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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 25 — Dec. 5, 2011
  • pp: 25672–25684
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ASE in thin disk lasers: theory and experiment

P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham  »View Author Affiliations


Optics Express, Vol. 19, Issue 25, pp. 25672-25684 (2011)
http://dx.doi.org/10.1364/OE.19.025672


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Abstract

We derive equations for the ASE intensity, decay time, and heat load. The crux of our development is frequency integration over the gain lineshape followed by a spatial integration over the emitters. These integrations result in a gain length that is determined from experiment. We measure the gain as a function of incident pump power for a multi-pass pumped Yb:YAG disk doped at 9.8 at.% with an anti-ASE cap. The incident pump powers are up to 3kW. Our fit to the measured gain is within 10% of the measured gain up to pump powers where the gain starts to flatten out and roll over. In this comparison we extract the gain length that turns out to be 43% of the pump spot size of 7mm.

© 2011 OSA

1. Introduction

There are two general approaches quantifying various fluorescence problems, such as spontaneous emission, amplified spontaneous emission (ASE), and peripherally one may include photon trapping. The first method is mostly analytic typified by refs [1

1. G. J. Linford, E. R. Peressini, W. R. Sooy, and M. L. Spaeth, “Very long lasers,” Appl. Opt. 13(2), 379–390 (1974). [CrossRef] [PubMed]

8

8. K. Contag, U. Brauch, S. Erhard, A. Giesen, I. Johannsen, M. Karszewski, and S. A. V. Christian, “Simulations of the lasing properties of a thin disk laser combining high ouptut power with good beam quality,” in Modeling and Simulation of High Power Laser Systems IV, U. O. Farrukh, S Basu,eds. Proc SPIE 2989, p. 23 (1991).

]. The second method is numerical like numerical ray tracing, based on a merging of analytics and ray tracing embodied in refs [9

9. K. Contag, M. Karszewski, C. Stewen, A. Giesen, and H. Hūgel, “Theoretical modelling and experimental investigations of the diode-pumped thin-disk Yb : YAG laser,” Quantum Electron. 29(8), 697–703 (1999). [CrossRef]

12

12. J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

]. All of these methods contain their own particular set of assumptions. The reason for distinct approaches is the complexity of these problem typified, in part by: a random distribution of emitters, the emitted frequency lineshape, polarization, random propagation directions, temperatures and the attendant cross section dependence, and stress induced variations. Additionally, the emitted fields are subject to the geometry, pumping volume, indices-of-refraction, surface shape, and so forth. These analytic models are typically limited to simple geometries.

In the following we derive an expression for the steady state ASE intensity incorporating the emission lineshape function. This leads to steady state equations for the ASE decay time, heating, energy storage, and the nonlinear gain; it is the latter that we compare with an experiment. This approach gives an analytic expression for the ASE intensity in terms of an effective ASE gain path length that is specific to a given disk geometry, gain volume, temperatures, and so forth. We determine this length by comparing the measured nonlinear gain as a function of input power with our analytic expression. Once the gain length is extracted we find the ASE intensity, decay time and heating for this particular thin disk experiment. Thus, the gain length condenses any particular experiment into one parameter. In our particular experiment this length is about 43% of the pump spot size. This number is commensurate with expectations in that it should be on the scale of the pump spot size [12

12. J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

]. This length is strongly dependent on the disk dimensions, the presence of an anti-ASE YAG cap, gain dimensions and surface reflectivities. As just mentioned the effective gain length,, is not a derived quantity, rather it is obtained through a comparison between our theoretical results and experimental results. Thus, this quantity is a result of post-experimental evaluations.

In order to make analytic progress on the steady state ASE problem we have employed several assumptions. In the order of the following development, we first assume the lineshape is Lorentizian. This assumption is certainly limiting since, in reality, the lineshape consists of several temperature dependent lines. But, it does reveal the effects of a limited bandwidth on ASE calculations. Additionally, we, and others, are faced with a spatial integration over all source points leaving expressions that are dependent on the observation point or independent of the detector position [3

3. N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission application to Nd:YAG lasers,” IEEE J. Quantum Electron. 35(1), 101–109 (1999). [CrossRef]

,12

12. J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

]. Without some assumption at this point the calculation is no longer analytic, but numerical. Here we sidestep all geometrical limitations such as disk size, multiple reflections, boundary conditions, by assuming that the radiation is from an amplified free space point source up to an effective gain length . Thus, the solid angle integrations which include boundaries and reflections are replaced by 4π; other approaches have relaxed this condition [12

12. J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

]. This assumption has a homogenizing effect on the derived ASE quantities making them position independent. Juxtaposed to this assumption is the assumption that the gain is independent of position. Even with these assumptions our theory is consistent with experiment and our temperature simulations.

