JOURNALIEEE
[11
ELECTRONICS,
OF QU.4NTUM
VOL. QE-8, N O . 3, MARCH 1972
REFERENCES
C. A. Burrus and R. W. Dawson,“Small-areahigh-current-
densityGaAs
electroluminescent diodesand
a method of
operationforimproveddegradation
characteristics,” A p p l .
Phys. Lett., vol. 17, pp. 97-99, 1970.
E21 C. A. BurrusandE.
A. Ulmer, Jr., “Efficient. small-area
GaAs-Gal-,Al,,4s
heterostructure electroluminescent
diodes
coupled toopt~ical fibers,” Proc. I E E E , vol. 5 9 , pp. 1263-1264,
Bug. 1971.
[31 C . A. BurrusandB.
I. Miller,“Small-area,double-heterostruckure aluminum-gallium arsenide
electroluminescent
diode
373
sources for optical-fiber transmissionlines,” Opt. Commun.,
to be published.
[41 J. J. Brophy,“Fluctuationsinluminescent junctions,” J . Appl.
Phys., vol. 38, pp. 2465-2469,
1967.
r.51 G. Guekos and M. J. 0. Strutt,“Current noise spectra of
GaAslaserdiodes
in the luminescence mode,” I E E E J .
QuantumElectron. (Corresp.), vol. QE-4, pp. 502-503, Aug.
1968.
E61 G. GuekosandM.
J. 0. Strutt,“Correlation between laser
emission voltage
noise and
noise in GaAsdiodes,”
CW laser
I E E E J. QuantumElectron.
(Corresp.), vol. QE-5, pp. 129130, Feb. 1969.
Astigmatically Compensated Cavities
for CW Dye Lasers
Abstract-An analysis is given of folded 3-mirror laser resonators
with an internal cell set
at
Brewster’s
angle. A method is described
tocompensatethe astigmatic distortionsintroducedby both the
internal
mirror
and
the
cell. This compensation is achieved for a
specific relation
between
cell thickness
and folding
angle.
It allows
the formation of a tight intracavity focus as required in applications
such as C W dye lasers.A discussion is given of the mode characteristics of compensated cavities and of the limitation
on
beam concentration set by the thicknessof the Brewster cell.
< r
2e
1
’
“>C
7;- - -
d2
-
\
R‘
*<y
Z
Fig. 1. Folded 3-mirror cavity with internal Brewster-angle cell.
I. INTRODUCTION
N CWdye-laserapplications
[1]-[3] there is a
need for cavity designs that provide an intracavity
focus where the modal beam is highly concentrated
(to typical beam diameters of 10 pm). This small spot
is demanded by pumpingrequirements. I n addition, i t
is desirable to have fairly
long cavity length (of typically 1 m) for tuning and mode-locking purposes. Cavities of thiskindhaveotherapplications.Oneexample
isacavity-dumpingsystem
[4] where an acoustooptic
deflector is positioned at the focus to allowhigh-speed
operation.Anotherapplicationisinthepassivemode
locking of low-gain lasers [5].
Cavity configurations that satisfy the seemingly contradictoryrequirements of longlengthandtight
focus
are resonators with internal
lenses [ 1j . An equivalent
systemisathree-mirrorresonatorsuchas
shown in
Fig. 1. This is oftenpreferred [2]-[5] because the reManuscript receivedOctober
27,1971.
The authors are with Bell Telephone Laboratories, Inc., Holmdel, N. J. 07733.
flection losses of thecentermirrorcaninpracticebe
kept to much smaller values than the Fresnel and bulk
losses of the internal lens. Such a folded cavity, however,
requires an oblique angle of incidence a t t h ecenter mirror
and introduces astigmatic distortions that limit the performance of the syst’em. In addition,theabovegroup
of applications requires an optical element to be inserted
at the focus (Le., the dye cell or the acoustooptic plate) This element is preferably inserted a t t h eBrewster angle
to minimize Fresnel losses. A Brewster window also has
astigmatic properties that can limit the performance of
the system. The effect of either astigmatic distortion becomes more severe the smaller the desired spot size. We
havefounditpossibleto
designa
three-mirrorlaser
reeonator with an internal Brewster-anglecell that is astigmatically compensated. The compensation is achieved
by offsetting the distortionsof t.he center mirror by those
of the Brewster cell. If, on the other hand, a. Brewsterangle cell is used in an “in line” resonator with
an internal lens [ l ], then only the astigmat.ism of the cell is.
present and no compensationis possible-
374
IEEE JOURNAL OF QUANTUM ELECTRONICS, MARCH
In the following, we will first discuss the properties of
small-spot cavities, then we will describe the astigmatic
properties of mirrors and Brewster-angle cells and how
they can be used t o compensate each other. This paper
is concluded byadiscussion of theproperties of compensated small-spot resonator systems.
