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JJAP PROOFS Design of End-Pumped Thin Rod Yb:YAG Laser Ampliﬁers

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J J A P P R O O F S
Design of End-Pumped Thin Rod Yb:YAG Laser Ampliﬁers
Sakae K
AWATO
Ã
and Takao K
OBAYASHI
Graduate School of Fiber Amenity Engineering, Fukui University, 3-9-1 Bunkyo, Fukui 910-8507, Japan
(Received September 2, 2002; accepted for publication January 8, 2003)
An end-pumped thin rod Yb:YAG laser ampliﬁer architecture is proposed for high-power and eﬃcient pulse ampliﬁcation of quasi-four-level lasers. A thin and long laser rod with low ion concentration is used and optimum conditions of the lasercrystal properties are discussed for eﬃcient ampliﬁcation of high-repetition-rate pulses. A high single-pass gain is expectedand the thermal birefringence loss is signiﬁcantly smaller than the single-pass gain. From these results, high optical conversioneﬃciency of more than 50% is expected for pulse ampliﬁcation in the master-oscillator and power-ampliﬁer (MOPA)system. [DOI: 10.1143/JJAP.42.dummy]
KEYWORDS: quasi-four-level laser, Yb:YAG, diode-pumped solid-state laser ampliﬁer, end-pumping, thin rod
1. Introduction
Yb:YAG material has been investigated in recent yearsfor use as high-power and ultrashort pulse lasers based on ahigh atomic quantum eﬃciency of 91% and a wide gainspectrum of 9.5nm full width at half maximum (FWHM).However, realization of high intensity pumping larger thanabout 10kW/cm
2
and eﬃcient cooling is required to reducethe re-absorption loss originating from the lower levelpopulation of the quasi-four-level system. Furthermore, highsingle-pass gain of the ampliﬁer is necessary for high energyextraction to overcome the optical loss in the ampliﬁer andthe temperature increase in the laser crystal by high intensitypumping causes unfavorable thermal eﬀects.Several designs of Ytterbium-material pumping architec-ture have been developed for high-power, high-eﬃciencyoscillation and ampliﬁcation, including thin disk,
1–5)
micro-chip,
6,7)
zigzag slab,
8–10)
rod
11–14)
and ﬁber
15,16)
structures.A thin disk laser was proposed by Giesen
et al.
,
1)
utilizing alongitudinal multi-pass pumping scheme to reduce the quasi-four-level re-absorption laser loss and the longitudinalcooling scheme to reduce thermo optic eﬀects. A continuouswave (CW) multimode output power of 1,070W wasachieved by the multi-disk laser with a high opticalconversion eﬃciency of 48%.
2)
The concept was alsoapplied to a regenerative ampliﬁer of picosecond pulses
5)
with average output power of 10.2W and optical conversioneﬃciency of 16%. The eﬃciency of the ampliﬁer is lowerthan that of the CW oscillator mainly due to the low single-pass gain of the thin disk Yb:YAG.Ytterbium-glass ﬁber ampliﬁers oﬀer a range of propertiessuch as high optical conversion eﬃciency that makes theman attractive eﬃcient source of ultrashort pulses, because of their high gain characteristics. In the ﬁber chirped-pulseampliﬁcation (CPA) system, an average output power of 5.5W has been achieved with a high eﬃciency of 37% and ahigh single-pass gain of 32dB.
15)
However, pulse energy of the ﬁber ampliﬁer is limited to about 1mJ, because of itssmall core diameter of 20–100
m
m
. For many scientiﬁc andindustrial applications, the development of high-powerultrashort pulse laser oscillators and ampliﬁers is requiredwith high-energy, high-eﬃciency and high-beam-qualityoutput in a compact structure.In the rod geometry, as the pump power increases thethermo-optical stress increases. It is diﬃcult to reduce thestress in the rod laser, as compared to the thin disk or slablaser. For example, the end-pumped rod laser was developedby Bibeau
et al.
with 2mm diameter,
13,14)
and the opticalconversion eﬃciency and the beam quality characteristicswere lower than those of the end-pumped thin disk andzigzag slab lasers.In this study, we analyze a new architecture of theYb:YAG ampliﬁer for ultrashort pulse ampliﬁcation withaverage output power of approximately 100W.