Finally, as has been done before, we treat the ASE flux as an additive cavity flux. Thus, it appears in the rate equations in the same fashion as the pumping and lasing fluxes through the populations, albeit spatial and frequency integrations.

Even though the anti-ASE cap is not a heat source, it never-the-less has a significant impact on the over all performance. Our experiments, explained in more detail in section 3 and Ref. 13

13. N. Vretenar and T. Carson, T. C. Newell, P. Peterson, “Thermal and stress characterization of various thin disk laser configurations at room temperature,” Solid State Lasers XX: Technology and Devices, Proc Vol. 7912 (2011)

, show that when the thin-disk is pumped at 3kW, a disk with an anti-ASE YAG cap has a maximum surface temperature of 120°C and the laser output is 1.4kW. However, an uncapped disk, with the same dimensions and doping, operating at the same pump power has a maximum surface temperature of 174°C with an output of 0.4kW [13

13. N. Vretenar and T. Carson, T. C. Newell, P. Peterson, “Thermal and stress characterization of various thin disk laser configurations at room temperature,” Solid State Lasers XX: Technology and Devices, Proc Vol. 7912 (2011)

].

Our gain experiment is for a water-impingement cooled thin-disk of radius 9mm, height of 200μm, and a pump beam spot size of 7mm with incident power up to 3kW; see ref [11

11. J. Speiser, “Thin disk laser – energy scaling,” Laser Phys. 19(2), 274–280 (2009). [CrossRef]

]. for more details on the pump architecture. Our experiment shows an exponential-like bending of the gain, as a function of input pump power away from the linear solution followed by a flattening of the gain, but does not show a complete gain rollover. This flattening of the gain is supposedly due to high order nonlinear processes. Our results reproduce the exponential-like gain up to the plateau for gains less than 6% or so.

2. Development of the ASE intensity

In this section we develop an equation for the ASE intensity and then apply it to decay times, heating, and the nonlinear gain. As mentioned our approach is based on incorporating the frequency content of the cross sections and gain, then showing how it impacts ASE intensity. Our analytics are rigorous as such, but rely on assumptions, which will be clearly noted as we proceed.

We begin with the equation for the emitted spontaneous flux in a frequency interval dνfrom a volume element dVwritten as [2

2. D. D. Lowenthal and J. M. Eagleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22(8), 1165–1173 (1986). [CrossRef]

,3

3. N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission application to Nd:YAG lasers,” IEEE J. Quantum Electron. 35(1), 101–109 (1999). [CrossRef]

]
dΦSE=(Λγe(ν)dν)N2τdV4πs2,
(1)
where the normalizationΛ1=γe(ν)dν, γe(ν) is the emission cross section, N2(r,z) is the upper lasing manifold population, andτ0 is the spontaneous decay time. The denominator 4πs2 expresses the point source characteristics of emission. Equation (1) contributes to the amplified spontaneous emission intensity via

dIASE=dΦSEhνexp(g(ν)ds).
(2)

The gain shape is g(ν). To continue we assume a homogeneous lineshape for the emission and absorption cross sections given by γe,a(ν)=γe,a(νs)L(ν), where the Lorentizian is

L(ν)=(Δν/2)2(ννs)2+(Δν/2)2.
(3)

Inserting this into Eq. (1) and integrating we find that the normalization Λ1=γe(νs)(Δνπ/2)where νs is the central the frequency and Δν is the HWHM. Next, assembling the above factors leads to the integrand
dIASE(r,z,ν)=hνΔν2πN2τ0dν(ννs)2+(Δν/2)2exp(β(ννs)2+(Δν/2)2)dV4πs2
(4)
where β=g0(Δν/2)2s, g0 is the gain that is assumed independent of position in the integration over ds. Since our concern is the frequency integration we are left with

νdν(ννs)2+(Δν/2)2exp(β(ννs)2+(Δν/2)2).
(5)

This can be integrated by a change of variable η=ννs followed by u=η2+(Δν/2)2 which gives

dIASE=hνsN2τ0dV4πs2I0(g0s/(2))exp(g0s/(2)).
(6)

I0 is the modified Bessel function of the first kind of order 0. The factor of one-half is a manifestation of the reduction in the gain due to the finite Lorentzian bandwidth.

