- Email: [email protected]
Modeling of Yb:YAG tuning curves
15 January
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
134 (1997) 15% 160
Modeling of Yb:YAG tuning curves P. Peterson ‘, M.P. Sharma 2, A. Gavrielides Nonlinrur
Optics Center of Technology,
Received
Phillips
Luborutory
PL / LIDN.
3550 Aherdrrn
Aw.
SE. Kirtlarrtl
AFR. NM X7/ 17-5776.
USA
I I March 1996; revised version received 28 August 1996; accepted 2 September 1996
Abstract We simulate experiment.
Yb:YAG laser
tuning
curves
using analytic equations for a two manifold laser. Our simulations
agree with
PACS: 42.55.R
1. Introduction A recent publication [l] reports a Yb:YAG laser experiment with a 0.4 mm thick crystal pumped at A,, = 940 nm. In that article the authors show the tuning curves for a four double-pass pump cavity with a reflectivity of r = 0.984 pumped with powers of 22, 26.5 W, and with a reflectivity of r = 0.84 pumped with a power of 30 W. Their major result is that Yb:YAG lasers have a rather large tuning range up to 38 nm depending on the resonator outcoupling and pump power. In this paper we simulate these tuning curves to within 12%, and then using these results as an anchor we make further predictions of the output power and tuning range. We are interested in this experiment because it presents an excellent opportunity to check our analytic formulas for lasing in two manifold systems. Usually, lasing is just
’ E-mail: [email protected]. ’ Permanent address: Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 8713 I, USA. W30-4018/97/$17.00 Copyright P/I SOO30-4018(96)00571-8
reported in the form of the linear power extraction for large outcouplings which does not present a rigorous check for analytics. However, the tuning curves are more of a challenge. Critical to modeling the tuning curves is an accurate description of the laser absorption and emission cross sections. We obtain the laser absorption cross section by digitizing existing data [2,3]. Following this, the emission cross section is obtained by using the reciprocity method of McCumber [4] and others [5]; more details will be given later. In the digitization process it is difficult to obtain an exact duplicate of the absorption cross section. This fact combined with the inherent errors in original absorption cross section measurement. estimated to be as large as 20%, contribute to our errors in fitting the experimental data [l]. These cross sections are then directly inserted into our multi-double-pass power extraction equations which we develop in the next section. The tunability of Yb:YAG, a rare earth element. is due to the transitions between the upper ‘Fs,,z manifold Stark levels and the lower ’ F7,: manifold Stark levels. Its broad tunability has been explained
0 1997 Elsevier Science B.V. All rights reserved
156
P. Peterson
et d/Optics
Communications
in terms of the strong coupling of the 13 4f electrons [I]. The tuning range also depends largely on three quantities. These are the emission cross section as a function of frequency, the cavity mirror outcoupling, and the absorbed pump power. The experiment shows three tuning curves for different pump powers and reflectivities and temperatures. After fitting these data we easily explore the tuning range for other cavity configurations. Thus, for example, we show that for fixed pump power the tuning range decreases as the mirror reflectivity decreases and that for fixed reflectivity the tuning range decreases as the pump power decreases.
134 (1997) 155-160
Casting the above description into equations the upper lasing manifold population rate equation is dni ~=(P,+P-)(~~N,-y,‘N,)-~
+(R++R_)(y,N,
-ye&)=
- $. (la)
Thus, N2 changes because of stimulated pump emission and absorption, decay through T, and stimulated laser emission and absorption, respectively. These yield the steady state gain YrN2 - Y*N, =N,{[P(z)++P(z)-lr-3d/7}
2. Theory We present a brief derivation of our basic formula, for more details we refer the reader to our previous publications [6-91. This theory describes lasing from a two manifold system in which the lower manifold, with concentration N,, is pumped up into the upper manifold, with concentration N2. Lasing occurs between the level (6) in the upper manifold, with concentration Nh = f,, N2, and the level (a) in the lower manifold, with concentration N, = f, N,. f, is given by f, = exp(- E,/kT)/Z, where E, is the energy of level (a> and 2, is the partition function for the lower manifold. The Boltzmann factor is k and T is the temperature. Likewise there is a similar expression for fh, and N, + N2 = N, which is the doping concentration. Also, N2 decays into N, at the rate l/7. The pump and laser intensities are coupled to the N, and N2 manifolds through the effective absorption and emission cross sections (y:, 7,‘) and (y,, y,>, respectively. The laser effective cross sections are related to the temperature independent spectroscopic cross section a, by y, = fOu, and ye = f,, u-~.The resonator is a plane wave Fabry-Perot with arbitrary outcoupling, however, Gaussian beams can be included by an adjustment of the area, as we do later. The multiple pump pass configuration requires the forward and reverse pump fluxes P+( z> in addition to the forward and reverse lasing fluxes R+(z).
