 8. Numerical results of the 3-D model (pump radius = 5mm, M = 4 ): (a) Total population inversion number N I vol( c ) of Nd : YVO 4 ; (b) al output laser power; (b) TEM 00 output laser power; (b) TEM 0 output laser power; (b3) TEM 0 output laser power; (b4) TEM output laser power; (c) pumping rate R ; ( c) clip of (c). From (a) to (a6): Population inversion density N at different time point. From (a) to (a6) : Clip of (a) to (a6): (a) time = 5µ s; (a) time = 5µ s; (a3) time = 0µ s. (a4) time = 40µ s; (a5) time = 00µ s; (a6) time = 00µ s. p I g I

48 48 Simulation. Pump radius = 5mm, mode number M = (the only mode is TEM 00 ) (a) (b)

49 STEADY STATES ANALYSIS OF THE SOLID STATE 49 (c) (d) Figure 9. Numerical results of the 3-D model (pump radius = 5mm, M = ): (a) Total population inversion number N I vol( c ) of Nd : I g I

50 50 YVO 4 ; (b) TEM 00 output laser power; (c) pumping rate R p; and (d) steady state of the population inversion density N. There is only one mode used in Simulation, therefore the result of al output laser power is much less precise than that in Simulation. The multimode simulation is much more capable of reflecting laser behaviours. Simulation 3. Pump radius = mm, mode number M = 4 (a)

51 STEADY STATES ANALYSIS OF THE SOLID STATE 5 (b) (b)

52 5 (b) (b3)

53 STEADY STATES ANALYSIS OF THE SOLID STATE 53 (b4) (c)

54 54 (d) Figure 0. Numerical results of the 3-D model (pump radius = mm, M = 4 ): (a) Total population inversion number N I vol( c ) of Nd : YVO 4 ; (b) al output laser power; (b) TEM 00 output laser power; (b) TEM 0 output laser power; (b3) TEM 0 output laser power; (b4) TEM output laser power; (c) pumping rate R p; and (d) steady state of the population inversion density N. When we use pump radius = mm, the output power of mode TEM 0, mode TEM 0, and mode TEM approaches zero quickly. I g I

55 STEADY STATES ANALYSIS OF THE SOLID STATE 55 Simulation 4. Pump radius = 0.5mm, mode number M = 4 (a) (b)

56 56 (b) (b)

57 STEADY STATES ANALYSIS OF THE SOLID STATE 57 (b3) (b4)

58 58 (c) (d) Figure. Numerical results of the 3-D model (pump radius = 0.5mm, M = 4 ): (a) Total population inversion number N I vol( c ) of Nd : I g I

59 STEADY STATES ANALYSIS OF THE SOLID STATE 59 YVO 4 ; (b) al output laser power; (b) TEM 00 output laser power; (b) TEM 0 output laser power; (b3) TEM 0 output laser power; (b4) TEM output laser power; (c) pumping rate R ; and (d) steady state of the population inversion density N. p When we use pump radius = 0.5mm, the output power of mode TEM 0, mode TEM 0, and mode TEM approaches zero quickly. However, mode TEM decreases to zero directly. Simulation 5. Pump power not large enough (a)

60 60 (b) (b)

61 STEADY STATES ANALYSIS OF THE SOLID STATE 6 (b) (b3)

62 6 (b4) (c)

63 STEADY STATES ANALYSIS OF THE SOLID STATE 63 (d) Figure. Numerical results of the 3-D model (pump power is not large enough, M = 4 ): (a) Total population inversion number N vol( c ) of Nd : YVO4 ; (b) al output laser power; (b) TEM 00 output laser power; (b) TEM 0 output laser power; (b3) TEM 0 output laser power; (b4) TEM output laser power; (c) pumping rate R p; and (d) steady state of the population inversion density N. I I g I R p ( x) is the driving force of the system. If the driving force is not strong enough, then Φ i will approach zero. Therefore, if the pump power ( P p ) is not large enough, then the al output power approaches zero quickly. Now, we show a table for these simulations:

64 64 Table. Integral U ( x) ~ ( C =.00e + 3) Ω g g mn Rp ( x) dx Rp ( x) + τ f N in simulations Simulation m = n = 0 m = 0, n = m =, n = 0 m = n = ~ ~ ~ ~.e + 5 > C 9.6e + 4 > C 9.6e + 4 > C 7.6e + 4 > C ~.e + 5 > C 3 ~ ~ ~ 3.63e + 5 > C.38e + 5 > C.38e + 5 > C > C ~ 4 ~ ~ ~ ~ 7.83e + 5 > C 7.44e + 3 > C 7.44e + 3 > C.e + 3 < C 5 ~ ~ ~ ~.e + 3 < C 9.6e + < C 9.6e + < C 7.6e + < C From the above table, we can see that () Simulation 5 has validated our Theorem 5.4. () In both Simulation and Simulation 3, all four integrals are ~ larger than C, but the stable steady state is different from each other = (3) In both Simulation 3 and Simulation 4, Φ >, Φ = Φ = Φ 0, but the process is different because of the different sign concerning. TEM Acknowledgement The research is funded by Erlangen Graduate School in Advanced Optical Technologies (SAOT) by the German Research Foundation (DFG) in the framework of the German excellence initiative and China Scholarship Council (CSC).

65 STEADY STATES ANALYSIS OF THE SOLID STATE 65 References [] R. L. Byer, Diode laser-pumped solid-state lasers, Science 39(484) (988), [] W. A. Clarkson, Thermal effects and their mitigation in end-pumped solid-state lasers, J. Phys. D Appl. Phys. 34 (00), [3] C. C. Davis, Lasers and Electro-Optics, Cambridge University Press, 000. [4] T. Y. Fan and R. L. Byer, Diode laser-pumped solid-state lasers, IEEE J. Quantum. Elect. 4 (988), [5] F. Feng and C. Pflaum, Efficient numerical methods for initial-value solid state laser problems, PAMM Proc. Appl. Math. Mech. (0), [6] A. Giesen, H. Huegel, A. Voss, K. Wittig, U. Brauch and H. Opower, Scalable concept for diode-pumped high-power solid-state lasers, Appl. Phys. B 58 (994), [7] G. Huber, C. Kraenkel and K. Petermann, Solid-state lasers status and future, J. Opt. Soc. Am. B 7() (00), B93-B05. [8] M. E. Innocenzi, H. T. Yura, C. L. Fincher and R. A. Fields, Thermal modelling of continouswave end-pumped solid-state lasers, Appl. Phys. Lett. 56 (990), [9] F. X. Kaertner, L. R. Brovelli, D. Kopf, M. Kamp, I. Calasso and U. Keller, Control of solid-state laser dynamics by semi-conductor devices, Opt. Eng. 34 (995), [0] W. Koechner, Solid-State Laser Engineering, Springer, 006. [] W. F. Krupke, Ytterbium solid-state lasers: The first decade, IEEE. Sel. Top. Quantum. Elect. 6 (000), [] T. H. Maiman, Stimulated optical radiation in ruby, Nature 87 (960), [3] K. Shimoda, Introduction to Laser Physics, Springer, 986. [4] O. Svelto, Principles of Lasers, Springer, 00. [5] M. Wohlmuth, C. Pflaum, K. Altmann, M. Paster and C. Hahn, Dynamic multimode analysis of q-switched solid-state laser cavities, Opt. Express. 7(0) (009), [6] M. Young, Optics and Lasers, Springer, 000.

66 66 Appendix A: Gaussian Modes used in the Simulations Gaussian modes are solutions to (A.)- the paraxial form of Helmholtz equation: U x + U y U ik z = 0. (A.) Gaussian modes represent the complex amplitude of the beam s electric field [3]. In our simulations, we use Hermite-Gaussian modes, which are typically designated as TEMmn, where m and n are the polynomial indices in the x and y directions (see Figure A. and Figure A.).

67 STEADY STATES ANALYSIS OF THE SOLID STATE 67 Figure A.. Representation of U with Hermite-Gaussian modes (for fixed z). Figure A.. Examples of Hermite-Gaussian modes. Hermite-Gaussian modes are [3]:

68 68 U ( x, y, z) w0 H w( z) ( x ( ) ) ( y Hn w x w( z) ) m, n = m where ik exp ( (,, )) ( )( ( ) ( ) ) i kz Φ m n z x + y +, w z R z H m ( X ) is order m Hermite polynomial. H m Then, we have [ m ] m m X d X ( ) m! ( ) ( )! ( )! ( ) m k X = e e = X m dx k m k. (A.) H ( X ), 0 = H ( X ) =, X H ( X ) = 4X, H ( X ) = 8X 3, 3 X k= 0 4 H ( X ) = 6X 48X, k U m, n ( x, y, z) = w0 H w ( z) m ( x ( ) ) ( y ( ) ) ( ( x + y ) Hn exp ). w z w z w ( z) (A.3) It can be proved that + + U ( ) ˆ m n x, y, z dxdy = C( m, n), m, n = 0,,,. (A.4), Integrals (A.4) are independent of z. g