17)
The basicstructure of the ampliﬁer module is a thin, long rod withtransverse size of less than about 1mm. A similar rodstructure has been found in the four-level laser materi-als.
18,19)
However, the basic conditions of the Yb:YAG rodstructure are diﬀerent from the conventional four-level lasersand new design principles are derived for the suppression of temperature increase and the optimization of the rod size todecrease the loss inherent in the quasi-four-level lasermaterials.Geometry and features of the thin rod Yb:YAG ampliﬁerare described in §2.1. Theoretical treatment of gain andeﬃciency is presented in §2.2 and thermal eﬀects of theampliﬁer are discussed in §2.3.
2. Design of End-Pumped Thin Rod Yb:YAG Ampli-ﬁers
2.1 Geometry and features of the thin rod Yb:YAGampliﬁer module
Schematic of the thin rod Yb:YAG ampliﬁer module isshown in Fig.
1
. Two CW laser diodes (LD) are used forpumping and the output beams are focused on the endsurfaces of the cylindrical thin rod Yb:YAG crystal. Thepump beam is tilted at a center axial angle
p
from the rodaxis to separate the laser beam and to increase the absorption
Fig. 1. Schematic of the end-pumped thin rod Yb:YAG ampliﬁer module.
p
is the center angle of the pump beam.
Ã
E-mail: kawato@fuee.fukui-u.ac.jpJpn. J. Appl. Phys. Vol. 42 (2003) pp. 0000–0000Part 1, No. 5A, May 2003
#
2003 The Japan Society of Applied Physics
02R09008
[1]
J J A P P R O O F S
pass length inside the rod. The transverse surface of the rodis coated with high-reﬂection ﬁlm to conﬁne the pump beamand to realize uniform pump intensity distribution averagedby multiple reﬂection inside the rod and the pump intensityis assumed to be transversely uniform. The cooling structureof the thin rod is much simpler than those of the thin disk and conventional rods. The rod is uniformly cooled throughthe transverse surface in contact with metal heat sinks cooledby water.One of the advantages of the thin rod structure is its highgain characteristics due to its long gain length compared tothe thin disk structure. It also exhibits high pulse energyoutput characteristics due to having a large gain areacompared to the ﬁber structure. The end-pumped thin rodstructure is suitable for eﬃcient cooling because of the largesurface area with small rod volume.
2.2 Unsaturated single-pass gain and storage eﬃciency
The repetitive pulse train with sub-ps to ns pulse width isassumed to be introduced into the ampliﬁer with timeinterval
r
. The unsaturated single-pass gain of the ampliﬁeris related to the unsaturated exponential gain
0
ð
r
;
Þ
as
20)
G
0
ð
r
;
Þ ¼
exp
½
0
ð
r
;
Þ
;
ð
1
Þ
where
r
is the radial position and
is the azimuth angle inthe cylindrical rod. The unsaturated exponential gain isrelated to the emission cross section
f
and the inversionpopulation density
Á
n
ð
r
;;
z
Þ
as
0
ð
r
;
Þ ¼
f
Z
L
0
Á
n
ð
r
;;
z
Þ
d
z
;
ð
2
Þ
where
z
is the longitudinal position inside the rod and
L
isthe rod length. Under the unsaturated condition of the quasi-four-level system, the inversion population density is
21,22)
Á
n
ð
r
;;
z
Þ ¼ f
f
a
ð
r
;;
z
Þ þ
f
b
ð
r
;;
z
Þg
p
P
p
f
r
p
ð
r
;;
z
Þ
h
p
À
f
0a
ð
r
;;
z
Þ
n
t
;
ð
3
Þ
where
n
t
is the concentration of the Yb laser ions,
f
is theﬂuorescence lifetime,
f
a
ð
r
;;
z
Þ
and
f
b
ð
r
;;
z
Þ
are thefractional population of the lower and the upper laser levels,respectively,
f
0a
ð
r
;;
z
Þ
is the initial value of the fractionalpopulation of the lower laser level,
p
is the pump quantumeﬃciency and
r
p
ð
r
;;
z
Þ
is the pump intensity distribution inthe rod. The absorption eﬃciency
a
is given by integratingthe pump intensity distribution
r
p
ð
r
;;
z
Þ
inside the rodvolume
V
as
a
¼
Z
V
r
p
ð
r
;;
z
Þ
d
V
;
ð
4
Þ
where
V
¼
SL
,
S
¼
r
20
is the rod cross section with radius
r
0
. For the case of transversely uniform pumping from twoend surfaces of the rod as shown in Fig.
