Computer Modelling and Numerical Simulation of the Solid State Diode Pumped Nd 3+ :YAG Laser with Intracavity Saturable Absorber

Copyright 2009 by YASHKIR CONSULTING LTD Computer Modelling and Numerical Simulation of the Solid State Diode Pumped Nd 3+ :YAG Laser with Intracavity Saturable Absorber Yuri Yashkir 1 Introduction The model of a laser is based on the following assumptions Laser consists of a laser output coupler (reflection R p, with the spherical surface radius of curvature R 0 ), an active crystal (rod 1 ) a c -long (refractive index n c 2 ), a passive Q-switching element (a saturable absorber crystal 3 with initial transparency T q, length a Cr ), and a HR back mirror (curvature radius R m ) The total length of the laser cavity is a The laser rod is pumped with a set of laser diodes Laser pump maximal power is P D max Laser active ion lifetime is n and the saturable absorber ion lifetime is q In this model it is assumed that all variables depend on r,t only ( r x, y,0, axis z is the cavity axis) Optical pump is given by D r (the two dimensional pump distribution) and t (pump pulse temporal shape) Methodology of laser intensity modeling and multi-dimentional light/matter interaction is based on previously published papers [ 1 7] 2 Basic model equations The laser cavity round trip time is denoted as T p, and the reflection coefficient of the output coupler (mirror) Iis R p Photon losses in the laser cavity (not related to the saturable absorber) are described by the photon lifetime constant ph It can be identified as follows During k round trips (time t=k T p ), the initial intracavity electromagnetic emission energy is reduced by a factor of R p t R p k From the obvious equation t T p =e ph we obtain p = T ph where parameter p = ln R p can be thought of as an average loss per p round-trip (it may be different for different laser cavity modes) This approximation is valid, obviously, if the laser power P t changes slowly compared to the cavity round trip time: max t d dt ln P t 1 T p For a laser pulse (pulse width p ) this criterion can be reduced to p T 1 Nd 3 :YAG, n =230 s 2 n c =1818 3 Cr 4+ :YAG, q =85 s 1

The optical pumping of the laser rod can be estimated as follows Diode pump energy at [ r,t ] in [d V, d t ] is given by the following expression: d 2 E D =P D t D r d V d t, where P D t is total pump power, D r is the spatial distribution defined by the configuration setup of pumping sources ( D r d V =1 ) The fraction u of the pump energy absorbed in the laser crystal is assumed known Then the number of photons absorbed dn D (same as number dn n of active ions excited) is equal to: dn n = u de h D D The relative population factor change is then equal to dn r, t = d N n = u de D dn 0 h D c 0 dv =u P t r dv dt D D h D c 0 dv where c 0 is the volume concentration of active ions Taking into account the spontaneous decay of excited ions (time constant n for relative population factor dn r, t = u n r, t P dt h D c D t D r 1 n r, t 0 n = u h D c 0 P D t D r dt, ) we arrive to the rate equation An estimation of the factor = u for a 1%-doped h D c N d 3 + :Y AG crystal ( c 0 =1 3 6 1 0 2 0 c m - 3 ) 0 pumped at D =800 nm, assuming u=1, is as follows: * =3 1 0 8 m 3 J 0 0 3 m m3 K W s For simplicity assume that the pump distribution function is uniform along laser crystal ( 0 z a, a=1 cm ) and it has the Gaussian shape in plane xy with the Gaussian diameter of w D = 1 mm : D x, y,z = 2 2 aw D e 4 x y 2 2 / w D Maximal value of D x, y,z is D 0,0, z = max D = 2 aw D 011 mm 3 Solution of the equation for n r,t asymptotically tends to 1 n = 1 1 P D max D n Assuming pump power P D =1 K W and D =230 µ we obtain n 043 Dependence of n on the pump power has a typical shape: 2