6d , _I_
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I
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1972
,
11. SMALL-SPOT
CAVITY
WITFI IKTERNAL
LENS
T o prepare t.he ground for the discussion of compensatedcavities we will recapitulate,inthissection,the
properties of resonators wit.h an interna,l lens. The properties of interest are the resonator stability, the location
of the beam focus or "beam waist", and the beam radius
at the waist. Expressions for t.hese quantities are available in the literature [6], but we will cast these into a
form that is geared towards the achievement of a tight
internal focus and that makes transparent the behavior
'of the system.
Fig. 2 shows a resonator with an int,ernal lens of focal
length f. The spacings between the lens and the mirrors
are dl and dz and the curvature radii of the mirrors are
Rl and R2. To achieve a small internal spot or focus, one
designs the system with a large spacing
d2 >> f and a
short spacing dl that is approximately dl s R1 f. The
small spot is produced in the vicinity
of the cent'er of
curvature of the mirror R1, Le., in the short leg of the
system.
T o analyze this system we can use the imaging rules
of [7]. These allow ustoreplacethecombination
of
mirror R z andthe lens byasingleequivalentmirror
RT2,which is indicated in Fig. 2. The result is an equivalent resonator that is empty and to which
we can applystandardformulas.Theimagingrulesgiveusthe
mirror spacing as
+
Fig. 2. Resonator with internal
lens
and
equivalent
empty
resona,tor.
In small-spot designs we usually have dz >> f and Rz z
00, which leadstostabilityranges
thatare approximately 2S = f"/&
The nextquantity of interestisthe
localtion of the
be'arn w8aistwhere the mode of the resonator is focused
to a small spot. For the distance
tl, from mirror R1 to
the waist, we have the expression [6]
+
t1 = d ( ~ ' , d ) / ( ~ ~R',
Aftm inserting (1)-(5)
$1
=
+6.(R1 + 6
(R1
(1)
.and the curvature radiusRrZ
we can rewrite this
(6)
in the form
6maJ
+ 26 -
(7)
I n inspecting this formula, it should be noted that the
waist location tl stays very much fixed a t Rl as one adjuststhemirror-lensspacing
dl throughthestability
range. We have
-
amin)/(Rl
&mar
d = dl - d2f/(dz - f),
- 24.
tl
=
=
&min
R1
=
-
6mas.
-
6min).
R,
X2/&.
(8)
At the limits of the range, the waist location is exactly
R1 and at the center of the range (i.e. for 26 = ,S
tl center deviatesonly a littlefrom R,. As anuIn small-spot systems of this type, the adjustment of mericalillustration,considertheparameters
of a systhe spacing dl is crit'ical, as the cavity is stable only over tern used in dye-laser experiments [ 2 ] ,where R1 = f =
asmallrange
of values(typicallya
few millimeters) 5 cm, dz = 190 em, and Rz = co. This resonator has a
near dl = Rl + f. It is, therefore, convenient to define
stability range of 28 = 1.3 mm. In the range center the
an adjustment measure S by
R1 by S2/R1 = 8.5 pm,
waistlocationdeviatesfrom
indeed,
a
small
deviation.
dl = R1 f
6.
(3)
The third quantity of interest is the beam diameter at
The stability range of the resonator extends from a adthe waist.The waist radiuswo is given by the formula[6]
justment value aminto a. maximum value s,,
which we
cancalculatefromtheusualstabilityconditions
[6] (~wo'/lX)~
= d(R1 - d)(R'z - d )
(i.e., d = R1 R f zand d = R 1 ) .We obtain
*(R1 Rz - d ) / ( R , R2 - 2d)2.
(9)
R'2
=
R2f2/(dz- f ) ( d z - Rz
-
f).