X([P(z)++P(z)-IL+ l/T +[R(z)++W-]y+}-’
(lb)
and pump absorption Y:N, - Y:& =N,{[R(z)++R(z)-]r+~d/7) X{[P(z)++P(z)-IT:+
l/T
+[R(z)++R(z)-]y+}-‘9
tic)
where yO,y,, y,&y,’ are combined to define y k = y, + yQ, r’+= r,’ f r: and I’= YLY, - y,'r,. The remaining differential equations describe the evolution of the forward and reverse field amplitudes and are given by dp+ = -i(y:N, dz dr+ -=+(y,N,-dz
- yfN,)p+-
ikpp+ ,
(2b)
yoN,)r+-ikrr+,
dp= +$( y:N, - y,‘N,)p_+ ik,p_ dz. dr-= -f(y,N,y,N,)r_+ik,r_. dz
(*a)
,
(2c) (*d)
Here the flux and their amplitudes are related by and R k = r +r f.. These equations are p,=p,p; constrained by the two point-boundary conditions on the laser fields. These conditions are obtained by first noting that the boundary conditions at z = 0 and
P. Peterson et al./Optic.s
C0mmunicution.s
z = 1 are r_(O) = r+(O) and r_(l) = fir+(l), where r is the laser intensity reflectivity. From Eqs. (2b),(2d) it is easy to show that R+( z)R_ ( z> = R+(O)R_(O) = R+(Z)R_(l) = R:(O) = rR:(l) or r+(O) = &r+(l) exp(ik,Z) which constitutes the two
point boundary conditions. Earlier we solved Eqs. (I), (2) for two pump configurations: that of a single pass pump [6], and when the pump fields are also constrained by a Fabry-Perot cavity 181.Furthermore, in the Fabry-Perot case the depletion of the pump can be obtained in one of two ways. Either the cavity two point boundary value problem can be solved, or the successively reflected and transmitted pump fields can be added once the single pass pump depletion equation is known. These two methods are equivalent and the second method is the standard FabryPerot analysis [lo]. Since we are interested in a finite number of passes we eventually follow the second method. We begin by treating the fields inside the cavity and later connect them through the boundary conditions to the incident and transmitted fluxes. The first step is to find the pump depletion equation at z = 0 and z = 1. This is found by multiplying Eq. (2a) by y+/( rp+) and Eq. (2b) by y\/( r r+ 1 and then subtracting the two equations. This leads to r; dr, __---Tr, d; NO
=T+i
Y+ dP+ Tp+
d=.
$kp-
This equation can be easily integrated and gives rise to Mr+(l)/r+(O)> and ln(p+(Z)/p+(O)) terms. The first natural logarithm can be replaced by the two point boundary condition derived above and the second term is the one we are seeking. Thus, depletion of the pump amplitude is given by P+(l) ~ P+ (0)
= exp( A/2)exp(i$), rNol
where A = - -
Y+
y: + --lnY+
1 \/;:
(4)
and the phase angle is the pump phase accumulated over the crystal length, 4 = - k,l. Note that this equation only relates the pump fields at the two cavity boundaries and is not yet related to the num-
134 (1997)
157
155-160
ber of pump passes. The corresponding p_ is P-(l) P- (0)
= exp( - A/2)exp(
equation
for
- i+) .
The next step is to find the cavity outcoupled laser power. We begin this portion of the calculation by using Eqs. (2a),(2b) to form the differential equaand R+= r+r;. After these tions for P+=p+p; two resulting equations are divided we find that dP+ -=dR+
-P+ R,
y:/~fr(R++R_) r( P++P_)
- yJ7
(6)
’
where the ratio on the right hand side arises from dividing Eq. (lc) by Eq. (lb). This equation is easily integrated [8] since the reverse variables can be eliminated through R+(z.)R_(z) = R+(O)R_(O) = R+(i)R_(I) and P+(z)P_(;) = P+(O)P_(O) = P+(l)P_ (I). Completing the integration and algebra
as outlined in Ref. [s] gives r[P-(0)
-P+(O)]
- (y~Y,/7)ln(P,(I)/P+(O))
= -[r(l-r)R+(f)+(y:/r)ln(l/r)].
