During 10/31/91 - 3/31/92, the following summarizes are major accomplishments: (1) numerical modeling of the four mirror astigmatically compensated, Z-fold cavity was performed; and (2) the simulation revealed several design parameters to be used for the construction of a femtosecond forsterite laser

Alfano, Robert R.

Petricevic, V.

Seas, A.

English

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NASA-CR-196529
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Tetravalent Chromium (Cr 4+) as Laser-Active Ion
for Tunable Solid-State Lasers
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A. Seas, V. Petri_:evi6, and R. R. Alfano
SEMI-ANNUAL PROGRESS REPORT
PRINCIPAL INVESTIGATOR: Prof. Robert R. Alfano
Period Covered: 10/31/91 - 3/31/92
Institute for Ultrafast Spectroscopy and Lasers
Physics Department, Room J-419
City College of New York
138 Street & Convent Avenue
New York, N.Y. 10031
GRANT NUMBER: NAG-l-1346
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ACCOMPLISHMENTS
During 10/31/91 - 3/31/92
the NASA grant: NAG- 1-1346
the following summarizes our major accomplishments made under
Numerical modeling of the four mirror astigmatically compensated, Z-fold cavity was
performed. The simulation revealed several design parameters to be used for the construction of a
femtosecond forsterite laser.
PUBLICATIONS AND PRESENTATIONS
During the period of this report the following paper was accepted for presentation:
it V. Petri_evi6, A. Seas, and R. R. Alfano, "Novel Cr4+-Based Tunable Solid-State Lasers",
Recent Advances in the Uses Of Light in Physics, Chemistry, Engineering and Medicine, at
City College of New York
2
RESEARCH PROGRAM
Modeling of a Four Mirror Astigmatically Compensated Cavity Design.
The major disadvantage of the three mirror cavity used for mode-locking of the forsterite
laser t,2 is that when is used with long samples the beam area inside the gain medium is not
uniform. This problem can be overcome by using a four mirror z-fold astigmatically compensated
cavity. The z-fold astigmatically compensated cavity design takes advantage of symmetry to force
the beam size in the xz and yz planes to be simultaneously minimum at the same spatial position in
the laser medium 3.
The z-fold four mirror cavity design is shown in figure 1. The four mirror cavity can be
thought of as a combination of two three mirror cavities as shown in figure 1. One three mirror
cavity consists of mirrors R 1, R 2, and a fiat mirror positioned at the image plane and the other
consists of mirrors R 4, R 3 and the fiat mirror at the image plane. Both cavities use half of the gain
medium. The folding angles of the mirrors R 2 and R 3 that compensate for astigmatism can be
estimated using eq.
= 2fsin0tan0 (1)
where: t - Half the thickness of the gain medium
f - Focal length of the folding mirrors
n - Index of refraction of the gain medium
0 - Angle of incidence of the folding mirrors
Numerical calculations were performed using eq. 1 to estimate the folding angle for mirrors R2 and
R 3 to achieve astigmatic compensation in the four mirror z-fold cavity. In the calculations it was
assumed that R2 = R 3. Figure 2 shows the folding angle as a function of the laser crystal thickness
in the case of the forsterite laser and assuming different values for the radius of curvature of the
folding mirrors.
R1 R2
i z B
x R3 R4
Fig. 1. Four mirror z-fold astigmatically compensated cavity design. 0 is the folding
angle of mirrors R2 and R 3.
35
30
oo 25
2o
v 15
<
5
| t | t |
0 1 6
R
I I I • I • • • , I • , • • I • • • • I • , , ,
2 3 4 5
Material Thickness (cm)
Fig. 2. Folding angle, 0, vs. forsterite crystal thickness for astigmatic compensation
in a four mirror cavity.
Other characteristics of the four mirror cavity can be determined by using the theory
presented in references 4 and 5.4, 5 A computer program was developed to simulate the beam size
4
in thefour mirror cavity usedin themode-lockingexperiments.For theprogramit wasassumed
thata 1cm longchromium-dopedforsteritecrystalwaspositionedexactlyat thecenterof thetwo
foldingmirrors R2 andR3.Theroundtrip ABCD matrix of thefour mirror cavity wascalculated
bychoosingareferenceplaneandfollowingthebeamfor acompleteroundtrip throughthecavity
(seefigure 3). Note that this processwasperformedtwice, oncefor the xz and oncefor the yz
planesincethefolding mirrorsandthebrewster-cutcrystalbehavedifferently in thetwoplanes
ReferencePlane
M1
R1
i
X
R 2
M2 M3_k
M5_M 4
M8 _-'_y6
M7M7
-- z _M 9 MlO
R3
MI1 OR 4
Fig. 3. Cavity configuration used in the numerical simulation.
The ABCD matrix for a complete round trip through the cavity is given by
Mt= M2M3M4M5M6MTM8M9MIoMllM10M9M8M7M6MsM4M3M2M1
where: M2, M 4, M 8, MI0 - ABCD matrices for traveling a distance in free space
M 1, MI1 - ABCD matrices for reflection from a flat mirror
M 9, M 3 - ABCD matrices for reflection from a curved mirror at an arbitrary angle of
incidence (Different for the xz and yz planes)
M5,M7- ABCD matricesfor interfacebetweenair andforsteritecrystal
M6 - ABCD matrix for travelingin the forsteritecrystal (Different for the xz andyz
planes)
The stability rangeof thefour mirror z-fold cavity designwasdeterminednext.This was
