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International Journal of Optics

Volume 2009 (2009), Article ID 730165, 8 pages

http://dx.doi.org/10.1155/2009/730165

## A Simple Approach for Enhancing the Output Performance of Solar-Pumped Solid-State Lasers

Centro de Física e Investigação Tecnológica (CEFITEC), Departamento de Física, Faculdade de Ciências e Tecnologia (FCT), Universidade Nova de Lisboa, Campus de Caparica, 2829-516 Caparica, Portugal

Received 19 September 2008; Accepted 2 December 2008

Academic Editor: David Coutts

Copyright © 2009 Dawei Liang and Rui Pereira. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A simple truncated fused silica elliptical cavity is proposed to enhance the output performance of solar-pumped solid-state lasers. The imaging property of the truncated elliptical cavity ensures an enhanced absorption distribution within an Nd:YAG rod. Optimum pumping parameters are found through ZEMAX nonsequential ray-tracing and LASCAD laser cavity analyses. Compared with the output laser performance of a 3D-compound parabolic concentrator-2D-compound parabolic concentrator (3D-CPC-2D-CPC) cavity, the truncated cavity provides 11% more multimode and 72.7% more laser powers. A laser beam of high beam quality can be produced efficiently. The standard tracking error for multimode laser power is also reduced to only 4.0% by the truncated cavity.

#### 1. Introduction

Solar-pumped lasers
convert broadband
solar radiation into coherent and narrowband laser emissions. Since the appearance of the first solar-pumped laser
[1], researchers have been attempting to improve the output performance such as
collection efficiency and beam quality [2–6]. Tracking error-induced laser power
decrease is also a major concern [7], since heliostat tracking error shifts the
center of absorption
inside the laser rod and the resonator stability depends on how well the sun is
tracked. The technologies of photovoltaics and diode-pumped solid-state lasers
have separately attracted much attention and achieved industrial maturity. The direct
pumping of a solid-state laser from sunlight to laser light saves two energy
conversion steps and is potentially more efficient. However, much less
consideration has been devoted to direct solar-pumped lasers and the technology
is currently much less mature. Solar-pumped laser is, however, much simpler and more reliable due to
the complete elimination of electrical power generation and power conditioning
equipment. Space-based applications for solar lasers hold the most promise.
6.7 W/m^{2} collection efficiency has been achieved by pumping the
Nd:YAG rod within the 3D-compound
parabolic concentrator-2D- compound parabolic concentrator (3D-CPC-2D-CPC) cavity [6]. The recent progress in high-efficiency and
economical solar-pumped laser with a Fresnel lens and a chromium codoped
ceramic laser rod has largely improved the collection efficiency to 18.7 W/m^{2} [8]. Researchers, including us, are now
strongly motivated to build a powerful solar-pumped laser for the efficient
reduction of magnesium (Mg) from magnesium oxide (MgO). Large amounts of heat
and hydrogen (H_{2}) are given off from the reaction of magnesium with
water. Mg can be an alternative to fossil fuel. But for a magnesium combustion
engine to function as a practical source of energy, the lasers need to be
powered by a renewable energy source, such as solar power.

The imaging property of an elliptical cavity has already been widely explored in solid-state lasers, contributing to the improvement of laser beam quality [9–11]. An input face of the truncated cavity is firstly positioned to the focal area of a primary parabolic mirror. By optical refraction at the air/fused silica interface, a concentrated focal spot with large rim angle is then converted into a divergent spot with small rim angle within the fused silica material, facilitating its further concentration into the laser rod by the truncated cavity. ZEMAX nonsequential ray-tracing and LASCAD laser cavity analyses confirm both the laser efficiency and the beam quality advantages of the truncated cavity. Large standard tracking error for multimode power from the 3D-CPC-2D-CPC cavity is considerably reduced by the truncated cavity.

