Download Two Photon Absorption, Nonlinear Refraction, And Optical

Two photon absorption, nonlinear refraction, and
and optical
optical limiting
in semiconductors
Eric W.
W. Van
Van Stryland
H. Vanherzeele
A.Woodall
M. A.
Woodall
Soileau
M. J. Soileau
Arthur L.
L. Smirl
Shekhar Guha
F. Boggess
Boggess
Thomas F.
North Texas
Texas State
State University
Center
Electronics
Center for Applied Quantum Electronics
Physics
Department of Physics
Denton, Texas
Texas 76203
76203
Abstract.Two
Two-photon
Abstract.
-photon absorption coefficients /32
ß2 of ten direct gap
gap semiconducsemiconductors with
band-gap
with band
-gap energy Eg
Egvarying
varying between
between 1.4 and 3.7 eV were measured
pirn and
and 0.53 pm
^m picosecond
picosecond pulses. j32
found to
to scale
scale as
as E93,
E~3 , as
using 1.06 pm
ß2 was found
predicted by
the samples
samples measured.
measured. Extension
Extension of
empirical
predicted
by theory
theory for the
of the empirical
f32 and Eg
Eg to
Eg =
relationship between ß2
agreeto InSb
InSb with
with Eg
= 0.2 eV also provides agreepreviously measured
measured values
values and the predicted ß2.
/32 . In addition, the
ment between previously
absolute values of ß2
p2 sre
are in
in excellent agreement (the average
average difference
difference being
being
<26%)
includes the
the effects
effects of
of nonparabolic
nonparabolic bands.
bands.
<26 %) with
with recent theory, which includes
induced in these materials was monitored and
The nonlinear refraction induced
and found
found
to agree
assumption that the selfself-refraction
agree well
well with the assumption
refraction originates
originates from
from the
the
two-photon-generated
observed selfself-defocusing
two -photon -generatedfree
free carriers.
carriers. The
The observed
defocusing yields
yields an
effective nonlinear
nonlinear index
index as
as much
much as
as two
two orders
orders of
of magnitude
magnitude larger
larger than
than CS2
CS2
for comparable
comparable irradiances.
irradiances. This
self-defocusing,
two-This selfdefocusing,inin conjunction
conjunction with two
photon absorption, was used
used to
to construct
construct aasimple,
simple,effective
effectiveoptical
opticallimiter
limiter that
that
has high
input irradiance
irradiance and
and low
low transmission
transmission at
at high
high
has
high transmission
transmission at
at low input
irradiance. The
The device is the optical
optical analog
analog of
of aa Zener diode.
input irradiance.
Subject terms:
terms: nonlinear
nonlinearoptics;
optics;two
two-photon
effects; index
index of
of refraction;
refraction;semiconductors.
semiconductors.
Subject
-photon effects;
Optical Engineering 24(4),
1985).
24(4), 613-623
613 -623 (July/August
(July /August 1985).
CONTENTS
1. Introduction
Introduction
1.
2. Theory
Theory
2.
3. Experiment
Experiment and data
3.
3.1.
Experiment
3.1. Experiment
3.2. Data
Data
3.2.
4.
Comparison of
of ß2
/32 values
values to theory
4. Comparison
Two-photon
theory
4.1. Two
-photon absorption theory
4.2. Comparison
4.2.
Comparison to
to theory
theory
5.
Self-refraction
5. Self
-refraction
Optical limiter
limiter
6. Optical
7.
Conclusion
7. Conclusion
Acknowledgments
8. Acknowledgments
References
9. References
1.
INTRODUCTION
1. INTRODUCTION
specific material
give rise to these high
high nonlinearinonlinearispecific
material parameters that give
ties. This
This predictive
predictive capability
capability is
is extremely
extremely important
important from
from the
the
ties.
of searching
searching for materials with large nonlinearities.
standpoint of
A study of the
the nonlinear
nonlinear optical properties of several
several semiconducsemiconducpresented here, and aa relationship
relationship between
between the
the two
two-photon
tors is presented
-photon
absorption
(2PA) coefficient
coefficient £t2
material properties
properties isis
absorption (2PA)
ß2 and
and other material
verified. Ten
materials were
were experimentally
experimentally studied
verified.
Ten different
different materials
studied for
photon energy1
energy^
band-gap
which the incident photon
w is less
less than
than the direct band
-gap
energy Eg
but greater
greater than
than EEg/2
thattwo
two-photon
absorption isis
energy
E but
/2 sosothat
-photon absorption
allowed J rBoth
Both 1.06
1.06 and 0.53°µm
O.SJ/xm picosecond
picosecond pulses
pulses were
were used
used in
in
allowed.
transmission experiments
ranging
transmission
experiments using
using semiconductors
semiconductorswith
with Eg
E ranging
1.4 to 3.7
3.7 eV.
eV. We
We find
find that
thatthe
the2PA
2PAcoefficient
coefficientß2/^isisgiven
given by
from 1.4
=
Ks/Ep
K v E f(2ftg»/Eg)
f(2$0/ Eg)
(1)
n2E3
g
ever-increasing
in light
light-wave
The ever
-increasing role of semiconductors
semiconductors in
-wave technology
characterization of
of the
the nonnonhas created a pressing demand for the characterization
linear
optical properties
properties of
of these
these materials.
materials. Semiconductors
Semiconductors are
are
linear optical
as elements
elements in nonlinear
nonlinear optical
optical devices
devices because
because of their
attractive as
large
nonlinearities. A
A careful
careful
large and
and potentially extremely fast optical nonlinearities.
study of these
these macroscopic
macroscopic nonlinearities should allow one to determine
micro­
mine the
the dependence
dependence of
of these
these nonlinearities
nonlinearities on
on fundamental
fundamental microscopic
scopic mechanical and
and electronic
electronic material
material properties (e.g., band gap,
carrier lifetime, carrier
base formed
formed by
by
carrier effective mass, etc.). The data
data base
this
information would
only to tabulate
tabulate the
the
this information
would then
then allow
allow one
one not
not only
materials
exhibit large
large nonlinearities
nonlinearities but
predict the
the
materials that
that exhibit
but also
also to predict
where K is
is aa material-independent
material- independent constant, n is the linear refractive
is nearly materialmaterial-independent
index, and EEp is
independent for a wide variety of
function f,f, whose
whose exact form
form depends
depends on the
semiconductors.* The function
structure, is
is a function only of the
assumed band structure,
the ratio
ratio of the
the photon
photon
energy ftcu
are optically
optically
energy
fiw to
to Eg
Eg,, which
which determines
determines the
the states
states that are
coupled. The scaling
scaling given
(1) agrees
coupled.
given by
by Eq.
Eq. (1)
agrees with
with the
the most recent
two-photon
absorption3-5
3 " 5 and allows for predictions of
of
theories for two
-photon absorption
coefficients for
materials at
at other
otherwavelengths
wavelengths given
given
2PA coefficients
for other materials
parameters. For
Forexample,
example,extension
extensionof
ofthis
this scaling
scaling
minimal material parameters.
10.6 Mm
fa of 6.8
W,
[see Eq. (21)]
(21)] to
to InSb (300 K) at 10.6
µm predicts a ß2
6.8 cm/
cm/ M
MW,
Specifically,
which is in
in excellent
excellent agreement
agreement with recent experiments. Specifically,
Miller et al.
W. Equation
(1) is therefore
al.66 obtained
obtained a value of 88 cm/
cm/ M
MW.
Equation (1)
therefore
Invited Paper
Paper NO-110
NO-110 received
received Jan.
Jan. 15,
15, 1985;
1985; revised
revised manuscript
manuscriptreceived
receivedFeb.
Feb.7,7,1985;
1985;
accepted
7, 1985;
1985; received
1985.
accepted for publication Feb. 7,
received by
by Managing
Managing Editor April 8, 1985.
© 1985
1985 Society
Engineers.
©
Society of
of Photo-Optical
Photo -Optical Instrumentation
Instrumentation Engineers.
*E ==2P2m
2P2 m/fi2
where PP isis the
the Kane
Kane momentum
momentum parameter
parameter and m
m is
is the
the
*E
/ft2,, where
mass (Ref. 2).
2).
electron mass
OPTICAL ENGINEERING
/ July/August
1985
/ Vol.
2424
No.
4 /4 / 613
OPTICAL
ENGINEERING
/ July /August
1985
/ Vol.
No.
613
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VAN
VAN STRYLAND,
STRYLAND,VANHERZEELE,
VANHERZEELE,WOODALL,
WOODALL,SOILEAU,
SOILEAU,SMIRL,
SMIRL,GUHA,
GUHA,BOGGESS
BOGGESS
valid over a range of 20 in band-gap
band -gap energy
energyfrom
from the
the infrared
infrared to the
visible.
visible. In
In addition,
addition, we
we find
find that
that the
the proportionality constant K
K as
calculated by Weiler
Weiler44 for
fornonparabolic
nonparabolic bands agrees
agrees with
with our
our experexperimentally determined K
K to
to within
within better
betterthan
than26
26%.
%.
We
We used
used the experimentally determined
determined 2PA coefficients
along
coefficients along
with
Drude theory
theory (modified
(modified to
toinclude
includeeffects
with a modified Drude
effects of
ofinter
inter-band transitions and band
band filling)
filling) to model the nonlinear refraction
in
in these
these semiconductors.
semiconductors. We quantitatively fit the predictions of this
theory to beam
beam propagation data obtained
obtained for
for CdSe
theory
CdSe and
and obtain
obtain
refraction isis assumed
excellent agreement when all of the nonlinear refraction
arise from
from the
the carrier
carrier generation.'
to arise
generation. 7 That
That is,
is, the
thecontributions
contributions
proportional
the photogenerated
photogenerated carrier
carrier density
proportional to the
density dominate
dominate the
the
electron nonlinear
nonlinear refractive
refractive index
usual bound electron
index changes.
changes. This has
previously shown
been previously
shown to be the case for one
-photon absorption
absorption in
one-photon
/im 8 and
and Si
Si at
at 11 µm.9
pm. 9 We find
materials such as InSb at 55 µm8
find that the
effective nonlinear
effective
nonlinear refraction can be two
two orders of magnitude larger
that for
for CS2
CS2 at comparable
comparable irradiances.
irradiances.
than that
combined effects
effects of
of two
two-photon
Finally, we utilized the cohibined
-photon absorption and nonlinear
nonlinear refraction
refraction in
in GaAs
GaAs totomake
makean
anirradiance
irradiance(flu
(flu-ence)
device. 10 This
ence) limiting device.03
Thisdevice
devicehas
has high
high linear
linear transmission at
low irradiance
irradiance (fluence)
(fluence) and
low
and low transmission
transmission at high
high irradiance
irradiance
(fluence).
high irradiances,
irradiances, laser
(fluence). At
At very high
-induced melting
melting is
is also
laser-induced
This device
device is passive,
involved in the limiting action. This
passive, has picosecond
turn-on
turn -on time,
time, and
and isis the
the optical
optical equivalent
equivalent of a Zener diode.
Sec. 2 we
we describe the model
model used
used and
and derive
In Sec.
derive the
the equations
equations
needed
needed to
to describe
describe both
both the
the nonlinear
nonlinear transmission
transmission and the nonlinear
refraction
semiconductors studied.
refraction observed
observed in
in the
the semiconductors
studied. In
In Sec.
Sec. 3 we
we
outline
experimental procedure
procedure used
used to determine
determine the
outline the
the experimental
the two
two-photon absorption
absorption coefficients
coefficients and present
present the
the experimentally
experimentally
photon
determined 2PA
2PA coefficients. Section 4 presents a comparison
comparison of the
2PA data to theory for parabolic and nonparabolic
nonparabolic bands
bands with
with and
without exciton corrections. In
In Sec.
Sec. 55 we
we present
present the
the experiments
experiments
propagationdata
datausing
usingthe
theresults
resultsof
ofSec.
Sec. 2.
and fits to the beam propagation
2. We
We
describe aa semiconductor
semiconductor optical
describe
optical limiter
limiterbased
basedon
on 2PA
2PA and
and its
its
design, operation, and uses
uses in
in Sec.
Sec. 6.
6.
design,
THEORY
2. THEORY
The experimental configuration used
used throughout
throughout this work was
was one
one
in which
which the sample
sample was
was very
very thin
thin compared
compared to the confocal beam
and moreover,
moreover, any
anyself
self-induced
beam phase
parameter, and
-induced beam
phase changes
changes
were small
effects in the sample were
small enough
enough that
that beam propagation
propagation effects
were
self-action
negligible (i.e., selfaction was
was"external,"as
"external," as described in Ref. 11).