1 Inversion population factor 08 n(0,0,z,inf) 06 04 02 0 0 05 1 15 2 25 3 Pump power, KW Figure 1 It is important to note that laser crystal saturation effect due to optical pumping is not very significant at pump power below 1KW Stimulated emission of the excited ions is based on the probability of an ion to emit a photon during time interval d t : db s = s I p h dt, where s is the emission cross-section of an ion, I p is the local light intensity, is the laser photon frequency Laser light intracavity consists of k m a x modes ( k N p photons in each k t h mode) Spatial distribution of photons is as follows: N p k r, where all mode functions are orthogonal: Hence, energy density in the vicinity of r,t is equal to h k max k =1 r n r d V = n N k p r, the intensity is thus equal to k m a x k I p =c h N p r The average (expected) number of stimulated transitions of excited ions (equal to k =1 the number of stimulated photons added) in an elementary volume d V d t is then as follows: k m a x d 2 N p = d 2 k N n =n r, t c 0 d V d B s =c c 0 s n r,t N p r d V d t k =1 It is convenient to introduce value k p k t = N p 2, where v c 0 v k =a c w k k is a k t h mode volume ( w k is the Gaussian mode width, and is the length of the active crystal) a c 3

The corresponding change of relative number of ions with downward transitions is equal then to dn r,t d 2 k N - =- n dt c 0 dv dt =c c n r, t max 0 s v k p k r k=1 We introduce now the parameter =c c 0 s and then we add the above expression to the rate equation for n r, t It is: k m a x = D P D t D r 1 n r,t n r,t v k p k t r n r, t k =1 n d n r,t d t where D =u/h c 0 Using this equation the total number of emitted photons per dt ( dn p = d d 2 N n ) can be calculates as V : k m a x d N p k =1 k m a x k d N p =c c 0 s k =1 N p k V n r,t r d V d t Assuming that photons in each mode cause stimulated emission into the same mode only, and that photons in each mode come from all elementary volume of the laser crystal we can write the following rate equation for p k t : d p k d t = p k n r,t r d V p p k T p Intracavity losses are accounted by the factor p The photon losses due to the absorption in a saturable absorber (during dt) are introduced as follows Absorption coefficient is equal to q =c q q (where c q and q are concentration and absorption crosssection of absorbing ions of the saturable absorber crystal) During one round trip (time T p ) the intracavity light density (photon flux) p k t r is reduced by the factor of e 2 qa q which is equivalent to the additional member of the rate equation for dp k dt r = p k r 2 a q q 1 q r,t T p Here a q is the length of the satuarable absorber, and q is the relative number of excited ions in the saturable absorber crystal (absorption takes place only from the ground state) It is convenient to use the following formula q a q lnt q (value T q is a one-path transparency of the crystal) Residual absorption of the crystal due to absorption from excited levels is introduced the same way: dp k dt r = p k r 2 a q q 1 q p T k r 2 a q q q p T p Integration by dv d 2 r gives, finally, the rate equation for photon dynamics d p k d t = p k n r, t r d V p k V T p 2 a p q [ 1 q V q r, t r d V q q r, t r d V ] V The rate equation for excitation dynamics of active ions of a saturable absorber is derived using the following considerations 4

The number of excited ions in a volume dv dt is dn q = de p h, where the absorbed light energy is the product of the light energy stored in the volume dv and the absorption rate during time dt : =[ k de h max p c 0 p k v k dv] [ q c q c 1 q dt ] k=1 Normalized (relative) inversion population of active ion of a saturable absorber dq dt = dn q c q dv is Introduce now parameter =c c 0 q c c 0 the time constant of q The final equation is k dq dt =c c 1 q max 0 q p k v k k k=1 c q q and take into account spontaneous decay of excited states with k dq dt = 1 q max k=1 p k v k q q Intensity of laser output emission for the k th mode can be estimated as corresponding intracavity photon energy divided by time necessary to let all photons escape through the output coupler : where Q=h c 0 P p k =N p k h 1 R p T 1 R p T h c 0 v k p k 1 R p Q v k p k, T The set of rate equations for multi-mode laser dynamics with an intra-cavity saturable element can be presented as follows: d n r,t d t d p k d t k m a x = D P D t D r 1 n r,t n r,t v k p k t r n r, t k =1 n = p k n r, t r d V p k V T p 2 a p q [ 1 q V q r, t r d V q q r, t r d V ] V k dq dt = 1 q max k=1 p k v k q q k The laser output power is P p = 1 R p Q v T k p k,[w ], where p D =u/h c 0, =c c 0 s, p = ln R p, q =c q q, q =c q, =c c 0 q, Q=h c 0,[ J/m 3 ] q a q lnt q, q a q ln T 5

