Note: Descriptions are shown in the official language in which they were submitted.
12~187;~7
-1- 35-EL-1620
FACE PUMPED RECTANGULAR SLAB LASER APPARATUS
HAVING AN IMPROVED OPTICAL RESONATOR CAVITY
Background of_the Invention
1. Field of the Invention:
The invention relates to a laser apparatus in
which the lasing medium is a slab of rectangular
cross section and to a laser apparatus in which the
optical cavity is optimized for use with a rectangular
slab.
2. Des.cription of the Prior Art:
Lasers employing a rectangular slab as the active
medium conventionally are face pumped, with the pump-
ing source being provided by a flash lamp having an
appropriate spectrum, placed adjacent two lateral faces.
In the conventional face pumped laser, the laser is
installed in an optical cavity with the end faces of
the slab providing the entrant surfaces for the laser
beam. Conventionally, the laser slab is cut in a
parallelogram or trapezoid configuration, as seen from
the side, with the angles of the faces being cut at
the Brewster angle for full acceptance of light of a
"P" polarization with a minimum loss. The beam is
introduced on the axis of the slab with the entrant
~2~ 3'7
-2- 35-EL-1620
beam being diffracted to the bo$tom lateral surface, re-
flected to an upper lateral surface and so forth until
it emerges at the other end face, normally in an
extension of the same line and colncident with it.
The output power of the fa~e pumped slab
laser is limited by the bulk of the laser slab and
its heat dissipating properties. In the interest
of increasing the output power~ the bulk of the laser
laser slab may be increased. Howe~er, if a square
10 cross section is retained, heat dissipation
performance worsens as the thickness of the material
through which hea~ must pass is increased. On the
other hand~ a rectangular cross section in which the
bulk of the slab is increased without increasing
15 the thickness through which heat must pass, improves
the heat dissipation performance. In principal,
the width dimensions could be significantly
i.crezsed and additional flash lamps and additional
cooling applied to the major lateral faces of the
20 slab to produce proportional increases in the
output beam. Having selected a slab of rectangular
~2~87~
-3- 35-EL-1620
cross section, one must provide efficient coupling
between the optical resonator and the laser slab
and at the same time maintain good output beam
quality.
Summary of the Invention
Accordingly, it is an object of the present
invention to provide a face pumped slab laser apparatus
having an improved power output.
It is another object of the invention to provide
10 a face pumped slab laser apparatus having improved
output efficiency.
It is still another object of the invention
to provide a face pumped slab laser apparatus using
a rectangular slab having an improved optical
15 resonator.
It is a further object of the invention to
provide a face pumped slab laser apparatus using a
rectangular slab having an optical resonator providing
higher performance and improved beam quality.
These and other objects of the invention are
achieved in a novel laser apparatus comprising a slab
of a gain medium of rectangular cross section,
optical pumping means, ~nd a novel resonant optical
~Z1~737
-4- 35-EL-1620
cavity. A plane parallel to the larger lateral face
of the slab passing through the optical axis defines
an "S" plane of the apparatus, and a plane parallel
to the smaller lateral face of the slab and passing
through to optical axis defines a "P" plane of the
apparatus. The optical pumping means may be coupled
to the larger lateral surface of the slab to
energize the beam.
The novel resonant optical cavity comprises a
first convex cylindrical mirror having a finite
radius of curvature (Rl) in the "S" plane, and a
second concave spherical mirror spaced a distance L
from the first mirror and having a radius of curvature
2-
lS The variables R~, R2 and L are selected to
effect stable resonator beam formation in the dimension
of the beam lying in the "P" plane, and unstable
resonator beam formation in the dimension of the beam
lying in the "S" plane. The design is optimized to
provide sufficient gain for lasing and an increased
effective active energy extraction volume in the slab
for maximum energy extraction.
7~
35-EL-1620
The apparatus further c~mprises a~ aperture
placed between the second mirr~r a~ he slab, the
edges being oriented in plane~ para~lel to ~he S
plane to improve the beam q~ ty to beam elements
lying in the P plane.
In a preferred emboa~ment, t~e aperture
provides a Fresnel number ~ grea~r than 1 and less
than 4 to beam elements l~ing in-the P plane, and
the apparatus (i.e. the slab) provides a Fresnel
number in excess of 1~ to beam elements lying in
the P plane correspoDa~g to a beao size of 13.5 to
3.5 millimeters with L = 1 meter.
