The time and spatial dynamics of the YAG:Nd3+/YAG:Cr4+ microchip laser emission

Andrey G. Okhrimchuk, Alexander V. Shestakov.

Elements of Laser Systems Co., 3 Vvedensky Str., Moscow 117342, Russia

ABSTRACT

Dynamics of the Q-switched microchip lasers pulse was investigated in experiment with high space and time resolution. It was found that spatial dynamics of the YAG:Nd3+/YAG:Cr4+ microchip laser emission could not be described by traditional formalism of transverse cavity modes. An alternative theoretical approach based on solution of the Maxwell scalar wave equation is presented. In our model we take into account the D2d local symmetry of the Cr4+ ions and the corresponding two types of saturable transitions.

Keywords: Microchip laser, Maxwell wave equations, Q-switch.

1. INTRODUCTION Last years microchip lasers found various applications in numerous fields of technologies, such as

micromachining, laser ranging, spectroscopy, because of its ability to produce high intensity light with robust

and reliable design and low cost. For practical applications, a vary important and attractive feature of

microchip lasers is a beam divergence close to the diffraction limit, like a standard single transverse mode

laser has. But at the same time, there are series of evidences, that a cross section profile of the microchip laser

beam with a planar cavity is very roughly described by traditional formalism of transverse modes, such as

Hermite-Guassian or Laguerre-Gaussian modes 1-5. But until now only the CW regime of operation was

carefully investigated with this concern, and it was found, that the traditional formalism is failed, when the

gain-guiding regime of laser operation dominates relative to the index guiding. Such regime takes place in

microchip lasers with planar cavity and high gain in the medium (the last circumstance is usually dealing with

low reflectivity output couplers). We expect that in addition to the above mentioned wave-guiding mechanism

there is a new one a in Q-switched microchip lasers due to space dependant absorption saturation. This

mechanism should act similarly to gain-guiding, because it contributes to an imaginary part of a dielectric

susceptibility, so as in the case of the gain-guiding. Thus for Q-switched microchip lasers discrepancy

between a real transverse profile of laser beam and a pictures, that traditional formalism of transverse modes

gives, should be even more pronounced. Moreover time-domain effects are added for Q-switched regime, and

this circumstance imports one more serious obstacle for implementation of formalism of the transverse

modes, because the last could not be formed during pulse development time.

In this paper we present results of experimental and theoretical investigations of the

YAG:Nd3+/YAG:Cr4+ microchip lasers. We have paid attention to laser beam profiles formation in transverse

space coordinate, and the time domain. Our special interest is concerned the lasers with high output energy,

thus with high absorption in the saturable absorber and high gain in the laser medium. We do not use the

transverse mode term at all, because of above mentioned futures of such Q-switched microchip lasers. Instead

of this we have obtained general numerical solution of the Maxwell scalar wave equation.

2. THE THEORY OF LASER PULSE DEVELOPMENT IN A Q-SWITCHED MICROCHIP LASER.

To obtain equations describing development of the field in the cavity of the Q-switched microchip laser we

start from the Maxwell scalar wave equation in paraxial approximation, following by approach have been

implemented for CW microchip lasers 1-4, and assuming a single longitudinal mode operation:

0

22

2 2

( )

0

( , , ) i kz t

EEc t

E r z t e ω

ε

−

∂∇ − =

∂≡ Ψ

(1),

where E(r,z,t) is the complex amplitude of the electric field (in polar coordinate, r is a radius, z - coordinate

along an optical axis), ε =ε(r,t) is a complex dielectric susceptibility, Ψ= Ψ(r,t) is a slowly varying amplitude

of the electric field E(r,z,t), k = n0k0 = n0ω0/c is a wave number, n0 is the unperturbed refractive index of the

medium (un-pumped medium), c is the speed of light, ω0 is the frequency of the longitudinal mode. Here we

restrict our analysis to zeros-order angular modes, thus azimuth dependences of the susceptibility ε and the

electric field E are omitted. Then, we neglect variations of ε and ψ along the optical z axis. For the dielectric

susceptibility ε we have 1:

2

00

( )( ) ( )2g rr n n r i

kε

= + ∆ −

(2),

where ∆n(r) is the perturbation of the refractive index due to the pump light, g(r) is the net gain ( difference

between gain and losses in the crystal). So as we neglect variation of the susceptibility ε along the optical

axis, we accept the mean-field limit, that is, we operate with the net g(r) gain averaged along the z axis of the

resonator:

( ) Nd s s

s

g l lg rl l

α−=

+ (3),

where l and ls are lengths of the YAG:Nd3+ and YAG:Cr4+ parts of the microchip laser correspondingly, gNg is

the averaged gain per unit length in YAG:Nd3+, αs is the averaged saturable absorption coefficient of

YAG:Cr4+ Q-switch.