11). In
this case
case the
Maxwell wave
this
the Maxwell
wave equation
equation for
for the
the propagation
propagation of the
electric
electric field
field EE can be written as
2ik
ôE
ôz =
w2
iwaµo E- -c-12 X(3) I E I2 E,
(2)
where X(3)
x^ denotes
denotesthe
the third-order
third -order nonlinear
nonlinear susceptibility and
iNe2
n 1\ /^T
— 4- iNe2
a =-(a(a++aeXN)
aex N) n
-+
V µo
^o
mehw
meh
w
(3)
Here, we
we have
have explicitly included
denotes the conductivity. Here,
included the possibility
throughthe
the term
term aeXN,
sibility of
ofphotogenerated
photogenerated carrier absorption
absorption through
aexN,
carrier cross
cross section
section (holes
(holes plus electrons) and
where aex is
is the total carrier
and
is the density
N is
density of these
these carriers.
carriers. Also,
Also, a is
usual residual
residual linear
linear
is the usual
(e.g., band
band-tail
absorption (e.g.,
-tail absorption,
absorption, impurity
impurity absorption, etc.),
etc.),
meh isis the
the reduced
reducedelectron
electron-hole
effective mass. Writing
and meh
-hole effective
Writing the
the
electric field as
E =
= Ae'4'
Aei<J>
E
(4)
with the irradiance given by I == (ne0C
(ne0 C/2)A2
with
/2)A2,, Eq.
Eq. (2)
(2) can
can be sepasepagiving
rated, giving
dI
-aI --ß2I2
£ = -ol
/3212 --aeXNI
aexNl
dz =
(5)
and
dcl)
dz
= ßiI -
(6)
/32 , thethe
two-photon
absorption
where 1632,
two -photon
absorptioncoefficient,
coefficient,isisproportional
proportional to
to
part of
of X(3)
x^ and
{ == wy/c
the imaginary part
and Pß1
wy /c isisproportional
proportional to the real
part of
ofX(3).
x (3^- 7y is
the more
moreusual
usualn2
part
is related
related to the
n2 by
by n2
n2 (esu) == en
en y/
y/40
40 rr,
TT,
where the right-hand
right -hand side
side of
of the
the equation
equation is
is in
in mks
mks system
systemunits.
units. yy!l in
in
(6) is
is given by
Eq. (6)
by
Mo1e2 CP
y\y, _= µoe2CP
2nmehw
2nmej1 a>
(7)
(7)
The parameter P is
is introduced
introduced here
here to
to account
account for
for contributions
contributions to
the nonlinear refraction proportional
proportionalto
toNex
Nex but not
not explained by
by the
Drude model. An example of such
such aa contribution
contribution is that arising from
interband transitions.
transitions.12
12
interband
The equation governing the carrier generation is
is
dN
dt
ft*
/3212
(8)
2Yiw
showing that for every two absorbed photons
-hole pair
photons one
oneelectron
electron-hole
is
is generated.
generated. This equation is valid only for pulses
pulses short
short enough that
recombination and diffusion can be
be neglected
neglected during the
the pulse.
pulse. We
We
assume this to be
be the
the case
case for
for our
ourpicosecond
picosecondpulses.13
pulses. 13
We note
note that Eq. (5)
(5) for
for the
the irradiance
irradiance is
is independent
independent of the phase
(6)]. This
assumption of
of aa thin
[Eq. (6)].
This is
is due
due to our assumption
thin sample
sample (i.e.,
(i.e., no
no
irradiance changes
changes due
due to
to nonlinear refraction within
within the material).
We
We can,
can, therefore, solve Eq. (5) simultaneously with Eq.
Eq. (8)
(8) for the
beam attenuation
in aa sample
sample of
of thickness L. These equations must
attenuation in
be
be solved
solved numerically
numerically unless
unlessthe
thecontribution
contribution to
to the
the absorption
absorption from
the
photogenerated carriers is
is negligible.
negligible. We
the photogenerated
We can estimate under
under
what conditions this is true by finding
finding the
the irradiance,
irradiance, denoted by
by 'Cr'
Icr ,
for which the
the carrier absorption is equal
equal to the
the multiphoton absorption.
approximate relation is
is found
limit of
of small
tion. An approximate
found in the limit
small total
as
absorption as
/2-Rqj
Jcr ~" 2 V2 tub
'Cr
aeXto(1 - R)
(9)
where R
R is
is the surface
surface reflectivity
reflectivityand
andtot0isisthe
theHW
H W11 // ee M (half
(half-width
-width
at 1/e
1 / eofofthe
the maximum
maximum inin irradiance)
irradiance) of
of the assumed
assumed Gaussian
Gaussian
temporal profile
profile pulses.
pulses. This result was first given
given by Bechtel
Bechtel and
Smith. 13Note
Notethat
thatthis
thiscritical
criticalirradiance
irradianceisis independent
Smith."
independent of
of02
/32 since
since
both the transmission change and the photogenerated carriers result
materials with
with small
small ß2,
/32 , which require
from 2PA. Thus, materials
require high
high incident
incident
irradiance to observe a transmission change, will
will be
be the most likely
likely
affected by
materials to be affected
by photogenerated
photogenerated carrier absorption.
The contribution to the change in transmission from these
these carricarriers is
is proportional
proportional to tot.
t~ l . Longer
Longer pulses
pulses of
of the
the same
ers
same irradiance
irradiance
more energy and, therefore, produce
contain more
produce more
more carriers.
carriers. We
We can
can
absorpdetermine if these
these carriers
carriers are
are contributing to the nonlinear absorpby measuring
measuring the change
change in transmission
transmission for
tion by
for different
different pulse
pulse
We show in Sec.
used are
widths. We
Sec. 33 that
that the irradiances
irradiances used
are well
well below Icr
that we can ignore
and that
ignore photogenerated
photogenerated carrier absorption.
absorption. This
This is
is the
reason for using picosecond pulses,
pulses, as
as discussed
discussed in
in Ref.
Ref. 13.
13. While
While we
find the carrier
carrier absorption
absorption to
to be
be negligible,
negligible, the
the refractive
refractive index
find
index
proportional to
to the
the carrier
carrierdensity
density[Eq.
[Eq. (6)]
(6)] isis definitely
definitely not
change proportional
not
negligible, as discussed in Sec.
negligible,
Sec. 5.
5.
The solution to Eq.
Eq. (5)
(5) with this assumption
assumption isis
I(z,r,t)
I(z r t) =
R)J I(o,r,t)
I(o,r,t)e~az
(1 -- R)i
e az
1 +
+q(z,r,t)
q(z, r, t)
(10)
( )
where
- -R)R)
(1 (1
-e~az)/a.
whereq(z,r,t)
q(z,r,t)==ftl(o,r,t)
ß21(o,r,t)(1 (1
-e)/a. Inside
Inside the
the
1, and behind the sample j = 22 since there are two surface
sample jj = 1,
614 / OPTICAL
/ OPTICALENGINEERING
ENGINEERING / /July/August
Vol. 24
July /August 1985
1985 // Vol.
24 No.
No. 44
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SEMICONDUCTORS
IN SEMICONDUCTORS
REFRACTION, AND OPTICAL LIMITING IN
TWO PHOTON ABSORPTION, NONLINEAR REFRACTION,
effect
The effect
sample. The
the sample.
of the
is equal to L, the length of
reflections. Also, z is
but
ignored but
been ignored
has been
absorption has
of the rear surface reflection on the absorption
/32
determinationofofß2
the determination
in the
errors in
is not expected to lead to significant errors
give the
used. 14 Equation
for the samples used.14
Equation (10)
(10) can
can be
be rearranged
rearranged to give
of a sample of
position of
each radial position
instantaneous transmission T' at each
length L as
[l+q(L,r,t)]e«L
[1
+ g(L,r,t)] eaL
T'" 1
of T'
Since
q(z, r, t) isis directly
directly proportional
proportional to
to I(o,r,t),
I(o, r, t), aa plot of
Since q(z,r,t)
1
intercept
whose intercept
line whose
straight line
yield a straight
should yield
versus incident irradiance
irradiance should
pulses
/32 . Experimentally, pulses
whose slope determines ß2.
and whose
determines aa and
requires
which requires
used, which
are used,
profiles are
temporal profiles
of Gaussian spatial and temporal
Eq. (10). Taking into account the
of Eq.
integrationof
temporal integration
spatial and temporal
temporal and spatial integrals,
find for the pulse transmission
integrals, we find
•s<
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g_-A
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v-H
s't-i-HAMP F
[ AUD I
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» J
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1
ATTEN.
ATTEN.
!
tp
MONITOR
MONITOR
(ID
(1 --R)2
(l
R)2
ENERGY
ENERGY
Nd:YAG
Nd:YAG
SYSTEM
LASER
LASER SYSTEM
n
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i
1—1
COLLIMATOR
COLLIMATOR
SAMPLE FF
SAMPLE
-Q- ------ -.{|- -- -Q—>-|
--- -CDxy ••''
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ENERGY
ENERGY
MONITOR
MONITOR
HeNe
HeNe
absorption
two-photon
the twomeasuring the
setup for measuring
Fig.
1. Experimental
photon absorption
Experimental setup
Fig. 1.
indicates
splitter indicates
beam splitter
the beam
before the
arrow before
The arrow
/um. The
1.06 µm.
/J2 at 1.06
coefficientsß2
coefficients
were
pulses were
/xmpulses
0.53 µm
second-harmonic
the secondofthe
position of
the position
harmonic crystal when 0.53
used.
by
given by
fluence, given
oo
00
2a(l
2a(1 -R)
- R)
=
T
T=
%/TT ß2(eaL _ 1)
-xz]
q(L,o,o)e~x2]
e
$dxln[l +q(L,o,o)
(12)
0
'<••'••>- ^
-
Io exp
to
(13)
)2
(1 --R) tit [1 + q(L,r,t)]
<J>(o,r,t)+
3>(L,r,t) = (1)(o,r,t)
(I)(L,r,t)
+ ß2 (1
02
(1 - R)2 71
t
5dt' Fl (t')
2fi
(14)
,
-00
where
- aBn[1 +q(L,r,t)]
F,(t) =
Fl(t)
+q(L,r,t)]
é al
1
e~aL
q(L,r,t)a I" _
_ q(L,r,t)q
+ q(I-, r, t) '
11 +q(L,r,t)J
11 -e~«L
-é aL [1L
(15)
the
describe the
completely describe
together completely
(14) together
and (14)
2) and
Equations (10)
(10) (j(j = 2)
Equations
electric field
field at
at the
the exit
exit plane
plane of
of the
the sample.
sample. From
From these solutions for
sample
the sample
outsidethe
positionoutside
anyposition
fieldatatany
4>(L,r,t),
andc(L,r,t),
I(L,r,t) and
thethe
field
I(L,r,t)
Huygens-Fresnel
usingthetheHuygens
determinedusing
then
canthen
z,r,t)can
(L +-f z,r,t)
bebe
determined
-Fresnel
15
formalismasas15
propagation formalism
propagation
27r
E L+ z,r,t)=-exp
z,r,t =
ex
J'
r irrr2
;rr \
Xz
J
¡ r'dr'EE (L,r',t
L,r',t-- -) exp (-XzX
X J r'dr'
00
c?.
J
z,r,t)| 2 dt
.
(17)
.
z
t7rr'2
Experimental results
results are
are compared
compared with
with numerical
numerical evaluations
evaluations of
of
Experimental
(17), as described in Sec. 5.
and / or spatial
spatial integrals
integrals of Eq.
Eq. (17),
(17) and/or
Eq. (17)
EXPERIMENT AND DATA
3.
3. EXPERIMENT
T"-11 versus
The resulting plot of T
versus II has
has a slight
slight downward curvature
the
both the
caused by these integrations, since at the higher irradiances both
the
of the
rear of
spatial and temporal profiles are broadened toward the rear
sample
(i.e., there
there is
is more
more 2PA
2PA at
at the middle,
middle, brightest
brightest part of the
sample (i.e.,
beam). Examples
function of I0
Io are
of TT"-11 from Eq. (12) as aa function
plots of
of plots
Examples of
3.
Sec. 3.
of Sec.