The laser output beam may be obtained from a
birefringent element and a polarizer installed within
the cavity or by use o a partially reflective end
mirxor. The apparatus is operable with reasonable
beam quality in either a long pulse mode ~r a
Q-switched short pulse mode. Outpu~s on ~he order
- of 1 joule per pulse have been obt~ined in the long
~0 pulse mode and over 0.65 joules us1ng Nd:YAG and
oYer 1 joule using Nd:glass in a Q-switched mode.
37
-6- 35-EB-1620
Brief Description of the Drawings
The novel and distinctive features of the
invention are set forth in the claims appended to
the present application. The invention itself,
however, together with further objects and advantages
thereof may best be understood by reference to the
following description and accompanying drawings in
which:
Figures lA and lB illustrate a stable/unstable
resonator containing a face pumped slab laser in a
laser apparatus in accordance with a first embodiment
of the inventionO The first embodiment operates in a
Q-switched mode by means of a Pockel cell. Figure lA
is a side elevation of the principal components and
Figure lB is a perspective view of the principal
components.
Figure lC illustrates a quarter wave plate of
a second embodiment of the invention in which a quarter
wave plate is substituted for the Pockel's cell of
Figures lA and lB to achieve long pulse mode operation
of the laser apparatus.
Figure 2 is a graph of the amplitude attenuation
factor (y) of the principal mode as a function of
magnification of the stable/unstable resonator of
~21~37
-7- 35-EL-1620
Figures lA and lB, in comparison to that of a
conventional unstable resonator.
Figure 3 is a graph of the output energy in
joules as a function of pump energy in joules of
s the stable/unstable resonator in comparison to that
of a conventional unstable resonator; and
Figure 4 is a graph of an output energy of the
~igures lA, lB embodiment as a function of stored
energy (Joules/cm3) with differing settings of the
Pockel's cell bias voltage, and providing a comparison
between long pulse mode and the Q-switched mode of
operation.
Description of the_Preferred Embodiments
Referring now to Figures lA and lBj a laser
l; apparatus in accordance with a first embodiment of the
invention is shown comprising a stable/unstable optical
resonator containing a gain medium in the form of a
slab, the apparatus being operated in a Q-switched mode.
The laser apparatus, which may be operated
either as an oscillator or as an amplitider, comprises
a slab 10 of the gain medium of rectangular cross
section, optical pumping means 11 arranged adjacent
the larger lateral surfaces of the slab, which includes
, ~.
l~,lS~37
-8- 35-EL-1620
a first convex cylindrical mirror 12, defining one
end of the optical cavity, and a second, concave
spherical member 13 defining the other end of the
optical cavity, and a Pockel's cell 14 and a
polarizer 15 for Q-switching the optical resonator to
effect operation in a Q-switched mode,the polarizer 15
also being the point at which the output beam of the
laser apparatus is derived.
The optical elements of the laser apparatus
are arranged along an op~ical axis (the Z axis) as
illus rated in both Figures lA and lB. In a left to
right se~uence, the Pockel's cell 14 is second, the
polarizer 15 is next, ~ollowed by the slab 10 and the
convex cylindrical mirror 12 is last. These five
elements are arranged at intervals which will be
further defined below. An optional aperture 30, with
vertical limits, may be provided between elements 13
and 14. The length of the cavity L, the radius R1 of
the convex cylindrical mirror 12 and the radius R2 of
the concave spherical mirror 13 define an optical
resonator in which stable operation is achieved in a
vertical dimension of the beam, the being being pre-
vented from expanding in the P dimension beyond the
aperture of the apparatus and unstable operation is
~chieved in a horizontal dimension, the beam being
121~3~ -
-9- 35-EL-1620
permitted to expand in the S dimension beyond the
aperture of the apparatus.
The P and S axes in Figures lA and lB, which
are established by the rotational orientation of the
slab 10 about the Z axis define both the vertical and
horizontal dimensions of the beam and thereby define
the rotational orientations of the non-polarization
sensitive members of the optical apparatus, but also
define polarization components, which define the
rotational orientations of the polarization sensitive
members of the laser apparatus. The larger lateral
faces of the slab 10 (the top and bottom faces in
Fig~res lA and lB) define the SZ plane, the SZ plane
being defined as the plane parallel to these faces
which passes through the optical axis of the apparatus.
The smaller lateral faces of the slab define the PZ
plane; the PZ plane being defined as the plane parallel
to these faces which passes through the optical axis
of the apparatus. Accordingly, for convenience in
discussing rays or a beam directed along the optical
axis of ~he laser, P polarization may be regarded as
being parallel to the PZ plane and S polarization may
be regarded as being parallel to the SZ plane.
