Modeling of diode pumped laser with pump dependent diffraction loss

15 August 2000

Optics Communications 182 Ž2000. 413–422 www.elsevier.comrlocateroptcom

Modeling of diode pumped laser with pump dependent diffraction loss Jan K. Jabczynski ´ ) Institute of Optoelectronics, Military UniÕersity of Technology, ul. Kaliskiego 2, 00 908 Warsaw, Poland Received 15 December 1999; received in revised form 21 June 2000; accepted 26 June 2000

Abstract The model of the end pumped laser with pump dependent diffraction loss is elaborated. The effects connected with thermal guiding caused by non-uniform pump density distribution are considered. Simple formulae describing output power in the fundamental mode are derived. The ABCD model of a cavity with internal lens and thermal lensing is applied to optimize the output characteristics of such a laser. Two types of cavities Žwith and without an internal lens. are analyzed and for both cases the best configurations, with respect to the maximum of output power, are found. The pump induced diffraction loss is found to be the main effect limiting output power in fundamental mode at higher pump levels. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Coherent pumping; Diode pumped solid state lasers; Thermal guiding

1. Introduction Although the theory of lasers pumped by laser beams was formulated much earlier w1–3x, its practical application has been started since advent of efficient high power laser diodes in the late 80’s. Despite great number of both theoretical and experimental works devoted to this topic w4–20x some important problems Že.g., role of joined gain and thermal guiding effects, optimum pump to mode beam sizes ratio, etc.. require additional elucidation till now. One of the main conclusions of the theory of laser pumped by laser beam was that the performance Ži.e.

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efficiency and beam quality. of such lasers depends on guiding effects connected with non-uniform pump density distribution inside active medium. In peculiarity, the ratio of averaged pump area to laser mode area defined in active medium is crucial for optimization of such lasers. Complete analysis of laser beam pumped lasers is possible for particular cases only using numerical models with 3D codes for solving the problems of energy transfer and beam propagation inside a cavity. For the given inversion density, proportional to the absorbed pump power density, the space-dependent rate equation model can be applied w4–6x enabling the analysis of steady state as well as simple time-dependent problems w11x. The system of space-dependent rate equations can be significantly simplified if the pump and laser mode beams have the radial Gaussian profiles. As a rule, the changes of the laser mode transverse profile

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J.K. Jabczynskir Optics Communications 182 (2000) 413–422 ´

inside active medium can be neglected, whereas the pump profile can change significantly inside it. The pump beam has much larger beam product than laser mode. In order to satisfy requirement of minimization of averaged pump area w7,8,12x the Rayleigh range of the pump beam should be shorter than active medium length. Thus, in majority of such models w4,8–10x the special functions Žcalled overlap integrals. are used in the final formulae on the output power. However, if we assume the known pump area averaged over a rod Žfor generally asymmetrical Gaussian pump beams the appropriate model was given in Ref. w12x., it is possible to obtain very simple formulae describing the output power in dependence on parameters of pumping beam and cavity w7,8x. Brussard and Laporta showed in Ref. w8x, assuming Gaussian shape of pump and mode beams and absence of reabsorption and diffraction losses, that the slope efficiency increases with increase in mode to pump area ratio. However, in cavities with physical apertures much wider than the laser mode width, diffraction loss occurs as a result of thermal and gain guiding effects, even for low pump densities. The gain guiding w13–16x and gain related effects w16x slightly modify the mode structure, changing the Rayleigh range of cavity. The complex Žwith non zero imaginary part. waveguide, created by gain and thermal guiding acts as a diaphragm of complex transmittance causing the discordance of mirror and wave front surfaces and results in diffraction loss. For single longitudinal mode microchip laser, the additional detuning loss occurs as a result of index guiding and anti-guiding effects w14–16x. It was demonstrated in numerous experimental works w17,18x that in contradiction to the conclusion of the Brussard–Laporta model w8x, optimum ratio of laser mode to pump beam areas should be less than 1. The simplest explanation of such a result is the increased role of thermally induced non-parabolic aberration for laser mode larger than pump beam. We intend to focus this paper on presentation of the simplified, phenomenological model of such a laser with thermally induced diffraction loss which can be applied to an analysis of multi-element cavities. In Section 2, simple formulae describing the output power of the end pumped laser and the model of thermal axicon aberration enabling estimation of diffraction loss, valid for such a laser, were pre-

sented. In Section 3, properties of two the simplest cavities were examined theoretically and experimentally and conclusions were derived in the last Section.