#### 2. Numerical Analysis of the 3D-CPC-2D-CPC Solar Laser System

The astigmatic corrected target
aligned (ACTA) solar concentrator system [6] provided an efficient pumping
approach, enabling the convenient placement of the laser system on a horizontal
optical table. The segmented primary mirror of 3.4 m diameter and 17 m radius of
curvature was mounted on a two-axis heliostat programmed to track the sun
continuously and project
the reflected sunlight toward
the entrance plane of a secondary concentrator via a plane folding mirror. The
double-stage secondary concentrator consisted of a first-stage 3D-CPC followed
by a second-stage 2D-CPC, as shown in Figure 1(a). Concentrated solar light at
half-angle cone entered the 100 mm diameter entrance aperture
of the 3D-CPC, which in turn funneled the light out of its 24 mm diameter exit
aperture at half-angle cone. The emitted light entered the mm^{2} aperture of the
2D-CPC, as illustrated in Figure 1(b), illuminating an antireflection end-coated
1.1% Nd:YAG laser rod, 6 mm in diameter and 72 mm in length. The laser rod was
mounted inside a quartz flow tube along the 2D-CPC axis. The laser resonator
design was plane-parallel. The rear mirrors were coated with a high reflectivity
dielectric coating of R 99.7%. Output mirrors
with different reflectivity, varying between 80% and 98%, were tested
individually. For the first mirror with 6.85 m^{2} collection area, 45 W laser
power was achieved by using a 95% reflectivity output mirror.

ZEMAX nonsequential ray tracing and LASCAD laser cavity software was used to evaluate numerically the laser performance of the 3D-CPC-2D-CPC solar laser system. The dimensions of the first-stage ACTA solar collector were utilized in the nonsequential ray tracing software. The profiles of both the 3D-CPC and the 2D-CPC were essential for designing both the axially symmetric 3D-CPC concentrator and the 2D-CPC pumping cavity. The 3D-CPC-2D-CPC cavity designed in AUTO-CAD software was then exported to ZEMAX software. The laser rod, the cooling water, and the flow tube were also dimensioned. Several aspects are important for the correct numerical analysis of the 3D-CPC-2D-CPC solar laser system.

*The Solar Spectra, the Tabulated Model of Absorption for Nd:YAG Material, and the Overlap between the Nd:YAG Absorption Spectra and the Solar Spectra*

The standard solar spectra [12] for one and a half air mass (AM1.5) were used as the reference data for consulting the spectral irradiance (W/m^{2}/nm) at
each wavelength. The irradiance cumulative integral of the whole solar spectra equals
to the typical terrestrial value of 900 W/m^{2}, which agrees well with
the experimental data in [6].

For 1.1%
Nd:YAG laser medium, 22 absorption peaks were defined in ZEMAX software. The central
wavelengths of these peaks are 527 nm, 531 nm, 568 nm, 578 nm, 586 nm, 592 nm, 732 nm,
736 nm, 743 nm, 746 nm, 753 nm, 758 nm, 790 nm, 793 nm, 803 nm, 805 nm, 808 nm, 811 nm,
815 nm, 820 nm, 865 nm, and 880 nm. The highest absorption coefficient reaches cm^{−1}, while the lowest
is about cm^{−1}. The averaged
FWHM absorption bandwidth of each peak is about 1 nm [13]. All the above peak
wavelengths and their respective absorption coefficients were added to the
glass catalog for
Nd:YAG material in ZEMAX software.

Solar
irradiance values corresponding to the above-mentioned 22 peak absorption wavelengths
for the Nd:YAG medium, ranging from 527 nm to 880 nm, could be consulted from the
standard solar spectra for AM1.5 and saved as source wavelength data in ZEMAX
software. In order to reduce the thermal load on the laser rod in the ACTA solar
laser system, the IR radiation which does not contribute to laser pumping is
filtered out by the filter residing between the folding mirror
and the entrance plane of the 3D-CPC-2D-CPC concentrator. The same filter
eliminates also the radiation below 500 nm to prevent the creation of color
centers within the rod. Only the source wavelength data within 527 nm–880 nm range are, therefore,
analyzed in ZEMAX software. The undesirable UV+Visble (less than 500 nm) and IR (more
than 880 nm) radiation were not included in the ray-tracing analysis. For the terrestrial insolation
value of 900 W/m^{2}, 6165 W solar power reaches the ACTA mirror with
6.85 m^{2} collection area. If 14% overlap between the Nd:YAG absorption
spectra and the solar spectra is considered [3], 864 W total absorbable solar power is then attributed
to a 3.4 m diameter circular light source, representing absorbable incoming
radiation to the ACTA mirror, in ZEMAX software. solar half-angle
is assumed in the ray-tracing analysis.