Returning now to a consideration of the operation
of the laser apparatus of Figures lA and lB, it may be
~2~1373~
-10- 35 EL-1620
assumed that the optical pumping means 11 produces
an "inversion" in the active medium of the laser slab
in which there are larger populations of electrons in
higher energy states than lower energy states. As a
result of the pumping, radiation begins within the
slab as a spontaneous emission from electronic trans-
itions between an excited higher and a lower energy
state. This radiation is of a precise wavelength
an~ of a potentially coherent nature.
The rays, which exit via the end faces 16 and
17 of the slab parallel to the optical axis of the
apparatus, are utilized in the optical cavity. The
internal path of these rays through the slab is best
seen in Figure lA. The slab, viewed from the side, is
1~ a parallelogram, with the end faces being established
at the "Brewster angle" to the optical axis of the
apparatus. The effect of a cut in this manner is to
ensure that all light entering the left face of the
slab 10 of P polarization will enter the slab without
reflection loss. In addition, a substantial percentage
(e.g. 20%) of light of S polarization will be reflected
away and lost. The trace of a ray directed along the
optical axis of the apparatus and assumed to enter the
slab 10 at the mid-point of the left face 16 is
~Z1~3~
-11- 35-EL-1620
illustrated in Figure 1~. The:antering ray is
initially reflected downward toward the bottom surface
of the slab (assuming the orientations of Figures lA
and lB), is next reflected upward, then downward, then
upward and so on successively untll a final downward
reflection occurs and the ray exits at the center of
the slab's right face 17, again directed along the
optical axis of the apparatus. ( A return ray proceed-
ing from right to left would retra~e the same path.)
In view of the polarization--selective action of
the end faces of the slab 10, the slab may be regarded
as optically coupled to rays of P polarization travers-
ing the optical resonator. As is well established,
an optical cavity or resonator conventionally consists
of two mirrors between which an active (lasing) slab
is installed and which serves to allow light to pass
recurrently through the active matesial to extract
energy from the excited states. With careful execution,
the rays produced in the slab form a ~oherent beam of
light of a single wavelength.
In the Figure 1 arrangement, rays which have
exited the right face of the slab lD proceed to the
to the right along the optical axis Df the apparatus
12~737
-12- 35-EL-1620
toward the convex cylindrical mirror 12, which has a
100% reflective coating, and is reflected leftwards to
the right face of the slab. The vertical polarization
marks 21 denote that the rays in this path are of a P
polarization ~and the lack of a small circle superimpos-
ed on these marks denotes the exclusion of significant
rays of an S polarization). The cylindrical mirror 12
is oriented in relation to the axes of the apparatus
such that a trace of the mirror in a plane parallel to
the PZ plane will be a straight line, while a trace of
the mirror in a plane parallel to the SZ plane will be
a circle having a radius Rl. The rays reflected left-
ward from the mirror 12 remain of a P polarization (21)
and now enter the slab 10 at its right face. The
lS rays exit at the left face of the slab, having retraced
the internal path illustrated in lA and remain of a
P polarization as indicated by the polari~ation marks
22. The rays continue leftward until they impinge on
the dielectric polarizer 15. The polarizer 15 is a
flat plate, which transmits (e.g. 99%) light polarized
parallel to the plane of incidence (a plane defined by
the incident ray and the projection of that ray upon
the face of the polarizer) and reflects (e.g. 99%)
121~37
-13- 35-EL-1620
light polarized perpendicular to the plane of incidence,
The polarizer is rotationally aligned upon the optical
axis of the apparatus such that its plane of incidence
is parallel to the PZ plane of the optical apparatus.
Thus, rays of substantially pure P polarization 22,
which enter the polarizer from the right, continue
throuyh the polarizer 15 to the left, with the vertical
marks 23 denoting P polarization, which continues for
the leftward rays until the Pockel's cell 14 is
encountered.
The Pockel's cell 14 i5 centered on the optical
axis of the apparatus in a prescribed rotational re-
lationship with the PZ, SZ planes, which will herein-
after be referred to as the "P" and "S" planes,
1~ respectively. The Pockel's cell forms an electronically
controlled optical "shutter" in the laser apparatus,
turning it on or off by use of an electro-optical
effect in its crystalline constituent subject to
voltage control. In the state illustrated in Figures lA
and lB, the Pockel's cell is electrically energized,
and has produced a net 90 rotation of the polarization
of the rays (23) (whose polarization is indicated by
vertical lines) proceeding leftward toward the
737
-1~- 35-EL-1620
Pockel's cell in relation to rays (25) (whose
polarization is indicated by circles) formed after
a reflection from mirror 13, and exiting the Pockel's
cell to the right, and proceeding toward the polarizer
15. When the Pockel's cell is in~this active state,
essentially all (e.g.99~) of the light impinging on
the polarizer 15 is ejected from the resonator as
shown at 26. The ejection of the light reduces the
optical "Q" of the resonant cavity, and produces
"Q-switching" to an off state.