In order to complete the spatial integration we must make some assumptions concerning the distribution of ASE rays. There have been several approaches to this problem [6

6. D. Kouznetsov and J.-F. Bisson, “Role of undoped cap in the scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25(3), 338–345 (2008). [CrossRef]

,12

12. J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

]. We take the tact that there are so many rays traveling in all directions reflected and transmitted at each surface that the volume element can be replaced by 4πs2ds, and that this is integrated up to an effective distance S. In an alternative ray trace approach Eq. (6) would be applied to each ray as it reflects around the disk and gain region until exiting a small area dA on the disk surface. Returning to Eq. (6) the remaining step requires the evaluation of

IASE(g)=hνsN2τ00SI0(g0s/(2))exp(g0s/(2))ds.
(7)

Referring to the integration tables in Mathematica the ASE intensity takes the form
IASE(g)=hνsΦASE(g)=hνsN2τ0Sexp(gS/2)(I0(gS/2)I1(gS/2)).
(8)
where ΦASE(g) is the ASE flux. Note, we have assumed that the population and the gain are spatially independent. In spite of the exponential divergence, Eq. (8) remains finite because of gain saturation or gain clamping. In subsection 2.3 this result will be incorporated into the gain equation which when compared to experiment yields S.

2.1 Decay time

We now address the effects of the ASE flux by considering the rate equation for the upper manifold. This allows us to extend our calculations to include the effective decay time, ASE heating, and the nonlinear gain. In the presence of ASE the upper manifold rate equation takes the form
dN2dt=(σaN1σeN2)P+(γaN1γeN2)IN2τ0(γe(r,z,λ)N2γa(r,z,λ)N1)dΦASE(r,z,λ)dν
(9)
wheredΦASE(r,z,λ) is the number of amplified fluorescence (ASE) photons per area and time per frequency interval (photon spectral flux density) and (P,I)are the pump and laser fluxes. σa,σe(γa,γe) are the absorption and emission cross section at the pump (laser) wavelength. Note, we are treating the ASE flux as an additional cavity flux coupled through the populations as has been done before [11

11. J. Speiser, “Thin disk laser – energy scaling,” Laser Phys. 19(2), 274–280 (2009). [CrossRef]

,12

12. J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

]. In general, the integrals in the last term are only solvable with the same assumptions as above: Lorentizian lineshape, emission independence of solid angle, and uniform gain . In order to be consistent with our earlier assumptions leading up to Eq. (8) we approximate the integrals in Eq. (9) as
dN2dt=(σaN1σeN2)P+(γaN1γeN2)IN2τ0gΦASE(g)+αASEΦASE(g);
(10)
this is a familiar equation [12

12. J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

]. Both the gaingand the absorptionαASE are at the ASE line center, which we assume is at the lasing frequency, see Ref. 13

13. N. Vretenar and T. Carson, T. C. Newell, P. Peterson, “Thermal and stress characterization of various thin disk laser configurations at room temperature,” Solid State Lasers XX: Technology and Devices, Proc Vol. 7912 (2011)

, and the ASE fluxΦASE(g) is given by Eq. (8). The first ASE term represents the gain of the ASE flux, which leads to an ASE decay time τASE. The last term is the absorption of the ASE flux and leads to heating, as we show below. In general, as the integrals in Eq. (9) indicate, there is an exchange between these two terms depending on the spectral content of the cross sections, the magnitude and spectral properties of the flux, the temperature, and the different upper level and lower level populations inside the pumped and unpumped volumes, and lasing regions.

We begin with the ASE gain term. By adding it to the spontaneous decay term we obtain
dN2dt=(σaN1σeN2)P+(γaN1γeN2)IN2τeff+αASEΦASE
(11)
where we define, and note the gain dependence, the effective and ASE decay rates and times as
Aeff(g)=1τeff=A0+AASE(g)=1τ0+1τASE(g).
(12a)
With

AASE(g)=gSτ0exp(gS/2)[I0(gS/2)I1(gS/2)]
(12b)

2.2 Steady state ASE heating

dN2dt=(σaN1σeN2)P+(γaN1γeN2)IN2τeff(g)+αASEΦASE(g)
(13)

This leads to the steady state heating through the heat load Q where
Q=hνp(σaN1σeN2)P+hνl(γaN1γeN2)IhνlN2τeff+αASEhνASEΦASE
(14)
where the ASE flux is again defined by Eq. (8). The pump and lasing frequencies are νp,νl, respectively.

Q˙ASE(g)=αpIpηA0Aeff(g)Sexp(gS/2)[I0(gS/2)I1(gS/2].
(16)

We later show that this term can be larger than the linear heating source seen in Eq. (15). Also, saturation can be included, however the algebra is a bit more labored.