*Optical Parameters for the Components of the ACTA Solar Laser System*

To
numerically analyze the ACTA solar laser system, 85% reflectivity for the first-stage
ACTA mirror, 90% reflectivity for the folding mirror, and 95% reflectivity for
all the other reflector surfaces of the 3D-CPC-2D-CPC cavity were used in ZEMAX
analysis. The quartz flow tube has 1 mm wall thickness. Three is a 1.5 mm gap between
the flow tube and the rod for cooling water. The side surface of both the rod
and the flow tube is
modeled as uncoated. The solar spectra absorption coefficients for both water
and the quartz flow tube are also defined accordingly. The 6.0 mm diameter cylindrical rod is divided
into a total of 40 000
zones. During ray tracing, the path length in each intercepted zone is found.
With this value and the absorption coefficients of the 22 peak absorption wavelengths,
the total power absorbed by the rod can be calculated by summing up the
absorbed pump radiations of all the zones within the rod. The absorbed powers
for the rods of other diameters are also analyzed.

Ray
propagation within the rod is three
dimensional. However, the number of rays propagating along the rod axis
direction is assumed equal on average. The ray distribution along the rod axis
is uniform as a result. As a typical illustration, we choose the central cross-sectional
absorption distribution perpendicular to the rod axis. The
gray-scale central cross-sectional
absorption distribution of the 6.0 mm diameter rod, pumped by the 3D-CPC-2D-CPC
cavity, is given in Figure 2. Black signifies near maximum absorption, whereas
white signifies little or no absorption. The 3D-CPC-2D-CPC cavity does not
provide a near-Gaussian absorption profile. There exists low absorption density
in the central core region. The ray-tracing analysis indicates that there is
also a strong lack of rotational symmetry and the maximum occurs on the sides
of the rod. The absorption distribution pumped by the 3D-CPC-2D-CPC cavity is also
represented by the absorbed flux/volume distributions along both the central
cross-section row and the central cross-section column of the rod, as shown in
Figure 2, indicating also a lack of absorbed flux in the central rod region.

The angular
tracking error of a commercial heliostat may reach 2 mrad [14]. For 8.5 m focal
distance, 2 mrad error corresponds to 17 mm transversal shift from the ideal
focal position of the ACTA mirror. If the focal spot moves in the direction
parallel to the laser rod axis, the strong pump absorption distribution will be
shifted correspondingly along the rod in the opposite axial direction, causing
only minor reduction in output laser power. If the focal spot moves in the
direction perpendicular to the rod axis (the transversal shift as we named),
the strong absorption distribution within the rod will be shifted laterally in
the direction perpendicular to the rod axis, causing significant reduction in
laser output power. It is, therefore, sensible to concentrate only on the
transversal shift of the focal spot in ray-tracing analysis. Pumped by the
concentrated light spot with 17 mm transversal shift along the input plane, the gray-scale
absorption distribution for the 6.0 mm diameter rod is illustrated in Figure 3. Significant
difference is found between the absorption distributions without and with tracking
errors, as illustrated in Figures 2 and 3, respectively. The transversal shift to
the right side causes the dislocation of absorbed pump power to the left bottom
side of the rod. Besides, the absorption distribution is neither rotationally
nor axially symmetric.

