The condition to match the pumped-volume's width with the laser mode-volume's waist in a diode four side pumped solid-state laser is analyzed by numeric solutions of the approximation equation. As a result, the dependences of pumped volume's width on pumping diode parameter and laser mode volume's waist on cavity parameter were found. Such a matching condition was simultaneously determined by cavity, pumping beam and laser rod parameters.%Key words: Laser diode side-pumped, Coherent pump, Excitred active center distribution

Luu, Mai Van

Quy, Ho Quang

Sau, Vu Ngoc

Vietnam Academy of Science and Technology: Journals Online

Communications in Physics, Vol. 20, No. 3 (2010), pp. 265-270
MATCHING OF PUMPED AND LASER MODE VOLUMES
INSIDE A DIODE SIDE-PUMPED SOLID-STATE LASER
HO QUANG QUY
Academy of Military Science and Technology
MAI VAN LUU AND VU NGOC SAU
Vinh University
Abstract. The condition to match the pumped-volume’s width with the laser mode-volume’s waist
in a diode four side pumped solid-state laser is analyzed by numeric solutions of the approximation
equation. As a result, the dependences of pumped volume’s width on pumping diode parameter
and laser mode volume’s waist on cavity parameter were found. Such a matching condition was
simultaneously determined by cavity, pumping beam and laser rod parameters.
I. INTRODUCTION
The diode side-pumped solid-state laser is one of advanced lasers and being paid
much attention in [1, 2]. High laser efficiency can be obtained because there is a good
spetial match between the emission spectrum of pump beam and the absorption spectrum
of solid-state laser medium. The optimum relation between the the pumped and the mode
volumes was obtained by theoretical analysis and experiment, which includes thermally
induced diffraction loss [3]. A good spatial match between the pumped and the laser mode
volumes can be obtained by using the end-pumping scheme, but the pump energy decreases
exponentially along the axial direction. Side-pumping geometry was used to achieve high
output laser powers [2]. For laser oscillator, one might use a near-Gaussian pumping
intensity distribution centered around the rod axis to help achieve TEM00 output; for
amplifier, one might use illumination to achieve the best radial uniformity [5]. In the
work [4], the relationship between the size of pump beam and mode volume (MV) was
analyzed by using space-dependent rate equations, which include the thermally induced
diffraction loss. In the work [6], we have shown that the excited active centers (EAC) in the
ovelapping area of pump intensities in a four sides configuration are spatially distributed as
near-Gaussian function. Moreover, in the work [7], we concluded that a suitable parameter
collection of laser diode’s beams (sparseness, location and azimuthal angle) could be chosen
so that the longitudinal distribution of EAC in the laser rod is homogeneous. Thus, by
four-side pumping, the laser rod is seem to be a graded-EAC rod with its EAC volume of
certain width. If EACV width matches with MV waist the pump efficiency is enhanced.
But, the matching condition depends on many parameters of the pump laser diodes and
the cavity.
In this work, we introduce the approximation equation describing the matching
condition between EACV width and MV waist. In. Sec. II, we investigate the dependence
266 HO QUANG QUY, MAI VAN LUU, AND VU NGOC SAU
of EACV width on pumping laser beam and that of MV waist on cavity parameters.
Finaly, the matching condition and its suitable parameter collection are discissed.
II. APPROXIMATION EQUATION FOR SPATIAL MATCHING
BETWEEN EACV WIDTH AND MV WAIST
As shown in previous work [6], the spatial distribution of four-side-pump intensity