We choose now typical material parameters for investigation of laser dynamics: c 0 =136 10 20 cm 3 ( Nd 3+ :YAG, 1% doping), s =28 10 20 cm 2 (the Nd emission cross section at 1064 nm), c q =644 10 17 cm 3 (the Cr density in Cr 4+ :YAG ), q =5 10 18 cm 2 (Cr absorption cross section at 1064 nm; from ground state), =159 10 19 cm 2 (Cr absorption cross section at 1064 nm; from upper state), the diode pumpe wavelength D =808 nm, the laser wavelength =1064 nm, the laser crystal length a c =20mm, the saturable absorber crystal length a q =5mm, D =003cm 3 J 1, the output coupler reflection R p =05 ( p =069 ), =1140ns 1, T q =02, T =095, q =322 cm 1, q =01 cm 1, =204 10 4 ns 1, Q=254 J cm 3 We assume also (for practical purposes) the following: all variables are constant along the cavity axis, the default mode volume is the same for all modes Theses assumptions are equivalent to: r = 1 a c x,y,k=d,1k max, v k =a c S ( S= w 0 2 /4 where w 0 is the Gaussian diameter of the TEM 00 mode of the cavity) With these assumptions the laser dynamics equations are: d n x, y, t = D d t a c p k t T p dq x, y, t dt k m a x P D t D x, y [1 n x, y, t ] n x, y, t S p k t x, y, t n x, y, t k =1 n d p k t d t 2 a p q [ 1 q S = p k t n x, y, t x, y d S - S q x, y, t x, y d S q q x, y, t x, y d S ] S k max q x, y,t = 1 q x, y,t S p k t x, y k =1 q k P p t = 1 R p Q a c S p k t T p D =u/h c 0, =c c 0 s, p = ln R p, q =c q q, q =c q, =c c 0 q, Q=h c 0 q a q lnt q, q a q ln T It is useful also to define the gain coefficient g t : g t = d ln p t dt 6

If g t =const during t [t 1, t 2 ], then p t 2 =p t 1 e g t 2 t 1 Assuming t 2 t 1 =T p /2 (time of a single path of laser photons through the laser cavity) we can define the gain factor: K a = 1 2 g T p In case of negligible intracavity losses we can estimate the gain coefficient for a k t h G k t = 1 2 g t T 1 k p 2 T p n r,t r d V V In a laser with cylindrical symmetry, the transverse mode pattern is given 4 by combination of a Gaussian beam profile and a Laguerre polynomial The modes are denoted TEM pl where p and l are integers labeling the radial mode and angular mode orders, respectively The intensity at a point [ r, ] (polar coordinates) is given by: I p l r, =I 0 l [L p l ] 2 c o s 2 l e, where ρ = 2r 2 /w 2, the parameter w is the spot size of the mode corresponding to the Gaussian beam radius, and L p l is the associate Laguerre polynomial of order p and index l: L n k x = 1 The first few Laguerre polynomials are: p d k d x [ L k n k x ]= 1 n! e x x k d n d x n e x x n k L p k x 0 1 1 x 1 k 1 2 2 x2 2 k 2 x k 1 k 2 1 3 3! x 3 3 k 3 x 2 3 k 2 k 3 x k 1 k 2 k 3 1 4 4! x 4 4 k 4 x 3 6 k 3 k 4 x 2 4 k 2 k 3 k 4 x k 1 k 2 k 3 k 4 The absorption parameter and the initial transmission T q e T t t = T 2 T q or =2 ln 1 T q of the saturable absorber are related as follows There is also an absorption parameter due to upward transitions from the excited quantum states of the laser crystal Corresponding absorption coefficient is =2 ln 1 T where T is the transmission of the bleached crystal Contrast of the saturable absorber is defined as K = 1 T q 1 T 4 See, for example, http://enwikipediaorg/wiki/transverse_mode 7