5(b) of
shown in Figs. 5(a) and 5(b)
propagabeam profile of the pulse and its propagaIn order to model
model the beam
1)
1 0) (j(j ==1)
(6) using Eq. (8) for N and Eq. ((10)
Eq. (6)
tion, we
integrate Eq.
we now integrate
phase:
thephase:
for the
expression for
an expression
obtain an
for the irradiance to obtain
+
F(L
F(L ++ z,r)
z,r) =
-00
where we have taken
1(o,r,t) =
co
Jo
J0 (
27rrr'
Xz
)
(16)
o
pulses, isis the
experiments, using
measure in
we measure
What
What we
in our experiments,
using short
short pulses,
the
3.1. Experiment
of
transmission of
the transmission
measured the
we measured
experiments we
ofexperiments
set of
first set
the first
In the
several semiconductors
semiconductors as
as aa function
function of incident
incident irradiance
irradiance to deterseveral
experimental
The experimental
coefficients. The
absorption coefficients.
nonlinear absorption
mine their nonlinear
mine
microwas a microused was
source used
laser source
1. The laser
Fig. 1.
in Fig.
shown in
arrangement isis shown
arrangement
processor- controlled, passively
passively mode-locked
mode -lockedNd:YAG
Nd:YAG laser
laser that
processor-controlled,
amplified pulses
pulses of
of energy
energyup
up to
to 77 mJ
mJ per pulse
pulse at
at
single amplified
produced single
could
width could
pulse width
The pulse
mode.* The
TEM 00 mode.*
1.06µm
operated in
in the TEMoo
jum when operated
1.06
selecting etalons
150 ps
be varied
40 and
and 150
ps (FWHM) by selecting
etalons of
between 40
varied between
was
pulse was
each pulse
ofeach
widthof
Thewidth
coupler. The
output coupler.
varying thickness as the output
the
ofthe
energyof
theenergy
squareofofthe
monitored
thesquare
ofthe
ratioRRof
the ratio
measuringthe
by measuring
monitored by
µm) pulse
pulse to
to the energy
energy of
of the
the second
second harmonic
harmonic
(1.06 pim)
fundamental (1.06
crystal. 16 This
LiIO3 crystal.16
in aa LilO3
produced in
was produced
(0.53 jum)
µm) pulse that was
This ratio is
(0.53
the
width, provided
pulse width,
laser pulse
the laser
directly proportional
proportional to
to the
provided that
that the
directly
by
calibrated by
was calibrated
unchanged. This
remains unchanged.
spatial
This ratio was
profile remains
spatial profile
second-background-free
nearlybackground
usingnearly
width using
pulse width
measuring the pulse
-free second
having aa
pulses having
onlypulses
acceptingonly
whileaccepting
scanswhile
harmonic
autocorrelationscans
harmonic autocorrelation
the preset
preset value.
value. To
To ensure
ensure that
that the ratio
15% of the
fixed ratio R within 15%
were
scans were
autocorrelation scans
width, autocorrelation
pulse width,
R was proportional to
the pulse
to the
ratio
indeed, the ratio
performed
for three
three output coupler etalons, and indeed,
performed for
scan isis
autocorrelation scan
such an autocorrelation
example of such
scaled
properly. An
An example
scaled properly.
shown in Fig. 22 along
along with
with the
the best
best Gaussian
Gaussian fit.
fit. The
The autocorrelation
autocorrelation
pulse width of 38
Gaussian pulse
to a Gaussian
width of 54
54 ps
ps(FWHM)
(FWHM) corresponds to
38
ps
(FWHM).
ps(FWHM).
temperature-tuned
required, aa temperature
was required,
When
0.53 jum
µm light was
-tuned CDA
When 0.53
(cesium
dihydrogen arsenate)
arsenate) crystal
crystal was
was placed
placed in
in the
the beam
beam at the
(cesium dihydrogen
was
p,m was
1.06 µm
at 1.06
Light at
1. Light
Fig. 1.
position
by the
the arrow in Fig.
indicated by
position indicated
100% dielectric reflecting mirrors.
two 100%
and two
polarizer and
blocked with a polarizer
second-harmonic
the second
Autocorrelation
-harmonic beam performed with
ofthe
scans of
Autocorrelationscans
crystal
dihydrogen phosphate) crystal
an angle
-tuned KDP (potassium
(potassium dihydrogen
angle-tuned
by
divided by
width divided
pm pulse width
1.06 µm
the 1.06
as the
scaled as
pulses scaled
these pulses
showed that
that these
Gaussian-shaped
for Gaussian
expected for
v2
%, as
-shaped pulses. Again
as expected
within1010%,
\/2totowithin
were as clean
data were
held fixed,
fixed, and
and the
the autocorrelation
autocorrelation data
the ratio R was held
2.
Fig. 2.
1.06 jum
as those shown for 1.06
µm in Fig.
the
used in the
were used
150 ps were
Two different
pulse widths
widths of 40
40 and 150
different pulse
the
Since the
/zm. Since
1.06 µm.
transmission
transmission experiments
experiments on
on each
each sample
sample at
at 1.06
change the pulse width were
to change
used to
etalons used
coupler etalons
output
were optically
optically
output coupler
well as the
as well
lineas
beamline
thebeam
plate,the
quartzplate,
rotatable quartz
contacted to a flat rotatable
in
change in
percent change
few percent
(Afew
fixed.(A
remainedfixed.
measured beam parameters
parametersremained
the
in the
self-focusing
slightselfbyslight
causedby
probablycaused
width probably
focusing in
spatial width
the beam spatial
Calif.
**Quantel
Quantelmodel
modelYG40,
YG40,Quantel
QuantelInternational,
International, Inc.,
Inc., Santa Clara, Calif.
4 /4 / 615
No.
2424
/ Vol.
1985
/ July/August
OPTICALENGINEERING
ENGINEERING
/ July /August
1985
/ Vol.
No.
615
OPTICAL
Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
STRYLAND, VANHERZEELE,
GUHA, BOGGESS
BOGGESS
VAN STRYLAND,
VANHERZEELE, WOODALL,
WOODALL, SOILEAU, SMIRL, GUHA,
1.2
1.0
1.0
0.8
si
L
0.5
\
0.0
00..
-80
-40
0
40
I
°'4
UJ
0.2
0.0
80
80'
-0.2
_
•A******^
-3
-1
DELAY (Ps)
|ps)
DELAY
Fig.
Autocorrelationscan
scanofofpulses
pulseshaving
havingaa FWHM
FWHM of
of 38
38 ps
ps as
as calcuFig. 2.
2. Autocorrelation
lated
Gaussian (solid line)
lated from the best fit
fit Gaussian
line) autocorrelation.
amplifier was
relative error
error bars
bars between
between
was taken into account.) The relative
one
transmission experiment
the next, where
one transmission
experiment and
and the
where only the pulse
pulse
width was changed, were very small. While at high irradiances
(a few
irradiances (a
G W/
) we
W /cm2
cm2)
wedid
didsee
seea asmall
smallpulse
pulsewidth
widthdependence
dependenceof
ofthe
thetransmistransmission in some samples, this difference was consistent with values for
free-carrier
sections (10
(10~-17
17 to 10
10~-18
18 cm2
). No
the free
-carrier cross sections
cm2).
No pulse
pulse width
at the
the low
low irradiance
irradiancelevels
levels (0.5
(0,5 GW
GW/cm2
dependence was observed at
/cm2
)um) used
used to
to extract
extractvalues
values of
of the
the 2PA
2PA coefficient.
at l1 µm)
coefficient. A
A calculation
of Icr
Sec. 22 for
fortypical
typicalsamples
samplesatat1µm
1 ^im
Icr from Sec.
gives
IcrIcr
---~5
5 GGW/cm2
W / cm2
gives
= 55 X 10
10~
for aex =
18 cm
-18
cm2.. In
In fact,
fact, Eq.
Eq. (9)
(9)considerably
considerably underestimates
underestimates
Icr
Icr isis several
several times
times
'Cr' From computer calculations, we find that Ier
larger, the difference
difference arising
larger,
arising mainly
mainly from
from the
the fact
fact that the
the spatial
spatial
averaging was ignored in the approximate expression.
irradiance averaging
expression. In
In
addition, at 0.5
0.5 µm
jum the
the contribution
contribution of
ofphotogenerated
photogenerated carrier
carrier
addition,
will be
be less
less than
than atat1µm
1 /xmsince
since'ha)
absorption will
ftcoincreases
increasesand
andGex
aex
decreases, both leading
leading to an increase in Icr.
Icr The
decreases,
The maximum experiexperimental irradiance
irradiance used
j32 from
the 0.53
0.53 jum
mental
used to
to extract
extract ß2
from the
µm data was,
was,
therefore, increased
increased to
to22 GGW/cm2
. The
therefore,
W / cm2.
Theabove
aboveexperimental
experimental considignoring free
free-carrier
calculating the
erations justify ignoring
-carrier absorption
absorption in calculating
transmitted irradiance
irradiance[Eq.
[Eq.(5)].13
(5)]. 13
transmitted
The spatial beam
beam profiles
profiles in both
both the
the horizontal
The
horizontal and
and vertical
vertical
directions were
pinhole at the
the
directions
were determined
determined by
by scanning
scanning aa 25
25 /xm
µm pinhole
of the sample.
sample. The
The beam
beam size
size was adjusted at the sample
sample by
by
position of
pairs of
of collimating
collimating lenses.
lenses. In all, four different spot sizes from
using pairs
1.5 mm (FWHM)
were used for the 1.06
1.06 pm
/zm data.
data. At
At 0.53
0.53
0.5 mm to 1.5
(FWHM) were
jum
size used
µm the beam size
used was
was 0.5
0.5 mm.
mm. In
In addition,
addition, the beam profiles
that there
there were
were no
no hot
hot spots,
were monitored on a vidicon to ensure that
spots,
or shot
shot-to-shot
spurious reflections, or
-to -shot beam
beamwidth
widthfluctuations.
fluctuations. Figure
Figure
^m beam of FWHM
33 shows
shows aa representative
representative pinhole
pinhole scan
scan for
for aa 1.06
1.06µm
1.50mm.
1.50 mm.
The incident energy
energy was continuously varied
varied using
using aa stepping
stepping-motor-controlled
half-wave
motorcontrolled rotating half
-wave plate
plate in
in combination with
with aa
fixed polarizer.
on the
the sample
sample
fixed
polarizer. This
This apparatus kept the polarization on
fixed and introduced no
no measurable
measurable beam
beam walk
walk with rotation
rotation angle.
angle.
Previous experience
experience indicates
indicates that
that other alternatives,
alternatives, e.g.,
e.g., rotating
rotating
calcite polarizers, may cause the beam to walk across the sample and
across the energyenergy-monitoring
monitoring detectors.
The choice of detectors,
detectors, as
as well
well as
as the
the detection
detection geometry,
geometry, was
was
also
critical. As discussed
discussed in
Sec. 5,
phase
also determined
determined to
to be critical.
in Sec.
5, the phase
aberrations introduced
introduced on
onthe
thebeam
beambybythe
thetwo
two-photon-generated
-photon -generated
free carriers cause considerable defocusing so that the beam profile at
the detector
detector varies with incident irradiance. Figure
Figure 44 shows an example of this
this defocusing
defocusing as
as observed in the near field behind a sample of
CdTe.
CdTe. As
As the
the irradiance is
is increased
increased the
the beam
beam broadens
broadens and breaks
x__.
7
0
1
POSITION(mm)
Fig. 3.
A pinhole
pinhole beam
beam scan
scan showing
a best
best fit Gaussian
Gaussian (solid
Fig.
3. A
showing a
(solid line)
line) of
of
FWHM 1.50mm.
FWHM
1.50 mm.
POSITION
Vidiconscans
scans(equivalent
(equivalenttotoaapinhole
pinholescan)
scan) in
in the
thenear
near field
Fig. 4. Vidicon
field of
of the
the
/urn beam
beam transmitted
transmitted through
through aa polycrystalline
polycrystalline sample
sample of
of CdTe
1.06 µm
CdTe
showing the defocusing for
for increasing
increasing irradiance.
irradiance.
as isis characteristic
characteristicfor
forselfself-defocusing.
17 Thus,
non-up, as
defocusing.17
Thus, any spatial non
uniformities in the detector response can lead to errors. Indeed, care
be exercised
exercised to
to ensure
ensure that
that"external"
"external"selfself-action
18 does
must be
action 18
does not
result
an occurrence
result in
in overfilling
overfillingthe
thetransmission
transmissiondetector
detector -an
occurrence that
that
could result
result in an overestimate
overestimate of ß2
/32 (Ref.
(Ref. 19)
19) or could
could result
result in
in
could
discussed in
optical limiting, as discussed
in Sec.
Sec. 6.
6. We
We found,
found, however,
however, that by
using large area detectors (1
(1 cm2)
cm2) with
withaameasured
measured spatial
spatial uniformity
uniformity
of better than
than 10
10%
%and
and placing
placing them
them as
as close
close as
as possible
possible to
to the
the sample
sample
(3 cm),
(3
cm), these
these effects
effects were
wereeliminated.
eliminated. The
The detectors
detectors were
were also
also deterdetermined to be
be linear
linear over
over their
their range
range of
of use
use and
andwere
wereabsolutely
absolutely
mined
calibrated with respect to a pyroelectric energy
energy monitor*
monitor* that was in
absorbing-type
turn checked against two others.
others. In addition, absorbing
-type neutral
neutral
density filters placed in front of
of these detectors
detectors were
were checked to
to have
have
transmission over
over a range at least a factor
linear transmission
factor of 10
10greater
greater than
than the
range used in these
these experiments.
experiments. Filters were never used
used to
to attenuate
attenuate
the beam prior to the sample. In addition, spike
spike filters transmitting
transmitting
only 1.06
1.06 pun
µm (0.53
(0.53 /zm)
µm) were
were placed
placed directly
directly in
in front
front of the detectors
reduce optical noise
noise from the flash
flash lamps.
lamps.
to reduce
*Gentec
ED-100,
*Gentec ED
-100,Gentec
GentecInc.,
Inc., Ste-Foy,
Ste -Foy,Quebec,
Quebec,Canada
Canada
6167
OPTICALENGINEERING
ENGINEERING // July/August
Vol. 24
24 No.
616
/ OPTICAL
July /August 1985 // Vol.