1
, the pumpintensity distribution
r
p
ð
r
;;
z
Þ
is simpliﬁed to
r
p
ð
z
Þ ¼
½
exp
ðÀ
z
Þ þ
exp
ðÀ
ð
z
À
L
ÞÞ
=
ð
2
S
Þ
;
ð
5
Þ
where
is the eﬀective absorption coeﬃcient for the pumpbeam.The spatial distribution of the fractional populations
f
a
and
f
b
is assumed to be uniform in the rod because thetemperature distribution is almost uniform in the rod as willbe shown in §2.3. Thus the unsaturated exponential gain isderived from eqs. (
2
) and (
3
) as
0
¼ ð
f
=
S
Þð
f
a
þ
f
b
Þ
a
p
N
p
½
1
À
l
N
t
=
ð
a
p
N
p
Þ
;
ð
6
Þ
where
N
t
¼
n
t
V
is the total number of ions in the rod and
N
p
is the number of pump photons incident within theﬂuorescence lifetime
f
N
p
¼
P
p
f
=
ð
h
p
Þ
;
ð
7
Þ
and the ratio of fractional populations is
l
¼
f
0a
=
ð
f
a
þ
f
b
Þ
:
ð
8
Þ
It is shown in eq. (
6
) that the gain of the quasi-four-levelsystems is decreased when the total number of ions
N
t
isextremely large, thus optimization is required.The saturation ﬂuence is given by
J
s
¼
h
L
=
½ð
f
a
þ
f
b
Þ
f
;
ð
9
Þ
where
h
L
is the laser photon energy. The storage ﬂuence of the ampliﬁer within the time interval
r
is
J
st
¼
0
J
s
½
1
À
exp
ðÀ
r
=
f
Þ
:
ð
10
Þ
The storage eﬃciency of the ampliﬁer
s
is deﬁned by theratio of the storage energy
SJ
st
and the pump energyintegrated within the time interval
r
as
s
¼
SJ
st
=
ð
P
p
r
Þ ¼
q
p
a
½
1
À
l
N
t
=
ð
a
p
N
p
ÞÂ ½
1
À
exp
ðÀ
r
=
f
Þð
f
=
r
Þ
:
ð
11
Þ
The storage eﬃciency depends on the total number of ions
N
t
. Therefore the maximum storage eﬃciency is obtained by
d
s
=
d
N
t
¼
0
. The optimum total number of ions is alsogiven by
d
s
=
d
N
t
¼
0
as
N
optt
¼
ln
½
B
p
N
p
=
l
=
B
;
ð
12
Þ
and
B
¼
a
=
ð
S
cos
p
Þ
;
ð
13
Þ
where
a
is the absorption cross section for the pumpwavelength. The optimum rod length is related to theoptimum number of ions and is given by
L
opt
¼
N
optt
=
ð
n
t
S
Þ
:
ð
14
Þ
The absorption eﬃciency is obtained from eqs. (
4
) and (
5
)and is approximated as
a
¼
1
À
exp
½À
N
t
B
:
ð
15
Þ
The optimum value of the absorption eﬃciency is obtainedby substituting eq. (
12
) into eq. (
15
) as
opta
¼
1
À
l
=
ð
B
p
N
p
Þ
:
ð
16
Þ
As shown in eq. (
15
), a small value of
B
with a small crosssection of the rod
S
and large angle of the pump beam
p
leads to the increase of the optimum absorption eﬃciency
opta
. By substituting eq. (
12
) into eqs. (
6
) and (
11
), themaximum unsaturated exponential gain is
max0
¼ ð
f
=
S
Þ
p
N
p
1
À
1
þ
ln
ð
B
p
N
p
=
l
Þ
B
p
N
p
=
l
;
ð
17
Þ
and the maximum storage eﬃciency is
02R09008