Q-switching operation depends upon the appearance
of birefringence in the Pockel's cell under the in-
fluence of an external electric fiel~, a property
which disappears when the field is removed. A bire-
fringent material exhibits two indices of refractionto light of orthogonal polarizations corresponding to
a fast and a slow axis of propagation. In the pre-
ferred orientatiGn, the planes defined by the electro-
optic axes of the birefringent material and the
normal passing through the center of the Pockel's cell
are oriented at about 45 to the P plane of the laser
apparatus. In other words, the fast (28~ and slow (29~
axes of the birefringent material are orie~ted at about
45 to the P polarization of the optical rays 23
~2~73~
-15- 35-EL-1620
incident from the right. Upon passage through the
Pockel's cell, the two orthogonal components of the
rays now travel along the same path but with the
trace of the P polarization on the P plane being phase
shifted 90 to the right along the Z axis, ahead of
the trace of the P polarization on the S plane. The
combination of the two mutually phase shifted compon-
ents results in a substantially circularly polarized
beam. The circularly polarized rays continue leftward
in the same circular polarization until they impinge
on the concave spherical mirror 13. The mirror 13
is centered on the optical axis of the apparatus and
aligned orthogonal thereto. The spherical mirror 13
reflects the rays incident thereon without disturbance
of the circular polarization (24). The circular
polarization continues as the reflected wave proceeds
to the right, returning to the Pockells cell 14. Upon
entering the Pockel's cell for a second time, the
rays encounter further birefringent action. The
component of the polarized rays projected on the P
plane again passes through the cell phase shifted
90 ahead of the component of the polarized rays pro-
jected on the S plane. The phase shift is now doubled
to 180, and the resultant vector, based on recombin-
ing the two component vectors, has now rotated 90 to
~L21~73'7
-16- 35-EL-1620
the horizontal P plane, as indicated by the
polarization circles 25. These rays (25) resulting
from Pockel's cell operation, now impinge on the
dielectric polarizer 15, which ejects substantially
all of the light (26) of S polarization to the output
optics, preventing laser operation.
The laser apparatus, assumiDg that the Pockel's
cell is not energized, lases - but without producing
an output. Assuming that the applied voltage is
reduced to zero, the Pockel's cell reverts to an
inactive state in which light of P polarization (23)
entering the Pockel cell from the right, exits the
Pockel's eellat the left remaining of a P polarization.
The light is next reflected from the mirror 13 (still
remalning of a P polarization) back into the cavity,
re-entering the Pockel's cell from the left, exlting
from the right, and continuing in a P polarization (23),
until it impinges on the lert race of the polarizer 15.
The polarizer 15 transmits all (99%) of the rightward
proceeding rays into the slab 10 and lasing continues
as if the Pockel's cell were not present. In this
setting or the Pockel's cell, substantially no light is
of the appropriate S polarization to be ejected to the
output. In this state, the laser, while operated a~
maximum power, produces negligible output.
~Z~73~7
-17- 35-EL-1620
Normal laser output occurs when an intermediate
voltage to that producing a 90~ phase rotation for
double transit and zero is applied to the Pockel's cell.
This produces an output polarization establishing a
specific percentage of feedback wi~hin the cavity and
-a specific percentage of light ejected to the output.
A typical Pockel's cell setting occurs with 40% of the
radiation incident on the polarizer 15 being ejected to
the output and 60% transmitted to the next element
within the resonator. This occurs with approximately
a 45 phase shift per transit (a 90 phase shift per
double transit) through the Pockel's cell, producing an
approximately circular polarization at the input to thf
polarizer.
In customary Q-switched operation, the pumping
means 11 is a flash lamp that is flashed several times
(e.g. 10) per second, each flash producing light peaking
after about 70 microseconds and having an overall duration
of about 100 microseconds. The stored energy in the
slab peaks about 120 microseconds after ignition of the
~lash lamp. The Q-switch is maintained in an off state
until the pumping has achieved maximum energy storage
in the slab, and then the Q-switch is operated to allow
lasing. Approximately 50 nanoseconds after the operation
~Z~73'7
-18- 35-EL-1620
of the Q-switch, a laser pulse of 20 to 30 nanoseconds
occurs. The Q-switch remains on until the intensity
of the laser beam has fallen to a small percentage of
the maximum intensity and then operates to prevent
further lasing. The Q-switch timing is set to prevent
the formation of two laser pulses rather than one, in
order to concentrate the laser output into a single
output pulse.