No. 4
Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
PHOTON ABSORPTION, NONLINEAR
NONLINEAR REFRACTION,
REFRACTION, AND OPTICAL
OPTICAL LIMITING
LIMITING IN
IN SEMICONDUCTORS
SEMICONDUCTORS
TWO PHOTON
TABLE I.I. Material Parameters
Parameters and
Materials Studied
Studied
and Two-Photon
Two- PhotonAbsorption
Absorption Coefficients
Coefficients of the Materials
Two-photon
/urn
Two
-photon absorption:
absorption: AA == 1.06 µm
Material
Form(a >
Form(a)
ZnTe< c)
ZnTelc)
CdSe<c>
CdSe(c)
w
W
CdTe<
d>
CdTe(d)
Z
Z
CdTe<
d>
CdTe(d)
Zp
Zp
CdS0.
5 Se05< c>
CdSo5$eo5(c)
CdS0.
25 Se075<c>
CdS025Seo(c)
w
W
w
W
GaAs<
e>
GaAs(e)
Z
Z
Z
Z
n(X)
n(a)
2.79<
c>
2.79(c)
2.56<
c>
2.56ík)
2.84<
c>
2.84(c)
Eg
(eV)
Eg(eV)
2.26< j)
2.26(')
11.74)))
.74<J>
Ep
(eV)< b>
Ep(eV)Ibl
ß2heor(cm/GW)
/32
theor (cm/GW)
/?fxp (cm/GW)
ß2xP(cm
/GW)
Eb/Eg
19.1
19.1
0.004(o)
0.004(°)
21
21
22
25.1
25.1
15
15
25.1
25.1
4.5
0.89
11.44ík)
.44< c>
11.44(k)
.44<c>
20.7
0.007(i)
0.0071')
0.003(o)
0.003(°)
20.7
0.003(0)
0.003101
1.93<
k>
1.93(k)
1.78<
k>
1.78)k)
21
21
0.01
0(l)
0.01011)
10
12.1
12.1
2.51<'>
2.51(1)
21
21
15
15
17.7
3.43(i)
3.431')
1.42<
1.42(')j >
25.7
0.008W
0.008(1)
0.003(0)
0.003 °1
23
19.7
j8fxP(cm/GW)
ßZxp(cm/GW)
ßZheor(cm/GW)
02theor (cm/GW)
2.84<c>
2.84(k)
2.45<">
2.451)
Two-photon
X = 0.53
0.53 µm
,
Two
-photon absorption: X
Form(a
Form(a)
n(X)
n (X)
ZnS<f>
ZnS(t)
ZnS<f>
ZnS1t)
Zp(clear)
Zp(yellow)
Zp(yellow)
2.40<
2.400)j)
2.40(i)
2.4011)
ZnSe<9>
ZnSe1g1
Zp
CdS<c>
CdS(c)
ZnO'
h>
Zn01h1
W
W
Material
2f>o>
215w== 2.34
2.34 eV
Eg
(eV)
Eg(eV)
18
18
18.6
2fio> = 4.68 eV
eV
2tiw
E(eV)
Eb/Eg
3.66<
3.66(')j >
20.4
0.01
0(i)
0.010(')
2.0
20.4
24.2
0.01
Of'*
0.0101'1
0.008(0)
0.00810)
3.5
3.5
2.70< j >
2.70111
3.66<
3.66(')j >
2.67<
2.67(')j >
2.60< j >
2.601i1
2.42<
c>
2.420)
21
21
0.01
2(i)
0.012)i)
5.5
5.5
2.10
2.10
4.27
4.87
2.05< j)
2.050)
3.20<
n)
3.20(n)
21
21
0.020<
0.02d i))
5.0
4.77
(a) ZZ == zinc
polycrystalline
(a)
zinc blende;
blende;W
W == wurtzite;
wurtzite; p = polycrystalline
(b) Values
the
(b)
Values taken
taken from
from Ref.
Ref. 20.
20. For
For values
values not
not listed in this Reference, the
21 eV
eV was assumed.
assumed.
value of 21
(c) Cleveland
(c)
Cleveland Crystals,
Crystals, Euclid, Ohio
(d) II-VI,
Pa.
(d)
II -VI, Inc.,
Inc., Saxonburg,
Saxonburg, Pa.
(e) Morgan
Tex.
(e)
Morgan Semiconductors, Garland, Tex.
(f)
(f) CVD
CVD Inc.,
Inc., Woburn,
Woburn, Mass.
(g)
(g) Raytheon
Raytheon Co.,
Co., Bedford,
Bedford, Mass.
3.2. Data
Data
3.2.
Table I lists the 10
10 samples used in these experiments. In
In all,
all, eight
eight
different
different materials
materials having
having either
either aa zinc
zinc blende
blende or
or wurtzite
wurtzite structure
structure
were investigated. All of the samples were II
II-VI
-VI materials
materials except for
GaAs, which is a III
III-V
sample used
used (0.5
(0.5 cm)
cm)
-V material.
material. The thickest sample
was
was over
over 100
100 times
timesthinner
thinner than
than the
the confocal
confocal beam
beam parameter (the
Rayleigh distance) used
used for this study.
study. Experiments were performed
on each single
single crystal
crystal sample
sample for
for two
two orthogonal
orthogonal directions of linear
polarization.
Within our experimental
experimental accuracy,
accuracy, no
polarization. Within
no anisotropy in
the
measured values
/32 was
was observed.
observed. A 15%
15% variation was
was
the measured
values for
for ß2
reported in Ref. 26
26 for
for room
room temperature
temperature CdTe; however, the optical
pulse
given. In addition, the
the absolute
absolute values
values of
of the
the
pulse width
width was
was not given.
2PA coefficients
coefficients reported
reported there
there were
were an
an order of magnitude larger
than those we
we measured.
measured. This
This may
may indicate
indicate the
the dominance
dominance of carrier
carrier
absorption
for long
long pulses.
pulses. In
In ZnTe
ZnTeand
andthe
thesingle
single-crystal
absorption for
-crystal CdTe, the
light
light propagation
propagation direction k was in the (110)
(110) direction,
direction, and in GaAs,
k was in the (111) direction.
InCdSe,
CdSe,CdS0
CdS05Sep
5 Se05,5 ,and
andCdS0.25Se0.75,
CdS0 25 Se0 75 ,
direction. In
k was parallel to the
the cc-axis,
-axis, while
while in
in CdS
CdS and
and ZnO, k was perpendicular to the c-axis.
c -axis.Examples
Examples of
ofdata
data used
used to
to extract
extract the 2PA coefficient
are shown
shown in Figs.
Figs. 5(a)
5(b) for 1.06
1.06 /urn
/urn, respecrespec5(a) and 5(b)
µm and 0.53 µm,
tively.
the average
average of
of five
five laser
laser firings.
firings. The solid
solid
tively. Each
Each data point isis the
line
W in
18 cm/
cm/ G
GW
line in
in Fig.
Fig. 5(a)
5(a) isis aa fit
fit for
for CdSe
CdSe using
using aa —
= 0 and /32
ß2 = 18
Eq. (12).
(12). The
The solid
solid line
line in
in Fig.
Fig. 5(b)
5(b) isisaasimilar
similarfit
fitfor
forZnSe
ZnSeusing
usinga.a ==
0.5 cm"
and ß2
/32 = 5.5 cm/ GW
0.5
cm I1 and
GW at 0.53
0.53 jum.
pm. In all samples the linear
absorption
value was
was unimportant
unimportant in
in the
the
absorption was
was small,
small, and
and its
its value
determination
samples of ZnS
ZnS the
the scattering
scattering was
was
In both samples
ß2.. In
determination of /32
significant,
significant, and
and this
this loss
loss mechanism
mechanism was
was included
included in
in the model
model as
Inthe
thelatter
lattercase,
case,the
theeffect
effect of
ofthe
the choice
choice of
of aa on
on ß2
/?2
linear absorption.
absorption. In
was less than
than 10
10%.
%.
these measurements
measurements of the 2PA
2PA coefficients
coefficients are
The results of these
given
given in
in the
the next
next to
to the
the last
last column
column of Table
Table I.I. The
The absolute
absolute error
bars on the values of ft2
±40%.
%. The
The relative
relative error
error
02 are estimated to be ±40
bars of ß2
/32 determined from one sample to the next are considerably
better, as observed by measuring
measuring all the samples in sequence for each
spot
size and
ZnS (yellow) listed
listed in
spot size
and each
each pulse
pulse width.
width. For example, ZnS(yellow)
5.5
5.5
Chemetals, Plainview,
Plainview, N.Y.
N.Y.
(h) Atomergic Chemetals,
(i) Ref.
Ref. 21
21
(i)
Ref. 22
(j) Ref.
Ref. 23
(k) Ref.
(l)These values obtained
uncton of composi­
(I)These
obtained by
by linear
linear extrapolation
extrapolation as aa ffuncton
composition between the known
known values
values for
forCdS
CdS and
and CdSe.
CdSe. See Ref.
Ref. 24.
24.
(n)
(n) Ref.
Ref. 25
(o) Ref. 4
Table II always
(t2 tnan
We conservatively
conservatively
always had
had aa larger ß2
than ZnS(clear). We
estimate these
these relative
relative error
error bars, which
which are
are important in determindependence of
ofß2
/32 as discussed in the next section,
ing the parametric
parametric dependence
be ±25
±25%.
to be
%.
second set of
of experiments
experiments (to be
be discussed
discussed in Sec.
Sec. 5) we
we
In the second
detector in
in Fig.
Fig. 11 by either a vidicon tube
replaced the transmission detector
interfaced with
with an optical multichannel analyzer or a 25
25 pm
µm pinhole
pinhole
placed at various radial positions and longitudinal distances from the
We then monitored the spatial beam
beam profile
profile of
of the transmittransmitsample. We
vidicon, or we
we monitored the pinhole
pinhole transmistransmisted beam using the vidicon,
irradiance. In
InSec.
Sec. 55 we
we describe the use
sion as aa function
function of incident irradiance.
(17) to theoretically fit both of
of these
these results.
results.
of Eq. (17)
4.
COMPARISON OF ß2
02 VALUES TO THEORY
4. COMPARISON
4.1.
Two-photon
4.1. Two
-photon absorption theory
We
comparisons in
in this
this section.
section. We
We first
first corn
com-We make
make three separate comparisons
pare
experimentally determined
pare our experimentally
determined /32
ß2 values
values with
with the
the theory
theory of
of
Refs. 3
3-5
Excellent agreement
agreement isis
Refs.
-5 using
using aa parabolic
parabolic band structure. Excellent
found for all materials except ZnTe. We then use the nonparabolic
theory and again find good agreement
agreement for
for all
all materials
materials except
except ZnTe.
ZnTe.
As
shown by Weiler,4
Weiler,4 the
differences between
As shown
the differences
between the
the parabolic
parabolic and
nonparabolic theories are minor so
so that this
this fit
fit is expected.
expected. We
We then
then
include exciton correction
correction factors,
factors, as
as given
given by
by Lee
Lee and
and Fan27
Fan 27 and as
calculated
when these
these are included,
included,
calculated by
by Weiler,4
Weiler,4and
and we
we find
find that
that when
ZnTe nearly fits
fits the dependencies
dependencies shown
shown by
by the
the other materials.
As.
Eg, and
and
As,stated
statedininRef.
Ref.5,5,the
theparametric
parametric dependence
dependence of /32
ß2 on n, Eg,
Ep
wasfirst
firstexplicitly
explicitly pointed
pointedout
outby
byPidgeon
Pidgeonetetal.,3
al.,3 although
although itit was
E was
present in the calculations of
of Basov
Basov et aí.28
al. 28 The
The band
band structure and
transitionscheme
schemeused
usedby
by Pidgeon
Pidgeonetetal.3
al. 3 is shown in Fig. 6.
transition
6. Calculations
using this
scheme have
parabolic and
tions using
this scheme
have been
been performed
performed for
for parabolic
nonparabolic bands. They found
ß2 =
4Tre4
v m c2
ff; f ( 2flw )
Eg
B
= 53.8
2
n E8
/2fto)\
2flcu
f ~F~ '
Eg /
\ ^g
(I8)
(18)
OPTICAL
/ July/August
1985
/ Vol.
617
OPTICALENGINEERING
ENGINEERING
/ July /August
1985
/ Vol.24
24No.