Further, we take unto account that amplification and losses on the scale of wavelength is small in

lasers, that is g/k0 << 1, and we assume the perturbation of refractive index is small too: ∆n<<1. Under these

conditions we obtain an approximation for (2) neglecting second-order terms of g and ∆n :

20 0 0 0( ) 2 ( ) ( )r n n n r in k g rε ≅ + ∆ − (4).

Thus gain and loss aspects of the problem are confined in the imaginary part of the dielectric susceptibility ε,

and perturbation of the real part of the susceptibility contains a waveguiding aspect.

Now it is convenient to introduce a coordinate system connected with a laser pulse:

1

1

cz z tn

r rT t

= −

= =

(5).

Such transformation allows us to derive an equation for the slowly varying amplitude Ψ with vanished

dependence upon the longitudinal coordinate z. Here we again accept the approximations: g/k0 << 1, ∆n<<1.

After substitution (4) in (1) and the coordinate transformation (5) we obtain:

2 2

0 2 20 0 0 0

( , ) ( ) ( ) 12 2

r T cg r n r iciT n n n r r r

ωω

∂Ψ ∆ ∂ Ψ ∂Ψ= Ψ + + + ∂ ∂ ∂

(6)

Above we considered the medium only. Now we add to the right part of (6) the term describing loss on an

output coupler, and obtain modification of (6):

2 2

0 2 20 0 0 0

( , ) ( ) ln 12 2 2r

r T cg r R n iciT n t n n r r r

ωω

∂Ψ ∆ ∂ Ψ ∂Ψ= Ψ + + + + ∂ ∂ ∂

(7),

where R is the reflection coefficient of the output coupler (defined for intensity I = (c/8π)Re(Ψ* Ψ ) ), tr is the

resonator round trip time:

( )02 s

r

n l lt

c+

= (8)

To get formulas for the net gain g we write rate equations for excitation populations in the mean-field limit:

dN INdt h

σν

= − (9),

dN INdt h

dN INdt h

ππ π

σσ σ

σν

σν

= =

(10),

where N=N(r,t) is inversion density in the Nd3+ ions system, σ is the effective emission cross section. Starting

here we assume that the YAG:Cr4+ crystal is used as saturable absorber to realized a Q-switched operation. In

the rate equations (10) we accept the model for the YAG:Cr4+ Q-switch, which takes into account two

electronic transitions in the four-fold coordinated Cr4+ ion, which are differed for each Cr4+ Q-switch center

by polarization 6. We believe, that the laser pulse has linear polarization coinciding with [100] crystal axis,

because of it corresponds to minimal losses in the YAG:Cr4+ Q-switch 6. In this model σπ is the differential

absorption cross section for the 3B1(3A2)→3A2(3T1) transition for polarization coinciding with the S4 local

symmetry axis of the Cr4+ Q-switch center, σσ is the differential absorption cross section for the 3B1(3A2)→3E2(3T2) transition for polarization perpendicular to the S4 local symmetry axis of the Cr4+ Q-switch

center, Nπ= Nπ(r,t) is ground state population of the Cr4+ ions, which the S4 local axis is parallel to

polarization of laser pulse, Nσ =Nσ(r,t) is ground state population of the Cr4+ ions, which the S4 local axis is

perpendicular to polarization of laser pulse. Here the absorption coefficient of the YAG:Cr4+ Q-switch αs and

the gain in the YAG:Nd3+ gNd are following:

0( , ) ( , ) 2 ( , )s r T N r T N r Tπ π σ σα σ σ α= + + (11)

( , ) ( , )Ndg r t N r Tσ= (12)

where α0 is the un-saturable loss coefficient in the YAG:Cr4+ Q-switch.