An optional item in the apparatus of Flgures lA
and 1~ is the aperture 30 illustrated in position
between the Pockel's cell 14 and the concave spherical
mirror 13 where it intercepts the collimated beam,
collimated by the mirror 13 before being returned via
the slab to the other mirror or ejected to the outpu~.
As illustrated,the apertur~ 30 is centered on the axis
of the optical system, and rotationally oriented such
that its upper and lower boundaries are parallel to the
S plane and bound the small stable axis of the optical
beam. The aperture normally has no limits parallel
to the P plane which might affect the unstable large
axis of the optical beam. The vertical opening is set
to insure optimum beam quality measured in the P plane
and has a size which establishes the vertical dimension
of the collimated beam as it passes to the output or
~2~8~73'7
-19- 35-EL-1620
back through the slab. Typically, this dimension
is 3.5 mm corresponding to approximately hal~ the
vertical dimension of the slab. The aperture reduces
any higher order modes which may be present in the P
plane and establishes an essentially Gaussian TEMoo
beam in that plane. The elimination of the highex
order modes tends to improve the quality of the wave-
front of the output beam in a composite manner affect-
ing both phase and amplitude. The introduction of a
lateral aperture (which is not herein suggested) along
the unstable axis of the resonator would have the
adverse affect of introducing additional fringes
(i.e. the Fresnel diffraction effect) into the output
beam and lowering the quality of the output beam.
I~ a practical example of the invention so far
described, the slab is a Nd:YAG slab laser having a
length of 139.37 mm, a width of ~15. mm, and a thickness
of ~8. mm with the ends cut at the Brewster angle.
The slab is placed within an optical cavity having a
length (L) of 1 meter. The concave spherical mirror 13
has a radius of curvature (Rl) of 6 meters and the
convex cylindrical mirror 12 has a radius of curvature
(R2p) of 4 meters.
lZ~L~373'7
-20- 35-EL-1620
The optical resonator cavity gives definition
to the laser beam originating from spontaneous emission
within the laser slab. As earlier stated, the optical
cavity is vertically stable, implying that the beam is
prevented from expanding vertically (P dimension)
beyond the aperture of the apparatus, and horizontally
unstable, implying that the beam is permitted to
e pand horizontally (S dimension), and elements beyond
the aperture of the apparatus are lost.
Beam formation may be visualized in the following
manner (assuming confocal operation). The beam
elements which precede from the slab toward the concave
mirror, even if diverging, are reflected backward
toward the slab in a substantially collimated condition.
The collimated beam, upon passing backward through the
slab (and further intensified) impinges on the cylin-
drical mirror where the vertical dimension of the beam
is held constant while the horizontal dimension of
the beam expands. Thus, the vertically collimated,
horiæontally expanding beam returns from the cylindrical
mirror via the slab (and further intensification) and
impinges again on the spherical mirror 13. The beam
is again recollimated in the spherical mirror and
returned via the slab to the cylindrical mirror for
121~3~37
-21- 35-EL-1620
further horizontal expansion. The process gradually
causes the lateral limits of the beam to expand past
the natural apertures of the optical system and to be
lost. The vertical limits of the beam, however, remain
at a substantially constant size.
The stable axis of the optical resonator falls
in the P plane. This implies that rays forming beam
elements on the vertical axis and lying in the P-
plane do not walk out of the aperture of the apparatus.
The stability of the resonator is defined for the
stable (P) axis by the quantity Gp, where for
stability Gp Sl. To beam eIements lying on the stable
axis in the P plane, the resonator consists of a flat
mirror 12 haviny a radius R2p = 6 meters, which
indicates that the stability factor, Gp Sl, Gp being
defined as:
Gp = (l - L/Rlp) (1 - L/R2p)
where R1p = ~ tbeing flat in the P plane)
R2p = 6 meters, and
L = l meter.
Substituting:
Gp = (l) (1 - l/6) = 0.833
and is stable.
12~ 37
-22- 35-EL-1620
The unstable axis of the optical resonator falls
in the S plane. This implies that rays forming beam
elements on the horizontal axis and lying in the S
plane do walk out of the aperture of the apparatus.
The effect of beam walkout is to cause some loss, but
also to improve the quality of the beam.