ZEMAX ray-tracing
code is used to maximize the absorbed pump flux within the rod. The absorbed
pump flux data from the ZEMAX analysis is then processed by LASCAD software. Output
couplers of different reflectivity, ranging from 80% to 98%, are tested
individually to maximize the multimode laser power. A plane-parallel laser
resonator of 450 mm length, an averaged solar pump wavelength of 660 nm, and a
round-trip loss of 5.0% [3] are used in the LASCAD analysis. For the 6 mm
diameter and 72 mm length Nd:YAG rod, 47 W laser power is achieved by adopting a 95%
reflectivity output coupler, matching well with the published experimental data
of 45 W [6]. This result validates also our estimate on the overlap between the solar
emission spectra and the Nd:YAG absorption spectra.

In LASCAD
simulation, 45 W output laser power is optimized to 57.3 W by using the 6 mm
diameter laser rod of only 50 mm, rather than 72 mm length. The round-trip loss is
consequently reduced from 5.0% to 3.6%. There is still enough space for both
mounting and cooling the Nd:YAG rod of 50 mm length through the 2D-CPC cavity of
33 mm length. To evaluate the laser performance of the 3D-CPC-2D-CPC cavity, the
Nd:YAG rods of 3, 4, 5, and 6 mm diameters are tested separately within the
3D-CPC-2D-CPC cavity. Output coupler of different reflectivity is also tested
separately to optimize the output laser power from each rod. The absorbed pump
powers, the multimode, and the TEM_{00} laser powers are listed in
Table 1. We define the standard tracking error as the power decrease resulting from a 2 mrad tracking error divided by
the laser power without tracking error. For the 17 mm
shift along the input plane of the 3D-CPC-2D-CPC cavity, the standard tracking
errors are given in italics in Table 1. With the increase of rod
diameter, there exist gradual increases in both absorbed pump power and
multimode laser power. The TEM_{00} laser power is, however, adversely
affected. A small diameter rod is effective in producing TEM_{00} laser
power. Relatively stable and large standard tracking errors are dominant
phenomena in the 3D-CPC-2D-CPC cavity. As rod diameter increases, there exists
a slow increase in standard tracking errors for output laser powers. The
reasons for these variations will be discussed in Section 4.

The
3D-CPC-2D-CPC cavity is commonly designed according to the edge-ray principle of
nonimaging optics [15]. The CPC is
used to convert the radiation from a large area homogenous source emitting
uniformly into small angles, to a small area homogenous output emitting
uniformly into large angles, thus the étendue is preserved. This preservation
implies that irradiance is larger at the output surface than at the input
surface, leading to a net concentration of the pump
radiation. The 3D-CPC-2D-CPC solar laser pumping system has, therefore, the
advantage of the efficient collection and concentration of pump power from the
large area input surface of the 3D-CPC concentrator to the small area rod surface
of the Nd:YAG medium. As seen from Figures 2 and 3, edge-ray design methods are,
however, too constrained to provide the desired absorption distribution within
the central core region of the laser rod, leading to reduced laser beam quality.
There are nevertheless numerous applications of when the strong concentration
of pump energy within the central laser rod volume is beneficial in laser
technology. It is, therefore, desirable to search for other cavity designs that
can produce efficient laser operation and laser beam of high beam quality.

#### 3. Truncated Fused Silica Elliptical Cavity

A heliostat with a plane mirror and a parabolic mirror are generally combined to pump a solar laser. In direct tracking mode, the parabolic mirror is mounted directly onto the heliostat. A laser head is mounted at the focal area of the parabolic mirror. Direct tracking features high collection efficiency. A small plane folding mirror can be applied to invert the incoming solar radiation so that the laser head can be installed a little more conveniently. In indirect tracking mode, a large plane mirror is mounted on a heliostat which redirects the incoming solar radiation to a stationary parabolic mirror. The laser head can be conveniently mounted at the focal area of the parabolic mirror. Easy laser resonator alignment is ensured. Both the heliostat and the parabolic mirrors can be easily installed on the same height [3]. Due to the large plane mirror required for redirecting the incoming rays from different incident angles, low collection efficiency is revealed by indirect tracking mode. The 3D-CPC-2D-CPC approach provides, on the other hand, advanced and high-cost technology for achieving the high collection efficiency. The simple and low-cost heliostat plane mirror and parabolic mirror combination is nevertheless still preferred in solar-pumped laser research. Due to the compactness of the truncated cavity and its consequent easy fitting in the focal area of the parabolic mirror, direct tracking mode is analyzed here to ensure high collection efficiency. Our analysis is also valid for the simple heliostat plane mirror and stationary parabolic mirror pumping approach.