2. Model of end pumped laser with pump dependent diffraction loss We intend to elaborate a simple theoretical model of end pumped laser enabling the analysis of the following effects: Ø pump dependent mode matching efficiency Ø several types of losses including pump dependent diffraction loss Ø geometry of cavity including thermal lensing. We restrict our consideration to the oscillators working in fundamental mode, moreover, we assume four-level scheme of laser action and near threshold approximation. In this case the total cavity round trip loss g can be divided into two components: the linear pump independent loss L lin and the pump dependent diffraction loss Ldiff Ž P .

g s L lin q Ldiff Ž P . .

Ž 1.

To fulfill energy conservation law in near threshold approximation at steady state condition, the total power dissipated inside cavity, as a result of linear and diffraction losses, should be equal to pump power above threshold multiplied by excitation efficiency w19x as follows

g Pint s h Ž P y Pth . ,

Ž 2.

where P is the incident pump power, Pth is the threshold pump power, and h is the excitation efficiency given by

h s hphm ,

Ž 3.

where hm is the mode matching efficiency and hp is the pumping efficiency expressed as

hp s habshdelh St ,

Ž 4.

where hdel is the delivering efficiency of pump radiation from pump source to active medium, habs is the absorption efficiency of pump radiation inside active medium, and h St s l prll is Stokes’ efficiency equal to ratio of the pump wavelength l p to the laser wavelength l l .

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The mode matching efficiency hm depends on spatial distribution of the absorbed pump density and laser mode profile inside active medium w4–10x. To estimate this dependence we use the simplified formula on hm . It was derived by Laporta and Brussard w8x for end pumping geometry and circularly symmetric beams of Gaussian intensity shapes for both pump and laser mode, as follows 1 q 2 mp

hm s

Ž1 q m p .

2

,

Ž 5.

where m p s A prA m , A m is the laser mode area and A p is the pump area averaged over active medium. In the same way we estimate the threshold pump power Pth w4,5,8x Pth s g

Isat

hp

Ž Am q Ap . ,

Ž 6.

where Isat is the saturation power density given by Isat s

hn l

st

,

Ž 7.

where h is the Planck’s constant, n l is the laser frequency, s is the emission cross section and t is the lifetime of upper laser level. To determine the output power we can use well known formula Žsee e.g. Ref. w19x. Pout s

TOC

g

h Ž P y Pth . ,

Ž 8.

where TOC is the output coupler transmission. Note, that in order to determine mode matching efficiency, threshold and output powers we should know the fundamental mode parameters and averaged pump area. Thus, in the first step of our analysis we should determine the cavity Rayleigh range being a function of the pump power via thermal lensing. The value of averaged pump area depends on pump beam parameters and the ratio of length of active medium to its absorption length Labs for a pump wavelength. According to the model presented in w12x the minimal value of pump area for long rods Žcomparing to absorption length. is given by A p G 2 Mp2l p Labs ,

Ž 9.

where Mp2 is the averaged beam quality parameter of a pump beam.