[2] Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 5A S. K
AWATO
and T. K
OBAYASHI
J J A P P R O O F S
maxs
¼
q
p
1
À
1
þ
ln
ð
B
p
N
p
=
l
Þ
B
p
N
p
=
l
1
À
exp
ðÀ
r
=
f
Þ
r
=
f
:
ð
18
Þ
From eqs. (
17
) and (
18
), one can see that large values of theparameter
BN
P
=
l
with a small cross section of the rod andlarge incident angle of the pump beam lead to increases inthe single-pass gain and the storage eﬃciency.In the following numerical analysis, the laser rodtemperature is assumed to be 300K with the pulse interval
r
¼
50
m
s and the center angle of the pump beam
p
¼
45
deg. For the Yb:YAG parameters, we assume
q
¼
1
:
0
,
l
¼
1
;
030
nm,
p
¼
940
nm,
f
¼
3
:
0
Â
10
À
20
cm
2
,
a
¼
7
:
7
Â
10
À
21
cm
2
and
f
¼
0
:
95
ms.The optimum number of ions
N
optt
and the optimumabsorption eﬃciency
opta
have been calculated as a functionof pump power
P
p
and the results are shown in Figs.
2
and
3
for several rod diameters. It is shown that the optimumnumber of ions
N
optt
increases with increasing pump powerand increasing rod diameter. The optimum absorptioneﬃciency
opta
also increases with the pump power butdecreases with increasing rod diameter. Therefore thestorage eﬃciency
s
and the unsaturated single-pass gain
G
0
are smaller for the large rod diameters as discussedabove.Figure
4
shows the pump power dependence of theunsaturated single-pass gain
G
0
for several rod diameters.High gain values are expected at high pump power.However, an extremely high gain value exceeding a factorof about 100 causes a parasitic oscillation and an ampliﬁedspontaneous emission, and some suppression techniquessuch as use of tapered rod with ﬂanged end-caps structuresbecome necessary as demonstrated in refs. 13 and 14.Therefore rods with diameter smaller than 0.5mm are notsuitable for high-power pumping. The pump power and therod diameter dependence of the storage eﬃciency
s
isshown in Fig.
5
. For the pump power above about 200W, a
0100200300400500
O p t i m u m n u m b e r o f i o n s
N
t o p t
Pump power
P
p
[W]
Rod diameter:
2
r
0
[mm]0.51.01.52.0
2.53.0
510
x10
18
Fig. 2. Optimum number of Yb ions
N
optt
as a function of pump power
P
p
for several rod diameters
2
r
0
.
0.60.70.80.910100200300400500
O p t i m u m a b s o r p t i o n e f f i c i e n c y
η
a o p t
Pump power
P
p
[W]
Rod diameter:
2
r
0
[mm]0.51.01.52.02.53.0
Fig. 3. Optimum absorption eﬃciency
opta
as a function of pump power
P
p
for several rod diameters
2
r
0
. Total number of Yb ions is assumed tobe optimized for pump power and rod diameters.