The elliptical cavity is based on the geometrical theorem that rays originating from one focal line of an elliptical cylinder are reflected to the other focal line. As a consequence of the preservation of angles, the portion of the elliptical reflector nearer the rod forms a reduced image of the pump source [16]. For the truncated elliptical cavity mounted in the focal area of the parabolic mirror, as shown in Figure 4(a), the concentrated rays of rim angle are firstly refracted into the divergent rays of rim angle within the fused silica cavity. As a result of the significant reduction in incident rim angle, from to , only the portion of the elliptical reflector near the crystal and the two end faces are finally used to focus tightly the pump radiations along the central core region of the laser rod.

Some
practical considerations are important for the ray-tracing analysis. 900 W/m^{2} solar irradiance is used as
ZEMAX analysis. By using the 3.4 diameter and 1.32 m focal distance parabolic mirror, mounted directly
on the two-axis heliostat in direct tracking mode, 6020 W solar power is
concentrated into a 13.6 mm FWHM diameter focal spot on the entrance plane of the
truncated cavity. The focal spot has a near-Gaussian profile with 19.8 W/mm^{2} maximum power density, resulting in a temperature high enough for melting
metal. It is, however, still far below the softening point
for fused silica material. Even without active cooling, fused silica light
guide and fibres have already been successfully used to transport high flux
solar radiation without any damage to the optical material [17, 18]. The utility
of fused silica material for the proposed approach is hence ensured. High
optical purity (99.9999%) fused silica material can be used to manufacture the truncated
cavity, as shown in Figure 4(b). The truncated input face is coplanar with the
first focal line and perpendicular to the major axis of the truncated elliptical
cavity. A 9 mm diameter cylindrical cooling channel is opened along the second
focal line. Both the input face and the inner surface of the cooling channel of
the cavity are optically polished to allow for the highest transparency. The
highly concentrated focal spot can easily damage antireflection coatings and
all the polished surfaces are therefore assumed as uncoated in ZEMAX analysis.
The curved elliptical cavity surface and the two parallel end faces are also optically polished
and gold coating is finally applied (not shown in Figure 4(b)). A specularly
reflective fused silica pump cavity is hence formed. The pump wavelengths above
500 nm are efficiently reflected to the rod, while the wavelengths below 500 nm
are poorly reflected and absorption occurs at the cavity walls. A small dielectric or metallic
multilayer plane folding mirror [19], for example, a cold mirror, is mounted near the focal area of the
parabolic mirror. Useful solar radiation (from 500 nm to 880 nm) is reflected to
the truncated cavity while infrared heat (larger than 880 nm) is transmitted away
from the folding mirror. The efficient cooling to the rod, the truncated cavity, and the plane
folding mirror are of
vital importance. The laser rod is mounted between two crystal holders,
supported by two water-cooled end blocks, between which the truncated cavity is
mounted.

The
maximum absorbed pump radiation and the most favorable absorption distribution within the rod
are two crucial factors for optimizing the truncated cavity. The matching between the
absorption spectra of 1.1% Nd:YAG
medium and the solar radiation spectra is analyzed, as mentioned in the previous section. Identical
to the 3D-CPC-2D-CPC system, 85% folding mirror angle-averaged reflectivity, 90%
parabolic mirror reflectivity, 95% cavity reflectivity are also utilized. The central cross-sectional absorption
distribution perpendicular to the rod axis is also examined. The cross-sectional gray-scale
absorption distribution for the 6 mm diameter Nd:YAG rod, pumped by the
truncated cavity, is given in Figure 5. Black signifies near maximum absorption,
whereas white signifies little or no absorption. The truncated cavity does provide a near-Gaussian absorption profile. The
ray-tracing analysis indicates that there is high degree of rotational symmetry
and the maximum absorption occurs along the central core region of the rod. The
cross-sectional absorption profiles pumped by the truncated cavity are
represented by the absorbed flux/volume distribution along both the central
cross-section row and the central cross-section column of the 6 mm diameter rod,
as shown in Figure 5, indicating the strong concentration of absorbed flux in
the central core region. The optimized truncated elliptical cavity has finally the
following dimensions: *a* = 25 mm, *b* = 19 mm, and *h* = 32 mm.