2.3 Small signal gain with ASE

Finally, we conclude our development with the transcendental equation for the gain as a function of the input pump flux with the gain length (S) as a parameter. Like the experiment in section (3), we choose a thin-disk quasi-three level Yb: YAG where the pump traverses the gain medium 16-times, see ref [11

11. J. Speiser, “Thin disk laser – energy scaling,” Laser Phys. 19(2), 274–280 (2009). [CrossRef]

]. Since the algebra is rather dense we refer the reader to the Appendix for definitions and details. For this material Eq. (A3) gives the forward (+) laser/amplifier intensity growth as
1Ii+dIi+dz=g(z)=(γeN2γaN1)=N0ΓPγa/τeffσ+P+1/τeff,
(17)
where Γ=σaγeσeγa, γ+=γa+γe and σ+=σa+σe. The pump absorption can be written as
1Pj+dPj+dz=(σaN1σeN2)=N0σa/τeffσ+P+1/τeff=σ+γ+(g(z)ΓN0σ+).
(18)
after the steady state populations, see Eq. (A2), are inserted into Eqs. (17-18). Dividing these two equations and rearranging gives

1Ii+dIi+dz(N0σaτeff)=1Pj+dPj+dz(N0ΓPγaτeff).
(19)

This equation has to be integrated for the flux as well as the sum over the forward and reverse fluxes P=k=1M(Pk+(z)+Pk(z)), where M is the number of bounces. The subscripts denote the pass. All the intervening algebra is relegated to the Appendix and referring Eqs. (A10,13,15,22) gives

N0σaτeff(gL)=N0ΓPabsγaN0τeffσ+γ+(gΓN0σ+)L.
(20)

Simplifying the terms and solving for the gain leads to

gL=γ+PabsτeffγaN0L=γ+PabsAeffγaN0L.
(21)

This equation can be used two ways: measure the gain as a function of the absorbed power and then solve for the effective decay time, or the just the opposite where the decay time is measured [14

14. A. Antoginini, K. Schumann, F. D. Amaro, F. Ç. Birben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelilcato, L. Julien, C.-Y. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y.-W. Liu, L. Ludhova, N. Moschüring, F. Ç. Mulhauser, T. Nebel, F. Ç. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk YbYAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009). [CrossRef]

].

Finally, we assemble the above equations, Eq. (19-21), and the equation for pump depletion Eqs. (A10, A22) to obtain the transcendental equation for the gain as a function of the input pump, and ASE gain length S as
gL=11+gSexp(gS/2))(I0(gS/2)I1(gS/2))γ+σ+PinPsat×{1RpM1exp[2M(ΓN0Lγ+σ+γ+gL]}γaN0L
(22)
where the saturation flux isPsat=A0/σ+. In the following section we solve this transcendental equation for the gain as a function of input pump power with (S) as a fitting parameter to fit the measured gain. The term in brackets is the absorbed multipass pump power after M passes for a gain region of length L with losses Rpon each bounce.

3. Experiment and comparison with theory

Our procedure is to first fit our gain equation Eq. (22) to the measured gain versus input pump power curve using the gain path length S as a parameter. Afterwards we will evaluate the subsequent equations for ASE decay time, power, and heating. We note that the effective gain length should be on the order of the disk radius, because the radial direction provides more gain than paths passing through the disk top. The gain curve for the uncapped and anti-ASE capped will be different due to the difference in ASE ray density in the gain volume. Thus, there is a different value for S for every different disk and pumping configuration. Specifically, (S) is shorter for the capped disk than for the uncapped.

Experimental small-signal gain measurements were conducted by reflecting a 10W Versadisk 1030nm single-mode laser source off the 0.2mm thick Yb:YAG disk. The light impinged at near-normal incidence, and the power is measured before and after reflection. Furthermore, the beam was expanded to approximately the same size as the 9mm diameter pump beam. The disk is pumped near 940nm in a 16-pass thin-disk laser cavity. There is no output coupler to form a laser resonator. The pump is incrementally ramped and an average input and output 1030nm power measured at each step. The gain follows from the ratio of the output to input power with a 2-pass length of 0.4mm

Figure (1)
Fig. 1 Gain (%) versus input pump power (kW).
shows the measured gain along the lasing axis [15

15. D. Albach, J.-C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009). [CrossRef] [PubMed]

] in percent versus the input pump power for power less than 3kW. This was measured in a pump probe configuration. The measured gain, the red dots in Fig. (1), shows a marked decrease from the linear case (S=0 in Eq. (22)), which is the black line. The blue line labeled nonlinear, is our transcendental ASE gain equation, Eq. (22) which is fit to within 10% of the measured value up to a gain of about 6%. At this point the measured gain rolls over into a flat top while our ASE equation just continues. The measured roll over is conjectured to be due to nonlinear processes. This fitting process establishes the gain length (S) at about 43% of the disk diameter, which is in the expected range. In passing we note that we have included the temperature dependence of the cross sections due to heating.