No.4 4/ / 617
Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
VAN STRYLAND,
VAN
STRYLAND,VANHERZEELE,
VANHERZEELE,WOODALL,
WOODALL,SOILEAU,
SOILEAU,SMIRL,
SMIRL,GUHA,
GUHA, BOGGESS
BOGGESS
3.2
2.4
116
ZnSe 0.53/am
i-
>k
1-6
(a)
21Ko
0.8
0.0
0.0
0.4
0.8
1.2
1.6
2.0
g
-
Zhu
IRRADIANCE (GW/cm 2 )
2.5i————i————i————i————i———
2.0
Fig. 6.
Band structure used
used in Refs.
Refs. 33 and
and 44 to
tocalculate
calculatetwotwo-photon
Fig.
6. Band
photon
absorption coefficients.
1.5
CdSe 1.06/um
(b)
As
shown in Ref.
Ref. 4,
4, the
the differences
differences between
between f for
for A«Eg
and
As shown
0«E and
A»Eg,
where0 is
A is
split-off
energyshown
shown in
in Fig.
Fig. 6,
6, are
are small.
small.
A» E , where
thethesplit
-off energy
is
The expression
expressionforforA»Eg
0» Eg is
1.0
fnp
(x),A»Eg
fnp(x),
0Eg =- 16
0.0
0.0
0.1
0.2
0.3
0.5
0.4
/x"
IRRADIANCE (GW cm2 )
where Ep
andEEg
andß2/£>inincm/
cm/GW
the last
last expression.
expression.
where
E and
areareinineVeVand
GW ininthe
Here, m is
is the
the electron
electron mass, e isis the
the electron
electron charge, c is
is the
the speed
speed of
of
light
refractive index.
index. The
The values
values of
ofEg,
Eg , E0,
Ep ,
light in
in vacuum, and nn is the refractive
and n for each material are listed in Table I.I. It is
is important to note
note
that for
for both
both parabolic
parabolic and
and nonparabolic
nonparabolic bands,
bands, the
the parametric
parametric
dependencies on n, Eg
Ep predicted by the theories are the same.
E ,, and Ep
same.
In addition,
addition,asaspointed
poinfed out in
in Ref.
Ref. 55 these
these dependencies
dependencies are indeinde­
In
pendent of crystal structure.
structure. The
The differences
differences lie
lie in the function f and
the
of fico
the ratio
ratio of
tut) to
to Eg
Eg(i.e.,
(i.e.,which
whichstates
statesare
are optically
optically coupled).
coupled). Weiler
Weiler44
corrected
Ref. 33 and
corrected an error in the calculation of f in Ref.
and obtained the
following
following expression
expression using
using parabolic bands:
= - 4 + 29 /2
(4+-
[(x
z
= 96.9 [F2(x)]
12
.
(19)
(19)
i TT
3
2x-1
7
+ 45
fnp (x),A«Eg =y
3
3/x~
X
X
3
2x
x
•+
(
(x
+ I)3 / 2
A
x 3x5
5 — (x4
(x4 + 2x2
2x2 +
-f 6)
6)
)3/2
.
(20)
(20)
-1)2
<
x5
(
55,
x4+2x2+6/I
— x2
4X +
—
x
16 X
16
.. (21)
(21)
results of
of these
these calculations
calculations of
ofthe
the excitonic
excitonicenhancement
enhancementgeX
ggx of
of/^
results
ß2 as
as
a function of
of Co/
fico/Eg
various values
values of
ofe.e.
Eg for various
4.2. Comparison
Comparison to theory
theory
4.2.
/(/ETTF2)
Eq.
Figure 88 shows
shows aa log-log
log -logplot
plotofofp2ß2scaled
scaledbybyn2n2/
( P F2) [see Eq.
(19)
F2] as
Eg . The
is aa least
(19) for
for F2]
as aa function
function of Eg.
The solid line is
least squares
squares fit to
the data (excluding ZnTe) for a line
line having
havingaaslope
slopeof
of—3
-3 to account
Eg 3 dependence of ß2.
fa. Clearly,
dependences
for the Eg3
Clearly, the parametric dependences
using this
fit the
the data
datavery
verywell
well except
except for
for ZnTe.
ZnTe.
using
this parabolic theory fit
This
single parameter
fit yields
yields aa 2PA
2PA coefficient
coefficient given
given by
by the
This single
parameter fit
the follow­
following
ing equation:
(3.1±0.5)X103
ß2 = (3.1
t0.5)X 103
fnp(x), 0«Eg = 5(x
x3
to values
values of
of ß2
/32 have been predicted by Lee and
Exciton corrections to
Fan. 27 Weiler
4 has
ratio ee
Fan.27
Weiler4
hasevaluated
evaluated these
these corrections
corrections in
in terms
terms of the
the ratio
binding energy
energy Eb
Eb to
to the
theband
band-gap
energy E2.
Eg. These
These
of the exciton binding
-gap energy
ratios are
are listed
listed in
in Table I.I. We
We reproduce
reproduce in
in Fig.
Fig. 77 (from
(from Rif'.
ReT. 4)
4) the
the
where
where F2
F2(x)
(x)isisthe
thesame
samefunction
functiondefined
definedininRef.
Ref.5.5.For
Fornonparabolic
nonparabolic
Weiler 4 found
bands, Weiler4
found for 0A«EP
<< Eg
32 (x - I)3 / 22
3
_6_
+1)3/2
I)3 / 2
6 (x +
X
Fig. 5.
Inverse transmission
transmission versus
(a) CdSe
CdSe at
Fig.
5. Inverse
versus incident
incident irradiance
irradiance for
for (a)
at
1.06 /*m
/urn. The solid lines are
Eq. (12).
µm and (b)
(b) ZnSe
ZnSe at
at 0.53
0.53 pm.
are fits
fits using Eq.
f(x) =
(x --1)3/2
I)3 / 2
(x
»
0.5
P
Ma))
(-E
/2fico\
n2
E3
n2E3
(22)
(22)
where again
areinineV
eV and
andß2
/32 isis in
in cm
cm/GW.
The values
values
where
again Ep
Ep and
and Eg
E are
/GW. The
predicted
predicted from
from Eq.
Eq. (22)
(2Z)are
are listed
listedininthe
the last
last column
column of
of Table
Table II for
for
each material.
material. The
The value
value of
ofthe
theconstant
constant[3.1
[3.1X103
Eq. (22)]
(22)]
each
X 103 ,in
in Eq.
predicted
by theory
theory from
from Eqs.
Eqs. (18)
(18) and
and(19)
(19) isis 5.21
5.21 X
X103
predicted by
103,, so
sothat
that the
the
absolute values
values of
of the experimentally
experimentally determined
determined ß2
/32 values
values are,
are, on
on
6187
OPTICALENGINEERING
ENGINEERING // July/August
Vol. 24 No.
No. 4
618
/ OPTICAL
July /August 1985
1985 // Vol.
4
Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
SEMICONDUCTORS
IN SEMICONDUCTORS
REFRACTION, AND
PHOTON ABSORPTION, NONLINEAR REFRACTION,
AND OPTICAL LIMITING IN
TWO PHOTON
10'
I
'
I
-e-0.01
E =0.01
-0.008
0.008
-0.006
0.006
-0.004
0.004
-0.002
0.002
-0.0005
0.0005
I
Saga
a)s
10°
100
0.5
I
0.6
j_
j_
0.7
0.1
0.8
I
0.9
1.0
10
GAP
ENERGY/ENERGY
PHOTON
PHOTON ENERGY
/ENERGY GAP
various
fkw/Eg
of hw
gex as aa function
factor
Fig. 7. Exciton
g8%
function of
/Eg for
for various
enhancementfactor
Excitonenhancement
band-gap
values
values of
of the
the exciton
exciton binding energy Eb to the band
-gap energy
energy Eg (repro4).
Ref. 4).
duced with
duced
with permission from Ref.
coefficients
absorption
two-photon
scaled
thescaled
ofthe
plot of
Fig. 9. A log
-log plot
twophoton absorption
coefficients
log-log
gap. The solid
energy gap.
versusenergy
band structure
structure (A«Eg)
(0«E9) versus
solid
nonparabolic band
for nonparabolic
ZnTe).
(omittingZnTe).
line is aa least
least squares
squaresfit
fit of
of the
the data
data to
to aa line
line of
of slope
slope-3
-3 (omitting
The
The dashed
dashed line
line isisthe
the theory
theory of
of Ref.
Ref. 4.
GaAs
1000
1000
-CdTe
CdSe
-
eó
CI
ZnTe
CdS.25$e.75
C2
CdS 5Se 5
100
ZnSe
CdS
100
1.06/um
1.061.im--r
-0.53µm
1.5
2.0
2.5
3.0
3.5
4.0
1.5
2.0
2.5
3.0
ZnS.
Zñ0
3.5
4.0
4
0
Eg(ev)
Eg)ev)
two-photon
scaled twothe scaled
log-log
Fig. 8.
Fig.
8. A log
-log plot of the
photon absorption coefficient
versus
versus energy
energy gap
gap assuming
assuming parabolic
parabolic band
band structure.
structure. The solid line is aa
ZnTe). The x's
(omittingZnTe).
3 (omitting
least
east squares
squaresfit
fit of
of the
the data
data to
to aa line
lineof
ofslope
slope -3
Ref. 13. Data
Data to
CdTe, CdSe, and ZnTe
GaAs, CdTe,
for GaAs,
shown for
ZnTe are
are data
data from
from Ref.
and to the
line were taken with 11 /urn
pm light and
the left of the vertical dotted line
M light.
right with 0.5 µm
1.7.
of 1.7.
the
the average,
average, low
low by
by aa factor
factor of
values with
/^ values
determined ß2
If we
If
we now
now compare
compare the
the experimentally
experimentally determined
with
results
the results
obtain the
we obtain
(20)], we
Eq.(20)],
the nonparabolic
nonparabolic theory
theory[A«E
[0 «Eg, Eq.
the
is
shown
shown in
in Fig.
Fig. 9.
9. While
While the
the overall
overall tit
Tittotothe
theparametric
parametric dependence
dependence is
is aa
line is
solid line
the solid
(again the
case (again
parabolic case
the parabolic
for the
as for
good as
as good
quite as
not
not quite
excluding ZnTe),
squares fit
least squares
least
fit to
to aa line
line of
of slope
slope—3,
-3, excluding
ZnTe), the
the absolute
absolute
this
by this
predicted by
lower than
26% lower
only 26%
average, only
the average,
values are,
values
are, on the
than predicted
(20).
and (20).
(18) and
Eqs. (18)
by Eqs.
predicted by
curve predicted
the curve
line isis the
dotted line
The dotted
theory. The
theory.
an
gives an
similar plot
A similar
plot (not
(not shown)
shown)using
usingEq.
Eq.(21)
(21)for
forA»Eg
0 »Eg gives
theory
between theory
difference between
the difference
case the
this case
in this
except in
fit, except
identical fit,
almost identical
almost
3.5%.
only3.5
reducedtotoonly
dataisis reduced
the data
of the
fit of
squares fit
least squares
and aa least
and
%. The
The actual
actual
although
theories, although
two theories,
the two
between the
in between
lies in
materials lies
these materials
for these
case for
case
the
in the
scatterin
forscatter
responsiblefor
partiallyresponsible
maybebepartially
andmay
Eg,and
closer to
closer
to A«
A «Eg,
/32).
in ß2).
change in
23% change
to 23%
(up to
data
data (up
in
given in
as given
excitons, as
/32 due
If
If we
we now
now include
include corrections
corrections to
to ß2
due to
to excitons,
the
used the
have used
we have
Here, we
10. Here,
Fig. 10.
of Fig.
curve of
the curve
we obtain the
7, we
Fig.
Fig. 7,
g^.
with gex.
scaling with
F2 ), except
(i.e., F2),
theory (i.e.,
parabolic
parabolic theory
except for
for the
the additional
additional scaling
bands. 27
parabolicbands.27
forparabolic
onlyfor
calculatedonly
beencalculated
hasbeen
factorhas
This
correction factor
This correction
we see
What
What we
see in
in Fig.
Fig. 10
10isisthat
thatmost
mostof
ofthe
thediscrepancy
discrepancy between
between theory
theory
by
removed by
been removed
has been
4.3)^ has
(gex ==4.3)4
ZnTe(gex
forZnTe
observedfor
and
experiment observed
and experiment
Fig. 10. A log-log
log -logplot
plot of
of the
the scaled
scaledtwo-photon
two- photonabsorption
absorption coefficients
coefficients
(gex ) for
enhancement (gex)
exciton enhancement
including exciton
for parabolic band structure
structure versus
energy gap.
the
to the
including
including excitonic
excitonic enhancement.
enhancement. However,
However, the
the overall
overall fit
fit to
is
fit) is
squares fit)
least squares
lineisis aa least
parametric dependence
straightline
the straight
(againthe
dependence(again
parametric
in
accuracy in
of accuracy
lack of
our lack
somewhat
somewhat worse.
worse. This
This may
may be
be attributed
attributed to
to our
is of
ggx is
materials, gex
calculating gex.