The 2 mrad angular tracking error is also used for the two-axis heliostat upon which a primary parabolic concentrator of 3.4 m diameter and 1.32 m focal distance are mounted, resulting in 2.656 mm transversal shift from the ideal focal position. Similar to the previous section, we concentrate here only on the standard tracking error analysis for the transversal shift of the focal spot. Figure 6 shows the gray-scale cross-sectional absorption distribution for the 6.0 mm diameter Nd:YAG rod. The symmetric absorption distribution in Figure 5 is slightly shifted away from the central core region.

The
optimized absorbed pump flux data from the ray-tracing analysis is then
processed by LASCAD software for optimizing the multimode laser power in a plane-parallel laser resonator of
450 mm length and 3.6% round-trip loss. The averaged solar pump wavelength of 660 nm is also
utilized.
Output coupler reflectivity is used for maximizing the multimode laser power. Nd:YAG
rods of different diameters are analyzed individually. The absorbed pump powers,
the multimode, and the TEM_{00} laser powers are listed in Table 2. The standard tracking error for
each laser power is listed below each respective power value. With small diameter rod,
high-order modes are suppressed by large diffraction losses and beam quality
improves. High pumping density or small rod surface is attractive but
challenging for the 3D-CPC-2D-CPC approach. The total absorbed pump powers within
small diameter rods are, however, largely improved by the truncated cavity, as
compared by both Tables 1 and 2.

#### 4. Comparisons between the 3D-CPC-2D-CPC Cavity and the Truncated Fused Silica Elliptical Cavity

##### 4.1. 3D-CPC-2D-CPC Cavity

The ray-tracing analysis indicates that the first-stage 3D-CPC causes the major transmission loss. Besides, the 17 mm transverse shift of focal spot along the entrance plane of the first 3D-CPC concentrator results in 6% extra transmission loss. Independent of the rod diameter used for capturing the pump radiation within the secondary 2D-CPC cavity, the 6% extra pump power has already been lost in the first-stage. This explains why the standard tracking error for the 3D-CPC-2D-CPC cavity remains relatively large and stable. Large diameter rod traps slightly more pump power, so there is a slow reduction in the standard tracking error, varying between 6.7% and 6.3%, as the rod diameter increases from 3 mm to 6 mm. The transverse shift of the focal spot increases the output angles of the rays exiting from the 3D-CPC concentrator. The incident angles of the rays entering the 2D-CPC cavity are also increased. By observing the edge-ray principle in nonimaging optics [15], the absorbed pump power within the rod is shifted further away from the central core region, as shown in Figure 3. The concentration of the absorbed pump power in the peripheral zones of the rod is increased. Only a large diameter rod is capable of absorbing all the tracking error-dependent pump radiations. The laser power from the large diameter rod is hence sensitive to the tracking errors. Small diameter laser rod captures less pump power, missing especially the tracking error-dependent pump radiation located away from the rod axis. Small diameter rod is therefore less sensitive to tracking errors. This explains why the tracking error for the laser powers in Table 1 increases slowly with the increase of rod diameters.

##### 4.2. Truncated Fused Silica Elliptical Cavity

Compared with the multimode performances of the 3D-CPC-2D-CPC cavity, as listed in Table 1, there exists a general improvement in multimode powers for the rod of all diameters when the truncated cavity is applied, as listed in Table 2. With the increase of rod diameter, the standard tracking error is reduced from 7.1% to 4.0%. Due to the imaging properties of the elliptical cavity, the concentrated focal spot without or with tracking errors can all be concentrated exactly or approximately in the central core region of the 6 mm diameter rod, as shown in Figures 5 and 6. The absorbed pump power and the output laser power for large diameter rod are less sensitive to the transversal movement of the focal spot. Any transversal shift of the spot can, however, cause significant variation of absorbed pump power for small diameter rod. This explains why the standard tracking error for the multimode laser powers is gradually reduced with the increase of rod diameter.