Now that (S) is determined, Fig. (2)
Fig. 2 Normalized effective decay time, see Eq. (12).
shows the effective decay time divided by the spontaneous decay time, see Eq. (12). The value of unity occurs when the gain is zero, at about 293W in Fig. (1). For powers greater than this we see an exponential like decrease in the effective decay time, and for input pump powers lower than 293W there is an increase in the decay rate associated with absorption and photon trapping.

Next, Fig. (3)
Fig. 3 Heat loads versus input pump power, see Eq. (15,16).
presents the heat load. We show the linear contribution, which is present when there is no ASE (the first half of Eq. (15)), and the nonlinear contribution given by the second half of Eq. (15), where αp=σa(T)N0. The ASE heating is dominant depending on the pumping power. However, if lasing causes the gain to be clamped at around 2% by the losses, then the ASE heating is about a factor of 7 greater. When this heat load is introduced into our COMSOL simulation [13

13. N. Vretenar and T. Carson, T. C. Newell, P. Peterson, “Thermal and stress characterization of various thin disk laser configurations at room temperature,” Solid State Lasers XX: Technology and Devices, Proc Vol. 7912 (2011)

] we obtain the measured surface temperature, given by the formula in the discussion of Fig. 1. Our temperature and structural COMSOL model includes: the convective cooling structure, bonding layers, gain and cap, heat loads and the attendent temperature dependent constants.

Figure 4
Fig. 4 Normalized upper state population. Black is spontaneous emission; red is ASE.
shows the upper lasing level population density normalized to the total population density in percent as a function of input pump power. From Eq. (A2) for the steady state unsaturated case N2=αpPinτeff (note σ+Pτeff=.15<<1 at 3kW), Fig. 4 is for both the spontaneous decay time (black line) and the effective decay time (red line). This clearly shows the dramatic effect of ASE from the linear case. This is major contributing factor to the gain shown Fig. (1). Figure 2 shows that as the pump power increases the effective decay rate increases. This impacts the upper state population by making it flatten, see Fig. 4, which then causes the gain decrease, see Fig. 1, and the ASE amplification to decrease, see Fig. 5
Fig. 5 ASE amplification, see Eq. (8) divided by input pump intensity.
. Figure 5 is Eq. (8) divided by the input pump power using the above formula for the upper state population.

Finally, as noted in our introduction, the ASE intensity shown in Eq. (8) is uniform and independent of the detector position. Clearly, the ASE is more intense exiting the rim than from the top of the disk. Thus, using this equation to calculate the exiting powers is illusive.

4. Summary and discussion

Our steady state development of the effects of ASE relied on a Lorentzian frequency integration followed by a spatial integration over a uniform distribution of amplified point source emitters without regard to reflections. This allowed us to write an analytic equation for the ASE intensity which in turn set the stage for effective decay times and ASE heating. Our solution for the gain agrees favorably with measurements.

There have been several designs to minimize ASE. The most useful is the use of an anti-ASE cap, which in effect decreases the ASE ray density in the gain region. Akin to this is the use of an edge bevel, which again reduces the ray density in the gain region and also decreases the standing wave parasitics. To this one can add an absorbing medium to the disk rim. Another straightforward option is to increase the disk diameter and to roughen the disk surface except for the AR, HR surfaces. These random phase bumps and depressions on the surface is intended to introduce random areas where there is a random distribution of surface normal’s, which inhibits rays from traversing the gain medium more than once. Thus, this would eliminate TIRs. A further advantage in the increase in disk diameter is that it smoothes and lowers the radial temperatures and so decreases the distortion and stresses.

Appendix

The solution to a multipass pump (M bounces) and a multipass laser/amplifier (N bounces) quasi-three level system is made tractable if the system is lossless, making all bounces equivalent, and there are no nonlinear processes in the rate equations. We begin with the forward (+) and reverse (-) differential equations for the jth of M pump passes and the ith of N laser/amplifier passes written as
1Pj±dPj±dz=(σaN1σ2N2),1Ii±dIi±dz=±(γeN2γaN1) (A1a,b)
where σa,σe(γa,γe) are the absorption and emission cross section at the pump (laser) wavelength and N0=N1+N2 where N0 is the total doping concentration. Equation (A1) immediately shows that the L.H.S. is independent of the index, and Pj+(z)Pj(z)=Pj+(0)Pj(0)=Pj+(L)Pj(L),and Ij+(z)Ij(z)=Ij+(0)Ij(0)=Ij+(L)Ij(L), which are oft quoted equations.