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2.1. Thermal axicon aberration model of pump induced diffraction loss To complete our model we should determine the dependence of diffraction loss on pump power. Although, as it was mentioned in Section 1, the diffraction loss occurs as a result of the joined gain and thermal guiding effects, we limit the scope of our considerations to diffraction loss induced by thermal aberrations of active medium because of two reasons. First, the gain guiding effects require much more tighten focusing of pump beam Žwith spot width of several dozens micrometers typical for microchips. and are negligible comparing to the thermal guiding effects for wider pump widths. The second reason is absence of gain guiding outside pump volume, where non-parabolic temperature profiles exist. To determine the pump dependent diffraction loss we should precisely examine the thermal lensing magnitude and non-parabolic thermally induced aberration contents. As a result of non-homogeneous temperature distribution inside active medium, caused by heat generation of absorbed pump power, the thermally induced distortion of active medium occurs. We are interested in end pumping schemes with a rod diameter significantly larger than the pump width. Because the rod diameter is much larger than pump and laser mode widths, the effect of diffraction loss caused by the rod aperture can be neglected. We can distinguish two regions of rod section: one region inside pump beam of the averaged radius Wp and the other outside it. For such a case the transverse temperature profile can be approximated by a parabolic-logarithmic shape w19–21x. The phase aberration can be divided into parabolic and non parabolic component. The first one causes occurrence of the paraxial thermal lensing of optical power Pt given by the following formula: Pt s

habs x T k h K c Ap

P,

Ž 10 .

where K c is the thermal conductivity of active medium, k h is the heat conversion factor and x T is the total temperature dispersion coefficient of active medium. The x T consists of temperature coefficient of refractive index d nrdT, the component caused by

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facet bending as a result of non homogenous thermal elongation and the part caused by stress induced birefringence of refractive index. According to theoretical and experimental investigations made by Pfister et al. w22x and confirmed by Agnesi et al. w23x, the role of stress induced birefringence and depolarization loss can be neglected for limited pump densities Žless than 5 kWrcm2 .. Thus the total temperature dispersion coefficient can be given:

xT s

dn dT

q Ž 1 q Õp . Ž n y 1 . a T ,

Ž 11 .

where a T is the linear expansion coefficient of active medium and Õ P is its Poisson ratio. The optical power Pt can be treated in terms of the Ist order optics as additional optical power of active medium, and it results in the changes of the Rayleigh range of a cavity in dependence on absorbed pump power but does not directly introduce additional loss. However, it is well known from practice, that with increase in heat load the output power and efficiency can decrease significantly till generation fading, especially for longer cavities. Thus, we intend to find a simple phenomenological model of non-parabolic thermally induced aberration. The shape and magnitude of thermal aberration depend on several parameters as pump density distribution, properties of thermal contacts, etc. The exact phase aberration function can be found through integrating optical path length over active medium with a thermally induced change of refractive index w10,20x or solving ray equation for such a type of thermally induced refractive index profile w21x. For high temperature differences of several dozens or hundreds Kelvins, typical for high heat load side pumped lasers, Hodgson and Weber proposed in Ref. w24x the model of radially variable thermal lens as a result of temperature depending changes of thermal conductivity of a rod. In such a case we have highly aberrated thermal lens described by the IVth order spherical aberration function even inside a pump volume. As it can be shown by means of a simple calculation for the absorbed pump power of 10 W and pump diameter of 0.5 mm the resulting temperature change is of the order of a few K thus, we can assume w22x that inside a pump area near perfect parabolic phase aberration occurs. However, outside a pump area there is a logarithmic temperature distri-

bution resulting in similar shape of a phase aberration function. It is proved experimentally w20,25x that the main lobe of output beam generated in such a laser is surrounded by the higher orders weak diffraction rings for higher heat load. Such intensity distributions are similar to Bessel beams distribution which can be generated by a convergent wavefront of a conical component w26x. Similar shape of wavefront distortion can be observed when a plane wave passes through glass axicon with a curved vertex. Thus, we propose to call it a thermal axicon aberration. Because a laser mode is located inside a pump beam and in the nearest exterior zone, the logarithmic profile can be approximated by a linear function. Moreover, we require that the phase function and its derivative should be continuous at the edge of a pump volume. Such assumptions lead to formulation of the following model of a thermal axicon aberration function WTA Ž x ,Ta . s Ta