If initial absorption changes (due to thicker crystal or increased Cr 4+ ion concentration) then we assume that changes proportionally to If the number of active centres in a saturable absorber changes (due to change either of their density or just of the size of the crystal), then both parameters ( and ) must be changed proportionally Dynamic transmission of the saturable absorber T q x, y can be obtained from the expression above replacing 1 q q We obtain T q r,t =e 1 2 1 q r, t q r, t ij ij 1 q T ij r,t q q T ij r,t This formula can be used to calculate transversal transmision of the saturable absorber as a function of time The following parameters are used in the above equations: the one round-trip loss for a k-mode: = log R k log[ 1 q p R p ], decay constants n (for spontaneous inverse population emission) and q (for saturable absorber inverse population lifetime); the gain parameter of the fundamental frequency in the activated laser crystal, excitation rate factor t for active laser ions caused by the diode pump, the parameter of the fundamental frequency absorption in a saturable absorber, and the parameter of the saturable absorber transition rate caused by photon absorption The fundamental frequency seeding emission p t has its origin in quantum fluctuations and is considered spatially uniform Initial conditions for the set of equations are: n r,0 = p r,0 =q r,0 =0 3 Determining laser cavity modes The set of optical modes to be used in the numerical model is defined by the Gaussian diameter w of the fundamental mode TEM 00 and by the list of higher order modes with indexes [p l] Parameter w is calculated on the basis of special algorithm where beam path inside the cavity is modelled by using the ABCD matrix method 5 A Gaussian beam at any point is defined by the complex parameter q : 1 q = 1 R i, w 2 where R is the beam wavefront curvature radius If q is defined, then w =1 / I 1 q and R =1 /R 1 q The propagation of the laser beam through intracavity components (left mirror, laser crystal, Q-switch crystal, right mirror) is calculated on the basis of the following formula: M 1 2, M 2 2 q B = q A, M q M A 1 1 q A M 2 1 where 1 q A is the beam parameter at the left mirror ( = 1 i 2 ), R q A R 1 w 1 is the radius of curvature of the 1 st 0 mirror, and w 0 is a zero iteration beam diameter Matrices are: =[ M M 11 M 12 M 21 M 22 ] = M R1 M a M R 2 ( transfer matrix from mirror 1 to mirror 2), M =[ R 1 0 - matrix for the spherical mirror, 2/ R 1] 5 See, for example, http://enwikipediaorg/wiki/ray_transfer_matrix 8

M a =[ 1 a n YAG 0 1 ] - matrix for intracavity crystals The beam travel in opposite direction is given by M =[ Finally, one round trip is defined by the matrix M 11 M 21 M =M M M 12 M 22 ] =M R 2 M a M R 1 Algorithm: 1 Set q 0 =1/ 1 R 1 i w 0 2, w 0 is an arbitrary zero iteration beam diameter 2 Repeat q j = q j 1, M, j=1,2, until w j w j 1, where is the required accuracy Using the Gaussian beam waist diameter w we can define all high-order transversal modes I p l r, for a given laser cavity Each optical mode may have aperture losses due to laser crystal limited diameter The relative loss for one path through the cavity for a k th mode can be calculated as follows = k r d 2 r r S At a given energy, the modes of higher order have lower peak intensities and higher losses due to increasing of mode transversal spread as indexes [p, l] icrease 2 The commonly used parameter for a multi mode beam quality can be defined as M = 2 d*, where w d * is n the 1/e th width of the transversal distribution of the beam intensity p k k r k =1 4 Examples of laser simulation We chose typical material parameters: Initial Q-switch absorption Residual Q-switch absorption =441 =004 Laser crystal gain parameter =20 Q-switch excitation parameter =80 Diode pump power parameter Energy scaling factor =036 Q=015 Output coupler reflection R p =05 Intracavity losses 6 q p =0005 6 This loss parameters is related to irregular (random) causes (such as impurities in the crystal, for example) Losses attributed to aperture effects (limited laser crystal diameter) are automatically calculated during simulation (depending on the mode transversal distribution) 9

