A new approach to compute overlap efficiency in axially pumped solid state lasers

A new approach to compute Overlap

efficiency in axially pumped Solid State

Lasers

Rakesh Kapoor, P. K. Mukhopadhyay, and Jogy George

Center for Advanced Technology, Indore 452 013, India

[email protected]

Abstract: In this work, we are reporting a new approach to computethe overlap efficiency of end pumped solid-state laser systems. Unlikeexisting methods in which the overlap integral is computed with a lin-earize approximation near the threshold, in this method the inverse ofthe overlap integral is computed numerically in the above thresholdregime for several values of circulating fields. Now by fitting a linearcurve to this data the overlap efficiency is obtained. The effect of thebeam quality factor is also taken into account. It is demonstrated thatthe linearized approximation near the threshold can give rise to 50%error in overlap efficiency. The method was used to estimate the overlapefficiency in different types of axially pumped lasers.c©1998 Optical Society of AmericaOCIS codes: (140.3480) Lasers, diode-pumped, (140.0140) Lasers and laser optics

References

1. D.G. Hall, R.J.Smith, and R.R.Rice, ” Pump size effects in Nd:YAG lasers,” Appl. Opt., vol.19,pp.3041-3043,1980.

2. D.G. Hall, ”Optimum mode size criterion for low gain lasers,” Appl. Opt., vol.20, pp.1579-1583,1981.

3. W.P.Risk, ”Modelling of longitudinally pumped solid state lasers exhibiting reabsorption losses,”J.Opt. Soc. Amer. B, vol.5,pp. 1412-1423,1988.

4. T.Y.Fan, and Antonio Sanchez, ”Pump source requirements for end pumped lasers,” IEEE J.Quantum Electron.,vol.QE - 26, pp. 311-316, 1990.

5. Paolo Laporta and Marcello Brussard, ”Design criteria for mode size optimization in diodepumped solid state lasers,” IEEE J. Quantum Electron.,vol.QE - 27, pp. 2319-2326, 1991.

6. C. Pfistner, P. Albers, H. P. Weber, ”Influence of spatial mode matching in end-pumped solidstate lasers,” Appl. Phys. Vol. B54, pp. 83-88, 1992.

7. Y.F. Chen, T.S. Liao, C.F. Kao, T.M. Huang, K.H. Lin, and S.C. Wang, ” Optimization of fibercoupled laser diode end pumped lasers: Influence of pump beam quality,” IEEE J. QuantumElectron.,vol.QE - 32, pp. 2010-2016, 1996.

8. A. E. Siegman ,”Lasers”, (University Science Book, Mill Valley, CA, 1986).

9. A. E. Siegman and Steven W. Townsend, ”Output beam propagation and beam quality from a

multimode stable-cavity laser,” IEEE J. Quantum Electron. Vol. 29, pp. 1212-1217, 1996.

1. Introduction

In the last decade, the development of efficient, high power diodes and new laser ma-terials has generated considerable interest in diode pumped solid state lasers. Axiallypumped lasers offer higher efficiency and better beam quality than side pumped lasers.The qualities of axially pumped lasers make them more suitable for applications in lasermedicine, high-density optical storage, color display, and optical testing. For optimumdesign of an axially pumped solid state laser, one of the most important parameter thatdetermines the laser efficiency and the output power is the spatial overlap between the

(C) 1999 OSA 13 September 1999 / Vol. 5, No. 6 / OPTICS EXPRESS 125#9110 - $15.00 US Received July 20, 1999; Revised September 06, 1999

pump beam and the laser mode. The influence of the pump and laser mode size on thelaser threshold and slope efficiency has been investigated by several authors [1, 2, 3, 4, 5].Most of these workers use an average pump spotsize. Although this approach simplifiesthe problem of obtaining analytical expressions for optimisation, the effect of some pa-rameters, such as the divergence of pump beam and waist location inside the crystal cannot be taken into account. In some reports the overlap integral is computed numerically[6, 7] and the pump beam divergence and waist location have been taken into account.But in all these works, the overlap integrals are based on a linearized approximation nearthe threshold [5] and this approximation is only valid if the circulating intensity in theresonator is negligible in comparison to the saturating intensity. In practical conditions,this assumption is not valid above the threshold regimes of pump power.