(7)
The last step consists of replacing ln( P, (II/P + (0)) with the magnitude of Eq. (4) and introducing the transmitted pump flux P, and incident pump flux P, through the flux conservation condition P_(O) P, (0) = P, - P,. Note that the pump enters and exits the cavity at the ; = 0 boundary. Completing these step gives 1 P, - P, = lnL+y+r 6
YJ%I~ +R y+r
0-r) ___ Out f
*
(8) The first two terms on the right hand side comprise the threshold and the last term is, of course, proportional to the the laser output flux. This equation is the conservation law for the net pump and laser photon flux and is almost the relation we are seeking. The final step relates the transmitted flux P, to the input pump flux P,. To do this we proceed with the usual Fabry-Perot type of analysis [lo] where the fields are described by successive reflections and propagations. That is, upon entering the cavity at : = 0 and traversing the laser medium to ; = I the
P. Peterson et d/Optics
1.58
Communicutions
first transit of the forward pump is assumed to be attenuated by P,,(1) = P,,+(O) exp(A), according to Eq. (4). Next at 2;= 1, the reverse and forward amplitudes satisfy p,,_(l) = p,,(l) since we assume that the mirror is perfectly reflecting. After this according to Eq. (5) the reverse field at z = 0 is P,,_ (0) = P,,_(I) exp( A) = P, exp(2 A). This process continues until the pump exits after four double-passes with the result that P, - P, = P,[ 1 - exp( 8 A)]. Inserting this into Eq. (8) gives
134 (1997)
155-160 10679
_
L
10624 -.-
‘5/Z
10327
/cm
E sE
:: N
z
0 c
5
zz
0 I-.
x
785
I612
2
F
56Vcm
7/Z 0
Rout= (1
--
p,[
_:+q
1-
exp(8A)l
1 1,; -Y+r (
+.L+y,NOZ
II.
Fig. 1. Yb:YAG energy level diagram.
(9)
In passing we mention that in Ref. [S], see Eq. (91, we showed that the two-pass Fabry-Perot analysis is identical to the pump cavity boundary value problem [8], i.e., P, - P, = P,[ 1 - exp( 2 A)]. Furthermore, using the above single-pass pump depletion equations one can show the equivalence of the pump two-point boundary value problem to the Fabry-Perot analysis with an infinite number of single-passes for arbitrary pump reflectivity r,,. In Eq. (9) the mirror intensity reflectivity for the laser is r and r + t + s = 1 where s is the mirror scattering loss. The cavity loss is given by L and is added phenomenologically; this approach agrees with the experiments [l 11. The crystal length is 1 and the output laser flux is R,,, and the input pump flux is P,. The flux is related to the power by: flux = intensity/photon energy = power/area/photon energy where [lo] area is nm2 for a flat top and (7rw2/2) for Gaussian beams. This simple scaling does not account for the sublinear behavior near threshold [12]. The factor of eight in Eq. (9) does not appear with the y,N,l term since this is the laser absorption loss which naturally occurs only over the length 1 not over 81. Finally, threshold is found by setting R,,, equal to zero in Eq. (9). If the pump threshold is then plotted as a function of length it displays a minimum pump threshold due to the onset of reabsorption [6,7]. This value of the length and the minimum threshold are strongly dependent on the factor of eight. That is, this model accounts for the increased laser reabsorption due to the increased pump absorption. This
completes the theoretical discussion and we now move to our simulations. Following the literature [ 1,2,11,13] the constants are: 76 = 7 X 10d2’ cm2, 76 > 7: for pumping at 940 nm; T = 1.2 ms; N, = 1.26 X 102’ cm-3 at a 8% doping [ 11; I = 0.04 cm; and w = 0.035 cm for the beam waist. Refs. [2,3] contain the absorption cross section of Yb:YAG at T = 300 K and we choose the later. We first digitize the absorption cross section Ye and then use the reciprocity method [4,5] to obtain the emission cross section y,(v). This method is embodied in the formula y”(v)=
Cdifiuij(v)dj,
(10)
ij
where fi = exp(- E,/kT)/Z,, Ei, and di are the Stark level energies and degeneracies in the lower manifold, denoted by the subscript i; likewise Z, is the lower manifold partition function. dj is the degeneracy of the Stark levels in the upper manifold, denoted by j, and aij( V) is the individual cross section between the ij levels and contains the lineshape information. For Yb:YAG the individual energy levels E, are not temperature sensitive [3]. Thus, Eq. (10) indicates that ~~y,( V> does not change much between 300 K and 330 K which is the temperature difference in the experiment [l]. An equation similar to Eq. (10) holds for the emission cross section y,(v) and by dividing these two equations one finds X( r~) = z,( v> $exp( u
EzL - hv)/kT,
(11)
P. Peterson
et d/Optics
Communications
Fig. 2. The emission spectra of Yb:YAG derived from the absorption spectra.