11010
2
0100200300400500
U n s a t u r a t e d s i n g l e - p a s s g a i n
G
0
Pump power
P
p
[W]
Rod diameter:
2
r
0
[mm]0.51.01.52.02.53.0
Fig. 4. Pump power dependence of the unsaturated single-pass gain
G
0
for several rod diameters
2
r
0
. Optimum number of ions is assumed.
00.20.40.60.810100200300400500
M a x i m u m s t o r a g e e f f i c i e n c y
η
s m a x
Pump power
P
p
[W]
Rod diameter:
2
r
0
[mm]0.51.01.52.02.53.0
Fig. 5. Dependence of the maximum storage eﬃciency
maxs
on pumppower
P
p
and rod diameter
2
r
0
. Input pulse interval of 50
m
s and theoptimum number of ions are assumed.
02R09008
Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 5A S. K
AWATO
and T. K
OBAYASHI
[3]
J J A P P R O O F S
high storage eﬃciency exceeding 70% is expected for rodswith diameter less than 1.0mm.The thin rod structure can also be used as the gain mediumof the multi-pass or regenerative ampliﬁer. The output pulseenergy is calculated based on the Frantz-Nodvik relation
23)
for these ampliﬁers. Complete spatial overlap of the inputand the ampliﬁed beams is assumed. For a pulse with aﬂuence
J
0
that is incident in the ampliﬁer, the gain issaturated when the gain equals the loss and the maximumﬂuence of the ampliﬁed pulse is given by
24)
J
max
¼
J
s
ln
ð
1
À
L
a
Þ
G
0
L
a
ð
1
À
L
a
Þ
G
0
L
a
" #( )
Â ½
1
À
exp
ðÀ
r
=
f
Þ þ
J
0
;
ð
19
Þ
where
L
a
is the single-pass transmission loss excluding thequasi-four-level loss of the laser rod. The ratio of themaximum extracted pulse energy to the storage energy givesthe maximum extraction eﬃciency
20)
maxe
¼
m
ð
J
max
À
J
0
Þ
=
J
st
;
ð
20
Þ
where
m
is the mode-matching eﬃciency between the pumpmode and the ampliﬁed laser mode distribution in the rod.The product of the maximum storage eﬃciency
maxs
and themaximum extraction eﬃciency
maxe
is the maximum opticalconversion eﬃciency
o
¼
maxs
maxe
:
ð
21
Þ
The pump power dependence of the maximum opticalconversion eﬃciency
o
for the regenerative and multi-passampliﬁer is calculated for several rod diameters assumingthe mode-matching eﬃciency of 100% and the results areshown in Fig.
6
. A high optical conversion eﬃciencyexceeding 50% is possible for rods with diameter less than1.0mm at a pump power above 150W. In this calculation,the single-pass loss
L
a
¼
10
% is assumed as a commonvalue in conventional regenerative ampliﬁers.The optical conversion eﬃciency is aﬀected by the mode-matching eﬃciency obtained by eqs. (
20
) and (
21
). Whenthe gain is saturated in the regenerative ampliﬁcation, themode-matching eﬃciency of the ampliﬁer is equal to that of the oscillator. Thus we will adopt the mode-matchingeﬃciency of the quasi-four-level laser oscillator with aTEM
00
laser beam of e
À
2
radius of
r
0
=
1
:
2
as discussed byTaira
et al.
22)
It is known that at the threshold pump power,the mode-matching eﬃciency decreases to lower than about50%, but then increases to 85% at high pump power inexcess of threefold greater than the threshold. From Fig.
6
,we can estimate that the optical conversion eﬃciency of theampliﬁer exceeds 50% for rods with diameter less than1.0mm at a pump power above about 250W. Therefore theeﬃciency of the thin rod ampliﬁer is much higher than thatof the Yb:YAG thin-disk regenerative ampliﬁers (16%).