Cavity roundtrip time T =0473 Diode maximal current I max D =125 A Diode threshold current I max D =25 A Active laser crystal ion lifetime n =230 s Q-switch crystal ion lifetime q =85 s Laser rod diameter a=3 mm Simulation examples will be presented for two case studies: Laser A Laser B Fundamental laser mode waist w=039 mm w=057 mm The simulation of the laser A without Q-switching element (free-running lasing) Figure 2 A typical random-pulsed lasing (presented here for the 1 st 100 µs Intensity fluctuations are caused both by basic laser dynamics and multimode competition processes 10

Figure 2 The beam quality parameter M 2 illustartes multimode lasing which starts from the TEM 00 mode (M 2 =1) followed by generation of higher order modes Figure 3 Competition of cavity modes during first 20 ms after reaching laser threshold Note that lasing starts from [00] mode and later several modes are generated simultaneously 11

Figure 4 Laser beam transversal distribution (averaged over the whole lasing time period) Note that transversal expansion to higher-order modes is limited by the laser rod diameter Results of the simulation (laser A: Q-switched regime) Results of the simulation of the laser A in its actual configuration (with the saturable absorber) are presented here The only one pulse is produced 12

Figure 5 A single laser 3-ns pulse is generated in the Q-switched regime Figure 6 Inverse population of the laser crystal and the Cr:YAG crystal versus time demonstrate slow increase of the gain until the lasing threshold, and rapid switch during laser pulse generation 13

Figure 7 Laser beam quality in the Q-switched regime is M 2 =1 (TEM 00 mode) during main laser output (18142 18145 µs) Deterioratiuon of the output beam quality is insignificant (it takes place when output intensity is less then 1% of its maximum) Laser pulse generated has purely Gaussian shape Figure 8 14

As expected, in Q-switched regime the only mode generated is the TEM 00 mode All modes of higher order either do not reach the threshold or are unable to bleach the saturating absorber Results of the simulation (laser B: free-running regime) Results of the simulation: Figure 9 A typical random-pulsed lasing (presented here for the 1 st 35 µs) Intensity fluctuations are caused both by basic laser dynamics and multi-mode competition processes 15

Figure 10 Mode intensities Competition of cavity modes during first 20 ms after reaching laser threshold Figure 11 Beam quality Competition of cavity modes during first 25 ms after reaching laser threshold 16

Figure 12 Laser average output beam shape Laser beam transversal distribution (averaged over the whole lasing time period) 17

Results of the simulation (laser B: Q-switched regime) Figure 13 Laser pulse in the Q-switched regime Figure 14 Laser pulse amplification from the quantum noise level Note the failed attempt of the [01] mode to enter the game 18

Figure 15 Laser output beam transversal distrtibution It is purely Gaussian 19

Results of the simulation (laser B: Q-switched regime with continuous pumping) The next numerical experiment is to illustrate how mode dynamics of a laser can change in case of a very long pump pulse In this case a train of Q-switched pulses will be produced Pulses will be separated by time periods sufficient to pump up the laser crystal to reach the lasing threshold Figure 16 Train of laser pulses in a Q-switched laser with a long pumping pulse It is indeed the case that pulses are produced with approximately 20 µs periods It is important to note that intensities of pulses fluctuate (in the range of rom 1 to 10) Explanation of this behaviour can be derived from the next graph 20

Figure 17 Mode amplitudes versus time in case of a long pumping pulse In case of a long pumping pulse every next laser Q-switched pulse belongs to a different cavity mode Every next pulse uses different cavity mode because the spatial distribution of the gain in the laser crystal changes with every next pulse As a result, the average beam quality deteriorates (in this example M 2 =246 for pump pulse duration of 150 µs ) The beam average transversal distribution is similar to the case of non-q-switched laser regime, but peak intensities are obviously much higher 21