In this work, we are reporting a new approach to calculate the overlap efficiencyin axially pumped cw or quasi cw solid state lasers. In this approach we applied thelinearized approximation in the above threshold regime. We first computed the overlapintegral numerically for different values of circulating intensity. Then inverse of theoverlap integral is plotted as a function of circulating field. Now by fitting a linear curveto this data, two coefficients are obtained and these coefficients are further used toestimate the overlap efficiency. We applied this method to calculate the mode overlap forboth circular and elliptic pump beams. We have tried to find out how the pump beamwith different beam quality-factor M2 can affect the efficiency of an axially pumpedSolid State Laser. We have also tried to find out the effect of pump waist size on theoverlap efficiency for a constant M2 value.

2. Theoretical Analysis

According to rate equation analysis, if Ip(z) is the incident pump intensity at any planeinside the gain medium and the reflectivity of the two end mirrors is equal to unity, thenin steady state, if we neglect the standing wave effects, the saturated gain coefficientg(z) at that plane can be given as[8]

g(z) =ηqαaIp(z)/Isat(1 + 2Icirc/Isat)

, (1)

Where αa is the absorption coefficient of the gain medium at the pump intensity and Isatis the saturation intensity of the gain medium, ηq is the product of quantum efficiencyand quantum defect of the system. In most of the four level solid state lasers the quantumefficiency is approximately equal to one therefore ηq is given by

ηq ≈ωm

ωp, (2)

where ωm is the laser frequency and ωp is the pump beam frequency. Eq.1 is valid onlyfor the plane wave approximation, in real lasers the pump intensity as well as the modeintensity inside the gain volume, is a function of position. Therefore, the gain coefficientinside the gain volume is also a function of position. If we neglect the saturation ofthe pump power absorption, then the spatial distribution of pump intensity inside anaxially pumped gain volume can be defined as

Ip(x, y, z) =Pp

Ap(z)fp(x, y, z)Exp(−αaz), (3)

and

Ap(z) =

∫ ∫fp(x, y, z)dxdy, (4)


where Pp is the incident pump power, fp(x, y, z) is the spatial profile function at a planez inside the gain medium and Ap(z) is the pump area at the same plane. If Pcirc isthe total circulating power inside the resonator, the Icirc(x, y, z) at any point inside thegain medium can similarly be defined as

Icirc(x, y, z) =Pcirc

Am(z)fm(x, y, z), (5)

where fm(x, y, z) is the spatial mode profile function at a plane z inside the gain mediumand Am(z) is the mode area at the same plane. If g(x, y, z) is the value of the saturatedgain coefficient at a particular position (x, y, z) inside the gain volume, then the changein circulating power in one round trip is defined as

∆Pcirc =

∫ ∫ ∫dI(x, y, z)dxdy = 2

∫ ∫ ∫ l0

g(x, y, z)Icirc(x, y, z)dzdxdy, (6)

where l is the length of the gain medium. Now for the small gain approximation thetotal saturated round trip gain G can be defined as

G =2∫ ∫ ∫

g(x, y, z)Icirc(x, y, z)dV

Pcirc(7)

Therefore, the total gain inside a resonator can be computed as follows

G = 2ηqαaP′p

∫ ∫ ∫Exp(−αaz)fp(x, y, z)fm(x, y, z)

Ap(z)Am(z)(1 +2P ′circfm(x,y,z)

Am(z))dV, (8)

The normalised circulating power P ′circ is defined in terms of saturation intensity Isatas

P ′p = Pp/Isat

P ′circ = Pcirc/Isat = 〈Am〉I′circ, (9)

where I ′circ is the normalized circulating intensity, 〈Am〉 is the average mode area insidethe gain medium and is defined as

〈Am〉 =1

l

∫ l0

∫ ∫fm(x, y, z)dxdydz, (10)

In steady state, the total gain inside a resonator is equal to total loss therefore

G = 2α0z + ln(1

R1R2) ≡ L+ T1 + T2, (11)

where R1 and R2 are the reflectivities of the two mirrors, and T1 and T2 are the cor-responding transmissions. α0 is the absorption coefficient at the lasing wavelength. Allthe intracavity losses are clubbed in term L. If T2 is the output coupler transmission,the output power from a laser is given as

Pout = T2Pcirc, (12)

With the help of Eq.8, Eq.11 and Eq.12, one can compute the laser output power.Eq.8 looks quite complicated to compute, as the knowledge of the pump distributionfunction and mode distribution function inside the gain volume is necessary. In most of


the practical conditions the output power of a laser changes linearly with input power,this is only possible if the total saturated round trip gain is approximated as follows

G = 2ηqαaP′p[

C

(1 + 2DI ′circ)], (13)

where C and D are constants. By comparing Eq.13 with Eq.8, the value of the integralin Eq.8 will be given as