The stability of the resonator is defined for
the unstable (S) axis by the quantity Gs, where the
stability Gs >l. To beam eleménts lying on the
unstable axis in the S plane, the resonator consists
of a curved convex mirror having a radius Rls = 4 meters,
a curved concave mirror 13 having a radius R2s =
6 meters, a curved concave mirror 13 having a radius
R2s = 6 meters, which indicates that the stability
factor Gs >l. Gs being defined as
G = (l - L/RlS) (l L/R2s)
= (l + l/4) (l - 1/6)
4 6 = 24 = 1.042
As noted above, the resonator is contocal in the
unstable axis. In particular:
Fls + F2s = L
with the quantities Fls = 3 m, F2s
and the common focus is placed outside the cavity.
One advantage of the confocal arrangement is that rays
ejected to the output as by the polarizer 15 from the
~2~E~73~7
-23- 35-E~-1620
concave spherical mirror are substantially colllmated
and do not require additional refocusing.
The magnification (M) of the optical cavity is
the ratio of the radii of curvature of the concave
mirror to the convex mirror: --
M = R2/Rl = 6m/4m = 1.where m is in meters.
The measure of the aperture size of the optical
cavity is the "Fresnel" number, which is equal to the
number of Fresnel fringes across the usable aperture
at the lasing wavelength. The beam size o~13.5 x 3.5 mm
corresponds to Fresnel numbers of
Nfl 1 /~
= 2.88 measured in the stable vertical axis
= 42.82 measured in the unstable horizontal
axis,
where al is half the aperture size and
~ is the wavelength of the radiation.
The beam produced by the apparatus of Figures lA
and lB is of rectangular cross section and visually
reassembles a horizontally magnified TEMoo mode. The
output beam is of high energy containing over .65 joule
for Q-switched operation. The estimated total volume
of the slab swept by the beam, based on a beam size
37
-24- 35-EL-1620
within the cavity of 13.5 x 3.~ millimeters, is about
6 cubic centime~. The unit has a Q-switched ex-
traction efficiency of over 50%, in contrast to a
more typical value of 30% to 40~.
S The increase in efficiency:of the use of
stored energy by virtue of the stable/unstable design
is evident from Figure 3. In Figure 3, the results
were experimentally obtained; the resonator was
an unstable confocal arrangement with a magnification
of 2 with a concave spherical mirror having a 4 meter
radius and a convex spherical mirror having a 2 meter
radius. The stable/unstable resonator employed a
2 meter radius convex cylindrical mirror, instead of
the 2 meter radius convex spherical mirror. The
resonator cavity length was 1 meter and apertures were
set to establish a beam size of 13.5 x 4 mm. Figure 3
illustrates that, assuming lo~g pulse mode operation
and pumping plotted through a range of 40-100 joules,
a conventional unstable resonator will produce an
output energy of from .08 to .65 joules. Under similar
conditions, the stable/unstable resonator will produce
output energy of from .15 to .95 joules, which is
significantly better. The efficiency of output energy
~2~37~7
-25- 35-EL-1620
utilization may ordinarily be somewhat further
increased as the magnification falls. Since this
produces some loss in beam divergence, the improve-
ment in output energy in reducing the magnification
to 1.5 or lower numbers must be reconciled with these
other factors.
The output energy in joules of the Figure 1~,
lB embodiment as a function of stored energy in
joules per cubic centimer is provided in Figure 4.
The magnification, as earlier noted, is 1.5 and the
device is operated in both the long pulse mode and
the Q-switched mode. In the long pulse mode the
separate Pockel's cell bias voltages were employed to
provide output reflectivities corresponding to 39%,
58% and 70%. The latter reading producing the highest
energy output. In the Q-switched mode, a setting
corresponding to 40% reflectivity was employed and at
the upper limit of operation, .2 of a joule p~r cc of
stored energy was achieved in a laser slab from
60 joules of pumping energy. As earlier noted, the
output energy was about .67 joules.
The design of the stable/unstable resonator
produces a larger output than comparable designs using
a rectangular slab while maintaining good beam quality.
~218737
-
-26- 35-EL-1620
It has been possible to generate a single transverse
beam whose dimension in one axis is comparable to the
-thickness of the slab. This is in part a property of
the slab geometry in which the rays propagating down
the slab are recurrently reflected between the major
lateral surfaces. In propagating the beam in success-
ive reflections between upper and lower lateral
surfaces of the slab, the thermal lensing effects are
cancelled out. The optical design of the cavity,
including the use of magnification (and unstable
operation) along one axis of the resonator, also con-
tributes to this larger output.