where Z,, is the upper manifold partition function and the zero-line energy EzL = 10327/cm is the energy separation between the lowest levels in the two manifolds [5]. In Eq. (11) the partition functions are formed with the energy level structure shown in Fig. 1. Digitizing the absorption cross section given in Ref. [3] and then applying Eq. (11) gives the emission cross section shown in Fig. 2. Both cross sections are inserted directly into Eq. (9) along with the above constants. We now present our simulations. According to the experiment [l] the temperature, pump power and reflectivities are fixed. Thus, we input the above data and vary the total cavity loss L and mirror scattering s until we obtain an acceptable fit to the experimental data. This leads us to a value of the total internal losses f. = 1.5% and s = 0.1% while the authors [l] quote total loss of less than 2%. Fig. 3 shows our simulations of the experimental data [ 11 for the three cases: r = 0.984, pump power P = 26.5 W, temperature T=(218+80) K; and P=22 W, T=(248C XO)K;andr=0.84. P=30WatT=(213+80)K. The additional (801 K is the authors’ estimate of the temperature rise due to beam heating [ 11. Our tuning curves shown in Fig. 3 fit their tuning range and maximum extracted powers to within 12%. Thus, we have pinned down the essential physics and we have now anchored our code to experiment. Next we present the tuning curves for fixed pump power, see Fig. 4, P, = 26.5 W and for various reflectivities
134 (1997)
155-160
Fig. K: (b) r = 0.984, PO = 26.5 W. T = 218 K; (c) r = 0.984. P, = 22 W. T = 248 K.
r = 0.984,0.9,0.8. Fig. 4 shows that as the transmission increases the tuning range dramatically decreases. The tuning range shrinks because the gain near 1048 nm drops below threshold as the losses increase. Finally, we set r = 0.84 and let P, equal 30,26.5,22 W. Fig. 5 shows that the tuning range changes very little and the output laser power naturally decreases as P, decreases. However, if only the cavity losses L increase the tuning range remains constant and the output laser power decreases; we do not show this graph. We have obtained the tuning range without considering thermal lensing within the
~““““““““““““““l”’ I’
"1
Fig. 4. Tuning curves for P, = .30 W. T = 213 K and (a) r = 0.985, (b) r = 0.9, Cc) Y = 0.8.
P. Peterson el d/Optics
Communications
134 (19971 155-160
experimentally obtained in Ref. [ll], can be simulated from Eq. ( 1).
3. Conclusion We have successfully simulated the tuning curves for Yb:YAG using a simple analytic model for a two manifold laser. Both the experiment and simulation show that the tuning range is strongly dependent on the cavity losses.
W.*.klwth
Fig.
5. Tuning
curves
for
r = 0.84
References
hm,
and (a)
P, = 30 W, (b)
P, = 26.5 W, (c) P, = 22 W.
crystal. Generally, the thermal focal length is inversely proportional to the absorbed power P,[ 1 - exp( 8 A)]. One can find the absorbed power as a function of wavelength for the (30 W, r = 84%) case and (26.5 W, r = 98.4%) case by inserting the emission cross section, shown in Fig. 2, into P,[ 1 - exp( 8A)]. This shows that the absorbed power at the (r = 84%, P, = 30 W, A = 1035 nm) point is 5% greater than that at the (r = 98.4%,P, = 26.5 W, A = 1055 nm) point. Thus, even though the input pump power changes, the absorbed power, and hence the thermally induced focal length, does not change appreciably. This is because the last term in A, (Y>/Y+ln(l/ fi)), changes as the reflectivity and laser emission cross section change. Consequently, the thermal gradients do not change much over the long-wavelength side for these pump powers and reflectivities. We also note that temperature dependence of the threshold and slope efficiency,
[l] U. Brauch, A. Giesen, M. Karszewski, Chr. Stewen and A. Voss, Optics Lett. 20 (19951 713. [2] P. Lacovara, H.K. Choi, C.A. Wang, R.L. Aggerwal and T. Fan, Optics Lett. 16 (19911 1089. 131 L.D. DeLoach, S.A. Payne, L.L. Chase, L.K. Smith, W.L. Kway and W.F. Krupke, IEEE J. Quantum Electron. QE-29 (19931 1179. 141 D.E. McCumber, Phys. Rev. 136 (1964) 954. [51 S.A. Payne, L.L. Chase, L.K. Smith, W.L. Kway and W.R. Krupke, IEEE J. Quantum Electron. QE-28 (1992) 2619. t61 P. Peterson, A. Gavrielides and P.M. Sharma, Optics Comm. 109 (1994) 282. 171 P. Peterson, A. Gavrielides and P.M. Sharma, Appl. Phys. B 61 (1995) 195. Ml P. Peterson, A. Gavrielides and P.M. Sharma, Optics Comm. 116 (1995) 123. [91 P. Peterson, A. Gavrielides and P.M. Shatma, Optical and Quantum Electronics 26 (1996) 695. [lOI A.E. Siegman. Lasers (University Science Books, Mill Valley, CA, 1986) pp. 415, 665. [III A. Giesen, H. Huge], A. Voss, K. Wittig, U. Brauch and H. Opower, Appl. Phys. B 58 (1994) 365. [121 T.Y. Fan and R.C. Byer, IEEE J. Quantum Electron. QE-23 (1987) 605. D.N. Vylegzhanin and A.A. Kaminskii, [131 G.A. Bogomolova, Sov. Phys. JETP 42 ( 1976) 440.