5)
2.3 Thermal eﬀects
Crystal temperature inﬂuences the lower laser levelpopulation and transmission loss in the quasi-four-levellaser material. It is assumed that the outer surface of the rodis eﬀectively cooled and kept at room temperature. Ingeneral, a long rod with low ion concentration is preferablefor eﬃcient cooling of the laser rod. For a long laser rod, theheat ﬂow in the longitudinal direction is extremely smallcompared to that in the radial direction and can be neglectedin the analysis. Based on the diﬀusion model of the radialheat ﬂow in the circular rod with the thermal conductivity
K
c
and the heat conversion eﬃciency deﬁned by
h
¼
1
À
p
q
,the temperature diﬀerence between the position of radius
r
and the outer surface
r
0
of the rod at longitudinal position
z
is
Á
T
c
ð
r
;
z
Þ ¼
h
a
P
p
4
K
c
r
p
ð
z
Þ
r
20
À
r
2
À Á
:
ð
22
Þ
The average value of the temperature increase over the rodvolume
V
is needed for the estimation of the quasi-four-levellaser loss and it is given by
Á
T
a
¼
1
V
Z
V
Á
T
c
ð
r
;
z
Þ
d
V
¼
h
a
P
p
=
ð
8
K
c
L
Þ
:
ð
23
Þ
The volume-averaged temperature increase
Á
T
a
as a func-tion of pump power is calculated for several rod diametersand the results are shown in Fig.
7
. In the calculation, weassume
K
c
¼
0
:
14
W/(cmK) and
h
¼
8
:
7
% for Yb:YAGcrystal. The average temperature increase is less than 3K even at a pump power of 500W for rods with diameter lessthan 1.5mm. The loss of the storage eﬃciency caused by thetemperature increase is estimated to be less than 1%.The maximum temperature increase can be measured atthe center position of the rod-end section and from eq. (
22
) itis estimated to be about 10K at pump power of 500W. Thetemperature increase of the laser rod with ion concentrationof around 1at.% is still low. It is shown that the coolingeﬃciency of the thin rod structure is high because of thewide cooling area.Thermal focal length for the polarization along the radial
r
or circular
direction is written as
25)
f
r
;
¼
2
K
c
S
h
a
P
p
d
n
0
=
d
T
þ
2
n
30
C
r
;
þ
l
ð
n
0
À
1
Þ
=
L
Â ÃÈ É
;
ð
24
Þ
where
n
0
is the refractive index,
is the thermal expansion
00.20.40.60.810100200300400500
O p t i c a l c o n v e r s i o n e f f i c i e n c y
η
o
Pump power
P
p
[W]
Single-pass loss:
L
i
= 10%
Rod diameter:
2
r
0
[mm]0.51.01.52.02.53.0
Fig. 6. Dependence of the maximum optical conversion eﬃciency
o
onpump power
P
p
and rod diameter
2
r
0
. Single-pass loss of 10% and theoptimum number of ions is assumed.
02R09008
[4] Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 5A S. K
AWATO
and T. K
OBAYASHI
J J A P P R O O F S
coeﬃcient,
C
r
and
C
are the photoelastic coeﬃcients in the[111] direction of the Yb:YAG crystal for radial
r
andcircular
polarizations, respectively, and
l
is the length of the rod-end section over which expansion occurs.The pump power dependence of the thermal focal lengthand the optimum rod length is calculated and the results areplotted in Fig.