½

x2 2 xy1

for x - 1 , for x G 1

Ž 12 .

where x s rrWp , r is the distance to the optical axis, and Ta is the magnitude of aberration given by Ta s

b T Wp2 Pt ll

,

Ž 13 .

where b T is the correction coefficient Žin our calculations we take b T s 1r2.. Note that the parabolic component of thermal axicon aberration describes paraxial thermal lensing effect occurring inside a pump area, whereas linear Žor conical. component describes net phase aberration outside a pump area. The influence of thermal axicon and the IVth order spherical aberration functions on beam propagation factor was investigated w27x. It should be noted that such a simple model is valid only for limited densities of heat sources. As it was shown in Ref. w28x the nonlinear dependence of thermal lensing optical power on pump power is observed for higher pump densities what is manifested in a highly aberrated shape of the wavefront inside pump area, too. Clarkson and co-workers proposed in Refs. w18,28x a strategy leading to diminution of influence of such thermal aberrations on laser efficiency and beam quality. Similar conclusion can be drawn from the analysis presented in the next section. Our approach

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consists in separation of paraxial thermal lensing, the magnitude of which is used for calculation of Rayleigh range of bare, ‘aberration-free’ cavity and estimation of diffraction loss caused by residual non-parabolic aberration. To determine the effect of non-parabolic phase aberration, the parabolic component was subtracted from WTA giving the following formula WnpTA Ž x ,Ta . s Ta

0 2 y Ž x y 1.

½

for x - 1 . for x G 1

Ž 14 .

Note, that non-parabolic component ŽFig. 1. has opposite sign than parabolic one and decreases the wavefront curvature for the outer region of a laser mode. For such the phase aberration function WnpTA we can determine the Strehl ratio of an incident Gaussian beam with the radius Wr w29x as follows 2

b

SR Ž a,b,Ta . s

H0

Fig. 2. Logarithmic diffraction loss in dependence on mode radius to pump radius ratio Wr r Wp for several thermal axicon aberration magnitudes.

exp Ž iWnpTA Ž ax ,Ta . y x 2 . xd x b

H0 exp Ž yx

2 2

,

. xd x Ž 15 .

Ldiff Ž a,b,Ta . s 1 y SR Ž a,b,Ta . .

where r2 a s WrrWp s my1 , b s R rodrWr , p

in numerical experiments that further simplifications of formula Ž15. are not allowed, because the magnitude Ta can be quite large. Knowing the Strehl ratio we can calculate the diffraction loss Ldiff w30x

Ž 16 .

Wr is the mode radius inside active medium, and R rod is the radius of active medium. We have found

Ž 17 .

The diffraction loss depends on pump power via dependence of Wr and Ta on the thermal lensing optical power Pt . It is interesting to examine the dependence of Ldiff on a ratio of the laser mode to the pump radii a ŽFig. 2.. Let us notice that logarithmic diffraction loss increases nonlinearly with increase of a. To achieve the efficient output power for a given pump power we should precisely examine the thermal lensing magnitude. Knowing its value, we have to find such cavity parameters which minimize diffraction loss for the given pump load. 3. Investigations of laser performance under high level of pump dependent aberrations

Fig. 1. Thermal axicon aberration vs. relative distance to optical axis.

Let us restrict our consideration to a simple cavity with the only one internal lens of the optical power Pf placed at the distance l 0 to the active medium and l 1 to the flat output coupler OC ŽFig. 3., respectively. The thermal lens of the optical power Pt is located in the middle of active medium. Such scheme is typical for folded cavity of V-shape. In particular

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Fig. 3. Scheme of cavity with internal lens of optical power Pf , and active medium AM of thermal lensing optical power Pt .