∫ ∫ ∫Exp(−αaz)fp(x, y, z)fm(x, y, z)

Ap(z)Am(z)[1 +2〈Am〉I′circfm(x,y,z)

Am(z)]dV =

C

1 + 2DI ′circ, (14)

Here the constant C will have inverse of length dimensions and D is a dimensionlessquantity. Now from Eq.11 and Eq.13, we get the following relation

2ηqαaP′p[

C

(1 + 2DP ′circ/〈Am〉] = (T + L), (15)

where T = T1 + T2. The circulating power, that must built up inside the resonator inorder to saturate the gain factor down to where it can just be equal to total cavitylosses, is given by

Pcirc =Isat〈Am〉

2D[2ηqαaP

′pC

T + L− 1], (16)

The output power from the laser will be given as

Pout =T2ηqαaPpC〈Am〉

D(T + L)−T2Isat〈Am〉

2D, (17)

If we define the overlap efficiency ηo as follows

ηo =αaC〈Am〉

ηaD, (18)

where ηa is the absorption efficiency of pump power inside the gain medium. The outputpower equation can be rewritten as

Pout =T2

(T + L)ηqηaηoPp −

T2Isat〈Am〉

2D, (19)

with the help of Eq.19, one can easily find the threshold power. It will correspond tothe pump power for which the output power is equal to zero. Therefore, the thresholdpower Pth can be given as

Pth =Isat(L+ T )

2ηqαaC, (20)

the expression for the slope efficiency m can also be obtained from Eq.19.

m =ηoηaηqT2

(L+ T ), (21)

it is clear from Eq.20 that for any given resonator conditions the threshold pump poweris proportional to the factor 1/αaC. If we compare this factor with the match functionwhich describes the spatial overlap of pump beam and resonator modes [7], then thevalue of the match function F can be given as

F = 2αaC〈Am〉, (22)


Considering a single transverse mode TEM00, fm(x, y, z) can be given as

fm(x, y, z) = Exp[−2x2

ω2mx(z)−2y2

ω2my(z)], (23)

where ωmx and ωmy are the radii of the beam along the x-axis and y-axis respectively.The pump profile function fp(x, y, z) for a diode laser can be written as

fp(x, y, z) = Exp[−2x2

ω2px(z)−2y2

ω2py(z)], (24)

where ωpx and ωpy are the radii of the beam along the x-axis and y-axis respectively.The expression for the beam radius can be given as

ωp(z) = ωp0[1 + (M2zλ/πω2p0)

2]1/2, (25)

where λ is the wavelength in the medium, ωp0 is the radius at the waist, and M2 is the

beam quality factor. The value ofM2 and ωp0 can be different for the x-axis and y-axis.For a particular beam, the beam quality factor M2 is a constant and is related to thewaist radius ω0 and far field divergence angle θ of a beam as [9]

ω0θ =M2λ

π, (26)

With the knowledge of the pump-beam quality factor, pump waist radii and modewaist radii along with their respective waist positions inside the resonator, the integralgiven in Eq.14 can be solved numerically for various values of normalised circulatingintensity I ′circ. In the linearized approximation near the threshold, both sides of Eq.14are expanded into a series. The values of the constant C and D are obtained only fromthe first two terms of the series. While in the proposed method no such approximationis made. The value of the inverse of the integral can be plotted as a function of I ′circ andby fitting a linear curve the values of the constant C and D can be obtained. There afterthe value of overlap efficiency and threshold power for a given setup can be computedwith the help of Eq.18 and Eq.20. Although by neglecting the higher order terms ofthe series in the linearized approximation near the threshold, the computed value ofthe threshold pump-power [5, 7] will not be affected much, but the same is not truefor overlap efficiency. The overlap efficiency computed with such approximations will bequite erroneous.

3. Simulations

To verify these facts an elliptic pump spot and an elliptic TEM00 mode profile wereconsidered. The absorption coefficient of a 0.5mm. thick gain medium is αa = 4.2mm

−1.The pump beam parameters along both axes were ωpx = 100µm,M

2x = 60, ωpy = 10µm,

and M2y = 2.5. Mode radii (1/e2) are ωmx = 40µm, and ωmy = 50µm. With these

parameters, we computed the variation of the match function and overlap efficiencywith the position of the pump-beam focal plane in the gain medium. The computationswere done with both the linearized approximations near the threshold and the proposedmethods. Fig.1(a) and Fig.1(b) show the variation of the match function and overlapefficiency with focal plane position. The threshold pump-power of a laser is inverselyproportional to the match function. Therefore, it can be seen that the values of thematch function obtained with both methods are almost the same. However the valuesof overlap efficiency computed with the linearized approximation near the threshold areabout 25% less than the values obtained with our method. This corresponds to about