Figure 1 and Table 1 indicate that the 3D-CPC-2D-CPC cavity is not effective in
providing the absorbed pump power within the central core region, producing
significantly less TEM_{00} power. Due to the large enhancement in the
absorbed power within the central core region, as shown in Figure 5, the strong
increase in TEM_{00} laser power is easily achieved by the truncated cavity.
The imaging property of the elliptical cavity signifies that any shift of the focal
spot along the input face causes the inevitable variation of the absorbed pump power
within the central core region. The standard tracking error for TEM_{00} laser power from the truncated cavity is hence large, varying from 11.5% to
9.5%, as shown in Table 2. The standard tracking error from the 3D-CPC-2D-CPC cavity is also relatively
large, varying from 8.6% to 9.2%, as shown in
Table 1.

The
truncated cavity constitutes an effective alternative to the 3D-CPC-2D-CPC
cavity. The laser output performance such as laser power and beam quality can
be significantly enhanced by this simple pumping approach. The standard tracking
error for the truncated cavity compares also favorably with the 3D-CPC-2D-CPC cavity. Table 3 summarizes the advantages
of the truncated cavity. The positive values in bold represent the enhancements
in the absorbed pump powers, in the multimode and, significantly, in the TEM_{00} laser powers. On the other hand, the negative values in italics represent the
reduction in standard tracking errors. There exist a general enhancement in
laser powers and an obvious reduction in standard tracking errors for multimode
laser powers. The positive values in italics represent the deficiency of the
truncated cavity, revealing slightly larger standard tracking errors for the
TEM_{00} power than those by the 3D-CPC-2D-CPC cavity.

The
*M ^{2}* factors for the 6 mm diameter rod, pumped by the truncated cavity,
are finally analyzed
by LASCAD software. and are achieved for the 450 mm resonator length, representing a near-symmetric
beam profile. The

*M*factors for the same rod, pumped by the 3D-CPC-2D-CPC cavity, are also analyzed. and are obtained for the same resonator length, revealing, however, a much more asymmetric beam profile. Laser beam brightness is the power divided by the product of beam spot area and its solid angle. That product is proportional to the product of for asymmetric beam. Thus, we define a figure of merit B, which is the laser power divided by . When the 6 mm diameter rod is pumped by the truncated cavity, 63.6 W laser power, and are obtained. The figure of merit B is 0.023 W. When the same rod is pumped by the 3D-CPC-2D-CPC cavity, 57.3 W laser power, and are achieved. The figure of merit B is then reduced to 0.016 W. The introduction of the truncated cavity favors, therefore, the efficient emission of high quality laser beam.

^{2}#### 5. Conclusions

The
truncated fused silica elliptical cavity is both simple and compact. ZEMAX nonsequential
ray-tracing and LASCAD laser cavity analysis softwares are used to numerically
calculate the laser output performance of the 3D-CPC-2D-CPC cavity, serving as
an excellent reference for the numerical evaluation of the proposed truncated cavity.
In terms of laser efficiency, Table 3 reveals a general enhancement in the absorbed
pump powers and the laser powers. The best choice is obviously the 6 mm diameter
rod. Compared with the optimized output performances of 3D-CPC-2D-CPC cavity,
the truncated cavity offers 3.6% more absorbed pump power, 11% more multimode
laser power and, significantly, 72.7% more TEM_{00} power. There exists
also 5.9% reduction in the standard tracking error for multimode laser power. Besides, the
truncated cavity improves largely the pump absorption distributions. There is considerably
more absorbed pump power within the central core region and the absorption
profile is more rotationally symmetric than the asymmetric absorption profile
pumped by the 3D-CPC-2D-CPC cavity. The truncated elliptical pumping approach is
hence effective for producing efficiently the laser beam of high beam quality.

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