1997
OPTICS COMMUNICATIONS ELSEVIER
Optics Communications
134 (1997) 15% 160
Modeling of Yb:YAG tuning curves P. Peterson ‘, M.P. Sharma 2, A. Gavrielides Nonlinrur
Optics Center of Technology,
Received
Phillips
Luborutory
PL / LIDN.
3550 Aherdrrn
Aw.
SE. Kirtlarrtl
AFR. NM X7/ 17-5776.
USA
I I March 1996; revised version received 28 August 1996; accepted 2 September 1996
Abstract We simulate experiment.
Yb:YAG laser
tuning
curves
using analytic equations for a two manifold laser. Our simulations
agree with
PACS: 42.55.R
1. Introduction A recent publication [l] reports a Yb:YAG laser experiment with a 0.4 mm thick crystal pumped at A,, = 940 nm. In that article the authors show the tuning curves for a four double-pass pump cavity with a reflectivity of r = 0.984 pumped with powers of 22, 26.5 W, and with a reflectivity of r = 0.84 pumped with a power of 30 W. Their major result is that Yb:YAG lasers have a rather large tuning range up to 38 nm depending on the resonator outcoupling and pump power. In this paper we simulate these tuning curves to within 12%, and then using these results as an anchor we make further predictions of the output power and tuning range. We are interested in this experiment because it presents an excellent opportunity to check our analytic formulas for lasing in two manifold systems. Usually, lasing is just
’ E-mail: [email protected]. ’ Permanent address: Department of Physics and Astronomy, University of New Mexico, Albuquerque, NM 8713 I, USA. W30-4018/97/$17.00 Copyright P/I SOO30-4018(96)00571-8
reported in the form of the linear power extraction for large outcouplings which does not present a rigorous check for analytics. However, the tuning curves are more of a challenge. Critical to modeling the tuning curves is an accurate description of the laser absorption and emission cross sections. We obtain the laser absorption cross section by digitizing existing data [2,3]. Following this, the emission cross section is obtained by using the reciprocity method of McCumber [4] and others [5]; more details will be given later. In the digitization process it is difficult to obtain an exact duplicate of the absorption cross section. This fact combined with the inherent errors in original absorption cross section measurement. estimated to be as large as 20%, contribute to our errors in fitting the experimental data [l]. These cross sections are then directly inserted into our multi-double-pass power extraction equations which we develop in the next section. The tunability of Yb:YAG, a rare earth element. is due to the transitions between the upper ‘Fs,,z manifold Stark levels and the lower ’ F7,: manifold Stark levels. Its broad tunability has been explained
0 1997 Elsevier Science B.V. All rights reserved
156
P. Peterson
et d/Optics
Communications
in terms of the strong coupling of the 13 4f electrons [I]. The tuning range also depends largely on three quantities. These are the emission cross section as a function of frequency, the cavity mirror outcoupling, and the absorbed pump power. The experiment shows three tuning curves for different pump powers and reflectivities and temperatures. After fitting these data we easily explore the tuning range for other cavity configurations. Thus, for example, we show that for fixed pump power the tuning range decreases as the mirror reflectivity decreases and that for fixed reflectivity the tuning range decreases as the pump power decreases.
134 (1997) 155-160
Casting the above description into equations the upper lasing manifold population rate equation is dni ~=(P,+P-)(~~N,-y,‘N,)-~
+(R++R_)(y,N,
-ye&)=
- $. (la)
Thus, N2 changes because of stimulated pump emission and absorption, decay through T, and stimulated laser emission and absorption, respectively. These yield the steady state gain YrN2 - Y*N, =N,{[P(z)++P(z)-lr-3d/7}
2. Theory We present a brief derivation of our basic formula, for more details we refer the reader to our previous publications [6-91. This theory describes lasing from a two manifold system in which the lower manifold, with concentration N,, is pumped up into the upper manifold, with concentration N2. Lasing occurs between the level (6) in the upper manifold, with concentration Nh = f,, N2, and the level (a) in the lower manifold, with concentration N, = f, N,. f, is given by f, = exp(- E,/kT)/Z, where E, is the energy of level (a> and 2, is the partition function for the lower manifold. The Boltzmann factor is k and T is the temperature. Likewise there is a similar expression for fh, and N, + N2 = N, which is the doping concentration. Also, N2 decays into N, at the rate l/7. The pump and laser intensities are coupled to the N, and N2 manifolds through the effective absorption and emission cross sections (y:, 7,‘) and (y,, y,>, respectively. The laser effective cross sections are related to the temperature independent spectroscopic cross section a, by y, = fOu, and ye = f,, u-~.The resonator is a plane wave Fabry-Perot with arbitrary outcoupling, however, Gaussian beams can be included by an adjustment of the area, as we do later. The multiple pump pass configuration requires the forward and reverse pump fluxes P+( z> in addition to the forward and reverse lasing fluxes R+(z).