Note that
that for
for most
of the
the order
order of
of 2,
2,
most materials,
ggx . Note
calculating
7).
(see Fig. 7).
well above the gap (see
even
when the
the coupled
coupled states are well
even when
Consequently,
Consequently, when
when excitonic
excitonic enhancement
enhancement is
is included
included for
for parabolic
parabolic
of 3.3
/32 values
bands, the
the absolute
absolute ß2
values predicted
predicted by
by theory
theory are
are aa factor
factor of
3.3
bands,
measured.
those measured.
larger
larger than
than those
consider­
is considerZnTe is
for ZnTe
enhancement for
exciton enhancement
the exciton
that the
reason that
The reason
two
jum, two
1.06 µm,
materials isis that
the other materials
ably
ably larger
larger than
than for
for the
that at 1.06
excitonic
where excitonic
gap, where
3.5% above the gap,
only 3.5%
couple states only
photons
photons couple
out,
point out,
We should
7. We
Fig. 7.
in Fig.
shown in
as shown
be greatest,
effects should
effects
should be
greatest, as
should point
however,
however, that
that we
we have
have only
only one
one material
material for
for which
which the
the coupling
coupling isis
larger
material are
this material
for this
bars for
this
this close
close to
to the
the gap,
gap, and
and the
the error
error bars
are larger
use
to use
us to
allowing us
easily, allowing
damagedeasily,
ZnTe damaged
materials. ZnTe
for the
than
than for
the other
other materials.
fitting
for fitting
small spot
aa small
spot size
size only
only and
and limiting
limiting the
the range
range of
of irradiances
irradiances for
overlap
is overlap
where there is
8), where
Fig. 8),
in Fig.
02.. On
On the
the other hand (as shown
shown in
/32
Bechtel and
by Bechtel
taken by
between
between the
the data
data taken
taken here
here and
and the
the data taken
/?2 for
large ß2
13 the
Smith,
Smith,13
theagreement
agreement isisexcellent.
excellent. They
They also
also obtained
obtained aa large
for
ZnTe.
ZnTe.
9,
8, 9,
Figs. 8,
in Figs.
shown in
and shown
Table II and
in Table
listed in
points listed
data points
Two
Two other
other data
an
gave an
sample gave
CdTe sample
polycrystalline CdTe
The polycrystalline
comment. The
require comment.
10 require
and 10
and
is
single-crystal
the single
lower than the
)32 lower
experimental ß2
experimental
-crystal CdTe
CdTe sample.
sample. It
It is
619
/ / 619
44
/ Vol.
1985
/ July/August
OPTICAL
OPTICALENGINEERING
ENGINEERING
/ July /August
1985
/ Vol.2424No.
No.
Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
VAN STRYLAND,
STRYLAND, VANHERZEELE,
VANHERZEELE,WOODALL,
WOODALL, SOILEAU,
SOILEAU, SMIRL, GUHA, BOGGESS
BOGGESS
designed as
as aa 10.6
10.6 µm
)um optical
opticalwindow
windowand
andisisdoped
dopedwith
with1017
10 l7 cm
cm~3
of
designed
-3 of
indium.
is not understood
understood at present
indium. ItIt is
present ifif or
or how
how these
these impurities
impurities
lower the
the measured
measured /32.
/32 . The
sample is
is aa chemical
chemical vapor
vapor
lower
The ZnS(yellow)
ZnS(yellow) sample
grown sample
sample that has
has aa yellow
yellow appearance caused
caused by
by
deposition grown
crystal lattice
lattice imperfections
imperfections that
annealed out by
by aa special
special
crystal
that can be annealed
heat
process. The
The ZnS(clear)
ZnS(clear) is
is the
the same
same starting
heat treatment
treatment process.
starting material
material
that has
has undergone
undergone this
this heat
heat treatment.
treatment. ItIt appears
appears water
water clear,
clear, and
and its
its
that
linear transmission
transmission cutoff
cutoffisis shifted
shifted into
into the
the UV.
U V. The
from
linear
The indication
indication from
the data
data presented
presented here
here is
is that
not only
only has
has the
the linear
linear absorption
absorption in
in
the
that not
the visible
visible and
near-UV
but the
the two
two-photon
absorp­
the
and near
-UV been
been reduced,
reduced, but
-photon absorption at
tion
at 0.5
0.5 pm
jum has
has also
also been
been reduced
reduced by
by this
this heat
heat treatment.
treatment. That
That is,
is,
defects may
may be
be contributing
to02.
/32 . This
This may
may also
also be
be true
true for
for the
the ZnTe
ZnTe
defects
contributing to
sample
since it
it damaged
damaged easily.
easily.
sample at
at 11 /urn
µm since
(b)
5. SELF
-REFRACTION
5.
SELF-REFRACTION
In
section we
beam propagation
propagation behind
behind
In this
this section
we present
present results
results for
for the
the beam
the sample,
sample, given
given the
irradiance and
and phase
phase distributions
distributionsofofEqs.
Eqs.(10)
(10)
the
the irradiance
(j =
2) and
(14). Equation
Equation(17)
(17) describes
describes the
thefluence
fluence atatany
anypoint
point(z(z,, r)
r)
(j
= 2)
and (14).
behind the
the sample.
sample. We
We present
present data
data here
here using
using aa beam
beam of
ofwidth
width 1.70
1.70
behind
mm (FWHM)
incidenton
on the
theCdSe
CdSesample
samplelisted
listed in
in Table
Table I,I, which
which is
is
mm
(FWHM) incident
placed near
near the
the beam
beam waist.
waist. Thus,
Thus, c(o,
4>(o, r,
in Eq.
Eq.
placed
r , t)t) isistaken
taken to
to be
be zero
zero in
(14). The
signal was
was monitored
monitoredatatdistances
distancesofof0.5
0.5 m
m and
and
(14).
The transmitted
transmitted signal
m behind
behind the
the sample,
sample, both
both of
ofwhich
which are
are near
near-field
regions. Figure
Figure
22 m
-field regions.
11 illustrates
11
illustrates the
the change
change in
in the
the beam
beam profile
profilewith
withirradiance
irradianceatat zz ==
0.5
pulses) as
as displayed
displayed on
on the
the vidicon.
vidicon. This
This
0.5 m
m (using
(using 92
92 ps
ps FWHM
FWHM pulses)
scan is
is aa narrow
slice through
the center
center of
of the
the beam
beam that
that isis equivaequiva­
scan
narrow slice
through the
lent to
to aa pinhole
pinhole scan.
scan. The
The dashed
dashed line
line isis aa fit
fit to
to Eq.
Eq. (17)
(17) as
lent
as aa function
function
ofrfor/32
18cm/GW,P
of
r for ß2 =
= 18
cm/ G W, P== 3,andy
3, and y =
= 00(i.e.,n2=0).
(i.e., n2 = 0). We
Wefind
find that
that ifif
present,
effects of
bound electronic
electronic contributions
non­
present, the
the effects
of bound
contributions to
to the nonlinear
refractive index
overshadowed by
by the photogenerated
photogenerated
linear refractive
index are overshadowed
carrier effects.
effects. This
This is
is confirmed
by our
other measurements,
measurements, which
which
carrier
confirmed by
our other
monitored
fluence transmitted
pinhole as
function of
of
monitored the
the fluence
transmitted by
by aa pinhole
as a function
irradiance.
irradiance.
Using aa 25
we
Using
25 /um
pm diameter
diameter pinhole
pinhole in
in front
front of aa photodetector, we
measured the
the on
on-axis
0) fluence
signal as
as aa
measured
-axis (r(r =
= 0)
fluence of
of the
the transmitted
transmitted signal
function
of the
the peak
peak on
on-axis
irradiance for
for two
two pulse
pulse widths
widths
function of
-axis input
input irradiance
and
distances. The
results are
shown in
Figs. 12
12 and 13.
13. To
and two distances.
The results
are shown
in Figs.
further
verify the
the validity
of the
the irradiance
irradiance dependence
dependence of
of
further verify
validity of
the theory,
theory, the
the transmitted
transmitted fluence
fluence at
at an
an off
off-axis
was also
measured. This
This
the
-axis point
point was
also measured.
is shown
shown in
in Fig.
Fig. 14.
14.
is
The
fits in
in Figs.
Figs. 12,
12, 13,
13, and
14 were
the
The theoretical
theoretical fits
and 14
were obtained
obtained from
from the
numerical evaluation
evaluation of
of FF given
given in
in Eq.
Eq. (17).
(17). Other
Other parameters
parameters(see
(see
numerical
Table 1)
I) used
were aa =— 0.2
0.2 cm
cm"-11 and
and meh
meh =
= 0.104
0.104
Table
used in
in the
the calculation
calculation were
m. 22 The
by two
two-photon
M.22
Thetotal
totaldensity
density NN of
of charge
charge carriers
carriers generated
generated by
-photon
absorption on
on-axis
can be
be calculated
by integration
integration of
of Eq.
Eq. (8).
(8). At
At aa
absorption
-axis can
calculated by
irradiance of
of 11 GW
GW/cm2
peak value
value of
of N
N
peak input irradiance
/cm2,, we
we obtain
obtain a peak
evaluated at
at the
theinput
inputplane
planeofofthe
thesample
sampletotobebe2 X
2X10
cnT"3
We
evaluated
101818 cm
-3.. We
find that
the best
best agreement
agreement between
between the
the experiments
find
that the
the theory
theory and
and the
experiments
is obtained
for PP ==3.5.
3.5. The
The fits
fits shown
shown in
in Figs.
Figs. 12,
12, 13,
13, and
14 are
is
obtained for
and 14
are for
for
values of
3.5 ±±1,1, with
with the
theexact
exactvalue
value of
ofPP adjusted
adjustedbetween
between4.5
4.5
values
of PP -=3.5
and 2.5
2.5 to
to obtain
obtain the
the best
best fit.
fit. Using
Using aa two
two-level
been
and
-level model,
model, PP has been
calculated
to be
be Egg
E2 /(E|
29 For
= 1.74
1.74 eV
eV at
room
calculated to
/(Eg-liV).
-t2w2).29
ForCdSe,
CdSe,Eg
E=
at room
temperature, and
and this
mis formula
formulapredicts
predictsPP ==22 at
at 11 µm.
/um. The
The peak
peak phase
phase
temperature,
change
change undergone
undergone by
by the
the beam,
beam, calculated
calculatedfrom
fromEq.
Eq. (14),
(14), isis (for
(for Io
I0 =
=
11 GW/cm2
) A<£
which is
is 1.3
1.3 wavelengths
wavelengths distortion.
We
GW /cm2)
0(1)==—8.1,
-8.1, which
distortion. We
have ignored
ignored the
the n2
n2 due
bound electronic
electronic effects
effects in
in these
these calculacalcula­
have
due to
to bound
tions. Even
Even ifif this
this nonlinearity
nonlinearityfor
forCdSe
CdSewere
wereasashigh
tions.
highasasthe
then2
n2 of
ofCS2
CS2
(i.e., 10
10~-11
n esu),
the phase
phase change
change
(i.e.,
esu),the
the maximum
maximum contribution
contribution to the
would be40n-Llwn2/
be407rLIcon2 /n0
for to
I0 =
= 11 GW/
GW/cm2
would
no,, which
which is0.2
is0.2 for
cm2.. This
This index
index
change also
also would
would be
be aa self
self-focusing
not aa defocusing
defocusing
change
-focusing effect
effect and
and not
effect as
the nonlinear
nonlinear refraction
refractionobserved
observedinin CdSe
CdSeisis
effect
as observed.
observed. Thus,
Thus, the
approximately 40
40 times
times larger
larger than
than in
in CS2
CS2 at
at this
this irradiance.
irradiance. Higher
Higher
approximately
irradiances give
give rapidly increasing
increasing values
values of
of defocusing
defocusing since
since the
the
irradiances
nonlinearity isis induced
of two
two increase
increase in
in
nonlinearity
induced by
by 2PA
2PA (i.e.,
(i.e., a factor of
irradiance gives
gives nearly
larger phase
phase distortion).
distortion).
irradiance
nearly aa factor
factor of
of four
four larger
OPTICAL LIMITER
LIMITER
6. OPTICAL
In
section we describe
describe a nonlinear
nonlinear optical
optical device
device (an optical
optical
In this section
power limiter)1O
limiter) 10 that
power
thatutilizes
utilizesboth
both two-photon
two -photon absorption,
absorption, as
as dis-
(a)
-..- i ---
I
-202
-2
2
0
TRANSVERSE
DISTANCE (mm)
TRANSVERSE DISTANCE
(mm]
4
Fig.
scan of beam
beam transmitted through the polycrystalline
polycrystalline
Fig. 11.