+
(2)
+ +
+
+
6min
= fz/(dz
amax
=
- R2
f2/(dz -
For the stability range2X we get
f>
*
-
f>
+
After inserting (1)-(5), we get, after some algebra,
(4)
KOGELNIK
375
e t Ul.: ASTIGMATICALLY COMPENSATED CAVITIES
Here the approximate expression is good a s long as R1
is large compared to the stability range 28. We see that
w o = 0 a t t h elimits of the stability range.As dl is tuned
away from these limits, wo increases rapidly reaching a
rather flat plateau that extends almost throughout the
range (see Fig. 4). I n t h e range center we have
( T W C / X ) , , ~ ~ ~ S2(1
~ : - S2/R12).
(11)
For R1 >> S the second term is negligibly small and we
can write for the confocal parameter
b of the mode
b
=
2awZ/X
M 28 k
5 f2/&
(12)
This implies that for a small-spot cavity, which
is adconpustednearthe
cent.er of itsstabilityrange,the
focal parameter of the modal beam is equal to the stability range.
Related to the confocal parameter is the far-field angle
of the beam, which is
Fig. 3. Brewster-angle cell geometry.Coordinatesandparameters used in analysis are shown.
using the propagation laws for Gaussian beams is indicated in Section V. Referring to the geometry shown in
Fig. 3 and a Brewster window of thickness t and refractive index n, one gets
+
4
=
x/awo
M
a.
f .4 M f dx/,xM
dm.
d,
(13)
From thisfollows the beam radius at the
lens
Wxena R5
d, = t d m / n 2
=
t d G / n 4 .
IV. COMPENSATION
OF THREE-MIRROR
CAVITIES
(14)
111.ASTIGMATICELEMENTS
I n a three-mirror cavity of the type shown in Fig. 1,
there are two optical elements that introduce astigmatic
distortions,thecentermirrorandtheBrewster-angle
cell. This
astigmatism
means
that
geometrical
ray
bundles in the (sagittal) xx plane behave differently than
ray bundles in the (tangential) yz plane. In this section
we will characterize the two optical elements within the
paraxial ray formalism. This will
give us the elements
of the corresponding raymatrices,whichcanthenbe
used to determine the properties of the resonator modes
in the Gaussian-beam approximation [6] (ABCD-law)
.
We consider, now, resonator systems of the typeshown
in Fig. 1, which have two internal astigmatic elements,
thecentermirror,andtheBrewster
cell. To analyze
such systems, the assumptions and the results of Section
I1 canbe used.However,
theastigmatismmakes
it
necessary to consider the ra'y and mode characteristics
( a ) andthetangential
(yz) planes
of thesagittal
separately. T o makethisdistinction,
we will use the
subscripts x and y. Following Section 11, two adjustment
measures 6, and 6, are defined by
dl,
dl,
+ + 6z = + dz
R1 + + 6, = dair+ d,
R1
fz
dair
(17)
where fs and f, arethe effective focallengths of the
center mirror, and d, and du are the effective distances
A) Mirror Astigmatism
of
the Brewster cell, as discussed in the previous section.
It is well known [SI that amirrorused
a t oblique
The
effective mirror-lens separation dl, and dl, are madeincidence focuses sagittal (zx)ray bundles a t a different
up
of
the respective cell distances d, and d, and of the
locat,ion Ohan tangential (yx) bundles.Thisis reflected
optical-path
distance in airclair, which is the same for the
in two different effective focal lengths f, and f,, which
x
and
the
y
cases.
are related to the actual focal length f of the mirror by
Let us, first, inspect the stability limits , , , ,S
and amin
and the stability range 2s. In the presence of astigmafz = f/cOS e
tism,thesequantitiesare,inprinciple,differentfor
x
f,= f . COS e,
(15) and y. Equations(4)and
(5) show thatthedominant
factor causing a change in these quantities
is the astigmawhere 8 is the angleof incidence asshown in Fig. 1.
fz appearing in these
tism of the center mirror. The factor
B ) Brewster-Cell A s t i p a t i s m
equations has to be modified to fGz and fg2. However, in
practice,thesefactorsdonotdifferverymuch,as
the
ABrewster cell orBrewster-anglewindowalsoacts
differently on the sagittal and the tangential ray bundles. practical angles of incidence 6 are very small. For 8 '=
This action can be expressed by two different
effective 6O, for example, we have cos2 6" = 0.99, which implies
distances d, and d,, which the rays have t o traverse [9]. that the mirror astigmatism leads to a one-percent inA precise definition of the concept of an effective distance crease in the sagittal stability range and limits and to a
is provided by the ray-matrix formalism. T h e distances corresponding decrease in the tangential quantities. The
quantities a,,,
amin, and 2s are,therefore,essentially
d, and d, can be derived from
Snell's law and also by
the same for x and y, which allows a considerable simusingHuygen'sprinciple
[9]. Yetanotherderivation
fY
IEEE JOURNAL OF QUAKTUM KLECTRONICS, MARCH
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3.2
WY
mm
I
1
10
I
0.5
1.5
2 .o
(Nt/R)X 10'
Fig. 5 . Compensationangle
f wx
0 = 0"
plification of our argument. It is important to note, however, that the x and y values of theselimits occur a t
differen,t values of clair.