X([P(z)++P(z)-IL+ l/T +[R(z)++W-]y+}-’
(lb)
and pump absorption Y:N, - Y:& =N,{[R(z)++R(z)-]r+~d/7) X{[P(z)++P(z)-IT:+
l/T
+[R(z)++R(z)-]y+}-‘9
tic)
where yO,y,, y,&y,’ are combined to define y k = y, + yQ, r’+= r,’ f r: and I’= YLY, - y,'r,. The remaining differential equations describe the evolution of the forward and reverse field amplitudes and are given by dp+ = -i(y:N, dz dr+ -=+(y,N,-dz
- yfN,)p+-
ikpp+ ,
(2b)
yoN,)r+-ikrr+,
dp= +$( y:N, - y,‘N,)p_+ ik,p_ dz. dr-= -f(y,N,y,N,)r_+ik,r_. dz
(*a)
,
(2c) (*d)
Here the flux and their amplitudes are related by and R k = r +r f.. These equations are p,=p,p; constrained by the two point-boundary conditions on the laser fields. These conditions are obtained by first noting that the boundary conditions at z = 0 and
P. Peterson et al./Optic.s
C0mmunicution.s
z = 1 are r_(O) = r+(O) and r_(l) = fir+(l), where r is the laser intensity reflectivity. From Eqs. (2b),(2d) it is easy to show that R+( z)R_ ( z> = R+(O)R_(O) = R+(Z)R_(l) = R:(O) = rR:(l) or r+(O) = &r+(l) exp(ik,Z) which constitutes the two
point boundary conditions. Earlier we solved Eqs. (I), (2) for two pump configurations: that of a single pass pump [6], and when the pump fields are also constrained by a Fabry-Perot cavity 181.Furthermore, in the Fabry-Perot case the depletion of the pump can be obtained in one of two ways. Either the cavity two point boundary value problem can be solved, or the successively reflected and transmitted pump fields can be added once the single pass pump depletion equation is known. These two methods are equivalent and the second method is the standard FabryPerot analysis [lo]. Since we are interested in a finite number of passes we eventually follow the second method. We begin by treating the fields inside the cavity and later connect them through the boundary conditions to the incident and transmitted fluxes. The first step is to find the pump depletion equation at z = 0 and z = 1. This is found by multiplying Eq. (2a) by y+/( rp+) and Eq. (2b) by y\/( r r+ 1 and then subtracting the two equations. This leads to r; dr, __---Tr, d; NO
=T+i
Y+ dP+ Tp+
d=.
$kp-
This equation can be easily integrated and gives rise to Mr+(l)/r+(O)> and ln(p+(Z)/p+(O)) terms. The first natural logarithm can be replaced by the two point boundary condition derived above and the second term is the one we are seeking. Thus, depletion of the pump amplitude is given by P+(l) ~ P+ (0)
= exp( A/2)exp(i$), rNol
where A = - -
Y+
y: + --lnY+
1 \/;:
(4)
and the phase angle is the pump phase accumulated over the crystal length, 4 = - k,l. Note that this equation only relates the pump fields at the two cavity boundaries and is not yet related to the num-
134 (1997)
157
155-160
ber of pump passes. The corresponding p_ is P-(l) P- (0)
= exp( - A/2)exp(
equation
for
- i+) .
The next step is to find the cavity outcoupled laser power. We begin this portion of the calculation by using Eqs. (2a),(2b) to form the differential equaand R+= r+r;. After these tions for P+=p+p; two resulting equations are divided we find that dP+ -=dR+
-P+ R,
y:/~fr(R++R_) r( P++P_)
- yJ7
(6)
’
where the ratio on the right hand side arises from dividing Eq. (lc) by Eq. (lb). This equation is easily integrated [8] since the reverse variables can be eliminated through R+(z.)R_(z) = R+(O)R_(O) = R+(i)R_(I) and P+(z)P_(;) = P+(O)P_(O) = P+(l)P_ (I). Completing the integration and algebra
as outlined in Ref. [s] gives r[P-(0)
-P+(O)]
- (y~Y,/7)ln(P,(I)/P+(O))
= -[r(l-r)R+(f)+(y:/r)ln(l/r)].