in overlapping area of laser rod’s cross-section is a near-Gausssian function. It means
that EAC’s density distribution in cross-section of laser rod is a near-Gaussian function
of radial distance too. With this assuming, the distribution of average EAC’s density can
be given by the following function
Qfourside(x, y) = BN
Ifourside(x, y)
hν¯
= QEAC0exp
[
−2(x
2 + y2)
W 2EAC
]
, (1)
where B is the Einstein’s coefficient, N is the density of active centers in laser rod,
QEAC0 is the maximum EAC’s density in rod axis, WEAC is the width of EACV, hν¯,
and Ifourside(x, y) intensity of exciting field, which is given in work [6] as follows:
Ifourside(ρ) = Ifourside(x, y)
= I0W
2
in0
{[
1
Win(y)
]2
exp
[
−2x2
W 2in(y)
]
+
[
1
Win(−y)
]2
exp
[
−2x2
W 2in(−y)
]
+
[
1
Win(x)
]2
exp
[
−2y2
W 2in(x)
]
+
[
1
Win(−x)
]2
exp
[
−2y2
W 2in(−x)
]}
, (2)
where all parameters of pump beam inside laser rod are given as:
Wino =MW0 is the beam waist inside laser rod, (3)
bin =M
2b is the confocal parameter inside laser rod, (4)
Win(x) =Win0
[
1 +
(
x
bin
)2] 12
is the width at x inside laser rod, (5)
M =
Mt√
1 + t
; t =
b
y0 − r0/(n− 1)
;Mt =
∣∣∣∣ r0y0(n− 1)− r0
∣∣∣∣ , (6)
r0 is the laser rod radiur, y0 is the distance between laser rod axis and w0 pump beam’s
waist, b is the confocal parameter of pumping beam, n is the refractive index of laser rod.
All above parameters are illustrated in Fig. 1.
Assuming four pumping laser diode arrays are symmetrical through rod axis, (2)
can be simplified to
Ifourside(ρ) = 4I0
{[
Win0
Win
(ρ)
]2
exp
[
− 2ρ
2
W 2in
(ρ)
]}
= 4I0G(ρ), (7)
where ρ =
√
x2 + y2 is the radial radius from rod axis.
MATCHING OF PUMP AND MODE VOLUMES INSIDE ... 267
Fig. 1. a- Cavity geometry for four side-pumping module; b- Cross-sectional ge-
ometry of Gaussian beam outside and inside laser rod.
From (1),(3),(4),(5),(6) and (7), the width of EACV is given by:
WEAC =
√√√√− 2ρ2
ln
(
BNIfourside(ρ)
hν¯QEAC0
) =
√√√√− 2ρ2
ln
(
BNI0
hν¯QEAC0
) + ln(G(ρ)). (8)
Since the maximum EAC density, QEAC0 depends on the maximum intensity I0, so it can
be chosen so that the first term in denominator under square-root to be zero and (8) is
reduced to
WEAC =
√
− 2ρ
2
ln(G(ρ))
. (9)
As shown in work [6], the distribution of overlopping intensity is near-Gaussian with
certain width (Fig. 2).
We approximately consider that pump intensity higher than contributes to exciting
process. Thus, the width of pump intensity distribution can be chosen as root of the
following equation {[
Win0
Win(ρ)
]2
exp
[
− 2ρ
2
W 2in(ρ)
]}
− 1/e = 0 (10)
and is noted as (root of equation (10) -RE)
RE(ρ1/e). (11)
Substituting (10)and (11) into (9), we have
WEAC = RE(ρ1/e). (12)
From (3), (4), (5), (6), (10), (11) and (12), we can see that the width of EACV depends on
r0, y0, n and the wavelength of pumping laser λp (in term of confocal parameter b). Now,
268 HO QUANG QUY, MAI VAN LUU, AND VU NGOC SAU
Fig. 2. Four-side-pump intensity distribution inside the laser rod [6]. ρ1/e is the
width at Imax=4I0/e.
we consider the EACV is placed in a confocal cavity of length, L and mirror’s curvature
radius, R. This cavity has mode TEM00, whose MV waist, WMODE is given by [8]
WMODE =
√
λL
2pi
√
L(R− L) (13)
where λL is the laser wavelength. From (10), (11), (12) and (13), the matching condition
is described by following equation
WEAC =WMODE or RE(ρ1/e) ≡WEAC) =
√
λL
2pi
√
L(R− L). (14)
The matching-condition equation (14) contains a single ”called” unknown variableWEAC .
Note that the left hand side (LHS), which is characterized by a ”called” pumping parameter
−WEAC , depends on the parameters of laser diode arrays and laser rod, and the right
hand side (RHS), which is characterized by a ”called” cavity parameter-WMODE, depends
on that of the cavity. If WEAC = WMODE ≡ W , i.e. matching condition satisfies, the