Theadjustmentmeasures
6, and 6, are,ingeneral,
,different.Thesemeasuresdependon
the adjustment of
dair,as indicated by (17). For a given clair, the resonator
willgenerallyoperatein
a differentportion o,f thex
stabilityrangethanthat
of the y stabilityrange.In
fact, it is possible that a resonator is stable in z and unstable in y, as illustrated in Fig. 4.
The purpose of an astigmatic compensation is to produceamaximumoverlap
of the 2- andy-st,ability
ranges,whichshouldleadto,adjustmentmeasuresand
mode characteristics that are more or less equal in x and
y (we shall see later that some differences remain even
after compensation). We can calculate the
difference in
theadjustmentmeasuresfrom(17)andobtain
=
(d, - d,) - (L - f,)
+
t(nz- 1) v'n2
1ln4
- f sin e tan e.
(18)
When the adjustment measures are equal, the resonator
operates in correspondingportions of the x and y stabilityranges andwehavecompensation.
According to
(18), this is assured when
=
2Nt
where
cavityparam-
~MICRoNS'
Fig. 4. Beam waist radiiandstabilityrangesfor
a folded
3-mirror ca.vity for various angles of incidence 0 at center
mirror. The horizontal scale measures the relative adjustment
of 2%fz and wI are thebeam waistradii inthe xz and yz
R1 = 5cm,
planes,respectively.
Thecavityparametersare
Ra 100, f = 5 cm, d, = 190 cm, dl 2 10 cm, t = 1.5 mm,
n = 1.45, and x = 0.57 ,urn.
8, - 6,
versusnormalized
eter N t / R .
=
2f sin e tan
e
R sin e tan 0
(19)
This shows that we can achieve compensation by
tradingt,he cell thickness t for the angle of incidence 8 on
the center mirror. Fig. 5. shows a normalized plot of the
relation between t and 0 leadingtocompensation (19).
The factor N is a function of t,he refractive index n only.
A selection of numericalvaluesfor
A T is given in the
following table.
n
N
1.3
0.396
1.4
0.430
1.5
0.445
1.6
0.449
1.7
0.446
It should be noted that the above compensationcan
only be achieved for the cell orientation shown in Fig. 1,
i.e., fora conmion plane of incidence,forbothcenter
mirror and Brewster cell. If the two planes of incidence
are oriented perpendicular to each other, then the astigmatic effects add and compensation is impossible.
V. PROPERTIES
OF COMPENSATED CAVITIES
I n this section we will invest,igate in further detail the
properties of compensat.ed three-mirror cavities and, in
particular, the characteristics of the modal beam inside
and outside the Brewstercell.
Fig. 4 illustrates the effects of compensationforour
numerical example from above, using values
n = 1.45.,t =
1.5 mm, and = 0.57 pm. I n this figure we plot the sagit'tal (w,)andthetangential (w,) beam-waistradiiasa
function of t.he relative adjustment of the distance d l .
From (10) w;.and w, arecalculatedandthe
region
for whichonegets
real wl valuesindicatesthestability range. We not'e that for 0 = 0 only one half of the
x-stabilityrangeoverlapsthe
y range.For 0 = 6.46"
compensation and good overlap of the st,ability ranges are
achieved. In the center of the common range, the waist
radii w, and w, are approximately the same and they are
relatively stationary with respect to a. change in dl.For
0 = 13" there is no stability overlap a t all, and the cavity
is always unstable in a t least one of the z or y coordinates.
Theapproximateequality
of wg and w, holdsquit,e
generally,notjustfortheabovenumericalexample.
KOQELNIK
377
et al. : ASTIGMATICALLY COMPENSATED CAVITIES
It followsfromtheequality
of thestabilityranges
(2&. z 2S,), which we havedemonstratedpreviously.