(7)
The last step consists of replacing ln( P, (II/P + (0)) with the magnitude of Eq. (4) and introducing the transmitted pump flux P, and incident pump flux P, through the flux conservation condition P_(O) P, (0) = P, - P,. Note that the pump enters and exits the cavity at the ; = 0 boundary. Completing these step gives 1 P, - P, = lnL+y+r 6
YJ%I~ +R y+r
0-r) ___ Out f
*
(8) The first two terms on the right hand side comprise the threshold and the last term is, of course, proportional to the the laser output flux. This equation is the conservation law for the net pump and laser photon flux and is almost the relation we are seeking. The final step relates the transmitted flux P, to the input pump flux P,. To do this we proceed with the usual Fabry-Perot type of analysis [lo] where the fields are described by successive reflections and propagations. That is, upon entering the cavity at : = 0 and traversing the laser medium to ; = I the
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first transit of the forward pump is assumed to be attenuated by P,,(1) = P,,+(O) exp(A), according to Eq. (4). Next at 2;= 1, the reverse and forward amplitudes satisfy p,,_(l) = p,,(l) since we assume that the mirror is perfectly reflecting. After this according to Eq. (5) the reverse field at z = 0 is P,,_ (0) = P,,_(I) exp( A) = P, exp(2 A). This process continues until the pump exits after four double-passes with the result that P, - P, = P,[ 1 - exp( 8 A)]. Inserting this into Eq. (8) gives
134 (1997)
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_
L
10624 -.-
‘5/Z
10327
/cm
E sE
:: N
z
0 c
5
zz
0 I-.
x
785
I612
2
F
56Vcm
7/Z 0
Rout= (1
--
p,[
_:+q
1-
exp(8A)l
1 1,; -Y+r (
+.L+y,NOZ
II.
Fig. 1. Yb:YAG energy level diagram.
(9)
In passing we mention that in Ref. [S], see Eq. (91, we showed that the two-pass Fabry-Perot analysis is identical to the pump cavity boundary value problem [8], i.e., P, - P, = P,[ 1 - exp( 2 A)]. Furthermore, using the above single-pass pump depletion equations one can show the equivalence of the pump two-point boundary value problem to the Fabry-Perot analysis with an infinite number of single-passes for arbitrary pump reflectivity r,,. In Eq. (9) the mirror intensity reflectivity for the laser is r and r + t + s = 1 where s is the mirror scattering loss. The cavity loss is given by L and is added phenomenologically; this approach agrees with the experiments [l 11. The crystal length is 1 and the output laser flux is R,,, and the input pump flux is P,. The flux is related to the power by: flux = intensity/photon energy = power/area/photon energy where [lo] area is nm2 for a flat top and (7rw2/2) for Gaussian beams. This simple scaling does not account for the sublinear behavior near threshold [12]. The factor of eight in Eq. (9) does not appear with the y,N,l term since this is the laser absorption loss which naturally occurs only over the length 1 not over 81. Finally, threshold is found by setting R,,, equal to zero in Eq. (9). If the pump threshold is then plotted as a function of length it displays a minimum pump threshold due to the onset of reabsorption [6,7]. This value of the length and the minimum threshold are strongly dependent on the factor of eight. That is, this model accounts for the increased laser reabsorption due to the increased pump absorption. This
completes the theoretical discussion and we now move to our simulations. Following the literature [ 1,2,11,13] the constants are: 76 = 7 X 10d2’ cm2, 76 > 7: for pumping at 940 nm; T = 1.2 ms; N, = 1.26 X 102’ cm-3 at a 8% doping [ 11; I = 0.04 cm; and w = 0.035 cm for the beam waist. Refs. [2,3] contain the absorption cross section of Yb:YAG at T = 300 K and we choose the later. We first digitize the absorption cross section Ye and then use the reciprocity method [4,5] to obtain the emission cross section y,(v). This method is embodied in the formula y”(v)=
Cdifiuij(v)dj,
(10)
ij
where fi = exp(- E,/kT)/Z,, Ei, and di are the Stark level energies and degeneracies in the lower manifold, denoted by the subscript i; likewise Z, is the lower manifold partition function. dj is the degeneracy of the Stark levels in the upper manifold, denoted by j, and aij( V) is the individual cross section between the ij levels and contains the lineshape information. For Yb:YAG the individual energy levels E, are not temperature sensitive [3]. Thus, Eq. (10) indicates that ~~y,( V> does not change much between 300 K and 330 K which is the temperature difference in the experiment [l]. An equation similar to Eq. (10) holds for the emission cross section y,(v) and by dividing these two equations one finds X( r~) = z,( v> $exp( u
EzL - hv)/kT,
(11)
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et d/Optics
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Fig. 2. The emission spectra of Yb:YAG derived from the absorption spectra.