In the pump-probe configuration the lower manifold rate equation is
N1=P(σaN1σeN2)+N2τeff
(A2)
where the sigma’s contain the temperature dependence through the partition functions. Inserting the steady state solutions of Eq. (A2) back into Eq. (A1) allows the gain equation to be written as
1Ii+dIi+dz=g(z)=(γeN2γaN1)=N0ΓPγa/τeffσ+P+1/τeff
(A3)
where Γ=σaγeσeγa,γ+=γa+γe and σ+=σa+σe. Likewise with a little algebra one can show that the forward pump equation can be written in terms of the gain as

1Pj+dPj+dz=(σaN1σeN2)=N0σa/τσ+P+1/τ=σ+γ+(g(z)ΓN0σ+).
(A4)

Equation (A3, 4) serves as the basis of our analysis. Dividing Eqs. (A3, A4) gives our working equation, see Eq. (19),

1Ii+dIi+dz(N0σaτeff)=1Pj+dPj+dz(N0ΓPγaτeff).
(A5)

It is this equation that must be integrated to obtain the gain equation for a multi-pass pump configuration. The second term is

dPj+Pj+k=1M(Pk+(z)+Pk(z)).
(A6)

On integration the first term in Eq. (A6) becomes
dPj+Pj+k=1MPk+(z)=k=1MdPj+=k=1M(Pk+(L)Pk+(0)),
(A7)
since the terms are independent of the index (j) as mentioned above. The second term can be simplified considerably by noting from the differential equations that

Pj+(z)Pj(z)=Cj=constant1Pj+(z)dPj+(z)dz=1Pj(z)dPj(z)dz.

Thus, the second term reduces to

dPj+Pj+k=1MPk(z)=k=1MdPjPjPk(z)=k=1MdPj=k=1M(Pk(L)Pk(0)).
(A8)

Combining Eqs. (A7, A8) gives

k=1M(Pk+(L)Pk+(0)Pk(L)+Pk(0)).
(A9)

The boundary condition at z=L is Pj+(L)=Pj(L) which eliminates the first and third terms. Upon expanding the remaining sum and employing the reflection identity Pj+(0)=RpPj1(0)forRp=1 gives for the second term in Eq. (A5) after integration

dPj+Pj+k=1M(Pk+(z)+Pk(z))=(P1nPout)=Pabs.
(A10)

Note, that if a lasing flux was present its evaluation would mimic the above, thus we can immediately write

dIi+Ii+k=1M(Ik+(z)+Ik(z))=(I1nIout)=Iamplified.
(A11)

We now turn to the first and last term in Eq. (A5), which serves to define the gain and the pump depletion. Upon integration of Eq. (A3) we obtain the growth for the forward flux and the depletion of the pump as
dIi+Ii+=gL,anddPj+Pj+=(σ+γ+gΓNoγ+)L.
(A12)
for a spatially independent gain. This equation along with Eq. (A10) is then returned to Eq. (20).

Usually in an experimental situation the unabsorbed pump can be measured or it is small enough to be ignored. However, if this is not possible then we calculate the absorbed pump power for a mirror reflectivity different from unity with the pump making M passes as above. To do this we first introduce the gain and depletion. From Eq. (A3) upon integrating we have
Ii+(L)Ii+(0)=exp(g(z)dz),
(A13)
and using the Ii+(z)Ii(z) conservation condition from above we get the double pass gain
Ii(0)Ii+(0)=exp(2(g(z)dz))G
(A14)
which serves to define the laser/amplifier double pass growth. And, from Eq. (A4) we have
Pj+(L)Pj+(0)=exp(σ+γ+g(z)dzΓN0γ+)
(A15)
which we cast into double pass depletion by again employing Pj+(z)Pj(z)=Pj+(0)Pj(0)=Pj+(1)Pj(1)=RpPj+(1)2. This equation combined with the pump reflection condition Pj+(0)=RpPj1(0) gives the depletion for a single bounce as
Pj(0)Pj+(0)=exp(2σ+γ+(g(z)dzΓN0σ+))D.
(A16)
where (D) expresses double pass pump depletion. This equation and the pump reflection condition lead to the stepping equation
Pi+(0)Pi(0)=RpD(Pi1+(0)Pi1(0)).
(A17)
which applied to Eq. (A4) gives

k=1M(Pk(0)Pk+(0))=Pin(1D)1(RD)M1RD.
(A18)