8
. Yb:YAG parameters are assumed to be
n
0
¼
1
:
82
,
d
n
0
=
d
T
¼
7
:
5
Â
10
À
6
K
À
1
,
¼
7
:
3
Â
10
À
6
K
À
1
,
C
r
¼
0
:
017
,
C
¼ À
0
:
0025
,
l
¼
2
r
0
, and
n
t
¼
0
:
5
at.%. Thethermal focusing, the length of which is more than twice theoptimum rod length
L
opt
, can be compensated by placingconcave or convex mirrors in the laser cavity. At high pumppower, the focal length is short and the short thin-rodstructure is eﬀective in preventing the reduction of themode-matching eﬃciency. Thus a thin rod with an ionconcentration of approximately 1.0at.% is useful for thecompensation of the thermal focusing eﬀects.If a thermal birefringent crystal is placed between apolarizer and an analyzer with parallel polarization direc-tion, the transmission loss due to the thermal birefringencefor the plane laser beam is calculated by integrating thetransmission loss distribution
L
b
ð
r
;
Þ
over the cross-sec-tional area of the rod as
26)
L
a
¼
1
S
Z
S
L
b
ð
r
;
Þ
d
S
¼
141
À
sinc
k
l
n
30
ð
C
À
C
r
Þ
h
a
P
p
K
c
!
;
ð
25
Þ
where
k
l
is the laser wave number.The pump power dependence of the thermal birefringencetransmission loss is calculated and the results are shown inFig.
9
. The loss depends on the absorbed pump power and itis estimated to be small compared to the single-pass gain of the thin rod ampliﬁer with diameter less than 1.5mm. Theloss can be reduced signiﬁcantly by using simple birefrin-gence compensation techniques.
27,28)
3. Conclusion
A thin rod Yb:YAG ampliﬁer has been designed for high-power and eﬃcient ultrashort pulse ampliﬁcation withaverage output power of approximately 100W. A thin andlong Yb:YAG crystal with low Yb ion concentration is usedand pumped from two end surfaces with a large angle of incidence. The optimum thin rod size is obtained at CWpump power of several hundreds W. The thin rod structurefeatures a high single-pass gain characteristic due to its longlength compared to the thin disk structure, and a high pulseenergy output characteristic due to its large cross sectioncompared to the ﬁber structure. Although this structure lookssimilar to the conventional thick rod laser with rod diameterlarger than 2mm,
11–14)
the thin rod model has the char-acteristics of higher pumping eﬃciency and high coolingeﬃciency. The pulse extraction eﬃciency and the storageeﬃciency of the thin rod laser are higher than the thick rodlaser. The gain can be chosen to ensure that suppression of parasitic oscillation and ampliﬁed spontaneous emission(ASE) without the tapered barrel structure occurs.
13)
The volume average temperature increase in the laser rodis estimated to be less than 3K and the increase of the quasi-four-level laser loss is maintained below 1%, thus it can be
012340100200300400500
A v e r a g e t e m p e r a t u r e i n c r e a s e
∆
T
a
[ K ]
Pump power
P
p
[W]
Rod diameter:
2
r
0
[mm]0.51.01.52.02.53.0
Fig. 7. Average temperature increase
Á
T
a
as a function of pump power
P
p
for several rod diameters. Ion concentration of 0.5at.% and the optimumnumber of ions are assumed.
01002003000100200300400500Pump power
P
p
[W]
T h e r m a l f o c a l l e n g t h
f
r
,
φ
[ m m ] O p t i m u m r o d l e n g t h
L
o p t
[ m m ]
f
φ
f
r
f
φ
f
r
f
φ
f
r
f
r
L
opt
0.51.01.52.0Rod diameter:
2
r
0
[mm]
Fig. 8. Optimum rod length
L
opt
and thermal focal length
f
r
;
as afunction of pump power
P
p
for several rod diameters. Ion concentrationof 0.5at.% and the optimum number of ions are assumed.
00.10.20100200300400500Pump power
P
p
[W]
Rod diameter:
2
r
0
[mm]0.51.01.5
D e p o l a r i z a t i o n l o s s
L
a
2.02.53.0
Fig. 9. Depolarization loss
L
a
as a function of pump power
P
p
for severalrod diameters. The optimum number of ions is assumed.
02R09008
Jpn. J. Appl. Phys. Vol. 42 (2003) Pt. 1, No. 5A S. K
AWATO
and T. K
OBAYASHI
[5]

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