Inserting this into (11), we get for the waist radii in the
center of t.he stability range
w, NN w,
=
(21)
wo-
I n addition, it follows from (8) that the effective distances tl, and tl, from the mirrorR1 to thex and y waists
.are approximaOely equal
t,, M t,,
NN
R,.
Next we have to consider the change in beam radius
w,,which takes place when a beam enters a Brewstercell.
This change is the consequence of the refractional change
of the direction of propagation and it'can be calculated
from a simple geometrical projection. For a beam with
radius w just before entering the cell, we get t.he in-cell
beam radii
(22)
The above relations are very simple and they specify
the beam in the short leg of the cavity. It must be kept
i n mind,however, that the aboveparametersrefer
to
the properties of the beam in air. For dye-laser applications, we also need to know the behavior of the beam
inside theBrewster cell, whichcontainsthedyeand
is required.Theremainder
of
whereahighintensity
this section is devoted to the investigation
of this behavior. Inparticular, we shallderivetheobtainable
beam concentration in thecell.
First we recall ( 1 6 ) , which tells us that the effective
distances in the Brewster
cell aremeasureddifferently
in x and y. Then we choose position of the cell, which is
such thatthe
beam
waists
occur
inside
the
cell.
The effective mirror-to-waist spacings are now made up
of two parts, the distance tairfrom the mirror to the cell
entrance, and the effective distance inside the cell from
the entrance to the waists
w,
=
w
w,
=
w . cos
*%/cos
*B
= nw,
(27)
where g R is the angle of incidencein air(Brewster's
angle), whichobeys tan gR = n,and gn is the corresponding angle in t.he cell medium.
As shown in Fig. 3 we use the parameter [ t,o measure
thepropagationdistancealongthebeamaxisinside
the cell. It is related to the cell-depth parameter (measured perpendicular to the
cell surface) by
t
=
r/cos
*,,
=
rdn2
+ l/n.
(28)
Now we apply the laws of beam expansion by diffraction
[6]. In air the beam radius w ( x ) expands as
w(2)
=
wodl
+ (x2/7rw:)2,
(29)
where z is measured from the beam waist and wo is the
waist radius. Assuming for the moment
that the beam
waist is located a t t h e cell entrance, we apply (27) and
obtainfortheexpansion
of beamradiiinsidethe
cell
+ (xt/-nnw:)'
n w o d l + (~t/7rn~w:)~,
w,(3-> = w 0 d 1
t,,
rZ and
=
tsir
+ r , d n 2 + 1/n4 w R,.
indicatethelocation of the twowaistsin
the cell measuredperpendiculartot,he
cell surfaceas
shown in Fig. 3. These waist locations change as a function of tair.However, it follows from ( 2 3 ) , thatthere
is a relation
T,
rY
NN n2r,
(24)
that is independent of the cell adjustment tair.
We now postulate a symmetrical configurationsuch
that the x and y waists are equidistant from the center
of the cell. This implies
7,
+
7,
(25)
= t.
This case of symmetry should be very close to a practical
optimum.Combining (24) and ( 2 5 ) we get
=
t/(n2
T,
=
tn2/(n2 I ) ,
AT = r , -
+
7,
= t(n2 -
+
(26)
where AT isthespacingbetweenthe
x and y waists
in the cell. For our numerical example of n = 1.45, we
find that T, = tJ3, rY
2t/3, and AT
t/3, i.e., waists
that are spaced by. about one third of the cell thickness.
=
K t )
=
(30)
= TWZW,
maw:
*
l)/(n2 l ) ,
=
where the reduced wavelength (X/n) in the cell medium
has been taken into account.
The reader will take note that at this pointwe have an
alternate devrivation of (16). Comparing (29) and (30)
effective path lengths can be assigned to the cell. In this
comparison, we have to use (28) and the result agrees
with the effective cell distances given in ( 1 6 ) . The difference between da and d, is seen t o be a direct consequence
of the geometrical change of w, on entry into the cell.
I n order to be able t o determine the beam intensity in
theBrewster cell, i t isnecessarytoknowthe
crosssectional beam areaA ([) = rrw,~,.
T o allow for different
waist locations [,and f, we write
=
+ I),
T,
w,@)
(23)
dl
41+ (x/mw,2)'({
+ (A/rn3w:)2(5.
-
sz)'
- tJ2
1)(3
where we have used (30). For the special case of symmetrically arranged waist locations we have found with
(26) and (28)
lz = t/nv'n2
t,
= tn/-.