The use o~ magnification to expand the beam
parallel to the major faces of the laser slab provides
a substantial increase in cross-section of the beam,
permits a significant increase in output power, but
also produces potentially offsetting losses in
efficiency and in beam quality. Accordingly, the
magnification increase must represent a compromise,
optimized for the particular application. In the two
designs herein disclosed, magnifications of l-l/2 and
2 in the unstable axis have been employed.
Figure 2 indicates the calculated increase in
the"amplitude attenuation factor" or eigenvalue (~)
~L2~7~7
-27- 35-EL-1620
associated with increasing magnifi~ation. The
diffraction loss per pass through the cavity is
(1 - ¦YI2), so that a "y" near 1 represents a low
loss condition and a y near "0" represents a very
high diffraction loss. Figure 2 illustrates a
comparison between a stable/unstable resonator and a
conventional unstable resonator. The magnification
is plotted over a range of 1 to 3-1/2. (The amplitude
attenuation factor of the dominant mode for a 4 milli-
meter wide stable resonator is almost 1 and would besubstantially independent of magnification.) The
amplitude attenuation factor for the dominant mode for
a 4 millimeter wide unstable resonator is shown to
decrease rapidly as the magnification increases.
Assuming a magnification of 2, for instance, a con-
ventional unstable resonator has an amplitude attenu-
ation factor of approximately .79. Assuming a
magnification of 1.5, a conventional unstable resonator
has a much better amplitude attenuation factor of .88.
At tAe same time, a stable/unstable resonator has
higher (and better) comparable values~ For a magnifica-
tion of 2, the attenuation factor is approximately
.86 and for a magnification of 1.5 it is .92~
~L2~737
-28- 35-EL-1620
The stability factor is defined by the variable
"G", whose independent variables are the mirror
positions and their curvatures. In the example where
the magnification is 1-1/2, "G" has a value of 1.25
indicating instability, since it is in excess of 1.
The implication arising from the stability factor is
that individual rays of the beam in multiple reflections
will be reflected beyond the boundaries of the optical
system and be lost.
The F-K diffraction theory describes the optical
cavity and provides guidance to the initial optical
design along either the stable or unstable axis. The
cavity may be regarded as sustaining a plurality of
simultaneous modes. These modes, which are known as
eigenfunctions, are well known both in visual appear-
ance and as to phase and amplitude. These modes are
characterized by three letter designations (Transverse
ElectroMagnetic wave) with mode order subscripts
(e.g. TE~oo, 01' 10' Two independent
parameters of the eigenfunction are the Fresnel number
and the G number earlier discussed. The relationship
of these modes to one another in the composite beam
is by the eigenvalue (y), a complex quantity denoted
the amplitude reduction factor (as noted earlier),
121~3'737
-29- 35-EL-1620
which indicates both the amplitude and the phase of
a given mode as a function of the parameters of the
optical resonator.
In a conventional beam, where the TEMoo mode
is desired, some degree of contribution may be expected
from secondary modes. The individual modes have
differing amplitudes and phases and thus where contri-
bution to the beam from differing modes occurs, both
phase and amplitude control of the individual modes
must take place in the interests of beam quality.
Ideally, the eigenvalue which appears as y in the
power loss expression l _ ~y]2 should be near unity
for the principal mode and near zero for undesired
higher order modes. The classical statement of the
beam formed in the stable optical cavity i5 thus a
contribution of a plurality of products: Yn Un, where
y is the eigenvalue, U is the eigenfunction and n the
subscript denoting the mode order in question.
In a well designed laser apparatus, the composite
beam is required to have a substantially constant
amplitude and substantially constant phase front.
Along the stable axis of the classical resonator, the
magnitude of the eigenvalue is reduced as its phase
lZ~737
.
-30- 35-EL-1620
changes as the order of eigenmodes increases, forming
a spiral locus in a complex plane, migrating to the
center thereof. The rate of decrease in magnitude of
the eigenvalue increases as the Fresnel number de-
creases. Thus, the mode selectivity increases. Forlarge Fresnel numbers, the eigenvalues migrate toward
a unity circle in the complex plane reducing mode
selectivity. Thus, the designer, while desiring to
employ larger Fresnel numbers for higher efficiency
and higher power, is usually held to smaller Fresnel
numbers and the use of strategic aperturing to suppress
undesired higher order modes. In the present embodi-
ments, the designer has been able to employ Fresnel
numbers as high as 4 along the stable axis, without
significant deterioration of the beam quality.
In relation to the unstable mode, both the
concept of an eigenmode and eigenvalue y are imperfect.