Finally, the last step is to obtain the output pump in terms of the input pump. To this end consider the sum

j=1Mln(Pj(0)Pj+(0))=lnP1(0)P1+(0)P2(0)P2+(0)P3(0)P3+(0)....PM1(0)PM1+(0)PM(0)PM+(0).
(A19)

This can be further reduced through the pump reflectivity condition as

j=1Mln(Pj(0)Pj+(0))=ln(1RpM1PM(0)P1+(0))=ln(1RpM1Pout(0)Pin(0)).
(A20)

Additionally, each term in the sum in Eq. (A18), or Eq. (A19), is independent of the index. Thus, we have

j=1Mln(Pj(0)Pj+(0))=Mln(Pj(0)Pj+(0)).
(A21)

Thus, Eqs. (A18-A20) give
ln(1RpM1Pout(L)Pin(0))=2Mσ+γ+(0Lg(z)dzΓN0σ+)
(A22a)
which we write as

Pout=(RpD)MRpPin.
(A22b)

Thus,
k=M(Pk(0)Pk+(0))=1D1RpD(PinRpPout)Rp0(PinPout)
(A23)
which can be written in terms of only the input pump through Eq. (A22). This completes our development of the pump absorption. In conclusion we mention that the above techniques can be used to obtain an analytic equation for the laser/amplifier extraction as a function of input pump power for M pump passes and N laser passes.

References and links

1.

G. J. Linford, E. R. Peressini, W. R. Sooy, and M. L. Spaeth, “Very long lasers,” Appl. Opt. 13(2), 379–390 (1974). [CrossRef] [PubMed]

2.

D. D. Lowenthal and J. M. Eagleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron. 22(8), 1165–1173 (1986). [CrossRef]

3.

N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission application to Nd:YAG lasers,” IEEE J. Quantum Electron. 35(1), 101–109 (1999). [CrossRef]

4.

J. Speiser, “Scaling of thin-disk lasers-influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

5.

D. Kouznetsov, J.-F. Bisson, J. Dong, and K.-I. Ueda, “Surface loss limit of the power scaling of a thin-disk laser,” J. Opt. Soc. Am. 23(6), 1074–1082 (2006). [CrossRef]

6.

D. Kouznetsov and J.-F. Bisson, “Role of undoped cap in the scaling of thin-disk lasers,” J. Opt. Soc. Am. B 25(3), 338–345 (2008). [CrossRef]

7.

D. Kouznetsov, J.-F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater. 31(5), 754–759 (2009). [CrossRef]

8.

K. Contag, U. Brauch, S. Erhard, A. Giesen, I. Johannsen, M. Karszewski, and S. A. V. Christian, “Simulations of the lasing properties of a thin disk laser combining high ouptut power with good beam quality,” in Modeling and Simulation of High Power Laser Systems IV, U. O. Farrukh, S Basu,eds. Proc SPIE 2989, p. 23 (1991).

9.

K. Contag, M. Karszewski, C. Stewen, A. Giesen, and H. Hūgel, “Theoretical modelling and experimental investigations of the diode-pumped thin-disk Yb : YAG laser,” Quantum Electron. 29(8), 697–703 (1999). [CrossRef]

10.

A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers:results and scaling laws,” IEEE J. Sel. Top. Quantum Electron. 13(3), 598–609 (2007). [CrossRef]

11.

J. Speiser, “Thin disk laser – energy scaling,” Laser Phys. 19(2), 274–280 (2009). [CrossRef]

12.

J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B 26(1), 26–35 (2009). [CrossRef]

13.

N. Vretenar and T. Carson, T. C. Newell, P. Peterson, “Thermal and stress characterization of various thin disk laser configurations at room temperature,” Solid State Lasers XX: Technology and Devices, Proc Vol. 7912 (2011)

N. Vretenar, “Room temperature and cryogenic Yb:YAG thin disk laser: single crystal and ceramic.” PhD dissertation, Center for High Technology Materials, University of New Mexico (2011).

14.

A. Antoginini, K. Schumann, F. D. Amaro, F. Ç. Birben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelilcato, L. Julien, C.-Y. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y.-W. Liu, L. Ludhova, N. Moschüring, F. Ç. Mulhauser, T. Nebel, F. Ç. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk YbYAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron. 45(8), 993–1005 (2009). [CrossRef]

15.