Solution of the system (9), (10) couples the Cr4+ ions populations with the Nd3+ ions inversion

density:

( , ) ( , )(0) (0)

( , ) ( , )(0) (0)

a

s

a

s

N r t N r tN N

N r t N r tN N

π

σ

π

σ

=

=

(13),

where:

a

a

ππ

σσ

σσσσ

=

= (14),

N(0)=Nini is the inversion population density at the threshold, Ns(0)=Ns is the 1/3 of the four-fold coordinated

Cr4+ ions concentration. Finally we substitute (13) in (11) and then (11) in (3), and obtain equation for the net

gain g(r,T), and together with equations (7), (9) they form a complete system, which numerical solution

describes development of a Q-switched laser pulse in time T and the space transverse coordinate r.

( )

2 2

0 2 20 0 0 0

0

( , ) ( ) ln ( ) 12 2 2

( , ) (c/8 )Re( * )

1 ( , ) ( , )( , ) ( , ) 2

r

a a

s s ss ini ini

r T cg r R n r iciT n t n n r r r

dN r T Ndt h

N r T N r Tg r T N r T l N l ll l N N

π σ

π σ

ωω

πσν

σ σ σ α

∂Ψ ∆ ∂ Ψ ∂Ψ = Ψ + + + + ∂ ∂ ∂

Ψ Ψ = −

= − − − +

(15).

Nature of the refractive index perturbation ∆n could be as electronic, so as thermal. Estimations of these

contributions give the result, that the electronic one could be neglected. For the the thermal contribution we

accept the quadratic approximation 2:

2

02

0

2n rnk ρ

∆ = (16),

where ρ0 is a parameter characterized the strength of thermal lens. It is a waist size of the Gaussian eigenmode

in the continuous wave regime, if the YAG:Cr4+ part is replaced by pure YAG crystal, and if the thermal lens

is the only guiding is presented in the cavity.

3. EXPERIMENT. We experimentally investigated the diffusion bonded YAG:Nd3+/YAG:Cr4 crystalline microchip laser,

operated at 1064 nm wavelength. It had a planar cavity configuration, a hight reflective mirror HR and an

output coupler OC were formed by dielectric coatings on the Nd-part and the Cr-part of YAG crystals

correspondingly. We have tested several microchips with different small signal Q-switch transmittances were

in the range 30…60%, and reflection coefficients of the output coupler were in the range 30…60%.

HR

YAG:Nd/Cr chip

OCpump fiberλ=808nm,

Single mode fiber,mooved in transverse coordinate

fast Ph/diodeOptical delay lineL=20m

slow Ph/diode

StroboscopicOscilloscope

trigger

signal

Fig.1. The experimental setup for characterizing of microchip laser beam.

The experimental setup for characterizing of microchip laser beam is shown in fig.1. The YAG:Nd3+

crystal was end pumped through HR mirror by the 808 nm emission of a laser diode array operated in the

regime of rectangular pulses of 250…350 µs duration, 25…30 W pick power and 50…300 Hz repetition rate.

The pump emission was delivered to the microchip laser by a SiO2/SiO2 fiber with core diameter equaled to

400 µm, and focused inside the laser crystal by a high aperture objective.

Pump intensity distribution was measured by the CCD camera at the absence of the laser crystal, and

then these data were used for calculation of the space distribution of the Nd3+ ions inversion density, assuming

4 cm-1 small signal absorption coefficient for pump emission and taking into account absorption saturation at

the 4I9/2 – 4F5/2 pump transition. The distribution of the inversion density was strain forward connected with

net gain at the threshold. Result of calculations of the net gain profile is drawn in fig.2.

0 100 200 300 4000

1G

ain

g Nd

Radius, r (µm)

Fig.2. Normalized gain profile, calculated from the measured pump intensity distribution.

Near field image of laser beam cross section was formed by a lens with the 1.5 mm focal length

(fig.1). Optical magnification factor was 8.3. Time-doman profiles of the laser pulse in the near field were

measured by the fast InGaAs photodiode pigtailed with single mode fiber of 10 µm core diameter. Own time-

domain response of the measured system was characterized in the test, made with the mode-locked

chromium-forsterite laser generating 0.2 fs pulses at 1.25 µm wavelength. The FWHM of the apparatus

function was 60 ps. The fiber input was scanned transversely to the optical axis, and thus series of space

resolved microchip laser pulse profiles was obtained in time-domain.