50% error. We also plotted in Fig.2 the variation of the inverse of the integral in Eq.14with respect to I ′circ for different pump beam position. It can be seen that in all the casesthe curves fit to the straight lines with a regression coefficient not less than 0.9986. Thisvalidates our assumption of the linear approximation for the overlap integral in Eq.13.Now we applied this method to a 0.5mm. thick axially pumped Nd : Y V O4 microchiplaser with absorption coefficient αa = 4.2mm

−1 and n = 2.165. The variation of theoverlap efficiency with respect to the mode radius was computed. The laser mode wastaken as a circular TEM00 mode. The pump-beam was considered to be elliptic with twodifferent spots sizes M2x = 40, and M

2y = 2.0. The results are shown in Fig.3. We have

also computed the variation of the overlap efficiency with respect to pump-beam spotsize in an axially pumped Nd : Y AG laser. A crystal of thickness 5mm with absorptioncoefficient αa = 0.6mm

−1 and n = 1.82 was considered. The computations were donefor a circular pump beam with different values of beam quality factor M2 and differentvalues of laser-mode waist radius ωm0. The results are shown in Fig.4. Values of overlapefficiency with M2 = 1 correspond nearly to the constant pump-beam spot along thegain medium.

4. Conclusion

We have demonstrated with computer simulations that the overlap efficiency computedwith the methods based on linearized approximation near the threshold can give a value25% less than the one computed in above threshold regimes. Our method is based on thecomputation of the total overlap integral for several values of circulating field intensity.The method also takes care of pump beam divergence and pump beam waist position.Therefore, the overlap efficiency computed with this method is more accurate. Althoughthe analysis is used for a four level system, but it is equally valid for a three level systemin linear pump power absorption regime.

-2 -1 0 1 20.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40 Proposed method Old method

Mat

ch fu

nctio

n F

Focal plane position in mm

(a)


-2 -1 0 1 2

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75 Proposed method Old method

Ove

rlap

effic

ienc

y η 0

Focal plane position in mm

(b)

Figure 1. Variation of (a) match function and (b) overlap efficiency withpump-beam waist position with respect to entrance plane of gain medium. Thepump beam profile is taken as an elliptic profile. The match function and overlapefficiency is computed with both, the linearize approximation near threshold andthe proposed method.

0 5 10 15 200.00.51.01.52.02.53.03.54.04.55.05.56.06.57.07.58.08.59.0

z = 2.0 mm

z = 1.6 mm

z = 1.2 mm

z = 0.8 mm

z = 0.0 mm

Inve

rse

of In

tegr

al in

a.u

.

Normalised Circulating Intensity I'circ

Figure 2. Variation of inverse of overlap integral with normalised circulating fieldintensity in different pump beam conditions. A linear fit to these plots gives thevalue of constants C and D. Solid lines are the linear fit to the plotted data. z isthe distance of pump beam waist from the entrance plane. The pump beam spot inthe gain medium increases with z.


50 100 150 200 250 3000.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

ωp = [100,10] µm

ωp = [200,20] µm

Ove

rlap

Effi

cien

cy η 0

Mode radius ωm in µm

Figure 3. Variation of match function with mode waist radius ωm0 for two differ-ent values of pump waist sizes. The pump beam profile is taken as an elliptic profileand ωp0 = [ωpx, ωpy].

0.0 0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.8

1.0

M2 = 250

M2 = 100

M2 = 50

M2 = 1

ωm0

= 100 µm

Ove

rlap

Effi

cien

cy η o

ωp0

(mm)

(a)

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

M2 = 100

M2 = 250

M2 = 50

M2 = 1

ωm0

= 300 µm

Ove

rlap

Effi

cien

cy η 0

ωp0

(mm)

(b)


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

M2 = 250

M2 = 100

M2 = 50

M2 = 1

ωm0

= 500 µm

Ove

rlap

Effi

cien

cy η 0

ωp0

(mm)

(c)

Figure 4. Variation of overlap efficiency η0 with pump waist radius for four differ-ent values of beam quality factorM2. (a) The mode waist radius ωm0 = 100µm, (b)The mode waist radius ωm0 = 300µm, (c) The mode waist radius ωm0 = 500µm.


A new approach to compute overlap efficiency in axially pumped solid state lasers

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