D. Albach, J.-C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express 17(5), 3792–3801 (2009). [CrossRef] [PubMed]

OCIS Codes
(140.3430) Lasers and laser optics : Laser theory
(140.3615) Lasers and laser optics : Lasers, ytterbium

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: August 31, 2011
Revised Manuscript: October 21, 2011
Manuscript Accepted: October 23, 2011
Published: December 1, 2011

Citation
P. Peterson, A. Gavrielides, T. C. Newell, N. Vretenar, and W. P. Latham, "ASE in thin disk lasers: theory and experiment," Opt. Express 19, 25672-25684 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-25-25672


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References

  1. G. J. Linford, E. R. Peressini, W. R. Sooy, and M. L. Spaeth, “Very long lasers,” Appl. Opt.13(2), 379–390 (1974). [CrossRef] [PubMed]
  2. D. D. Lowenthal and J. M. Eagleston, “ASE effects in small aspect ratio laser oscillators and amplifiers with nonsaturable absorption,” IEEE J. Quantum Electron.22(8), 1165–1173 (1986). [CrossRef]
  3. N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission application to Nd:YAG lasers,” IEEE J. Quantum Electron.35(1), 101–109 (1999). [CrossRef]
  4. J. Speiser, “Scaling of thin-disk lasers-influence of amplified spontaneous emission,” J. Opt. Soc. Am. B26(1), 26–35 (2009). [CrossRef]
  5. D. Kouznetsov, J.-F. Bisson, J. Dong, and K.-I. Ueda, “Surface loss limit of the power scaling of a thin-disk laser,” J. Opt. Soc. Am.23(6), 1074–1082 (2006). [CrossRef]
  6. D. Kouznetsov and J.-F. Bisson, “Role of undoped cap in the scaling of thin-disk lasers,” J. Opt. Soc. Am. B25(3), 338–345 (2008). [CrossRef]
  7. D. Kouznetsov, J.-F. Bisson, and K. Ueda, “Scaling laws of disk lasers,” Opt. Mater.31(5), 754–759 (2009). [CrossRef]
  8. K. Contag, U. Brauch, S. Erhard, A. Giesen, I. Johannsen, M. Karszewski, and S. A. V. Christian, “Simulations of the lasing properties of a thin disk laser combining high ouptut power with good beam quality,” in Modeling and Simulation of High Power Laser Systems IV, U. O. Farrukh, S Basu,eds. Proc SPIE 2989, p. 23 (1991).
  9. K. Contag, M. Karszewski, C. Stewen, A. Giesen, and H. Hūgel, “Theoretical modelling and experimental investigations of the diode-pumped thin-disk Yb : YAG laser,” Quantum Electron.29(8), 697–703 (1999). [CrossRef]
  10. A. Giesen and J. Speiser, “Fifteen years of work on thin-disk lasers:results and scaling laws,” IEEE J. Sel. Top. Quantum Electron.13(3), 598–609 (2007). [CrossRef]
  11. J. Speiser, “Thin disk laser – energy scaling,” Laser Phys.19(2), 274–280 (2009). [CrossRef]
  12. J. Speiser, “Scaling of thin-disk lasers – influence of amplified spontaneous emission,” J. Opt. Soc. Am. B26(1), 26–35 (2009). [CrossRef]
  13. N. Vretenar and T. Carson, T. C. Newell, P. Peterson, “Thermal and stress characterization of various thin disk laser configurations at room temperature,” Solid State Lasers XX: Technology and Devices, Proc Vol. 7912 (2011)
  14. N. Vretenar, “Room temperature and cryogenic Yb:YAG thin disk laser: single crystal and ceramic.” PhD dissertation, Center for High Technology Materials, University of New Mexico (2011).
  15. A. Antoginini, K. Schumann, F. D. Amaro, F. Ç. Birben, A. Dax, A. Giesen, T. Graf, T. W. Hänsch, P. Indelilcato, L. Julien, C.-Y. Kao, P. E. Knowles, F. Kottmann, E. Le Bigot, Y.-W. Liu, L. Ludhova, N. Moschüring, F. Ç. Mulhauser, T. Nebel, F. Ç. Nez, P. Rabinowitz, C. Schwob, D. Taqqu, and R. Pohl, “Thin-disk YbYAG oscillator-amplifier laser, ASE, and effective Yb:YAG lifetime,” IEEE J. Quantum Electron.45(8), 993–1005 (2009). [CrossRef]
  16. D. Albach, J.-C. Chanteloup, and G. Touzé, “Influence of ASE on the gain distribution in large size, high gain Yb3+:YAG slabs,” Opt. Express17(5), 3792–3801 (2009). [CrossRef] [PubMed]

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