In fig.3 we present detailed experimental and theoretical results for the microchip with following

parameters. The reflection coefficient of the output coupler was 30 %. Concentration of Nd3+ ions was 1 at.%.

Length of the YAG:Nd3+ crystal was 4 mm, length of the YAG:Cr4+ crystal was 2.7 mm, thus total length of

the laser cavity was 6.7 mm. The low signal transmittance of YAG:Cr4+ Q-switch was 30%, ultimate contrast

of the Q-switch was 11 6. Pump pick power was 28 W, the pump pulse duration was 340 µs, and the repetition

rate was 50 Hz. The emission spectrum of the laser was tested with double-grating polychromator. Resolution

of the polychromator was better, than 0.01 nm. It was established, that the microchip laser operated in a single

longitudinal mode.

0 200 400 600 800 1000 12000

200

400

600

time (ps)

Tran

sver

se c

oord

inat

e (µ

m)

200 400 600 800 10000

200

400

600

time (ps)Tr

ansv

erse

coo

rdin

ate

(µm

)

a) b)

Fig. 3. Intensity distribution of the laser pulse in time and space domains. a) experiment, b) calculations with ρ0= 140 µm.

4. COMPARISON OF EXPERIMENTAL RESULTS WITH CALCULATIONS.

Experimental results demonstrated that the transverse profile of the laser beam dramatically changed

in time. This fact made implementation of the transverse modes formalism ineffective for the Q-switched

microchip lasers. Typically, the transverse profile had nearly the Gaussian form at the front of laser pulse.

Then region of maximal intensity spread out from the optical axis, and a nearly flat top or even dip were

observed at the center of the beam, when the pulse reached maximal intensity. The transverse profiles were

depended upon parameters of the microchip laser (most of all upon the reflection coefficient of the output

coupler and the transmittance of the Q-switch). The shoulder could be observed at the center of the pulse tail.

This feature did not necessary present, and depends on pump conditions.

In numerical experiments conducted for solution of the equation system (15) we found, that pulse

shape in space and time domains was very sensitive to initial gain profile and strength of the thermal lens. In

fig.4 we presented results of numerical calculations for hypothetical Gaussian gain profile. The waist size of

pumped region (200 µm) is same, as for more real gain profile, presented in fig.2. One could see by

comparing of fig.3b and fig.4, that intensity distributions are principally differed. That is we observed delay in

pulse development at the periphery of the laser beam in the case close to the experiment, and for the Gaussian

gains profile the laser pulse shown more compact structure. This fact approved an important role of gain-

guiding for lasers investigated.

200 400 600 800 10000

200

400

600 FieldGaussPum

time (ps)

Tran

sver

se c

oord

inat

e (µ

m)

Fig.4. Calculated intensity distribution of the laser pulse in time and space domains for Gaussian hypothetical gain

profile with waist size of 200 µm. Other laser parameters are the same as for calculations presented in fig.3b.

So as the thermal lens can not be exactly be determined in the experiment, we made series of

numerical calculations with the parameter, characterizing thermal lens ρ, was in the range 10…1000 µm. It

was found that ρ0=140 µm gave the best agreement with the experiment (fig.3). The gain profile at the

threshold for these calculations was taken from the results of measurements of pump intensity distribution

(fig. 2.).

Intensity distributions shown in fig.3 demonstrate that theoretical prediction is close to the

experiment. Both pictures correspond to pulse duration (FWHM) around 250 ps. Some discrepancy in sizes of

the laser beam should be explained by inefficient characterization of the gain profile at the laser threshold.

Particularly, we did not take into account up-conversion processes in Nd3+ ions system and amplified

spontaneous emission. All these processes could deform distribution of inversion density, particularly to make

it more flat at the top.

5. CONCLUSIONS. We have constructed a mathematical model which describes pulse development of the Q-switched

microchip laser in the time and space domains. It was found that adequate characterization of the Q-switched

microchip laser pulse is inconsistent with the transverse mode formalism. The proposed model overcomes this

problem, it avoids concept of a transverse modes, and predicts behavior of the laser pulse in space and time

domains. Numerical calculations demonstrate that results are very sensitive to the gain profile, and thus

appropriate focusing of pump emission in the laser crystal is very important for generation of high intensity

laser pulses with divergence close to the diffraction limit.

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