The mode description is not exact since after a finite
number (on the order of 10 for Q-switched operation) of
reflections, a significant number of rays will be
reflected out of the system and lost. The quantity "y"
thus represents a mixture of diffraction losses and
beam "walkout". The mode structure is susceptible of
~Z~3~737
-31- 35~EL-1620
numerical integration, however, and indicates that
there is not an orderly pattern of dependency of the
eigenvalues upon the Fresnel numbers. Thus, while
one might expect larger Fresnel numbers to be un-
satisfactory in mode suppression, Fresnel numbersas large as 43 have been achieved with careful use of
magnification as earlier discussed.
The present stable/unstable resonator may be
operated to produce a good quality beam in either the
Q-switched short pulse mode illustrated in Figures lA
and lB or in a long pulse mode. In Q-switched opera-
tion, a stable resonator may in fact depend on the
time required to establish a mode to advantage to
suppress undesired higher order modes. In the present
stable/unstable resonator, Lmprovements in power with
reasonable beam quality have been practical in the
Q-switched, short pulse mode. ~In the long pulse mode,
this improvement is particularly great due to the off
axis beam wal~out of the design.
The invention has been used with Nd:YAG, as well
as Nd glass slab laser materials. The design contem-
plates operation with both high gain and moderate gain
laser materials, the typical high gain materials
lZ1~3'7
-32- 35-EL-1620
permitting magnifications from above 1 to about three
before losses become excessive~ Lower gain materials
would require lower magnifications.
Figure lC illustrates a convenient variation
of the first embodiment in whic~;a quarter-wave
plate 31 may be substituted for *he Pockel's cell 14
of Figures lA and lB to achieve long pulse operation
of the laser apparatus. The quarter-wave plate 31
is rotated to a desired angle in relation to the P,S
coordinate axes to reflect a desired amount of the
beam ~reflected from the concave spherical mirror 13)
to the output and to transmit a desired amount of the
beam back to the slab for recirculation within the
cavity.
In the arrangements so far described, the
output beam is derived by a dielectric polarizer in
association with a Pockel's cell or a quarter-wave
plate installed within the optical cavity. The
quarter-wave plate intercepts the beam within the
cavity and causes a portion of it to be ejected out
of the cavity to form an output be2m and a portion of
it to be transmitted on to the next element within
the cavity for recirculation within the cavity. If
12~ 73~7
-33- 35-EL-1620
the dielectric polarizer derives its bea~ from re-
flection from the concave spherical mirror (13), a
collimated beam is derived which simplifies the
output optics. One can, of course, derive the beam
from the spherical mirror if cylindrical correction
is introduced or one may employ a less than 100g
reflective sur~ace for the end mirrors and derive the
output beam from transmission through one of the
mirrors 12, 13. In this last case, the percentage
of reflection and the percentage of transmission of
the end mirror determines the amount of optical
energy which remains in the cavity and the amount
which is ejected to form the output beam.
In the arrangements so far described, the
primary coordinates of the optical system are defined
in relation to the slab 10. Ordinarily, the beam
which enters the slab at its center point of one end
face and which exits the other end of the slab at its
center point defines the optical axis of the laser
apparatus. Ordinarily, an extension of the incident
ray will be in approximate coincidence with the
emergent ray and thus to a first approximation the
optical axis of the system is a straight line. The
121~3~3'7
-34- 35-E~-1620
axis of the optical system need not be a straight
line, but may be bent or folded, so long as orienta-
tions orthogonal to the optical axis are maintained.
The use of a slab having Brewster angle ends, for
polarization selection, thus defines the P or S
polarization, which is used to apportion energy between
the output beam of the cavity and the beam recirculated
within the cavity. The lateral surfaces of the slab
are assumed for purposes of this discussion to be flat
with the major lateral surfaces parallel, the minor
lateral surfaces parallel, and the major lateral
surfaces lying in planes oriented 90 with respect to
the minor lateral surfaces. These lateral surfaces,
as earlier noted, define the P and the S planes.
Since the actual polarization settings are relatively
approximate, some imprecision in these requirements
may be tolerated without significant impairment of the
function of the laser apparatus.
The use of a srewster angle at the ends o the
slab laser permits a highly efficient entrance of the
beam into the slab with very little beam loss and
facilitates polarization of the resulting laser beam.
The Brewster angle construction permits the slab to
7~'7
-35- 35-EL-1620
operate at higher output power levels and provides
a beam polarizer useful in Q-switched operation in
which the polarizer itself has a high power capability.