+I
(32)
378
IEEE JOURNAL O F QUANTUM ELECTRONICS, MARCH
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1972
t
f 57.5
-
I
m
m
P
'0 1.0 -
x
x
2
N
E
-
( r l -
-a5
.8 -
4
W
WAIST
LOCATIONS
1
U
f-5
W
LL
a
IT-
a
I.
a .6
I
-
4
W
m
W
m
I
.4
-
.2
-
p-CELL
REGION-q
I
I
II
DISTANCE
V
I
05
I
I
I
1.5
C E L LT H I C K N E S S
(Cml
(a)
Fig. 7. Minimumbeamarea
versus cell thickness forvarious
focal lengths of the center mirror. Other cavity parameters are
same as in Fig. 4.
A
14
-
12
-
'2 I O
-
hyperbola. For large values of f the beam focus (minimum A ) occurs approximatelyin cell center,shifting
-
m
X
N
s
I
-
a
W
Lz
4
z
6 -
W
m
4 -
2 -
I i -
C E L L REGION
q-
DISTANCE
Fig. 6. Beam area versus distance (y) in the Brewstercell for
a compensated cavity with various focal lengths of the center
mirror. (a) t = I mm. (b) t = IO mm.Othercavityparameters are same as in Fig. 4.
Fig. 6 illustrates the variation of t,he beam area A as
a function of the distance [ in the cell for the symmetricalcaseandfortwodifferent
cell thicknesses t. The
cavity parameters are thoseused in our previous numerical examples with several values for the focal length
f
of the center mirror. Fig. 6 ( a j shows the situation for a
1-mm-thick cell. The waist locationslrn
and &, are marked
on the plot. We note that A varies approximately as a
somewhattowards Cs (thesmallwaistradius)
for decreasing valuesof f.
Fig. 6(bj shows a similar plot for a
cell thickness of
t = 1 cm. There we notice the limiting effect of the cell
thickness on theachievablebeamconcent,ration.
As
before, the beam concentration tends t o improve as f is
decreased. B u t a t values lower than f z 5 em, a double
minimum develops and the beam area in the cell center
increases again. The minima occur near the waist locabetions Crn and bU where the beam areas seem to have
come insensit.ive tochangesin
f . For this case, f = 5
cm is a better choice than f = 2.5 ern and we do not gain
in beam concentration by reducing f beyond that value.
Inspecting (31j, the twoextrehesforthebehavior
illustratedabovearefoundtodependontherelative
magnitudes of the confocal bean1 parameter (i.e., the
stability ranges) and thecell thickness t. For X z r w o 2 / ~
>> t , we obt'ain for ( 5 = $)
A FZ m w O 2E nAX = nAf2/dz,
(33)
i.e., abeamarea
that isapproximatelyequaltothe
product between wavelength and stability range.
In the
limit of large cell thickness t >> 8, we get (also a t 4 = &)
A NN M(n" - l ) / n 3 d n 3+ I,
(34)
which tells us that the achievable minimum beam area
is now limited by the cell thickness. For n G 1.45,this
limit is approximately A = 0.2ht.
Fig. 7 illustrates how the minimum beam area varies
IEEE JOURNAL OF QUANTUM
ELECTRONICS, VOL. QE-S, NO.
3,
MARCH
379
1972
as a funct.ion of cell thickness for various values
of f. The
dashed line indicates the large t limit of (34). The (exactly)calculatedintersections
of the curveswiththe
ordinate are very nearly a t t h evalues predicted by (33).
VI. CONCLUSIONS
We have shown t h a t it is possible to designfolded
three-mirror cavities such t h a t t h eastigmatic distortions
of the center mirror are compensatedby the astigmatism
of the internal Brewster cell. T h e compensation leads t.0
beamcharacteristicsthatarequitestablerelativeto
small adjustments of the length of the short leg of the
cavity and it allows the achievement of small internal
beam spots. However, even with compensation, the thickness of the internal cell limits the minimum attainable
beam area t o about 0.2Xt, which is usually a much more
severe restriction than that posed by the F numbers of
the cavity mirrors. The results that we have derived are
applicable to any tightly focused system that requires
the use of either a Brewster-angle element or an off-axis
mirror. I n particular,thegreatbeamconcentration
achievablewiththecompensatedthree-mirrorcavities
have made this system of considerable interest for CW
dye-laser applications.