doneby determiningtheseparationbetweenmirrors R2andR3for which theself consistentand
perturbationstablerequirementsweresatisfied.Thefour mirror cavity is stablefor -+ 1mmfrom
optimumseparationbetweenmirrorsR2andR3whichis similar to thethreemirror cavity design.
When the stability rangewasdetermined,thedistancebetweenthe folding mirrors was
chosenaroundthecenterof thestabilityrangeandanotherprogramwaswritten to calculatethesize
of thebeamwaistatanypoint insidetheresonator.This wasdoneby movingthereferenceplane
throughout thecavity andcalculatingthetotalABCD matrix for thenewreferenceplane.Thenthe
beamwaistatthereferenceplanewasestimatedusingeq.2.
where
2 = IBl__  x1 -m21 (2)
A+D
m-
2
The results of the calculations are shown in figures 4 and 5. Figure 4 shows the size of the
beam waist as a function of the position in the cavity while figure 5 is a magnified version of figure
4 showing the behavior of the beam inside the forsterite crystal. Considering figure 4 the beam
waist at zero or at mirror R 1 has a value of about 600 I.tm and is the same for the xz and yz planes.
This fact can serve as an indication when aligning the four mirror cavity. A nice round output beam
is indicative of astigmatic compensation while an elliptical beam at the output indicates that the
folding angle is off the optimum value. The beam waist increases from 600 lam to 800 lam as it
moves 85 cm from mirror R 1 to the folding mirror R 2. Still at mirror R 2 there is good agreement in
6
thesizein thexz andyz planes.Mirror R2hasa radiusof curvatureof 10cm, forcing thebeamto
focus5 cm awayfrom themirror insidetheforsteritecrystal.As it canbeseenfrom figure5 the
beamsinsidetheforsteritelaserhavetheirminimumwaistatexactlythesamespot. Oncethebeam
comesout of theforsteritecrystalexpandsrapidly up to mirror R3 which collimates the beam.
Finally thebeammovesfrom mirror R3to R4whereit getsreflectedandthesametransformations
arerepeated.
_, 800
b
_, 600
400
E
200
0
R
R
Forsterite Crystal
Plane
--yz Plane
0 30 60 90 120 150 180
Cavity Length (cm)
Fig. 4. Beam waist vs. position in the four mirror, z-fold, astigmatically cavity
design.
oV
.,i,.a
.,,=_
o
200
150
100
50
0
89
xz Plane
--yz Plane //
• , . I • , , • I . , , , I | • | • i • . . , i • •
89.5 90 90.5 91 91.5
Cavity Length (cm)
Fig. 5. Beam waist vs. position in the forsterite crystal for the four mirror, z-fold,
astigmatically cavity design.
In order to be able to compare the results obtained from the analysis of the four mirror
cavity design with the three mirror cavity the same computer program was modified to correspond
to the three mirror cavity. For this program a 1 cm long brewster-cut forsterite crystal was placed
in a three mirror cavity with R 1 = 5 cm, R = 10 cm, R2 = flat and the same procedure as with the
four mirror cavity was repeated. The results of this simulation are shown in figures 6 and 7.
Figure 6 shows the size of the beam waist as a function of the position in the cavity while figure 7
is a magnified version of figure 6 showing the behavior of the beam inside the forsterite crystal.
By comparing figures 5 with 7 it is clear that a four mirror z-fold cavity design has better
mode characteristics as compared to the three mirror cavity. The four mirror cavity compensates for
the astigmatic effects introduced by the brewster forsterite crystal and also offers a uniform mode
area in the forsterite crystal. The cavity mode can be easily matched with the pump mode. In the
three mirror cavity design the minimum beam waists for the two planes are formed at different
positions in the forsterite crystal resulting in a non uniform mode area making hard to match pump
8
andcavitymodes.Sub-100-I.tmbeamsizeis achievedusingeithercavityconfiguration.
Fig. 6.
1200
_" 1000
"_' 800
600
-xz .... il
200 :. --Yz Plane Fo_terit¢ Crysta_
0
0 30
"R t
60 90 120 150
Cavity Length (cm)
180
Beam waist vs. position in the three mirror astigmatically compensated cavity
design.
400
tt_
o 300
t_,t
V
_ 200
• i,,,q
_ 100
o
.o_-xz Plane /
__ --yz Plane /
\\_ ¢"
......... ! • . , .... • • ! • ........ I • • . .
173.5 174.5 175.5 176.5
Cavity Length (era)
Fig. 7. Beam behavior inside the forsterite crystal in the three mirror astigmatically
compensated cavity design.
REFERENCES
,
.
,
o
5,
A. Seas, V. Petri_evir, and R. R. Alfano, "CW Mode-Locked Operation of Chromium-Doped
Forsterite Laser", OSA Proceedings on Advanced Solid-State Lasers, G. Dube and L. Chase,
eds. (Optical Society of America, Washington, D.C., 1991), Vol. 10, pp. 69-71.
A. Seas, V. Petri_evir, and R. R. Alfano, "Continuous-Wave Mode-Locked Operation of
Chromium-Doped Forsterite Laser", Optics Letters, Vol. 16, pp. 1668 (1991).
W. D. Johnston, P. K. Runge, "An improved Astigmatically Compensated Resonator for CW
Dye Lasers", IEEE J. Quantum Electron., Correspondence, pp. 724 (1972).
L. W. Casperson, "Mode stability of lasers and periodic optical systems", IEEE J. Quantum
Electron., Vol. QE-10, pp. 629 (1974).
A. E. Siegman, "Lasers", University Science Books, Mill Valey, California, chapter 21
(1986).
10


Tetravalent Chromium (Cr(4+)) as Laser-Active Ion for Tunable Solid-State Lasers

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