11. Vidicon scan
sample
sample of
of CdSe
CdSe at
at (a)
(a)high
high irradiance
irradiance (1(1 GW/cm2
GW /cm2)) and
and(b)
(b)low
low irradiance
irradiance
(0.3 GW
GW/cm2
m behind
behind the
the sample
sample (near
(near field). The
The
/cm2)) at aa distance
distance of 0.5 m
pulse
pulse width
width used
beam profiles
profiles are normalized to
used was 92 ps FWHM. The beam
have
have the
the same
same on-axis
on -axisfluence.
fluence. The
The dashed
dashed line
line is
is experimental
experimental and the
solid
using Eq.
Eq. (17).
solid line is
is aa theoretical
theoretical fit
fit using
cussed
Sees. 33 and 4,
4, and
and the
theassociated
associated nonlinear
nonlinear refraction
refraction
cussed in
in Secs.
discussed
discussed in
in Sec.
Sec. 5.
5. This
This completely
completely passive
passive device
device has
has aa high
high trans­
transmission for
it clamps
at
mission
for low
low input
input irradiance
irradiance (fluence),
(fluence), but
but it
clamps the
the output
output at
aa constant
constant irradiance
above aa predetermined
irradiance (fluence)
(fluence) above
predetermined input.
input. Such
Such aa
device
restrict the
the irradiance
irradiance
device can
can be
be used
used as
as aa protective
protective element
element to
to restrict
(fluence) of
of aa pulse
pulse incident
incident upon
upon sensitive
sensitive optical
optical components
components or
(fluence)
or as
as
a regulator to smooth
smooth optical
optical transients.
transients. This
This device
device is the optical
analog
of aa Zener
Zener diode.
diode.
analog of
Optical
limiting by
absorption in
Optical limiting
by nonlinear
nonlinear absorption
in semiconductors
semiconductors was
was
proposed and
and demonstrated
demonstratedininthe
thelate
late1960s.30
1960s. 30 "-32
32 Moreover,
proposed
Moreover, non­
nonlinear
linear refraction
refraction combined
combined with
with spatial
spatial filtering
filtering has
has been
been used
used to
to
demonstrateoptical
opticallimiting
limiting
liquid33 - 34and
andgas
gas-filled
cells. 35 Here
Here
demonstrate
in in
liquid
-33.34
-filled cells.35
we demonstrate
optical limiting
limiting in
in GaAs.
GaAs. Below
Below the
the melting
we
demonstrate optical
melting threshthresh­
old,
what we
we have
have done
differently isis to
use not only
old, what
done differently
to use
only nonlinear
nonlinear
GaAs but
but nonlinear
nonlinear refraction
refraction together
together with
with
absorption (2PA) in GaAs
spatial
construct a more
more effective
effective device.
device. Above
Above the
spatial filtering
filtering to
to construct
melting
we also
also take
take advantage
advantageofofaasolid
solid-to-liquid
phase
melting threshold,
threshold, we
-to- liquid phase
transition,
which affects
affects the
the reflectivity
reflectivity and
transmission.
transition, which
and transmission.
The geometry
geometry we
we used
limiting is
is shown
Fig. 15.
15. A
The
used for
for optical
optical limiting
shown in
in Fig.
A
single
1.06 ^m
pulse was
was focused
focused to
100 µm
/um
single 40
40 ps
ps (FWHM)
(FWHM) 1.06
µm pulse
to a 100
(FWHM)spot
spotatatthe
thesurface
surfaceofofthe
theGaAs
GaAswith
withaa465
465 mm
mm focal
focal length
length
(FWHM)
lens
beam was
was collected
collected and
and collimated
collimatedby
byaa381
381
lens Lj.
L1. The
The transmitted
transmitted beam
mm
mm focal
focal length
length lens
lens L2
L2placed
placed one
one focal
focal length
length behind
behind the
the sample.
sample.
The recollimated
recollimated beam
beam then
then passed
passed through
through aa 22 mm
mm diameter
diameter aperaper­
The
ture placed
placed one
one focal
focal length
length beyond
L2 and
of aa
beyond L2
and directly
directly in
in front of
photodiode.
sample used
sample
photodiode. The
The sample
used was
was nearly
nearly identical
identical to
to the sample
listed in
in Table
Table I.
listed
I. The
The limiting
limiting capabilities
capabilities of
of the
the device
device are
are shown
shown by
by
the
triangles in
16. At
/itJ), the
the device
device
the triangles
in Fig.
Fig. 16.
At low
low input
input energies
energies(<0.5
(<0.5 pi),
response
was linear
response was
linear and
and was
was consistent
consistent with
with the
the 45%
45% linear
linear transmis­
transmission
the 73%
73% transmission
of the
the pinhole.
pinhole. Above
Above10
10
sion of
of the
the GaAs
GaAs and
and the
transmission of
/zJ input,
input, the
the output
output energy
energy was
was essentially
essentially clamped
µJ
clamped at
at 11 )uJ.
µJ. The
device
device continued
continued to
to limit
limit for
for input
input energies
energies greater
greater than
than 11 mJ.
mJ. Over
Over
the
full range of operation,
operation, the
the system
system transmission
transmission was
was reduced
reduced
the full
from
33% to
input enerener­
from 33%
to 0.1%.
0.1 %.Notice
Noticethat
thatregulation
regulation continued
continued for
for input
gies
the GaAs
GaAs single
single-shot
of 0.9
0.9 JJ/cm2
gies far
far above
above the
-shot melting
melting threshold
threshold of
/cm2
(indicated by
the arrow
arrow in
in Fig.
Fig. 16).
I6). Above
Above the
the melting
melting threshold,
threshold, the
the
(indicated
by the
GaAs
before each
each firing
firing so
each pulse
pulse irradiated
irradiated
GaAs was
was translated
translated before
so that
that each
virgin
virgin material.
material.
Below
for melting,
melting, as
energy was
Below the
the irradiance
irradiance required
required for
as the
the input
input energy
was
increased, the
increased,
2PA. In
the transmission
transmission of
of the
the GaAs
GaAs was
was reduced
reduced by
by 2PA.
In this
this
620 // OPTICAL
Vol. 24
4
620
OPTICAL ENGINEERING
ENGINEERING//July/August
July /August 1985
1985 // Vol.
24 No.
No. 4
Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
SEMICONDUCTORS
IN SEMICONDUCTORS
LIMITING IN
OPTICAL LIMITING
REFRACTION, AND
NONLINEAR REFRACTION,
AND OPTICAL
TWO PHOTON ABSORPTION, NONLINEAR
la)
XX
lb)
0
0.00
0
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
cm2 )
INCIDENT
INCIDENT IRRADIANCE
IRRADIANCE (GW
(GW /cm2)
0.5
0.5
cm2 ]
(GW /cm2)
IRRADIANCE (GW
INCIDENT IRRADIANCE
INCIDENT
irradiance at
incident irradiance
Fig.
-axis fluence as aa function
function of incident
on-axis
Transmittedon
Fig. 12. Transmitted
and (b) 43 ps
pulses and
FWHM pulses
ps FWHM
(a) 92 ps
sample: (a)
the sample:
behind the
distance of 0.5 m behind
aa distance
Eq. (17).
pulses. The
FWHM pulses.
FWHM
The solid
solid lines
lines are
are numerically
numerically calculated
calculated from Eq.
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
i
as a
off-axis as
measured at
Fig.
Fig. 14.
14. Transmitted fluence measured
at aa point
point 1.1
1.1 mm off-axis
sample:
the sample:
behind the
m behind
0.5 m
of 0.5
distance of
ataa distance
irradianceat
incidentirradiance
ofincident
function of
function
are
lines are
solid lines
The solid
pulses. The
FWHM pulses.
ps FWHM
and (b) 43 ps
pulses and
(a) 92 ps FWHM pulses
(a)
Eq. (17).
numerically
numerically calculated
calculated from Eq.
f2
H
A
GaAs
1
2mm
_L
V
L2
A
Fig.
Fig. 15.
15. Schematic of GaAs
GaAs optical
optical limiter.
limiter.
10*
102
.
»';i-/.
.
.
f ..:'V
10'
101
• ••**
0.0
i
i
i
0.1
0.1
0.2
0.2
0.3
0.3
0.4
0.4
cm2 ]
INCIDENT
INCIDENT IRRADIANCE
IRRADIANCE (GW
(GW /cm2)
irradiance at
incident irradiance
of incident
on-axis
Transmittedon
Fig. 13. Transmitted
-axis fluence as aa function
function of
and (b) 43 ps
pulses and
FWHM pulses
ps FWHM
92 ps
(a) 92
sample: (a)
the sample:
behind the
m behind
2.0 m
a distance of 2.0
Eq. (17).
FWHM pulses.
The solid
solid lines
lines are
are numerically
numerically calculated
calculated from Eq.
pulses. The
i5 ir" "
= 10o»
A
*
lo-'i
GaAs
GaAs
10 -1
by the
regime,
regime, the
the amplitude
amplitude of
of the
the spatial
spatial beam
beam profile
profile transmitted
transmitted by
the
by
distorted by
was distorted
phase was
the phase
solely by
GaAs was
was distorted
distorted solely
by 2PA,
2PA, and
and the
GaAs
2PA-generated
the 2PAindex
index changes
changes associated
associated with
with the
generated free
free carriers,
carriers, as
as
self-diffraction
discussed
discussed in
in Sees.
Secs. 22 and
and 4.
4. The
The selfdiffraction associated
associated with
with this
this
effective pinhole
the effective
reduced the
phase and
and amplitude
amplitude distortion
distortion reduced
pinhole trans­
transphase
reflectivity
the reflectivity
in the
increase in
the increase
melting threshold,
mission. Above
Above the
the melting
threshold, the
mission.
GaAs
the GaAs
reduced the
further reduced
region further
molten region
absorptivity of the molten
and
and absorptivity
the
of the
profile of
amplitude profile
distorted the amplitude
transmission
transmission and
and further
further distorted
beam.
transmitted
transmitted beam.
reflection to
nonlinear reflection
of 2PA
The
The contribution
contribution of
2PA and
and nonlinear
to the
the limiting
limiting
action
action can
can be
be separated
separated from
fromthe
thecontribution
contribution of
of nonlinear
nonlinear refraction
refraction
by
carefully collecting
collecting all
all of
ofthe
the transmitted
transmitted energy
energy with
with the
the pinhole
pinhole
by carefully
Aperture
No Aperture
• No
Aperture
2mm Aperture
A 2mm
ETH
10-2:
10-
TH
(o)
1
*
i
40
40
ill
80
80
120
120
i
i
160
160
200
200
240
'I 240
ENERGY
ENERGYININ|/MJ)
(µl)
(circles) the 2 mm
(triangles) and without
response with
Fig. 16.
16. Device
with (triangles)
without (circles)
Device response
Fig.
aperture in place. ETH
ETH represents the single-shot
single -shot melting
melting threshold.
threshold.
circles
the circles
by the
shown by
are shown
results are
These results
16. These
Fig. 16.
in Fig.
shown in
as shown
removed, as
removed,
by
dominated by
is dominated
Below ETH
16. Below
Fig. 16.
in
in Fig.
ETH,, the
the nonlinear
nonlinear transmission
transmission is
GW. 10 The
cm/GW.IO
26 cm/
coefficientofof26
2PA coefficient
by aa 2PA
fit by
be fit
can be
and can
A and
2P
2PA
The value
value of
of
621
/ Vol.
1985
/ July/August
OPTICALENGINEERING
ENGINEERING
/ July /August
1985
/ Vol.2424No.
No.4 4/ / 621
OPTICAL
Downloaded From: http://opticalengineering.spiedigitallibrary.org/ on 09/21/2016 Terms of Use: http://spiedigitallibrary.org/ss/termsofuse.aspx
VAN STRYLAND,
STRYLAND, VANHERZEELE,
BOGGESS
VANHERZEELE,WOODALL,
WOODALL, SOILEAU,
SOILEAU, SMIRL, GUHA, BOGGESS
26
26 cm/GW
cm/ GW reported
reported in
in Ref.
Ref. 10
10 for
for the
the 2PA
2PA coefficient
coefficient ß2
fa of
of GaAs
GaAs is
is
nearly
23 cm/
cm/GW
nearly identical
identical to
to the value of 23
GW shown
shown in
in Table
Table I and
indicates the
the confidence
confidence in
in the
the values
values of
of ß2.
/32 . These
These measurements
measurements of
of
indicates
j32 on
GaAs were
were made
made independently
independently with
with aa nearly
nearly identical
identical laser
laser
ß2
on GaAs
system. Above
Above ETH,
ETH , the
system.
the problem
problem becomes
becomes much
much more
more complicated
complicated
is outside
outside the scope
scope of this paper. In
In this
this regime,
regime, the central
and is
region
pulse that arrives
arrives after
after melting
melting is
is initiated
initiated isis heavily
heavily
region of
of the
the pulse
attenuated and
GaAs. In addition,
and reflected
reflected by the molten layer of GaAs.
there is
is considerable
of material
material for
for fluences
fluences more
there
considerable evaporation
evaporation of
more than
than
— 10% above
above the
the melting
melting threshold.
threshold. It
It isis clear,
clear, however,
however, that the
the
-10%
transition from
from below
below to
to above
above threshold
threshold is
is aa smooth
smooth one.
one. That
That is,
is,
transition
there is
is no
discontinuity in
in limiter
limiter response
response at
threshold.
there
no discontinuity
at threshold.