Figure 18 Average beam cross-section in case of Q-switched laser with a 150-µs pump pulse length If losses of all modes except TEM 00 are made high artificially, then the only mode will be TEM 00 with the regular period of 14 µs) 5 Laser modes and thermal lens effect Pumping of the laser crystal creates thermal lens which modifies the cavity optical modes We assume that a lens with optical strength f 1 is located between the laser crystal and the passive Q-switch crystal The ABCD matrices must be changed as follows M =M R1 M a M th M R 2, M =M R2 M th M a M R 1, 1 0 f 1 is the matrix associated with the thermal lens The optical strength of the thermal lens is roughly proportional to the average absorbed pump power where M th =[ 1] Illustration of the thermal lens effect in case of initial non-stable optical cavities is presented in Figure 19 Laser A The flat/concave 1800mm cavity ( R 1 =, R 2 = 18 m ) is non-stable Thermal lens effect provides stabilization of the cavity mode when pump absorption is high enough to cause focusing with f 181 m The change of thermal lens focus from 175 m to 180 m increases mode diameter from 08mm to 16mm and mode losses - from 007% to 26% per round trip Laser B The convex/concave cavity ( R 1 =1 m, R 2 = 07 m ) is not stable In this case thermal lens must be such that f 263 m The change of thermal lens focus from 250 m to 261 m causes mode diameter increase from 075 mm to 11 mm, and mode losses grow from 004% to 25% per round trip 22

Figure 19 Dependence of the laser mode waist size and mode losses versus thermal lens focus f It is important to keep pump power stable because power drift of just 2% can increase mode losses dramatically and even shift to the range of optical mode instability Summary Stimulated emission in the Nd:YAG laser with a saturable absorber Cr:YAG is modelled as a superposition of interacting optical modes which are stable in a given optical cavity The interaction of the active laser crystal and a passive Q-switch with diode pump and optical cavity modes is modelled with account of transversal two-dimentional variation of the pump field and of all participating optical modes Each elementary volume of an active laser crystal and the Q-switching crystal interacts with all modes (this way modes are nonlinearly linked to each other), and every optical mode interacts with the whole laser crystal This interaction is given by rate equations Total number in this set of equations is given by the following formula 2 2 N e =2 k max j max where k max is number of cavity modes and j max is the total number of elementary volumes representing intracavity crystals Most of simulation experiments presented in this paper were done with N e =2 10 5 2 ( k max =20, j max =100 100 ) Special numerical tools were used to accelerate calculations without loss of precision Simulation experiments for two laser devices provide quantitative validation of some experimentally observed effects (domination of the fundamental mode in a Q-swithed regime, multimode operation without Q switch) and predict performance of a laser with long pump pulses (trains of laser Q-switched pulses with pule-to-pulse mode switching) 23

Referencies 1 Experimental and theoretical study of the laser micro-machining of glass using highrepetition-rate ultrafast laser Yuri Yashkir and Qiang Liu Solid State Lasers and Amplifiers II, edited by Alphan Sennaroglu, Proc of SPIE Vol 6190, 61900V, (2006) 2 Pulse shape control in a dual cavity laser: numerical modeling Yuri Yashkir Solid State Lasers and Amplifiers II, edited by Alphan Sennaroglu, Proc of SPIE Vol 6190, 619018, (2006) 3 Numerical modeling of the phase-conjugate laser with an intra-cavity stimulated Brillouin scattering mirror: Q-switching mode Yuri Yashkir Solid State Lasers and Amplifiers II, edited by Alphan Sennaroglu, Proc of SPIE Vol 6190, 61901A, (2006) 4 Numerical modeling of the intra cavity stimulated Raman scattering as a source of sub nanosecond optical pulses Yuri M Yashkir, Yuriy Yu Yashkir Proceedings of SPIE - 5460, "Solid State Lasers and Amplifiers", Alphan Sennaroglu, James G Fujimoto, Clifford R Pollock, Editors, September 2004, pp 220-227 5 Numerical simulation of the laser dynamics and laser/matter interactions, and its applications for laser micro-machining YYashkir, M Nantel, B Hockley International Journal of Applied Electromagnetics and Mechanics, v19, 373-377 (2004) 6 Laser micromachining with passively Q-switched intra cavity harmonic generator YYashkir SPIE TD01, 462-466 (2002) 7 Passively Q-switched 157-µm intra cavity optical parametric oscillator YYashkir, Henry M van Driel Applied Optics 38, 2554-2559 (1999) 24