REFERENCES
M.HercherandH.
A. Pike. “Tunable dve laser confisurations,” Opt.Commun., vol. 3 , pp. 65-67, Gar. 1971. ”
R. L. Kohn, C. V. Shank, E. P. Ippen, and A. Dienes, “An
intracavity pumped CW dye laser,”-Opt. Commun., vol. 3, pp.
177-178, May 1971 * dso, A. Dienes, E. P. Ippen, and C. V.
Shank, “High-effidency tunable CW dye laser,’’ I E E E J .
Quantum Electron, this issue, p. 388.
A . Dienes, E. P. Ippen, and C. V. Shank, “A mode-locked
CW dye laser,” A p p l . Phys. Lett., vol. 19, pp. 258-260, Oct.
1971.
D. Maydan, “Fast modulator for extraction of internal laser
power,” J . App2. Phys., vol. 41, pp. 1552-1559, Mar. 1970.
P. K. Runge, “Mode-locking of He-Nelasers with saturable
organic dyes,” Opt. Commun., vol. 3, pp. 434-436, Aug. 1971.
H. Kogelnik and T. Li, “Laser beams and resonators,’’ AppZ.
Opt., ~051.5, pp. 1550-1567, Oct. 1966.
H.Kogelnik, “Imaging of optical modes-resonators with
internal lenses,” Bell Syst. Tech. J., vol. 44, pp. 455-494, Mar.
1965.
F. A. Jenkins and H. E. White, Fundamentab of Optics.
New York:McGraw-Hill, 1957, p. ‘95.
D. C. Hanna,“AstigmaticGaussian
beams produced by
axially asymmetric lasercavities,” I E E E J . QuantumElectron., V O ~ .&E-5, pp. 483-488, Oct. 1969.
Correspondence
Experimental Tests of Proposed Mechanisms for Gradual
Degradation of GaAsDouble-HeterostructureInjection
Lasers
D. H. NEWMAN, S. RITCHIE,
AND
S. O’HARA
Absiruci-Degradation rates of double-heterostructe GaAs
lasers have been measuredand found to vary superlinearly with the
injected current density above threshold. The results are discussed
in the context of the Gold-Weisberg phonon-kick and the extended
Longini field-inhibited diffusion degradation mechanisms.
The purpose of this correspondenceis
to comment on
the difficulties of interpreting the experimentalresultson
gradualdegradation of GaAslasers in terms of either the
or the extendedLonginifieldGold-Weisbergphonon-kick
inhibited diffusionmecha.nisms [l]. Othermechanisms,including the effect of the environment andtransport of impuritiesonorfromsurfaces,
may alsoexplainexperimental
results. The objectivehereis
to considerspecifically
the
feasibility of the two mechanisms mentioned above and about
Manuscript received Au ust 20, 1971: revised November 10,1971. The devices
described in this oorresponfencewere supplied by P. R. Selway andA. R. Goodwin
of Standard Telecommunication Laboratories, Harlow, England, under
a Ministry
of Defense (Navy Department) contract.
The authors are with the British Post Office Research Department, Martlesham
England.
Suffolk,
Ipswich,
Heath,
whichextensivediscussion
has taken place intheliterature
C11-[51.
It does not appear to be generally appreciated in the &erature ondegradation that theories of laseractioninperfect
homogeneous p-n junctiondiodes imply that the gain in the
activeregion satura,tes at the threshold current and remains
constant for steady currents abovethreshold [6], [7]. The
round trip gainmust in fa.ct be unity regardless of current
density above threshold for steady-state conditions to prevail.
As the gain is determined by the separation of the quasi-Fermi
levels for electronsand holes [ 8 ] , thelatter becomefixed
relativet.0 the bandedges. Thus in a perfecthomogeneous
p-n diode laser, all the excess current injected above threshold
goes into the stimulated emissionprocess. Also the electron
and hole densities inthe conduction and valence bands respectivelyremain constant above the threshold.Hence:
1) any electron-holerecombinationprocess
(other than
stimulated emission) is held at a fixed rate for all injection
current densities greater than threshold;
2) the voltage drop a t the junctionremainsconstant
above the threshold current density; the current can increase
because of the shortenedlifetime for electron-holerecombination at the junction due to the stimulatedemission
process.
It follows that if either the Gold-Weisbergphonon-kick
mechanism
(dependent
rate
on the
of nonradiative
recom-