case, for Pf s 0, we have simple cavity with thermal lensing solely. The round trip ABCD matrix parameters Žstarting from the output coupler plane OC. for such a cavity are given by A s y Ž x y l eqv Pt . y l eqv Pf , B s Ž 2 x y l eqv Pt . l eqv , C s yy Ž 2 Pf y yPt . , D s xy y Ž Pf q yPt . l eqv ,

Ž 18 .

where

thermal lensing function Pt in dependence on pump power P. The magnitude of thermal lensing depends on several technical parameters such as: diameter of pump caustics, sizes, dopant level and quality of active medium, sizes and quality of heat sink contacts, etc. To be close to our experiments, we assume that Pt w1rmx s 0.7P wWx. We have obtained such approximate relation between incident pump power and optical power of thermal lens for Nd:YAG rod Ž3 mm diameter and 5 mm length. pumped by 10 W fiber coupled diode with pump beam diameter of about 0.7 mm. Moreover, we take the value of passive linear loss equal to 0.03 and OC transmission equal to 0.15. 3.1. InÕestigations of caÕity with thermal lensing solely In such a case we want to examine the dependence of cavity parameters and output power on cavity length and pump power. The dependences of output power on cavity length for several pump levels are shown in Fig. 4. The optimal cavity length decreases with increase of pump power. For our particular case Ži.e., the given thermal lensing dependence on pump power. for the pump power of 12 W

x s 1 y Pf l 0 , y s 1 y Pf l 1 , l eqv s l 0 q l 1 y Pf l 1 l 0 .

Ž 19 . Further, using well known ABCD method w30,31x, we can determine the Rayleigh range of the cavity Z R and the mode magnification m as follows ZR s

ms

pWc2

ll Wr

ž / Wo

s

)

l eqv Ž 2 x y l eqv Pt . y Ž 2 Pf y yPt .

2

s x 2 q yl eqv

,

2 Pf y yPt 2 x y l eqv Pt

Ž 20 .

.

Ž 21 .

Let us notice that both radii Wr and Wo depend on the pump power via the optical power of the thermal lensing Pt . Our particular task is to find the optimal cavity configuration for a pump unit emitting up to 15 W. Let us assume that the pump beam radius Wp , averaged over active medium, is equal to 0.35 mm. To start analysis we should known, moreover, the exact

Fig. 4. Dependence of output power on cavity length for several pump powers of 5 W Žcontinuous curve., 10 W Ždashed curve. and 15 W Ždotted curve.; cavity with thermal lensing solely.

J.K. Jabczynskir Optics Communications 182 (2000) 413–422 ´

Fig. 5. Dependence of output power on pump power for several cavity lengths of 20 mm Žcontinuous curve., 80 mm Ždashed curve. and 110 mm Ždotted curve.; cavity with thermal lensing solely.

the optimal cavity length is 80 mm. The change of cavity length of 5 mm Žabout 5% of length. causes drop of maximum power of about 5% in our case. The output power dependences on pump power for several cavity lengths were shown in Fig. 5. The

419

shape of output power dependence on pump power depends on a cavity length and can differ significantly from linear dependence predicted by elementary theory of end pumped laser. One of the major properties of such characteristics, frequently observed in practice, is the ‘saturation’ of output power for longer cavities as a result of increase of diffraction loss. As it was shown in Fig. 6, the mode matching efficiency increases, however, stronger increase in diffraction loss causes the diminution of output power for cavity length longer than 100 mm. Note, that the above characteristics describe the fundamental mode output. To compare these theoretical results with experimental ones we measured the same power dependences in a real laser simultaneously with M 2 parameter for each pump level ŽFig. 7.. The Nd:YAG rod of 3 mm diameter and the length of 5 mm was used as the active medium and the pump diameter averaged over rod was 0.7 mm. According to the estimation proposed by Siegman w30x the ratio of the pump to mode area m p was taken as the theoretical value of M 2 . It was shown in Fig. 7 the optimal cavity length corresponds to pump to mode ratio about 2. For such length we achieve in experiment near fundamental mode and maximum of output power. The output power starts to fade for

Fig. 6. Dependence of pump to mode ratio Žsolid line., mode matching efficiency Ždashed line. and diffraction loss Ždotted line. on cavity length for pump power 12 W; cavity with thermal lensing only.