where Z,, is the upper manifold partition function and the zero-line energy EzL = 10327/cm is the energy separation between the lowest levels in the two manifolds [5]. In Eq. (11) the partition functions are formed with the energy level structure shown in Fig. 1. Digitizing the absorption cross section given in Ref. [3] and then applying Eq. (11) gives the emission cross section shown in Fig. 2. Both cross sections are inserted directly into Eq. (9) along with the above constants. We now present our simulations. According to the experiment [l] the temperature, pump power and reflectivities are fixed. Thus, we input the above data and vary the total cavity loss L and mirror scattering s until we obtain an acceptable fit to the experimental data. This leads us to a value of the total internal losses f. = 1.5% and s = 0.1% while the authors [l] quote total loss of less than 2%. Fig. 3 shows our simulations of the experimental data [ 11 for the three cases: r = 0.984, pump power P = 26.5 W, temperature T=(218+80) K; and P=22 W, T=(248C XO)K;andr=0.84. P=30WatT=(213+80)K. The additional (801 K is the authors’ estimate of the temperature rise due to beam heating [ 11. Our tuning curves shown in Fig. 3 fit their tuning range and maximum extracted powers to within 12%. Thus, we have pinned down the essential physics and we have now anchored our code to experiment. Next we present the tuning curves for fixed pump power, see Fig. 4, P, = 26.5 W and for various reflectivities
134 (1997)
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Fig. K: (b) r = 0.984, PO = 26.5 W. T = 218 K; (c) r = 0.984. P, = 22 W. T = 248 K.
r = 0.984,0.9,0.8. Fig. 4 shows that as the transmission increases the tuning range dramatically decreases. The tuning range shrinks because the gain near 1048 nm drops below threshold as the losses increase. Finally, we set r = 0.84 and let P, equal 30,26.5,22 W. Fig. 5 shows that the tuning range changes very little and the output laser power naturally decreases as P, decreases. However, if only the cavity losses L increase the tuning range remains constant and the output laser power decreases; we do not show this graph. We have obtained the tuning range without considering thermal lensing within the
~““““““““““““““l”’ I’
"1
Fig. 4. Tuning curves for P, = .30 W. T = 213 K and (a) r = 0.985, (b) r = 0.9, Cc) Y = 0.8.
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134 (19971 155-160
experimentally obtained in Ref. [ll], can be simulated from Eq. ( 1).
3. Conclusion We have successfully simulated the tuning curves for Yb:YAG using a simple analytic model for a two manifold laser. Both the experiment and simulation show that the tuning range is strongly dependent on the cavity losses.
W.*.klwth
Fig.
5. Tuning
curves
for
r = 0.84
References
hm,
and (a)
P, = 30 W, (b)
P, = 26.5 W, (c) P, = 22 W.
crystal. Generally, the thermal focal length is inversely proportional to the absorbed power P,[ 1 - exp( 8 A)]. One can find the absorbed power as a function of wavelength for the (30 W, r = 84%) case and (26.5 W, r = 98.4%) case by inserting the emission cross section, shown in Fig. 2, into P,[ 1 - exp( 8A)]. This shows that the absorbed power at the (r = 84%, P, = 30 W, A = 1035 nm) point is 5% greater than that at the (r = 98.4%,P, = 26.5 W, A = 1055 nm) point. Thus, even though the input pump power changes, the absorbed power, and hence the thermally induced focal length, does not change appreciably. This is because the last term in A, (Y>/Y+ln(l/ fi)), changes as the reflectivity and laser emission cross section change. Consequently, the thermal gradients do not change much over the long-wavelength side for these pump powers and reflectivities. We also note that temperature dependence of the threshold and slope efficiency,
[l] U. Brauch, A. Giesen, M. Karszewski, Chr. Stewen and A. Voss, Optics Lett. 20 (19951 713. [2] P. Lacovara, H.K. Choi, C.A. Wang, R.L. Aggerwal and T. Fan, Optics Lett. 16 (19911 1089. 131 L.D. DeLoach, S.A. Payne, L.L. Chase, L.K. Smith, W.L. Kway and W.F. Krupke, IEEE J. Quantum Electron. QE-29 (19931 1179. 141 D.E. McCumber, Phys. Rev. 136 (1964) 954. [51 S.A. Payne, L.L. Chase, L.K. Smith, W.L. Kway and W.R. Krupke, IEEE J. Quantum Electron. QE-28 (1992) 2619. t61 P. Peterson, A. Gavrielides and P.M. Sharma, Optics Comm. 109 (1994) 282. 171 P. Peterson, A. Gavrielides and P.M. Sharma, Appl. Phys. B 61 (1995) 195. Ml P. Peterson, A. Gavrielides and P.M. Sharma, Optics Comm. 116 (1995) 123. [91 P. Peterson, A. Gavrielides and P.M. Shatma, Optical and Quantum Electronics 26 (1996) 695. [lOI A.E. Siegman. Lasers (University Science Books, Mill Valley, CA, 1986) pp. 415, 665. [III A. Giesen, H. Huge], A. Voss, K. Wittig, U. Brauch and H. Opower, Appl. Phys. B 58 (1994) 365. [121 T.Y. Fan and R.C. Byer, IEEE J. Quantum Electron. QE-23 (1987) 605. D.N. Vylegzhanin and A.A. Kaminskii, [131 G.A. Bogomolova, Sov. Phys. JETP 42 ( 1976) 440.