It is
is interesting
compare the
the contributions
contributionsofof2PA
2PAand
andself
self-dif­
It
interesting to
to compare
-difinput energy
energy of 80
fraction just below the melting threshold.
threshold. At an input
/uJ, 2PA
2PA acting
acting alone
alone has
has reduced
reducedthe
thetransmission
transmissionby
byaafactor
factorofoffive.
five.
µJ,
On
the other
other hand,
hand, the
the combined
combinedeffects
effects of
of 2PA
2PA and
and selfself-diffraction
On the
diffraction
have reduced
30. Thus,
present
have
reduced the
the transmission by a factor of 30.
Thus, the present
configuration
is aa considerable
configuration is
considerable improvement
improvement over
over limiters
limiters that
that uti­
utilize 2PA
exclusively.
lize
2PA exclusively.
We
the present
present switch
switch with
with the
the Si
Si device
device
We should
should also
also contrast the
demonstrated
in Ref.
Ref. 36
36 that
that utilizes
utilizes indirect
indirect absorption,
absorption,free
free-carrier
demonstrated in
-carrier
absorption,
self-diffraction,
andsolid
solid-to-liquid
phase change
change to
to limit
limit
absorption, selfdiffraction, and
-to- liquid phase
energetic
/zm. The
The nonlinear
nonlinear absorption
absorption in
in Si
Si at
at 11 µm
jum is
is
energetic pulses
pulsesatat 11µm.
strictly fluence-dependent,
is independent
independent
strictly
fluence- dependent, and
and the
the device
device operation
operation is
of pulse
pulse width
width for
for pulse
pulse widths
widths shorter
shorter than the carrier recombinarecombina­
tion time.
time. The
The present
present device
device isis considerably
tion
considerably more
more complicated.
complicated. The
The
nonlinear absorption, which
which isis dominated
dominated by
by 2PA,
2PA,isisirradiance
irradiance-nonlinear
dependent, while
while the
arises from
dependent,
the nonlinear
nonlinear refractive
refractive index
index that
that arises
from the
the
2PA-generated
time-integrated
persists
2PA
-generated free
free carriers
carriers is
is aa time
-integrated effect
effect that
that persists
the duration
duration of
ofthe
thecarrier
carrierlifetime.
lifetime. Nevertheless,
Nevertheless, the
the carriers
carriers
for the
be generated
generated without
without aa sufficiently
sufficiently intense
intense pulse,
pulse, which
which in
cannot be
practice
practice restricts
restricts this
this device
deviceto
to operation
operation with
with short
short pulses.
pulses. However,
However,
the
the limiting
limiting pulse
pulse energy
energy can
can be
be varied
varied considerably
considerably by
by changing
changing the
the
geometry
geometry (e.g.,
(e.g., using
using aa very
very short
short focal
focal length
length lens
lens L1
Lj in
in Fig.
Fig. 15
15 will
will
lower
lower the
the limiting
limiting energy).
energy). An
An advantage
advantage of
of the
the present
present device
device (and
(and
2PA-based
limiters in
in general)
general) over
over the
the Si
Si device
device isis its
its higher
higher
2PA
-based optical
optical limiters
linear
advantage of
of
linear transmission
transmission at
at 11 /zm.
µm. Another
Another more
more important
important advantage
2PA-based
offer. For
For
2PA
-basedlimiters
limitersisisthe
the broader
broader bandpass
bandpass that
that they offer.
example, the
the GaAs
GaAs device
device should
should function
function for
for wavelengths
wavelengths between
between
example,
approximately 0.9
0.9 and
and 1.7
1.7 µm,
/zm, where
where 2PA
2PA is
is the
absorp­
approximately
the dominant
dominant absorption
process.
tion process.
7.
CONCLUSION
7. CONCLUSION
The material
material parameter
parameter dependence
dependence found
wide variety
variety of
The
found for
for aa wide
semiconductors, as
semiconductors,
as discussed
discussed in
in Sec.
Sec. 3,
3, allows
allows us
us to
to predict, with
reasonable confidence,
confidence, the
two-photon
coefficient of
reasonable
the two
-photon absorption
absorption coefficient
of
other materials
materials at
at other
otherwavelengths.
wavelengths. This
This includes,
includes, for
for example,
example,
other
mixed ternary
Thus, the
the 2PA
2PA at
at aa particular
wavelength
mixed
ternary compounds.
compounds. Thus,
particular wavelength
can
be tailored
tailored for
for aa specific
specific application.
application.
can be
The fact
fact that
that this
this scaling
scaling fits
fits the
the data
data so
so well
well implies
implies that
that all
all of
of the
the
The
important
important material
material parameters
parametershave
have been
been included
included in
in the
the theory
theory and
and
that
other contributions
contributionstotoß2
fa (e.g.,
(e.g., higher
higher bands)
bands) cause
cause small
small effects.
that other
effects.
One
possible deviation
the predicted
predicted scaling
scaling is
is the
the effect
effect of
of
One possible
deviation from
from the
excitons.
material (ZnTe)
(ZnTe) that indicates
indicates that such
such
excitons. We
We have
have one
one material
effects may
If we
we attempt
to extend
extend this
this theory
theory into
into the
the
effects
may be
be important.
important. If
attempt to
UV,
the predictions
predictions are
are in
in general
general considerably
considerably lower
lower
UV, we
we find
find that
that the
than
in the
however, aa general
general trend
trend can
can be
be found.
If the
the
than in
the experiments;
experiments; however,
found. If
coupled
are well
well above
gap, the
the deviations
deviations are
are relatively
relatively
coupled states
states are
above the
the gap,
small,
order of
of aa factor
of four
four or
or five.
five. As
As the
get
small, on
on the
the order
factor of
the coupled
coupled states
states get
close
exam­
close to
to the
the gap,
gap, however,
however, the
the deviation
deviation increases
increases rapidly.
rapidly. For
For example,
Rbl at
at 266
266 nm
nm where
where 2ftw/
2fico/Eg
1.47, the
the experimentally
experimentally
ple, in
in RbI
E ==1.47,
37 is
measured
2PA coefficient
coefficient of
of 2.49
measured 2PA
2.49 cm/bw
cm/kiW 37
is 3.7
3.7 times
times larger
larger than
than
predicted
by Eq.
Eq. (22).
(22). At
At aa wavelength
wavelength of
of 355
355 nm
nm where
where 2fiw/
2fto>/ Eg
predicted by
Eg==
1.10,
coefficient of
of 5.08
5.08 cm/GW
is 14
14
37 is
1.10, however,
however, the
the measured
measured 2PA
2PA coefficient
cm/ GW 37
times
predicted. This
This trend
trend is
is maintained
maintained for
for the
the limited
limited
times larger
larger than
than predicted.
data available.37
Considering color
color centers
centers as
as excitons
available. 37 Considering
excitons with large
large
binding
binding energies,
energies, such
such large
large correction
correction factors
factors may
may be
be accounted
accounted for.
for.
The
study. A
A
The role
role of
of excitons
excitons and
and color
color centers
centers in
in 2PA
2PA needs
needs further
further study.
study
wavelength dependence
study of the wavelength
dependence of
of 2PA
2PA near
near the
the gap using a
10*
i
104
aa
103
103 -®
•®
^
i
bb
f hh
©.
f
êu
CD
i
dd
©
®
°e
10
102 -
' <D
i
esi
N
Qi
OO
8g
c
k ——®
nn
©
m
10 1
101
00
65
65
70
70
75
75
80
80
85
85
Year
Year
Fig. 17.
A semilogarithmic
semilogarithmic plot
plot of
ofthe
thereported
reported twotwo-photon
absorption
Fig.
17. A
photon absorption
coefficientsfor
forGaAs
GaAs(lower
(lower-case
letters) and
and CdSe
CdSe (upper
(upper-case
coefficients
-case letters)
-case letters)
versus year:
year: (a)
(a) Ref.
Ref. 28,
28, (b)
(b) Ref.
Ref. 32.
32, (c)
(c) Ref.
Ref. 31, (d) Ref. 38, (e) Ref. 39, (f)
(g) Ref. 41, (h)
(h) Ref. 27,
27. (i) Ref. 26, (k)
(k) Ref. 42, (I)
(I) Ref. 43, (m)
(m)
Ref. 41, (g)
31, (C)
(C) Ref.
Ref 45,
45, (D)
(D) Ref.
Ref. 13,
13,(E)
(E)
Ref. 44, (n) this work, (A) Ref. 28, (B) Ref. 31,
Ref. 46, (F)
(F) this work.
continuously tunable
tunable laser
laser should
should help
help clarify
clarify the
the role
role of
of excitons.
excitons.
continuously
It is
is interesting
interesting to
to look
look at
at the
the history
history of
the measurement
measurement of
It
of the
of 2PA
2PA
coefficients.
17 shows
coefficients. Figure
Figure 17
shows experimental
experimental values
values of
of fa
ß2 reported
reported for
for
GaAs
CdSe for
1966. This
GaAs and
and CdSe
for years
years beginning
beginning with
with 1966.
This figure
figure illustrates
illustrates
the
values as
as both
the lasers
lasers and
and the
the
the fairly
fairly steady
steady decrease
decrease in
in reported
reported values
both the
experimental
techniques were
were refined.
of the
the earlier
earlier data
data were
were
experimental techniques
refined. Much
Much of
obtained
with nanosecond
nanosecond pulsed
pulsed lasers
lasers where
where photogenerated
car­
obtained with
photogenerated carrier
absorption is expected
expected to dominate.
dominate. Additionally,
Additionally, in some
some
rier absorption
instances
instances multimode
multimode lasers
lasers were
were used.
used. An
An additional experimental
problem
previously recognized
overestimate
problem not
not previously
recognized that
that can
can lead
lead to
to an
an overestimate
of
fa is
the extreme
extreme defocusing
defocusing present
present may
may allow
allow some
some of
of the
the
of 02
is that
that the
transmitted
light to
to go
go undetected.
undetected.
transmitted light
While
While this
this defocusing
defocusing may
may be
be an
an experimental
experimental difficulty
difficulty in
in deter­
determining
nonlinear optical
optical devices
devices
mining fa,
ß2, itit can
can be
be extremely
extremely useful
useful for
for nonlinear
such
such as
as the
the limiter
limiter discussed
discussed in
in Sec.
Sec. 6.6. We
Wefound
found that
that this
this defocusing
defocusing
was
quantitatively explained
the nonlinear
nonlinear
was quantitatively
explained by
by attributing
attributing all
all of the
refraction
buildup of
of excited
excited carriers,
carriers, although the
the simple
simple
refraction to
to the buildup
Drude theory
theory had
had to
to be
be modified
modified to
transitions.
Drude
to allow
allow for
for interband
interband transitions.
These
may actually
These fits
fits indicate
indicatethat
that interband
interband transitions
transitions may
actually dominate
dominate
the
refractive index
index change
change in
in these
In addiaddi­
the refractive
these two-photon
two -photon absorbers.
absorbers. In
tion,
we demonstrated
simple but
but effective
effective optical
tion, we
demonstrated aa simple
optical limiter
limiter based
based on
on
two-photon
the associated
associated selfself-defocusing,
two -photon absorption,
absorption, the
defocusing, and
and spatial
spatial
filtering.
By choosing
parame­
filtering. By
choosing aa substance
substance with
with the
the proper
proper material
material parameters
and aa specific
specific geometry,
geometry, this
to suit
suit aa
ters and
this limiting
limiting can
can be
be tailored
tailored to
specific
specific need.
need. Clearly,
Clearly,further
further applications
applications of
of the
the combined
combined action
action of
of
2PA
and selfself-refraction
devices have
2PA and
refraction in
in nonlinear
nonlinear optical
optical devices
have been
been and
and
will
will be
be found.
found.
.
8.
ACKNOWLEDGMENTS
8. ACKNOWLEDGMENTS
The authors
authors are
are most
most grateful
grateful to
to B.
B. S.
The
S. Wherrett
Wherrett for
for many
many stimulating
stimulating
and
lengthy discussions.
discussions. This
and lengthy
This research
research was
was supported
supported with
with funds
funds
from
Science Foundation
(ECS #8310625),
#8310625), the
from the
the National
National Science
Foundation (ECS
the Office
Office of
of
Naval
Research, the
the Defense
Defense Advanced
Advanced Research
Research Projects
Projects Agency,
Agency,
Naval Research,
and
Robert A.
A. Welch
Welch Foundation.
Foundation.
and The
The Robert
9.
REFERENCES
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TWO PHOTON
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