equation (14) may be graphically solved by plotting its LHS and RHS versusWand finding
the intersections. Every intersection point corresponds to Wwith a distinct collection of
pumping and cavity parameters.
III. SIMULATED RESULTS AND DISCUSSION
We consider a solid-state laser generating at the wavelength of 1.06µm pumped by
four 0.83µm laser diodes. The spatial distribution of pumping laser diodes is assumed
a Gaussian beam characterized by a waist W0 = (5 × 10−2 and 10.10−2)mm. The laser
rod has its refractive index, n=1.78, and its radius of 4.0, 3.0, 2.0, 1.5 and 1.0mm. The
location of the pump beam is changed from 1mm to 20 mm, and the curvature radius of
the mirror is changed from 1m to 20m.
MATCHING OF PUMP AND MODE VOLUMES INSIDE ... 269
Fig. 3. Graphical construction for solving the matching-condition equation (10).
The left- and right-hand sides are plotted as function. The intersection points are
the solutions. (A) W0 = 5.10
−2 mm and (B) W0 = 10.10
−2 mm.
As illustrated in Fig. 3 (A) forW0 = 5.10
−2mm and Fig. 3 (B) forW0 = 10.10
−2mm,
the LHS has five branches corresponding to the laser rod radiurs, r0 = 4.0(a), 3.0(b),
2.0(c), 1.5(d) and 1.0(e) mm and the RHS has five branches corresponding to the cavity
lengths, L = 20(5), 40(4), 60(3), 80(2), and 100(1) mm, respectively. From Fig. 3, we can
see that: first, for every LHS branch there are many RHS branches that the matching
condition satisfies; second, to have larger width WEAC , the pumping beam waist of laser
diode should be located near the foci of laser rod, i.e., y0 ≈ f = r0/(n− 1) corresponds to
that the cavity length and the curvature mirror radius must be chosen longer as possible;
third, every intersection point corresponds to a matching volume with distinct collection
of parameters, for example, to have an intersection point marked by circle, a collection of
y0 = 13.5mm, r0 = 4.0mm,W0 = 5.10
−2, L = 20mm, and R = 13.75m... must be chosen.
Finally, the matching of pump volume (or EAC volume) with mode volume inside laser
rod will be appeared when a collections of pumping parameters (W0, y0, λp) and cavity
parameters (L,R, λL) are chosen suitably with a given collection of laser rod parameters
(r0, n).
270 HO QUANG QUY, MAI VAN LUU, AND VU NGOC SAU
IV. CONCLUSION
The approximation equation describing the spatial matching condition between the
EACV width and the MV waist of a solid-state laser pumped by four-side laser diodes is
considered. This equation is numerically resolved to find the pumping parameter-EACV’s
width and cavity-MV’s waist characteristics. The intersection points of those characteris-
tical curves are the matching values. Based on the matching values, the distinct collection
of main parameters of pumping laser diodes (beam waist, location) and cavity (length,
curvature radius) can be suitably chosen to enhance pumping efficiency.
REFERENCES
[1] B. J. Comaskey, et al., IEEE J. Quant. Electron. 28 (1992) 992-996.
[2] N. Hodgson, S Dong, and Q. Lu, Opt. Lett. 18 (1993) 1727-1729.
[3] R. J. St. Pierre et al, IEEE J. Sel.Top. Quant. Electron. 3 (1997) 53-58.
[4] W. Xie et al., Applied Optics, 39 (2000) 5482-5487.
[5] T. Y. Fan, and R. L. Byer, IEEE J. Quant. Electron. 24 (1988) 895-912.
[6] H. Q. Quy et al., Commun. in Phys. 18 (2009) 39-44.
[7] H. Q. Quy et al., Advances in OPS and Appl. (2009) 237-241.
[8] A. M. Prochorov, Laser Handbook,V.2, Sovietskoe Radio, Moscow, 1978.
Received 30 September 2009.


Matching of Pumped and Laser Mode Volumes Inside  a Diode Side-pumped Solid-state Laser

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Matching of Pumped and Laser Mode Volumes Inside a Diode Side-pumped Solid-state Laser

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