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Fig. 7. Dependence of output power in fundamental mode, Žcontinuous curve-theory, triangles experiment points.; pump to mode ratio Ždashed curve., circles: experimental M 2 points. on cavity length for 10 W; cavity with thermal lensing only.

longer cavities till breaking out the generation for a cavity length corresponding to the end of its stability range. It is evident that decrease in output power is caused by increase in laser mode width inside active medium and resulting increase in diffraction loss ŽFig. 6..

3.2. InÕestigations of caÕity with thermal lensing and internal lens

Fig. 8. Dependence of output power on pump power for several cavity lengths of 330 mm Žcontinuous curve., 360 mm Ždashed curve. and 390 mm Ždotted curve.; cavity with thermal lensing and internal lens of Pf s10 my1 .

Fig. 9. Dependence of output power on cavity length for several pump powers of 5 W Žcontinuous curve., 10 W Ždashed curve. and 15 W Ždotted curve.; cavity with thermal lensing and internal lens of Pf s10 my1 .

The short length of cavity, required for maximum output power at high pump levels, can be a serious drawback in design of several types of laser e.g.

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Fig. 10. Dependence of pump to mode ratio Žsolid line., mode matching efficiency Ždashed line. and diffraction loss Ždotted line. on cavity length for pump power 12 W; cavity with thermal lensing and internal lens of Pf s 10 my1 .

intracavity frequency doubled or mode locked lasers. Thus, we tried to find such cavity configuration with folding spherical mirror Žequivalent to internal lens. which can be optimized with respect to the output power. Let us consider the cavity with folding mirror of the curvature radius equal to 200 mm Ži.e., Pf s 10 my1 .. As a result of step by step optimization it was found that the cavity length should be approximately equal to 380 mm for pump power of 10 W. The output power characteristics vs. pump power depend on the cavity length ŽFig. 8., similarly as for cavity with thermal lensing only. The dependences of output power on cavity length for three pump levels were shown in Fig. 9. We can see that for "10 mm deviations of cavity length from the optimal value the output power falls not more than 10%. Thus, such a cavity scheme, similarly as the previous one, is also insensitive to cavity length variation. As it was shown in Fig. 10, the diffraction loss achieves its minimum for the cavity length equal to 380 mm. For such a case the pump to mode ratio was about 2.5, whereas the mode matching efficiency is about 0.5.

4. Conclusions The model of end pumped laser with pump dependent diffraction loss was presented. The effects

connected with mode matching, thermal lensing, pump induced diffraction loss, and parameters of cavity were included as well. The analytical model of pump induced diffraction loss, assuming thermal axicon phase aberration, was proposed. The ABCD model of cavity with internal lens and thermal lensing was derived to optimize output characteristics. It was shown, that the appropriate ratio of fundamental mode radius to pump radius, in respect to minimization of pump induced diffraction loss, should be less than 1. Two types of cavities Žwith and without internal lens. were analyzed assuming given dependence of thermal lensing on pump power. The best configurations, with respect to maximum of output power, were found for two types of cavities. The optimized length of cavity of 80 mm, with thermal lensing solely determined from our model, was confirmed by the experiment. The cavity with internal folding mirror of 200 mm radius and optimized length was shown to have the same performance as the previous one. The pump induced diffraction loss was found to be the main factor limiting performance of such lasers for high pump levels. To optimize output characteristics and beam quality of such lasers it is necessary to determine from experiments the thermally induced pump aberrations for given pump parameters. The elaborated model can be a useful tool for analysis and optimization of several types of coherently pumped laser. The same

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approach was applied lately to optimization of intracavity doubled laser w32x. Acknowledgements This work was supported by the Polish Committee for Science Research ŽKBN. under the project 8T11B05417. References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x

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