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numerical investigation of thermal effects in end pumped Cr4 forsterite lasers

numerical investigation of thermal effects in end pumped Cr4 forsterite lasers
numerical investigation of thermal effects in end pumped Cr4 forsterite lasers

Experimental and Numerical Investigation of Thermal Effects in End-Pumped

Cr

TABLE I

N AMES ,U NITS ,

AND

R EPRESENTATIVE V ALUES OF THE O PTICAL AND T HERMAL P ARAMETERS

WHICH

C HARACTERIZE THE C R 4+:F ORSTERITE G AIN M

EDIUM

is indicated,the parameter values were determined in this study.As discussed in Section III-A,a systematic experimental study further investigated the temperature sensitivity of the threshold pump power in three

Cr :forsterite crystals with different cross-sectional sizes and/or different values

of

,and other relevant

resonator parameters for each sample are listed in Table II.Sections III-A and III-B further describe experimental procedures employed to determine the thermal coef?cients of the ?uorescence lifetime and the stimulated absorption cross-section.Experimentally measured threshold data obtained with the three

Cr :forsterite crystals were then analyzed by using the stimulated emission cross

section

which mini-mizes the incident threshold pump power in room-temperature

Cr :forsterite lasers.At a crystal boundary temperature of

15

was determined to be 0.64cm

for a 2-cm-long

Cr :forsterite crystal,corresponding to an unsaturated power absorption of 72%.Section IV presents additional data which delineate the role of heat conductivity in the enhancement of thermal loading and which estimate the maximum boundary temperature at which lasing can be obtained in a hypothetical

Cr :forsterite crystal with the optimum absorption coef?cient.

II.T HEORY

A.Model Equations

As schematically shown in Fig.1,vibronic coupling makes

Cr :forsterite a four-level laser system suitable for low-threshold CW operation [14].As an inevitable drawback

of

Fig.1.The four-level laser scheme of the Cr 4+:forsterite gain medium.

the electron–phonon interaction,however,nonradiative pro-cesses (shown as dashed arrows in Fig.1)also in?uence the population dynamics and,in particular,give rise to the temperature dependence of the ?uorescence

lifetime of

state

are assumed to occur at much faster rates,so that

population densities in

levels

is Planck’s

constant,

,

and

where

according

to

(3)

where

(4)

where

and are the

absorption and emission saturation intensities,respectively,

and

is the saturated differential absorption coef?cient.The spatial variation in the saturation intensities stems from the temperature dependence

of .Similarly,the cavity

intensity

is the maximum extractable small-signal differential gain coef?cient

and

is the saturation

intensity which is evaluated by using the value of the ?uores-cence

lifetime

at the crystal boundary

temperature

and are the average and initial saturation parameters,given,respectively,

by

(8)

In general,the transverse variation of the pump and the cavity intensities complicates the solution of (4)and (5).One observes,however,that a signi?cant interaction between the two beams will occur only at those locations where appreciable overlap between the two Gaussian modes exists.The analysis may therefore be reduced to a one-dimensional (1-D)model by further assuming that the effective pump

intensity

which interacts with the cavity beam can be approximated

as

(9)

where

is the mode-matching function which quanti?es the degree of overlap between the Gaussian-shaped pump beam and the fundamental

TEM mode of the cavity at a particular longitudinal

coordinate

can be expressed

as

and are the respective pump and cavity

beam spot-size functions.

B.Temperature Calculations

In addition to providing optical gain for the circulating cavity intensity,the pump beam also induces local thermal gradients during its passage through the crystal.The resulting temperature pro?le inside the crystal can be calculated by solving the heat equation.In the temperature calculations,two approximations were made.First,because the rectangular cross section of each laser crystal was very large compared with the dimensions of the pump and the cavity beams,edge effects were neglected and the medium was assumed to be radially symmetric,represented by an effective

radius that gives the same cross-sectional area.Since the pump beam is assumed to have a Gaussian intensity distribution,temperature calculations can then be performed by solving the cylindrically symmetric

heat equation.The value

of

is listed for each sample in Table II.Second,the longitudinal derivative in the heat equation was neglected.This is readily justi?ed by observing that in typical experimental con?gurations,the periphery of the crystal is actively cooled and as a result,convective and radiative heat losses that occur at the

input

therefore

satis?es

(11)

where

per unit volume is given in terms of

the saturated differential absorption

coef?cient

and the beam

intensity

is needed to account for the

temperature dependence of the ?uorescence lifetime in the threshold calculations.As described in [15]and [17],the heat source was approximated as a square function to obtain an

analytical expression

for

:and the radial

width

of the

square heat source is given

by

for different levels of the incident pump

power

C.

Here,and the parameter values of sample

3listed in Table II were used.In addition,the pump and the cavity beam parameters listed in Table II were estimated by employing the well-known methods of Gaussian beam propagation [19].Note that because of saturation effects,the temperature rise at a given point does not increase linearly with the incident power especially near the beam waist location (e.g.,

at

,

where

fully accounts for all the crucial

factors which affect the oscillation threshold:the amount

of

feedback

,material

losses ,saturation/thermal

loading

).To see the role of

thermal loading,suppose that the local temperature inside the crystal increases.This could be brought about by an increase

TABLE II

M ATERIAL AND R ESONATOR P ARAMETERS OF THE T HREE C R4+:F ORSTERITE C RYSTALS U SED IN THE E XPERIMENTAL AND T HEORETICAL S TUDY

in the boundary temperature,pump power,or absorption

coef?cient.The accompanying reduction in the?uorescence

lifetime will increase the saturation intensity and will

therefore result in a larger threshold pump power since a larger

pump intensity distribution

.As a concrete example,Fig.3

shows the calculated variation of

,cm

and the parameter values of sample3given in Table II were

used.Note that the calculation clearly shows the detrimental

effect of thermal loading since the incident threshold pump

power required for oscillation(i.e.,the value of

)increases with

m,pumped the symmetrical

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